Properties

Label 9.4
Level 9
Weight 4
Dimension 5
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 24
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(9))\).

Total New Old
Modular forms 13 10 3
Cusp forms 5 5 0
Eisenstein series 8 5 3

Trace form

\( 5q - 3q^{2} - 3q^{3} - 13q^{4} - 15q^{5} + 9q^{6} + 13q^{7} + 66q^{8} + 45q^{9} + O(q^{10}) \) \( 5q - 3q^{2} - 3q^{3} - 13q^{4} - 15q^{5} + 9q^{6} + 13q^{7} + 66q^{8} + 45q^{9} + 12q^{10} - 66q^{11} - 156q^{12} - 59q^{13} - 60q^{14} + 27q^{15} + 71q^{16} + 198q^{17} + 216q^{18} - 98q^{19} + 12q^{20} + 21q^{21} + 33q^{22} - 33q^{23} - 99q^{24} - 4q^{25} - 528q^{26} - 432q^{27} + 172q^{28} + 51q^{29} + 288q^{30} + 265q^{31} + 423q^{32} + 198q^{33} - 297q^{34} + 6q^{35} - 225q^{36} + 10q^{37} + 561q^{38} + 759q^{39} - 264q^{40} - 132q^{41} - 486q^{42} - 608q^{43} - 462q^{44} - 675q^{45} - 528q^{46} - 399q^{47} - 21q^{48} + 570q^{49} + 429q^{50} + 297q^{51} + 1330q^{52} + 108q^{53} + 1215q^{54} + 1254q^{55} - 66q^{56} - 1221q^{57} + 60q^{58} - 798q^{59} - 36q^{60} - 257q^{61} + 228q^{62} + 603q^{63} - 1966q^{64} - 165q^{65} - 990q^{66} - 1868q^{67} - 693q^{68} + 891q^{69} - 318q^{70} + 2736q^{71} + 891q^{72} + 280q^{73} - 816q^{74} - 363q^{75} + 1081q^{76} + 165q^{77} - 990q^{78} + 1687q^{79} + 192q^{80} - 567q^{81} + 3630q^{82} - 813q^{83} + 642q^{84} - 594q^{85} - 33q^{86} - 153q^{87} - 1221q^{88} - 792q^{89} - 756q^{90} - 2962q^{91} + 858q^{92} - 213q^{93} - 2100q^{94} + 132q^{95} + 1080q^{96} - 2066q^{97} - 846q^{98} + 297q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.4.a \(\chi_{9}(1, \cdot)\) 9.4.a.a 1 1
9.4.c \(\chi_{9}(4, \cdot)\) 9.4.c.a 4 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T^{2} \))(\( 1 + 3 T - T^{2} - 18 T^{3} - 36 T^{4} - 144 T^{5} - 64 T^{6} + 1536 T^{7} + 4096 T^{8} \))
$3$ (\( 1 + 3 T - 18 T^{2} + 81 T^{3} + 729 T^{4} \))
$5$ (\( 1 + 125 T^{2} \))(\( 1 + 15 T - 73 T^{2} + 720 T^{3} + 45054 T^{4} + 90000 T^{5} - 1140625 T^{6} + 29296875 T^{7} + 244140625 T^{8} \))
$7$ (\( 1 - 20 T + 343 T^{2} \))(\( 1 + 7 T - 575 T^{2} - 434 T^{3} + 254920 T^{4} - 148862 T^{5} - 67648175 T^{6} + 282475249 T^{7} + 13841287201 T^{8} \))
$11$ (\( 1 + 1331 T^{2} \))(\( 1 + 66 T + 1133 T^{2} + 37026 T^{3} + 2818332 T^{4} + 49281606 T^{5} + 2007178613 T^{6} + 155624547606 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 + 70 T + 2197 T^{2} \))(\( 1 - 11 T - 2447 T^{2} + 20086 T^{3} + 1501978 T^{4} + 44128942 T^{5} - 11811201623 T^{6} - 116649493103 T^{7} + 23298085122481 T^{8} \))
$17$ (\( 1 + 4913 T^{2} \))(\( ( 1 - 99 T + 11608 T^{2} - 486387 T^{3} + 24137569 T^{4} )^{2} \))
$19$ (\( 1 - 56 T + 6859 T^{2} \))(\( ( 1 + 77 T + 9186 T^{2} + 528143 T^{3} + 47045881 T^{4} )^{2} \))
$23$ (\( 1 + 12167 T^{2} \))(\( 1 + 33 T - 20539 T^{2} - 89298 T^{3} + 306484632 T^{4} - 1086488766 T^{5} - 3040509124171 T^{6} + 59438037828279 T^{7} + 21914624432020321 T^{8} \))
