Properties

Label 3.33.b
Level 3
Weight 33
Character orbit b
Rep. character \(\chi_{3}(2,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 1
Sturm bound 11
Trace bound 0

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 33 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(11\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{33}(3, [\chi])\).

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

Trace form

\( 10q - 21387150q^{3} - 25792034864q^{4} - 2424530788848q^{6} - 5568062418940q^{7} - 790123604155542q^{9} + O(q^{10}) \) \( 10q - 21387150q^{3} - 25792034864q^{4} - 2424530788848q^{6} - 5568062418940q^{7} - 790123604155542q^{9} - 7003812786596640q^{10} - 279007424380502640q^{12} + 567697557679805780q^{13} - 20342205597863242080q^{15} + 93452609752238437504q^{16} - 546532439317269948000q^{18} + 571007688520350419876q^{19} - 202809296674597359852q^{21} - 7726803521259943913760q^{22} + 40966888307328818303616q^{24} - 68909708350780834128950q^{25} - 72967926462080465281230q^{27} + 205937256056975756717600q^{28} - 1107420825536697507876000q^{30} + 2337947500037502285593540q^{31} + 2522449071689961111334560q^{33} - 6060358314194999366692224q^{34} + 19880961335883894024618192q^{36} - 35307903416851790686359340q^{37} - 31624005377757923978634972q^{39} + 103824872020641394026996480q^{40} - 141365101025586253616004960q^{42} + 20133134465218107006792740q^{43} + 324508325822765233105320000q^{45} - 1081312096716791246752963776q^{46} + 3981901930430244196770990720q^{48} - 2767041101974767237639509154q^{49} + 1958591008210563208705802112q^{51} - 5054272042223836449280397920q^{52} + 8609682620485865645134310448q^{54} - 11870374622399979665591304000q^{55} + 40568829971233106153298510900q^{57} - 126691224865576416891245282400q^{58} + 279833341447916827662753642240q^{60} - 196667345182223458343232398380q^{61} + 349775312774222889759843110340q^{63} - 738816306154787409411488427008q^{64} + 939413077861687418842822458720q^{66} - 442550741350999501345614861340q^{67} + 1096443581430237499419700598208q^{69} - 3084260511067452830890360584000q^{70} + 5478956649034785201935062114560q^{72} - 4530286149796751043896742263020q^{73} + 5689646867484953837977805651250q^{75} - 10804853453030293252172855440096q^{76} + 9245617327241370897996048982560q^{78} - 2115110787448996851533482223164q^{79} - 454653247801524309158818752630q^{81} + 6754240700896495824250705604160q^{82} - 10589261014798101343390846457952q^{84} + 8783873758902640422771709896960q^{85} - 12893796544943502316879779198240q^{87} + 15416255855088684041150976480000q^{88} - 76437416541013126109366518121760q^{90} + 59544256621898576615810233543816q^{91} - 105131728169400940473110940107820q^{93} + 301803894289991891760064059517056q^{94} - 522680019947351351608967407401984q^{96} + 390148965222370670128665882607700q^{97} - 333895893109019690643343683610560q^{99} + O(q^{100}) \)

