Properties

Label 108.5.d
Level 108
Weight 5
Character orbit d
Rep. character \(\chi_{108}(55,\cdot)\)
Character field \(\Q\)
Dimension 32
Newform subspaces 2
Sturm bound 90
Trace bound 4

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Defining parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(90\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(108, [\chi])\).

Total New Old
Modular forms 78 32 46
Cusp forms 66 32 34
Eisenstein series 12 0 12

Trace form

\( 32q + 14q^{4} + O(q^{10}) \) \( 32q + 14q^{4} - 26q^{10} - 176q^{13} - 118q^{16} + 1122q^{22} + 4368q^{25} + 2154q^{28} - 1016q^{34} + 3280q^{37} - 3542q^{40} - 1788q^{46} - 8800q^{49} - 2948q^{52} - 9092q^{58} - 4400q^{61} + 5738q^{64} + 12522q^{70} + 8320q^{73} - 41472q^{76} - 38396q^{82} - 8032q^{85} + 18774q^{88} + 48084q^{94} + 7936q^{97} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(108, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
108.5.d.a \(16\) \(11.164\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{7}+\cdots)q^{5}+\cdots\)
108.5.d.b \(16\) \(11.164\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{7}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{7}+\beta _{9}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(108, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(108, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 7 T^{2} + 76 T^{4} + 304 T^{6} + 94720 T^{8} + 77824 T^{10} + 4980736 T^{12} + 117440512 T^{14} + 4294967296 T^{16} \))(\( 1 - 14 T^{2} + 76 T^{4} - 992 T^{6} - 2816 T^{8} - 253952 T^{10} + 4980736 T^{12} - 234881024 T^{14} + 4294967296 T^{16} \))
$3$ 1
$5$ (\( ( 1 + 2092 T^{2} + 2149522 T^{4} + 1622274208 T^{6} + 1066877108443 T^{8} + 633700862500000 T^{10} + 327991027832031250 T^{12} + \)\(12\!\cdots\!00\)\( T^{14} + \)\(23\!\cdots\!25\)\( T^{16} )^{2} \))(\( ( 1 + 1816 T^{2} + 1373884 T^{4} + 428961832 T^{6} + 50788147846 T^{8} + 167563215625000 T^{10} + 209638061523437500 T^{12} + \)\(10\!\cdots\!00\)\( T^{14} + \)\(23\!\cdots\!25\)\( T^{16} )^{2} \))
$7$ (\( ( 1 - 9380 T^{2} + 36105250 T^{4} - 73168453568 T^{6} + 129372615567787 T^{8} - 421801574297259968 T^{10} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(17\!\cdots\!80\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 7628 T^{2} + 28548058 T^{4} - 90794922320 T^{6} + 251781425444899 T^{8} - 523414658985258320 T^{10} + \)\(94\!\cdots\!58\)\( T^{12} - \)\(14\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \))
$11$ (\( ( 1 - 48044 T^{2} + 1167571642 T^{4} - 20151290866448 T^{6} + 304446182867375107 T^{8} - \)\(43\!\cdots\!88\)\( T^{10} + \)\(53\!\cdots\!62\)\( T^{12} - \)\(47\!\cdots\!04\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 66488 T^{2} + 2449218172 T^{4} - 58905964133768 T^{6} + 1015821535494509830 T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(11\!\cdots\!92\)\( T^{12} - \)\(65\!\cdots\!08\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} )^{2} \))
$13$ (\( ( 1 + 88 T + 63508 T^{2} + 5077384 T^{3} + 2650180198 T^{4} + 145015164424 T^{5} + 51805426629268 T^{6} + 2050231490778328 T^{7} + 665416609183179841 T^{8} )^{4} \))(\( ( 1 - 44 T + 67090 T^{2} - 966368 T^{3} + 2192145307 T^{4} - 27600436448 T^{5} + 54727374071890 T^{6} - 1025115745389164 T^{7} + 665416609183179841 T^{8} )^{4} \))
