LMFDB - Embedded newform 8.20.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,20,Mod(1,8)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 20, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.3053357245$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1453}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 363 $ x^2 - x - 363
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-18.5591$ of defining polynomial
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22637.5 q^{3} -996804. q^{5} -7.24780e7 q^{7} -6.49805e8 q^{9} +O(q^{10})$	$q+22637.5 q^{3} -996804. q^{5} -7.24780e7 q^{7} -6.49805e8 q^{9} +4.59555e9 q^{11} -7.68575e9 q^{13} -2.25652e10 q^{15} -6.70499e11 q^{17} -6.03766e11 q^{19} -1.64072e12 q^{21} -1.41235e13 q^{23} -1.80799e13 q^{25} -4.10207e13 q^{27} +1.75566e13 q^{29} +7.97149e13 q^{31} +1.04032e14 q^{33} +7.22464e13 q^{35} +1.12013e15 q^{37} -1.73986e14 q^{39} +3.00183e15 q^{41} -3.41815e15 q^{43} +6.47728e14 q^{45} -1.14830e16 q^{47} -6.14583e15 q^{49} -1.51784e16 q^{51} +2.49535e16 q^{53} -4.58086e15 q^{55} -1.36678e16 q^{57} +7.22986e15 q^{59} +1.29766e17 q^{61} +4.70966e16 q^{63} +7.66119e15 q^{65} +2.74592e17 q^{67} -3.19720e17 q^{69} +1.56270e17 q^{71} -8.21165e17 q^{73} -4.09283e17 q^{75} -3.33076e17 q^{77} +2.09830e17 q^{79} -1.73363e17 q^{81} +3.15127e17 q^{83} +6.68357e17 q^{85} +3.97438e17 q^{87} +2.78627e18 q^{89} +5.57048e17 q^{91} +1.80455e18 q^{93} +6.01837e17 q^{95} +7.58373e18 q^{97} -2.98621e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9}+O(q^{10}) $ 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9	$ 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9} - 7163787608 q^{11} - 10126923604 q^{13} - 134958171120 q^{15} - 72045078940 q^{17} - 3120480472232 q^{19} - 9778613729472 q^{21} - 14759207090288 q^{23} - 32209737998450 q^{25} - 52683939132624 q^{27} - 30249539245044 q^{29} - 123389562777920 q^{31} + 698460345860448 q^{33} + 430192267170720 q^{35} + 20\!\cdots\!24 q^{37}+ \cdots - 19\!\cdots\!52 q^{99}+O(q^{100}) $ 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9 - 7163787608 * q^11 - 10126923604 * q^13 - 134958171120 * q^15 - 72045078940 * q^17 - 3120480472232 * q^19 - 9778613729472 * q^21 - 14759207090288 * q^23 - 32209737998450 * q^25 - 52683939132624 * q^27 - 30249539245044 * q^29 - 123389562777920 * q^31 + 698460345860448 * q^33 + 430192267170720 * q^35 + 2015393170174524 * q^37 - 50586287718576 * q^39 + 2540784959504244 * q^41 - 5633655093389464 * q^43 + 3744938810129580 * q^45 - 21948339587130336 * q^47 + 8372586663064786 * q^49 - 45429993200591760 * q^51 - 9418125066904676 * q^53 - 30726853722003280 * q^55 + 113550921657554592 * q^57 + 98542449590407624 * q^59 + 10292145377839820 * q^61 + 271352203396831728 * q^63 + 2233432360014760 * q^65 + 75753628003984504 * q^67 - 287583943397508672 * q^69 + 17407052566713776 * q^71 - 857508255059832268 * q^73 + 304974862137280200 * q^75 - 2226194512471875648 * q^77 - 226291921444855072 * q^79 - 1202809709767720302 * q^81 + 767515701460985048 * q^83 + 1998974022815558200 * q^85 + 2814015710341120464 * q^87 + 6092545894435174548 * q^89 + 164047654392377376 * q^91 + 12071374787086705920 * q^93 - 4993887710656339120 * q^95 + 1548148249522347076 * q^97 - 19366859521479728952 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	22637.5	0.664013	0.332007	−	0.943277i	$-0.392274\pi$
$3$	22637.5	0.664013	0.332007	+	0.943277i	$0.392274\pi$
$4$	0	0
$5$	−996804.	−0.228242	−0.114121	−	0.993467i	$-0.536405\pi$
$5$	−996804.	−0.228242	−0.114121	+	0.993467i	$0.536405\pi$
$6$	0	0
$7$	−7.24780e7	−0.678852	−0.339426	−	0.940633i	$-0.610233\pi$
$7$	−7.24780e7	−0.678852	−0.339426	+	0.940633i	$0.610233\pi$
$8$	0	0
$9$	−6.49805e8	−0.559087
$10$	0	0
$11$	4.59555e9	0.587634	0.293817	−	0.955862i	$-0.405074\pi$
$11$	4.59555e9	0.587634	0.293817	+	0.955862i	$0.405074\pi$
$12$	0	0
$13$	−7.68575e9	−0.201013	−0.100507	−	0.994936i	$-0.532046\pi$
$13$	−7.68575e9	−0.201013	−0.100507	+	0.994936i	$0.532046\pi$
$14$	0	0
$15$	−2.25652e10	−0.151555
$16$	0	0
$17$	−6.70499e11	−1.37130	−0.685652	−	0.727930i	$-0.740483\pi$
$17$	−6.70499e11	−1.37130	−0.685652	+	0.727930i	$0.740483\pi$
$18$	0	0
$19$	−6.03766e11	−0.429249	−0.214625	−	0.976697i	$-0.568853\pi$
$19$	−6.03766e11	−0.429249	−0.214625	+	0.976697i	$0.568853\pi$
$20$	0	0
$21$	−1.64072e12	−0.450767
$22$	0	0
$23$	−1.41235e13	−1.63503	−0.817517	−	0.575904i	$-0.804650\pi$
$23$	−1.41235e13	−1.63503	−0.817517	+	0.575904i	$0.804650\pi$
$24$	0	0
$25$	−1.80799e13	−0.947906
$26$	0	0
$27$	−4.10207e13	−1.03525
$28$	0	0
$29$	1.75566e13	0.224729	0.112365	−	0.993667i	$-0.464158\pi$
$29$	1.75566e13	0.224729	0.112365	+	0.993667i	$0.464158\pi$
$30$	0	0
$31$	7.97149e13	0.541506	0.270753	−	0.962649i	$-0.412727\pi$
$31$	7.97149e13	0.541506	0.270753	+	0.962649i	$0.412727\pi$
$32$	0	0
$33$	1.04032e14	0.390197
$34$	0	0
$35$	7.22464e13	0.154942
$36$	0	0
$37$	1.12013e15	1.41694	0.708469	−	0.705742i	$-0.249386\pi$
$37$	1.12013e15	1.41694	0.708469	+	0.705742i	$0.249386\pi$
$38$	0	0
$39$	−1.73986e14	−0.133475
$40$	0	0
$41$	3.00183e15	1.43199	0.715995	−	0.698106i	$-0.245974\pi$
$41$	3.00183e15	1.43199	0.715995	+	0.698106i	$0.245974\pi$
$42$	0	0