$29$ (\( 1 + 24389 T^{2} \))(\( 1 - 51 T - 46819 T^{2} - 32742 T^{3} + 1784077290 T^{4} - 798544638 T^{5} - 27849033065899 T^{6} - 739864444769319 T^{7} + 353814783205469041 T^{8} \))
$31$ (\( 1 - 308 T + 29791 T^{2} \))(\( 1 + 43 T - 58121 T^{2} + 16684 T^{3} + 2653813660 T^{4} + 497033044 T^{5} - 51582601443401 T^{6} + 1136903752908853 T^{7} + 787662783788549761 T^{8} \))
$37$ (\( 1 - 110 T + 50653 T^{2} \))(\( ( 1 + 50 T + 77874 T^{2} + 2532650 T^{3} + 2565726409 T^{4} )^{2} \))
$41$ (\( 1 + 68921 T^{2} \))(\( 1 + 132 T - 45541 T^{2} - 9883764 T^{3} - 1986392520 T^{4} - 681198898644 T^{5} - 216324497239381 T^{6} + 43214415340002852 T^{7} + 22563490300366186081 T^{8} \))
$43$ (\( 1 + 520 T + 79507 T^{2} \))(\( 1 + 88 T - 152909 T^{2} + 144232 T^{3} + 18872321152 T^{4} + 11467453624 T^{5} - 966593302459541 T^{6} + 44228149850442184 T^{7} + 39959630797262576401 T^{8} \))
$47$ (\( 1 + 103823 T^{2} \))(\( 1 + 399 T - 19927 T^{2} - 11378682 T^{3} + 4778899632 T^{4} - 1181368901286 T^{5} - 214797423860983 T^{6} + 446533058768004033 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))
$53$ (\( 1 + 148877 T^{2} \))(\( ( 1 - 54 T + 81970 T^{2} - 8039358 T^{3} + 22164361129 T^{4} )^{2} \))
$59$ (\( 1 + 205379 T^{2} \))(\( 1 + 798 T + 219437 T^{2} + 5273982 T^{3} + 1228510332 T^{4} + 1083165149178 T^{5} + 9255969760580117 T^{6} + 6913070663286641322 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 - 182 T + 226981 T^{2} \))(\( 1 + 439 T - 218465 T^{2} - 18778664 T^{3} + 73809546934 T^{4} - 4262399933384 T^{5} - 11255398584775865 T^{6} + 5133730134754187899 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 + 880 T + 300763 T^{2} \))(\( 1 + 988 T + 166519 T^{2} + 205601812 T^{3} + 271446260584 T^{4} + 61837417782556 T^{5} + 15063039340399711 T^{6} + 26880055983539407636 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))
$71$ (\( 1 + 357911 T^{2} \))(\( ( 1 - 1368 T + 1012606 T^{2} - 489622248 T^{3} + 128100283921 T^{4} )^{2} \))
$73$ (\( 1 - 1190 T + 389017 T^{2} \))(\( ( 1 + 455 T + 342636 T^{2} + 177002735 T^{3} + 151334226289 T^{4} )^{2} \))
$79$ (\( 1 - 884 T + 493039 T^{2} \))(\( 1 - 803 T + 285247 T^{2} + 503092348 T^{3} - 431718608228 T^{4} + 248044148165572 T^{5} + 69339967424998687 T^{6} - 96240831574042510157 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))
$83$ (\( 1 + 571787 T^{2} \))(\( 1 + 813 T - 553393 T^{2} + 57550644 T^{3} + 769801212072 T^{4} + 32906710080828 T^{5} - 180926514039791017 T^{6} + \)\(15\!\cdots\!39\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$89$ (\( 1 + 704969 T^{2} \))(\( ( 1 + 396 T + 1406374 T^{2} + 279167724 T^{3} + 496981290961 T^{4} )^{2} \))
$97$ (\( 1 + 1330 T + 912673 T^{2} \))(\( 1 + 736 T - 1333241 T^{2} + 36498976 T^{3} + 2188025435632 T^{4} + 33311629922848 T^{5} - 1110552428823544889 T^{6} + \)\(55\!\cdots\!12\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))
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