LMFDB - Embedded newform 6012.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 26x^{8} + 82x^{7} + 211x^{6} - 340x^{5} - 593x^{4} + 192x^{3} + 423x^{2} + 126x + 9 $ x^10 - 4x^9 - 26x^8 + 82x^7 + 211x^6 - 340x^5 - 593x^4 + 192x^3 + 423x^2 + 126*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.79792$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.67303 q^{5} +4.79792 q^{7} +O(q^{10})$	$q-3.67303 q^{5} +4.79792 q^{7} -1.91517 q^{11} +3.05652 q^{13} +2.92806 q^{17} -8.17555 q^{19} -6.97269 q^{23} +8.49112 q^{25} +3.87744 q^{29} +0.104588 q^{31} -17.6229 q^{35} -1.37862 q^{37} +10.4874 q^{41} -5.08108 q^{43} -10.8230 q^{47} +16.0200 q^{49} -7.42211 q^{53} +7.03447 q^{55} -0.535117 q^{59} +8.80494 q^{61} -11.2267 q^{65} -8.50182 q^{67} +5.00918 q^{71} +8.80655 q^{73} -9.18882 q^{77} -6.76692 q^{79} -10.3540 q^{83} -10.7548 q^{85} -12.2737 q^{89} +14.6649 q^{91} +30.0290 q^{95} -12.4388 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{5} + 4 q^{7}+O(q^{10}) $ 10 * q - 6 * q^5 + 4 * q^7	$ 10 q - 6 q^{5} + 4 q^{7} - 8 q^{11} - 2 q^{13} - 6 q^{17} - 20 q^{23} + 24 q^{25} - 8 q^{29} - 4 q^{31} - 4 q^{37} + 14 q^{41} + 20 q^{43} - 48 q^{47} - 2 q^{49} - 22 q^{53} - 6 q^{55} - 2 q^{59} - 8 q^{61} - 28 q^{65} - 6 q^{67} - 20 q^{71} + 20 q^{73} - 24 q^{77} - 4 q^{79} - 46 q^{83} - 18 q^{85} + 8 q^{89} + 28 q^{91} - 36 q^{95} - 34 q^{97}+O(q^{100}) $ 10 * q - 6 * q^5 + 4 * q^7 - 8 * q^11 - 2 * q^13 - 6 * q^17 - 20 * q^23 + 24 * q^25 - 8 * q^29 - 4 * q^31 - 4 * q^37 + 14 * q^41 + 20 * q^43 - 48 * q^47 - 2 * q^49 - 22 * q^53 - 6 * q^55 - 2 * q^59 - 8 * q^61 - 28 * q^65 - 6 * q^67 - 20 * q^71 + 20 * q^73 - 24 * q^77 - 4 * q^79 - 46 * q^83 - 18 * q^85 + 8 * q^89 + 28 * q^91 - 36 * q^95 - 34 * 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
$5$	−3.67303	−1.64263	−0.821314	−	0.570477i	$-0.806758\pi$
$5$	−3.67303	−1.64263	−0.821314	+	0.570477i	$0.806758\pi$
$6$	0	0
$7$	4.79792	1.81344	0.906721	−	0.421731i	$-0.138577\pi$
$7$	4.79792	1.81344	0.906721	+	0.421731i	$0.138577\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.91517	−0.577445	−0.288723	−	0.957413i	$-0.593231\pi$
$11$	−1.91517	−0.577445	−0.288723	+	0.957413i	$0.593231\pi$
$12$	0	0
$13$	3.05652	0.847726	0.423863	−	0.905726i	$-0.360674\pi$
$13$	3.05652	0.847726	0.423863	+	0.905726i	$0.360674\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.92806	0.710158	0.355079	−	0.934836i	$-0.384454\pi$
$17$	2.92806	0.710158	0.355079	+	0.934836i	$0.384454\pi$
$18$	0	0
$19$	−8.17555	−1.87560	−0.937800	−	0.347177i	$-0.887140\pi$
$19$	−8.17555	−1.87560	−0.937800	+	0.347177i	$0.887140\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.97269	−1.45391	−0.726953	−	0.686687i	$-0.759064\pi$
$23$	−6.97269	−1.45391	−0.726953	+	0.686687i	$0.759064\pi$
$24$	0	0
$25$	8.49112	1.69822
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.87744	0.720022	0.360011	−	0.932948i	$-0.382773\pi$
$29$	3.87744	0.720022	0.360011	+	0.932948i	$0.382773\pi$
$30$	0	0
$31$	0.104588	0.0187845	0.00939226	−	0.999956i	$-0.497010\pi$
$31$	0.104588	0.0187845	0.00939226	+	0.999956i	$0.497010\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−17.6229	−2.97881
$36$	0	0
$37$	−1.37862	−0.226644	−0.113322	−	0.993558i	$-0.536149\pi$
$37$	−1.37862	−0.226644	−0.113322	+	0.993558i	$0.536149\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.4874	1.63786	0.818930	−	0.573893i	$-0.194567\pi$
$41$	10.4874	1.63786	0.818930	+	0.573893i	$0.194567\pi$
$42$	0	0
$43$	−5.08108	−0.774857	−0.387429	−	0.921900i	$-0.626637\pi$
$43$	−5.08108	−0.774857	−0.387429	+	0.921900i	$0.626637\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.8230	−1.57870	−0.789352	−	0.613941i	$-0.789583\pi$
$47$	−10.8230	−1.57870	−0.789352	+	0.613941i	$0.789583\pi$
$48$	0	0
$49$	16.0200	2.28857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.42211	−1.01950	−0.509752	−	0.860321i	$-0.670263\pi$
$53$	−7.42211	−1.01950	−0.509752	+	0.860321i	$0.670263\pi$
$54$	0	0
$55$	7.03447	0.948527
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.535117	−0.0696662	−0.0348331	−	0.999393i	$-0.511090\pi$
$59$	−0.535117	−0.0696662	−0.0348331	+	0.999393i	$0.511090\pi$
$60$	0	0
$61$	8.80494	1.12736	0.563679	−	0.825994i	$-0.309386\pi$
$61$	8.80494	1.12736	0.563679	+	0.825994i	$0.309386\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.2267	−1.39250
$66$	0	0
$67$	−8.50182	−1.03866	−0.519331	−	0.854573i	$-0.673819\pi$
$67$	−8.50182	−1.03866	−0.519331	+	0.854573i	$0.673819\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.00918	0.594480	0.297240	−	0.954803i	$-0.403934\pi$
