Properties

Label 32.6
Level 32
Weight 6
Dimension 85
Nonzero newspaces 3
Newform subspaces 6
Sturm bound 384
Trace bound 1

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Defining parameters

Level: \( N \) = \( 32\( 32 = 2^{5} \) \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 6 \)
Sturm bound: \(384\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(32))\).

Total New Old
Modular forms 176 95 81
Cusp forms 144 85 59
Eisenstein series 32 10 22

Trace form

\( 85q - 4q^{2} - 4q^{3} - 4q^{4} + 34q^{5} - 4q^{6} - 100q^{7} - 4q^{8} + 281q^{9} + O(q^{10}) \) \( 85q - 4q^{2} - 4q^{3} - 4q^{4} + 34q^{5} - 4q^{6} - 100q^{7} - 4q^{8} + 281q^{9} - 204q^{10} - 4q^{11} - 1588q^{12} + 106q^{13} + 2476q^{14} + 416q^{15} + 4176q^{16} + 1810q^{17} - 1624q^{18} - 4q^{19} - 7604q^{20} - 5892q^{21} - 12384q^{22} - 668q^{23} + 21456q^{24} - 2725q^{25} + 12976q^{26} - 7468q^{27} - 2184q^{28} + 3674q^{29} - 32316q^{30} + 35984q^{31} - 18584q^{32} + 10824q^{33} - 6256q^{34} - 4780q^{35} + 65584q^{36} - 7262q^{37} + 404q^{38} - 80012q^{39} - 28952q^{40} - 35338q^{41} - 53624q^{42} + 32068q^{43} + 3716q^{44} + 64182q^{45} + 63324q^{46} + 54720q^{47} + 112368q^{48} + 67617q^{49} - 4524q^{50} - 19912q^{51} + 18468q^{52} - 125950q^{53} + 1312q^{54} + 24572q^{55} - 80984q^{56} - 135028q^{57} - 84576q^{58} - 28964q^{59} - 46088q^{60} + 204666q^{61} + 63480q^{62} - 5328q^{63} - 49192q^{64} + 93948q^{65} - 145012q^{66} - 61164q^{67} - 151216q^{68} - 271204q^{69} - 56296q^{70} - 62852q^{71} + 27092q^{72} - 80346q^{73} + 213100q^{74} + 205744q^{75} + 255996q^{76} + 180428q^{77} + 83508q^{78} + 247872q^{79} - 97096q^{80} + 299713q^{81} + 435196q^{82} - 329244q^{83} + 597472q^{84} - 118628q^{85} + 269888q^{86} - 590060q^{87} - 199840q^{88} - 259338q^{89} - 706840q^{90} + 200108q^{91} - 650328q^{92} + 151600q^{93} - 261592q^{94} + 837336q^{95} - 501376q^{96} + 260058q^{97} - 395624q^{98} + 338544q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(32))\)

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
32.6.a \(\chi_{32}(1, \cdot)\) 32.6.a.a 1 1
32.6.a.b 1
32.6.a.c 1
32.6.a.d 2
32.6.b \(\chi_{32}(17, \cdot)\) 32.6.b.a 4 1
32.6.e \(\chi_{32}(9, \cdot)\) None 0 2
32.6.g \(\chi_{32}(5, \cdot)\) 32.6.g.a 76 4

