# Properties

 Label 175.10.a Level $175$ Weight $10$ Character orbit 175.a Rep. character $\chi_{175}(1,\cdot)$ Character field $\Q$ Dimension $85$ Newform subspaces $13$ Sturm bound $200$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 175.a (trivial) Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$200$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_0(175))$$.

Total New Old
Modular forms 186 85 101
Cusp forms 174 85 89
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$5$$$$7$$FrickeDim
$$+$$$$+$$$$+$$$$19$$
$$+$$$$-$$$$-$$$$22$$
$$-$$$$+$$$$-$$$$23$$
$$-$$$$-$$$$+$$$$21$$
Plus space$$+$$$$40$$
Minus space$$-$$$$45$$

## Trace form

 $$85 q - 17 q^{2} + 294 q^{3} + 22273 q^{4} - 11386 q^{6} + 2401 q^{7} - 7905 q^{8} + 522233 q^{9} + O(q^{10})$$ $$85 q - 17 q^{2} + 294 q^{3} + 22273 q^{4} - 11386 q^{6} + 2401 q^{7} - 7905 q^{8} + 522233 q^{9} - 54432 q^{11} + 480838 q^{12} - 48136 q^{13} - 88837 q^{14} + 5786121 q^{16} + 711878 q^{17} - 2153581 q^{18} - 640366 q^{19} - 408170 q^{21} + 2328236 q^{22} - 695176 q^{23} - 8763666 q^{24} - 3625100 q^{26} + 4282140 q^{27} + 5226977 q^{28} - 12919774 q^{29} - 356700 q^{31} + 2428023 q^{32} - 9834072 q^{33} + 10780618 q^{34} + 121310165 q^{36} - 34821542 q^{37} - 28892630 q^{38} + 53917784 q^{39} + 39695746 q^{41} - 8926918 q^{42} + 78756864 q^{43} + 10878162 q^{44} - 130524998 q^{46} + 87054668 q^{47} + 175660814 q^{48} + 490008085 q^{49} + 251405632 q^{51} + 98290868 q^{52} + 107791454 q^{53} - 350769592 q^{54} - 112316379 q^{56} + 128646980 q^{57} - 484167470 q^{58} - 517888282 q^{59} - 243558728 q^{61} + 529325916 q^{62} + 166900713 q^{63} + 1343015567 q^{64} + 1377678316 q^{66} - 526332 q^{67} - 511592234 q^{68} + 484724056 q^{69} - 774052540 q^{71} - 848550385 q^{72} + 317883234 q^{73} + 578853308 q^{74} + 340259166 q^{76} + 168242872 q^{77} - 370287692 q^{78} + 881409660 q^{79} + 1567163561 q^{81} - 1592705654 q^{82} + 989257114 q^{83} - 1017394938 q^{84} + 3261763934 q^{86} - 3212160700 q^{87} - 2388172120 q^{88} - 548591634 q^{89} + 326948972 q^{91} + 2656347768 q^{92} + 722379888 q^{93} + 8394540344 q^{94} + 3715282870 q^{96} + 2220934678 q^{97} - 98001617 q^{98} + 3534938932 q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_0(175))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 7
175.10.a.a $1$ $90.131$ $$\Q$$ None $$-28$$ $$116$$ $$0$$ $$-2401$$ $+$ $+$ $$q-28q^{2}+116q^{3}+272q^{4}-3248q^{6}+\cdots$$
175.10.a.b $2$ $90.131$ $$\Q(\sqrt{193})$$ None $$6$$ $$86$$ $$0$$ $$4802$$ $+$ $-$ $$q+(3+\beta )q^{2}+(43-11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots$$
175.10.a.c $2$ $90.131$ $$\Q(\sqrt{2})$$ None $$24$$ $$174$$ $$0$$ $$-4802$$ $+$ $+$ $$q+(12+\beta )q^{2}+(87+54\beta )q^{3}+(-360+\cdots)q^{4}+\cdots$$
175.10.a.d $3$ $90.131$ $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ None $$-21$$ $$-84$$ $$0$$ $$-7203$$ $+$ $+$ $$q+(-7+\beta _{2})q^{2}+(-28+\beta _{1}+\beta _{2})q^{3}+\cdots$$
175.10.a.e $4$ $90.131$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$19$$ $$18$$ $$0$$ $$9604$$ $+$ $-$ $$q+(5+\beta _{1})q^{2}+(4-\beta _{2})q^{3}+(435+9\beta _{1}+\cdots)q^{4}+\cdots$$
175.10.a.f $5$ $90.131$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$-2$$ $$-140$$ $$0$$ $$-12005$$ $+$ $+$ $$q-\beta _{1}q^{2}+(-28-\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots$$
175.10.a.g $6$ $90.131$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$-15$$ $$124$$ $$0$$ $$14406$$ $+$ $-$ $$q+(-3+\beta _{1})q^{2}+(20+\beta _{1}-\beta _{2})q^{3}+\cdots$$
175.10.a.h $8$ $90.131$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-27$$ $$69$$ $$0$$ $$19208$$ $-$ $-$ $$q+(-3-\beta _{1})q^{2}+(8+\beta _{1}-\beta _{2})q^{3}+\cdots$$
175.10.a.i $8$ $90.131$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$27$$ $$-69$$ $$0$$ $$-19208$$ $+$ $+$ $$q+(3+\beta _{1})q^{2}+(-8-\beta _{1}+\beta _{2})q^{3}+\cdots$$
175.10.a.j $10$ $90.131$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-22$$ $$77$$ $$0$$ $$-24010$$ $-$ $+$ $$q+(-2-\beta _{1})q^{2}+(8+\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots$$
175.10.a.k $10$ $90.131$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$22$$ $$-77$$ $$0$$ $$24010$$ $+$ $-$ $$q+(2+\beta _{1})q^{2}+(-8-\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots$$
175.10.a.l $13$ $90.131$ $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ None $$-32$$ $$-158$$ $$0$$ $$31213$$ $-$ $-$ $$q+(-2-\beta _{1})q^{2}+(-12+\beta _{3})q^{3}+(208+\cdots)q^{4}+\cdots$$
175.10.a.m $13$ $90.131$ $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ None $$32$$ $$158$$ $$0$$ $$-31213$$ $-$ $+$ $$q+(2+\beta _{1})q^{2}+(12-\beta _{3})q^{3}+(208+5\beta _{1}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_0(175))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_0(175)) \simeq$$ $$S_{10}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 3}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(25))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(35))$$$$^{\oplus 2}$$