$71$	5.00918	0.594480	0.297240	+	0.954803i	$0.403934\pi$
$72$	0	0
$73$	8.80655	1.03073	0.515364	−	0.856971i	$-0.327657\pi$
$73$	8.80655	1.03073	0.515364	+	0.856971i	$0.327657\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.18882	−1.04716
$78$	0	0
$79$	−6.76692	−0.761338	−0.380669	−	0.924711i	$-0.624306\pi$
$79$	−6.76692	−0.761338	−0.380669	+	0.924711i	$0.624306\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.3540	−1.13650	−0.568250	−	0.822856i	$-0.692379\pi$
$83$	−10.3540	−1.13650	−0.568250	+	0.822856i	$0.692379\pi$
$84$	0	0
$85$	−10.7548	−1.16653
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.2737	−1.30101	−0.650503	−	0.759503i	$-0.725442\pi$
$89$	−12.2737	−1.30101	−0.650503	+	0.759503i	$0.725442\pi$
$90$	0	0
$91$	14.6649	1.53730
$92$	0	0
$93$	0	0
$94$	0	0
$95$	30.0290	3.08091
$96$	0	0
$97$	−12.4388	−1.26297	−0.631483	−	0.775390i	$-0.717553\pi$
$97$	−12.4388	−1.26297	−0.631483	+	0.775390i	$0.717553\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.6036	−1.15460	−0.577299	−	0.816533i	$-0.695893\pi$
$101$	−11.6036	−1.15460	−0.577299	+	0.816533i	$0.695893\pi$
$102$	0	0
$103$	5.02122	0.494756	0.247378	−	0.968919i	$-0.420431\pi$
$103$	5.02122	0.494756	0.247378	+	0.968919i	$0.420431\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.503150	−0.0486413	−0.0243207	−	0.999704i	$-0.507742\pi$
$107$	−0.503150	−0.0486413	−0.0243207	+	0.999704i	$0.507742\pi$
$108$	0	0
$109$	7.18686	0.688376	0.344188	−	0.938901i	$-0.388154\pi$
$109$	7.18686	0.688376	0.344188	+	0.938901i	$0.388154\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.1132	1.32766	0.663831	−	0.747883i	$-0.268930\pi$
$113$	14.1132	1.32766	0.663831	+	0.747883i	$0.268930\pi$
$114$	0	0
$115$	25.6109	2.38823
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.0486	1.28783
$120$	0	0
$121$	−7.33213	−0.666557
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.8230	−1.14692
$126$	0	0
$127$	14.7666	1.31032	0.655160	−	0.755490i	$-0.272601\pi$
$127$	14.7666	1.31032	0.655160	+	0.755490i	$0.272601\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.61825	−0.840350	−0.420175	−	0.907443i	$-0.638031\pi$
$131$	−9.61825	−0.840350	−0.420175	+	0.907443i	$0.638031\pi$
$132$	0	0
$133$	−39.2256	−3.40129
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.8778	1.27110	0.635549	−	0.772060i	$-0.280774\pi$
$137$	14.8778	1.27110	0.635549	+	0.772060i	$0.280774\pi$
$138$	0	0
$139$	14.8016	1.25545	0.627727	−	0.778434i	$-0.283986\pi$
$139$	14.8016	1.25545	0.627727	+	0.778434i	$0.283986\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.85375	−0.489515
$144$	0	0
$145$	−14.2419	−1.18273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.96434	0.570541	0.285270	−	0.958447i	$-0.407917\pi$
$149$	6.96434	0.570541	0.285270	+	0.958447i	$0.407917\pi$
$150$	0	0
$151$	−12.3659	−1.00633	−0.503163	−	0.864192i	$-0.667830\pi$
$151$	−12.3659	−1.00633	−0.503163	+	0.864192i	$0.667830\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.384154	−0.0308560
$156$	0	0
$157$	−5.22168	−0.416735	−0.208368	−	0.978051i	$-0.566815\pi$
$157$	−5.22168	−0.416735	−0.208368	+	0.978051i	$0.566815\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−33.4544	−2.63657
$162$	0	0
$163$	−22.1034	−1.73127	−0.865637	−	0.500672i	$-0.833086\pi$
$163$	−22.1034	−1.73127	−0.865637	+	0.500672i	$0.833086\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−3.65769	−0.281361
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.66845	0.202878	0.101439	−	0.994842i	$-0.467655\pi$
$173$	2.66845	0.202878	0.101439	+	0.994842i	$0.467655\pi$
$174$	0	0
$175$	40.7397	3.07963
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.2726	−1.06678	−0.533392	−	0.845868i	$-0.679083\pi$
$179$	−14.2726	−1.06678	−0.533392	+	0.845868i	$0.679083\pi$
$180$	0	0
$181$	0.0242324	0.00180118	0.000900589	−	1.00000i	$-0.499713\pi$
$181$	0.0242324	0.00180118	0.000900589		1.00000i	$0.499713\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.06372	0.372292
$186$	0	0
$187$	−5.60772	−0.410077
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.07687	0.367350	0.183675	−	0.982987i	$-0.441201\pi$
$191$	5.07687	0.367350	0.183675	+	0.982987i	$0.441201\pi$
$192$	0	0
$193$	−21.3648	−1.53788	−0.768938	−	0.639324i	$-0.779214\pi$
$193$	−21.3648	−1.53788	−0.768938	+	0.639324i	$0.779214\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.27277	0.304422	0.152211	−	0.988348i	$-0.451361\pi$
$197$	4.27277	0.304422	0.152211	+	0.988348i	$0.451361\pi$
$198$	0	0
