Properties

Label 23.9
Level 23
Weight 9
Dimension 165
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 396
Trace bound 1

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Defining parameters

Level: \( N \) = \( 23 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(396\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(23))\).

Total New Old
Modular forms 187 187 0
Cusp forms 165 165 0
Eisenstein series 22 22 0

Trace form

\( 165q - 11q^{2} - 11q^{3} - 11q^{4} - 11q^{5} - 11q^{6} - 11q^{7} - 11q^{8} - 11q^{9} + O(q^{10}) \) \( 165q - 11q^{2} - 11q^{3} - 11q^{4} - 11q^{5} - 11q^{6} - 11q^{7} - 11q^{8} - 11q^{9} - 11q^{10} - 11q^{11} - 11q^{12} - 11q^{13} - 11q^{14} - 130218q^{15} - 261899q^{16} + 246004q^{17} + 872949q^{18} - 113894q^{19} - 1571339q^{20} - 1081058q^{21} + 957319q^{23} + 4553450q^{24} + 1166275q^{25} - 177419q^{26} - 2952356q^{27} - 6167051q^{28} - 1109504q^{29} + 2511861q^{30} + 2936230q^{31} + 5955829q^{32} - 5636796q^{33} + 6374511q^{34} - 10230011q^{35} - 11186516q^{36} + 9285749q^{37} + 22442794q^{38} + 14170453q^{39} + 4234989q^{40} - 5773163q^{41} - 51900761q^{42} - 21835627q^{43} - 24763398q^{44} + 32424381q^{46} + 29161418q^{47} + 68005421q^{48} + 41607445q^{49} + 18046864q^{50} - 14740715q^{51} - 100920061q^{52} - 28488779q^{53} - 57299715q^{54} - 69529317q^{55} - 99295328q^{56} + 72285400q^{57} + 194073374q^{58} + 103041697q^{59} + 67737659q^{60} + 47508681q^{61} - 1951499q^{62} - 62854836q^{63} - 171573259q^{64} - 173044652q^{65} - 172762656q^{66} - 25488397q^{67} + 69676134q^{69} + 208147434q^{70} + 166935604q^{71} + 406513162q^{72} + 53013719q^{73} + 329575444q^{74} - 58151258q^{75} - 610693402q^{76} - 497077394q^{77} - 556884141q^{78} - 108193503q^{79} + 214101118q^{80} + 811655801q^{81} + 568273695q^{82} + 421863970q^{83} + 585479675q^{84} - 297208417q^{85} - 1051470365q^{86} - 819724235q^{87} - 546014403q^{88} - 331520123q^{89} - 592624846q^{90} + 300035626q^{92} + 760407626q^{93} + 763896320q^{94} + 1148773186q^{95} + 1544287096q^{96} - 137375018q^{97} - 441866579q^{98} - 653311670q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(23))\)

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
23.9.b \(\chi_{23}(22, \cdot)\) 23.9.b.a 3 1
23.9.b.b 12
23.9.d \(\chi_{23}(5, \cdot)\) 23.9.d.a 150 10

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 1951 T^{3} + 16777216 T^{6} \))(\( ( 1 + 4 T + 686 T^{2} + 1872 T^{3} + 260512 T^{4} + 395648 T^{5} + 71703424 T^{6} + 101285888 T^{7} + 17072914432 T^{8} + 31406948352 T^{9} + 2946347565056 T^{10} + 4398046511104 T^{11} + 281474976710656 T^{12} )^{2} \))
