Properties

Label 8.20.a.b
Level 8
Weight 20
Character orbit 8.a
Self dual yes
Analytic conductor 18.305
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3053357245\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 2519 x + 43659\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 7911 - \beta_{1} ) q^{3} + ( 713429 - 70 \beta_{1} + \beta_{2} ) q^{5} + ( 18618154 - 2722 \beta_{1} - 20 \beta_{2} ) q^{7} + ( 215591919 - 21956 \beta_{1} + 150 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 7911 - \beta_{1} ) q^{3} + ( 713429 - 70 \beta_{1} + \beta_{2} ) q^{5} + ( 18618154 - 2722 \beta_{1} - 20 \beta_{2} ) q^{7} + ( 215591919 - 21956 \beta_{1} + 150 \beta_{2} ) q^{9} + ( -99151979 + 63333 \beta_{1} - 360 \beta_{2} ) q^{11} + ( -4954589763 + 1369882 \beta_{1} - 1615 \beta_{2} ) q^{13} + ( 97547157054 - 5829910 \beta_{1} + 12276 \beta_{2} ) q^{15} + ( 267776864140 + 1887484 \beta_{1} - 15370 \beta_{2} ) q^{17} + ( 1070753896939 + 8077147 \beta_{1} - 101080 \beta_{2} ) q^{19} + ( 3730764546324 + 25817976 \beta_{1} + 372780 \beta_{2} ) q^{21} + ( 8316140088174 + 88928218 \beta_{1} + 112820 \beta_{2} ) q^{23} + ( 24113633854987 - 538445480 \beta_{1} - 2830372 \beta_{2} ) q^{25} + ( 21364107232062 + 18297878 \beta_{1} + 3559800 \beta_{2} ) q^{27} + ( 25888630013633 + 1239207634 \beta_{1} + 10262525 \beta_{2} ) q^{29} + ( -82810767991560 + 597433112 \beta_{1} - 28652000 \beta_{2} ) q^{31} + ( -84024721099314 + 2476663124 \beta_{1} - 10139310 \beta_{2} ) q^{33} + ( -461454980102684 - 13281410860 \beta_{1} + 107486704 \beta_{2} ) q^{35} + ( -138286237201687 - 7528018766 \beta_{1} - 62935315 \beta_{2} ) q^{37} + ( -1840692212529498 + 30869912018 \beta_{1} - 208350540 \beta_{2} ) q^{39} + ( 939574252603102 + 14862604952 \beta_{1} + 260455420 \beta_{2} ) q^{41} + ( -2221090480684363 + 40602131309 \beta_{1} + 124451920 \beta_{2} ) q^{43} + ( 7608373704953061 - 148810711670 \beta_{1} - 265972791 \beta_{2} ) q^{45} + ( 500210422580428 - 133848474700 \beta_{1} + 225462040 \beta_{2} ) q^{47} + ( 13213584127233233 + 371294308848 \beta_{1} - 618686760 \beta_{2} ) q^{49} + ( -361621376282970 - 177737853890 \beta_{1} - 310419720 \beta_{2} ) q^{51} + ( 18689081009440473 + 99868809010 \beta_{1} + 1877847725 \beta_{2} ) q^{53} + ( -18939413522672806 + 383997647150 \beta_{1} + 1010165836 \beta_{2} ) q^{55} + ( -2136147609531726 - 539513269844 \beta_{1} - 1391090130 \beta_{2} ) q^{57} + ( -51439318139589743 + 691356272217 \beta_{1} - 7788999840 \beta_{2} ) q^{59} + ( -44997198970745035 - 2780969311222 \beta_{1} + 1309891465 \beta_{2} ) q^{61} + ( -26144428277991294 - 1745298490410 \beta_{1} + 20034590220 \beta_{2} ) q^{63} + ( -187966850995212512 + 7766979173640 \beta_{1} - 4717422828 \beta_{2} ) q^{65} + ( 50342944062644463 + 3361346559743 \beta_{1} - 11630042840 \beta_{2} ) q^{67} + ( -51194273240697156 - 7533465669784 \beta_{1} - 13138864380 \beta_{2} ) q^{69} + ( 403514856373642538 - 2684735998098 \beta_{1} - 35539345380 \beta_{2} ) q^{71} + ( -27293891104567556 + 5474791886508 \beta_{1} + 74922153390 \beta_{2} ) q^{73} + ( 899432685208461537 - 19977236292455 \beta_{1} + 75740081328 \beta_{2} ) q^{75} + ( 32410846285929244 + 2546433188264 \beta_{1} - 62097485660 \beta_{2} ) q^{77} + ( 479664642636798084 + 34669544655436 \beta_{1} - 114419142280 \beta_{2} ) q^{79} + ( -106219021458074961 - 10302332459900 \beta_{1} - 170761696950 \beta_{2} ) q^{81} + ( -327821367763737149 + 25744231091131 \beta_{1} + 256261270400 \beta_{2} ) q^{83} + ( -539510402881407450 - 6954020917540 \beta_{1} + 318702521350 \beta_{2} ) q^{85} + ( -1426786918155990522 - 50902371514878 \beta_{1} - 167654900700 \beta_{2} ) q^{87} + ( -437597677067787636 - 64495714179348 \beta_{1} - 1914528690 \beta_{2} ) q^{89} + ( -3829371634623831212 - 27713537850460 \beta_{1} - 764920780400 \beta_{2} ) q^{91} + ( -1436155531565376240 + 209629915861600 \beta_{1} - 140500918800 \beta_{2} ) q^{93} + ( -3642044827469186394 - 20290339711150 \beta_{1} + 1424589549364 \beta_{2} ) q^{95} + ( 4677769063243700932 - 61680013256660 \beta_{1} - 97149847570 \beta_{2} ) q^{97} + ( -3805278927355985871 + 87109073527993 \beta_{1} + 28907244960 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 23732q^{3} + 2140218q^{5} + 55851720q^{7} + 646753951q^{9} + O(q^{10}) \) \( 3q + 23732q^{3} + 2140218q^{5} + 55851720q^{7} + 646753951q^{9} - 297392964q^{11} - 14862401022q^{13} + 292635653528q^{15} + 803332464534q^{17} + 3212269666884q^{19} + 11192319829728q^{21} + 24948509305560q^{23} + 72340360289109q^{25} + 64092343553864q^{27} + 77667139511058q^{29} - 248431735193568q^{31} - 252071696774128q^{33} - 1384378114232208q^{35} - 414866302559142q^{37} - 5522045976027016q^{39} + 2818737880869678q^{41} - 6663230715469860q^{43} + 22824972038174722q^{45} + 1500497644728624q^{47} + 39641123057321787q^{49} - 1085042177122520q^{51} + 56067344774978154q^{53} - 56817855560205432q^{55} - 6408983732955152q^{57} - 154317270851496852q^{59} - 134994376571654862q^{61} - 78435010097874072q^{63} - 563892790723886724q^{65} + 151032181904450292q^{67} - 153590366326625632q^{69} + 1210541848845584136q^{71} - 81876123599662770q^{73} + 2698278154129173484q^{75} + 97235023193490336q^{77} + 1439028483035907408q^{79} - 318667537468381733q^{81} - 983438102798849916q^{83} - 1618537843962618540q^{85} - 4280411824494387144q^{87} - 1312857528832070946q^{89} - 11488143382330124496q^{91} - 4308257105281185920q^{93} - 10926153348157720968q^{95} + 14033245412567998566q^{97} - 11415749644087184660q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 2519 x + 43659\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 64 \nu^{2} + 1664 \nu - 108053 \)
\(\beta_{2}\)\(=\)\( -896 \nu^{2} + 124160 \nu + 1463595 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 14 \beta_{1} + 49147\)\()/147456\)
\(\nu^{2}\)\(=\)\((\)\(-13 \beta_{2} + 970 \beta_{1} + 123838145\)\()/73728\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
37.1696
−56.8359
20.6663
0 −34307.5 0 2.59881e6 0 −1.93114e8 0 1.47441e7 0
1.2 0 3798.34 0 −8.06198e6 0 1.77174e8 0 −1.14783e9 0
1.3 0 54241.2 0 7.60339e6 0 7.17920e7 0 1.77984e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.20.a.b 3
3.b odd 2 1 72.20.a.f 3
4.b odd 2 1 16.20.a.f 3
8.b even 2 1 64.20.a.l 3
8.d odd 2 1 64.20.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.20.a.b 3 1.a even 1 1 trivial
16.20.a.f 3 4.b odd 2 1
64.20.a.l 3 8.b even 2 1
64.20.a.m 3 8.d odd 2 1
72.20.a.f 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 23732 T_{3}^{2} - 1785165264 T_{3} + \)\(70\!\cdots\!20\)\( \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 23732 T + 1701619137 T^{2} - 48097327472568 T^{3} + 1977726354444893979 T^{4} - \)\(32\!\cdots\!48\)\( T^{5} + \)\(15\!\cdots\!63\)\( T^{6} \)
$5$ \( 1 - 2140218 T - 5269684108605 T^{2} + 77660066501171142500 T^{3} - \)\(10\!\cdots\!25\)\( T^{4} - \)\(77\!\cdots\!50\)\( T^{5} + \)\(69\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 55851720 T - 1162511437121979 T^{2} + \)\(11\!\cdots\!68\)\( T^{3} - \)\(13\!\cdots\!97\)\( T^{4} - \)\(72\!\cdots\!80\)\( T^{5} + \)\(14\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + 297392964 T + \)\(16\!\cdots\!17\)\( T^{2} + \)\(48\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!47\)\( T^{4} + \)\(11\!\cdots\!84\)\( T^{5} + \)\(22\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 + 14862401022 T + \)\(61\!\cdots\!51\)\( T^{2} + \)\(91\!\cdots\!52\)\( T^{3} + \)\(90\!\cdots\!27\)\( T^{4} + \)\(31\!\cdots\!38\)\( T^{5} + \)\(31\!\cdots\!33\)\( T^{6} \)
