Properties

Label 390.2.e
Level $390$
Weight $2$
Character orbit 390.e
Rep. character $\chi_{390}(79,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $5$
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.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(168\)
Trace bound: \(10\)
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

\( 12 q - 12 q^{4} + 8 q^{5} - 4 q^{6} - 12 q^{9} + O(q^{10}) \) \( 12 q - 12 q^{4} + 8 q^{5} - 4 q^{6} - 12 q^{9} + 4 q^{10} - 16 q^{11} + 4 q^{15} + 12 q^{16} + 8 q^{19} - 8 q^{20} - 8 q^{21} + 4 q^{24} + 4 q^{25} + 8 q^{30} + 8 q^{31} + 16 q^{34} - 24 q^{35} + 12 q^{36} - 4 q^{40} + 16 q^{41} + 16 q^{44} - 8 q^{45} - 32 q^{46} - 12 q^{49} + 8 q^{51} + 4 q^{54} + 16 q^{55} + 16 q^{59} - 4 q^{60} - 40 q^{61} - 12 q^{64} - 8 q^{65} - 8 q^{69} - 8 q^{70} - 16 q^{74} - 8 q^{76} + 32 q^{79} + 8 q^{80} + 12 q^{81} + 8 q^{84} - 24 q^{85} + 32 q^{86} + 32 q^{89} - 4 q^{90} - 16 q^{94} + 24 q^{95} - 4 q^{96} + 16 q^{99} + O(q^{100}) \)

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.e.a 390.e 5.b $2$ $3.114$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+iq^{3}-q^{4}+(-2-i)q^{5}+\cdots\)
390.2.e.b 390.e 5.b $2$ $3.114$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+iq^{3}-q^{4}+(2-i)q^{5}-q^{6}+\cdots\)
390.2.e.c 390.e 5.b $2$ $3.114$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-iq^{3}-q^{4}+(2+i)q^{5}+q^{6}+\cdots\)
390.2.e.d 390.e 5.b $2$ $3.114$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-iq^{3}-q^{4}+(2-i)q^{5}+q^{6}+\cdots\)
390.2.e.e 390.e 5.b $4$ $3.114$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-\beta _{2}q^{5}-q^{6}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(390, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)