Properties

Label 9.5.d
Level 9
Weight 5
Character orbit d
Rep. character \(\chi_{9}(2,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 6
Newform subspaces 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 9.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(9, [\chi])\).

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

Trace form

\( 6q - 3q^{2} - 3q^{3} + 15q^{4} - 12q^{5} - 99q^{6} + 12q^{7} + 99q^{9} + O(q^{10}) \) \( 6q - 3q^{2} - 3q^{3} + 15q^{4} - 12q^{5} - 99q^{6} + 12q^{7} + 99q^{9} - 36q^{10} + 483q^{11} + 330q^{12} - 6q^{13} - 1146q^{14} - 1026q^{15} + 15q^{16} + 1404q^{18} - 258q^{19} + 1614q^{20} + 480q^{21} - 369q^{22} - 282q^{23} - 1449q^{24} - 273q^{25} + 54q^{27} + 1308q^{28} - 1056q^{29} - 1278q^{30} + 1290q^{31} - 1161q^{32} + 279q^{33} + 513q^{34} - 2385q^{36} + 12q^{37} - 789q^{38} + 1974q^{39} - 1314q^{40} + 7629q^{41} + 9612q^{42} - 285q^{43} - 4212q^{45} - 5760q^{46} - 9642q^{47} - 6771q^{48} - 1863q^{49} + 3027q^{50} + 2457q^{51} - 240q^{52} - 405q^{54} + 2016q^{55} - 462q^{56} + 5367q^{57} + 6462q^{58} + 6225q^{59} + 7470q^{60} + 3630q^{61} - 7578q^{63} + 15450q^{64} - 7158q^{65} - 13734q^{66} - 5055q^{67} - 10503q^{68} - 13878q^{69} - 9684q^{70} + 8451q^{72} - 14622q^{73} + 26454q^{74} + 21021q^{75} - 4047q^{76} + 2580q^{77} - 12060q^{78} + 4764q^{79} + 18387q^{81} - 9702q^{82} - 1866q^{83} - 6486q^{84} + 12366q^{85} - 37731q^{86} - 21564q^{87} + 14787q^{88} + 20790q^{90} + 34836q^{91} + 33636q^{92} + 19254q^{93} - 12708q^{94} - 13362q^{95} - 3672q^{96} - 28959q^{97} - 9126q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(9, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
9.5.d.a \(6\) \(0.930\) 6.0.39400128.1 None \(-3\) \(-3\) \(-12\) \(12\) \(q+\beta _{2}q^{2}+(-3+\beta _{1}-3\beta _{3}+\beta _{4})q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 21 T^{2} + 54 T^{3} + 150 T^{4} + 1284 T^{5} + 556 T^{6} + 20544 T^{7} + 38400 T^{8} + 221184 T^{9} + 1376256 T^{10} + 3145728 T^{11} + 16777216 T^{12} \)
$3$ \( 1 + 3 T - 45 T^{2} - 162 T^{3} - 3645 T^{4} + 19683 T^{5} + 531441 T^{6} \)
$5$ \( 1 + 12 T + 1146 T^{2} + 13176 T^{3} + 624786 T^{4} + 24253584 T^{5} + 478052638 T^{6} + 15158490000 T^{7} + 244057031250 T^{8} + 3216796875000 T^{9} + 174865722656250 T^{10} + 1144409179687500 T^{11} + 59604644775390625 T^{12} \)
$7$ \( 1 - 12 T - 2598 T^{2} - 158692 T^{3} + 1689750 T^{4} + 261282744 T^{5} + 12756026130 T^{6} + 627339868344 T^{7} + 9741072489750 T^{8} - 2196501548501092 T^{9} - 86339153619823398 T^{10} - 957507195571344012 T^{11} + \)\(19\!\cdots\!01\)\( T^{12} \)
$11$ \( 1 - 483 T + 146469 T^{2} - 33184998 T^{3} + 6185106393 T^{4} - 950746069515 T^{5} + 124654869741274 T^{6} - 13919873203769115 T^{7} + 1325832485269426233 T^{8} - \)\(10\!\cdots\!58\)\( T^{9} + \)\(67\!\cdots\!09\)\( T^{10} - \)\(32\!\cdots\!83\)\( T^{11} + \)\(98\!\cdots\!41\)\( T^{12} \)
