# Properties

 Label 3150.2.b Level 3150 Weight 2 Character orbit b Rep. character $$\chi_{3150}(251,\cdot)$$ Character field $$\Q$$ Dimension 48 Newform subspaces 6 Sturm bound 1440 Trace bound 46

# Related objects

## Defining parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$1440$$ Trace bound: $$46$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$17$$, $$43$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3150, [\chi])$$.

Total New Old
Modular forms 768 48 720
Cusp forms 672 48 624
Eisenstein series 96 0 96

## Trace form

 $$48q - 48q^{4} + 8q^{7} + O(q^{10})$$ $$48q - 48q^{4} + 8q^{7} + 48q^{16} - 8q^{28} + 16q^{37} + 32q^{43} + 16q^{46} - 48q^{49} + 16q^{58} - 48q^{64} - 64q^{67} - 96q^{79} + 32q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3150, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3150.2.b.a $$8$$ $$25.153$$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{4}+(-\beta _{3}+\beta _{5})q^{7}-\beta _{1}q^{8}+\cdots$$
3150.2.b.b $$8$$ $$25.153$$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{4}+(-\beta _{3}-\beta _{5})q^{7}-\beta _{1}q^{8}+\cdots$$
3150.2.b.c $$8$$ $$25.153$$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{4}+(-\beta _{2}+\beta _{7})q^{7}-\beta _{1}q^{8}+\cdots$$
3150.2.b.d $$8$$ $$25.153$$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-q^{4}+(\beta _{3}+\beta _{5})q^{7}+\beta _{1}q^{8}+\cdots$$
3150.2.b.e $$8$$ $$25.153$$ 8.0.7442857984.4 None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{3}q^{2}-q^{4}-\beta _{6}q^{7}+\beta _{3}q^{8}+(-\beta _{5}+\cdots)q^{11}+\cdots$$
3150.2.b.f $$8$$ $$25.153$$ 8.0.7442857984.4 None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{3}q^{2}-q^{4}+\beta _{5}q^{7}+\beta _{3}q^{8}+(-\beta _{5}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3150, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3150, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(630, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1050, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1575, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)
$3$ ()()()()()()
$5$ ()()()()()()
$7$ ($$( 1 + 6 T^{2} + 49 T^{4} )^{2}$$)($$( 1 + 6 T^{2} + 49 T^{4} )^{2}$$)($$( 1 - 4 T^{2} + 49 T^{4} )^{2}$$)($$( 1 + 6 T^{2} + 49 T^{4} )^{2}$$)($$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 28 T^{5} + 294 T^{6} - 1372 T^{7} + 2401 T^{8}$$)($$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 28 T^{5} + 294 T^{6} - 1372 T^{7} + 2401 T^{8}$$)
$11$ ($$( 1 - 20 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{4}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{4}$$)($$1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16}$$)($$1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16}$$)
$13$ ($$( 1 - 22 T^{2} + 259 T^{4} - 3718 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 6 T^{2} + 169 T^{4} )^{4}$$)($$( 1 - 16 T^{2} + 169 T^{4} )^{4}$$)($$( 1 - 22 T^{2} + 259 T^{4} - 3718 T^{6} + 28561 T^{8} )^{2}$$)($$1 - 44 T^{2} + 872 T^{4} - 9828 T^{6} + 102190 T^{8} - 1660932 T^{10} + 24905192 T^{12} - 212379596 T^{14} + 815730721 T^{16}$$)($$1 - 44 T^{2} + 872 T^{4} - 9828 T^{6} + 102190 T^{8} - 1660932 T^{10} + 24905192 T^{12} - 212379596 T^{14} + 815730721 T^{16}$$)
$17$ ($$( 1 + 29 T^{2} + 289 T^{4} )^{4}$$)($$( 1 + 24 T^{2} + 289 T^{4} )^{4}$$)($$( 1 + 14 T^{2} + 289 T^{4} )^{4}$$)($$( 1 + 29 T^{2} + 289 T^{4} )^{4}$$)($$( 1 + 18 T^{2} - 120 T^{3} - 38 T^{4} - 2040 T^{5} + 5202 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 18 T^{2} + 120 T^{3} - 38 T^{4} + 2040 T^{5} + 5202 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 16 T^{2} - 14 T^{4} - 5776 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 28 T^{2} + 361 T^{4} )^{4}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 16 T^{2} - 14 T^{4} - 5776 T^{6} + 130321 T^{8} )^{2}$$)($$1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16}$$)($$1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16}$$)
