Properties

Label 27.4
Level 27
Weight 4
Dimension 56
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 216
Trace bound 2

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Defining parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(216\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(27))\).

Total New Old
Modular forms 96 72 24
Cusp forms 66 56 10
Eisenstein series 30 16 14

Trace form

\( 56q - 3q^{2} - 6q^{3} + 11q^{4} + 21q^{5} - 18q^{6} - 41q^{7} - 141q^{8} - 54q^{9} + O(q^{10}) \) \( 56q - 3q^{2} - 6q^{3} + 11q^{4} + 21q^{5} - 18q^{6} - 41q^{7} - 141q^{8} - 54q^{9} - 45q^{10} + 123q^{11} + 147q^{12} + 103q^{13} + 111q^{14} - 36q^{15} - 205q^{16} - 405q^{17} - 639q^{18} - 95q^{19} - 609q^{20} - 138q^{21} + 27q^{22} + 435q^{23} + 1170q^{24} + 425q^{25} + 2442q^{26} + 1125q^{27} + 490q^{28} + 429q^{29} + 459q^{30} - 437q^{31} - 1071q^{32} - 639q^{33} - 9q^{34} - 1263q^{35} - 2088q^{36} - 797q^{37} - 2085q^{38} - 768q^{39} - 621q^{40} - 1599q^{41} - 3078q^{42} + 175q^{43} - 1749q^{44} - 360q^{45} + 873q^{46} + 1383q^{47} + 2289q^{48} + 33q^{49} + 3930q^{50} + 2655q^{51} - 41q^{52} + 2628q^{53} + 5454q^{54} + 216q^{55} + 5973q^{56} + 3426q^{57} + 1647q^{58} + 3036q^{59} + 1314q^{60} + 391q^{61} - 2346q^{62} - 2610q^{63} - 2239q^{64} - 6825q^{65} - 11115q^{66} - 3191q^{67} - 10476q^{68} - 6138q^{69} - 4185q^{70} - 5841q^{71} - 36q^{72} + 715q^{73} + 4359q^{74} + 2604q^{75} + 5539q^{76} + 4557q^{77} + 6066q^{78} + 3631q^{79} + 9678q^{80} + 3438q^{81} + 1134q^{82} + 4281q^{83} + 7674q^{84} + 5823q^{85} + 3657q^{86} + 2880q^{87} + 2547q^{88} - 4410q^{89} - 12510q^{90} - 1657q^{91} - 19311q^{92} - 11802q^{93} - 9045q^{94} - 10245q^{95} - 14094q^{96} - 4685q^{97} - 3546q^{98} - 1242q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(27))\)

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
27.4.a \(\chi_{27}(1, \cdot)\) 27.4.a.a 1 1
27.4.a.b 1
27.4.a.c 2
27.4.c \(\chi_{27}(10, \cdot)\) 27.4.c.a 4 2
27.4.e \(\chi_{27}(4, \cdot)\) 27.4.e.a 48 6

