2. Preliminaries

In this section, we recall some basic facts and definitions for modular forms (for more details, see [Reference Koblitz11, Reference Ono12]).

First, we define the matrix groups

$$ \begin{align*} \text{SL}_2(\mathbb{Z})&:=\bigg\{\begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \mathbb{Z}, ad-bc=1\bigg\},\\ \Gamma_0(N)&:=\bigg\{\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \text{SL}_2(\mathbb{Z}) : c\equiv 0 \pmod N\bigg\},\\ \Gamma_1(N)&:=\bigg\{\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \Gamma_0(N) : a\equiv d\equiv 1 \pmod N\bigg\}, \end{align*} $$

and

$$ \begin{align*} \Gamma(N):=\bigg\{\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \text{SL}_2(\mathbb{Z}) : a\equiv d\equiv 1 \pmod N \text{ and } b\equiv c\equiv 0 \pmod N \bigg\}, \end{align*} $$

where N is a positive integer. A subgroup $\Gamma $ of $\text {SL}_2(\mathbb {Z})$ is called a congruence subgroup if $\Gamma (N)\subseteq \Gamma $ for some N, and the smallest N with this property is called the level of $\Gamma $ . For instance, $\Gamma _0(N)$ and $\Gamma _1(N)$ are congruence subgroups of level N.

Let $\mathbb {H}$ denote the upper half of the complex plane. The group

$$ \begin{align*} \text{GL}_2^{+}(\mathbb{R}):=\bigg\{\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \text{SL}_2(\mathbb{Z}) : a, b, c, d \in \mathbb{R} \text{ and } ad-bc>0 \bigg\} \end{align*} $$

acts on $\mathbb {H}$ by

$$ \begin{align*}\begin{bmatrix} a & b \\ c & d \end{bmatrix} z = \dfrac{az+b}{cz+d}.\end{align*} $$

We identify $\infty $ with ${1}/{0}$ and define

$$ \begin{align*}\begin{bmatrix} a & b \\ c & d \end{bmatrix} \dfrac{r}{s} = \dfrac{ar+bs}{cr+ds},\end{align*} $$

where ${r}/{s} \in \mathbb {Q} \cup \{ \infty \}$ . This gives an action of $\text {GL}_2^{+}(\mathbb {R})$ on the extended upper half plane $\mathbb {H}^*=\mathbb {H}\cup \mathbb {Q}\cup \{\infty \}$ . Suppose that $\Gamma $ is a congruence subgroup of $\text {SL}_2(\mathbb {Z})$ . A cusp of $\Gamma $ is an equivalence class in $\mathbb {P}^1=\mathbb {Q}\cup \{\infty \}$ under the action of $\Gamma $ .

The group $\text {GL}_2^{+}(\mathbb {R})$ also acts on functions $f:\mathbb {H}\rightarrow \mathbb {C}$ . In particular, if $\gamma =[\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}] \in \text {GL}_2^{+}(\mathbb {R})$ , $f(z)$ is a meromorphic function on $\mathbb {H}$ and $\ell $ is an integer, we define the slash operator $|_{\ell }$ by

$$ \begin{align*} (f|_{\ell}\gamma)(z):=(\text{det}(\gamma))^{\ell/2}(cz+d)^{-\ell}f(\gamma z). \end{align*} $$

For a positive integer $\ell $ , the complex vector space of modular forms of weight $\ell $ with respect to a congruence subgroup $\Gamma $ is denoted by $M_{\ell }(\Gamma )$ .

Definition 2.2 [Reference Ono12, Definition 1.15].

If $\chi $ is a Dirichlet character modulo N, then we say that a modular form $f \in M_{\ell }(\Gamma _{1}(N))$ has Nebentypus character $\chi $ if

$$ \begin{align*} f\bigg(\dfrac{az+b}{cz+d}\bigg) = \chi(d)(cz+d)^{\ell}f(z) \end{align*} $$

for all $z\in \mathbb {H}$ and all $[\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}] \in \Gamma _{0}(N)$ . The space of such modular forms is denoted by $M_{\ell }(\Gamma _{0}(N),\chi )$ .

