# Properties

 Label 336.2.q Level 336 Weight 2 Character orbit q Rep. character $$\chi_{336}(193,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 16 Newform subspaces 7 Sturm bound 128 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$7$$ Sturm bound: $$128$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 152 16 136
Cusp forms 104 16 88
Eisenstein series 48 0 48

## Trace form

 $$16q - 2q^{3} - 2q^{7} - 8q^{9} + O(q^{10})$$ $$16q - 2q^{3} - 2q^{7} - 8q^{9} + 4q^{11} - 10q^{19} + 16q^{23} - 12q^{25} + 4q^{27} + 16q^{29} + 14q^{31} - 4q^{33} + 36q^{35} - 8q^{37} + 2q^{39} + 16q^{41} - 12q^{43} - 12q^{47} - 16q^{49} - 16q^{53} - 56q^{55} - 8q^{57} - 16q^{59} - 8q^{61} - 2q^{63} - 8q^{65} - 6q^{67} + 24q^{71} + 12q^{73} - 14q^{75} + 8q^{77} + 18q^{79} - 8q^{81} - 24q^{83} - 16q^{85} + 12q^{87} + 16q^{89} + 54q^{91} + 8q^{93} - 36q^{95} + 8q^{97} - 8q^{99} + O(q^{100})$$

