# Properties

 Label 54.4 Level 54 Weight 4 Dimension 64 Nonzero newspaces 3 Newform subspaces 8 Sturm bound 648 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newform subspaces: $$8$$ Sturm bound: $$648$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(54))$$.

Total New Old
Modular forms 273 64 209
Cusp forms 213 64 149
Eisenstein series 60 0 60

## Trace form

 $$64q - 2q^{2} + 4q^{4} - 42q^{5} + 12q^{6} + 56q^{7} + 40q^{8} + 96q^{9} + O(q^{10})$$ $$64q - 2q^{2} + 4q^{4} - 42q^{5} + 12q^{6} + 56q^{7} + 40q^{8} + 96q^{9} + 60q^{10} - 90q^{11} - 12q^{12} - 142q^{13} - 232q^{14} - 72q^{15} + 16q^{16} + 282q^{17} + 276q^{18} + 20q^{19} + 120q^{20} + 204q^{21} + 48q^{22} - 396q^{23} + 19q^{25} - 544q^{26} - 1377q^{27} + 80q^{28} - 414q^{29} - 504q^{30} + 236q^{31} - 32q^{32} + 135q^{33} - 900q^{34} - 84q^{35} + 600q^{36} - 142q^{37} + 350q^{38} + 696q^{39} + 240q^{40} + 2226q^{41} + 912q^{42} + 1028q^{43} + 576q^{44} + 1674q^{45} + 672q^{46} + 822q^{47} + 192q^{48} + 417q^{49} - 1406q^{50} - 3186q^{51} - 568q^{52} - 4116q^{53} - 1008q^{54} - 306q^{55} - 928q^{56} - 3141q^{57} - 204q^{58} - 135q^{59} + 1008q^{60} + 1082q^{61} + 3152q^{62} + 3936q^{63} - 1088q^{64} + 8970q^{65} + 4320q^{66} + 3908q^{67} + 876q^{68} + 6120q^{69} + 1308q^{70} + 552q^{71} - 192q^{72} - 3166q^{73} - 3940q^{74} - 7998q^{75} - 2260q^{76} - 11016q^{77} - 5688q^{78} - 8476q^{79} - 384q^{80} - 6156q^{81} - 2172q^{82} - 4560q^{83} - 2160q^{84} - 7236q^{85} - 3532q^{86} - 2178q^{87} - 456q^{88} + 6753q^{89} + 1260q^{90} + 4234q^{91} + 3312q^{92} + 9696q^{93} + 6996q^{94} + 15918q^{95} + 960q^{96} + 9506q^{97} + 2754q^{98} + 6282q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$

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
54.4.a $$\chi_{54}(1, \cdot)$$ 54.4.a.a 1 1
54.4.a.b 1
54.4.a.c 1
54.4.a.d 1
54.4.c $$\chi_{54}(19, \cdot)$$ 54.4.c.a 2 2
54.4.c.b 4
54.4.e $$\chi_{54}(7, \cdot)$$ 54.4.e.a 24 6
54.4.e.b 30

