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Properties

Label 390.2.bd
Level $390$
Weight $2$
Character orbit 390.bd
Rep. character $\chi_{390}(7,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $56$
Newform subspaces $3$
Sturm bound $168$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bd (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 3 \)
Sturm bound: \(168\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(390, [\chi])\).

Total New Old
Modular forms 368 56 312
Cusp forms 304 56 248
Eisenstein series 64 0 64

Trace form

\( 56 q + 28 q^{4} + 4 q^{5} + O(q^{10}) \) \( 56 q + 28 q^{4} + 4 q^{5} - 8 q^{11} + 12 q^{13} + 8 q^{15} - 28 q^{16} + 8 q^{17} + 8 q^{18} - 40 q^{19} - 4 q^{20} + 16 q^{21} + 16 q^{23} + 36 q^{25} + 24 q^{31} - 24 q^{33} + 36 q^{34} + 48 q^{37} + 8 q^{39} - 12 q^{41} - 8 q^{43} + 8 q^{44} + 8 q^{46} - 48 q^{47} - 36 q^{49} + 32 q^{50} + 12 q^{53} - 88 q^{55} - 24 q^{56} - 20 q^{58} - 8 q^{59} - 8 q^{60} - 32 q^{61} - 72 q^{62} - 56 q^{64} + 24 q^{65} + 16 q^{66} - 72 q^{67} + 4 q^{68} + 16 q^{69} - 16 q^{70} - 32 q^{71} + 4 q^{72} - 60 q^{74} - 32 q^{76} - 48 q^{77} - 16 q^{78} - 8 q^{80} + 28 q^{81} + 44 q^{82} - 48 q^{83} - 16 q^{84} + 4 q^{85} + 36 q^{89} - 4 q^{90} - 48 q^{91} + 32 q^{92} - 24 q^{94} - 48 q^{95} + 144 q^{97} + 96 q^{98} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(390, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
390.2.bd.a 390.bd 65.t $8$ $3.114$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{2}q^{2}+\zeta_{24}^{5}q^{3}+\zeta_{24}^{4}q^{4}+\cdots\)
390.2.bd.b 390.bd 65.t $16$ $3.114$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(4\) \(4\) $\mathrm{SU}(2)[C_{12}]$ \(q+(\beta _{5}-\beta _{13})q^{2}+\beta _{2}q^{3}+(1+\beta _{14}+\cdots)q^{4}+\cdots\)
390.2.bd.c 390.bd 65.t $32$ $3.114$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(390, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(390, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)