LMFDB - Embedded newform 6012.2.a.i.1.6

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:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.419033$ 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)

$f(q)$	$=$	$q+0.419033 q^{5} -4.23221 q^{7} +O(q^{10})$	$q+0.419033 q^{5} -4.23221 q^{7} +2.07280 q^{11} -5.80837 q^{13} +0.0751145 q^{17} -3.35395 q^{19} -4.32486 q^{23} -4.82441 q^{25} -8.99097 q^{29} +6.91570 q^{31} -1.77344 q^{35} +6.31045 q^{37} -4.28666 q^{41} -1.06448 q^{43} +9.61226 q^{47} +10.9116 q^{49} +14.0744 q^{53} +0.868572 q^{55} -3.20748 q^{59} +11.7562 q^{61} -2.43390 q^{65} -5.13807 q^{67} +0.458914 q^{71} +11.5309 q^{73} -8.77254 q^{77} -4.93027 q^{79} +9.02732 q^{83} +0.0314754 q^{85} -13.3953 q^{89} +24.5823 q^{91} -1.40541 q^{95} +10.7908 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - q^5 + 2 * q^7	$ 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) $ 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * 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
$5$	0.419033	0.187397	0.0936986	−	0.995601i	$-0.470131\pi$
$5$	0.419033	0.187397	0.0936986	+	0.995601i	$0.470131\pi$
$6$	0	0
$7$	−4.23221	−1.59963	−0.799813	−	0.600249i	$-0.795068\pi$
$7$	−4.23221	−1.59963	−0.799813	+	0.600249i	$0.795068\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.07280	0.624973	0.312487	−	0.949922i	$-0.398838\pi$
$11$	2.07280	0.624973	0.312487	+	0.949922i	$0.398838\pi$
$12$	0	0
$13$	−5.80837	−1.61095	−0.805476	−	0.592628i	$-0.798090\pi$
$13$	−5.80837	−1.61095	−0.805476	+	0.592628i	$0.798090\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.0751145	0.0182179	0.00910897	−	0.999959i	$-0.497100\pi$
$17$	0.0751145	0.0182179	0.00910897	+	0.999959i	$0.497100\pi$
$18$	0	0
$19$	−3.35395	−0.769448	−0.384724	−	0.923032i	$-0.625703\pi$
$19$	−3.35395	−0.769448	−0.384724	+	0.923032i	$0.625703\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.32486	−0.901796	−0.450898	−	0.892576i	$-0.648896\pi$
$23$	−4.32486	−0.901796	−0.450898	+	0.892576i	$0.648896\pi$
$24$	0	0
$25$	−4.82441	−0.964882
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.99097	−1.66958	−0.834791	−	0.550567i	$-0.814411\pi$
$29$	−8.99097	−1.66958	−0.834791	+	0.550567i	$0.814411\pi$
$30$	0	0
$31$	6.91570	1.24210	0.621048	−	0.783773i	$-0.286707\pi$
$31$	6.91570	1.24210	0.621048	+	0.783773i	$0.286707\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.77344	−0.299765
$36$	0	0
$37$	6.31045	1.03743	0.518716	−	0.854947i	$-0.326410\pi$
$37$	6.31045	1.03743	0.518716	+	0.854947i	$0.326410\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.28666	−0.669464	−0.334732	−	0.942313i	$-0.608646\pi$
$41$	−4.28666	−0.669464	−0.334732	+	0.942313i	$0.608646\pi$
$42$	0	0
$43$	−1.06448	−0.162332	−0.0811659	−	0.996701i	$-0.525864\pi$
$43$	−1.06448	−0.162332	−0.0811659	+	0.996701i	$0.525864\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.61226	1.40209	0.701046	−	0.713116i	$-0.252717\pi$
$47$	9.61226	1.40209	0.701046	+	0.713116i	$0.252717\pi$
$48$	0	0
$49$	10.9116	1.55880
$50$	0	0
$51$	0	0
$52$	0	0
$53$	14.0744	1.93327	0.966636	−	0.256155i	$-0.0824557\pi$
$53$	14.0744	1.93327	0.966636	+	0.256155i	$0.0824557\pi$
$54$	0	0
$55$	0.868572	0.117118
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.20748	−0.417578	−0.208789	−	0.977961i	$-0.566952\pi$
$59$	−3.20748	−0.417578	−0.208789	+	0.977961i	$0.566952\pi$
$60$	0	0
$61$	11.7562	1.50523	0.752616	−	0.658460i	$-0.228792\pi$
$61$	11.7562	1.50523	0.752616	+	0.658460i	$0.228792\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.43390	−0.301888
$66$	0	0
$67$	−5.13807	−0.627715	−0.313858	−	0.949470i	$-0.601621\pi$
$67$	−5.13807	−0.627715	−0.313858	+	0.949470i	$0.601621\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.458914	0.0544630	0.0272315	−	0.999629i	$-0.491331\pi$
$71$	0.458914	0.0544630	0.0272315	+	0.999629i	$0.491331\pi$
$72$	0	0
$73$	11.5309	1.34960	0.674798	−	0.738003i	$-0.264231\pi$
$73$	11.5309	1.34960	0.674798	+	0.738003i	$0.264231\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.77254	−0.999724
$78$	0	0
$79$	−4.93027	−0.554698	−0.277349	−	0.960769i	$-0.589456\pi$
$79$	−4.93027	−0.554698	−0.277349	+	0.960769i	$0.589456\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.02732	0.990877	0.495439	−	0.868643i	$-0.335007\pi$
$83$	9.02732	0.990877	0.495439	+	0.868643i	$0.335007\pi$
$84$	0	0
