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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

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

Display 100 coefficients 10 coefficients

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

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

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