$17$ (\( ( 1 + 239464 T^{2} + 41438060380 T^{4} + 5059782297489112 T^{6} + \)\(47\!\cdots\!02\)\( T^{8} + \)\(35\!\cdots\!92\)\( T^{10} + \)\(20\!\cdots\!80\)\( T^{12} + \)\(81\!\cdots\!44\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 + 422296 T^{2} + 87803746876 T^{4} + 11972376766816936 T^{6} + \)\(11\!\cdots\!30\)\( T^{8} + \)\(83\!\cdots\!76\)\( T^{10} + \)\(42\!\cdots\!56\)\( T^{12} + \)\(14\!\cdots\!16\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} )^{2} \))
$19$ (\( ( 1 - 680120 T^{2} + 233557156252 T^{4} - 51394020863660552 T^{6} + \)\(79\!\cdots\!98\)\( T^{8} - \)\(87\!\cdots\!32\)\( T^{10} + \)\(67\!\cdots\!12\)\( T^{12} - \)\(33\!\cdots\!20\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 298556 T^{2} + 58331731402 T^{4} - 10017587005489136 T^{6} + \)\(14\!\cdots\!39\)\( T^{8} - \)\(17\!\cdots\!76\)\( T^{10} + \)\(16\!\cdots\!62\)\( T^{12} - \)\(14\!\cdots\!76\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} )^{2} \))
$23$ (\( ( 1 - 675608 T^{2} + 314533671388 T^{4} - 115630955446337192 T^{6} + \)\(35\!\cdots\!18\)\( T^{8} - \)\(90\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!68\)\( T^{12} - \)\(32\!\cdots\!28\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 1184696 T^{2} + 509745640444 T^{4} - 84604026939000968 T^{6} + \)\(71\!\cdots\!22\)\( T^{8} - \)\(66\!\cdots\!08\)\( T^{10} + \)\(31\!\cdots\!84\)\( T^{12} - \)\(56\!\cdots\!36\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} )^{2} \))
$29$ (\( ( 1 + 2998744 T^{2} + 4863922457692 T^{4} + 5416371986440228072 T^{6} + \)\(44\!\cdots\!14\)\( T^{8} + \)\(27\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!32\)\( T^{12} + \)\(37\!\cdots\!64\)\( T^{14} + \)\(62\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 + 1223560 T^{2} + 603752953756 T^{4} - 126384872678073416 T^{6} - \)\(34\!\cdots\!30\)\( T^{8} - \)\(63\!\cdots\!76\)\( T^{10} + \)\(15\!\cdots\!76\)\( T^{12} + \)\(15\!\cdots\!60\)\( T^{14} + \)\(62\!\cdots\!41\)\( T^{16} )^{2} \))
$31$ (\( ( 1 - 1328756 T^{2} + 2683287954034 T^{4} - 2424947225563311392 T^{6} + \)\(29\!\cdots\!59\)\( T^{8} - \)\(20\!\cdots\!72\)\( T^{10} + \)\(19\!\cdots\!54\)\( T^{12} - \)\(82\!\cdots\!76\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 4495112 T^{2} + 10589743228444 T^{4} - 16346503421024387384 T^{6} + \)\(17\!\cdots\!66\)\( T^{8} - \)\(13\!\cdots\!44\)\( T^{10} + \)\(77\!\cdots\!64\)\( T^{12} - \)\(27\!\cdots\!52\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{16} )^{2} \))
$37$ (\( ( 1 - 800 T + 1852732 T^{2} - 3784975328 T^{3} + 1716048721222 T^{4} - 7093653145699808 T^{5} + 6507683083621962172 T^{6} - \)\(52\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{4} \))(\( ( 1 - 20 T + 5679442 T^{2} - 534101024 T^{3} + 14604995159131 T^{4} - 1000991309240864 T^{5} + 19948923334735992082 T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{4} \))
