Properties

Label 42.3
Level 42
Weight 3
Dimension 24
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 288
Trace bound 4

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Defining parameters

Level: \( N \) = \( 42\( 42 = 2 \cdot 3 \cdot 7 \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(288\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(42))\).

Total New Old
Modular forms 120 24 96
Cusp forms 72 24 48
Eisenstein series 48 0 48

Trace form

\( 24q + 6q^{3} + 8q^{4} + 12q^{5} - 16q^{7} + 18q^{9} + O(q^{10}) \) \( 24q + 6q^{3} + 8q^{4} + 12q^{5} - 16q^{7} + 18q^{9} - 48q^{10} - 36q^{11} - 12q^{12} - 28q^{13} - 84q^{15} - 48q^{17} - 24q^{18} - 40q^{19} - 18q^{21} + 48q^{23} + 24q^{24} + 132q^{25} + 96q^{26} + 252q^{27} + 72q^{28} + 120q^{29} + 132q^{30} + 8q^{31} - 102q^{33} - 132q^{35} - 72q^{36} + 28q^{37} + 24q^{38} - 84q^{39} + 96q^{40} - 72q^{42} + 64q^{43} - 72q^{44} - 6q^{45} - 72q^{46} - 132q^{47} - 132q^{49} - 96q^{50} - 90q^{51} - 136q^{52} - 96q^{53} - 144q^{54} - 504q^{55} - 96q^{56} - 252q^{57} - 384q^{58} - 24q^{59} - 108q^{60} - 16q^{61} + 282q^{63} - 64q^{64} + 420q^{65} + 336q^{66} + 552q^{67} + 96q^{68} + 252q^{69} + 504q^{70} + 192q^{71} + 48q^{72} + 380q^{73} + 240q^{74} + 360q^{75} + 392q^{76} + 144q^{77} + 288q^{78} + 144q^{79} - 48q^{80} - 42q^{81} + 240q^{82} - 12q^{84} + 120q^{85} - 264q^{86} - 324q^{87} - 144q^{88} - 72q^{89} - 672q^{90} - 448q^{91} - 48q^{92} - 546q^{93} - 456q^{94} - 372q^{95} - 48q^{96} - 448q^{97} - 192q^{98} + 84q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(42))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
42.3.b \(\chi_{42}(29, \cdot)\) 42.3.b.a 4 1
42.3.c \(\chi_{42}(13, \cdot)\) 42.3.c.a 4 1
42.3.g \(\chi_{42}(19, \cdot)\) 42.3.g.a 4 2
42.3.h \(\chi_{42}(11, \cdot)\) 42.3.h.a 4 2
42.3.h.b 8

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(42)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))
$3$ (\( 1 - 10 T^{2} + 81 T^{4} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )^{2} \))(\( 1 - 2 T - 5 T^{2} - 18 T^{3} + 81 T^{4} \))(\( 1 + 2 T + 7 T^{2} - 42 T^{3} - 108 T^{4} - 378 T^{5} + 567 T^{6} + 1458 T^{7} + 6561 T^{8} \))
$5$ (\( 1 - 56 T^{2} + 1586 T^{4} - 35000 T^{6} + 390625 T^{8} \))(\( 1 - 64 T^{2} + 1986 T^{4} - 40000 T^{6} + 390625 T^{8} \))(\( 1 - 12 T + 86 T^{2} - 456 T^{3} + 2019 T^{4} - 11400 T^{5} + 53750 T^{6} - 187500 T^{7} + 390625 T^{8} \))(\( 1 - 22 T^{2} - 141 T^{4} - 13750 T^{6} + 390625 T^{8} \))(\( 1 + 62 T^{2} + 1985 T^{4} + 37758 T^{6} + 662756 T^{8} + 23598750 T^{10} + 775390625 T^{12} + 15136718750 T^{14} + 152587890625 T^{16} \))
