# Properties

 Label 1050.2.b Level $1050$ Weight $2$ Character orbit 1050.b Rep. character $\chi_{1050}(251,\cdot)$ Character field $\Q$ Dimension $52$ Newform subspaces $6$ Sturm bound $480$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$480$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$, $$17$$, $$37$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1050, [\chi])$$.

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

## Trace form

 $$52q - 52q^{4} - 8q^{7} + O(q^{10})$$ $$52q - 52q^{4} - 8q^{7} + 52q^{16} + 4q^{18} + 8q^{21} + 8q^{28} - 16q^{37} + 36q^{39} + 16q^{42} - 48q^{43} + 16q^{46} + 52q^{49} + 16q^{51} - 36q^{57} - 32q^{58} - 20q^{63} - 52q^{64} + 64q^{67} - 4q^{72} + 4q^{78} + 104q^{79} + 20q^{81} - 8q^{84} - 16q^{91} - 40q^{93} + 32q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1050, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1050.2.b.a $$4$$ $$8.384$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$-2$$ $$0$$ $$-6$$ $$q+\beta _{2}q^{2}+(-1+\beta _{3})q^{3}-q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots$$
1050.2.b.b $$4$$ $$8.384$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{2}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{1}q^{6}+(1+\cdots)q^{7}+\cdots$$
1050.2.b.c $$4$$ $$8.384$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$2$$ $$0$$ $$-6$$ $$q+\beta _{2}q^{2}+(1-\beta _{3})q^{3}-q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{6}+\cdots$$
1050.2.b.d $$12$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{4}q^{2}-\beta _{9}q^{3}-q^{4}+\beta _{8}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots$$
1050.2.b.e $$12$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{4}q^{2}-\beta _{9}q^{3}-q^{4}-\beta _{8}q^{6}+(\beta _{7}+\cdots)q^{7}+\cdots$$
1050.2.b.f $$16$$ $$8.384$$ 16.0.$$\cdots$$.7 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{10}q^{2}-\beta _{7}q^{3}-q^{4}-\beta _{2}q^{6}+\beta _{9}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1050, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1050, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{6}$$)($$( 1 + T^{2} )^{6}$$)($$( 1 + T^{2} )^{8}$$)
$3$ ($$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$)($$1 + 9 T^{4}$$)($$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$)($$1 + 4 T^{4} - 30 T^{6} + 36 T^{8} + 729 T^{12}$$)($$1 + 4 T^{4} - 30 T^{6} + 36 T^{8} + 729 T^{12}$$)($$( 1 - 10 T^{4} + 81 T^{8} )^{2}$$)
$5$ 1
$7$ ($$1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4}$$)($$( 1 + 2 T - 5 T^{2} - 36 T^{3} - 35 T^{4} + 98 T^{5} + 343 T^{6} )^{2}$$)($$( 1 - 2 T - 5 T^{2} + 36 T^{3} - 35 T^{4} - 98 T^{5} + 343 T^{6} )^{2}$$)($$( 1 - 4 T^{2} - 10 T^{4} - 196 T^{6} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 11 T^{2} )^{4}$$)($$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 38720 T^{8} - 292820 T^{10} + 1771561 T^{12} )^{2}$$)($$( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 38720 T^{8} - 292820 T^{10} + 1771561 T^{12} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{8}( 1 + 6 T + 11 T^{2} )^{8}$$)
$13$ ($$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$)($$( 1 - 20 T^{2} + 169 T^{4} )^{2}$$)($$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$)($$( 1 - 15 T^{2} + 559 T^{4} - 5110 T^{6} + 94471 T^{8} - 428415 T^{10} + 4826809 T^{12} )^{2}$$)($$( 1 - 15 T^{2} + 559 T^{4} - 5110 T^{6} + 94471 T^{8} - 428415 T^{10} + 4826809 T^{12} )^{2}$$)($$( 1 - 40 T^{2} + 710 T^{4} - 6760 T^{6} + 28561 T^{8} )^{4}$$)
$17$ ($$( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 29 T^{2} + 494 T^{4} + 5297 T^{6} + 142766 T^{8} + 2422109 T^{10} + 24137569 T^{12} )^{2}$$)($$( 1 + 29 T^{2} + 494 T^{4} + 5297 T^{6} + 142766 T^{8} + 2422109 T^{10} + 24137569 T^{12} )^{2}$$)($$( 1 + 44 T^{2} + 950 T^{4} + 12716 T^{6} + 83521 T^{8} )^{4}$$)
