Properties

 Label 336.2.s Level 336 Weight 2 Character orbit s Rep. character $$\chi_{336}(155,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 96 Newform subspaces 4 Sturm bound 128 Trace bound 2

Related objects

Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.s (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$48$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$128$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(336, [\chi])$$.

Total New Old
Modular forms 136 96 40
Cusp forms 120 96 24
Eisenstein series 16 0 16

Trace form

 $$96q + 12q^{6} + O(q^{10})$$ $$96q + 12q^{6} - 8q^{10} + 12q^{12} - 24q^{16} + 16q^{19} - 16q^{22} - 32q^{24} - 24q^{27} - 48q^{30} - 8q^{34} - 8q^{36} - 48q^{39} - 32q^{43} + 88q^{46} + 52q^{48} + 96q^{49} + 48q^{52} - 64q^{55} - 88q^{58} + 8q^{60} - 32q^{61} - 72q^{64} - 92q^{66} - 32q^{67} - 24q^{70} - 88q^{72} + 56q^{75} - 24q^{76} - 40q^{78} - 32q^{82} - 32q^{85} + 112q^{87} + 136q^{88} - 8q^{90} - 48q^{93} + 48q^{94} - 32q^{96} + 64q^{99} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(336, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
336.2.s.a $$4$$ $$2.683$$ $$\Q(\zeta_{8})$$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$ $$q+(-1+\zeta_{8})q^{2}+(-1+\zeta_{8}^{2})q^{3}-2\zeta_{8}q^{4}+\cdots$$
336.2.s.b $$4$$ $$2.683$$ $$\Q(\zeta_{8})$$ None $$4$$ $$0$$ $$4$$ $$-4$$ $$q+(1+\zeta_{8})q^{2}+(-\zeta_{8}-\zeta_{8}^{3})q^{3}+2\zeta_{8}q^{4}+\cdots$$
336.2.s.c $$40$$ $$2.683$$ None $$0$$ $$4$$ $$0$$ $$-40$$
336.2.s.d $$48$$ $$2.683$$ None $$0$$ $$0$$ $$0$$ $$48$$

Decomposition of $$S_{2}^{\mathrm{old}}(336, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(336, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 2}$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T + 2 T^{2} )^{2}$$)($$( 1 - 2 T + 2 T^{2} )^{2}$$)
$3$ ($$( 1 + 2 T + 3 T^{2} )^{2}$$)($$1 - 2 T^{2} + 9 T^{4}$$)
$5$ ($$1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 60 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8}$$)($$1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 60 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8}$$)
$7$ ($$( 1 + T )^{4}$$)($$( 1 + T )^{4}$$)
$11$ ($$1 + 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 132 T^{5} + 968 T^{6} + 5324 T^{7} + 14641 T^{8}$$)($$1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 132 T^{5} + 968 T^{6} - 5324 T^{7} + 14641 T^{8}$$)
$13$ ($$1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8}$$)($$1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8}$$)
$17$ ($$1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8}$$)($$1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8}$$)
$19$ ($$1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 1292 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8}$$)($$1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 1292 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8}$$)
$23$ ($$1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8}$$)($$1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8}$$)
$29$ ($$( 1 - 4 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$)($$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 4 T + 29 T^{2} )^{2}$$)
$31$ ($$1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8}$$)($$1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8}$$)
$37$ ($$1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8}$$)($$1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8}$$)
$41$ ($$( 1 + 12 T + 86 T^{2} + 492 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 4988 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8}$$)($$1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 4988 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8}$$)
$47$ ($$( 1 + 62 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 62 T^{2} + 2209 T^{4} )^{2}$$)
$53$ ($$1 + 12 T + 72 T^{2} + 660 T^{3} + 6046 T^{4} + 34980 T^{5} + 202248 T^{6} + 1786524 T^{7} + 7890481 T^{8}$$)($$1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 34980 T^{5} + 202248 T^{6} - 1786524 T^{7} + 7890481 T^{8}$$)
$59$ ($$( 1 + 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} )$$)($$( 1 - 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} )$$)
$61$ ($$1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8}$$)($$1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8}$$)
$67$ ($$( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2}$$)
$71$ ($$( 1 - 106 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 106 T^{2} + 5041 T^{4} )^{2}$$)
$73$ ($$1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8}$$)($$1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8}$$)
$79$ ($$( 1 - 154 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 154 T^{2} + 6241 T^{4} )^{2}$$)
$83$ ($$1 + 4 T + 8 T^{2} - 444 T^{3} - 12994 T^{4} - 36852 T^{5} + 55112 T^{6} + 2287148 T^{7} + 47458321 T^{8}$$)($$1 - 4 T + 8 T^{2} + 444 T^{3} - 12994 T^{4} + 36852 T^{5} + 55112 T^{6} - 2287148 T^{7} + 47458321 T^{8}$$)
$89$ ($$( 1 + 20 T + 246 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 20 T + 246 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 + 2 T + 97 T^{2} )^{4}$$)($$( 1 + 2 T + 97 T^{2} )^{4}$$)