Properties

Label 390.2.a
Level $390$
Weight $2$
Character orbit 390.a
Rep. character $\chi_{390}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $8$
Sturm bound $168$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(168\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(11\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(390))\).

Total New Old
Modular forms 92 9 83
Cusp forms 77 9 68
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(8\)

Trace form

\( 9 q + q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 9 q^{9} + O(q^{10}) \) \( 9 q + q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 9 q^{9} + q^{10} + 12 q^{11} + q^{12} - 3 q^{13} + q^{15} + 9 q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + q^{20} + 8 q^{21} + 4 q^{22} + 8 q^{23} + q^{24} + 9 q^{25} + q^{26} + q^{27} + 8 q^{28} - 10 q^{29} + q^{30} + q^{32} - 12 q^{33} + 2 q^{34} - 8 q^{35} + 9 q^{36} - 18 q^{37} - 4 q^{38} + q^{39} + q^{40} - 6 q^{41} - 8 q^{42} + 12 q^{43} + 12 q^{44} + q^{45} + 8 q^{46} - 16 q^{47} + q^{48} - 15 q^{49} + q^{50} - 6 q^{51} - 3 q^{52} - 18 q^{53} + q^{54} + 4 q^{55} + 12 q^{57} - 10 q^{58} - 20 q^{59} + q^{60} - 2 q^{61} - 8 q^{62} + 8 q^{63} + 9 q^{64} + q^{65} - 4 q^{66} - 4 q^{67} - 6 q^{68} - 16 q^{69} + 8 q^{71} + q^{72} - 22 q^{73} - 26 q^{74} + q^{75} + 4 q^{76} - 32 q^{77} - 3 q^{78} - 16 q^{79} + q^{80} + 9 q^{81} - 30 q^{82} - 12 q^{83} + 8 q^{84} - 6 q^{85} - 4 q^{86} - 10 q^{87} + 4 q^{88} - 54 q^{89} + q^{90} + 8 q^{92} + 8 q^{93} - 16 q^{94} + 4 q^{95} + q^{96} - 30 q^{97} - 7 q^{98} + 12 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(390))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 13
390.2.a.a 390.a 1.a $1$ $3.114$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
390.2.a.b 390.a 1.a $1$ $3.114$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
390.2.a.c 390.a 1.a $1$ $3.114$ \(\Q\) None \(-1\) \(1\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
390.2.a.d 390.a 1.a $1$ $3.114$ \(\Q\) None \(-1\) \(1\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.e 390.a 1.a $1$ $3.114$ \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.f 390.a 1.a $1$ $3.114$ \(\Q\) None \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
390.2.a.g 390.a 1.a $1$ $3.114$ \(\Q\) None \(1\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
390.2.a.h 390.a 1.a $2$ $3.114$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(390))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(390)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 2}\)