# Properties

 Label 252.2.bi Level 252 Weight 2 Character orbit bi Rep. character $$\chi_{252}(139,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 88 Newform subspaces 3 Sturm bound 96 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.bi (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$252$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(252, [\chi])$$.

Total New Old
Modular forms 104 104 0
Cusp forms 88 88 0
Eisenstein series 16 16 0

## Trace form

 $$88q - 2q^{2} - 2q^{4} - 8q^{8} - 8q^{9} + O(q^{10})$$ $$88q - 2q^{2} - 2q^{4} - 8q^{8} - 8q^{9} - 2q^{16} - 10q^{18} - 2q^{21} - 10q^{22} + 24q^{25} - 12q^{28} - 12q^{29} - 40q^{30} - 32q^{32} + 34q^{36} - 16q^{37} - 2q^{42} - 68q^{44} - 16q^{46} - 2q^{49} - 4q^{50} - 16q^{53} + 12q^{56} - 24q^{57} + 6q^{58} + 32q^{60} - 8q^{64} - 52q^{65} - 16q^{70} - 52q^{72} + 64q^{74} - 22q^{77} - 32q^{81} + 62q^{84} + 16q^{85} - 56q^{86} + 14q^{88} - 10q^{92} - 12q^{93} + 64q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(252, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
252.2.bi.a $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$-6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(2\zeta_{12}+\cdots)q^{3}+\cdots$$
252.2.bi.b $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-2\zeta_{12}+\cdots)q^{3}+\cdots$$
252.2.bi.c $$80$$ $$2.012$$ None $$-6$$ $$0$$ $$0$$ $$0$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$)($$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$)
$3$ ($$( 1 - 3 T^{2} )^{2}$$)($$( 1 - 3 T^{2} )^{2}$$)
$5$ ($$( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2}$$)($$( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2}$$)
$7$ ($$1 - 13 T^{2} + 49 T^{4}$$)($$1 + 2 T^{2} + 49 T^{4}$$)
$11$ ($$1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8}$$)($$1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8}$$)
$13$ ($$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2}$$)
$17$ ($$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - 10 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 361 T^{4} )^{2}$$)
$23$ ($$1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8}$$)($$1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8}$$)
$29$ ($$( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} )$$)($$( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} )$$)
$37$ ($$( 1 + 37 T^{2} )^{4}$$)($$( 1 + 37 T^{2} )^{4}$$)
$41$ ($$( 1 - 15 T + 116 T^{2} - 615 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 15 T + 116 T^{2} + 615 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)
$47$ ($$1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8}$$)($$1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8}$$)
$53$ ($$( 1 - 4 T + 53 T^{2} )^{4}$$)($$( 1 - 4 T + 53 T^{2} )^{4}$$)
$59$ ($$1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8}$$)($$1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8}$$)
$61$ ($$( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 9 T + 88 T^{2} - 549 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 - 91 T^{2} + 3792 T^{4} - 408499 T^{6} + 20151121 T^{8}$$)($$1 - 91 T^{2} + 3792 T^{4} - 408499 T^{6} + 20151121 T^{8}$$)
$71$ ($$( 1 - 126 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 126 T^{2} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$)
$79$ ($$1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8}$$)($$1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8}$$)
$83$ ($$1 - 163 T^{2} + 19680 T^{4} - 1122907 T^{6} + 47458321 T^{8}$$)($$1 - 163 T^{2} + 19680 T^{4} - 1122907 T^{6} + 47458321 T^{8}$$)
$89$ ($$( 1 + 122 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 122 T^{2} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 - 5 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 5 T + 97 T^{2} )^{2}$$)