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Properties

Label 58.9.c
Level $58$
Weight $9$
Character orbit 58.c
Rep. character $\chi_{58}(17,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $40$
Newform subspaces $2$
Sturm bound $67$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 58.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 29 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(67\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(58, [\chi])\).

Total New Old
Modular forms 124 40 84
Cusp forms 116 40 76
Eisenstein series 8 0 8

Trace form

\( 40 q + O(q^{10}) \) \( 40 q - 15360 q^{10} - 22776 q^{11} - 27648 q^{14} + 156916 q^{15} - 655360 q^{16} + 94560 q^{17} + 328972 q^{19} + 86016 q^{20} + 428944 q^{21} - 992712 q^{23} - 131072 q^{24} - 2453864 q^{25} + 32256 q^{26} + 2534676 q^{27} + 251844 q^{29} - 1795584 q^{30} - 416240 q^{31} - 9903104 q^{36} - 4355864 q^{37} - 10031116 q^{39} - 1966080 q^{40} - 8392608 q^{41} + 5376440 q^{43} + 2915328 q^{44} - 14853944 q^{45} + 2516480 q^{46} - 16343256 q^{47} + 5258824 q^{49} + 21485568 q^{50} - 18350080 q^{52} + 6243216 q^{53} + 6588928 q^{54} + 1314900 q^{55} + 3538944 q^{56} + 19523840 q^{58} - 49163016 q^{59} + 20085248 q^{60} + 41379472 q^{61} + 161583864 q^{65} - 74191872 q^{66} - 12103680 q^{68} + 275512 q^{69} - 23170560 q^{70} + 58192168 q^{73} + 35043840 q^{74} + 81727444 q^{75} - 42108416 q^{76} - 150645504 q^{77} - 36863488 q^{78} + 24665368 q^{79} + 71338424 q^{81} + 116544512 q^{82} + 368548152 q^{83} - 54904832 q^{84} - 172258432 q^{85} + 56674584 q^{87} + 131442648 q^{89} - 74817536 q^{90} - 240069120 q^{94} - 28308216 q^{95} + 415961944 q^{97} - 122136576 q^{98} + 543517908 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(58, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
58.9.c.a 58.c 29.c $20$ $23.628$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(-160\) \(-32\) \(0\) \(1728\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-8+8\beta _{1})q^{2}+(-2+2\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
58.9.c.b 58.c 29.c $20$ $23.628$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(160\) \(32\) \(0\) \(-1728\) $\mathrm{SU}(2)[C_{4}]$ \(q+(8+8\beta _{3})q^{2}+(2-\beta _{2}+2\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(58, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(58, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(29, [\chi])\)\(^{\oplus 2}\)