# Properties

 Label 504.2.p Level 504 Weight 2 Character orbit p Rep. character $$\chi_{504}(307,\cdot)$$ Character field $$\Q$$ Dimension 38 Newform subspaces 7 Sturm bound 192 Trace bound 14

# Related objects

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.p (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$56$$ Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$192$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$5$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(504, [\chi])$$.

Total New Old
Modular forms 104 42 62
Cusp forms 88 38 50
Eisenstein series 16 4 12

## Trace form

 $$38q + q^{2} - 3q^{4} + 7q^{8} + O(q^{10})$$ $$38q + q^{2} - 3q^{4} + 7q^{8} + 8q^{11} + 11q^{14} - 11q^{16} + 8q^{22} + 26q^{25} - 13q^{28} - 9q^{32} - 12q^{44} + 6q^{46} - 10q^{49} + 43q^{50} - 35q^{56} - 16q^{58} - 39q^{64} + 24q^{65} + 16q^{67} - 48q^{70} - 28q^{74} - 4q^{86} + 8q^{88} - 24q^{91} - 62q^{92} - 39q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(504, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
504.2.p.a $$2$$ $$4.024$$ $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{-7})$$ $$-1$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta )q^{2}+(-1-\beta )q^{4}+(-1+\cdots)q^{7}+\cdots$$
504.2.p.b $$4$$ $$4.024$$ $$\Q(\zeta_{12})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\zeta_{12})q^{2}-2\zeta_{12}q^{4}+2\zeta_{12}^{3}q^{5}+\cdots$$
504.2.p.c $$4$$ $$4.024$$ $$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Q(\sqrt{-42})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-2q^{4}-\beta _{3}q^{7}-2\beta _{1}q^{8}+\cdots$$
504.2.p.d $$4$$ $$4.024$$ $$\Q(i, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}+(2-\beta _{2})q^{4}+(-1+2\beta _{2}+\cdots)q^{7}+\cdots$$
504.2.p.e $$4$$ $$4.024$$ $$\Q(\zeta_{12})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(1+\zeta_{12})q^{2}+2\zeta_{12}q^{4}-2\zeta_{12}^{3}q^{5}+\cdots$$
504.2.p.f $$4$$ $$4.024$$ $$\Q(i, \sqrt{6})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{1})q^{2}-2\beta _{1}q^{4}+\beta _{3}q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots$$
504.2.p.g $$16$$ $$4.024$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$-2$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}+\beta _{3}q^{4}+\beta _{5}q^{5}+\beta _{8}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(504, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(504, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T + 2 T^{2}$$)($$( 1 + 2 T + 2 T^{2} )^{2}$$)($$( 1 + 2 T^{2} )^{2}$$)($$1 - 3 T^{2} + 4 T^{4}$$)($$( 1 - 2 T + 2 T^{2} )^{2}$$)($$( 1 - 2 T + 2 T^{2} )^{2}$$)($$( 1 + T - 2 T^{3} - 4 T^{4} - 4 T^{5} + 8 T^{7} + 16 T^{8} )^{2}$$)
$3$ 1
$5$ ($$( 1 + 5 T^{2} )^{2}$$)($$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 5 T^{2} )^{4}$$)($$( 1 + 5 T^{2} )^{4}$$)($$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 140 T^{4} + 832 T^{6} + 4326 T^{8} + 20800 T^{10} + 87500 T^{12} + 250000 T^{14} + 390625 T^{16} )^{2}$$)
$7$ ($$1 + 7 T^{2}$$)($$1 + 2 T^{2} + 49 T^{4}$$)($$( 1 - 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$1 + 2 T^{2} + 49 T^{4}$$)($$1 - 10 T^{2} + 49 T^{4}$$)($$1 + 4 T^{2} + 68 T^{4} + 572 T^{6} + 3382 T^{8} + 28028 T^{10} + 163268 T^{12} + 470596 T^{14} + 5764801 T^{16}$$)
$11$ ($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{4}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 - 6 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{4}$$)($$( 1 + 2 T + 11 T^{2} )^{4}$$)($$( 1 - 2 T + 20 T^{2} - 6 T^{3} + 182 T^{4} - 66 T^{5} + 2420 T^{6} - 2662 T^{7} + 14641 T^{8} )^{4}$$)
$13$ ($$( 1 + 13 T^{2} )^{2}$$)($$( 1 + 14 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 169 T^{4} )^{2}$$)($$( 1 + 13 T^{2} )^{4}$$)($$( 1 + 14 T^{2} + 169 T^{4} )^{2}$$)($$( 1 + 20 T^{2} + 169 T^{4} )^{2}$$)($$( 1 + 36 T^{2} + 692 T^{4} + 12604 T^{6} + 195894 T^{8} + 2130076 T^{10} + 19764212 T^{12} + 173765124 T^{14} + 815730721 T^{16} )^{2}$$)
$17$ ($$( 1 - 17 T^{2} )^{2}$$)($$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 17 T^{2} )^{4}$$)($$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 72 T^{2} + 2652 T^{4} - 66024 T^{6} + 1253414 T^{8} - 19080936 T^{10} + 221497692 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 19 T^{2} )^{2}$$)($$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 19 T^{2} )^{4}$$)($$( 1 - 19 T^{2} )^{4}$$)($$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 1532 T^{4} - 36808 T^{6} + 804262 T^{8} - 13287688 T^{10} + 199651772 T^{12} - 2634569336 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$)($$( 1 - 42 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 42 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 88 T^{2} + 3548 T^{4} - 96952 T^{6} + 2287782 T^{8} - 51287608 T^{10} + 992875868 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$)($$( 1 + 6 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 6 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$)($$( 1 - 164 T^{2} + 12980 T^{4} - 650812 T^{6} + 22500790 T^{8} - 547332892 T^{10} + 9180507380 T^{12} - 97551024644 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 50 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 31 T^{2} )^{4}$$)($$( 1 + 50 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 68 T^{2} + 5252 T^{4} + 204988 T^{6} + 