Properties

Label 390.2.b
Level $390$
Weight $2$
Character orbit 390.b
Rep. character $\chi_{390}(181,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $4$
Sturm bound $168$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(168\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 92 12 80
Cusp forms 76 12 64
Eisenstein series 16 0 16

Trace form

\( 12 q - 4 q^{3} - 12 q^{4} + 12 q^{9} + O(q^{10}) \) \( 12 q - 4 q^{3} - 12 q^{4} + 12 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{13} + 12 q^{16} + 8 q^{17} - 12 q^{25} + 4 q^{26} - 4 q^{27} + 8 q^{29} + 4 q^{30} + 8 q^{35} - 12 q^{36} - 24 q^{38} + 8 q^{39} - 4 q^{40} - 8 q^{42} + 16 q^{43} - 4 q^{48} - 44 q^{49} + 8 q^{51} - 4 q^{52} - 8 q^{53} + 24 q^{61} - 8 q^{62} - 12 q^{64} - 4 q^{65} + 8 q^{66} - 8 q^{68} - 8 q^{69} + 8 q^{74} + 4 q^{75} - 64 q^{77} + 8 q^{78} + 12 q^{81} + 8 q^{82} + 8 q^{87} + 4 q^{90} + 40 q^{91} + 16 q^{94} + 32 q^{95} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
390.2.b.a 390.b 13.b $2$ $3.114$ \(\Q(\sqrt{-1}) \) None 390.2.b.a \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}-q^{3}-q^{4}+iq^{5}+iq^{6}+iq^{8}+\cdots\)
390.2.b.b 390.b 13.b $2$ $3.114$ \(\Q(\sqrt{-1}) \) None 390.2.b.b \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}-q^{3}-q^{4}+iq^{5}+iq^{6}+iq^{8}+\cdots\)
390.2.b.c 390.b 13.b $4$ $3.114$ \(\Q(i, \sqrt{13})\) None 390.2.b.c \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-q^{3}-q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)
390.2.b.d 390.b 13.b $4$ $3.114$ \(\Q(i, \sqrt{17})\) None 390.2.b.d \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+q^{3}-q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(390, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)