Properties

Label 33.2
Level 33
Weight 2
Dimension 19
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 160
Trace bound 2

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Defining parameters

Level: \( N \) = \( 33\( 33 = 3 \cdot 11 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(160\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(33))\).

Total New Old
Modular forms 60 39 21
Cusp forms 21 19 2
Eisenstein series 39 20 19

Trace form

\( 19q - 3q^{2} - 6q^{3} - 17q^{4} - 6q^{5} - 3q^{6} - 8q^{7} + 5q^{8} + 4q^{9} + O(q^{10}) \) \( 19q - 3q^{2} - 6q^{3} - 17q^{4} - 6q^{5} - 3q^{6} - 8q^{7} + 5q^{8} + 4q^{9} + 2q^{10} - q^{11} + 3q^{12} - 14q^{13} + 6q^{14} + 9q^{15} + 19q^{16} + 12q^{17} + 7q^{18} + 8q^{20} + 12q^{21} + 17q^{22} - 4q^{23} + 5q^{24} - q^{25} + 8q^{26} - 21q^{27} - 16q^{28} - 10q^{29} - 18q^{30} - 22q^{31} - 43q^{32} - 26q^{33} - 34q^{34} - 8q^{35} - 7q^{36} - 8q^{37} - 10q^{38} + 6q^{39} + 10q^{40} + 18q^{41} + 6q^{42} + 16q^{43} + 33q^{44} + 29q^{45} + 48q^{46} + 22q^{47} + 39q^{48} + 43q^{49} + 17q^{50} + 32q^{51} + 22q^{52} + 16q^{53} + 7q^{54} + 4q^{55} - 30q^{57} - 10q^{58} - 30q^{59} - 22q^{60} - 22q^{61} - 36q^{62} - 28q^{63} - 37q^{64} - 24q^{65} - 43q^{66} - 38q^{67} - 26q^{68} - 19q^{69} - 4q^{70} - 12q^{71} - 25q^{72} + 6q^{73} - 24q^{74} + 9q^{75} + 40q^{76} + 2q^{77} - 2q^{78} + 40q^{79} + 24q^{80} + 4q^{81} - 26q^{82} + 46q^{83} + 4q^{84} + 12q^{85} + 8q^{86} + 30q^{87} - 15q^{88} - 10q^{89} + 42q^{90} - 12q^{91} + 2q^{92} + 43q^{93} - 4q^{94} + 30q^{95} - 3q^{96} + 12q^{97} + 49q^{98} + 14q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(33))\)

