Properties

Label 58.2.d
Level $58$
Weight $2$
Character orbit 58.d
Rep. character $\chi_{58}(7,\cdot)$
Character field $\Q(\zeta_{7})$
Dimension $18$
Newform subspaces $2$
Sturm bound $15$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 29 \)
Character field: \(\Q(\zeta_{7})\)
Newform subspaces: \( 2 \)
Sturm bound: \(15\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(58, [\chi])\).

Total New Old
Modular forms 54 18 36
Cusp forms 30 18 12
Eisenstein series 24 0 24

Trace form

\( 18 q - q^{2} - 3 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{7} - q^{8} - 17 q^{9} + O(q^{10}) \) \( 18 q - q^{2} - 3 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{7} - q^{8} - 17 q^{9} - 10 q^{10} - 8 q^{11} + 12 q^{13} - 8 q^{14} - 18 q^{15} - 3 q^{16} - 22 q^{17} - 5 q^{18} - 12 q^{19} + 3 q^{20} + 2 q^{21} + 4 q^{22} + 28 q^{23} + 8 q^{24} + 11 q^{25} + 11 q^{26} + 48 q^{27} - 4 q^{28} + q^{29} + 56 q^{30} - 4 q^{31} - q^{32} + 30 q^{33} + 5 q^{34} + 4 q^{35} - 3 q^{36} + 18 q^{37} - 14 q^{38} - 20 q^{39} + 11 q^{40} - 42 q^{41} - 28 q^{42} - 12 q^{43} + 20 q^{44} - 23 q^{45} - 28 q^{46} - 30 q^{47} + 19 q^{49} + 5 q^{50} - 20 q^{51} - 16 q^{52} - 9 q^{53} - 12 q^{54} + 12 q^{55} - 8 q^{56} - 20 q^{57} - 15 q^{58} - 18 q^{60} + 26 q^{61} - 26 q^{62} + 2 q^{63} - 3 q^{64} - 11 q^{65} - 48 q^{66} + 4 q^{67} - 8 q^{68} - 8 q^{69} - 12 q^{70} + 26 q^{71} - 5 q^{72} - 17 q^{73} + 44 q^{74} + 24 q^{75} + 16 q^{76} + 22 q^{77} + 60 q^{78} - 60 q^{79} - 4 q^{80} + 65 q^{81} + 10 q^{82} - 4 q^{83} + 44 q^{84} + 12 q^{85} + 44 q^{86} + 20 q^{87} + 4 q^{88} + 52 q^{89} - 24 q^{90} + 60 q^{91} + 28 q^{92} - 66 q^{93} + 20 q^{94} + 80 q^{95} - 6 q^{96} + q^{97} + 55 q^{98} + 40 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(58, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
58.2.d.a 58.d 29.d $6$ $0.463$ \(\Q(\zeta_{14})\) None 58.2.d.a \(1\) \(3\) \(-4\) \(-5\) $\mathrm{SU}(2)[C_{7}]$ \(q-\zeta_{14}^{4}q^{2}+(1-\zeta_{14}+\zeta_{14}^{4}-\zeta_{14}^{5})q^{3}+\cdots\)
58.2.d.b 58.d 29.d $12$ $0.463$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 58.2.d.b \(-2\) \(-3\) \(0\) \(1\) $\mathrm{SU}(2)[C_{7}]$ \(q+\beta _{2}q^{2}+\beta _{11}q^{3}+\beta _{6}q^{4}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(58, [\chi])\) into lower level spaces

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