Properties

Label 8.14.b
Level 8
Weight 14
Character orbit b
Rep. character \(\chi_{8}(5,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 2
Sturm bound 14
Trace bound 1

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{14}(8, [\chi])\).

Total New Old
Modular forms 14 14 0
Cusp forms 12 12 0
Eisenstein series 2 2 0

Trace form

\( 12q - 2q^{2} - 8556q^{4} + 58444q^{6} + 235296q^{7} + 1076872q^{8} - 5314412q^{9} + O(q^{10}) \) \( 12q - 2q^{2} - 8556q^{4} + 58444q^{6} + 235296q^{7} + 1076872q^{8} - 5314412q^{9} - 7752648q^{10} - 8536664q^{12} + 21101872q^{14} - 42373856q^{15} - 62847216q^{16} - 49714600q^{17} + 262778834q^{18} + 381237904q^{20} + 196775484q^{22} - 149850784q^{23} + 1739438480q^{24} - 1654349124q^{25} - 685452248q^{26} - 978617568q^{28} + 539961392q^{30} - 10881769344q^{31} - 8956909792q^{32} + 3397961328q^{33} + 8857451100q^{34} + 18075363660q^{36} - 23710846420q^{38} + 79057948000q^{39} - 6505554528q^{40} + 12231942520q^{41} - 3443304224q^{42} - 28647791928q^{44} - 51061300848q^{46} - 146392008000q^{47} - 161587694304q^{48} + 50764315692q^{49} + 246819442102q^{50} + 441779820720q^{52} - 639572716168q^{54} - 4023386208q^{55} - 699100837568q^{56} - 351867765712q^{57} + 992195908104q^{58} + 1614284565792q^{60} - 1019628353344q^{62} + 389587709920q^{63} - 1464077986752q^{64} + 173779028800q^{65} + 2946347707608q^{66} + 2850998874984q^{68} - 5505591528768q^{70} - 1892582134752q^{71} - 8538615449032q^{72} + 324055378296q^{73} + 9399657781240q^{74} + 10892359157736q^{76} - 16235130104944q^{78} + 4716009102144q^{79} - 16787669526720q^{80} + 410958020828q^{81} + 14910525285612q^{82} + 26784826987072q^{84} - 24718490913284q^{86} + 352970975712q^{87} - 26046521626416q^{88} + 5301429525304q^{89} + 43218339424520q^{90} + 34108342295904q^{92} - 37727942061792q^{94} - 15785048234464q^{95} - 62733552180160q^{96} - 10954301425896q^{97} + 61714328751438q^{98} + O(q^{100}) \)

Decomposition of \(S_{14}^{\mathrm{new}}(8, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
8.14.b.a \(2\) \(8.578\) \(\Q(\sqrt{-79}) \) None \(-112\) \(0\) \(0\) \(-351664\) \(q+(-56-4\beta )q^{2}+129\beta q^{3}+(-1920+\cdots)q^{4}+\cdots\)
8.14.b.b \(10\) \(8.578\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(110\) \(0\) \(0\) \(586960\) \(q+(11+\beta _{1})q^{2}+(3\beta _{1}+\beta _{3})q^{3}+(-472+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 112 T + 8192 T^{2} \))(\( 1 - 110 T + 8408 T^{2} - 390976 T^{3} - 935936 T^{4} + 3612475392 T^{5} - 7667187712 T^{6} - 26237955211264 T^{7} + 4622346883170304 T^{8} - 495395959010754560 T^{9} + 36893488147419103232 T^{10} \))
$3$ (\( 1 + 2069910 T^{2} + 2541865828329 T^{4} \))(\( 1 - 8978642 T^{2} + 41923886773365 T^{4} - \)\(13\!\cdots\!32\)\( T^{6} + \)\(31\!\cdots\!22\)\( T^{8} - \)\(56\!\cdots\!92\)\( T^{10} + \)\(78\!\cdots\!38\)\( T^{12} - \)\(85\!\cdots\!12\)\( T^{14} + \)\(68\!\cdots\!85\)\( T^{16} - \)\(37\!\cdots\!02\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \))
