Properties

Label 210.2
Level 210
Weight 2
Dimension 249
Nonzero newspaces 12
Newform subspaces 32
Sturm bound 4608
Trace bound 4

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Defining parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 32 \)
Sturm bound: \(4608\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(210))\).

Total New Old
Modular forms 1344 249 1095
Cusp forms 961 249 712
Eisenstein series 383 0 383

Trace form

\( 249 q + q^{2} + 9 q^{3} + 9 q^{4} + 21 q^{5} + 17 q^{6} + 33 q^{7} + q^{8} + 17 q^{9} + O(q^{10}) \) \( 249 q + q^{2} + 9 q^{3} + 9 q^{4} + 21 q^{5} + 17 q^{6} + 33 q^{7} + q^{8} + 17 q^{9} + 5 q^{10} + 28 q^{11} - 7 q^{12} + 14 q^{13} - 15 q^{14} - 19 q^{15} + q^{16} - 6 q^{17} - 39 q^{18} + 20 q^{19} - 7 q^{20} - 27 q^{21} - 4 q^{22} - 7 q^{24} - 7 q^{25} - 10 q^{26} - 63 q^{27} - 23 q^{28} - 66 q^{29} - 31 q^{30} - 16 q^{31} + q^{32} - 76 q^{33} - 62 q^{34} - 111 q^{35} - 31 q^{36} - 122 q^{37} - 52 q^{38} - 106 q^{39} - 11 q^{40} - 38 q^{41} - 27 q^{42} - 20 q^{43} - 4 q^{44} - 51 q^{45} - 24 q^{46} - 24 q^{47} - 7 q^{48} + 25 q^{49} + 33 q^{50} - 30 q^{51} - 34 q^{52} + 30 q^{53} - 31 q^{54} - 32 q^{55} + 17 q^{56} + 28 q^{57} - 26 q^{58} - 20 q^{59} + q^{60} - 66 q^{61} + 32 q^{62} - 55 q^{63} + 9 q^{64} - 46 q^{65} - 36 q^{66} - 172 q^{67} - 6 q^{68} - 56 q^{69} + q^{70} - 72 q^{71} + 41 q^{72} - 126 q^{73} + 30 q^{74} - 111 q^{75} - 12 q^{76} - 12 q^{77} + 38 q^{78} - 48 q^{79} + 21 q^{80} + 49 q^{81} + 122 q^{82} + 36 q^{83} + 45 q^{84} + 66 q^{85} + 52 q^{86} + 110 q^{87} + 20 q^{88} + 26 q^{89} + 85 q^{90} + 110 q^{91} + 24 q^{92} + 160 q^{93} + 128 q^{94} + 80 q^{95} + q^{96} + 74 q^{97} + q^{98} + 188 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(210))\)

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
210.2.a \(\chi_{210}(1, \cdot)\) 210.2.a.a 1 1
210.2.a.b 1
210.2.a.c 1
210.2.a.d 1
210.2.a.e 1
210.2.b \(\chi_{210}(41, \cdot)\) 210.2.b.a 4 1
210.2.b.b 4
210.2.d \(\chi_{210}(209, \cdot)\) 210.2.d.a 8 1
210.2.d.b 8
210.2.g \(\chi_{210}(169, \cdot)\) 210.2.g.a 2 1
210.2.g.b 2
210.2.i \(\chi_{210}(121, \cdot)\) 210.2.i.a 2 2
210.2.i.b 2
210.2.i.c 2
210.2.i.d 2
210.2.j \(\chi_{210}(113, \cdot)\) 210.2.j.a 12 2
210.2.j.b 12
210.2.m \(\chi_{210}(13, \cdot)\) 210.2.m.a 8 2
210.2.m.b 8
210.2.n \(\chi_{210}(79, \cdot)\) 210.2.n.a 4 2
210.2.n.b 12
210.2.r \(\chi_{210}(101, \cdot)\) 210.2.r.a 12 2
210.2.r.b 12
210.2.t \(\chi_{210}(59, \cdot)\) 210.2.t.a 4 2
210.2.t.b 4
210.2.t.c 4
210.2.t.d 4
210.2.t.e 8
210.2.t.f 8
210.2.u \(\chi_{210}(73, \cdot)\) 210.2.u.a 16 4
210.2.u.b 16
210.2.x \(\chi_{210}(23, \cdot)\) 210.2.x.a 64 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(210))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(210)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 2}\)