↑

Properties

Label 40.9
Level 40
Weight 9
Dimension 194
Nonzero newspaces 4
Newform subspaces 7
Sturm bound 864
Trace bound 1

Downloads

Learn more

Defining parameters

Level: \( N \) = \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 7 \)
Sturm bound: \(864\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(40))\).

Total New Old
Modular forms 408 206 202
Cusp forms 360 194 166
Eisenstein series 48 12 36

Trace form

\( 194 q - 8 q^{2} + 660 q^{4} - 336 q^{5} + 1472 q^{6} - 164 q^{7} + 5452 q^{8} - 21874 q^{9} + O(q^{10}) \) \( 194 q - 8 q^{2} + 660 q^{4} - 336 q^{5} + 1472 q^{6} - 164 q^{7} + 5452 q^{8} - 21874 q^{9} + 4076 q^{10} - 27748 q^{11} + 11016 q^{12} - 4760 q^{13} + 34544 q^{14} - 111076 q^{15} - 153304 q^{16} + 57956 q^{17} + 202436 q^{18} + 335100 q^{19} - 345904 q^{20} - 792736 q^{21} + 230800 q^{22} + 687356 q^{23} - 291064 q^{24} - 2309198 q^{25} - 3078968 q^{26} - 2605152 q^{27} + 2917320 q^{28} + 1778160 q^{30} + 1829176 q^{31} - 4733048 q^{32} - 3614112 q^{33} - 6770284 q^{34} - 2741092 q^{35} + 9223712 q^{36} + 4667960 q^{37} + 12740816 q^{38} - 3084524 q^{40} - 1591500 q^{41} - 19281504 q^{42} + 1015488 q^{43} + 33461120 q^{44} - 1242296 q^{45} + 1911312 q^{46} + 28771196 q^{47} - 33903800 q^{48} + 13781354 q^{49} - 28806044 q^{50} - 36874560 q^{51} + 13837716 q^{52} + 13804920 q^{53} + 43843672 q^{54} + 31146848 q^{55} + 53957392 q^{56} - 22315400 q^{57} + 21153836 q^{58} - 2130820 q^{59} - 43678440 q^{60} + 41075424 q^{61} - 104005984 q^{62} - 20422396 q^{63} - 1325280 q^{64} - 30520496 q^{65} + 1212784 q^{66} - 71527680 q^{67} + 137020500 q^{68} + 143121640 q^{70} + 93815224 q^{71} + 86182612 q^{72} + 80854516 q^{73} - 46961476 q^{74} - 139476256 q^{75} - 255130576 q^{76} + 28417440 q^{77} - 483750536 q^{78} + 215900376 q^{80} - 153453198 q^{81} + 350277196 q^{82} - 47701920 q^{83} + 208762720 q^{84} + 21997032 q^{85} - 275686480 q^{86} - 445846408 q^{87} - 287764160 q^{88} + 211142300 q^{89} - 631424644 q^{90} + 352303992 q^{91} - 263239920 q^{92} + 106186080 q^{93} + 277397064 q^{94} + 642323608 q^{95} + 1633636240 q^{96} + 17465204 q^{97} + 256655216 q^{98} - 1234678788 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(40))\)

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
40.9.b \(\chi_{40}(31, \cdot)\) None 0 1
40.9.e \(\chi_{40}(19, \cdot)\) 40.9.e.a 1 1
40.9.e.b 1
40.9.e.c 44
40.9.g \(\chi_{40}(11, \cdot)\) 40.9.g.a 32 1
40.9.h \(\chi_{40}(39, \cdot)\) None 0 1
40.9.i \(\chi_{40}(13, \cdot)\) 40.9.i.a 92 2
40.9.l \(\chi_{40}(17, \cdot)\) 40.9.l.a 12 2
40.9.l.b 12

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(40))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(40)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 1}\)