$5$ (\( 1 - 1931729850 T^{2} + 1490116119384765625 T^{4} \))(\( 1 - 4565314338 T^{2} + 10147469641592691461 T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!50\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!50\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!25\)\( T^{16} - \)\(22\!\cdots\!50\)\( T^{18} + \)\(73\!\cdots\!25\)\( T^{20} \))
$7$ (\( ( 1 + 175832 T + 96889010407 T^{2} )^{2} \))(\( ( 1 - 293480 T + 239610643203 T^{2} - 95417056996483936 T^{3} + \)\(32\!\cdots\!74\)\( T^{4} - \)\(13\!\cdots\!28\)\( T^{5} + \)\(31\!\cdots\!18\)\( T^{6} - \)\(89\!\cdots\!64\)\( T^{7} + \)\(21\!\cdots\!29\)\( T^{8} - \)\(25\!\cdots\!80\)\( T^{9} + \)\(85\!\cdots\!07\)\( T^{10} )^{2} \))
$11$ (\( 1 - 62100824999962 T^{2} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 - 179340783809858 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(50\!\cdots\!78\)\( T^{8} - \)\(19\!\cdots\!64\)\( T^{10} + \)\(60\!\cdots\!58\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!21\)\( T^{16} - \)\(36\!\cdots\!78\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \))
$13$ (\( 1 + 360392330876150 T^{2} + \)\(91\!\cdots\!09\)\( T^{4} \))(\( 1 - 1881700477395890 T^{2} + \)\(17\!\cdots\!13\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(48\!\cdots\!02\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!18\)\( T^{12} - \)\(91\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!77\)\( T^{16} - \)\(13\!\cdots\!90\)\( T^{18} + \)\(64\!\cdots\!49\)\( T^{20} \))
$17$ (\( ( 1 + 133520302 T + 9904578032905937 T^{2} )^{2} \))(\( ( 1 - 108663002 T + 32276400633367229 T^{2} - \)\(23\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!42\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!54\)\( T^{6} - \)\(23\!\cdots\!96\)\( T^{7} + \)\(31\!\cdots\!37\)\( T^{8} - \)\(10\!\cdots\!22\)\( T^{9} + \)\(95\!\cdots\!57\)\( T^{10} )^{2} \))
$19$ (\( 1 - 82896428399326282 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 162780863064290354 T^{2} + \)\(13\!\cdots\!97\)\( T^{4} - \)\(80\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!66\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{10} + \)\(69\!\cdots\!46\)\( T^{12} - \)\(25\!\cdots\!96\)\( T^{14} + \)\(75\!\cdots\!77\)\( T^{16} - \)\(15\!\cdots\!34\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \))
$23$ (\( ( 1 + 35585416 T + 504036361936467383 T^{2} )^{2} \))(\( ( 1 + 39339976 T + 1242751119340586771 T^{2} + \)\(42\!\cdots\!92\)\( T^{3} + \)\(76\!\cdots\!02\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(38\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!88\)\( T^{7} + \)\(15\!\cdots\!77\)\( T^{8} + \)\(25\!\cdots\!96\)\( T^{9} + \)\(32\!\cdots\!43\)\( T^{10} )^{2} \))
$29$ (\( 1 - 18002148940349036842 T^{2} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 - 60047671171855484498 T^{2} + \)\(18\!\cdots\!41\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{6} + \)\(58\!\cdots\!78\)\( T^{8} - \)\(67\!\cdots\!84\)\( T^{10} + \)\(61\!\cdots\!38\)\( T^{12} - \)\(42\!\cdots\!60\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} - \)\(73\!\cdots\!38\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \))
