# Properties

 Label 26.3 Level 26 Weight 3 Dimension 14 Nonzero newspaces 2 Newform subspaces 3 Sturm bound 126 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$26\( 26 = 2 \cdot 13$$ \) Weight: $$k$$ = $$3$$ Nonzero newspaces: $$2$$ Newform subspaces: $$3$$ Sturm bound: $$126$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(26))$$.

Total New Old
Modular forms 54 14 40
Cusp forms 30 14 16
Eisenstein series 24 0 24

## Trace form

 $$14q - 20q^{7} - 12q^{8} - 48q^{9} + O(q^{10})$$ $$14q - 20q^{7} - 12q^{8} - 48q^{9} - 30q^{10} - 12q^{11} + 24q^{13} + 24q^{14} + 72q^{15} + 16q^{16} + 36q^{17} + 90q^{18} + 40q^{19} + 24q^{20} - 60q^{21} - 24q^{23} + 72q^{27} - 40q^{28} - 156q^{29} - 192q^{30} - 124q^{31} + 12q^{33} - 96q^{34} - 60q^{35} - 48q^{36} + 8q^{37} + 114q^{41} + 144q^{42} + 12q^{43} + 144q^{44} + 330q^{45} + 192q^{46} + 276q^{47} + 300q^{49} + 246q^{50} + 20q^{52} - 240q^{53} - 360q^{54} - 228q^{55} - 96q^{56} - 396q^{57} - 162q^{58} - 84q^{59} - 96q^{60} + 78q^{61} - 72q^{62} - 228q^{63} - 342q^{65} + 96q^{66} - 32q^{67} + 12q^{68} + 132q^{69} + 216q^{70} - 120q^{71} + 192q^{72} + 64q^{73} + 78q^{74} + 120q^{75} - 80q^{76} + 216q^{78} - 120q^{79} + 48q^{80} + 132q^{81} - 330q^{82} + 108q^{83} - 168q^{84} - 30q^{85} - 96q^{86} + 264q^{87} + 216q^{89} + 388q^{91} + 48q^{92} + 492q^{93} + 240q^{94} + 672q^{95} + 128q^{97} - 240q^{98} + 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(26))$$

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
26.3.d $$\chi_{26}(5, \cdot)$$ 26.3.d.a 2 2
26.3.f $$\chi_{26}(7, \cdot)$$ 26.3.f.a 4 4
26.3.f.b 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(26))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(26)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T + 2 T^{2}$$)($$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$)($$( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$)
$3$ ($$( 1 + 9 T^{2} )^{2}$$)($$1 - 15 T^{2} + 144 T^{4} - 1215 T^{6} + 6561 T^{8}$$)($$1 + 3 T^{2} - 24 T^{3} - 87 T^{4} + 72 T^{5} - 54 T^{6} + 1800 T^{7} + 630 T^{8} + 16200 T^{9} - 4374 T^{10} + 52488 T^{11} - 570807 T^{12} - 1417176 T^{13} + 1594323 T^{14} + 43046721 T^{16}$$)
$5$ ($$1 + 6 T + 18 T^{2} + 150 T^{3} + 625 T^{4}$$)($$1 + 686 T^{4} + 390625 T^{8}$$)($$1 - 6 T + 18 T^{2} - 240 T^{3} + 445 T^{4} + 2988 T^{5} + 2862 T^{6} + 36666 T^{7} - 716556 T^{8} + 916650 T^{9} + 1788750 T^{10} + 46687500 T^{11} + 173828125 T^{12} - 2343750000 T^{13} + 4394531250 T^{14} - 36621093750 T^{15} + 152587890625 T^{16}$$)
$7$ ($$1 - 4 T + 8 T^{2} - 196 T^{3} + 2401 T^{4}$$)($$1 + 22 T + 221 T^{2} + 1362 T^{3} + 8024 T^{4} + 66738 T^{5} + 530621 T^{6} + 2588278 T^{7} + 5764801 T^{8}$$)($$1 + 2 T - 127 T^{2} - 434 T^{3} + 8255 T^{4} + 22156 T^{5} - 385248 T^{6} - 453708 T^{7} + 19028482 T^{8} - 22231692 T^{9} - 924980448 T^{10} + 2606631244 T^{11} + 47588432255 T^{12} - 122594258066 T^{13} - 1757843474527 T^{14} + 1356446145698 T^{15} + 33232930569601 T^{16}$$)