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(27))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(27)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 8 T^{2} \))(\( 1 - 3 T + 8 T^{2} \))(\( 1 - 2 T^{2} + 64 T^{4} \))(\( 1 - 3 T - T^{2} + 18 T^{3} - 36 T^{4} + 144 T^{5} - 64 T^{6} - 1536 T^{7} + 4096 T^{8} \))
$3$ 1
$5$ (\( 1 + 15 T + 125 T^{2} \))(\( 1 - 15 T + 125 T^{2} \))(\( 1 - 38 T^{2} + 15625 T^{4} \))(\( 1 - 15 T - 73 T^{2} - 720 T^{3} + 45054 T^{4} - 90000 T^{5} - 1140625 T^{6} - 29296875 T^{7} + 244140625 T^{8} \))
$7$ (\( 1 + 25 T + 343 T^{2} \))(\( 1 + 25 T + 343 T^{2} \))(\( ( 1 - 11 T + 343 T^{2} )^{2} \))(\( 1 + 7 T - 575 T^{2} - 434 T^{3} + 254920 T^{4} - 148862 T^{5} - 67648175 T^{6} + 282475249 T^{7} + 13841287201 T^{8} \))
$11$ (\( 1 - 15 T + 1331 T^{2} \))(\( 1 + 15 T + 1331 T^{2} \))(\( 1 + 2374 T^{2} + 1771561 T^{4} \))(\( 1 - 66 T + 1133 T^{2} - 37026 T^{3} + 2818332 T^{4} - 49281606 T^{5} + 2007178613 T^{6} - 155624547606 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 - 20 T + 2197 T^{2} \))(\( 1 - 20 T + 2197 T^{2} \))(\( ( 1 - 29 T + 2197 T^{2} )^{2} \))(\( 1 - 11 T - 2447 T^{2} + 20086 T^{3} + 1501978 T^{4} + 44128942 T^{5} - 11811201623 T^{6} - 116649493103 T^{7} + 23298085122481 T^{8} \))
$17$ (\( 1 + 72 T + 4913 T^{2} \))(\( 1 - 72 T + 4913 T^{2} \))(\( 1 + 7234 T^{2} + 24137569 T^{4} \))(\( ( 1 + 99 T + 11608 T^{2} + 486387 T^{3} + 24137569 T^{4} )^{2} \))
$19$ (\( 1 - 2 T + 6859 T^{2} \))(\( 1 - 2 T + 6859 T^{2} \))(\( ( 1 - 29 T + 6859 T^{2} )^{2} \))(\( ( 1 + 77 T + 9186 T^{2} + 528143 T^{3} + 47045881 T^{4} )^{2} \))
$23$ (\( 1 + 114 T + 12167 T^{2} \))(\( 1 - 114 T + 12167 T^{2} \))(\( 1 + 17134 T^{2} + 148035889 T^{4} \))(\( 1 - 33 T - 20539 T^{2} + 89298 T^{3} + 306484632 T^{4} + 1086488766 T^{5} - 3040509124171 T^{6} - 59438037828279 T^{7} + 21914624432020321 T^{8} \))
$29$ (\( 1 + 30 T + 24389 T^{2} \))(\( 1 - 30 T + 24389 T^{2} \))(\( 1 - 24950 T^{2} + 594823321 T^{4} \))(\( 1 + 51 T - 46819 T^{2} + 32742 T^{3} + 1784077290 T^{4} + 798544638 T^{5} - 27849033065899 T^{6} + 739864444769319 T^{7} + 353814783205469041 T^{8} \))
$31$ (\( 1 - 101 T + 29791 T^{2} \))(\( 1 - 101 T + 29791 T^{2} \))(\( ( 1 + 268 T + 29791 T^{2} )^{2} \))(\( 1 + 43 T - 58121 T^{2} + 16684 T^{3} + 2653813660 T^{4} + 497033044 T^{5} - 51582601443401 T^{6} + 1136903752908853 T^{7} + 787662783788549761 T^{8} \))
$37$ (\( 1 + 430 T + 50653 T^{2} \))(\( 1 + 430 T + 50653 T^{2} \))(\( ( 1 - 83 T + 50653 T^{2} )^{2} \))(\( ( 1 + 50 T + 77874 T^{2} + 2532650 T^{3} + 2565726409 T^{4} )^{2} \))
$41$ (\( 1 - 30 T + 68921 T^{2} \))(\( 1 + 30 T + 68921 T^{2} \))(\( 1 + 64114 T^{2} + 4750104241 T^{4} \))(\( 1 - 132 T - 45541 T^{2} + 9883764 T^{3} - 1986392520 T^{4} + 681198898644 T^{5} - 216324497239381 T^{6} - 43214415340002852 T^{7} + 22563490300366186081 T^{8} \))
$43$ (\( 1 - 110 T + 79507 T^{2} \))(\( 1 - 110 T + 79507 T^{2} \))(\( ( 1 + 232 T + 79507 T^{2} )^{2} \))(\( 1 + 88 T - 152909 T^{2} + 144232 T^{3} + 18872321152 T^{4} + 11467453624 T^{5} - 966593302459541 T^{6} + 44228149850442184 T^{7} + 39959630797262576401 T^{8} \))
