Properties

Label 18.4
Level 18
Weight 4
Dimension 7
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 72
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(72\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(18))\).

Total New Old
Modular forms 35 7 28
Cusp forms 19 7 12
Eisenstein series 16 0 16

Trace form

\( 7q + 4q^{2} + 3q^{3} - 8q^{4} + 12q^{5} - 18q^{6} - 4q^{7} - 8q^{8} - 105q^{9} + O(q^{10}) \) \( 7q + 4q^{2} + 3q^{3} - 8q^{4} + 12q^{5} - 18q^{6} - 4q^{7} - 8q^{8} - 105q^{9} - 12q^{10} + 27q^{11} + 24q^{12} + 14q^{13} + 68q^{14} + 90q^{15} - 32q^{16} + 48q^{17} + 156q^{18} + 230q^{19} + 48q^{20} - 36q^{21} - 42q^{22} - 432q^{23} - 24q^{24} - 308q^{25} - 316q^{26} - 160q^{28} - 378q^{29} - 288q^{30} - 94q^{31} + 64q^{32} + 765q^{33} + 342q^{34} + 1428q^{35} + 516q^{36} + 446q^{37} + 362q^{38} - 582q^{39} - 48q^{40} - 249q^{41} - 816q^{42} + 77q^{43} - 360q^{44} - 702q^{45} + 168q^{46} - 564q^{47} - 144q^{48} - 672q^{49} + 436q^{50} - 153q^{51} + 56q^{52} + 330q^{53} + 954q^{54} - 1332q^{55} + 272q^{56} + 987q^{57} + 192q^{58} + 333q^{59} - 936q^{60} + 320q^{61} - 1840q^{62} - 1794q^{63} + 448q^{64} + 186q^{65} + 288q^{66} + 2471q^{67} + 660q^{68} + 1494q^{69} + 408q^{70} - 1104q^{71} - 24q^{72} - 40q^{73} + 1364q^{74} + 3231q^{75} - 340q^{76} + 900q^{77} + 132q^{78} - 2002q^{79} - 672q^{80} + 315q^{81} - 3000q^{82} + 354q^{83} + 744q^{84} - 648q^{85} - 190q^{86} - 3204q^{87} - 168q^{88} - 3894q^{89} - 432q^{90} + 1900q^{91} - 1728q^{92} - 2634q^{93} + 804q^{94} - 2136q^{95} + 384q^{96} + 2183q^{97} + 2430q^{98} + 1152q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)

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
18.4.a \(\chi_{18}(1, \cdot)\) 18.4.a.a 1 1
18.4.c \(\chi_{18}(7, \cdot)\) 18.4.c.a 2 2
18.4.c.b 4