$43$	−3.41815e15	−1.03715	−0.518574	−	0.855033i	$-0.673537\pi$
$43$	−3.41815e15	−1.03715	−0.518574	+	0.855033i	$0.673537\pi$
$44$	0	0
$45$	6.47728e14	0.127607
$46$	0	0
$47$	−1.14830e16	−1.49667	−0.748336	−	0.663320i	$-0.769147\pi$
$47$	−1.14830e16	−1.49667	−0.748336	+	0.663320i	$0.769147\pi$
$48$	0	0
$49$	−6.14583e15	−0.539160
$50$	0	0
$51$	−1.51784e16	−0.910563
$52$	0	0
$53$	2.49535e16	1.03875	0.519374	−	0.854547i	$-0.326165\pi$
$53$	2.49535e16	1.03875	0.519374	+	0.854547i	$0.326165\pi$
$54$	0	0
$55$	−4.58086e15	−0.134122
$56$	0	0
$57$	−1.36678e16	−0.285027
$58$	0	0
$59$	7.22986e15	0.108652	0.0543258	−	0.998523i	$-0.482699\pi$
$59$	7.22986e15	0.108652	0.0543258	+	0.998523i	$0.482699\pi$
$60$	0	0
$61$	1.29766e17	1.42078	0.710391	−	0.703808i	$-0.248518\pi$
$61$	1.29766e17	1.42078	0.710391	+	0.703808i	$0.248518\pi$
$62$	0	0
$63$	4.70966e16	0.379537
$64$	0	0
$65$	7.66119e15	0.0458796
$66$	0	0
$67$	2.74592e17	1.23304	0.616520	−	0.787340i	$-0.288542\pi$
$67$	2.74592e17	1.23304	0.616520	+	0.787340i	$0.288542\pi$
$68$	0	0
$69$	−3.19720e17	−1.08568
$70$	0	0
$71$	1.56270e17	0.404504	0.202252	−	0.979334i	$-0.435174\pi$
$71$	1.56270e17	0.404504	0.202252	+	0.979334i	$0.435174\pi$
$72$	0	0
$73$	−8.21165e17	−1.63254	−0.816269	−	0.577672i	$-0.803961\pi$
$73$	−8.21165e17	−1.63254	−0.816269	+	0.577672i	$0.803961\pi$
$74$	0	0
$75$	−4.09283e17	−0.629422
$76$	0	0
$77$	−3.33076e17	−0.398916
$78$	0	0
$79$	2.09830e17	0.196974	0.0984871	−	0.995138i	$-0.468600\pi$
$79$	2.09830e17	0.196974	0.0984871	+	0.995138i	$0.468600\pi$
$80$	0	0
$81$	−1.73363e17	−0.128336
$82$	0	0
$83$	3.15127e17	0.185030	0.0925152	−	0.995711i	$-0.470509\pi$
$83$	3.15127e17	0.185030	0.0925152	+	0.995711i	$0.470509\pi$
$84$	0	0
$85$	6.68357e17	0.312989
$86$	0	0
$87$	3.97438e17	0.149223
$88$	0	0
$89$	2.78627e18	0.842981	0.421491	−	0.906833i	$-0.361507\pi$
$89$	2.78627e18	0.842981	0.421491	+	0.906833i	$0.361507\pi$
$90$	0	0
$91$	5.57048e17	0.136458
$92$	0	0
$93$	1.80455e18	0.359567
$94$	0	0
$95$	6.01837e17	0.0979725
$96$	0	0
$97$	7.58373e18	1.01286	0.506432	−	0.862280i	$-0.330964\pi$
$97$	7.58373e18	1.01286	0.506432	+	0.862280i	$0.330964\pi$
$98$	0	0
$99$	−2.98621e18	−0.328538
$100$	0	0
$101$	−3.62374e18	−0.329689	−0.164845	−	0.986320i	$-0.552712\pi$
$101$	−3.62374e18	−0.329689	−0.164845	+	0.986320i	$0.552712\pi$
$102$	0	0
$103$	−2.07112e19	−1.56405	−0.782026	−	0.623246i	$-0.785814\pi$
$103$	−2.07112e19	−1.56405	−0.782026	+	0.623246i	$0.785814\pi$
$104$	0	0
$105$	1.63548e18	0.102884
$106$	0	0
$107$	−2.42775e19	−1.27661	−0.638305	−	0.769784i	$-0.720364\pi$
$107$	−2.42775e19	−1.27661	−0.638305	+	0.769784i	$0.720364\pi$
$108$	0	0
$109$	−1.19033e19	−0.524946	−0.262473	−	0.964939i	$-0.584538\pi$
$109$	−1.19033e19	−0.524946	−0.262473	+	0.964939i	$0.584538\pi$
$110$	0	0
$111$	2.53569e19	0.940866
$112$	0	0
$113$	−2.65952e18	−0.0832833	−0.0416417	−	0.999133i	$-0.513259\pi$
$113$	−2.65952e18	−0.0832833	−0.0416417	+	0.999133i	$0.513259\pi$
$114$	0	0
$115$	1.40783e19	0.373183
$116$	0	0
$117$	4.99424e18	0.112384
$118$	0	0
$119$	4.85965e19	0.930912
$120$	0	0
$121$	−4.00400e19	−0.654687
$122$	0	0
$123$	6.79540e19	0.950860
$124$	0	0
$125$	3.70346e19	0.444593
$126$	0	0
$127$	6.57941e19	0.679284	0.339642	−	0.940555i	$-0.389694\pi$
$127$	6.57941e19	0.679284	0.339642	+	0.940555i	$0.389694\pi$
$128$	0	0
$129$	−7.73784e19	−0.688680
$130$	0	0
$131$	−3.73472e19	−0.287198	−0.143599	−	0.989636i	$-0.545868\pi$
$131$	−3.73472e19	−0.287198	−0.143599	+	0.989636i	$0.545868\pi$
$132$	0	0
$133$	4.37598e19	0.291397
$134$	0	0
$135$	4.08896e19	0.236288
$136$	0	0
$137$	−3.14142e18	−0.0157863	−0.00789315	−	0.999969i	$-0.502512\pi$
$137$	−3.14142e18	−0.0157863	−0.00789315	+	0.999969i	$0.502512\pi$
$138$	0	0
$139$	−3.82616e20	−1.67542	−0.837709	−	0.546117i	$-0.816105\pi$
$139$	−3.82616e20	−1.67542	−0.837709	+	0.546117i	$0.816105\pi$
$140$	0	0
$141$	−2.59947e20	−0.993809
$142$	0	0
$143$	−3.53202e19	−0.118122
$144$	0	0
$145$	−1.75005e19	−0.0512926
$146$	0	0
$147$	−1.39126e20	−0.358009
$148$	0	0
$149$	3.27921e20	0.742163	0.371081	−	0.928600i	$-0.378987\pi$
$149$	3.27921e20	0.742163	0.371081	+	0.928600i	$0.378987\pi$
$150$	0	0
$151$	1.20975e20	0.241220	0.120610	−	0.992700i	$-0.461515\pi$
$151$	1.20975e20	0.241220	0.120610	+	0.992700i	$0.461515\pi$
$152$	0	0
$153$	4.35694e20	0.766677
$154$	0	0
$155$	−7.94601e19	−0.123594
$156$	0	0
$157$	5.68747e20	0.783199	0.391599	−	0.920136i	$-0.371922\pi$
$157$	5.68747e20	0.783199	0.391599	+	0.920136i	$0.371922\pi$
$158$	0	0
$159$	5.64884e20	0.689742
$160$	0	0
$161$	1.02364e21	1.10995
$162$	0	0
$163$	−1.77617e21	−1.71279	−0.856393	−	0.516325i	$-0.827300\pi$
$163$	−1.77617e21	−1.71279	−0.856393	+	0.516325i	$0.827300\pi$
$164$	0	0
$165$	−1.03699e20	−0.0890591
$166$	0	0
$167$	1.55998e21	1.19485	0.597424	−	0.801926i	$-0.296191\pi$
$167$	1.55998e21	1.19485	0.597424	+	0.801926i	$0.296191\pi$
$168$	0	0
$169$	−1.40285e21	−0.959594
$170$	0	0
$171$	3.92330e20	0.239987
$172$	0	0
$173$	−1.55034e21	−0.849159	−0.424580	−	0.905391i	$-0.639578\pi$
$173$	−1.55034e21	−0.849159	−0.424580	+	0.905391i	$0.639578\pi$
$174$	0	0
$175$	1.31039e21	0.643488
$176$	0	0
$177$	1.63666e20	0.0721461
$178$	0	0
$179$	−2.89048e21	−1.14516	−0.572579	−	0.819849i	$-0.694057\pi$