The relevant modular forms for the results of this paper arise from eta-quotients. The Dedekind eta-function $\eta (z)$ is defined by

$$ \begin{align*} \eta(z):=q^{1/24}(q;q)_{\infty}=q^{1/24}\displaystyle\prod_{n=1}^{\infty}(1-q^n), \end{align*} $$

where $q:=e^{2\pi iz}$ and $z\in \mathbb {H}$ . A function $f(z)$ is called an eta-quotient if it is of the form

$$ \begin{align*} f(z)=\displaystyle\prod_{\delta|N}\eta(\delta z)^{r_{\delta}}, \end{align*} $$

where N is a positive integer and $r_{\delta }$ is an integer. Now, we recall two important theorems from [Reference Ono12, page 18] that will be used later.

Theorem 2.3 [Reference Ono12, Theorem 1.64].

If $f(z)=\prod _{\delta |N}\eta (\delta z)^{r_{\delta }}$ is an eta-quotient such that $\ell ={1}/{2}\sum _{\delta |N}r_{\delta } \in \mathbb {Z}$ , $\sum _{\delta |N}\delta r_{\delta }\equiv 0 \pmod {24}$ and $\sum _{\delta |N}({N}/{\delta }) r_{\delta }\equiv 0 \pmod {24}$ , then $f(z)$ satisfies

$$ \begin{align*} f\bigg(\dfrac{az+b}{cz+d}\bigg)=\chi(d)(cz+d)^{\ell}f(z) \end{align*} $$

for every $[\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}] \in \Gamma _{0}(N)$ . Here the character $\chi $ is defined by

$$ \begin{align*}\chi(d) := \bigg(\dfrac{(-1)^{\ell}s}{d}\bigg), \quad\text{where}\ s:= \displaystyle\prod_{\delta|N}\delta^{r_{\delta}}.\end{align*} $$

Let f be an eta-quotient that satisfies the conditions of Theorem 2.3 and suppose that the associated weight $\ell $ is a positive integer. If $f(z)$ is holomorphic at all the cusps of $\Gamma _{0}(N)$ , then $f(z) \in M_{\ell }(\Gamma _{0}(N), \chi )$ . The necessary criterion for determining orders of an eta-quotient at cusps is given by the following theorem.

Theorem 2.4 [Reference Ono12, Theorem 1.64].

Let c, d and N be positive integers with $d \mid N$ and $\gcd (c,d)=1$ . If f is an eta-quotient satisfying the conditions of Theorem 2.3 for N, then the order of vanishing of $f(z)$ at the cusp $c/d$ is

$$ \begin{align*} \dfrac{N}{24}\displaystyle\sum_{\delta|N}\dfrac{\gcd(d,\delta)^2r_{\delta}}{\gcd(d,N/d)d\delta}. \end{align*} $$

We now recall a deep theorem of Serre [see Reference Ono12, page 43] that will be used in proving Theorems 1.2 and 1.3.

Theorem 2.5 [Reference Ono12, page 43].

Let $g(z) \in M_k(\Gamma _0(N),\chi )$ have Fourier expansion

$$ \begin{align*} g(z)=\displaystyle\sum_{n=0}^\infty b(n)q^n \in \mathbb{Z}[[q]]. \end{align*} $$

Then for a positive integer r, there is a constant $\alpha> 0$ such that

$$ \begin{align*} \#\{0<n\leq X: b(n)\not\equiv 0 \pmod r\}=\mathcal{O}\bigg(\dfrac{X}{(\log X)^\alpha}\bigg). \end{align*} $$

Equivalently,

$$ \begin{align*} \displaystyle{\lim_{X \to \infty}} \dfrac{\#\{0<n\leq X: b(n)\not\equiv 0 \pmod r\}}{X}=0. \end{align*} $$

