## Defining parameters

 Level: $$N$$ = $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$24$$ Sturm bound: $$470400$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1225))$$.

Total New Old
Modular forms 178080 157010 21070
Cusp forms 174720 155021 19699
Eisenstein series 3360 1989 1371

## Trace form

 $$155021q - 197q^{2} - 197q^{3} - 221q^{4} - 245q^{5} - 383q^{6} - 258q^{7} - 307q^{8} - 75q^{9} + O(q^{10})$$ $$155021q - 197q^{2} - 197q^{3} - 221q^{4} - 245q^{5} - 383q^{6} - 258q^{7} - 307q^{8} - 75q^{9} - 270q^{10} - 397q^{11} - 327q^{12} - 135q^{13} - 258q^{14} - 432q^{15} - 581q^{16} - 733q^{17} - 803q^{18} - 213q^{19} - 70q^{20} - 150q^{21} + 627q^{22} - 29q^{23} - 2129q^{24} - 899q^{25} - 4161q^{26} - 2699q^{27} - 1194q^{28} + 179q^{29} + 1486q^{30} + 1899q^{31} + 6863q^{32} + 6279q^{33} + 5103q^{34} + 936q^{35} + 8769q^{36} + 1332q^{37} - 305q^{38} - 1281q^{39} - 1724q^{40} - 3711q^{41} - 3279q^{42} - 3629q^{43} - 6689q^{44} - 3517q^{45} - 3399q^{46} - 985q^{47} + 3690q^{48} + 3648q^{49} + 3732q^{50} + 5517q^{51} + 7163q^{52} + 3556q^{53} - 5281q^{54} - 6098q^{55} - 11460q^{56} - 20857q^{57} - 29929q^{58} - 16817q^{59} - 16750q^{60} - 9129q^{61} - 12509q^{62} + 96q^{63} + 9339q^{64} + 4861q^{65} + 31289q^{66} + 24791q^{67} + 39129q^{68} + 34273q^{69} + 12132q^{70} + 17187q^{71} + 53649q^{72} + 25491q^{73} + 35563q^{74} + 13108q^{75} + 17795q^{76} + 1404q^{77} + 5q^{78} - 8401q^{79} - 4802q^{80} - 38181q^{81} - 61886q^{82} - 40507q^{83} - 51834q^{84} - 26493q^{85} - 72472q^{86} - 62323q^{87} - 83674q^{88} - 40194q^{89} - 33936q^{90} - 2661q^{91} - 26080q^{92} + 16483q^{93} + 23226q^{94} + 9258q^{95} + 61304q^{96} + 36928q^{97} + 45108q^{98} + 57144q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1225))$$

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
1225.4.a $$\chi_{1225}(1, \cdot)$$ 1225.4.a.a 1 1
1225.4.a.b 1
1225.4.a.c 1
1225.4.a.d 1
1225.4.a.e 1
1225.4.a.f 1
1225.4.a.g 1
1225.4.a.h 1
1225.4.a.i 1
1225.4.a.j 1
1225.4.a.k 1
1225.4.a.l 1
1225.4.a.m 2
1225.4.a.n 2
1225.4.a.o 2
1225.4.a.p 2
1225.4.a.q 2
1225.4.a.r 2
1225.4.a.s 2
1225.4.a.t 2
1225.4.a.u 2
1225.4.a.v 2
1225.4.a.w 2
1225.4.a.x 2
1225.4.a.y 3
1225.4.a.z 4
1225.4.a.ba 4
1225.4.a.bb 4
1225.4.a.bc 4
1225.4.a.bd 4
1225.4.a.be 5
1225.4.a.bf 5
1225.4.a.bg 5
1225.4.a.bh 5
1225.4.a.bi 6
1225.4.a.bj 6
1225.4.a.bk 8
1225.4.a.bl 8
1225.4.a.bm 8
1225.4.a.bn 8
1225.4.a.bo 8
1225.4.a.bp 10
1225.4.a.bq 10
1225.4.a.br 12
1225.4.a.bs 12
1225.4.a.bt 12
1225.4.b $$\chi_{1225}(99, \cdot)$$ n/a 180 1
1225.4.e $$\chi_{1225}(226, \cdot)$$ n/a 368 2
1225.4.f $$\chi_{1225}(293, \cdot)$$ n/a 352 2
1225.4.h $$\chi_{1225}(246, \cdot)$$ n/a 1212 4
1225.4.k $$\chi_{1225}(324, \cdot)$$ n/a 352 2
1225.4.l $$\chi_{1225}(176, \cdot)$$ n/a 1578 6
1225.4.o $$\chi_{1225}(344, \cdot)$$ n/a 1208 4
1225.4.p $$\chi_{1225}(68, \cdot)$$ n/a 704 4
1225.4.t $$\chi_{1225}(274, \cdot)$$ n/a 1500 6
1225.4.u $$\chi_{1225}(116, \cdot)$$ n/a 2368 8
1225.4.w $$\chi_{1225}(48, \cdot)$$ n/a 2368 8
1225.4.x $$\chi_{1225}(51, \cdot)$$ n/a 3156 12
1225.4.z $$\chi_{1225}(118, \cdot)$$ n/a 3000 12
1225.4.ba $$\chi_{1225}(79, \cdot)$$ n/a 2368 8
1225.4.bd $$\chi_{1225}(36, \cdot)$$ n/a 10032 24
1225.4.be $$\chi_{1225}(74, \cdot)$$ n/a 3000 12
1225.4.bi $$\chi_{1225}(117, \cdot)$$ n/a 4736 16
1225.4.bj $$\chi_{1225}(29, \cdot)$$ n/a 10032 24
1225.4.bn $$\chi_{1225}(82, \cdot)$$ n/a 6000 24
1225.4.bo $$\chi_{1225}(11, \cdot)$$ n/a 20064 48
1225.4.bp $$\chi_{1225}(13, \cdot)$$ n/a 20064 48
1225.4.bt $$\chi_{1225}(4, \cdot)$$ n/a 20064 48
1225.4.bu $$\chi_{1225}(3, \cdot)$$ n/a 40128 96

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1225))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1225)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(245))$$$$^{\oplus 2}$$