LMFDB - Embedded newform 920.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,2,Mod(1,920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.34623698596$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} -1.00000 q^{5} +1.56155 q^{7} +3.56155 q^{9} -2.00000 q^{11} +0.561553 q^{13} +2.56155 q^{15} +5.56155 q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.43845 q^{27} +0.123106 q^{29} -8.12311 q^{31} +5.12311 q^{33} -1.56155 q^{35} -3.56155 q^{37} -1.43845 q^{39} -4.12311 q^{41} -10.2462 q^{43} -3.56155 q^{45} +3.68466 q^{47} -4.56155 q^{49} -14.2462 q^{51} +4.43845 q^{53} +2.00000 q^{55} +5.12311 q^{57} -5.56155 q^{59} -9.12311 q^{61} +5.56155 q^{63} -0.561553 q^{65} -11.5616 q^{67} +2.56155 q^{69} -5.00000 q^{71} +3.43845 q^{73} -2.56155 q^{75} -3.12311 q^{77} -9.12311 q^{79} -7.00000 q^{81} -4.68466 q^{83} -5.56155 q^{85} -0.315342 q^{87} +8.00000 q^{89} +0.876894 q^{91} +20.8078 q^{93} +2.00000 q^{95} -3.12311 q^{97} -7.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} - 4 q^{19} - 8 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 8 q^{29} - 8 q^{31} + 2 q^{33} + q^{35} - 3 q^{37} - 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 7 * q^17 - 4 * q^19 - 8 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 8 * q^29 - 8 * q^31 + 2 * q^33 + q^35 - 3 * q^37 - 7 * q^39 - 4 * q^43 - 3 * q^45 - 5 * q^47 - 5 * q^49 - 12 * q^51 + 13 * q^53 + 4 * q^55 + 2 * q^57 - 7 * q^59 - 10 * q^61 + 7 * q^63 + 3 * q^65 - 19 * q^67 + q^69 - 10 * q^71 + 11 * q^73 - q^75 + 2 * q^77 - 10 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 - 13 * q^87 + 16 * q^89 + 10 * q^91 + 21 * q^93 + 4 * q^95 + 2 * q^97 - 6 * q^99

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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	5.56155	1.34887	0.674437	−	0.738332i	$-0.264386\pi$
$17$	5.56155	1.34887	0.674437	+	0.738332i	$0.264386\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	0.123106	0.0228601	0.0114301	−	0.999935i	$-0.496362\pi$
$29$	0.123106	0.0228601	0.0114301	+	0.999935i	$0.496362\pi$
$30$	0	0
$31$	−8.12311	−1.45895	−0.729476	−	0.684006i	$-0.760236\pi$
$31$	−8.12311	−1.45895	−0.729476	+	0.684006i	$0.760236\pi$
$32$	0	0
$33$	5.12311	0.891818
$34$	0	0
$35$	−1.56155	−0.263951
$36$	0	0
$37$	−3.56155	−0.585516	−0.292758	−	0.956187i	$-0.594573\pi$
$37$	−3.56155	−0.585516	−0.292758	+	0.956187i	$0.594573\pi$
$38$	0	0
$39$	−1.43845	−0.230336
$40$	0	0
$41$	−4.12311	−0.643921	−0.321960	−	0.946753i	$-0.604342\pi$
$41$	−4.12311	−0.643921	−0.321960	+	0.946753i	$0.604342\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	3.68466	0.537463	0.268731	−	0.963215i	$-0.413396\pi$
$47$	3.68466	0.537463	0.268731	+	0.963215i	$0.413396\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−14.2462	−1.99487
$52$	0	0
$53$	4.43845	0.609668	0.304834	−	0.952406i	$-0.401399\pi$
$53$	4.43845	0.609668	0.304834	+	0.952406i	$0.401399\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	5.12311	0.678572
$58$	0	0
$59$	−5.56155	−0.724053	−0.362026	−	0.932168i	$-0.617915\pi$
$59$	−5.56155	−0.724053	−0.362026	+	0.932168i	$0.617915\pi$
$60$	0	0
$61$	−9.12311	−1.16809	−0.584047	−	0.811720i	$-0.698532\pi$
$61$	−9.12311	−1.16809	−0.584047	+	0.811720i	$0.698532\pi$
$62$	0	0
$63$	5.56155	0.700690
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	−11.5616	−1.41247	−0.706234	−	0.707978i	$-0.749607\pi$
$67$	−11.5616	−1.41247	−0.706234	+	0.707978i	$0.749607\pi$
$68$	0	0
$69$	2.56155	0.308375
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	3.43845	0.402440	0.201220	−	0.979546i	$-0.435509\pi$
$73$	3.43845	0.402440	0.201220	+	0.979546i	$0.435509\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	−3.12311	−0.355911
$78$	0	0
$79$	−9.12311	−1.02643	−0.513215	−	0.858260i	$-0.671546\pi$
$79$	−9.12311	−1.02643	−0.513215	+	0.858260i	$0.671546\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−4.68466	−0.514208	−0.257104	−	0.966384i	$-0.582768\pi$
$83$	−4.68466	−0.514208	−0.257104	+	0.966384i	$0.582768\pi$
$84$	0	0
$85$	−5.56155	−0.603235
$86$	0	0
$87$	−0.315342	−0.0338082
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0.876894	0.0919235
$92$	0	0
$93$	20.8078	2.15766
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−3.12311	−0.317103	−0.158552	−	0.987351i	$-0.550682\pi$
$97$	−3.12311	−0.317103	−0.158552	+	0.987351i	$0.550682\pi$
$98$	0	0
