Properties

Label 8.20.a
Level 8
Weight 20
Character orbit a
Rep. character \(\chi_{8}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 2
Sturm bound 20
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(8))\).

Total New Old
Modular forms 21 5 16
Cusp forms 17 5 12
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(3\)
\(-\)\(2\)

Trace form

\( 5q - 4180q^{3} + 3366838q^{5} + 144362232q^{7} + 1389940489q^{9} + O(q^{10}) \) \( 5q - 4180q^{3} + 3366838q^{5} + 144362232q^{7} + 1389940489q^{9} - 7461180572q^{11} - 24989324626q^{13} + 157677482408q^{15} + 731287385594q^{17} + 91789194652q^{19} + 1413706100256q^{21} + 10189302215272q^{23} + 40130622290659q^{25} + 11408404421240q^{27} + 47417600266014q^{29} - 371821297971488q^{31} + 446388649086320q^{33} - 954185847061488q^{35} + 1600526867615382q^{37} - 5572632263745592q^{39} + 5359522840373922q^{41} - 12296885808859324q^{43} + 26569910848304302q^{45} - 20447841942401712q^{47} + 48013709720386573q^{49} - 46515035377714280q^{51} + 46649219708073478q^{53} - 87544709282208712q^{55} + 107141937924599440q^{57} - 55774821261089228q^{59} - 124702231193815042q^{61} + 192917193298957656q^{63} - 561659358363871964q^{65} + 226785809908434796q^{67} - 441174309724134304q^{69} + 1227948901412297912q^{71} - 939384378659495038q^{73} + 3003253016266453684q^{75} - 2128959489278385312q^{77} + 1212736561591052336q^{79} - 1521477247236102035q^{81} - 215922401337864868q^{83} + 380436178852939660q^{85} - 1466396114153266680q^{87} + 4779688365603103602q^{89} - 11324095727937747120q^{91} + 7763117681805520000q^{93} - 15920041058814060088q^{95} + 15581393662090345642q^{97} - 30782609165566913612q^{99} + O(q^{100}) \)

