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Properties

Label 234.4.a
Level $234$
Weight $4$
Character orbit 234.a
Rep. character $\chi_{234}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $13$
Sturm bound $168$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 234.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(168\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(234))\).

Total New Old
Modular forms 134 15 119
Cusp forms 118 15 103
Eisenstein series 16 0 16

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

\(2\)\(3\)\(13\)FrickeDim
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(1\)
Plus space\(+\)\(9\)
Minus space\(-\)\(6\)

Trace form

\( 15 q - 2 q^{2} + 60 q^{4} + 22 q^{5} - 20 q^{7} - 8 q^{8} + O(q^{10}) \) \( 15 q - 2 q^{2} + 60 q^{4} + 22 q^{5} - 20 q^{7} - 8 q^{8} + 24 q^{10} - 28 q^{11} + 13 q^{13} + 108 q^{14} + 240 q^{16} - 120 q^{17} + 168 q^{19} + 88 q^{20} + 32 q^{22} + 172 q^{23} + 347 q^{25} - 78 q^{26} - 80 q^{28} + 118 q^{29} + 284 q^{31} - 32 q^{32} - 284 q^{34} + 370 q^{35} - 294 q^{37} + 400 q^{38} + 96 q^{40} + 750 q^{41} - 86 q^{43} - 112 q^{44} + 1160 q^{46} + 404 q^{47} + 1041 q^{49} - 414 q^{50} + 52 q^{52} - 750 q^{53} - 1104 q^{55} + 432 q^{56} + 356 q^{58} - 1976 q^{59} - 1274 q^{61} + 240 q^{62} + 960 q^{64} - 676 q^{65} - 2668 q^{67} - 480 q^{68} + 1128 q^{70} + 308 q^{71} - 726 q^{73} + 760 q^{74} + 672 q^{76} + 1928 q^{77} - 636 q^{79} + 352 q^{80} - 380 q^{82} - 1452 q^{83} + 712 q^{85} + 1296 q^{86} + 128 q^{88} - 378 q^{89} + 806 q^{91} + 688 q^{92} - 2260 q^{94} - 4260 q^{95} + 398 q^{97} - 2898 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(234))\) 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 3 13
234.4.a.a 234.a 1.a $1$ $13.806$ \(\Q\) None \(-2\) \(0\) \(-17\) \(-35\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-17q^{5}-35q^{7}-8q^{8}+\cdots\)
234.4.a.b 234.a 1.a $1$ $13.806$ \(\Q\) None \(-2\) \(0\) \(-6\) \(20\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-6q^{5}+20q^{7}-8q^{8}+\cdots\)
234.4.a.c 234.a 1.a $1$ $13.806$ \(\Q\) None \(-2\) \(0\) \(-4\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-4q^{5}+4q^{7}-8q^{8}+\cdots\)
234.4.a.d 234.a 1.a $1$ $13.806$ \(\Q\) None \(-2\) \(0\) \(2\) \(-26\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+2q^{5}-26q^{7}-8q^{8}+\cdots\)
234.4.a.e 234.a 1.a $1$ $13.806$ \(\Q\) None \(-2\) \(0\) \(18\) \(20\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+18q^{5}+20q^{7}-8q^{8}+\cdots\)
234.4.a.f 234.a 1.a $1$ $13.806$ \(\Q\) None \(-2\) \(0\) \(20\) \(-32\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+20q^{5}-2^{5}q^{7}-8q^{8}+\cdots\)
234.4.a.g 234.a 1.a $1$ $13.806$ \(\Q\) None \(2\) \(0\) \(-11\) \(19\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-11q^{5}+19q^{7}+8q^{8}+\cdots\)
234.4.a.h 234.a 1.a $1$ $13.806$ \(\Q\) None \(2\) \(0\) \(-10\) \(-8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-10q^{5}-8q^{7}+8q^{8}+\cdots\)
234.4.a.i 234.a 1.a $1$ $13.806$ \(\Q\) None \(2\) \(0\) \(-2\) \(-26\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-2q^{5}-26q^{7}+8q^{8}+\cdots\)
234.4.a.j 234.a 1.a $1$ $13.806$ \(\Q\) None \(2\) \(0\) \(16\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+2^{4}q^{5}-8q^{7}+8q^{8}+\cdots\)
234.4.a.k 234.a 1.a $1$ $13.806$ \(\Q\) None \(2\) \(0\) \(16\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+2^{4}q^{5}+28q^{7}+8q^{8}+\cdots\)
234.4.a.l 234.a 1.a $2$ $13.806$ \(\Q(\sqrt{22}) \) None \(-4\) \(0\) \(-8\) \(12\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-4+\beta )q^{5}+(6-\beta )q^{7}+\cdots\)
234.4.a.m 234.a 1.a $2$ $13.806$ \(\Q(\sqrt{22}) \) None \(4\) \(0\) \(8\) \(12\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(4+\beta )q^{5}+(6+\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(234))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(234)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)