# Properties

 Label 525.2.bf Level 525 Weight 2 Character orbit bf Rep. character $$\chi_{525}(32,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 176 Newform subspaces 7 Sturm bound 160 Trace bound 21

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## Defining parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bf (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$105$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$7$$ Sturm bound: $$160$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$2$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(525, [\chi])$$.

Total New Old
Modular forms 368 208 160
Cusp forms 272 176 96
Eisenstein series 96 32 64

## Trace form

 $$176q + 2q^{3} + 12q^{7} + O(q^{10})$$ $$176q + 2q^{3} + 12q^{7} + 10q^{12} + 16q^{13} + 56q^{16} - 14q^{18} + 32q^{21} + 8q^{22} - 40q^{27} + 60q^{28} - 16q^{31} + 4q^{33} - 160q^{36} - 4q^{37} - 14q^{42} - 16q^{43} + 40q^{46} - 44q^{48} - 28q^{51} - 36q^{52} + 88q^{57} - 56q^{58} - 28q^{61} - 44q^{63} - 92q^{66} - 12q^{67} + 34q^{72} - 52q^{73} - 256q^{76} + 120q^{78} - 52q^{81} - 104q^{82} + 46q^{87} - 12q^{91} + 44q^{93} - 204q^{96} + 120q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(525, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
525.2.bf.a $$8$$ $$4.192$$ $$\Q(\zeta_{24})$$ None $$-4$$ $$4$$ $$0$$ $$0$$ $$q+(-1+\zeta_{24}^{2}+\zeta_{24}^{4})q^{2}+(-\zeta_{24}+\cdots)q^{3}+\cdots$$
525.2.bf.b $$8$$ $$4.192$$ $$\Q(\zeta_{24})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\zeta_{24}-\zeta_{24}^{5})q^{3}+2\zeta_{24}^{2}q^{4}+(-2\zeta_{24}^{3}+\cdots)q^{7}+\cdots$$
525.2.bf.c $$8$$ $$4.192$$ $$\Q(\zeta_{24})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\zeta_{24}-\zeta_{24}^{5})q^{3}+2\zeta_{24}^{2}q^{4}+(3\zeta_{24}^{3}+\cdots)q^{7}+\cdots$$
525.2.bf.d $$8$$ $$4.192$$ $$\Q(\zeta_{24})$$ None $$4$$ $$-4$$ $$0$$ $$0$$ $$q+(1-\zeta_{24}^{2}-\zeta_{24}^{4})q^{2}+(\zeta_{24}+\zeta_{24}^{2}+\cdots)q^{3}+\cdots$$
525.2.bf.e $$16$$ $$4.192$$ 16.0.$$\cdots$$.4 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{5}+\beta _{6}+\beta _{9})q^{2}+\beta _{15}q^{3}-3\beta _{7}q^{4}+\cdots$$
525.2.bf.f $$48$$ $$4.192$$ None $$0$$ $$2$$ $$0$$ $$12$$
525.2.bf.g $$80$$ $$4.192$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(525, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(525, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T + 2 T^{2} )^{4}( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$)($$( 1 - 2 T + 2 T^{2} )^{4}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$)($$( 1 + 7 T^{4} + 33 T^{8} + 112 T^{12} + 256 T^{16} )^{2}$$)
$3$ ($$1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 24 T^{5} + 72 T^{6} - 108 T^{7} + 81 T^{8}$$)($$1 - 9 T^{4} + 81 T^{8}$$)($$1 - 9 T^{4} + 81 T^{8}$$)($$1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 24 T^{5} + 72 T^{6} + 108 T^{7} + 81 T^{8}$$)($$1 + 17 T^{4} + 208 T^{8} + 1377 T^{12} + 6561 T^{16}$$)
$5$ 1
$7$ ($$1 + 23 T^{4} + 2401 T^{8}$$)($$1 + 71 T^{4} + 2401 T^{8}$$)($$1 - 94 T^{4} + 2401 T^{8}$$)($$1 + 23 T^{4} + 2401 T^{8}$$)($$( 1 - 49 T^{4} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 + 11 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 11 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 + 11 T^{2} + 121 T^{4} )^{8}$$)
$13$ ($$( 1 - 337 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 + 191 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 + 146 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 - 337 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 - 334 T^{4} + 28561 T^{8} )^{4}$$)
$17$ ($$( 1 + 2 T + 2 T^{2} - 64 T^{3} - 353 T^{4} - 1088 T^{5} + 578 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2}$$)($$( 1 - 289 T^{4} + 83521 T^{8} )^{2}$$)($$( 1 - 289 T^{4} + 83521 T^{8} )^{2}$$)($$( 1 - 2 T + 2 T^{2} + 64 T^{3} - 353 T^{4} + 1088 T^{5} + 578 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2}$$)($$( 1 + 382 T^{4} + 62403 T^{8} + 31905022 T^{12} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 11 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 795 T^{4} + 12274 T^{6} + 130321 T^{8} )^{4}$$)
$23$ ($$( 1 + 6 T + 18 T^{2} - 168 T^{3} - 1033 T^{4} - 3864 T^{5} + 9522 T^{6} + 73002 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 - 529 T^{4} + 279841 T^{8} )^{2}$$)($$( 1 - 529 T^{4} + 279841 T^{8} )^{2}$$)($$( 1 - 6 T + 18 T^{2} + 168 T^{3} - 1033 T^{4} + 3864 T^{5} + 9522 T^{6} - 73002 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 + 1057 T^{4} + 837408 T^{8} + 295791937 T^{12} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 + 8 T^{2} + 841 T^{4} )^{4}$$)($$( 1 + 29 T^{2} )^{8}$$)($$( 1 + 29 T^{2} )^{8}$$)($$( 1 + 8 T^{2} + 841 T^{4} )^{4}$$)($$( 1 + 23 T^{2} + 841 T^{4} )^{8}$$)
$31$ ($$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{4}$$)($$( 1 - 4 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$)($$( 1 - 4 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$)($$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{4}$$)($$( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{8}$$)