Finally, we recall the definition of Hecke operators. Let m be a positive integer and $f(z)=\sum _{n=0}^{\infty }a(n)q^n \in M_{\ell }(\Gamma _{0}(N),\chi )$ . Then the action of the Hecke operator $T_{m}$ on $f(z)$ is defined by

$$ \begin{align*} f(z)|T_m:=\displaystyle\sum_{n=0}^{\infty}\bigg(\displaystyle\sum_{d|\text{gcd}(n,m)}\chi(d)d^{\ell-1}a\bigg(\dfrac{nm}{d^2}\bigg)\bigg)q^n. \end{align*} $$

In particular, if $m=p$ is prime, then

(2.1) $$ \begin{align} f(z)|T_p:=\displaystyle\sum_{n=0}^{\infty}\bigg(a(pn)+\chi(p)p^{\ell-1}a\bigg(\dfrac{n}{p}\bigg)\bigg)q^n. \end{align} $$

We note that $a(n)=0$ unless n is a nonnegative integer.

3. Proof of Theorem 1.2

Putting $t=3^{\alpha }m$ in (1.2),

(3.1) $$ \begin{align} \displaystyle\sum_{n=0}^{\infty}\overline{a}_{3^{\alpha}m}(n)q^n=\dfrac{f_2f_{3^{\alpha}m}^{2\cdot3^{\alpha}m}}{f_1^2f_{2\cdot3^{\alpha}m}^{3^{\alpha}m}}. \end{align} $$

We define

$$ \begin{align*} A_{\alpha,m}(z):=\dfrac{\eta^2\,(2^33^{\alpha+1}mz)}{\eta\,(2^43^{\alpha+1}mz)}. \end{align*} $$

For any prime p and positive integer j, $(q;q)_{\infty }^{p^j}\equiv (q^p;q^p)_{\infty }^{p^{j-1}}\pmod {p^j}$ . Hence, for any integer $k\geq 1$ ,

(3.2) $$ \begin{align} A^{2^k}_{\alpha,m}(z)=\dfrac{\eta^{2^{k+1}}(2^33^{\alpha+1}mz)}{\eta^{2^k}(2^43^{\alpha+1}mz)}\equiv 1\pmod{2^{k+1}}. \end{align} $$

Next we define

$$ \begin{align*} B_{\alpha,m,k}(z) :=\dfrac{\eta\,(48z)\eta^{2\cdot3^{\alpha}m}(2^33^{\alpha+1}mz)}{\eta^2(24z)\eta^{3^{\alpha}m}(2^43^{\alpha+1}mz)}A^{2^k}_{\alpha,m}(z) =\dfrac{\eta\,(48z)\eta^{2\cdot3^{\alpha}m+2^{k+1}}(2^33^{\alpha+1}mz)}{\eta^2(24z)\eta^{3^{\alpha}m+2^k}(2^43^{\alpha+1}mz)}. \end{align*} $$

In view of (3.1) and (3.2),

(3.3) $$ \begin{align} B_{\alpha,m,k}(z)\equiv\dfrac{\eta\,(48z)\eta^{2\cdot3^{\alpha}m}(2^33^{\alpha+1}mz)}{\eta^2(24z)\eta^{3^{\alpha}m}(2^43^{\alpha+1}mz)} \equiv\dfrac{f_{48}f_{2^33^{\alpha+1}m}^{2\cdot3^{\alpha}m}}{f_{24}^2f_{2^4\cdot3^{\alpha+1}m}^{3^{\alpha}m}} \equiv\displaystyle\sum_{n=0}^{\infty}\overline{a}_{3^{\alpha}m}(n)q^{24n}\pmod{2^{k+1}}. \end{align} $$

Next, we will show that $B_{\alpha ,m,k}(z)$ is a modular form. Applying Theorem 2.3, we find that the level of $B_{\alpha ,m,k}(z)$ is $N=2^43^{\alpha +1}mM$ , where M is the smallest positive integer such that

$$ \begin{align*} 2^43^{\alpha+1}mM\bigg(\dfrac{-2}{24}+\dfrac{1}{48}+\dfrac{2\cdot3^{\alpha}m+2^{k+1}}{2^33^{\alpha+1}m}+\dfrac{-3^{\alpha}m-2^k}{2^43^{\alpha+1}m}\bigg)\equiv 0\pmod{24}, \end{align*} $$

which implies

$$ \begin{align*} 3\cdot2^kM\equiv 0\pmod{24}. \end{align*} $$

Therefore, $M=4$ and the level of $B_{\alpha ,m,k}(z)$ is $N=2^6 3^{\alpha +1}m$ .

