LMFDB - Embedded newform 7840.2.a.cd.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7840,2,Mod(1,7840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7840 = 2^{5} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7840.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:	$62.6027151847$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.170145936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 22x^{2} - 9 $ x^6 - 10x^4 + 22x^2 - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.728018$ of defining polynomial
Character	$\chi$	$=$	7840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.39276 q^{3} -1.00000 q^{5} +8.51082 q^{9} +O(q^{10})$	$q+3.39276 q^{3} -1.00000 q^{5} +8.51082 q^{9} -0.589491 q^{11} -1.57084 q^{13} -3.39276 q^{15} +6.93998 q^{17} +4.73999 q^{19} -4.25931 q^{23} +1.00000 q^{25} +18.6969 q^{27} -0.570840 q^{29} -7.06257 q^{31} -2.00000 q^{33} +8.51082 q^{37} -5.32948 q^{39} +1.00000 q^{41} -6.30483 q^{43} -8.51082 q^{45} -0.589491 q^{47} +23.5457 q^{51} +3.57084 q^{53} +0.589491 q^{55} +16.0817 q^{57} -0.277055 q^{59} +8.45080 q^{61} +1.57084 q^{65} -11.3573 q^{67} -14.4508 q^{69} +10.9360 q^{71} +10.0817 q^{73} +3.39276 q^{75} +15.0271 q^{79} +37.9016 q^{81} +1.65967 q^{83} -6.93998 q^{85} -1.93672 q^{87} -13.5108 q^{89} -23.9616 q^{93} -4.73999 q^{95} -7.87996 q^{97} -5.01705 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} + 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^5 + 10 * q^9	$ 6 q - 6 q^{5} + 10 q^{9} - 2 q^{13} + 8 q^{17} + 6 q^{25} + 4 q^{29} - 12 q^{33} + 10 q^{37} + 6 q^{41} - 10 q^{45} + 14 q^{53} + 48 q^{57} - 24 q^{61} + 2 q^{65} - 12 q^{69} + 12 q^{73} + 78 q^{81} - 8 q^{85} - 40 q^{89} - 28 q^{93} + 20 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 + 10 * q^9 - 2 * q^13 + 8 * q^17 + 6 * q^25 + 4 * q^29 - 12 * q^33 + 10 * q^37 + 6 * q^41 - 10 * q^45 + 14 * q^53 + 48 * q^57 - 24 * q^61 + 2 * q^65 - 12 * q^69 + 12 * q^73 + 78 * q^81 - 8 * q^85 - 40 * q^89 - 28 * q^93 + 20 * q^97

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
$3$	3.39276	1.95881	0.979405	−	0.201903i	$-0.0647127\pi$
$3$	3.39276	1.95881	0.979405	+	0.201903i	$0.0647127\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.51082	2.83694
$10$	0	0
$11$	−0.589491	−0.177738	−0.0888690	−	0.996043i	$-0.528325\pi$
$11$	−0.589491	−0.177738	−0.0888690	+	0.996043i	$0.528325\pi$
$12$	0	0
$13$	−1.57084	−0.435673	−0.217836	−	0.975985i	$-0.569900\pi$
$13$	−1.57084	−0.435673	−0.217836	+	0.975985i	$0.569900\pi$
$14$	0	0
$15$	−3.39276	−0.876007
$16$	0	0
$17$	6.93998	1.68319	0.841596	−	0.540107i	$-0.181616\pi$
$17$	6.93998	1.68319	0.841596	+	0.540107i	$0.181616\pi$
$18$	0	0
$19$	4.73999	1.08743	0.543715	−	0.839270i	$-0.317017\pi$
$19$	4.73999	1.08743	0.543715	+	0.839270i	$0.317017\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.25931	−0.888127	−0.444063	−	0.895995i	$-0.646463\pi$
$23$	−4.25931	−0.888127	−0.444063	+	0.895995i	$0.646463\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	18.6969	3.59822
$28$	0	0
$29$	−0.570840	−0.106002	−0.0530012	−	0.998594i	$-0.516879\pi$
$29$	−0.570840	−0.106002	−0.0530012	+	0.998594i	$0.516879\pi$
$30$	0	0
$31$	−7.06257	−1.26848	−0.634238	−	0.773138i	$-0.718686\pi$
$31$	−7.06257	−1.26848	−0.634238	+	0.773138i	$0.718686\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.51082	1.39917	0.699585	−	0.714549i	$-0.253368\pi$
$37$	8.51082	1.39917	0.699585	+	0.714549i	$0.253368\pi$
$38$	0	0
$39$	−5.32948	−0.853400
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	−6.30483	−0.961478	−0.480739	−	0.876864i	$-0.659632\pi$
$43$	−6.30483	−0.961478	−0.480739	+	0.876864i	$0.659632\pi$
$44$	0	0
$45$	−8.51082	−1.26872
$46$	0	0
$47$	−0.589491	−0.0859860	−0.0429930	−	0.999075i	$-0.513689\pi$
$47$	−0.589491	−0.0859860	−0.0429930	+	0.999075i	$0.513689\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	23.5457	3.29706
$52$	0	0
$53$	3.57084	0.490493	0.245246	−	0.969461i	$-0.421131\pi$
$53$	3.57084	0.490493	0.245246	+	0.969461i	$0.421131\pi$
$54$	0	0
$55$	0.589491	0.0794869
$56$	0	0
$57$	16.0817	2.13007
$58$	0	0
$59$	−0.277055	−0.0360694	−0.0180347	−	0.999837i	$-0.505741\pi$
$59$	−0.277055	−0.0360694	−0.0180347	+	0.999837i	$0.505741\pi$
$60$	0	0
$61$	8.45080	1.08201	0.541007	−	0.841018i	$-0.318043\pi$
$61$	8.45080	1.08201	0.541007	+	0.841018i	$0.318043\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.57084	0.194839
$66$	0	0
$67$	−11.3573	−1.38751	−0.693755	−	0.720211i	$-0.744045\pi$
$67$	−11.3573	−1.38751	−0.693755	+	0.720211i	$0.744045\pi$
$68$	0	0
$69$	−14.4508	−1.73967
$70$	0	0
