Properties

Label 32.9.d
Level $32$
Weight $9$
Character orbit 32.d
Rep. character $\chi_{32}(15,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $2$
Sturm bound $36$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 32.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(32, [\chi])\).

Total New Old
Modular forms 36 9 27
Cusp forms 28 7 21
Eisenstein series 8 2 6

Trace form

\( 7 q + 2 q^{3} + 10933 q^{9} + O(q^{10}) \) \( 7 q + 2 q^{3} + 10933 q^{9} - 19774 q^{11} - 38642 q^{17} + 167554 q^{19} - 57305 q^{25} - 12412 q^{27} + 813212 q^{33} - 1989120 q^{35} - 281906 q^{41} + 4455106 q^{43} - 6890681 q^{49} - 1219580 q^{51} + 5066012 q^{57} - 1065406 q^{59} + 27060480 q^{65} - 33905918 q^{67} - 21917362 q^{73} + 112794050 q^{75} - 57515933 q^{81} - 174856318 q^{83} + 46653646 q^{89} + 273971712 q^{91} + 47775374 q^{97} - 445152506 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(32, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
32.9.d.a 32.d 8.d $1$ $13.036$ \(\Q\) \(\Q(\sqrt{-2}) \) 8.9.d.a \(0\) \(-34\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-34q^{3}-5405q^{9}+27166q^{11}+\cdots\)
32.9.d.b 32.d 8.d $6$ $13.036$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 8.9.d.b \(0\) \(36\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(6+\beta _{1})q^{3}-\beta _{2}q^{5}+(-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(32, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(32, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 2}\)