$23$ ($$( 1 - 45 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 30 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 10 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 45 T^{2} + 529 T^{4} )^{4}$$)($$1 - 80 T^{2} + 3740 T^{4} - 123312 T^{6} + 3185414 T^{8} - 65232048 T^{10} + 1046605340 T^{12} - 11842871120 T^{14} + 78310985281 T^{16}$$)($$1 - 80 T^{2} + 3740 T^{4} - 123312 T^{6} + 3185414 T^{8} - 65232048 T^{10} + 1046605340 T^{12} - 11842871120 T^{14} + 78310985281 T^{16}$$)
$29$ ($$( 1 - 30 T^{2} + 107 T^{4} - 25230 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 50 T^{2} + 841 T^{4} )^{4}$$)($$( 1 - 50 T^{2} + 841 T^{4} )^{4}$$)($$( 1 - 30 T^{2} + 107 T^{4} - 25230 T^{6} + 707281 T^{8} )^{2}$$)($$1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16}$$)($$1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16}$$)
$31$ ($$( 1 - 34 T^{2} + 1411 T^{4} - 32674 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 22 T^{2} + 961 T^{4} )^{4}$$)($$( 1 - 31 T^{2} )^{8}$$)($$( 1 - 34 T^{2} + 1411 T^{4} - 32674 T^{6} + 923521 T^{8} )^{2}$$)($$1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16}$$)($$1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16}$$)
$37$ ($$( 1 + 66 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 - 24 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 + 56 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 + 66 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 - 4 T + 84 T^{2} - 532 T^{3} + 3602 T^{4} - 19684 T^{5} + 114996 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 - 4 T + 84 T^{2} - 532 T^{3} + 3602 T^{4} - 19684 T^{5} + 114996 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 + 74 T^{2} + 3931 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 62 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 - 8 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 + 74 T^{2} + 3931 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 4 T + 142 T^{2} + 468 T^{3} + 8354 T^{4} + 19188 T^{5} + 238702 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 - 4 T + 142 T^{2} - 468 T^{3} + 8354 T^{4} - 19188 T^{5} + 238702 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 + 10 T + 109 T^{2} + 430 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 + 84 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 + 14 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 - 8 T + 146 T^{2} - 984 T^{3} + 8842 T^{4} - 42312 T^{5} + 269954 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 8 T + 146 T^{2} - 984 T^{3} + 8842 T^{4} - 42312 T^{5} + 269954 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$( 1 + 128 T^{2} + 7714 T^{4} + 282752 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 + 4 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 74 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 128 T^{2} + 7714 T^{4} + 282752 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 - 20 T + 166 T^{2} - 396 T^{3} - 838 T^{4} - 18612 T^{5} + 366694 T^{6} - 2076460 T^{7} + 4879681 T^{8} )^{2}$$)($$( 1 + 20 T + 166 T^{2} + 396 T^{3} - 838 T^{4} + 18612 T^{5} + 366694 T^{6} + 2076460 T^{7} + 4879681 T^{8} )^{2}$$)
$53$ ($$( 1 - 110 T^{2} + 8443 T^{4} - 308990 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 - 14 T + 53 T^{2} )^{4}( 1 + 14 T + 53 T^{2} )^{4}$$)($$( 1 - 70 T^{2} + 2809 T^{4} )^{4}$$)($$( 1 - 110 T^{2} + 8443 T^{4} - 308990 T^{6} + 7890481 T^{8} )^{2}$$)($$1 - 304 T^{2} + 43772 T^{4} - 3946320 T^{6} + 247218022 T^{8} - 11085212880 T^{10} + 345382134332 T^{12} - 6737965783216 T^{14} + 62259690411361 T^{16}$$)($$1 - 304 T^{2} + 43772 T^{4} - 3946320 T^{6} + 247218022 T^{8} - 11085212880 T^{10} + 345382134332 T^{12} - 6737965783216 T^{14} + 62259690411361 T^{16}$$)
