# Properties

 Label 8400.2 Level 8400 Weight 2 Dimension 684980 Nonzero newspaces 112 Sturm bound 7372800

## Defining parameters

 Level: $$N$$ = $$8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$112$$ Sturm bound: $$7372800$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(8400))$$.

Total New Old
Modular forms 1862016 688672 1173344
Cusp forms 1824385 684980 1139405
Eisenstein series 37631 3692 33939

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(8400))$$

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
8400.2.a $$\chi_{8400}(1, \cdot)$$ 8400.2.a.a 1 1
8400.2.a.b 1
8400.2.a.c 1
8400.2.a.d 1
8400.2.a.e 1
8400.2.a.f 1
8400.2.a.g 1
8400.2.a.h 1
8400.2.a.i 1
8400.2.a.j 1
8400.2.a.k 1
8400.2.a.l 1
8400.2.a.m 1
8400.2.a.n 1
8400.2.a.o 1
8400.2.a.p 1
8400.2.a.q 1
8400.2.a.r 1
8400.2.a.s 1
8400.2.a.t 1
8400.2.a.u 1
8400.2.a.v 1
8400.2.a.w 1
8400.2.a.x 1
8400.2.a.y 1
8400.2.a.z 1
8400.2.a.ba 1
8400.2.a.bb 1
8400.2.a.bc 1
8400.2.a.bd 1
8400.2.a.be 1
8400.2.a.bf 1
8400.2.a.bg 1
8400.2.a.bh 1
8400.2.a.bi 1
8400.2.a.bj 1
8400.2.a.bk 1
8400.2.a.bl 1
8400.2.a.bm 1
8400.2.a.bn 1
8400.2.a.bo 1
8400.2.a.bp 1
8400.2.a.bq 1
8400.2.a.br 1
8400.2.a.bs 1
8400.2.a.bt 1
8400.2.a.bu 1
8400.2.a.bv 1
8400.2.a.bw 1
8400.2.a.bx 1
8400.2.a.by 1
8400.2.a.bz 1
8400.2.a.ca 1
8400.2.a.cb 1
8400.2.a.cc 1
8400.2.a.cd 1
8400.2.a.ce 1
8400.2.a.cf 1
8400.2.a.cg 1
8400.2.a.ch 1
8400.2.a.ci 1
8400.2.a.cj 1
8400.2.a.ck 1
8400.2.a.cl 1
8400.2.a.cm 1
8400.2.a.cn 1
8400.2.a.co 1
8400.2.a.cp 1
8400.2.a.cq 1
8400.2.a.cr 1
8400.2.a.cs 1
8400.2.a.ct 1
8400.2.a.cu 2
8400.2.a.cv 2
8400.2.a.cw 2
8400.2.a.cx 2
8400.2.a.cy 2
8400.2.a.cz 2
8400.2.a.da 2
8400.2.a.db 2
8400.2.a.dc 2
8400.2.a.dd 2
8400.2.a.de 2
8400.2.a.df 2
8400.2.a.dg 3
8400.2.a.dh 3
8400.2.a.di 3
8400.2.a.dj 3
8400.2.a.dk 3
8400.2.a.dl 3
8400.2.d $$\chi_{8400}(7951, \cdot)$$ n/a 152 1
8400.2.e $$\chi_{8400}(1751, \cdot)$$ None 0 1
8400.2.f $$\chi_{8400}(7601, \cdot)$$ n/a 298 1
8400.2.g $$\chi_{8400}(4201, \cdot)$$ None 0 1
8400.2.j $$\chi_{8400}(1849, \cdot)$$ None 0 1
8400.2.k $$\chi_{8400}(5249, \cdot)$$ n/a 284 1
8400.2.p $$\chi_{8400}(7799, \cdot)$$ None 0 1
8400.2.q $$\chi_{8400}(5599, \cdot)$$ n/a 144 1
8400.2.t $$\chi_{8400}(6049, \cdot)$$ n/a 108 1
8400.2.u $$\chi_{8400}(1049, \cdot)$$ None 0 1
8400.2.v $$\chi_{8400}(3599, \cdot)$$ n/a 216 1
8400.2.w $$\chi_{8400}(1399, \cdot)$$ None 0 1
8400.2.z $$\chi_{8400}(3751, \cdot)$$ None 0 1
8400.2.ba $$\chi_{8400}(5951, \cdot)$$ n/a 228 1
8400.2.bf $$\chi_{8400}(3401, \cdot)$$ None 0 1
8400.2.bg $$\chi_{8400}(1201, \cdot)$$ n/a 304 2
8400.2.bj $$\chi_{8400}(7657, \cdot)$$ None 0 2
8400.2.bk $$\chi_{8400}(1457, \cdot)$$ n/a 432 2
