Properties

Label 117.4.b
Level $117$
Weight $4$
Character orbit 117.b
Rep. character $\chi_{117}(64,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $5$
Sturm bound $56$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(56\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(117, [\chi])\).

Total New Old
Modular forms 46 18 28
Cusp forms 38 16 22
Eisenstein series 8 2 6

Trace form

\( 16 q - 58 q^{4} + O(q^{10}) \) \( 16 q - 58 q^{4} + 102 q^{10} + 38 q^{13} + 174 q^{14} + 154 q^{16} - 222 q^{17} + 432 q^{22} - 156 q^{23} - 226 q^{25} + 198 q^{26} + 720 q^{29} - 714 q^{35} + 12 q^{38} - 822 q^{40} + 350 q^{43} - 554 q^{49} + 292 q^{52} + 708 q^{53} - 2112 q^{55} - 2358 q^{56} - 1604 q^{61} - 1008 q^{62} + 2054 q^{64} - 570 q^{65} + 5694 q^{68} - 2658 q^{74} - 2040 q^{77} + 1180 q^{79} + 960 q^{82} - 1344 q^{88} + 390 q^{91} + 7884 q^{92} + 522 q^{94} - 1620 q^{95} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(117, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
117.4.b.a 117.b 13.b $2$ $6.903$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-q^{4}-3iq^{5}+5iq^{7}+7iq^{8}+\cdots\)
117.4.b.b 117.b 13.b $2$ $6.903$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+8q^{4}+\zeta_{6}q^{7}+(35-\zeta_{6})q^{13}+2^{6}q^{16}+\cdots\)
117.4.b.c 117.b 13.b $4$ $6.903$ 4.0.8112.1 \(\Q(\sqrt{-39}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}+(-8-\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{5}+\cdots\)
117.4.b.d 117.b 13.b $4$ $6.903$ 4.0.5054412.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}-6\beta _{1}q^{7}+\cdots\)
117.4.b.e 117.b 13.b $4$ $6.903$ 4.0.1362828.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-4+\beta _{3})q^{4}+(2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(117, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(117, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)