$199$	5.23438	0.371056	0.185528	−	0.982639i	$-0.440601\pi$
$199$	5.23438	0.371056	0.185528	+	0.982639i	$0.440601\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.6036	1.30572
$204$	0	0
$205$	−38.5206	−2.69039
$206$	0	0
$207$	0	0
$208$	0	0
$209$	15.6576	1.08306
$210$	0	0
$211$	2.86156	0.196998	0.0984989	−	0.995137i	$-0.468596\pi$
$211$	2.86156	0.196998	0.0984989	+	0.995137i	$0.468596\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.6629	1.27280
$216$	0	0
$217$	0.501804	0.0340646
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.94966	0.602019
$222$	0	0
$223$	−26.0317	−1.74321	−0.871607	−	0.490206i	$-0.836922\pi$
$223$	−26.0317	−1.74321	−0.871607	+	0.490206i	$0.836922\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.28114	−0.350522	−0.175261	−	0.984522i	$-0.556077\pi$
$227$	−5.28114	−0.350522	−0.175261	+	0.984522i	$0.556077\pi$
$228$	0	0
$229$	−12.6859	−0.838310	−0.419155	−	0.907915i	$-0.637674\pi$
$229$	−12.6859	−0.838310	−0.419155	+	0.907915i	$0.637674\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.9135	1.69765	0.848825	−	0.528673i	$-0.177310\pi$
$233$	25.9135	1.69765	0.848825	+	0.528673i	$0.177310\pi$
$234$	0	0
$235$	39.7533	2.59322
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.76237	0.502106	0.251053	−	0.967973i	$-0.419223\pi$
$239$	7.76237	0.502106	0.251053	+	0.967973i	$0.419223\pi$
$240$	0	0
$241$	9.75954	0.628667	0.314334	−	0.949313i	$-0.398219\pi$
$241$	9.75954	0.628667	0.314334	+	0.949313i	$0.398219\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−58.8419	−3.75927
$246$	0	0
$247$	−24.9887	−1.58999
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.7710	−1.50041	−0.750205	−	0.661205i	$-0.770045\pi$
$251$	−23.7710	−1.50041	−0.750205	+	0.661205i	$0.770045\pi$
$252$	0	0
$253$	13.3539	0.839551
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.00848	−0.561934	−0.280967	−	0.959717i	$-0.590655\pi$
$257$	−9.00848	−0.561934	−0.280967	+	0.959717i	$0.590655\pi$
$258$	0	0
$259$	−6.61452	−0.411006
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.7018	−1.58484	−0.792421	−	0.609975i	$-0.791180\pi$
$263$	−25.7018	−1.58484	−0.792421	+	0.609975i	$0.791180\pi$
$264$	0	0
$265$	27.2616	1.67467
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.12529	0.251523	0.125762	−	0.992060i	$-0.459863\pi$
$269$	4.12529	0.251523	0.125762	+	0.992060i	$0.459863\pi$
$270$	0	0
$271$	−30.4509	−1.84976	−0.924879	−	0.380260i	$-0.875834\pi$
$271$	−30.4509	−1.84976	−0.924879	+	0.380260i	$0.875834\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.2619	−0.980631
$276$	0	0
$277$	4.32001	0.259564	0.129782	−	0.991543i	$-0.458572\pi$
$277$	4.32001	0.259564	0.129782	+	0.991543i	$0.458572\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.3504	−1.03504	−0.517519	−	0.855672i	$-0.673144\pi$
$281$	−17.3504	−1.03504	−0.517519	+	0.855672i	$0.673144\pi$
$282$	0	0
$283$	−25.3295	−1.50568	−0.752840	−	0.658204i	$-0.771317\pi$
$283$	−25.3295	−1.50568	−0.752840	+	0.658204i	$0.771317\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	50.3178	2.97017
$288$	0	0
$289$	−8.42648	−0.495675
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.6214	1.20472	0.602358	−	0.798226i	$-0.294228\pi$
$293$	20.6214	1.20472	0.602358	+	0.798226i	$0.294228\pi$
$294$	0	0
$295$	1.96550	0.114436
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.3121	−1.23251
$300$	0	0
$301$	−24.3786	−1.40516
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−32.3408	−1.85183
$306$	0	0
$307$	15.7715	0.900125	0.450062	−	0.892997i	$-0.351402\pi$
$307$	15.7715	0.900125	0.450062	+	0.892997i	$0.351402\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−23.2654	−1.31926	−0.659629	−	0.751592i	$-0.729286\pi$
$311$	−23.2654	−1.31926	−0.659629	+	0.751592i	$0.729286\pi$
$312$	0	0
$313$	−6.86653	−0.388119	−0.194060	−	0.980990i	$-0.562166\pi$
$313$	−6.86653	−0.388119	−0.194060	+	0.980990i	$0.562166\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.6632	−0.992065	−0.496032	−	0.868304i	$-0.665210\pi$
$317$	−17.6632	−0.992065	−0.496032	+	0.868304i	$0.665210\pi$
$318$	0	0
$319$	−7.42595	−0.415773
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−23.9385	−1.33197
$324$	0	0
$325$	25.9533	1.43963
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−51.9281	−2.86289
$330$	0	0
$331$	19.6966	1.08262	0.541312	−	0.840822i	$-0.317928\pi$
$331$	19.6966	1.08262	0.541312	+	0.840822i	$0.317928\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	31.2274	1.70613
$336$	0	0
$337$	−26.6703	−1.45282	−0.726411	−	0.687261i	$-0.758813\pi$
