LMFDB - Embedded newform 8712.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, 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("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.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:	$69.5656702409$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{5} +4.23607 q^{7} +O(q^{10})$	$q-0.381966 q^{5} +4.23607 q^{7} -2.23607 q^{13} -1.85410 q^{17} -0.381966 q^{19} -6.23607 q^{23} -4.85410 q^{25} +0.472136 q^{29} +8.61803 q^{31} -1.61803 q^{35} +3.76393 q^{37} -5.00000 q^{41} -11.4721 q^{43} -0.145898 q^{47} +10.9443 q^{49} +9.56231 q^{53} -14.5623 q^{59} +6.85410 q^{61} +0.854102 q^{65} -4.56231 q^{67} -12.5623 q^{71} -3.70820 q^{73} -14.2361 q^{79} -1.94427 q^{83} +0.708204 q^{85} -3.47214 q^{89} -9.47214 q^{91} +0.145898 q^{95} -6.85410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 3 * q^5 + 4 * q^7	$ 2 q - 3 q^{5} + 4 q^{7} + 3 q^{17} - 3 q^{19} - 8 q^{23} - 3 q^{25} - 8 q^{29} + 15 q^{31} - q^{35} + 12 q^{37} - 10 q^{41} - 14 q^{43} - 7 q^{47} + 4 q^{49} - q^{53} - 9 q^{59} + 7 q^{61} - 5 q^{65} + 11 q^{67} - 5 q^{71} + 6 q^{73} - 24 q^{79} + 14 q^{83} - 12 q^{85} + 2 q^{89} - 10 q^{91} + 7 q^{95} - 7 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 + 4 * q^7 + 3 * q^17 - 3 * q^19 - 8 * q^23 - 3 * q^25 - 8 * q^29 + 15 * q^31 - q^35 + 12 * q^37 - 10 * q^41 - 14 * q^43 - 7 * q^47 + 4 * q^49 - q^53 - 9 * q^59 + 7 * q^61 - 5 * q^65 + 11 * q^67 - 5 * q^71 + 6 * q^73 - 24 * q^79 + 14 * q^83 - 12 * q^85 + 2 * q^89 - 10 * q^91 + 7 * q^95 - 7 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.381966	−0.170820	−0.0854102	−	0.996346i	$-0.527220\pi$
$5$	−0.381966	−0.170820	−0.0854102	+	0.996346i	$0.527220\pi$
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.23607	−0.620174	−0.310087	−	0.950708i	$-0.600358\pi$
$13$	−2.23607	−0.620174	−0.310087	+	0.950708i	$0.600358\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.85410	−0.449686	−0.224843	−	0.974395i	$-0.572187\pi$
$17$	−1.85410	−0.449686	−0.224843	+	0.974395i	$0.572187\pi$
$18$	0	0
$19$	−0.381966	−0.0876290	−0.0438145	−	0.999040i	$-0.513951\pi$
$19$	−0.381966	−0.0876290	−0.0438145	+	0.999040i	$0.513951\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.23607	−1.30031	−0.650155	−	0.759802i	$-0.725296\pi$
$23$	−6.23607	−1.30031	−0.650155	+	0.759802i	$0.725296\pi$
$24$	0	0
$25$	−4.85410	−0.970820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	8.61803	1.54784	0.773922	−	0.633281i	$-0.218292\pi$
$31$	8.61803	1.54784	0.773922	+	0.633281i	$0.218292\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.61803	−0.273498
$36$	0	0
$37$	3.76393	0.618787	0.309393	−	0.950934i	$-0.399874\pi$
$37$	3.76393	0.618787	0.309393	+	0.950934i	$0.399874\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−11.4721	−1.74948	−0.874742	−	0.484589i	$-0.838969\pi$
$43$	−11.4721	−1.74948	−0.874742	+	0.484589i	$0.838969\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.145898	−0.0212814	−0.0106407	−	0.999943i	$-0.503387\pi$
$47$	−0.145898	−0.0212814	−0.0106407	+	0.999943i	$0.503387\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.56231	1.31348	0.656742	−	0.754115i	$-0.271934\pi$
$53$	9.56231	1.31348	0.656742	+	0.754115i	$0.271934\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.5623	−1.89585	−0.947925	−	0.318493i	$-0.896823\pi$
$59$	−14.5623	−1.89585	−0.947925	+	0.318493i	$0.896823\pi$
$60$	0	0
$61$	6.85410	0.877578	0.438789	−	0.898590i	$-0.355408\pi$
$61$	6.85410	0.877578	0.438789	+	0.898590i	$0.355408\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.854102	0.105938
$66$	0	0
$67$	−4.56231	−0.557374	−0.278687	−	0.960382i	$-0.589899\pi$
$67$	−4.56231	−0.557374	−0.278687	+	0.960382i	$0.589899\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.5623	−1.49087	−0.745436	−	0.666578i	$-0.767759\pi$
$71$	−12.5623	−1.49087	−0.745436	+	0.666578i	$0.767759\pi$
$72$	0	0
$73$	−3.70820	−0.434012	−0.217006	−	0.976170i	$-0.569629\pi$
$73$	−3.70820	−0.434012	−0.217006	+	0.976170i	$0.569629\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.2361	−1.60168	−0.800841	−	0.598877i	$-0.795614\pi$
$79$	−14.2361	−1.60168	−0.800841	+	0.598877i	$0.795614\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.94427	−0.213412	−0.106706	−	0.994291i	$-0.534030\pi$
$83$	−1.94427	−0.213412	−0.106706	+	0.994291i	$0.534030\pi$