$99$	−7.12311	−0.715899
$100$	0	0
$101$	−9.80776	−0.975909	−0.487954	−	0.872869i	$-0.662257\pi$
$101$	−9.80776	−0.975909	−0.487954	+	0.872869i	$0.662257\pi$
$102$	0	0
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	10.6847	1.03292	0.516462	−	0.856310i	$-0.327249\pi$
$107$	10.6847	1.03292	0.516462	+	0.856310i	$0.327249\pi$
$108$	0	0
$109$	7.12311	0.682270	0.341135	−	0.940014i	$-0.389189\pi$
$109$	7.12311	0.682270	0.341135	+	0.940014i	$0.389189\pi$
$110$	0	0
$111$	9.12311	0.865927
$112$	0	0
$113$	9.56155	0.899475	0.449738	−	0.893161i	$-0.351517\pi$
$113$	9.56155	0.899475	0.449738	+	0.893161i	$0.351517\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	8.68466	0.796121
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	10.5616	0.952303
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.807764	0.0716775	0.0358387	−	0.999358i	$-0.488590\pi$
$127$	0.807764	0.0716775	0.0358387	+	0.999358i	$0.488590\pi$
$128$	0	0
$129$	26.2462	2.31085
$130$	0	0
$131$	9.93087	0.867664	0.433832	−	0.900994i	$-0.357161\pi$
$131$	9.93087	0.867664	0.433832	+	0.900994i	$0.357161\pi$
$132$	0	0
$133$	−3.12311	−0.270808
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	−19.6155	−1.67587	−0.837934	−	0.545772i	$-0.816237\pi$
$137$	−19.6155	−1.67587	−0.837934	+	0.545772i	$0.816237\pi$
$138$	0	0
$139$	14.3693	1.21879	0.609395	−	0.792867i	$-0.291412\pi$
$139$	14.3693	1.21879	0.609395	+	0.792867i	$0.291412\pi$
$140$	0	0
$141$	−9.43845	−0.794861
$142$	0	0
$143$	−1.12311	−0.0939188
$144$	0	0
$145$	−0.123106	−0.0102234
$146$	0	0
$147$	11.6847	0.963734
$148$	0	0
$149$	9.12311	0.747394	0.373697	−	0.927551i	$-0.378090\pi$
$149$	9.12311	0.747394	0.373697	+	0.927551i	$0.378090\pi$
$150$	0	0
$151$	−2.56155	−0.208456	−0.104228	−	0.994553i	$-0.533237\pi$
$151$	−2.56155	−0.208456	−0.104228	+	0.994553i	$0.533237\pi$
$152$	0	0
$153$	19.8078	1.60136
$154$	0	0
$155$	8.12311	0.652464
$156$	0	0
$157$	−3.31534	−0.264593	−0.132297	−	0.991210i	$-0.542235\pi$
$157$	−3.31534	−0.264593	−0.132297	+	0.991210i	$0.542235\pi$
$158$	0	0
$159$	−11.3693	−0.901645
$160$	0	0
$161$	−1.56155	−0.123068
$162$	0	0
$163$	5.68466	0.445257	0.222628	−	0.974903i	$-0.428536\pi$
$163$	5.68466	0.445257	0.222628	+	0.974903i	$0.428536\pi$
$164$	0	0
$165$	−5.12311	−0.398833
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	−7.12311	−0.544718
$172$	0	0
$173$	1.12311	0.0853881	0.0426941	−	0.999088i	$-0.486406\pi$
$173$	1.12311	0.0853881	0.0426941	+	0.999088i	$0.486406\pi$
$174$	0	0
$175$	1.56155	0.118042
$176$	0	0
$177$	14.2462	1.07081
$178$	0	0
$179$	−19.6847	−1.47130	−0.735650	−	0.677362i	$-0.763123\pi$
$179$	−19.6847	−1.47130	−0.735650	+	0.677362i	$0.763123\pi$
$180$	0	0
$181$	−17.1231	−1.27275	−0.636375	−	0.771380i	$-0.719567\pi$
$181$	−17.1231	−1.27275	−0.636375	+	0.771380i	$0.719567\pi$
$182$	0	0
$183$	23.3693	1.72751
$184$	0	0
$185$	3.56155	0.261851
$186$	0	0
$187$	−11.1231	−0.813402
$188$	0	0
$189$	−2.24621	−0.163388
$190$	0	0
$191$	5.12311	0.370695	0.185347	−	0.982673i	$-0.440659\pi$
$191$	5.12311	0.370695	0.185347	+	0.982673i	$0.440659\pi$
$192$	0	0
$193$	−9.93087	−0.714840	−0.357420	−	0.933944i	$-0.616343\pi$
$193$	−9.93087	−0.714840	−0.357420	+	0.933944i	$0.616343\pi$
$194$	0	0
$195$	1.43845	0.103009
$196$	0	0
$197$	−2.80776	−0.200045	−0.100022	−	0.994985i	$-0.531891\pi$
$197$	−2.80776	−0.200045	−0.100022	+	0.994985i	$0.531891\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	29.6155	2.08892
$202$	0	0
$203$	0.192236	0.0134923
$204$	0	0
$205$	4.12311	0.287970
$206$	0	0
$207$	−3.56155	−0.247545
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	18.9309	1.30325	0.651627	−	0.758539i	$-0.274087\pi$
$211$	18.9309	1.30325	0.651627	+	0.758539i	$0.274087\pi$
$212$	0	0
$213$	12.8078	0.877574
$214$	0	0
$215$	10.2462	0.698786
$216$	0	0
$217$	−12.6847	−0.861091
$218$	0	0
$219$	−8.80776	−0.595174
$220$	0	0
$221$	3.12311	0.210083
$222$	0	0
$223$	27.1231	1.81630	0.908149	−	0.418648i	$-0.137496\pi$
$223$	27.1231	1.81630	0.908149	+	0.418648i	$0.137496\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	1.75379	0.116403	0.0582015	−	0.998305i	$-0.481463\pi$
$227$	1.75379	0.116403	0.0582015	+	0.998305i	$0.481463\pi$
$228$	0	0
$229$	−22.2462	−1.47007	−0.735036	−	0.678029i	$-0.762835\pi$
$229$	−22.2462	−1.47007	−0.735036	+	0.678029i	$0.762835\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	−25.3002	−1.65747	−0.828735	−	0.559641i	$-0.810939\pi$
