Properties

Label 17.4.c
Level 17
Weight 4
Character orbit c
Rep. character \(\chi_{17}(4,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 8
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 17.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 17 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(17, [\chi])\).

Total New Old
Modular forms 12 12 0
Cusp forms 8 8 0
Eisenstein series 4 4 0

Trace form

\( 8q - 36q^{4} + 14q^{5} + 22q^{6} + 2q^{7} + O(q^{10}) \) \( 8q - 36q^{4} + 14q^{5} + 22q^{6} + 2q^{7} + 78q^{10} - 108q^{11} - 174q^{12} - 88q^{13} + 108q^{14} + 420q^{16} - 10q^{17} + 428q^{18} - 306q^{20} - 260q^{21} + 30q^{22} - 22q^{23} - 862q^{24} + 540q^{27} - 764q^{28} + 46q^{29} - 120q^{30} + 610q^{31} + 816q^{33} + 1002q^{34} + 1172q^{35} - 574q^{37} - 768q^{38} - 844q^{39} - 342q^{40} - 968q^{41} + 550q^{44} - 1154q^{45} - 944q^{46} - 368q^{47} + 2494q^{48} + 468q^{50} + 296q^{51} + 2564q^{52} - 1592q^{54} - 1996q^{55} + 684q^{56} - 300q^{57} + 266q^{58} + 1258q^{61} - 2516q^{62} + 122q^{63} - 3044q^{64} + 628q^{65} + 764q^{67} + 1914q^{68} + 1812q^{69} + 1266q^{71} + 1404q^{72} - 1732q^{73} + 1538q^{74} + 1292q^{75} - 2836q^{78} + 914q^{79} + 498q^{80} + 280q^{81} - 280q^{82} - 2952q^{84} - 2498q^{85} - 4244q^{86} + 442q^{88} - 2156q^{89} + 2478q^{90} - 1632q^{91} - 1768q^{92} + 1484q^{95} + 3998q^{96} + 1836q^{97} + 6728q^{98} - 2088q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(17, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
17.4.c.a \(8\) \(1.003\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(14\) \(2\) \(q+(\beta _{1}-\beta _{3})q^{2}+\beta _{4}q^{3}+(-5-\beta _{2}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 14 T^{2} + 65 T^{4} - 48 T^{6} - 1152 T^{8} - 3072 T^{10} + 266240 T^{12} - 3670016 T^{14} + 16777216 T^{16} \)
$3$ \( 1 - 180 T^{3} - 124 T^{4} + 4500 T^{5} + 16200 T^{6} + 1800 T^{7} - 880154 T^{8} + 48600 T^{9} + 11809800 T^{10} + 88573500 T^{11} - 65898684 T^{12} - 2582803260 T^{13} + 282429536481 T^{16} \)
$5$ \( 1 - 14 T + 98 T^{2} - 1742 T^{3} + 12064 T^{4} + 83238 T^{5} - 830322 T^{6} + 22258486 T^{7} - 523179234 T^{8} + 2782310750 T^{9} - 12973781250 T^{10} + 162574218750 T^{11} + 2945312500000 T^{12} - 53161621093750 T^{13} + 373840332031250 T^{14} - 6675720214843750 T^{15} + 59604644775390625 T^{16} \)
$7$ \( 1 - 2 T + 2 T^{2} - 10226 T^{3} + 81296 T^{4} + 1358798 T^{5} + 49405350 T^{6} - 812471938 T^{7} - 20043782946 T^{8} - 278677874734 T^{9} + 5812490022150 T^{10} + 54832400484386 T^{11} + 1125241284292496 T^{12} - 48548564000677118 T^{13} + 3256827195820898 T^{14} - 1117091728166568014 T^{15} + \)\(19\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + 108 T + 5832 T^{2} + 285984 T^{3} + 14525412 T^{4} + 631823112 T^{5} + 24418117440 T^{6} + 981926711892 T^{7} + 38344333256678 T^{8} + 1306944453528252 T^{9} + 43258184550123840 T^{10} + 1489805848060834392 T^{11} + 45586965204363734052 T^{12} + \)\(11\!\cdots\!84\)\( T^{13} + \)\(32\!\cdots\!92\)\( T^{14} + \)\(79\!\cdots\!88\)\( T^{15} + \)\(98\!\cdots\!41\)\( T^{16} \)
$13$ \( ( 1 + 44 T + 6452 T^{2} + 185644 T^{3} + 18227830 T^{4} + 407859868 T^{5} + 31142571668 T^{6} + 466597972412 T^{7} + 23298085122481 T^{8} )^{2} \)