$71$	10.9360	1.29787	0.648934	−	0.760845i	$-0.275215\pi$
$71$	10.9360	1.29787	0.648934	+	0.760845i	$0.275215\pi$
$72$	0	0
$73$	10.0817	1.17997	0.589985	−	0.807415i	$-0.299134\pi$
$73$	10.0817	1.17997	0.589985	+	0.807415i	$0.299134\pi$
$74$	0	0
$75$	3.39276	0.391762
$76$	0	0
$77$	0	0
$78$	0	0
$79$	15.0271	1.69068	0.845339	−	0.534230i	$-0.179398\pi$
$79$	15.0271	1.69068	0.845339	+	0.534230i	$0.179398\pi$
$80$	0	0
$81$	37.9016	4.21129
$82$	0	0
$83$	1.65967	0.182172	0.0910862	−	0.995843i	$-0.470966\pi$
$83$	1.65967	0.182172	0.0910862	+	0.995843i	$0.470966\pi$
$84$	0	0
$85$	−6.93998	−0.752747
$86$	0	0
$87$	−1.93672	−0.207639
$88$	0	0
$89$	−13.5108	−1.43214	−0.716072	−	0.698026i	$-0.754062\pi$
$89$	−13.5108	−1.43214	−0.716072	+	0.698026i	$0.754062\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−23.9616	−2.48470
$94$	0	0
$95$	−4.73999	−0.486313
$96$	0	0
$97$	−7.87996	−0.800089	−0.400044	−	0.916496i	$-0.631005\pi$
$97$	−7.87996	−0.800089	−0.400044	+	0.916496i	$0.631005\pi$
$98$	0	0
$99$	−5.01705	−0.504232
$100$	0	0
$101$	2.57084	0.255808	0.127904	−	0.991787i	$-0.459175\pi$
$101$	2.57084	0.255808	0.127904	+	0.991787i	$0.459175\pi$
$102$	0	0
$103$	0.975347	0.0961038	0.0480519	−	0.998845i	$-0.484699\pi$
$103$	0.975347	0.0961038	0.0480519	+	0.998845i	$0.484699\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.99930	−0.869995	−0.434998	−	0.900432i	$-0.643251\pi$
$107$	−8.99930	−0.869995	−0.434998	+	0.900432i	$0.643251\pi$
$108$	0	0
$109$	−13.5925	−1.30192	−0.650962	−	0.759111i	$-0.725634\pi$
$109$	−13.5925	−1.30192	−0.650962	+	0.759111i	$0.725634\pi$
$110$	0	0
$111$	28.8752	2.74071
$112$	0	0
$113$	16.9400	1.59358	0.796790	−	0.604257i	$-0.206530\pi$
$113$	16.9400	1.59358	0.796790	+	0.604257i	$0.206530\pi$
$114$	0	0
$115$	4.25931	0.397182
$116$	0	0
$117$	−13.3691	−1.23598
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.6525	−0.968409
$122$	0	0
$123$	3.39276	0.305915
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.3395	1.36116	0.680581	−	0.732673i	$-0.261728\pi$
$127$	15.3395	1.36116	0.680581	+	0.732673i	$0.261728\pi$
$128$	0	0
$129$	−21.3908	−1.88335
$130$	0	0
$131$	6.82090	0.595945	0.297972	−	0.954574i	$-0.403690\pi$
$131$	6.82090	0.595945	0.297972	+	0.954574i	$0.403690\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−18.6969	−1.60917
$136$	0	0
$137$	−5.87996	−0.502359	−0.251179	−	0.967941i	$-0.580818\pi$
$137$	−5.87996	−0.502359	−0.251179	+	0.967941i	$0.580818\pi$
$138$	0	0
$139$	9.69759	0.822539	0.411269	−	0.911514i	$-0.365086\pi$
$139$	9.69759	0.822539	0.411269	+	0.911514i	$0.365086\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	0.925996	0.0774356
$144$	0	0
$145$	0.570840	0.0474057
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.3091	1.09033	0.545163	−	0.838330i	$-0.316468\pi$
$149$	13.3091	1.09033	0.545163	+	0.838330i	$0.316468\pi$
$150$	0	0
$151$	−8.24156	−0.670688	−0.335344	−	0.942096i	$-0.608853\pi$
$151$	−8.24156	−0.670688	−0.335344	+	0.942096i	$0.608853\pi$
$152$	0	0
$153$	59.0649	4.77512
$154$	0	0
$155$	7.06257	0.567280
$156$	0	0
$157$	11.3691	0.907356	0.453678	−	0.891166i	$-0.350112\pi$
$157$	11.3691	0.907356	0.453678	+	0.891166i	$0.350112\pi$
$158$	0	0
$159$	12.1150	0.960782
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.961377	0.0753008	0.0376504	−	0.999291i	$-0.488013\pi$
$163$	0.961377	0.0753008	0.0376504	+	0.999291i	$0.488013\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	6.39966	0.495221	0.247610	−	0.968860i	$-0.420355\pi$
$167$	6.39966	0.495221	0.247610	+	0.968860i	$0.420355\pi$
$168$	0	0
$169$	−10.5325	−0.810189
$170$	0	0
$171$	40.3412	3.08497
$172$	0	0
$173$	−16.5108	−1.25529	−0.627647	−	0.778498i	$-0.715982\pi$
$173$	−16.5108	−1.25529	−0.627647	+	0.778498i	$0.715982\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.939980	−0.0706532
$178$	0	0
$179$	0.312436	0.0233526	0.0116763	−	0.999932i	$-0.496283\pi$
$179$	0.312436	0.0233526	0.0116763	+	0.999932i	$0.496283\pi$
$180$	0	0
$181$	11.5925	0.861662	0.430831	−	0.902433i	$-0.358220\pi$
$181$	11.5925	0.861662	0.430831	+	0.902433i	$0.358220\pi$
$182$	0	0
$183$	28.6715	2.11946
$184$	0	0
$185$	−8.51082	−0.625728
$186$	0	0
$187$	−4.09105	−0.299167
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.33963	−0.531077	−0.265538	−	0.964100i	$-0.585550\pi$
$191$	−7.33963	−0.531077	−0.265538	+	0.964100i	$0.585550\pi$
$192$	0	0
$193$	−7.02164	−0.505429	−0.252714	−	0.967541i	$-0.581323\pi$
$193$	−7.02164	−0.505429	−0.252714	+	0.967541i	$0.581323\pi$
$194$	0	0
$195$	5.32948	0.381652
$196$	0	0
$197$	0.630860	0.0449469	0.0224735	−	0.999747i	$-0.492846\pi$
$197$	0.630860	0.0449469	0.0224735	+	0.999747i	$0.492846\pi$
