# Properties

 Label 300.3 Level 300 Weight 3 Dimension 1895 Nonzero newspaces 12 Newform subspaces 39 Sturm bound 14400 Trace bound 7

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## Defining parameters

 Level: $$N$$ = $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$39$$ Sturm bound: $$14400$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(300))$$.

Total New Old
Modular forms 5080 1975 3105
Cusp forms 4520 1895 2625
Eisenstein series 560 80 480

## Trace form

 $$1895 q - 2 q^{2} + q^{3} - 24 q^{4} - 12 q^{5} - 28 q^{6} - 54 q^{7} - 8 q^{8} - 57 q^{9} + O(q^{10})$$ $$1895 q - 2 q^{2} + q^{3} - 24 q^{4} - 12 q^{5} - 28 q^{6} - 54 q^{7} - 8 q^{8} - 57 q^{9} + 32 q^{11} + 66 q^{12} - 34 q^{13} + 216 q^{14} + 14 q^{15} + 172 q^{16} - 120 q^{17} + 44 q^{18} - 94 q^{19} - 20 q^{20} + 154 q^{21} - 76 q^{22} + 40 q^{23} + 184 q^{25} - 228 q^{26} + 169 q^{27} - 52 q^{28} + 308 q^{29} - 6 q^{30} + 58 q^{31} + 248 q^{32} + 72 q^{33} + 280 q^{34} + 164 q^{35} - 198 q^{36} - 74 q^{37} + 524 q^{38} - 54 q^{39} + 664 q^{40} - 28 q^{41} + 74 q^{42} + 274 q^{43} + 92 q^{44} + 272 q^{45} - 492 q^{46} + 160 q^{47} - 512 q^{48} + 117 q^{49} - 364 q^{50} + 368 q^{51} - 888 q^{52} - 220 q^{53} - 678 q^{54} - 272 q^{55} - 584 q^{56} - 322 q^{57} - 1388 q^{58} - 800 q^{59} - 1006 q^{60} - 474 q^{61} - 1260 q^{62} - 176 q^{63} - 552 q^{64} - 120 q^{65} + 678 q^{66} - 414 q^{67} + 488 q^{68} - 14 q^{69} + 468 q^{70} + 160 q^{71} + 1214 q^{72} + 1110 q^{73} + 828 q^{74} + 98 q^{75} + 1304 q^{76} + 688 q^{77} + 1160 q^{78} + 1138 q^{79} + 356 q^{80} - 97 q^{81} + 328 q^{82} + 1120 q^{83} - 298 q^{84} + 1904 q^{85} - 384 q^{86} + 70 q^{87} - 1780 q^{88} + 784 q^{89} - 970 q^{90} - 540 q^{91} - 2264 q^{92} - 1208 q^{93} - 2900 q^{94} - 80 q^{95} - 1918 q^{96} - 1274 q^{97} - 1594 q^{98} - 1120 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(300))$$

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

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
300.3.b $$\chi_{300}(149, \cdot)$$ 300.3.b.a 2 1
300.3.b.b 2
300.3.b.c 4
300.3.b.d 4
300.3.c $$\chi_{300}(151, \cdot)$$ 300.3.c.a 2 1
300.3.c.b 2
300.3.c.c 2
300.3.c.d 8
300.3.c.e 8
300.3.c.f 8
300.3.c.g 8
300.3.f $$\chi_{300}(199, \cdot)$$ 300.3.f.a 4 1
300.3.f.b 16
300.3.f.c 16
300.3.g $$\chi_{300}(101, \cdot)$$ 300.3.g.a 1 1
300.3.g.b 1
300.3.g.c 1
300.3.g.d 2
300.3.g.e 2
300.3.g.f 2
300.3.g.g 2
300.3.g.h 2
300.3.k $$\chi_{300}(157, \cdot)$$ 300.3.k.a 4 2
300.3.k.b 4
300.3.k.c 4
300.3.l $$\chi_{300}(107, \cdot)$$ 300.3.l.a 4 2
300.3.l.b 4
300.3.l.c 4
300.3.l.d 4
300.3.l.e 8
300.3.l.f 8
300.3.l.g 40
300.3.l.h 64
300.3.p $$\chi_{300}(31, \cdot)$$ 300.3.p.a 240 4
300.3.q $$\chi_{300}(29, \cdot)$$ 300.3.q.a 80 4
300.3.s $$\chi_{300}(41, \cdot)$$ 300.3.s.a 80 4
300.3.t $$\chi_{300}(19, \cdot)$$ 300.3.t.a 240 4
300.3.u $$\chi_{300}(23, \cdot)$$ 300.3.u.a 928 8
300.3.v $$\chi_{300}(13, \cdot)$$ 300.3.v.a 80 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(300))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(300)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 2}$$