$3$ (\( 1 + 1062686 T^{3} + 282429536481 T^{6} \))(\( ( 1 + 36 T + 18858 T^{2} + 150336 T^{3} + 194326002 T^{4} + 582082848 T^{5} + 1546769185470 T^{6} + 3819045565728 T^{7} + 8365097191139442 T^{8} + 42459326796407616 T^{9} + 34944254721368017578 T^{10} + \)\(43\!\cdots\!36\)\( T^{11} + \)\(79\!\cdots\!61\)\( T^{12} )^{2} \))
$5$ (\( ( 1 - 625 T )^{3}( 1 + 625 T )^{3} \))(\( 1 - 1252944 T^{2} + 1158086837718 T^{4} - 786015312888325360 T^{6} + \)\(44\!\cdots\!75\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{10} + \)\(89\!\cdots\!00\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!75\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{18} + \)\(62\!\cdots\!50\)\( T^{20} - \)\(10\!\cdots\!00\)\( T^{22} + \)\(12\!\cdots\!25\)\( T^{24} \))
$7$ (\( ( 1 - 2401 T )^{3}( 1 + 2401 T )^{3} \))(\( 1 - 28683480 T^{2} + 464926261176678 T^{4} - \)\(54\!\cdots\!36\)\( T^{6} + \)\(49\!\cdots\!51\)\( T^{8} - \)\(37\!\cdots\!64\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{12} - \)\(12\!\cdots\!64\)\( T^{14} + \)\(55\!\cdots\!51\)\( T^{16} - \)\(19\!\cdots\!36\)\( T^{18} + \)\(56\!\cdots\!78\)\( T^{20} - \)\(11\!\cdots\!80\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \))
$11$ (\( ( 1 - 14641 T )^{3}( 1 + 14641 T )^{3} \))(\( 1 - 1090255224 T^{2} + 648198991532402886 T^{4} - \)\(27\!\cdots\!80\)\( T^{6} + \)\(90\!\cdots\!55\)\( T^{8} - \)\(25\!\cdots\!52\)\( T^{10} + \)\(58\!\cdots\!48\)\( T^{12} - \)\(11\!\cdots\!72\)\( T^{14} + \)\(19\!\cdots\!55\)\( T^{16} - \)\(26\!\cdots\!80\)\( T^{18} + \)\(28\!\cdots\!26\)\( T^{20} - \)\(22\!\cdots\!24\)\( T^{22} + \)\(94\!\cdots\!61\)\( T^{24} \))
$13$ (\( 1 - 25363320370274 T^{3} + \)\(54\!\cdots\!61\)\( T^{6} \))(\( ( 1 - 16848 T + 2306491824 T^{2} - 29722304486212 T^{3} + 3362430981673413072 T^{4} - \)\(32\!\cdots\!92\)\( T^{5} + \)\(30\!\cdots\!58\)\( T^{6} - \)\(26\!\cdots\!32\)\( T^{7} + \)\(22\!\cdots\!52\)\( T^{8} - \)\(16\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!44\)\( T^{10} - \)\(60\!\cdots\!48\)\( T^{11} + \)\(29\!\cdots\!21\)\( T^{12} )^{2} \))
$17$ (\( ( 1 - 83521 T )^{3}( 1 + 83521 T )^{3} \))(\( 1 - 7587348816 T^{2} + \)\(13\!\cdots\!74\)\( T^{4} - \)\(62\!\cdots\!24\)\( T^{6} + \)\(27\!\cdots\!19\)\( T^{8} - \)\(29\!\cdots\!20\)\( T^{10} - \)\(14\!\cdots\!08\)\( T^{12} - \)\(14\!\cdots\!20\)\( T^{14} + \)\(65\!\cdots\!59\)\( T^{16} - \)\(71\!\cdots\!84\)\( T^{18} + \)\(76\!\cdots\!54\)\( T^{20} - \)\(20\!\cdots\!16\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))
