Properties

Label 36.2
Level 36
Weight 2
Dimension 13
Nonzero newspaces 4
Newform subspaces 4
Sturm bound 144
Trace bound 4

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Defining parameters

Level: \( N \) = \( 36\( 36 = 2^{2} \cdot 3^{2} \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(36))\).

Total New Old
Modular forms 56 21 35
Cusp forms 17 13 4
Eisenstein series 39 8 31

Trace form

\( 13q - 3q^{2} - 5q^{4} - 9q^{5} - 3q^{6} - 3q^{7} - 12q^{9} + O(q^{10}) \) \( 13q - 3q^{2} - 5q^{4} - 9q^{5} - 3q^{6} - 3q^{7} - 12q^{9} - 4q^{10} - 3q^{11} + 6q^{12} - 7q^{13} + 12q^{14} + 9q^{15} + 7q^{16} + 12q^{17} + 18q^{18} + 18q^{20} - 3q^{21} + 3q^{22} + 3q^{23} + 3q^{24} - 9q^{25} - 12q^{28} + 3q^{29} - 18q^{30} - 9q^{31} - 33q^{32} + 15q^{33} - 13q^{34} - 6q^{35} - 33q^{36} - 10q^{37} - 27q^{38} - 3q^{39} + 2q^{40} + 21q^{41} - 18q^{42} + 9q^{43} + 15q^{45} + 12q^{46} + 9q^{47} + 21q^{48} + 19q^{49} + 21q^{50} + 32q^{52} - 12q^{53} + 39q^{54} + 18q^{55} + 18q^{56} + 6q^{57} + 32q^{58} + 3q^{59} + 6q^{60} + 5q^{61} - 3q^{63} + 10q^{64} - 27q^{65} - 24q^{66} - 9q^{67} - 15q^{68} - 39q^{69} - 6q^{70} - 24q^{71} - 21q^{72} - 58q^{73} - 30q^{74} - 12q^{75} - 3q^{76} - 27q^{77} - 12q^{78} - 15q^{79} - 12q^{81} + 14q^{82} + 9q^{83} + 30q^{84} + 10q^{85} + 21q^{86} - 9q^{87} - 21q^{88} + 12q^{89} + 6q^{90} - 6q^{91} + 24q^{92} + 45q^{93} - 18q^{94} + 12q^{95} + 12q^{96} + 23q^{97} + 9q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)

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
36.2.a \(\chi_{36}(1, \cdot)\) 36.2.a.a 1 1
36.2.b \(\chi_{36}(35, \cdot)\) 36.2.b.a 2 1
36.2.e \(\chi_{36}(13, \cdot)\) 36.2.e.a 2 2
36.2.h \(\chi_{36}(11, \cdot)\) 36.2.h.a 8 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(36))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(36)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 6 T^{4} + 12 T^{5} + 20 T^{6} + 24 T^{7} + 16 T^{8} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} + 12 T^{4} + 27 T^{6} + 81 T^{8} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( ( 1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 120 T^{5} + 275 T^{6} + 375 T^{7} + 625 T^{8} )^{2} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( 1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 76342 T^{10} + 434581 T^{12} + 2235331 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + 11 T^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 863456 T^{10} + 8184319 T^{12} - 56689952 T^{14} + 214358881 T^{16} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 + T - 17 T^{2} - 8 T^{3} + 142 T^{4} - 104 T^{5} - 2873 T^{6} + 2197 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} + 1500 T^{4} - 17629 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 8 T + 19 T^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 49 T^{2} + 1248 T^{4} - 17689 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 60040442 T^{10} + 950619877 T^{12} - 11398763453 T^{14} + 78310985281 T^{16} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 40 T^{2} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( ( 1 - 3 T + 59 T^{2} - 168 T^{3} + 2382 T^{4} - 4872 T^{5} + 49619 T^{6} - 73167 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 12790910 T^{10} + 1242135745 T^{12} + 48812702455 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 42 T^{2} + 74 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 - 80 T^{2} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( ( 1 - 12 T + 131 T^{2} - 996 T^{3} + 7176 T^{4} - 40836 T^{5} + 220211 T^{6} - 827052 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 117270976 T^{10} - 2027348993 T^{12} + 404567235136 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 428735974 T^{10} + 10017985093 T^{12} - 571298412437 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 - 56 T^{2} + 2809 T^{4} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 625549624 T^{10} - 7476411737 T^{12} - 2362109883896 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 - 14 T + 61 T^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 488 T^{5} - 420473 T^{6} + 226981 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 + 16 T + 67 T^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 4573034080 T^{10} + 206649745855 T^{12} + 14473341147040 T^{14} + 406067677556641 T^{16} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 140 T^{2} + 10230 T^{4} + 705740 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 + 10 T + 73 T^{2} \))(\( ( 1 + 16 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - T + 138 T^{2} - 73 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( ( 1 - 79 T^{2} )^{2} \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 2367024070 T^{10} - 99517456955 T^{12} + 27955057384915 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 18577166738 T^{10} + 1084944676381 T^{12} - 72253822514549 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 184 T^{2} + 21006 T^{4} - 1457464 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 14 T + 97 T^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( 1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4} \))(\( ( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 25414 T^{5} - 555131 T^{6} - 1825346 T^{7} + 88529281 T^{8} )^{2} \))
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