$85$	0.0314754	0.00341399
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.3953	−1.41990	−0.709948	−	0.704254i	$-0.751282\pi$
$89$	−13.3953	−1.41990	−0.709948	+	0.704254i	$0.751282\pi$
$90$	0	0
$91$	24.5823	2.57692
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.40541	−0.144192
$96$	0	0
$97$	10.7908	1.09564	0.547818	−	0.836598i	$-0.315459\pi$
$97$	10.7908	1.09564	0.547818	+	0.836598i	$0.315459\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.08758	0.108218	0.0541092	−	0.998535i	$-0.482768\pi$
$101$	1.08758	0.108218	0.0541092	+	0.998535i	$0.482768\pi$
$102$	0	0
$103$	8.55080	0.842535	0.421268	−	0.906936i	$-0.361585\pi$
$103$	8.55080	0.842535	0.421268	+	0.906936i	$0.361585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.28422	−0.124150	−0.0620752	−	0.998071i	$-0.519772\pi$
$107$	−1.28422	−0.124150	−0.0620752	+	0.998071i	$0.519772\pi$
$108$	0	0
$109$	−2.12627	−0.203660	−0.101830	−	0.994802i	$-0.532470\pi$
$109$	−2.12627	−0.203660	−0.101830	+	0.994802i	$0.532470\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.70039	−0.912536	−0.456268	−	0.889842i	$-0.650814\pi$
$113$	−9.70039	−0.912536	−0.456268	+	0.889842i	$0.650814\pi$
$114$	0	0
$115$	−1.81226	−0.168994
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.317900	−0.0291419
$120$	0	0
$121$	−6.70349	−0.609408
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.11675	−0.368213
$126$	0	0
$127$	−7.62835	−0.676907	−0.338454	−	0.940983i	$-0.609904\pi$
$127$	−7.62835	−0.676907	−0.338454	+	0.940983i	$0.609904\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.58643	−0.225977	−0.112989	−	0.993596i	$-0.536042\pi$
$131$	−2.58643	−0.225977	−0.112989	+	0.993596i	$0.536042\pi$
$132$	0	0
$133$	14.1946	1.23083
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.21345	0.359979	0.179990	−	0.983668i	$-0.442394\pi$
$137$	4.21345	0.359979	0.179990	+	0.983668i	$0.442394\pi$
$138$	0	0
$139$	−7.30842	−0.619892	−0.309946	−	0.950754i	$-0.600311\pi$
$139$	−7.30842	−0.619892	−0.309946	+	0.950754i	$0.600311\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.0396	−1.00680
$144$	0	0
$145$	−3.76751	−0.312875
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.66285	−0.709689	−0.354844	−	0.934925i	$-0.615466\pi$
$149$	−8.66285	−0.709689	−0.354844	+	0.934925i	$0.615466\pi$
$150$	0	0
$151$	7.01818	0.571132	0.285566	−	0.958359i	$-0.407818\pi$
$151$	7.01818	0.571132	0.285566	+	0.958359i	$0.407818\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.89790	0.232765
$156$	0	0
$157$	13.1769	1.05163	0.525814	−	0.850599i	$-0.323761\pi$
$157$	13.1769	1.05163	0.525814	+	0.850599i	$0.323761\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.3037	1.44254
$162$	0	0
$163$	8.40709	0.658494	0.329247	−	0.944244i	$-0.393205\pi$
$163$	8.40709	0.658494	0.329247	+	0.944244i	$0.393205\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	20.7372	1.59517
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.397914	−0.0302529	−0.0151264	−	0.999886i	$-0.504815\pi$
$173$	−0.397914	−0.0302529	−0.0151264	+	0.999886i	$0.504815\pi$
$174$	0	0
$175$	20.4179	1.54345
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.3288	1.66893	0.834466	−	0.551059i	$-0.185776\pi$
$179$	22.3288	1.66893	0.834466	+	0.551059i	$0.185776\pi$
$180$	0	0
$181$	−6.82249	−0.507112	−0.253556	−	0.967321i	$-0.581600\pi$
$181$	−6.82249	−0.507112	−0.253556	+	0.967321i	$0.581600\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.64428	0.194412
$186$	0	0
$187$	0.155697	0.0113857
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.9092	0.789361	0.394681	−	0.918818i	$-0.370855\pi$
$191$	10.9092	0.789361	0.394681	+	0.918818i	$0.370855\pi$
$192$	0	0
$193$	−9.78214	−0.704134	−0.352067	−	0.935975i	$-0.614521\pi$
$193$	−9.78214	−0.704134	−0.352067	+	0.935975i	$0.614521\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.2135	−1.08391	−0.541957	−	0.840406i	$-0.682316\pi$
$197$	−15.2135	−1.08391	−0.541957	+	0.840406i	$0.682316\pi$
$198$	0	0
$199$	21.8623	1.54978	0.774888	−	0.632099i	$-0.217806\pi$
$199$	21.8623	1.54978	0.774888	+	0.632099i	$0.217806\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	38.0517	2.67071
$204$	0	0
$205$	−1.79625	−0.125456
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.95207	−0.480885
$210$	0	0
$211$	5.76231	0.396693	0.198347	−	0.980132i	$-0.436443\pi$
