Properties

Label 108.6.e
Level 108
Weight 6
Character orbit e
Rep. character \(\chi_{108}(37,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 10
Newform subspaces 1
Sturm bound 108
Trace bound 0

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 108.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 1 \)
Sturm bound: \(108\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(108, [\chi])\).

Total New Old
Modular forms 198 10 188
Cusp forms 162 10 152
Eisenstein series 36 0 36

Trace form

\( 10q + 21q^{5} + 29q^{7} + O(q^{10}) \) \( 10q + 21q^{5} + 29q^{7} - 177q^{11} - 181q^{13} - 2280q^{17} - 832q^{19} - 399q^{23} - 4778q^{25} + 6033q^{29} + 2759q^{31} - 37146q^{35} - 15172q^{37} + 18435q^{41} + 1469q^{43} + 25155q^{47} - 4056q^{49} - 116844q^{53} + 14778q^{55} + 90537q^{59} + 1403q^{61} + 148407q^{65} + 13907q^{67} - 229368q^{71} + 15200q^{73} + 211983q^{77} + 29993q^{79} + 228951q^{83} - 49662q^{85} - 598332q^{89} + 124930q^{91} + 394764q^{95} + 40541q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(108, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
108.6.e.a \(10\) \(17.321\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(21\) \(29\) \(q+(4-4\beta _{1}+\beta _{7})q^{5}+(6\beta _{1}+\beta _{2}+\beta _{7}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(108, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(108, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 21 T - 5203 T^{2} + 519930 T^{3} + 14035794 T^{4} - 2854822770 T^{5} + 76722872007 T^{6} + 9761967315441 T^{7} - 599011867854189 T^{8} - 11924309583255600 T^{9} + 2533145723872694124 T^{10} - 37263467447673750000 T^{11} - \)\(58\!\cdots\!25\)\( T^{12} + \)\(29\!\cdots\!25\)\( T^{13} + \)\(73\!\cdots\!75\)\( T^{14} - \)\(85\!\cdots\!50\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} + \)\(15\!\cdots\!50\)\( T^{17} - \)\(47\!\cdots\!75\)\( T^{18} - \)\(59\!\cdots\!25\)\( T^{19} + \)\(88\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 - 29 T - 39569 T^{2} + 3762444 T^{3} + 440397336 T^{4} - 77352503496 T^{5} - 2769093584103 T^{6} - 560172784984473 T^{7} + 238615372451780007 T^{8} + 16031898530170676332 T^{9} - \)\(69\!\cdots\!60\)\( T^{10} + \)\(26\!\cdots\!24\)\( T^{11} + \)\(67\!\cdots\!43\)\( T^{12} - \)\(26\!\cdots\!39\)\( T^{13} - \)\(22\!\cdots\!03\)\( T^{14} - \)\(10\!\cdots\!72\)\( T^{15} + \)\(99\!\cdots\!64\)\( T^{16} + \)\(14\!\cdots\!92\)\( T^{17} - \)\(25\!\cdots\!69\)\( T^{18} - \)\(31\!\cdots\!03\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 + 177 T - 396232 T^{2} + 71434269 T^{3} + 104816625882 T^{4} - 33726096455301 T^{5} - 6913987980717606 T^{6} + 8552599160812456257 T^{7} - \)\(10\!\cdots\!51\)\( T^{8} - \)\(50\!\cdots\!82\)\( T^{9} + \)\(47\!\cdots\!16\)\( T^{10} - \)\(81\!\cdots\!82\)\( T^{11} - \)\(27\!\cdots\!51\)\( T^{12} + \)\(35\!\cdots\!07\)\( T^{13} - \)\(46\!\cdots\!06\)\( T^{14} - \)\(36\!\cdots\!51\)\( T^{15} + \)\(18\!\cdots\!82\)\( T^{16} + \)\(20\!\cdots\!19\)\( T^{17} - \)\(17\!\cdots\!32\)\( T^{18} + \)\(12\!