Properties

Label 232.4.a
Level $232$
Weight $4$
Character orbit 232.a
Rep. character $\chi_{232}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $5$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(232))\).

Total New Old
Modular forms 94 21 73
Cusp forms 86 21 65
Eisenstein series 8 0 8

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

\(2\)\(29\)FrickeDim.
\(+\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(11\)
Minus space\(-\)\(10\)

Trace form

\( 21 q + 8 q^{3} - 36 q^{7} + 219 q^{9} + O(q^{10}) \) \( 21 q + 8 q^{3} - 36 q^{7} + 219 q^{9} + 88 q^{11} + 36 q^{13} - 100 q^{15} - 182 q^{17} - 88 q^{19} - 184 q^{21} + 188 q^{23} + 453 q^{25} - 304 q^{27} - 87 q^{29} + 240 q^{31} - 82 q^{33} - 76 q^{35} - 110 q^{37} + 68 q^{39} + 46 q^{41} + 264 q^{43} + 142 q^{45} - 1136 q^{47} + 1365 q^{49} + 220 q^{51} + 976 q^{53} - 1220 q^{55} + 1072 q^{57} + 56 q^{59} - 906 q^{61} - 232 q^{63} + 162 q^{65} - 1052 q^{67} + 636 q^{69} - 304 q^{71} + 214 q^{73} + 56 q^{75} + 1672 q^{77} + 640 q^{79} + 1725 q^{81} - 1560 q^{83} + 3224 q^{85} - 4546 q^{89} - 2548 q^{91} - 2294 q^{93} + 2680 q^{95} - 274 q^{97} + 1520 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(232))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 29
232.4.a.a 232.a 1.a $3$ $13.688$ 3.3.4481.1 None \(0\) \(3\) \(11\) \(-38\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(4+\beta _{1}+\beta _{2})q^{5}+(-13+\cdots)q^{7}+\cdots\)
232.4.a.b 232.a 1.a $3$ $13.688$ 3.3.229.1 None \(0\) \(6\) \(4\) \(16\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-3\beta _{2})q^{3}+(1+3\beta _{1}-4\beta _{2})q^{5}+\cdots\)
232.4.a.c 232.a 1.a $4$ $13.688$ 4.4.225792.1 None \(0\) \(0\) \(-20\) \(-8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{1}-\beta _{3})q^{3}+(-5+\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
232.4.a.d 232.a 1.a $5$ $13.688$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(4\) \(10\) \(32\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(2+\beta _{3})q^{5}+(6+\beta _{2}+\cdots)q^{7}+\cdots\)
232.4.a.e 232.a 1.a $6$ $13.688$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-5\) \(-5\) \(-38\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{3}+(-1-\beta _{3}-\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(232))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(232)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 2}\)