The representatives for the cusps of $\Gamma _0(2^63^{\alpha +1}m)$ are given by fractions $c/d$ where $d \mid 2^63^{\alpha +1}m$ and $\gcd (c, 2^63^{\alpha +1}m) = 1$ (see [Reference Cotron, Michaelsen, Stamm and Zhu5, Proposition 2.1]). By Theorem 2.4, $B_{\alpha ,m,k}(z)$ is holomorphic at a cusp $c/d$ if and only if

$$ \begin{align*} -2 \dfrac{\gcd (d,24)^2}{24} & + \dfrac{\gcd (d,48)^2}{48} \\ &+(3^\alpha m+2^k)\bigg(2\dfrac{ \gcd (d,2^3 3^{\alpha+1} m)^2}{2^3 3^{\alpha+1} m}-\dfrac{\gcd (d,2^4 3^{\alpha+1} m)^2}{2^4 3^{\alpha+1} m}\bigg)\geq 0. \end{align*} $$

Equivalently, $B_{\alpha ,m,k}(z)$ is holomorphic at a cusp $c/d$ if and only if

$$ \begin{align*} L:=3^{\alpha}m(-4G_1+G_2+4G_3-1)+2^k(4G_3-1)\geq 0, \end{align*} $$

where

$$ \begin{align*}G_1=\dfrac{\gcd (d,24)^2}{\gcd (d,2^4 3^{\alpha+1} m)^2}, \quad G_2=\dfrac{\gcd (d,48)^2}{\gcd (d,2^43^{\alpha+1} m)^2}, \quad G_3=\dfrac{\gcd (d,2^3 3^{\alpha+1} m)^2}{\gcd (d,2^4 3^{\alpha+1} m)^2}.\end{align*} $$

Let d be a divisor of $2^6 3^{\alpha +1} m$ . We can write $d = 2^{r_1}3^{r_2}t$ where $0 \leq r_1 \leq 6$ , $0 \leq r_2 \leq \alpha + 1$ and $t \mid m$ . We now consider two cases depending on $r_1$ .

Case 1: $0\leq r_1\leq 3$ , $0\leq r_2\leq \alpha +1$ . Then $G_1=G_2$ , ${1}/{3^{2\alpha }t^2}\leq G_1\leq 1$ and $G_3=1$ . Therefore, $L= 3^{\alpha +1}m(1-G_1)+3\cdot 2^k\geq 3\cdot 2^k.$

Case 2: $4\leq r_1\leq 6$ , $0\leq r_2\leq \alpha +1$ . Then $G_2=4G_1$ , ${1}/{4\cdot 3^{2\alpha }t^2}\leq G_1\leq {1}/{4}$ and $G_3={1}/{4}$ , which implies $L= 0.$

Hence, $B_{\alpha ,m,k}(z)$ is holomorphic at every cusp $c/d$ . The weight of $B_{\alpha ,m,k}(z)$ is $\ell = {1}/{2} (3^\alpha m+2^k-1)$ which is a positive integer and the associated character is given by

$$ \begin{align*} \chi_1(\bullet)=\bigg(\dfrac{(-1)^\ell 3^{(\alpha+1) (3^\alpha m+2^k)-1} m^{3^\alpha m+2^k}}{\bullet}\bigg). \end{align*} $$

Thus, $B_{\alpha ,m,k}(z)\in M_\ell (\Gamma _0(N),\chi )$ where $\ell $ , N and $\chi $ are as above. By Theorem 2.5, the Fourier coefficients of $B_{\alpha ,m,k}(z)$ are almost all divisible by $r=2^k$ . From (3.3), this holds for $\overline {a}_{3^{\alpha }m}(n)$ also. This completes the proof of Theorem 1.2.