\cdots\!27\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} + \)\(98\!\cdots\!51\)\( T^{8} + \)\(77\!\cdots\!72\)\( T^{9} - \)\(41\!\cdots\!24\)\( T^{10} + \)\(28\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!99\)\( T^{12} - \)\(41\!\cdots\!29\)\( T^{13} - \)\(36\!\cdots\!73\)\( T^{14} - \)\(59\!\cdots\!66\)\( T^{15} + \)\(11\!\cdots\!26\)\( T^{16} + \)\(14\!\cdots\!74\)\( T^{17} - \)\(36\!\cdots\!31\)\( T^{18} + \)\(24\!\cdots\!33\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} \)
$17$ \( ( 1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 14296356292995309833 T^{6} + \)\(69\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} + \)\(46\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!57\)\( T^{10} )^{2} \)
$19$ \( ( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(93\!\cdots\!99\)\( T^{10} )^{2} \)
$23$ \( 1 + 399 T - 16077241 T^{2} + 38108825820 T^{3} + 155650662506976 T^{4} - 562944417983120520 T^{5} - 48522958516353490863 T^{6} + \)\(48\!\cdots\!51\)\( T^{7} - \)\(71\!\cdots\!41\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(81\!\cdots\!80\)\( T^{10} - \)\(74\!\cdots\!36\)\( T^{11} - \)\(29\!\cdots\!09\)\( T^{12} + \)\(12\!\cdots\!57\)\( T^{13} - \)\(83\!\cdots\!63\)\( T^{14} - \)\(62\!\cdots\!60\)\( T^{15} + \)\(11\!\cdots\!24\)\( T^{16} + \)\(17\!\cdots\!40\)\( T^{17} - \)\(47\!\cdots\!41\)\( T^{18} + \)\(75\!\cdots\!57\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 - 6033 T + 3652157 T^{2} - 31641196734 T^{3} + 283528398607854 T^{4} - 668469168127712358 T^{5} + \)\(64\!\cdots\!39\)\( T^{6} - \)\(82\!\cdots\!55\)\( T^{7} - \)\(90\!\cdots\!17\)\( T^{8} - \)\(14\!\cdots\!60\)\( T^{9} + \)\(58\!\cdots\!16\)\( T^{10} - \)\(29\!\cdots\!40\)\( T^{11} - \)\(38\!\cdots\!17\)\( T^{12} - \)\(71\!\cdots\!95\)\( T^{13} + \)\(11\!\cdots\!39\)\( T^{14} - \)\(24\!\cdots\!42\)\( T^{15} + \)\(21\!\cdots\!54\)\( T^{16} - \)\(48\!\cdots\!66\)\( T^{17} + \)\(11\!\cdots\!57\)\( T^{18} - \)\(38\!\cdots\!17\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 - 2759 T - 54902477 T^{2} - 189444651072 T^{3} + 2052158291804100 T^{4} + 13274031992302596720 T^{5} - \)\(32\!\cdots\!47\)\( T^{6} - \)\(42\!\cdots\!39\)\( T^{7} - \)\(13\!\cdots\!49\)\( T^{8} + \)\(36\!\cdots\!48\)\( T^{9} + \)\(63\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{11} - \)\(11\!\cdots\!49\)\( T^{12} - \)\(99\!\cdots\!89\)\( T^{13} - \)\(21\!\cdots\!47\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!00\)\( T^{16} - \)\(29\!\cdots\!72\)\( T^{17} - \)\(24\!\cdots\!77\)\( T^{18} - \)\(35\!\cdots\!09\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$37$ \( ( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + \)\(13\!\cdots\!82\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(67\!\cdots\!49\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{2} \)