$17$ \( 1 + 10 T - 2720 T^{2} - 2890 T^{3} + 45425598 T^{4} - 14198570 T^{5} - 65654187680 T^{6} + 1185878764970 T^{7} + 582622237229761 T^{8} \)
$19$ \( 1 - 49548 T^{2} + 1108248628 T^{4} - 14581622201236 T^{6} + 123306344393137910 T^{8} - \)\(68\!\cdots\!16\)\( T^{10} + \)\(24\!\cdots\!08\)\( T^{12} - \)\(51\!\cdots\!68\)\( T^{14} + \)\(48\!\cdots\!21\)\( T^{16} \)
$23$ \( 1 + 22 T + 242 T^{2} + 305150 T^{3} + 283710864 T^{4} + 1994580286 T^{5} + 21780998454 T^{6} + 34099874325206 T^{7} + 50076554997930590 T^{8} + 414893170914781402 T^{9} + 3224369469445515606 T^{10} + \)\(35\!\cdots\!18\)\( T^{11} + \)\(62\!\cdots\!44\)\( T^{12} + \)\(81\!\cdots\!50\)\( T^{13} + \)\(78\!\cdots\!98\)\( T^{14} + \)\(86\!\cdots\!06\)\( T^{15} + \)\(48\!\cdots\!41\)\( T^{16} \)
$29$ \( 1 - 46 T + 1058 T^{2} - 350894 T^{3} - 1469341632 T^{4} + 48677544854 T^{5} - 623040317010 T^{6} - 326680376209530 T^{7} + 1059567541310947486 T^{8} - 7967407695374227170 T^{9} - \)\(37\!\cdots\!10\)\( T^{10} + \)\(70\!\cdots\!26\)\( T^{11} - \)\(51\!\cdots\!12\)\( T^{12} - \)\(30\!\cdots\!06\)\( T^{13} + \)\(22\!\cdots\!38\)\( T^{14} - \)\(23\!\cdots\!34\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
$31$ \( 1 - 610 T + 186050 T^{2} - 45370570 T^{3} + 7376742336 T^{4} - 319696487050 T^{5} - 148183743449850 T^{6} + 60089442929313630 T^{7} - 13665887910277786754 T^{8} + \)\(17\!\cdots\!30\)\( T^{9} - \)\(13\!\cdots\!50\)\( T^{10} - \)\(84\!\cdots\!50\)\( T^{11} + \)\(58\!\cdots\!96\)\( T^{12} - \)\(10\!\cdots\!70\)\( T^{13} + \)\(13\!\cdots\!50\)\( T^{14} - \)\(12\!\cdots\!10\)\( T^{15} + \)\(62\!\cdots\!21\)\( T^{16} \)
$37$ \( 1 + 574 T + 164738 T^{2} + 36289990 T^{3} + 6240461504 T^{4} + 1192660509522 T^{5} + 315027672319726 T^{6} + 82684080067897738 T^{7} + 20284670290250127070 T^{8} + \)\(41\!\cdots\!14\)\( T^{9} + \)\(80\!\cdots\!34\)\( T^{10} + \)\(15\!\cdots\!94\)\( T^{11} + \)\(41\!\cdots\!24\)\( T^{12} + \)\(12\!\cdots\!70\)\( T^{13} + \)\(27\!\cdots\!02\)\( T^{14} + \)\(49\!\cdots\!38\)\( T^{15} + \)\(43\!\cdots\!61\)\( T^{16} \)
$41$ \( 1 + 968 T + 468512 T^{2} + 198428824 T^{3} + 85385967612 T^{4} + 30688635230792 T^{5} + 9389247542584800 T^{6} + 2850342553409293272 T^{7} + \)\(81\!\cdots\!98\)\( T^{8} + \)\(19\!\cdots\!12\)\( T^{9} + \)\(44\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!12\)\( T^{11} + \)\(19\!\cdots\!72\)\( T^{12} + \)\(30\!\cdots\!24\)\( T^{13} + \)\(50\!\cdots\!52\)\( T^{14} + \)\(71\!\cdots\!88\)\( T^{15} + \)\(50\!\cdots\!61\)\( T^{16} \)
$43$ \( 1 - 524936 T^{2} + 127553384540 T^{4} - 18699551386799032 T^{6} + \)\(18\!\cdots\!98\)\( T^{8} - \)\(11\!\cdots\!68\)\( T^{10} + \)\(50\!\cdots\!40\)\( T^{12} - \)\(13\!\cdots\!64\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \)
$47$ \( ( 1 + 184 T + 303628 T^{2} + 45624920 T^{3} + 43219349926 T^{4} + 4736916069160 T^{5} + 3272871591913612 T^{6} + 205920007050909128 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} )^{2} \)
$53$ \( 1 - 429476 T^{2} + 132292785860 T^{4} - 29887976589939132 T^{6} + \)\(48\!\cdots\!18\)\( T^{8} - \)\(66\!\cdots\!28\)\( T^{10} + \)\(64\!\cdots\!60\)\( T^{12} - \)\(46\!\cdots\!64\)\( T^{14} + \)\(24\!\cdots\!81\)\( T^{16} \)
