# Properties

 Label 1575.2.b Level 1575 Weight 2 Character orbit b Rep. character $$\chi_{1575}(251,\cdot)$$ Character field $$\Q$$ Dimension 52 Newform subspaces 8 Sturm bound 480 Trace bound 22

# Related objects

## Defining parameters

 Level: $$N$$ = $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1575.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$480$$ Trace bound: $$22$$ Distinguishing $$T_p$$: $$2$$, $$37$$, $$47$$, $$67$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1575, [\chi])$$.

Total New Old
Modular forms 264 52 212
Cusp forms 216 52 164
Eisenstein series 48 0 48

## Trace form

 $$52q - 56q^{4} - 8q^{7} + O(q^{10})$$ $$52q - 56q^{4} - 8q^{7} + 60q^{16} + 20q^{22} + 28q^{28} - 16q^{37} - 16q^{43} - 52q^{46} + 32q^{49} + 44q^{58} - 120q^{64} + 48q^{67} + 112q^{79} - 20q^{88} - 28q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1575, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1575.2.b.a $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}+\beta _{2}q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots$$
1575.2.b.b $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}+\beta _{3})q^{7}+\cdots$$
1575.2.b.c $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{1}-\beta _{3})q^{7}+\cdots$$
1575.2.b.d $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q-\beta _{1}q^{2}-\beta _{2}q^{7}-2\beta _{1}q^{8}-\beta _{1}q^{11}+\cdots$$
1575.2.b.e $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{7}+2\beta _{1}q^{8}-\beta _{1}q^{11}+\cdots$$
1575.2.b.f $$8$$ $$12.576$$ 8.0.$$\cdots$$.3 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}-\beta _{4}q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots$$
1575.2.b.g $$8$$ $$12.576$$ 8.0.$$\cdots$$.3 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}+\beta _{4}q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots$$
1575.2.b.h $$16$$ $$12.576$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}+(-1-\beta _{1})q^{4}-\beta _{8}q^{7}+(-\beta _{2}+\cdots)q^{8}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1575, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1575, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{4} + 16 T^{8}$$)($$1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8}$$)($$1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8}$$)($$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$)($$1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16}$$)($$1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16}$$)($$( 1 - 2 T^{2} + 3 T^{4} - 8 T^{6} + 16 T^{8} )^{4}$$)
$3$ 1
$5$ 1
$7$ ($$( 1 - 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 + T + 7 T^{2} )^{2}$$)($$( 1 - T + 7 T^{2} )^{2}$$)($$( 1 - 7 T^{2} )^{4}$$)($$( 1 - 7 T^{2} )^{4}$$)($$( 1 + 4 T^{2} + 6 T^{4} + 196 T^{6} + 2401 T^{8} )^{2}$$)
$11$ ($$1 - 206 T^{4} + 14641 T^{8}$$)($$1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8}$$)($$1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$)($$1 + 206 T^{4} + 27795 T^{8} + 3016046 T^{12} + 214358881 T^{16}$$)($$1 + 206 T^{4} + 27795 T^{8} + 3016046 T^{12} + 214358881 T^{16}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{8}$$)
$13$ ($$( 1 - 13 T^{2} )^{4}$$)($$1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8}$$)($$1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8}$$)($$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2}$$)($$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2}$$)($$( 1 - 13 T^{2} )^{8}$$)($$( 1 - 13 T^{2} )^{8}$$)($$( 1 - 20 T^{2} + 222 T^{4} - 3380 T^{6} + 28561 T^{8} )^{4}$$)
$17$ ($$( 1 + 17 T^{2} )^{4}$$)($$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 17 T^{2} )^{8}$$)($$( 1 + 17 T^{2} )^{8}$$)($$( 1 + 4 T^{2} + 558 T^{4} + 1156 T^{6} + 83521 T^{8} )^{4}$$)
$19$ ($$( 1 - 19 T^{2} )^{4}$$)($$1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8}$$)($$1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8}$$)($$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$)($$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 4 T^{2} + 510 T^{4} - 1444 T^{6} + 130321 T^{8} )^{4}$$)
$23$ ($$1 - 734 T^{4} + 279841 T^{8}$$)($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 4 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 4 T^{2} + 529 T^{4} )^{2}$$)($$1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16}$$)($$1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16}$$)($$( 1 - 56 T^{2} + 1818 T^{4} - 29624 T^{6} + 279841 T^{8} )^{4}$$)
$29$ ($$1 + 1234 T^{4} + 707281 T^{8}$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$)($$1 - 1234 T^{4} + 815475 T^{8} - 872784754 T^{12} + 500246412961 T^{16}$$)($$1 - 1234 T^{4} + 815475 T^{8} - 872784754 T^{12} + 500246412961 T^{16}$$)($$( 1 - 16 T^{2} + 1362 T^{4} - 13456 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 - 31 T^{2} )^{4}$$)($$1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8}$$)($$1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8}$$)($$( 1 - 35 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 35 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 31 T^{2} )^{8}$$)($$( 1 - 31 T^{2} )^{8}$$)($$( 1 - 52 T^{2} + 1422 T^{4} - 49972 T^{6} + 923521 T^{8} )^{4}$$)