$233$	−25.3002	−1.65747	−0.828735	+	0.559641i	$0.810939\pi$
$234$	0	0
$235$	−3.68466	−0.240361
$236$	0	0
$237$	23.3693	1.51800
$238$	0	0
$239$	−16.1231	−1.04292	−0.521459	−	0.853277i	$-0.674612\pi$
$239$	−16.1231	−1.04292	−0.521459	+	0.853277i	$0.674612\pi$
$240$	0	0
$241$	14.4924	0.933539	0.466769	−	0.884379i	$-0.345418\pi$
$241$	14.4924	0.933539	0.466769	+	0.884379i	$0.345418\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	4.56155	0.291427
$246$	0	0
$247$	−1.12311	−0.0714615
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	0.876894	0.0553491	0.0276745	−	0.999617i	$-0.491190\pi$
$251$	0.876894	0.0553491	0.0276745	+	0.999617i	$0.491190\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	14.2462	0.892132
$256$	0	0
$257$	11.6847	0.728869	0.364434	−	0.931229i	$-0.381262\pi$
$257$	11.6847	0.728869	0.364434	+	0.931229i	$0.381262\pi$
$258$	0	0
$259$	−5.56155	−0.345578
$260$	0	0
$261$	0.438447	0.0271392
$262$	0	0
$263$	−26.6847	−1.64545	−0.822723	−	0.568442i	$-0.807546\pi$
$263$	−26.6847	−1.64545	−0.822723	+	0.568442i	$0.807546\pi$
$264$	0	0
$265$	−4.43845	−0.272652
$266$	0	0
$267$	−20.4924	−1.25412
$268$	0	0
$269$	5.49242	0.334879	0.167439	−	0.985882i	$-0.446450\pi$
$269$	5.49242	0.334879	0.167439	+	0.985882i	$0.446450\pi$
$270$	0	0
$271$	−21.1771	−1.28642	−0.643208	−	0.765691i	$-0.722397\pi$
$271$	−21.1771	−1.28642	−0.643208	+	0.765691i	$0.722397\pi$
$272$	0	0
$273$	−2.24621	−0.135947
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−24.5616	−1.47576	−0.737880	−	0.674932i	$-0.764173\pi$
$277$	−24.5616	−1.47576	−0.737880	+	0.674932i	$0.764173\pi$
$278$	0	0
$279$	−28.9309	−1.73205
$280$	0	0
$281$	−0.876894	−0.0523111	−0.0261556	−	0.999658i	$-0.508327\pi$
$281$	−0.876894	−0.0523111	−0.0261556	+	0.999658i	$0.508327\pi$
$282$	0	0
$283$	−24.9309	−1.48199	−0.740993	−	0.671513i	$-0.765645\pi$
$283$	−24.9309	−1.48199	−0.740993	+	0.671513i	$0.765645\pi$
$284$	0	0
$285$	−5.12311	−0.303467
$286$	0	0
$287$	−6.43845	−0.380050
$288$	0	0
$289$	13.9309	0.819463
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	10.4384	0.609821	0.304910	−	0.952381i	$-0.401374\pi$
$293$	10.4384	0.609821	0.304910	+	0.952381i	$0.401374\pi$
$294$	0	0
$295$	5.56155	0.323806
$296$	0	0
$297$	2.87689	0.166934
$298$	0	0
$299$	−0.561553	−0.0324754
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	25.1231	1.44328
$304$	0	0
$305$	9.12311	0.522388
$306$	0	0
$307$	9.36932	0.534735	0.267368	−	0.963595i	$-0.413846\pi$
$307$	9.36932	0.534735	0.267368	+	0.963595i	$0.413846\pi$
$308$	0	0
$309$	−5.75379	−0.327322
$310$	0	0
$311$	−24.8078	−1.40672	−0.703360	−	0.710834i	$-0.748318\pi$
$311$	−24.8078	−1.40672	−0.703360	+	0.710834i	$0.748318\pi$
$312$	0	0
$313$	14.9309	0.843943	0.421971	−	0.906609i	$-0.361338\pi$
$313$	14.9309	0.843943	0.421971	+	0.906609i	$0.361338\pi$
$314$	0	0
$315$	−5.56155	−0.313358
$316$	0	0
$317$	0.630683	0.0354227	0.0177113	−	0.999843i	$-0.494362\pi$
$317$	0.630683	0.0354227	0.0177113	+	0.999843i	$0.494362\pi$
$318$	0	0
$319$	−0.246211	−0.0137852
$320$	0	0
$321$	−27.3693	−1.52761
$322$	0	0
$323$	−11.1231	−0.618906
$324$	0	0
$325$	0.561553	0.0311493
$326$	0	0
$327$	−18.2462	−1.00902
$328$	0	0
$329$	5.75379	0.317217
$330$	0	0
$331$	29.4924	1.62105	0.810525	−	0.585704i	$-0.199182\pi$
$331$	29.4924	1.62105	0.810525	+	0.585704i	$0.199182\pi$
$332$	0	0
$333$	−12.6847	−0.695115
$334$	0	0
$335$	11.5616	0.631675
$336$	0	0
$337$	4.87689	0.265661	0.132831	−	0.991139i	$-0.457593\pi$
$337$	4.87689	0.265661	0.132831	+	0.991139i	$0.457593\pi$
$338$	0	0
$339$	−24.4924	−1.33025
$340$	0	0
$341$	16.2462	0.879782
$342$	0	0
$343$	−18.0540	−0.974823
$344$	0	0
$345$	−2.56155	−0.137909
$346$	0	0
$347$	2.24621	0.120583	0.0602915	−	0.998181i	$-0.480797\pi$
$347$	2.24621	0.120583	0.0602915	+	0.998181i	$0.480797\pi$
$348$	0	0
$349$	4.12311	0.220705	0.110352	−	0.993893i	$-0.464802\pi$
$349$	4.12311	0.220705	0.110352	+	0.993893i	$0.464802\pi$
$350$	0	0
$351$	−0.807764	−0.0431153
$352$	0	0
$353$	24.8078	1.32038	0.660192	−	0.751097i	$-0.270475\pi$
$353$	24.8078	1.32038	0.660192	+	0.751097i	$0.270475\pi$
$354$	0	0
$355$	5.00000	0.265372
$356$	0	0
$357$	−22.2462	−1.17739
$358$	0	0
$359$	2.87689	0.151837	0.0759183	−	0.997114i	$-0.475811\pi$
$359$	2.87689	0.151837	0.0759183	+	0.997114i	$0.475811\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	17.9309	0.941127
