# Properties

 Label 1050.2.g Level 1050 Weight 2 Character orbit g Rep. character $$\chi_{1050}(799,\cdot)$$ Character field $$\Q$$ Dimension 20 Newform subspaces 10 Sturm bound 480 Trace bound 14

# Related objects

## Defining parameters

 Level: $$N$$ = $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1050.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$5$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$480$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$17$$, $$19$$

## Dimensions

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

Total New Old
Modular forms 264 20 244
Cusp forms 216 20 196
Eisenstein series 48 0 48

## Trace form

 $$20q - 20q^{4} - 20q^{9} + O(q^{10})$$ $$20q - 20q^{4} - 20q^{9} + 20q^{16} - 16q^{19} + 4q^{21} + 24q^{26} - 40q^{29} - 16q^{31} - 8q^{34} + 20q^{36} + 16q^{39} + 24q^{41} + 8q^{46} - 20q^{49} + 8q^{51} - 16q^{59} + 24q^{61} - 20q^{64} - 16q^{66} - 16q^{69} + 16q^{71} - 8q^{74} + 16q^{76} + 16q^{79} + 20q^{81} - 4q^{84} - 48q^{86} - 8q^{89} + 8q^{91} + 48q^{94} + O(q^{100})$$

## 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.g.a $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}-iq^{7}-iq^{8}+\cdots$$
1050.2.g.b $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}-iq^{7}+iq^{8}+\cdots$$
1050.2.g.c $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+iq^{7}+iq^{8}+\cdots$$
1050.2.g.d $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}-iq^{7}+iq^{8}+\cdots$$
1050.2.g.e $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}-iq^{7}-iq^{8}+\cdots$$
1050.2.g.f $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}-iq^{7}+iq^{8}+\cdots$$
1050.2.g.g $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+iq^{7}+iq^{8}+\cdots$$
1050.2.g.h $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}-iq^{7}+iq^{8}+\cdots$$
1050.2.g.i $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}-iq^{7}+iq^{8}+\cdots$$
1050.2.g.j $$2$$ $$8.384$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+iq^{7}+iq^{8}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1050, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$3$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$5$ 1
$7$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$11$ ($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)
$13$ ($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 + 23 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 25 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 25 T^{2} + 169 T^{4}$$)($$1 - 25 T^{2} + 169 T^{4}$$)
$17$ ($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 + 15 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$1 - 33 T^{2} + 289 T^{4}$$)
$19$ ($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)
$23$ ($$1 + 18 T^{2} + 529 T^{4}$$)($$1 - 21 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$1 - 37 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 45 T^{2} + 529 T^{4}$$)($$1 + 3 T^{2} + 529 T^{4}$$)
$29$ ($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 + 9 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + 3 T + 29 T^{2} )^{2}$$)($$( 1 + 10 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 - 5 T + 29 T^{2} )^{2}$$)($$( 1 + T + 29 T^{2} )^{2}$$)
$31$ ($$( 1 + 31 T^{2} )^{2}$$)($$( 1 - T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 - 5 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{2}$$)($$( 1 - 3 T + 31 T^{2} )^{2}$$)
$37$ ($$1 + 26 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 38 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 38 T^{2} + 1369 T^{4}$$)
$41$ ($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 11 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 - 9 T + 41 T^{2} )^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 - 7 T + 41 T^{2} )^{2}$$)($$( 1 + 3 T + 41 T^{2} )^{2}$$)
$43$ ($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 77 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 + 58 T^{2} + 1849 T^{4}$$)($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 + 35 T^{2} + 1849 T^{4}$$)($$1 - 85 T^{2} + 1849 T^{4}$$)
$47$ ($$( 1 - 47 T^{2} )^{2}$$)($$1 - 78 T^{2} + 2209 T^{4}$$)($$( 1 - 47 T^{2} )^{2}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$( 1 - 47 T^{2} )^{2}$$)($$1 - 78 T^{2} + 2209 T^{4}$$)($$( 1 - 47 T^{2} )^{2}$$)($$1 + 50 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 + 50 T^{2} + 2209 T^{4}$$)
$53$ ($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 97 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 25 T^{2} + 2809 T^{4}$$)($$1 - 6 T^{2} + 2809 T^{4}$$)($$1 - 6 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 105 T^{2} + 2809 T^{4}$$)($$1 + 15 T^{2} + 2809 T^{4}$$)
$59$ ($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 + 7 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 + 9 T + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 - 5 T + 59 T^{2} )^{2}$$)($$( 1 - 3 T + 59 T^{2} )^{2}$$)
$61$ ($$( 1 - 6 T + 61 T^{2} )^{2}$$)($$( 1 + 5 T + 61 T^{2} )^{2}$$)($$( 1 + 10 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{2}$$)($$( 1 - 11 T + 61 T^{2} )^{2}$$)($$( 1 + 6 T + 61 T^{2} )^{2}$$)($$( 1 - 14 T + 61 T^{2} )^{2}$$)($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 + 3 T + 61 T^{2} )^{2}$$)($$( 1 - 5 T + 61 T^{2} )^{2}$$)
$67$ ($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$( 1 - 67 T^{2} )^{2}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)
$71$ ($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 4 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 4 T + 71 T^{2} )^{2}$$)
$73$ ($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 + 50 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 + 50 T^{2} + 5329 T^{4}$$)($$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$)($$1 + 50 T^{2} + 5329 T^{4}$$)
$79$ ($$( 1 + 79 T^{2} )^{2}$$)($$( 1 - 6 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 - 10 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 16 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 + 10 T + 79 T^{2} )^{2}$$)($$( 1 - 2 T + 79 T^{2} )^{2}$$)
$83$ ($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 85 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 85 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 45 T^{2} + 6889 T^{4}$$)($$1 - 157 T^{2} + 6889 T^{4}$$)
$89$ ($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 14 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)
$97$ ($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)