$211$	5.76231	0.396693	0.198347	+	0.980132i	$0.436443\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.446052	−0.0304205
$216$	0	0
$217$	−29.2687	−1.98689
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.436293	−0.0293482
$222$	0	0
$223$	13.1968	0.883722	0.441861	−	0.897083i	$-0.354318\pi$
$223$	13.1968	0.883722	0.441861	+	0.897083i	$0.354318\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.5195	−1.49467	−0.747335	−	0.664447i	$-0.768667\pi$
$227$	−22.5195	−1.49467	−0.747335	+	0.664447i	$0.768667\pi$
$228$	0	0
$229$	5.47130	0.361554	0.180777	−	0.983524i	$-0.442139\pi$
$229$	5.47130	0.361554	0.180777	+	0.983524i	$0.442139\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.9451	1.43767	0.718837	−	0.695178i	$-0.244675\pi$
$233$	21.9451	1.43767	0.718837	+	0.695178i	$0.244675\pi$
$234$	0	0
$235$	4.02785	0.262748
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.51693	0.227491	0.113745	−	0.993510i	$-0.463715\pi$
$239$	3.51693	0.227491	0.113745	+	0.993510i	$0.463715\pi$
$240$	0	0
$241$	−21.6462	−1.39435	−0.697177	−	0.716899i	$-0.745561\pi$
$241$	−21.6462	−1.39435	−0.697177	+	0.716899i	$0.745561\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.57233	0.292115
$246$	0	0
$247$	19.4810	1.23955
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.1241	1.20710	0.603551	−	0.797324i	$-0.293752\pi$
$251$	19.1241	1.20710	0.603551	+	0.797324i	$0.293752\pi$
$252$	0	0
$253$	−8.96458	−0.563598
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.7536	−1.73122	−0.865612	−	0.500716i	$-0.833070\pi$
$257$	−27.7536	−1.73122	−0.865612	+	0.500716i	$0.833070\pi$
$258$	0	0
$259$	−26.7072	−1.65950
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.701597	0.0432623	0.0216312	−	0.999766i	$-0.493114\pi$
$263$	0.701597	0.0432623	0.0216312	+	0.999766i	$0.493114\pi$
$264$	0	0
$265$	5.89765	0.362290
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.5331	0.642213	0.321107	−	0.947043i	$-0.395945\pi$
$269$	10.5331	0.642213	0.321107	+	0.947043i	$0.395945\pi$
$270$	0	0
$271$	24.9096	1.51315	0.756576	−	0.653906i	$-0.226871\pi$
$271$	24.9096	1.51315	0.756576	+	0.653906i	$0.226871\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.0001	−0.603026
$276$	0	0
$277$	15.3100	0.919891	0.459945	−	0.887947i	$-0.347869\pi$
$277$	15.3100	0.919891	0.459945	+	0.887947i	$0.347869\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.2673	−1.20904	−0.604522	−	0.796589i	$-0.706636\pi$
$281$	−20.2673	−1.20904	−0.604522	+	0.796589i	$0.706636\pi$
$282$	0	0
$283$	29.3380	1.74396	0.871982	−	0.489538i	$-0.162835\pi$
$283$	29.3380	1.74396	0.871982	+	0.489538i	$0.162835\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.1421	1.07089
$288$	0	0
$289$	−16.9944	−0.999668
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.29085	−0.484357	−0.242178	−	0.970232i	$-0.577862\pi$
$293$	−8.29085	−0.484357	−0.242178	+	0.970232i	$0.577862\pi$
$294$	0	0
$295$	−1.34404	−0.0782529
$296$	0	0
$297$	0	0
$298$	0	0
$299$	25.1204	1.45275
$300$	0	0
$301$	4.50511	0.259670
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.92625	0.282076
$306$	0	0
$307$	−4.92451	−0.281057	−0.140528	−	0.990077i	$-0.544880\pi$
$307$	−4.92451	−0.281057	−0.140528	+	0.990077i	$0.544880\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.34593	−0.416549	−0.208275	−	0.978070i	$-0.566785\pi$
$311$	−7.34593	−0.416549	−0.208275	+	0.978070i	$0.566785\pi$
$312$	0	0
$313$	10.3083	0.582657	0.291328	−	0.956623i	$-0.405903\pi$
$313$	10.3083	0.582657	0.291328	+	0.956623i	$0.405903\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.5279	1.15296	0.576481	−	0.817110i	$-0.304425\pi$
$317$	20.5279	1.15296	0.576481	+	0.817110i	$0.304425\pi$
$318$	0	0
$319$	−18.6365	−1.04344
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.251930	−0.0140178
$324$	0	0
$325$	28.0220	1.55438
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−40.6811	−2.24282
$330$	0	0
$331$	−1.88498	−0.103608	−0.0518039	−	0.998657i	$-0.516497\pi$
$331$	−1.88498	−0.103608	−0.0518039	+	0.998657i	$0.516497\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.15302	−0.117632
$336$	0	0
$337$	−8.26097	−0.450004	−0.225002	−	0.974358i	$-0.572239\pi$
$337$	−8.26097	−0.450004	−0.225002	+	0.974358i	$0.572239\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.3349	0.776277
$342$	0	0
$343$	−16.5548	−0.893877
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.7422	1.70401	0.852004	−	0.523535i	$-0.175387\pi$
