# Properties

 Label 504.2.c Level 504 Weight 2 Character orbit c Rep. character $$\chi_{504}(253,\cdot)$$ Character field $$\Q$$ Dimension 30 Newform subspaces 6 Sturm bound 192 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$8$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$192$$ Trace bound: $$7$$ 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 30 74
Cusp forms 88 30 58
Eisenstein series 16 0 16

## Trace form

 $$30q - q^{2} + 5q^{4} - 2q^{7} + 5q^{8} + O(q^{10})$$ $$30q - q^{2} + 5q^{4} - 2q^{7} + 5q^{8} + 4q^{10} + q^{14} + 5q^{16} - 4q^{17} - 24q^{20} - 18q^{22} + 16q^{23} - 26q^{25} - q^{28} + 16q^{31} + 29q^{32} - 2q^{34} - 10q^{38} - 4q^{40} + 12q^{41} + 14q^{44} - 12q^{46} + 30q^{49} - 13q^{50} - 52q^{52} - 32q^{55} + 7q^{56} + 32q^{58} + 16q^{62} + 17q^{64} + 16q^{65} + 26q^{68} - 12q^{70} + 24q^{71} - 20q^{73} - 12q^{74} + 22q^{76} - 56q^{80} + 14q^{82} - 46q^{86} + 34q^{88} - 20q^{89} - 40q^{92} - 24q^{94} - 72q^{95} - 4q^{97} - q^{98} + O(q^{100})$$

## 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.c.a $$2$$ $$4.024$$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta q^{2}-2q^{4}-\beta q^{5}+q^{7}-2\beta q^{8}+\cdots$$
504.2.c.b $$4$$ $$4.024$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{4}+\cdots$$
504.2.c.c $$4$$ $$4.024$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+\beta _{2}q^{5}+q^{7}+\cdots$$
504.2.c.d $$4$$ $$4.024$$ 4.0.2312.1 None $$1$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}-q^{7}+\cdots$$
504.2.c.e $$8$$ $$4.024$$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{5}q^{5}-q^{7}+\beta _{3}q^{8}+\cdots$$
504.2.c.f $$8$$ $$4.024$$ 8.0.386672896.3 None $$0$$ $$0$$ $$0$$ $$8$$ $$q-\beta _{3}q^{2}-\beta _{2}q^{4}+(-\beta _{2}+\beta _{6})q^{5}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(504, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T^{2}$$)($$1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4}$$)($$1 - 2 T^{2} + 4 T^{4}$$)($$1 - T - 2 T^{3} + 4 T^{4}$$)($$1 - T^{2} - 2 T^{4} - 4 T^{6} + 16 T^{8}$$)($$1 - T^{2} - 2 T^{3} + 2 T^{4} - 4 T^{5} - 4 T^{6} + 16 T^{8}$$)
$3$ 1
$5$ ($$1 - 8 T^{2} + 25 T^{4}$$)($$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$)($$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$)($$1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8}$$)($$( 1 - 6 T^{2} + 18 T^{4} - 150 T^{6} + 625 T^{8} )^{2}$$)($$1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 4200 T^{10} + 10000 T^{12} - 125000 T^{14} + 390625 T^{16}$$)
$7$ ($$( 1 - T )^{2}$$)($$( 1 + T )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T )^{8}$$)($$( 1 - T )^{8}$$)
$11$ ($$( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$)($$1 - 20 T^{2} + 234 T^{4} - 2420 T^{6} + 14641 T^{8}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$)($$1 - 24 T^{2} + 318 T^{4} - 2904 T^{6} + 14641 T^{8}$$)($$( 1 - 18 T^{2} + 282 T^{4} - 2178 T^{6} + 14641 T^{8} )^{2}$$)($$1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 1005752 T^{10} + 8667472 T^{12} - 42517464 T^{14} + 214358881 T^{16}$$)
$13$ ($$1 - 8 T^{2} + 169 T^{4}$$)($$1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8}$$)($$( 1 - 13 T^{2} )^{4}$$)($$1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8}$$)($$( 1 - 8 T^{2} + 190 T^{4} - 1352 T^{6} + 28561 T^{8} )^{2}$$)($$1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 4512976 T^{10} + 38271740 T^{12} - 231686832 T^{14} + 815730721 T^{16}$$)
