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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 13
390.2.a.a \(1\) \(3.114\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
390.2.a.b \(1\) \(3.114\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
390.2.a.c \(1\) \(3.114\) \(\Q\) None \(-1\) \(1\) \(-1\) \(4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
390.2.a.d \(1\) \(3.114\) \(\Q\) None \(-1\) \(1\) \(1\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.e \(1\) \(3.114\) \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.f \(1\) \(3.114\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
390.2.a.g \(1\) \(3.114\) \(\Q\) None \(1\) \(1\) \(-1\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
390.2.a.h \(2\) \(3.114\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)

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

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