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Properties

Label 75.13
Level 75
Weight 13
Dimension 1639
Nonzero newspaces 6
Sturm bound 5200
Trace bound 3

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 13 \)
Nonzero newspaces: \( 6 \)
Sturm bound: \(5200\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(\Gamma_1(75))\).

Total New Old
Modular forms 2456 1681 775
Cusp forms 2344 1639 705
Eisenstein series 112 42 70

Trace form

\( 1639 q + 13 q^{3} - 4564 q^{4} - 8496 q^{5} - 299618 q^{6} + 1045386 q^{7} - 1912680 q^{8} - 447183 q^{9} + O(q^{10}) \) \( 1639 q + 13 q^{3} - 4564 q^{4} - 8496 q^{5} - 299618 q^{6} + 1045386 q^{7} - 1912680 q^{8} - 447183 q^{9} + 6157856 q^{10} - 4560192 q^{11} - 6376962 q^{12} + 35856666 q^{13} - 41406744 q^{15} - 41933732 q^{16} - 105001200 q^{17} - 236000570 q^{18} + 398335394 q^{19} - 309737844 q^{20} + 241643724 q^{21} + 140870140 q^{22} - 271059840 q^{23} + 485404828 q^{24} - 1692847816 q^{25} + 2444478624 q^{26} - 909788867 q^{27} + 1864934076 q^{28} + 2347171200 q^{29} - 2798573726 q^{30} - 203401694 q^{31} - 17388299880 q^{32} + 4106673270 q^{33} + 14571559868 q^{34} - 3431845440 q^{35} - 14229663714 q^{36} - 17891653974 q^{37} - 10940513220 q^{38} - 39975980460 q^{39} + 1480926072 q^{40} + 42972898752 q^{41} + 88564316650 q^{42} - 3951573894 q^{43} - 164876096700 q^{44} - 65939734306 q^{45} - 79260004412 q^{46} + 181630958400 q^{47} + 185188385148 q^{48} + 96891366417 q^{49} - 264791221884 q^{50} - 50949912996 q^{51} - 229757376904 q^{52} + 61956478080 q^{53} + 49843754504 q^{54} + 203767701004 q^{55} + 188469428040 q^{56} + 288488572676 q^{57} - 378192304520 q^{58} - 460017835200 q^{59} - 701268570166 q^{60} - 301522569086 q^{61} + 685626733620 q^{62} - 281034903724 q^{63} + 109595265504 q^{64} + 720217472472 q^{65} + 726072373742 q^{66} - 427693481574 q^{67} - 1273217604480 q^{68} - 1139560788386 q^{69} + 320280404500 q^{70} + 966700347264 q^{71} + 250414772190 q^{72} + 1319631827466 q^{73} - 307920366584 q^{75} - 3581009530264 q^{76} - 2160645157440 q^{77} + 217070329280 q^{78} + 2173383834674 q^{79} + 3264958513836 q^{80} + 3524717582889 q^{81} + 425827679020 q^{82} - 770992819200 q^{83} - 5296453965130 q^{84} - 5896090868992 q^{85} + 3143105283528 q^{86} + 6318525218870 q^{87} + 14672556614860 q^{88} + 3799729791000 q^{89} - 3784901332794 q^{90} - 5492469601736 q^{91} - 16524906064200 q^{92} - 6554996941964 q^{93} + 1436331646748 q^{94} + 5946477374016 q^{95} + 2405422966562 q^{96} + 23585023689546 q^{97} + 12403014541560 q^{98} - 1115948248980 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{13}^{\mathrm{new}}(\Gamma_1(75))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
75.13.c \(\chi_{75}(26, \cdot)\) 75.13.c.a 1 1
75.13.c.b 2
75.13.c.c 2
75.13.c.d 16
75.13.c.e 16
75.13.c.f 16
75.13.c.g 20
75.13.d \(\chi_{75}(74, \cdot)\) 75.13.d.a 2 1
75.13.d.b 4
75.13.d.c 32
75.13.d.d 32
75.13.f \(\chi_{75}(7, \cdot)\) 75.13.f.a 16 2
75.13.f.b 16
75.13.f.c 16
75.13.f.d 24
75.13.h \(\chi_{75}(14, \cdot)\) n/a 472 4
75.13.j \(\chi_{75}(11, \cdot)\) n/a 472 4
75.13.k \(\chi_{75}(13, \cdot)\) n/a 480 8

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{13}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces

\( S_{13}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{13}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 1}\)