$37$ ($$( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 - 529 T^{4} + 1874161 T^{8} )$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 - 529 T^{4} + 1874161 T^{8} )$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 + 2591 T^{4} + 1874161 T^{8} )$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 - 529 T^{4} + 1874161 T^{8} )$$)($$( 1 + 1294 T^{4} - 199725 T^{8} + 2425164334 T^{12} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 32 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 - 41 T^{2} )^{8}$$)($$( 1 - 41 T^{2} )^{8}$$)($$( 1 - 32 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 - 47 T^{2} + 1681 T^{4} )^{8}$$)
$43$ ($$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$)($$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$)($$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$)($$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$)($$( 1 + 2543 T^{4} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 - 8 T + 32 T^{2} + 496 T^{3} - 4193 T^{4} + 23312 T^{5} + 70688 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2}$$)($$( 1 - 2209 T^{4} + 4879681 T^{8} )^{2}$$)($$( 1 - 2209 T^{4} + 4879681 T^{8} )^{2}$$)($$( 1 + 8 T + 32 T^{2} - 496 T^{3} - 4193 T^{4} - 23312 T^{5} + 70688 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2}$$)($$( 1 + 4222 T^{4} + 12945603 T^{8} + 20602013182 T^{12} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 4 T + 8 T^{2} + 392 T^{3} - 3593 T^{4} + 20776 T^{5} + 22472 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{2}$$)($$( 1 - 2809 T^{4} + 7890481 T^{8} )^{2}$$)($$( 1 - 2809 T^{4} + 7890481 T^{8} )^{2}$$)($$( 1 + 4 T + 8 T^{2} - 392 T^{3} - 3593 T^{4} - 20776 T^{5} + 22472 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2}$$)($$( 1 - 1778 T^{4} - 4729197 T^{8} - 14029275218 T^{12} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 59 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 - 59 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 22 T^{2} - 2997 T^{4} + 76582 T^{6} + 12117361 T^{8} )^{4}$$)
$61$ ($$( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 + T + 61 T^{2} )^{4}( 1 + 13 T + 61 T^{2} )^{4}$$)($$( 1 - 14 T + 61 T^{2} )^{4}( 1 + 13 T + 61 T^{2} )^{4}$$)($$( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 - 11 T + 60 T^{2} - 671 T^{3} + 3721 T^{4} )^{8}$$)
$67$ ($$1 - 8711 T^{4} + 55730400 T^{8} - 175536415031 T^{12} + 406067677556641 T^{16}$$)($$( 1 + 2903 T^{4} + 20151121 T^{8} )( 1 + 5906 T^{4} + 20151121 T^{8} )$$)($$( 1 - 8809 T^{4} + 20151121 T^{8} )( 1 + 2903 T^{4} + 20151121 T^{8} )$$)($$1 - 8711 T^{4} + 55730400 T^{8} - 175536415031 T^{12} + 406067677556641 T^{16}$$)($$( 1 - 7151 T^{4} + 30985680 T^{8} - 144100666271 T^{12} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 - 92 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 71 T^{2} )^{8}$$)($$( 1 - 71 T^{2} )^{8}$$)($$( 1 - 92 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 12 T + 71 T^{2} )^{8}( 1 + 12 T + 71 T^{2} )^{8}$$)
$73$ ($$( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} )$$)($$( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} )$$)($$( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 - 1249 T^{4} + 28398241 T^{8} )$$)($$( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} )$$)($$( 1 - 3266 T^{4} - 17731485 T^{8} - 92748655106 T^{12} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 + 109 T^{2} + 5640 T^{4} + 680269 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 142 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 6241 T^{4} )^{2}( 1 + 11 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 109 T^{2} + 5640 T^{4} + 680269 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 + 154 T^{2} + 17475 T^{4} + 961114 T^{6} + 38950081 T^{8} )^{4}$$)
$83$ ($$( 1 + 2 T + 2 T^{2} + 166 T^{3} + 6889 T^{4} )^{4}$$)($$( 1 + 6889 T^{4} )^{4}$$)($$( 1 + 6889 T^{4} )^{4}$$)($$( 1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4} )^{4}$$)($$( 1 + 12143 T^{4} + 47458321 T^{8} )^{4}$$)
$89$ ($$( 1 + 64 T^{2} - 3825 T^{4} + 506944 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 89 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 - 89 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 64 T^{2} - 3825 T^{4} + 506944 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 143 T^{2} + 12528 T^{4} - 1132703 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 9409 T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 2328 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 18814 T^{4} + 88529281 T^{8} )^{2}$$)($$( 1 + 9071 T^{4} + 88529281 T^{8} )^{2}$$)($$( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 9409 T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 2328 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 12094 T^{4} + 88529281 T^{8} )^{4}$$)
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