$11$ ($$1 - 12 T + 72 T^{2} - 1452 T^{3} + 14641 T^{4}$$)($$1 + 6 T + 45 T^{2} - 1710 T^{3} - 16024 T^{4} - 206910 T^{5} + 658845 T^{6} + 10629366 T^{7} + 214358881 T^{8}$$)($$1 + 18 T + 105 T^{2} - 3114 T^{3} - 59393 T^{4} - 565908 T^{5} + 897792 T^{6} + 64810956 T^{7} + 1103342370 T^{8} + 7842125676 T^{9} + 13144572672 T^{10} - 1002540542388 T^{11} - 12731417019233 T^{12} - 80769140207514 T^{13} + 329534979555705 T^{14} + 6835497004498338 T^{15} + 45949729863572161 T^{16}$$)
$13$ ($$1 + 169 T^{2}$$)($$1 + 12 T + 182 T^{2} + 2028 T^{3} + 28561 T^{4}$$)($$1 - 36 T + 589 T^{2} - 7824 T^{3} + 106236 T^{4} - 1322256 T^{5} + 16822429 T^{6} - 173765124 T^{7} + 815730721 T^{8}$$)
$17$ ($$1 - 542 T^{2} + 83521 T^{4}$$)($$1 - 78 T + 3077 T^{2} - 81822 T^{3} + 1602972 T^{4} - 23646558 T^{5} + 256994117 T^{6} - 1882730382 T^{7} + 6975757441 T^{8}$$)($$1 + 42 T + 1726 T^{2} + 47796 T^{3} + 1303465 T^{4} + 30227796 T^{5} + 646060522 T^{6} + 12441458526 T^{7} + 218725843324 T^{8} + 3595581514014 T^{9} + 53959620857962 T^{10} + 729625511667924 T^{11} + 9092655672833065 T^{12} + 96356444465860404 T^{13} + 1005605981458567486 T^{14} + 7071868715494839018 T^{15} + 48661191875666868481 T^{16}$$)
$19$ ($$1 - 52 T + 1352 T^{2} - 18772 T^{3} + 130321 T^{4}$$)($$1 + 58 T + 1325 T^{2} + 12942 T^{3} + 74504 T^{4} + 4672062 T^{5} + 172675325 T^{6} + 2728661098 T^{7} + 16983563041 T^{8}$$)($$1 - 46 T + 977 T^{2} - 19730 T^{3} + 483311 T^{4} - 8783612 T^{5} + 118611072 T^{6} - 1680839364 T^{7} + 28608116338 T^{8} - 606783010404 T^{9} + 15457513514112 T^{10} - 413232764902172 T^{11} + 8208342836908751 T^{12} - 120965937266413730 T^{13} + 2162408675927639297 T^{14} - 36754307546012669566 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$)
$23$ ($$1 - 482 T^{2} + 279841 T^{4}$$)($$1 - 12 T + 677 T^{2} - 7548 T^{3} + 141192 T^{4} - 3992892 T^{5} + 189452357 T^{6} - 1776430668 T^{7} + 78310985281 T^{8}$$)($$1 + 36 T + 2131 T^{2} + 61164 T^{3} + 2152429 T^{4} + 44226000 T^{5} + 1303450510 T^{6} + 22452294672 T^{7} + 648751140166 T^{8} + 11877263881488 T^{9} + 364758894168910 T^{10} + 6547035226914000 T^{11} + 168558835737397549 T^{12} + 2533811131871627436 T^{13} + 46700064664635304051 T^{14} +$$$$41\!\cdots\!24$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$)
$29$ ($$( 1 + 48 T + 841 T^{2} )^{2}$$)($$1 + 54 T + 1273 T^{2} - 2106 T^{3} - 460188 T^{4} - 1771146 T^{5} + 900368713 T^{6} + 32120459334 T^{7} + 500246412961 T^{8}$$)($$1 + 6 T - 2398 T^{2} - 14772 T^{3} + 2997253 T^{4} + 14035500 T^{5} - 3325399546 T^{6} - 4535559294 T^{7} + 3230740374436 T^{8} - 3814405366254 T^{9} - 2351991916294426 T^{10} + 8348642721895500 T^{11} + 1499365061986596133 T^{12} - 6214687250310569172 T^{13} -$$$$84\!\cdots\!18$$$$T^{14} +$$$$17\!\cdots\!86$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16}$$)