8400.2.bl $$\chi_{8400}(1807, \cdot)$$ n/a 216 2
8400.2.bm $$\chi_{8400}(5207, \cdot)$$ None 0 2
8400.2.bp $$\chi_{8400}(4243, \cdot)$$ n/a 864 2
8400.2.bs $$\chi_{8400}(1357, \cdot)$$ n/a 1152 2
8400.2.bu $$\chi_{8400}(3557, \cdot)$$ n/a 1728 2
8400.2.bv $$\chi_{8400}(3107, \cdot)$$ n/a 2288 2
8400.2.bx $$\chi_{8400}(3149, \cdot)$$ n/a 2288 2
8400.2.ca $$\chi_{8400}(3851, \cdot)$$ n/a 1824 2
8400.2.cb $$\chi_{8400}(3949, \cdot)$$ n/a 864 2
8400.2.ce $$\chi_{8400}(1651, \cdot)$$ n/a 1216 2
8400.2.cg $$\chi_{8400}(2101, \cdot)$$ n/a 912 2
8400.2.ch $$\chi_{8400}(3499, \cdot)$$ n/a 1152 2
8400.2.ck $$\chi_{8400}(1301, \cdot)$$ n/a 2408 2
8400.2.cl $$\chi_{8400}(1499, \cdot)$$ n/a 1728 2
8400.2.co $$\chi_{8400}(1693, \cdot)$$ n/a 1152 2
8400.2.cp $$\chi_{8400}(43, \cdot)$$ n/a 864 2
8400.2.cr $$\chi_{8400}(3443, \cdot)$$ n/a 2288 2
8400.2.cu $$\chi_{8400}(3893, \cdot)$$ n/a 1728 2
8400.2.cx $$\chi_{8400}(6007, \cdot)$$ None 0 2
8400.2.cy $$\chi_{8400}(1007, \cdot)$$ n/a 576 2
8400.2.cz $$\chi_{8400}(3457, \cdot)$$ n/a 288 2
8400.2.da $$\chi_{8400}(5657, \cdot)$$ None 0 2
8400.2.dd $$\chi_{8400}(1681, \cdot)$$ n/a 720 4
8400.2.dg $$\chi_{8400}(199, \cdot)$$ None 0 2
8400.2.dh $$\chi_{8400}(1199, \cdot)$$ n/a 576 2
8400.2.di $$\chi_{8400}(3449, \cdot)$$ None 0 2
8400.2.dj $$\chi_{8400}(3649, \cdot)$$ n/a 288 2
8400.2.dm $$\chi_{8400}(2201, \cdot)$$ None 0 2
8400.2.dr $$\chi_{8400}(3551, \cdot)$$ n/a 608 2
8400.2.ds $$\chi_{8400}(2551, \cdot)$$ None 0 2
8400.2.dv $$\chi_{8400}(1801, \cdot)$$ None 0 2
8400.2.dw $$\chi_{8400}(1601, \cdot)$$ n/a 596 2
8400.2.dx $$\chi_{8400}(2951, \cdot)$$ None 0 2
8400.2.dy $$\chi_{8400}(1951, \cdot)$$ n/a 304 2
8400.2.eb $$\chi_{8400}(4399, \cdot)$$ n/a 288 2
8400.2.ec $$\chi_{8400}(599, \cdot)$$ None 0 2
8400.2.eh $$\chi_{8400}(4049, \cdot)$$ n/a 568 2
8400.2.ei $$\chi_{8400}(3049, \cdot)$$ None 0 2
8400.2.ej $$\chi_{8400}(41, \cdot)$$ None 0 4
8400.2.eo $$\chi_{8400}(911, \cdot)$$ n/a 1440 4
8400.2.ep $$\chi_{8400}(391, \cdot)$$ None 0 4
8400.2.es $$\chi_{8400}(3079, \cdot)$$ None 0 4
8400.2.et $$\chi_{8400}(239, \cdot)$$ n/a 1440 4
8400.2.eu $$\chi_{8400}(2729, \cdot)$$ None 0 4
8400.2.ev $$\chi_{8400}(1009, \cdot)$$ n/a 720 4
8400.2.ey $$\chi_{8400}(559, \cdot)$$ n/a 960 4
8400.2.ez $$\chi_{8400}(1079, \cdot)$$ None 0 4
8400.2.fe $$\chi_{8400}(209, \cdot)$$ n/a 1904 4
8400.2.ff $$\chi_{8400}(169, \cdot)$$ None 0 4
8400.2.fi $$\chi_{8400}(841, \cdot)$$ None 0 4
8400.2.fj $$\chi_{8400}(881, \cdot)$$ n/a 1904 4
8400.2.fk $$\chi_{8400}(71, \cdot)$$ None 0 4
8400.2.fl $$\chi_{8400}(1231, \cdot)$$ n/a 960 4
8400.2.fo $$\chi_{8400}(3257, \cdot)$$ None 0 4
