Properties

Label 28.3.g
Level 28
Weight 3
Character orbit g
Rep. character \(\chi_{28}(11,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 12
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 28.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(28, [\chi])\).

Total New Old
Modular forms 20 20 0
Cusp forms 12 12 0
Eisenstein series 8 8 0

Trace form

\( 12q - 2q^{2} - 4q^{4} - 2q^{5} - 12q^{6} - 8q^{8} + 4q^{9} + O(q^{10}) \) \( 12q - 2q^{2} - 4q^{4} - 2q^{5} - 12q^{6} - 8q^{8} + 4q^{9} - 2q^{10} - 24q^{12} - 24q^{13} + 2q^{14} + 16q^{16} - 2q^{17} + 56q^{18} + 152q^{20} - 78q^{21} + 44q^{22} - 44q^{24} + 56q^{26} + 8q^{28} + 72q^{29} - 74q^{30} - 112q^{32} - 14q^{33} - 316q^{34} - 160q^{36} + 86q^{37} - 2q^{38} - 148q^{40} + 8q^{41} + 68q^{42} + 64q^{44} + 156q^{45} + 162q^{46} + 512q^{48} + 108q^{49} + 208q^{50} - 64q^{52} - 74q^{53} + 182q^{54} + 16q^{56} - 220q^{57} - 176q^{58} - 232q^{60} + 86q^{61} - 532q^{62} - 160q^{64} - 140q^{65} + 102q^{66} - 68q^{68} - 300q^{69} + 90q^{70} + 152q^{72} - 234q^{73} + 290q^{74} + 576q^{76} - 262q^{77} + 64q^{78} + 146q^{81} + 272q^{82} - 28q^{84} + 268q^{85} - 16q^{86} - 188q^{88} + 6q^{89} - 640q^{90} - 448q^{92} + 162q^{93} + 102q^{94} - 320q^{96} + 744q^{97} - 190q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(28, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
28.3.g.a \(12\) \(0.763\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-2\) \(0\) \(-2\) \(0\) \(q+\beta _{6}q^{2}+\beta _{4}q^{3}+(-\beta _{1}+\beta _{5}+\beta _{9}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} + 8 T^{3} + 8 T^{4} + 32 T^{5} + 80 T^{6} + 128 T^{7} + 128 T^{8} + 512 T^{9} + 1024 T^{10} + 2048 T^{11} + 4096 T^{12} \)
$3$ \( 1 + 25 T^{2} + 267 T^{4} + 1932 T^{6} + 11441 T^{8} - 29653 T^{10} - 1105946 T^{12} - 2401893 T^{14} + 75064401 T^{16} + 1026744012 T^{18} + 11493474507 T^{20} + 87169610025 T^{22} + 282429536481 T^{24} \)
$5$ \( ( 1 + T - 37 T^{2} + 92 T^{3} + 521 T^{4} - 2301 T^{5} - 4746 T^{6} - 57525 T^{7} + 325625 T^{8} + 1437500 T^{9} - 14453125 T^{10} + 9765625 T^{11} + 244140625 T^{12} )^{2} \)
$7$ \( 1 - 54 T^{2} + 1071 T^{4} + 6860 T^{6} + 2571471 T^{8} - 311299254 T^{10} + 13841287201 T^{12} \)
$11$ \( 1 + 625 T^{2} + 219387 T^{4} + 53661164 T^{6} + 10139548961 T^{8} + 1568956103859 T^{10} + 205124285853862 T^{12} + 22971086316599619 T^{14} + 2173502369124672641 T^{16} + \)\(16\!\cdots\!44\)\( T^{18} + \)\(10\!\cdots\!07\)\( T^{20} + \)\(42\!\cdots\!25\)\( T^{22} + \)\(98\!\cdots\!41\)\( T^{24} \)
$13$ \( ( 1 + 6 T + 431 T^{2} + 1748 T^{3} + 72839 T^{4} + 171366 T^{5} + 4826809 T^{6} )^{4} \)
$17$ \( ( 1 + T - 357 T^{2} - 9268 T^{3} + 19889 T^{4} + 1599651 T^{5} + 34317190 T^{6} + 462299139 T^{7} + 1661149169 T^{8} - 223706989492 T^{9} - 2490345406437 T^{10} + 2015993900449 T^{11} + 582622237229761 T^{12} )^{2} \)
$19$ \( 1 + 1361 T^{2} + 926139 T^{4} + 455129644 T^{6} + 189034671233 T^{8} + 67541974765395 T^{10} + 23345482892884198 T^{12} + 8802137693401041795 T^{14} + \)\(32\!\cdots\!53\)\( T^{16} + \)\(10\!\cdots\!84\)\( T^{18} + \)\(26\!\cdots\!59\)\( T^{20} + \)\(51\!\cdots\!61\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \)
$23$ \( 1 + 1529 T^{2} + 1564875 T^{4} + 666601324 T^{6} - 12750232255 T^{8} - 342590421568629 T^{10} - 240073619368345850 T^{12} - 95870846162186707989 T^{14} - \)\(99\!\cdots\!55\)\( T^{16} + \)\(14\!\cdots\!04\)\( T^{18} + \)\(95\!\cdots\!75\)\( T^{20} + \)\(26\!\cdots\!29\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( ( 1 - 18 T + 1791 T^{2} - 34684 T^{3} + 1506231 T^{4} - 12731058 T^{5} + 594823321 T^{6} )^{4} \)
