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Properties

Label 390.2.bb
Level $390$
Weight $2$
Character orbit 390.bb
Rep. character $\chi_{390}(121,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $3$
Sturm bound $168$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bb (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(168\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(7\)

Dimensions

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

Total New Old
Modular forms 184 16 168
Cusp forms 152 16 136
Eisenstein series 32 0 32

Trace form

\( 16 q + 8 q^{4} - 8 q^{9} + O(q^{10}) \) \( 16 q + 8 q^{4} - 8 q^{9} - 4 q^{10} + 12 q^{11} - 8 q^{13} + 24 q^{14} - 8 q^{16} + 16 q^{17} + 12 q^{19} - 16 q^{25} - 16 q^{26} - 8 q^{29} - 4 q^{30} + 4 q^{35} + 8 q^{36} + 24 q^{37} - 4 q^{39} - 8 q^{40} + 8 q^{42} - 8 q^{43} - 12 q^{46} + 12 q^{49} - 32 q^{51} + 8 q^{52} - 16 q^{53} - 12 q^{55} + 12 q^{56} - 24 q^{58} + 24 q^{59} - 8 q^{61} + 8 q^{62} - 16 q^{64} - 20 q^{65} - 8 q^{66} + 24 q^{67} - 16 q^{68} + 8 q^{69} - 72 q^{71} + 4 q^{74} + 12 q^{76} + 16 q^{77} + 16 q^{78} - 8 q^{79} - 8 q^{81} - 8 q^{82} - 8 q^{87} + 12 q^{89} + 8 q^{90} + 20 q^{91} + 48 q^{93} - 16 q^{94} + 16 q^{95} + 48 q^{98} + O(q^{100}) \)

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.bb.a 390.bb 13.e $4$ $3.114$ \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
390.2.bb.b 390.bb 13.e $4$ $3.114$ \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
390.2.bb.c 390.bb 13.e $8$ $3.114$ 8.0.\(\cdots\).1 None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{2}-\beta _{6}q^{3}+(1-\beta _{6})q^{4}-\beta _{2}q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(390, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)