$31$ ($$1 + 28 T + 392 T^{2} + 26908 T^{3} + 923521 T^{4}$$)($$1 + 128 T + 8192 T^{2} + 365952 T^{3} + 12745358 T^{4} + 351679872 T^{5} + 7565484032 T^{6} + 113600471168 T^{7} + 852891037441 T^{8}$$)($$1 - 32 T + 512 T^{2} - 24592 T^{3} + 1881968 T^{4} - 48317872 T^{5} + 884987520 T^{6} - 50977297056 T^{7} + 2940963189598 T^{8} - 48989182470816 T^{9} + 817304559457920 T^{10} - 42882289258086832 T^{11} + 1605113639950763888 T^{12} - 20156298833431858192 T^{13} +$$$$40\!\cdots\!32$$$$T^{14} -$$$$24\!\cdots\!72$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$)
$37$ ($$( 1 - 37 T )^{2}( 1 + 1369 T^{2} )$$)($$1 - 40 T + 401 T^{2} + 52236 T^{3} - 3248140 T^{4} + 71511084 T^{5} + 751538561 T^{6} - 102629056360 T^{7} + 3512479453921 T^{8}$$)($$1 + 106 T + 8342 T^{2} + 393068 T^{3} + 15188585 T^{4} + 276977564 T^{5} - 1726711374 T^{6} - 651456022674 T^{7} - 28349233687076 T^{8} - 891843295040706 T^{9} - 3236135115407214 T^{10} + 710648650655287676 T^{11} + 53349592746632691785 T^{12} +$$$$18\!\cdots\!32$$$$T^{13} +$$$$54\!\cdots\!02$$$$T^{14} +$$$$95\!\cdots\!34$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$)
$41$ ($$1 + 18 T + 162 T^{2} + 30258 T^{3} + 2825761 T^{4}$$)($$1 + 1521 T^{2} + 91572 T^{3} + 840356 T^{4} + 153932532 T^{5} + 4297982481 T^{6} + 7984925229121 T^{8}$$)($$1 - 132 T + 10686 T^{2} - 553992 T^{3} + 19988317 T^{4} - 213687144 T^{5} - 26569998546 T^{6} + 2518966024308 T^{7} - 124070043071700 T^{8} + 4234381886861748 T^{9} - 75080465661343506 T^{10} - 1015036208961577704 T^{11} +$$$$15\!\cdots\!57$$$$T^{12} -$$$$74\!\cdots\!92$$$$T^{13} +$$$$24\!\cdots\!66$$$$T^{14} -$$$$50\!\cdots\!52$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16}$$)
$43$ ($$1 - 2402 T^{2} + 3418801 T^{4}$$)($$1 - 120 T + 8177 T^{2} - 405240 T^{3} + 16860528 T^{4} - 749288760 T^{5} + 27955535777 T^{6} - 758563565880 T^{7} + 11688200277601 T^{8}$$)($$1 + 108 T + 10879 T^{2} + 755028 T^{3} + 48271669 T^{4} + 2548436904 T^{5} + 130030600102 T^{6} + 5877290089008 T^{7} + 263654368009582 T^{8} + 10867109374575792 T^{9} + 444548745659317702 T^{10} + 16109594877653560296 T^{11} +$$$$56\!\cdots\!69$$$$T^{12} +$$$$16\!\cdots\!72$$$$T^{13} +$$$$43\!\cdots\!79$$$$T^{14} +$$$$79\!\cdots\!92$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$)
$47$ ($$1 - 84 T + 3528 T^{2} - 185556 T^{3} + 4879681 T^{4}$$)($$( 1 - 66 T + 2178 T^{2} - 145794 T^{3} + 4879681 T^{4} )^{2}$$)($$1 - 60 T + 1800 T^{2} - 154932 T^{3} + 21951088 T^{4} - 867532140 T^{5} + 24541932312 T^{6} - 2003115348324 T^{7} + 163148147612766 T^{8} - 4424881804447716 T^{9} + 119756800806152472 T^{10} - 9351315741888174060 T^{11} +$$$$52\!\cdots\!68$$$$T^{12} -$$$$81\!\cdots\!68$$$$T^{13} +$$$$20\!\cdots\!00$$$$T^{14} -$$$$15\!\cdots\!40$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$)
