Properties

Label 350.2
Level 350
Weight 2
Dimension 1058
Nonzero newspaces 12
Newforms 48
Sturm bound 14400
Trace bound 4

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newforms: \( 48 \)
Sturm bound: \(14400\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(350))\).

Total New Old
Modular forms 3936 1058 2878
Cusp forms 3265 1058 2207
Eisenstein series 671 0 671

Trace form

\(1058q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 14q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(1058q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 14q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut 10q^{10} \) \(\mathstrut +\mathstrut 36q^{11} \) \(\mathstrut +\mathstrut 10q^{12} \) \(\mathstrut +\mathstrut 38q^{13} \) \(\mathstrut +\mathstrut 14q^{14} \) \(\mathstrut +\mathstrut 40q^{15} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut +\mathstrut 20q^{17} \) \(\mathstrut -\mathstrut 12q^{18} \) \(\mathstrut -\mathstrut 18q^{19} \) \(\mathstrut +\mathstrut 18q^{21} \) \(\mathstrut -\mathstrut 44q^{22} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 26q^{24} \) \(\mathstrut -\mathstrut 94q^{25} \) \(\mathstrut -\mathstrut 42q^{26} \) \(\mathstrut -\mathstrut 140q^{27} \) \(\mathstrut -\mathstrut 52q^{28} \) \(\mathstrut -\mathstrut 92q^{29} \) \(\mathstrut -\mathstrut 112q^{30} \) \(\mathstrut -\mathstrut 84q^{31} \) \(\mathstrut -\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 176q^{33} \) \(\mathstrut -\mathstrut 98q^{34} \) \(\mathstrut -\mathstrut 52q^{35} \) \(\mathstrut -\mathstrut 56q^{36} \) \(\mathstrut -\mathstrut 38q^{37} \) \(\mathstrut -\mathstrut 38q^{38} \) \(\mathstrut -\mathstrut 144q^{39} \) \(\mathstrut +\mathstrut 10q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 82q^{42} \) \(\mathstrut -\mathstrut 36q^{43} \) \(\mathstrut -\mathstrut 12q^{44} \) \(\mathstrut -\mathstrut 102q^{45} \) \(\mathstrut +\mathstrut 24q^{46} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 10q^{48} \) \(\mathstrut +\mathstrut 90q^{49} \) \(\mathstrut +\mathstrut 50q^{50} \) \(\mathstrut +\mathstrut 84q^{51} \) \(\mathstrut +\mathstrut 38q^{52} \) \(\mathstrut +\mathstrut 78q^{53} \) \(\mathstrut +\mathstrut 116q^{54} \) \(\mathstrut +\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 14q^{56} \) \(\mathstrut -\mathstrut 20q^{57} \) \(\mathstrut +\mathstrut 96q^{58} \) \(\mathstrut -\mathstrut 42q^{59} \) \(\mathstrut -\mathstrut 70q^{61} \) \(\mathstrut -\mathstrut 20q^{62} \) \(\mathstrut -\mathstrut 136q^{63} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut -\mathstrut 142q^{65} \) \(\mathstrut +\mathstrut 24q^{66} \) \(\mathstrut -\mathstrut 152q^{67} \) \(\mathstrut -\mathstrut 60q^{68} \) \(\mathstrut -\mathstrut 184q^{69} \) \(\mathstrut -\mathstrut 40q^{70} \) \(\mathstrut -\mathstrut 40q^{71} \) \(\mathstrut +\mathstrut 38q^{72} \) \(\mathstrut -\mathstrut 128q^{73} \) \(\mathstrut -\mathstrut 88q^{74} \) \(\mathstrut -\mathstrut 216q^{75} \) \(\mathstrut -\mathstrut 18q^{76} \) \(\mathstrut -\mathstrut 124q^{77} \) \(\mathstrut -\mathstrut 40q^{78} \) \(\mathstrut -\mathstrut 104q^{79} \) \(\mathstrut +\mathstrut 10q^{80} \) \(\mathstrut +\mathstrut 48q^{81} \) \(\mathstrut -\mathstrut 40q^{82} \) \(\mathstrut -\mathstrut 170q^{83} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 78q^{85} \) \(\mathstrut -\mathstrut 44q^{86} \) \(\mathstrut -\mathstrut 100q^{87} \) \(\mathstrut -\mathstrut 12q^{88} \) \(\mathstrut -\mathstrut 186q^{89} \) \(\mathstrut -\mathstrut 86q^{90} \) \(\mathstrut -\mathstrut 66q^{91} \) \(\mathstrut -\mathstrut 64q^{92} \) \(\mathstrut -\mathstrut 64q^{93} \) \(\mathstrut -\mathstrut 36q^{94} \) \(\mathstrut -\mathstrut 72q^{95} \) \(\mathstrut -\mathstrut 34q^{96} \) \(\mathstrut -\mathstrut 44q^{97} \) \(\mathstrut -\mathstrut 190q^{98} \) \(\mathstrut -\mathstrut 36q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(350))\)

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
350.2.a \(\chi_{350}(1, \cdot)\) 350.2.a.a 1 1
350.2.a.b 1
350.2.a.c 1
350.2.a.d 1
350.2.a.e 1
350.2.a.f 1
350.2.a.g 2
350.2.a.h 2
350.2.c \(\chi_{350}(99, \cdot)\) 350.2.c.a 2 1
350.2.c.b 2
350.2.c.c 2
350.2.c.d 2
350.2.e \(\chi_{350}(51, \cdot)\) 350.2.e.a 2 2
350.2.e.b 2
350.2.e.c 2
350.2.e.d 2
350.2.e.e 2
350.2.e.f 2
350.2.e.g 2
350.2.e.h 2
350.2.e.i 2
350.2.e.j 2
350.2.e.k 2
350.2.e.l 2
350.2.g \(\chi_{350}(293, \cdot)\) 350.2.g.a 8 2
350.2.g.b 16
350.2.h \(\chi_{350}(71, \cdot)\) 350.2.h.a 8 4
350.2.h.b 12
350.2.h.c 16
350.2.h.d 20
350.2.j \(\chi_{350}(149, \cdot)\) 350.2.j.a 4 2
350.2.j.b 4
350.2.j.c 4
350.2.j.d 4
350.2.j.e 4
350.2.j.f 4
350.2.m \(\chi_{350}(29, \cdot)\) 350.2.m.a 24 4
350.2.m.b 40
350.2.o \(\chi_{350}(143, \cdot)\) 350.2.o.a 8 4
350.2.o.b 8
350.2.o.c 16
350.2.o.d 16
350.2.q \(\chi_{350}(11, \cdot)\) 350.2.q.a 8 8
350.2.q.b 72
350.2.q.c 80
350.2.r \(\chi_{350}(13, \cdot)\) 350.2.r.a 160 8
350.2.u \(\chi_{350}(9, \cdot)\) 350.2.u.a 160 8
350.2.x \(\chi_{350}(3, \cdot)\) 350.2.x.a 320 16

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(350)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(175))\)\(^{\oplus 2}\)