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(54))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(54)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T$$)($$1 + 2 T$$)($$1 - 2 T$$)($$1 - 2 T$$)($$1 - 2 T + 4 T^{2}$$)($$( 1 + 2 T + 4 T^{2} )^{2}$$)
$3$ 1
$5$ ($$1 + 12 T + 125 T^{2}$$)($$1 + 3 T + 125 T^{2}$$)($$1 - 3 T + 125 T^{2}$$)($$1 - 12 T + 125 T^{2}$$)($$1 + 9 T - 44 T^{2} + 1125 T^{3} + 15625 T^{4}$$)($$1 + 9 T + 47 T^{2} - 1944 T^{3} - 24594 T^{4} - 243000 T^{5} + 734375 T^{6} + 17578125 T^{7} + 244140625 T^{8}$$)
$7$ ($$1 + 7 T + 343 T^{2}$$)($$1 - 29 T + 343 T^{2}$$)($$1 - 29 T + 343 T^{2}$$)($$1 + 7 T + 343 T^{2}$$)($$1 - 31 T + 618 T^{2} - 10633 T^{3} + 117649 T^{4}$$)($$1 + 19 T - 179 T^{2} - 2774 T^{3} + 50128 T^{4} - 951482 T^{5} - 21059171 T^{6} + 766718533 T^{7} + 13841287201 T^{8}$$)
$11$ ($$1 + 60 T + 1331 T^{2}$$)($$1 - 57 T + 1331 T^{2}$$)($$1 + 57 T + 1331 T^{2}$$)($$1 - 60 T + 1331 T^{2}$$)($$1 + 15 T - 1106 T^{2} + 19965 T^{3} + 1771561 T^{4}$$)($$1 + 24 T - 1285 T^{2} - 19224 T^{3} + 925104 T^{4} - 25587144 T^{5} - 2276455885 T^{6} + 56590744584 T^{7} + 3138428376721 T^{8}$$)
$13$ ($$1 + 79 T + 2197 T^{2}$$)($$1 - 20 T + 2197 T^{2}$$)($$1 - 20 T + 2197 T^{2}$$)($$1 + 79 T + 2197 T^{2}$$)($$1 - 37 T - 828 T^{2} - 81289 T^{3} + 4826809 T^{4}$$)($$1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 93007798 T^{5} - 6598247903 T^{6} + 646874461753 T^{7} + 23298085122481 T^{8}$$)
$17$ ($$1 - 108 T + 4913 T^{2}$$)($$1 - 72 T + 4913 T^{2}$$)($$1 + 72 T + 4913 T^{2}$$)($$1 + 108 T + 4913 T^{2}$$)($$( 1 - 42 T + 4913 T^{2} )^{2}$$)($$( 1 + 3 T + 9592 T^{2} + 14739 T^{3} + 24137569 T^{4} )^{2}$$)
$19$ ($$1 - 11 T + 6859 T^{2}$$)($$1 + 106 T + 6859 T^{2}$$)($$1 + 106 T + 6859 T^{2}$$)($$1 - 11 T + 6859 T^{2}$$)($$( 1 + 28 T + 6859 T^{2} )^{2}$$)($$( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} )^{2}$$)
$23$ ($$1 - 132 T + 12167 T^{2}$$)($$1 + 174 T + 12167 T^{2}$$)($$1 - 174 T + 12167 T^{2}$$)($$1 + 132 T + 12167 T^{2}$$)($$1 - 195 T + 25858 T^{2} - 2372565 T^{3} + 148035889 T^{4}$$)($$1 - 69 T - 20527 T^{2} - 65826 T^{3} + 433519968 T^{4} - 800904942 T^{5} - 3038732693503 T^{6} - 124279533640947 T^{7} + 21914624432020321 T^{8}$$)
$29$ ($$1 + 96 T + 24389 T^{2}$$)($$1 - 210 T + 24389 T^{2}$$)($$1 + 210 T + 24389 T^{2}$$)($$1 - 96 T + 24389 T^{2}$$)($$1 - 111 T - 12068 T^{2} - 2707179 T^{3} + 594823321 T^{4}$$)($$1 - 237 T + 4925 T^{2} - 584442 T^{3} + 661218474 T^{4} - 14253955938 T^{5} + 2929504855925 T^{6} - 3438193596280953 T^{7} + 353814783205469041 T^{8}$$)
$31$ ($$1 - 20 T + 29791 T^{2}$$)($$1 - 47 T + 29791 T^{2}$$)($$1 - 47 T + 29791 T^{2}$$)($$1 - 20 T + 29791 T^{2}$$)($$1 - 205 T + 12234 T^{2} - 6107155 T^{3} + 887503681 T^{4}$$)($$1 + 211 T - 24065 T^{2} + 1899844 T^{3} + 2490210604 T^{4} + 56598252604 T^{5} - 21357776083265 T^{6} + 5578760275901581 T^{7} + 787662783788549761 T^{8}$$)
$37$ ($$1 + 169 T + 50653 T^{2}$$)($$1 - 2 T + 50653 T^{2}$$)($$1 - 2 T + 50653 T^{2}$$)($$1 + 169 T + 50653 T^{2}$$)($$( 1 + 166 T + 50653 T^{2} )^{2}$$)($$( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} )^{2}$$)
$41$ ($$1 + 192 T + 68921 T^{2}$$)($$1 - 6 T + 68921 T^{2}$$)($$1 + 6 T + 68921 T^{2}$$)($$1 - 192 T + 68921 T^{2}$$)($$1 + 261 T - 800 T^{2} + 17988381 T^{3} + 4750104241 T^{4}$$)($$1 - 468 T + 27371 T^{2} - 25183548 T^{3} + 16885415064 T^{4} - 1735675311708 T^{5} + 130015103180411 T^{6} - 153214745296373748 T^{7} + 22563490300366186081 T^{8}$$)
$43$ ($$1 - 488 T + 79507 T^{2}$$)($$1 - 218 T + 79507 T^{2}$$)($$1 - 218 T + 79507 T^{2}$$)($$1 - 488 T + 79507 T^{2}$$)($$1 - 43 T - 77658 T^{2} - 3418801 T^{3} + 6321363049 T^{4}$$)($$1 - 86 T - 17387 T^{2} + 11543866 T^{3} - 6295199732 T^{4} + 917818154062 T^{5} - 109909539332963 T^{6} - 43222964626568498 T^{7} + 39959630797262576401 T^{8}$$)