$19$ ($$1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8}$$)($$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$)($$1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8}$$)($$( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 1491652 T^{8} - 12510816 T^{10} + 47045881 T^{12} )^{2}$$)($$( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 1491652 T^{8} - 12510816 T^{10} + 47045881 T^{12} )^{2}$$)($$( 1 - 24 T^{2} + 838 T^{4} - 8664 T^{6} + 130321 T^{8} )^{4}$$)
$23$ ($$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 73 T^{2} + 2415 T^{4} - 58154 T^{6} + 1277535 T^{8} - 20428393 T^{10} + 148035889 T^{12} )^{2}$$)($$( 1 - 73 T^{2} + 2415 T^{4} - 58154 T^{6} + 1277535 T^{8} - 20428393 T^{10} + 148035889 T^{12} )^{2}$$)($$( 1 - 28 T^{2} + 806 T^{4} - 14812 T^{6} + 279841 T^{8} )^{4}$$)
$29$ ($$1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8}$$)($$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$)($$1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8}$$)($$( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 4654935 T^{8} - 77093629 T^{10} + 594823321 T^{12} )^{2}$$)($$( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 4654935 T^{8} - 77093629 T^{10} + 594823321 T^{12} )^{2}$$)($$( 1 - 52 T^{2} + 1350 T^{4} - 43732 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8}$$)($$( 1 - 31 T^{2} )^{4}$$)($$1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8}$$)($$( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2520703 T^{8} - 47099571 T^{10} + 887503681 T^{12} )^{2}$$)($$( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2520703 T^{8} - 47099571 T^{10} + 887503681 T^{12} )^{2}$$)($$( 1 - 84 T^{2} + 3574 T^{4} - 80724 T^{6} + 923521 T^{8} )^{4}$$)
$37$ ($$( 1 + 6 T + 78 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{4}$$)($$( 1 + 6 T + 78 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 2 T + 37 T^{2} + 300 T^{3} + 1369 T^{4} + 2738 T^{5} + 50653 T^{6} )^{4}$$)($$( 1 - 2 T + 37 T^{2} - 300 T^{3} + 1369 T^{4} - 2738 T^{5} + 50653 T^{6} )^{4}$$)($$( 1 + 20 T^{2} + 38 T^{4} + 27380 T^{6} + 1874161 T^{8} )^{4}$$)
$41$ ($$( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 58 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 3 T^{2} + 3166 T^{4} - 943 T^{6} + 5322046 T^{8} - 8477283 T^{10} + 4750104241 T^{12} )^{2}$$)($$( 1 - 3 T^{2} + 3166 T^{4} - 943 T^{6} + 5322046 T^{8} - 8477283 T^{10} + 4750104241 T^{12} )^{2}$$)($$( 1 + 84 T^{2} + 4678 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 + 8 T + 22 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 43 T^{2} )^{4}$$)($$( 1 + 8 T + 22 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 3 T + 109 T^{2} - 266 T^{3} + 4687 T^{4} - 5547 T^{5} + 79507 T^{6} )^{4}$$)($$( 1 + 3 T + 109 T^{2} + 266 T^{3} + 4687 T^{4} + 5547 T^{5} + 79507 T^{6} )^{4}$$)($$( 1 + 44 T^{2} + 2390 T^{4} + 81356 T^{6} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 4 T + 78 T^{2} - 188 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 78 T^{2} + 7315 T^{4} + 322124 T^{6} + 16158835 T^{8} + 380615118 T^{10} + 10779215329 T^{12} )^{2}$$)($$( 1 + 78 T^{2} + 7315 T^{4} + 322124 T^{6} + 16158835 T^{8} + 380615118 T^{10} + 10779215329 T^{12} )^{2}$$)($$( 1 + 108 T^{2} + 6886 T^{4} + 238572 T^{6} + 4879681 T^{8} )^{4}$$)
$53$ ($$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$)($$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$)($$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$)($$( 1 - 165 T^{2} + 13615 T^{4} - 813650 T^{6} + 38244535 T^{8} - 1301929365 T^{10} + 22164361129 T^{12} )^{2}$$)($$( 1 - 165 T^{2} + 13615 T^{4} - 813650 T^{6} + 38244535 T^{8} - 1301929365 T^{10} + 22164361129 T^{12} )^{2}$$)($$( 1 + 20 T^{2} + 3926 T^{4} + 56180 T^{6} + 7890481 T^{8} )^{4}$$)
