# Properties

 Label 27.2 Level 27 Weight 2 Dimension 13 Nonzero newspaces 2 Newform subspaces 2 Sturm bound 108 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$108$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(27))$$.

Total New Old
Modular forms 42 29 13
Cusp forms 13 13 0
Eisenstein series 29 16 13

## Trace form

 $$13q - 6q^{2} - 6q^{3} - 8q^{4} - 3q^{5} - 7q^{7} + 6q^{8} + O(q^{10})$$ $$13q - 6q^{2} - 6q^{3} - 8q^{4} - 3q^{5} - 7q^{7} + 6q^{8} - 3q^{10} + 3q^{11} + 12q^{12} - q^{13} + 15q^{14} + 9q^{15} + 4q^{16} + 9q^{17} + 9q^{18} - 10q^{19} - 3q^{20} - 12q^{21} + 3q^{22} - 12q^{23} - 18q^{24} - 2q^{25} - 30q^{26} - 9q^{27} - 10q^{28} - 6q^{29} - 9q^{30} - q^{31} + 9q^{34} + 12q^{35} + 18q^{36} + 8q^{37} + 42q^{38} + 33q^{39} + 21q^{40} + 15q^{41} + 18q^{42} + 11q^{43} + 3q^{44} - 9q^{45} - 3q^{46} - 15q^{47} - 15q^{48} + 6q^{49} - 33q^{50} - 18q^{51} - q^{52} - 18q^{53} - 54q^{54} - 12q^{55} - 33q^{56} - 3q^{57} + 21q^{58} - 12q^{59} + 11q^{61} - 12q^{62} + 9q^{63} + 4q^{64} + 3q^{65} - 9q^{66} - 10q^{67} + 9q^{68} + 9q^{69} - 15q^{70} + 27q^{71} + 18q^{72} - q^{73} + 33q^{74} + 39q^{75} - 34q^{76} + 15q^{77} + 18q^{78} - 25q^{79} + 42q^{80} + 36q^{81} - 12q^{82} + 39q^{83} + 6q^{84} - 27q^{85} + 51q^{86} + 9q^{87} - 30q^{88} + 9q^{89} + 18q^{90} + q^{91} - 39q^{92} - 39q^{93} - 15q^{94} - 33q^{95} - 16q^{97} - 45q^{98} - 27q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$

We only show spaces with even 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
27.2.a $$\chi_{27}(1, \cdot)$$ 27.2.a.a 1 1
27.2.c $$\chi_{27}(10, \cdot)$$ None 0 2
27.2.e $$\chi_{27}(4, \cdot)$$ 27.2.e.a 12 6

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T^{2}$$)($$( 1 + 3 T + 3 T^{2} - 3 T^{4} - 6 T^{5} - 11 T^{6} - 12 T^{7} - 12 T^{8} + 48 T^{10} + 96 T^{11} + 64 T^{12} )( 1 + 3 T + 9 T^{2} + 18 T^{3} + 36 T^{4} + 57 T^{5} + 91 T^{6} + 114 T^{7} + 144 T^{8} + 144 T^{9} + 144 T^{10} + 96 T^{11} + 64 T^{12} )$$)
$3$ ()($$1 + 6 T + 18 T^{2} + 39 T^{3} + 63 T^{4} + 81 T^{5} + 117 T^{6} + 243 T^{7} + 567 T^{8} + 1053 T^{9} + 1458 T^{10} + 1458 T^{11} + 729 T^{12}$$)
$5$ ($$1 + 5 T^{2}$$)($$1 + 3 T + 3 T^{2} + 18 T^{3} + 87 T^{4} + 147 T^{5} + 323 T^{6} + 1368 T^{7} + 3096 T^{8} + 5562 T^{9} + 16272 T^{10} + 41310 T^{11} + 82629 T^{12} + 206550 T^{13} + 406800 T^{14} + 695250 T^{15} + 1935000 T^{16} + 4275000 T^{17} + 5046875 T^{18} + 11484375 T^{19} + 33984375 T^{20} + 35156250 T^{21} + 29296875 T^{22} + 146484375 T^{23} + 244140625 T^{24}$$)
$7$ ($$1 + T + 7 T^{2}$$)($$1 + 6 T + 12 T^{2} - 11 T^{3} - 213 T^{4} - 678 T^{5} - 224 T^{6} + 3942 T^{7} + 15255 T^{8} + 25135 T^{9} - 22044 T^{10} - 210732 T^{11} - 647141 T^{12} - 1475124 T^{13} - 1080156 T^{14} + 8621305 T^{15} + 36627255 T^{16} + 66253194 T^{17} - 26353376 T^{18} - 558362154 T^{19} - 1227902613 T^{20} - 443889677 T^{21} + 3389702988 T^{22} + 11863960458 T^{23} + 13841287201 T^{24}$$)