$7$ (\( ( 1 - 7 T^{2} )^{2} \))(\( 1 - 8 T + 42 T^{2} - 392 T^{3} + 2401 T^{4} \))(\( 1 + 10 T + 51 T^{2} + 490 T^{3} + 2401 T^{4} \))(\( ( 1 + 7 T + 49 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 2401 T^{4} )^{2} \))
$11$ (\( 1 - 272 T^{2} + 36578 T^{4} - 3982352 T^{6} + 214358881 T^{8} \))(\( ( 1 + 12 T + 260 T^{2} + 1452 T^{3} + 14641 T^{4} )^{2} \))(\( ( 1 + 6 T - 85 T^{2} + 726 T^{3} + 14641 T^{4} )^{2} \))(\( ( 1 - 11 T + 121 T^{2} )^{2}( 1 + 11 T + 121 T^{2} )^{2} \))(\( 1 - 70 T^{2} - 23407 T^{4} + 68250 T^{6} + 515186468 T^{8} + 999248250 T^{10} - 5017498327567 T^{12} - 219689986370470 T^{14} + 45949729863572161 T^{16} \))
$13$ (\( ( 1 - 20 T + 326 T^{2} - 3380 T^{3} + 28561 T^{4} )^{2} \))(\( 1 - 244 T^{2} + 53574 T^{4} - 6968884 T^{6} + 815730721 T^{8} \))(\( 1 + 98 T^{2} + 54915 T^{4} + 2798978 T^{6} + 815730721 T^{8} \))(\( ( 1 + T + 169 T^{2} )^{4} \))(\( ( 1 + 16 T + 314 T^{2} + 2704 T^{3} + 28561 T^{4} )^{4} \))
$17$ (\( 1 - 440 T^{2} + 204242 T^{4} - 36749240 T^{6} + 6975757441 T^{8} \))(\( 1 + 320 T^{2} + 157794 T^{4} + 26726720 T^{6} + 6975757441 T^{8} \))(\( 1 + 48 T + 1514 T^{2} + 35808 T^{3} + 694947 T^{4} + 10348512 T^{5} + 126450794 T^{6} + 1158603312 T^{7} + 6975757441 T^{8} \))(\( 1 + 506 T^{2} + 172515 T^{4} + 42261626 T^{6} + 6975757441 T^{8} \))(\( 1 + 1118 T^{2} + 770753 T^{4} + 348960222 T^{6} + 118234187396 T^{8} + 29145506701662 T^{10} + 5376585974923073 T^{12} + 651371661222872798 T^{14} + 48661191875666868481 T^{16} \))
$19$ (\( ( 1 + 16 T + 361 T^{2} )^{4} \))(\( 1 - 652 T^{2} + 348486 T^{4} - 84969292 T^{6} + 16983563041 T^{8} \))(\( 1 + 42 T + 1433 T^{2} + 35490 T^{3} + 795972 T^{4} + 12811890 T^{5} + 186749993 T^{6} + 1975927002 T^{7} + 16983563041 T^{8} \))(\( ( 1 - 31 T + 600 T^{2} - 11191 T^{3} + 130321 T^{4} )^{2} \))(\( ( 1 - 2 T - 521 T^{2} + 394 T^{3} + 143860 T^{4} + 142234 T^{5} - 67897241 T^{6} - 94091762 T^{7} + 16983563041 T^{8} )^{2} \))
$23$ (\( 1 + 688 T^{2} + 666818 T^{4} + 192530608 T^{6} + 78310985281 T^{8} \))(\( ( 1 - 12 T + 932 T^{2} - 6348 T^{3} + 279841 T^{4} )^{2} \))(\( 1 - 24 T + 22 T^{2} + 12096 T^{3} - 277629 T^{4} + 6398784 T^{5} + 6156502 T^{6} - 3552861336 T^{7} + 78310985281 T^{8} \))(\( 1 + 986 T^{2} + 692355 T^{4} + 275923226 T^{6} + 78310985281 T^{8} \))(\( 1 + 1370 T^{2} + 955793 T^{4} + 495152250 T^{6} + 244893547268 T^{8} + 138563900792250 T^{10} + 74849091554682833 T^{12} + 30023035471867839770 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \))