$337$	−26.6703	−1.45282	−0.726411	+	0.687261i	$0.758813\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.200303	−0.0108470
$342$	0	0
$343$	43.2772	2.33675
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.58809	−0.246302	−0.123151	−	0.992388i	$-0.539300\pi$
$347$	−4.58809	−0.246302	−0.123151	+	0.992388i	$0.539300\pi$
$348$	0	0
$349$	1.69727	0.0908528	0.0454264	−	0.998968i	$-0.485535\pi$
$349$	1.69727	0.0908528	0.0454264	+	0.998968i	$0.485535\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.4077	−1.29909	−0.649545	−	0.760323i	$-0.725041\pi$
$353$	−24.4077	−1.29909	−0.649545	+	0.760323i	$0.725041\pi$
$354$	0	0
$355$	−18.3988	−0.976510
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.322263	−0.0170084	−0.00850420	−	0.999964i	$-0.502707\pi$
$359$	−0.322263	−0.0170084	−0.00850420	+	0.999964i	$0.502707\pi$
$360$	0	0
$361$	47.8396	2.51787
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−32.3467	−1.69310
$366$	0	0
$367$	−11.5461	−0.602702	−0.301351	−	0.953513i	$-0.597438\pi$
$367$	−11.5461	−0.602702	−0.301351	+	0.953513i	$0.597438\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−35.6106	−1.84881
$372$	0	0
$373$	−18.1800	−0.941327	−0.470663	−	0.882313i	$-0.655985\pi$
$373$	−18.1800	−0.941327	−0.470663	+	0.882313i	$0.655985\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.8515	0.610381
$378$	0	0
$379$	−28.3720	−1.45737	−0.728685	−	0.684849i	$-0.759868\pi$
$379$	−28.3720	−1.45737	−0.728685	+	0.684849i	$0.759868\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.4525	−1.65825	−0.829123	−	0.559066i	$-0.811160\pi$
$383$	−32.4525	−1.65825	−0.829123	+	0.559066i	$0.811160\pi$
$384$	0	0
$385$	33.7508	1.72010
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.7445	−0.747578	−0.373789	−	0.927514i	$-0.621942\pi$
$389$	−14.7445	−0.747578	−0.373789	+	0.927514i	$0.621942\pi$
$390$	0	0
$391$	−20.4164	−1.03250
$392$	0	0
$393$	0	0
$394$	0	0
$395$	24.8551	1.25060
$396$	0	0
$397$	−23.2101	−1.16488	−0.582441	−	0.812873i	$-0.697902\pi$
$397$	−23.2101	−1.16488	−0.582441	+	0.812873i	$0.697902\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.3934	0.968461	0.484230	−	0.874941i	$-0.339100\pi$
$401$	19.3934	0.968461	0.484230	+	0.874941i	$0.339100\pi$
$402$	0	0
$403$	0.319675	0.0159241
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.64030	0.130875
$408$	0	0
$409$	−35.0781	−1.73450	−0.867251	−	0.497872i	$-0.834115\pi$
$409$	−35.0781	−1.73450	−0.867251	+	0.497872i	$0.834115\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.56744	−0.126336
$414$	0	0
$415$	38.0306	1.86685
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.76721	0.477159	0.238580	−	0.971123i	$-0.423318\pi$
$419$	9.76721	0.477159	0.238580	+	0.971123i	$0.423318\pi$
$420$	0	0
$421$	−9.52914	−0.464422	−0.232211	−	0.972665i	$-0.574596\pi$
$421$	−9.52914	−0.464422	−0.232211	+	0.972665i	$0.574596\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	24.8625	1.20601
$426$	0	0
$427$	42.2454	2.04440
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.20574	−0.0580786	−0.0290393	−	0.999578i	$-0.509245\pi$
$431$	−1.20574	−0.0580786	−0.0290393	+	0.999578i	$0.509245\pi$
$432$	0	0
$433$	−4.13190	−0.198567	−0.0992833	−	0.995059i	$-0.531655\pi$
$433$	−4.13190	−0.198567	−0.0992833	+	0.995059i	$0.531655\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	57.0055	2.72694
$438$	0	0
$439$	−12.8595	−0.613751	−0.306875	−	0.951750i	$-0.599283\pi$
$439$	−12.8595	−0.613751	−0.306875	+	0.951750i	$0.599283\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.9304	−1.61208	−0.806041	−	0.591860i	$-0.798394\pi$
$443$	−33.9304	−1.61208	−0.806041	+	0.591860i	$0.798394\pi$
$444$	0	0
$445$	45.0815	2.13707
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.9490	−0.847068	−0.423534	−	0.905880i	$-0.639210\pi$
$449$	−17.9490	−0.847068	−0.423534	+	0.905880i	$0.639210\pi$
$450$	0	0
$451$	−20.0852	−0.945775
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−53.8646	−2.52521
$456$	0	0
$457$	−7.58770	−0.354938	−0.177469	−	0.984126i	$-0.556791\pi$
$457$	−7.58770	−0.354938	−0.177469	+	0.984126i	$0.556791\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.5987	1.51827	0.759136	−	0.650932i	$-0.225622\pi$
$461$	32.5987	1.51827	0.759136	+	0.650932i	$0.225622\pi$
$462$	0	0
$463$	16.4807	0.765924	0.382962	−	0.923764i	$-0.374904\pi$
$463$	16.4807	0.765924	0.382962	+	0.923764i	$0.374904\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.43107	0.390143	0.195072	−	0.980789i	$-0.437506\pi$
$467$	8.43107	0.390143	0.195072	+	0.980789i	$0.437506\pi$
