Properties

Label 414.2.f
Level $414$
Weight $2$
Character orbit 414.f
Rep. character $\chi_{414}(137,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $48$
Newform subspaces $1$
Sturm bound $144$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 207 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(144\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(414, [\chi])\).

Total New Old
Modular forms 152 48 104
Cusp forms 136 48 88
Eisenstein series 16 0 16

Trace form

\( 48 q + 4 q^{3} + 24 q^{4} - 8 q^{6} + O(q^{10}) \) \( 48 q + 4 q^{3} + 24 q^{4} - 8 q^{6} - 4 q^{12} - 24 q^{16} + 16 q^{18} - 6 q^{23} - 4 q^{24} - 12 q^{25} - 20 q^{27} + 48 q^{29} + 12 q^{31} - 24 q^{39} + 60 q^{41} - 12 q^{46} - 48 q^{47} - 8 q^{48} + 12 q^{49} - 24 q^{50} + 8 q^{54} + 24 q^{55} + 72 q^{59} - 48 q^{64} - 40 q^{69} - 12 q^{70} + 8 q^{72} - 56 q^{75} + 12 q^{77} - 8 q^{78} - 80 q^{81} - 112 q^{87} - 6 q^{92} + 20 q^{93} + 12 q^{94} - 12 q^{95} + 4 q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(414, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
414.2.f.a 414.f 207.g $48$ $3.306$ None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(414, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(414, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(207, [\chi])\)\(^{\oplus 2}\)