$13$ \( 1 + 6 T - 68352 T^{2} - 1958440 T^{3} + 2716582572 T^{4} + 62453281302 T^{5} - 85607901136722 T^{6} + 1783728167266422 T^{7} + 2215999860113594412 T^{8} - 45627901827271689640 T^{9} - \)\(45\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!06\)\( T^{11} + \)\(54\!\cdots\!61\)\( T^{12} \)
$17$ \( 1 - 346011 T^{2} + 53617620939 T^{4} - 5294214329064626 T^{6} + \)\(37\!\cdots\!99\)\( T^{8} - \)\(16\!\cdots\!91\)\( T^{10} + \)\(33\!\cdots\!21\)\( T^{12} \)
$19$ \( ( 1 + 129 T + 372939 T^{2} + 32427790 T^{3} + 48601783419 T^{4} + 2190879632289 T^{5} + 2213314919066161 T^{6} )^{2} \)
$23$ \( 1 + 282 T + 648876 T^{2} + 175507776 T^{3} + 227497773480 T^{4} + 72727641623526 T^{5} + 70224828144985186 T^{6} + 20352175959569139366 T^{7} + \)\(17\!\cdots\!80\)\( T^{8} + \)\(38\!\cdots\!96\)\( T^{9} + \)\(39\!\cdots\!36\)\( T^{10} + \)\(48\!\cdots\!82\)\( T^{11} + \)\(48\!\cdots\!41\)\( T^{12} \)
$29$ \( 1 + 1056 T + 1968402 T^{2} + 1686104640 T^{3} + 1760514009810 T^{4} + 1015827714042684 T^{5} + 1221310573219134286 T^{6} + \)\(71\!\cdots\!04\)\( T^{7} + \)\(88\!\cdots\!10\)\( T^{8} + \)\(59\!\cdots\!40\)\( T^{9} + \)\(49\!\cdots\!42\)\( T^{10} + \)\(18\!\cdots\!56\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \)
$31$ \( 1 - 1290 T - 570606 T^{2} + 1377538196 T^{3} + 182213722890 T^{4} - 535899775714218 T^{5} + 81606055489213290 T^{6} - \)\(49\!\cdots\!78\)\( T^{7} + \)\(15\!\cdots\!90\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{9} - \)\(41\!\cdots\!86\)\( T^{10} - \)\(86\!\cdots\!90\)\( T^{11} + \)\(62\!\cdots\!21\)\( T^{12} \)
$37$ \( ( 1 - 6 T + 3399531 T^{2} + 1254253444 T^{3} + 6371268418491 T^{4} - 21074876723526 T^{5} + 6582952005840035281 T^{6} )^{2} \)
$41$ \( 1 - 7629 T + 33279051 T^{2} - 105879107016 T^{3} + 271025936843037 T^{4} - 581054677368191211 T^{5} + \)\(10\!\cdots\!02\)\( T^{6} - \)\(16\!\cdots\!71\)\( T^{7} + \)\(21\!\cdots\!77\)\( T^{8} - \)\(23\!\cdots\!96\)\( T^{9} + \)\(21\!\cdots\!91\)\( T^{10} - \)\(13\!\cdots\!29\)\( T^{11} + \)\(50\!\cdots\!61\)\( T^{12} \)
$43$ \( 1 + 285 T - 4416855 T^{2} + 6696709682 T^{3} + 5626286140005 T^{4} - 17612350767962595 T^{5} + 17128129577980595658 T^{6} - \)\(60\!\cdots\!95\)\( T^{7} + \)\(65\!\cdots\!05\)\( T^{8} + \)\(26\!\cdots\!82\)\( T^{9} - \)\(60\!\cdots\!55\)\( T^{10} + \)\(13\!\cdots\!85\)\( T^{11} + \)\(15\!\cdots\!01\)\( T^{12} \)
$47$ \( 1 + 9642 T + 52651236 T^{2} + 208863538416 T^{3} + 680262116291880 T^{4} + 1905117042381356886 T^{5} + \)\(45\!\cdots\!66\)\( T^{6} + \)\(92\!\cdots\!66\)\( T^{7} + \)\(16\!\cdots\!80\)\( T^{8} + \)\(24\!\cdots\!56\)\( T^{9} + \)\(29\!\cdots\!56\)\( T^{10} + \)\(26\!\cdots\!42\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \)