$179$	−2.89048e21	−1.14516	−0.572579	+	0.819849i	$0.694057\pi$
$180$	0	0
$181$	−5.15105e21	−1.83633	−0.918163	−	0.396204i	$-0.870327\pi$
$181$	−5.15105e21	−1.83633	−0.918163	+	0.396204i	$0.870327\pi$
$182$	0	0
$183$	2.93758e21	0.943418
$184$	0	0
$185$	−1.11655e21	−0.323404
$186$	0	0
$187$	−3.08131e21	−0.805824
$188$	0	0
$189$	2.97310e21	0.702784
$190$	0	0
$191$	7.17061e21	1.53369	0.766847	−	0.641830i	$-0.221824\pi$
$191$	7.17061e21	1.53369	0.766847	+	0.641830i	$0.221824\pi$
$192$	0	0
$193$	−3.25664e21	−0.630921	−0.315461	−	0.948939i	$-0.602159\pi$
$193$	−3.25664e21	−0.630921	−0.315461	+	0.948939i	$0.602159\pi$
$194$	0	0
$195$	1.73430e20	0.0304646
$196$	0	0
$197$	1.19004e22	1.89729	0.948643	−	0.316349i	$-0.102457\pi$
$197$	1.19004e22	1.89729	0.948643	+	0.316349i	$0.102457\pi$
$198$	0	0
$199$	−2.18989e21	−0.317188	−0.158594	−	0.987344i	$-0.550696\pi$
$199$	−2.18989e21	−0.317188	−0.158594	+	0.987344i	$0.550696\pi$
$200$	0	0
$201$	6.21607e21	0.818754
$202$	0	0
$203$	−1.27247e21	−0.152558
$204$	0	0
$205$	−2.99224e21	−0.326840
$206$	0	0
$207$	9.17750e21	0.914126
$208$	0	0
$209$	−2.77464e21	−0.252241
$210$	0	0
$211$	−2.32425e22	−1.93019	−0.965096	−	0.261898i	$-0.915652\pi$
$211$	−2.32425e22	−1.93019	−0.965096	+	0.261898i	$0.915652\pi$
$212$	0	0
$213$	3.53757e21	0.268596
$214$	0	0
$215$	3.40723e21	0.236720
$216$	0	0
$217$	−5.77758e21	−0.367602
$218$	0	0
$219$	−1.85891e22	−1.08403
$220$	0	0
$221$	5.15329e21	0.275650
$222$	0	0
$223$	−3.36614e21	−0.165286	−0.0826429	−	0.996579i	$-0.526336\pi$
$223$	−3.36614e21	−0.165286	−0.0826429	+	0.996579i	$0.526336\pi$
$224$	0	0
$225$	1.17484e22	0.529961
$226$	0	0
$227$	−3.44209e21	−0.142750	−0.0713751	−	0.997450i	$-0.522739\pi$
$227$	−3.44209e21	−0.142750	−0.0713751	+	0.997450i	$0.522739\pi$
$228$	0	0
$229$	−3.04661e22	−1.16246	−0.581232	−	0.813738i	$-0.697429\pi$
$229$	−3.04661e22	−1.16246	−0.581232	+	0.813738i	$0.697429\pi$
$230$	0	0
$231$	−7.54002e21	−0.264886
$232$	0	0
$233$	1.01497e22	0.328527	0.164264	−	0.986416i	$-0.447475\pi$
$233$	1.01497e22	0.328527	0.164264	+	0.986416i	$0.447475\pi$
$234$	0	0
$235$	1.14463e22	0.341603
$236$	0	0
$237$	4.75002e21	0.130793
$238$	0	0
$239$	5.94716e22	1.51192	0.755961	−	0.654616i	$-0.227170\pi$
$239$	5.94716e22	1.51192	0.755961	+	0.654616i	$0.227170\pi$
$240$	0	0
$241$	−6.71172e21	−0.157642	−0.0788210	−	0.996889i	$-0.525116\pi$
$241$	−6.71172e21	−0.157642	−0.0788210	+	0.996889i	$0.525116\pi$
$242$	0	0
$243$	4.37522e22	0.950037
$244$	0	0
$245$	6.12619e21	0.123059
$246$	0	0
$247$	4.64040e21	0.0862847
$248$	0	0
$249$	7.13368e21	0.122863
$250$	0	0
$251$	6.83985e22	1.09181	0.545904	−	0.837848i	$-0.316186\pi$
$251$	6.83985e22	1.09181	0.545904	+	0.837848i	$0.316186\pi$
$252$	0	0
$253$	−6.49051e22	−0.960802
$254$	0	0
$255$	1.51299e22	0.207828
$256$	0	0
$257$	−3.76051e22	−0.479603	−0.239801	−	0.970822i	$-0.577082\pi$
$257$	−3.76051e22	−0.479603	−0.239801	+	0.970822i	$0.577082\pi$
$258$	0	0
$259$	−8.11846e22	−0.961892
$260$	0	0
$261$	−1.14084e22	−0.125643
$262$	0	0
$263$	−1.77095e23	−1.81396	−0.906980	−	0.421173i	$-0.861618\pi$
$263$	−1.77095e23	−1.81396	−0.906980	+	0.421173i	$0.861618\pi$
$264$	0	0
$265$	−2.48737e22	−0.237086
$266$	0	0
$267$	6.30742e22	0.559751
$268$	0	0
$269$	1.19248e23	0.985837	0.492919	−	0.870075i	$-0.335930\pi$
$269$	1.19248e23	0.985837	0.492919	+	0.870075i	$0.335930\pi$
$270$	0	0
$271$	−8.67879e22	−0.668730	−0.334365	−	0.942444i	$-0.608522\pi$
$271$	−8.67879e22	−0.668730	−0.334365	+	0.942444i	$0.608522\pi$
$272$	0	0
$273$	1.26102e22	0.0906100
$274$	0	0
$275$	−8.30869e22	−0.557021
$276$	0	0
$277$	7.99312e21	0.0500217	0.0250109	−	0.999687i	$-0.492038\pi$
$277$	7.99312e21	0.0500217	0.0250109	+	0.999687i	$0.492038\pi$
$278$	0	0
$279$	−5.17991e22	−0.302749
$280$	0	0
$281$	4.90889e22	0.268085	0.134043	−	0.990976i	$-0.457204\pi$
$281$	4.90889e22	0.268085	0.134043	+	0.990976i	$0.457204\pi$
$282$	0	0
$283$	2.70773e23	1.38240	0.691201	−	0.722662i	$-0.257082\pi$
$283$	2.70773e23	1.38240	0.691201	+	0.722662i	$0.257082\pi$
$284$	0	0
$285$	1.36241e22	0.0650550
$286$	0	0
$287$	−2.17567e23	−0.972109
$288$	0	0
$289$	2.10497e23	0.880473
$290$	0	0
$291$	1.71677e23	0.672555
$292$	0	0
$293$	2.47805e23	0.909635	0.454818	−	0.890585i	$-0.349704\pi$
$293$	2.47805e23	0.909635	0.454818	+	0.890585i	$0.349704\pi$
$294$	0	0
$295$	−7.20676e21	−0.0247988
$296$	0	0
$297$	−1.88512e23	−0.608350
$298$	0	0
$299$	1.08549e23	0.328664
$300$	0	0
$301$	2.47741e23	0.704070
$302$	0	0
$303$	−8.20325e22	−0.218918
$304$	0	0
$305$	−1.29351e23	−0.324281
$306$	0	0
$307$	2.71564e23	0.639820	0.319910	−	0.947448i	$-0.396347\pi$
$307$	2.71564e23	0.639820	0.319910	+	0.947448i	$0.396347\pi$
$308$	0	0
$309$	−4.68850e23	−1.03855
$310$	0	0
$311$	−4.17401e23	−0.869620	−0.434810	−	0.900522i	$-0.643184\pi$
$311$	−4.17401e23	−0.869620	−0.434810	+	0.900522i	$0.643184\pi$
$312$	0	0
$313$	−2.21476e23	−0.434166	−0.217083	−	0.976153i	$-0.569654\pi$
$313$	−2.21476e23	−0.434166	−0.217083	+	0.976153i	$0.569654\pi$
$314$	0	0
$315$	−4.69461e22	−0.0866261
$316$	0	0
$317$	−8.33402e23	−1.44808	−0.724038	−	0.689760i	$-0.757716\pi$
$317$	−8.33402e23	−1.44808	−0.724038	+	0.689760i	$0.757716\pi$
$318$	0	0
$319$	8.06823e22	0.132059
$320$	0	0
$321$	−5.49582e23	−0.847686
$322$	0	0
$323$	4.04825e23	0.588631