$364$	0	0
$365$	−3.43845	−0.179977
$366$	0	0
$367$	−31.8078	−1.66035	−0.830176	−	0.557502i	$-0.811760\pi$
$367$	−31.8078	−1.66035	−0.830176	+	0.557502i	$0.811760\pi$
$368$	0	0
$369$	−14.6847	−0.764453
$370$	0	0
$371$	6.93087	0.359833
$372$	0	0
$373$	−24.7386	−1.28092	−0.640459	−	0.767992i	$-0.721256\pi$
$373$	−24.7386	−1.28092	−0.640459	+	0.767992i	$0.721256\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	0.0691303	0.00356039
$378$	0	0
$379$	32.9848	1.69432	0.847159	−	0.531340i	$-0.178311\pi$
$379$	32.9848	1.69432	0.847159	+	0.531340i	$0.178311\pi$
$380$	0	0
$381$	−2.06913	−0.106005
$382$	0	0
$383$	14.9309	0.762932	0.381466	−	0.924383i	$-0.375419\pi$
$383$	14.9309	0.762932	0.381466	+	0.924383i	$0.375419\pi$
$384$	0	0
$385$	3.12311	0.159168
$386$	0	0
$387$	−36.4924	−1.85501
$388$	0	0
$389$	−27.1231	−1.37520	−0.687598	−	0.726092i	$-0.741335\pi$
$389$	−27.1231	−1.37520	−0.687598	+	0.726092i	$0.741335\pi$
$390$	0	0
$391$	−5.56155	−0.281260
$392$	0	0
$393$	−25.4384	−1.28320
$394$	0	0
$395$	9.12311	0.459033
$396$	0	0
$397$	24.1771	1.21341	0.606706	−	0.794926i	$-0.292490\pi$
$397$	24.1771	1.21341	0.606706	+	0.794926i	$0.292490\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	16.7386	0.835887	0.417944	−	0.908473i	$-0.362751\pi$
$401$	16.7386	0.835887	0.417944	+	0.908473i	$0.362751\pi$
$402$	0	0
$403$	−4.56155	−0.227227
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	7.12311	0.353079
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	50.2462	2.47846
$412$	0	0
$413$	−8.68466	−0.427344
$414$	0	0
$415$	4.68466	0.229961
$416$	0	0
$417$	−36.8078	−1.80248
$418$	0	0
$419$	19.6155	0.958281	0.479141	−	0.877738i	$-0.340948\pi$
$419$	19.6155	0.958281	0.479141	+	0.877738i	$0.340948\pi$
$420$	0	0
$421$	−39.1231	−1.90674	−0.953372	−	0.301798i	$-0.902413\pi$
$421$	−39.1231	−1.90674	−0.953372	+	0.301798i	$0.902413\pi$
$422$	0	0
$423$	13.1231	0.638067
$424$	0	0
$425$	5.56155	0.269775
$426$	0	0
$427$	−14.2462	−0.689422
$428$	0	0
$429$	2.87689	0.138898
$430$	0	0
$431$	13.7538	0.662497	0.331248	−	0.943544i	$-0.392530\pi$
$431$	13.7538	0.662497	0.331248	+	0.943544i	$0.392530\pi$
$432$	0	0
$433$	−1.31534	−0.0632113	−0.0316056	−	0.999500i	$-0.510062\pi$
$433$	−1.31534	−0.0632113	−0.0316056	+	0.999500i	$0.510062\pi$
$434$	0	0
$435$	0.315342	0.0151195
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	16.8078	0.802191	0.401095	−	0.916036i	$-0.368630\pi$
$439$	16.8078	0.802191	0.401095	+	0.916036i	$0.368630\pi$
$440$	0	0
$441$	−16.2462	−0.773629
$442$	0	0
$443$	−15.0540	−0.715236	−0.357618	−	0.933868i	$-0.616411\pi$
$443$	−15.0540	−0.715236	−0.357618	+	0.933868i	$0.616411\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	−23.3693	−1.10533
$448$	0	0
$449$	30.6847	1.44810	0.724049	−	0.689748i	$-0.242279\pi$
$449$	30.6847	1.44810	0.724049	+	0.689748i	$0.242279\pi$
$450$	0	0
$451$	8.24621	0.388299
$452$	0	0
$453$	6.56155	0.308289
$454$	0	0
$455$	−0.876894	−0.0411094
$456$	0	0
$457$	10.9309	0.511325	0.255662	−	0.966766i	$-0.417706\pi$
$457$	10.9309	0.511325	0.255662	+	0.966766i	$0.417706\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	6.17708	0.287695	0.143848	−	0.989600i	$-0.454052\pi$
$461$	6.17708	0.287695	0.143848	+	0.989600i	$0.454052\pi$
$462$	0	0
$463$	−20.8769	−0.970232	−0.485116	−	0.874450i	$-0.661223\pi$
$463$	−20.8769	−0.970232	−0.485116	+	0.874450i	$0.661223\pi$
$464$	0	0
$465$	−20.8078	−0.964937
$466$	0	0
$467$	2.68466	0.124231	0.0621156	−	0.998069i	$-0.480215\pi$
$467$	2.68466	0.124231	0.0621156	+	0.998069i	$0.480215\pi$
$468$	0	0
$469$	−18.0540	−0.833655
$470$	0	0
$471$	8.49242	0.391310
$472$	0	0
$473$	20.4924	0.942243
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	15.8078	0.723788
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	3.12311	0.141813
$486$	0	0
$487$	−11.6847	−0.529482	−0.264741	−	0.964319i	$-0.585287\pi$
$487$	−11.6847	−0.529482	−0.264741	+	0.964319i	$0.585287\pi$
$488$	0	0
$489$	−14.5616	−0.658496
$490$	0	0
$491$	28.6155	1.29140	0.645700	−	0.763591i	$-0.276566\pi$
$491$	28.6155	1.29140	0.645700	+	0.763591i	$0.276566\pi$
$492$	0	0
$493$	0.684658	0.0308355
$494$	0	0
$495$	7.12311	0.320160
$496$	0	0
$497$	−7.80776	−0.350226
$498$	0	0
$499$	9.73863	0.435961	0.217981	−	0.975953i	$-0.430053\pi$
$499$	9.73863	0.435961	0.217981	+	0.975953i	$0.430053\pi$
$500$	0	0
