# Properties

 Label 175.2.h Level 175 Weight 2 Character orbit h Rep. character $$\chi_{175}(36,\cdot)$$ Character field $$\Q(\zeta_{5})$$ Dimension 64 Newform subspaces 3 Sturm bound 40 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 175.h (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$25$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$3$$ Sturm bound: $$40$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 88 64 24
Cusp forms 72 64 8
Eisenstein series 16 0 16

## Trace form

 $$64q - 2q^{2} - 4q^{3} - 18q^{4} + 4q^{5} - 6q^{8} - 24q^{9} + O(q^{10})$$ $$64q - 2q^{2} - 4q^{3} - 18q^{4} + 4q^{5} - 6q^{8} - 24q^{9} - 14q^{10} - 8q^{11} + 8q^{12} - 24q^{13} - 4q^{14} + 2q^{15} + 2q^{16} - 16q^{17} + 4q^{18} + 12q^{19} + 8q^{20} - 4q^{21} + 26q^{22} - 6q^{23} - 28q^{24} + 22q^{25} - 32q^{26} + 2q^{27} - 22q^{29} - 32q^{30} - 12q^{31} + 16q^{32} - 16q^{33} + 30q^{34} - 2q^{35} - 16q^{36} + 28q^{37} - 10q^{38} - 4q^{39} + 64q^{40} - 24q^{41} + 30q^{42} - 44q^{43} + 22q^{44} - 10q^{45} + 4q^{46} + 38q^{47} + 78q^{48} + 64q^{49} - 4q^{50} + 44q^{51} - 92q^{52} - 34q^{53} + 54q^{54} + 6q^{55} - 12q^{56} - 68q^{57} + 66q^{58} - 6q^{59} + 12q^{60} - 16q^{61} - 46q^{62} + 12q^{63} - 6q^{64} + 60q^{65} + 74q^{66} + 2q^{67} - 116q^{68} - 18q^{69} - 16q^{70} - 32q^{71} - 50q^{72} - 28q^{73} + 40q^{74} - 100q^{75} - 52q^{76} + 12q^{77} + 142q^{78} - 14q^{80} + 2q^{81} - 28q^{82} - 74q^{83} - 12q^{84} + 38q^{85} + 20q^{86} + 82q^{87} - 44q^{88} + 28q^{89} - 202q^{90} - 8q^{91} + 8q^{92} + 16q^{93} + 48q^{94} + 58q^{95} - 12q^{96} - 58q^{97} - 2q^{98} + 20q^{99} + O(q^{100})$$

## 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.h.a $$4$$ $$1.397$$ $$\Q(\zeta_{10})$$ None $$-5$$ $$-4$$ $$-5$$ $$-4$$ $$q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$$
175.2.h.b $$28$$ $$1.397$$ None $$6$$ $$4$$ $$8$$ $$-28$$
175.2.h.c $$32$$ $$1.397$$ None $$-3$$ $$-4$$ $$1$$ $$32$$

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8}$$)
$3$ ($$1 + 4 T + 13 T^{2} + 30 T^{3} + 61 T^{4} + 90 T^{5} + 117 T^{6} + 108 T^{7} + 81 T^{8}$$)
$5$ ($$1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4}$$)
$7$ ($$( 1 + T )^{4}$$)
$11$ ($$1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 396 T^{5} - 847 T^{6} + 2662 T^{7} + 14641 T^{8}$$)
$13$ ($$1 - 3 T^{2} + 40 T^{3} + 129 T^{4} + 520 T^{5} - 507 T^{6} + 28561 T^{8}$$)
$17$ ($$1 + 12 T + 77 T^{2} + 420 T^{3} + 1981 T^{4} + 7140 T^{5} + 22253 T^{6} + 58956 T^{7} + 83521 T^{8}$$)
$19$ ($$1 + 2 T - 15 T^{2} - 68 T^{3} + 149 T^{4} - 1292 T^{5} - 5415 T^{6} + 13718 T^{7} + 130321 T^{8}$$)
$23$ ($$1 + 16 T + 113 T^{2} + 570 T^{3} + 2741 T^{4} + 13110 T^{5} + 59777 T^{6} + 194672 T^{7} + 279841 T^{8}$$)
$29$ ($$1 - 2 T - 25 T^{2} - 32 T^{3} + 929 T^{4} - 928 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8}$$)
$31$ ($$1 + 16 T + 105 T^{2} + 554 T^{3} + 3269 T^{4} + 17174 T^{5} + 100905 T^{6} + 476656 T^{7} + 923521 T^{8}$$)
$37$ ($$1 + 3 T - 33 T^{2} - 35 T^{3} + 1296 T^{4} - 1295 T^{5} - 45177 T^{6} + 151959 T^{7} + 1874161 T^{8}$$)
$41$ ($$1 + 8 T - 7 T^{2} + 36 T^{3} + 1925 T^{4} + 1476 T^{5} - 11767 T^{6} + 551368 T^{7} + 2825761 T^{8}$$)
$43$ ($$( 1 + 10 T + 106 T^{2} + 430 T^{3} + 1849 T^{4} )^{2}$$)
$47$ ($$1 + 16 T + 49 T^{2} - 628 T^{3} - 7311 T^{4} - 29516 T^{5} + 108241 T^{6} + 1661168 T^{7} + 4879681 T^{8}$$)
$53$ ($$1 - 17 T + 71 T^{2} + 849 T^{3} - 12256 T^{4} + 44997 T^{5} + 199439 T^{6} - 2530909 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 - 19 T^{2} - 390 T^{3} + 2701 T^{4} - 23010 T^{5} - 66139 T^{6} + 12117361 T^{8}$$)
$61$ ($$1 + 8 T - 27 T^{2} - 584 T^{3} - 2575 T^{4} - 35624 T^{5} - 100467 T^{6} + 1815848 T^{7} + 13845841 T^{8}$$)
$67$ ($$1 - 6 T + 69 T^{2} - 382 T^{3} + 1329 T^{4} - 25594 T^{5} + 309741 T^{6} - 1804578 T^{7} + 20151121 T^{8}$$)
$71$ ($$1 + 20 T + 169 T^{2} + 1600 T^{3} + 17121 T^{4} + 113600 T^{5} + 851929 T^{6} + 7158220 T^{7} + 25411681 T^{8}$$)
$73$ ($$1 - 12 T + 21 T^{2} - 476 T^{3} + 9429 T^{4} - 34748 T^{5} + 111909 T^{6} - 4668204 T^{7} + 28398241 T^{8}$$)
$79$ ($$1 - 6 T + 57 T^{2} - 118 T^{3} + 105 T^{4} - 9322 T^{5} + 355737 T^{6} - 2958234 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 - 12 T - 19 T^{2} + 204 T^{3} + 4489 T^{4} + 16932 T^{5} - 130891 T^{6} - 6861444 T^{7} + 47458321 T^{8}$$)
$89$ ($$1 + 3 T - 35 T^{2} + 693 T^{3} + 9604 T^{4} + 61677 T^{5} - 277235 T^{6} + 2114907 T^{7} + 62742241 T^{8}$$)
$97$ ($$1 + 36 T + 549 T^{2} + 5092 T^{3} + 43869 T^{4} + 493924 T^{5} + 5165541 T^{6} + 32856228 T^{7} + 88529281 T^{8}$$)