$29$ (\( 1 - 1652 T^{2} + 1917638 T^{4} - 1168428212 T^{6} + 500246412961 T^{8} \))(\( ( 1 - 30 T + 841 T^{2} )^{4} \))(\( ( 1 + 530 T^{2} + 707281 T^{4} )^{2} \))(\( ( 1 - 1394 T^{2} + 707281 T^{4} )^{2} \))(\( ( 1 - 764 T^{2} + 680486 T^{4} - 540362684 T^{6} + 500246412961 T^{8} )^{2} \))
$31$ (\( ( 1 + 64 T + 2246 T^{2} + 61504 T^{3} + 923521 T^{4} )^{2} \))(\( 1 - 1252 T^{2} + 745926 T^{4} - 1156248292 T^{6} + 852891037441 T^{8} \))(\( 1 - 102 T + 6041 T^{2} - 262446 T^{3} + 9029556 T^{4} - 252210606 T^{5} + 5578990361 T^{6} - 90525375462 T^{7} + 852891037441 T^{8} \))(\( ( 1 - 7 T - 912 T^{2} - 6727 T^{3} + 923521 T^{4} )^{2} \))(\( ( 1 - 10 T - 1297 T^{2} + 5250 T^{3} + 931988 T^{4} + 5045250 T^{5} - 1197806737 T^{6} - 8875036810 T^{7} + 852891037441 T^{8} )^{2} \))
$37$ (\( ( 1 - 20 T + 1369 T^{2} )^{4} \))(\( ( 1 + 40 T + 546 T^{2} + 54760 T^{3} + 1874161 T^{4} )^{2} \))(\( 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 5029706 T^{5} - 3911374007 T^{6} - 56445980998 T^{7} + 3512479453921 T^{8} \))(\( ( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{2} \))(\( ( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{4} \))
$41$ (\( 1 - 856 T^{2} - 2330094 T^{4} - 2418851416 T^{6} + 7984925229121 T^{8} \))(\( 1 - 4960 T^{2} + 11110434 T^{4} - 14015774560 T^{6} + 7984925229121 T^{8} \))(\( 1 - 2476 T^{2} + 6405414 T^{4} - 6996584236 T^{6} + 7984925229121 T^{8} \))(\( ( 1 - 2210 T^{2} + 2825761 T^{4} )^{2} \))(\( ( 1 - 5788 T^{2} + 13912710 T^{4} - 16355504668 T^{6} + 7984925229121 T^{8} )^{2} \))
$43$ (\( ( 1 - 40 T + 3090 T^{2} - 73960 T^{3} + 3418801 T^{4} )^{2} \))(\( ( 1 + 64 T + 2922 T^{2} + 118336 T^{3} + 3418801 T^{4} )^{2} \))(\( ( 1 - 14 T + 3675 T^{2} - 25886 T^{3} + 3418801 T^{4} )^{2} \))(\( ( 1 + 31 T + 1849 T^{2} )^{4} \))(\( ( 1 - 52 T + 3582 T^{2} - 96148 T^{3} + 3418801 T^{4} )^{4} \))
$47$ (\( ( 1 - 4346 T^{2} + 4879681 T^{4} )^{2} \))(\( 1 - 3940 T^{2} + 12737094 T^{4} - 19225943140 T^{6} + 23811286661761 T^{8} \))(\( 1 + 132 T + 11654 T^{2} + 771672 T^{3} + 42125907 T^{4} + 1704623448 T^{5} + 56867802374 T^{6} + 1422856423428 T^{7} + 23811286661761 T^{8} \))(\( 1 + 2618 T^{2} + 1974243 T^{4} + 12775004858 T^{6} + 23811286661761 T^{8} \))(\( 1 + 538 T^{2} - 7938479 T^{4} - 823914182 T^{6} + 42475035844804 T^{8} - 4020438379535942 T^{10} - \)\(18\!\cdots\!19\)\( T^{12} + \)\(62\!\cdots\!58\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \))