8400.2.fp $$\chi_{8400}(2257, \cdot)$$ n/a 576 4
8400.2.fu $$\chi_{8400}(143, \cdot)$$ n/a 1152 4
8400.2.fv $$\chi_{8400}(3607, \cdot)$$ None 0 4
8400.2.fw $$\chi_{8400}(1643, \cdot)$$ n/a 4576 4
8400.2.fz $$\chi_{8400}(893, \cdot)$$ n/a 4576 4
8400.2.gb $$\chi_{8400}(157, \cdot)$$ n/a 2304 4
8400.2.gc $$\chi_{8400}(907, \cdot)$$ n/a 2304 4
8400.2.ge $$\chi_{8400}(101, \cdot)$$ n/a 4816 4
8400.2.gh $$\chi_{8400}(2699, \cdot)$$ n/a 4576 4
8400.2.gi $$\chi_{8400}(3301, \cdot)$$ n/a 2432 4
8400.2.gl $$\chi_{8400}(1699, \cdot)$$ n/a 2304 4
8400.2.gn $$\chi_{8400}(949, \cdot)$$ n/a 2304 4
8400.2.go $$\chi_{8400}(451, \cdot)$$ n/a 2432 4
8400.2.gr $$\chi_{8400}(1349, \cdot)$$ n/a 4576 4
8400.2.gs $$\chi_{8400}(851, \cdot)$$ n/a 4816 4
8400.2.gv $$\chi_{8400}(557, \cdot)$$ n/a 4576 4
8400.2.gw $$\chi_{8400}(1307, \cdot)$$ n/a 4576 4
8400.2.gy $$\chi_{8400}(1243, \cdot)$$ n/a 2304 4
8400.2.hb $$\chi_{8400}(493, \cdot)$$ n/a 2304 4
8400.2.hc $$\chi_{8400}(4007, \cdot)$$ None 0 4
8400.2.hd $$\chi_{8400}(3007, \cdot)$$ n/a 576 4
8400.2.hi $$\chi_{8400}(2657, \cdot)$$ n/a 1136 4
8400.2.hj $$\chi_{8400}(1657, \cdot)$$ None 0 4
8400.2.hk $$\chi_{8400}(961, \cdot)$$ n/a 1920 8
8400.2.hn $$\chi_{8400}(617, \cdot)$$ None 0 8
8400.2.ho $$\chi_{8400}(97, \cdot)$$ n/a 1920 8
8400.2.hp $$\chi_{8400}(2687, \cdot)$$ n/a 3840 8
8400.2.hq $$\chi_{8400}(967, \cdot)$$ None 0 8
8400.2.ht $$\chi_{8400}(533, \cdot)$$ n/a 11520 8
8400.2.hw $$\chi_{8400}(83, \cdot)$$ n/a 15296 8
8400.2.hy $$\chi_{8400}(547, \cdot)$$ n/a 5760 8
8400.2.hz $$\chi_{8400}(13, \cdot)$$ n/a 7680 8
8400.2.ic $$\chi_{8400}(659, \cdot)$$ n/a 11520 8
8400.2.id $$\chi_{8400}(461, \cdot)$$ n/a 15296 8
8400.2.ig $$\chi_{8400}(139, \cdot)$$ n/a 7680 8
8400.2.ih $$\chi_{8400}(421, \cdot)$$ n/a 5760 8
8400.2.ij $$\chi_{8400}(811, \cdot)$$ n/a 7680 8
8400.2.im $$\chi_{8400}(589, \cdot)$$ n/a 5760 8
8400.2.in $$\chi_{8400}(491, \cdot)$$ n/a 11520 8
8400.2.iq $$\chi_{8400}(629, \cdot)$$ n/a 15296 8
8400.2.is $$\chi_{8400}(923, \cdot)$$ n/a 15296 8
8400.2.it $$\chi_{8400}(197, \cdot)$$ n/a 11520 8
8400.2.iv $$\chi_{8400}(853, \cdot)$$ n/a 7680 8
8400.2.iy $$\chi_{8400}(883, \cdot)$$ n/a 5760 8
8400.2.jb $$\chi_{8400}(167, \cdot)$$ None 0 8
8400.2.jc $$\chi_{8400}(127, \cdot)$$ n/a 1440 8
8400.2.jd $$\chi_{8400}(113, \cdot)$$ n/a 2880 8
8400.2.je $$\chi_{8400}(937, \cdot)$$ None 0 8
8400.2.jh $$\chi_{8400}(1129, \cdot)$$ None 0 8
8400.2.ji $$\chi_{8400}(689, \cdot)$$ n/a 3808 8
8400.2.jn $$\chi_{8400}(359, \cdot)$$ None 0 8
8400.2.jo $$\chi_{8400}(1039, \cdot)$$ n/a 1920 8
8400.2.jr $$\chi_{8400}(31, \cdot)$$ n/a 1920 8
8400.2.js $$\chi_{8400}(1031, \cdot)$$ None 0 8