$53$ ($$( 1 - 30 T + 2809 T^{2} )^{2}$$)($$( 1 + 84 T + 5930 T^{2} + 235956 T^{3} + 7890481 T^{4} )^{2}$$)($$( 1 + 66 T + 5917 T^{2} + 184818 T^{3} + 12226368 T^{4} + 519153762 T^{5} + 46687976077 T^{6} + 1462847834514 T^{7} + 62259690411361 T^{8} )^{2}$$)
$59$ ($$1 + 108 T + 5832 T^{2} + 375948 T^{3} + 12117361 T^{4}$$)($$1 - 6 T + 1305 T^{2} - 208794 T^{3} - 5098528 T^{4} - 726811914 T^{5} + 15813156105 T^{6} - 253083201846 T^{7} + 146830437604321 T^{8}$$)($$1 - 18 T + 5565 T^{2} - 318294 T^{3} + 23254759 T^{4} - 1693659348 T^{5} + 98133474288 T^{6} - 8885846915100 T^{7} + 327983542866810 T^{8} - 30931633111463100 T^{9} + 1189118734131913968 T^{10} - 71439455104708126068 T^{11} +$$$$34\!\cdots\!39$$$$T^{12} -$$$$16\!\cdots\!94$$$$T^{13} +$$$$99\!\cdots\!65$$$$T^{14} -$$$$11\!\cdots\!98$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$)
$61$ ($$( 1 + 18 T + 3721 T^{2} )^{2}$$)($$1 - 78 T - 2771 T^{2} - 110214 T^{3} + 41926620 T^{4} - 410106294 T^{5} - 38366825411 T^{6} - 4018589200158 T^{7} + 191707312997281 T^{8}$$)($$1 - 36 T - 11878 T^{2} + 285336 T^{3} + 88972381 T^{4} - 1311266160 T^{5} - 475974647662 T^{6} + 1883345164380 T^{7} + 2030181383859340 T^{8} + 7007927356657980 T^{9} - 6590269291559073742 T^{10} - 67556923450110923760 T^{11} +$$$$17\!\cdots\!61$$$$T^{12} +$$$$20\!\cdots\!36$$$$T^{13} -$$$$31\!\cdots\!38$$$$T^{14} -$$$$35\!\cdots\!76$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$)
$67$ ($$1 + 44 T + 968 T^{2} + 197516 T^{3} + 20151121 T^{4}$$)($$1 - 86 T + 4985 T^{2} - 290202 T^{3} + 7709888 T^{4} - 1302716778 T^{5} + 100453338185 T^{6} - 7779420866534 T^{7} + 406067677556641 T^{8}$$)($$1 + 74 T + 15917 T^{2} + 695782 T^{3} + 97615703 T^{4} + 493359916 T^{5} + 227553513504 T^{6} - 21342931471668 T^{7} + 178591314167674 T^{8} - 95808419376317652 T^{9} + 4585458384594237984 T^{10} + 44628539828393737804 T^{11} +$$$$39\!\cdots\!23$$$$T^{12} +$$$$12\!\cdots\!18$$$$T^{13} +$$$$13\!\cdots\!37$$$$T^{14} +$$$$27\!\cdots\!46$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16}$$)
$71$ ($$1 - 12 T + 72 T^{2} - 60492 T^{3} + 25411681 T^{4}$$)($$1 - 42 T + 3357 T^{2} - 375150 T^{3} + 3657944 T^{4} - 1891131150 T^{5} + 85307013117 T^{6} - 5380211924682 T^{7} + 645753531245761 T^{8}$$)($$1 + 174 T + 14793 T^{2} + 561450 T^{3} - 2330369 T^{4} - 584834196 T^{5} + 83898109968 T^{6} + 14951669162460 T^{7} + 1250020614182514 T^{8} + 75371364247960860 T^{9} + 2131992007009736208 T^{10} - 74917426554309762516 T^{11} -$$$$15\!\cdots\!09$$$$T^{12} +$$$$18\!\cdots\!50$$$$T^{13} +$$$$24\!\cdots\!13$$$$T^{14} +$$$$14\!\cdots\!94$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16}$$)
