# Properties

 Label 350.2.q Level 350 Weight 2 Character orbit q Rep. character $$\chi_{350}(11,\cdot)$$ Character field $$\Q(\zeta_{15})$$ Dimension 160 Newform subspaces 3 Sturm bound 120 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.q (of order $$15$$ and degree $$8$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$175$$ Character field: $$\Q(\zeta_{15})$$ Newform subspaces: $$3$$ Sturm bound: $$120$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(350, [\chi])$$.

Total New Old
Modular forms 512 160 352
Cusp forms 448 160 288
Eisenstein series 64 0 64

## Trace form

 $$160q + 20q^{4} - 2q^{5} - 8q^{6} + 8q^{7} + 20q^{9} + O(q^{10})$$ $$160q + 20q^{4} - 2q^{5} - 8q^{6} + 8q^{7} + 20q^{9} - 2q^{10} + 6q^{11} + 48q^{15} + 20q^{16} - 24q^{17} + 8q^{18} + 4q^{19} + 4q^{20} + 12q^{21} + 32q^{22} + 2q^{23} - 16q^{24} + 4q^{25} - 48q^{26} + 96q^{27} - 14q^{28} + 24q^{29} - 18q^{30} + 6q^{31} + 26q^{33} - 16q^{34} - 16q^{35} - 40q^{36} - 4q^{37} - 8q^{38} - 32q^{39} - 2q^{40} - 68q^{41} - 14q^{42} - 40q^{43} - 4q^{44} + 36q^{45} - 12q^{46} + 20q^{47} + 24q^{49} - 12q^{51} - 56q^{53} + 16q^{54} - 16q^{55} - 136q^{57} + 24q^{59} + 6q^{60} + 8q^{61} - 72q^{62} - 136q^{63} - 40q^{64} - 2q^{65} + 56q^{68} - 116q^{69} + 34q^{70} - 68q^{71} + 8q^{72} + 12q^{73} + 16q^{74} - 60q^{75} + 32q^{76} - 72q^{77} + 16q^{78} - 8q^{79} - 2q^{80} - 24q^{81} + 64q^{82} - 128q^{83} - 18q^{84} - 4q^{85} - 12q^{86} + 24q^{87} + 14q^{88} + 54q^{89} - 160q^{90} - 44q^{91} + 16q^{92} - 52q^{93} - 36q^{95} + 4q^{96} - 84q^{97} - 64q^{98} - 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(350, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
350.2.q.a $$8$$ $$2.795$$ $$\Q(\zeta_{15})$$ None $$-1$$ $$-3$$ $$0$$ $$-4$$ $$q-\zeta_{15}q^{2}+(1-\zeta_{15}^{2}+\zeta_{15}^{3}-\zeta_{15}^{4}+\cdots)q^{3}+\cdots$$
350.2.q.b $$72$$ $$2.795$$ None $$-9$$ $$1$$ $$0$$ $$8$$
350.2.q.c $$80$$ $$2.795$$ None $$10$$ $$2$$ $$-2$$ $$4$$