8400.2.jt $$\chi_{8400}(1361, \cdot)$$ n/a 3808 8
8400.2.ju $$\chi_{8400}(121, \cdot)$$ None 0 8
8400.2.jx $$\chi_{8400}(871, \cdot)$$ None 0 8
8400.2.jy $$\chi_{8400}(191, \cdot)$$ n/a 3840 8
8400.2.kd $$\chi_{8400}(521, \cdot)$$ None 0 8
8400.2.kg $$\chi_{8400}(289, \cdot)$$ n/a 1920 8
8400.2.kh $$\chi_{8400}(89, \cdot)$$ None 0 8
8400.2.ki $$\chi_{8400}(1439, \cdot)$$ n/a 3840 8
8400.2.kj $$\chi_{8400}(439, \cdot)$$ None 0 8
8400.2.km $$\chi_{8400}(73, \cdot)$$ None 0 16
8400.2.kn $$\chi_{8400}(737, \cdot)$$ n/a 7616 16
8400.2.ks $$\chi_{8400}(1087, \cdot)$$ n/a 3840 16
8400.2.kt $$\chi_{8400}(647, \cdot)$$ None 0 16
8400.2.ku $$\chi_{8400}(733, \cdot)$$ n/a 15360 16
8400.2.kx $$\chi_{8400}(67, \cdot)$$ n/a 15360 16
8400.2.kz $$\chi_{8400}(563, \cdot)$$ n/a 30592 16
8400.2.la $$\chi_{8400}(53, \cdot)$$ n/a 30592 16
8400.2.ld $$\chi_{8400}(11, \cdot)$$ n/a 30592 16
8400.2.le $$\chi_{8400}(269, \cdot)$$ n/a 30592 16
8400.2.lh $$\chi_{8400}(691, \cdot)$$ n/a 15360 16
8400.2.li $$\chi_{8400}(109, \cdot)$$ n/a 15360 16
8400.2.lk $$\chi_{8400}(19, \cdot)$$ n/a 15360 16
8400.2.ln $$\chi_{8400}(541, \cdot)$$ n/a 15360 16
8400.2.lo $$\chi_{8400}(179, \cdot)$$ n/a 30592 16
8400.2.lr $$\chi_{8400}(341, \cdot)$$ n/a 30592 16
8400.2.lt $$\chi_{8400}(163, \cdot)$$ n/a 15360 16
8400.2.lu $$\chi_{8400}(397, \cdot)$$ n/a 15360 16
8400.2.lw $$\chi_{8400}(653, \cdot)$$ n/a 30592 16
8400.2.lz $$\chi_{8400}(227, \cdot)$$ n/a 30592 16
8400.2.ma $$\chi_{8400}(247, \cdot)$$ None 0 16
8400.2.mb $$\chi_{8400}(47, \cdot)$$ n/a 7680 16
8400.2.mg $$\chi_{8400}(577, \cdot)$$ n/a 3840 16
8400.2.mh $$\chi_{8400}(137, \cdot)$$ None 0 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(8400))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(8400)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 20}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 15}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 20}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 20}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(350))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(420))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(525))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(560))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(700))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(840))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1050))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1400))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1680))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2800))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4200))$$$$^{\oplus 2}$$