LMFDB - Embedded newform 4016.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.138917.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 2x^{2} + 3x + 1 $ x^5 - 6x^3 - 2x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.50123$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54179 q^{3} +3.81582 q^{5} -2.11419 q^{7} +3.46068 q^{9} +O(q^{10})$	$q+2.54179 q^{3} +3.81582 q^{5} -2.11419 q^{7} +3.46068 q^{9} +0.101432 q^{11} +1.74218 q^{13} +9.69900 q^{15} -1.86048 q^{17} +5.94159 q^{19} -5.37382 q^{21} -6.27187 q^{23} +9.56047 q^{25} +1.17095 q^{27} +4.57076 q^{29} +6.24753 q^{31} +0.257818 q^{33} -8.06736 q^{35} +9.81711 q^{37} +4.42826 q^{39} -1.83614 q^{41} -5.43989 q^{43} +13.2053 q^{45} -2.44200 q^{47} -2.53020 q^{49} -4.72896 q^{51} -1.44282 q^{53} +0.387045 q^{55} +15.1023 q^{57} -8.05923 q^{59} +7.70064 q^{61} -7.31654 q^{63} +6.64785 q^{65} +14.5153 q^{67} -15.9418 q^{69} -11.5924 q^{71} +9.82066 q^{73} +24.3007 q^{75} -0.214446 q^{77} +9.98148 q^{79} -7.40573 q^{81} -2.61542 q^{83} -7.09927 q^{85} +11.6179 q^{87} -4.93696 q^{89} -3.68330 q^{91} +15.8799 q^{93} +22.6720 q^{95} -8.14100 q^{97} +0.351023 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9	$ 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9} - 7 q^{11} + 4 q^{13} - 3 q^{15} + 6 q^{17} + 10 q^{19} + 2 q^{21} - 13 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{29} + 13 q^{31} + 6 q^{33} - 13 q^{35} + 16 q^{37} + 16 q^{39} + 6 q^{41} + 8 q^{43} + 26 q^{45} - 29 q^{47} + 4 q^{49} + 7 q^{51} + 25 q^{53} - q^{55} + 19 q^{57} - 11 q^{59} + 11 q^{61} - 15 q^{63} + 7 q^{65} + 6 q^{67} - 12 q^{69} - 15 q^{71} - 9 q^{73} + 25 q^{75} + 12 q^{77} + 29 q^{79} - 7 q^{81} + 9 q^{83} - 7 q^{85} + 11 q^{87} - 9 q^{89} + 34 q^{91} - 8 q^{93} + 13 q^{95} - 6 q^{97} + q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9 - 7 * q^11 + 4 * q^13 - 3 * q^15 + 6 * q^17 + 10 * q^19 + 2 * q^21 - 13 * q^23 + 6 * q^25 - 8 * q^27 + 4 * q^29 + 13 * q^31 + 6 * q^33 - 13 * q^35 + 16 * q^37 + 16 * q^39 + 6 * q^41 + 8 * q^43 + 26 * q^45 - 29 * q^47 + 4 * q^49 + 7 * q^51 + 25 * q^53 - q^55 + 19 * q^57 - 11 * q^59 + 11 * q^61 - 15 * q^63 + 7 * q^65 + 6 * q^67 - 12 * q^69 - 15 * q^71 - 9 * q^73 + 25 * q^75 + 12 * q^77 + 29 * q^79 - 7 * q^81 + 9 * q^83 - 7 * q^85 + 11 * q^87 - 9 * q^89 + 34 * q^91 - 8 * q^93 + 13 * q^95 - 6 * q^97 + q^99

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
$3$	2.54179	1.46750	0.733751	−	0.679419i	$-0.237768\pi$
$3$	2.54179	1.46750	0.733751	+	0.679419i	$0.237768\pi$
$4$	0	0
$5$	3.81582	1.70649	0.853243	−	0.521514i	$-0.174632\pi$
$5$	3.81582	1.70649	0.853243	+	0.521514i	$0.174632\pi$
$6$	0	0
$7$	−2.11419	−0.799088	−0.399544	−	0.916714i	$-0.630832\pi$
$7$	−2.11419	−0.799088	−0.399544	+	0.916714i	$0.630832\pi$
$8$	0	0
$9$	3.46068	1.15356
$10$	0	0
$11$	0.101432	0.0305828	0.0152914	−	0.999883i	$-0.495132\pi$
$11$	0.101432	0.0305828	0.0152914	+	0.999883i	$0.495132\pi$
$12$	0	0
$13$	1.74218	0.483194	0.241597	−	0.970377i	$-0.422329\pi$
$13$	1.74218	0.483194	0.241597	+	0.970377i	$0.422329\pi$
$14$	0	0
$15$	9.69900	2.50427
$16$	0	0
$17$	−1.86048	−0.451234	−0.225617	−	0.974216i	$-0.572440\pi$
$17$	−1.86048	−0.451234	−0.225617	+	0.974216i	$0.572440\pi$
$18$	0	0
$19$	5.94159	1.36309	0.681547	−	0.731774i	$-0.261307\pi$
$19$	5.94159	1.36309	0.681547	+	0.731774i	$0.261307\pi$
$20$	0	0
$21$	−5.37382	−1.17266
$22$	0	0
$23$	−6.27187	−1.30778	−0.653888	−	0.756592i	$-0.726863\pi$
$23$	−6.27187	−1.30778	−0.653888	+	0.756592i	$0.726863\pi$
$24$	0	0
$25$	9.56047	1.91209
$26$	0	0
$27$	1.17095	0.225350
$28$	0	0
$29$	4.57076	0.848768	0.424384	−	0.905482i	$-0.360491\pi$
$29$	4.57076	0.848768	0.424384	+	0.905482i	$0.360491\pi$
$30$	0	0
$31$	6.24753	1.12209	0.561045	−	0.827785i	$-0.310400\pi$
$31$	6.24753	1.12209	0.561045	+	0.827785i	$0.310400\pi$
$32$	0	0
$33$	0.257818	0.0448804
$34$	0	0
$35$	−8.06736	−1.36363
$36$	0	0
$37$	9.81711	1.61392	0.806962	−	0.590604i	$-0.201110\pi$
$37$	9.81711	1.61392	0.806962	+	0.590604i	$0.201110\pi$
$38$	0	0
$39$	4.42826	0.709088
$40$	0	0
$41$	−1.83614	−0.286758	−0.143379	−	0.989668i	$-0.545797\pi$
$41$	−1.83614	−0.286758	−0.143379	+	0.989668i	$0.545797\pi$
$42$	0	0
$43$	−5.43989	−0.829575	−0.414787	−	0.909918i	$-0.636144\pi$
$43$	−5.43989	−0.829575	−0.414787	+	0.909918i	$0.636144\pi$
$44$	0	0
$45$	13.2053	1.96853
$46$	0	0
$47$	−2.44200	−0.356202	−0.178101	−	0.984012i	$-0.556995\pi$
