Properties

Label 676.2.l
Level $676$
Weight $2$
Character orbit 676.l
Rep. character $\chi_{676}(19,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $268$
Newform subspaces $14$
Sturm bound $182$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.l (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 52 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 14 \)
Sturm bound: \(182\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(17\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(676, [\chi])\).

Total New Old
Modular forms 420 348 72
Cusp forms 308 268 40
Eisenstein series 112 80 32

Trace form

\( 268 q + 4 q^{2} + 6 q^{4} + 6 q^{5} + 14 q^{6} - 2 q^{8} + 98 q^{9} + O(q^{10}) \) \( 268 q + 4 q^{2} + 6 q^{4} + 6 q^{5} + 14 q^{6} - 2 q^{8} + 98 q^{9} + 6 q^{10} - 56 q^{14} - 14 q^{16} + 12 q^{17} - 6 q^{18} - 14 q^{20} + 28 q^{21} + 8 q^{22} - 10 q^{24} - 12 q^{28} + 4 q^{29} - 42 q^{30} - 36 q^{32} + 20 q^{33} - 10 q^{34} + 6 q^{36} - 10 q^{37} + 32 q^{40} - 20 q^{41} + 48 q^{42} + 8 q^{44} - 20 q^{45} + 46 q^{46} - 82 q^{48} - 60 q^{49} + 6 q^{50} - 108 q^{53} + 16 q^{54} + 60 q^{56} - 12 q^{57} + 74 q^{58} + 24 q^{60} - 22 q^{61} + 18 q^{62} - 128 q^{66} - 128 q^{68} + 12 q^{69} - 28 q^{70} - 68 q^{72} - 42 q^{73} - 2 q^{74} - 22 q^{76} - 68 q^{80} + 18 q^{81} - 54 q^{82} - 84 q^{84} + 22 q^{85} - 16 q^{86} - 36 q^{88} + 58 q^{89} - 52 q^{92} + 92 q^{93} + 70 q^{94} + 72 q^{96} + 38 q^{97} + 16 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(676, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
676.2.l.a 676.l 52.l $4$ $5.398$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(-2\) \(0\) \(-12\) \(0\) $\mathrm{U}(1)[D_{12}]$ \(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
676.2.l.b 676.l 52.l $4$ $5.398$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-13}) \) \(-2\) \(0\) \(0\) \(-2\) $\mathrm{U}(1)[D_{12}]$ \(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}-2\zeta_{12}q^{4}+\cdots\)
676.2.l.c 676.l 52.l $4$ $5.398$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(-2\) \(0\) \(6\) \(0\) $\mathrm{U}(1)[D_{12}]$ \(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
676.2.l.d 676.l 52.l $4$ $5.398$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(-6\) \(0\) $\mathrm{U}(1)[D_{12}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
676.2.l.e 676.l 52.l $4$ $5.398$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(-6\) \(0\) $\mathrm{U}(1)[D_{12}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
676.2.l.f 676.l 52.l $4$ $5.398$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-13}) \) \(2\) \(0\) \(0\) \(2\) $\mathrm{U}(1)[D_{12}]$ \(q+(-\zeta_{12}+\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}-2\zeta_{12}q^{4}+\cdots\)
676.2.l.g 676.l 52.l $4$ $5.398$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(12\) \(0\) $\mathrm{U}(1)[D_{12}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
676.2.l.h 676.l 52.l $16$ $5.398$ 16.0.\(\cdots\).2 None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta _{2}+\beta _{9})q^{2}+(-2\beta _{1}-\beta _{11}-\beta _{13}+\cdots)q^{3}+\cdots\)
676.2.l.i 676.l 52.l $16$ $5.398$ 16.0.\(\cdots\).1 None \(-4\) \(0\) \(12\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{9}q^{2}+(-\beta _{1}+\beta _{3}+\beta _{6}+\beta _{9}+\beta _{12}+\cdots)q^{3}+\cdots\)
676.2.l.j 676.l 52.l $16$ $5.398$ 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{1}q^{2}+(2\beta _{2}+\beta _{5})q^{3}+(\beta _{10}+\beta _{11}+\cdots)q^{4}+\cdots\)
676.2.l.k 676.l 52.l $16$ $5.398$ 16.0.\(\cdots\).1 None \(2\) \(0\) \(12\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{12}q^{2}+(\beta _{3}+\beta _{12}-\beta _{13}+\beta _{14}+\cdots)q^{3}+\cdots\)
676.2.l.l 676.l 52.l $16$ $5.398$ 16.0.\(\cdots\).2 None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(\beta _{2}-\beta _{9})q^{2}+(-2\beta _{1}-\beta _{11}-\beta _{13}+\cdots)q^{3}+\cdots\)
676.2.l.m 676.l 52.l $16$ $5.398$ 16.0.\(\cdots\).1 None \(4\) \(0\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q-\beta _{9}q^{2}+(-\beta _{1}+\beta _{3}+\beta _{6}+\beta _{9}+\beta _{12}+\cdots)q^{3}+\cdots\)
676.2.l.n 676.l 52.l $144$ $5.398$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(676, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(676, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 2}\)