$41$ (\( ( 1 + 14045080 T^{2} + 100482710328412 T^{4} + \)\(46\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!54\)\( T^{8} + \)\(37\!\cdots\!64\)\( T^{10} + \)\(64\!\cdots\!92\)\( T^{12} + \)\(71\!\cdots\!80\)\( T^{14} + \)\(40\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 + 8498440 T^{2} + 39457953574684 T^{4} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(47\!\cdots\!42\)\( T^{8} + \)\(11\!\cdots\!28\)\( T^{10} + \)\(25\!\cdots\!44\)\( T^{12} + \)\(43\!\cdots\!40\)\( T^{14} + \)\(40\!\cdots\!81\)\( T^{16} )^{2} \))
$43$ (\( ( 1 - 11990648 T^{2} + 69115070774812 T^{4} - \)\(29\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(34\!\cdots\!24\)\( T^{10} + \)\(94\!\cdots\!12\)\( T^{12} - \)\(19\!\cdots\!48\)\( T^{14} + \)\(18\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 16979720 T^{2} + 140301823474588 T^{4} - \)\(77\!\cdots\!28\)\( T^{6} + \)\(30\!\cdots\!14\)\( T^{8} - \)\(90\!\cdots\!28\)\( T^{10} + \)\(19\!\cdots\!88\)\( T^{12} - \)\(27\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!01\)\( T^{16} )^{2} \))
$47$ (\( ( 1 - 27701336 T^{2} + 374201000701276 T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(75\!\cdots\!28\)\( T^{10} + \)\(21\!\cdots\!96\)\( T^{12} - \)\(37\!\cdots\!16\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 11004728 T^{2} + 98757031754236 T^{4} - \)\(67\!\cdots\!20\)\( T^{6} + \)\(36\!\cdots\!06\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{10} + \)\(55\!\cdots\!56\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} )^{2} \))
$53$ (\( ( 1 + 35837884 T^{2} + 582188420839618 T^{4} + \)\(60\!\cdots\!16\)\( T^{6} + \)\(50\!\cdots\!79\)\( T^{8} + \)\(37\!\cdots\!76\)\( T^{10} + \)\(22\!\cdots\!78\)\( T^{12} + \)\(86\!\cdots\!04\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 + 25978312 T^{2} + 440128249758748 T^{4} + \)\(51\!\cdots\!52\)\( T^{6} + \)\(46\!\cdots\!86\)\( T^{8} + \)\(32\!\cdots\!72\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} + \)\(62\!\cdots\!72\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} )^{2} \))
$59$ (\( ( 1 - 70312136 T^{2} + 2419955263154716 T^{4} - \)\(51\!\cdots\!68\)\( T^{6} + \)\(75\!\cdots\!74\)\( T^{8} - \)\(76\!\cdots\!28\)\( T^{10} + \)\(52\!\cdots\!56\)\( T^{12} - \)\(22\!\cdots\!96\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 - 31826744 T^{2} + 538590143907196 T^{4} - \)\(92\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!74\)\( T^{8} - \)\(13\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!84\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} )^{2} \))
$61$ (\( ( 1 + 688 T + 35342980 T^{2} + 28994874640 T^{3} + 632810106882118 T^{4} + 401458424080372240 T^{5} + \)\(67\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{4} \))(\( ( 1 + 412 T + 9607138 T^{2} + 3267599488 T^{3} + 216368249231659 T^{4} + 45242662962529408 T^{5} + \)\(18\!\cdots\!78\)\( T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{4} \))
$67$ (\( ( 1 - 29579000 T^{2} + 1274223737389084 T^{4} - \)\(30\!\cdots\!28\)\( T^{6} + \)\(73\!\cdots\!58\)\( T^{8} - \)\(12\!\cdots\!48\)\( T^{10} + \)\(21\!\cdots\!04\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 38213180 T^{2} + 1702311028453834 T^{4} - \)\(38\!\cdots\!28\)\( T^{6} + \)\(99\!\cdots\!27\)\( T^{8} - \)\(15\!\cdots\!48\)\( T^{10} + \)\(28\!\cdots\!54\)\( T^{12} - \)\(25\!\cdots\!80\)\( T^{14} + \)\(27\!\cdots\!61\)\( T^{16} )^{2} \))