$59$ ($$( 1 + 46 T^{2} + 5691 T^{4} + 160126 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 98 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 + 28 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 + 46 T^{2} + 5691 T^{4} + 160126 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 42 T^{2} + 312 T^{3} + 3106 T^{4} + 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 42 T^{2} - 312 T^{3} + 3106 T^{4} - 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 - 214 T^{2} + 18691 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 32 T^{2} + 3721 T^{4} )^{4}$$)($$( 1 - 8 T + 61 T^{2} )^{4}( 1 + 8 T + 61 T^{2} )^{4}$$)($$( 1 - 214 T^{2} + 18691 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2}$$)($$1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16}$$)($$1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16}$$)
$67$ ($$( 1 + 67 T^{2} )^{8}$$)($$( 1 + 84 T^{2} + 4489 T^{4} )^{4}$$)($$( 1 + 67 T^{2} )^{8}$$)($$( 1 + 67 T^{2} )^{8}$$)($$( 1 + 16 T + 218 T^{2} + 1728 T^{3} + 15338 T^{4} + 115776 T^{5} + 978602 T^{6} + 4812208 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 16 T + 218 T^{2} + 1728 T^{3} + 15338 T^{4} + 115776 T^{5} + 978602 T^{6} + 4812208 T^{7} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 - 20 T^{2} - 2618 T^{4} - 100820 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 140 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 110 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 20 T^{2} - 2618 T^{4} - 100820 T^{6} + 25411681 T^{8} )^{2}$$)($$1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16}$$)($$1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16}$$)
$73$ ($$( 1 - 72 T^{2} + 4754 T^{4} - 383688 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 + 34 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 106 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 72 T^{2} + 4754 T^{4} - 383688 T^{6} + 28398241 T^{8} )^{2}$$)($$1 - 232 T^{2} + 30332 T^{4} - 3074136 T^{6} + 255245510 T^{8} - 16382070744 T^{10} + 861375446012 T^{12} - 35109540499048 T^{14} + 806460091894081 T^{16}$$)($$1 - 232 T^{2} + 30332 T^{4} - 3074136 T^{6} + 255245510 T^{8} - 16382070744 T^{10} + 861375446012 T^{12} - 35109540499048 T^{14} + 806460091894081 T^{16}$$)
$79$ ($$( 1 + 12 T + 144 T^{2} + 948 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 6 T + 79 T^{2} )^{8}$$)($$( 1 - 4 T + 79 T^{2} )^{8}$$)($$( 1 + 12 T + 144 T^{2} + 948 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 - 4 T + 48 T^{2} - 532 T^{3} + 9470 T^{4} - 42028 T^{5} + 299568 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2}$$)($$( 1 - 4 T + 48 T^{2} - 532 T^{3} + 9470 T^{4} - 42028 T^{5} + 299568 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2}$$)
$83$ ($$( 1 + 62 T^{2} + 9739 T^{4} + 427118 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 6 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 + 86 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 + 62 T^{2} + 9739 T^{4} + 427118 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 8 T + 244 T^{2} + 1800 T^{3} + 27878 T^{4} + 149400 T^{5} + 1680916 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2}$$)($$( 1 - 8 T + 244 T^{2} - 1800 T^{3} + 27878 T^{4} - 149400 T^{5} + 1680916 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 + 116 T^{2} + 6406 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 158 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 88 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 116 T^{2} + 6406 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 4 T + 118 T^{2} - 1308 T^{3} - 670 T^{4} - 116412 T^{5} + 934678 T^{6} + 2819876 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 - 4 T + 118 T^{2} + 1308 T^{3} - 670 T^{4} + 116412 T^{5} + 934678 T^{6} - 2819876 T^{7} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 - 48 T^{2} - 9406 T^{4} - 451632 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 97 T^{2} )^{8}$$)($$( 1 - 34 T^{2} + 9409 T^{4} )^{4}$$)($$( 1 - 48 T^{2} - 9406 T^{4} - 451632 T^{6} + 88529281 T^{8} )^{2}$$)($$1 - 208 T^{2} + 38300 T^{4} - 4059696 T^{6} + 468613574 T^{8} - 38197679664 T^{10} + 3390671462300 T^{12} - 173258177025232 T^{14} + 7837433594376961 T^{16}$$)($$1 - 208 T^{2} + 38300 T^{4} - 4059696 T^{6} + 468613574 T^{8} - 38197679664 T^{10} + 3390671462300 T^{12} - 173258177025232 T^{14} + 7837433594376961 T^{16}$$)