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(8))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
8.20.a.a \(2\) \(18.305\) \(\Q(\sqrt{1453}) \) None \(0\) \(-27912\) \(1226620\) \(88510512\) \(-\) \(q+(-13956-\beta )q^{3}+(613310+44\beta )q^{5}+\cdots\)
8.20.a.b \(3\) \(18.305\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(23732\) \(2140218\) \(55851720\) \(+\) \(q+(7911-\beta _{1})q^{3}+(713429-70\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(8))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(8)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 27912 T + 1180208070 T^{2} + 32441042066904 T^{3} + 1350851717672992089 T^{4} \))(\( 1 - 23732 T + 1701619137 T^{2} - 48097327472568 T^{3} + 1977726354444893979 T^{4} - \)\(32\!\cdots\!48\)\( T^{5} + \)\(15\!\cdots\!63\)\( T^{6} \))
$5$ (\( 1 - 1226620 T + 35930653639550 T^{2} - 23395919799804687500 T^{3} + \)\(36\!\cdots\!25\)\( T^{4} \))(\( 1 - 2140218 T - 5269684108605 T^{2} + 77660066501171142500 T^{3} - \)\(10\!\cdots\!25\)\( T^{4} - \)\(77\!\cdots\!50\)\( T^{5} + \)\(69\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 - 88510512 T + 11129657221091822 T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!49\)\( T^{4} \))(\( 1 - 55851720 T - 1162511437121979 T^{2} + \)\(11\!\cdots\!68\)\( T^{3} - \)\(13\!\cdots\!97\)\( T^{4} - \)\(72\!\cdots\!80\)\( T^{5} + \)\(14\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 + 7163787608 T + 68277596800784135798 T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + \)\(37\!\cdots\!81\)\( T^{4} \))(\( 1 + 297392964 T + \)\(16\!\cdots\!17\)\( T^{2} + \)\(48\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!47\)\( T^{4} + \)\(11\!\cdots\!84\)\( T^{5} + \)\(22\!\cdots\!71\)\( T^{6} \))
$13$ (\( 1 + 10126923604 T + \)\(29\!\cdots\!58\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!29\)\( T^{4} \))(\( 1 + 14862401022 T + \)\(61\!\cdots\!51\)\( T^{2} + \)\(91\!\cdots\!52\)\( T^{3} + \)\(90\!\cdots\!27\)\( T^{4} + \)\(31\!\cdots\!38\)\( T^{5} + \)\(31\!\cdots\!33\)\( T^{6} \))
$17$ (\( 1 + 72045078940 T + \)\(76\!\cdots\!06\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(57\!\cdots\!09\)\( T^{4} \))(\( 1 - 803332464534 T + \)\(91\!\cdots\!99\)\( T^{2} - \)\(39\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!47\)\( T^{4} - \)\(45\!\cdots\!06\)\( T^{5} + \)\(13\!\cdots\!77\)\( T^{6} \))
$19$ (\( 1 + 3120480472232 T + \)\(54\!\cdots\!14\)\( T^{2} + \)\(61\!\cdots\!28\)\( T^{3} + \)\(39\!\cdots\!41\)\( T^{4} \))(\( 1 - 3212269666884 T + \)\(86\!\cdots\!81\)\( T^{2} - \)\(13\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(77\!\cdots\!39\)\( T^{6} \))
$23$ (\( 1 + 14759207090288 T + \)\(15\!\cdots\!10\)\( T^{2} + \)\(11\!\cdots\!56\)\( T^{3} + \)\(55\!\cdots\!69\)\( T^{4} \))(\( 1 - 24948509305560 T + \)\(41\!\cdots\!93\)\( T^{2} - \)\(41\!\cdots\!72\)\( T^{3} + \)\(30\!\cdots\!91\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(41\!\cdots\!03\)\( T^{6} \))
$29$ (\( 1 + 30249539245044 T + \)\(11\!\cdots\!22\)\( T^{2} + \)\(18\!\cdots\!36\)\( T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( 1 - 77667139511058 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(90\!\cdots\!52\)\( T^{3} + \)\(70\!\cdots\!47\)\( T^{4} - \)\(28\!\cdots\!38\)\( T^{5} + \)\(22\!\cdots\!09\)\( T^{6} \))
$31$ (\( 1 + 123389562777920 T + \)\(27\!\cdots\!42\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!41\)\( T^{4} \))(\( 1 + 248431735193568 T + \)\(40\!\cdots\!53\)\( T^{2} + \)\(38\!\cdots\!56\)\( T^{3} + \)\(87\!\cdots\!63\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!11\)\( T^{6} \))
$37$ (\( 1 - 2015393170174524 T + \)\(22\!\cdots\!90\)\( T^{2} - \)\(12\!\cdots\!52\)\( T^{3} + \)\(39\!\cdots\!29\)\( T^{4} \))(\( 1 + 414866302559142 T + \)\(16\!\cdots\!15\)\( T^{2} + \)\(51\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!18\)\( T^{5} + \)\(24\!\cdots\!17\)\( T^{6} \))
$41$ (\( 1 - 2540784959504244 T + \)\(74\!\cdots\!06\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))(\( 1 - 2818737880869678 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(20\!\cdots\!72\)\( T^{3} + \)\(51\!\cdots\!43\)\( T^{4} - \)\(54\!\cdots\!38\)\( T^{5} + \)\(84\!\cdots\!81\)\( T^{6} \))