$59$ ($$( 1 + 98 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 32 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 98 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 64701347 T^{8} + 2169007619 T^{10} + 42180533641 T^{12} )^{2}$$)($$( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 64701347 T^{8} + 2169007619 T^{10} + 42180533641 T^{12} )^{2}$$)($$( 1 + 216 T^{2} + 18598 T^{4} + 751896 T^{6} + 12117361 T^{8} )^{4}$$)
$61$ ($$1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8}$$)($$( 1 + 28 T^{2} + 3721 T^{4} )^{2}$$)($$1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8}$$)($$( 1 - 183 T^{2} + 12451 T^{4} - 609898 T^{6} + 46330171 T^{8} - 2533788903 T^{10} + 51520374361 T^{12} )^{2}$$)($$( 1 - 183 T^{2} + 12451 T^{4} - 609898 T^{6} + 46330171 T^{8} - 2533788903 T^{10} + 51520374361 T^{12} )^{2}$$)($$( 1 - 136 T^{2} + 9798 T^{4} - 506056 T^{6} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 - 12 T + 67 T^{2} )^{4}$$)($$( 1 + 8 T + 67 T^{2} )^{4}$$)($$( 1 - 12 T + 67 T^{2} )^{4}$$)($$( 1 + 16 T + 258 T^{2} + 2108 T^{3} + 17286 T^{4} + 71824 T^{5} + 300763 T^{6} )^{4}$$)($$( 1 - 16 T + 258 T^{2} - 2108 T^{3} + 17286 T^{4} - 71824 T^{5} + 300763 T^{6} )^{4}$$)($$( 1 + 140 T^{2} + 12086 T^{4} + 628460 T^{6} + 20151121 T^{8} )^{4}$$)
$71$ ($$1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8}$$)($$( 1 - 71 T^{2} )^{4}$$)($$1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8}$$)($$( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 109223347 T^{8} - 5336453010 T^{10} + 128100283921 T^{12} )^{2}$$)($$( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 109223347 T^{8} - 5336453010 T^{10} + 128100283921 T^{12} )^{2}$$)($$( 1 - 28 T^{2} + 9270 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8}$$)($$( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2}$$)($$1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8}$$)($$( 1 - 332 T^{2} + 52356 T^{4} - 4849846 T^{6} + 279005124 T^{8} - 9428216012 T^{10} + 151334226289 T^{12} )^{2}$$)($$( 1 - 332 T^{2} + 52356 T^{4} - 4849846 T^{6} + 279005124 T^{8} - 9428216012 T^{10} + 151334226289 T^{12} )^{2}$$)($$( 1 - 252 T^{2} + 26422 T^{4} - 1342908 T^{6} + 28398241 T^{8} )^{4}$$)
$79$ ($$( 1 + 78 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 10 T + 79 T^{2} )^{4}$$)($$( 1 + 78 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 14 T + 193 T^{2} - 2172 T^{3} + 15247 T^{4} - 87374 T^{5} + 493039 T^{6} )^{4}$$)($$( 1 - 14 T + 193 T^{2} - 2172 T^{3} + 15247 T^{4} - 87374 T^{5} + 493039 T^{6} )^{4}$$)($$( 1 - 4 T + 134 T^{2} - 316 T^{3} + 6241 T^{4} )^{8}$$)
$83$ ($$( 1 + 2 T - 78 T^{2} + 166 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 160 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 2 T - 78 T^{2} - 166 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 309 T^{2} + 44338 T^{4} + 4238477 T^{6} + 305444482 T^{8} + 14664621189 T^{10} + 326940373369 T^{12} )^{2}$$)($$( 1 + 309 T^{2} + 44338 T^{4} + 4238477 T^{6} + 305444482 T^{8} + 14664621189 T^{10} + 326940373369 T^{12} )^{2}$$)($$( 1 + 224 T^{2} + 26294 T^{4} + 1543136 T^{6} + 47458321 T^{8} )^{4}$$)
$89$ ($$( 1 - 20 T + 258 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 459703156 T^{8} + 21834299868 T^{10} + 496981290961 T^{12} )^{2}$$)($$( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 459703156 T^{8} + 21834299868 T^{10} + 496981290961 T^{12} )^{2}$$)($$( 1 - 60 T^{2} + 14950 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8}$$)($$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$)($$1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8}$$)($$( 1 - 158 T^{2} + 21375 T^{4} - 2986564 T^{6} + 201117375 T^{8} - 13987626398 T^{10} + 832972004929 T^{12} )^{2}$$)($$( 1 - 158 T^{2} + 21375 T^{4} - 2986564 T^{6} + 201117375 T^{8} - 13987626398 T^{10} + 832972004929 T^{12} )^{2}$$)($$( 1 - 108 T^{2} + 16246 T^{4} - 1016172 T^{6} + 88529281 T^{8} )^{4}$$)