$17$ ($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 8 T^{2} + 34 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{4}$$)($$( 1 - 6 T^{2} + 218 T^{4} - 1734 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 2 T + 38 T^{2} + 70 T^{3} + 706 T^{4} + 1190 T^{5} + 10982 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2}$$)
$19$ ($$1 - 20 T^{2} + 361 T^{4}$$)($$1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8}$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$)($$1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8}$$)($$( 1 + 558 T^{4} + 130321 T^{8} )^{2}$$)($$1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 15317952 T^{10} + 262205852 T^{12} - 3010936384 T^{14} + 16983563041 T^{16}$$)
$23$ ($$( 1 - 6 T + 23 T^{2} )^{2}$$)($$( 1 - 10 T + 68 T^{2} - 230 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 40 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 34 T^{4} + 1058 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 + 6 T + 74 T^{2} + 334 T^{3} + 2282 T^{4} + 7682 T^{5} + 39146 T^{6} + 73002 T^{7} + 279841 T^{8} )^{2}$$)
$29$ ($$1 - 50 T^{2} + 841 T^{4}$$)($$( 1 - 46 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$)($$1 - 76 T^{2} + 2854 T^{4} - 63916 T^{6} + 707281 T^{8}$$)($$( 1 - 64 T^{2} + 2542 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2}$$)($$1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 29374448 T^{10} + 2034140156 T^{12} - 9517173136 T^{14} + 500246412961 T^{16}$$)
$31$ ($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 2 T + 31 T^{2} )^{4}$$)($$( 1 - 2 T + 31 T^{2} )^{4}$$)($$( 1 + 4 T - 2 T^{2} + 124 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 2 T + 22 T^{2} - 62 T^{3} + 961 T^{4} )^{4}$$)($$( 1 - 4 T + 80 T^{2} - 244 T^{3} + 3294 T^{4} - 7564 T^{5} + 76880 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2}$$)
$37$ ($$1 - 2 T^{2} + 1369 T^{4}$$)($$1 - 44 T^{2} + 2454 T^{4} - 60236 T^{6} + 1874161 T^{8}$$)($$( 1 - 62 T^{2} + 1369 T^{4} )^{2}$$)($$1 - 108 T^{2} + 5382 T^{4} - 147852 T^{6} + 1874161 T^{8}$$)($$( 1 - 72 T^{2} + 3870 T^{4} - 98568 T^{6} + 1874161 T^{8} )^{2}$$)($$1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1609768768 T^{10} + 35076797276 T^{12} - 492619470528 T^{14} + 3512479453921 T^{16}$$)
$41$ ($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 8 T^{2} - 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 8 T + 30 T^{2} - 328 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 74 T^{2} + 3706 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 8774 T^{5} + 225254 T^{6} - 137842 T^{7} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} )$$)($$1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8}$$)($$( 1 - 8 T + 43 T^{2} )^{2}( 1 + 8 T + 43 T^{2} )^{2}$$)($$1 - 152 T^{2} + 9406 T^{4} - 281048 T^{6} + 3418801 T^{8}$$)($$( 1 - 68 T^{2} + 2230 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2}$$)($$1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1898035480 T^{10} + 53593124476 T^{12} - 1061988992232 T^{14} + 11688200277601 T^{16}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 - 2 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{8}$$)($$( 1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 6016 T^{5} + 97196 T^{6} + 4879681 T^{8} )^{2}$$)
$53$ ($$1 - 74 T^{2} + 2809 T^{4}$$)($$( 1 + 2 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 74 T^{2} + 2809 T^{4} )^{2}$$)($$1 - 132 T^{2} + 8886 T^{4} - 370788 T^{6} + 7890481 T^{8}$$)($$( 1 - 96 T^{2} + 7758 T^{4} - 269664 T^{6} + 7890481 T^{8} )^{2}$$)($$1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 14121899184 T^{10} + 415070862524 T^{12} - 7447225339344 T^{14} + 62259690411361 T^{16}$$)
