Properties

Label 40.2
Level 40
Weight 2
Dimension 19
Nonzero newspaces 5
Newform subspaces 5
Sturm bound 192
Trace bound 2

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Defining parameters

Level: \( N \) = \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 5 \)
Sturm bound: \(192\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(40))\).

Total New Old
Modular forms 72 31 41
Cusp forms 25 19 6
Eisenstein series 47 12 35

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)

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
40.2.a \(\chi_{40}(1, \cdot)\) 40.2.a.a 1 1
40.2.c \(\chi_{40}(9, \cdot)\) 40.2.c.a 2 1
40.2.d \(\chi_{40}(21, \cdot)\) 40.2.d.a 4 1
40.2.f \(\chi_{40}(29, \cdot)\) 40.2.f.a 4 1
40.2.j \(\chi_{40}(7, \cdot)\) None 0 2
40.2.k \(\chi_{40}(3, \cdot)\) 40.2.k.a 8 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(40))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(40)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 4 T^{2} + 10 T^{4} - 36 T^{6} + 81 T^{8} \))(\( ( 1 + 4 T^{2} + 9 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 2 T^{3} - 2 T^{4} + 6 T^{5} + 18 T^{6} + 54 T^{7} + 81 T^{8} )^{2} \))
$5$ (\( 1 - T \))(\( 1 + 2 T + 5 T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 30 T^{4} + 625 T^{8} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( ( 1 + 2 T + 12 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 49 T^{4} )^{2} \))(\( 1 - 24 T^{4} + 3326 T^{8} - 57624 T^{12} + 5764801 T^{16} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 18 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 2 T + 18 T^{2} + 22 T^{3} + 121 T^{4} )^{4} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( 1 - 4 T^{4} - 24794 T^{8} - 114244 T^{12} + 815730721 T^{16} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4} )^{4} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( 1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 34 T^{2} + 361 T^{4} )^{4} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 - 42 T^{2} + 529 T^{4} \))(\( ( 1 + 2 T + 20 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 40 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 984 T^{4} + 581246 T^{8} - 275363544 T^{12} + 78310985281 T^{16} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 + 76 T^{2} + 2806 T^{4} + 63916 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 54 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{4} \))(\( ( 1 - 24 T^{2} + 1566 T^{4} - 23064 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} + 1369 T^{4} )^{2} \))(\( 1 + 2876 T^{4} + 4505446 T^{8} + 5390087036 T^{12} + 3512479453921 T^{16} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 4 T + 74 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 2 T + 38 T^{2} + 82 T^{3} + 1681 T^{4} )^{4} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( 1 - 68 T^{2} + 4266 T^{4} - 125732 T^{6} + 3418801 T^{8} \))(\( ( 1 + 68 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 14 T + 98 T^{2} - 910 T^{3} + 7966 T^{4} - 39130 T^{5} + 181202 T^{6} - 1113098 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( ( 1 - 10 T + 116 T^{2} - 470 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 40 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 3544 T^{4} + 6827326 T^{8} - 17293589464 T^{12} + 23811286661761 T^{16} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \))(\( 1 - 60 T^{2} + 3446 T^{4} - 168540 T^{6} + 7890481 T^{8} \))(\( ( 1 + 74 T^{2} + 2809 T^{4} )^{2} \))(\( 1 + 7356 T^{4} + 26359526 T^{8} + 58042378236 T^{12} + 62259690411361 T^{16} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( 1 - 180 T^{2} + 14294 T^{4} - 626580 T^{6} + 12117361 T^{8} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 164 T^{2} + 12406 T^{4} - 570884 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( 1 - 140 T^{2} + 11574 T^{4} - 520940 T^{6} + 13845841 T^{8} \))(\( ( 1 - 110 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 144 T^{2} + 10206 T^{4} - 535824 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 - 100 T^{2} + 10506 T^{4} - 448900 T^{6} + 20151121 T^{8} \))(\( ( 1 + 116 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 14 T + 98 T^{2} + 1246 T^{3} + 15358 T^{4} + 83482 T^{5} + 439922 T^{6} + 4210682 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 134 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{4} \))(\( ( 1 - 104 T^{2} + 11166 T^{4} - 524264 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 82 T^{2} + 5329 T^{4} \))(\( ( 1 - 8 T + 150 T^{2} - 584 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 122 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} + 72 T^{3} - 6562 T^{4} + 5256 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 79 T^{2} \))(\( ( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 16 T + 174 T^{2} + 1264 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{4} \))(\( ( 1 + 156 T^{2} + 13446 T^{4} + 973596 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 + 16 T + 83 T^{2} \))(\( 1 - 162 T^{2} + 6889 T^{4} \))(\( 1 - 308 T^{2} + 37386 T^{4} - 2121812 T^{6} + 47458321 T^{8} \))(\( ( 1 + 68 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 22 T + 242 T^{2} + 3102 T^{3} + 36398 T^{4} + 257466 T^{5} + 1667138 T^{6} + 12579314 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 4 T + 134 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{4} \))(\( ( 1 - 308 T^{2} + 39238 T^{4} - 2439668 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 + 62 T^{2} + 9409 T^{4} \))(\( ( 1 + 8 T + 102 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 760 T^{3} + 18046 T^{4} - 73720 T^{5} + 301088 T^{6} - 7301384 T^{7} + 88529281 T^{8} )^{2} \))
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