$11$ ($$1 + 11 T^{2}$$)($$1 - 3 T - 15 T^{2} + 126 T^{3} - 201 T^{4} - 1488 T^{5} + 7145 T^{6} - 1530 T^{7} - 61974 T^{8} + 202716 T^{9} - 19692 T^{10} - 1304451 T^{11} + 4526883 T^{12} - 14348961 T^{13} - 2382732 T^{14} + 269814996 T^{15} - 907361334 T^{16} - 246408030 T^{17} + 12657803345 T^{18} - 28996910448 T^{19} - 43086135081 T^{20} + 297101409066 T^{21} - 389061369015 T^{22} - 855935011833 T^{23} + 3138428376721 T^{24}$$)
$13$ ($$1 - 5 T + 13 T^{2}$$)($$1 + 6 T + 48 T^{2} + 214 T^{3} + 1488 T^{4} + 5928 T^{5} + 32329 T^{6} + 112023 T^{7} + 560277 T^{8} + 1799710 T^{9} + 8467593 T^{10} + 25055985 T^{11} + 112181629 T^{12} + 325727805 T^{13} + 1431023217 T^{14} + 3953962870 T^{15} + 16002071397 T^{16} + 41593355739 T^{17} + 156045908161 T^{18} + 371973208776 T^{19} + 1213807312848 T^{20} + 2269362865822 T^{21} + 6617207608752 T^{22} + 10752962364222 T^{23} + 23298085122481 T^{24}$$)
$17$ ($$1 + 17 T^{2}$$)($$1 - 9 T - 30 T^{2} + 423 T^{3} + 1029 T^{4} - 14184 T^{5} - 23521 T^{6} + 296649 T^{7} + 637560 T^{8} - 4620213 T^{9} - 12537675 T^{10} + 28264410 T^{11} + 250681641 T^{12} + 480494970 T^{13} - 3623388075 T^{14} - 22699106469 T^{15} + 53249648760 T^{16} + 421199159193 T^{17} - 567739760449 T^{18} - 5820243737832 T^{19} + 7178054406789 T^{20} + 50162671758231 T^{21} - 60479817013470 T^{22} - 308447066768697 T^{23} + 582622237229761 T^{24}$$)
$19$ ($$1 + 7 T + 19 T^{2}$$)($$1 + 3 T - 75 T^{2} - 242 T^{3} + 3012 T^{4} + 9714 T^{5} - 85589 T^{6} - 257166 T^{7} + 1946502 T^{8} + 4391737 T^{9} - 39399504 T^{10} - 33490578 T^{11} + 763159453 T^{12} - 636320982 T^{13} - 14223220944 T^{14} + 30122924083 T^{15} + 253670087142 T^{16} - 636768475434 T^{17} - 4026609908909 T^{18} + 8683070072646 T^{19} + 51154491879492 T^{20} - 78090422862518 T^{21} - 459829969335075 T^{22} + 349470776694657 T^{23} + 2213314919066161 T^{24}$$)
$23$ ($$1 + 23 T^{2}$$)($$1 + 12 T + 48 T^{2} + 153 T^{3} - 336 T^{4} - 12228 T^{5} - 51922 T^{6} - 116820 T^{7} - 165330 T^{8} + 4324509 T^{9} + 14509764 T^{10} - 2453454 T^{11} + 107316369 T^{12} - 56429442 T^{13} + 7675665156 T^{14} + 52616301003 T^{15} - 46266112530 T^{16} - 751893589260 T^{17} - 7686319428658 T^{18} - 41634205565916 T^{19} - 26312491054416 T^{20} + 275576357203839 T^{21} + 1988472538255152 T^{22} + 11433717094967124 T^{23} + 21914624432020321 T^{24}$$)
$29$ ($$1 + 29 T^{2}$$)($$1 + 6 T + 21 T^{2} + 252 T^{3} + 249 T^{4} - 984 T^{5} + 18431 T^{6} + 29592 T^{7} + 680634 T^{8} + 5882274 T^{9} + 10161684 T^{10} + 17557326 T^{11} + 254066229 T^{12} + 509162454 T^{13} + 8545976244 T^{14} + 143462780586 T^{15} + 481399496154 T^{16} + 606965921208 T^{17} + 10963188629351 T^{18} - 16973878288056 T^{19} + 124561356827289 T^{20} + 3655800785918988 T^{21} + 8834851899304221 T^{22} + 73203058594234974 T^{23} + 353814783205469041 T^{24}$$)
