# Properties

 Label 504.2.bk Level 504 Weight 2 Character orbit bk Rep. character $$\chi_{504}(19,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 76 Newform subspaces 3 Sturm bound 192 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.bk (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$56$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$192$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

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

Total New Old
Modular forms 208 84 124
Cusp forms 176 76 100
Eisenstein series 32 8 24

## Trace form

 $$76q + 2q^{2} - 4q^{8} + O(q^{10})$$ $$76q + 2q^{2} - 4q^{8} + 6q^{10} - 2q^{11} + 4q^{14} - 4q^{16} + 6q^{17} - 6q^{19} - 8q^{22} - 32q^{25} + 24q^{26} - 14q^{28} + 12q^{32} + 6q^{35} + 42q^{38} - 18q^{40} - 30q^{44} + 6q^{46} + 4q^{49} - 28q^{50} - 60q^{52} - 16q^{56} - 26q^{58} + 54q^{59} + 48q^{64} + 12q^{65} + 14q^{67} - 60q^{68} + 18q^{70} - 30q^{73} + 22q^{74} - 72q^{80} - 78q^{82} - 50q^{86} - 38q^{88} - 18q^{89} - 24q^{91} + 68q^{92} - 48q^{94} - 54q^{98} + O(q^{100})$$

## 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.bk.a $$12$$ $$4.024$$ 12.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-\beta _{3}+\beta _{9}-\beta _{10}+\beta _{11})q^{4}+\cdots$$
504.2.bk.b $$32$$ $$4.024$$ None $$0$$ $$0$$ $$0$$ $$0$$
504.2.bk.c $$32$$ $$4.024$$ None $$2$$ $$0$$ $$0$$ $$0$$