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(18))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T \))(\( 1 + 2 T + 4 T^{2} \))(\( ( 1 - 2 T + 4 T^{2} )^{2} \))
$3$ (\( 1 + 27 T^{2} \))(\( 1 - 3 T + 30 T^{2} - 81 T^{3} + 729 T^{4} \))
$5$ (\( 1 + 6 T + 125 T^{2} \))(\( 1 - 9 T - 44 T^{2} - 1125 T^{3} + 15625 T^{4} \))(\( 1 - 9 T + 47 T^{2} + 1944 T^{3} - 24594 T^{4} + 243000 T^{5} + 734375 T^{6} - 17578125 T^{7} + 244140625 T^{8} \))
$7$ (\( 1 + 16 T + 343 T^{2} \))(\( 1 - 31 T + 618 T^{2} - 10633 T^{3} + 117649 T^{4} \))(\( 1 + 19 T - 179 T^{2} - 2774 T^{3} + 50128 T^{4} - 951482 T^{5} - 21059171 T^{6} + 766718533 T^{7} + 13841287201 T^{8} \))
$11$ (\( 1 + 12 T + 1331 T^{2} \))(\( 1 - 15 T - 1106 T^{2} - 19965 T^{3} + 1771561 T^{4} \))(\( 1 - 24 T - 1285 T^{2} + 19224 T^{3} + 925104 T^{4} + 25587144 T^{5} - 2276455885 T^{6} - 56590744584 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 - 38 T + 2197 T^{2} \))(\( 1 - 37 T - 828 T^{2} - 81289 T^{3} + 4826809 T^{4} \))(\( 1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 93007798 T^{5} - 6598247903 T^{6} + 646874461753 T^{7} + 23298085122481 T^{8} \))
$17$ (\( 1 - 126 T + 4913 T^{2} \))(\( ( 1 + 42 T + 4913 T^{2} )^{2} \))(\( ( 1 - 3 T + 9592 T^{2} - 14739 T^{3} + 24137569 T^{4} )^{2} \))
$19$ (\( 1 - 20 T + 6859 T^{2} \))(\( ( 1 + 28 T + 6859 T^{2} )^{2} \))(\( ( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} )^{2} \))
$23$ (\( 1 + 168 T + 12167 T^{2} \))(\( 1 + 195 T + 25858 T^{2} + 2372565 T^{3} + 148035889 T^{4} \))(\( 1 + 69 T - 20527 T^{2} + 65826 T^{3} + 433519968 T^{4} + 800904942 T^{5} - 3038732693503 T^{6} + 124279533640947 T^{7} + 21914624432020321 T^{8} \))
$29$ (\( 1 + 30 T + 24389 T^{2} \))(\( 1 + 111 T - 12068 T^{2} + 2707179 T^{3} + 594823321 T^{4} \))(\( 1 + 237 T + 4925 T^{2} + 584442 T^{3} + 661218474 T^{4} + 14253955938 T^{5} + 2929504855925 T^{6} + 3438193596280953 T^{7} + 353814783205469041 T^{8} \))
$31$ (\( 1 + 88 T + 29791 T^{2} \))(\( 1 - 205 T + 12234 T^{2} - 6107155 T^{3} + 887503681 T^{4} \))(\( 1 + 211 T - 24065 T^{2} + 1899844 T^{3} + 2490210604 T^{4} + 56598252604 T^{5} - 21357776083265 T^{6} + 5578760275901581 T^{7} + 787662783788549761 T^{8} \))
$37$ (\( 1 - 254 T + 50653 T^{2} \))(\( ( 1 + 166 T + 50653 T^{2} )^{2} \))(\( ( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} )^{2} \))
$41$ (\( 1 + 42 T + 68921 T^{2} \))(\( 1 - 261 T - 800 T^{2} - 17988381 T^{3} + 4750104241 T^{4} \))(\( 1 + 468 T + 27371 T^{2} + 25183548 T^{3} + 16885415064 T^{4} + 1735675311708 T^{5} + 130015103180411 T^{6} + 153214745296373748 T^{7} + 22563490300366186081 T^{8} \))
$43$ (\( 1 + 52 T + 79507 T^{2} \))(\( 1 - 43 T - 77658 T^{2} - 3418801 T^{3} + 6321363049 T^{4} \))(\( 1 - 86 T - 17387 T^{2} + 11543866 T^{3} - 6295199732 T^{4} + 917818154062 T^{5} - 109909539332963 T^{6} - 43222964626568498 T^{7} + 39959630797262576401 T^{8} \))
$47$ (\( 1 - 96 T + 103823 T^{2} \))(\( 1 + 177 T - 72494 T^{2} + 18376671 T^{3} + 10779215329 T^{4} \))(\( 1 + 483 T + 20477 T^{2} + 2495178 T^{3} + 10288968168 T^{4} + 259056865494 T^{5} + 220725992291933 T^{6} + 540540018508636461 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))
$53$ (\( 1 + 198 T + 148877 T^{2} \))(\( ( 1 - 114 T + 148877 T^{2} )^{2} \))(\( ( 1 - 150 T + 257074 T^{2} - 22331550 T^{3} + 22164361129 T^{4} )^{2} \))
$59$ (\( 1 - 660 T + 205379 T^{2} \))(\( 1 + 159 T - 180098 T^{2} + 32655261 T^{3} + 42180533641 T^{4} \))(\( 1 + 168 T - 388645 T^{2} + 1026648 T^{3} + 125802612624 T^{4} + 210851939592 T^{5} - 16393253496906445 T^{6} + 1455383297534029752 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 + 538 T + 226981 T^{2} \))(\( 1 + 191 T - 190500 T^{2} + 43353371 T^{3} + 51520374361 T^{4} \))(\( 1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 52845547546136 T^{5} + 21870141314372695 T^{6} - 12267159251383013909 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 - 884 T + 300763 T^{2} \))(\( 1 - 421 T - 123522 T^{2} - 126621223 T^{3} + 90458382169 T^{4} \))(\( 1 - 1166 T + 452161 T^{2} - 356643254 T^{3} + 324003162628 T^{4} - 107265095002802 T^{5} + 40901752539917209 T^{6} - 31722819106079908202 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))
$71$ (\( 1 + 792 T + 357911 T^{2} \))(\( ( 1 - 156 T + 357911 T^{2} )^{2} \))(\( ( 1 + 312 T + 498238 T^{2} + 111668232 T^{3} + 128100283921 T^{4} )^{2} \))
$73$ (\( 1 - 218 T + 389017 T^{2} \))(\( ( 1 - 182 T + 389017 T^{2} )^{2} \))(\( ( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} )^{2} \))
$79$ (\( 1 + 520 T + 493039 T^{2} \))(\( 1 + 1133 T + 790650 T^{2} + 558613187 T^{3} + 243087455521 T^{4} \))(\( 1 + 349 T - 854801 T^{2} - 3307124 T^{3} + 650611367644 T^{4} - 1630541109836 T^{5} - 207791400066806321 T^{6} + 41828206997933793331 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))
$83$ (\( 1 - 492 T + 571787 T^{2} \))(\( 1 - 1083 T + 601102 T^{2} - 619245321 T^{3} + 326940373369 T^{4} \))(\( 1 + 1221 T + 78743 T^{2} + 327867804 T^{3} + 814636885368 T^{4} + 187470548045748 T^{5} + 25744265820195167 T^{6} + \)\(22\!\cdots\!63\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$89$ (\( 1 + 810 T + 704969 T^{2} \))(\( ( 1 + 1050 T + 704969 T^{2} )^{2} \))(\( ( 1 + 492 T + 1092454 T^{2} + 346844748 T^{3} + 496981290961 T^{4} )^{2} \))
$97$ (\( 1 - 1154 T + 912673 T^{2} \))(\( 1 - 901 T - 100872 T^{2} - 822318373 T^{3} + 832972004929 T^{4} \))(\( 1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 12879524553856 T^{5} - 1414980373408956377 T^{6} - 97309575507784347776 T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))
show more
show less