Properties

Label 390.2.l
Level $390$
Weight $2$
Character orbit 390.l
Rep. character $\chi_{390}(53,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $48$
Newform subspaces $4$
Sturm bound $168$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(168\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(7\), \(17\)

Dimensions

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

Total New Old
Modular forms 184 48 136
Cusp forms 152 48 104
Eisenstein series 32 0 32

Trace form

\( 48 q + 8 q^{3} + 8 q^{6} + 8 q^{7} + O(q^{10}) \) \( 48 q + 8 q^{3} + 8 q^{6} + 8 q^{7} - 8 q^{10} - 8 q^{12} - 16 q^{15} - 48 q^{16} + 8 q^{22} + 32 q^{25} + 20 q^{27} + 8 q^{28} + 12 q^{30} - 8 q^{33} - 8 q^{36} - 64 q^{37} - 20 q^{42} - 64 q^{43} + 56 q^{45} - 48 q^{46} - 8 q^{48} + 48 q^{51} - 8 q^{55} + 8 q^{57} + 56 q^{58} - 8 q^{60} + 64 q^{61} - 24 q^{63} - 32 q^{66} + 16 q^{67} + 24 q^{70} + 24 q^{73} - 52 q^{75} - 88 q^{81} - 16 q^{85} - 32 q^{87} + 8 q^{88} - 4 q^{90} - 16 q^{91} + 16 q^{93} - 8 q^{96} + 88 q^{97} + O(q^{100}) \)

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.l.a 390.l 15.e $4$ $3.114$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(\zeta_{8}+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.l.b 390.l 15.e $4$ $3.114$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1-\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.l.c 390.l 15.e $20$ $3.114$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{9}q^{2}+\beta _{1}q^{3}-\beta _{11}q^{4}+(\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots\)
390.2.l.d 390.l 15.e $20$ $3.114$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{4}q^{2}+\beta _{12}q^{3}+\beta _{11}q^{4}+(-\beta _{5}+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(390, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)