## 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

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T^{3} + 8 T^{6} )^{2}$$)
$3$ 1
$5$ ($$1 - 15 T^{2} + 99 T^{4} - 412 T^{6} + 1641 T^{8} - 7989 T^{10} + 39846 T^{12} - 199725 T^{14} + 1025625 T^{16} - 6437500 T^{18} + 38671875 T^{20} - 146484375 T^{22} + 244140625 T^{24}$$)
$7$ ($$1 + 6 T^{2} - 33 T^{4} - 700 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12}$$)
$11$ ($$( 1 - 3 T - 21 T^{2} + 28 T^{3} + 393 T^{4} - 153 T^{5} - 4846 T^{6} - 1683 T^{7} + 47553 T^{8} + 37268 T^{9} - 307461 T^{10} - 483153 T^{11} + 1771561 T^{12} )^{2}$$)
$13$ ($$( 1 + 42 T^{2} + 903 T^{4} + 13340 T^{6} + 152607 T^{8} + 1199562 T^{10} + 4826809 T^{12} )^{2}$$)
$17$ ($$( 1 - 3 T + 39 T^{2} - 108 T^{3} + 729 T^{4} - 753 T^{5} + 12014 T^{6} - 12801 T^{7} + 210681 T^{8} - 530604 T^{9} + 3257319 T^{10} - 4259571 T^{11} + 24137569 T^{12} )^{2}$$)
$19$ ($$( 1 + 3 T + 39 T^{2} + 108 T^{3} + 705 T^{4} + 2265 T^{5} + 12706 T^{6} + 43035 T^{7} + 254505 T^{8} + 740772 T^{9} + 5082519 T^{10} + 7428297 T^{11} + 47045881 T^{12} )^{2}$$)
$23$ ($$1 + 69 T^{2} + 1863 T^{4} + 44136 T^{6} + 1489365 T^{8} + 33541107 T^{10} + 591428630 T^{12} + 17743245603 T^{14} + 416785390965 T^{16} + 6533711996904 T^{18} + 145893365578503 T^{20} + 2858429273741781 T^{22} + 21914624432020321 T^{24}$$)
$29$ ($$( 1 - 126 T^{2} + 7623 T^{4} - 278868 T^{6} + 6410943 T^{8} - 89117406 T^{10} + 594823321 T^{12} )^{2}$$)
$31$ ($$1 - 87 T^{2} + 3867 T^{4} - 78332 T^{6} - 455595 T^{8} + 87513459 T^{10} - 3532899090 T^{12} + 84100434099 T^{14} - 420751549995 T^{16} - 69519938340092 T^{18} + 3298129641784347 T^{20} - 71307660967329687 T^{22} + 787662783788549761 T^{24}$$)
$37$ ($$1 + 105 T^{2} + 3339 T^{4} + 152764 T^{6} + 14203497 T^{8} + 501483003 T^{10} + 10611705558 T^{12} + 686530231107 T^{14} + 26619640141017 T^{16} + 391950629144476 T^{18} + 11728168896642219 T^{20} + 504901359103874145 T^{22} + 6582952005840035281 T^{24}$$)
$41$ ($$( 1 - 102 T^{2} + 6783 T^{4} - 298852 T^{6} + 11402223 T^{8} - 288227622 T^{10} + 4750104241 T^{12} )^{2}$$)
$43$ ($$( 1 + 43 T^{2} )^{12}$$)
$47$ ($$1 - 159 T^{2} + 13131 T^{4} - 642364 T^{6} + 18269013 T^{8} - 74583429 T^{10} - 12209351154 T^{12} - 164754794661 T^{14} + 89146955624853 T^{16} - 6924179875597756 T^{18} + 312666005155583691 T^{20} - 8363262025496977791 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$)
$53$ ($$1 + 105 T^{2} - 693 T^{4} - 10308 T^{6} + 39259017 T^{8} + 1185419067 T^{10} - 43539019882 T^{12} + 3329842159203 T^{14} + 309772527717177 T^{16} - 228470234517732 T^{18} - 43145965455073173 T^{20} + 18363184388378870145 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$)
$59$ ($$( 1 + 21 T + 309 T^{2} + 3402 T^{3} + 29601 T^{4} + 225789 T^{5} + 1760606 T^{6} + 13321551 T^{7} + 103041081 T^{8} + 698699358 T^{9} + 3744264549 T^{10} + 15013410279 T^{11} + 42180533641 T^{12} )^{2}$$)
$61$ ($$1 - 159 T^{2} + 7419 T^{4} - 352004 T^{6} + 54786249 T^{8} - 3286587357 T^{10} + 113418284406 T^{12} - 12229391555397 T^{14} + 758561692640409 T^{16} - 18135377856569444 T^{18} + 1422276555126827739 T^{20} -$$$$11\!\cdots\!59$$$$T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$)
$67$ ($$( 1 - 15 T - 3 T^{2} + 142 T^{3} + 9993 T^{4} - 15123 T^{5} - 719466 T^{6} - 1013241 T^{7} + 44858577 T^{8} + 42708346 T^{9} - 60453363 T^{10} - 20251876605 T^{11} + 90458382169 T^{12} )^{2}$$)
$71$ ($$( 1 - 16 T + 71 T^{2} )^{6}( 1 + 16 T + 71 T^{2} )^{6}$$)
$73$ ($$( 1 - 9 T + 183 T^{2} - 1404 T^{3} + 16701 T^{4} - 147195 T^{5} + 1376278 T^{6} - 10745235 T^{7} + 88999629 T^{8} - 546179868 T^{9} + 5196878103 T^{10} - 18657644337 T^{11} + 151334226289 T^{12} )^{2}$$)
$79$ ($$1 + 261 T^{2} + 30519 T^{4} + 2829272 T^{6} + 267713397 T^{8} + 18988265187 T^{10} + 1205673992502 T^{12} + 118505763032067 T^{14} + 10427458497935157 T^{16} + 687760531456810712 T^{18} + 46300643769538335159 T^{20} +$$$$24\!\cdots\!61$$$$T^{22} +$$$$59\!\cdots\!41$$$$T^{24}$$)
$83$ ($$( 1 - 270 T^{2} + 28455 T^{4} - 2146852 T^{6} + 196026495 T^{8} - 12813746670 T^{10} + 326940373369 T^{12} )^{2}$$)
$89$ ($$( 1 + 3 T + 92 T^{2} + 267 T^{3} + 7921 T^{4} )^{6}$$)
$97$ ($$( 1 - 222 T^{2} + 20703 T^{4} - 1529300 T^{6} + 194794527 T^{8} - 19653500382 T^{10} + 832972004929 T^{12} )^{2}$$)