$501$	−20.4924	−0.915534
$502$	0	0
$503$	19.5616	0.872207	0.436103	−	0.899897i	$-0.356358\pi$
$503$	19.5616	0.872207	0.436103	+	0.899897i	$0.356358\pi$
$504$	0	0
$505$	9.80776	0.436440
$506$	0	0
$507$	32.4924	1.44304
$508$	0	0
$509$	−28.5616	−1.26597	−0.632984	−	0.774165i	$-0.718170\pi$
$509$	−28.5616	−1.26597	−0.632984	+	0.774165i	$0.718170\pi$
$510$	0	0
$511$	5.36932	0.237525
$512$	0	0
$513$	2.87689	0.127018
$514$	0	0
$515$	−2.24621	−0.0989799
$516$	0	0
$517$	−7.36932	−0.324102
$518$	0	0
$519$	−2.87689	−0.126282
$520$	0	0
$521$	−33.8617	−1.48351	−0.741755	−	0.670671i	$-0.766006\pi$
$521$	−33.8617	−1.48351	−0.741755	+	0.670671i	$0.766006\pi$
$522$	0	0
$523$	−6.24621	−0.273128	−0.136564	−	0.990631i	$-0.543606\pi$
$523$	−6.24621	−0.273128	−0.136564	+	0.990631i	$0.543606\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−45.1771	−1.96794
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−19.8078	−0.859584
$532$	0	0
$533$	−2.31534	−0.100289
$534$	0	0
$535$	−10.6847	−0.461938
$536$	0	0
$537$	50.4233	2.17593
$538$	0	0
$539$	9.12311	0.392960
$540$	0	0
$541$	−21.0540	−0.905181	−0.452591	−	0.891718i	$-0.649500\pi$
$541$	−21.0540	−0.905181	−0.452591	+	0.891718i	$0.649500\pi$
$542$	0	0
$543$	43.8617	1.88229
$544$	0	0
$545$	−7.12311	−0.305120
$546$	0	0
$547$	−19.6847	−0.841655	−0.420828	−	0.907141i	$-0.638260\pi$
$547$	−19.6847	−0.841655	−0.420828	+	0.907141i	$0.638260\pi$
$548$	0	0
$549$	−32.4924	−1.38674
$550$	0	0
$551$	−0.246211	−0.0104890
$552$	0	0
$553$	−14.2462	−0.605811
$554$	0	0
$555$	−9.12311	−0.387254
$556$	0	0
$557$	32.3002	1.36860	0.684301	−	0.729199i	$-0.260107\pi$
$557$	32.3002	1.36860	0.684301	+	0.729199i	$0.260107\pi$
$558$	0	0
$559$	−5.75379	−0.243359
$560$	0	0
$561$	28.4924	1.20295
$562$	0	0
$563$	−31.8078	−1.34054	−0.670269	−	0.742118i	$-0.733821\pi$
$563$	−31.8078	−1.34054	−0.670269	+	0.742118i	$0.733821\pi$
$564$	0	0
$565$	−9.56155	−0.402258
$566$	0	0
$567$	−10.9309	−0.459053
$568$	0	0
$569$	−4.73863	−0.198654	−0.0993269	−	0.995055i	$-0.531669\pi$
$569$	−4.73863	−0.198654	−0.0993269	+	0.995055i	$0.531669\pi$
$570$	0	0
$571$	18.7386	0.784187	0.392094	−	0.919925i	$-0.371751\pi$
$571$	18.7386	0.784187	0.392094	+	0.919925i	$0.371751\pi$
$572$	0	0
$573$	−13.1231	−0.548226
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−19.6847	−0.819483	−0.409742	−	0.912202i	$-0.634381\pi$
$577$	−19.6847	−0.819483	−0.409742	+	0.912202i	$0.634381\pi$
$578$	0	0
$579$	25.4384	1.05719
$580$	0	0
$581$	−7.31534	−0.303492
$582$	0	0
$583$	−8.87689	−0.367643
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	43.3002	1.78719	0.893595	−	0.448874i	$-0.148175\pi$
$587$	43.3002	1.78719	0.893595	+	0.448874i	$0.148175\pi$
$588$	0	0
$589$	16.2462	0.669413
$590$	0	0
$591$	7.19224	0.295849
$592$	0	0
$593$	48.3542	1.98567	0.992834	−	0.119504i	$-0.0381304\pi$
$593$	48.3542	1.98567	0.992834	+	0.119504i	$0.0381304\pi$
$594$	0	0
$595$	−8.68466	−0.356036
$596$	0	0
$597$	−46.1080	−1.88707
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−40.8617	−1.66679	−0.833393	−	0.552682i	$-0.813605\pi$
$601$	−40.8617	−1.66679	−0.833393	+	0.552682i	$0.813605\pi$
$602$	0	0
$603$	−41.1771	−1.67686
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	0	0
$609$	−0.492423	−0.0199540
$610$	0	0
$611$	2.06913	0.0837081
$612$	0	0
$613$	31.6155	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$613$	31.6155	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$614$	0	0
$615$	−10.5616	−0.425883
$616$	0	0
$617$	46.3002	1.86398	0.931988	−	0.362490i	$-0.118073\pi$
$617$	46.3002	1.86398	0.931988	+	0.362490i	$0.118073\pi$
$618$	0	0
$619$	11.5076	0.462529	0.231264	−	0.972891i	$-0.425714\pi$
$619$	11.5076	0.462529	0.231264	+	0.972891i	$0.425714\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	12.4924	0.500498
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.2462	−0.409194
$628$	0	0
$629$	−19.8078	−0.789787
$630$	0	0
$631$	−34.7386	−1.38292	−0.691462	−	0.722413i	$-0.743033\pi$
$631$	−34.7386	−1.38292	−0.691462	+	0.722413i	$0.743033\pi$
$632$	0	0
$633$	−48.4924	−1.92740
$634$	0	0
$635$	−0.807764	−0.0320551
$636$	0	0
$637$	−2.56155	−0.101492
$638$	0	0
$639$	−17.8078	−0.704464
$640$	0	0
$641$	11.1231	0.439336	0.219668	−	0.975575i	$-0.429503\pi$
$641$	11.1231	0.439336	0.219668	+	0.975575i	$0.429503\pi$
$642$	0	0
$643$	23.1771	0.914015	0.457007	−	0.889463i	$-0.348921\pi$
