# Properties

 Label 8002.2 Level 8002 Weight 2 Dimension 666999 Nonzero newspaces 20 Sturm bound 8.004e+06

## Defining parameters

 Level: $$N$$ = $$8002 = 2 \cdot 4001$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$20$$ Sturm bound: $$8004000$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(8002))$$.

Total New Old
Modular forms 2005000 666999 1338001
Cusp forms 1997001 666999 1330002
Eisenstein series 7999 0 7999

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(8002))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
8002.2.a $$\chi_{8002}(1, \cdot)$$ 8002.2.a.a 1 1
8002.2.a.b 1
8002.2.a.c 1
8002.2.a.d 69
8002.2.a.e 77
8002.2.a.f 89
8002.2.a.g 95
8002.2.b $$\chi_{8002}(8001, \cdot)$$ n/a 334 1
8002.2.c $$\chi_{8002}(899, \cdot)$$ n/a 668 2
8002.2.d $$\chi_{8002}(1401, \cdot)$$ n/a 1336 4
8002.2.e $$\chi_{8002}(2915, \cdot)$$ n/a 1336 4
8002.2.f $$\chi_{8002}(3099, \cdot)$$ n/a 1336 4
8002.2.g $$\chi_{8002}(1115, \cdot)$$ n/a 2664 8
8002.2.h $$\chi_{8002}(1305, \cdot)$$ n/a 2672 8
8002.2.i $$\chi_{8002}(201, \cdot)$$ n/a 6680 20
8002.2.k $$\chi_{8002}(673, \cdot)$$ n/a 5344 16
8002.2.l $$\chi_{8002}(625, \cdot)$$ n/a 6680 20
8002.2.m $$\chi_{8002}(121, \cdot)$$ n/a 10656 32
8002.2.n $$\chi_{8002}(25, \cdot)$$ n/a 13360 40
8002.2.o $$\chi_{8002}(35, \cdot)$$ n/a 33400 100
8002.2.q $$\chi_{8002}(5, \cdot)$$ n/a 26720 80
8002.2.r $$\chi_{8002}(65, \cdot)$$ n/a 33400 100
8002.2.s $$\chi_{8002}(59, \cdot)$$ n/a 53280 160
8002.2.t $$\chi_{8002}(49, \cdot)$$ n/a 66800 200
8002.2.v $$\chi_{8002}(7, \cdot)$$ n/a 133600 400
8002.2.w $$\chi_{8002}(9, \cdot)$$ n/a 266400 800

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(8002))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(8002)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(4001))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T$$)($$1 - T$$)($$1 - T$$)
$3$ ($$1 + T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 2 T + 3 T^{2}$$)
$5$ ($$1 - T + 5 T^{2}$$)($$1 + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)
$7$ ($$1 + 7 T^{2}$$)($$1 + 7 T^{2}$$)($$1 + 7 T^{2}$$)
$11$ ($$1 + 11 T^{2}$$)($$1 + 2 T + 11 T^{2}$$)($$1 + 6 T + 11 T^{2}$$)
$13$ ($$1 - 2 T + 13 T^{2}$$)($$1 - 4 T + 13 T^{2}$$)($$1 - 2 T + 13 T^{2}$$)
$17$ ($$1 - 4 T + 17 T^{2}$$)($$1 + 17 T^{2}$$)($$1 + 2 T + 17 T^{2}$$)
$19$ ($$1 + 4 T + 19 T^{2}$$)($$1 - T + 19 T^{2}$$)($$1 - 8 T + 19 T^{2}$$)
$23$ ($$1 - T + 23 T^{2}$$)($$1 - 3 T + 23 T^{2}$$)($$1 - 4 T + 23 T^{2}$$)
$29$ ($$1 + 6 T + 29 T^{2}$$)($$1 + 4 T + 29 T^{2}$$)($$1 + 6 T + 29 T^{2}$$)
$31$ ($$1 + 6 T + 31 T^{2}$$)($$1 + 31 T^{2}$$)($$1 + 31 T^{2}$$)
$37$ ($$1 + 6 T + 37 T^{2}$$)($$1 - 6 T + 37 T^{2}$$)($$1 + 37 T^{2}$$)
$41$ ($$1 - 10 T + 41 T^{2}$$)($$1 + 8 T + 41 T^{2}$$)($$1 + 2 T + 41 T^{2}$$)
$43$ ($$1 + 9 T + 43 T^{2}$$)($$1 + 12 T + 43 T^{2}$$)($$1 + 6 T + 43 T^{2}$$)
$47$ ($$1 - 6 T + 47 T^{2}$$)($$1 + 10 T + 47 T^{2}$$)($$1 + 47 T^{2}$$)
$53$ ($$1 + 4 T + 53 T^{2}$$)($$1 + 13 T + 53 T^{2}$$)($$1 + 4 T + 53 T^{2}$$)
$59$ ($$1 + 59 T^{2}$$)($$1 - 7 T + 59 T^{2}$$)($$1 + 59 T^{2}$$)
$61$ ($$1 + 9 T + 61 T^{2}$$)($$1 - 12 T + 61 T^{2}$$)($$1 + 6 T + 61 T^{2}$$)
$67$ ($$1 + T + 67 T^{2}$$)($$1 + 16 T + 67 T^{2}$$)($$1 - 2 T + 67 T^{2}$$)
$71$ ($$1 + 2 T + 71 T^{2}$$)($$1 - 8 T + 71 T^{2}$$)($$1 + 8 T + 71 T^{2}$$)
$73$ ($$1 - 4 T + 73 T^{2}$$)($$1 - 4 T + 73 T^{2}$$)($$1 - 10 T + 73 T^{2}$$)
$79$ ($$1 + 10 T + 79 T^{2}$$)($$1 - 10 T + 79 T^{2}$$)($$1 + 16 T + 79 T^{2}$$)
$83$ ($$1 - 12 T + 83 T^{2}$$)($$1 + 4 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)
$89$ ($$1 - 7 T + 89 T^{2}$$)($$1 - 15 T + 89 T^{2}$$)($$1 + 2 T + 89 T^{2}$$)
$97$ ($$1 + T + 97 T^{2}$$)($$1 - 10 T + 97 T^{2}$$)($$1 - 2 T + 97 T^{2}$$)