$84$	0	0
$85$	0.708204	0.0768155
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.47214	−0.368046	−0.184023	−	0.982922i	$-0.558912\pi$
$89$	−3.47214	−0.368046	−0.184023	+	0.982922i	$0.558912\pi$
$90$	0	0
$91$	−9.47214	−0.992950
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.145898	0.0149688
$96$	0	0
$97$	−6.85410	−0.695929	−0.347964	−	0.937508i	$-0.613127\pi$
$97$	−6.85410	−0.695929	−0.347964	+	0.937508i	$0.613127\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.527864	0.0525244	0.0262622	−	0.999655i	$-0.491640\pi$
$101$	0.527864	0.0525244	0.0262622	+	0.999655i	$0.491640\pi$
$102$	0	0
$103$	−3.52786	−0.347611	−0.173805	−	0.984780i	$-0.555606\pi$
$103$	−3.52786	−0.347611	−0.173805	+	0.984780i	$0.555606\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.9443	1.15470	0.577348	−	0.816498i	$-0.304088\pi$
$107$	11.9443	1.15470	0.577348	+	0.816498i	$0.304088\pi$
$108$	0	0
$109$	−17.8885	−1.71341	−0.856706	−	0.515805i	$-0.827493\pi$
$109$	−17.8885	−1.71341	−0.856706	+	0.515805i	$0.827493\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.2361	−1.71550	−0.857752	−	0.514063i	$-0.828140\pi$
$113$	−18.2361	−1.71550	−0.857752	+	0.514063i	$0.828140\pi$
$114$	0	0
$115$	2.38197	0.222119
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.85410	−0.719984
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.76393	0.336656
$126$	0	0
$127$	−0.763932	−0.0677880	−0.0338940	−	0.999425i	$-0.510791\pi$
$127$	−0.763932	−0.0677880	−0.0338940	+	0.999425i	$0.510791\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.32624	0.203244	0.101622	−	0.994823i	$-0.467597\pi$
$131$	2.32624	0.203244	0.101622	+	0.994823i	$0.467597\pi$
$132$	0	0
$133$	−1.61803	−0.140301
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.47214	−0.467516	−0.233758	−	0.972295i	$-0.575102\pi$
$137$	−5.47214	−0.467516	−0.233758	+	0.972295i	$0.575102\pi$
$138$	0	0
$139$	18.6180	1.57916	0.789581	−	0.613647i	$-0.210298\pi$
$139$	18.6180	1.57916	0.789581	+	0.613647i	$0.210298\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.180340	−0.0149764
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.47214	0.612141	0.306071	−	0.952009i	$-0.400986\pi$
$149$	7.47214	0.612141	0.306071	+	0.952009i	$0.400986\pi$
$150$	0	0
$151$	13.4164	1.09181	0.545906	−	0.837846i	$-0.316186\pi$
$151$	13.4164	1.09181	0.545906	+	0.837846i	$0.316186\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.29180	−0.264403
$156$	0	0
$157$	16.1803	1.29133	0.645666	−	0.763620i	$-0.276580\pi$
$157$	16.1803	1.29133	0.645666	+	0.763620i	$0.276580\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−26.4164	−2.08190
$162$	0	0
$163$	−15.3820	−1.20481	−0.602404	−	0.798191i	$-0.705790\pi$
$163$	−15.3820	−1.20481	−0.602404	+	0.798191i	$0.705790\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.3262	1.80504	0.902519	−	0.430650i	$-0.141715\pi$
$167$	23.3262	1.80504	0.902519	+	0.430650i	$0.141715\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.0344	−1.29510	−0.647552	−	0.762021i	$-0.724207\pi$
$173$	−17.0344	−1.29510	−0.647552	+	0.762021i	$0.724207\pi$
$174$	0	0
$175$	−20.5623	−1.55436
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.4164	1.67548	0.837740	−	0.546069i	$-0.183876\pi$
$179$	22.4164	1.67548	0.837740	+	0.546069i	$0.183876\pi$
$180$	0	0
$181$	14.8885	1.10666	0.553328	−	0.832963i	$-0.313357\pi$
$181$	14.8885	1.10666	0.553328	+	0.832963i	$0.313357\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.43769	−0.105701
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.76393	−0.417063	−0.208532	−	0.978016i	$-0.566868\pi$
$191$	−5.76393	−0.417063	−0.208532	+	0.978016i	$0.566868\pi$
$192$	0	0
$193$	0.326238	0.0234831	0.0117416	−	0.999931i	$-0.496262\pi$
$193$	0.326238	0.0234831	0.0117416	+	0.999931i	$0.496262\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.14590	−0.580371	−0.290186	−	0.956970i	$-0.593717\pi$
$197$	−8.14590	−0.580371	−0.290186	+	0.956970i	$0.593717\pi$
$198$	0	0
$199$	9.47214	0.671462	0.335731	−	0.941958i	$-0.391017\pi$
$199$	9.47214	0.671462	0.335731	+	0.941958i	$0.391017\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	1.90983	0.133388
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−3.38197	−0.232824	−0.116412	−	0.993201i	$-0.537139\pi$
$211$	−3.38197	−0.232824	−0.116412	+	0.993201i	$0.537139\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.38197	0.298848
