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Properties

Label 690.2.d
Level $690$
Weight $2$
Character orbit 690.d
Rep. character $\chi_{690}(139,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $4$
Sturm bound $288$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(288\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(690, [\chi])\).

Total New Old
Modular forms 152 20 132
Cusp forms 136 20 116
Eisenstein series 16 0 16

Trace form

\( 20 q - 20 q^{4} - 4 q^{6} - 20 q^{9} + O(q^{10}) \) \( 20 q - 20 q^{4} - 4 q^{6} - 20 q^{9} + 4 q^{10} + 4 q^{15} + 20 q^{16} - 16 q^{19} + 16 q^{21} + 4 q^{24} - 4 q^{25} + 16 q^{26} - 24 q^{29} + 16 q^{31} - 16 q^{34} + 24 q^{35} + 20 q^{36} - 16 q^{39} - 4 q^{40} - 24 q^{41} - 8 q^{46} - 20 q^{49} - 8 q^{50} + 4 q^{54} + 24 q^{55} + 40 q^{59} - 4 q^{60} - 20 q^{64} + 16 q^{65} + 8 q^{69} - 8 q^{70} - 56 q^{71} + 16 q^{75} + 16 q^{76} - 24 q^{79} + 20 q^{81} - 16 q^{84} + 32 q^{86} + 16 q^{89} - 4 q^{90} - 16 q^{91} + 32 q^{94} - 40 q^{95} - 4 q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(690, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
690.2.d.a 690.d 5.b $4$ $5.510$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{3}q^{5}+q^{6}+\cdots\)
690.2.d.b 690.d 5.b $4$ $5.510$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}-\zeta_{8}^{2}q^{3}-q^{4}+(2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots\)
690.2.d.c 690.d 5.b $6$ $5.510$ 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{3}q^{5}-q^{6}+\cdots\)
690.2.d.d 690.d 5.b $6$ $5.510$ 6.0.5161984.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{4}q^{5}-q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(690, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(690, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(115, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(230, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(345, [\chi])\)\(^{\oplus 2}\)