$47$	−2.44200	−0.356202	−0.178101	+	0.984012i	$0.556995\pi$
$48$	0	0
$49$	−2.53020	−0.361458
$50$	0	0
$51$	−4.72896	−0.662186
$52$	0	0
$53$	−1.44282	−0.198187	−0.0990936	−	0.995078i	$-0.531594\pi$
$53$	−1.44282	−0.198187	−0.0990936	+	0.995078i	$0.531594\pi$
$54$	0	0
$55$	0.387045	0.0521892
$56$	0	0
$57$	15.1023	2.00034
$58$	0	0
$59$	−8.05923	−1.04922	−0.524611	−	0.851342i	$-0.675789\pi$
$59$	−8.05923	−1.04922	−0.524611	+	0.851342i	$0.675789\pi$
$60$	0	0
$61$	7.70064	0.985966	0.492983	−	0.870039i	$-0.335906\pi$
$61$	7.70064	0.985966	0.492983	+	0.870039i	$0.335906\pi$
$62$	0	0
$63$	−7.31654	−0.921797
$64$	0	0
$65$	6.64785	0.824564
$66$	0	0
$67$	14.5153	1.77333	0.886665	−	0.462412i	$-0.153016\pi$
$67$	14.5153	1.77333	0.886665	+	0.462412i	$0.153016\pi$
$68$	0	0
$69$	−15.9418	−1.91916
$70$	0	0
$71$	−11.5924	−1.37576	−0.687881	−	0.725823i	$-0.741459\pi$
$71$	−11.5924	−1.37576	−0.687881	+	0.725823i	$0.741459\pi$
$72$	0	0
$73$	9.82066	1.14942	0.574710	−	0.818357i	$-0.305115\pi$
$73$	9.82066	1.14942	0.574710	+	0.818357i	$0.305115\pi$
$74$	0	0
$75$	24.3007	2.80600
$76$	0	0
$77$	−0.214446	−0.0244384
$78$	0	0
$79$	9.98148	1.12300	0.561502	−	0.827475i	$-0.310224\pi$
$79$	9.98148	1.12300	0.561502	+	0.827475i	$0.310224\pi$
$80$	0	0
$81$	−7.40573	−0.822859
$82$	0	0
$83$	−2.61542	−0.287080	−0.143540	−	0.989645i	$-0.545849\pi$
$83$	−2.61542	−0.287080	−0.143540	+	0.989645i	$0.545849\pi$
$84$	0	0
$85$	−7.09927	−0.770024
$86$	0	0
$87$	11.6179	1.24557
$88$	0	0
$89$	−4.93696	−0.523317	−0.261658	−	0.965161i	$-0.584269\pi$
$89$	−4.93696	−0.523317	−0.261658	+	0.965161i	$0.584269\pi$
$90$	0	0
$91$	−3.68330	−0.386115
$92$	0	0
$93$	15.8799	1.64667
$94$	0	0
$95$	22.6720	2.32610
$96$	0	0
$97$	−8.14100	−0.826593	−0.413297	−	0.910596i	$-0.635623\pi$
$97$	−8.14100	−0.826593	−0.413297	+	0.910596i	$0.635623\pi$
$98$	0	0
$99$	0.351023	0.0352792
$100$	0	0
$101$	−12.7077	−1.26447	−0.632234	−	0.774778i	$-0.717862\pi$
$101$	−12.7077	−1.26447	−0.632234	+	0.774778i	$0.717862\pi$
$102$	0	0
$103$	−7.72624	−0.761289	−0.380644	−	0.924722i	$-0.624298\pi$
$103$	−7.72624	−0.761289	−0.380644	+	0.924722i	$0.624298\pi$
$104$	0	0
$105$	−20.5055	−2.00113
$106$	0	0
$107$	−8.65027	−0.836253	−0.418127	−	0.908389i	$-0.637313\pi$
$107$	−8.65027	−0.836253	−0.418127	+	0.908389i	$0.637313\pi$
$108$	0	0
$109$	2.22669	0.213278	0.106639	−	0.994298i	$-0.465991\pi$
$109$	2.22669	0.213278	0.106639	+	0.994298i	$0.465991\pi$
$110$	0	0
$111$	24.9530	2.36843
$112$	0	0
$113$	20.5097	1.92939	0.964696	−	0.263366i	$-0.0848328\pi$
$113$	20.5097	1.92939	0.964696	+	0.263366i	$0.0848328\pi$
$114$	0	0
$115$	−23.9323	−2.23170
$116$	0	0
$117$	6.02914	0.557394
$118$	0	0
$119$	3.93342	0.360576
$120$	0	0
$121$	−10.9897	−0.999065
$122$	0	0
$123$	−4.66709	−0.420817
$124$	0	0
$125$	17.4019	1.55648
$126$	0	0
$127$	20.0648	1.78047	0.890233	−	0.455506i	$-0.150542\pi$
$127$	20.0648	1.78047	0.890233	+	0.455506i	$0.150542\pi$
$128$	0	0
$129$	−13.8270	−1.21740
$130$	0	0
$131$	−9.63657	−0.841951	−0.420976	−	0.907072i	$-0.638312\pi$
$131$	−9.63657	−0.841951	−0.420976	+	0.907072i	$0.638312\pi$
$132$	0	0
$133$	−12.5616	−1.08923
$134$	0	0
$135$	4.46815	0.384557
$136$	0	0
$137$	10.3870	0.887425	0.443713	−	0.896169i	$-0.353661\pi$
$137$	10.3870	0.887425	0.443713	+	0.896169i	$0.353661\pi$
$138$	0	0
$139$	2.95698	0.250808	0.125404	−	0.992106i	$-0.459977\pi$
$139$	2.95698	0.250808	0.125404	+	0.992106i	$0.459977\pi$
$140$	0	0
$141$	−6.20704	−0.522727
$142$	0	0
$143$	0.176713	0.0147775
$144$	0	0
$145$	17.4412	1.44841
$146$	0	0
$147$	−6.43124	−0.530440
$148$	0	0
$149$	17.2334	1.41182	0.705909	−	0.708303i	$-0.250539\pi$
$149$	17.2334	1.41182	0.705909	+	0.708303i	$0.250539\pi$
$150$	0	0
$151$	13.7412	1.11825	0.559123	−	0.829085i	$-0.311138\pi$
$151$	13.7412	1.11825	0.559123	+	0.829085i	$0.311138\pi$
$152$	0	0
$153$	−6.43854	−0.520525
$154$	0	0
$155$	23.8394	1.91483
$156$	0	0
$157$	−9.72624	−0.776238	−0.388119	−	0.921609i	$-0.626875\pi$
$157$	−9.72624	−0.776238	−0.388119	+	0.921609i	$0.626875\pi$
$158$	0	0
$159$	−3.66735	−0.290840
$160$	0	0
$161$	13.2599	1.04503
$162$	0	0
$163$	−21.0880	−1.65174	−0.825871	−	0.563858i	$-0.809316\pi$
$163$	−21.0880	−1.65174	−0.825871	+	0.563858i	$0.809316\pi$
$164$	0	0
$165$	0.983787	0.0765877
$166$	0	0
$167$	8.34834	0.646014	0.323007	−	0.946397i	$-0.395306\pi$
$167$	8.34834	0.646014	0.323007	+	0.946397i	$0.395306\pi$
$168$	0	0
$169$	−9.96480	−0.766523
$170$	0	0
$171$	20.5620	1.57241
$172$	0	0
