# Properties

 Label 350.2.g Level 350 Weight 2 Character orbit g Rep. character $$\chi_{350}(293,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 24 Newform subspaces 2 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.g (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ 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 144 24 120
Cusp forms 96 24 72
Eisenstein series 48 0 48

## Trace form

 $$24q + 8q^{7} + O(q^{10})$$ $$24q + 8q^{7} - 24q^{16} - 8q^{18} + 8q^{21} + 16q^{22} + 8q^{23} - 8q^{28} - 56q^{36} - 32q^{37} - 32q^{43} + 32q^{46} + 64q^{51} + 32q^{53} + 16q^{56} + 8q^{57} + 16q^{58} - 16q^{67} + 32q^{71} - 8q^{72} + 16q^{77} + 40q^{78} - 88q^{81} - 16q^{88} - 112q^{91} + 8q^{92} + 16q^{93} - 32q^{98} + O(q^{100})$$

## 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.g.a $$8$$ $$2.795$$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\zeta_{16}q^{2}+\zeta_{16}^{2}q^{3}+\zeta_{16}^{3}q^{4}+(\zeta_{16}^{4}+\cdots)q^{6}+\cdots$$
350.2.g.b $$16$$ $$2.795$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+\beta _{3}q^{3}+\beta _{10}q^{4}+\beta _{15}q^{6}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$3$ ($$( 1 - 8 T^{2} + 32 T^{4} - 72 T^{6} + 81 T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 72 T^{6} + 81 T^{8} )$$)($$( 1 - 10 T^{4} + 139 T^{8} - 810 T^{12} + 6561 T^{16} )^{2}$$)
$5$ 1
$7$ ($$1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 616 T^{5} + 1568 T^{6} - 2744 T^{7} + 2401 T^{8}$$)($$1 + 28 T^{4} + 3270 T^{8} + 67228 T^{12} + 5764801 T^{16}$$)
$11$ ($$( 1 + 14 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 19 T^{2} + 121 T^{4} )^{8}$$)
$13$ ($$( 1 - 240 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 - 260 T^{4} + 30822 T^{8} - 7425860 T^{12} + 815730721 T^{16} )^{2}$$)
$17$ ($$1 - 252 T^{4} + 17030 T^{8} - 21047292 T^{12} + 6975757441 T^{16}$$)($$( 1 - 602 T^{4} + 184635 T^{8} - 50279642 T^{12} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 + 24 T^{2} + 288 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 34 T^{2} + 903 T^{4} + 12274 T^{6} + 130321 T^{8} )^{4}$$)
$23$ ($$( 1 - 4 T + 8 T^{2} - 84 T^{3} + 878 T^{4} - 1932 T^{5} + 4232 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 + 28 T^{4} + 171078 T^{8} + 7835548 T^{12} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 92 T^{2} + 3690 T^{4} - 77372 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 - 108 T^{2} + 4806 T^{4} - 103788 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 28 T^{2} + 1146 T^{4} - 26908 T^{6} + 923521 T^{8} )^{4}$$)
$37$ ($$( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 7666 T^{4} + 38480 T^{5} + 175232 T^{6} + 810448 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 - 644 T^{4} - 1762266 T^{8} - 1206959684 T^{12} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 148 T^{2} + 8806 T^{4} - 248788 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 38 T^{2} + 2751 T^{4} - 63878 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 + 8 T + 32 T^{2} + 344 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 + 1778 T^{4} + 3418801 T^{8} )^{4}$$)
$47$ ($$1 + 6020 T^{4} + 17872774 T^{8} + 29375679620 T^{12} + 23811286661761 T^{16}$$)($$( 1 - 2660 T^{4} + 3301254 T^{8} - 12979951460 T^{12} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 16 T + 128 T^{2} - 784 T^{3} + 4786 T^{4} - 41552 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} )^{2}$$)($$( 1 + 6268 T^{4} + 25462950 T^{8} + 49457534908 T^{12} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 136 T^{2} + 10336 T^{4} + 473416 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 59 T^{2} )^{16}$$)
$61$ ($$( 1 - 96 T^{2} + 8688 T^{4} - 357216 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 76 T^{2} + 7158 T^{4} - 282796 T^{6} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 + 8 T + 32 T^{2} - 552 T^{3} - 8974 T^{4} - 36984 T^{5} + 143648 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 - 2642 T^{4} + 1066035 T^{8} - 53239261682 T^{12} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 6 T + 124 T^{2} - 426 T^{3} + 5041 T^{4} )^{8}$$)
$73$ ($$1 + 1220 T^{4} + 55730374 T^{8} + 34645854020 T^{12} + 806460091894081 T^{16}$$)($$( 1 - 11930 T^{4} + 77910459 T^{8} - 338791015130 T^{12} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 208 T^{2} + 22498 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 154 T^{2} + 6241 T^{4} )^{8}$$)
$83$ ($$1 - 4224 T^{4} + 99283874 T^{8} - 200463947904 T^{12} + 2252292232139041 T^{16}$$)($$( 1 - 4394 T^{4} + 58495659 T^{8} - 208531862474 T^{12} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 60 T^{2} + 10470 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 230 T^{2} + 28095 T^{4} + 1821830 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$1 - 11900 T^{4} + 108781062 T^{8} - 1053498443900 T^{12} + 7837433594376961 T^{16}$$)($$( 1 - 380 T^{4} + 60281862 T^{8} - 33641126780 T^{12} + 7837433594376961 T^{16} )^{2}$$)