$468$	0	0
$469$	−40.7910	−1.88355
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.73112	0.447437
$474$	0	0
$475$	−69.4196	−3.18519
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.64192	−0.257786	−0.128893	−	0.991659i	$-0.541142\pi$
$479$	−5.64192	−0.257786	−0.128893	+	0.991659i	$0.541142\pi$
$480$	0	0
$481$	−4.21379	−0.192132
$482$	0	0
$483$	0	0
$484$	0	0
$485$	45.6879	2.07458
$486$	0	0
$487$	14.4101	0.652986	0.326493	−	0.945200i	$-0.394133\pi$
$487$	14.4101	0.652986	0.326493	+	0.945200i	$0.394133\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.98281	0.315130	0.157565	−	0.987509i	$-0.449636\pi$
$491$	6.98281	0.315130	0.157565	+	0.987509i	$0.449636\pi$
$492$	0	0
$493$	11.3534	0.511329
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0336	1.07806
$498$	0	0
$499$	17.0286	0.762305	0.381153	−	0.924512i	$-0.375527\pi$
$499$	17.0286	0.762305	0.381153	+	0.924512i	$0.375527\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.01078	0.178832	0.0894159	−	0.995994i	$-0.471500\pi$
$503$	4.01078	0.178832	0.0894159	+	0.995994i	$0.471500\pi$
$504$	0	0
$505$	42.6202	1.89657
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−33.8481	−1.50029	−0.750145	−	0.661273i	$-0.770016\pi$
$509$	−33.8481	−1.50029	−0.750145	+	0.661273i	$0.770016\pi$
$510$	0	0
$511$	42.2531	1.86917
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−18.4431	−0.812700
$516$	0	0
$517$	20.7280	0.911615
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.4116	0.719005	0.359503	−	0.933144i	$-0.382946\pi$
$521$	16.4116	0.719005	0.359503	+	0.933144i	$0.382946\pi$
$522$	0	0
$523$	21.8213	0.954180	0.477090	−	0.878855i	$-0.341692\pi$
$523$	21.8213	0.954180	0.477090	+	0.878855i	$0.341692\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.306239	0.0133400
$528$	0	0
$529$	25.6184	1.11384
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32.0550	1.38846
$534$	0	0
$535$	1.84808	0.0798996
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−30.6810	−1.32152
$540$	0	0
$541$	−4.06569	−0.174798	−0.0873988	−	0.996173i	$-0.527855\pi$
$541$	−4.06569	−0.174798	−0.0873988	+	0.996173i	$0.527855\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−26.3975	−1.13075
$546$	0	0
$547$	−2.67799	−0.114502	−0.0572512	−	0.998360i	$-0.518234\pi$
$547$	−2.67799	−0.114502	−0.0572512	+	0.998360i	$0.518234\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−31.7002	−1.35047
$552$	0	0
$553$	−32.4671	−1.38064
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.7081	0.453716	0.226858	−	0.973928i	$-0.427155\pi$
$557$	10.7081	0.453716	0.226858	+	0.973928i	$0.427155\pi$
$558$	0	0
$559$	−15.5304	−0.656866
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.0422	1.35042	0.675209	−	0.737626i	$-0.264053\pi$
$563$	32.0422	1.35042	0.675209	+	0.737626i	$0.264053\pi$
$564$	0	0
$565$	−51.8383	−2.18085
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.2153	0.595936	0.297968	−	0.954576i	$-0.403691\pi$
$569$	14.2153	0.595936	0.297968	+	0.954576i	$0.403691\pi$
$570$	0	0
$571$	36.3290	1.52032	0.760160	−	0.649735i	$-0.225120\pi$
$571$	36.3290	1.52032	0.760160	+	0.649735i	$0.225120\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−59.2059	−2.46906
$576$	0	0
$577$	43.7193	1.82006	0.910030	−	0.414542i	$-0.136058\pi$
$577$	43.7193	1.82006	0.910030	+	0.414542i	$0.136058\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−49.6777	−2.06098
$582$	0	0
$583$	14.2146	0.588708
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.5724	1.63333	0.816665	−	0.577112i	$-0.195821\pi$
$587$	39.5724	1.63333	0.816665	+	0.577112i	$0.195821\pi$
$588$	0	0
$589$	−0.855063	−0.0352322
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.9804	−1.23115	−0.615573	−	0.788080i	$-0.711076\pi$
$593$	−29.9804	−1.23115	−0.615573	+	0.788080i	$0.711076\pi$
$594$	0	0
$595$	−51.6008	−2.11543
$596$	0	0
$597$	0	0
$598$	0	0
$599$	47.5354	1.94224	0.971121	−	0.238586i	$-0.0766838\pi$
$599$	47.5354	1.94224	0.971121	+	0.238586i	$0.0766838\pi$
$600$	0	0
$601$	9.82335	0.400703	0.200351	−	0.979724i	$-0.435792\pi$
$601$	9.82335	0.400703	0.200351	+	0.979724i	$0.435792\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	26.9311	1.09490
$606$	0	0
$607$	8.83155	0.358462	0.179231	−	0.983807i	$-0.442639\pi$
$607$	8.83155	0.358462	0.179231	+	0.983807i	$0.442639\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−33.0808	−1.33831
$612$	0	0
$613$	−8.93013	−0.360685	−0.180342	−	0.983604i	$-0.557721\pi$
$613$	−8.93013	−0.360685	−0.180342	+	0.983604i	$0.557721\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.31064	−0.0930230	−0.0465115	−	0.998918i	$-0.514810\pi$