$19$ (\( ( 1 - 130321 T )^{3}( 1 + 130321 T )^{3} \))(\( 1 - 63712758540 T^{2} + \)\(19\!\cdots\!06\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(82\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!52\)\( T^{10} + \)\(37\!\cdots\!80\)\( T^{12} - \)\(54\!\cdots\!12\)\( T^{14} + \)\(68\!\cdots\!95\)\( T^{16} - \)\(98\!\cdots\!20\)\( T^{18} + \)\(13\!\cdots\!26\)\( T^{20} - \)\(12\!\cdots\!40\)\( T^{22} + \)\(57\!\cdots\!81\)\( T^{24} \))
$23$ (\( ( 1 - 279841 T )^{3} \))(\( 1 + 634540 T + 382644867050 T^{2} + 148402364888283324 T^{3} + \)\(59\!\cdots\!59\)\( T^{4} + \)\(18\!\cdots\!16\)\( T^{5} + \)\(57\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!96\)\( T^{7} + \)\(36\!\cdots\!99\)\( T^{8} + \)\(71\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!50\)\( T^{10} + \)\(18\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!81\)\( T^{12} \))
$29$ (\( 1 + 647932355939762206 T^{3} + \)\(12\!\cdots\!81\)\( T^{6} \))(\( ( 1 - 1591748 T + 2758044324752 T^{2} - 2767871314362870492 T^{3} + \)\(29\!\cdots\!04\)\( T^{4} - \)\(21\!\cdots\!76\)\( T^{5} + \)\(17\!\cdots\!74\)\( T^{6} - \)\(10\!\cdots\!36\)\( T^{7} + \)\(72\!\cdots\!84\)\( T^{8} - \)\(34\!\cdots\!52\)\( T^{9} + \)\(17\!\cdots\!32\)\( T^{10} - \)\(49\!\cdots\!48\)\( T^{11} + \)\(15\!\cdots\!61\)\( T^{12} )^{2} \))
$31$ (\( 1 - 1237087799571624194 T^{3} + \)\(62\!\cdots\!21\)\( T^{6} \))(\( ( 1 + 1079544 T + 4272246529746 T^{2} + 3758401203560198984 T^{3} + \)\(81\!\cdots\!10\)\( T^{4} + \)\(57\!\cdots\!40\)\( T^{5} + \)\(89\!\cdots\!10\)\( T^{6} + \)\(48\!\cdots\!40\)\( T^{7} + \)\(59\!\cdots\!10\)\( T^{8} + \)\(23\!\cdots\!64\)\( T^{9} + \)\(22\!\cdots\!06\)\( T^{10} + \)\(48\!\cdots\!44\)\( T^{11} + \)\(38\!\cdots\!41\)\( T^{12} )^{2} \))
$37$ (\( ( 1 - 1874161 T )^{3}( 1 + 1874161 T )^{3} \))(\( 1 - 2560647284976 T^{2} + \)\(29\!\cdots\!54\)\( T^{4} - \)\(97\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!39\)\( T^{8} - \)\(23\!\cdots\!60\)\( T^{10} + \)\(57\!\cdots\!12\)\( T^{12} - \)\(28\!\cdots\!60\)\( T^{14} + \)\(66\!\cdots\!59\)\( T^{16} - \)\(18\!\cdots\!04\)\( T^{18} + \)\(67\!\cdots\!94\)\( T^{20} - \)\(73\!\cdots\!76\)\( T^{22} + \)\(35\!\cdots\!41\)\( T^{24} \))
$41$ (\( 1 - 12576527614080568514 T^{3} + \)\(50\!\cdots\!61\)\( T^{6} \))(\( ( 1 - 2241188 T + 16519077369968 T^{2} - 12193422124448170236 T^{3} + \)\(17\!\cdots\!40\)\( T^{4} - \)\(23\!\cdots\!60\)\( T^{5} + \)\(20\!\cdots\!50\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!40\)\( T^{8} - \)\(62\!\cdots\!96\)\( T^{9} + \)\(67\!\cdots\!08\)\( T^{10} - \)\(72\!\cdots\!88\)\( T^{11} + \)\(25\!\cdots\!21\)\( T^{12} )^{2} \))