$173$	−14.9169	−1.13411	−0.567055	−	0.823680i	$-0.691917\pi$
$173$	−14.9169	−1.13411	−0.567055	+	0.823680i	$0.691917\pi$
$174$	0	0
$175$	−20.2126	−1.52793
$176$	0	0
$177$	−20.4849	−1.53974
$178$	0	0
$179$	5.63518	0.421193	0.210597	−	0.977573i	$-0.432459\pi$
$179$	5.63518	0.421193	0.210597	+	0.977573i	$0.432459\pi$
$180$	0	0
$181$	0.645898	0.0480092	0.0240046	−	0.999712i	$-0.492358\pi$
$181$	0.645898	0.0480092	0.0240046	+	0.999712i	$0.492358\pi$
$182$	0	0
$183$	19.5734	1.44691
$184$	0	0
$185$	37.4603	2.75414
$186$	0	0
$187$	−0.188712	−0.0138000
$188$	0	0
$189$	−2.47562	−0.180075
$190$	0	0
$191$	−22.5921	−1.63471	−0.817354	−	0.576136i	$-0.804560\pi$
$191$	−22.5921	−1.63471	−0.817354	+	0.576136i	$0.804560\pi$
$192$	0	0
$193$	−12.4273	−0.894538	−0.447269	−	0.894400i	$-0.647603\pi$
$193$	−12.4273	−0.894538	−0.447269	+	0.894400i	$0.647603\pi$
$194$	0	0
$195$	16.8974	1.21005
$196$	0	0
$197$	−14.5297	−1.03520	−0.517599	−	0.855623i	$-0.673174\pi$
$197$	−14.5297	−1.03520	−0.517599	+	0.855623i	$0.673174\pi$
$198$	0	0
$199$	4.37191	0.309916	0.154958	−	0.987921i	$-0.450476\pi$
$199$	4.37191	0.309916	0.154958	+	0.987921i	$0.450476\pi$
$200$	0	0
$201$	36.8949	2.60236
$202$	0	0
$203$	−9.66345	−0.678241
$204$	0	0
$205$	−7.00639	−0.489348
$206$	0	0
$207$	−21.7049	−1.50860
$208$	0	0
$209$	0.602666	0.0416873
$210$	0	0
$211$	20.7745	1.43018	0.715089	−	0.699033i	$-0.246386\pi$
$211$	20.7745	1.43018	0.715089	+	0.699033i	$0.246386\pi$
$212$	0	0
$213$	−29.4654	−2.01893
$214$	0	0
$215$	−20.7576	−1.41566
$216$	0	0
$217$	−13.2085	−0.896649
$218$	0	0
$219$	24.9620	1.68678
$220$	0	0
$221$	−3.24130	−0.218034
$222$	0	0
$223$	2.92005	0.195541	0.0977705	−	0.995209i	$-0.468829\pi$
$223$	2.92005	0.195541	0.0977705	+	0.995209i	$0.468829\pi$
$224$	0	0
$225$	33.0857	2.20572
$226$	0	0
$227$	−12.8562	−0.853297	−0.426648	−	0.904418i	$-0.640306\pi$
$227$	−12.8562	−0.853297	−0.426648	+	0.904418i	$0.640306\pi$
$228$	0	0
$229$	−18.0053	−1.18982	−0.594912	−	0.803791i	$-0.702813\pi$
$229$	−18.0053	−1.18982	−0.594912	+	0.803791i	$0.702813\pi$
$230$	0	0
$231$	−0.545076	−0.0358634
$232$	0	0
$233$	−17.1650	−1.12452	−0.562258	−	0.826962i	$-0.690067\pi$
$233$	−17.1650	−1.12452	−0.562258	+	0.826962i	$0.690067\pi$
$234$	0	0
$235$	−9.31823	−0.607854
$236$	0	0
$237$	25.3708	1.64801
$238$	0	0
$239$	14.1055	0.912408	0.456204	−	0.889875i	$-0.349209\pi$
$239$	14.1055	0.912408	0.456204	+	0.889875i	$0.349209\pi$
$240$	0	0
$241$	−8.40699	−0.541542	−0.270771	−	0.962644i	$-0.587279\pi$
$241$	−8.40699	−0.541542	−0.270771	+	0.962644i	$0.587279\pi$
$242$	0	0
$243$	−22.3366	−1.43290
$244$	0	0
$245$	−9.65480	−0.616823
$246$	0	0
$247$	10.3513	0.658639
$248$	0	0
$249$	−6.64785	−0.421290
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.636167	−0.0399955
$254$	0	0
$255$	−18.0448	−1.13001
$256$	0	0
$257$	−28.9030	−1.80292	−0.901460	−	0.432863i	$-0.857503\pi$
$257$	−28.9030	−1.80292	−0.901460	+	0.432863i	$0.857503\pi$
$258$	0	0
$259$	−20.7552	−1.28967
$260$	0	0
$261$	15.8179	0.979106
$262$	0	0
$263$	19.7404	1.21724	0.608622	−	0.793460i	$-0.291723\pi$
$263$	19.7404	1.21724	0.608622	+	0.793460i	$0.291723\pi$
$264$	0	0
$265$	−5.50555	−0.338204
$266$	0	0
$267$	−12.5487	−0.767968
$268$	0	0
$269$	−11.7553	−0.716736	−0.358368	−	0.933580i	$-0.616667\pi$
$269$	−11.7553	−0.716736	−0.358368	+	0.933580i	$0.616667\pi$
$270$	0	0
$271$	4.39876	0.267206	0.133603	−	0.991035i	$-0.457345\pi$
$271$	4.39876	0.267206	0.133603	+	0.991035i	$0.457345\pi$
$272$	0	0
$273$	−9.36217	−0.566624
$274$	0	0
$275$	0.969736	0.0584773
$276$	0	0
$277$	−9.97716	−0.599470	−0.299735	−	0.954023i	$-0.596898\pi$
$277$	−9.97716	−0.599470	−0.299735	+	0.954023i	$0.596898\pi$
$278$	0	0
$279$	21.6207	1.29440
$280$	0	0
$281$	22.7696	1.35832	0.679160	−	0.733990i	$-0.262344\pi$
$281$	22.7696	1.35832	0.679160	+	0.733990i	$0.262344\pi$
$282$	0	0
$283$	−31.5691	−1.87659	−0.938293	−	0.345841i	$-0.887594\pi$
$283$	−31.5691	−1.87659	−0.938293	+	0.345841i	$0.887594\pi$
$284$	0	0
$285$	57.6275	3.41356
$286$	0	0
$287$	3.88196	0.229145
$288$	0	0
$289$	−13.5386	−0.796388
$290$	0	0
$291$	−20.6927	−1.21303
$292$	0	0
$293$	−15.9976	−0.934591	−0.467296	−	0.884101i	$-0.654772\pi$
$293$	−15.9976	−0.934591	−0.467296	+	0.884101i	$0.654772\pi$
$294$	0	0
$295$	−30.7526	−1.79048
$296$	0	0
$297$	0.118772	0.00689185
$298$	0	0
$299$	−10.9267	−0.631910
$300$	0	0
$301$	11.5009	0.662904
$302$	0	0
$303$	−32.3004	−1.85561
$304$	0	0
$305$	29.3843	1.68254