4. Proof of Theorem 1.3

We proceed along the same lines as in the proof of Theorem 1.2. Here we define

$$ \begin{align*} C_{\alpha,m}(z):=\dfrac{\eta^3(2^43^{\alpha+1}mz)}{\eta(2^43^{\alpha+2}mz)}. \end{align*} $$

By the binomial theorem, for any integer $k\geq 1$ ,

(4.1) $$ \begin{align} C^{3^k}_{\alpha,m}(z)=\dfrac{\eta^{3^{k+1}}(2^43^{\alpha+1}mz)}{\eta^{3^k}(2^43^{\alpha+2}mz)}\equiv 1\pmod{3^{k+1}}. \end{align} $$

Next we define

$$ \begin{align*} D_{\alpha,m,k}(z) & :=\dfrac{\eta\,(48z)\eta^{2\cdot3^{\alpha}m}(2^33^{\alpha+1}mz)}{\eta^2(24z)\eta^{3^{\alpha}m}(2^43^{\alpha+1}mz)}C^{3^k}_{\alpha,m}(z) \\ & =\dfrac{\eta\,(48z)\eta^{2\cdot3^{\alpha}m}(2^33^{\alpha+1}mz)\eta^{3^{k+1}-3^{\alpha}m}(2^43^{\alpha+1}mz)}{\eta^2(24z)\eta^{3^{k}}(2^43^{\alpha+2}mz)}. \end{align*} $$

From (3.1) and (4.1),

(4.2) $$ \begin{align} D_{\alpha,m,k}(z)\equiv\dfrac{\eta\,(48z)\eta^{2\cdot3^{\alpha}m}(2^33^{\alpha+1}mz)}{\eta^2(24z)\eta^{3^{\alpha}m}(2^43^{\alpha+1}mz)} \equiv\dfrac{f_{48}f_{2^33^{\alpha+1}m}^{2\cdot3^{\alpha}m}}{f_{24}^2f_{2^4\cdot3^{\alpha+1}m}^{3^{\alpha}m}} \equiv\displaystyle\sum_{n=0}^{\infty}\overline{a}_{3^{\alpha}m}(n)q^{24n}\pmod{3^{k+1}}. \end{align} $$

We now prove that $D_{\alpha ,m,k}(z)$ is a modular form. Applying Theorem 2.3, we find that the level of $D_{\alpha ,m,k}(z)$ is $N=2^43^{\alpha +2}mM$ , where M is the smallest positive integer such that

$$ \begin{align*} 2^43^{\alpha+2}mM\bigg(\dfrac{-2}{24}+\dfrac{1}{48}+\dfrac{2\cdot3^{\alpha}m}{2^33^{\alpha+1}m}+\dfrac{3^{k+1}-3^{\alpha}m}{2^43^{\alpha+1}m}+\dfrac{-3^k}{2^43^{\alpha+2}m}\bigg)\equiv 0\pmod{24}, \end{align*} $$

which gives $8\cdot 3^kM\equiv 0\pmod {24}$ . Therefore, $M=1$ and the level of $D_{\alpha ,m,k}(z)$ is $N=2^4 3^{\alpha +2}m$ .

The representatives for the cusps of $\Gamma _0(2^43^{\alpha +2}m)$ are given by fractions $c/d$ where $d \mid 2^43^{\alpha +2}m$ and $\gcd (c, 2^43^{\alpha +2}m) = 1$ . From Theorem 2.4, $D_{\alpha ,m,k}(z)$ is holomorphic at a cusp $c/d$ if and only if

$$ \begin{align*} -2 \dfrac{\gcd (d,24)^2}{24} & + \dfrac{\gcd (d,48)^2}{48}+2\cdot 3^{\alpha} m \dfrac{\gcd (d,2^33^{\alpha+1} m)^2}{2^33^{\alpha+1} m} \\ & +(3^{k+1}-3^{\alpha} m) \dfrac{\gcd (d,2^4 3^{\alpha+1} m)^2}{2^43^{\alpha+1} m} -3^k~ \dfrac{\gcd (d,2^4 3^{\alpha+2} m)^2}{2^43^{\alpha+2} m} \geq 0. \end{align*} $$