$71$ (\( ( 1 - 86649128 T^{2} + 3567527319446236 T^{4} - \)\(11\!\cdots\!36\)\( T^{6} + \)\(32\!\cdots\!90\)\( T^{8} - \)\(74\!\cdots\!96\)\( T^{10} + \)\(14\!\cdots\!56\)\( T^{12} - \)\(23\!\cdots\!68\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 102214088 T^{2} + 3903098982427036 T^{4} - \)\(62\!\cdots\!96\)\( T^{6} + \)\(63\!\cdots\!58\)\( T^{8} - \)\(40\!\cdots\!56\)\( T^{10} + \)\(16\!\cdots\!56\)\( T^{12} - \)\(27\!\cdots\!28\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} )^{2} \))
$73$ (\( ( 1 - 2060 T + 55108594 T^{2} - 101224244576 T^{3} + 2192760100160827 T^{4} - 2874590492512190816 T^{5} + \)\(44\!\cdots\!14\)\( T^{6} - \)\(47\!\cdots\!60\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{4} \))(\( ( 1 - 20 T + 67167226 T^{2} + 662878096 T^{3} + 2724353301452611 T^{4} + 18824571923829136 T^{5} + \)\(54\!\cdots\!06\)\( T^{6} - \)\(45\!\cdots\!20\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{4} \))
$79$ (\( ( 1 - 162885800 T^{2} + 12409200314583772 T^{4} - \)\(63\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!06\)\( T^{8} - \)\(96\!\cdots\!96\)\( T^{10} + \)\(28\!\cdots\!12\)\( T^{12} - \)\(56\!\cdots\!00\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 164533964 T^{2} + 15212503116962266 T^{4} - \)\(95\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!71\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{10} + \)\(35\!\cdots\!86\)\( T^{12} - \)\(57\!\cdots\!84\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} )^{2} \))
$83$ (\( ( 1 - 140155436 T^{2} + 8287556309932090 T^{4} - \)\(42\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!03\)\( T^{8} - \)\(95\!\cdots\!72\)\( T^{10} + \)\(42\!\cdots\!90\)\( T^{12} - \)\(16\!\cdots\!56\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 185955656 T^{2} + 18958921421058076 T^{4} - \)\(13\!\cdots\!16\)\( T^{6} + \)\(73\!\cdots\!26\)\( T^{8} - \)\(30\!\cdots\!56\)\( T^{10} + \)\(96\!\cdots\!56\)\( T^{12} - \)\(21\!\cdots\!76\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \))
$89$ (\( ( 1 + 257253400 T^{2} + 38619499284608860 T^{4} + \)\(38\!\cdots\!44\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{8} + \)\(15\!\cdots\!64\)\( T^{10} + \)\(59\!\cdots\!60\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 + 103821208 T^{2} + 12140769019811644 T^{4} + \)\(81\!\cdots\!28\)\( T^{6} + \)\(66\!\cdots\!06\)\( T^{8} + \)\(32\!\cdots\!68\)\( T^{10} + \)\(18\!\cdots\!84\)\( T^{12} + \)\(63\!\cdots\!28\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} )^{2} \))
$97$ (\( ( 1 + 1732 T + 297359818 T^{2} + 428249450896 T^{3} + 37002869299122643 T^{4} + 37912615976467685776 T^{5} + \)\(23\!\cdots\!98\)\( T^{6} + \)\(12\!\cdots\!12\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{4} \))(\( ( 1 - 3716 T + 259176202 T^{2} - 799969063952 T^{3} + 30273937496959507 T^{4} - 70820686053913578512 T^{5} + \)\(20\!\cdots\!22\)\( T^{6} - \)\(25\!\cdots\!56\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{4} \))
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