$306$	0	0
$307$	−5.13270	−0.292939	−0.146469	−	0.989215i	$-0.546791\pi$
$307$	−5.13270	−0.292939	−0.146469	+	0.989215i	$0.546791\pi$
$308$	0	0
$309$	−19.6384	−1.11719
$310$	0	0
$311$	−12.3253	−0.698902	−0.349451	−	0.936955i	$-0.613632\pi$
$311$	−12.3253	−0.698902	−0.349451	+	0.936955i	$0.613632\pi$
$312$	0	0
$313$	27.9751	1.58125	0.790624	−	0.612303i	$-0.209757\pi$
$313$	27.9751	1.58125	0.790624	+	0.612303i	$0.209757\pi$
$314$	0	0
$315$	−27.9186	−1.57303
$316$	0	0
$317$	−9.62715	−0.540715	−0.270357	−	0.962760i	$-0.587142\pi$
$317$	−9.62715	−0.540715	−0.270357	+	0.962760i	$0.587142\pi$
$318$	0	0
$319$	0.463620	0.0259577
$320$	0	0
$321$	−21.9871	−1.22720
$322$	0	0
$323$	−11.0542	−0.615074
$324$	0	0
$325$	16.6561	0.923913
$326$	0	0
$327$	5.65976	0.312986
$328$	0	0
$329$	5.16285	0.284637
$330$	0	0
$331$	21.2278	1.16678	0.583392	−	0.812191i	$-0.301725\pi$
$331$	21.2278	1.16678	0.583392	+	0.812191i	$0.301725\pi$
$332$	0	0
$333$	33.9739	1.86176
$334$	0	0
$335$	55.3879	3.02616
$336$	0	0
$337$	−29.6952	−1.61760	−0.808801	−	0.588082i	$-0.799883\pi$
$337$	−29.6952	−1.61760	−0.808801	+	0.588082i	$0.799883\pi$
$338$	0	0
$339$	52.1313	2.83139
$340$	0	0
$341$	0.633698	0.0343167
$342$	0	0
$343$	20.1487	1.08793
$344$	0	0
$345$	−60.8308	−3.27502
$346$	0	0
$347$	−16.2876	−0.874365	−0.437182	−	0.899373i	$-0.644024\pi$
$347$	−16.2876	−0.874365	−0.437182	+	0.899373i	$0.644024\pi$
$348$	0	0
$349$	−5.96365	−0.319227	−0.159613	−	0.987180i	$-0.551025\pi$
$349$	−5.96365	−0.319227	−0.159613	+	0.987180i	$0.551025\pi$
$350$	0	0
$351$	2.04002	0.108888
$352$	0	0
$353$	−30.4914	−1.62289	−0.811446	−	0.584427i	$-0.801319\pi$
$353$	−30.4914	−1.62289	−0.811446	+	0.584427i	$0.801319\pi$
$354$	0	0
$355$	−44.2344	−2.34772
$356$	0	0
$357$	9.99790	0.529145
$358$	0	0
$359$	11.5459	0.609369	0.304685	−	0.952453i	$-0.401449\pi$
$359$	11.5459	0.609369	0.304685	+	0.952453i	$0.401449\pi$
$360$	0	0
$361$	16.3025	0.858026
$362$	0	0
$363$	−27.9335	−1.46613
$364$	0	0
$365$	37.4738	1.96147
$366$	0	0
$367$	−0.855037	−0.0446326	−0.0223163	−	0.999751i	$-0.507104\pi$
$367$	−0.855037	−0.0446326	−0.0223163	+	0.999751i	$0.507104\pi$
$368$	0	0
$369$	−6.35431	−0.330792
$370$	0	0
$371$	3.05040	0.158369
$372$	0	0
$373$	25.9662	1.34448	0.672240	−	0.740333i	$-0.265332\pi$
$373$	25.9662	1.34448	0.672240	+	0.740333i	$0.265332\pi$
$374$	0	0
$375$	44.2320	2.28413
$376$	0	0
$377$	7.96309	0.410120
$378$	0	0
$379$	−36.7838	−1.88946	−0.944729	−	0.327852i	$-0.893675\pi$
$379$	−36.7838	−1.88946	−0.944729	+	0.327852i	$0.893675\pi$
$380$	0	0
$381$	51.0005	2.61284
$382$	0	0
$383$	33.8652	1.73043	0.865216	−	0.501400i	$-0.167181\pi$
$383$	33.8652	1.73043	0.865216	+	0.501400i	$0.167181\pi$
$384$	0	0
$385$	−0.818287	−0.0417038
$386$	0	0
$387$	−18.8257	−0.956965
$388$	0	0
$389$	8.87865	0.450165	0.225083	−	0.974340i	$-0.427735\pi$
$389$	8.87865	0.450165	0.225083	+	0.974340i	$0.427735\pi$
$390$	0	0
$391$	11.6687	0.590112
$392$	0	0
$393$	−24.4941	−1.23556
$394$	0	0
$395$	38.0875	1.91639
$396$	0	0
$397$	4.46902	0.224294	0.112147	−	0.993692i	$-0.464227\pi$
$397$	4.46902	0.224294	0.112147	+	0.993692i	$0.464227\pi$
$398$	0	0
$399$	−31.9290	−1.59845
$400$	0	0
$401$	20.5134	1.02439	0.512194	−	0.858870i	$-0.328833\pi$
$401$	20.5134	1.02439	0.512194	+	0.858870i	$0.328833\pi$
$402$	0	0
$403$	10.8843	0.542187
$404$	0	0
$405$	−28.2589	−1.40420
$406$	0	0
$407$	0.995768	0.0493584
$408$	0	0
$409$	−15.8005	−0.781284	−0.390642	−	0.920543i	$-0.627747\pi$
$409$	−15.8005	−0.781284	−0.390642	+	0.920543i	$0.627747\pi$
$410$	0	0
$411$	26.4017	1.30230
$412$	0	0
$413$	17.0387	0.838422
$414$	0	0
$415$	−9.97998	−0.489898
$416$	0	0
$417$	7.51601	0.368061
$418$	0	0
$419$	−16.2896	−0.795800	−0.397900	−	0.917429i	$-0.630261\pi$
$419$	−16.2896	−0.795800	−0.397900	+	0.917429i	$0.630261\pi$
$420$	0	0
$421$	22.3954	1.09149	0.545743	−	0.837952i	$-0.316247\pi$
$421$	22.3954	1.09149	0.545743	+	0.837952i	$0.316247\pi$
$422$	0	0
$423$	−8.45098	−0.410901
$424$	0	0
$425$	−17.7871	−0.862801
$426$	0	0
$427$	−16.2806	−0.787874
$428$	0	0
$429$	0.449166	0.0216859
$430$	0	0
$431$	−8.34607	−0.402016	−0.201008	−	0.979590i	$-0.564422\pi$
$431$	−8.34607	−0.402016	−0.201008	+	0.979590i	$0.564422\pi$
$432$	0	0
$433$	−29.7595	−1.43015	−0.715076	−	0.699047i	$-0.753608\pi$
$433$	−29.7595	−1.43015	−0.715076	+	0.699047i	$0.753608\pi$
$434$	0	0
$435$	44.3318	2.12555
$436$	0	0
$437$	−37.2649	−1.78262
$438$	0	0