$617$	−2.31064	−0.0930230	−0.0465115	+	0.998918i	$0.514810\pi$
$618$	0	0
$619$	−7.92409	−0.318496	−0.159248	−	0.987239i	$-0.550907\pi$
$619$	−7.92409	−0.318496	−0.159248	+	0.987239i	$0.550907\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−58.8881	−2.35930
$624$	0	0
$625$	4.64353	0.185741
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.03669	−0.160953
$630$	0	0
$631$	−16.2418	−0.646576	−0.323288	−	0.946301i	$-0.604788\pi$
$631$	−16.2418	−0.646576	−0.323288	+	0.946301i	$0.604788\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−54.2379	−2.15237
$636$	0	0
$637$	48.9654	1.94008
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11.3414	−0.447959	−0.223979	−	0.974594i	$-0.571905\pi$
$641$	−11.3414	−0.447959	−0.223979	+	0.974594i	$0.571905\pi$
$642$	0	0
$643$	46.5575	1.83605	0.918024	−	0.396524i	$-0.129784\pi$
$643$	46.5575	1.83605	0.918024	+	0.396524i	$0.129784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.5379	0.610859	0.305430	−	0.952215i	$-0.401200\pi$
$647$	15.5379	0.610859	0.305430	+	0.952215i	$0.401200\pi$
$648$	0	0
$649$	1.02484	0.0402284
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.9961	1.36950	0.684752	−	0.728776i	$-0.259911\pi$
$653$	34.9961	1.36950	0.684752	+	0.728776i	$0.259911\pi$
$654$	0	0
$655$	35.3281	1.38038
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.96335	−0.154390	−0.0771952	−	0.997016i	$-0.524596\pi$
$659$	−3.96335	−0.154390	−0.0771952	+	0.997016i	$0.524596\pi$
$660$	0	0
$661$	−34.0321	−1.32370	−0.661848	−	0.749638i	$-0.730227\pi$
$661$	−34.0321	−1.32370	−0.661848	+	0.749638i	$0.730227\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	144.077	5.58705
$666$	0	0
$667$	−27.0361	−1.04684
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.8630	−0.650987
$672$	0	0
$673$	−11.1268	−0.428907	−0.214454	−	0.976734i	$-0.568797\pi$
$673$	−11.1268	−0.428907	−0.214454	+	0.976734i	$0.568797\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.1494	1.77367	0.886833	−	0.462090i	$-0.152900\pi$
$677$	46.1494	1.77367	0.886833	+	0.462090i	$0.152900\pi$
$678$	0	0
$679$	−59.6802	−2.29032
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.9645	−1.14656	−0.573280	−	0.819360i	$-0.694329\pi$
$683$	−29.9645	−1.14656	−0.573280	+	0.819360i	$0.694329\pi$
$684$	0	0
$685$	−54.6467	−2.08794
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.6858	−0.864260
$690$	0	0
$691$	29.1144	1.10756	0.553782	−	0.832662i	$-0.313184\pi$
$691$	29.1144	1.10756	0.553782	+	0.832662i	$0.313184\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−54.3666	−2.06224
$696$	0	0
$697$	30.7078	1.16314
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−14.2895	−0.539706	−0.269853	−	0.962901i	$-0.586975\pi$
$701$	−14.2895	−0.539706	−0.269853	+	0.962901i	$0.586975\pi$
$702$	0	0
$703$	11.2710	0.425094
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−55.6729	−2.09380
$708$	0	0
$709$	−20.7699	−0.780029	−0.390015	−	0.920809i	$-0.627530\pi$
$709$	−20.7699	−0.780029	−0.390015	+	0.920809i	$0.627530\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.729258	−0.0273109
$714$	0	0
$715$	21.5010	0.804091
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.1175	0.787549	0.393774	−	0.919207i	$-0.371169\pi$
$719$	21.1175	0.787549	0.393774	+	0.919207i	$0.371169\pi$
$720$	0	0
$721$	24.0914	0.897211
$722$	0	0
$723$	0	0
$724$	0	0
$725$	32.9238	1.22276
$726$	0	0
$727$	2.28911	0.0848982	0.0424491	−	0.999099i	$-0.486484\pi$
$727$	2.28911	0.0848982	0.0424491	+	0.999099i	$0.486484\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14.8777	−0.550271
$732$	0	0
$733$	41.4761	1.53196	0.765978	−	0.642867i	$-0.222255\pi$
$733$	41.4761	1.53196	0.765978	+	0.642867i	$0.222255\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.2824	0.599770
$738$	0	0
$739$	−10.6238	−0.390804	−0.195402	−	0.980723i	$-0.562601\pi$
$739$	−10.6238	−0.390804	−0.195402	+	0.980723i	$0.562601\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.1811	0.556942	0.278471	−	0.960445i	$-0.410172\pi$
$743$	15.1811	0.556942	0.278471	+	0.960445i	$0.410172\pi$
$744$	0	0
$745$	−25.5802	−0.937186
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.41407	−0.0882082
$750$	0	0
$751$	26.1855	0.955523	0.477761	−	0.878490i	$-0.341448\pi$
$751$	26.1855	0.955523	0.477761	+	0.878490i	$0.341448\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	45.4204	1.65302
$756$	0	0
$757$	13.1148	0.476664	0.238332	−	0.971184i	$-0.423399\pi$