$31$ (\( ( 1 + 5765001568 T + 24417546297445042591 T^{2} )^{2} \))(\( ( 1 - 324116896 T + 94759416869201467419 T^{2} - \)\(76\!\cdots\!28\)\( T^{3} + \)\(40\!\cdots\!58\)\( T^{4} - \)\(32\!\cdots\!48\)\( T^{5} + \)\(98\!\cdots\!78\)\( T^{6} - \)\(45\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!49\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(86\!\cdots\!51\)\( T^{10} )^{2} \))
$37$ (\( 1 - \)\(31\!\cdots\!70\)\( T^{2} + \)\(59\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(85\!\cdots\!14\)\( T^{2} + \)\(35\!\cdots\!29\)\( T^{4} - \)\(83\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!78\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{10} + \)\(67\!\cdots\!02\)\( T^{12} - \)\(29\!\cdots\!44\)\( T^{14} + \)\(73\!\cdots\!41\)\( T^{16} - \)\(10\!\cdots\!54\)\( T^{18} + \)\(73\!\cdots\!49\)\( T^{20} \))
$41$ (\( ( 1 + 23546348918 T + \)\(92\!\cdots\!21\)\( T^{2} )^{2} \))(\( ( 1 - 29662320178 T + \)\(14\!\cdots\!89\)\( T^{2} - \)\(15\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!62\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(97\!\cdots\!02\)\( T^{6} - \)\(13\!\cdots\!56\)\( T^{7} + \)\(11\!\cdots\!29\)\( T^{8} - \)\(21\!\cdots\!18\)\( T^{9} + \)\(67\!\cdots\!01\)\( T^{10} )^{2} \))
$43$ (\( 1 - \)\(32\!\cdots\!30\)\( T^{2} + \)\(29\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(61\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!05\)\( T^{4} - \)\(25\!\cdots\!04\)\( T^{6} + \)\(52\!\cdots\!62\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{10} + \)\(15\!\cdots\!38\)\( T^{12} - \)\(22\!\cdots\!04\)\( T^{14} + \)\(39\!\cdots\!45\)\( T^{16} - \)\(46\!\cdots\!14\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \))
$47$ (\( ( 1 + 68107736592 T + \)\(54\!\cdots\!27\)\( T^{2} )^{2} \))(\( ( 1 + 5088267408 T + \)\(10\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(79\!\cdots\!70\)\( T^{4} + \)\(90\!\cdots\!04\)\( T^{5} + \)\(43\!\cdots\!90\)\( T^{6} + \)\(48\!\cdots\!12\)\( T^{7} + \)\(17\!\cdots\!69\)\( T^{8} + \)\(45\!\cdots\!28\)\( T^{9} + \)\(48\!\cdots\!07\)\( T^{10} )^{2} \))
$53$ (\( 1 - \)\(24\!\cdots\!30\)\( T^{2} + \)\(67\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(15\!\cdots\!98\)\( T^{2} + \)\(12\!\cdots\!49\)\( T^{4} - \)\(63\!\cdots\!52\)\( T^{6} + \)\(24\!\cdots\!06\)\( T^{8} - \)\(72\!\cdots\!20\)\( T^{10} + \)\(16\!\cdots\!74\)\( T^{12} - \)\(29\!\cdots\!32\)\( T^{14} + \)\(37\!\cdots\!61\)\( T^{16} - \)\(32\!\cdots\!38\)\( T^{18} + \)\(14\!\cdots\!49\)\( T^{20} \))
$59$ (\( 1 - \)\(19\!\cdots\!22\)\( T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(54\!\cdots\!82\)\( T^{2} + \)\(16\!\cdots\!77\)\( T^{4} - \)\(32\!\cdots\!52\)\( T^{6} + \)\(49\!\cdots\!86\)\( T^{8} - \)\(58\!\cdots\!80\)\( T^{10} + \)\(54\!\cdots\!26\)\( T^{12} - \)\(39\!\cdots\!12\)\( T^{14} + \)\(21\!\cdots\!17\)\( T^{16} - \)\(80\!\cdots\!02\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} \))
