# Properties

 Label 2100.2.k Level 2100 Weight 2 Character orbit k Rep. character $$\chi_{2100}(1849,\cdot)$$ Character field $$\Q$$ Dimension 20 Newform subspaces 10 Sturm bound 960 Trace bound 19

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.k (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$960$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(2100, [\chi])$$.

Total New Old
Modular forms 516 20 496
Cusp forms 444 20 424
Eisenstein series 72 0 72

## Trace form

 $$20q - 20q^{9} + O(q^{10})$$ $$20q - 20q^{9} + 8q^{19} + 4q^{21} - 16q^{29} - 16q^{31} + 16q^{39} + 24q^{41} - 20q^{49} + 8q^{51} + 8q^{59} + 24q^{61} + 8q^{69} + 16q^{71} + 40q^{79} + 20q^{81} + 16q^{89} + 8q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(2100, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
2100.2.k.a $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}-6q^{11}+2iq^{13}+\cdots$$
2100.2.k.b $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}-3q^{11}-4iq^{13}+\cdots$$
2100.2.k.c $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-2q^{11}-4iq^{13}+\cdots$$
2100.2.k.d $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+iq^{7}-q^{9}-q^{11}-2iq^{13}+\cdots$$
2100.2.k.e $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}-iq^{7}-q^{9}-q^{11}+2iq^{13}+\cdots$$
2100.2.k.f $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}-iq^{7}-q^{9}+q^{11}+4iq^{13}+\cdots$$
2100.2.k.g $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}+2q^{11}-4iq^{13}+\cdots$$
2100.2.k.h $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}+2q^{11}+4iq^{13}+\cdots$$
2100.2.k.i $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}+2q^{11}+6iq^{13}+\cdots$$
2100.2.k.j $$2$$ $$16.769$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}+6q^{11}-4iq^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(2100, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(2100, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(140, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(420, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(700, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1050, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$5$ 1
$7$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$11$ ($$( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 + 3 T + 11 T^{2} )^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$( 1 + T + 11 T^{2} )^{2}$$)($$( 1 + T + 11 T^{2} )^{2}$$)($$( 1 - T + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{2}$$)
$13$ ($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)
$17$ ($$( 1 - 17 T^{2} )^{2}$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 - 18 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)
$19$ ($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 + 6 T + 19 T^{2} )^{2}$$)($$( 1 - 2 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 6 T + 19 T^{2} )^{2}$$)($$( 1 - 2 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)
$23$ ($$1 - 10 T^{2} + 529 T^{4}$$)($$1 - 37 T^{2} + 529 T^{4}$$)($$1 - 30 T^{2} + 529 T^{4}$$)($$1 - 45 T^{2} + 529 T^{4}$$)($$1 - 45 T^{2} + 529 T^{4}$$)($$1 + 3 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$1 - 30 T^{2} + 529 T^{4}$$)($$1 - 42 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)
$29$ ($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + 3 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + T + 29 T^{2} )^{2}$$)($$( 1 + T + 29 T^{2} )^{2}$$)($$( 1 - 9 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)
$31$ ($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 10 T + 31 T^{2} )^{2}$$)($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 - 6 T + 31 T^{2} )^{2}$$)($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 - 10 T + 31 T^{2} )^{2}$$)($$( 1 + 6 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 10 T + 31 T^{2} )^{2}$$)
$37$ ($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 25 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 - 25 T^{2} + 1369 T^{4}$$)($$1 + 7 T^{2} + 1369 T^{4}$$)($$1 - 73 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 38 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)
$41$ ($$( 1 - 12 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 - 8 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)
$43$ ($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 + 58 T^{2} + 1849 T^{4}$$)($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 - 5 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)
$47$ ($$1 + 50 T^{2} + 2209 T^{4}$$)($$1 + 50 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 - 90 T^{2} + 2209 T^{4}$$)($$1 - 58 T^{2} + 2209 T^{4}$$)($$1 - 78 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$( 1 - 47 T^{2} )^{2}$$)($$1 + 50 T^{2} + 2209 T^{4}$$)($$( 1 - 47 T^{2} )^{2}$$)
$53$ ($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} )$$)($$1 - 102 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$)($$1 - 42 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)
$59$ ($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 - 8 T + 59 T^{2} )^{2}$$)($$( 1 + 10 T + 59 T^{2} )^{2}$$)($$( 1 - 6 T + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 - 8 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 8 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)
$61$ ($$( 1 + 10 T + 61 T^{2} )^{2}$$)($$( 1 + 4 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{2}$$)($$( 1 + 61 T^{2} )^{2}$$)($$( 1 - 8 T + 61 T^{2} )^{2}$$)($$( 1 - 4 T + 61 T^{2} )^{2}$$)($$( 1 - 6 T + 61 T^{2} )^{2}$$)($$( 1 + 10 T + 61 T^{2} )^{2}$$)($$( 1 - 6 T + 61 T^{2} )^{2}$$)($$( 1 - 14 T + 61 T^{2} )^{2}$$)
$67$ ($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 85 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 125 T^{2} + 4489 T^{4}$$)($$1 - 125 T^{2} + 4489 T^{4}$$)($$1 - 53 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)
$71$ ($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 - 9 T + 71 T^{2} )^{2}$$)($$( 1 + 10 T + 71 T^{2} )^{2}$$)($$( 1 + 9 T + 71 T^{2} )^{2}$$)($$( 1 - 7 T + 71 T^{2} )^{2}$$)($$( 1 - 5 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 + 14 T + 71 T^{2} )^{2}$$)($$( 1 - 14 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)
$73$ ($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)($$( 1 - 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )( 1 + 6 T + 73 T^{2} )$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 2 T^{2} + 5329 T^{4}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)
$79$ ($$( 1 - 4 T + 79 T^{2} )^{2}$$)($$( 1 + 17 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 + T + 79 T^{2} )^{2}$$)($$( 1 + T + 79 T^{2} )^{2}$$)($$( 1 - 15 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 12 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)
$83$ ($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 162 T^{2} + 6889 T^{4}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)
$89$ ($$( 1 + 12 T + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 + 8 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)
$97$ ($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 + 62 T^{2} + 9409 T^{4}$$)