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What is the most efficient way to determine the coefficients {a_{1}, a_{2}, ..., a_{21}} such that \Delta^{21} \mid T_{3} \equiv \sum_{i = 1}^{21} c_{i} \Delta^{i} (mod 2)?

I tried finding the q-series expansion of LHS and accordingly kept subtracting powers of \Delta to find the scalars, but is there a better way to do this?

By \Delta(z), I mean the Ramanujan Delta function which is the 24th power of Dedekind eta function.

What is the most efficient way to determine the coefficients {a_{1}, $$a_{1}, a_{2}, ..., a_{21}} a_{21}$$ such that \Delta^{21} $$\Delta^{21} \mid T_{3} \equiv \sum_{i = 1}^{21} c_{i} \Delta^{i} (mod 2)? \ (\text{mod }2)\ ?$$

I tried finding the q-series $q$-series expansion of LHS and accordingly kept subtracting powers of \Delta $\Delta$ to find the scalars, but is there a better way to do this?

By \Delta(z), $\Delta(z)$, I mean the Ramanujan Delta function which is the 24th $24$th power of Dedekind eta function. function.

What is the most efficient way to determine the coefficients $$a_{1}, a_{2}, ..., a_{21}$$ such that $$\Delta^{21} \mid T_{3} \equiv \sum_{i = 1}^{21} c_{i} \Delta^{i} \ (\text{mod }2)\ ?$$

I tried finding the $q$-series expansion of LHS and accordingly kept subtracting powers of $\Delta$ to find the scalars, but is there a better way to do this?

By $\Delta(z)$, I mean the Ramanujan Delta function which is the $24$th power of Dedekind eta function.

What is the most efficient way to determine the coefficients $$a_{1}, a_{2}, ..., a_{21}$$ such that $$\Delta^{21} \mid T_{3} \equiv \sum_{i = 1}^{21} c_{i} a_{i} \Delta^{i} \ (\text{mod }2)\ ?$$

I tried finding the $q$-series expansion of LHS and accordingly kept subtracting powers of $\Delta$ to find the scalars, but is there a better way to do this?

By $\Delta(z)$, I mean the Ramanujan Delta function which is the $24$th power of Dedekind eta function.

For a prime $p$ and $z \in \mathbb{H}$, we have

$(f \mid T_{p})(z) = (T_{p}(f))(z) = p^{k - 1} f(pz) + \frac{1}{p} \:\displaystyle{\sum_{b = 0}^{p - 1}} \: f \left(\frac{z + b}{p} \right)$.

Furthermore, in my case,

$f(z) = \Delta(z) = q \: \displaystyle{\prod_{n \geq 1} (1 - q^{n})^{24}}$ where $q = e^{2 \pi i z}$ and $z \in \mathbb{H}$.

What is the most efficient way to determine the coefficients $$a_{1}, a_{2}, ..., a_{21}$$ such that $$\Delta^{21} \mid T_{3} \equiv \sum_{i = 1}^{21} a_{i} \Delta^{i} \ (\text{mod }2)\ ?$$

I tried finding the $q$-series expansion of LHS and accordingly kept subtracting powers of $\Delta$ to find the scalars, but is there a better way to do this?

By $\Delta(z)$, I mean the Ramanujan Delta function which is the $24$th power of Dedekind eta function.

For a prime $p$ and $z \in \mathbb{H}$, we have

$(f \mid T_{p})(z) = (T_{p}(f))(z) = p^{k - 1} f(pz) + \frac{1}{p} \:\displaystyle{\sum_{b = 0}^{p - 1}} \: f \left(\frac{z + b}{p} \right)$.

Furthermore, in my case,

$f(z) = \Delta(z) = q \: \displaystyle{\prod_{n \geq 1} (1 - q^{n})^{24}}$ where $q = e^{2 \pi i z}$ and $z \in \mathbb{H}$.

What is the most efficient way to determine the coefficients $$a_{1}, a_{2}, ..., a_{21}$$ such that $$\Delta^{21} \mid T_{3} \equiv \sum_{i = 1}^{21} a_{i} \Delta^{i} \ (\text{mod }2)\ ?$$

I tried finding the $q$-series expansion of LHS and accordingly kept subtracting powers of $\Delta$ to find the scalars, but is there a better way to do this?

By $\Delta(z)$, I mean the Ramanujan Delta function which is the $24$th power of Dedekind eta function.

For a prime $p$ and $z \in \mathbb{H}$, we have

$(f \mid T_{p})(z) = (T_{p}(f))(z) = p^{k - 1} f(pz) + \frac{1}{p} \:\displaystyle{\sum_{b = 0}^{p - 1}} \: f \left(\frac{z + b}{p} \right)$.

Furthermore, in my case,

$f(z) = \Delta(z) = q \: \displaystyle{\prod_{n \geq 1} (1 - q^{n})^{24}}$ where $q = e^{2 \pi i z}$ and $z \in \mathbb{H}$.