$59$ ($$1 - 116 T^{2} + 3481 T^{4}$$)($$1 - 12 T^{2} - 5290 T^{4} - 41772 T^{6} + 12117361 T^{8}$$)($$( 1 + 10 T^{2} + 3481 T^{4} )^{2}$$)($$1 - 178 T^{2} + 14050 T^{4} - 619618 T^{6} + 12117361 T^{8}$$)($$( 1 - 124 T^{2} + 8182 T^{4} - 431644 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 102 T^{2} + 3481 T^{4} )^{4}$$)
$61$ ($$1 + 40 T^{2} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 + 70 T^{2} + 3721 T^{4} )^{2}$$)($$1 - 230 T^{2} + 20650 T^{4} - 855830 T^{6} + 13845841 T^{8}$$)($$( 1 - 200 T^{2} + 17278 T^{4} - 744200 T^{6} + 13845841 T^{8} )^{2}$$)($$1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 11277964016 T^{10} + 491748888956 T^{12} - 14013541826192 T^{14} + 191707312997281 T^{16}$$)
$67$ ($$( 1 - 67 T^{2} )^{2}$$)($$( 1 - 98 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2}$$)($$1 + 112 T^{2} + 8782 T^{4} + 502768 T^{6} + 20151121 T^{8}$$)($$( 1 - 164 T^{2} + 13078 T^{4} - 736196 T^{6} + 20151121 T^{8} )^{2}$$)($$1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 30450108008 T^{10} + 1208986655516 T^{12} - 31117683466136 T^{14} + 406067677556641 T^{16}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 18 T + 220 T^{2} + 1278 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 88 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{4}$$)($$( 1 + 210 T^{2} + 20738 T^{4} + 1058610 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 14 T + 194 T^{2} - 1686 T^{3} + 14330 T^{4} - 119706 T^{5} + 977954 T^{6} - 5010754 T^{7} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 - 24 T + 278 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{4}$$)($$( 1 + 6 T + 73 T^{2} )^{4}$$)($$( 1 + 6 T + 73 T^{2} )^{8}$$)($$( 1 + 4 T + 92 T^{2} - 580 T^{3} + 166 T^{4} - 42340 T^{5} + 490268 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 54 T^{2} + 316 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{4}$$)($$( 1 + 79 T^{2} )^{4}$$)($$( 1 + 79 T^{2} )^{8}$$)($$( 1 + 20 T + 340 T^{2} + 3972 T^{3} + 38678 T^{4} + 313788 T^{5} + 2121940 T^{6} + 9860780 T^{7} + 38950081 T^{8} )^{2}$$)
$83$ ($$1 + 76 T^{2} + 6889 T^{4}$$)($$1 - 300 T^{2} + 36086 T^{4} - 2066700 T^{6} + 47458321 T^{8}$$)($$( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2}$$)($$1 - 210 T^{2} + 23970 T^{4} - 1446690 T^{6} + 47458321 T^{8}$$)($$( 1 - 276 T^{2} + 32166 T^{4} - 1901364 T^{6} + 47458321 T^{8} )^{2}$$)($$1 - 248 T^{2} + 44156 T^{4} - 5204488 T^{6} + 499369126 T^{8} - 35853717832 T^{10} + 2095569622076 T^{12} - 81081212595512 T^{14} + 2252292232139041 T^{16}$$)
$89$ ($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 200 T^{2} + 890 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 16 T + 174 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 266 T^{2} + 32506 T^{4} + 2106986 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 43966 T^{5} + 1568358 T^{6} + 7049690 T^{7} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 + 16 T + 246 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 10 T + 97 T^{2} )^{4}$$)($$( 1 - 8 T + 142 T^{2} - 776 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 8 T + 46 T^{2} - 776 T^{3} + 9409 T^{4} )^{4}$$)($$( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 249484 T^{5} + 2822700 T^{6} - 18253460 T^{7} + 88529281 T^{8} )^{2}$$)