$73$ ($$1 - 34 T + 578 T^{2} - 181186 T^{3} + 28398241 T^{4}$$)($$1 + 136 T + 9248 T^{2} + 444312 T^{3} + 17094734 T^{4} + 2367738648 T^{5} + 262626932768 T^{6} + 20581454775304 T^{7} + 806460091894081 T^{8}$$)($$1 - 166 T + 13778 T^{2} - 1402664 T^{3} + 172017437 T^{4} - 14047512068 T^{5} + 945563904750 T^{6} - 83152659919950 T^{7} + 7211040518499604 T^{8} - 443120524713413550 T^{9} + 26852351647991544750 T^{10} -$$$$21\!\cdots\!52$$$$T^{11} +$$$$13\!\cdots\!97$$$$T^{12} -$$$$60\!\cdots\!36$$$$T^{13} +$$$$31\!\cdots\!38$$$$T^{14} -$$$$20\!\cdots\!94$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$)
$79$ ($$( 1 + 108 T + 6241 T^{2} )^{2}$$)($$( 1 - 96 T + 10898 T^{2} - 599136 T^{3} + 38950081 T^{4} )^{2}$$)($$( 1 + 48 T + 10084 T^{2} + 724368 T^{3} + 50280774 T^{4} + 4520780688 T^{5} + 392772616804 T^{6} + 11668197865008 T^{7} + 1517108809906561 T^{8} )^{2}$$)
$83$ ($$1 - 156 T + 12168 T^{2} - 1074684 T^{3} + 47458321 T^{4}$$)($$1 - 192 T + 18432 T^{2} - 1598016 T^{3} + 136488302 T^{4} - 11008732224 T^{5} + 874751772672 T^{6} - 62772551686848 T^{7} + 2252292232139041 T^{8}$$)($$1 + 240 T + 28800 T^{2} + 2896512 T^{3} + 298194640 T^{4} + 28074339648 T^{5} + 2344726766592 T^{6} + 177350340390480 T^{7} + 13560551099314782 T^{8} + 1221766494950016720 T^{9} +$$$$11\!\cdots\!32$$$$T^{10} +$$$$91\!\cdots\!12$$$$T^{11} +$$$$67\!\cdots\!40$$$$T^{12} +$$$$44\!\cdots\!88$$$$T^{13} +$$$$30\!\cdots\!00$$$$T^{14} +$$$$17\!\cdots\!60$$$$T^{15} +$$$$50\!\cdots\!81$$$$T^{16}$$)
$89$ ($$1 + 18 T + 162 T^{2} + 142578 T^{3} + 62742241 T^{4}$$)($$1 + 60 T + 5661 T^{2} - 927072 T^{3} - 53599444 T^{4} - 7343337312 T^{5} + 355183826301 T^{6} + 29818877457660 T^{7} + 3936588805702081 T^{8}$$)($$1 - 294 T + 63735 T^{2} - 10176306 T^{3} + 1386303445 T^{4} - 163006886868 T^{5} + 17292939386730 T^{6} - 1689166620371832 T^{7} + 154185854226666234 T^{8} - 13379888799965281272 T^{9} +$$$$10\!\cdots\!30$$$$T^{10} -$$$$81\!\cdots\!48$$$$T^{11} +$$$$54\!\cdots\!45$$$$T^{12} -$$$$31\!\cdots\!06$$$$T^{13} +$$$$15\!\cdots\!35$$$$T^{14} -$$$$57\!\cdots\!54$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$)
$97$ ($$1 + 94 T + 4418 T^{2} + 884446 T^{3} + 88529281 T^{4}$$)($$1 - 280 T + 24089 T^{2} + 773796 T^{3} - 268596364 T^{4} + 7280646564 T^{5} + 2132581850009 T^{6} - 233232161380120 T^{7} + 7837433594376961 T^{8}$$)($$1 + 58 T + 14363 T^{2} + 998 T^{3} - 4472611 T^{4} - 1301881588 T^{5} - 229071912798 T^{6} + 178692170625312 T^{7} + 7600464823934194 T^{8} + 1681314633413560608 T^{9} - 20279571737301638238 T^{10} -$$$$10\!\cdots\!52$$$$T^{11} -$$$$35\!\cdots\!71$$$$T^{12} +$$$$73\!\cdots\!02$$$$T^{13} +$$$$99\!\cdots\!83$$$$T^{14} +$$$$37\!\cdots\!02$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$)