Properties

Label 1690.2.d
Level $1690$
Weight $2$
Character orbit 1690.d
Rep. character $\chi_{1690}(1351,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $12$
Sturm bound $546$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(546\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1690, [\chi])\).

Total New Old
Modular forms 302 54 248
Cusp forms 246 54 192
Eisenstein series 56 0 56

Trace form

\( 54q - 4q^{3} - 54q^{4} + 54q^{9} + O(q^{10}) \) \( 54q - 4q^{3} - 54q^{4} + 54q^{9} + 2q^{10} + 4q^{12} + 4q^{14} + 54q^{16} + 8q^{17} + 4q^{22} - 54q^{25} + 8q^{27} - 8q^{29} + 4q^{30} - 54q^{36} - 4q^{38} - 2q^{40} + 20q^{43} - 4q^{48} - 82q^{49} + 16q^{51} + 4q^{53} + 8q^{55} - 4q^{56} - 8q^{61} - 16q^{62} - 54q^{64} + 8q^{66} - 8q^{68} - 16q^{69} - 4q^{74} + 4q^{75} - 8q^{77} + 32q^{79} + 86q^{81} - 8q^{82} - 16q^{87} - 4q^{88} - 6q^{90} + 36q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1690, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1690.2.d.a \(2\) \(13.495\) \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(0\) \(0\) \(q+iq^{2}-2q^{3}-q^{4}+iq^{5}-2iq^{6}+\cdots\)
1690.2.d.b \(2\) \(13.495\) \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(0\) \(0\) \(q+iq^{2}-2q^{3}-q^{4}-iq^{5}-2iq^{6}+\cdots\)
1690.2.d.c \(2\) \(13.495\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+iq^{5}+3iq^{7}-iq^{8}+\cdots\)
1690.2.d.d \(2\) \(13.495\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+iq^{5}-iq^{8}-3q^{9}+\cdots\)
1690.2.d.e \(2\) \(13.495\) \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(0\) \(0\) \(q+iq^{2}+2q^{3}-q^{4}-iq^{5}+2iq^{6}+\cdots\)
1690.2.d.f \(4\) \(13.495\) \(\Q(\zeta_{12})\) None \(0\) \(-4\) \(0\) \(0\) \(q-\zeta_{12}q^{2}+(-1+\zeta_{12}^{3})q^{3}-q^{4}+\cdots\)
1690.2.d.g \(4\) \(13.495\) \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{1}q^{5}-\beta _{2}q^{6}+\cdots\)
1690.2.d.h \(4\) \(13.495\) \(\Q(\zeta_{12})\) None \(0\) \(4\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(1+\zeta_{12}^{3})q^{3}-q^{4}-\zeta_{12}q^{5}+\cdots\)
1690.2.d.i \(6\) \(13.495\) 6.0.153664.1 None \(0\) \(-2\) \(0\) \(0\) \(q+\beta _{5}q^{2}-\beta _{2}q^{3}-q^{4}-\beta _{5}q^{5}+(\beta _{1}+\cdots)q^{6}+\cdots\)
1690.2.d.j \(6\) \(13.495\) 6.0.153664.1 None \(0\) \(2\) \(0\) \(0\) \(q-\beta _{5}q^{2}+\beta _{4}q^{3}-q^{4}+\beta _{5}q^{5}-\beta _{1}q^{6}+\cdots\)
1690.2.d.k \(8\) \(13.495\) 8.0.22581504.2 None \(0\) \(4\) \(0\) \(0\) \(q-\beta _{5}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{5}q^{5}+\beta _{3}q^{6}+\cdots\)
1690.2.d.l \(12\) \(13.495\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-4\) \(0\) \(0\) \(q+\beta _{2}q^{2}+(-1+\beta _{5}+\beta _{8})q^{3}-q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1690, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1690, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(845, [\chi])\)\(^{\oplus 2}\)