$198$	0	0
$199$	0.554109	0.0392798	0.0196399	−	0.999807i	$-0.493748\pi$
$199$	0.554109	0.0392798	0.0196399	+	0.999807i	$0.493748\pi$
$200$	0	0
$201$	−38.5325	−2.71787
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	0	0
$207$	−36.2502	−2.51956
$208$	0	0
$209$	−2.79418	−0.193278
$210$	0	0
$211$	−15.9531	−1.09825	−0.549127	−	0.835739i	$-0.685040\pi$
$211$	−15.9531	−1.09825	−0.549127	+	0.835739i	$0.685040\pi$
$212$	0	0
$213$	37.1033	2.54228
$214$	0	0
$215$	6.30483	0.429986
$216$	0	0
$217$	0	0
$218$	0	0
$219$	34.2047	2.31134
$220$	0	0
$221$	−10.9016	−0.733321
$222$	0	0
$223$	−7.33963	−0.491498	−0.245749	−	0.969334i	$-0.579034\pi$
$223$	−7.33963	−0.491498	−0.245749	+	0.969334i	$0.579034\pi$
$224$	0	0
$225$	8.51082	0.567388
$226$	0	0
$227$	−12.0556	−0.800155	−0.400078	−	0.916481i	$-0.631017\pi$
$227$	−12.0556	−0.800155	−0.400078	+	0.916481i	$0.631017\pi$
$228$	0	0
$229$	−9.14168	−0.604099	−0.302050	−	0.953292i	$-0.597671\pi$
$229$	−9.14168	−0.604099	−0.302050	+	0.953292i	$0.597671\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.1033	−1.25150	−0.625749	−	0.780024i	$-0.715207\pi$
$233$	−19.1033	−1.25150	−0.625749	+	0.780024i	$0.715207\pi$
$234$	0	0
$235$	0.589491	0.0384541
$236$	0	0
$237$	50.9833	3.31172
$238$	0	0
$239$	21.8126	1.41094	0.705470	−	0.708740i	$-0.250736\pi$
$239$	21.8126	1.41094	0.705470	+	0.708740i	$0.250736\pi$
$240$	0	0
$241$	3.36914	0.217025	0.108513	−	0.994095i	$-0.465391\pi$
$241$	3.36914	0.217025	0.108513	+	0.994095i	$0.465391\pi$
$242$	0	0
$243$	72.5004	4.65090
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.44577	−0.473763
$248$	0	0
$249$	5.63086	0.356841
$250$	0	0
$251$	−16.7955	−1.06013	−0.530063	−	0.847958i	$-0.677832\pi$
$251$	−16.7955	−1.06013	−0.530063	+	0.847958i	$0.677832\pi$
$252$	0	0
$253$	2.51082	0.157854
$254$	0	0
$255$	−23.5457	−1.47449
$256$	0	0
$257$	25.8800	1.61435	0.807174	−	0.590314i	$-0.200996\pi$
$257$	25.8800	1.61435	0.807174	+	0.590314i	$0.200996\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.85832	−0.300722
$262$	0	0
$263$	20.7778	1.28121	0.640607	−	0.767869i	$-0.278683\pi$
$263$	20.7778	1.28121	0.640607	+	0.767869i	$0.278683\pi$
$264$	0	0
$265$	−3.57084	−0.219355
$266$	0	0
$267$	−45.8390	−2.80530
$268$	0	0
$269$	−20.6141	−1.25687	−0.628433	−	0.777864i	$-0.716303\pi$
$269$	−20.6141	−1.25687	−0.628433	+	0.777864i	$0.716303\pi$
$270$	0	0
$271$	−30.3312	−1.84249	−0.921245	−	0.388983i	$-0.872826\pi$
$271$	−30.3312	−1.84249	−0.921245	+	0.388983i	$0.872826\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.589491	−0.0355476
$276$	0	0
$277$	16.8199	1.01061	0.505306	−	0.862940i	$-0.331380\pi$
$277$	16.8199	1.01061	0.505306	+	0.862940i	$0.331380\pi$
$278$	0	0
$279$	−60.1083	−3.59859
$280$	0	0
$281$	23.3691	1.39409	0.697043	−	0.717029i	$-0.254499\pi$
$281$	23.3691	1.39409	0.697043	+	0.717029i	$0.254499\pi$
$282$	0	0
$283$	−22.0897	−1.31309	−0.656547	−	0.754285i	$-0.727984\pi$
$283$	−22.0897	−1.31309	−0.656547	+	0.754285i	$0.727984\pi$
$284$	0	0
$285$	−16.0817	−0.952595
$286$	0	0
$287$	0	0
$288$	0	0
$289$	31.1633	1.83314
$290$	0	0
$291$	−26.7348	−1.56722
$292$	0	0
$293$	−21.5325	−1.25794	−0.628970	−	0.777430i	$-0.716523\pi$
$293$	−21.5325	−1.25794	−0.628970	+	0.777430i	$0.716523\pi$
$294$	0	0
$295$	0.277055	0.0161307
$296$	0	0
$297$	−11.0216	−0.639540
$298$	0	0
$299$	6.69069	0.386932
$300$	0	0
$301$	0	0
$302$	0	0
$303$	8.72224	0.501080
$304$	0	0
$305$	−8.45080	−0.483891
$306$	0	0
$307$	20.8372	1.18924	0.594622	−	0.804005i	$-0.297302\pi$
$307$	20.8372	1.18924	0.594622	+	0.804005i	$0.297302\pi$
$308$	0	0
$309$	3.30912	0.188249
$310$	0	0
$311$	−10.9360	−0.620125	−0.310062	−	0.950716i	$-0.600350\pi$
$311$	−10.9360	−0.620125	−0.310062	+	0.950716i	$0.600350\pi$
$312$	0	0
$313$	26.0433	1.47205	0.736027	−	0.676953i	$-0.236700\pi$
$313$	26.0433	1.47205	0.736027	+	0.676953i	$0.236700\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.93998	−0.165126	−0.0825629	−	0.996586i	$-0.526311\pi$
$317$	−2.93998	−0.165126	−0.0825629	+	0.996586i	$0.526311\pi$
$318$	0	0
$319$	0.336505	0.0188407
$320$	0	0
$321$	−30.5325	−1.70416
$322$	0	0
$323$	32.8955	1.83035
$324$	0	0
$325$	−1.57084	−0.0871345
$326$	0	0
$327$	−46.1160	−2.55022
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.79242	−0.538240	−0.269120	−	0.963107i	$-0.586733\pi$
$331$	−9.79242	−0.538240	−0.269120	+	0.963107i	$0.586733\pi$
$332$	0	0
$333$	72.4341	3.96936
$334$	0	0
$335$	11.3573	0.620513
$336$	0	0
$337$	−4.08166	−0.222342	−0.111171	−	0.993801i	$-0.535460\pi$
$337$	−4.08166	−0.222342	−0.111171	+	0.993801i	$0.535460\pi$
$338$	0	0
$339$	57.4733	3.12152