$53$ (\( ( 1 - 3026 T^{2} + 7890481 T^{4} )^{2} \))(\( ( 1 + 108 T + 8246 T^{2} + 303372 T^{3} + 7890481 T^{4} )^{2} \))(\( 1 - 120 T + 5830 T^{2} - 354240 T^{3} + 25104819 T^{4} - 995060160 T^{5} + 46001504230 T^{6} - 2659723335480 T^{7} + 62259690411361 T^{8} \))(\( 1 + 4970 T^{2} + 16810419 T^{4} + 39215690570 T^{6} + 62259690411361 T^{8} \))(\( 1 + 5854 T^{2} + 10947457 T^{4} + 44144411038 T^{6} + 211248030023524 T^{8} + 348320636551529278 T^{10} + \)\(68\!\cdots\!77\)\( T^{12} + \)\(28\!\cdots\!14\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} \))
$59$ (\( 1 - 10532 T^{2} + 49098278 T^{4} - 127620046052 T^{6} + 146830437604321 T^{8} \))(\( 1 - 4420 T^{2} + 6539622 T^{4} - 53558735620 T^{6} + 146830437604321 T^{8} \))(\( 1 + 24 T + 6026 T^{2} + 140016 T^{3} + 22586547 T^{4} + 487395696 T^{5} + 73019217386 T^{6} + 1012332807384 T^{7} + 146830437604321 T^{8} \))(\( 1 + 6890 T^{2} + 35354739 T^{4} + 83488617290 T^{6} + 146830437604321 T^{8} \))(\( 1 + 746 T^{2} - 10773535 T^{4} - 9626884566 T^{6} - 25203887215996 T^{8} - 116652435591550326 T^{10} - \)\(15\!\cdots\!35\)\( T^{12} + \)\(13\!\cdots\!26\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))
$61$ (\( ( 1 + 28 T + 4838 T^{2} + 104188 T^{3} + 13845841 T^{4} )^{2} \))(\( 1 - 14596 T^{2} + 80934054 T^{4} - 202093895236 T^{6} + 191707312997281 T^{8} \))(\( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 2006660880 T^{5} + 127630962338 T^{6} + 3709466953992 T^{7} + 191707312997281 T^{8} \))(\( ( 1 + 50 T - 1221 T^{2} + 186050 T^{3} + 13845841 T^{4} )^{2} \))(\( ( 1 - 106 T + 1337 T^{2} - 260442 T^{3} + 42335204 T^{4} - 969104682 T^{5} + 18511889417 T^{6} - 5461159682266 T^{7} + 191707312997281 T^{8} )^{2} \))
$67$ (\( ( 1 - 120 T + 12466 T^{2} - 538680 T^{3} + 20151121 T^{4} )^{2} \))(\( ( 1 - 88 T + 10626 T^{2} - 395032 T^{3} + 20151121 T^{4} )^{2} \))(\( 1 - 110 T + 3625 T^{2} + 55330 T^{3} - 2642396 T^{4} + 248376370 T^{5} + 73047813625 T^{6} - 9950422038590 T^{7} + 406067677556641 T^{8} \))(\( ( 1 + 65 T - 264 T^{2} + 291785 T^{3} + 20151121 T^{4} )^{2} \))(\( ( 1 - 78 T - 697 T^{2} + 171366 T^{3} - 1480236 T^{4} + 769261974 T^{5} - 14045331337 T^{6} - 7055753809182 T^{7} + 406067677556641 T^{8} )^{2} \))
$71$ (\( 1 - 12176 T^{2} + 74167106 T^{4} - 309412627856 T^{6} + 645753531245761 T^{8} \))(\( ( 1 + 60 T + 4484 T^{2} + 302460 T^{3} + 25411681 T^{4} )^{2} \))(\( ( 1 - 156 T + 12638 T^{2} - 786396 T^{3} + 25411681 T^{4} )^{2} \))(\( ( 1 - 6554 T^{2} + 25411681 T^{4} )^{2} \))(\( ( 1 - 1316 T^{2} + 44745734 T^{4} - 33441772196 T^{6} + 645753531245761 T^{8} )^{2} \))