$37$ ($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 + 4 T + 66 T^{2} + 148 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 4 T + 66 T^{2} + 148 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 8 T + 37 T^{2} )^{4}$$)($$( 1 + 8 T + 37 T^{2} )^{4}$$)($$( 1 + 38 T^{2} + 75 T^{4} + 52022 T^{6} + 1874161 T^{8} )^{2}$$)($$( 1 + 38 T^{2} + 75 T^{4} + 52022 T^{6} + 1874161 T^{8} )^{2}$$)($$( 1 + 76 T^{2} + 3318 T^{4} + 104044 T^{6} + 1874161 T^{8} )^{4}$$)
$41$ ($$( 1 + 41 T^{2} )^{4}$$)($$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 41 T^{2} )^{8}$$)($$( 1 + 41 T^{2} )^{8}$$)($$( 1 + 44 T^{2} + 3246 T^{4} + 73964 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 + 58 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 42 T^{2} + 172 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 42 T^{2} + 172 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 5 T + 43 T^{2} )^{4}$$)($$( 1 + 5 T + 43 T^{2} )^{4}$$)($$( 1 - 58 T^{2} + 1515 T^{4} - 107242 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 58 T^{2} + 1515 T^{4} - 107242 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 + 100 T^{2} + 6102 T^{4} + 184900 T^{6} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 12 T + 118 T^{2} + 564 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 - 12 T + 118 T^{2} - 564 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 40 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 40 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{8}$$)($$( 1 + 47 T^{2} )^{8}$$)($$( 1 + 124 T^{2} + 7398 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{4}$$)
$53$ ($$1 - 5582 T^{4} + 7890481 T^{8}$$)($$1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8}$$)($$1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8}$$)($$( 1 - 74 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 74 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$)($$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$)($$( 1 - 104 T^{2} + 6378 T^{4} - 292136 T^{6} + 7890481 T^{8} )^{4}$$)
$59$ ($$( 1 + 59 T^{2} )^{4}$$)($$( 1 - 12 T + 142 T^{2} - 708 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 12 T + 142 T^{2} + 708 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 64 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 64 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 59 T^{2} )^{8}$$)($$( 1 + 59 T^{2} )^{8}$$)($$( 1 + 92 T^{2} + 4374 T^{4} + 320252 T^{6} + 12117361 T^{8} )^{4}$$)
$61$ ($$( 1 - 61 T^{2} )^{4}$$)($$1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8}$$)($$1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8}$$)($$( 1 - 95 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 95 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 61 T^{2} )^{8}$$)($$( 1 - 61 T^{2} )^{8}$$)($$( 1 - 52 T^{2} + 6582 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 - 4 T + 67 T^{2} )^{4}$$)($$( 1 - 8 T + 102 T^{2} - 536 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 - 8 T + 102 T^{2} - 536 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 7 T + 67 T^{2} )^{4}$$)($$( 1 - 7 T + 67 T^{2} )^{4}$$)($$( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 + 124 T^{2} + 12438 T^{4} + 556636 T^{6} + 20151121 T^{8} )^{4}$$)
$71$ ($$1 + 2914 T^{4} + 25411681 T^{8}$$)($$1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8}$$)($$1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8}$$)($$( 1 - 134 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 134 T^{2} + 5041 T^{4} )^{2}$$)($$1 - 2914 T^{4} - 16920285 T^{8} - 74049638434 T^{12} + 645753531245761 T^{16}$$)($$1 - 2914 T^{4} - 16920285 T^{8} - 74049638434 T^{12} + 645753531245761 T^{16}$$)($$( 1 - 184 T^{2} + 18162 T^{4} - 927544 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 - 73 T^{2} )^{4}$$)($$1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8}$$)($$1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8}$$)($$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 73 T^{2} )^{8}$$)($$( 1 - 73 T^{2} )^{8}$$)($$( 1 - 164 T^{2} + 13326 T^{4} - 873956 T^{6} + 28398241 T^{8} )^{4}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )^{4}$$)($$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 14 T + 79 T^{2} )^{4}$$)($$( 1 - 14 T + 79 T^{2} )^{4}$$)($$( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 - 4 T + 79 T^{2} )^{16}$$)
$83$ ($$( 1 + 83 T^{2} )^{4}$$)($$( 1 - 12 T + 190 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 12 T + 190 T^{2} + 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 112 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 112 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 83 T^{2} )^{8}$$)($$( 1 + 83 T^{2} )^{8}$$)($$( 1 + 268 T^{2} + 30870 T^{4} + 1846252 T^{6} + 47458321 T^{8} )^{4}$$)
$89$ ($$( 1 + 89 T^{2} )^{4}$$)($$( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 12 T + 166 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{8}$$)($$( 1 + 89 T^{2} )^{8}$$)($$( 1 + 140 T^{2} + 18798 T^{4} + 1108940 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$( 1 - 97 T^{2} )^{4}$$)($$1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8}$$)($$1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8}$$)($$( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2}$$)($$( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2}$$)($$( 1 - 97 T^{2} )^{8}$$)($$( 1 - 97 T^{2} )^{8}$$)($$( 1 - 356 T^{2} + 50286 T^{4} - 3349604 T^{6} + 88529281 T^{8} )^{4}$$)