## Defining parameters

 Level: $$N$$ = $$10000 = 2^{4} \cdot 5^{4}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$28$$ Sturm bound: $$12000000$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(10000))$$.

Total New Old
Modular forms 3015400 1558656 1456744
Cusp forms 2984601 1551744 1432857
Eisenstein series 30799 6912 23887

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(10000))$$

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
10000.2.a $$\chi_{10000}(1, \cdot)$$ 10000.2.a.a 2 1
10000.2.a.b 2
10000.2.a.c 2
10000.2.a.d 2
10000.2.a.e 2
10000.2.a.f 2
10000.2.a.g 2
10000.2.a.h 2
10000.2.a.i 2
10000.2.a.j 2
10000.2.a.k 2
10000.2.a.l 2
10000.2.a.m 2
10000.2.a.n 2
10000.2.a.o 4
10000.2.a.p 4
10000.2.a.q 4
10000.2.a.r 4
10000.2.a.s 4
10000.2.a.t 4
10000.2.a.u 4
10000.2.a.v 4
10000.2.a.w 4
10000.2.a.x 4
10000.2.a.y 4
10000.2.a.z 4
10000.2.a.ba 4
10000.2.a.bb 4
10000.2.a.bc 6
10000.2.a.bd 6
10000.2.a.be 8
10000.2.a.bf 8
10000.2.a.bg 8
10000.2.a.bh 8
10000.2.a.bi 8
10000.2.a.bj 8
10000.2.a.bk 8
10000.2.a.bl 8
10000.2.a.bm 8
10000.2.a.bn 8
10000.2.a.bo 12
10000.2.a.bp 12
10000.2.a.bq 16
10000.2.a.br 16
10000.2.c $$\chi_{10000}(1249, \cdot)$$ n/a 232 1
10000.2.d $$\chi_{10000}(5001, \cdot)$$ None 0 1
10000.2.f $$\chi_{10000}(6249, \cdot)$$ None 0 1
10000.2.j $$\chi_{10000}(443, \cdot)$$ n/a 1888 2
10000.2.l $$\chi_{10000}(2501, \cdot)$$ n/a 1888 2
10000.2.n $$\chi_{10000}(2943, \cdot)$$ n/a 480 2
10000.2.o $$\chi_{10000}(807, \cdot)$$ None 0 2
10000.2.q $$\chi_{10000}(3749, \cdot)$$ n/a 1888 2
10000.2.s $$\chi_{10000}(3307, \cdot)$$ n/a 1888 2
10000.2.u $$\chi_{10000}(2001, \cdot)$$ n/a 936 4
10000.2.w $$\chi_{10000}(249, \cdot)$$ None 0 4
10000.2.y $$\chi_{10000}(3249, \cdot)$$ n/a 936 4
10000.2.bb $$\chi_{10000}(1001, \cdot)$$ None 0 4
10000.2.bd $$\chi_{10000}(1307, \cdot)$$ n/a 7584 8
10000.2.be $$\chi_{10000}(501, \cdot)$$ n/a 7584 8
10000.2.bh $$\chi_{10000}(1943, \cdot)$$ None 0 8
10000.2.bi $$\chi_{10000}(943, \cdot)$$ n/a 1920 8
10000.2.bl $$\chi_{10000}(749, \cdot)$$ n/a 7584 8
10000.2.bm $$\chi_{10000}(307, \cdot)$$ n/a 7584 8
10000.2.bo $$\chi_{10000}(401, \cdot)$$ n/a 4440 20
10000.2.br $$\chi_{10000}(201, \cdot)$$ None 0 20
10000.2.bt $$\chi_{10000}(649, \cdot)$$ None 0 20
10000.2.bu $$\chi_{10000}(49, \cdot)$$ n/a 4440 20
10000.2.bx $$\chi_{10000}(149, \cdot)$$ n/a 35760 40
10000.2.by $$\chi_{10000}(43, \cdot)$$ n/a 35760 40
10000.2.cb $$\chi_{10000}(7, \cdot)$$ None 0 40
10000.2.cc $$\chi_{10000}(143, \cdot)$$ n/a 9000 40
10000.2.cf $$\chi_{10000}(107, \cdot)$$ n/a 35760 40
10000.2.cg $$\chi_{10000}(101, \cdot)$$ n/a 35760 40
10000.2.ci $$\chi_{10000}(81, \cdot)$$ n/a 37400 100
10000.2.cl $$\chi_{10000}(41, \cdot)$$ None 0 100
10000.2.cm $$\chi_{10000}(129, \cdot)$$ n/a 37400 100
10000.2.cp $$\chi_{10000}(9, \cdot)$$ None 0 100
10000.2.cq $$\chi_{10000}(29, \cdot)$$ n/a 299600 200
10000.2.ct $$\chi_{10000}(67, \cdot)$$ n/a 299600 200
10000.2.cu $$\chi_{10000}(3, \cdot)$$ n/a 299600 200
10000.2.cx $$\chi_{10000}(23, \cdot)$$ None 0 200
10000.2.cy $$\chi_{10000}(47, \cdot)$$ n/a 75000 200
10000.2.da $$\chi_{10000}(21, \cdot)$$ n/a 299600 200

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(10000))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(10000)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 15}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(125))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(250))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(500))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(625))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1000))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1250))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2000))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2500))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5000))$$$$^{\oplus 2}$$