Equivalently, $D_{\alpha ,m,k}(z)$ is holomorphic at a cusp $c/d$ if and only if

$$ \begin{align*} L:=3^{\alpha+1}m(-4G_1+G_2+4G_3-G_4)+3^{k}(9G_4-1)\geq 0, \end{align*} $$

where

$$ \begin{align*} & G_1=\dfrac{\gcd (d,24)^2}{\gcd (d,2^4 3^{\alpha+2} m)^2}, \quad G_2=\dfrac{\gcd (d,48)^2}{\gcd (d,2^4 3^{\alpha+2} m)^2}, \\ & G_3=\dfrac{\gcd (d,2^3 3^{\alpha+1} m)^2}{\gcd (d,2^4 3^{\alpha+2} m)^2}, \quad G_4=\dfrac{\gcd (d,2^4 3^{\alpha+1} m)^2}{\gcd (d,2^4 3^{\alpha+2} m)^2}. \end{align*} $$

Let d be a divisor of $2^43^{\alpha +2} m$ . We write $d = 2^{r_1}3^{r_2}t$ where $0 \leq r_1 \leq 4$ , $0 \leq r_2 \leq \alpha + 2$ and $t \mid m$ . We now consider four cases depending on the values of $r_1$ and $r_2$ .

Case 1: $0{\kern-1pt}\leq{\kern-1pt} r_1{\kern-1pt}\leq{\kern-1pt} 3$ , $0\leq r_2\leq \alpha +1$ . Then $G_1{\kern-1pt}={\kern-1pt}G_2$ , ${1}/{3^{2\alpha }t^2}\leq G_1\leq 1$ and $G_3= G_4=1$ . Hence, we have $L= 3^{\alpha +2}m(1-G_1)+8\cdot 3^k\geq 8\cdot 3^k.$

Case 2: $0\leq r_1\leq 3$ , $r_2= \alpha +2$ . Then $G_1=G_2$ , ${1}/{3^{2(\alpha +1)}t^2}\leq G_1\leq {1}/{3^{2(\alpha +1)}}$ and $G_3=G_4={1}/{9}$ . Therefore, $L= 3^{\alpha +2}m({1}/{9}-G_1)\geq 0.$

Case 3: Let $r_1=4$ , $0\leq r_2 \leq \alpha +1$ . Then $G_2=4G_1$ , ${1}/{4\cdot 3^{(\alpha +1)}t^2}\leq G_1\leq {1}/{4}$ , $G_4=4G_3$ and $G_3={1}/{4}$ . Hence, we have $L=8\cdot 3^k.$

Case 4: Let $r_1=4$ , $ r_2 =\alpha +2$ . Then $G_2=4G_1$ , ${1}/{4\cdot 3^{(\alpha +1)}t^2}\leq G_1\leq {1}/{4\cdot 3^{2(\alpha +1)}}$ , $G_4=4G_3$ and $G_3={1}/{36}$ . Therefore, $L=0.$

Therefore, $D_{\alpha ,m,k}(z)$ is holomorphic at every cusp $c/d$ . The weight of $D_{\alpha ,m,k}(z)$ is $\ell = {1}/{2}{(3^\alpha m-1)}+3^k$ which is a positive integer and the associated character is given by

$$ \begin{align*} \chi_2(\bullet)=\bigg(\dfrac{(-1)^\ell 3^{2 \alpha 3^k+3^\alpha \alpha m+3^\alpha m+3^k-1} m^{3^\alpha m+2\cdot 3^k}}{\bullet}\bigg). \end{align*} $$

Thus, $D_{\alpha ,m,k}(z)\in M_\ell (\Gamma _0(N),\chi )$ where $\ell $ , N and $\chi $ are as above. By Theorem 2.5, the Fourier coefficients of $D_{\alpha ,m,k}(z)$ are almost all divisible by $r=3^k$ . From (4.2), this holds for $\overline {a}_{3^{\alpha }m}(n)$ also. This completes the proof of Theorem 1.3.

