Properties

Label 13.4
Level 13
Weight 4
Dimension 15
Nonzero newspaces 4
Newform subspaces 7
Sturm bound 56
Trace bound 1

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 7 \)
Sturm bound: \(56\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(13))\).

Total New Old
Modular forms 27 25 2
Cusp forms 15 15 0
Eisenstein series 12 10 2

Trace form

\( 15q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} - 42q^{7} - 78q^{8} - 6q^{9} + O(q^{10}) \) \( 15q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} - 42q^{7} - 78q^{8} - 6q^{9} + 84q^{10} + 54q^{11} + 228q^{12} + 138q^{13} + 108q^{14} + 66q^{15} - 6q^{16} - 195q^{17} - 834q^{18} - 450q^{19} - 522q^{20} - 6q^{21} + 384q^{22} + 222q^{23} + 1050q^{24} + 363q^{25} + 744q^{26} + 816q^{27} - 60q^{28} - 435q^{29} - 582q^{30} - 318q^{31} - 660q^{32} - 606q^{33} - 702q^{34} - 6q^{35} - 162q^{36} + 177q^{37} + 1356q^{38} + 66q^{39} - 228q^{40} - 477q^{41} - 636q^{42} - 426q^{43} + 228q^{44} + 279q^{45} + 822q^{46} + 126q^{47} - 390q^{48} + 966q^{49} - 102q^{50} - 396q^{51} - 336q^{52} + 936q^{53} + 720q^{54} + 1398q^{55} + 996q^{56} + 1158q^{57} + 1020q^{58} + 486q^{59} + 1152q^{60} - 1137q^{61} - 2622q^{62} - 3186q^{63} - 4692q^{64} - 4065q^{65} - 4152q^{66} - 918q^{67} + 1320q^{68} + 2022q^{69} + 1980q^{70} + 1938q^{71} + 2940q^{72} + 2154q^{73} + 2568q^{74} + 612q^{75} - 1842q^{76} + 372q^{77} + 2940q^{78} - 972q^{79} + 3384q^{80} + 456q^{81} + 3552q^{82} + 3426q^{83} + 2268q^{84} + 1011q^{85} - 2808q^{86} - 3906q^{87} - 444q^{88} - 3726q^{89} - 6936q^{90} - 2730q^{91} - 4812q^{92} - 4194q^{93} - 1722q^{94} - 1962q^{95} + 1548q^{96} + 1806q^{97} - 2730q^{98} + 5190q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(13))\)

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
13.4.a \(\chi_{13}(1, \cdot)\) 13.4.a.a 1 1
13.4.a.b 2
13.4.b \(\chi_{13}(12, \cdot)\) 13.4.b.a 2 1
13.4.c \(\chi_{13}(3, \cdot)\) 13.4.c.a 2 2
13.4.c.b 4
13.4.e \(\chi_{13}(4, \cdot)\) 13.4.e.a 2 2
13.4.e.b 2

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 5 T + 8 T^{2} \))(\( 1 - T + 12 T^{2} - 8 T^{3} + 64 T^{4} \))(\( 1 - 7 T^{2} + 64 T^{4} \))(\( 1 + 4 T + 8 T^{2} + 32 T^{3} + 64 T^{4} \))(\( 1 - 5 T + 7 T^{2} - 10 T^{3} + 60 T^{4} - 80 T^{5} + 448 T^{6} - 2560 T^{7} + 4096 T^{8} \))(\( 1 + 6 T + 20 T^{2} + 48 T^{3} + 64 T^{4} \))(\( 1 - 3 T + 11 T^{2} - 24 T^{3} + 64 T^{4} \))
$3$ (\( 1 + 7 T + 27 T^{2} \))(\( 1 - 5 T + 22 T^{2} - 135 T^{3} + 729 T^{4} \))(\( ( 1 + T + 27 T^{2} )^{2} \))(\( 1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4} \))(\( 1 + 5 T + 3 T^{2} - 160 T^{3} - 920 T^{4} - 4320 T^{5} + 2187 T^{6} + 98415 T^{7} + 531441 T^{8} \))(\( 1 - 7 T + 22 T^{2} - 189 T^{3} + 729 T^{4} \))(\( 1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4} \))
