# 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 $10$

# Related objects

## 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 Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
676.2.l.a $4$ $5.398$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$-2$$ $$0$$ $$-12$$ $$0$$ $$q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
676.2.l.b $4$ $5.398$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-13})$$ $$-2$$ $$0$$ $$0$$ $$-2$$ $$q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}-2\zeta_{12}q^{4}+\cdots$$
676.2.l.c $4$ $5.398$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$-2$$ $$0$$ $$6$$ $$0$$ $$q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
676.2.l.d $4$ $5.398$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$2$$ $$0$$ $$-6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
676.2.l.e $4$ $5.398$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$2$$ $$0$$ $$-6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
676.2.l.f $4$ $5.398$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-13})$$ $$2$$ $$0$$ $$0$$ $$2$$ $$q+(-\zeta_{12}+\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}-2\zeta_{12}q^{4}+\cdots$$
676.2.l.g $4$ $5.398$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$2$$ $$0$$ $$12$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
676.2.l.h $16$ $5.398$ 16.0.$$\cdots$$.2 None $$-4$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{2}+\beta _{9})q^{2}+(-2\beta _{1}-\beta _{11}-\beta _{13}+\cdots)q^{3}+\cdots$$
676.2.l.i $16$ $5.398$ 16.0.$$\cdots$$.1 None $$-4$$ $$0$$ $$12$$ $$0$$ $$q+\beta _{9}q^{2}+(-\beta _{1}+\beta _{3}+\beta _{6}+\beta _{9}+\beta _{12}+\cdots)q^{3}+\cdots$$
676.2.l.j $16$ $5.398$ 16.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(2\beta _{2}+\beta _{5})q^{3}+(\beta _{10}+\beta _{11}+\cdots)q^{4}+\cdots$$
676.2.l.k $16$ $5.398$ 16.0.$$\cdots$$.1 None $$2$$ $$0$$ $$12$$ $$0$$ $$q+\beta _{12}q^{2}+(\beta _{3}+\beta _{12}-\beta _{13}+\beta _{14}+\cdots)q^{3}+\cdots$$
676.2.l.l $16$ $5.398$ 16.0.$$\cdots$$.2 None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{2}-\beta _{9})q^{2}+(-2\beta _{1}-\beta _{11}-\beta _{13}+\cdots)q^{3}+\cdots$$
676.2.l.m $16$ $5.398$ 16.0.$$\cdots$$.1 None $$4$$ $$0$$ $$-12$$ $$0$$ $$q-\beta _{9}q^{2}+(-\beta _{1}+\beta _{3}+\beta _{6}+\beta _{9}+\beta _{12}+\cdots)q^{3}+\cdots$$
676.2.l.n $144$ $5.398$ None $$0$$ $$0$$ $$0$$ $$0$$

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

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