$439$	5.42837	0.259082	0.129541	−	0.991574i	$-0.458650\pi$
$439$	5.42837	0.259082	0.129541	+	0.991574i	$0.458650\pi$
$440$	0	0
$441$	−8.75623	−0.416963
$442$	0	0
$443$	19.0028	0.902852	0.451426	−	0.892308i	$-0.350915\pi$
$443$	19.0028	0.902852	0.451426	+	0.892308i	$0.350915\pi$
$444$	0	0
$445$	−18.8385	−0.893032
$446$	0	0
$447$	43.8037	2.07184
$448$	0	0
$449$	−8.96165	−0.422926	−0.211463	−	0.977386i	$-0.567823\pi$
$449$	−8.96165	−0.422926	−0.211463	+	0.977386i	$0.567823\pi$
$450$	0	0
$451$	−0.186243	−0.00876986
$452$	0	0
$453$	34.9273	1.64103
$454$	0	0
$455$	−14.0548	−0.658900
$456$	0	0
$457$	31.8051	1.48778	0.743890	−	0.668302i	$-0.232978\pi$
$457$	31.8051	1.48778	0.743890	+	0.668302i	$0.232978\pi$
$458$	0	0
$459$	−2.17854	−0.101686
$460$	0	0
$461$	10.6668	0.496804	0.248402	−	0.968657i	$-0.420095\pi$
$461$	10.6668	0.496804	0.248402	+	0.968657i	$0.420095\pi$
$462$	0	0
$463$	−10.2115	−0.474568	−0.237284	−	0.971440i	$-0.576257\pi$
$463$	−10.2115	−0.474568	−0.237284	+	0.971440i	$0.576257\pi$
$464$	0	0
$465$	60.5948	2.81002
$466$	0	0
$467$	33.7807	1.56319	0.781593	−	0.623789i	$-0.214408\pi$
$467$	33.7807	1.56319	0.781593	+	0.623789i	$0.214408\pi$
$468$	0	0
$469$	−30.6882	−1.41705
$470$	0	0
$471$	−24.7220	−1.13913
$472$	0	0
$473$	−0.551777	−0.0253708
$474$	0	0
$475$	56.8044	2.60636
$476$	0	0
$477$	−4.99315	−0.228621
$478$	0	0
$479$	21.9198	1.00154	0.500770	−	0.865580i	$-0.333050\pi$
$479$	21.9198	1.00154	0.500770	+	0.865580i	$0.333050\pi$
$480$	0	0
$481$	17.1032	0.779839
$482$	0	0
$483$	33.7039	1.53358
$484$	0	0
$485$	−31.0646	−1.41057
$486$	0	0
$487$	5.28289	0.239391	0.119695	−	0.992811i	$-0.461808\pi$
$487$	5.28289	0.239391	0.119695	+	0.992811i	$0.461808\pi$
$488$	0	0
$489$	−53.6013	−2.42394
$490$	0	0
$491$	25.0601	1.13095	0.565473	−	0.824767i	$-0.308693\pi$
$491$	25.0601	1.13095	0.565473	+	0.824767i	$0.308693\pi$
$492$	0	0
$493$	−8.50382	−0.382993
$494$	0	0
$495$	1.33944	0.0602034
$496$	0	0
$497$	24.5085	1.09936
$498$	0	0
$499$	27.3581	1.22472	0.612359	−	0.790580i	$-0.290221\pi$
$499$	27.3581	1.22472	0.612359	+	0.790580i	$0.290221\pi$
$500$	0	0
$501$	21.2197	0.948027
$502$	0	0
$503$	24.7815	1.10495	0.552477	−	0.833528i	$-0.313683\pi$
$503$	24.7815	1.10495	0.552477	+	0.833528i	$0.313683\pi$
$504$	0	0
$505$	−48.4904	−2.15780
$506$	0	0
$507$	−25.3284	−1.12487
$508$	0	0
$509$	−37.1176	−1.64521	−0.822605	−	0.568613i	$-0.807480\pi$
$509$	−37.1176	−1.64521	−0.822605	+	0.568613i	$0.807480\pi$
$510$	0	0
$511$	−20.7627	−0.918489
$512$	0	0
$513$	6.95733	0.307174
$514$	0	0
$515$	−29.4819	−1.29913
$516$	0	0
$517$	−0.247696	−0.0108937
$518$	0	0
$519$	−37.9155	−1.66431
$520$	0	0
$521$	−4.67964	−0.205019	−0.102509	−	0.994732i	$-0.532687\pi$
$521$	−4.67964	−0.205019	−0.102509	+	0.994732i	$0.532687\pi$
$522$	0	0
$523$	−6.75196	−0.295243	−0.147621	−	0.989044i	$-0.547162\pi$
$523$	−6.75196	−0.295243	−0.147621	+	0.989044i	$0.547162\pi$
$524$	0	0
$525$	−51.3762	−2.24224
$526$	0	0
$527$	−11.6234	−0.506325
$528$	0	0
$529$	16.3363	0.710276
$530$	0	0
$531$	−27.8904	−1.21034
$532$	0	0
$533$	−3.19890	−0.138560
$534$	0	0
$535$	−33.0079	−1.42705
$536$	0	0
$537$	14.3234	0.618102
$538$	0	0
$539$	−0.256643	−0.0110544
$540$	0	0
$541$	25.6739	1.10381	0.551903	−	0.833909i	$-0.313902\pi$
$541$	25.6739	1.10381	0.551903	+	0.833909i	$0.313902\pi$
$542$	0	0
$543$	1.64174	0.0704536
$544$	0	0
$545$	8.49663	0.363956
$546$	0	0
$547$	−40.0451	−1.71220	−0.856102	−	0.516807i	$-0.827121\pi$
$547$	−40.0451	−1.71220	−0.856102	+	0.516807i	$0.827121\pi$
$548$	0	0
$549$	26.6495	1.13737
$550$	0	0
$551$	27.1576	1.15695
$552$	0	0
$553$	−21.1027	−0.897380
$554$	0	0
$555$	95.2161	4.04170
$556$	0	0
$557$	26.9643	1.14251	0.571257	−	0.820771i	$-0.306456\pi$
$557$	26.9643	1.14251	0.571257	+	0.820771i	$0.306456\pi$
$558$	0	0
$559$	−9.47727	−0.400846
$560$	0	0
$561$	−0.479666	−0.0202515
$562$	0	0
$563$	−43.8445	−1.84782	−0.923912	−	0.382605i	$-0.875027\pi$
$563$	−43.8445	−1.84782	−0.923912	+	0.382605i	$0.875027\pi$
$564$	0	0
$565$	78.2613	3.29248
$566$	0	0
$567$	15.6571	0.657537
$568$	0	0
$569$	−33.2627	−1.39444	−0.697222	−	0.716855i	$-0.745581\pi$
$569$	−33.2627	−1.39444	−0.697222	+	0.716855i	$0.745581\pi$
$570$	0	0
$571$	−13.5024	−0.565058	−0.282529	−	0.959259i	$-0.591173\pi$
$571$	−13.5024	−0.565058	−0.282529	+	0.959259i	$0.591173\pi$
$572$	0	0
$573$	−57.4243	−2.39894
$574$	0	0
$575$	−59.9620	−2.50059
$576$	0	0
$577$	−12.4818	−0.519625	−0.259812	−	0.965659i	$-0.583661\pi$
