# Properties

 Label 225.2.f Level 225 Weight 2 Character orbit f Rep. character $$\chi_{225}(107,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 12 Newform subspaces 2 Sturm bound 60 Trace bound 1

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## Defining parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 225.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$60$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(225, [\chi])$$.

Total New Old
Modular forms 84 12 72
Cusp forms 36 12 24
Eisenstein series 48 0 48

## Trace form

 $$12q + 8q^{7} + O(q^{10})$$ $$12q + 8q^{7} - 4q^{13} - 28q^{16} - 8q^{22} - 8q^{28} - 8q^{31} - 4q^{37} + 32q^{43} + 32q^{46} - 4q^{52} - 12q^{58} - 24q^{61} - 16q^{67} - 4q^{73} + 160q^{76} + 4q^{82} + 24q^{88} - 88q^{91} + 44q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(225, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
225.2.f.a $$4$$ $$1.797$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{4}+(2-2\zeta_{8}^{2})q^{7}+\cdots$$
225.2.f.b $$8$$ $$1.797$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{6}q^{2}-4\zeta_{24}^{2}q^{4}+\zeta_{24}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(225, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(225, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{4} + 16 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$)
$3$ 1
$5$ 1
$7$ ($$( 1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4} )^{2}$$)($$( 1 + 23 T^{4} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 - 4 T^{2} + 121 T^{4} )^{4}$$)
$13$ ($$( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2}$$)($$( 1 - 337 T^{4} + 28561 T^{8} )^{2}$$)
$17$ ($$1 - 254 T^{4} + 83521 T^{8}$$)($$( 1 + 206 T^{4} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 19 T^{2} )^{4}$$)($$( 1 - 13 T^{2} + 361 T^{4} )^{4}$$)
$23$ ($$1 - 158 T^{4} + 279841 T^{8}$$)($$( 1 + 542 T^{4} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 + 40 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 40 T^{2} + 841 T^{4} )^{4}$$)
$31$ ($$( 1 + 4 T + 31 T^{2} )^{4}$$)($$( 1 - T + 31 T^{2} )^{8}$$)
$37$ ($$( 1 + 2 T + 2 T^{2} + 74 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 1106 T^{4} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 - 80 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 1681 T^{4} )^{4}$$)
$43$ ($$( 1 - 16 T + 128 T^{2} - 688 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 3191 T^{4} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 - 3518 T^{4} + 4879681 T^{8}$$)($$( 1 - 2818 T^{4} + 4879681 T^{8} )^{2}$$)
$53$ ($$( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} )$$)($$( 1 - 5518 T^{4} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + 46 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 44 T^{2} + 3481 T^{4} )^{4}$$)
$61$ ($$( 1 - 8 T + 61 T^{2} )^{4}$$)($$( 1 + 7 T + 61 T^{2} )^{8}$$)
$67$ ($$( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 2471 T^{4} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 70 T^{2} + 5041 T^{4} )^{4}$$)
$73$ ($$( 1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 7298 T^{4} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 - 14 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 154 T^{2} + 6241 T^{4} )^{4}$$)
$83$ ($$1 + 8722 T^{4} + 47458321 T^{8}$$)($$( 1 + 11822 T^{4} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 + 16 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 106 T^{2} + 7921 T^{4} )^{4}$$)
$97$ ($$( 1 - 22 T + 242 T^{2} - 2134 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 16609 T^{4} + 88529281 T^{8} )^{2}$$)
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