$31$ \( 1 + 1537 T^{2} + 64443 T^{4} - 657806548 T^{6} - 61517921455 T^{8} - 398864870588157 T^{10} - 927701155689421178 T^{12} - \)\(36\!\cdots\!97\)\( T^{14} - \)\(52\!\cdots\!55\)\( T^{16} - \)\(51\!\cdots\!28\)\( T^{18} + \)\(46\!\cdots\!83\)\( T^{20} + \)\(10\!\cdots\!37\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( ( 1 - 43 T - 997 T^{2} + 25300 T^{3} + 1028465 T^{4} + 44336247 T^{5} - 3970184154 T^{6} + 60696322143 T^{7} + 1927508992865 T^{8} + 64912878147700 T^{9} - 3501942015559237 T^{10} - 206769128013967507 T^{11} + 6582952005840035281 T^{12} )^{2} \)
$41$ \( ( 1 - 2 T + 2711 T^{2} + 37572 T^{3} + 4557191 T^{4} - 5651522 T^{5} + 4750104241 T^{6} )^{4} \)
$43$ \( ( 1 - 4118 T^{2} + 5077855 T^{4} - 2161925300 T^{6} + 17360175751855 T^{8} - 48132008743160918 T^{10} + 39959630797262576401 T^{12} )^{2} \)
$47$ \( 1 + 8513 T^{2} + 34636347 T^{4} + 114973411756 T^{6} + 365269331235857 T^{8} + 977402417239620291 T^{10} + \)\(22\!\cdots\!94\)\( T^{12} + \)\(47\!\cdots\!71\)\( T^{14} + \)\(86\!\cdots\!77\)\( T^{16} + \)\(13\!\cdots\!96\)\( T^{18} + \)\(19\!\cdots\!87\)\( T^{20} + \)\(23\!\cdots\!13\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( ( 1 + 37 T - 6101 T^{2} - 98492 T^{3} + 27882353 T^{4} + 139373047 T^{5} - 88083945850 T^{6} + 391498889023 T^{7} + 220005176581793 T^{8} - 2183012256317468 T^{9} - 379846371199713461 T^{10} + 6470836403523982813 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \)
$59$ \( 1 + 8857 T^{2} + 19615883 T^{4} + 62334311372 T^{6} + 716687290054865 T^{8} + 2166290925668370603 T^{10} + \)\(27\!\cdots\!42\)\( T^{12} + \)\(26\!\cdots\!83\)\( T^{14} + \)\(10\!\cdots\!65\)\( T^{16} + \)\(11\!\cdots\!32\)\( T^{18} + \)\(42\!\cdots\!03\)\( T^{20} + \)\(23\!\cdots\!57\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \)
$61$ \( ( 1 - 43 T - 7509 T^{2} + 86404 T^{3} + 42241889 T^{4} + 112159143 T^{5} - 188685362330 T^{6} + 417344171103 T^{7} + 584874478633649 T^{8} + 4451566426287844 T^{9} - 1439530213296583029 T^{10} - 30673745201503951843 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \)
$67$ \( 1 + 22377 T^{2} + 274481131 T^{4} + 2446555048844 T^{6} + 17357874347837969 T^{8} + \)\(10\!\cdots\!23\)\( T^{10} + \)\(49\!\cdots\!50\)\( T^{12} + \)\(20\!\cdots\!83\)\( T^{14} + \)\(70\!\cdots\!29\)\( T^{16} + \)\(20\!\cdots\!84\)\( T^{18} + \)\(45\!\cdots\!11\)\( T^{20} + \)\(74\!\cdots\!77\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \)
$71$ \( ( 1 - 18598 T^{2} + 175808431 T^{4} - 1076604135508 T^{6} + 4467587765682511 T^{8} - 12009724174108663078 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( ( 1 + 117 T - 6509 T^{2} - 220196 T^{3} + 149197385 T^{4} + 4225343807 T^{5} - 591649860826 T^{6} + 22516857147503 T^{7} + 4236943295799785 T^{8} - 33323191291932644 T^{9} - 5249248738138573229 T^{10} + \)\(50\!\cdots\!33\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \)
$79$ \( 1 + 19737 T^{2} + 240943563 T^{4} + 1326301878892 T^{6} + 439440672698145 T^{8} - 79155928359774252117 T^{10} - \)\(68\!\cdots\!18\)\( T^{12} - \)\(30\!\cdots\!77\)\( T^{14} + \)\(66\!\cdots\!45\)\( T^{16} + \)\(78\!\cdots\!72\)\( T^{18} + \)\(55\!\cdots\!23\)\( T^{20} + \)\(17\!\cdots\!37\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( ( 1 - 38390 T^{2} + 633108223 T^{4} - 5732720590964 T^{6} + 30046253274873583 T^{8} - 86465498791817783990 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \)
$89$ \( ( 1 - 3 T - 22925 T^{2} + 11788 T^{3} + 344187257 T^{4} - 22390969 T^{5} - 3171173759098 T^{6} - 177358865449 T^{7} + 21595079827822937 T^{8} + 5858415457848268 T^{9} - 90246298370720206925 T^{10} - 93545159789898550803 T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \)
$97$ \( ( 1 - 186 T + 27543 T^{2} - 2505868 T^{3} + 259152087 T^{4} - 16466446266 T^{5} + 832972004929 T^{6} )^{4} \)
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