$347$	31.7422	1.70401	0.852004	+	0.523535i	$0.175387\pi$
$348$	0	0
$349$	3.31362	0.177374	0.0886870	−	0.996060i	$-0.471733\pi$
$349$	3.31362	0.177374	0.0886870	+	0.996060i	$0.471733\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.8238	1.74704	0.873518	−	0.486791i	$-0.161833\pi$
$353$	32.8238	1.74704	0.873518	+	0.486791i	$0.161833\pi$
$354$	0	0
$355$	0.192300	0.0102062
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.55562	−0.451549	−0.225774	−	0.974180i	$-0.572491\pi$
$359$	−8.55562	−0.451549	−0.225774	+	0.974180i	$0.572491\pi$
$360$	0	0
$361$	−7.75103	−0.407949
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.83184	0.252910
$366$	0	0
$367$	21.2074	1.10702	0.553508	−	0.832844i	$-0.313289\pi$
$367$	21.2074	1.10702	0.553508	+	0.832844i	$0.313289\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−59.5660	−3.09251
$372$	0	0
$373$	−4.30504	−0.222906	−0.111453	−	0.993770i	$-0.535551\pi$
$373$	−4.30504	−0.222906	−0.111453	+	0.993770i	$0.535551\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	52.2229	2.68962
$378$	0	0
$379$	−27.9843	−1.43746	−0.718729	−	0.695290i	$-0.755276\pi$
$379$	−27.9843	−1.43746	−0.718729	+	0.695290i	$0.755276\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.66131	0.0848890	0.0424445	−	0.999099i	$-0.486485\pi$
$383$	1.66131	0.0848890	0.0424445	+	0.999099i	$0.486485\pi$
$384$	0	0
$385$	−3.67598	−0.187345
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.71849	0.137833	0.0689165	−	0.997622i	$-0.478046\pi$
$389$	2.71849	0.137833	0.0689165	+	0.997622i	$0.478046\pi$
$390$	0	0
$391$	−0.324860	−0.0164289
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.06594	−0.103949
$396$	0	0
$397$	−17.0954	−0.857993	−0.428996	−	0.903306i	$-0.641133\pi$
$397$	−17.0954	−0.857993	−0.428996	+	0.903306i	$0.641133\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.89575	−0.244482	−0.122241	−	0.992500i	$-0.539008\pi$
$401$	−4.89575	−0.244482	−0.122241	+	0.992500i	$0.539008\pi$
$402$	0	0
$403$	−40.1690	−2.00096
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.0803	0.648367
$408$	0	0
$409$	10.6040	0.524332	0.262166	−	0.965023i	$-0.415563\pi$
$409$	10.6040	0.524332	0.262166	+	0.965023i	$0.415563\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.5747	0.667968
$414$	0	0
$415$	3.78274	0.185688
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.5290	−0.807492	−0.403746	−	0.914871i	$-0.632292\pi$
$419$	−16.5290	−0.807492	−0.403746	+	0.914871i	$0.632292\pi$
$420$	0	0
$421$	−4.87460	−0.237573	−0.118787	−	0.992920i	$-0.537900\pi$
$421$	−4.87460	−0.237573	−0.118787	+	0.992920i	$0.537900\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.362383	−0.0175782
$426$	0	0
$427$	−49.7549	−2.40781
$428$	0	0
$429$	0	0
$430$	0	0
$431$	38.9911	1.87814	0.939069	−	0.343730i	$-0.111690\pi$
$431$	38.9911	1.87814	0.939069	+	0.343730i	$0.111690\pi$
$432$	0	0
$433$	−22.1079	−1.06244	−0.531218	−	0.847235i	$-0.678266\pi$
$433$	−22.1079	−1.06244	−0.531218	+	0.847235i	$0.678266\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.5054	0.693885
$438$	0	0
$439$	1.62530	0.0775714	0.0387857	−	0.999248i	$-0.487651\pi$
$439$	1.62530	0.0775714	0.0387857	+	0.999248i	$0.487651\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.4447	0.876333	0.438167	−	0.898894i	$-0.355628\pi$
$443$	18.4447	0.876333	0.438167	+	0.898894i	$0.355628\pi$
$444$	0	0
$445$	−5.61306	−0.266085
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.6369	−1.16269	−0.581344	−	0.813658i	$-0.697473\pi$
$449$	−24.6369	−1.16269	−0.581344	+	0.813658i	$0.697473\pi$
$450$	0	0
$451$	−8.88541	−0.418397
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.3008	0.482908
$456$	0	0
$457$	1.21824	0.0569869	0.0284934	−	0.999594i	$-0.490929\pi$
$457$	1.21824	0.0569869	0.0284934	+	0.999594i	$0.490929\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−38.3645	−1.78681	−0.893406	−	0.449250i	$-0.851691\pi$
$461$	−38.3645	−1.78681	−0.893406	+	0.449250i	$0.851691\pi$
$462$	0	0
$463$	26.0196	1.20923	0.604617	−	0.796517i	$-0.293326\pi$
$463$	26.0196	1.20923	0.604617	+	0.796517i	$0.293326\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.06356	−0.373137	−0.186569	−	0.982442i	$-0.559737\pi$
$467$	−8.06356	−0.373137	−0.186569	+	0.982442i	$0.559737\pi$
$468$	0	0
$469$	21.7454	1.00411
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.20646	−0.101453
$474$	0	0
