Properties

Label 690.2.i
Level $690$
Weight $2$
Character orbit 690.i
Rep. character $\chi_{690}(47,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $88$
Newform subspaces $6$
Sturm bound $288$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.i (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

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

Total New Old
Modular forms 304 88 216
Cusp forms 272 88 184
Eisenstein series 32 0 32

Trace form

\( 88q + 8q^{7} + O(q^{10}) \) \( 88q + 8q^{7} - 8q^{10} + 16q^{13} - 24q^{15} - 88q^{16} - 16q^{21} - 8q^{22} + 8q^{28} + 24q^{30} + 32q^{31} + 16q^{33} + 16q^{36} + 16q^{37} - 16q^{40} - 8q^{42} + 16q^{45} + 48q^{51} - 16q^{52} + 40q^{55} + 48q^{57} + 40q^{58} - 32q^{61} - 64q^{63} - 48q^{66} - 32q^{67} + 8q^{70} - 8q^{73} - 16q^{75} - 32q^{76} + 24q^{78} + 16q^{81} + 64q^{82} - 80q^{85} - 32q^{87} - 8q^{88} - 56q^{90} + 96q^{91} + 88q^{93} - 24q^{97} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
690.2.i.a \(4\) \(5.510\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(-4\) \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
690.2.i.b \(4\) \(5.510\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(12\) \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
690.2.i.c \(8\) \(5.510\) \(\Q(\zeta_{24})\) None \(0\) \(-8\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{2}+(-1+\zeta_{24}+\zeta_{24}^{3})q^{3}+\cdots\)
690.2.i.d \(8\) \(5.510\) 8.0.40960000.1 None \(0\) \(-8\) \(0\) \(0\) \(q-\beta _{6}q^{2}+(-1+\beta _{1}+\beta _{3})q^{3}-\beta _{3}q^{4}+\cdots\)
690.2.i.e \(32\) \(5.510\) None \(0\) \(4\) \(0\) \(-8\)
690.2.i.f \(32\) \(5.510\) None \(0\) \(4\) \(0\) \(8\)

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

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