Properties

Label 36.22.a
Level $36$
Weight $22$
Character orbit 36.a
Rep. character $\chi_{36}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $4$
Sturm bound $132$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 36.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(132\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(36))\).

Total New Old
Modular forms 132 9 123
Cusp forms 120 9 111
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)FrickeDim
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(5\)
Minus space\(-\)\(4\)

Trace form

\( 9 q - 31249134 q^{5} - 43147656 q^{7} - 33213025548 q^{11} - 89073033570 q^{13} - 4793865260634 q^{17} - 19611560500236 q^{19} + 161916007099176 q^{23} + 11\!\cdots\!91 q^{25} + 617422502303226 q^{29} + 41\!\cdots\!88 q^{31}+ \cdots + 12\!\cdots\!18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(36))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
36.22.a.a 36.a 1.a $1$ $100.612$ \(\Q\) None 12.22.a.a \(0\) \(0\) \(11268090\) \(281914136\) $-$ $-$ $\mathrm{SU}(2)$ \(q+11268090q^{5}+281914136q^{7}+\cdots\)
36.22.a.b 36.a 1.a $2$ $100.612$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 12.22.a.b \(0\) \(0\) \(-28827900\) \(509669728\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-14413950-5\beta )q^{5}+(254834864+\cdots)q^{7}+\cdots\)
36.22.a.c 36.a 1.a $2$ $100.612$ \(\Q(\sqrt{2161}) \) None 4.22.a.a \(0\) \(0\) \(-13689324\) \(-260508080\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-6844662-68\beta )q^{5}+(-130254040+\cdots)q^{7}+\cdots\)
36.22.a.d 36.a 1.a $4$ $100.612$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 36.22.a.d \(0\) \(0\) \(0\) \(-574223440\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+(-143555860-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(36))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(36)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 9}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)