$5$ (\( 1 + 7 T + 125 T^{2} \))(\( 1 + 3 T + 248 T^{2} + 375 T^{3} + 15625 T^{4} \))(\( 1 - 169 T^{2} + 15625 T^{4} \))(\( ( 1 - 17 T + 125 T^{2} )^{2} \))(\( ( 1 + 15 T + 200 T^{2} + 1875 T^{3} + 15625 T^{4} )^{2} \))(\( 1 - 58 T^{2} + 15625 T^{4} \))(\( 1 - 247 T^{2} + 15625 T^{4} \))
$7$ (\( 1 + 13 T + 343 T^{2} \))(\( 1 + 9 T + 192 T^{2} + 3087 T^{3} + 117649 T^{4} \))(\( 1 - 461 T^{2} + 117649 T^{4} \))(\( ( 1 - 17 T + 343 T^{2} )( 1 + 37 T + 343 T^{2} ) \))(\( 1 + 15 T - 513 T^{2} + 780 T^{3} + 349820 T^{4} + 267540 T^{5} - 60353937 T^{6} + 605304105 T^{7} + 13841287201 T^{8} \))(\( 1 - 39 T + 850 T^{2} - 13377 T^{3} + 117649 T^{4} \))(\( 1 + 24 T + 535 T^{2} + 8232 T^{3} + 117649 T^{4} \))
$11$ (\( 1 + 26 T + 1331 T^{2} \))(\( 1 - 80 T + 3650 T^{2} - 106480 T^{3} + 1771561 T^{4} \))(\( 1 - 358 T^{2} + 1771561 T^{4} \))(\( 1 - 32 T - 307 T^{2} - 42592 T^{3} + 1771561 T^{4} \))(\( 1 + 17 T - 1489 T^{2} - 15028 T^{3} + 1005064 T^{4} - 20002268 T^{5} - 2637854329 T^{6} + 40085110747 T^{7} + 3138428376721 T^{8} \))(\( 1 + 39 T + 1838 T^{2} + 51909 T^{3} + 1771561 T^{4} \))(\( 1 - 24 T + 1523 T^{2} - 31944 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 - 13 T \))(\( ( 1 + 13 T )^{2} \))(\( 1 - 52 T + 2197 T^{2} \))(\( 1 + 91 T + 2197 T^{2} \))(\( 1 - 125 T + 7956 T^{2} - 274625 T^{3} + 4826809 T^{4} \))(\( 1 + 26 T + 2197 T^{2} \))(\( 1 - 91 T + 2197 T^{2} \))
$17$ (\( 1 - 77 T + 4913 T^{2} \))(\( 1 - 19 T + 8688 T^{2} - 93347 T^{3} + 24137569 T^{4} \))(\( ( 1 + 45 T + 4913 T^{2} )^{2} \))(\( 1 - 13 T - 4744 T^{2} - 63869 T^{3} + 24137569 T^{4} \))(\( 1 + 70 T - 5063 T^{2} + 9590 T^{3} + 51050100 T^{4} + 47115670 T^{5} - 122208511847 T^{6} + 8301151354790 T^{7} + 582622237229761 T^{8} \))(\( 1 + 27 T - 4184 T^{2} + 132651 T^{3} + 24137569 T^{4} \))(\( 1 + 117 T + 8776 T^{2} + 574821 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 + 126 T + 6859 T^{2} \))(\( 1 + 84 T + 11130 T^{2} + 576156 T^{3} + 47045881 T^{4} \))(\( 1 - 13682 T^{2} + 47045881 T^{4} \))(\( 1 + 30 T - 5959 T^{2} + 205770 T^{3} + 47045881 T^{4} \))(\( 1 - 141 T + 1299 T^{2} - 685824 T^{3} + 161881064 T^{4} - 4704066816 T^{5} + 61112599419 T^{6} - 45498965386839 T^{7} + 2213314919066161 T^{8} \))(\( 1 + 153 T + 14662 T^{2} + 1049427 T^{3} + 47045881 T^{4} \))(\( 1 + 198 T + 19927 T^{2} + 1358082 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 + 96 T + 12167 T^{2} \))(\( 1 - 196 T + 33326 T^{2} - 2384732 T^{3} + 148035889 T^{4} \))(\( ( 1 - 162 T + 12167 T^{2} )^{2} \))(\( 1 + 78 T - 6083 T^{2} + 