# Properties

 Label 637.2.e Level $637$ Weight $2$ Character orbit 637.e Rep. character $\chi_{637}(79,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $80$ Newform subspaces $15$ Sturm bound $130$ Trace bound $5$

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## Defining parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$15$$ Sturm bound: $$130$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(637, [\chi])$$.

Total New Old
Modular forms 148 80 68
Cusp forms 116 80 36
Eisenstein series 32 0 32

## Trace form

 $$80 q + 4 q^{2} - 36 q^{4} + 2 q^{5} - 12 q^{8} - 44 q^{9} + O(q^{10})$$ $$80 q + 4 q^{2} - 36 q^{4} + 2 q^{5} - 12 q^{8} - 44 q^{9} - 10 q^{10} + 6 q^{11} - 10 q^{12} - 4 q^{13} + 20 q^{15} - 12 q^{16} - 4 q^{17} + 10 q^{18} + 8 q^{19} - 32 q^{20} - 68 q^{22} - 2 q^{23} + 20 q^{24} - 34 q^{25} + 6 q^{26} - 4 q^{29} + 30 q^{30} + 4 q^{31} + 46 q^{32} + 8 q^{33} + 32 q^{34} + 4 q^{36} + 24 q^{37} - 26 q^{38} + 8 q^{40} - 28 q^{41} + 12 q^{43} + 18 q^{44} - 32 q^{45} + 52 q^{46} + 2 q^{47} + 76 q^{48} - 76 q^{50} - 20 q^{51} + 6 q^{52} + 26 q^{53} + 18 q^{54} - 30 q^{58} + 10 q^{59} - 24 q^{60} - 12 q^{61} + 76 q^{62} - 28 q^{64} + 4 q^{65} + 6 q^{66} + 24 q^{67} - 18 q^{68} - 56 q^{69} - 60 q^{71} + 28 q^{72} + 22 q^{73} - 10 q^{74} - 20 q^{75} - 48 q^{76} + 20 q^{78} + 34 q^{79} + 70 q^{80} - 72 q^{81} + 44 q^{82} - 12 q^{83} - 60 q^{85} + 14 q^{86} - 28 q^{87} - 4 q^{89} + 88 q^{90} + 140 q^{92} + 66 q^{93} + 46 q^{94} + 22 q^{95} - 22 q^{96} + 68 q^{97} + 72 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(637, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
637.2.e.a $2$ $5.086$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-3q^{8}+3\zeta_{6}q^{9}+\cdots$$
637.2.e.b $2$ $5.086$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$-3$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots$$
637.2.e.c $2$ $5.086$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$3$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots$$
637.2.e.d $2$ $5.086$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-3$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots$$
637.2.e.e $2$ $5.086$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$3$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots$$
637.2.e.f $4$ $5.086$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(-3+\beta _{1}+\cdots)q^{5}+\cdots$$
637.2.e.g $4$ $5.086$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(3-\beta _{1}+\cdots)q^{5}+\cdots$$
637.2.e.h $4$ $5.086$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$3$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{1}+\beta _{3})q^{2}+(2\beta _{1}+2\beta _{2}-\beta _{3})q^{3}+\cdots$$
637.2.e.i $6$ $5.086$ 6.0.2696112.1 None $$-1$$ $$-2$$ $$2$$ $$0$$ $$q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4}+\beta _{5})q^{3}+\cdots$$
637.2.e.j $6$ $5.086$ 6.0.2696112.1 None $$-1$$ $$2$$ $$-2$$ $$0$$ $$q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}-\beta _{5})q^{3}+\cdots$$
637.2.e.k $6$ $5.086$ 6.0.4406832.1 None $$2$$ $$-4$$ $$-5$$ $$0$$ $$q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(-1+\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots$$
637.2.e.l $6$ $5.086$ 6.0.4406832.1 None $$2$$ $$4$$ $$5$$ $$0$$ $$q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(1-\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots$$
637.2.e.m $10$ $5.086$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-4$$ $$0$$ $$2$$ $$0$$ $$q+(\beta _{1}-\beta _{7})q^{2}+(\beta _{4}-\beta _{9})q^{3}+(-2+\cdots)q^{4}+\cdots$$
637.2.e.n $12$ $5.086$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-8$$ $$-6$$ $$0$$ $$q+(-\beta _{3}+\beta _{11})q^{2}+(\beta _{1}+\beta _{2}-\beta _{8}+\cdots)q^{3}+\cdots$$
637.2.e.o $12$ $5.086$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$8$$ $$6$$ $$0$$ $$q+(-\beta _{3}+\beta _{11})q^{2}+(-\beta _{1}-\beta _{2}+\beta _{8}+\cdots)q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(637, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(637, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(91, [\chi])$$$$^{\oplus 2}$$