$577$	−12.4818	−0.519625	−0.259812	+	0.965659i	$0.583661\pi$
$578$	0	0
$579$	−31.5876	−1.31274
$580$	0	0
$581$	5.52950	0.229402
$582$	0	0
$583$	−0.146348	−0.00606113
$584$	0	0
$585$	23.0061	0.951185
$586$	0	0
$587$	−10.0177	−0.413476	−0.206738	−	0.978396i	$-0.566285\pi$
$587$	−10.0177	−0.413476	−0.206738	+	0.978396i	$0.566285\pi$
$588$	0	0
$589$	37.1203	1.52951
$590$	0	0
$591$	−36.9314	−1.51915
$592$	0	0
$593$	29.5015	1.21148	0.605740	−	0.795663i	$-0.292877\pi$
$593$	29.5015	1.21148	0.605740	+	0.795663i	$0.292877\pi$
$594$	0	0
$595$	15.0092	0.615317
$596$	0	0
$597$	11.1125	0.454803
$598$	0	0
$599$	6.03971	0.246776	0.123388	−	0.992359i	$-0.460624\pi$
$599$	6.03971	0.246776	0.123388	+	0.992359i	$0.460624\pi$
$600$	0	0
$601$	−25.6442	−1.04605	−0.523024	−	0.852318i	$-0.675196\pi$
$601$	−25.6442	−1.04605	−0.523024	+	0.852318i	$0.675196\pi$
$602$	0	0
$603$	50.2329	2.04564
$604$	0	0
$605$	−41.9347	−1.70489
$606$	0	0
$607$	−31.9515	−1.29687	−0.648435	−	0.761270i	$-0.724576\pi$
$607$	−31.9515	−1.29687	−0.648435	+	0.761270i	$0.724576\pi$
$608$	0	0
$609$	−24.5624	−0.995320
$610$	0	0
$611$	−4.25441	−0.172115
$612$	0	0
$613$	28.1816	1.13824	0.569122	−	0.822253i	$-0.307283\pi$
$613$	28.1816	1.13824	0.569122	+	0.822253i	$0.307283\pi$
$614$	0	0
$615$	−17.8088	−0.718119
$616$	0	0
$617$	23.0360	0.927393	0.463696	−	0.885994i	$-0.346523\pi$
$617$	23.0360	0.927393	0.463696	+	0.885994i	$0.346523\pi$
$618$	0	0
$619$	−31.3518	−1.26014	−0.630068	−	0.776540i	$-0.716973\pi$
$619$	−31.3518	−1.26014	−0.630068	+	0.776540i	$0.716973\pi$
$620$	0	0
$621$	−7.34408	−0.294708
$622$	0	0
$623$	10.4377	0.418176
$624$	0	0
$625$	18.6002	0.744009
$626$	0	0
$627$	1.53185	0.0611762
$628$	0	0
$629$	−18.2646	−0.728257
$630$	0	0
$631$	23.1791	0.922745	0.461372	−	0.887207i	$-0.347357\pi$
$631$	23.1791	0.922745	0.461372	+	0.887207i	$0.347357\pi$
$632$	0	0
$633$	52.8045	2.09879
$634$	0	0
$635$	76.5637	3.03834
$636$	0	0
$637$	−4.40808	−0.174654
$638$	0	0
$639$	−40.1175	−1.58702
$640$	0	0
$641$	6.03049	0.238190	0.119095	−	0.992883i	$-0.462001\pi$
$641$	6.03049	0.238190	0.119095	+	0.992883i	$0.462001\pi$
$642$	0	0
$643$	−3.71015	−0.146314	−0.0731571	−	0.997320i	$-0.523307\pi$
$643$	−3.71015	−0.146314	−0.0731571	+	0.997320i	$0.523307\pi$
$644$	0	0
$645$	−52.7614	−2.07748
$646$	0	0
$647$	−30.2794	−1.19040	−0.595202	−	0.803576i	$-0.702928\pi$
$647$	−30.2794	−1.19040	−0.595202	+	0.803576i	$0.702928\pi$
$648$	0	0
$649$	−0.817463	−0.0320882
$650$	0	0
$651$	−33.5731	−1.31583
$652$	0	0
$653$	25.3995	0.993959	0.496979	−	0.867762i	$-0.334442\pi$
$653$	25.3995	0.993959	0.496979	+	0.867762i	$0.334442\pi$
$654$	0	0
$655$	−36.7714	−1.43678
$656$	0	0
$657$	33.9862	1.32593
$658$	0	0
$659$	7.36423	0.286870	0.143435	−	0.989660i	$-0.454185\pi$
$659$	7.36423	0.286870	0.143435	+	0.989660i	$0.454185\pi$
$660$	0	0
$661$	28.3522	1.10277	0.551386	−	0.834250i	$-0.314099\pi$
$661$	28.3522	1.10277	0.551386	+	0.834250i	$0.314099\pi$
$662$	0	0
$663$	−8.23870	−0.319965
$664$	0	0
$665$	−47.9329	−1.85876
$666$	0	0
$667$	−28.6672	−1.11000
$668$	0	0
$669$	7.42214	0.286957
$670$	0	0
$671$	0.781090	0.0301536
$672$	0	0
$673$	8.26712	0.318674	0.159337	−	0.987224i	$-0.449064\pi$
$673$	8.26712	0.318674	0.159337	+	0.987224i	$0.449064\pi$
$674$	0	0
$675$	11.1949	0.430891
$676$	0	0
$677$	−23.8172	−0.915370	−0.457685	−	0.889114i	$-0.651321\pi$
$677$	−23.8172	−0.915370	−0.457685	+	0.889114i	$0.651321\pi$
$678$	0	0
$679$	17.2116	0.660521
$680$	0	0
$681$	−32.6777	−1.25221
$682$	0	0
$683$	−5.14106	−0.196717	−0.0983586	−	0.995151i	$-0.531359\pi$
$683$	−5.14106	−0.196717	−0.0983586	+	0.995151i	$0.531359\pi$
$684$	0	0
$685$	39.6351	1.51438
$686$	0	0
$687$	−45.7656	−1.74607
$688$	0	0
$689$	−2.51366	−0.0957629
$690$	0	0
$691$	49.1272	1.86889	0.934444	−	0.356109i	$-0.115897\pi$
$691$	49.1272	1.86889	0.934444	+	0.356109i	$0.115897\pi$
$692$	0	0
$693$	−0.742129	−0.0281912
$694$	0	0
$695$	11.2833	0.428000
$696$	0	0
$697$	3.41612	0.129395
$698$	0	0
$699$	−43.6297	−1.65023
$700$	0	0
$701$	10.4092	0.393149	0.196574	−	0.980489i	$-0.437018\pi$
$701$	10.4092	0.393149	0.196574	+	0.980489i	$0.437018\pi$
$702$	0	0
$703$	58.3293	2.19993
$704$	0	0
$705$	−23.6850	−0.892027
$706$	0	0
$707$	26.8666	1.01042
$708$	0	0
$709$	18.2887	0.686845	0.343423	−	0.939181i	$-0.388414\pi$
$709$	18.2887	0.686845	0.343423	+	0.939181i	$0.388414\pi$
$710$	0	0
$711$	34.5427	1.29545
$712$	0	0
$713$	−39.1837	−1.46744
$714$	0	0
$715$	0.674303	0.0252175
$716$	0	0
