# Properties

 Label 504.2.bf Level 504 Weight 2 Character orbit bf Rep. character $$\chi_{504}(115,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 184 Newform subspaces 2 Sturm bound 192 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.bf (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$504$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$192$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

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

Total New Old
Modular forms 200 200 0
Cusp forms 184 184 0
Eisenstein series 16 16 0

## Trace form

 $$184q - 2q^{2} - 6q^{3} - 2q^{4} - 6q^{6} - 8q^{8} - 2q^{9} + O(q^{10})$$ $$184q - 2q^{2} - 6q^{3} - 2q^{4} - 6q^{6} - 8q^{8} - 2q^{9} - 6q^{10} + 2q^{11} + 12q^{12} + 2q^{14} - 2q^{16} - 12q^{17} - 4q^{18} - 12q^{19} + 24q^{20} - 6q^{22} - 12q^{24} - 74q^{25} - 5q^{30} - 42q^{32} - 6q^{33} - 6q^{34} - 18q^{35} + 2q^{36} - 33q^{38} + 12q^{40} + 10q^{42} - 4q^{43} - 21q^{44} + 2q^{46} - 9q^{48} - 2q^{49} + 19q^{50} + 6q^{51} + 21q^{52} - 51q^{54} - 2q^{56} - 20q^{57} + 5q^{58} + 25q^{60} - 8q^{64} + 36q^{65} + 36q^{66} - 4q^{67} + 12q^{68} + 21q^{70} - 4q^{72} - 12q^{73} + 47q^{74} - 6q^{75} - 12q^{76} - 39q^{78} - 63q^{80} + 14q^{81} - 12q^{82} + 60q^{83} + 7q^{84} - 31q^{86} + 9q^{88} - 36q^{89} - 33q^{90} + 20q^{91} - 32q^{92} - 45q^{96} + 27q^{98} - 26q^{99} + O(q^{100})$$

## 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.bf.a $$4$$ $$4.024$$ $$\Q(\zeta_{12})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(1+\zeta_{12}^{3})q^{2}+(1-2\zeta_{12}^{2})q^{3}+2\zeta_{12}^{3}q^{4}+\cdots$$
504.2.bf.b $$180$$ $$4.024$$ None $$-6$$ $$-6$$ $$0$$ $$0$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T + 2 T^{2} )^{2}$$)
$3$ ($$( 1 + 3 T^{2} )^{2}$$)
$5$ ($$1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8}$$)
$7$ ($$1 + 2 T^{2} + 49 T^{4}$$)
$11$ ($$( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$( 1 - 22 T^{2} + 169 T^{4} )( 1 - T^{2} + 169 T^{4} )$$)
$17$ ($$( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$1 - 3 T^{2} - 520 T^{4} - 1587 T^{6} + 279841 T^{8}$$)
$29$ ($$1 + 9 T^{2} - 760 T^{4} + 7569 T^{6} + 707281 T^{8}$$)
$31$ ($$( 1 + 14 T^{2} + 961 T^{4} )^{2}$$)
$37$ ($$( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} )$$)
$41$ ($$( 1 - 21 T + 188 T^{2} - 861 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$( 1 + 9 T + 38 T^{2} + 387 T^{3} + 1849 T^{4} )^{2}$$)
$47$ ($$( 1 + 82 T^{2} + 2209 T^{4} )^{2}$$)
$53$ ($$1 - 15 T^{2} - 2584 T^{4} - 42135 T^{6} + 7890481 T^{8}$$)
$59$ ($$( 1 - 70 T^{2} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 + 110 T^{2} + 3721 T^{4} )^{2}$$)
$67$ ($$( 1 + 8 T + 67 T^{2} )^{4}$$)
$71$ ($$( 1 - 42 T^{2} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$)
$83$ ($$( 1 - 27 T + 326 T^{2} - 2241 T^{3} + 6889 T^{4} )^{2}$$)
$89$ ($$( 1 - 15 T + 164 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4} )^{2}$$)