$340$	0	0
$341$	4.16332	0.225456
$342$	0	0
$343$	0	0
$344$	0	0
$345$	14.4508	0.778005
$346$	0	0
$347$	8.03792	0.431498	0.215749	−	0.976449i	$-0.430781\pi$
$347$	8.03792	0.431498	0.215749	+	0.976449i	$0.430781\pi$
$348$	0	0
$349$	12.5708	0.672902	0.336451	−	0.941701i	$-0.390773\pi$
$349$	12.5708	0.672902	0.336451	+	0.941701i	$0.390773\pi$
$350$	0	0
$351$	−29.3698	−1.56765
$352$	0	0
$353$	14.8199	0.788786	0.394393	−	0.918942i	$-0.370955\pi$
$353$	14.8199	0.788786	0.394393	+	0.918942i	$0.370955\pi$
$354$	0	0
$355$	−10.9360	−0.580424
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.1536	−0.588666	−0.294333	−	0.955703i	$-0.595097\pi$
$359$	−11.1536	−0.588666	−0.294333	+	0.955703i	$0.595097\pi$
$360$	0	0
$361$	3.46754	0.182502
$362$	0	0
$363$	−36.1414	−1.89693
$364$	0	0
$365$	−10.0817	−0.527698
$366$	0	0
$367$	8.56796	0.447244	0.223622	−	0.974676i	$-0.428212\pi$
$367$	8.56796	0.447244	0.223622	+	0.974676i	$0.428212\pi$
$368$	0	0
$369$	8.51082	0.443056
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.858319	−0.0444421	−0.0222210	−	0.999753i	$-0.507074\pi$
$373$	−0.858319	−0.0444421	−0.0222210	+	0.999753i	$0.507074\pi$
$374$	0	0
$375$	−3.39276	−0.175201
$376$	0	0
$377$	0.896699	0.0461823
$378$	0	0
$379$	28.3451	1.45599	0.727996	−	0.685582i	$-0.240452\pi$
$379$	28.3451	1.45599	0.727996	+	0.685582i	$0.240452\pi$
$380$	0	0
$381$	52.0433	2.66626
$382$	0	0
$383$	−8.75763	−0.447494	−0.223747	−	0.974647i	$-0.571829\pi$
$383$	−8.75763	−0.447494	−0.223747	+	0.974647i	$0.571829\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−53.6593	−2.72765
$388$	0	0
$389$	36.9016	1.87099	0.935493	−	0.353346i	$-0.114956\pi$
$389$	36.9016	1.87099	0.935493	+	0.353346i	$0.114956\pi$
$390$	0	0
$391$	−29.5595	−1.49489
$392$	0	0
$393$	23.1417	1.16734
$394$	0	0
$395$	−15.0271	−0.756094
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.14168	0.306701	0.153350	−	0.988172i	$-0.450994\pi$
$401$	6.14168	0.306701	0.153350	+	0.988172i	$0.450994\pi$
$402$	0	0
$403$	11.0942	0.552640
$404$	0	0
$405$	−37.9016	−1.88335
$406$	0	0
$407$	−5.01705	−0.248686
$408$	0	0
$409$	−17.5108	−0.865854	−0.432927	−	0.901429i	$-0.642519\pi$
$409$	−17.5108	−0.865854	−0.432927	+	0.901429i	$0.642519\pi$
$410$	0	0
$411$	−19.9493	−0.984026
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.65967	−0.0814700
$416$	0	0
$417$	32.9016	1.61120
$418$	0	0
$419$	−6.19603	−0.302696	−0.151348	−	0.988481i	$-0.548361\pi$
$419$	−6.19603	−0.302696	−0.151348	+	0.988481i	$0.548361\pi$
$420$	0	0
$421$	4.73416	0.230729	0.115364	−	0.993323i	$-0.463196\pi$
$421$	4.73416	0.230729	0.115364	+	0.993323i	$0.463196\pi$
$422$	0	0
$423$	−5.01705	−0.243937
$424$	0	0
$425$	6.93998	0.336638
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.14168	0.151682
$430$	0	0
$431$	26.7348	1.28777	0.643885	−	0.765122i	$-0.277321\pi$
$431$	26.7348	1.28777	0.643885	+	0.765122i	$0.277321\pi$
$432$	0	0
$433$	−18.8199	−0.904429	−0.452214	−	0.891909i	$-0.649366\pi$
$433$	−18.8199	−0.904429	−0.452214	+	0.891909i	$0.649366\pi$
$434$	0	0
$435$	1.93672	0.0928588
$436$	0	0
$437$	−20.1891	−0.965775
$438$	0	0
$439$	−15.3041	−0.730426	−0.365213	−	0.930924i	$-0.619004\pi$
$439$	−15.3041	−0.730426	−0.365213	+	0.930924i	$0.619004\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.12585	−0.243537	−0.121768	−	0.992559i	$-0.538856\pi$
$443$	−5.12585	−0.243537	−0.121768	+	0.992559i	$0.538856\pi$
$444$	0	0
$445$	13.5108	0.640474
$446$	0	0
$447$	45.1546	2.13574
$448$	0	0
$449$	11.1200	0.524787	0.262394	−	0.964961i	$-0.415488\pi$
$449$	11.1200	0.524787	0.262394	+	0.964961i	$0.415488\pi$
$450$	0	0
$451$	−0.589491	−0.0277580
$452$	0	0
$453$	−27.9616	−1.31375
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.97836	0.326434	0.163217	−	0.986590i	$-0.447813\pi$
$457$	6.97836	0.326434	0.163217	+	0.986590i	$0.447813\pi$
$458$	0	0
$459$	129.756	6.05649
$460$	0	0
$461$	−0.120040	−0.00559083	−0.00279541	−	0.999996i	$-0.500890\pi$
$461$	−0.120040	−0.00559083	−0.00279541	+	0.999996i	$0.500890\pi$
$462$	0	0
$463$	7.57864	0.352209	0.176105	−	0.984371i	$-0.443650\pi$
$463$	7.57864	0.352209	0.176105	+	0.984371i	$0.443650\pi$
$464$	0	0
$465$	23.9616	1.11119
$466$	0	0
$467$	2.98549	0.138152	0.0690761	−	0.997611i	$-0.477995\pi$
$467$	2.98549	0.138152	0.0690761	+	0.997611i	$0.477995\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	38.5728	1.77734
$472$	0	0
$473$	3.71664	0.170891
$474$	0	0
$475$	4.73999	0.217486
$476$	0	0
$477$	30.3908	1.39150
$478$	0	0
$479$	6.50847	0.297379	0.148690	−	0.988884i	$-0.452494\pi$
$479$	6.50847	0.297379	0.148690	+	0.988884i	$0.452494\pi$
$480$	0	0
$481$	−13.3691	−0.609580