$717$	35.8531	1.33896
$718$	0	0
$719$	−17.1166	−0.638343	−0.319171	−	0.947697i	$-0.603405\pi$
$719$	−17.1166	−0.638343	−0.319171	+	0.947697i	$0.603405\pi$
$720$	0	0
$721$	16.3347	0.608337
$722$	0	0
$723$	−21.3688	−0.794713
$724$	0	0
$725$	43.6986	1.62292
$726$	0	0
$727$	−2.72217	−0.100960	−0.0504799	−	0.998725i	$-0.516075\pi$
$727$	−2.72217	−0.100960	−0.0504799	+	0.998725i	$0.516075\pi$
$728$	0	0
$729$	−34.5578	−1.27992
$730$	0	0
$731$	10.1208	0.374332
$732$	0	0
$733$	5.37588	0.198563	0.0992813	−	0.995059i	$-0.468346\pi$
$733$	5.37588	0.198563	0.0992813	+	0.995059i	$0.468346\pi$
$734$	0	0
$735$	−24.5404	−0.905188
$736$	0	0
$737$	1.47232	0.0542335
$738$	0	0
$739$	49.0946	1.80597	0.902987	−	0.429667i	$-0.141369\pi$
$739$	49.0946	1.80597	0.902987	+	0.429667i	$0.141369\pi$
$740$	0	0
$741$	26.3109	0.966554
$742$	0	0
$743$	18.8504	0.691554	0.345777	−	0.938317i	$-0.387615\pi$
$743$	18.8504	0.691554	0.345777	+	0.938317i	$0.387615\pi$
$744$	0	0
$745$	65.7597	2.40925
$746$	0	0
$747$	−9.05115	−0.331164
$748$	0	0
$749$	18.2883	0.668240
$750$	0	0
$751$	−17.7881	−0.649097	−0.324549	−	0.945869i	$-0.605212\pi$
$751$	−17.7881	−0.649097	−0.324549	+	0.945869i	$0.605212\pi$
$752$	0	0
$753$	−2.54179	−0.0926279
$754$	0	0
$755$	52.4341	1.90827
$756$	0	0
$757$	−7.48829	−0.272167	−0.136083	−	0.990697i	$-0.543451\pi$
$757$	−7.48829	−0.272167	−0.136083	+	0.990697i	$0.543451\pi$
$758$	0	0
$759$	−1.61700	−0.0586934
$760$	0	0
$761$	−21.3336	−0.773343	−0.386671	−	0.922218i	$-0.626375\pi$
$761$	−21.3336	−0.773343	−0.386671	+	0.922218i	$0.626375\pi$
$762$	0	0
$763$	−4.70763	−0.170428
$764$	0	0
$765$	−24.5683	−0.888269
$766$	0	0
$767$	−14.0407	−0.506979
$768$	0	0
$769$	7.53727	0.271801	0.135901	−	0.990722i	$-0.456607\pi$
$769$	7.53727	0.271801	0.135901	+	0.990722i	$0.456607\pi$
$770$	0	0
$771$	−73.4653	−2.64579
$772$	0	0
$773$	14.1658	0.509510	0.254755	−	0.967006i	$-0.418005\pi$
$773$	14.1658	0.509510	0.254755	+	0.967006i	$0.418005\pi$
$774$	0	0
$775$	59.7293	2.14554
$776$	0	0
$777$	−52.7554	−1.89259
$778$	0	0
$779$	−10.9096	−0.390878
$780$	0	0
$781$	−1.17584	−0.0420747
$782$	0	0
$783$	5.35215	0.191270
$784$	0	0
$785$	−37.1135	−1.32464
$786$	0	0
$787$	22.2196	0.792043	0.396021	−	0.918241i	$-0.370391\pi$
$787$	22.2196	0.792043	0.396021	+	0.918241i	$0.370391\pi$
$788$	0	0
$789$	50.1758	1.78631
$790$	0	0
$791$	−43.3614	−1.54175
$792$	0	0
$793$	13.4159	0.476413
$794$	0	0
$795$	−13.9939	−0.496314
$796$	0	0
$797$	−31.5959	−1.11918	−0.559592	−	0.828768i	$-0.689042\pi$
$797$	−31.5959	−1.11918	−0.559592	+	0.828768i	$0.689042\pi$
$798$	0	0
$799$	4.54330	0.160730
$800$	0	0
$801$	−17.0852	−0.603677
$802$	0	0
$803$	0.996127	0.0351526
$804$	0	0
$805$	50.5974	1.78333
$806$	0	0
$807$	−29.8796	−1.05181
$808$	0	0
$809$	−7.18671	−0.252671	−0.126336	−	0.991988i	$-0.540322\pi$
$809$	−7.18671	−0.252671	−0.126336	+	0.991988i	$0.540322\pi$
$810$	0	0
$811$	44.5834	1.56554	0.782768	−	0.622314i	$-0.213807\pi$
$811$	44.5834	1.56554	0.782768	+	0.622314i	$0.213807\pi$
$812$	0	0
$813$	11.1807	0.392125
$814$	0	0
$815$	−80.4681	−2.81868
$816$	0	0
$817$	−32.3216	−1.13079
$818$	0	0
$819$	−12.7467	−0.445407
$820$	0	0
$821$	0.397876	0.0138860	0.00694299	−	0.999976i	$-0.497790\pi$
$821$	0.397876	0.0138860	0.00694299	+	0.999976i	$0.497790\pi$
$822$	0	0
$823$	8.86415	0.308985	0.154493	−	0.987994i	$-0.450626\pi$
$823$	8.86415	0.308985	0.154493	+	0.987994i	$0.450626\pi$
$824$	0	0
$825$	2.46486	0.0858155
$826$	0	0
$827$	−27.7115	−0.963624	−0.481812	−	0.876275i	$-0.660021\pi$
$827$	−27.7115	−0.963624	−0.481812	+	0.876275i	$0.660021\pi$
$828$	0	0
$829$	6.25339	0.217189	0.108595	−	0.994086i	$-0.465365\pi$
$829$	6.25339	0.217189	0.108595	+	0.994086i	$0.465365\pi$
$830$	0	0
$831$	−25.3598	−0.879722
$832$	0	0
$833$	4.70741	0.163102
$834$	0	0
$835$	31.8558	1.10241
$836$	0	0
$837$	7.31557	0.252863
$838$	0	0
$839$	−31.5902	−1.09062	−0.545308	−	0.838236i	$-0.683587\pi$
$839$	−31.5902	−1.09062	−0.545308	+	0.838236i	$0.683587\pi$
$840$	0	0
$841$	−8.10818	−0.279592
$842$	0	0
$843$	57.8754	1.99334
$844$	0	0
$845$	−38.0239	−1.30806
$846$	0	0
$847$	23.2343	0.798341
$848$	0	0
$849$	−80.2419	−2.75389
$850$	0	0
$851$	−61.5716	−2.11065
$852$	0	0
$853$	−54.8742	−1.87886	−0.939428	−	0.342747i	$-0.888642\pi$
$853$	−54.8742	−1.87886	−0.939428	+	0.342747i	$0.888642\pi$
$854$	0	0
$855$	78.4607	2.68330
$856$	0	0
$857$	−28.3681	−0.969037	−0.484519	−	0.874781i	$-0.661005\pi$
