# Properties

 Label 450.2.f Level 450 Weight 2 Character orbit f Rep. character $$\chi_{450}(107,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 12 Newform subspaces 3 Sturm bound 180 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 450.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$3$$ Sturm bound: $$180$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(450, [\chi])$$.

Total New Old
Modular forms 228 12 216
Cusp forms 132 12 120
Eisenstein series 96 0 96

## Trace form

 $$12q - 8q^{7} + O(q^{10})$$ $$12q - 8q^{7} + 12q^{13} - 12q^{16} - 8q^{22} - 8q^{28} + 48q^{31} + 12q^{37} - 32q^{46} - 12q^{52} - 4q^{58} - 80q^{61} + 16q^{67} - 4q^{73} - 16q^{76} + 28q^{82} - 8q^{88} + 96q^{91} + 12q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(450, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
450.2.f.a $$4$$ $$3.593$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-3+3\zeta_{8}^{2})q^{7}+\cdots$$
450.2.f.b $$4$$ $$3.593$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-2+2\zeta_{8}^{2})q^{7}+\cdots$$
450.2.f.c $$4$$ $$3.593$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(3-3\zeta_{8}^{2})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(450, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(450, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)
$3$ ()()()
$5$ ()()()
$7$ ($$( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4} )^{2}$$)($$( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} )^{2}$$)
$11$ ($$( 1 - 4 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 - 4 T^{2} + 121 T^{4} )^{2}$$)
$13$ ($$( 1 + 6 T + 18 T^{2} + 78 T^{3} + 169 T^{4} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} )^{2}$$)
$17$ ($$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )$$)($$1 - 254 T^{4} + 83521 T^{8}$$)($$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )$$)
$19$ ($$( 1 - 34 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 34 T^{2} + 361 T^{4} )^{2}$$)
$23$ ($$1 - 958 T^{4} + 279841 T^{8}$$)($$1 - 158 T^{4} + 279841 T^{8}$$)($$1 - 958 T^{4} + 279841 T^{8}$$)
$29$ ($$( 1 - 14 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 56 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )^{4}$$)($$( 1 - 4 T + 31 T^{2} )^{4}$$)($$( 1 - 4 T + 31 T^{2} )^{4}$$)
$37$ ($$( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$)
$41$ ($$( 1 - 64 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 64 T^{2} + 1681 T^{4} )^{2}$$)
$43$ ($$( 1 + 1849 T^{4} )^{2}$$)($$( 1 + 1849 T^{4} )^{2}$$)($$( 1 + 1849 T^{4} )^{2}$$)
$47$ ($$( 1 + 2209 T^{4} )^{2}$$)($$( 1 + 2209 T^{4} )^{2}$$)($$( 1 + 2209 T^{4} )^{2}$$)
$53$ ($$1 - 718 T^{4} + 7890481 T^{8}$$)($$( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} )$$)($$1 - 718 T^{4} + 7890481 T^{8}$$)
$59$ ($$( 1 + 100 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 110 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 100 T^{2} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 + 10 T + 61 T^{2} )^{4}$$)($$( 1 + 61 T^{2} )^{4}$$)($$( 1 + 10 T + 61 T^{2} )^{4}$$)
$67$ ($$( 1 - 12 T + 72 T^{2} - 804 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 12 T + 72 T^{2} + 804 T^{3} + 4489 T^{4} )^{2}$$)
$71$ ($$( 1 - 70 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 70 T^{2} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 - 12 T + 72 T^{2} - 876 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 12 T + 72 T^{2} + 876 T^{3} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 - 94 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 6241 T^{4} )^{2}$$)
$83$ ($$1 - 13294 T^{4} + 47458321 T^{8}$$)($$1 - 13294 T^{4} + 47458321 T^{8}$$)($$1 - 13294 T^{4} + 47458321 T^{8}$$)
$89$ ($$( 1 + 160 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 80 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 160 T^{2} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 582 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 24 T + 288 T^{2} + 2328 T^{3} + 9409 T^{4} )^{2}$$)