5. Proof of Theorem 1.4

First we recall the following result of Ono and Taguchi [Reference Ono and Taguchi13] on the nilpotency of Hecke operators.

Theorem 5.1 [Reference Ono and Taguchi13, Theorem 1.3(3)].

Let n be a nonnegative integer and k be a positive integer. Let $\chi $ be a quadratic Dirichlet character of conductor $9\cdot 2^a$ . Then there is an integer $c \geq 0$ such that for every $f(z) \in M_k(\Gamma _0(9\cdot 2^a),\chi ) \cap \mathbb {Z}[[q]]$ and every $t\geq 1$ ,

$$ \begin{align*} f(z)|T_{p_1}|T_{p_2}|\cdots|T_{p_{c+t}}\equiv 0\pmod{2^t} \end{align*} $$

whenever the primes $p_1, \ldots , p_{c+t}$ are coprime to $6$ .

We apply this theorem to the modular form $B_{1,1,k}(z)$ to prove Theorem 1.4. Putting $\alpha =1$ and $m=1$ in (3.3), we find that

$$ \begin{align*} B_{1,1,k}(z)\equiv\displaystyle\sum_{n=0}^{\infty}\overline{a}_{3}(n)q^{24n}\pmod{2^{k+1}}, \end{align*} $$

which yields

(5.1) $$ \begin{align} B_{1,1,k}(z):=\displaystyle\sum_{n=0}^{\infty}F_k(n)q^{n}\equiv\displaystyle\sum_{n=0}^{\infty}\overline{a}_{3}\bigg(\dfrac{n}{24}\bigg)q^{n}\pmod{2^{k+1}}. \end{align} $$

Now, $B_{1,1,k}(z)\in M_{2^{k-1}+1}(\Gamma _0(9\cdot 2^6),\chi _3)$ for $k\geq 1$ where $\chi _3$ is the associated character (which is $\chi _1$ evaluated at $\alpha =1$ and $m=1$ ). In view of Theorem 5.1, there is an integer $c\geq 0$ such that for any $d\geq 1$ ,

$$ \begin{align*} B_{1,1,k}(z)\mid T_{p_1}\mid T_{p_2}\mid\cdots\mid T_{p_{c+d}}\equiv 0\pmod{2^d} \end{align*} $$

whenever $p_1,\ldots , p_{c+d}$ are coprime to 6. It follows from the definition of Hecke operators that if $p_1,\ldots , p_{c+d}$ are distinct primes and if n is coprime to $p_1\cdots p_{c+d}$ , then

(5.2) $$ \begin{align} F_k(p_1\cdots p_{c+d}\cdot n)\equiv 0\pmod{2^d}. \end{align} $$

Combining (5.1) and (5.2) completes the proof of the theorem.

6. Concluding remarks

Theorems 1.2 and 1.3 of this paper and [Reference Gireesh, Ray and Shivashankar7, Theorem 1.8] give the arithmetic densities of $\overline {a}_t(n)$ for odd t, but similar techniques cannot be used to obtain the arithmetic density of $\overline {a}_t(n)$ when t is even. It would be interesting to study the arithmetic density of $\overline {a}_t(n)$ for even values of t.

Computational evidence suggests that there are Ramanujan type congruences for $\overline {a}_t(n)$ modulo powers of 2, 3 and other primes $\geq 5$ for various t that are not covered by the results of [Reference Bandyopadhyay and Baruah2, Reference Gireesh, Ray and Shivashankar7]. It would be of interest to find new congruences for $\overline {a}_t(n)$ .

Asymptotic formulae for partition functions and other related functions have been widely studied in the literature. For instance, the asymptotic formulae for $p(n)$ and $c_t(n)$ were obtained by Hardy and Ramanujan [Reference Hardy and Ramanujan9] and Anderson [Reference Anderson1], respectively. It will be desirable to find an asymptotic formula for $\overline {a}_t(n)$ .

Some relations connecting $\overline {a}_t(n)$ and $c_t(n)$ have been discussed in [Reference Bandyopadhyay and Baruah2]. A combinatorial treatment for $\overline {a}_t(n)$ might reveal more interesting partition theoretic connections.