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.87996	0.357811
$486$	0	0
$487$	−32.0048	−1.45028	−0.725139	−	0.688603i	$-0.758225\pi$
$487$	−32.0048	−1.45028	−0.725139	+	0.688603i	$0.758225\pi$
$488$	0	0
$489$	3.26172	0.147500
$490$	0	0
$491$	−20.6336	−0.931182	−0.465591	−	0.885000i	$-0.654158\pi$
$491$	−20.6336	−0.931182	−0.465591	+	0.885000i	$0.654158\pi$
$492$	0	0
$493$	−3.96162	−0.178422
$494$	0	0
$495$	5.01705	0.225500
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16.8904	0.756117	0.378059	−	0.925782i	$-0.376592\pi$
$499$	16.8904	0.756117	0.378059	+	0.925782i	$0.376592\pi$
$500$	0	0
$501$	21.7125	0.970044
$502$	0	0
$503$	−14.9537	−0.666751	−0.333375	−	0.942794i	$-0.608188\pi$
$503$	−14.9537	−0.666751	−0.333375	+	0.942794i	$0.608188\pi$
$504$	0	0
$505$	−2.57084	−0.114401
$506$	0	0
$507$	−35.7341	−1.58701
$508$	0	0
$509$	−9.30912	−0.412619	−0.206310	−	0.978487i	$-0.566145\pi$
$509$	−9.30912	−0.412619	−0.206310	+	0.978487i	$0.566145\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	88.6231	3.91281
$514$	0	0
$515$	−0.975347	−0.0429789
$516$	0	0
$517$	0.347499	0.0152830
$518$	0	0
$519$	−56.0172	−2.45888
$520$	0	0
$521$	10.6741	0.467643	0.233821	−	0.972280i	$-0.424877\pi$
$521$	10.6741	0.467643	0.233821	+	0.972280i	$0.424877\pi$
$522$	0	0
$523$	−25.6266	−1.12057	−0.560286	−	0.828299i	$-0.689309\pi$
$523$	−25.6266	−1.12057	−0.560286	+	0.828299i	$0.689309\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−49.0141	−2.13509
$528$	0	0
$529$	−4.85832	−0.211231
$530$	0	0
$531$	−2.35796	−0.102327
$532$	0	0
$533$	−1.57084	−0.0680406
$534$	0	0
$535$	8.99930	0.389074
$536$	0	0
$537$	1.06002	0.0457432
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.69088	−0.201677	−0.100838	−	0.994903i	$-0.532152\pi$
$541$	−4.69088	−0.201677	−0.100838	+	0.994903i	$0.532152\pi$
$542$	0	0
$543$	39.3305	1.68783
$544$	0	0
$545$	13.5925	0.582238
$546$	0	0
$547$	10.5148	0.449580	0.224790	−	0.974407i	$-0.427830\pi$
$547$	10.5148	0.449580	0.224790	+	0.974407i	$0.427830\pi$
$548$	0	0
$549$	71.9232	3.06961
$550$	0	0
$551$	−2.70578	−0.115270
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−28.8752	−1.22568
$556$	0	0
$557$	−21.2491	−0.900353	−0.450177	−	0.892940i	$-0.648639\pi$
$557$	−21.2491	−0.900353	−0.450177	+	0.892940i	$0.648639\pi$
$558$	0	0
$559$	9.90388	0.418890
$560$	0	0
$561$	−13.8800	−0.586012
$562$	0	0
$563$	−12.5362	−0.528340	−0.264170	−	0.964476i	$-0.585098\pi$
$563$	−12.5362	−0.528340	−0.264170	+	0.964476i	$0.585098\pi$
$564$	0	0
$565$	−16.9400	−0.712670
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.39078	−0.351760	−0.175880	−	0.984412i	$-0.556277\pi$
$569$	−8.39078	−0.351760	−0.175880	+	0.984412i	$0.556277\pi$
$570$	0	0
$571$	−16.2061	−0.678202	−0.339101	−	0.940750i	$-0.610123\pi$
$571$	−16.2061	−0.678202	−0.339101	+	0.940750i	$0.610123\pi$
$572$	0	0
$573$	−24.9016	−1.04028
$574$	0	0
$575$	−4.25931	−0.177625
$576$	0	0
$577$	11.4250	0.475631	0.237815	−	0.971310i	$-0.423569\pi$
$577$	11.4250	0.475631	0.237815	+	0.971310i	$0.423569\pi$
$578$	0	0
$579$	−23.8227	−0.990039
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.10498	−0.0871792
$584$	0	0
$585$	13.3691	0.552746
$586$	0	0
$587$	−26.2996	−1.08550	−0.542750	−	0.839894i	$-0.682617\pi$
$587$	−26.2996	−1.08550	−0.542750	+	0.839894i	$0.682617\pi$
$588$	0	0
$589$	−33.4766	−1.37938
$590$	0	0
$591$	2.14036	0.0880426
$592$	0	0
$593$	26.8199	1.10136	0.550681	−	0.834715i	$-0.314368\pi$
$593$	26.8199	1.10136	0.550681	+	0.834715i	$0.314368\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.87996	0.0769416
$598$	0	0
$599$	−29.9834	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$599$	−29.9834	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−96.6596	−3.93628
$604$	0	0
$605$	10.6525	0.433086
$606$	0	0
$607$	−35.4217	−1.43772	−0.718861	−	0.695154i	$-0.755336\pi$
$607$	−35.4217	−1.43772	−0.718861	+	0.695154i	$0.755336\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.925996	0.0374618
$612$	0	0
$613$	−21.1674	−0.854945	−0.427472	−	0.904028i	$-0.640596\pi$
$613$	−21.1674	−0.854945	−0.427472	+	0.904028i	$0.640596\pi$
$614$	0	0
$615$	−3.39276	−0.136809
$616$	0	0
$617$	−23.0649	−0.928559	−0.464279	−	0.885689i	$-0.653687\pi$
$617$	−23.0649	−0.928559	−0.464279	+	0.885689i	$0.653687\pi$
$618$	0	0
$619$	−38.7550	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$619$	−38.7550	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$620$	0	0
$621$	−79.6358	−3.19567
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−9.47999	−0.378594
$628$	0	0
$629$	59.0649	2.35507
$630$	0	0