$324$	0	0
$325$	1.38957e23	0.190542
$326$	0	0
$327$	−2.69460e23	−0.348571
$328$	0	0
$329$	8.32267e23	1.01602
$330$	0	0
$331$	1.25972e24	1.45181	0.725904	−	0.687796i	$-0.241422\pi$
$331$	1.25972e24	1.45181	0.725904	+	0.687796i	$0.241422\pi$
$332$	0	0
$333$	−7.27864e23	−0.792191
$334$	0	0
$335$	−2.73714e23	−0.281431
$336$	0	0
$337$	8.60548e23	0.836163	0.418082	−	0.908409i	$-0.362703\pi$
$337$	8.60548e23	0.836163	0.418082	+	0.908409i	$0.362703\pi$
$338$	0	0
$339$	−6.02049e22	−0.0553012
$340$	0	0
$341$	3.66333e23	0.318207
$342$	0	0
$343$	1.27161e24	1.04486
$344$	0	0
$345$	3.18698e23	0.247798
$346$	0	0
$347$	−2.55690e24	−1.88184	−0.940922	−	0.338623i	$-0.890039\pi$
$347$	−2.55690e24	−1.88184	−0.940922	+	0.338623i	$0.890039\pi$
$348$	0	0
$349$	1.73272e24	1.20750	0.603751	−	0.797173i	$-0.293672\pi$
$349$	1.73272e24	1.20750	0.603751	+	0.797173i	$0.293672\pi$
$350$	0	0
$351$	3.15275e23	0.208100
$352$	0	0
$353$	7.44110e23	0.465348	0.232674	−	0.972555i	$-0.425253\pi$
$353$	7.44110e23	0.465348	0.232674	+	0.972555i	$0.425253\pi$
$354$	0	0
$355$	−1.55771e23	−0.0923246
$356$	0	0
$357$	1.10010e24	0.618138
$358$	0	0
$359$	−1.67777e24	−0.893995	−0.446997	−	0.894535i	$-0.647507\pi$
$359$	−1.67777e24	−0.893995	−0.446997	+	0.894535i	$0.647507\pi$
$360$	0	0
$361$	−1.61389e24	−0.815745
$362$	0	0
$363$	−9.06407e23	−0.434720
$364$	0	0
$365$	8.18541e23	0.372613
$366$	0	0
$367$	1.06458e24	0.460096	0.230048	−	0.973179i	$-0.426112\pi$
$367$	1.06458e24	0.460096	0.230048	+	0.973179i	$0.426112\pi$
$368$	0	0
$369$	−1.95060e24	−0.800606
$370$	0	0
$371$	−1.80858e24	−0.705156
$372$	0	0
$373$	−3.55316e24	−1.31638	−0.658189	−	0.752853i	$-0.728677\pi$
$373$	−3.55316e24	−1.31638	−0.658189	+	0.752853i	$0.728677\pi$
$374$	0	0
$375$	8.38372e23	0.295216
$376$	0	0
$377$	−1.34936e23	−0.0451736
$378$	0	0
$379$	−2.96331e24	−0.943418	−0.471709	−	0.881754i	$-0.656363\pi$
$379$	−2.96331e24	−0.943418	−0.471709	+	0.881754i	$0.656363\pi$
$380$	0	0
$381$	1.48941e24	0.451054
$382$	0	0
$383$	4.14872e22	0.0119544	0.00597718	−	0.999982i	$-0.498097\pi$
$383$	4.14872e22	0.0119544	0.00597718	+	0.999982i	$0.498097\pi$
$384$	0	0
$385$	3.32012e23	0.0910493
$386$	0	0
$387$	2.22113e24	0.579855
$388$	0	0
$389$	1.32009e24	0.328158	0.164079	−	0.986447i	$-0.447535\pi$
$389$	1.32009e24	0.328158	0.164079	+	0.986447i	$0.447535\pi$
$390$	0	0
$391$	9.46978e24	2.24213
$392$	0	0
$393$	−8.45448e23	−0.190703
$394$	0	0
$395$	−2.09159e23	−0.0449577
$396$	0	0
$397$	−2.49987e24	−0.512162	−0.256081	−	0.966655i	$-0.582431\pi$
$397$	−2.49987e24	−0.512162	−0.256081	+	0.966655i	$0.582431\pi$
$398$	0	0
$399$	9.90612e23	0.193491
$400$	0	0
$401$	5.22882e24	0.973939	0.486970	−	0.873419i	$-0.338102\pi$
$401$	5.22882e24	0.973939	0.486970	+	0.873419i	$0.338102\pi$
$402$	0	0
$403$	−6.12669e23	−0.108850
$404$	0	0
$405$	1.72809e23	0.0292916
$406$	0	0
$407$	5.14760e24	0.832641
$408$	0	0
$409$	−3.59253e24	−0.554663	−0.277331	−	0.960774i	$-0.589450\pi$
$409$	−3.59253e24	−0.554663	−0.277331	+	0.960774i	$0.589450\pi$
$410$	0	0
$411$	−7.11139e22	−0.0104823
$412$	0	0
$413$	−5.24006e23	−0.0737583
$414$	0	0
$415$	−3.14120e23	−0.0422317
$416$	0	0
$417$	−8.66148e24	−1.11250
$418$	0	0
$419$	−7.64100e24	−0.937815	−0.468907	−	0.883247i	$-0.655352\pi$
$419$	−7.64100e24	−0.937815	−0.468907	+	0.883247i	$0.655352\pi$
$420$	0	0
$421$	−6.63037e24	−0.777782	−0.388891	−	0.921284i	$-0.627142\pi$
$421$	−6.63037e24	−0.777782	−0.388891	+	0.921284i	$0.627142\pi$
$422$	0	0
$423$	7.46172e24	0.836769
$424$	0	0
$425$	1.21225e25	1.29987
$426$	0	0
$427$	−9.40518e24	−0.964500
$428$	0	0
$429$	−7.99562e23	−0.0784347
$430$	0	0
$431$	−8.83021e24	−0.828776	−0.414388	−	0.910100i	$-0.636004\pi$
$431$	−8.83021e24	−0.828776	−0.414388	+	0.910100i	$0.636004\pi$
$432$	0	0
$433$	−1.89602e24	−0.170298	−0.0851488	−	0.996368i	$-0.527137\pi$
$433$	−1.89602e24	−0.170298	−0.0851488	+	0.996368i	$0.527137\pi$
$434$	0	0
$435$	−3.96168e23	−0.0340590
$436$	0	0
$437$	8.52727e24	0.701837
$438$	0	0
$439$	−2.03577e25	−1.60441	−0.802206	−	0.597047i	$-0.796340\pi$
$439$	−2.03577e25	−1.60441	−0.802206	+	0.597047i	$0.796340\pi$
$440$	0	0
$441$	3.99359e24	0.301437
$442$	0	0
$443$	−9.63050e24	−0.696327	−0.348164	−	0.937434i	$-0.613195\pi$
$443$	−9.63050e24	−0.696327	−0.348164	+	0.937434i	$0.613195\pi$
$444$	0	0
$445$	−2.77736e24	−0.192403
$446$	0	0
$447$	7.42330e24	0.492806
$448$	0	0
$449$	2.24366e25	1.42764	0.713818	−	0.700331i	$-0.246964\pi$
$449$	2.24366e25	1.42764	0.713818	+	0.700331i	$0.246964\pi$
$450$	0	0
$451$	1.37951e25	0.841485
$452$	0	0
$453$	2.73856e24	0.160173
$454$	0	0
$455$	−5.55268e23	−0.0311454
$456$	0	0
$457$	−1.43601e25	−0.772597	−0.386299	−	0.922374i	$-0.626247\pi$
$457$	−1.43601e25	−0.772597	−0.386299	+	0.922374i	$0.626247\pi$
$458$	0	0
$459$	2.75043e25	1.41965
$460$	0	0
$461$	3.20527e25	1.58747	0.793736	−	0.608262i	$-0.208133\pi$
$461$	3.20527e25	1.58747	0.793736	+	0.608262i	$0.208133\pi$
$462$	0	0
$463$	4.99392e24	0.237368	0.118684	−	0.992932i	$-0.462132\pi$
$463$	4.99392e24	0.237368	0.118684	+	0.992932i	$0.462132\pi$
$464$	0	0
$465$	−1.79878e24	−0.0820682
$466$	0	0
$467$	−8.24714e24	−0.361238	−0.180619	−	0.983553i	$-0.557810\pi$
$467$	−8.24714e24	−0.361238	−0.180619	+	0.983553i	$0.557810\pi$
$468$	0	0
$469$	−1.99019e25	−0.837051