$53$ \( 1 - 20649822 T^{2} + 267970470920223 T^{4} - \)\(22\!\cdots\!72\)\( T^{6} + \)\(16\!\cdots\!03\)\( T^{8} - \)\(80\!\cdots\!62\)\( T^{10} + \)\(24\!\cdots\!81\)\( T^{12} \)
$59$ \( 1 - 6225 T + 34368069 T^{2} - 133533682650 T^{3} + 415161196812177 T^{4} - 1308007205289880689 T^{5} + \)\(36\!\cdots\!46\)\( T^{6} - \)\(15\!\cdots\!29\)\( T^{7} + \)\(60\!\cdots\!17\)\( T^{8} - \)\(23\!\cdots\!50\)\( T^{9} + \)\(74\!\cdots\!29\)\( T^{10} - \)\(16\!\cdots\!25\)\( T^{11} + \)\(31\!\cdots\!61\)\( T^{12} \)
$61$ \( 1 - 3630 T - 31672896 T^{2} + 38334701264 T^{3} + 1025452935692820 T^{4} - 698091809532688782 T^{5} - \)\(14\!\cdots\!30\)\( T^{6} - \)\(96\!\cdots\!62\)\( T^{7} + \)\(19\!\cdots\!20\)\( T^{8} + \)\(10\!\cdots\!44\)\( T^{9} - \)\(11\!\cdots\!56\)\( T^{10} - \)\(18\!\cdots\!30\)\( T^{11} + \)\(70\!\cdots\!41\)\( T^{12} \)
$67$ \( 1 + 5055 T - 13352055 T^{2} + 5710009118 T^{3} + 246985082323965 T^{4} - 2158812454985978265 T^{5} - \)\(15\!\cdots\!22\)\( T^{6} - \)\(43\!\cdots\!65\)\( T^{7} + \)\(10\!\cdots\!65\)\( T^{8} + \)\(46\!\cdots\!98\)\( T^{9} - \)\(22\!\cdots\!55\)\( T^{10} + \)\(16\!\cdots\!55\)\( T^{11} + \)\(66\!\cdots\!21\)\( T^{12} \)
$71$ \( 1 - 87967842 T^{2} + 4070274316914543 T^{4} - \)\(12\!\cdots\!12\)\( T^{6} + \)\(26\!\cdots\!23\)\( T^{8} - \)\(36\!\cdots\!82\)\( T^{10} + \)\(26\!\cdots\!81\)\( T^{12} \)
$73$ \( ( 1 + 7311 T + 93929019 T^{2} + 399497247430 T^{3} + 2667418918455579 T^{4} + 5896029731837626191 T^{5} + \)\(22\!\cdots\!21\)\( T^{6} )^{2} \)
$79$ \( 1 - 4764 T - 88341198 T^{2} + 209973394004 T^{3} + 6244109393857182 T^{4} - 7045428820224395952 T^{5} - \)\(25\!\cdots\!90\)\( T^{6} - \)\(27\!\cdots\!12\)\( T^{7} + \)\(94\!\cdots\!02\)\( T^{8} + \)\(12\!\cdots\!64\)\( T^{9} - \)\(20\!\cdots\!58\)\( T^{10} - \)\(42\!\cdots\!64\)\( T^{11} + \)\(34\!\cdots\!81\)\( T^{12} \)
$83$ \( 1 + 1866 T + 10594740 T^{2} + 17604008208 T^{3} + 123292807619064 T^{4} + 3108136851733394166 T^{5} - \)\(11\!\cdots\!18\)\( T^{6} + \)\(14\!\cdots\!86\)\( T^{7} + \)\(27\!\cdots\!24\)\( T^{8} + \)\(18\!\cdots\!88\)\( T^{9} + \)\(53\!\cdots\!40\)\( T^{10} + \)\(44\!\cdots\!66\)\( T^{11} + \)\(11\!\cdots\!21\)\( T^{12} \)
$89$ \( 1 - 92525118 T^{2} + 4141804686688959 T^{4} - \)\(27\!\cdots\!72\)\( T^{6} + \)\(16\!\cdots\!79\)\( T^{8} - \)\(14\!\cdots\!98\)\( T^{10} + \)\(61\!\cdots\!41\)\( T^{12} \)
$97$ \( 1 + 28959 T + 315999867 T^{2} + 3434149167476 T^{3} + 57450550006007937 T^{4} + \)\(60\!\cdots\!37\)\( T^{5} + \)\(48\!\cdots\!90\)\( T^{6} + \)\(53\!\cdots\!97\)\( T^{7} + \)\(45\!\cdots\!57\)\( T^{8} + \)\(23\!\cdots\!16\)\( T^{9} + \)\(19\!\cdots\!07\)\( T^{10} + \)\(15\!\cdots\!59\)\( T^{11} + \)\(48\!\cdots\!81\)\( T^{12} \)
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