## Decomposition of $$S_{2}^{\mathrm{old}}(350, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8}$$)
$3$ ($$1 + 3 T + 8 T^{2} + 5 T^{3} - 28 T^{5} + 24 T^{6} + 110 T^{7} + 409 T^{8} + 330 T^{9} + 216 T^{10} - 756 T^{11} + 1215 T^{13} + 5832 T^{14} + 6561 T^{15} + 6561 T^{16}$$)
$5$ ($$( 1 + 5 T^{2} + 25 T^{4} )^{2}$$)
$7$ ($$( 1 + T + 7 T^{2} )^{4}$$)
$11$ ($$1 - 8 T + 41 T^{2} - 52 T^{3} - 274 T^{4} + 2156 T^{5} - 2211 T^{6} - 17666 T^{7} + 119427 T^{8} - 194326 T^{9} - 267531 T^{10} + 2869636 T^{11} - 4011634 T^{12} - 8374652 T^{13} + 72634001 T^{14} - 155897368 T^{15} + 214358881 T^{16}$$)
$13$ ($$( 1 - 10 T + 27 T^{2} + 40 T^{3} - 391 T^{4} + 520 T^{5} + 4563 T^{6} - 21970 T^{7} + 28561 T^{8} )^{2}$$)
$17$ ($$1 - 14 T + 77 T^{2} - 190 T^{3} + 380 T^{4} - 3796 T^{5} + 27351 T^{6} - 109670 T^{7} + 386499 T^{8} - 1864390 T^{9} + 7904439 T^{10} - 18649748 T^{11} + 31737980 T^{12} - 269772830 T^{13} + 1858592813 T^{14} - 5744741422 T^{15} + 6975757441 T^{16}$$)
$19$ ($$1 + 6 T + 19 T^{2} + 126 T^{3} + 756 T^{4} + 4308 T^{5} + 20501 T^{6} + 90468 T^{7} + 412487 T^{8} + 1718892 T^{9} + 7400861 T^{10} + 29548572 T^{11} + 98522676 T^{12} + 311988474 T^{13} + 893871739 T^{14} + 5363230434 T^{15} + 16983563041 T^{16}$$)
$23$ ($$( 1 - 20 T + 217 T^{2} - 1600 T^{3} + 8769 T^{4} - 36800 T^{5} + 114793 T^{6} - 243340 T^{7} + 279841 T^{8} )( 1 + 13 T + 46 T^{2} - 61 T^{3} - 891 T^{4} - 1403 T^{5} + 24334 T^{6} + 158171 T^{7} + 279841 T^{8} )$$)
$29$ ($$( 1 - 11 T + 32 T^{2} - 113 T^{3} + 1115 T^{4} - 3277 T^{5} + 26912 T^{6} - 268279 T^{7} + 707281 T^{8} )^{2}$$)
$31$ ($$1 - 6 T + 31 T^{2} + 318 T^{3} - 2898 T^{4} + 16284 T^{5} - 1873 T^{6} - 480582 T^{7} + 3789491 T^{8} - 14898042 T^{9} - 1799953 T^{10} + 485116644 T^{11} - 2676363858 T^{12} + 9104070018 T^{13} + 27512614111 T^{14} - 165075684666 T^{15} + 852891037441 T^{16}$$)
$37$ ($$1 + T + 32 T^{2} - 245 T^{3} - 590 T^{4} - 2246 T^{5} - 32624 T^{6} + 408180 T^{7} - 1426941 T^{8} + 15102660 T^{9} - 44662256 T^{10} - 113766638 T^{11} - 1105754990 T^{12} - 16989269465 T^{13} + 82103245088 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16}$$)
$41$ ($$( 1 + 8 T + 73 T^{2} + 556 T^{3} + 2205 T^{4} + 22796 T^{5} + 122713 T^{6} + 551368 T^{7} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 + 81 T^{2} + 1849 T^{4} )^{4}$$)
$47$ ($$1 - 2 T - 13 T^{2} - 294 T^{3} - 212 T^{4} + 11364 T^{5} + 91289 T^{6} - 391562 T^{7} - 4902677 T^{8} - 18403414 T^{9} + 201657401 T^{10} + 1179844572 T^{11} - 1034492372 T^{12} - 67427432058 T^{13} - 140129799277 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$1 - 7 T + 83 T^{2} - 180 T^{3} + 1240 T^{4} + 23277 T^{5} - 22906 T^{6} + 667850 T^{7} + 3038809 T^{8} + 35396050 T^{9} - 64342954 T^{10} + 3465409929 T^{11} + 9784196440 T^{12} - 75275188740 T^{13} + 1839641973707 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16}$$)
$59$ ($$1 + 15 T + 94 T^{2} - 825 T^{3} - 17940 T^{4} - 160230 T^{5} - 233584 T^{6} + 8132460 T^{7} + 106709249 T^{8} + 479815140 T^{9} - 813105904 T^{10} - 32907877170 T^{11} - 217385456340 T^{12} - 589812546675 T^{13} + 3964970162254 T^{14} + 37329772272285 T^{15} + 146830437604321 T^{16}$$)
$61$ ($$1 - 11 T + 131 T^{2} - 1292 T^{3} + 13072 T^{4} - 79991 T^{5} + 834742 T^{6} - 4920222 T^{7} + 38414121 T^{8} - 300133542 T^{9} + 3106074982 T^{10} - 18156437171 T^{11} + 180992833552 T^{12} - 1091218420892 T^{13} + 6749169041291 T^{14} - 34570171196231 T^{15} + 191707312997281 T^{16}$$)
$67$ ($$1 + T + 67 T^{2} + 200 T^{3} + 200 T^{4} - 8711 T^{5} - 282874 T^{6} - 1742200 T^{7} - 21893321 T^{8} - 116727400 T^{9} - 1269821386 T^{10} - 2619946493 T^{11} + 4030224200 T^{12} + 270025021400 T^{13} + 6060711605323 T^{14} + 6060711605323 T^{15} + 406067677556641 T^{16}$$)
$71$ ($$( 1 - 13 T + 218 T^{2} - 1831 T^{3} + 21125 T^{4} - 130001 T^{5} + 1098938 T^{6} - 4652843 T^{7} + 25411681 T^{8} )^{2}$$)
$73$ ($$1 - 4 T + 43 T^{2} - 1052 T^{3} + 2828 T^{4} + 28448 T^{5} + 223961 T^{6} + 1434004 T^{7} - 44342897 T^{8} + 104682292 T^{9} + 1193488169 T^{10} + 11066755616 T^{11} + 80310225548 T^{12} - 2180871315836 T^{13} + 6507371730427 T^{14} - 44189594076388 T^{15} + 806460091894081 T^{16}$$)
$79$ ($$1 + 20 T + 239 T^{2} + 20 T^{3} - 25040 T^{4} - 374440 T^{5} - 1011119 T^{6} + 19593600 T^{7} + 350987679 T^{8} + 1547894400 T^{9} - 6310393679 T^{10} - 184613523160 T^{11} - 975310028240 T^{12} + 61541127980 T^{13} + 58097901869519 T^{14} + 384078179723180 T^{15} + 1517108809906561 T^{16}$$)
$83$ ($$( 1 - 13 T - 4 T^{2} + 1151 T^{3} - 11791 T^{4} + 95533 T^{5} - 27556 T^{6} - 7433231 T^{7} + 47458321 T^{8} )^{2}$$)
$89$ ($$1 - T - 136 T^{2} - 891 T^{3} + 7966 T^{4} + 107562 T^{5} + 814376 T^{6} - 6768808 T^{7} - 97658633 T^{8} - 602423912 T^{9} + 6450672296 T^{10} + 75827875578 T^{11} + 499804691806 T^{12} - 4975396969059 T^{13} - 67589455570696 T^{14} - 44231334895529 T^{15} + 3936588805702081 T^{16}$$)
$97$ ($$( 1 - 4 T - T^{2} + 682 T^{3} + 829 T^{4} + 66154 T^{5} - 9409 T^{6} - 3650692 T^{7} + 88529281 T^{8} )^{2}$$)