# Properties

 Label 975.2.bc Level $975$ Weight $2$ Character orbit 975.bc Rep. character $\chi_{975}(751,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $90$ Newform subspaces $14$ Sturm bound $280$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.bc (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$14$$ Sturm bound: $$280$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(975, [\chi])$$.

Total New Old
Modular forms 304 90 214
Cusp forms 256 90 166
Eisenstein series 48 0 48

## Trace form

 $$90q + q^{3} + 46q^{4} - 3q^{7} - 45q^{9} + O(q^{10})$$ $$90q + q^{3} + 46q^{4} - 3q^{7} - 45q^{9} + 18q^{11} + 12q^{12} - 5q^{13} + 16q^{14} - 36q^{16} + 12q^{17} - 24q^{19} - 12q^{22} - 2q^{23} + 52q^{26} - 2q^{27} + 6q^{28} + 18q^{29} - 60q^{32} + 6q^{33} + 46q^{36} + 12q^{37} - 8q^{39} - 48q^{41} + 20q^{42} - 19q^{43} - 84q^{46} + 20q^{48} + 74q^{49} - 16q^{51} - 26q^{52} + 8q^{53} + 4q^{56} + 24q^{58} + 54q^{59} - 37q^{61} + 52q^{62} + 3q^{63} - 80q^{64} + 32q^{66} + 15q^{67} - 24q^{68} + 6q^{69} - 42q^{71} - 4q^{74} - 48q^{76} - 52q^{77} + 8q^{78} + 42q^{79} - 45q^{81} + 16q^{82} + 30q^{84} + 10q^{87} - 8q^{88} - 84q^{89} - 39q^{91} + 16q^{92} + 3q^{93} - 40q^{94} + 27q^{97} + 96q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(975, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
975.2.bc.a $$2$$ $$7.785$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$-1$$ $$0$$ $$0$$ $$q+(-1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
975.2.bc.b $$2$$ $$7.785$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$1$$ $$0$$ $$-3$$ $$q+(-1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
975.2.bc.c $$2$$ $$7.785$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$3$$ $$q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(2-\zeta_{6})q^{7}+\cdots$$
975.2.bc.d $$2$$ $$7.785$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$6$$ $$q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(4-2\zeta_{6})q^{7}+\cdots$$
975.2.bc.e $$2$$ $$7.785$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$0$$ $$-6$$ $$q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-4+2\zeta_{6})q^{7}+\cdots$$
975.2.bc.f $$2$$ $$7.785$$ $$\Q(\sqrt{-3})$$ None $$3$$ $$-1$$ $$0$$ $$3$$ $$q+(1+\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
975.2.bc.g $$2$$ $$7.785$$ $$\Q(\sqrt{-3})$$ None $$3$$ $$1$$ $$0$$ $$0$$ $$q+(1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
975.2.bc.h $$4$$ $$7.785$$ $$\Q(\zeta_{12})$$ None $$6$$ $$2$$ $$0$$ $$12$$ $$q+(2-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{2}q^{3}+\cdots$$
975.2.bc.i $$8$$ $$7.785$$ 8.0.56070144.2 None $$-6$$ $$4$$ $$0$$ $$-6$$ $$q+(-1-\beta _{2}+\beta _{5}+\beta _{6})q^{2}+\beta _{6}q^{3}+\cdots$$
975.2.bc.j $$8$$ $$7.785$$ 8.0.191102976.5 None $$0$$ $$-4$$ $$0$$ $$-12$$ $$q+(-\beta _{5}-\beta _{6})q^{2}-\beta _{1}q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots$$
975.2.bc.k $$12$$ $$7.785$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$-6$$ $$0$$ $$3$$ $$q+(\beta _{1}-\beta _{3})q^{2}+\beta _{6}q^{3}+(1-\beta _{2}+\beta _{6}+\cdots)q^{4}+\cdots$$
975.2.bc.l $$12$$ $$7.785$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$6$$ $$0$$ $$-3$$ $$q+\beta _{3}q^{2}+(1+\beta _{6})q^{3}+(-\beta _{6}-\beta _{10}+\cdots)q^{4}+\cdots$$
975.2.bc.m $$16$$ $$7.785$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$-8$$ $$0$$ $$-6$$ $$q+(-\beta _{1}-\beta _{5})q^{2}+(-1-\beta _{3})q^{3}+(-\beta _{3}+\cdots)q^{4}+\cdots$$
975.2.bc.n $$16$$ $$7.785$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$8$$ $$0$$ $$6$$ $$q+\beta _{5}q^{2}-\beta _{3}q^{3}+(1-\beta _{2}+\beta _{3}+\beta _{13}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(975, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(975, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(195, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(325, [\chi])$$$$^{\oplus 2}$$