$470$	0	0
$471$	1.28750e25	0.520054
$472$	0	0
$473$	−1.57083e25	−0.609463
$474$	0	0
$475$	1.09160e25	0.406888
$476$	0	0
$477$	−1.62149e25	−0.580750
$478$	0	0
$479$	−2.71323e25	−0.933899	−0.466950	−	0.884284i	$-0.654647\pi$
$479$	−2.71323e25	−0.933899	−0.466950	+	0.884284i	$0.654647\pi$
$480$	0	0
$481$	−8.60902e24	−0.284823
$482$	0	0
$483$	2.31727e25	0.737019
$484$	0	0
$485$	−7.55949e24	−0.231178
$486$	0	0
$487$	−5.00346e25	−1.47145	−0.735725	−	0.677281i	$-0.763158\pi$
$487$	−5.00346e25	−1.47145	−0.735725	+	0.677281i	$0.763158\pi$
$488$	0	0
$489$	−4.02081e25	−1.13731
$490$	0	0
$491$	3.80743e25	1.03600	0.517998	−	0.855382i	$-0.326677\pi$
$491$	3.80743e25	1.03600	0.517998	+	0.855382i	$0.326677\pi$
$492$	0	0
$493$	−1.17717e25	−0.308172
$494$	0	0
$495$	2.97667e24	0.0749861
$496$	0	0
$497$	−1.13262e25	−0.274598
$498$	0	0
$499$	1.94991e25	0.455049	0.227525	−	0.973772i	$-0.426937\pi$
$499$	1.94991e25	0.455049	0.227525	+	0.973772i	$0.426937\pi$
$500$	0	0
$501$	3.53141e25	0.793394
$502$	0	0
$503$	−4.11673e25	−0.890546	−0.445273	−	0.895395i	$-0.646893\pi$
$503$	−4.11673e25	−0.890546	−0.445273	+	0.895395i	$0.646893\pi$
$504$	0	0
$505$	3.61216e24	0.0752488
$506$	0	0
$507$	−3.17570e25	−0.637183
$508$	0	0
$509$	6.45581e24	0.124776	0.0623881	−	0.998052i	$-0.480128\pi$
$509$	6.45581e24	0.124776	0.0623881	+	0.998052i	$0.480128\pi$
$510$	0	0
$511$	5.95164e25	1.10825
$512$	0	0
$513$	2.47669e25	0.444382
$514$	0	0
$515$	2.06450e25	0.356982
$516$	0	0
$517$	−5.27708e25	−0.879495
$518$	0	0
$519$	−3.50958e25	−0.563853
$520$	0	0
$521$	7.61781e25	1.17997	0.589986	−	0.807413i	$-0.299133\pi$
$521$	7.61781e25	1.17997	0.589986	+	0.807413i	$0.299133\pi$
$522$	0	0
$523$	7.54461e25	1.12686	0.563431	−	0.826163i	$-0.309481\pi$
$523$	7.54461e25	1.12686	0.563431	+	0.826163i	$0.309481\pi$
$524$	0	0
$525$	2.96640e25	0.427284
$526$	0	0
$527$	−5.34488e25	−0.742569
$528$	0	0
$529$	1.24857e26	1.67334
$530$	0	0
$531$	−4.69800e24	−0.0607456
$532$	0	0
$533$	−2.30713e25	−0.287849
$534$	0	0
$535$	2.41999e25	0.291376
$536$	0	0
$537$	−6.54332e25	−0.760400
$538$	0	0
$539$	−2.82435e25	−0.316829
$540$	0	0
$541$	1.27935e26	1.38553	0.692763	−	0.721165i	$-0.256393\pi$
$541$	1.27935e26	1.38553	0.692763	+	0.721165i	$0.256393\pi$
$542$	0	0
$543$	−1.16607e26	−1.21934
$544$	0	0
$545$	1.18652e25	0.119814
$546$	0	0
$547$	−1.03216e26	−1.00662	−0.503310	−	0.864106i	$-0.667885\pi$
$547$	−1.03216e26	−1.00662	−0.503310	+	0.864106i	$0.667885\pi$
$548$	0	0
$549$	−8.43225e25	−0.794340
$550$	0	0
$551$	−1.06001e25	−0.0964649
$552$	0	0
$553$	−1.52080e25	−0.133716
$554$	0	0
$555$	−2.52759e25	−0.214745
$556$	0	0
$557$	−1.15924e26	−0.951811	−0.475905	−	0.879497i	$-0.657880\pi$
$557$	−1.15924e26	−0.951811	−0.475905	+	0.879497i	$0.657880\pi$
$558$	0	0
$559$	2.62710e25	0.208480
$560$	0	0
$561$	−6.97532e25	−0.535078
$562$	0	0
$563$	−6.69128e25	−0.496226	−0.248113	−	0.968731i	$-0.579810\pi$
$563$	−6.69128e25	−0.496226	−0.248113	+	0.968731i	$0.579810\pi$
$564$	0	0
$565$	2.65102e24	0.0190087
$566$	0	0
$567$	1.25650e25	0.0871210
$568$	0	0
$569$	−1.98116e26	−1.32847	−0.664236	−	0.747523i	$-0.731243\pi$
$569$	−1.98116e26	−1.32847	−0.664236	+	0.747523i	$0.731243\pi$
$570$	0	0
$571$	2.67644e26	1.73586	0.867930	−	0.496687i	$-0.165450\pi$
$571$	2.67644e26	1.73586	0.867930	+	0.496687i	$0.165450\pi$
$572$	0	0
$573$	1.62325e26	1.01839
$574$	0	0
$575$	2.55350e26	1.54986
$576$	0	0
$577$	−1.86979e25	−0.109805	−0.0549027	−	0.998492i	$-0.517485\pi$
$577$	−1.86979e25	−0.109805	−0.0549027	+	0.998492i	$0.517485\pi$
$578$	0	0
$579$	−7.37222e25	−0.418940
$580$	0	0
$581$	−2.28398e25	−0.125608
$582$	0	0
$583$	1.14675e26	0.610403
$584$	0	0
$585$	−4.97828e24	−0.0256507
$586$	0	0
$587$	−1.83068e26	−0.913167	−0.456583	−	0.889681i	$-0.650927\pi$
$587$	−1.83068e26	−0.913167	−0.456583	+	0.889681i	$0.650927\pi$
$588$	0	0
$589$	−4.81291e25	−0.232441
$590$	0	0
$591$	2.69396e26	1.25982
$592$	0	0
$593$	9.12500e25	0.413250	0.206625	−	0.978420i	$-0.433752\pi$
$593$	9.12500e25	0.413250	0.206625	+	0.978420i	$0.433752\pi$
$594$	0	0
$595$	−4.84412e25	−0.212473
$596$	0	0
$597$	−4.95735e25	−0.210617
$598$	0	0
$599$	2.37311e26	0.976705	0.488353	−	0.872646i	$-0.337598\pi$
$599$	2.37311e26	0.976705	0.488353	+	0.872646i	$0.337598\pi$
$600$	0	0
$601$	2.29560e26	0.915355	0.457677	−	0.889118i	$-0.348681\pi$
$601$	2.29560e26	0.915355	0.457677	+	0.889118i	$0.348681\pi$
$602$	0	0
$603$	−1.78431e26	−0.689376
$604$	0	0
$605$	3.99121e25	0.149427
$606$	0	0
$607$	−1.10962e26	−0.402608	−0.201304	−	0.979529i	$-0.564518\pi$
$607$	−1.10962e26	−0.402608	−0.201304	+	0.979529i	$0.564518\pi$
$608$	0	0
$609$	−2.88055e25	−0.101300
$610$	0	0
$611$	8.82556e25	0.300851
$612$	0	0
$613$	−1.27027e25	−0.0419779	−0.0209890	−	0.999780i	$-0.506681\pi$
$613$	−1.27027e25	−0.0419779	−0.0209890	+	0.999780i	$0.506681\pi$
$614$	0	0
$615$	−6.77368e25	−0.217026
$616$	0	0
$617$	−4.22914e26	−1.31384	−0.656921	−	0.753960i	$-0.728141\pi$
$617$	−4.22914e26	−1.31384	−0.656921	+	0.753960i	$0.728141\pi$
$618$	0	0
$619$	−5.49720e26	−1.65608	−0.828039	−	0.560671i	$-0.810543\pi$
$619$	−5.49720e26	−1.65608	−0.828039	+	0.560671i	$0.810543\pi$
$620$	0	0
$621$	5.79354e26	1.69268
$622$	0	0
$623$	−2.01943e26	−0.572259
$624$	0	0
$625$	3.07930e26	0.846431
