# Properties

 Label 175.2.f Level 175 Weight 2 Character orbit f Rep. character $$\chi_{175}(118,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 20 Newform subspaces 4 Sturm bound 40 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ = $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 175.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$40$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(175, [\chi])$$.

Total New Old
Modular forms 52 28 24
Cusp forms 28 20 8
Eisenstein series 24 8 16

## Trace form

 $$20q + 4q^{2} - 4q^{7} + 8q^{8} + O(q^{10})$$ $$20q + 4q^{2} - 4q^{7} + 8q^{8} - 16q^{11} - 48q^{16} - 8q^{18} + 8q^{21} - 4q^{22} - 8q^{23} + 104q^{36} + 24q^{37} - 20q^{42} + 12q^{43} - 52q^{46} - 64q^{51} - 4q^{53} + 68q^{56} - 20q^{57} - 12q^{58} + 8q^{63} + 4q^{67} + 24q^{71} + 16q^{72} + 4q^{77} + 20q^{78} - 44q^{81} - 132q^{86} - 8q^{88} + 48q^{91} - 20q^{93} + 12q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(175, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
175.2.f.a $$4$$ $$1.397$$ $$\Q(i, \sqrt{14})$$ $$\Q(\sqrt{-35})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-2\beta _{2}q^{4}-\beta _{3}q^{7}+4\beta _{2}q^{9}+\cdots$$
175.2.f.b $$4$$ $$1.397$$ $$\Q(i, \sqrt{14})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+5\beta _{2}q^{4}-\beta _{1}q^{7}+3\beta _{3}q^{8}+\cdots$$
175.2.f.c $$4$$ $$1.397$$ $$\Q(i, \sqrt{10})$$ None $$4$$ $$0$$ $$0$$ $$-4$$ $$q+(1+\beta _{2})q^{2}+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{6}+\cdots$$
175.2.f.d $$8$$ $$1.397$$ 8.0.$$\cdots$$.1 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{4}+(\beta _{1}+\beta _{3})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(175, [\chi])$$ into lower level spaces

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2}$$)($$1 + T^{4} + 16 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}$$)($$1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16}$$)
$3$ ($$1 - 17 T^{4} + 81 T^{8}$$)($$( 1 + 9 T^{4} )^{2}$$)($$1 - 17 T^{4} + 81 T^{8}$$)($$( 1 + 9 T^{4} )^{4}$$)
$5$ 1
$7$ ($$1 + 49 T^{4}$$)($$1 + 49 T^{4}$$)($$1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4}$$)($$( 1 + 49 T^{4} )^{2}$$)
$11$ ($$( 1 + 3 T + 11 T^{2} )^{4}$$)($$( 1 - 4 T + 11 T^{2} )^{4}$$)($$( 1 + T + 11 T^{2} )^{4}$$)($$( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{4}$$)
$13$ ($$1 + 23 T^{4} + 28561 T^{8}$$)($$( 1 + 169 T^{4} )^{2}$$)($$1 + 103 T^{4} + 28561 T^{8}$$)($$( 1 + 169 T^{4} )^{4}$$)
$17$ ($$1 + 263 T^{4} + 83521 T^{8}$$)($$( 1 + 289 T^{4} )^{2}$$)($$1 + 263 T^{4} + 83521 T^{8}$$)($$( 1 + 289 T^{4} )^{4}$$)
$19$ ($$( 1 + 19 T^{2} )^{4}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 19 T^{2} )^{8}$$)
$23$ ($$( 1 + 529 T^{4} )^{2}$$)($$1 - 734 T^{4} + 279841 T^{8}$$)($$( 1 + 4 T + 8 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$)($$1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16}$$)
$29$ ($$( 1 + 23 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 54 T^{2} + 2075 T^{4} + 45414 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 31 T^{2} )^{8}$$)
$37$ ($$( 1 + 1369 T^{4} )^{2}$$)($$1 - 1294 T^{4} + 1874161 T^{8}$$)($$( 1 - 12 T + 72 T^{2} - 444 T^{3} + 1369 T^{4} )^{2}$$)($$1 + 1294 T^{4} - 199725 T^{8} + 2425164334 T^{12} + 3512479453921 T^{16}$$)
$41$ ($$( 1 - 41 T^{2} )^{4}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 + 8 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 41 T^{2} )^{8}$$)
$43$ ($$( 1 + 1849 T^{4} )^{2}$$)($$1 - 334 T^{4} + 3418801 T^{8}$$)($$( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{2}$$)($$1 + 334 T^{4} - 3307245 T^{8} + 1141879534 T^{12} + 11688200277601 T^{16}$$)
$47$ ($$1 - 3457 T^{4} + 4879681 T^{8}$$)($$( 1 + 2209 T^{4} )^{2}$$)($$1 - 2017 T^{4} + 4879681 T^{8}$$)($$( 1 + 2209 T^{4} )^{4}$$)
$53$ ($$( 1 + 2809 T^{4} )^{2}$$)($$1 - 5582 T^{4} + 7890481 T^{8}$$)($$( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 28 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 59 T^{2} )^{8}$$)
$61$ ($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 82 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 61 T^{2} )^{8}$$)
$67$ ($$( 1 + 4489 T^{4} )^{2}$$)($$1 + 4946 T^{4} + 20151121 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$)($$1 - 4946 T^{4} + 4311795 T^{8} - 99667444466 T^{12} + 406067677556641 T^{16}$$)
$71$ ($$( 1 - 12 T + 71 T^{2} )^{4}$$)($$( 1 + 16 T + 71 T^{2} )^{4}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 - 16 T + 185 T^{2} - 1136 T^{3} + 5041 T^{4} )^{4}$$)
$73$ ($$1 - 9502 T^{4} + 28398241 T^{8}$$)($$( 1 + 5329 T^{4} )^{2}$$)($$( 1 + 5329 T^{4} )^{2}$$)($$( 1 + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 - 157 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 11 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 94 T^{2} + 2595 T^{4} + 586654 T^{6} + 38950081 T^{8} )^{2}$$)
$83$ ($$1 - 6382 T^{4} + 47458321 T^{8}$$)($$( 1 + 6889 T^{4} )^{2}$$)($$1 + 7538 T^{4} + 47458321 T^{8}$$)($$( 1 + 6889 T^{4} )^{4}$$)
$89$ ($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 138 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{8}$$)
$97$ ($$1 + 3383 T^{4} + 88529281 T^{8}$$)($$( 1 + 9409 T^{4} )^{2}$$)($$1 + 16903 T^{4} + 88529281 T^{8}$$)($$( 1 + 9409 T^{4} )^{4}$$)