$59$ \( 1 - 803128 T^{2} + 306274528508 T^{4} - 82679168899041416 T^{6} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(34\!\cdots\!56\)\( T^{10} + \)\(54\!\cdots\!48\)\( T^{12} - \)\(60\!\cdots\!88\)\( T^{14} + \)\(31\!\cdots\!61\)\( T^{16} \)
$61$ \( 1 - 1258 T + 791282 T^{2} - 465239298 T^{3} + 289545560288 T^{4} - 163355761361878 T^{5} + 84613159959199710 T^{6} - 47961410201311305310 T^{7} + \)\(25\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!10\)\( T^{9} + \)\(43\!\cdots\!10\)\( T^{10} - \)\(19\!\cdots\!98\)\( T^{11} + \)\(76\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!98\)\( T^{13} + \)\(10\!\cdots\!42\)\( T^{14} - \)\(39\!\cdots\!38\)\( T^{15} + \)\(70\!\cdots\!41\)\( T^{16} \)
$67$ \( ( 1 - 382 T + 518844 T^{2} + 127863586 T^{3} + 50809502070 T^{4} + 38456635716118 T^{5} + 46933788838092636 T^{6} - 10392896139384669754 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \)
$71$ \( 1 - 1266 T + 801378 T^{2} - 406667714 T^{3} + 257625327744 T^{4} - 188037640245138 T^{5} + 114289697458506374 T^{6} - 42774297641864818786 T^{7} + \)\(12\!\cdots\!34\)\( T^{8} - \)\(15\!\cdots\!46\)\( T^{9} + \)\(14\!\cdots\!54\)\( T^{10} - \)\(86\!\cdots\!78\)\( T^{11} + \)\(42\!\cdots\!04\)\( T^{12} - \)\(23\!\cdots\!14\)\( T^{13} + \)\(16\!\cdots\!58\)\( T^{14} - \)\(95\!\cdots\!86\)\( T^{15} + \)\(26\!\cdots\!81\)\( T^{16} \)
$73$ \( 1 + 1732 T + 1499912 T^{2} + 1036616588 T^{3} + 594980634828 T^{4} + 355861503522212 T^{5} + 261220505412716920 T^{6} + \)\(20\!\cdots\!24\)\( T^{7} + \)\(15\!\cdots\!50\)\( T^{8} + \)\(81\!\cdots\!08\)\( T^{9} + \)\(39\!\cdots\!80\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!88\)\( T^{12} + \)\(92\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!36\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$79$ \( 1 - 914 T + 417698 T^{2} + 360102614 T^{3} + 55218179088 T^{4} - 435936491163994 T^{5} + 440218376260007590 T^{6} - 25818998319394847330 T^{7} - \)\(81\!\cdots\!34\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{9} + \)\(10\!\cdots\!90\)\( T^{10} - \)\(52\!\cdots\!86\)\( T^{11} + \)\(32\!\cdots\!08\)\( T^{12} + \)\(10\!\cdots\!86\)\( T^{13} + \)\(59\!\cdots\!78\)\( T^{14} - \)\(64\!\cdots\!06\)\( T^{15} + \)\(34\!\cdots\!81\)\( T^{16} \)
$83$ \( 1 - 2421080 T^{2} + 2978675688956 T^{4} - 2441080253881110760 T^{6} + \)\(15\!\cdots\!06\)\( T^{8} - \)\(79\!\cdots\!40\)\( T^{10} + \)\(31\!\cdots\!16\)\( T^{12} - \)\(84\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \)
$89$ \( ( 1 + 1078 T + 2962392 T^{2} + 2155154546 T^{3} + 3159947923854 T^{4} + 1519317145139074 T^{5} + 1472253400492538712 T^{6} + \)\(37\!\cdots\!02\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \)
$97$ \( 1 - 1836 T + 1685448 T^{2} - 1048047300 T^{3} + 1167661514444 T^{4} - 1645617931539948 T^{5} + 1602523329629378616 T^{6} - \)\(45\!\cdots\!92\)\( T^{7} - \)\(36\!\cdots\!30\)\( T^{8} - \)\(41\!\cdots\!16\)\( T^{9} + \)\(13\!\cdots\!64\)\( T^{10} - \)\(12\!\cdots\!16\)\( T^{11} + \)\(81\!\cdots\!04\)\( T^{12} - \)\(66\!\cdots\!00\)\( T^{13} + \)\(97\!\cdots\!72\)\( T^{14} - \)\(96\!\cdots\!92\)\( T^{15} + \)\(48\!\cdots\!81\)\( T^{16} \)
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