$216$	0	0
$217$	36.5066	2.47823
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.14590	0.278883
$222$	0	0
$223$	−12.5279	−0.838928	−0.419464	−	0.907772i	$-0.637782\pi$
$223$	−12.5279	−0.838928	−0.419464	+	0.907772i	$0.637782\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.5967	−1.36705	−0.683527	−	0.729925i	$-0.739555\pi$
$227$	−20.5967	−1.36705	−0.683527	+	0.729925i	$0.739555\pi$
$228$	0	0
$229$	−11.8885	−0.785617	−0.392809	−	0.919620i	$-0.628496\pi$
$229$	−11.8885	−0.785617	−0.392809	+	0.919620i	$0.628496\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.0902	−1.31615	−0.658075	−	0.752952i	$-0.728629\pi$
$233$	−20.0902	−1.31615	−0.658075	+	0.752952i	$0.728629\pi$
$234$	0	0
$235$	0.0557281	0.00363530
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.0344	0.907813	0.453906	−	0.891049i	$-0.350030\pi$
$239$	14.0344	0.907813	0.453906	+	0.891049i	$0.350030\pi$
$240$	0	0
$241$	30.0689	1.93691	0.968454	−	0.249194i	$-0.0801657\pi$
$241$	30.0689	1.93691	0.968454	+	0.249194i	$0.0801657\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.18034	−0.267072
$246$	0	0
$247$	0.854102	0.0543452
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.14590	−0.135448	−0.0677239	−	0.997704i	$-0.521574\pi$
$251$	−2.14590	−0.135448	−0.0677239	+	0.997704i	$0.521574\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.7984	−1.11023	−0.555116	−	0.831773i	$-0.687326\pi$
$257$	−17.7984	−1.11023	−0.555116	+	0.831773i	$0.687326\pi$
$258$	0	0
$259$	15.9443	0.990729
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.8541	0.669293	0.334646	−	0.942344i	$-0.391383\pi$
$263$	10.8541	0.669293	0.334646	+	0.942344i	$0.391383\pi$
$264$	0	0
$265$	−3.65248	−0.224370
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.41641	0.0863599	0.0431800	−	0.999067i	$-0.486251\pi$
$269$	1.41641	0.0863599	0.0431800	+	0.999067i	$0.486251\pi$
$270$	0	0
$271$	−17.6180	−1.07022	−0.535110	−	0.844783i	$-0.679730\pi$
$271$	−17.6180	−1.07022	−0.535110	+	0.844783i	$0.679730\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.14590	0.128935	0.0644673	−	0.997920i	$-0.479465\pi$
$277$	2.14590	0.128935	0.0644673	+	0.997920i	$0.479465\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.5967	1.04973	0.524867	−	0.851184i	$-0.324115\pi$
$281$	17.5967	1.04973	0.524867	+	0.851184i	$0.324115\pi$
$282$	0	0
$283$	15.7082	0.933756	0.466878	−	0.884322i	$-0.345379\pi$
$283$	15.7082	0.933756	0.466878	+	0.884322i	$0.345379\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−21.1803	−1.25024
$288$	0	0
$289$	−13.5623	−0.797783
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.3607	1.59843	0.799214	−	0.601047i	$-0.205249\pi$
$293$	27.3607	1.59843	0.799214	+	0.601047i	$0.205249\pi$
$294$	0	0
$295$	5.56231	0.323850
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.9443	0.806418
$300$	0	0
$301$	−48.5967	−2.80107
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.61803	−0.149908
$306$	0	0
$307$	−10.7984	−0.616296	−0.308148	−	0.951338i	$-0.599709\pi$
$307$	−10.7984	−0.616296	−0.308148	+	0.951338i	$0.599709\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−14.2361	−0.807253	−0.403627	−	0.914924i	$-0.632251\pi$
$311$	−14.2361	−0.807253	−0.403627	+	0.914924i	$0.632251\pi$
$312$	0	0
$313$	8.88854	0.502410	0.251205	−	0.967934i	$-0.419173\pi$
$313$	8.88854	0.502410	0.251205	+	0.967934i	$0.419173\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.94427	0.109201	0.0546006	−	0.998508i	$-0.482611\pi$
$317$	1.94427	0.109201	0.0546006	+	0.998508i	$0.482611\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.708204	0.0394055
$324$	0	0
$325$	10.8541	0.602077
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.618034	−0.0340733
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.74265	0.0952109
$336$	0	0
$337$	10.2918	0.560630	0.280315	−	0.959908i	$-0.409561\pi$
$337$	10.2918	0.560630	0.280315	+	0.959908i	$0.409561\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−12.2361	−0.654982	−0.327491	−	0.944854i	$-0.606203\pi$
$349$	−12.2361	−0.654982	−0.327491	+	0.944854i	$0.606203\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.4721	−0.770274	−0.385137	−	0.922859i	$-0.625846\pi$
$353$	−14.4721	−0.770274	−0.385137	+	0.922859i	$0.625846\pi$
$354$	0	0
$355$	4.79837	0.254671