$47$ (\( 1 - 330 T + 103823 T^{2} \))(\( 1 + 330 T + 103823 T^{2} \))(\( 1 + 55294 T^{2} + 10779215329 T^{4} \))(\( 1 - 399 T - 19927 T^{2} + 11378682 T^{3} + 4778899632 T^{4} + 1181368901286 T^{5} - 214797423860983 T^{6} - 446533058768004033 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))
$53$ (\( 1 + 621 T + 148877 T^{2} \))(\( 1 - 621 T + 148877 T^{2} \))(\( 1 + 204442 T^{2} + 22164361129 T^{4} \))(\( ( 1 + 54 T + 81970 T^{2} + 8039358 T^{3} + 22164361129 T^{4} )^{2} \))
$59$ (\( 1 - 660 T + 205379 T^{2} \))(\( 1 + 660 T + 205379 T^{2} \))(\( 1 + 327526 T^{2} + 42180533641 T^{4} \))(\( 1 - 798 T + 219437 T^{2} - 5273982 T^{3} + 1228510332 T^{4} - 1083165149178 T^{5} + 9255969760580117 T^{6} - 6913070663286641322 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 + 376 T + 226981 T^{2} \))(\( 1 + 376 T + 226981 T^{2} \))(\( ( 1 - 767 T + 226981 T^{2} )^{2} \))(\( 1 + 439 T - 218465 T^{2} - 18778664 T^{3} + 73809546934 T^{4} - 4262399933384 T^{5} - 11255398584775865 T^{6} + 5133730134754187899 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 + 250 T + 300763 T^{2} \))(\( 1 + 250 T + 300763 T^{2} \))(\( ( 1 + 511 T + 300763 T^{2} )^{2} \))(\( 1 + 988 T + 166519 T^{2} + 205601812 T^{3} + 271446260584 T^{4} + 61837417782556 T^{5} + 15063039340399711 T^{6} + 26880055983539407636 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))
$71$ (\( 1 - 360 T + 357911 T^{2} \))(\( 1 + 360 T + 357911 T^{2} \))(\( 1 + 207790 T^{2} + 128100283921 T^{4} \))(\( ( 1 + 1368 T + 1012606 T^{2} + 489622248 T^{3} + 128100283921 T^{4} )^{2} \))
$73$ (\( 1 - 785 T + 389017 T^{2} \))(\( 1 - 785 T + 389017 T^{2} \))(\( ( 1 - 137 T + 389017 T^{2} )^{2} \))(\( ( 1 + 455 T + 342636 T^{2} + 177002735 T^{3} + 151334226289 T^{4} )^{2} \))
$79$ (\( 1 - 488 T + 493039 T^{2} \))(\( 1 - 488 T + 493039 T^{2} \))(\( ( 1 + 475 T + 493039 T^{2} )^{2} \))(\( 1 - 803 T + 285247 T^{2} + 503092348 T^{3} - 431718608228 T^{4} + 248044148165572 T^{5} + 69339967424998687 T^{6} - 96240831574042510157 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))
$83$ (\( 1 + 489 T + 571787 T^{2} \))(\( 1 - 489 T + 571787 T^{2} \))(\( 1 + 810646 T^{2} + 326940373369 T^{4} \))(\( 1 - 813 T - 553393 T^{2} - 57550644 T^{3} + 769801212072 T^{4} - 32906710080828 T^{5} - 180926514039791017 T^{6} - \)\(15\!\cdots\!39\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$89$ (\( 1 - 450 T + 704969 T^{2} \))(\( 1 + 450 T + 704969 T^{2} \))(\( 1 + 1345138 T^{2} + 496981290961 T^{4} \))(\( ( 1 - 396 T + 1406374 T^{2} - 279167724 T^{3} + 496981290961 T^{4} )^{2} \))
$97$ (\( 1 + 1105 T + 912673 T^{2} \))(\( 1 + 1105 T + 912673 T^{2} \))(\( ( 1 - 821 T + 912673 T^{2} )^{2} \))(\( 1 + 736 T - 1333241 T^{2} + 36498976 T^{3} + 2188025435632 T^{4} + 33311629922848 T^{5} - 1110552428823544889 T^{6} + \)\(55\!\cdots\!12\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))
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