# 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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