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.70820	0.406823	0.203412	−	0.979093i	$-0.434797\pi$
$359$	7.70820	0.406823	0.203412	+	0.979093i	$0.434797\pi$
$360$	0	0
$361$	−18.8541	−0.992321
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.41641	0.0741382
$366$	0	0
$367$	−21.3262	−1.11322	−0.556610	−	0.830774i	$-0.687898\pi$
$367$	−21.3262	−1.11322	−0.556610	+	0.830774i	$0.687898\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	40.5066	2.10300
$372$	0	0
$373$	1.94427	0.100671	0.0503353	−	0.998732i	$-0.483971\pi$
$373$	1.94427	0.100671	0.0503353	+	0.998732i	$0.483971\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.05573	−0.0543728
$378$	0	0
$379$	5.18034	0.266096	0.133048	−	0.991110i	$-0.457524\pi$
$379$	5.18034	0.266096	0.133048	+	0.991110i	$0.457524\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.2918	0.985765	0.492882	−	0.870096i	$-0.335943\pi$
$383$	19.2918	0.985765	0.492882	+	0.870096i	$0.335943\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−34.7984	−1.76435	−0.882174	−	0.470924i	$-0.843921\pi$
$389$	−34.7984	−1.76435	−0.882174	+	0.470924i	$0.843921\pi$
$390$	0	0
$391$	11.5623	0.584731
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.43769	0.273600
$396$	0	0
$397$	−21.2918	−1.06860	−0.534302	−	0.845293i	$-0.679426\pi$
$397$	−21.2918	−1.06860	−0.534302	+	0.845293i	$0.679426\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.4508	1.67046	0.835228	−	0.549904i	$-0.185336\pi$
$401$	33.4508	1.67046	0.835228	+	0.549904i	$0.185336\pi$
$402$	0	0
$403$	−19.2705	−0.959932
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−33.8885	−1.67568	−0.837840	−	0.545915i	$-0.816182\pi$
$409$	−33.8885	−1.67568	−0.837840	+	0.545915i	$0.816182\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−61.6869	−3.03541
$414$	0	0
$415$	0.742646	0.0364550
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.4508	0.999089	0.499545	−	0.866288i	$-0.333501\pi$
$419$	20.4508	0.999089	0.499545	+	0.866288i	$0.333501\pi$
$420$	0	0
$421$	22.7426	1.10841	0.554204	−	0.832381i	$-0.313023\pi$
$421$	22.7426	1.10841	0.554204	+	0.832381i	$0.313023\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.00000	0.436564
$426$	0	0
$427$	29.0344	1.40508
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.9787	−1.68487	−0.842433	−	0.538802i	$-0.818877\pi$
$431$	−34.9787	−1.68487	−0.842433	+	0.538802i	$0.818877\pi$
$432$	0	0
$433$	−30.9443	−1.48709	−0.743543	−	0.668688i	$-0.766856\pi$
$433$	−30.9443	−1.48709	−0.743543	+	0.668688i	$0.766856\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.38197	0.113945
$438$	0	0
$439$	−29.4721	−1.40663	−0.703314	−	0.710879i	$-0.748297\pi$
$439$	−29.4721	−1.40663	−0.703314	+	0.710879i	$0.748297\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.1803	−1.43391	−0.716956	−	0.697119i	$-0.754465\pi$
$443$	−30.1803	−1.43391	−0.716956	+	0.697119i	$0.754465\pi$
$444$	0	0
$445$	1.32624	0.0628697
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−35.3050	−1.66614	−0.833072	−	0.553165i	$-0.813420\pi$
$449$	−35.3050	−1.66614	−0.833072	+	0.553165i	$0.813420\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.61803	0.169616
$456$	0	0
$457$	−1.74265	−0.0815175	−0.0407588	−	0.999169i	$-0.512978\pi$
$457$	−1.74265	−0.0815175	−0.0407588	+	0.999169i	$0.512978\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.0902	−1.02884	−0.514421	−	0.857538i	$-0.671993\pi$
$461$	−22.0902	−1.02884	−0.514421	+	0.857538i	$0.671993\pi$
$462$	0	0
$463$	−31.4508	−1.46164	−0.730822	−	0.682568i	$-0.760863\pi$
$463$	−31.4508	−1.46164	−0.730822	+	0.682568i	$0.760863\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.41641	−0.296916	−0.148458	−	0.988919i	$-0.547431\pi$
$467$	−6.41641	−0.296916	−0.148458	+	0.988919i	$0.547431\pi$
$468$	0	0
$469$	−19.3262	−0.892403
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.85410	0.0850720
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.7984	1.49860	0.749298	−	0.662233i	$-0.230391\pi$
$479$	32.7984	1.49860	0.749298	+	0.662233i	$0.230391\pi$
$480$	0	0
$481$	−8.41641	−0.383755
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.61803	0.118879
$486$	0	0
$487$	−25.9443	−1.17565	−0.587824	−	0.808989i	$-0.700015\pi$
$487$	−25.9443	−1.17565	−0.587824	+	0.808989i	$0.700015\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.9098	1.21442	0.607212	−	0.794540i	$-0.292288\pi$