949026 T^{3} + 148035889 T^{4} \))(\( 1 + 145 T - 3937 T^{2} + 91060 T^{3} + 219254380 T^{4} + 1107927020 T^{5} - 582817294993 T^{6} + 261167135912135 T^{7} + 21914624432020321 T^{8} \))(\( 1 + 57 T - 8918 T^{2} + 693519 T^{3} + 148035889 T^{4} \))(\( 1 - 78 T - 6083 T^{2} - 949026 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 + 82 T + 24389 T^{2} \))(\( 1 + 44 T + 10094 T^{2} + 1073116 T^{3} + 594823321 T^{4} \))(\( ( 1 + 144 T + 24389 T^{2} )^{2} \))(\( 1 + 197 T + 14420 T^{2} + 4804633 T^{3} + 594823321 T^{4} \))(\( 1 + 34 T - 32611 T^{2} - 510374 T^{3} + 517193284 T^{4} - 12447511486 T^{5} - 19397783321131 T^{6} + 493242963179546 T^{7} + 353814783205469041 T^{8} \))(\( 1 - 69 T - 19628 T^{2} - 1682841 T^{3} + 594823321 T^{4} \))(\( 1 - 141 T - 4508 T^{2} - 3438849 T^{3} + 594823321 T^{4} \))
$31$ (\( 1 - 196 T + 29791 T^{2} \))(\( 1 + 86 T + 56518 T^{2} + 2562026 T^{3} + 887503681 T^{4} \))(\( 1 + 10114 T^{2} + 887503681 T^{4} \))(\( ( 1 + 74 T + 29791 T^{2} )^{2} \))(\( ( 1 + 140 T + 21982 T^{2} + 4170740 T^{3} + 887503681 T^{4} )^{2} \))(\( 1 - 54290 T^{2} + 887503681 T^{4} \))(\( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \))
$37$ (\( 1 + 131 T + 50653 T^{2} \))(\( 1 - 209 T + 112120 T^{2} - 10586477 T^{3} + 2565726409 T^{4} \))(\( 1 - 9497 T^{2} + 2565726409 T^{4} \))(\( 1 - 227 T + 876 T^{2} - 11498231 T^{3} + 2565726409 T^{4} \))(\( 1 - 190 T - 66003 T^{2} - 151430 T^{3} + 6030722900 T^{4} - 7670383790 T^{5} - 169345640173227 T^{6} - 24692730561064630 T^{7} + 6582952005840035281 T^{8} \))(\( 1 + 69 T + 52240 T^{2} + 3495057 T^{3} + 2565726409 T^{4} \))(\( 1 + 249 T + 71320 T^{2} + 12612597 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 - 336 T + 68921 T^{2} \))(\( 1 + 230 T + 149010 T^{2} + 15851830 T^{3} + 4750104241 T^{4} \))(\( 1 - 100978 T^{2} + 4750104241 T^{4} \))(\( 1 - 165 T - 41696 T^{2} - 11371965 T^{3} + 4750104241 T^{4} \))(\( 1 + 538 T + 80941 T^{2} + 38015618 T^{3} + 18774626844 T^{4} + 2620074408178 T^{5} + 384478187370781 T^{6} + 176131480703951018 T^{7} + 22563490300366186081 T^{8} \))(\( 1 + 681 T + 223508 T^{2} + 46935201 T^{3} + 4750104241 T^{4} \))(\( 1 - 471 T + 142868 T^{2} - 32461791 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 + 201 T + 79507 T^{2} \))(\( 1 - 287 T + 92698 T^{2} - 22818509 T^{3} + 6321363049 T^{4} \))(\( ( 1 + 97 T + 79507 T^{2} )^{2} \))(\( 1 - 156 T - 55171 T^{2} - 12403092 T^{3} + 6321363049 T^{4} \))(\( 1 + 455 T + 36243 T^{2} + 5354440 T^{3} + 6385191800 T^{4} + 425715461080 T^{5} + 229105160984907 T^{6} + 228679638431263565 T^{7} + 39959630797262576401 T^{8} \))(\( 1 - 85 T - 72282 T^{2} - 6758095 T^{3} + 