Properties

Label 18.5
Level 18
Weight 5
Dimension 8
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 90
Trace bound 0

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Defining parameters

Level: \( N \) = \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(90\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(18))\).

Total New Old
Modular forms 44 8 36
Cusp forms 28 8 20
Eisenstein series 16 0 16

Trace form

\( 8q + 6q^{3} + 32q^{4} + 18q^{5} + 48q^{6} - 26q^{7} - 78q^{9} + O(q^{10}) \) \( 8q + 6q^{3} + 32q^{4} + 18q^{5} + 48q^{6} - 26q^{7} - 78q^{9} - 720q^{11} - 144q^{12} + 10q^{13} + 288q^{14} + 1134q^{15} - 256q^{16} - 384q^{18} + 100q^{19} + 144q^{20} + 438q^{21} + 336q^{22} + 1278q^{23} + 384q^{24} + 794q^{25} - 1296q^{27} - 416q^{28} - 1854q^{29} - 3456q^{30} - 1478q^{31} - 3384q^{33} - 96q^{34} + 1056q^{36} - 32q^{37} + 6768q^{38} + 5274q^{39} - 36q^{41} + 2592q^{42} - 68q^{43} + 3402q^{45} + 2112q^{46} + 2214q^{47} - 1536q^{48} + 2442q^{49} - 15552q^{50} - 12006q^{51} - 80q^{52} + 7056q^{54} - 3996q^{55} + 2304q^{56} + 10902q^{57} - 2400q^{58} + 9108q^{59} + 6480q^{60} - 4478q^{61} - 6654q^{63} - 4096q^{64} - 22554q^{65} - 19872q^{66} + 7504q^{67} - 11088q^{68} - 5994q^{69} + 6048q^{70} + 5376q^{72} + 20716q^{73} + 15264q^{74} + 16590q^{75} + 400q^{76} + 34434q^{77} + 24096q^{78} - 6050q^{79} - 21150q^{81} + 1152q^{82} - 3834q^{83} - 9600q^{84} - 16092q^{85} - 12528q^{86} + 10170q^{87} - 2688q^{88} + 2592q^{90} - 45868q^{91} + 10224q^{92} - 10926q^{93} + 672q^{94} + 20880q^{95} + 31336q^{97} - 22338q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(18))\)

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
18.5.b \(\chi_{18}(17, \cdot)\) None 0 1
18.5.d \(\chi_{18}(5, \cdot)\) 18.5.d.a 8 2