$643$	23.1771	0.914015	0.457007	+	0.889463i	$0.348921\pi$
$644$	0	0
$645$	−26.2462	−1.03344
$646$	0	0
$647$	−34.1771	−1.34364	−0.671820	−	0.740715i	$-0.734487\pi$
$647$	−34.1771	−1.34364	−0.671820	+	0.740715i	$0.734487\pi$
$648$	0	0
$649$	11.1231	0.436620
$650$	0	0
$651$	32.4924	1.27348
$652$	0	0
$653$	−32.4233	−1.26882	−0.634411	−	0.772996i	$-0.718757\pi$
$653$	−32.4233	−1.26882	−0.634411	+	0.772996i	$0.718757\pi$
$654$	0	0
$655$	−9.93087	−0.388031
$656$	0	0
$657$	12.2462	0.477770
$658$	0	0
$659$	11.6155	0.452477	0.226238	−	0.974072i	$-0.427357\pi$
$659$	11.6155	0.452477	0.226238	+	0.974072i	$0.427357\pi$
$660$	0	0
$661$	44.2462	1.72098	0.860489	−	0.509469i	$-0.170158\pi$
$661$	44.2462	1.72098	0.860489	+	0.509469i	$0.170158\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	3.12311	0.121109
$666$	0	0
$667$	−0.123106	−0.00476667
$668$	0	0
$669$	−69.4773	−2.68615
$670$	0	0
$671$	18.2462	0.704387
$672$	0	0
$673$	−36.8078	−1.41884	−0.709418	−	0.704788i	$-0.751042\pi$
$673$	−36.8078	−1.41884	−0.709418	+	0.704788i	$0.751042\pi$
$674$	0	0
$675$	−1.43845	−0.0553659
$676$	0	0
$677$	−6.68466	−0.256912	−0.128456	−	0.991715i	$-0.541002\pi$
$677$	−6.68466	−0.256912	−0.128456	+	0.991715i	$0.541002\pi$
$678$	0	0
$679$	−4.87689	−0.187158
$680$	0	0
$681$	−4.49242	−0.172150
$682$	0	0
$683$	−21.9309	−0.839161	−0.419581	−	0.907718i	$-0.637823\pi$
$683$	−21.9309	−0.839161	−0.419581	+	0.907718i	$0.637823\pi$
$684$	0	0
$685$	19.6155	0.749471
$686$	0	0
$687$	56.9848	2.17411
$688$	0	0
$689$	2.49242	0.0949537
$690$	0	0
$691$	32.4924	1.23607	0.618035	−	0.786151i	$-0.287929\pi$
$691$	32.4924	1.23607	0.618035	+	0.786151i	$0.287929\pi$
$692$	0	0
$693$	−11.1231	−0.422532
$694$	0	0
$695$	−14.3693	−0.545059
$696$	0	0
$697$	−22.9309	−0.868569
$698$	0	0
$699$	64.8078	2.45125
$700$	0	0
$701$	26.2462	0.991306	0.495653	−	0.868521i	$-0.334929\pi$
$701$	26.2462	0.991306	0.495653	+	0.868521i	$0.334929\pi$
$702$	0	0
$703$	7.12311	0.268653
$704$	0	0
$705$	9.43845	0.355472
$706$	0	0
$707$	−15.3153	−0.575993
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	−32.4924	−1.21856
$712$	0	0
$713$	8.12311	0.304213
$714$	0	0
$715$	1.12311	0.0420018
$716$	0	0
$717$	41.3002	1.54238
$718$	0	0
$719$	8.68466	0.323883	0.161942	−	0.986800i	$-0.448224\pi$
$719$	8.68466	0.323883	0.161942	+	0.986800i	$0.448224\pi$
$720$	0	0
$721$	3.50758	0.130629
$722$	0	0
$723$	−37.1231	−1.38062
$724$	0	0
$725$	0.123106	0.00457203
$726$	0	0
$727$	15.5616	0.577146	0.288573	−	0.957458i	$-0.406819\pi$
$727$	15.5616	0.577146	0.288573	+	0.957458i	$0.406819\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−56.9848	−2.10766
$732$	0	0
$733$	−5.17708	−0.191220	−0.0956099	−	0.995419i	$-0.530480\pi$
$733$	−5.17708	−0.191220	−0.0956099	+	0.995419i	$0.530480\pi$
$734$	0	0
$735$	−11.6847	−0.430995
$736$	0	0
$737$	23.1231	0.851751
$738$	0	0
$739$	−31.4924	−1.15847	−0.579234	−	0.815161i	$-0.696648\pi$
$739$	−31.4924	−1.15847	−0.579234	+	0.815161i	$0.696648\pi$
$740$	0	0
$741$	2.87689	0.105685
$742$	0	0
$743$	−29.7538	−1.09156	−0.545780	−	0.837928i	$-0.683767\pi$
$743$	−29.7538	−1.09156	−0.545780	+	0.837928i	$0.683767\pi$
$744$	0	0
$745$	−9.12311	−0.334245
$746$	0	0
$747$	−16.6847	−0.610460
$748$	0	0
$749$	16.6847	0.609644
$750$	0	0
$751$	−46.3542	−1.69149	−0.845744	−	0.533589i	$-0.820843\pi$
$751$	−46.3542	−1.69149	−0.845744	+	0.533589i	$0.820843\pi$
$752$	0	0
$753$	−2.24621	−0.0818565
$754$	0	0
$755$	2.56155	0.0932245
$756$	0	0
$757$	44.5464	1.61907	0.809533	−	0.587074i	$-0.199720\pi$
$757$	44.5464	1.61907	0.809533	+	0.587074i	$0.199720\pi$
$758$	0	0
$759$	−5.12311	−0.185957
$760$	0	0
$761$	−43.9848	−1.59445	−0.797225	−	0.603683i	$-0.793699\pi$
$761$	−43.9848	−1.59445	−0.797225	+	0.603683i	$0.793699\pi$
$762$	0	0
$763$	11.1231	0.402683
$764$	0	0
$765$	−19.8078	−0.716151
$766$	0	0
$767$	−3.12311	−0.112769
$768$	0	0
$769$	33.6155	1.21221	0.606103	−	0.795386i	$-0.292732\pi$
$769$	33.6155	1.21221	0.606103	+	0.795386i	$0.292732\pi$
$770$	0	0
$771$	−29.9309	−1.07793
$772$	0	0
$773$	−20.8769	−0.750890	−0.375445	−	0.926845i	$-0.622510\pi$
$773$	−20.8769	−0.750890	−0.375445	+	0.926845i	$0.622510\pi$
$774$	0	0
$775$	−8.12311	−0.291791
$776$	0	0
$777$	14.2462	0.511080
$778$	0	0
$779$	8.24621	0.295451
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	−0.177081	−0.00632836
$784$	0	0
$785$	3.31534	0.118330
$786$	0	0
