# Properties

 Label 756.2.bj Level 756 Weight 2 Character orbit bj Rep. character $$\chi_{756}(451,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 88 Newform subspaces 2 Sturm bound 288 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.bj (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$252$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$288$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(756, [\chi])$$.

Total New Old
Modular forms 312 104 208
Cusp forms 264 88 176
Eisenstein series 48 16 32

## Trace form

 $$88q + 2q^{2} - 2q^{4} + 6q^{5} + 8q^{8} + O(q^{10})$$ $$88q + 2q^{2} - 2q^{4} + 6q^{5} + 8q^{8} - 6q^{10} - 12q^{14} - 2q^{16} + 12q^{17} + 24q^{20} + 2q^{22} + 30q^{25} + 12q^{26} + 2q^{32} - 6q^{34} - 4q^{37} - 27q^{38} - 18q^{40} + 5q^{44} + 2q^{46} - 2q^{49} + 31q^{50} - 27q^{52} + 4q^{53} - 54q^{56} - 3q^{58} - 8q^{64} + 28q^{65} + 18q^{68} + 5q^{70} - 12q^{73} + 17q^{74} + 12q^{76} + 22q^{77} - 51q^{80} - 12q^{82} - 14q^{85} - 13q^{86} - 7q^{88} + 72q^{89} - 44q^{92} + 5q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(756, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
756.2.bj.a $$4$$ $$6.037$$ $$\Q(\zeta_{12})$$ None $$4$$ $$0$$ $$12$$ $$0$$ $$q+(1+\zeta_{12}^{3})q^{2}+2\zeta_{12}^{3}q^{4}+(4-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
756.2.bj.b $$84$$ $$6.037$$ None $$-2$$ $$0$$ $$-6$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(756, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(756, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T + 2 T^{2} )^{2}$$)
$3$ 1
$5$ ($$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )^{2}$$)
$7$ ($$1 - 13 T^{2} + 49 T^{4}$$)
$11$ ($$1 + 18 T^{2} + 203 T^{4} + 2178 T^{6} + 14641 T^{8}$$)
$13$ ($$( 1 + 2 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2}$$)
$17$ ($$( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} )^{2}$$)
$19$ ($$1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8}$$)
$23$ ($$1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8}$$)
$29$ ($$( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 + 35 T^{2} + 961 T^{4} )^{2}$$)
$37$ ($$( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2}$$)
$41$ ($$( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8}$$)
$47$ ($$( 1 + 91 T^{2} + 2209 T^{4} )^{2}$$)
$53$ ($$( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2}$$)
$59$ ($$( 1 + 115 T^{2} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 - 95 T^{2} + 3721 T^{4} )^{2}$$)
$67$ ($$( 1 - 53 T^{2} + 4489 T^{4} )^{2}$$)
$71$ ($$( 1 - 138 T^{2} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 - 149 T^{2} + 6241 T^{4} )^{2}$$)
$83$ ($$1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8}$$)
$89$ ($$( 1 + 15 T + 164 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} )^{2}$$)