Properties

Label 13.4.c
Level 13
Weight 4
Character orbit c
Rep. character \(\chi_{13}(3,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 6
Newform subspaces 2
Sturm bound 4
Trace bound 1

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\( 6q + q^{2} - 7q^{3} - 13q^{4} + 4q^{5} + 30q^{6} - 35q^{7} - 30q^{8} - 12q^{9} + O(q^{10}) \) \( 6q + q^{2} - 7q^{3} - 13q^{4} + 4q^{5} + 30q^{6} - 35q^{7} - 30q^{8} - 12q^{9} - 63q^{10} + 15q^{11} + 312q^{12} + 34q^{13} + 68q^{14} - 124q^{15} + 7q^{16} - 57q^{17} - 614q^{18} + 111q^{19} - 311q^{20} + 206q^{21} + 298q^{22} - 223q^{23} + 216q^{24} + 478q^{25} + 237q^{26} + 470q^{27} - 240q^{28} - 231q^{29} + 4q^{30} - 428q^{31} + 151q^{32} - 361q^{33} - 182q^{34} - 270q^{35} - 541q^{36} + 417q^{37} + 860q^{38} - 51q^{39} - 370q^{40} - 373q^{41} + 210q^{42} - 299q^{43} + 848q^{44} + 16q^{45} - 230q^{46} - 204q^{47} + 368q^{48} + 508q^{49} - 1106q^{50} - 518q^{51} - 102q^{52} + 1276q^{53} + 1314q^{54} + 34q^{55} + 172q^{56} - 330q^{57} - 193q^{58} + 1673q^{59} + 144q^{60} - 647q^{61} + 796q^{62} + 70q^{63} - 3566q^{64} - 2102q^{65} - 3708q^{66} - 387q^{67} + 609q^{68} + 323q^{69} + 2560q^{70} - 781q^{71} + 1155q^{72} + 1600q^{73} + 59q^{74} + 1397q^{75} - 100q^{76} - 770q^{77} + 3278q^{78} + 328q^{79} + 2153q^{80} - 543q^{81} + 2175q^{82} + 1776q^{83} - 1540q^{84} + 1426q^{85} - 5172q^{86} - 2009q^{87} + 1020q^{88} - 655q^{89} - 4578q^{90} - 767q^{91} - 832q^{92} - 2052q^{93} - 392q^{94} - 1780q^{95} + 2816q^{96} + 177q^{97} - 1513q^{98} + 5892q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
13.4.c.a \(2\) \(0.767\) \(\Q(\sqrt{-3}) \) None \(-4\) \(-2\) \(34\) \(-20\) \(q+(-4+4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\cdots\)
13.4.c.b \(4\) \(0.767\) \(\Q(\sqrt{-3}, \sqrt{17})\) None \(5\) \(-5\) \(-30\) \(-15\) \(q+(2-\beta _{1}-2\beta _{2}-\beta _{3})q^{2}+(-1+3\beta _{1}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T + 8 T^{2} + 32 T^{3} + 64 T^{4} \))(\( 1 - 5 T + 7 T^{2} - 10 T^{3} + 60 T^{4} - 80 T^{5} + 448 T^{6} - 2560 T^{7} + 4096 T^{8} \))
$3$ (\( 1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4} \))(\( 1 + 5 T + 3 T^{2} - 160 T^{3} - 920 T^{4} - 4320 T^{5} + 2187 T^{6} + 98415 T^{7} + 531441 T^{8} \))
$5$ (\( ( 1 - 17 T + 125 T^{2} )^{2} \))(\( ( 1 + 15 T + 200 T^{2} + 1875 T^{3} + 15625 T^{4} )^{2} \))
$7$ (\( ( 1 - 17 T + 343 T^{2} )( 1 + 37 T + 343 T^{2} ) \))(\( 1 + 15 T - 513 T^{2} + 780 T^{3} + 349820 T^{4} + 267540 T^{5} - 60353937 T^{6} + 605304105 T^{7} + 13841287201 T^{8} \))
$11$ (\( 1 - 32 T - 307 T^{2} - 42592 T^{3} + 1771561 T^{4} \))(\( 1 + 17 T - 1489 T^{2} - 15028 T^{3} + 1005064 T^{4} - 20002268 T^{5} - 2637854329 T^{6} + 40085110747 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 + 91 T + 2197 T^{2} \))(\( 1 - 125 T + 7956 T^{2} - 274625 T^{3} + 4826809 T^{4} \))
$17$ (\( 1 - 13 T - 4744 T^{2} - 63869 T^{3} + 24137569 T^{4} \))(\( 1 + 70 T - 5063 T^{2} + 9590 T^{3} + 51050100 T^{4} + 47115670 T^{5} - 122208511847 T^{6} + 8301151354790 T^{7} + 582622237229761 T^{8} \))
$19$ (\( 1 + 30 T - 5959 T^{2} + 205770 T^{3} + 47045881 T^{4} \))(\( 1 - 141 T + 1299 T^{2} - 685824 T^{3} + 161881064 T^{4} - 4704066816 T^{5} + 61112599419 T^{6} - 45498965386839 T^{7} + 2213314919066161 T^{8} \))
$23$ (\( 1 + 78 T - 6083 T^{2} + 949026 T^{3} + 148035889 T^{4} \))(\( 1 + 145 T - 3937 T^{2} + 91060 T^{3} + 219254380 T^{4} + 1107927020 T^{5} - 582817294993 T^{6} + 261167135912135 T^{7} + 21914624432020321 T^{8} \))