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(18))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(18)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T^{2} + 64 T^{4} )^{2} \)
$3$ \( 1 - 6 T + 57 T^{2} + 162 T^{3} + 4212 T^{4} + 13122 T^{5} + 373977 T^{6} - 3188646 T^{7} + 43046721 T^{8} \)
$5$ \( 1 - 18 T + 1015 T^{2} - 16326 T^{3} + 558925 T^{4} - 5405940 T^{5} - 314464142 T^{6} + 3511352520 T^{7} - 306662326214 T^{8} + 2194595325000 T^{9} - 122837555468750 T^{10} - 1319809570312500 T^{11} + 85285186767578125 T^{12} - 1556968688964843750 T^{13} + 60498714447021484375 T^{14} - \)\(67\!\cdots\!50\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 + 26 T - 5685 T^{2} + 19510 T^{3} + 19835513 T^{4} - 334899084 T^{5} - 45037679042 T^{6} + 451520935736 T^{7} + 90278690161986 T^{8} + 1084101766702136 T^{9} - 259633257179000642 T^{10} - 4635434404995823884 T^{11} + \)\(65\!\cdots\!13\)\( T^{12} + \)\(15\!\cdots\!10\)\( T^{13} - \)\(10\!\cdots\!85\)\( T^{14} + \)\(11\!\cdots\!26\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + 720 T + 274582 T^{2} + 73283040 T^{3} + 15179596969 T^{4} + 2592718024176 T^{5} + 383108185780198 T^{6} + 51080491982360160 T^{7} + 6354846385614434932 T^{8} + \)\(74\!\cdots\!60\)\( T^{9} + \)\(82\!\cdots\!38\)\( T^{10} + \)\(81\!\cdots\!96\)\( T^{11} + \)\(69\!\cdots\!09\)\( T^{12} + \)\(49\!\cdots\!40\)\( T^{13} + \)\(27\!\cdots\!62\)\( T^{14} + \)\(10\!\cdots\!20\)\( T^{15} + \)\(21\!\cdots\!21\)\( T^{16} \)
$13$ \( 1 - 10 T - 34545 T^{2} - 8013734 T^{3} + 152683301 T^{4} + 264027759132 T^{5} + 34691703939994 T^{6} - 3939272844487792 T^{7} - 880372532161082934 T^{8} - \)\(11\!\cdots\!12\)\( T^{9} + \)\(28\!\cdots\!74\)\( T^{10} + \)\(61\!\cdots\!92\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} - \)\(15\!\cdots\!34\)\( T^{13} - \)\(18\!\cdots\!45\)\( T^{14} - \)\(15\!\cdots\!10\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 249218 T^{2} + 30482460289 T^{4} - 3595961342423810 T^{6} + \)\(36\!\cdots\!56\)\( T^{8} - \)\(25\!\cdots\!10\)\( T^{10} + \)\(14\!\cdots\!09\)\( T^{12} - \)\(84\!\cdots\!78\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} \)
$19$ \( ( 1 - 50 T + 226153 T^{2} - 12428354 T^{3} + 33132078964 T^{4} - 1619675521634 T^{5} + 3840883732411273 T^{6} - 110665745953308050 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \)
$23$ \( 1 - 1278 T + 1655467 T^{2} - 1419907842 T^{3} + 1186049018761 T^{4} - 817434220578732 T^{5} + 550327618738318606 T^{6} - \)\(32\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!30\)\( T^{8} - \)\(89\!\cdots\!96\)\( T^{9} + \)\(43\!\cdots\!86\)\( T^{10} - \)\(17\!\cdots\!72\)\( T^{11} + \)\(72\!\cdots\!21\)\( T^{12} - \)\(24\!\cdots\!42\)\( T^{13} + \)\(79\!\cdots\!47\)\( T^{14} - \)\(17\!\cdots\!18\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 + 1854 T + 3116071 T^{2} + 3652934346 T^{3} + 3878113467181 T^{4} + 4083633101435148 T^{5} + 4044828167734324498 T^{6} + \)\(39\!\cdots\!28\)\( T^{7} + \)\(34\!\cdots\!30\)\( T^{8} + \)\(28\!\cdots\!68\)\( T^{9} + \)\(20\!\cdots\!78\)\( T^{10} + \)\(14\!\cdots\!68\)\( T^{11} + \)\(97\!\cdots\!01\)\( T^{12} + \)\(64\!\cdots\!46\)\( T^{13} + \)\(39\!\cdots\!51\)\( T^{14} + \)\(16\!\cdots\!94\)\( T^{15} + \)\(62\!\cdots\!41\)\( T^{16} \)
$31$ \( 1 + 1478 T - 273609 T^{2} - 14717462 T^{3} + 1239820588133 T^{4} - 59691395608740 T^{5} - 148444619433025670 T^{6} - \)\(17\!\cdots\!40\)\( T^{7} - \)\(10\!\cdots\!22\)\( T^{8} - \)\(16\!\cdots\!40\)\( T^{9} - \)\(12\!\cdots\!70\)\( T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(90\!\cdots\!73\)\( T^{12} - \)\(98\!\cdots\!62\)\( T^{13} - \)\(16\!\cdots\!89\)\( T^{14} + \)\(84\!\cdots\!98\)\( T^{15} + \)\(52\!\cdots\!61\)\( T^{16} \)
$37$ \( ( 1 + 16 T + 5152060 T^{2} + 1816976752 T^{3} + 11981316770374 T^{4} + 3405306966505072 T^{5} + 18096504895368227260 T^{6} + \)\(10\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \)
$41$ \( 1 + 36 T + 10981366 T^{2} + 395313624 T^{3} + 74491912981249 T^{4} + 2021455678408920 T^{5} + \)\(33\!\cdots\!50\)\( T^{6} + \)\(74\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!68\)\( T^{8} + \)\(21\!\cdots\!60\)\( T^{9} + \)\(26\!\cdots\!50\)\( T^{10} + \)\(45\!\cdots\!20\)\( T^{11} + \)\(47\!\cdots\!09\)\( T^{12} + \)\(71\!\cdots\!24\)\( T^{13} + \)\(55\!\cdots\!26\)\( T^{14} + \)\(51\!\cdots\!56\)\( T^{15} + \)\(40\!\cdots\!81\)\( T^{16} \)