$626$	0	0
$627$	−6.28108e25	−0.167492
$628$	0	0
$629$	−7.51045e26	−1.94305
$630$	0	0
$631$	2.83709e25	0.0712187	0.0356093	−	0.999366i	$-0.488663\pi$
$631$	2.83709e25	0.0712187	0.0356093	+	0.999366i	$0.488663\pi$
$632$	0	0
$633$	−5.26153e26	−1.28167
$634$	0	0
$635$	−6.55838e25	−0.155041
$636$	0	0
$637$	4.72353e25	0.108378
$638$	0	0
$639$	−1.01545e26	−0.226153
$640$	0	0
$641$	8.77075e26	1.89620	0.948102	−	0.317965i	$-0.102999\pi$
$641$	8.77075e26	1.89620	0.948102	+	0.317965i	$0.102999\pi$
$642$	0	0
$643$	−4.19589e26	−0.880683	−0.440341	−	0.897830i	$-0.645143\pi$
$643$	−4.19589e26	−0.880683	−0.440341	+	0.897830i	$0.645143\pi$
$644$	0	0
$645$	7.71311e25	0.157185
$646$	0	0
$647$	3.89932e26	0.771611	0.385805	−	0.922580i	$-0.373924\pi$
$647$	3.89932e26	0.771611	0.385805	+	0.922580i	$0.373924\pi$
$648$	0	0
$649$	3.32252e25	0.0638473
$650$	0	0
$651$	−1.30790e26	−0.244093
$652$	0	0
$653$	5.24023e26	0.949895	0.474947	−	0.880014i	$-0.342467\pi$
$653$	5.24023e26	0.949895	0.474947	+	0.880014i	$0.342467\pi$
$654$	0	0
$655$	3.72279e25	0.0655505
$656$	0	0
$657$	5.33597e26	0.912730
$658$	0	0
$659$	7.67459e26	1.27539	0.637696	−	0.770288i	$-0.279888\pi$
$659$	7.67459e26	1.27539	0.637696	+	0.770288i	$0.279888\pi$
$660$	0	0
$661$	−1.32790e26	−0.214413	−0.107206	−	0.994237i	$-0.534191\pi$
$661$	−1.32790e26	−0.214413	−0.107206	+	0.994237i	$0.534191\pi$
$662$	0	0
$663$	1.16658e26	0.183035
$664$	0	0
$665$	−4.36199e25	−0.0665088
$666$	0	0
$667$	−2.47960e26	−0.367440
$668$	0	0
$669$	−7.62010e25	−0.109752
$670$	0	0
$671$	5.96346e26	0.834899
$672$	0	0
$673$	−3.37509e26	−0.459349	−0.229674	−	0.973268i	$-0.573766\pi$
$673$	−3.37509e26	−0.459349	−0.229674	+	0.973268i	$0.573766\pi$
$674$	0	0
$675$	7.41648e26	0.981323
$676$	0	0
$677$	−1.12451e27	−1.44667	−0.723336	−	0.690496i	$-0.757392\pi$
$677$	−1.12451e27	−1.44667	−0.723336	+	0.690496i	$0.757392\pi$
$678$	0	0
$679$	−5.49654e26	−0.687585
$680$	0	0
$681$	−7.79203e25	−0.0947880
$682$	0	0
$683$	7.05871e26	0.835081	0.417540	−	0.908658i	$-0.362892\pi$
$683$	7.05871e26	0.835081	0.417540	+	0.908658i	$0.362892\pi$
$684$	0	0
$685$	3.13138e24	0.00360309
$686$	0	0
$687$	−6.89676e26	−0.771891
$688$	0	0
$689$	−1.91786e26	−0.208802
$690$	0	0
$691$	−4.70572e25	−0.0498408	−0.0249204	−	0.999689i	$-0.507933\pi$
$691$	−4.70572e25	−0.0498408	−0.0249204	+	0.999689i	$0.507933\pi$
$692$	0	0
$693$	2.16435e26	0.223029
$694$	0	0
$695$	3.81394e26	0.382400
$696$	0	0
$697$	−2.01273e27	−1.96369
$698$	0	0
$699$	2.29764e26	0.218147
$700$	0	0
$701$	5.38593e25	0.0497668	0.0248834	−	0.999690i	$-0.492079\pi$
$701$	5.38593e25	0.0497668	0.0248834	+	0.999690i	$0.492079\pi$
$702$	0	0
$703$	−6.76295e26	−0.608220
$704$	0	0
$705$	2.59116e26	0.226829
$706$	0	0
$707$	2.62642e26	0.223810
$708$	0	0
$709$	−2.80682e26	−0.232849	−0.116425	−	0.993200i	$-0.537143\pi$
$709$	−2.80682e26	−0.232849	−0.116425	+	0.993200i	$0.537143\pi$
$710$	0	0
$711$	−1.36348e26	−0.110126
$712$	0	0
$713$	−1.12585e27	−0.885381
$714$	0	0
$715$	3.52074e25	0.0269604
$716$	0	0
$717$	1.34629e27	1.00394
$718$	0	0
$719$	−1.82032e27	−1.32198	−0.660988	−	0.750397i	$-0.729862\pi$
$719$	−1.82032e27	−1.32198	−0.660988	+	0.750397i	$0.729862\pi$
$720$	0	0
$721$	1.50111e27	1.06176
$722$	0	0
$723$	−1.51937e26	−0.104676
$724$	0	0
$725$	−3.17421e26	−0.213022
$726$	0	0
$727$	−1.47227e27	−0.962522	−0.481261	−	0.876577i	$-0.659821\pi$
$727$	−1.47227e27	−0.962522	−0.481261	+	0.876577i	$0.659821\pi$
$728$	0	0
$729$	1.19193e27	0.759173
$730$	0	0
$731$	2.29187e27	1.42224
$732$	0	0
$733$	2.23845e27	1.35350	0.676751	−	0.736212i	$-0.263387\pi$
$733$	2.23845e27	1.35350	0.676751	+	0.736212i	$0.263387\pi$
$734$	0	0
$735$	1.38682e26	0.0817127
$736$	0	0
$737$	1.26190e27	0.724576
$738$	0	0
$739$	4.64717e26	0.260056	0.130028	−	0.991510i	$-0.458493\pi$
$739$	4.64717e26	0.260056	0.130028	+	0.991510i	$0.458493\pi$
$740$	0	0
$741$	1.05047e26	0.0572942
$742$	0	0
$743$	5.63164e26	0.299393	0.149696	−	0.988732i	$-0.452170\pi$
$743$	5.63164e26	0.299393	0.149696	+	0.988732i	$0.452170\pi$
$744$	0	0
$745$	−3.26873e26	−0.169392
$746$	0	0
$747$	−2.04771e26	−0.103448
$748$	0	0
$749$	1.75959e27	0.866629
$750$	0	0
$751$	−6.04104e26	−0.290090	−0.145045	−	0.989425i	$-0.546333\pi$
$751$	−6.04104e26	−0.290090	−0.145045	+	0.989425i	$0.546333\pi$
$752$	0	0
$753$	1.54837e27	0.724975
$754$	0	0
$755$	−1.20588e26	−0.0550564
$756$	0	0
$757$	−3.91726e27	−1.74410	−0.872051	−	0.489415i	$-0.837211\pi$
$757$	−3.91726e27	−1.74410	−0.872051	+	0.489415i	$0.837211\pi$
$758$	0	0
$759$	−1.46929e27	−0.637985
$760$	0	0
$761$	−3.21509e27	−1.36157	−0.680783	−	0.732486i	$-0.738360\pi$
$761$	−3.21509e27	−1.36157	−0.680783	+	0.732486i	$0.738360\pi$
$762$	0	0
$763$	8.62725e26	0.356360
$764$	0	0
$765$	−4.34301e26	−0.174988
$766$	0	0
$767$	−5.55669e25	−0.0218404
$768$	0	0
$769$	3.87198e27	1.48468	0.742339	−	0.670024i	$-0.233716\pi$
$769$	3.87198e27	1.48468	0.742339	+	0.670024i	$0.233716\pi$
$770$	0	0
$771$	−8.51285e26	−0.318463
$772$	0	0
$773$	−2.22257e27	−0.811240	−0.405620	−	0.914042i	$-0.632944\pi$
$773$	−2.22257e27	−0.811240	−0.405620	+	0.914042i	$0.632944\pi$
$774$	0	0
$775$	−1.44123e27	−0.513297
$776$	0	0
$777$	−1.83782e27	−0.638709
$778$	0	0
$779$	−1.81240e27	−0.614680
$780$	0	0
$781$	7.18148e26	0.237700
