# Properties

 Label 175.4.b Level $175$ Weight $4$ Character orbit 175.b Rep. character $\chi_{175}(99,\cdot)$ Character field $\Q$ Dimension $26$ Newform subspaces $6$ Sturm bound $80$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$80$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(175, [\chi])$$.

Total New Old
Modular forms 66 26 40
Cusp forms 54 26 28
Eisenstein series 12 0 12

## Trace form

 $$26q - 120q^{4} - 24q^{6} - 294q^{9} + O(q^{10})$$ $$26q - 120q^{4} - 24q^{6} - 294q^{9} - 80q^{11} + 56q^{14} + 392q^{16} - 96q^{19} - 196q^{21} - 116q^{24} + 452q^{26} + 276q^{29} - 168q^{31} - 1328q^{34} + 2196q^{36} - 560q^{39} + 1568q^{41} + 2726q^{44} - 1526q^{46} - 1274q^{49} - 4036q^{51} + 3868q^{54} - 714q^{56} - 92q^{59} + 3664q^{61} + 1314q^{64} - 2500q^{66} - 3048q^{69} - 3588q^{71} - 8318q^{74} + 1596q^{76} + 5300q^{79} + 1522q^{81} + 2800q^{84} + 2486q^{86} - 1984q^{89} - 224q^{91} + 7108q^{94} - 13428q^{96} - 52q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(175, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
175.4.b.a $$2$$ $$10.325$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+8iq^{3}+7q^{4}-8q^{6}+7iq^{7}+\cdots$$
175.4.b.b $$2$$ $$10.325$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-2iq^{3}+7q^{4}+2q^{6}+7iq^{7}+\cdots$$
175.4.b.c $$4$$ $$10.325$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(4\zeta_{8}-\zeta_{8}^{2})q^{2}+(-\zeta_{8}-4\zeta_{8}^{2})q^{3}+\cdots$$
175.4.b.d $$4$$ $$10.325$$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{1}-2\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots$$
175.4.b.e $$6$$ $$10.325$$ 6.0.3299353600.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{2})q^{2}+(-\beta _{1}+\beta _{2}+\beta _{5})q^{3}+\cdots$$
175.4.b.f $$8$$ $$10.325$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}+(-\beta _{2}-\beta _{5})q^{3}+(-9+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(175, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(175, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 2}$$