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Properties

Label 413.2.i
Level $413$
Weight $2$
Character orbit 413.i
Rep. character $\chi_{413}(15,\cdot)$
Character field $\Q(\zeta_{29})$
Dimension $840$
Newform subspaces $2$
Sturm bound $80$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 413 = 7 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 413.i (of order \(29\) and degree \(28\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 59 \)
Character field: \(\Q(\zeta_{29})\)
Newform subspaces: \( 2 \)
Sturm bound: \(80\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(413, [\chi])\).

Total New Old
Modular forms 1176 840 336
Cusp forms 1064 840 224
Eisenstein series 112 0 112

Trace form

\( 840 q - 2 q^{2} - 8 q^{3} - 30 q^{4} - 4 q^{5} - 16 q^{6} - 18 q^{8} - 46 q^{9} + O(q^{10}) \) \( 840 q - 2 q^{2} - 8 q^{3} - 30 q^{4} - 4 q^{5} - 16 q^{6} - 18 q^{8} - 46 q^{9} - 16 q^{10} - 16 q^{11} - 56 q^{12} - 4 q^{13} - 4 q^{14} - 28 q^{15} - 46 q^{16} - 20 q^{17} - 30 q^{18} - 16 q^{19} - 16 q^{20} - 28 q^{22} - 36 q^{23} - 36 q^{24} - 46 q^{25} - 12 q^{26} - 68 q^{27} - 8 q^{29} - 32 q^{30} - 32 q^{31} - 38 q^{32} - 44 q^{33} - 28 q^{34} - 4 q^{35} - 86 q^{36} - 20 q^{37} - 40 q^{38} - 80 q^{39} - 76 q^{40} - 56 q^{41} - 4 q^{42} - 64 q^{43} - 76 q^{44} - 34 q^{45} - 64 q^{46} + 2 q^{47} + 128 q^{48} - 30 q^{49} - 102 q^{50} + 156 q^{51} + 132 q^{52} - 68 q^{53} + 278 q^{54} + 86 q^{55} - 12 q^{56} - 100 q^{57} - 16 q^{58} + 56 q^{59} - 156 q^{60} + 92 q^{61} + 64 q^{62} - 8 q^{63} - 146 q^{64} + 82 q^{65} + 278 q^{66} - 84 q^{67} + 68 q^{68} + 116 q^{69} - 28 q^{70} - 60 q^{71} + 234 q^{72} - 26 q^{73} - 88 q^{74} - 94 q^{75} - 140 q^{76} - 16 q^{77} - 160 q^{78} - 76 q^{79} - 124 q^{80} - 214 q^{81} - 168 q^{82} - 44 q^{83} - 40 q^{85} - 116 q^{86} - 72 q^{87} - 228 q^{88} - 64 q^{89} - 148 q^{90} - 16 q^{91} - 176 q^{92} - 152 q^{93} - 184 q^{94} - 72 q^{95} - 128 q^{96} - 108 q^{97} - 2 q^{98} + 46 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(413, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
413.2.i.a 413.i 59.c $420$ $3.298$ None 413.2.i.a \(-3\) \(-4\) \(-4\) \(-15\) $\mathrm{SU}(2)[C_{29}]$
413.2.i.b 413.i 59.c $420$ $3.298$ None 413.2.i.b \(1\) \(-4\) \(0\) \(15\) $\mathrm{SU}(2)[C_{29}]$

Decomposition of \(S_{2}^{\mathrm{old}}(413, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(413, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(59, [\chi])\)\(^{\oplus 2}\)