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
33.2.a \(\chi_{33}(1, \cdot)\) 33.2.a.a 1 1
33.2.d \(\chi_{33}(32, \cdot)\) 33.2.d.a 2 1
33.2.e \(\chi_{33}(4, \cdot)\) 33.2.e.a 4 4
33.2.e.b 4
33.2.f \(\chi_{33}(2, \cdot)\) 33.2.f.a 8 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(33))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(33)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( ( 1 + 2 T^{2} )^{2} \))(\( 1 + 3 T + 2 T^{2} + T^{4} + 8 T^{6} + 24 T^{7} + 16 T^{8} \))(\( 1 + T + 4 T^{2} + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 + T^{2} + 2 T^{4} + 8 T^{6} + 25 T^{8} + 32 T^{10} + 32 T^{12} + 64 T^{14} + 256 T^{16} \))
$3$ (\( 1 + T \))(\( 1 - T + 3 T^{2} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + 6 T + 13 T^{2} + 10 T^{3} + T^{4} + 30 T^{5} + 117 T^{6} + 162 T^{7} + 81 T^{8} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( ( 1 - 3 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} ) \))(\( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8} \))(\( 1 - T^{2} + 51 T^{4} + 109 T^{6} + 1136 T^{8} + 2725 T^{10} + 31875 T^{12} - 15625 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 105 T^{5} + 98 T^{6} + 1029 T^{7} + 2401 T^{8} \))(\( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 91 T^{5} - 294 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( ( 1 + 5 T + 12 T^{2} - 5 T^{3} - 51 T^{4} - 35 T^{5} + 588 T^{6} + 1715 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 - T \))(\( 1 + 11 T^{2} \))(\( 1 - 9 T + 41 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4} \))(\( 1 + 19 T^{2} + 301 T^{4} + 2299 T^{6} + 14641 T^{8} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 1495 T^{5} + 3042 T^{6} + 19773 T^{7} + 28561 T^{8} \))(\( 1 - 7 T + 6 T^{2} + 49 T^{3} - 181 T^{4} + 637 T^{5} + 1014 T^{6} - 15379 T^{7} + 28561 T^{8} \))(\( ( 1 + 5 T + 18 T^{2} - 5 T^{3} - 21 T^{4} - 65 T^{5} + 3042 T^{6} + 10985 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 340 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} \))(\( 1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 3060 T^{5} + 10693 T^{6} - 58956 T^{7} + 83521 T^{8} \))(\( 1 - 34 T^{2} + 267 T^{4} + 8548 T^{6} - 277795 T^{8} + 2470372 T^{10} + 22300107 T^{12} - 820677346 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 + 19 T^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( ( 1 - 10 T + 69 T^{2} - 410 T^{3} + 1911 T^{4} - 7790 T^{5} + 24909 T^{6} - 68590 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( ( 1 - 9 T + 23 T^{2} )( 1 + 9 T + 23 T^{2} ) \))(\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1363 T^{4} - 26450 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 5800 T^{5} + 26071 T^{6} + 243890 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 - 58 T^{2} + 1523 T^{4} - 6056 T^{6} - 679595 T^{8} - 5093096 T^{10} + 1077188963 T^{12} - 34499752618 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 1426 T^{5} + 2883 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 63 T^{2} + 434 T^{3} + 3255 T^{4} + 13454 T^{5} + 60543 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( ( 1 + 10 T + 9 T^{2} - 130 T^{3} - 409 T^{4} - 4030 T^{5} + 8649 T^{6} + 297910 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 5735 T^{5} - 24642 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( 1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 11359 T^{5} - 8214 T^{6} - 455877 T^{7} + 1874161 T^{8} \))(\( ( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 7215 T^{5} - 38332 T^{6} + 151959 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 - 23 T + 208 T^{2} - 961 T^{3} + 3975 T^{4} - 39401 T^{5} + 349648 T^{6} - 1585183 T^{7} + 2825761 T^{8} \))(\( 1 + 3 T - 22 T^{2} + 171 T^{3} + 2215 T^{4} + 7011 T^{5} - 36982 T^{6} + 206763 T^{7} + 2825761 T^{8} \))(\( 1 - 87 T^{2} + 4308 T^{4} - 137029 T^{6} + 5286975 T^{8} - 230345749 T^{10} + 12173378388 T^{12} - 413259068967 T^{14} + 7984925229121 T^{16} \))