$41$ \( 1 - 18435 T - 117679042 T^{2} + 4344492069675 T^{3} - 505249106564622 T^{4} - \)\(52\!\cdots\!97\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!43\)\( T^{7} - \)\(31\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!42\)\( T^{9} + \)\(29\!\cdots\!40\)\( T^{10} - \)\(11\!\cdots\!42\)\( T^{11} - \)\(41\!\cdots\!03\)\( T^{12} + \)\(62\!\cdots\!43\)\( T^{13} + \)\(30\!\cdots\!40\)\( T^{14} - \)\(10\!\cdots\!97\)\( T^{15} - \)\(12\!\cdots\!22\)\( T^{16} + \)\(12\!\cdots\!75\)\( T^{17} - \)\(38\!\cdots\!42\)\( T^{18} - \)\(69\!\cdots\!35\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} \)
$43$ \( 1 - 1469 T - 271863536 T^{2} + 4016430594327 T^{3} + 12129147672135834 T^{4} - \)\(75\!\cdots\!27\)\( T^{5} + \)\(55\!\cdots\!62\)\( T^{6} - \)\(12\!\cdots\!57\)\( T^{7} - \)\(29\!\cdots\!39\)\( T^{8} + \)\(61\!\cdots\!62\)\( T^{9} - \)\(11\!\cdots\!84\)\( T^{10} + \)\(90\!\cdots\!66\)\( T^{11} - \)\(64\!\cdots\!11\)\( T^{12} - \)\(38\!\cdots\!99\)\( T^{13} + \)\(26\!\cdots\!62\)\( T^{14} - \)\(52\!\cdots\!61\)\( T^{15} + \)\(12\!\cdots\!66\)\( T^{16} + \)\(59\!\cdots\!89\)\( T^{17} - \)\(59\!\cdots\!36\)\( T^{18} - \)\(47\!\cdots\!67\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} \)
$47$ \( 1 - 25155 T - 401246233 T^{2} + 14349179861244 T^{3} + 97557609874842960 T^{4} - \)\(41\!\cdots\!12\)\( T^{5} - \)\(25\!\cdots\!27\)\( T^{6} + \)\(55\!\cdots\!85\)\( T^{7} + \)\(12\!\cdots\!11\)\( T^{8} - \)\(42\!\cdots\!56\)\( T^{9} - \)\(37\!\cdots\!60\)\( T^{10} - \)\(96\!\cdots\!92\)\( T^{11} + \)\(67\!\cdots\!39\)\( T^{12} + \)\(67\!\cdots\!55\)\( T^{13} - \)\(71\!\cdots\!27\)\( T^{14} - \)\(26\!\cdots\!84\)\( T^{15} + \)\(14\!\cdots\!40\)\( T^{16} + \)\(47\!\cdots\!92\)\( T^{17} - \)\(30\!\cdots\!33\)\( T^{18} - \)\(44\!\cdots\!85\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} \)
$53$ \( ( 1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} + \)\(71\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} + \)\(17\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} )^{2} \)
$59$ \( 1 - 90537 T + 2831117840 T^{2} - 13805150996349 T^{3} - 966660594685472478 T^{4} + \)\(10\!\cdots\!09\)\( T^{5} + \)\(69\!\cdots\!78\)\( T^{6} - \)\(39\!\cdots\!93\)\( T^{7} + \)\(84\!\cdots\!01\)\( T^{8} + \)\(17\!\cdots\!74\)\( T^{9} - \)\(12\!\cdots\!20\)\( T^{10} + \)\(12\!\cdots\!26\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} - \)\(14\!\cdots\!07\)\( T^{13} + \)\(18\!\cdots\!78\)\( T^{14} + \)\(19\!\cdots\!91\)\( T^{15} - \)\(12\!\cdots\!78\)\( T^{16} - \)\(13\!\cdots\!51\)\( T^{17} + \)\(19\!\cdots\!40\)\( T^{18} - \)\(44\!\cdots\!63\)\( T^{19} + \)\(34\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} + \)\(54\!\cdots\!90\)\( T^{5} - \)\(10\!\cdots\!33\)\( T^{6} - \)\(70\!\cdots\!89\)\( T^{7} + \)\(11\!\cdots\!67\)\( T^{8} + \)\(32\!\cdots\!04\)\( T^{9} - \)\(10\!\cdots\!12\)\( T^{10} + \)\(27\!\cdots\!04\)\( T^{11} + \)\(80\!\cdots\!67\)\( T^{12} - \)\(42\!\cdots\!89\)\( T^{13} - \)\(51\!\cdots\!33\)\( T^{14} + \)\(23\!\cdots\!90\)\( T^{15} + \)\(25\!\cdots\!98\)\( T^{16} - \)\(13\!\cdots\!46\)\( T^{17} - \)\(91\!\cdots\!83\)\( T^{18} - \)\(30\!\cdots\!03\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 - 13907 T - 3876685544 T^{2} - 77425491657903 T^{3} + 10014688417385231130 T^{4} + \)\(30\!\cdots\!39\)\( T^{5} - \)\(79\!