Decomposition of \(S_{33}^{\mathrm{new}}(3, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3.33.b.a \(10\) \(19.460\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(-21387150\) \(0\) \(-5\!\cdots\!40\) \(q+\beta _{1}q^{2}+(-2138715+35\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8578819048 T^{2} + 41128902250647155712 T^{4} - \)\(86\!\cdots\!08\)\( T^{6} + \)\(27\!\cdots\!52\)\( T^{8} - \)\(90\!\cdots\!68\)\( T^{10} + \)\(50\!\cdots\!32\)\( T^{12} - \)\(29\!\cdots\!48\)\( T^{14} + \)\(25\!\cdots\!52\)\( T^{16} - \)\(99\!\cdots\!28\)\( T^{18} + \)\(21\!\cdots\!76\)\( T^{20} \)
$3$ \( 1 + 21387150 T + 623766894639021 T^{2} + \)\(34\!\cdots\!60\)\( T^{3} + \)\(81\!\cdots\!78\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!98\)\( T^{6} + \)\(11\!\cdots\!60\)\( T^{7} + \)\(39\!\cdots\!41\)\( T^{8} + \)\(25\!\cdots\!50\)\( T^{9} + \)\(21\!\cdots\!01\)\( T^{10} \)
$5$ \( 1 - \)\(81\!\cdots\!50\)\( T^{2} + \)\(40\!\cdots\!25\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(47\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!50\)\( T^{12} - \)\(45\!\cdots\!00\)\( T^{14} + \)\(65\!\cdots\!25\)\( T^{16} - \)\(70\!\cdots\!50\)\( T^{18} + \)\(46\!\cdots\!25\)\( T^{20} \)
$7$ \( ( 1 + 2784031209470 T + \)\(34\!\cdots\!41\)\( T^{2} + \)\(42\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!18\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(68\!\cdots\!18\)\( T^{6} + \)\(51\!\cdots\!80\)\( T^{7} + \)\(46\!\cdots\!41\)\( T^{8} + \)\(41\!\cdots\!70\)\( T^{9} + \)\(16\!\cdots\!01\)\( T^{10} )^{2} \)
$11$ \( 1 - \)\(76\!\cdots\!70\)\( T^{2} + \)\(32\!\cdots\!25\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!70\)\( T^{8} - \)\(70\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!70\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(29\!\cdots\!25\)\( T^{16} - \)\(30\!\cdots\!70\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \)
$13$ \( ( 1 - 283848778839902890 T + \)\(16\!\cdots\!61\)\( T^{2} - \)\(26\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!18\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!58\)\( T^{6} - \)\(52\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} )^{2} \)
$17$ \( 1 - \)\(18\!\cdots\!38\)\( T^{2} + \)\(17\!\cdots\!77\)\( T^{4} - \)\(95\!\cdots\!28\)\( T^{6} + \)\(36\!\cdots\!42\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(20\!\cdots\!82\)\( T^{12} - \)\(30\!\cdots\!48\)\( T^{14} + \)\(30\!\cdots\!97\)\( T^{16} - \)\(18\!\cdots\!78\)\( T^{18} + \)\(55\!\cdots\!01\)\( T^{20} \)
$19$ \( ( 1 - \)\(28\!\cdots\!38\)\( T + \)\(24\!\cdots\!97\)\( T^{2} - \)\(48\!\cdots\!68\)\( T^{3} + \)\(33\!\cdots\!62\)\( T^{4} - \)\(57\!\cdots\!08\)\( T^{5} + \)\(27\!\cdots\!82\)\( T^{6} - \)\(33\!\cdots\!28\)\( T^{7} + \)\(14\!\cdots\!57\)\( T^{8} - \)\(13\!\cdots\!58\)\( T^{9} + \)\(39\!\cdots\!01\)\( T^{10} )^{2} \)
$23$ \( 1 - \)\(14\!\cdots\!38\)\( T^{2} + \)\(82\!\cdots\!37\)\( T^{4} - \)\(18\!\cdots\!48\)\( T^{6} - \)\(25\!\cdots\!98\)\( T^{8} + \)\(26\!\cdots\!92\)\( T^{10} - \)\(35\!\cdots\!18\)\( T^{12} - \)\(36\!\cdots\!88\)\( T^{14} + \)\(23\!\cdots\!77\)\( T^{16} - \)\(58\!\cdots\!18\)\( T^{18} + \)\(56\!\cdots\!01\)\( T^{20} \)
$29$ \( 1 - \)\(40\!\cdots\!50\)\( T^{2} + \)\(80\!\cdots\!65\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(98\!\cdots\!50\)\( T^{8} - \)\(70\!\cdots\!52\)\( T^{10} + \)\(38\!\cdots\!50\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(48\!\cdots\!65\)\( T^{16} - \)\(96\!\cdots\!50\)\( T^{18} + \)\(92\!\cdots\!01\)\( T^{20} \)
$31$ \( ( 1 - \)\(11\!\cdots\!70\)\( T + \)\(20\!\cdots\!25\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!70\)\( T^{4} - \)\(56\!\cdots\!52\)\( T^{5} + \)\(68\!\cdots\!70\)\( T^{6} - \)\(34\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!25\)\( T^{8} - \)\(91\!\cdots\!70\)\( T^{9} + \)\(41\!\cdots\!01\)\( T^{10} )^{2} \)
$37$ \( ( 1 + \)\(17\!\cdots\!70\)\( T + \)\(55\!\cdots\!61\)\( T^{2} + \)\(85\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!18\)\( T^{4} + \)\(18\!\cdots\!80\)\( T^{5} + \)\(21\!\cdots\!58\)\( T^{6} + \)\(19\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!01\)\( T^{8} + \)\(94\!\cdots\!70\)\( T^{9} + \)\(81\!\cdots\!01\)\( T^{10} )^{2} \)