$29$ (\( 1 + 197 T + 14420 T^{2} + 4804633 T^{3} + 594823321 T^{4} \))(\( 1 + 34 T - 32611 T^{2} - 510374 T^{3} + 517193284 T^{4} - 12447511486 T^{5} - 19397783321131 T^{6} + 493242963179546 T^{7} + 353814783205469041 T^{8} \))
$31$ (\( ( 1 + 74 T + 29791 T^{2} )^{2} \))(\( ( 1 + 140 T + 21982 T^{2} + 4170740 T^{3} + 887503681 T^{4} )^{2} \))
$37$ (\( 1 - 227 T + 876 T^{2} - 11498231 T^{3} + 2565726409 T^{4} \))(\( 1 - 190 T - 66003 T^{2} - 151430 T^{3} + 6030722900 T^{4} - 7670383790 T^{5} - 169345640173227 T^{6} - 24692730561064630 T^{7} + 6582952005840035281 T^{8} \))
$41$ (\( 1 - 165 T - 41696 T^{2} - 11371965 T^{3} + 4750104241 T^{4} \))(\( 1 + 538 T + 80941 T^{2} + 38015618 T^{3} + 18774626844 T^{4} + 2620074408178 T^{5} + 384478187370781 T^{6} + 176131480703951018 T^{7} + 22563490300366186081 T^{8} \))
$43$ (\( 1 - 156 T - 55171 T^{2} - 12403092 T^{3} + 6321363049 T^{4} \))(\( 1 + 455 T + 36243 T^{2} + 5354440 T^{3} + 6385191800 T^{4} + 425715461080 T^{5} + 229105160984907 T^{6} + 228679638431263565 T^{7} + 39959630797262576401 T^{8} \))
$47$ (\( ( 1 + 162 T + 103823 T^{2} )^{2} \))(\( ( 1 - 60 T + 125246 T^{2} - 6229380 T^{3} + 10779215329 T^{4} )^{2} \))
$53$ (\( ( 1 - 93 T + 148877 T^{2} )^{2} \))(\( ( 1 - 545 T + 256304 T^{2} - 81137965 T^{3} + 22164361129 T^{4} )^{2} \))
$59$ (\( 1 - 864 T + 541117 T^{2} - 177447456 T^{3} + 42180533641 T^{4} \))(\( 1 - 809 T + 92959 T^{2} - 121968076 T^{3} + 138709769544 T^{4} - 25049681480804 T^{5} + 3921060226733719 T^{6} - 7008363617291845651 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} \))(\( 1 + 502 T - 94959 T^{2} - 53713498 T^{3} + 11662829084 T^{4} - 12191943489538 T^{5} - 4892323228946199 T^{6} + 5870461338602738782 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 + 862 T + 442281 T^{2} + 259257706 T^{3} + 90458382169 T^{4} \))(\( 1 - 475 T - 397113 T^{2} - 10075700 T^{3} + 229484582600 T^{4} - 3030397759100 T^{5} - 35922199518278097 T^{6} - 12923103838240099825 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))
$71$ (\( 1 + 654 T + 69805 T^{2} + 234073794 T^{3} + 128100283921 T^{4} \))(\( 1 + 127 T - 656869 T^{2} - 5438648 T^{3} + 319053277564 T^{4} - 1946551944328 T^{5} - 84145105398903349 T^{6} + 5822759591243026937 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \))
$73$ (\( ( 1 - 215 T + 389017 T^{2} )^{2} \))(\( ( 1 - 585 T + 832884 T^{2} - 227574945 T^{3} + 151334226289 T^{4} )^{2} \))
$79$ (\( ( 1 + 76 T + 493039 T^{2} )^{2} \))(\( ( 1 - 240 T + 993678 T^{2} - 118329360 T^{3} + 243087455521 T^{4} )^{2} \))
$83$ (\( ( 1 - 628 T + 571787 T^{2} )^{2} \))(\( ( 1 - 260 T + 1117974 T^{2} - 148664620 T^{3} + 326940373369 T^{4} )^{2} \))
$89$ (\( 1 - 266 T - 634213 T^{2} - 187521754 T^{3} + 496981290961 T^{4} \))(\( 1 + 921 T - 707351 T^{2} + 134147334 T^{3} + 1324901569974 T^{4} + 94569711902646 T^{5} - 351540213142554311 T^{6} + \)\(32\!\cdots\!89\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \))
$97$ (\( 1 + 238 T - 856029 T^{2} + 217216174 T^{3} + 832972004929 T^{4} \))(\( 1 - 415 T - 761343 T^{2} + 370087870 T^{3} - 118607901730 T^{4} + 337769206576510 T^{5} - 634177405148659647 T^{6} - \)\(31\!\cdots\!55\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))
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