\cdots\!54\)\( T^{6} - \)\(69\!\cdots\!51\)\( T^{7} - \)\(33\!\cdots\!67\)\( T^{8} + \)\(37\!\cdots\!46\)\( T^{9} + \)\(22\!\cdots\!76\)\( T^{10} + \)\(51\!\cdots\!22\)\( T^{11} - \)\(60\!\cdots\!83\)\( T^{12} - \)\(16\!\cdots\!93\)\( T^{13} - \)\(26\!\cdots\!54\)\( T^{14} + \)\(13\!\cdots\!73\)\( T^{15} + \)\(60\!\cdots\!70\)\( T^{16} - \)\(63\!\cdots\!29\)\( T^{17} - \)\(42\!\cdots\!44\)\( T^{18} - \)\(20\!\cdots\!49\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} \)
$71$ \( ( 1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} + \)\(77\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!50\)\( T^{6} + \)\(13\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!09\)\( T^{8} + \)\(12\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} )^{2} \)
$73$ \( ( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!33\)\( T^{6} - \)\(13\!\cdots\!86\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} )^{2} \)
$79$ \( 1 - 29993 T - 5352351629 T^{2} - 358913063028768 T^{3} + 26234825811851125236 T^{4} + \)\(21\!\cdots\!52\)\( T^{5} + \)\(27\!\cdots\!85\)\( T^{6} - \)\(84\!\cdots\!45\)\( T^{7} - \)\(35\!\cdots\!45\)\( T^{8} + \)\(81\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!20\)\( T^{11} - \)\(33\!\cdots\!45\)\( T^{12} - \)\(24\!\cdots\!55\)\( T^{13} + \)\(24\!\cdots\!85\)\( T^{14} + \)\(60\!\cdots\!48\)\( T^{15} + \)\(22\!\cdots\!36\)\( T^{16} - \)\(93\!\cdots\!32\)\( T^{17} - \)\(43\!\cdots\!29\)\( T^{18} - \)\(74\!\cdots\!07\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 - 228951 T + 21403431983 T^{2} - 1202282302650156 T^{3} + 62567029919071222368 T^{4} - \)\(36\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!01\)\( T^{6} + \)\(11\!\cdots\!41\)\( T^{7} - \)\(18\!\cdots\!73\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{9} - \)\(88\!\cdots\!72\)\( T^{10} + \)\(55\!\cdots\!12\)\( T^{11} - \)\(28\!\cdots\!77\)\( T^{12} + \)\(67\!\cdots\!87\)\( T^{13} + \)\(28\!\cdots\!01\)\( T^{14} - \)\(34\!\cdots\!24\)\( T^{15} + \)\(23\!\cdots\!32\)\( T^{16} - \)\(17\!\cdots\!92\)\( T^{17} + \)\(12\!\cdots\!83\)\( T^{18} - \)\(52\!\cdots\!93\)\( T^{19} + \)\(89\!\cdots\!49\)\( T^{20} \)
$89$ \( ( 1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} + \)\(55\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} + \)\(20\!\cdots\!88\)\( T^{7} + \)\(91\!\cdots\!69\)\( T^{8} + \)\(29\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!49\)\( T^{10} )^{2} \)
$97$ \( 1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} - \)\(30\!\cdots\!95\)\( T^{5} + \)\(10\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!29\)\( T^{7} - \)\(20\!\cdots\!15\)\( T^{8} - \)\(65\!\cdots\!06\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} - \)\(56\!\cdots\!42\)\( T^{11} - \)\(15\!\cdots\!35\)\( T^{12} + \)\(13\!\cdots\!97\)\( T^{13} + \)\(56\!\cdots\!20\)\( T^{14} - \)\(14\!\cdots\!15\)\( T^{15} + \)\(39\!\cdots\!90\)\( T^{16} + \)\(77\!\cdots\!49\)\( T^{17} - \)\(52\!\cdots\!38\)\( T^{18} - \)\(10\!\cdots\!37\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} \)
show more
show less