$61$ (\( 1 - \)\(14\!\cdots\!62\)\( T^{2} + \)\(26\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(10\!\cdots\!62\)\( T^{2} + \)\(47\!\cdots\!37\)\( T^{4} - \)\(13\!\cdots\!12\)\( T^{6} + \)\(29\!\cdots\!66\)\( T^{8} - \)\(51\!\cdots\!60\)\( T^{10} + \)\(77\!\cdots\!26\)\( T^{12} - \)\(96\!\cdots\!52\)\( T^{14} + \)\(86\!\cdots\!97\)\( T^{16} - \)\(48\!\cdots\!42\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \))
$67$ (\( 1 - \)\(95\!\cdots\!30\)\( T^{2} + \)\(30\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(19\!\cdots\!42\)\( T^{2} + \)\(23\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(71\!\cdots\!60\)\( T^{10} + \)\(37\!\cdots\!34\)\( T^{12} - \)\(17\!\cdots\!68\)\( T^{14} + \)\(64\!\cdots\!41\)\( T^{16} - \)\(16\!\cdots\!82\)\( T^{18} + \)\(24\!\cdots\!49\)\( T^{20} \))
$71$ (\( ( 1 + 1309471657368 T + \)\(11\!\cdots\!11\)\( T^{2} )^{2} \))(\( ( 1 - 363180589992 T + \)\(34\!\cdots\!95\)\( T^{2} - \)\(14\!\cdots\!16\)\( T^{3} + \)\(61\!\cdots\!94\)\( T^{4} - \)\(22\!\cdots\!64\)\( T^{5} + \)\(71\!\cdots\!34\)\( T^{6} - \)\(20\!\cdots\!36\)\( T^{7} + \)\(53\!\cdots\!45\)\( T^{8} - \)\(66\!\cdots\!72\)\( T^{9} + \)\(21\!\cdots\!51\)\( T^{10} )^{2} \))
$73$ (\( ( 1 - 478647871914 T + \)\(16\!\cdots\!33\)\( T^{2} )^{2} \))(\( ( 1 + 316620182766 T + \)\(64\!\cdots\!53\)\( T^{2} + \)\(24\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(60\!\cdots\!52\)\( T^{5} + \)\(31\!\cdots\!82\)\( T^{6} + \)\(67\!\cdots\!76\)\( T^{7} + \)\(29\!\cdots\!61\)\( T^{8} + \)\(24\!\cdots\!86\)\( T^{9} + \)\(13\!\cdots\!93\)\( T^{10} )^{2} \))
$79$ (\( ( 1 + 364547231600 T + \)\(46\!\cdots\!39\)\( T^{2} )^{2} \))(\( ( 1 - 2722551782672 T + \)\(12\!\cdots\!19\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(55\!\cdots\!14\)\( T^{4} - \)\(94\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!46\)\( T^{6} - \)\(49\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!99\)\( T^{10} )^{2} \))
$83$ (\( 1 - \)\(16\!\cdots\!70\)\( T^{2} + \)\(78\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(43\!\cdots\!74\)\( T^{2} + \)\(87\!\cdots\!65\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!62\)\( T^{8} - \)\(14\!\cdots\!44\)\( T^{10} + \)\(11\!\cdots\!78\)\( T^{12} - \)\(76\!\cdots\!04\)\( T^{14} + \)\(42\!\cdots\!85\)\( T^{16} - \)\(16\!\cdots\!54\)\( T^{18} + \)\(30\!\cdots\!49\)\( T^{20} \))
$89$ (\( ( 1 + 102457641350 T + \)\(21\!\cdots\!69\)\( T^{2} )^{2} \))(\( ( 1 - 2753172404002 T + \)\(62\!\cdots\!29\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!54\)\( T^{4} - \)\(47\!\cdots\!48\)\( T^{5} + \)\(47\!\cdots\!26\)\( T^{6} - \)\(70\!\cdots\!24\)\( T^{7} + \)\(66\!\cdots\!61\)\( T^{8} - \)\(64\!\cdots\!42\)\( T^{9} + \)\(51\!\cdots\!49\)\( T^{10} )^{2} \))
$97$ (\( ( 1 + 6157717373342 T + \)\(67\!\cdots\!77\)\( T^{2} )^{2} \))(\( ( 1 - 680566660394 T + \)\(17\!\cdots\!53\)\( T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!90\)\( T^{6} + \)\(53\!\cdots\!32\)\( T^{7} + \)\(54\!\cdots\!49\)\( T^{8} - \)\(13\!\cdots\!54\)\( T^{9} + \)\(13\!\cdots\!57\)\( T^{10} )^{2} \))
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