8264182 T^{8} + 196993468 T^{10} + 4850332292 T^{12} + 60350250308 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 6 T + 37 T^{2} )( 1 + 6 T + 37 T^{2} )$$)($$( 1 - 58 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 37 T^{2} )^{4}$$)($$( 1 - 6 T + 37 T^{2} )^{2}( 1 + 6 T + 37 T^{2} )^{2}$$)($$( 1 - 58 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 232 T^{2} + 25148 T^{4} - 1670040 T^{6} + 74486502 T^{8} - 2286284760 T^{10} + 47131400828 T^{12} - 595248526888 T^{14} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 41 T^{2} )^{2}$$)($$( 1 + 26 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 + 26 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 - 120 T^{2} + 10204 T^{4} - 627512 T^{6} + 28679654 T^{8} - 1054847672 T^{10} + 28834065244 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 12 T + 43 T^{2} )^{2}$$)($$( 1 - 6 T + 43 T^{2} )^{4}$$)($$( 1 - 2 T + 43 T^{2} )^{4}$$)($$( 1 + 12 T + 43 T^{2} )^{4}$$)($$( 1 - 6 T + 43 T^{2} )^{4}$$)($$( 1 + 6 T + 43 T^{2} )^{4}$$)($$( 1 + 2 T + 120 T^{2} + 250 T^{3} + 6782 T^{4} + 10750 T^{5} + 221880 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 120 T^{2} + 10396 T^{4} + 591944 T^{6} + 31424966 T^{8} + 1307604296 T^{10} + 50729163676 T^{12} + 1293505839480 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 10 T + 53 T^{2} )( 1 + 10 T + 53 T^{2} )$$)($$( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2}$$)($$( 1 + 22 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 6 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2}$$)($$( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2}$$)($$( 1 - 260 T^{2} + 33908 T^{4} - 2877468 T^{6} + 176791542 T^{8} - 8082807612 T^{10} + 267550429748 T^{12} - 5762733893540 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 - 59 T^{2} )^{2}$$)($$( 1 - 70 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 106 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 59 T^{2} )^{4}$$)($$( 1 - 70 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 112 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 264 T^{2} + 34076 T^{4} - 3023352 T^{6} + 203740198 T^{8} - 10524288312 T^{10} + 412911193436 T^{12} - 11135660881224 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 61 T^{2} )^{2}$$)($$( 1 + 14 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 94 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 61 T^{2} )^{4}$$)($$( 1 + 14 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 68 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 324 T^{2} + 46708 T^{4} + 4167580 T^{6} + 280236854 T^{8} + 15507565180 T^{10} + 646711541428 T^{12} + 16692601292964 T^{14} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 - 4 T + 67 T^{2} )^{2}$$)($$( 1 - 10 T + 67 T^{2} )^{4}$$)($$( 1 + 10 T + 67 T^{2} )^{4}$$)($$( 1 - 4 T + 67 T^{2} )^{4}$$)($$( 1 - 10 T + 67 T^{2} )^{4}$$)($$( 1 + 2 T + 67 T^{2} )^{4}$$)($$( 1 + 10 T + 208 T^{2} + 1122 T^{3} + 16622 T^{4} + 75174 T^{5} + 933712 T^{6} + 3007630 T^{7} + 20151121 T^{8} )^{4}$$)
$71$ ($$( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} )$$)($$( 1 - 138 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 58 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 114 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 138 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 42 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 456 T^{2} + 94300 T^{4} - 11817416 T^{6} + 1002933542 T^{8} - 59571594056 T^{10} + 2396321518300 T^{12} - 58413729467976 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$( 1 - 73 T^{2} )^{2}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 + 70 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 328 T^{2} + 54492 T^{4} - 6043256 T^{6} + 500200454 T^{8} - 32204511224 T^{10} + 1547476948572 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$)($$( 1 - 14 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 79 T^{2} )^{4}$$)($$( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 14 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 444 T^{2} + 95492 T^{4} - 12990660 T^{6} + 1221053302 T^{8} - 81074709060 T^{10} + 3719421134852 T^{12} - 107930830251324 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 - 83 T^{2} )^{2}$$)($$( 1 - 118 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 58 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 83 T^{2} )^{4}$$)($$( 1 - 118 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 160 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 120 T^{2} + 20284 T^{4} - 1531784 T^{6} + 164981030 T^{8} - 10552459976 T^{10} + 962644583164 T^{12} - 39232844804280 T^{14} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 89 T^{2} )^{2}$$)($$( 1 - 70 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 122 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 89 T^{2} )^{4}$$)($$( 1 - 70 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 312 T^{2} + 51484 T^{4} - 5696504 T^{6} + 530357222 T^{8} - 45122008184 T^{10} + 3230221535644 T^{12} - 155058162779832 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 - 97 T^{2} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 232 T^{2} + 17244 T^{4} + 409768 T^{6} - 157308730 T^{8} + 3855507112 T^{10} + 1526598921564 T^{12} - 193249505143528 T^{14} + 7837433594376961 T^{16} )^{2}$$)