$41$ \( 1 - \)\(30\!\cdots\!70\)\( T^{2} + \)\(43\!\cdots\!25\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{10} + \)\(42\!\cdots\!70\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{14} + \)\(19\!\cdots\!25\)\( T^{16} - \)\(22\!\cdots\!70\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \)
$43$ \( ( 1 - \)\(10\!\cdots\!70\)\( T + \)\(53\!\cdots\!61\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!78\)\( T^{4} + \)\(39\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!78\)\( T^{6} + \)\(45\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!61\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{9} + \)\(22\!\cdots\!01\)\( T^{10} )^{2} \)
$47$ \( 1 - \)\(12\!\cdots\!38\)\( T^{2} + \)\(90\!\cdots\!97\)\( T^{4} - \)\(47\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!82\)\( T^{8} - \)\(70\!\cdots\!68\)\( T^{10} + \)\(20\!\cdots\!42\)\( T^{12} - \)\(51\!\cdots\!88\)\( T^{14} + \)\(99\!\cdots\!77\)\( T^{16} - \)\(14\!\cdots\!98\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \)
$53$ \( 1 - \)\(91\!\cdots\!38\)\( T^{2} + \)\(41\!\cdots\!97\)\( T^{4} - \)\(12\!\cdots\!08\)\( T^{6} + \)\(26\!\cdots\!82\)\( T^{8} - \)\(44\!\cdots\!68\)\( T^{10} + \)\(59\!\cdots\!42\)\( T^{12} - \)\(62\!\cdots\!88\)\( T^{14} + \)\(47\!\cdots\!77\)\( T^{16} - \)\(23\!\cdots\!98\)\( T^{18} + \)\(58\!\cdots\!01\)\( T^{20} \)
$59$ \( 1 - \)\(20\!\cdots\!70\)\( T^{2} + \)\(24\!\cdots\!25\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!70\)\( T^{8} - \)\(67\!\cdots\!52\)\( T^{10} + \)\(28\!\cdots\!70\)\( T^{12} - \)\(93\!\cdots\!00\)\( T^{14} + \)\(24\!\cdots\!25\)\( T^{16} - \)\(44\!\cdots\!70\)\( T^{18} + \)\(47\!\cdots\!01\)\( T^{20} \)
$61$ \( ( 1 + \)\(98\!\cdots\!90\)\( T + \)\(77\!\cdots\!45\)\( T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} + \)\(87\!\cdots\!48\)\( T^{5} + \)\(29\!\cdots\!10\)\( T^{6} + \)\(82\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!45\)\( T^{8} + \)\(32\!\cdots\!90\)\( T^{9} + \)\(44\!\cdots\!01\)\( T^{10} )^{2} \)
$67$ \( ( 1 + \)\(22\!\cdots\!70\)\( T + \)\(54\!\cdots\!81\)\( T^{2} + \)\(38\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!18\)\( T^{4} - \)\(69\!\cdots\!60\)\( T^{5} + \)\(67\!\cdots\!98\)\( T^{6} + \)\(28\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} + \)\(12\!\cdots\!70\)\( T^{9} + \)\(14\!\cdots\!01\)\( T^{10} )^{2} \)
$71$ \( 1 - \)\(55\!\cdots\!50\)\( T^{2} + \)\(19\!\cdots\!65\)\( T^{4} - \)\(53\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(21\!\cdots\!52\)\( T^{10} + \)\(35\!\cdots\!50\)\( T^{12} - \)\(48\!\cdots\!40\)\( T^{14} + \)\(53\!\cdots\!65\)\( T^{16} - \)\(46\!\cdots\!50\)\( T^{18} + \)\(25\!\cdots\!01\)\( T^{20} \)
$73$ \( ( 1 + \)\(22\!\cdots\!10\)\( T + \)\(38\!\cdots\!81\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} + \)\(28\!\cdots\!20\)\( T^{5} + \)\(16\!\cdots\!58\)\( T^{6} + \)\(77\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!41\)\( T^{8} + \)\(72\!\cdots\!10\)\( T^{9} + \)\(13\!\cdots\!01\)\( T^{10} )^{2} \)
$79$ \( ( 1 + \)\(10\!\cdots\!82\)\( T + \)\(11\!\cdots\!17\)\( T^{2} + \)\(49\!\cdots\!32\)\( T^{3} + \)\(61\!\cdots\!22\)\( T^{4} + \)\(63\!\cdots\!92\)\( T^{5} + \)\(32\!\cdots\!02\)\( T^{6} + \)\(13\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!57\)\( T^{8} + \)\(83\!\cdots\!02\)\( T^{9} + \)\(41\!\cdots\!01\)\( T^{10} )^{2} \)
$83$ \( 1 - \)\(14\!\cdots\!98\)\( T^{2} + \)\(78\!\cdots\!77\)\( T^{4} - \)\(15\!\cdots\!88\)\( T^{6} - \)\(27\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!32\)\( T^{10} - \)\(18\!\cdots\!18\)\( T^{12} - \)\(67\!\cdots\!08\)\( T^{14} + \)\(22\!\cdots\!97\)\( T^{16} - \)\(27\!\cdots\!38\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \)
$89$ \( 1 - \)\(19\!\cdots\!70\)\( T^{2} + \)\(17\!\cdots\!25\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(21\!\cdots\!70\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!25\)\( T^{16} - \)\(21\!\cdots\!70\)\( T^{18} + \)\(63\!\cdots\!01\)\( T^{20} \)
$97$ \( ( 1 - \)\(19\!\cdots\!50\)\( T + \)\(26\!\cdots\!61\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} + \)\(74\!\cdots\!18\)\( T^{6} - \)\(34\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} - \)\(39\!\cdots\!50\)\( T^{9} + \)\(76\!\cdots\!01\)\( T^{10} )^{2} \)
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