$631$	17.8797	0.711779	0.355890	−	0.934528i	$-0.384178\pi$
$631$	17.8797	0.711779	0.355890	+	0.934528i	$0.384178\pi$
$632$	0	0
$633$	−54.1249	−2.15127
$634$	0	0
$635$	−15.3395	−0.608730
$636$	0	0
$637$	0	0
$638$	0	0
$639$	93.0745	3.68197
$640$	0	0
$641$	21.2834	0.840642	0.420321	−	0.907375i	$-0.361917\pi$
$641$	21.2834	0.840642	0.420321	+	0.907375i	$0.361917\pi$
$642$	0	0
$643$	−27.4786	−1.08365	−0.541825	−	0.840491i	$-0.682266\pi$
$643$	−27.4786	−1.08365	−0.541825	+	0.840491i	$0.682266\pi$
$644$	0	0
$645$	21.3908	0.842261
$646$	0	0
$647$	−36.0465	−1.41714	−0.708568	−	0.705643i	$-0.750658\pi$
$647$	−36.0465	−1.41714	−0.708568	+	0.705643i	$0.750658\pi$
$648$	0	0
$649$	0.163321	0.00641091
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.3308	1.14780	0.573901	−	0.818925i	$-0.305430\pi$
$653$	29.3308	1.14780	0.573901	+	0.818925i	$0.305430\pi$
$654$	0	0
$655$	−6.82090	−0.266515
$656$	0	0
$657$	85.8032	3.34750
$658$	0	0
$659$	−0.831164	−0.0323775	−0.0161888	−	0.999869i	$-0.505153\pi$
$659$	−0.831164	−0.0323775	−0.0161888	+	0.999869i	$0.505153\pi$
$660$	0	0
$661$	−37.7558	−1.46853	−0.734265	−	0.678863i	$-0.762473\pi$
$661$	−37.7558	−1.46853	−0.734265	+	0.678863i	$0.762473\pi$
$662$	0	0
$663$	−36.9865	−1.43644
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.43138	0.0941435
$668$	0	0
$669$	−24.9016	−0.962751
$670$	0	0
$671$	−4.98167	−0.192315
$672$	0	0
$673$	21.0216	0.810325	0.405162	−	0.914245i	$-0.367215\pi$
$673$	21.0216	0.810325	0.405162	+	0.914245i	$0.367215\pi$
$674$	0	0
$675$	18.6969	0.719644
$676$	0	0
$677$	36.0690	1.38625	0.693123	−	0.720819i	$-0.256234\pi$
$677$	36.0690	1.38625	0.693123	+	0.720819i	$0.256234\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−40.9016	−1.56735
$682$	0	0
$683$	−23.9669	−0.917069	−0.458534	−	0.888677i	$-0.651625\pi$
$683$	−23.9669	−0.917069	−0.458534	+	0.888677i	$0.651625\pi$
$684$	0	0
$685$	5.87996	0.224662
$686$	0	0
$687$	−31.0155	−1.18332
$688$	0	0
$689$	−5.60922	−0.213694
$690$	0	0
$691$	21.4648	0.816559	0.408279	−	0.912857i	$-0.366129\pi$
$691$	21.4648	0.816559	0.408279	+	0.912857i	$0.366129\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.69759	−0.367851
$696$	0	0
$697$	6.93998	0.262870
$698$	0	0
$699$	−64.8129	−2.45145
$700$	0	0
$701$	8.16744	0.308480	0.154240	−	0.988033i	$-0.450707\pi$
$701$	8.16744	0.308480	0.154240	+	0.988033i	$0.450707\pi$
$702$	0	0
$703$	40.3412	1.52150
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−42.6141	−1.60041	−0.800203	−	0.599729i	$-0.795275\pi$
$709$	−42.6141	−1.60041	−0.800203	+	0.599729i	$0.795275\pi$
$710$	0	0
$711$	127.893	4.79635
$712$	0	0
$713$	30.0817	1.12657
$714$	0	0
$715$	−0.925996	−0.0346303
$716$	0	0
$717$	74.0049	2.76376
$718$	0	0
$719$	36.2856	1.35322	0.676612	−	0.736340i	$-0.263448\pi$
$719$	36.2856	1.35322	0.676612	+	0.736340i	$0.263448\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.4307	0.425112
$724$	0	0
$725$	−0.570840	−0.0212005
$726$	0	0
$727$	−1.22833	−0.0455563	−0.0227782	−	0.999741i	$-0.507251\pi$
$727$	−1.22833	−0.0455563	−0.0227782	+	0.999741i	$0.507251\pi$
$728$	0	0
$729$	132.272	4.89895
$730$	0	0
$731$	−43.7554	−1.61835
$732$	0	0
$733$	2.51082	0.0927393	0.0463696	−	0.998924i	$-0.485235\pi$
$733$	2.51082	0.0927393	0.0463696	+	0.998924i	$0.485235\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.69500	0.246613
$738$	0	0
$739$	46.5019	1.71060	0.855300	−	0.518134i	$-0.173373\pi$
$739$	46.5019	1.71060	0.855300	+	0.518134i	$0.173373\pi$
$740$	0	0
$741$	−25.2617	−0.928013
$742$	0	0
$743$	−11.8165	−0.433507	−0.216753	−	0.976226i	$-0.569547\pi$
$743$	−11.8165	−0.433507	−0.216753	+	0.976226i	$0.569547\pi$
$744$	0	0
$745$	−13.3091	−0.487608
$746$	0	0
$747$	14.1251	0.516812
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−9.47999	−0.345930	−0.172965	−	0.984928i	$-0.555335\pi$
$751$	−9.47999	−0.345930	−0.172965	+	0.984928i	$0.555335\pi$
$752$	0	0
$753$	−56.9833	−2.07659
$754$	0	0
$755$	8.24156	0.299941
$756$	0	0
$757$	−5.62674	−0.204507	−0.102254	−	0.994758i	$-0.532605\pi$
$757$	−5.62674	−0.204507	−0.102254	+	0.994758i	$0.532605\pi$
$758$	0	0
$759$	8.51861	0.309206
$760$	0	0
$761$	−13.4892	−0.488982	−0.244491	−	0.969652i	$-0.578621\pi$
$761$	−13.4892	−0.488982	−0.244491	+	0.969652i	$0.578621\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−59.0649	−2.13550
$766$	0	0
$767$	0.435209	0.0157145
$768$	0	0
$769$	19.5325	0.704359	0.352179	−	0.935933i	$-0.385441\pi$
$769$	19.5325	0.704359	0.352179	+	0.935933i	$0.385441\pi$
$770$	0	0
$771$	87.8045	3.16220
$772$	0	0
$773$	−24.8758	−0.894722	−0.447361	−	0.894354i	$-0.647636\pi$
$773$	−24.8758	−0.894722	−0.447361	+	0.894354i	$0.647636\pi$
