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Properties

Label 416.2.bg
Level $416$
Weight $2$
Character orbit 416.bg
Rep. character $\chi_{416}(77,\cdot)$
Character field $\Q(\zeta_{8})$
Dimension $216$
Newform subspaces $1$
Sturm bound $112$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.bg (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 416 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 1 \)
Sturm bound: \(112\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(416, [\chi])\).

Total New Old
Modular forms 232 232 0
Cusp forms 216 216 0
Eisenstein series 16 16 0

Trace form

\( 216 q - 8 q^{3} - 8 q^{4} - 8 q^{9} + O(q^{10}) \) \( 216 q - 8 q^{3} - 8 q^{4} - 8 q^{9} - 8 q^{10} - 24 q^{12} - 4 q^{13} + 24 q^{14} - 8 q^{16} + 8 q^{22} - 8 q^{23} - 8 q^{25} + 16 q^{26} - 56 q^{27} - 8 q^{29} - 24 q^{30} - 8 q^{35} + 16 q^{36} - 48 q^{38} - 28 q^{39} + 32 q^{40} + 32 q^{42} - 8 q^{43} - 112 q^{48} + 16 q^{51} - 12 q^{52} - 8 q^{53} - 40 q^{55} - 48 q^{56} - 8 q^{61} + 56 q^{62} - 8 q^{64} - 8 q^{65} - 56 q^{66} - 8 q^{69} - 8 q^{74} - 16 q^{75} - 8 q^{77} + 84 q^{78} - 88 q^{82} - 8 q^{87} + 72 q^{88} + 88 q^{90} + 44 q^{91} - 80 q^{92} + 24 q^{94} + 112 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(416, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
416.2.bg.a 416.bg 416.ag $216$ $3.322$ None \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{8}]$