Properties

Label 210.2
Level 210
Weight 2
Dimension 249
Nonzero newspaces 12
Newforms 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: \( 12 \)
Newforms: \( 32 \)
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

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

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

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