$31$ ($$1 + 4 T + 31 T^{2}$$)($$1 - 3 T + 84 T^{2} - 434 T^{3} + 5601 T^{4} - 30963 T^{5} + 266473 T^{6} - 1627992 T^{7} + 11453211 T^{8} - 69240287 T^{9} + 408317577 T^{10} - 2527882269 T^{11} + 13547586181 T^{12} - 78364350339 T^{13} + 392393191497 T^{14} - 2062737390017 T^{15} + 10577280875931 T^{16} - 46608028794792 T^{17} + 236495768387113 T^{18} - 851873070718893 T^{19} + 4777042700707041 T^{20} - 11474796017731214 T^{21} + 68848776106387284 T^{22} - 76225430689214493 T^{23} + 787662783788549761 T^{24}$$)
$37$ ($$1 - 11 T + 37 T^{2}$$)($$1 + 3 T - 156 T^{2} - 107 T^{3} + 13731 T^{4} - 9132 T^{5} - 864755 T^{6} + 641043 T^{7} + 43249536 T^{8} - 18536771 T^{9} - 1953626739 T^{10} + 269355786 T^{11} + 78884071369 T^{12} + 9966164082 T^{13} - 2674515005691 T^{14} - 938943061463 T^{15} + 81056593639296 T^{16} + 44452458227151 T^{17} - 2218724740814795 T^{18} - 866917901978556 T^{19} + 48229855381789251 T^{20} - 13905906158073239 T^{21} - 750139162097184444 T^{22} + 533752865338381239 T^{23} + 6582952005840035281 T^{24}$$)
$41$ ($$1 + 41 T^{2}$$)($$1 - 15 T + 93 T^{2} - 90 T^{3} - 2460 T^{4} + 12513 T^{5} + 27971 T^{6} - 441396 T^{7} + 3206862 T^{8} - 12736494 T^{9} - 41813613 T^{10} + 1731506832 T^{11} - 16389887967 T^{12} + 70991780112 T^{13} - 70288683453 T^{14} - 877811902974 T^{15} + 9061825571982 T^{16} - 51138463696596 T^{17} + 132865165725011 T^{18} + 2436960229072953 T^{19} - 19642916063637660 T^{20} - 29464374095456490 T^{21} + 1248307315844173293 T^{22} - 8254935475743726615 T^{23} + 22563490300366186081 T^{24}$$)
$43$ ($$1 - 8 T + 43 T^{2}$$)($$1 - 3 T - 60 T^{2} + 16 T^{3} + 606 T^{4} + 5874 T^{5} + 128269 T^{6} - 73818 T^{7} - 5307417 T^{8} - 22910987 T^{9} - 79924800 T^{10} + 580797228 T^{11} + 14492410483 T^{12} + 24974280804 T^{13} - 147780955200 T^{14} - 1821583843409 T^{15} - 18145002547017 T^{16} - 10851869245374 T^{17} + 810834916932181 T^{18} + 1596662521642518 T^{19} + 7083049368226206 T^{20} + 8041481790989488 T^{21} - 1296688938797054940 T^{22} - 2787881218413668121 T^{23} + 39959630797262576401 T^{24}$$)
$47$ ($$1 + 47 T^{2}$$)($$1 + 15 T + 111 T^{2} + 873 T^{3} + 6828 T^{4} + 69612 T^{5} + 654227 T^{6} + 4732173 T^{7} + 31522707 T^{8} + 170376048 T^{9} + 1258782219 T^{10} + 11485670769 T^{11} + 82734051465 T^{12} + 539826526143 T^{13} + 2780649921771 T^{14} + 17688952431504 T^{15} + 153820754416467 T^{16} + 1085300249810211 T^{17} + 7052053707045683 T^{18} + 35267048661670356 T^{19} + 162583465326504108 T^{20} + 977000903018715591 T^{21} + 5838503678177135439 T^{22} + 37082388226260184545 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$)