$475$	16.1808	0.742427
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.6001	1.44385	0.721923	−	0.691973i	$-0.243258\pi$
$479$	31.6001	1.44385	0.721923	+	0.691973i	$0.243258\pi$
$480$	0	0
$481$	−36.6534	−1.67125
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.52168	0.205319
$486$	0	0
$487$	36.5964	1.65834	0.829170	−	0.558996i	$-0.188813\pi$
$487$	36.5964	1.65834	0.829170	+	0.558996i	$0.188813\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.1135	0.817451	0.408726	−	0.912657i	$-0.365973\pi$
$491$	18.1135	0.817451	0.408726	+	0.912657i	$0.365973\pi$
$492$	0	0
$493$	−0.675352	−0.0304163
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.94222	−0.0871205
$498$	0	0
$499$	11.8702	0.531383	0.265691	−	0.964058i	$-0.414400\pi$
$499$	11.8702	0.531383	0.265691	+	0.964058i	$0.414400\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.76481	0.123277	0.0616384	−	0.998099i	$-0.480367\pi$
$503$	2.76481	0.123277	0.0616384	+	0.998099i	$0.480367\pi$
$504$	0	0
$505$	0.455732	0.0202798
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.9838	−0.752796	−0.376398	−	0.926458i	$-0.622837\pi$
$509$	−16.9838	−0.752796	−0.376398	+	0.926458i	$0.622837\pi$
$510$	0	0
$511$	−48.8014	−2.15885
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.58306	0.157889
$516$	0	0
$517$	19.9243	0.876270
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.9133	−0.916229	−0.458115	−	0.888893i	$-0.651475\pi$
$521$	−20.9133	−0.916229	−0.458115	+	0.888893i	$0.651475\pi$
$522$	0	0
$523$	−8.41272	−0.367862	−0.183931	−	0.982939i	$-0.558882\pi$
$523$	−8.41272	−0.367862	−0.183931	+	0.982939i	$0.558882\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.519469	0.0226284
$528$	0	0
$529$	−4.29559	−0.186765
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.8985	1.07848
$534$	0	0
$535$	−0.538131	−0.0232654
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22.6176	0.974211
$540$	0	0
$541$	−11.9997	−0.515907	−0.257954	−	0.966157i	$-0.583048\pi$
$541$	−11.9997	−0.515907	−0.257954	+	0.966157i	$0.583048\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.890978	−0.0381653
$546$	0	0
$547$	−33.6543	−1.43896	−0.719478	−	0.694516i	$-0.755619\pi$
$547$	−33.6543	−1.43896	−0.719478	+	0.694516i	$0.755619\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	30.1552	1.28466
$552$	0	0
$553$	20.8659	0.887310
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.94121	−0.166994	−0.0834972	−	0.996508i	$-0.526609\pi$
$557$	−3.94121	−0.166994	−0.0834972	+	0.996508i	$0.526609\pi$
$558$	0	0
$559$	6.18290	0.261509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.1203	1.64872	0.824361	−	0.566064i	$-0.191534\pi$
$563$	39.1203	1.64872	0.824361	+	0.566064i	$0.191534\pi$
$564$	0	0
$565$	−4.06478	−0.171007
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.3792	−1.27356	−0.636781	−	0.771045i	$-0.719734\pi$
$569$	−30.3792	−1.27356	−0.636781	+	0.771045i	$0.719734\pi$
$570$	0	0
$571$	14.1764	0.593263	0.296631	−	0.954992i	$-0.404137\pi$
$571$	14.1764	0.593263	0.296631	+	0.954992i	$0.404137\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	20.8649	0.870127
$576$	0	0
$577$	1.21416	0.0505461	0.0252731	−	0.999681i	$-0.491954\pi$
$577$	1.21416	0.0505461	0.0252731	+	0.999681i	$0.491954\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−38.2055	−1.58503
$582$	0	0
$583$	29.1735	1.20824
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−47.1128	−1.94455	−0.972277	−	0.233833i	$-0.924873\pi$
$587$	−47.1128	−1.94455	−0.972277	+	0.233833i	$0.924873\pi$
$588$	0	0
$589$	−23.1949	−0.955729
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.1617	0.910074	0.455037	−	0.890473i	$-0.349626\pi$
$593$	22.1617	0.910074	0.455037	+	0.890473i	$0.349626\pi$
$594$	0	0
$595$	−0.133211	−0.00546111
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.3831	−0.873690	−0.436845	−	0.899537i	$-0.643904\pi$
$599$	−21.3831	−0.873690	−0.436845	+	0.899537i	$0.643904\pi$
$600$	0	0
$601$	37.6389	1.53532	0.767662	−	0.640855i	$-0.221420\pi$
$601$	37.6389	1.53532	0.767662	+	0.640855i	$0.221420\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.80898	−0.114201
$606$	0	0
$607$	−13.2663	−0.538463	−0.269231	−	0.963076i	$-0.586770\pi$
$607$	−13.2663	−0.538463	−0.269231	+	0.963076i	$0.586770\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−55.8316	−2.25870
$612$	0	0
$613$	28.5709	1.15397	0.576984	−	0.816756i	$-0.304230\pi$