$491$	26.9098	1.21442	0.607212	+	0.794540i	$0.292288\pi$
$492$	0	0
$493$	−0.875388	−0.0394255
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−53.2148	−2.38701
$498$	0	0
$499$	1.90983	0.0854957	0.0427479	−	0.999086i	$-0.486389\pi$
$499$	1.90983	0.0854957	0.0427479	+	0.999086i	$0.486389\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.23607	0.144289	0.0721446	−	0.997394i	$-0.477016\pi$
$503$	3.23607	0.144289	0.0721446	+	0.997394i	$0.477016\pi$
$504$	0	0
$505$	−0.201626	−0.00897224
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.8541	0.702721	0.351360	−	0.936240i	$-0.385719\pi$
$509$	15.8541	0.702721	0.351360	+	0.936240i	$0.385719\pi$
$510$	0	0
$511$	−15.7082	−0.694890
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.34752	0.0593790
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	0	0
$523$	6.43769	0.281501	0.140750	−	0.990045i	$-0.455049\pi$
$523$	6.43769	0.281501	0.140750	+	0.990045i	$0.455049\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.9787	−0.696044
$528$	0	0
$529$	15.8885	0.690806
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.1803	0.484274
$534$	0	0
$535$	−4.56231	−0.197246
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.03444	0.302434	0.151217	−	0.988501i	$-0.451681\pi$
$541$	7.03444	0.302434	0.151217	+	0.988501i	$0.451681\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.83282	0.292686
$546$	0	0
$547$	16.1459	0.690349	0.345174	−	0.938539i	$-0.387820\pi$
$547$	16.1459	0.690349	0.345174	+	0.938539i	$0.387820\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.180340	−0.00768274
$552$	0	0
$553$	−60.3050	−2.56443
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.3262	0.734136	0.367068	−	0.930194i	$-0.380362\pi$
$557$	17.3262	0.734136	0.367068	+	0.930194i	$0.380362\pi$
$558$	0	0
$559$	25.6525	1.08498
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.2361	1.27430	0.637149	−	0.770741i	$-0.280114\pi$
$563$	30.2361	1.27430	0.637149	+	0.770741i	$0.280114\pi$
$564$	0	0
$565$	6.96556	0.293043
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−33.7082	−1.41312	−0.706561	−	0.707652i	$-0.749754\pi$
$569$	−33.7082	−1.41312	−0.706561	+	0.707652i	$0.749754\pi$
$570$	0	0
$571$	23.1459	0.968626	0.484313	−	0.874895i	$-0.339070\pi$
$571$	23.1459	0.968626	0.484313	+	0.874895i	$0.339070\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	30.2705	1.26237
$576$	0	0
$577$	−28.1803	−1.17316	−0.586581	−	0.809890i	$-0.699527\pi$
$577$	−28.1803	−1.17316	−0.586581	+	0.809890i	$0.699527\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.23607	−0.341690
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.7082	0.524524	0.262262	−	0.964997i	$-0.415532\pi$
$587$	12.7082	0.524524	0.262262	+	0.964997i	$0.415532\pi$
$588$	0	0
$589$	−3.29180	−0.135636
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−38.5623	−1.58356	−0.791782	−	0.610804i	$-0.790846\pi$
$593$	−38.5623	−1.58356	−0.791782	+	0.610804i	$0.790846\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.2918	0.583947	0.291973	−	0.956426i	$-0.405688\pi$
$599$	14.2918	0.583947	0.291973	+	0.956426i	$0.405688\pi$
$600$	0	0
$601$	13.9443	0.568799	0.284399	−	0.958706i	$-0.408206\pi$
$601$	13.9443	0.568799	0.284399	+	0.958706i	$0.408206\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.6180	−0.512150	−0.256075	−	0.966657i	$-0.582429\pi$
$607$	−12.6180	−0.512150	−0.256075	+	0.966657i	$0.582429\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.326238	0.0131982
$612$	0	0
$613$	−5.81966	−0.235054	−0.117527	−	0.993070i	$-0.537497\pi$
$613$	−5.81966	−0.235054	−0.117527	+	0.993070i	$0.537497\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.5279	0.504353	0.252176	−	0.967681i	$-0.418854\pi$
$617$	12.5279	0.504353	0.252176	+	0.967681i	$0.418854\pi$
$618$	0	0
$619$	−16.5279	−0.664311	−0.332155	−	0.943225i	$-0.607776\pi$
$619$	−16.5279	−0.664311	−0.332155	+	0.943225i	$0.607776\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.7082	−0.589272
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.97871	−0.278260
$630$	0	0
$631$	26.5066	1.05521	0.527605	−	0.849490i	$-0.323090\pi$
$631$	26.5066	1.05521	0.527605	+	0.849490i	$0.323090\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.291796	0.0115796
$636$	0	0
$637$	−24.4721	−0.969621
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.7984	−0.702994	−0.351497	−	0.936189i	$-0.614327\pi$
