Properties

Label 690.2.a
Level $690$
Weight $2$
Character orbit 690.a
Rep. character $\chi_{690}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $12$
Sturm bound $288$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(690))\).

Total New Old
Modular forms 152 13 139
Cusp forms 137 13 124
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.

\(2\)\(3\)\(5\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(10\)

Trace form

\( 13q + q^{2} - 3q^{3} + 13q^{4} + q^{5} + q^{6} + q^{8} + 13q^{9} + O(q^{10}) \) \( 13q + q^{2} - 3q^{3} + 13q^{4} + q^{5} + q^{6} + q^{8} + 13q^{9} + q^{10} + 12q^{11} - 3q^{12} - 2q^{13} - 8q^{14} + q^{15} + 13q^{16} + 2q^{17} + q^{18} + 12q^{19} + q^{20} - 4q^{22} + q^{23} + q^{24} + 13q^{25} + 14q^{26} - 3q^{27} + 6q^{29} + q^{30} + q^{32} - 4q^{33} + 18q^{34} + 13q^{36} - 2q^{37} + 4q^{38} - 10q^{39} + q^{40} + 18q^{41} + 8q^{42} - 12q^{43} + 12q^{44} + q^{45} + q^{46} - 3q^{48} - 11q^{49} + q^{50} + 2q^{51} - 2q^{52} - 10q^{53} + q^{54} + 12q^{55} - 8q^{56} - 4q^{57} + 22q^{58} + 4q^{59} + q^{60} + 6q^{61} + 13q^{64} + 14q^{65} + 12q^{66} - 4q^{67} + 2q^{68} + q^{69} + 32q^{71} + q^{72} + 34q^{73} - 26q^{74} - 3q^{75} + 12q^{76} - 48q^{77} + 6q^{78} - 24q^{79} + q^{80} + 13q^{81} - 22q^{82} - 28q^{83} + 2q^{85} - 20q^{86} - 2q^{87} - 4q^{88} - 38q^{89} + q^{90} + 32q^{91} + q^{92} - 32q^{93} - 32q^{94} + 4q^{95} + q^{96} - 22q^{97} - 7q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(690))\) 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 23
690.2.a.a \(1\) \(5.510\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
690.2.a.b \(1\) \(5.510\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(4\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
690.2.a.c \(1\) \(5.510\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
690.2.a.d \(1\) \(5.510\) \(\Q\) None \(-1\) \(-1\) \(1\) \(4\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
690.2.a.e \(1\) \(5.510\) \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
690.2.a.f \(1\) \(5.510\) \(\Q\) None \(-1\) \(1\) \(1\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
690.2.a.g \(1\) \(5.510\) \(\Q\) None \(1\) \(-1\) \(-1\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-2q^{7}+\cdots\)
690.2.a.h \(1\) \(5.510\) \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
690.2.a.i \(1\) \(5.510\) \(\Q\) None \(1\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
690.2.a.j \(1\) \(5.510\) \(\Q\) None \(1\) \(1\) \(1\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)
690.2.a.k \(1\) \(5.510\) \(\Q\) None \(1\) \(1\) \(1\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)
690.2.a.l \(2\) \(5.510\) \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(2\) \(-2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(690))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(690)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 2}\)