$613$	28.5709	1.15397	0.576984	+	0.816756i	$0.304230\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.5762	−0.868627	−0.434314	−	0.900762i	$-0.643009\pi$
$617$	−21.5762	−0.868627	−0.434314	+	0.900762i	$0.643009\pi$
$618$	0	0
$619$	14.3494	0.576752	0.288376	−	0.957517i	$-0.406885\pi$
$619$	14.3494	0.576752	0.288376	+	0.957517i	$0.406885\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	56.6917	2.27130
$624$	0	0
$625$	22.3970	0.895880
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.474006	0.0188999
$630$	0	0
$631$	28.1153	1.11925	0.559626	−	0.828745i	$-0.310945\pi$
$631$	28.1153	1.11925	0.559626	+	0.828745i	$0.310945\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.19653	−0.126850
$636$	0	0
$637$	−63.3788	−2.51116
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.1878	1.27134	0.635672	−	0.771959i	$-0.280723\pi$
$641$	32.1878	1.27134	0.635672	+	0.771959i	$0.280723\pi$
$642$	0	0
$643$	−14.6701	−0.578534	−0.289267	−	0.957249i	$-0.593411\pi$
$643$	−14.6701	−0.578534	−0.289267	+	0.957249i	$0.593411\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.94159	−0.194274	−0.0971370	−	0.995271i	$-0.530969\pi$
$647$	−4.94159	−0.194274	−0.0971370	+	0.995271i	$0.530969\pi$
$648$	0	0
$649$	−6.64846	−0.260975
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.9200	−0.622996	−0.311498	−	0.950247i	$-0.600831\pi$
$653$	−15.9200	−0.622996	−0.311498	+	0.950247i	$0.600831\pi$
$654$	0	0
$655$	−1.08380	−0.0423475
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.4683	0.641513	0.320757	−	0.947162i	$-0.396063\pi$
$659$	16.4683	0.641513	0.320757	+	0.947162i	$0.396063\pi$
$660$	0	0
$661$	29.8115	1.15953	0.579767	−	0.814783i	$-0.303144\pi$
$661$	29.8115	1.15953	0.579767	+	0.814783i	$0.303144\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.94801	0.230654
$666$	0	0
$667$	38.8847	1.50562
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24.3683	0.940730
$672$	0	0
$673$	20.2024	0.778744	0.389372	−	0.921081i	$-0.372692\pi$
$673$	20.2024	0.778744	0.389372	+	0.921081i	$0.372692\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.25637	−0.163586	−0.0817929	−	0.996649i	$-0.526065\pi$
$677$	−4.25637	−0.163586	−0.0817929	+	0.996649i	$0.526065\pi$
$678$	0	0
$679$	−45.6688	−1.75261
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−37.8779	−1.44936	−0.724679	−	0.689087i	$-0.758012\pi$
$683$	−37.8779	−1.44936	−0.724679	+	0.689087i	$0.758012\pi$
$684$	0	0
$685$	1.76557	0.0674591
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−81.7495	−3.11441
$690$	0	0
$691$	43.0240	1.63671	0.818355	−	0.574713i	$-0.194886\pi$
$691$	43.0240	1.63671	0.818355	+	0.574713i	$0.194886\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.06247	−0.116166
$696$	0	0
$697$	−0.321990	−0.0121963
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34.3231	−1.29637	−0.648184	−	0.761484i	$-0.724471\pi$
$701$	−34.3231	−1.29637	−0.648184	+	0.761484i	$0.724471\pi$
$702$	0	0
$703$	−21.1649	−0.798250
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.60288	−0.173109
$708$	0	0
$709$	35.4476	1.33126	0.665631	−	0.746281i	$-0.268162\pi$
$709$	35.4476	1.33126	0.665631	+	0.746281i	$0.268162\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−29.9094	−1.12012
$714$	0	0
$715$	−5.04499	−0.188672
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.9032	−1.37626	−0.688129	−	0.725589i	$-0.741568\pi$
$719$	−36.9032	−1.37626	−0.688129	+	0.725589i	$0.741568\pi$
$720$	0	0
$721$	−36.1888	−1.34774
$722$	0	0
$723$	0	0
$724$	0	0
$725$	43.3761	1.61095
$726$	0	0
$727$	1.47138	0.0545705	0.0272852	−	0.999628i	$-0.491314\pi$
$727$	1.47138	0.0545705	0.0272852	+	0.999628i	$0.491314\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.0799579	−0.00295735
$732$	0	0
$733$	−21.8906	−0.808548	−0.404274	−	0.914638i	$-0.632476\pi$
$733$	−21.8906	−0.808548	−0.404274	+	0.914638i	$0.632476\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.6502	−0.392305
$738$	0	0
$739$	−19.8564	−0.730429	−0.365215	−	0.930923i	$-0.619004\pi$
$739$	−19.8564	−0.730429	−0.365215	+	0.930923i	$0.619004\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.7189	−1.64058	−0.820288	−	0.571950i	$-0.806187\pi$
$743$	−44.7189	−1.64058	−0.820288	+	0.571950i	$0.806187\pi$
$744$	0	0
$745$	−3.63002	−0.132994
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.43510	0.198594
$750$	0	0