$641$	−17.7984	−0.702994	−0.351497	+	0.936189i	$0.614327\pi$
$642$	0	0
$643$	−14.9098	−0.587986	−0.293993	−	0.955808i	$-0.594984\pi$
$643$	−14.9098	−0.587986	−0.293993	+	0.955808i	$0.594984\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.3951	1.58810	0.794048	−	0.607855i	$-0.207970\pi$
$647$	40.3951	1.58810	0.794048	+	0.607855i	$0.207970\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.7426	0.694323	0.347162	−	0.937805i	$-0.387146\pi$
$653$	17.7426	0.694323	0.347162	+	0.937805i	$0.387146\pi$
$654$	0	0
$655$	−0.888544	−0.0347183
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−19.1246	−0.744989	−0.372495	−	0.928034i	$-0.621497\pi$
$659$	−19.1246	−0.744989	−0.372495	+	0.928034i	$0.621497\pi$
$660$	0	0
$661$	−8.14590	−0.316839	−0.158419	−	0.987372i	$-0.550640\pi$
$661$	−8.14590	−0.316839	−0.158419	+	0.987372i	$0.550640\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.618034	0.0239663
$666$	0	0
$667$	−2.94427	−0.114003
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8.52786	0.328725	0.164363	−	0.986400i	$-0.447443\pi$
$673$	8.52786	0.328725	0.164363	+	0.986400i	$0.447443\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.4164	0.438768	0.219384	−	0.975639i	$-0.429595\pi$
$677$	11.4164	0.438768	0.219384	+	0.975639i	$0.429595\pi$
$678$	0	0
$679$	−29.0344	−1.11424
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.4853	−1.31954	−0.659772	−	0.751466i	$-0.729347\pi$
$683$	−34.4853	−1.31954	−0.659772	+	0.751466i	$0.729347\pi$
$684$	0	0
$685$	2.09017	0.0798613
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−21.3820	−0.814588
$690$	0	0
$691$	13.5967	0.517245	0.258622	−	0.965979i	$-0.416731\pi$
$691$	13.5967	0.517245	0.258622	+	0.965979i	$0.416731\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.11146	−0.269753
$696$	0	0
$697$	9.27051	0.351146
$698$	0	0
$699$	0	0
$700$	0	0
$701$	32.9230	1.24348	0.621742	−	0.783222i	$-0.286425\pi$
$701$	32.9230	1.24348	0.621742	+	0.783222i	$0.286425\pi$
$702$	0	0
$703$	−1.43769	−0.0542237
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.23607	0.0840960
$708$	0	0
$709$	−10.3262	−0.387810	−0.193905	−	0.981020i	$-0.562115\pi$
$709$	−10.3262	−0.387810	−0.193905	+	0.981020i	$0.562115\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−53.7426	−2.01268
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.1803	1.16283	0.581415	−	0.813607i	$-0.302499\pi$
$719$	31.1803	1.16283	0.581415	+	0.813607i	$0.302499\pi$
$720$	0	0
$721$	−14.9443	−0.556554
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.29180	−0.0851152
$726$	0	0
$727$	−31.7426	−1.17727	−0.588635	−	0.808399i	$-0.700334\pi$
$727$	−31.7426	−1.17727	−0.588635	+	0.808399i	$0.700334\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21.2705	0.786718
$732$	0	0
$733$	−36.0689	−1.33223	−0.666117	−	0.745847i	$-0.732045\pi$
$733$	−36.0689	−1.33223	−0.666117	+	0.745847i	$0.732045\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−9.06888	−0.333604	−0.166802	−	0.985990i	$-0.553344\pi$
$739$	−9.06888	−0.333604	−0.166802	+	0.985990i	$0.553344\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.94427	0.291447	0.145724	−	0.989325i	$-0.453449\pi$
$743$	7.94427	0.291447	0.145724	+	0.989325i	$0.453449\pi$
$744$	0	0
$745$	−2.85410	−0.104566
$746$	0	0
$747$	0	0
$748$	0	0
$749$	50.5967	1.84876
$750$	0	0
$751$	4.09017	0.149252	0.0746262	−	0.997212i	$-0.476224\pi$
$751$	4.09017	0.149252	0.0746262	+	0.997212i	$0.476224\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.12461	−0.186504
$756$	0	0
$757$	−8.52786	−0.309950	−0.154975	−	0.987918i	$-0.549530\pi$
$757$	−8.52786	−0.309950	−0.154975	+	0.987918i	$0.549530\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.94427	0.360480	0.180240	−	0.983623i	$-0.442313\pi$
$761$	9.94427	0.360480	0.180240	+	0.983623i	$0.442313\pi$
$762$	0	0
$763$	−75.7771	−2.74331
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.5623	1.17576
$768$	0	0
$769$	−26.5623	−0.957861	−0.478931	−	0.877853i	$-0.658975\pi$
$769$	−26.5623	−0.957861	−0.478931	+	0.877853i	$0.658975\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.83282	−0.281727	−0.140863	−	0.990029i	$-0.544988\pi$
$773$	−7.83282	−0.281727	−0.140863	+	0.990029i	$0.544988\pi$
$774$	0	0
$775$	−41.8328	−1.50268
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.90983	0.0684268
