# Properties

 Label 175.7.d Level $175$ Weight $7$ Character orbit 175.d Rep. character $\chi_{175}(76,\cdot)$ Character field $\Q$ Dimension $73$ Newform subspaces $9$ Sturm bound $140$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 175.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$140$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(175, [\chi])$$.

Total New Old
Modular forms 126 79 47
Cusp forms 114 73 41
Eisenstein series 12 6 6

## Trace form

 $$73 q - 3 q^{2} + 2197 q^{4} + 477 q^{7} - 1071 q^{8} - 14631 q^{9} + O(q^{10})$$ $$73 q - 3 q^{2} + 2197 q^{4} + 477 q^{7} - 1071 q^{8} - 14631 q^{9} - 2358 q^{11} + 6869 q^{14} + 65805 q^{16} + 27313 q^{18} + 3876 q^{21} + 27166 q^{22} - 10422 q^{23} + 87361 q^{28} - 85790 q^{29} - 244047 q^{32} - 221475 q^{36} + 128802 q^{37} + 123432 q^{39} - 168540 q^{42} - 174462 q^{43} - 507496 q^{44} + 230816 q^{46} + 219173 q^{49} - 43608 q^{51} + 515586 q^{53} + 326251 q^{56} + 522720 q^{57} - 97946 q^{58} - 426547 q^{63} + 1160931 q^{64} - 1389038 q^{67} - 814158 q^{71} + 2991041 q^{72} + 3894664 q^{74} + 1087482 q^{77} + 2045340 q^{78} + 388330 q^{79} + 1671105 q^{81} - 928104 q^{84} - 260864 q^{86} + 1607450 q^{88} + 228476 q^{91} - 3382422 q^{92} - 1633200 q^{93} - 175227 q^{98} + 2894442 q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(175, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
175.7.d.a $1$ $40.259$ $$\Q$$ $$\Q(\sqrt{-7})$$ $$-9$$ $$0$$ $$0$$ $$343$$ $$q-9q^{2}+17q^{4}+7^{3}q^{7}+423q^{8}+\cdots$$
175.7.d.b $2$ $40.259$ $$\Q(\sqrt{21})$$ $$\Q(\sqrt{-7})$$ $$-9$$ $$0$$ $$0$$ $$-686$$ $$q+(-5-\beta )q^{2}+(92+9\beta )q^{4}-7^{3}q^{7}+\cdots$$
175.7.d.c $2$ $40.259$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-35})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-26iq^{3}-2^{6}q^{4}-7^{3}iq^{7}+53q^{9}+\cdots$$
175.7.d.d $2$ $40.259$ $$\Q(\sqrt{21})$$ $$\Q(\sqrt{-7})$$ $$9$$ $$0$$ $$0$$ $$686$$ $$q+(5+\beta )q^{2}+(92+9\beta )q^{4}+7^{3}q^{7}+\cdots$$
175.7.d.e $2$ $40.259$ $$\Q(\sqrt{-510})$$ None $$16$$ $$0$$ $$0$$ $$-266$$ $$q+8q^{2}-\beta q^{3}-8\beta q^{6}+(-133-7\beta )q^{7}+\cdots$$
175.7.d.f $14$ $40.259$ $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ None $$-4$$ $$0$$ $$0$$ $$-526$$ $$q+\beta _{2}q^{2}+\beta _{1}q^{3}+(20-2\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$$
175.7.d.g $14$ $40.259$ $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ None $$4$$ $$0$$ $$0$$ $$526$$ $$q-\beta _{2}q^{2}+\beta _{1}q^{3}+(20-2\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$$
175.7.d.h $16$ $40.259$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-10$$ $$0$$ $$0$$ $$400$$ $$q+(-1+\beta _{1})q^{2}-\beta _{3}q^{3}+(38+\beta _{2}+\cdots)q^{4}+\cdots$$
175.7.d.i $20$ $40.259$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{10}q^{2}-\beta _{11}q^{3}+(39-\beta _{1})q^{4}+\cdots$$

## Decomposition of $$S_{7}^{\mathrm{old}}(175, [\chi])$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(175, [\chi]) \simeq$$ $$S_{7}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 2}$$