6321363049 T^{4} \))(\( 1 + 104 T - 68691 T^{2} + 8268728 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 + 105 T + 103823 T^{2} \))(\( 1 - 435 T + 192728 T^{2} - 45163005 T^{3} + 10779215329 T^{4} \))(\( 1 - 195325 T^{2} + 10779215329 T^{4} \))(\( ( 1 + 162 T + 103823 T^{2} )^{2} \))(\( ( 1 - 60 T + 125246 T^{2} - 6229380 T^{3} + 10779215329 T^{4} )^{2} \))(\( 1 - 90034 T^{2} + 10779215329 T^{4} \))(\( 1 - 116818 T^{2} + 10779215329 T^{4} \))
$53$ (\( 1 + 432 T + 148877 T^{2} \))(\( 1 + 118 T + 297410 T^{2} + 17567486 T^{3} + 22164361129 T^{4} \))(\( ( 1 + 414 T + 148877 T^{2} )^{2} \))(\( ( 1 - 93 T + 148877 T^{2} )^{2} \))(\( ( 1 - 545 T + 256304 T^{2} - 81137965 T^{3} + 22164361129 T^{4} )^{2} \))(\( ( 1 - 426 T + 148877 T^{2} )^{2} \))(\( ( 1 - 93 T + 148877 T^{2} )^{2} \))
$59$ (\( 1 + 294 T + 205379 T^{2} \))(\( 1 + 368 T + 379266 T^{2} + 75579472 T^{3} + 42180533641 T^{4} \))(\( 1 - 138274 T^{2} + 42180533641 T^{4} \))(\( 1 - 864 T + 541117 T^{2} - 177447456 T^{3} + 42180533641 T^{4} \))(\( 1 - 809 T + 92959 T^{2} - 121968076 T^{3} + 138709769544 T^{4} - 25049681480804 T^{5} + 3921060226733719 T^{6} - 7008363617291845651 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 + 33 T + 205742 T^{2} + 6777507 T^{3} + 42180533641 T^{4} \))(\( 1 + 492 T + 286067 T^{2} + 101046468 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 + 56 T + 226981 T^{2} \))(\( 1 + 1058 T + 580378 T^{2} + 240145898 T^{3} + 51520374361 T^{4} \))(\( ( 1 - 376 T + 226981 T^{2} )^{2} \))(\( 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} \))(\( 1 + 502 T - 94959 T^{2} - 53713498 T^{3} + 11662829084 T^{4} - 12191943489538 T^{5} - 4892323228946199 T^{6} + 5870461338602738782 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 - 17 T - 226692 T^{2} - 3858677 T^{3} + 51520374361 T^{4} \))(\( 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 - 478 T + 300763 T^{2} \))(\( 1 - 68 T + 373930 T^{2} - 20451884 T^{3} + 90458382169 T^{4} \))(\( 1 - 600230 T^{2} + 90458382169 T^{4} \))(\( 1 + 862 T + 442281 T^{2} + 259257706 T^{3} + 90458382169 T^{4} \))(\( 1 - 475 T - 397113 T^{2} - 10075700 T^{3} + 229484582600 T^{4} - 3030397759100 T^{5} - 35922199518278097 T^{6} - 12923103838240099825 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 285 T + 327838 T^{2} - 85717455 T^{3} + 90458382169 T^{4} \))(\( 1 + 1362 T + 919111 T^{2} + 409639206 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 - 9 T + 357911 T^{2} \))(\( 1 + 131 T + 493328 T^{2} + 46886341 T^{3} + 128100283921 T^{4} \))(\( 1 - 588373 T^{2} + 128100283921 T^{4} \))(\( 1 + 654 T + 69805 T^{2} + 234073794 T^{3} + 128100283921 T^{4} \))(\( 1 + 127 T - 656869 T^{2} - 5438648 T^{3} + 319053277564 T^{4} - 1946551944328 T^{5} - 84145105398903349 T^{6} + 5822759591243026937 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \))(\( 1 - 1011 T + 698618 T^{2} - 361848021 T^{3} + 128100283921 T^{4} \))(\( 1 - 1830 T + 1474211 T^{2} - 654977130 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 - 98 T + 389017 T^{2} \))(\( 1 - 456 T + 542718 T^{2} - 177391752 T^{3} + 151334226289 T^{4} \))(\( ( 1 - 592 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} ) \))(\( ( 1 - 215 T + 389017 T^{2} )^{2} \))(\( ( 1 - 585 T + 832884 T^{2} - 227574945 T^{3} + 151334226289 T^{4} )^{2} \))(\( 1 + 231166 T^{2} + 151334226289 T^{4} \))(\( 1 - 567359 T^{2} + 151334226289 T^{4} \))
$79$ (\( 1 - 1304 T + 493039 T^{2} \))(\( 1 + 1008 T + 1233294 T^{2} + 496983312 T^{3} + 243087455521 T^{4} \))(\( ( 1 + 830 T + 493039 T^{2} )^{2} \))(\( ( 1 + 76 T + 493039 T^{2} )^{2} \))(\( ( 1 - 240 T + 993678 T^{2} - 118329360 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 + 1244 T + 493039 T^{2} )^{2} \))(\( ( 1 - 1276 T + 493039 T^{2} )^{2} \))
$83$ (\( 1 + 308 T + 571787 T^{2} \))(\( 1 - 1958 T + 1961238 T^{2} - 1119558946 T^{3} + 326940373369 T^{4} \))(\( 1 - 951730 T^{2} + 326940373369 T^{4} \))(\( ( 1 - 628 T + 571787 T^{2} )^{2} \))(\( ( 1 - 260 T + 1117974 T^{2} - 148664620 T^{3} + 326940373369 T^{4} )^{2} \))(\( 1 - 962026 T^{2} + 326940373369 T^{4} \))(\( 1 - 519766 T^{2} + 326940373369 T^{4} \))
$89$ (\( 1 + 1190 T + 704969 T^{2} \))(\( 1 + 720 T + 899726 T^{2} + 507577680 T^{3} + 496981290961 T^{4} \))(\( 1 - 1218094 T^{2} + 496981290961 T^{4} \))(\( 1 - 266 T - 634213 T^{2} - 187521754 T^{3} + 496981290961 T^{4} \))(\( 1 + 921 T - 707351 T^{2} + 134147334 T^{3} + 1324901569974 T^{4} + 94569711902646 T^{5} - 351540213142554311 T^{6} + \)\(32\!\cdots\!89\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \))(\( 1 - 531 T + 798956 T^{2} - 374338539 T^{3} + 496981290961 T^{4} \))(\( 1 + 1692 T + 1659257 T^{2} + 1192807548 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 - 70 T + 912673 T^{2} \))(\( 1 + 928 T + 943870 T^{2} + 846960544 T^{3} + 832972004929 T^{4} \))(\( 1 - 1099442 T^{2} + 832972004929 T^{4} \))(\( 1 + 238 T - 856029 T^{2} + 217216174 T^{3} + 832972004929 T^{4} \))(\( 1 - 415 T - 761343 T^{2} + 370087870 T^{3} - 118607901730 T^{4} + 337769206576510 T^{5} - 634177405148659647 T^{6} - \)\(31\!\cdots\!55\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 2139 T + 2437780 T^{2} - 1952207547 T^{3} + 832972004929 T^{4} \))(\( 1 - 348 T + 953041 T^{2} - 317610204 T^{3} + 832972004929 T^{4} \))
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