$43$ (\( ( 1 - 3418801 T )^{3}( 1 + 3418801 T )^{3} \))(\( 1 - 79466299556316 T^{2} + \)\(31\!\cdots\!74\)\( T^{4} - \)\(81\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!19\)\( T^{8} - \)\(24\!\cdots\!40\)\( T^{10} + \)\(31\!\cdots\!92\)\( T^{12} - \)\(33\!\cdots\!40\)\( T^{14} + \)\(29\!\cdots\!19\)\( T^{16} - \)\(20\!\cdots\!84\)\( T^{18} + \)\(10\!\cdots\!74\)\( T^{20} - \)\(37\!\cdots\!16\)\( T^{22} + \)\(65\!\cdots\!01\)\( T^{24} \))
$47$ (\( 1 - \)\(19\!\cdots\!54\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{6} \))(\( ( 1 - 890432 T + 77095643518730 T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!42\)\( T^{4} - \)\(44\!\cdots\!32\)\( T^{5} + \)\(98\!\cdots\!46\)\( T^{6} - \)\(10\!\cdots\!52\)\( T^{7} + \)\(19\!\cdots\!82\)\( T^{8} - \)\(15\!\cdots\!60\)\( T^{9} + \)\(24\!\cdots\!30\)\( T^{10} - \)\(68\!\cdots\!32\)\( T^{11} + \)\(18\!\cdots\!61\)\( T^{12} )^{2} \))
$53$ (\( ( 1 - 7890481 T )^{3}( 1 + 7890481 T )^{3} \))(\( 1 - 411904206361308 T^{2} + \)\(81\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!35\)\( T^{8} - \)\(87\!\cdots\!92\)\( T^{10} + \)\(59\!\cdots\!16\)\( T^{12} - \)\(33\!\cdots\!32\)\( T^{14} + \)\(15\!\cdots\!35\)\( T^{16} - \)\(61\!\cdots\!60\)\( T^{18} + \)\(18\!\cdots\!66\)\( T^{20} - \)\(36\!\cdots\!08\)\( T^{22} + \)\(33\!\cdots\!21\)\( T^{24} \))
$59$ (\( ( 1 - 15279074 T + 146830437604321 T^{2} )^{3} \))(\( ( 1 + 6904828 T + 568221694536170 T^{2} + \)\(43\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!59\)\( T^{4} + \)\(11\!\cdots\!08\)\( T^{5} + \)\(29\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!68\)\( T^{7} + \)\(34\!\cdots\!19\)\( T^{8} + \)\(13\!\cdots\!64\)\( T^{9} + \)\(26\!\cdots\!70\)\( T^{10} + \)\(47\!\cdots\!28\)\( T^{11} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))
$61$ (\( ( 1 - 13845841 T )^{3}( 1 + 13845841 T )^{3} \))(\( 1 - 1102038048908460 T^{2} + \)\(55\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(34\!\cdots\!95\)\( T^{8} - \)\(55\!\cdots\!52\)\( T^{10} + \)\(92\!\cdots\!20\)\( T^{12} - \)\(20\!\cdots\!72\)\( T^{14} + \)\(46\!\cdots\!95\)\( T^{16} - \)\(83\!\cdots\!20\)\( T^{18} + \)\(10\!\cdots\!26\)\( T^{20} - \)\(73\!\cdots\!60\)\( T^{22} + \)\(24\!\cdots\!61\)\( T^{24} \))
$67$ (\( ( 1 - 20151121 T )^{3}( 1 + 20151121 T )^{3} \))(\( 1 - 1467797187606936 T^{2} + \)\(96\!\cdots\!54\)\( T^{4} - \)\(44\!\cdots\!24\)\( T^{6} + \)\(21\!\cdots\!19\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{10} + \)\(57\!\cdots\!32\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{14} + \)\(57\!\cdots\!59\)\( T^{16} - \)\(20\!\cdots\!84\)\( T^{18} + \)\(71\!\cdots\!34\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{22} + \)\(20\!\cdots\!81\)\( T^{24} \))