$73$ (\( ( 1 - 60 T + 9766 T^{2} - 319740 T^{3} + 28398241 T^{4} )^{2} \))(\( 1 - 13108 T^{2} + 98972646 T^{4} - 372244143028 T^{6} + 806460091894081 T^{8} \))(\( 1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 499785594 T^{5} + 81588146393 T^{6} + 9988058935074 T^{7} + 806460091894081 T^{8} \))(\( ( 1 - 143 T + 5329 T^{2} )^{2}( 1 + 46 T + 5329 T^{2} )^{2} \))(\( ( 1 - 66 T - 7039 T^{2} - 48642 T^{3} + 78234660 T^{4} - 259213218 T^{5} - 199895218399 T^{6} - 9988058935074 T^{7} + 806460091894081 T^{8} )^{2} \))
$79$ (\( ( 1 + 64 T + 10706 T^{2} + 399424 T^{3} + 38950081 T^{4} )^{2} \))(\( ( 1 - 64 T + 3138 T^{2} - 399424 T^{3} + 38950081 T^{4} )^{2} \))(\( 1 + 10 T - 3695 T^{2} - 86870 T^{3} - 25172156 T^{4} - 542155670 T^{5} - 143920549295 T^{6} + 2430874555210 T^{7} + 1517108809906561 T^{8} \))(\( ( 1 - 103 T + 4368 T^{2} - 642823 T^{3} + 38950081 T^{4} )^{2} \))(\( ( 1 + 26 T - 5617 T^{2} - 160914 T^{3} - 3567148 T^{4} - 1004264274 T^{5} - 218782604977 T^{6} + 6320273843546 T^{7} + 1517108809906561 T^{8} )^{2} \))
$83$ (\( 1 - 6356 T^{2} - 6983674 T^{4} - 301645088276 T^{6} + 2252292232139041 T^{8} \))(\( 1 - 16612 T^{2} + 134415078 T^{4} - 788377628452 T^{6} + 2252292232139041 T^{8} \))(\( 1 - 9628 T^{2} + 107694438 T^{4} - 456928714588 T^{6} + 2252292232139041 T^{8} \))(\( ( 1 - 11978 T^{2} + 47458321 T^{4} )^{2} \))(\( ( 1 - 18404 T^{2} + 168689894 T^{4} - 873422939684 T^{6} + 2252292232139041 T^{8} )^{2} \))
$89$ (\( 1 - 27832 T^{2} + 318232338 T^{4} - 1746242051512 T^{6} + 3936588805702081 T^{8} \))(\( 1 - 8896 T^{2} + 52680354 T^{4} - 558154975936 T^{6} + 3936588805702081 T^{8} \))(\( ( 1 + 36 T + 8353 T^{2} + 285156 T^{3} + 62742241 T^{4} )^{2} \))(\( 1 + 1730 T^{2} - 59749341 T^{4} + 108544076930 T^{6} + 3936588805702081 T^{8} \))(\( 1 + 2542 T^{2} + 3560113 T^{4} - 311605556402 T^{6} - 4333594781349404 T^{8} - 19550830916713376882 T^{10} + \)\(14\!\cdots\!53\)\( T^{12} + \)\(62\!\cdots\!82\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))
$97$ (\( ( 1 + 180 T + 26470 T^{2} + 1693620 T^{3} + 88529281 T^{4} )^{2} \))(\( 1 - 24820 T^{2} + 317127462 T^{4} - 2197296754420 T^{6} + 7837433594376961 T^{8} \))(\( 1 - 36580 T^{2} + 511416774 T^{4} - 3238401098980 T^{6} + 7837433594376961 T^{8} \))(\( ( 1 + 166 T + 9409 T^{2} )^{4} \))(\( ( 1 - 144 T + 21802 T^{2} - 1354896 T^{3} + 88529281 T^{4} )^{4} \))
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