$757$	13.1148	0.476664	0.238332	+	0.971184i	$0.423399\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17.6038	−0.638137	−0.319068	−	0.947732i	$-0.603370\pi$
$761$	−17.6038	−0.638137	−0.319068	+	0.947732i	$0.603370\pi$
$762$	0	0
$763$	34.4819	1.24833
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.63559	−0.0590579
$768$	0	0
$769$	4.53272	0.163454	0.0817270	−	0.996655i	$-0.473956\pi$
$769$	4.53272	0.163454	0.0817270	+	0.996655i	$0.473956\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	46.4514	1.67074	0.835370	−	0.549689i	$-0.185254\pi$
$773$	46.4514	1.67074	0.835370	+	0.549689i	$0.185254\pi$
$774$	0	0
$775$	0.888068	0.0319003
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−85.7405	−3.07197
$780$	0	0
$781$	−9.59343	−0.343280
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.1794	0.684541
$786$	0	0
$787$	−24.1583	−0.861151	−0.430576	−	0.902555i	$-0.641689\pi$
$787$	−24.1583	−0.861151	−0.430576	+	0.902555i	$0.641689\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	67.7141	2.40764
$792$	0	0
$793$	26.9125	0.955690
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−43.8645	−1.55376	−0.776881	−	0.629648i	$-0.783199\pi$
$797$	−43.8645	−1.55376	−0.776881	+	0.629648i	$0.783199\pi$
$798$	0	0
$799$	−31.6905	−1.12113
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.8660	−0.595189
$804$	0	0
$805$	122.879	4.33091
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.5251	0.686465	0.343232	−	0.939251i	$-0.388478\pi$
$809$	19.5251	0.686465	0.343232	+	0.939251i	$0.388478\pi$
$810$	0	0
$811$	47.3540	1.66282	0.831411	−	0.555658i	$-0.187534\pi$
$811$	47.3540	1.66282	0.831411	+	0.555658i	$0.187534\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	81.1865	2.84384
$816$	0	0
$817$	41.5406	1.45332
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.8789	−0.658878	−0.329439	−	0.944177i	$-0.606860\pi$
$821$	−18.8789	−0.658878	−0.329439	+	0.944177i	$0.606860\pi$
$822$	0	0
$823$	2.05916	0.0717779	0.0358890	−	0.999356i	$-0.488574\pi$
$823$	2.05916	0.0717779	0.0358890	+	0.999356i	$0.488574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.9398	1.87567	0.937836	−	0.347080i	$-0.112827\pi$
$827$	53.9398	1.87567	0.937836	+	0.347080i	$0.112827\pi$
$828$	0	0
$829$	−19.3325	−0.671446	−0.335723	−	0.941961i	$-0.608981\pi$
$829$	−19.3325	−0.671446	−0.335723	+	0.941961i	$0.608981\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	46.9075	1.62525
$834$	0	0
$835$	−3.67303	−0.127110
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39.0863	1.34941	0.674705	−	0.738088i	$-0.264271\pi$
$839$	39.0863	1.34941	0.674705	+	0.738088i	$0.264271\pi$
$840$	0	0
$841$	−13.9655	−0.481569
$842$	0	0
$843$	0	0
$844$	0	0
$845$	13.4348	0.462171
$846$	0	0
$847$	−35.1789	−1.20876
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.61271	0.329519
$852$	0	0
$853$	26.0284	0.891194	0.445597	−	0.895234i	$-0.352991\pi$
$853$	26.0284	0.891194	0.445597	+	0.895234i	$0.352991\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	55.2462	1.88717	0.943587	−	0.331126i	$-0.107429\pi$
$857$	55.2462	1.88717	0.943587	+	0.331126i	$0.107429\pi$
$858$	0	0
$859$	24.2278	0.826641	0.413320	−	0.910586i	$-0.364369\pi$
$859$	24.2278	0.826641	0.413320	+	0.910586i	$0.364369\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−53.9889	−1.83780	−0.918902	−	0.394486i	$-0.870923\pi$
$863$	−53.9889	−1.83780	−0.918902	+	0.394486i	$0.870923\pi$
$864$	0	0
$865$	−9.80127	−0.333253
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.9598	0.439631
$870$	0	0
$871$	−25.9860	−0.880501
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−61.5236	−2.07988
$876$	0	0
$877$	25.8149	0.871708	0.435854	−	0.900017i	$-0.356446\pi$
$877$	25.8149	0.871708	0.435854	+	0.900017i	$0.356446\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.25617	−0.0423215	−0.0211608	−	0.999776i	$-0.506736\pi$
$881$	−1.25617	−0.0423215	−0.0211608	+	0.999776i	$0.506736\pi$
$882$	0	0
$883$	−35.2995	−1.18792	−0.593962	−	0.804493i	$-0.702437\pi$
$883$	−35.2995	−1.18792	−0.593962	+	0.804493i	$0.702437\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	56.0079	1.88056	0.940280	−	0.340402i	$-0.110563\pi$
$887$	56.0079	1.88056	0.940280	+	0.340402i	$0.110563\pi$
$888$	0	0
$889$	70.8487	2.37619
$890$	0	0
$891$	0	0
$892$	0	0
$893$	88.4843	2.96102
$894$	0	0
$895$	52.4237	1.75233
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.405533	0.0135253
$900$	0	0
$901$	−21.7324	−0.724010
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.0890061	−0.00295866
$906$	0	0
$907$	11.4839	0.381317	0.190658	−	0.981656i	$-0.438938\pi$