$751$	31.9142	1.16457	0.582283	−	0.812986i	$-0.302160\pi$
$751$	31.9142	1.16457	0.582283	+	0.812986i	$0.302160\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.94085	0.107028
$756$	0	0
$757$	24.7634	0.900039	0.450020	−	0.893019i	$-0.351417\pi$
$757$	24.7634	0.900039	0.450020	+	0.893019i	$0.351417\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39.2201	−1.42173	−0.710863	−	0.703330i	$-0.751696\pi$
$761$	−39.2201	−1.42173	−0.710863	+	0.703330i	$0.751696\pi$
$762$	0	0
$763$	8.99884	0.325780
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.6302	0.672698
$768$	0	0
$769$	−48.0481	−1.73266	−0.866329	−	0.499473i	$-0.833527\pi$
$769$	−48.0481	−1.73266	−0.866329	+	0.499473i	$0.833527\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	54.4339	1.95785	0.978925	−	0.204219i	$-0.0654654\pi$
$773$	54.4339	1.95785	0.978925	+	0.204219i	$0.0654654\pi$
$774$	0	0
$775$	−33.3642	−1.19848
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.3772	0.515118
$780$	0	0
$781$	0.951237	0.0340380
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.52154	0.197072
$786$	0	0
$787$	−26.5959	−0.948042	−0.474021	−	0.880513i	$-0.657198\pi$
$787$	−26.5959	−0.948042	−0.474021	+	0.880513i	$0.657198\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	41.0541	1.45972
$792$	0	0
$793$	−68.2846	−2.42486
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.0860	1.03028	0.515140	−	0.857106i	$-0.327740\pi$
$797$	29.0860	1.03028	0.515140	+	0.857106i	$0.327740\pi$
$798$	0	0
$799$	0.722020	0.0255432
$800$	0	0
$801$	0	0
$802$	0	0
$803$	23.9014	0.843461
$804$	0	0
$805$	7.66986	0.270327
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.9699	1.40527	0.702634	−	0.711552i	$-0.252007\pi$
$809$	39.9699	1.40527	0.702634	+	0.711552i	$0.252007\pi$
$810$	0	0
$811$	−13.5883	−0.477149	−0.238574	−	0.971124i	$-0.576680\pi$
$811$	−13.5883	−0.477149	−0.238574	+	0.971124i	$0.576680\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.52285	0.123400
$816$	0	0
$817$	3.57021	0.124906
$818$	0	0
$819$	0	0
$820$	0	0
$821$	49.0067	1.71035	0.855173	−	0.518343i	$-0.173451\pi$
$821$	49.0067	1.71035	0.855173	+	0.518343i	$0.173451\pi$
$822$	0	0
$823$	44.0006	1.53376	0.766882	−	0.641788i	$-0.221807\pi$
$823$	44.0006	1.53376	0.766882	+	0.641788i	$0.221807\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	49.9051	1.73537	0.867685	−	0.497114i	$-0.165607\pi$
$827$	49.9051	1.73537	0.867685	+	0.497114i	$0.165607\pi$
$828$	0	0
$829$	−18.3711	−0.638054	−0.319027	−	0.947746i	$-0.603356\pi$
$829$	−18.3711	−0.638054	−0.319027	+	0.947746i	$0.603356\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.819621	0.0283982
$834$	0	0
$835$	0.419033	0.0145012
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.8506	−0.650795	−0.325398	−	0.945577i	$-0.605498\pi$
$839$	−18.8506	−0.650795	−0.325398	+	0.945577i	$0.605498\pi$
$840$	0	0
$841$	51.8376	1.78750
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.68956	0.298930
$846$	0	0
$847$	28.3706	0.974825
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−27.2918	−0.935551
$852$	0	0
$853$	−32.3912	−1.10905	−0.554526	−	0.832166i	$-0.687100\pi$
$853$	−32.3912	−1.10905	−0.554526	+	0.832166i	$0.687100\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.9280	1.32975	0.664877	−	0.746953i	$-0.268484\pi$
$857$	38.9280	1.32975	0.664877	+	0.746953i	$0.268484\pi$
$858$	0	0
$859$	−20.0265	−0.683295	−0.341648	−	0.939828i	$-0.610985\pi$
$859$	−20.0265	−0.683295	−0.341648	+	0.939828i	$0.610985\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.32903	0.317564	0.158782	−	0.987314i	$-0.449243\pi$
$863$	9.32903	0.317564	0.158782	+	0.987314i	$0.449243\pi$
$864$	0	0
$865$	−0.166739	−0.00566930
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.2195	−0.346672
$870$	0	0
$871$	29.8438	1.01122
$872$	0	0
$873$	0	0
$874$	0	0
$875$	17.4230	0.589004
$876$	0	0
$877$	55.3456	1.86889	0.934444	−	0.356111i	$-0.115898\pi$
$877$	55.3456	1.86889	0.934444	+	0.356111i	$0.115898\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.8218	0.566740	0.283370	−	0.959011i	$-0.408548\pi$
$881$	16.8218	0.566740	0.283370	+	0.959011i	$0.408548\pi$
$882$	0	0
$883$	−34.5574	−1.16295	−0.581474	−	0.813565i	$-0.697524\pi$
$883$	−34.5574	−1.16295	−0.581474	+	0.813565i	$0.697524\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51.9595	−1.74463	−0.872314	−	0.488945i	$-0.837382\pi$