$47$ ($$1 + 204 T + 103823 T^{2}$$)($$1 + 474 T + 103823 T^{2}$$)($$1 - 474 T + 103823 T^{2}$$)($$1 - 204 T + 103823 T^{2}$$)($$1 - 177 T - 72494 T^{2} - 18376671 T^{3} + 10779215329 T^{4}$$)($$1 - 483 T + 20477 T^{2} - 2495178 T^{3} + 10288968168 T^{4} - 259056865494 T^{5} + 220725992291933 T^{6} - 540540018508636461 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)
$53$ ($$1 + 360 T + 148877 T^{2}$$)($$1 + 81 T + 148877 T^{2}$$)($$1 - 81 T + 148877 T^{2}$$)($$1 - 360 T + 148877 T^{2}$$)($$( 1 + 114 T + 148877 T^{2} )^{2}$$)($$( 1 + 150 T + 257074 T^{2} + 22331550 T^{3} + 22164361129 T^{4} )^{2}$$)
$59$ ($$1 + 156 T + 205379 T^{2}$$)($$1 + 84 T + 205379 T^{2}$$)($$1 - 84 T + 205379 T^{2}$$)($$1 - 156 T + 205379 T^{2}$$)($$1 - 159 T - 180098 T^{2} - 32655261 T^{3} + 42180533641 T^{4}$$)($$1 - 168 T - 388645 T^{2} - 1026648 T^{3} + 125802612624 T^{4} - 210851939592 T^{5} - 16393253496906445 T^{6} - 1455383297534029752 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)
$61$ ($$1 - 83 T + 226981 T^{2}$$)($$1 - 56 T + 226981 T^{2}$$)($$1 - 56 T + 226981 T^{2}$$)($$1 - 83 T + 226981 T^{2}$$)($$1 + 191 T - 190500 T^{2} + 43353371 T^{3} + 51520374361 T^{4}$$)($$1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 52845547546136 T^{5} + 21870141314372695 T^{6} - 12267159251383013909 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)
$67$ ($$1 - 47 T + 300763 T^{2}$$)($$1 + 142 T + 300763 T^{2}$$)($$1 + 142 T + 300763 T^{2}$$)($$1 - 47 T + 300763 T^{2}$$)($$1 - 421 T - 123522 T^{2} - 126621223 T^{3} + 90458382169 T^{4}$$)($$1 - 1166 T + 452161 T^{2} - 356643254 T^{3} + 324003162628 T^{4} - 107265095002802 T^{5} + 40901752539917209 T^{6} - 31722819106079908202 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)
$71$ ($$1 + 216 T + 357911 T^{2}$$)($$1 + 360 T + 357911 T^{2}$$)($$1 - 360 T + 357911 T^{2}$$)($$1 - 216 T + 357911 T^{2}$$)($$( 1 + 156 T + 357911 T^{2} )^{2}$$)($$( 1 - 312 T + 498238 T^{2} - 111668232 T^{3} + 128100283921 T^{4} )^{2}$$)
$73$ ($$1 + 511 T + 389017 T^{2}$$)($$1 + 1159 T + 389017 T^{2}$$)($$1 + 1159 T + 389017 T^{2}$$)($$1 + 511 T + 389017 T^{2}$$)($$( 1 - 182 T + 389017 T^{2} )^{2}$$)($$( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} )^{2}$$)
$79$ ($$1 + 529 T + 493039 T^{2}$$)($$1 + 160 T + 493039 T^{2}$$)($$1 + 160 T + 493039 T^{2}$$)($$1 + 529 T + 493039 T^{2}$$)($$1 + 1133 T + 790650 T^{2} + 558613187 T^{3} + 243087455521 T^{4}$$)($$1 + 349 T - 854801 T^{2} - 3307124 T^{3} + 650611367644 T^{4} - 1630541109836 T^{5} - 207791400066806321 T^{6} + 41828206997933793331 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)
$83$ ($$1 - 1128 T + 571787 T^{2}$$)($$1 + 735 T + 571787 T^{2}$$)($$1 - 735 T + 571787 T^{2}$$)($$1 + 1128 T + 571787 T^{2}$$)($$1 + 1083 T + 601102 T^{2} + 619245321 T^{3} + 326940373369 T^{4}$$)($$1 - 1221 T + 78743 T^{2} - 327867804 T^{3} + 814636885368 T^{4} - 187470548045748 T^{5} + 25744265820195167 T^{6} -$$$$22\!\cdots\!63$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$)
$89$ ($$1 + 36 T + 704969 T^{2}$$)($$1 - 954 T + 704969 T^{2}$$)($$1 + 954 T + 704969 T^{2}$$)($$1 - 36 T + 704969 T^{2}$$)($$( 1 - 1050 T + 704969 T^{2} )^{2}$$)($$( 1 - 492 T + 1092454 T^{2} - 346844748 T^{3} + 496981290961 T^{4} )^{2}$$)
$97$ ($$1 - 605 T + 912673 T^{2}$$)($$1 - 191 T + 912673 T^{2}$$)($$1 - 191 T + 912673 T^{2}$$)($$1 - 605 T + 912673 T^{2}$$)($$1 - 901 T - 100872 T^{2} - 822318373 T^{3} + 832972004929 T^{4}$$)($$1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 12879524553856 T^{5} - 1414980373408956377 T^{6} - 97309575507784347776 T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)