$43$ \( 1 + 68 T - 10748538 T^{2} + 2055805384 T^{3} + 65375958867329 T^{4} - 16226764390392264 T^{5} - \)\(28\!\cdots\!82\)\( T^{6} + \)\(26\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!76\)\( T^{8} + \)\(91\!\cdots\!32\)\( T^{9} - \)\(33\!\cdots\!82\)\( T^{10} - \)\(64\!\cdots\!64\)\( T^{11} + \)\(89\!\cdots\!29\)\( T^{12} + \)\(96\!\cdots\!84\)\( T^{13} - \)\(17\!\cdots\!38\)\( T^{14} + \)\(37\!\cdots\!68\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 2214 T + 16322779 T^{2} - 32521107258 T^{3} + 136095840713065 T^{4} - 267869759907619980 T^{5} + \)\(91\!\cdots\!74\)\( T^{6} - \)\(16\!\cdots\!08\)\( T^{7} + \)\(50\!\cdots\!62\)\( T^{8} - \)\(82\!\cdots\!48\)\( T^{9} + \)\(21\!\cdots\!14\)\( T^{10} - \)\(31\!\cdots\!80\)\( T^{11} + \)\(77\!\cdots\!65\)\( T^{12} - \)\(89\!\cdots\!58\)\( T^{13} + \)\(22\!\cdots\!99\)\( T^{14} - \)\(14\!\cdots\!54\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 47149064 T^{2} + 1030375014657436 T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(86\!\cdots\!20\)\( T^{10} + \)\(39\!\cdots\!56\)\( T^{12} - \)\(11\!\cdots\!84\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - 9108 T + 65431078 T^{2} - 344092862520 T^{3} + 1463362600081057 T^{4} - 4890955357569963960 T^{5} + \)\(14\!\cdots\!50\)\( T^{6} - \)\(36\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!92\)\( T^{8} - \)\(44\!\cdots\!72\)\( T^{9} + \)\(21\!\cdots\!50\)\( T^{10} - \)\(87\!\cdots\!60\)\( T^{11} + \)\(31\!\cdots\!37\)\( T^{12} - \)\(89\!\cdots\!20\)\( T^{13} + \)\(20\!\cdots\!58\)\( T^{14} - \)\(34\!\cdots\!68\)\( T^{15} + \)\(46\!\cdots\!81\)\( T^{16} \)
$61$ \( 1 + 4478 T - 125937 T^{2} - 125289669758 T^{3} - 568176190027387 T^{4} - 742254823045377828 T^{5} + \)\(27\!\cdots\!38\)\( T^{6} + \)\(22\!\cdots\!92\)\( T^{7} + \)\(73\!\cdots\!38\)\( T^{8} + \)\(30\!\cdots\!72\)\( T^{9} + \)\(52\!\cdots\!78\)\( T^{10} - \)\(19\!\cdots\!88\)\( T^{11} - \)\(20\!\cdots\!07\)\( T^{12} - \)\(63\!\cdots\!58\)\( T^{13} - \)\(88\!\cdots\!17\)\( T^{14} + \)\(43\!\cdots\!18\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \)
$67$ \( 1 - 7504 T - 22347594 T^{2} + 226068030400 T^{3} + 738296226968777 T^{4} - 4521766862440091376 T^{5} - \)\(18\!\cdots\!90\)\( T^{6} + \)\(18\!\cdots\!56\)\( T^{7} + \)\(57\!\cdots\!20\)\( T^{8} + \)\(38\!\cdots\!76\)\( T^{9} - \)\(76\!\cdots\!90\)\( T^{10} - \)\(37\!\cdots\!36\)\( T^{11} + \)\(12\!\cdots\!37\)\( T^{12} + \)\(75\!\cdots\!00\)\( T^{13} - \)\(14\!\cdots\!74\)\( T^{14} - \)\(10\!\cdots\!64\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$71$ \( 1 - 125290160 T^{2} + 8106610376011420 T^{4} - \)\(34\!\cdots\!88\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(21\!\cdots\!68\)\( T^{10} + \)\(33\!\cdots\!20\)\( T^{12} - \)\(33\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \)
$73$ \( ( 1 - 10358 T + 106803217 T^{2} - 769838088062 T^{3} + 4439859073965124 T^{4} - 21862047555763898942 T^{5} + \)\(86\!\cdots\!77\)\( T^{6} - \)\(23\!\cdots\!18\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \)
$79$ \( 1 + 6050 T - 42146805 T^{2} + 267987207070 T^{3} + 3321952223354537 T^{4} - 14945323617952326060 T^{5} + \)\(48\!\cdots\!70\)\( T^{6} + \)\(67\!\cdots\!20\)\( T^{7} - \)\(27\!\cdots\!02\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{9} + \)\(73\!\cdots\!70\)\( T^{10} - \)\(88\!\cdots\!60\)\( T^{11} + \)\(76\!\cdots\!77\)\( T^{12} + \)\(24\!\cdots\!70\)\( T^{13} - \)\(14\!\cdots\!05\)\( T^{14} + \)\(82\!\cdots\!50\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 + 3834 T + 163867483 T^{2} + 609481897254 T^{3} + 15130998622630921 T^{4} + 49888358310056197140 T^{5} + \)\(10\!\cdots\!30\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(54\!\cdots\!10\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(23\!\cdots\!30\)\( T^{10} + \)\(53\!\cdots\!40\)\( T^{11} + \)\(76\!\cdots\!01\)\( T^{12} + \)\(14\!\cdots\!54\)\( T^{13} + \)\(18\!\cdots\!43\)\( T^{14} + \)\(20\!\cdots\!94\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - 206196488 T^{2} + 19314405141094684 T^{4} - \)\(14\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!86\)\( T^{8} - \)\(57\!\cdots\!60\)\( T^{10} + \)\(29\!\cdots\!24\)\( T^{12} - \)\(12\!\cdots\!08\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - 31336 T + 307771782 T^{2} - 2149219106864 T^{3} + 52690258168920329 T^{4} - \)\(68\!\cdots\!16\)\( T^{5} + \)\(38\!\cdots\!30\)\( T^{6} - \)\(46\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!04\)\( T^{8} - \)\(40\!\cdots\!00\)\( T^{9} + \)\(30\!\cdots\!30\)\( T^{10} - \)\(47\!\cdots\!56\)\( T^{11} + \)\(32\!\cdots\!09\)\( T^{12} - \)\(11\!\cdots\!64\)\( T^{13} + \)\(14\!\cdots\!42\)\( T^{14} - \)\(13\!\cdots\!96\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
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