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.18034	−0.220586
$786$	0	0
$787$	45.0132	1.60455	0.802273	−	0.596958i	$-0.203624\pi$
$787$	45.0132	1.60455	0.802273	+	0.596958i	$0.203624\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−77.2492	−2.74667
$792$	0	0
$793$	−15.3262	−0.544251
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.70820	0.202195	0.101097	−	0.994877i	$-0.467765\pi$
$797$	5.70820	0.202195	0.101097	+	0.994877i	$0.467765\pi$
$798$	0	0
$799$	0.270510	0.00956995
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	10.0902	0.355632
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.2705	−0.361092	−0.180546	−	0.983567i	$-0.557786\pi$
$809$	−10.2705	−0.361092	−0.180546	+	0.983567i	$0.557786\pi$
$810$	0	0
$811$	−0.326238	−0.0114558	−0.00572788	−	0.999984i	$-0.501823\pi$
$811$	−0.326238	−0.0114558	−0.00572788	+	0.999984i	$0.501823\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.87539	0.205806
$816$	0	0
$817$	4.38197	0.153306
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.0000	1.08191	0.540954	−	0.841052i	$-0.318063\pi$
$821$	31.0000	1.08191	0.540954	+	0.841052i	$0.318063\pi$
$822$	0	0
$823$	−41.5410	−1.44803	−0.724014	−	0.689785i	$-0.757705\pi$
$823$	−41.5410	−1.44803	−0.724014	+	0.689785i	$0.757705\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.7082	0.824415	0.412208	−	0.911090i	$-0.364758\pi$
$827$	23.7082	0.824415	0.412208	+	0.911090i	$0.364758\pi$
$828$	0	0
$829$	−24.2148	−0.841014	−0.420507	−	0.907289i	$-0.638148\pi$
$829$	−24.2148	−0.841014	−0.420507	+	0.907289i	$0.638148\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20.2918	−0.703069
$834$	0	0
$835$	−8.90983	−0.308337
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.4721	−1.50082	−0.750412	−	0.660970i	$-0.770145\pi$
$839$	−43.4721	−1.50082	−0.750412	+	0.660970i	$0.770145\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.05573	0.105120
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−23.4721	−0.804614
$852$	0	0
$853$	−28.5279	−0.976775	−0.488388	−	0.872627i	$-0.662415\pi$
$853$	−28.5279	−0.976775	−0.488388	+	0.872627i	$0.662415\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.8885	0.781858	0.390929	−	0.920421i	$-0.372154\pi$
$857$	22.8885	0.781858	0.390929	+	0.920421i	$0.372154\pi$
$858$	0	0
$859$	19.6525	0.670534	0.335267	−	0.942123i	$-0.391174\pi$
$859$	19.6525	0.670534	0.335267	+	0.942123i	$0.391174\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.3050	−0.520987	−0.260493	−	0.965476i	$-0.583885\pi$
$863$	−15.3050	−0.520987	−0.260493	+	0.965476i	$0.583885\pi$
$864$	0	0
$865$	6.50658	0.221230
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	10.2016	0.345669
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15.9443	0.539015
$876$	0	0
$877$	11.8328	0.399566	0.199783	−	0.979840i	$-0.435976\pi$
$877$	11.8328	0.399566	0.199783	+	0.979840i	$0.435976\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.85410	−0.264611	−0.132306	−	0.991209i	$-0.542238\pi$
$881$	−7.85410	−0.264611	−0.132306	+	0.991209i	$0.542238\pi$
$882$	0	0
$883$	33.7771	1.13669	0.568345	−	0.822791i	$-0.307584\pi$
$883$	33.7771	1.13669	0.568345	+	0.822791i	$0.307584\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.1803	0.576859	0.288430	−	0.957501i	$-0.406867\pi$
$887$	17.1803	0.576859	0.288430	+	0.957501i	$0.406867\pi$
$888$	0	0
$889$	−3.23607	−0.108534
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.0557281	0.00186487
$894$	0	0
$895$	−8.56231	−0.286206
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.06888	0.135705
$900$	0	0
$901$	−17.7295	−0.590655
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.68692	−0.189040
$906$	0	0
$907$	5.43769	0.180556	0.0902778	−	0.995917i	$-0.471224\pi$
$907$	5.43769	0.180556	0.0902778	+	0.995917i	$0.471224\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.7082	0.421042	0.210521	−	0.977589i	$-0.432484\pi$
$911$	12.7082	0.421042	0.210521	+	0.977589i	$0.432484\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.85410	0.325411
$918$	0	0
$919$	−2.88854	−0.0952843	−0.0476421	−	0.998864i	$-0.515171\pi$
$919$	−2.88854	−0.0952843	−0.0476421	+	0.998864i	$0.515171\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.0902	0.924599
$924$	0	0
$925$	−18.2705	−0.600731
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.05573	−0.0674463	−0.0337231	−	0.999431i	$-0.510736\pi$