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(32))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(32)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 8 T + 243 T^{2} \))(\( 1 + 243 T^{2} \))(\( 1 - 8 T + 243 T^{2} \))(\( 1 - 282 T^{2} + 59049 T^{4} \))(\( 1 - 404 T^{2} + 84150 T^{4} - 23855796 T^{6} + 3486784401 T^{8} \))
$5$ (\( 1 - 14 T + 3125 T^{2} \))(\( 1 + 82 T + 3125 T^{2} \))(\( 1 - 14 T + 3125 T^{2} \))(\( ( 1 - 46 T + 3125 T^{2} )^{2} \))(\( 1 - 7028 T^{2} + 24404246 T^{4} - 68632812500 T^{6} + 95367431640625 T^{8} \))
$7$ (\( 1 + 208 T + 16807 T^{2} \))(\( 1 + 16807 T^{2} \))(\( 1 - 208 T + 16807 T^{2} \))(\( 1 + 5966 T^{2} + 282475249 T^{4} \))(\( ( 1 + 48 T + 15502 T^{2} + 806736 T^{3} + 282475249 T^{4} )^{2} \))
$11$ (\( 1 + 536 T + 161051 T^{2} \))(\( 1 + 161051 T^{2} \))(\( 1 - 536 T + 161051 T^{2} \))(\( 1 + 315190 T^{2} + 25937424601 T^{4} \))(\( 1 - 296436 T^{2} + 49128544726 T^{4} - 7688786399022036 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} \))
$13$ (\( 1 - 694 T + 371293 T^{2} \))(\( 1 + 1194 T + 371293 T^{2} \))(\( 1 - 694 T + 371293 T^{2} \))(\( ( 1 + 42 T + 371293 T^{2} )^{2} \))(\( 1 - 894228 T^{2} + 396323515894 T^{4} - 123276923449147572 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))
$17$ (\( 1 + 1278 T + 1419857 T^{2} \))(\( 1 - 2242 T + 1419857 T^{2} \))(\( 1 + 1278 T + 1419857 T^{2} \))(\( ( 1 - 962 T + 1419857 T^{2} )^{2} \))(\( ( 1 - 100 T + 2767462 T^{2} - 141985700 T^{3} + 2015993900449 T^{4} )^{2} \))
$19$ (\( 1 - 1112 T + 2476099 T^{2} \))(\( 1 + 2476099 T^{2} \))(\( 1 + 1112 T + 2476099 T^{2} \))(\( 1 + 632198 T^{2} + 6131066257801 T^{4} \))(\( 1 - 6794580 T^{2} + 21506967947254 T^{4} - 41658020173929518580 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} \))
$23$ (\( 1 - 3216 T + 6436343 T^{2} \))(\( 1 + 6436343 T^{2} \))(\( 1 + 3216 T + 6436343 T^{2} \))(\( 1 + 2891758 T^{2} + 41426511213649 T^{4} \))(\( ( 1 + 1168 T + 10055470 T^{2} + 7517648624 T^{3} + 41426511213649 T^{4} )^{2} \))
$29$ (\( 1 - 2918 T + 20511149 T^{2} \))(\( 1 - 2950 T + 20511149 T^{2} \))(\( 1 - 2918 T + 20511149 T^{2} \))(\( ( 1 + 2554 T + 20511149 T^{2} )^{2} \))(\( 1 - 31255380 T^{2} + 976386653995702 T^{4} - \)\(13\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))
$31$ (\( 1 + 2624 T + 28629151 T^{2} \))(\( 1 + 28629151 T^{2} \))(\( 1 - 2624 T + 28629151 T^{2} \))(\( 1 + 53276990 T^{2} + 819628286980801 T^{4} \))(\( ( 1 - 6464 T + 65013054 T^{2} - 185058832064 T^{3} + 819628286980801 T^{4} )^{2} \))
$37$ (\( 1 + 9458 T + 69343957 T^{2} \))(\( 1 + 12242 T + 69343957 T^{2} \))(\( 1 + 9458 T + 69343957 T^{2} \))(\( ( 1 - 11950 T + 69343957 T^{2} )^{2} \))(\( 1 - 241262580 T^{2} + 24149916431784598 T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))
$41$ (\( 1 - 170 T + 115856201 T^{2} \))(\( 1 + 20950 T + 115856201 T^{2} \))(\( 1 - 170 T + 115856201 T^{2} \))(\( ( 1 + 5078 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 2284 T + 146603254 T^{2} + 264615563084 T^{3} + 13422659310152401 T^{4} )^{2} \))
