Properties

Label 75.6
Level 75
Weight 6
Dimension 671
Nonzero newspaces 6
Newform subspaces 26
Sturm bound 2400
Trace bound 1

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 26 \)
Sturm bound: \(2400\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(75))\).

Total New Old
Modular forms 1056 711 345
Cusp forms 944 671 273
Eisenstein series 112 40 72

Trace form

\( 671q + 10q^{2} - q^{3} - 208q^{4} - 126q^{5} + 456q^{6} + 724q^{7} - 48q^{8} - 253q^{9} + O(q^{10}) \) \( 671q + 10q^{2} - q^{3} - 208q^{4} - 126q^{5} + 456q^{6} + 724q^{7} - 48q^{8} - 253q^{9} - 1784q^{10} - 644q^{11} + 770q^{12} + 3874q^{13} + 11232q^{14} + 3386q^{15} + 2124q^{16} - 4138q^{17} - 14440q^{18} - 22664q^{19} - 23844q^{20} - 3390q^{21} + 15604q^{22} + 20432q^{23} + 22248q^{24} + 32894q^{25} + 23532q^{26} + 26639q^{27} - 12772q^{28} - 52418q^{29} - 21406q^{30} + 20180q^{31} - 21896q^{32} + 7262q^{33} + 47064q^{34} + 41900q^{35} - 144526q^{36} - 73036q^{37} + 15244q^{38} - 50100q^{39} - 155048q^{40} - 33590q^{41} - 55998q^{42} - 5112q^{43} + 57948q^{44} + 40144q^{45} + 283620q^{46} + 191216q^{47} + 260992q^{48} + 235183q^{49} + 329996q^{50} - 191182q^{51} - 289048q^{52} - 93848q^{53} - 93882q^{54} - 216956q^{55} - 166920q^{56} + 16234q^{57} - 86292q^{58} - 70276q^{59} - 292246q^{60} + 283258q^{61} + 215708q^{62} + 187804q^{63} + 254424q^{64} + 56422q^{65} - 253674q^{66} - 147728q^{67} - 492040q^{68} - 315668q^{69} - 204380q^{70} + 235112q^{71} + 46422q^{72} + 480382q^{73} + 401612q^{74} - 164994q^{75} + 445816q^{76} - 127392q^{77} - 815312q^{78} - 501460q^{79} - 164684q^{80} - 658909q^{81} + 620608q^{82} + 405220q^{83} + 652838q^{84} - 569402q^{85} - 626144q^{86} + 307578q^{87} - 725588q^{88} - 447264q^{89} - 267994q^{90} + 359660q^{91} - 182456q^{92} - 187184q^{93} + 1286940q^{94} + 622936q^{95} - 539310q^{96} + 1113582q^{97} + 1861186q^{98} + 56700q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(75))\)

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
75.6.a \(\chi_{75}(1, \cdot)\) 75.6.a.a 1 1
75.6.a.b 1
75.6.a.c 1
75.6.a.d 1
75.6.a.e 1
75.6.a.f 2
75.6.a.g 2
75.6.a.h 2
75.6.a.i 2
75.6.a.j 2
75.6.b \(\chi_{75}(49, \cdot)\) 75.6.b.a 2 1
75.6.b.b 2
75.6.b.c 2
75.6.b.d 2
75.6.b.e 4
75.6.b.f 4
75.6.e \(\chi_{75}(32, \cdot)\) 75.6.e.a 4 2
75.6.e.b 4
75.6.e.c 8
75.6.e.d 8
75.6.e.e 16
75.6.e.f 16
75.6.g \(\chi_{75}(16, \cdot)\) 75.6.g.a 52 4
75.6.g.b 52
75.6.i \(\chi_{75}(4, \cdot)\) 75.6.i.a 96 4
75.6.l \(\chi_{75}(2, \cdot)\) 75.6.l.a 384 8

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)