# Properties

 Label 525.2.d Level $525$ Weight $2$ Character orbit 525.d Rep. character $\chi_{525}(274,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $5$ Sturm bound $160$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$160$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 92 16 76
Cusp forms 68 16 52
Eisenstein series 24 0 24

## Trace form

 $$16q - 20q^{4} + 8q^{6} - 16q^{9} + O(q^{10})$$ $$16q - 20q^{4} + 8q^{6} - 16q^{9} + 12q^{16} - 8q^{19} - 4q^{21} - 24q^{24} - 64q^{26} + 40q^{29} + 32q^{31} - 8q^{34} + 20q^{36} - 48q^{41} + 68q^{44} - 4q^{46} - 16q^{49} - 8q^{54} - 12q^{56} + 8q^{59} + 24q^{64} - 24q^{69} - 16q^{71} - 44q^{74} + 40q^{76} + 16q^{81} + 12q^{84} + 20q^{86} + 56q^{89} - 16q^{91} - 16q^{94} + 16q^{96} + O(q^{100})$$

## 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.d.a $$2$$ $$4.192$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}+q^{4}-q^{6}+iq^{7}+3iq^{8}+\cdots$$
525.2.d.b $$2$$ $$4.192$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}+q^{4}+q^{6}+iq^{7}+3iq^{8}+\cdots$$
525.2.d.c $$4$$ $$4.192$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}-\beta _{1}q^{3}-3q^{4}+\beta _{3}q^{6}-\beta _{1}q^{7}+\cdots$$
525.2.d.d $$4$$ $$4.192$$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-2+\beta _{3})q^{4}+(1+\cdots)q^{6}+\cdots$$
525.2.d.e $$4$$ $$4.192$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{3})q^{2}+\beta _{3}q^{3}+3\beta _{2}q^{4}+(1+\cdots)q^{6}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(525, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 3 T^{2} + 4 T^{4}$$)($$1 - 3 T^{2} + 4 T^{4}$$)($$( 1 + T^{2} + 4 T^{4} )^{2}$$)($$1 - T^{2} + 5 T^{4} - 4 T^{6} + 16 T^{8}$$)($$1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8}$$)
$3$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)
$5$ 1
$7$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)
$11$ ($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 3 T + 11 T^{2} )^{4}$$)($$( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$( 1 - 6 T^{2} + 169 T^{4} )^{2}$$)($$1 - 24 T^{2} + 430 T^{4} - 4056 T^{6} + 28561 T^{8}$$)($$1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8}$$)
$17$ ($$1 + 2 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)($$1 - 40 T^{2} + 926 T^{4} - 11560 T^{6} + 83521 T^{8}$$)($$1 + 24 T^{2} + 542 T^{4} + 6936 T^{6} + 83521 T^{8}$$)
$19$ ($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 8 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 6 T + 34 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$( 1 - 23 T^{2} )^{2}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$)($$1 + 14 T^{2} + 899 T^{4} + 7406 T^{6} + 279841 T^{8}$$)($$( 1 - 21 T^{2} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{4}$$)($$( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 4 T + 17 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 42 T^{2} + 961 T^{4} )^{2}$$)
$37$ ($$1 - 38 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 20 T^{2} + 1558 T^{4} + 27380 T^{6} + 1874161 T^{8}$$)($$1 - 26 T^{2} + 1035 T^{4} - 35594 T^{6} + 1874161 T^{8}$$)($$1 - 106 T^{2} + 5467 T^{4} - 145114 T^{6} + 1874161 T^{8}$$)
$41$ ($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{4}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 8 T + 41 T^{2} )^{4}$$)
$43$ ($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$( 1 - 6 T^{2} + 1849 T^{4} )^{2}$$)($$1 - 74 T^{2} + 3195 T^{4} - 136826 T^{6} + 3418801 T^{8}$$)($$1 - 90 T^{2} + 5003 T^{4} - 166410 T^{6} + 3418801 T^{8}$$)
$47$ ($$( 1 - 47 T^{2} )^{2}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 + 4 T^{2} - 698 T^{4} + 8836 T^{6} + 4879681 T^{8}$$)($$1 - 64 T^{2} + 2894 T^{4} - 141376 T^{6} + 4879681 T^{8}$$)($$1 - 128 T^{2} + 8014 T^{4} - 282752 T^{6} + 4879681 T^{8}$$)
$53$ ($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 6 T^{2} + 2809 T^{4}$$)($$1 - 44 T^{2} + 982 T^{4} - 123596 T^{6} + 7890481 T^{8}$$)($$1 - 76 T^{2} + 3734 T^{4} - 213484 T^{6} + 7890481 T^{8}$$)($$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$)
$59$ ($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 14 T + 154 T^{2} - 826 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - 6 T + 122 T^{2} - 354 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 + 2 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{4}$$)($$( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$( 1 - 118 T^{2} + 4489 T^{4} )^{2}$$)($$1 + 46 T^{2} + 2019 T^{4} + 206494 T^{6} + 20151121 T^{8}$$)($$1 - 146 T^{2} + 11427 T^{4} - 655394 T^{6} + 20151121 T^{8}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 3 T + 71 T^{2} )^{4}$$)($$( 1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 124 T^{2} + 9382 T^{4} - 660796 T^{6} + 28398241 T^{8}$$)($$1 - 248 T^{2} + 25566 T^{4} - 1321592 T^{6} + 28398241 T^{8}$$)($$1 - 280 T^{2} + 30238 T^{4} - 1492120 T^{6} + 28398241 T^{8}$$)
$79$ ($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 8 T + 57 T^{2} - 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T + 169 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)
$83$ ($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 44 T^{2} - 6218 T^{4} - 303116 T^{6} + 47458321 T^{8}$$)($$1 - 196 T^{2} + 20054 T^{4} - 1350244 T^{6} + 47458321 T^{8}$$)($$1 - 164 T^{2} + 15382 T^{4} - 1129796 T^{6} + 47458321 T^{8}$$)
$89$ ($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 2 T + 89 T^{2} )^{4}$$)($$( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 2 T + 134 T^{2} + 178 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$)($$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$)($$1 - 316 T^{2} + 42502 T^{4} - 2973244 T^{6} + 88529281 T^{8}$$)($$1 - 156 T^{2} + 11590 T^{4} - 1467804 T^{6} + 88529281 T^{8}$$)($$1 - 220 T^{2} + 25798 T^{4} - 2069980 T^{6} + 88529281 T^{8}$$)