$71$ (\( 1 - \)\(28\!\cdots\!94\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{6} \))(\( ( 1 - 25659548 T + 3196727240846618 T^{2} - \)\(58\!\cdots\!96\)\( T^{3} + \)\(44\!\cdots\!50\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!30\)\( T^{6} - \)\(39\!\cdots\!40\)\( T^{7} + \)\(18\!\cdots\!50\)\( T^{8} - \)\(15\!\cdots\!76\)\( T^{9} + \)\(55\!\cdots\!38\)\( T^{10} - \)\(28\!\cdots\!48\)\( T^{11} + \)\(72\!\cdots\!61\)\( T^{12} )^{2} \))
$73$ (\( 1 - \)\(39\!\cdots\!54\)\( T^{3} + \)\(52\!\cdots\!41\)\( T^{6} \))(\( ( 1 + 483912 T + 3135447356213544 T^{2} - \)\(39\!\cdots\!72\)\( T^{3} + \)\(38\!\cdots\!92\)\( T^{4} - \)\(99\!\cdots\!72\)\( T^{5} + \)\(32\!\cdots\!38\)\( T^{6} - \)\(79\!\cdots\!32\)\( T^{7} + \)\(25\!\cdots\!12\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{9} + \)\(13\!\cdots\!24\)\( T^{10} + \)\(16\!\cdots\!12\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} )^{2} \))
$79$ (\( ( 1 - 38950081 T )^{3}( 1 + 38950081 T )^{3} \))(\( 1 - 11450094913133100 T^{2} + \)\(63\!\cdots\!86\)\( T^{4} - \)\(23\!\cdots\!40\)\( T^{6} + \)\(62\!\cdots\!15\)\( T^{8} - \)\(13\!\cdots\!92\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{12} - \)\(30\!\cdots\!32\)\( T^{14} + \)\(33\!\cdots\!15\)\( T^{16} - \)\(28\!\cdots\!40\)\( T^{18} + \)\(17\!\cdots\!66\)\( T^{20} - \)\(73\!\cdots\!00\)\( T^{22} + \)\(14\!\cdots\!21\)\( T^{24} \))
$83$ (\( ( 1 - 47458321 T )^{3}( 1 + 47458321 T )^{3} \))(\( 1 - 12609715658286840 T^{2} + \)\(83\!\cdots\!78\)\( T^{4} - \)\(37\!\cdots\!56\)\( T^{6} + \)\(13\!\cdots\!11\)\( T^{8} - \)\(37\!\cdots\!24\)\( T^{10} + \)\(91\!\cdots\!20\)\( T^{12} - \)\(19\!\cdots\!44\)\( T^{14} + \)\(34\!\cdots\!71\)\( T^{16} - \)\(49\!\cdots\!96\)\( T^{18} + \)\(55\!\cdots\!38\)\( T^{20} - \)\(42\!\cdots\!40\)\( T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))
$89$ (\( ( 1 - 62742241 T )^{3}( 1 + 62742241 T )^{3} \))(\( 1 - 23373378959233404 T^{2} + \)\(28\!\cdots\!66\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(16\!\cdots\!95\)\( T^{8} - \)\(86\!\cdots\!32\)\( T^{10} + \)\(37\!\cdots\!08\)\( T^{12} - \)\(13\!\cdots\!52\)\( T^{14} + \)\(39\!\cdots\!95\)\( T^{16} - \)\(92\!\cdots\!20\)\( T^{18} + \)\(16\!\cdots\!06\)\( T^{20} - \)\(20\!\cdots\!04\)\( T^{22} + \)\(13\!\cdots\!61\)\( T^{24} \))
$97$ (\( ( 1 - 88529281 T )^{3}( 1 + 88529281 T )^{3} \))(\( 1 - 29843143059888816 T^{2} + \)\(42\!\cdots\!14\)\( T^{4} - \)\(43\!\cdots\!04\)\( T^{6} + \)\(38\!\cdots\!79\)\( T^{8} - \)\(35\!\cdots\!00\)\( T^{10} + \)\(30\!\cdots\!92\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(14\!\cdots\!39\)\( T^{16} - \)\(99\!\cdots\!44\)\( T^{18} + \)\(61\!\cdots\!34\)\( T^{20} - \)\(26\!\cdots\!16\)\( T^{22} + \)\(53\!\cdots\!21\)\( T^{24} \))
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