$53$ ($$1 + 53 T^{2}$$)($$( 1 + 9 T + 210 T^{2} + 1872 T^{3} + 23856 T^{4} + 168327 T^{5} + 1634317 T^{6} + 8921331 T^{7} + 67011504 T^{8} + 278697744 T^{9} + 1657001010 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} )^{2}$$)
$59$ ($$1 + 59 T^{2}$$)($$1 + 12 T + 192 T^{2} + 2349 T^{3} + 25089 T^{4} + 223824 T^{5} + 1972808 T^{6} + 12709350 T^{7} + 71501877 T^{8} + 339681357 T^{9} - 107943444 T^{10} - 13247965206 T^{11} - 122980417173 T^{12} - 781629947154 T^{13} - 375751128564 T^{14} + 69763417419303 T^{15} + 866414055786597 T^{16} + 9086223139495650 T^{17} + 83214094211233928 T^{18} + 557019929938127856 T^{19} + 3683828849054809569 T^{20} + 20349377178020451711 T^{21} + 98134416633723148992 T^{22} +$$$$36\!\cdots\!08$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$)
$61$ ($$1 + T + 61 T^{2}$$)($$1 - 12 T - 51 T^{2} + 583 T^{3} + 2127 T^{4} + 45474 T^{5} - 455363 T^{6} - 2399139 T^{7} + 11507670 T^{8} + 53383966 T^{9} + 844033821 T^{10} - 2578276122 T^{11} - 56950876769 T^{12} - 157274843442 T^{13} + 3140649847941 T^{14} + 12117145986646 T^{15} + 159333369100470 T^{16} - 2026303924984839 T^{17} - 23460472230148043 T^{18} + 142913087725218954 T^{19} + 407761454745216687 T^{20} + 6817687172122304203 T^{21} - 36380488494807012651 T^{22} -$$$$52\!\cdots\!32$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$)
$67$ ($$1 - 5 T + 67 T^{2}$$)($$1 + 15 T + 255 T^{2} + 2968 T^{3} + 36174 T^{4} + 397221 T^{5} + 4107115 T^{6} + 41367024 T^{7} + 386429292 T^{8} + 3556146616 T^{9} + 31289775603 T^{10} + 264760435272 T^{11} + 2216964278029 T^{12} + 17738949163224 T^{13} + 140459802681867 T^{14} + 1069557324668008 T^{15} + 7786983421036332 T^{16} + 55850657704271568 T^{17} + 371522978282032435 T^{18} + 2407441924578007383 T^{19} + 14689092167933931534 T^{20} + 80748994088203402696 T^{21} +$$$$46\!\cdots\!95$$$$T^{22} +$$$$18\!\cdots\!45$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$)
$71$ ($$1 + 71 T^{2}$$)($$1 - 27 T + 78 T^{2} + 2565 T^{3} + 13071 T^{4} - 524664 T^{5} - 751711 T^{6} + 30297321 T^{7} + 410765508 T^{8} - 3391054713 T^{9} - 30034133541 T^{10} + 13624108308 T^{11} + 3600759258249 T^{12} + 967311689868 T^{13} - 151402067180181 T^{14} - 1213695783384543 T^{15} + 10438242055098948 T^{16} + 54663315804868671 T^{17} - 96294392526538831 T^{18} - 4771882122782055624 T^{19} + 8440644406913342031 T^{20} +$$$$11\!\cdots\!15$$$$T^{21} +$$$$25\!\cdots\!78$$$$T^{22} -$$$$62\!\cdots\!17$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$)