$887$	−51.9595	−1.74463	−0.872314	+	0.488945i	$0.837382\pi$
$888$	0	0
$889$	32.2848	1.08280
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−32.2390	−1.07884
$894$	0	0
$895$	9.35650	0.312753
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−62.1788	−2.07378
$900$	0	0
$901$	1.05719	0.0352202
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.85885	−0.0950313
$906$	0	0
$907$	28.4886	0.945949	0.472975	−	0.881076i	$-0.343180\pi$
$907$	28.4886	0.945949	0.472975	+	0.881076i	$0.343180\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41.6638	−1.38038	−0.690191	−	0.723627i	$-0.742473\pi$
$911$	−41.6638	−1.38038	−0.690191	+	0.723627i	$0.742473\pi$
$912$	0	0
$913$	18.7119	0.619272
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.9463	0.361479
$918$	0	0
$919$	51.9148	1.71251	0.856255	−	0.516553i	$-0.172785\pi$
$919$	51.9148	1.71251	0.856255	+	0.516553i	$0.172785\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.66554	−0.0877374
$924$	0	0
$925$	−30.4442	−1.00100
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.5365	0.706588	0.353294	−	0.935512i	$-0.385061\pi$
$929$	21.5365	0.706588	0.353294	+	0.935512i	$0.385061\pi$
$930$	0	0
$931$	−36.5970	−1.19942
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.0652423	0.00213365
$936$	0	0
$937$	56.3298	1.84021	0.920107	−	0.391667i	$-0.128101\pi$
$937$	56.3298	1.84021	0.920107	+	0.391667i	$0.128101\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.6528	−0.477667	−0.238833	−	0.971061i	$-0.576765\pi$
$941$	−14.6528	−0.477667	−0.238833	+	0.971061i	$0.576765\pi$
$942$	0	0
$943$	18.5392	0.603720
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.7078	0.607921	0.303960	−	0.952685i	$-0.401691\pi$
$947$	18.7078	0.607921	0.303960	+	0.952685i	$0.401691\pi$
$948$	0	0
$949$	−66.9760	−2.17413
$950$	0	0
$951$	0	0
$952$	0	0
$953$	57.5762	1.86507	0.932537	−	0.361074i	$-0.117590\pi$
$953$	57.5762	1.86507	0.932537	+	0.361074i	$0.117590\pi$
$954$	0	0
$955$	4.57131	0.147924
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.8322	−0.575832
$960$	0	0
$961$	16.8269	0.542802
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.09904	−0.131953
$966$	0	0
$967$	46.5575	1.49719	0.748594	−	0.663028i	$-0.230729\pi$
$967$	46.5575	1.49719	0.748594	+	0.663028i	$0.230729\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−57.5407	−1.84657	−0.923284	−	0.384119i	$-0.874505\pi$
$971$	−57.5407	−1.84657	−0.923284	+	0.384119i	$0.874505\pi$
$972$	0	0
$973$	30.9308	0.991595
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.6548	−0.980735	−0.490367	−	0.871516i	$-0.663137\pi$
$977$	−30.6548	−0.980735	−0.490367	+	0.871516i	$0.663137\pi$
$978$	0	0
$979$	−27.7658	−0.887397
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−51.5579	−1.64444	−0.822221	−	0.569168i	$-0.807265\pi$
$983$	−51.5579	−1.64444	−0.822221	+	0.569168i	$0.807265\pi$
$984$	0	0
$985$	−6.37494	−0.203122
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.60373	0.146390
$990$	0	0
$991$	31.9100	1.01365	0.506827	−	0.862048i	$-0.330818\pi$
$991$	31.9100	1.01365	0.506827	+	0.862048i	$0.330818\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.16101	0.290424
$996$	0	0
$997$	12.2665	0.388485	0.194242	−	0.980954i	$-0.437775\pi$
$997$	12.2665	0.388485	0.194242	+	0.980954i	$0.437775\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.i.1.6		9
3.2	odd	2		2004.2.a.c.1.4	✓	9
12.11	even	2		8016.2.a.bc.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.4	✓	9	3.2	odd	2
6012.2.a.i.1.6		9	1.1	even	1	trivial
8016.2.a.bc.1.4		9	12.11	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.419033 q^{5} -4.23221 q^{7} +O(q^{10})\)	\(q+0.419033 q^{5} -4.23221 q^{7} +2.07280 q^{11} -5.80837 q^{13} +0.0751145 q^{17} -3.35395 q^{19} -4.32486 q^{23} -4.82441 q^{25} -8.99097 q^{29} +6.91570 q^{31} -1.77344 q^{35} +6.31045 q^{37} -4.28666 q^{41} -1.06448 q^{43} +9.61226 q^{47} +10.9116 q^{49} +14.0744 q^{53} +0.868572 q^{55} -3.20748 q^{59} +11.7562 q^{61} -2.43390 q^{65} -5.13807 q^{67} +0.458914 q^{71} +11.5309 q^{73} -8.77254 q^{77} -4.93027 q^{79} +9.02732 q^{83} +0.0314754 q^{85} -13.3953 q^{89} +24.5823 q^{91} -1.40541 q^{95} +10.7908 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - q^5 + 2 * q^7	\( 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) \) 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

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