$782$	0	0
$783$	−7.20184e26	−0.232652
$784$	0	0
$785$	−5.66929e26	−0.178759
$786$	0	0
$787$	3.32013e27	1.02187	0.510934	−	0.859620i	$-0.329300\pi$
$787$	3.32013e27	1.02187	0.510934	+	0.859620i	$0.329300\pi$
$788$	0	0
$789$	−4.00900e27	−1.20449
$790$	0	0
$791$	1.92757e26	0.0565370
$792$	0	0
$793$	−9.97349e26	−0.285596
$794$	0	0
$795$	−5.63079e26	−0.157428
$796$	0	0
$797$	3.80480e27	1.03867	0.519335	−	0.854571i	$-0.326180\pi$
$797$	3.80480e27	1.03867	0.519335	+	0.854571i	$0.326180\pi$
$798$	0	0
$799$	7.69936e27	2.05239
$800$	0	0
$801$	−1.81053e27	−0.471300
$802$	0	0
$803$	−3.77370e27	−0.959334
$804$	0	0
$805$	−1.02037e27	−0.253336
$806$	0	0
$807$	2.69948e27	0.654609
$808$	0	0
$809$	−6.75938e26	−0.160102	−0.0800508	−	0.996791i	$-0.525508\pi$
$809$	−6.75938e26	−0.160102	−0.0800508	+	0.996791i	$0.525508\pi$
$810$	0	0
$811$	1.86888e27	0.432398	0.216199	−	0.976349i	$-0.430634\pi$
$811$	1.86888e27	0.432398	0.216199	+	0.976349i	$0.430634\pi$
$812$	0	0
$813$	−1.96466e27	−0.444045
$814$	0	0
$815$	1.77050e27	0.390929
$816$	0	0
$817$	2.06376e27	0.445195
$818$	0	0
$819$	−3.61973e26	−0.0762919
$820$	0	0
$821$	1.78616e27	0.367841	0.183921	−	0.982941i	$-0.441121\pi$
$821$	1.78616e27	0.367841	0.183921	+	0.982941i	$0.441121\pi$
$822$	0	0
$823$	4.84426e27	0.974831	0.487415	−	0.873170i	$-0.337940\pi$
$823$	4.84426e27	0.974831	0.487415	+	0.873170i	$0.337940\pi$
$824$	0	0
$825$	−1.88088e27	−0.369870
$826$	0	0
$827$	7.46068e26	0.143376	0.0716879	−	0.997427i	$-0.477161\pi$
$827$	7.46068e26	0.143376	0.0716879	+	0.997427i	$0.477161\pi$
$828$	0	0
$829$	9.45101e27	1.77505	0.887524	−	0.460761i	$-0.152424\pi$
$829$	9.45101e27	1.77505	0.887524	+	0.460761i	$0.152424\pi$
$830$	0	0
$831$	1.80944e26	0.0332151
$832$	0	0
$833$	4.12077e27	0.739352
$834$	0	0
$835$	−1.55499e27	−0.272714
$836$	0	0
$837$	−3.26996e27	−0.560596
$838$	0	0
$839$	−8.87983e27	−1.48822	−0.744108	−	0.668059i	$-0.767125\pi$
$839$	−8.87983e27	−1.48822	−0.744108	+	0.668059i	$0.767125\pi$
$840$	0	0
$841$	−5.79503e27	−0.949497
$842$	0	0
$843$	1.11125e27	0.178012
$844$	0	0
$845$	1.39837e27	0.219019
$846$	0	0
$847$	2.90202e27	0.444435
$848$	0	0
$849$	6.12963e27	0.917934
$850$	0	0
$851$	−1.58201e28	−2.31674
$852$	0	0
$853$	−8.81716e26	−0.126274	−0.0631369	−	0.998005i	$-0.520110\pi$
$853$	−8.81716e26	−0.126274	−0.0631369	+	0.998005i	$0.520110\pi$
$854$	0	0
$855$	−3.91076e26	−0.0547751
$856$	0	0
$857$	−1.68989e26	−0.0231495	−0.0115747	−	0.999933i	$-0.503684\pi$
$857$	−1.68989e26	−0.0231495	−0.0115747	+	0.999933i	$0.503684\pi$
$858$	0	0
$859$	−1.45204e28	−1.94556	−0.972779	−	0.231735i	$-0.925560\pi$
$859$	−1.45204e28	−1.94556	−0.972779	+	0.231735i	$0.925560\pi$
$860$	0	0
$861$	−4.92517e27	−0.645493
$862$	0	0
$863$	1.45753e27	0.186860	0.0934299	−	0.995626i	$-0.470217\pi$
$863$	1.45753e27	0.186860	0.0934299	+	0.995626i	$0.470217\pi$
$864$	0	0
$865$	1.54539e27	0.193813
$866$	0	0
$867$	4.76512e27	0.584645
$868$	0	0
$869$	9.64282e26	0.115749
$870$	0	0
$871$	−2.11044e27	−0.247857
$872$	0	0
$873$	−4.92794e27	−0.566279
$874$	0	0
$875$	−2.68420e27	−0.301813
$876$	0	0
$877$	8.44834e27	0.929555	0.464777	−	0.885428i	$-0.346134\pi$
$877$	8.44834e27	0.929555	0.464777	+	0.885428i	$0.346134\pi$
$878$	0	0
$879$	5.60968e27	0.604010
$880$	0	0
$881$	3.79803e26	0.0400209	0.0200105	−	0.999800i	$-0.493630\pi$
$881$	3.79803e26	0.0400209	0.0200105	+	0.999800i	$0.493630\pi$
$882$	0	0
$883$	1.64147e27	0.169280	0.0846400	−	0.996412i	$-0.473026\pi$
$883$	1.64147e27	0.169280	0.0846400	+	0.996412i	$0.473026\pi$
$884$	0	0
$885$	−1.63143e26	−0.0164667
$886$	0	0
$887$	1.53135e28	1.51286	0.756432	−	0.654072i	$-0.226941\pi$
$887$	1.53135e28	1.51286	0.756432	+	0.654072i	$0.226941\pi$
$888$	0	0
$889$	−4.76863e27	−0.461133
$890$	0	0
$891$	−7.96696e26	−0.0754144
$892$	0	0
$893$	6.93306e27	0.642445
$894$	0	0
$895$	2.88124e27	0.261373
$896$	0	0
$897$	2.45729e27	0.218237
$898$	0	0
$899$	1.39952e27	0.121692
$900$	0	0
$901$	−1.67313e28	−1.42444
$902$	0	0
$903$	5.60823e27	0.467511
$904$	0	0
$905$	5.13459e27	0.419126
$906$	0	0
$907$	2.27504e28	1.81853	0.909265	−	0.416218i	$-0.136645\pi$
$907$	2.27504e28	1.81853	0.909265	+	0.416218i	$0.136645\pi$
$908$	0	0
$909$	2.35473e27	0.184325
$910$	0	0
$911$	−1.90023e28	−1.45674	−0.728369	−	0.685185i	$-0.759721\pi$
$911$	−1.90023e28	−1.45674	−0.728369	+	0.685185i	$0.759721\pi$
$912$	0	0
$913$	1.44818e27	0.108730
$914$	0	0
$915$	−2.92819e27	−0.215327
$916$	0	0
$917$	2.70685e27	0.194965
$918$	0	0
$919$	7.01793e27	0.495122	0.247561	−	0.968872i	$-0.420371\pi$
$919$	7.01793e27	0.495122	0.247561	+	0.968872i	$0.420371\pi$
$920$	0	0
$921$	6.14753e27	0.424849
$922$	0	0
$923$	−1.20106e27	−0.0813106
$924$	0	0
$925$	−2.02518e28	−1.34312
$926$	0	0
$927$	1.34582e28	0.874441
$928$	0	0
$929$	1.04079e28	0.662545	0.331273	−	0.943535i	$-0.392522\pi$
$929$	1.04079e28	0.662545	0.331273	+	0.943535i	$0.392522\pi$
$930$	0	0
$931$	3.71064e27	0.231434
$932$	0	0
$933$	−9.44891e27	−0.577439
$934$	0	0
$935$	3.07146e27	0.183923
$936$	0	0
$937$	−2.36290e28	−1.38650	−0.693249	−	0.720698i	$-0.743821\pi$
$937$	−2.36290e28	−1.38650	−0.693249	+	0.720698i	$0.743821\pi$
$938$	0	0
$939$	−5.01366e27	−0.288292
$940$	0	0
$941$	−4.44433e27	−0.250441	−0.125220	−	0.992129i	$-0.539964\pi$