$73$ ($$1 + 7 T + 73 T^{2}$$)($$1 - 6 T - 228 T^{2} + 2296 T^{3} + 24945 T^{4} - 381255 T^{5} - 980072 T^{6} + 40200363 T^{7} - 102286134 T^{8} - 2648934335 T^{9} + 21743689350 T^{10} + 78452536893 T^{11} - 2017821540323 T^{12} + 5727035193189 T^{13} + 115872120546150 T^{14} - 1030480488198695 T^{15} - 2904746284290294 T^{16} + 83338230563588259 T^{17} - 148318437827512808 T^{18} - 4211875922398326735 T^{19} + 20117146992297850545 T^{20} +$$$$13\!\cdots\!48$$$$T^{21} -$$$$97\!\cdots\!72$$$$T^{22} -$$$$18\!\cdots\!62$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$)
$79$ ($$1 - 17 T + 79 T^{2}$$)($$1 + 42 T + 813 T^{2} + 9520 T^{3} + 72840 T^{4} + 356811 T^{5} + 973207 T^{6} - 893781 T^{7} - 62793603 T^{8} - 1355379536 T^{9} - 23955645108 T^{10} - 329298299862 T^{11} - 3388931313773 T^{12} - 26014565689098 T^{13} - 149507181119028 T^{14} - 668254971049904 T^{15} - 2445815923131843 T^{16} - 2750214545354619 T^{17} + 236574413325225847 T^{18} + 6852165969260378949 T^{19} +$$$$11\!\cdots\!40$$$$T^{20} +$$$$11\!\cdots\!80$$$$T^{21} +$$$$76\!\cdots\!13$$$$T^{22} +$$$$31\!\cdots\!18$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$)
$83$ ($$1 + 83 T^{2}$$)($$1 - 39 T + 912 T^{2} - 16200 T^{3} + 251079 T^{4} - 3515997 T^{5} + 45358019 T^{6} - 541131408 T^{7} + 6100532325 T^{8} - 65514800025 T^{9} + 671478204717 T^{10} - 6541571603403 T^{11} + 60933732837525 T^{12} - 542950443082449 T^{13} + 4625813352295413 T^{14} - 37460510961894675 T^{15} + 289521021350726325 T^{16} - 2131538609315815344 T^{17} + 14829367667138196011 T^{18} - 95410273871375563119 T^{19} +$$$$56\!\cdots\!39$$$$T^{20} -$$$$30\!\cdots\!00$$$$T^{21} +$$$$14\!\cdots\!88$$$$T^{22} -$$$$50\!\cdots\!13$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$)
$89$ ($$1 + 89 T^{2}$$)($$1 - 9 T - 273 T^{2} + 2772 T^{3} + 38802 T^{4} - 449316 T^{5} - 3561871 T^{6} + 54551502 T^{7} + 157767516 T^{8} - 4371660207 T^{9} + 3816883044 T^{10} + 152630961444 T^{11} - 900621732009 T^{12} + 13584155568516 T^{13} + 30233530591524 T^{14} - 3081884924468583 T^{15} + 9898687510843356 T^{16} + 304618830200242398 T^{17} - 1770183247816548031 T^{18} - 19873846469919508164 T^{19} +$$$$15\!\cdots\!62$$$$T^{20} +$$$$97\!\cdots\!48$$$$T^{21} -$$$$85\!\cdots\!73$$$$T^{22} -$$$$24\!\cdots\!01$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$)
$97$ ($$1 + 19 T + 97 T^{2}$$)($$1 - 3 T + 102 T^{2} - 1010 T^{3} + 15132 T^{4} - 13512 T^{5} + 1127323 T^{6} - 7762230 T^{7} + 23311719 T^{8} + 575063737 T^{9} + 1596748254 T^{10} + 54554445012 T^{11} - 1453572795209 T^{12} + 5291781166164 T^{13} + 15023804321886 T^{14} + 524845146039001 T^{15} + 2063769721944039 T^{16} - 66656910163093110 T^{17} + 939028499512575067 T^{18} - 1091746419868262856 T^{19} +$$$$11\!\cdots\!52$$$$T^{20} -$$$$76\!\cdots\!70$$$$T^{21} +$$$$75\!\cdots\!98$$$$T^{22} -$$$$21\!\cdots\!59$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$)