$774$	0	0
$775$	−7.06257	−0.253695
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.73999	0.169828
$780$	0	0
$781$	−6.44668	−0.230680
$782$	0	0
$783$	−10.6729	−0.381420
$784$	0	0
$785$	−11.3691	−0.405782
$786$	0	0
$787$	−38.2817	−1.36460	−0.682298	−	0.731074i	$-0.739019\pi$
$787$	−38.2817	−1.36460	−0.682298	+	0.731074i	$0.739019\pi$
$788$	0	0
$789$	70.4941	2.50966
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−13.2749	−0.471404
$794$	0	0
$795$	−12.1150	−0.429675
$796$	0	0
$797$	−24.7816	−0.877808	−0.438904	−	0.898534i	$-0.644633\pi$
$797$	−24.7816	−0.877808	−0.438904	+	0.898534i	$0.644633\pi$
$798$	0	0
$799$	−4.09105	−0.144731
$800$	0	0
$801$	−114.988	−4.06291
$802$	0	0
$803$	−5.94304	−0.209725
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−69.9388	−2.46196
$808$	0	0
$809$	31.4982	1.10742	0.553709	−	0.832710i	$-0.313212\pi$
$809$	31.4982	1.10742	0.553709	+	0.832710i	$0.313212\pi$
$810$	0	0
$811$	6.69069	0.234942	0.117471	−	0.993076i	$-0.462521\pi$
$811$	6.69069	0.234942	0.117471	+	0.993076i	$0.462521\pi$
$812$	0	0
$813$	−102.907	−3.60909
$814$	0	0
$815$	−0.961377	−0.0336756
$816$	0	0
$817$	−29.8849	−1.04554
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.5966	−0.893327	−0.446664	−	0.894702i	$-0.647388\pi$
$821$	−25.5966	−0.893327	−0.446664	+	0.894702i	$0.647388\pi$
$822$	0	0
$823$	33.9136	1.18215	0.591077	−	0.806615i	$-0.298703\pi$
$823$	33.9136	1.18215	0.591077	+	0.806615i	$0.298703\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	1.58891	0.0552517	0.0276259	−	0.999618i	$-0.491205\pi$
$827$	1.58891	0.0552517	0.0276259	+	0.999618i	$0.491205\pi$
$828$	0	0
$829$	8.12004	0.282021	0.141010	−	0.990008i	$-0.454965\pi$
$829$	8.12004	0.282021	0.141010	+	0.990008i	$0.454965\pi$
$830$	0	0
$831$	57.0660	1.97960
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.39966	−0.221469
$836$	0	0
$837$	−132.048	−4.56425
$838$	0	0
$839$	−55.1748	−1.90484	−0.952422	−	0.304781i	$-0.901417\pi$
$839$	−55.1748	−1.90484	−0.952422	+	0.304781i	$0.901417\pi$
$840$	0	0
$841$	−28.6741	−0.988763
$842$	0	0
$843$	79.2859	2.73075
$844$	0	0
$845$	10.5325	0.362328
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−74.9449	−2.57210
$850$	0	0
$851$	−36.2502	−1.24264
$852$	0	0
$853$	−9.85420	−0.337401	−0.168701	−	0.985667i	$-0.553957\pi$
$853$	−9.85420	−0.337401	−0.168701	+	0.985667i	$0.553957\pi$
$854$	0	0
$855$	−40.3412	−1.37964
$856$	0	0
$857$	−46.3266	−1.58249	−0.791244	−	0.611501i	$-0.790566\pi$
$857$	−46.3266	−1.58249	−0.791244	+	0.611501i	$0.790566\pi$
$858$	0	0
$859$	−37.2469	−1.27085	−0.635425	−	0.772163i	$-0.719175\pi$
$859$	−37.2469	−1.27085	−0.635425	+	0.772163i	$0.719175\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.0586	0.580683	0.290341	−	0.956923i	$-0.406231\pi$
$863$	17.0586	0.580683	0.290341	+	0.956923i	$0.406231\pi$
$864$	0	0
$865$	16.5108	0.561385
$866$	0	0
$867$	105.730	3.59077
$868$	0	0
$869$	−8.85832	−0.300498
$870$	0	0
$871$	17.8404	0.604500
$872$	0	0
$873$	−67.0649	−2.26980
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−49.6141	−1.67535	−0.837675	−	0.546169i	$-0.816086\pi$
$877$	−49.6141	−1.67535	−0.837675	+	0.546169i	$0.816086\pi$
$878$	0	0
$879$	−73.0545	−2.46407
$880$	0	0
$881$	−7.85832	−0.264754	−0.132377	−	0.991199i	$-0.542261\pi$
$881$	−7.85832	−0.264754	−0.132377	+	0.991199i	$0.542261\pi$
$882$	0	0
$883$	−9.69759	−0.326350	−0.163175	−	0.986597i	$-0.552174\pi$
$883$	−9.69759	−0.326350	−0.163175	+	0.986597i	$0.552174\pi$
$884$	0	0
$885$	0.939980	0.0315971
$886$	0	0
$887$	−14.1112	−0.473807	−0.236904	−	0.971533i	$-0.576133\pi$
$887$	−14.1112	−0.473807	−0.236904	+	0.971533i	$0.576133\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−22.3426	−0.748506
$892$	0	0
$893$	−2.79418	−0.0935037
$894$	0	0
$895$	−0.312436	−0.0104436
$896$	0	0
$897$	22.6999	0.757928
$898$	0	0
$899$	4.03160	0.134461
$900$	0	0
$901$	24.7816	0.825593
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.5925	−0.385347
$906$	0	0
$907$	−21.3914	−0.710288	−0.355144	−	0.934812i	$-0.615568\pi$
$907$	−21.3914	−0.710288	−0.355144	+	0.934812i	$0.615568\pi$
$908$	0	0
$909$	21.8800	0.725712
$910$	0	0
$911$	33.8568	1.12173	0.560864	−	0.827908i	$-0.310469\pi$
$911$	33.8568	1.12173	0.560864	+	0.827908i	$0.310469\pi$
$912$	0	0
$913$	−0.978360	−0.0323790
$914$	0	0
$915$	−28.6715	−0.947852
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−56.2943	−1.85698	−0.928489	−	0.371360i	$-0.878892\pi$
$919$	−56.2943	−1.85698	−0.928489	+	0.371360i	$0.878892\pi$
$920$	0	0
$921$	70.6958	2.32951
$922$	0	0
$923$	−17.1787	−0.565445
$924$	0	0
$925$	8.51082	0.279834
$926$	0	0