$941$	−4.44433e27	−0.250441	−0.125220	+	0.992129i	$0.539964\pi$
$942$	0	0
$943$	−4.23963e28	−2.34135
$944$	0	0
$945$	−2.96360e27	−0.160405
$946$	0	0
$947$	1.77081e28	0.939390	0.469695	−	0.882829i	$-0.344364\pi$
$947$	1.77081e28	0.939390	0.469695	+	0.882829i	$0.344364\pi$
$948$	0	0
$949$	6.31127e27	0.328162
$950$	0	0
$951$	−1.88661e28	−0.961542
$952$	0	0
$953$	4.35882e27	0.217764	0.108882	−	0.994055i	$-0.465273\pi$
$953$	4.35882e27	0.217764	0.108882	+	0.994055i	$0.465273\pi$
$954$	0	0
$955$	−7.14769e27	−0.350053
$956$	0	0
$957$	1.82645e27	0.0876886
$958$	0	0
$959$	2.27684e26	0.0107166
$960$	0	0
$961$	−1.53162e28	−0.706771
$962$	0	0
$963$	1.57756e28	0.713735
$964$	0	0
$965$	3.24623e27	0.144003
$966$	0	0
$967$	−7.37579e27	−0.320817	−0.160408	−	0.987051i	$-0.551281\pi$
$967$	−7.37579e27	−0.320817	−0.160408	+	0.987051i	$0.551281\pi$
$968$	0	0
$969$	9.16422e27	0.390859
$970$	0	0
$971$	−1.32649e28	−0.554780	−0.277390	−	0.960757i	$-0.589469\pi$
$971$	−1.32649e28	−0.554780	−0.277390	+	0.960757i	$0.589469\pi$
$972$	0	0
$973$	2.77313e28	1.13736
$974$	0	0
$975$	3.14565e27	0.126522
$976$	0	0
$977$	1.57429e28	0.620991	0.310496	−	0.950575i	$-0.399505\pi$
$977$	1.57429e28	0.620991	0.310496	+	0.950575i	$0.399505\pi$
$978$	0	0
$979$	1.28044e28	0.495364
$980$	0	0
$981$	7.73480e27	0.293490
$982$	0	0
$983$	−2.23838e27	−0.0833059	−0.0416529	−	0.999132i	$-0.513262\pi$
$983$	−2.23838e27	−0.0833059	−0.0416529	+	0.999132i	$0.513262\pi$
$984$	0	0
$985$	−1.18624e28	−0.433040
$986$	0	0
$987$	1.88404e28	0.674649
$988$	0	0
$989$	4.82761e28	1.69577
$990$	0	0
$991$	−2.72873e27	−0.0940288	−0.0470144	−	0.998894i	$-0.514971\pi$
$991$	−2.72873e27	−0.0940288	−0.0470144	+	0.998894i	$0.514971\pi$
$992$	0	0
$993$	2.85170e28	0.964019
$994$	0	0
$995$	2.18289e27	0.0723956
$996$	0	0
$997$	1.56740e28	0.510008	0.255004	−	0.966940i	$-0.417923\pi$
$997$	1.56740e28	0.510008	0.255004	+	0.966940i	$0.417923\pi$
$998$	0	0
$999$	−4.59484e28	−1.46689

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.20.a.a.1.2	✓	2
3.2	odd	2		72.20.a.a.1.2		2
4.3	odd	2		16.20.a.e.1.1		2
8.3	odd	2		64.20.a.j.1.2		2
8.5	even	2		64.20.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.a.a.1.2	✓	2	1.1	even	1	trivial
16.20.a.e.1.1		2	4.3	odd	2
64.20.a.j.1.2		2	8.3	odd	2
64.20.a.k.1.1		2	8.5	even	2
72.20.a.a.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+22637.5 q^{3} -996804. q^{5} -7.24780e7 q^{7} -6.49805e8 q^{9} +O(q^{10})\)	\(q+22637.5 q^{3} -996804. q^{5} -7.24780e7 q^{7} -6.49805e8 q^{9} +4.59555e9 q^{11} -7.68575e9 q^{13} -2.25652e10 q^{15} -6.70499e11 q^{17} -6.03766e11 q^{19} -1.64072e12 q^{21} -1.41235e13 q^{23} -1.80799e13 q^{25} -4.10207e13 q^{27} +1.75566e13 q^{29} +7.97149e13 q^{31} +1.04032e14 q^{33} +7.22464e13 q^{35} +1.12013e15 q^{37} -1.73986e14 q^{39} +3.00183e15 q^{41} -3.41815e15 q^{43} +6.47728e14 q^{45} -1.14830e16 q^{47} -6.14583e15 q^{49} -1.51784e16 q^{51} +2.49535e16 q^{53} -4.58086e15 q^{55} -1.36678e16 q^{57} +7.22986e15 q^{59} +1.29766e17 q^{61} +4.70966e16 q^{63} +7.66119e15 q^{65} +2.74592e17 q^{67} -3.19720e17 q^{69} +1.56270e17 q^{71} -8.21165e17 q^{73} -4.09283e17 q^{75} -3.33076e17 q^{77} +2.09830e17 q^{79} -1.73363e17 q^{81} +3.15127e17 q^{83} +6.68357e17 q^{85} +3.97438e17 q^{87} +2.78627e18 q^{89} +5.57048e17 q^{91} +1.80455e18 q^{93} +6.01837e17 q^{95} +7.58373e18 q^{97} -2.98621e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9}+O(q^{10}) \) 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9	\( 2 q - 27912 q^{3} + 1226620 q^{5} + 88510512 q^{7} + 743186538 q^{9} - 7163787608 q^{11} - 10126923604 q^{13} - 134958171120 q^{15} - 72045078940 q^{17} - 3120480472232 q^{19} - 9778613729472 q^{21} - 14759207090288 q^{23} - 32209737998450 q^{25} - 52683939132624 q^{27} - 30249539245044 q^{29} - 123389562777920 q^{31} + 698460345860448 q^{33} + 430192267170720 q^{35} + 20\!\cdots\!24 q^{37}+ \cdots - 19\!\cdots\!52 q^{99}+O(q^{100}) \) 2 * q - 27912 * q^3 + 1226620 * q^5 + 88510512 * q^7 + 743186538 * q^9 - 7163787608 * q^11 - 10126923604 * q^13 - 134958171120 * q^15 - 72045078940 * q^17 - 3120480472232 * q^19 - 9778613729472 * q^21 - 14759207090288 * q^23 - 32209737998450 * q^25 - 52683939132624 * q^27 - 30249539245044 * q^29 - 123389562777920 * q^31 + 698460345860448 * q^33 + 430192267170720 * q^35 + 2015393170174524 * q^37 - 50586287718576 * q^39 + 2540784959504244 * q^41 - 5633655093389464 * q^43 + 3744938810129580 * q^45 - 21948339587130336 * q^47 + 8372586663064786 * q^49 - 45429993200591760 * q^51 - 9418125066904676 * q^53 - 30726853722003280 * q^55 + 113550921657554592 * q^57 + 98542449590407624 * q^59 + 10292145377839820 * q^61 + 271352203396831728 * q^63 + 2233432360014760 * q^65 + 75753628003984504 * q^67 - 287583943397508672 * q^69 + 17407052566713776 * q^71 - 857508255059832268 * q^73 + 304974862137280200 * q^75 - 2226194512471875648 * q^77 - 226291921444855072 * q^79 - 1202809709767720302 * q^81 + 767515701460985048 * q^83 + 1998974022815558200 * q^85 + 2814015710341120464 * q^87 + 6092545894435174548 * q^89 + 164047654392377376 * q^91 + 12071374787086705920 * q^93 - 4993887710656339120 * q^95 + 1548148249522347076 * q^97 - 19366859521479728952 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more