## 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.q.a $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-2$$ $$-5$$ $$q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$$
336.2.q.b $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$-1$$ $$q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(1-3\zeta_{6})q^{7}+\cdots$$
336.2.q.c $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$2$$ $$-1$$ $$q+(-1+\zeta_{6})q^{3}+2\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots$$
336.2.q.d $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-3$$ $$-5$$ $$q+(1-\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(-3+\zeta_{6})q^{7}+\cdots$$
336.2.q.e $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$1$$ $$-1$$ $$q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(1-3\zeta_{6})q^{7}+\cdots$$
336.2.q.f $$2$$ $$2.683$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$2$$ $$5$$ $$q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots$$
336.2.q.g $$4$$ $$2.683$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$-2$$ $$1$$ $$6$$ $$q+(-1+\beta _{2})q^{3}+(-1+2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(336, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()()()()()()
$3$ ($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T + T^{2} )^{2}$$)
$5$ ($$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$)($$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$)($$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$)($$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$)($$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$)($$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$)($$( 1 - T + 5 T^{2} )^{2}( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )$$)
$7$ ($$1 + 5 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$( 1 - 3 T + 7 T^{2} )^{2}$$)
$11$ ($$1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4}$$)($$1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4}$$)($$1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 154 T^{5} - 847 T^{6} + 1331 T^{7} + 14641 T^{8}$$)
$13$ ($$( 1 + 3 T + 13 T^{2} )^{2}$$)($$( 1 + 13 T^{2} )^{2}$$)($$( 1 + 3 T + 13 T^{2} )^{2}$$)($$( 1 + 4 T + 13 T^{2} )^{2}$$)($$( 1 - 4 T + 13 T^{2} )^{2}$$)($$( 1 - T + 13 T^{2} )^{2}$$)($$( 1 - 5 T + 18 T^{2} - 65 T^{3} + 169 T^{4} )^{2}$$)
$17$ ($$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$)($$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$)($$1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4}$$)($$1 - 17 T^{2} + 289 T^{4}$$)($$1 - 17 T^{2} + 289 T^{4}$$)($$1 - 17 T^{2} + 289 T^{4}$$)($$( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} )^{2}$$)
$19$ ($$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$)($$( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$)($$( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$)($$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$1 + 5 T - 5 T^{2} - 40 T^{3} + 64 T^{4} - 760 T^{5} - 1805 T^{6} + 34295 T^{7} + 130321 T^{8}$$)
$23$ ($$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$)($$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$)($$1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 + 4 T + 29 T^{2} )^{2}$$)($$( 1 + 5 T + 29 T^{2} )^{2}$$)($$( 1 - 4 T + 29 T^{2} )^{2}$$)($$( 1 - 9 T + 29 T^{2} )^{2}$$)($$( 1 + 3 T + 29 T^{2} )^{2}$$)($$( 1 - 4 T + 29 T^{2} )^{2}$$)($$( 1 - 3 T + 46 T^{2} - 87 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4}$$)($$1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4}$$)($$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$)($$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$)($$1 - 9 T + 50 T^{2} - 279 T^{3} + 961 T^{4}$$)($$( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{2}$$)
$37$ ($$1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4}$$)($$1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4}$$)($$( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)($$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$)($$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 1332 T^{5} - 72557 T^{6} + 151959 T^{7} + 1874161 T^{8}$$)
$41$ ($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 - 8 T + 41 T^{2} )^{2}$$)($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 34 T^{2} - 246 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$( 1 - T + 43 T^{2} )^{2}$$)($$( 1 + 2 T + 43 T^{2} )^{2}$$)($$( 1 + 11 T + 43 T^{2} )^{2}$$)($$( 1 - 10 T + 43 T^{2} )^{2}$$)($$( 1 + 6 T + 43 T^{2} )^{2}$$)($$( 1 + 5 T + 43 T^{2} )^{2}$$)($$( 1 - 7 T + 84 T^{2} - 301 T^{3} + 1849 T^{4} )^{2}$$)
$47$ ($$1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4}$$)($$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4}$$)($$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4}$$)($$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2}$$)
$53$ ($$1 + 8 T + 11 T^{2} + 424 T^{3} + 2809 T^{4}$$)($$1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4}$$)($$1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4}$$)($$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$)($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4}$$)($$1 + 11 T - T^{2} + 176 T^{3} + 5662 T^{4} + 9328 T^{5} - 2809 T^{6} + 1637647 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4}$$)($$1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$1 + 5 T - 34 T^{2} + 295 T^{3} + 3481 T^{4}$$)($$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$)($$1 - 5 T - 85 T^{2} + 40 T^{3} + 7144 T^{4} + 2360 T^{5} - 295885 T^{6} - 1026895 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4}$$)($$1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4}$$)($$( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 + 15 T + 158 T^{2} + 1005 T^{3} + 4489 T^{4}$$)($$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$)($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4}$$)($$1 - 6 T - 31 T^{2} - 402 T^{3} + 4489 T^{4}$$)($$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$1 - 7 T - 83 T^{2} + 14 T^{3} + 9652 T^{4} + 938 T^{5} - 372587 T^{6} - 2105341 T^{7} + 20151121 T^{8}$$)
$71$ ($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{2}$$)($$( 1 - 10 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 10 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{4}$$)
$73$ ($$1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4}$$)($$( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4}$$)($$1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4}$$)($$1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4}$$)($$1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4}$$)($$1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 1022 T^{5} - 698099 T^{6} - 389017 T^{7} + 28398241 T^{8}$$)
$79$ ($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4}$$)($$1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 - 8 T - 53 T^{2} + 328 T^{3} + 2392 T^{4} + 25912 T^{5} - 330773 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$)
$83$ ($$( 1 + 6 T + 83 T^{2} )^{2}$$)($$( 1 - 7 T + 83 T^{2} )^{2}$$)($$( 1 + 2 T + 83 T^{2} )^{2}$$)($$( 1 - 9 T + 83 T^{2} )^{2}$$)($$( 1 + 7 T + 83 T^{2} )^{2}$$)($$( 1 + 6 T + 83 T^{2} )^{2}$$)($$( 1 + 7 T + 164 T^{2} + 581 T^{3} + 6889 T^{4} )^{2}$$)
$89$ ($$1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4}$$)($$1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4}$$)($$1 - 89 T^{2} + 7921 T^{4}$$)($$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$)($$1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4}$$)($$1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4}$$)($$1 - 6 T - 94 T^{2} + 288 T^{3} + 5775 T^{4} + 25632 T^{5} - 744574 T^{6} - 4229814 T^{7} + 62742241 T^{8}$$)
$97$ ($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 7 T + 97 T^{2} )^{2}$$)($$( 1 - 10 T + 97 T^{2} )^{2}$$)($$( 1 + T + 97 T^{2} )^{2}$$)($$( 1 + 17 T + 97 T^{2} )^{2}$$)($$( 1 + 6 T + 97 T^{2} )^{2}$$)($$( 1 - 25 T + 336 T^{2} - 2425 T^{3} + 9409 T^{4} )^{2}$$)