## Defining parameters

 Level: $$N$$ = $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$1267200$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3450))$$.

Total New Old
Modular forms 321728 72722 249006
Cusp forms 311873 72722 239151
Eisenstein series 9855 0 9855

## Trace form

 $$72722q - 6q^{2} - 14q^{3} - 6q^{4} - 20q^{5} - 14q^{6} - 48q^{7} - 6q^{8} - 6q^{9} + O(q^{10})$$ $$72722q - 6q^{2} - 14q^{3} - 6q^{4} - 20q^{5} - 14q^{6} - 48q^{7} - 6q^{8} - 6q^{9} - 4q^{10} - 8q^{11} + 18q^{12} - 20q^{13} + 16q^{14} + 8q^{15} - 6q^{16} + 24q^{17} + 2q^{18} + 60q^{19} + 16q^{20} - 2q^{21} + 76q^{22} - 32q^{23} + 26q^{24} + 172q^{25} - 48q^{26} - 80q^{27} - 12q^{28} + 32q^{29} + 40q^{30} - 12q^{31} + 14q^{32} + 18q^{33} + 56q^{34} + 16q^{35} - 6q^{36} - 40q^{37} - 24q^{38} + 64q^{39} + 12q^{40} - 56q^{41} + 16q^{42} + 64q^{43} - 40q^{44} + 140q^{45} + 24q^{46} - 24q^{47} + 18q^{48} - 6q^{49} - 36q^{50} + 16q^{51} - 20q^{52} + 4q^{53} + 8q^{54} + 48q^{55} - 16q^{56} + 62q^{57} - 116q^{58} - 84q^{59} - 64q^{60} - 164q^{61} - 96q^{62} - 58q^{63} - 6q^{64} - 148q^{65} - 8q^{66} - 216q^{67} - 12q^{68} - 70q^{69} - 96q^{70} - 48q^{71} + 6q^{72} - 220q^{73} - 68q^{74} - 328q^{75} + 8q^{76} - 192q^{77} - 170q^{78} - 172q^{79} - 20q^{80} + 6q^{81} - 124q^{82} - 84q^{83} - 114q^{84} - 68q^{85} - 40q^{86} - 190q^{87} - 40q^{88} - 52q^{89} - 108q^{90} - 88q^{91} - 24q^{92} + 40q^{93} - 160q^{94} + 144q^{95} + 18q^{96} + 820q^{97} + 466q^{98} + 202q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3450))$$

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
3450.2.a $$\chi_{3450}(1, \cdot)$$ 3450.2.a.a 1 1
3450.2.a.b 1
3450.2.a.c 1
3450.2.a.d 1
3450.2.a.e 1
3450.2.a.f 1
3450.2.a.g 1
3450.2.a.h 1
3450.2.a.i 1
3450.2.a.j 1
3450.2.a.k 1
3450.2.a.l 1
3450.2.a.m 1
3450.2.a.n 1
3450.2.a.o 1
3450.2.a.p 1
3450.2.a.q 1
3450.2.a.r 1
3450.2.a.s 1
3450.2.a.t 1
3450.2.a.u 1
3450.2.a.v 1
3450.2.a.w 1
3450.2.a.x 1
3450.2.a.y 1
3450.2.a.z 1
3450.2.a.ba 1
3450.2.a.bb 1
3450.2.a.bc 2
3450.2.a.bd 2
3450.2.a.be 2
3450.2.a.bf 2
3450.2.a.bg 2
3450.2.a.bh 2
3450.2.a.bi 2
3450.2.a.bj 2
3450.2.a.bk 2
3450.2.a.bl 2
3450.2.a.bm 2
3450.2.a.bn 2
3450.2.a.bo 3
3450.2.a.bp 3
3450.2.a.bq 3
3450.2.a.br 3
3450.2.a.bs 3
3450.2.a.bt 3
3450.2.d $$\chi_{3450}(2899, \cdot)$$ 3450.2.d.a 2 1
3450.2.d.b 2
3450.2.d.c 2
3450.2.d.d 2
3450.2.d.e 2
3450.2.d.f 2
3450.2.d.g 2
3450.2.d.h 2
3450.2.d.i 2
3450.2.d.j 2
3450.2.d.k 2
3450.2.d.l 2
3450.2.d.m 2
3450.2.d.n 2
3450.2.d.o 2
3450.2.d.p 2
3450.2.d.q 2
3450.2.d.r 2
3450.2.d.s 2
3450.2.d.t 2
3450.2.d.u 2
3450.2.d.v 4
3450.2.d.w 4
3450.2.d.x 4
3450.2.d.y 4
3450.2.d.z 4
3450.2.d.ba 6
3450.2.e $$\chi_{3450}(551, \cdot)$$ n/a 152 1
3450.2.h $$\chi_{3450}(3449, \cdot)$$ n/a 144 1
3450.2.i $$\chi_{3450}(2393, \cdot)$$ n/a 264 2
3450.2.j $$\chi_{3450}(643, \cdot)$$ n/a 144 2
3450.2.m $$\chi_{3450}(691, \cdot)$$ n/a 448 4
3450.2.n $$\chi_{3450}(1241, \cdot)$$ n/a 960 4
3450.2.o $$\chi_{3450}(139, \cdot)$$ n/a 432 4
3450.2.r $$\chi_{3450}(689, \cdot)$$ n/a 960 4
3450.2.u $$\chi_{3450}(151, \cdot)$$ n/a 760 10
3450.2.x $$\chi_{3450}(367, \cdot)$$ n/a 960 8
3450.2.y $$\chi_{3450}(47, \cdot)$$ n/a 1760 8
3450.2.z $$\chi_{3450}(149, \cdot)$$ n/a 1440 10
3450.2.bc $$\chi_{3450}(251, \cdot)$$ n/a 1520 10
3450.2.bd $$\chi_{3450}(49, \cdot)$$ n/a 720 10
3450.2.bi $$\chi_{3450}(7, \cdot)$$ n/a 1440 20
3450.2.bj $$\chi_{3450}(257, \cdot)$$ n/a 2880 20
3450.2.bk $$\chi_{3450}(31, \cdot)$$ n/a 4800 40
3450.2.bn $$\chi_{3450}(89, \cdot)$$ n/a 9600 40
3450.2.bq $$\chi_{3450}(169, \cdot)$$ n/a 4800 40
3450.2.br $$\chi_{3450}(11, \cdot)$$ n/a 9600 40
3450.2.bs $$\chi_{3450}(77, \cdot)$$ n/a 19200 80
3450.2.bt $$\chi_{3450}(37, \cdot)$$ n/a 9600 80

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3450))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3450)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(69))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(115))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(138))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(230))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(345))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(575))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(690))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1150))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1725))$$$$^{\oplus 2}$$