$929$	−2.05573	−0.0674463	−0.0337231	+	0.999431i	$0.510736\pi$
$930$	0	0
$931$	−4.18034	−0.137005
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.70820	0.153810	0.0769052	−	0.997038i	$-0.475496\pi$
$937$	4.70820	0.153810	0.0769052	+	0.997038i	$0.475496\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.21478	−0.202596	−0.101298	−	0.994856i	$-0.532300\pi$
$941$	−6.21478	−0.202596	−0.101298	+	0.994856i	$0.532300\pi$
$942$	0	0
$943$	31.1803	1.01537
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.72949	−0.251175	−0.125587	−	0.992083i	$-0.540082\pi$
$947$	−7.72949	−0.251175	−0.125587	+	0.992083i	$0.540082\pi$
$948$	0	0
$949$	8.29180	0.269163
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−18.6525	−0.604213	−0.302106	−	0.953274i	$-0.597690\pi$
$953$	−18.6525	−0.604213	−0.302106	+	0.953274i	$0.597690\pi$
$954$	0	0
$955$	2.20163	0.0712429
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23.1803	−0.748532
$960$	0	0
$961$	43.2705	1.39582
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.124612	−0.00401140
$966$	0	0
$967$	22.3820	0.719756	0.359878	−	0.932999i	$-0.382818\pi$
$967$	22.3820	0.719756	0.359878	+	0.932999i	$0.382818\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.3050	1.13299	0.566495	−	0.824065i	$-0.308299\pi$
$971$	35.3050	1.13299	0.566495	+	0.824065i	$0.308299\pi$
$972$	0	0
$973$	78.8673	2.52837
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.41641	0.205279	0.102640	−	0.994719i	$-0.467271\pi$
$977$	6.41641	0.205279	0.102640	+	0.994719i	$0.467271\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37.7771	1.20490	0.602451	−	0.798156i	$-0.294191\pi$
$983$	37.7771	1.20490	0.602451	+	0.798156i	$0.294191\pi$
$984$	0	0
$985$	3.11146	0.0991392
$986$	0	0
$987$	0	0
$988$	0	0
$989$	71.5410	2.27487
$990$	0	0
$991$	1.32624	0.0421293	0.0210647	−	0.999778i	$-0.493294\pi$
$991$	1.32624	0.0421293	0.0210647	+	0.999778i	$0.493294\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.61803	−0.114699
$996$	0	0
$997$	−26.2016	−0.829814	−0.414907	−	0.909864i	$-0.636186\pi$
$997$	−26.2016	−0.829814	−0.414907	+	0.909864i	$0.636186\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bc.1.2		2
3.2	odd	2		2904.2.a.ba.1.1		2
11.2	odd	10		792.2.r.d.433.1		4
11.6	odd	10		792.2.r.d.289.1		4
11.10	odd	2		8712.2.a.ba.1.2		2
12.11	even	2		5808.2.a.bv.1.1		2
33.2	even	10		264.2.q.b.169.1	yes	4
33.17	even	10		264.2.q.b.25.1	✓	4
33.32	even	2		2904.2.a.z.1.1		2
132.35	odd	10		528.2.y.h.433.1		4
132.83	odd	10		528.2.y.h.289.1		4
132.131	odd	2		5808.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.q.b.25.1	✓	4	33.17	even	10
264.2.q.b.169.1	yes	4	33.2	even	10
528.2.y.h.289.1		4	132.83	odd	10
528.2.y.h.433.1		4	132.35	odd	10
792.2.r.d.289.1		4	11.6	odd	10
792.2.r.d.433.1		4	11.2	odd	10
2904.2.a.z.1.1		2	33.32	even	2
2904.2.a.ba.1.1		2	3.2	odd	2
5808.2.a.bv.1.1		2	12.11	even	2
5808.2.a.bw.1.1		2	132.131	odd	2
8712.2.a.ba.1.2		2	11.10	odd	2
8712.2.a.bc.1.2		2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{5} +4.23607 q^{7} +O(q^{10})\)	\(q-0.381966 q^{5} +4.23607 q^{7} -2.23607 q^{13} -1.85410 q^{17} -0.381966 q^{19} -6.23607 q^{23} -4.85410 q^{25} +0.472136 q^{29} +8.61803 q^{31} -1.61803 q^{35} +3.76393 q^{37} -5.00000 q^{41} -11.4721 q^{43} -0.145898 q^{47} +10.9443 q^{49} +9.56231 q^{53} -14.5623 q^{59} +6.85410 q^{61} +0.854102 q^{65} -4.56231 q^{67} -12.5623 q^{71} -3.70820 q^{73} -14.2361 q^{79} -1.94427 q^{83} +0.708204 q^{85} -3.47214 q^{89} -9.47214 q^{91} +0.145898 q^{95} -6.85410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 3 * q^5 + 4 * q^7	\( 2 q - 3 q^{5} + 4 q^{7} + 3 q^{17} - 3 q^{19} - 8 q^{23} - 3 q^{25} - 8 q^{29} + 15 q^{31} - q^{35} + 12 q^{37} - 10 q^{41} - 14 q^{43} - 7 q^{47} + 4 q^{49} - q^{53} - 9 q^{59} + 7 q^{61} - 5 q^{65} + 11 q^{67} - 5 q^{71} + 6 q^{73} - 24 q^{79} + 14 q^{83} - 12 q^{85} + 2 q^{89} - 10 q^{91} + 7 q^{95} - 7 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 + 4 * q^7 + 3 * q^17 - 3 * q^19 - 8 * q^23 - 3 * q^25 - 8 * q^29 + 15 * q^31 - q^35 + 12 * q^37 - 10 * q^41 - 14 * q^43 - 7 * q^47 + 4 * q^49 - q^53 - 9 * q^59 + 7 * q^61 - 5 * q^65 + 11 * q^67 - 5 * q^71 + 6 * q^73 - 24 * q^79 + 14 * q^83 - 12 * q^85 + 2 * q^89 - 10 * q^91 + 7 * q^95 - 7 * q^97

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