$43$ (\( 1 + 43 T^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 122 T^{2} + 6919 T^{4} - 225578 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( ( 1 - 12 T + 47 T^{2} )( 1 + 12 T + 47 T^{2} ) \))(\( 1 + 3 T - 43 T^{2} - 45 T^{3} + 2116 T^{4} - 2115 T^{5} - 94987 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( 1 - 17 T + 67 T^{2} + 335 T^{3} - 4344 T^{4} + 15745 T^{5} + 148003 T^{6} - 1764991 T^{7} + 4879681 T^{8} \))(\( 1 + 15 T^{2} - 969 T^{4} - 93835 T^{6} + 1517976 T^{8} - 207281515 T^{10} - 4728410889 T^{12} + 161688229935 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( ( 1 - 6 T + 53 T^{2} )( 1 + 6 T + 53 T^{2} ) \))(\( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 6360 T^{5} + 64607 T^{6} - 893262 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T - 47 T^{2} + 160 T^{3} + 2121 T^{4} + 8480 T^{5} - 132023 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 70 T^{2} + 6051 T^{4} + 452660 T^{6} + 19250861 T^{8} + 1271521940 T^{10} + 47745300531 T^{12} + 1551505279030 T^{14} + 62259690411361 T^{16} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( ( 1 - 15 T + 59 T^{2} )( 1 + 15 T + 59 T^{2} ) \))(\( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T + 17 T^{2} - 222 T^{3} - 1685 T^{4} - 13098 T^{5} + 59177 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 122 T^{2} + 9663 T^{4} + 604444 T^{6} + 40851365 T^{8} + 2104069564 T^{10} + 117090059343 T^{12} + 5146025104202 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( 1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 26901 T^{5} - 26047 T^{6} - 680943 T^{7} + 13845841 T^{8} \))(\( 1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 156099 T^{5} + 911645 T^{6} + 4766601 T^{7} + 13845841 T^{8} \))(\( ( 1 + 5 T + 61 T^{2} - 305 T^{3} + 796 T^{4} - 18605 T^{5} + 226981 T^{6} + 1134905 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( ( 1 + 13 T + 67 T^{2} )^{2} \))(\( ( 1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + T + 73 T^{2} + 67 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 - 3 T + 71 T^{2} )( 1 + 3 T + 71 T^{2} ) \))(\( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 58149 T^{5} + 1275373 T^{6} + 9663597 T^{7} + 25411681 T^{8} \))(\( 1 - 15 T + 119 T^{2} - 1455 T^{3} + 17296 T^{4} - 103305 T^{5} + 599879 T^{6} - 5368665 T^{7} + 25411681 T^{8} \))(\( 1 - 13 T^{2} + 5103 T^{4} - 100031 T^{6} + 31068680 T^{8} - 504256271 T^{10} + 129675808143 T^{12} - 1665303690973 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 9490 T^{5} - 303753 T^{6} - 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 62050 T^{5} + 335727 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( ( 1 + 73 T^{2} - 360 T^{3} + 2449 T^{4} - 26280 T^{5} + 389017 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( ( 1 - 79 T^{2} )^{2} \))(\( 1 - 5 T + 6 T^{2} - 715 T^{3} + 9821 T^{4} - 56485 T^{5} + 37446 T^{6} - 2465195 T^{7} + 38950081 T^{8} \))(\( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 32153 T^{5} + 262122 T^{6} + 5423429 T^{7} + 38950081 T^{8} \))(\( ( 1 - 25 T + 304 T^{2} - 2435 T^{3} + 19501 T^{4} - 192365 T^{5} + 1897264 T^{6} - 12325975 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 - 21 T + 88 T^{2} + 915 T^{3} - 13199 T^{4} + 75945 T^{5} + 606232 T^{6} - 12007527 T^{7} + 47458321 T^{8} \))(\( 1 - 13 T - 14 T^{2} + 421 T^{3} + 1449 T^{4} + 34943 T^{5} - 96446 T^{6} - 7433231 T^{7} + 47458321 T^{8} \))(\( 1 + 149 T^{2} + 16452 T^{4} + 1844407 T^{6} + 200466815 T^{8} + 12706119823 T^{10} + 780784297092 T^{12} + 48714115631981 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 9 T + 89 T^{2} )( 1 + 9 T + 89 T^{2} ) \))(\( ( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 12 T + 209 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 266 T^{2} + 31531 T^{4} - 2106986 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( ( 1 - 17 T + 97 T^{2} )^{2} \))(\( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 645535 T^{5} + 5052633 T^{6} + 30118209 T^{7} + 88529281 T^{8} \))(\( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 74205 T^{5} - 404587 T^{6} - 2738019 T^{7} + 88529281 T^{8} \))(\( ( 1 - 3 T - 63 T^{2} + 835 T^{3} + 4236 T^{4} + 80995 T^{5} - 592767 T^{6} - 2738019 T^{7} + 88529281 T^{8} )^{2} \))
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