$907$	11.4839	0.381317	0.190658	+	0.981656i	$0.438938\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42.3400	−1.40279	−0.701394	−	0.712774i	$-0.747439\pi$
$911$	−42.3400	−1.40279	−0.701394	+	0.712774i	$0.747439\pi$
$912$	0	0
$913$	19.8297	0.656267
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−46.1476	−1.52393
$918$	0	0
$919$	−29.5614	−0.975139	−0.487570	−	0.873084i	$-0.662116\pi$
$919$	−29.5614	−0.975139	−0.487570	+	0.873084i	$0.662116\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.3107	0.503956
$924$	0	0
$925$	−11.7061	−0.384893
$926$	0	0
$927$	0	0
$928$	0	0
$929$	31.2597	1.02560	0.512799	−	0.858509i	$-0.328609\pi$
$929$	31.2597	1.02560	0.512799	+	0.858509i	$0.328609\pi$
$930$	0	0
$931$	−130.972	−4.29244
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20.5973	0.673604
$936$	0	0
$937$	−43.8181	−1.43147	−0.715737	−	0.698370i	$-0.753909\pi$
$937$	−43.8181	−1.43147	−0.715737	+	0.698370i	$0.753909\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.7803	−0.416626	−0.208313	−	0.978062i	$-0.566797\pi$
$941$	−12.7803	−0.416626	−0.208313	+	0.978062i	$0.566797\pi$
$942$	0	0
$943$	−73.1255	−2.38130
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.16371	0.265285	0.132642	−	0.991164i	$-0.457654\pi$
$947$	8.16371	0.265285	0.132642	+	0.991164i	$0.457654\pi$
$948$	0	0
$949$	26.9174	0.873775
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.5228	0.340866	0.170433	−	0.985369i	$-0.445483\pi$
$953$	10.5228	0.340866	0.170433	+	0.985369i	$0.445483\pi$
$954$	0	0
$955$	−18.6475	−0.603419
$956$	0	0
$957$	0	0
$958$	0	0
$959$	71.3826	2.30506
$960$	0	0
$961$	−30.9891	−0.999647
$962$	0	0
$963$	0	0
$964$	0	0
$965$	78.4736	2.52616
$966$	0	0
$967$	26.5223	0.852901	0.426451	−	0.904511i	$-0.359764\pi$
$967$	26.5223	0.852901	0.426451	+	0.904511i	$0.359764\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.83362	0.219301	0.109651	−	0.993970i	$-0.465027\pi$
$971$	6.83362	0.219301	0.109651	+	0.993970i	$0.465027\pi$
$972$	0	0
$973$	71.0167	2.27669
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.8956	−0.604522	−0.302261	−	0.953225i	$-0.597741\pi$
$977$	−18.8956	−0.604522	−0.302261	+	0.953225i	$0.597741\pi$
$978$	0	0
$979$	23.5062	0.751260
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−29.3473	−0.936034	−0.468017	−	0.883719i	$-0.655031\pi$
$983$	−29.3473	−0.936034	−0.468017	+	0.883719i	$0.655031\pi$
$984$	0	0
$985$	−15.6940	−0.500052
$986$	0	0
$987$	0	0
$988$	0	0
$989$	35.4288	1.12657
$990$	0	0
$991$	−0.388734	−0.0123486	−0.00617428	−	0.999981i	$-0.501965\pi$
$991$	−0.388734	−0.0123486	−0.00617428	+	0.999981i	$0.501965\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−19.2260	−0.609506
$996$	0	0
$997$	−21.2097	−0.671720	−0.335860	−	0.941912i	$-0.609027\pi$
$997$	−21.2097	−0.671720	−0.335860	+	0.941912i	$0.609027\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.j.1.2	✓	10
3.2	odd	2		6012.2.a.k.1.9	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.j.1.2	✓	10	1.1	even	1	trivial
6012.2.a.k.1.9	yes	10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-3.67303 q^{5} +4.79792 q^{7} +O(q^{10})\)	\(q-3.67303 q^{5} +4.79792 q^{7} -1.91517 q^{11} +3.05652 q^{13} +2.92806 q^{17} -8.17555 q^{19} -6.97269 q^{23} +8.49112 q^{25} +3.87744 q^{29} +0.104588 q^{31} -17.6229 q^{35} -1.37862 q^{37} +10.4874 q^{41} -5.08108 q^{43} -10.8230 q^{47} +16.0200 q^{49} -7.42211 q^{53} +7.03447 q^{55} -0.535117 q^{59} +8.80494 q^{61} -11.2267 q^{65} -8.50182 q^{67} +5.00918 q^{71} +8.80655 q^{73} -9.18882 q^{77} -6.76692 q^{79} -10.3540 q^{83} -10.7548 q^{85} -12.2737 q^{89} +14.6649 q^{91} +30.0290 q^{95} -12.4388 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{5} + 4 q^{7}+O(q^{10}) \) 10 * q - 6 * q^5 + 4 * q^7	\( 10 q - 6 q^{5} + 4 q^{7} - 8 q^{11} - 2 q^{13} - 6 q^{17} - 20 q^{23} + 24 q^{25} - 8 q^{29} - 4 q^{31} - 4 q^{37} + 14 q^{41} + 20 q^{43} - 48 q^{47} - 2 q^{49} - 22 q^{53} - 6 q^{55} - 2 q^{59} - 8 q^{61} - 28 q^{65} - 6 q^{67} - 20 q^{71} + 20 q^{73} - 24 q^{77} - 4 q^{79} - 46 q^{83} - 18 q^{85} + 8 q^{89} + 28 q^{91} - 36 q^{95} - 34 q^{97}+O(q^{100}) \) 10 * q - 6 * q^5 + 4 * q^7 - 8 * q^11 - 2 * q^13 - 6 * q^17 - 20 * q^23 + 24 * q^25 - 8 * q^29 - 4 * q^31 - 4 * q^37 + 14 * q^41 + 20 * q^43 - 48 * q^47 - 2 * q^49 - 22 * q^53 - 6 * q^55 - 2 * q^59 - 8 * q^61 - 28 * q^65 - 6 * q^67 - 20 * q^71 + 20 * q^73 - 24 * q^77 - 4 * q^79 - 46 * q^83 - 18 * q^85 + 8 * q^89 + 28 * q^91 - 36 * q^95 - 34 * q^97

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