$787$	20.3002	0.723624	0.361812	−	0.932251i	$-0.382158\pi$
$787$	20.3002	0.723624	0.361812	+	0.932251i	$0.382158\pi$
$788$	0	0
$789$	68.3542	2.43347
$790$	0	0
$791$	14.9309	0.530881
$792$	0	0
$793$	−5.12311	−0.181927
$794$	0	0
$795$	11.3693	0.403228
$796$	0	0
$797$	45.8078	1.62259	0.811297	−	0.584634i	$-0.198762\pi$
$797$	45.8078	1.62259	0.811297	+	0.584634i	$0.198762\pi$
$798$	0	0
$799$	20.4924	0.724970
$800$	0	0
$801$	28.4924	1.00673
$802$	0	0
$803$	−6.87689	−0.242680
$804$	0	0
$805$	1.56155	0.0550375
$806$	0	0
$807$	−14.0691	−0.495257
$808$	0	0
$809$	30.1922	1.06150	0.530751	−	0.847528i	$-0.321910\pi$
$809$	30.1922	1.06150	0.530751	+	0.847528i	$0.321910\pi$
$810$	0	0
$811$	−10.3693	−0.364116	−0.182058	−	0.983288i	$-0.558276\pi$
$811$	−10.3693	−0.364116	−0.182058	+	0.983288i	$0.558276\pi$
$812$	0	0
$813$	54.2462	1.90250
$814$	0	0
$815$	−5.68466	−0.199125
$816$	0	0
$817$	20.4924	0.716939
$818$	0	0
$819$	3.12311	0.109130
$820$	0	0
$821$	5.50758	0.192216	0.0961079	−	0.995371i	$-0.469361\pi$
$821$	5.50758	0.192216	0.0961079	+	0.995371i	$0.469361\pi$
$822$	0	0
$823$	12.1771	0.424466	0.212233	−	0.977219i	$-0.431926\pi$
$823$	12.1771	0.424466	0.212233	+	0.977219i	$0.431926\pi$
$824$	0	0
$825$	5.12311	0.178364
$826$	0	0
$827$	40.6847	1.41474	0.707372	−	0.706841i	$-0.249881\pi$
$827$	40.6847	1.41474	0.707372	+	0.706841i	$0.249881\pi$
$828$	0	0
$829$	−16.4384	−0.570931	−0.285465	−	0.958389i	$-0.592148\pi$
$829$	−16.4384	−0.570931	−0.285465	+	0.958389i	$0.592148\pi$
$830$	0	0
$831$	62.9157	2.18252
$832$	0	0
$833$	−25.3693	−0.878995
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	11.6847	0.403881
$838$	0	0
$839$	7.61553	0.262917	0.131459	−	0.991322i	$-0.458034\pi$
$839$	7.61553	0.262917	0.131459	+	0.991322i	$0.458034\pi$
$840$	0	0
$841$	−28.9848	−0.999477
$842$	0	0
$843$	2.24621	0.0773636
$844$	0	0
$845$	12.6847	0.436366
$846$	0	0
$847$	−10.9309	−0.375589
$848$	0	0
$849$	63.8617	2.19173
$850$	0	0
$851$	3.56155	0.122088
$852$	0	0
$853$	18.4924	0.633168	0.316584	−	0.948564i	$-0.397464\pi$
$853$	18.4924	0.633168	0.316584	+	0.948564i	$0.397464\pi$
$854$	0	0
$855$	7.12311	0.243605
$856$	0	0
$857$	29.5464	1.00929	0.504643	−	0.863328i	$-0.331624\pi$
$857$	29.5464	1.00929	0.504643	+	0.863328i	$0.331624\pi$
$858$	0	0
$859$	24.3693	0.831470	0.415735	−	0.909486i	$-0.363524\pi$
$859$	24.3693	0.831470	0.415735	+	0.909486i	$0.363524\pi$
$860$	0	0
$861$	16.4924	0.562060
$862$	0	0
$863$	21.6847	0.738154	0.369077	−	0.929399i	$-0.379674\pi$
$863$	21.6847	0.738154	0.369077	+	0.929399i	$0.379674\pi$
$864$	0	0
$865$	−1.12311	−0.0381867
$866$	0	0
$867$	−35.6847	−1.21191
$868$	0	0
$869$	18.2462	0.618960
$870$	0	0
$871$	−6.49242	−0.219987
$872$	0	0
$873$	−11.1231	−0.376460
$874$	0	0
$875$	−1.56155	−0.0527901
$876$	0	0
$877$	51.4773	1.73826	0.869132	−	0.494580i	$-0.164678\pi$
$877$	51.4773	1.73826	0.869132	+	0.494580i	$0.164678\pi$
$878$	0	0
$879$	−26.7386	−0.901872
$880$	0	0
$881$	−26.7386	−0.900847	−0.450424	−	0.892815i	$-0.648727\pi$
$881$	−26.7386	−0.900847	−0.450424	+	0.892815i	$0.648727\pi$
$882$	0	0
$883$	−20.9848	−0.706196	−0.353098	−	0.935586i	$-0.614872\pi$
$883$	−20.9848	−0.706196	−0.353098	+	0.935586i	$0.614872\pi$
$884$	0	0
$885$	−14.2462	−0.478881
$886$	0	0
$887$	−2.17708	−0.0730992	−0.0365496	−	0.999332i	$-0.511637\pi$
$887$	−2.17708	−0.0730992	−0.0365496	+	0.999332i	$0.511637\pi$
$888$	0	0
$889$	1.26137	0.0423049
$890$	0	0
$891$	14.0000	0.469018
$892$	0	0
$893$	−7.36932	−0.246605
$894$	0	0
$895$	19.6847	0.657986
$896$	0	0
$897$	1.43845	0.0480284
$898$	0	0
$899$	−1.00000	−0.0333519
$900$	0	0
$901$	24.6847	0.822365
$902$	0	0
$903$	40.9848	1.36389
$904$	0	0
$905$	17.1231	0.569191
$906$	0	0
$907$	37.4233	1.24262	0.621310	−	0.783565i	$-0.286601\pi$
$907$	37.4233	1.24262	0.621310	+	0.783565i	$0.286601\pi$
$908$	0	0
$909$	−34.9309	−1.15858
$910$	0	0
$911$	14.8769	0.492894	0.246447	−	0.969156i	$-0.420737\pi$
$911$	14.8769	0.492894	0.246447	+	0.969156i	$0.420737\pi$
$912$	0	0
$913$	9.36932	0.310079
$914$	0	0
$915$	−23.3693	−0.772566
$916$	0	0
$917$	15.5076	0.512105
$918$	0	0
$919$	12.4924	0.412087	0.206043	−	0.978543i	$-0.433941\pi$
$919$	12.4924	0.412087	0.206043	+	0.978543i	$0.433941\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	−2.80776	−0.0924187
$924$	0	0
$925$	−3.56155	−0.117103
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	49.7386	1.63187	0.815936	−	0.578142i	$-0.196222\pi$