$17$ \( 1 - 803332464534 T + \)\(91\!\cdots\!99\)\( T^{2} - \)\(39\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!47\)\( T^{4} - \)\(45\!\cdots\!06\)\( T^{5} + \)\(13\!\cdots\!77\)\( T^{6} \)
$19$ \( 1 - 3212269666884 T + \)\(86\!\cdots\!81\)\( T^{2} - \)\(13\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(77\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - 24948509305560 T + \)\(41\!\cdots\!93\)\( T^{2} - \)\(41\!\cdots\!72\)\( T^{3} + \)\(30\!\cdots\!91\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(41\!\cdots\!03\)\( T^{6} \)
$29$ \( 1 - 77667139511058 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(90\!\cdots\!52\)\( T^{3} + \)\(70\!\cdots\!47\)\( T^{4} - \)\(28\!\cdots\!38\)\( T^{5} + \)\(22\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 + 248431735193568 T + \)\(40\!\cdots\!53\)\( T^{2} + \)\(38\!\cdots\!56\)\( T^{3} + \)\(87\!\cdots\!63\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 + 414866302559142 T + \)\(16\!\cdots\!15\)\( T^{2} + \)\(51\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!18\)\( T^{5} + \)\(24\!\cdots\!17\)\( T^{6} \)
$41$ \( 1 - 2818737880869678 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(20\!\cdots\!72\)\( T^{3} + \)\(51\!\cdots\!43\)\( T^{4} - \)\(54\!\cdots\!38\)\( T^{5} + \)\(84\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 + 6663230715469860 T + \)\(43\!\cdots\!69\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(47\!\cdots\!83\)\( T^{4} + \)\(78\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - 1500497644728624 T + \)\(13\!\cdots\!41\)\( T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(81\!\cdots\!03\)\( T^{4} - \)\(51\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!87\)\( T^{6} \)
$53$ \( 1 - 56067344774978154 T + \)\(25\!\cdots\!23\)\( T^{2} - \)\(66\!\cdots\!68\)\( T^{3} + \)\(14\!\cdots\!91\)\( T^{4} - \)\(18\!\cdots\!06\)\( T^{5} + \)\(19\!\cdots\!13\)\( T^{6} \)
$59$ \( 1 + 154317270851496852 T + \)\(16\!\cdots\!97\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!83\)\( T^{4} + \)\(30\!\cdots\!92\)\( T^{5} + \)\(86\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + 134994376571654862 T + \)\(15\!\cdots\!43\)\( T^{2} + \)\(13\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} + \)\(93\!\cdots\!22\)\( T^{5} + \)\(58\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 - 151032181904450292 T + \)\(12\!\cdots\!49\)\( T^{2} - \)\(11\!\cdots\!96\)\( T^{3} + \)\(62\!\cdots\!47\)\( T^{4} - \)\(37\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \)
$71$ \( 1 - 1210541848845584136 T + \)\(85\!\cdots\!77\)\( T^{2} - \)\(39\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!87\)\( T^{4} - \)\(26\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + 81876123599662770 T + \)\(39\!\cdots\!23\)\( T^{2} + \)\(38\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!51\)\( T^{4} + \)\(52\!\cdots\!30\)\( T^{5} + \)\(16\!\cdots\!53\)\( T^{6} \)
$79$ \( 1 - 1439028483035907408 T + \)\(10\!\cdots\!53\)\( T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!07\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 + 983438102798849916 T + \)\(41\!\cdots\!01\)\( T^{2} + \)\(28\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!47\)\( T^{4} + \)\(82\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!23\)\( T^{6} \)
$89$ \( 1 + 1312857528832070946 T + \)\(25\!\cdots\!31\)\( T^{2} + \)\(23\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!79\)\( T^{4} + \)\(15\!\cdots\!26\)\( T^{5} + \)\(13\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 - 14033245412567998566 T + \)\(22\!\cdots\!51\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!83\)\( T^{4} - \)\(44\!\cdots\!74\)\( T^{5} + \)\(17\!\cdots\!37\)\( T^{6} \)
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