Properties

Label 21.4
Level 21
Weight 4
Dimension 30
Nonzero newspaces 4
Newforms 8
Sturm bound 128
Trace bound 1

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 4 \)
Newforms: \( 8 \)
Sturm bound: \(128\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(21))\).

Total New Old
Modular forms 60 42 18
Cusp forms 36 30 6
Eisenstein series 24 12 12

Trace form

\(30q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 24q^{5} \) \(\mathstrut -\mathstrut 36q^{6} \) \(\mathstrut -\mathstrut 54q^{7} \) \(\mathstrut +\mathstrut 54q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(30q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 24q^{5} \) \(\mathstrut -\mathstrut 36q^{6} \) \(\mathstrut -\mathstrut 54q^{7} \) \(\mathstrut +\mathstrut 54q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut +\mathstrut 48q^{10} \) \(\mathstrut -\mathstrut 96q^{12} \) \(\mathstrut -\mathstrut 84q^{13} \) \(\mathstrut -\mathstrut 312q^{14} \) \(\mathstrut -\mathstrut 198q^{15} \) \(\mathstrut -\mathstrut 126q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 270q^{18} \) \(\mathstrut +\mathstrut 558q^{19} \) \(\mathstrut +\mathstrut 1044q^{20} \) \(\mathstrut +\mathstrut 585q^{21} \) \(\mathstrut +\mathstrut 432q^{22} \) \(\mathstrut -\mathstrut 168q^{23} \) \(\mathstrut +\mathstrut 144q^{24} \) \(\mathstrut -\mathstrut 426q^{25} \) \(\mathstrut -\mathstrut 294q^{26} \) \(\mathstrut -\mathstrut 108q^{27} \) \(\mathstrut -\mathstrut 1110q^{28} \) \(\mathstrut -\mathstrut 648q^{29} \) \(\mathstrut -\mathstrut 1530q^{30} \) \(\mathstrut -\mathstrut 750q^{31} \) \(\mathstrut -\mathstrut 1218q^{32} \) \(\mathstrut -\mathstrut 801q^{33} \) \(\mathstrut -\mathstrut 276q^{34} \) \(\mathstrut +\mathstrut 84q^{35} \) \(\mathstrut +\mathstrut 258q^{36} \) \(\mathstrut +\mathstrut 1422q^{37} \) \(\mathstrut +\mathstrut 1710q^{38} \) \(\mathstrut +\mathstrut 1212q^{39} \) \(\mathstrut +\mathstrut 1416q^{40} \) \(\mathstrut +\mathstrut 192q^{41} \) \(\mathstrut +\mathstrut 2322q^{42} \) \(\mathstrut +\mathstrut 1152q^{43} \) \(\mathstrut +\mathstrut 2184q^{44} \) \(\mathstrut +\mathstrut 2151q^{45} \) \(\mathstrut +\mathstrut 708q^{46} \) \(\mathstrut +\mathstrut 792q^{47} \) \(\mathstrut -\mathstrut 1056q^{48} \) \(\mathstrut -\mathstrut 2346q^{49} \) \(\mathstrut -\mathstrut 3258q^{50} \) \(\mathstrut -\mathstrut 3069q^{51} \) \(\mathstrut -\mathstrut 4680q^{52} \) \(\mathstrut -\mathstrut 2100q^{53} \) \(\mathstrut -\mathstrut 4158q^{54} \) \(\mathstrut -\mathstrut 1020q^{55} \) \(\mathstrut +\mathstrut 690q^{56} \) \(\mathstrut -\mathstrut 150q^{57} \) \(\mathstrut +\mathstrut 3528q^{58} \) \(\mathstrut -\mathstrut 336q^{59} \) \(\mathstrut +\mathstrut 3060q^{60} \) \(\mathstrut +\mathstrut 3138q^{61} \) \(\mathstrut +\mathstrut 1812q^{62} \) \(\mathstrut +\mathstrut 2283q^{63} \) \(\mathstrut +\mathstrut 4278q^{64} \) \(\mathstrut +\mathstrut 504q^{65} \) \(\mathstrut +\mathstrut 882q^{66} \) \(\mathstrut +\mathstrut 918q^{67} \) \(\mathstrut -\mathstrut 1536q^{68} \) \(\mathstrut -\mathstrut 1656q^{69} \) \(\mathstrut -\mathstrut 3504q^{70} \) \(\mathstrut -\mathstrut 492q^{71} \) \(\mathstrut -\mathstrut 4050q^{72} \) \(\mathstrut -\mathstrut 4326q^{73} \) \(\mathstrut -\mathstrut 2730q^{74} \) \(\mathstrut -\mathstrut 2304q^{75} \) \(\mathstrut -\mathstrut 5544q^{76} \) \(\mathstrut -\mathstrut 180q^{77} \) \(\mathstrut +\mathstrut 504q^{78} \) \(\mathstrut -\mathstrut 510q^{79} \) \(\mathstrut +\mathstrut 3096q^{80} \) \(\mathstrut +\mathstrut 3987q^{81} \) \(\mathstrut +\mathstrut 7488q^{82} \) \(\mathstrut +\mathstrut 4200q^{83} \) \(\mathstrut +\mathstrut 7956q^{84} \) \(\mathstrut +\mathstrut 4668q^{85} \) \(\mathstrut +\mathstrut 5166q^{86} \) \(\mathstrut +\mathstrut 1656q^{87} \) \(\mathstrut -\mathstrut 1812q^{88} \) \(\mathstrut -\mathstrut 1752q^{89} \) \(\mathstrut -\mathstrut 1080q^{90} \) \(\mathstrut +\mathstrut 804q^{91} \) \(\mathstrut -\mathstrut 4704q^{92} \) \(\mathstrut -\mathstrut 633q^{93} \) \(\mathstrut +\mathstrut 504q^{94} \) \(\mathstrut +\mathstrut 1344q^{95} \) \(\mathstrut -\mathstrut 2700q^{96} \) \(\mathstrut +\mathstrut 300q^{97} \) \(\mathstrut -\mathstrut 5982q^{98} \) \(\mathstrut -\mathstrut 4446q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(21))\)

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
21.4.a \(\chi_{21}(1, \cdot)\) 21.4.a.a 1 1
21.4.a.b 1
21.4.a.c 2
21.4.c \(\chi_{21}(20, \cdot)\) 21.4.c.a 2 1
21.4.c.b 4
21.4.e \(\chi_{21}(4, \cdot)\) 21.4.e.a 2 2
21.4.e.b 6
21.4.g \(\chi_{21}(5, \cdot)\) 21.4.g.a 12 2

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(21))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(21)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)