# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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