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Properties

Label 690.3.k
Level $690$
Weight $3$
Character orbit 690.k
Rep. character $\chi_{690}(277,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $88$
Newform subspaces $2$
Sturm bound $432$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 690.k (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(432\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(690, [\chi])\).

Total New Old
Modular forms 592 88 504
Cusp forms 560 88 472
Eisenstein series 32 0 32

Trace form

\( 88 q - 8 q^{2} - 16 q^{5} - 16 q^{7} + 16 q^{8} + O(q^{10}) \) \( 88 q - 8 q^{2} - 16 q^{5} - 16 q^{7} + 16 q^{8} - 8 q^{10} - 8 q^{13} + 48 q^{15} - 352 q^{16} + 24 q^{17} - 24 q^{18} - 16 q^{20} - 96 q^{21} + 64 q^{22} + 24 q^{25} + 80 q^{26} + 32 q^{28} + 176 q^{31} + 32 q^{32} - 48 q^{33} + 48 q^{35} + 528 q^{36} + 88 q^{37} - 16 q^{40} - 208 q^{41} - 96 q^{42} + 24 q^{45} - 368 q^{47} - 120 q^{50} - 16 q^{52} + 152 q^{53} - 416 q^{55} + 96 q^{57} - 128 q^{58} + 64 q^{61} + 128 q^{62} - 48 q^{63} + 600 q^{65} - 32 q^{67} - 48 q^{68} + 128 q^{70} - 144 q^{71} - 48 q^{72} + 520 q^{73} - 288 q^{75} + 64 q^{76} + 176 q^{77} - 96 q^{78} + 64 q^{80} - 792 q^{81} - 128 q^{82} - 576 q^{83} - 296 q^{85} - 192 q^{86} + 144 q^{87} - 128 q^{88} + 24 q^{90} + 288 q^{91} + 480 q^{95} + 504 q^{97} + 760 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(690, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
690.3.k.a 690.k 5.c $40$ $18.801$ None \(40\) \(0\) \(-8\) \(-8\) $\mathrm{SU}(2)[C_{4}]$
690.3.k.b 690.k 5.c $48$ $18.801$ None \(-48\) \(0\) \(-8\) \(-8\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{3}^{\mathrm{old}}(690, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(690, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)