$927$	8.30101	0.272641
$928$	0	0
$929$	−24.9232	−0.817705	−0.408853	−	0.912600i	$-0.634071\pi$
$929$	−24.9232	−0.817705	−0.408853	+	0.912600i	$0.634071\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−37.1033	−1.21471
$934$	0	0
$935$	4.09105	0.133792
$936$	0	0
$937$	−5.02164	−0.164050	−0.0820249	−	0.996630i	$-0.526139\pi$
$937$	−5.02164	−0.164050	−0.0820249	+	0.996630i	$0.526139\pi$
$938$	0	0
$939$	88.3586	2.88347
$940$	0	0
$941$	−32.9016	−1.07256	−0.536281	−	0.844040i	$-0.680171\pi$
$941$	−32.9016	−1.07256	−0.536281	+	0.844040i	$0.680171\pi$
$942$	0	0
$943$	−4.25931	−0.138702
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.1845	−0.785892	−0.392946	−	0.919562i	$-0.628544\pi$
$947$	−24.1845	−0.785892	−0.392946	+	0.919562i	$0.628544\pi$
$948$	0	0
$949$	−15.8367	−0.514080
$950$	0	0
$951$	−9.97465	−0.323450
$952$	0	0
$953$	53.8800	1.74534	0.872672	−	0.488308i	$-0.162386\pi$
$953$	53.8800	1.74534	0.872672	+	0.488308i	$0.162386\pi$
$954$	0	0
$955$	7.33963	0.237505
$956$	0	0
$957$	1.14168	0.0369053
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.8800	0.609031
$962$	0	0
$963$	−76.5914	−2.46812
$964$	0	0
$965$	7.02164	0.226035
$966$	0	0
$967$	25.7595	0.828369	0.414184	−	0.910193i	$-0.364067\pi$
$967$	25.7595	0.828369	0.414184	+	0.910193i	$0.364067\pi$
$968$	0	0
$969$	111.606	3.58531
$970$	0	0
$971$	−30.9801	−0.994200	−0.497100	−	0.867693i	$-0.665602\pi$
$971$	−30.9801	−0.994200	−0.497100	+	0.867693i	$0.665602\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5.32948	−0.170680
$976$	0	0
$977$	−24.1200	−0.771668	−0.385834	−	0.922568i	$-0.626086\pi$
$977$	−24.1200	−0.771668	−0.385834	+	0.922568i	$0.626086\pi$
$978$	0	0
$979$	7.96450	0.254547
$980$	0	0
$981$	−115.683	−3.69348
$982$	0	0
$983$	24.2794	0.774391	0.387196	−	0.921998i	$-0.373444\pi$
$983$	24.2794	0.774391	0.387196	+	0.921998i	$0.373444\pi$
$984$	0	0
$985$	−0.630860	−0.0201009
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.8542	0.853914
$990$	0	0
$991$	−60.0970	−1.90904	−0.954522	−	0.298141i	$-0.903634\pi$
$991$	−60.0970	−1.90904	−0.954522	+	0.298141i	$0.903634\pi$
$992$	0	0
$993$	−33.2233	−1.05431
$994$	0	0
$995$	−0.554109	−0.0175664
$996$	0	0
$997$	−44.2450	−1.40125	−0.700626	−	0.713528i	$-0.747096\pi$
$997$	−44.2450	−1.40125	−0.700626	+	0.713528i	$0.747096\pi$
$998$	0	0
$999$	159.126	5.03452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7840.2.a.cd.1.6		6
4.3	odd	2	inner	7840.2.a.cd.1.1		6
7.3	odd	6		1120.2.q.k.961.6	yes	12
7.5	odd	6		1120.2.q.k.641.6	yes	12
7.6	odd	2		7840.2.a.ce.1.1		6
28.3	even	6		1120.2.q.k.961.1	yes	12
28.19	even	6		1120.2.q.k.641.1	✓	12
28.27	even	2		7840.2.a.ce.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.q.k.641.1	✓	12	28.19	even	6
1120.2.q.k.641.6	yes	12	7.5	odd	6
1120.2.q.k.961.1	yes	12	28.3	even	6
1120.2.q.k.961.6	yes	12	7.3	odd	6
7840.2.a.cd.1.1		6	4.3	odd	2	inner
7840.2.a.cd.1.6		6	1.1	even	1	trivial
7840.2.a.ce.1.1		6	7.6	odd	2
7840.2.a.ce.1.6		6	28.27	even	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:	\( N \)	\(=\)	\( 7840 = 2^{5} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7840.a (trivial)

\(f(q)\)	\(=\)	\(q+3.39276 q^{3} -1.00000 q^{5} +8.51082 q^{9} +O(q^{10})\)	\(q+3.39276 q^{3} -1.00000 q^{5} +8.51082 q^{9} -0.589491 q^{11} -1.57084 q^{13} -3.39276 q^{15} +6.93998 q^{17} +4.73999 q^{19} -4.25931 q^{23} +1.00000 q^{25} +18.6969 q^{27} -0.570840 q^{29} -7.06257 q^{31} -2.00000 q^{33} +8.51082 q^{37} -5.32948 q^{39} +1.00000 q^{41} -6.30483 q^{43} -8.51082 q^{45} -0.589491 q^{47} +23.5457 q^{51} +3.57084 q^{53} +0.589491 q^{55} +16.0817 q^{57} -0.277055 q^{59} +8.45080 q^{61} +1.57084 q^{65} -11.3573 q^{67} -14.4508 q^{69} +10.9360 q^{71} +10.0817 q^{73} +3.39276 q^{75} +15.0271 q^{79} +37.9016 q^{81} +1.65967 q^{83} -6.93998 q^{85} -1.93672 q^{87} -13.5108 q^{89} -23.9616 q^{93} -4.73999 q^{95} -7.87996 q^{97} -5.01705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} + 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^5 + 10 * q^9	\( 6 q - 6 q^{5} + 10 q^{9} - 2 q^{13} + 8 q^{17} + 6 q^{25} + 4 q^{29} - 12 q^{33} + 10 q^{37} + 6 q^{41} - 10 q^{45} + 14 q^{53} + 48 q^{57} - 24 q^{61} + 2 q^{65} - 12 q^{69} + 12 q^{73} + 78 q^{81} - 8 q^{85} - 40 q^{89} - 28 q^{93} + 20 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 + 10 * q^9 - 2 * q^13 + 8 * q^17 + 6 * q^25 + 4 * q^29 - 12 * q^33 + 10 * q^37 + 6 * q^41 - 10 * q^45 + 14 * q^53 + 48 * q^57 - 24 * q^61 + 2 * q^65 - 12 * q^69 + 12 * q^73 + 78 * q^81 - 8 * q^85 - 40 * q^89 - 28 * q^93 + 20 * q^97

Introduction

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