$929$	49.7386	1.63187	0.815936	+	0.578142i	$0.196222\pi$
$930$	0	0
$931$	9.12311	0.298998
$932$	0	0
$933$	63.5464	2.08042
$934$	0	0
$935$	11.1231	0.363764
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−38.2462	−1.24812
$940$	0	0
$941$	−0.492423	−0.0160525	−0.00802626	−	0.999968i	$-0.502555\pi$
$941$	−0.492423	−0.0160525	−0.00802626	+	0.999968i	$0.502555\pi$
$942$	0	0
$943$	4.12311	0.134267
$944$	0	0
$945$	2.24621	0.0730693
$946$	0	0
$947$	13.4384	0.436691	0.218345	−	0.975872i	$-0.429934\pi$
$947$	13.4384	0.436691	0.218345	+	0.975872i	$0.429934\pi$
$948$	0	0
$949$	1.93087	0.0626787
$950$	0	0
$951$	−1.61553	−0.0523871
$952$	0	0
$953$	−27.7538	−0.899033	−0.449517	−	0.893272i	$-0.648404\pi$
$953$	−27.7538	−0.899033	−0.449517	+	0.893272i	$0.648404\pi$
$954$	0	0
$955$	−5.12311	−0.165780
$956$	0	0
$957$	0.630683	0.0203871
$958$	0	0
$959$	−30.6307	−0.989116
$960$	0	0
$961$	34.9848	1.12854
$962$	0	0
$963$	38.0540	1.22627
$964$	0	0
$965$	9.93087	0.319686
$966$	0	0
$967$	−36.1771	−1.16338	−0.581688	−	0.813412i	$-0.697608\pi$
$967$	−36.1771	−1.16338	−0.581688	+	0.813412i	$0.697608\pi$
$968$	0	0
$969$	28.4924	0.915308
$970$	0	0
$971$	34.1080	1.09458	0.547288	−	0.836944i	$-0.315660\pi$
$971$	34.1080	1.09458	0.547288	+	0.836944i	$0.315660\pi$
$972$	0	0
$973$	22.4384	0.719344
$974$	0	0
$975$	−1.43845	−0.0460672
$976$	0	0
$977$	41.3153	1.32179	0.660897	−	0.750476i	$-0.270176\pi$
$977$	41.3153	1.32179	0.660897	+	0.750476i	$0.270176\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	25.3693	0.809980
$982$	0	0
$983$	−41.0388	−1.30894	−0.654468	−	0.756090i	$-0.727107\pi$
$983$	−41.0388	−1.30894	−0.654468	+	0.756090i	$0.727107\pi$
$984$	0	0
$985$	2.80776	0.0894628
$986$	0	0
$987$	−14.7386	−0.469136
$988$	0	0
$989$	10.2462	0.325811
$990$	0	0
$991$	18.4384	0.585717	0.292858	−	0.956156i	$-0.405394\pi$
$991$	18.4384	0.585717	0.292858	+	0.956156i	$0.405394\pi$
$992$	0	0
$993$	−75.5464	−2.39739
$994$	0	0
$995$	−18.0000	−0.570638
$996$	0	0
$997$	−46.1080	−1.46025	−0.730127	−	0.683312i	$-0.760539\pi$
$997$	−46.1080	−1.46025	−0.730127	+	0.683312i	$0.760539\pi$
$998$	0	0
$999$	5.12311	0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.e.1.1	✓	2
3.2	odd	2		8280.2.a.bf.1.2		2
4.3	odd	2		1840.2.a.o.1.2		2
5.2	odd	4		4600.2.e.n.4049.4		4
5.3	odd	4		4600.2.e.n.4049.1		4
5.4	even	2		4600.2.a.t.1.2		2
8.3	odd	2		7360.2.a.bl.1.1		2
8.5	even	2		7360.2.a.bp.1.2		2
20.19	odd	2		9200.2.a.bq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.e.1.1	✓	2	1.1	even	1	trivial
1840.2.a.o.1.2		2	4.3	odd	2
4600.2.a.t.1.2		2	5.4	even	2
4600.2.e.n.4049.1		4	5.3	odd	4
4600.2.e.n.4049.4		4	5.2	odd	4
7360.2.a.bl.1.1		2	8.3	odd	2
7360.2.a.bp.1.2		2	8.5	even	2
8280.2.a.bf.1.2		2	3.2	odd	2
9200.2.a.bq.1.1		2	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} -1.00000 q^{5} +1.56155 q^{7} +3.56155 q^{9} -2.00000 q^{11} +0.561553 q^{13} +2.56155 q^{15} +5.56155 q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.43845 q^{27} +0.123106 q^{29} -8.12311 q^{31} +5.12311 q^{33} -1.56155 q^{35} -3.56155 q^{37} -1.43845 q^{39} -4.12311 q^{41} -10.2462 q^{43} -3.56155 q^{45} +3.68466 q^{47} -4.56155 q^{49} -14.2462 q^{51} +4.43845 q^{53} +2.00000 q^{55} +5.12311 q^{57} -5.56155 q^{59} -9.12311 q^{61} +5.56155 q^{63} -0.561553 q^{65} -11.5616 q^{67} +2.56155 q^{69} -5.00000 q^{71} +3.43845 q^{73} -2.56155 q^{75} -3.12311 q^{77} -9.12311 q^{79} -7.00000 q^{81} -4.68466 q^{83} -5.56155 q^{85} -0.315342 q^{87} +8.00000 q^{89} +0.876894 q^{91} +20.8078 q^{93} +2.00000 q^{95} -3.12311 q^{97} -7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} - 4 q^{19} - 8 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 8 q^{29} - 8 q^{31} + 2 q^{33} + q^{35} - 3 q^{37} - 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 7 * q^17 - 4 * q^19 - 8 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 8 * q^29 - 8 * q^31 + 2 * q^33 + q^35 - 3 * q^37 - 7 * q^39 - 4 * q^43 - 3 * q^45 - 5 * q^47 - 5 * q^49 - 12 * q^51 + 13 * q^53 + 4 * q^55 + 2 * q^57 - 7 * q^59 - 10 * q^61 + 7 * q^63 + 3 * q^65 - 19 * q^67 + q^69 - 10 * q^71 + 11 * q^73 - q^75 + 2 * q^77 - 10 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 - 13 * q^87 + 16 * q^89 + 10 * q^91 + 21 * q^93 + 4 * q^95 + 2 * q^97 - 6 * q^99

Introduction

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