$43$ (\( 1 + 19928 T + 147008443 T^{2} \))(\( 1 + 147008443 T^{2} \))(\( 1 - 19928 T + 147008443 T^{2} \))(\( 1 + 136416374 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 466346868 T^{2} + 96250708269010006 T^{4} - \)\(10\!\cdots\!32\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - 32 T + 229345007 T^{2} \))(\( 1 + 229345007 T^{2} \))(\( 1 + 32 T + 229345007 T^{2} \))(\( 1 + 307289566 T^{2} + 52599132235830049 T^{4} \))(\( ( 1 - 27360 T + 591338206 T^{2} - 6274879391520 T^{3} + 52599132235830049 T^{4} )^{2} \))
$53$ (\( 1 + 22178 T + 418195493 T^{2} \))(\( 1 - 7294 T + 418195493 T^{2} \))(\( 1 + 22178 T + 418195493 T^{2} \))(\( ( 1 + 19714 T + 418195493 T^{2} )^{2} \))(\( 1 - 1039152180 T^{2} + 595616955270391126 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))
$59$ (\( 1 - 41480 T + 714924299 T^{2} \))(\( 1 + 714924299 T^{2} \))(\( 1 + 41480 T + 714924299 T^{2} \))(\( 1 + 1350713110 T^{2} + 511116753300641401 T^{4} \))(\( 1 - 1537424180 T^{2} + 1399694789142612374 T^{4} - \)\(78\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))
$61$ (\( 1 - 15462 T + 844596301 T^{2} \))(\( 1 - 18950 T + 844596301 T^{2} \))(\( 1 - 15462 T + 844596301 T^{2} \))(\( ( 1 - 29318 T + 844596301 T^{2} )^{2} \))(\( 1 + 741098540 T^{2} + 1478044222094100534 T^{4} + \)\(52\!\cdots\!40\)\( T^{6} + \)\(50\!\cdots\!01\)\( T^{8} \))
$67$ (\( 1 + 20744 T + 1350125107 T^{2} \))(\( 1 + 1350125107 T^{2} \))(\( 1 - 20744 T + 1350125107 T^{2} \))(\( 1 + 2415413606 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 1366835860 T^{2} + 3274116308996825526 T^{4} - \)\(24\!\cdots\!40\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))
$71$ (\( 1 - 28592 T + 1804229351 T^{2} \))(\( 1 + 1804229351 T^{2} \))(\( 1 + 28592 T + 1804229351 T^{2} \))(\( 1 - 2948789810 T^{2} + 3255243551009881201 T^{4} \))(\( ( 1 + 103344 T + 6217736974 T^{2} + 186456278049744 T^{3} + 3255243551009881201 T^{4} )^{2} \))
$73$ (\( 1 + 53670 T + 2073071593 T^{2} \))(\( 1 + 88806 T + 2073071593 T^{2} \))(\( 1 + 53670 T + 2073071593 T^{2} \))(\( ( 1 - 37914 T + 2073071593 T^{2} )^{2} \))(\( ( 1 - 19988 T + 2541602870 T^{2} - 41436555000884 T^{3} + 4297625829703557649 T^{4} )^{2} \))
$79$ (\( 1 + 69152 T + 3077056399 T^{2} \))(\( 1 + 3077056399 T^{2} \))(\( 1 - 69152 T + 3077056399 T^{2} \))(\( 1 - 1729880290 T^{2} + 9468276082626847201 T^{4} \))(\( ( 1 - 123936 T + 9855929374 T^{2} - 381358061866464 T^{3} + 9468276082626847201 T^{4} )^{2} \))
$83$ (\( 1 + 37800 T + 3939040643 T^{2} \))(\( 1 + 3939040643 T^{2} \))(\( 1 - 37800 T + 3939040643 T^{2} \))(\( 1 + 6331666438 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 10047855188 T^{2} + 55381071937674414326 T^{4} - \)\(15\!\cdots\!12\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))
$89$ (\( 1 + 126806 T + 5584059449 T^{2} \))(\( 1 - 51050 T + 5584059449 T^{2} \))(\( 1 + 126806 T + 5584059449 T^{2} \))(\( ( 1 - 13930 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 42316 T + 4292401174 T^{2} + 236295059643884 T^{3} + 31181719929966183601 T^{4} )^{2} \))
$97$ (\( 1 - 62290 T + 8587340257 T^{2} \))(\( 1 + 92142 T + 8587340257 T^{2} \))(\( 1 - 62290 T + 8587340257 T^{2} \))(\( ( 1 - 163602 T + 8587340257 T^{2} )^{2} \))(\( ( 1 + 49788 T + 16391371462 T^{2} + 427546496715516 T^{3} + 73742412689492826049 T^{4} )^{2} \))
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