$43$ (\( 1 + 5633655093389464 T + \)\(29\!\cdots\!38\)\( T^{2} + \)\(61\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \))(\( 1 + 6663230715469860 T + \)\(43\!\cdots\!69\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(47\!\cdots\!83\)\( T^{4} + \)\(78\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 + 21948339587130336 T + \)\(23\!\cdots\!90\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(34\!\cdots\!89\)\( T^{4} \))(\( 1 - 1500497644728624 T + \)\(13\!\cdots\!41\)\( T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(81\!\cdots\!03\)\( T^{4} - \)\(51\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!87\)\( T^{6} \))
$53$ (\( 1 + 9418125066904676 T + \)\(29\!\cdots\!78\)\( T^{2} + \)\(54\!\cdots\!92\)\( T^{3} + \)\(33\!\cdots\!89\)\( T^{4} \))(\( 1 - 56067344774978154 T + \)\(25\!\cdots\!23\)\( T^{2} - \)\(66\!\cdots\!68\)\( T^{3} + \)\(14\!\cdots\!91\)\( T^{4} - \)\(18\!\cdots\!06\)\( T^{5} + \)\(19\!\cdots\!13\)\( T^{6} \))
$59$ (\( 1 - 98542449590407624 T + \)\(95\!\cdots\!22\)\( T^{2} - \)\(43\!\cdots\!36\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))(\( 1 + 154317270851496852 T + \)\(16\!\cdots\!97\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!83\)\( T^{4} + \)\(30\!\cdots\!92\)\( T^{5} + \)\(86\!\cdots\!19\)\( T^{6} \))
$61$ (\( 1 - 10292145377839820 T + \)\(11\!\cdots\!82\)\( T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(69\!\cdots\!81\)\( T^{4} \))(\( 1 + 134994376571654862 T + \)\(15\!\cdots\!43\)\( T^{2} + \)\(13\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} + \)\(93\!\cdots\!22\)\( T^{5} + \)\(58\!\cdots\!21\)\( T^{6} \))
$67$ (\( 1 - 75753628003984504 T + \)\(44\!\cdots\!10\)\( T^{2} - \)\(37\!\cdots\!12\)\( T^{3} + \)\(24\!\cdots\!09\)\( T^{4} \))(\( 1 - 151032181904450292 T + \)\(12\!\cdots\!49\)\( T^{2} - \)\(11\!\cdots\!96\)\( T^{3} + \)\(62\!\cdots\!47\)\( T^{4} - \)\(37\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \))
$71$ (\( 1 - 17407052566713776 T + \)\(27\!\cdots\!06\)\( T^{2} - \)\(25\!\cdots\!56\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))(\( 1 - 1210541848845584136 T + \)\(85\!\cdots\!77\)\( T^{2} - \)\(39\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!87\)\( T^{4} - \)\(26\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!91\)\( T^{6} \))
$73$ (\( 1 + 857508255059832268 T + \)\(53\!\cdots\!30\)\( T^{2} + \)\(21\!\cdots\!16\)\( T^{3} + \)\(64\!\cdots\!69\)\( T^{4} \))(\( 1 + 81876123599662770 T + \)\(39\!\cdots\!23\)\( T^{2} + \)\(38\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!51\)\( T^{4} + \)\(52\!\cdots\!30\)\( T^{5} + \)\(16\!\cdots\!53\)\( T^{6} \))
$79$ (\( 1 + 226291921444855072 T + \)\(21\!\cdots\!34\)\( T^{2} + \)\(25\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} \))(\( 1 - 1439028483035907408 T + \)\(10\!\cdots\!53\)\( T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!07\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!59\)\( T^{6} \))
$83$ (\( 1 - 767515701460985048 T + \)\(59\!\cdots\!70\)\( T^{2} - \)\(22\!\cdots\!56\)\( T^{3} + \)\(84\!\cdots\!09\)\( T^{4} \))(\( 1 + 983438102798849916 T + \)\(41\!\cdots\!01\)\( T^{2} + \)\(28\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!47\)\( T^{4} + \)\(82\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!23\)\( T^{6} \))
$89$ (\( 1 - 6092545894435174548 T + \)\(31\!\cdots\!94\)\( T^{2} - \)\(66\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \))(\( 1 + 1312857528832070946 T + \)\(25\!\cdots\!31\)\( T^{2} + \)\(23\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!79\)\( T^{4} + \)\(15\!\cdots\!26\)\( T^{5} + \)\(13\!\cdots\!29\)\( T^{6} \))
$97$ (\( 1 - 1548148249522347076 T + \)\(66\!\cdots\!10\)\( T^{2} - \)\(86\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!89\)\( T^{4} \))(\( 1 - 14033245412567998566 T + \)\(22\!\cdots\!51\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!83\)\( T^{4} - \)\(44\!\cdots\!74\)\( T^{5} + \)\(17\!\cdots\!37\)\( T^{6} \))
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