Properties

Label 117.4.a
Level $117$
Weight $4$
Character orbit 117.a
Rep. character $\chi_{117}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $7$
Sturm bound $56$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(56\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(117))\).

Total New Old
Modular forms 46 15 31
Cusp forms 38 15 23
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.

\(3\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(9\)
Minus space\(-\)\(6\)

Trace form

\( 15 q + 82 q^{4} - 6 q^{5} + 2 q^{7} + O(q^{10}) \) \( 15 q + 82 q^{4} - 6 q^{5} + 2 q^{7} - 54 q^{10} + 42 q^{11} - 13 q^{13} + 78 q^{14} + 218 q^{16} - 36 q^{17} - 162 q^{19} + 192 q^{20} - 16 q^{22} + 36 q^{23} + 635 q^{25} + 78 q^{26} + 888 q^{28} - 294 q^{29} - 730 q^{31} + 180 q^{32} - 184 q^{34} - 366 q^{35} + 78 q^{37} - 732 q^{38} - 1550 q^{40} - 402 q^{41} + 806 q^{43} + 600 q^{44} - 1996 q^{46} - 282 q^{47} - 91 q^{49} + 144 q^{50} - 624 q^{52} + 834 q^{53} + 452 q^{55} + 390 q^{56} - 1172 q^{58} + 1734 q^{59} + 1622 q^{61} + 48 q^{62} + 3626 q^{64} + 468 q^{65} + 2178 q^{67} - 2634 q^{68} - 3388 q^{70} + 1098 q^{71} - 1870 q^{73} - 3210 q^{74} - 1664 q^{76} - 1812 q^{77} - 1096 q^{79} + 1392 q^{80} - 168 q^{82} + 1422 q^{83} - 712 q^{85} - 5004 q^{86} - 456 q^{88} + 534 q^{89} - 676 q^{91} + 204 q^{92} + 5138 q^{94} + 3720 q^{95} + 1758 q^{97} + 5424 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13
117.4.a.a 117.a 1.a $1$ $6.903$ \(\Q\) None 39.4.a.a \(0\) \(0\) \(12\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{4}+12q^{5}+2q^{7}+6^{2}q^{11}+13q^{13}+\cdots\)
117.4.a.b 117.a 1.a $1$ $6.903$ \(\Q\) None 13.4.a.a \(5\) \(0\) \(7\) \(-13\) $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{2}+17q^{4}+7q^{5}-13q^{7}+45q^{8}+\cdots\)
117.4.a.c 117.a 1.a $2$ $6.903$ \(\Q(\sqrt{14}) \) None 39.4.a.b \(-2\) \(0\) \(-24\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(7-2\beta )q^{4}+(-12+\cdots)q^{5}+\cdots\)
117.4.a.d 117.a 1.a $2$ $6.903$ \(\Q(\sqrt{17}) \) None 13.4.a.b \(-1\) \(0\) \(3\) \(-9\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-4+\beta )q^{4}+(2-\beta )q^{5}+(-10+\cdots)q^{7}+\cdots\)
117.4.a.e 117.a 1.a $2$ $6.903$ \(\Q(\sqrt{7}) \) None 117.4.a.e \(0\) \(0\) \(0\) \(-44\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{4}-4\beta q^{5}-22q^{7}-9\beta q^{8}+\cdots\)
117.4.a.f 117.a 1.a $3$ $6.903$ 3.3.3144.1 None 39.4.a.c \(-2\) \(0\) \(-4\) \(30\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
117.4.a.g 117.a 1.a $4$ $6.903$ 4.4.1520092.1 None 117.4.a.g \(0\) \(0\) \(0\) \(36\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(14+\beta _{2})q^{4}-\beta _{3}q^{5}+(8+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(117))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(117)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)