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Properties

Label 1690.2.l
Level $1690$
Weight $2$
Character orbit 1690.l
Rep. character $\chi_{1690}(361,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $100$
Newform subspaces $14$
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.l (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 14 \)
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 604 100 504
Cusp forms 492 100 392
Eisenstein series 112 0 112

Trace form

\( 100 q + 50 q^{4} - 50 q^{9} + O(q^{10}) \) \( 100 q + 50 q^{4} - 50 q^{9} - 2 q^{10} + 6 q^{11} + 20 q^{14} - 12 q^{15} - 50 q^{16} + 4 q^{17} + 18 q^{19} - 4 q^{22} - 100 q^{25} - 24 q^{27} - 16 q^{29} - 4 q^{30} + 60 q^{33} + 6 q^{35} + 50 q^{36} + 12 q^{37} + 16 q^{38} - 4 q^{40} + 24 q^{41} + 12 q^{42} - 12 q^{46} + 56 q^{49} - 40 q^{51} - 40 q^{53} - 36 q^{54} - 20 q^{55} + 10 q^{56} - 12 q^{58} - 12 q^{59} - 36 q^{61} - 20 q^{62} - 48 q^{63} - 100 q^{64} + 40 q^{66} - 24 q^{67} - 4 q^{68} + 4 q^{69} - 24 q^{72} - 26 q^{74} + 18 q^{76} + 8 q^{77} + 8 q^{79} - 34 q^{81} + 8 q^{82} - 12 q^{84} + 36 q^{85} - 32 q^{87} + 4 q^{88} + 42 q^{89} - 12 q^{90} - 36 q^{93} - 18 q^{94} - 36 q^{97} + 48 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1690.2.l.a 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}-2\zeta_{12}^{2}q^{3}+(1+\cdots)q^{4}+\cdots\)
1690.2.l.b 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(-2+2\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
1690.2.l.c 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
1690.2.l.d 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
1690.2.l.e 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2})q^{4}+\cdots\)
1690.2.l.f 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2})q^{4}+\cdots\)
1690.2.l.g 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
1690.2.l.h 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+2\zeta_{12}^{2}q^{3}+\cdots\)
1690.2.l.i 1690.l 13.e $4$ $13.495$ \(\Q(\zeta_{12})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(2-2\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
1690.2.l.j 1690.l 13.e $8$ $13.495$ 8.0.22581504.2 None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(\beta _{2}-\beta _{4}-\beta _{7})q^{3}+(1-\beta _{4}+\cdots)q^{4}+\cdots\)
1690.2.l.k 1690.l 13.e $8$ $13.495$ 8.0.3317760000.2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{1}q^{2}+(-\beta _{4}+\beta _{7})q^{3}+\beta _{2}q^{4}+\cdots\)
1690.2.l.l 1690.l 13.e $12$ $13.495$ 12.0.\(\cdots\).1 None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{10}q^{2}+(-1+\beta _{3}+\beta _{5}-\beta _{9})q^{3}+\cdots\)
1690.2.l.m 1690.l 13.e $12$ $13.495$ 12.0.\(\cdots\).1 None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{6}q^{2}-\beta _{9}q^{3}+\beta _{7}q^{4}+(\beta _{6}-\beta _{10}+\cdots)q^{5}+\cdots\)
1690.2.l.n 1690.l 13.e $24$ $13.495$ None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

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

\( S_{2}^{\mathrm{old}}(1690, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(845, [\chi])\)\(^{\oplus 2}\)