$857$	−28.3681	−0.969037	−0.484519	+	0.874781i	$0.661005\pi$
$858$	0	0
$859$	−8.96825	−0.305993	−0.152996	−	0.988227i	$-0.548892\pi$
$859$	−8.96825	−0.305993	−0.152996	+	0.988227i	$0.548892\pi$
$860$	0	0
$861$	9.86711	0.336270
$862$	0	0
$863$	−40.2605	−1.37048	−0.685241	−	0.728316i	$-0.740303\pi$
$863$	−40.2605	−1.37048	−0.685241	+	0.728316i	$0.740303\pi$
$864$	0	0
$865$	−56.9201	−1.93534
$866$	0	0
$867$	−34.4122	−1.16870
$868$	0	0
$869$	1.01244	0.0343447
$870$	0	0
$871$	25.2884	0.856863
$872$	0	0
$873$	−28.1734	−0.953525
$874$	0	0
$875$	−36.7910	−1.24376
$876$	0	0
$877$	2.73260	0.0922735	0.0461367	−	0.998935i	$-0.485309\pi$
$877$	2.73260	0.0922735	0.0461367	+	0.998935i	$0.485309\pi$
$878$	0	0
$879$	−40.6626	−1.37151
$880$	0	0
$881$	40.1919	1.35410	0.677050	−	0.735937i	$-0.263258\pi$
$881$	40.1919	1.35410	0.677050	+	0.735937i	$0.263258\pi$
$882$	0	0
$883$	29.7033	0.999596	0.499798	−	0.866142i	$-0.333407\pi$
$883$	29.7033	0.999596	0.499798	+	0.866142i	$0.333407\pi$
$884$	0	0
$885$	−78.1665	−2.62754
$886$	0	0
$887$	−9.64865	−0.323970	−0.161985	−	0.986793i	$-0.551790\pi$
$887$	−9.64865	−0.323970	−0.161985	+	0.986793i	$0.551790\pi$
$888$	0	0
$889$	−42.4208	−1.42275
$890$	0	0
$891$	−0.751176	−0.0251654
$892$	0	0
$893$	−14.5094	−0.485537
$894$	0	0
$895$	21.5028	0.718760
$896$	0	0
$897$	−27.7734	−0.927328
$898$	0	0
$899$	28.5559	0.952394
$900$	0	0
$901$	2.68435	0.0894287
$902$	0	0
$903$	29.2330	0.972812
$904$	0	0
$905$	2.46463	0.0819271
$906$	0	0
$907$	−32.9608	−1.09444	−0.547222	−	0.836987i	$-0.684315\pi$
$907$	−32.9608	−1.09444	−0.547222	+	0.836987i	$0.684315\pi$
$908$	0	0
$909$	−43.9774	−1.45864
$910$	0	0
$911$	4.75581	0.157567	0.0787835	−	0.996892i	$-0.474896\pi$
$911$	4.75581	0.157567	0.0787835	+	0.996892i	$0.474896\pi$
$912$	0	0
$913$	−0.265287	−0.00877972
$914$	0	0
$915$	74.6885	2.46913
$916$	0	0
$917$	20.3735	0.672794
$918$	0	0
$919$	−0.971293	−0.0320400	−0.0160200	−	0.999872i	$-0.505100\pi$
$919$	−0.971293	−0.0320400	−0.0160200	+	0.999872i	$0.505100\pi$
$920$	0	0
$921$	−13.0462	−0.429888
$922$	0	0
$923$	−20.1960	−0.664760
$924$	0	0
$925$	93.8562	3.08597
$926$	0	0
$927$	−26.7380	−0.878193
$928$	0	0
$929$	15.5414	0.509896	0.254948	−	0.966955i	$-0.417942\pi$
$929$	15.5414	0.509896	0.254948	+	0.966955i	$0.417942\pi$
$930$	0	0
$931$	−15.0334	−0.492701
$932$	0	0
$933$	−31.3282	−1.02564
$934$	0	0
$935$	−0.720092	−0.0235495
$936$	0	0
$937$	−39.7125	−1.29735	−0.648676	−	0.761065i	$-0.724677\pi$
$937$	−39.7125	−1.29735	−0.648676	+	0.761065i	$0.724677\pi$
$938$	0	0
$939$	71.1068	2.32048
$940$	0	0
$941$	33.3208	1.08623	0.543113	−	0.839660i	$-0.317246\pi$
$941$	33.3208	1.08623	0.543113	+	0.839660i	$0.317246\pi$
$942$	0	0
$943$	11.5161	0.375014
$944$	0	0
$945$	−9.44652	−0.307295
$946$	0	0
$947$	42.1521	1.36976	0.684880	−	0.728656i	$-0.259855\pi$
$947$	42.1521	1.36976	0.684880	+	0.728656i	$0.259855\pi$
$948$	0	0
$949$	17.1094	0.555394
$950$	0	0
$951$	−24.4702	−0.793500
$952$	0	0
$953$	−9.21579	−0.298529	−0.149264	−	0.988797i	$-0.547691\pi$
$953$	−9.21579	−0.298529	−0.149264	+	0.988797i	$0.547691\pi$
$954$	0	0
$955$	−86.2074	−2.78961
$956$	0	0
$957$	1.17842	0.0380930
$958$	0	0
$959$	−21.9602	−0.709131
$960$	0	0
$961$	8.03163	0.259085
$962$	0	0
$963$	−29.9358	−0.964669
$964$	0	0
$965$	−47.4204	−1.52652
$966$	0	0
$967$	45.4649	1.46205	0.731026	−	0.682349i	$-0.239042\pi$
$967$	45.4649	1.46205	0.731026	+	0.682349i	$0.239042\pi$
$968$	0	0
$969$	−28.0975	−0.902622
$970$	0	0
$971$	−18.6873	−0.599704	−0.299852	−	0.953986i	$-0.596937\pi$
$971$	−18.6873	−0.599704	−0.299852	+	0.953986i	$0.596937\pi$
$972$	0	0
$973$	−6.25161	−0.200417
$974$	0	0
$975$	42.3362	1.35584
$976$	0	0
$977$	−6.11986	−0.195792	−0.0978959	−	0.995197i	$-0.531211\pi$
$977$	−6.11986	−0.195792	−0.0978959	+	0.995197i	$0.531211\pi$
$978$	0	0
$979$	−0.500765	−0.0160045
$980$	0	0
$981$	7.70585	0.246029
$982$	0	0
$983$	−27.8190	−0.887287	−0.443643	−	0.896203i	$-0.646314\pi$
$983$	−27.8190	−0.887287	−0.443643	+	0.896203i	$0.646314\pi$
$984$	0	0
$985$	−55.4427	−1.76655
$986$	0	0
$987$	13.1229	0.417705
$988$	0	0
$989$	34.1183	1.08490
$990$	0	0
$991$	−58.4093	−1.85543	−0.927716	−	0.373288i	$-0.878230\pi$
$991$	−58.4093	−1.85543	−0.927716	+	0.373288i	$0.878230\pi$
$992$	0	0
$993$	53.9564	1.71226
$994$	0	0
$995$	16.6824	0.528868
$996$	0	0
$997$	−28.8018	−0.912162	−0.456081	−	0.889938i	$-0.650747\pi$
$997$	−28.8018	−0.912162	−0.456081	+	0.889938i	$0.650747\pi$
$998$	0	0
$999$	11.4954	0.363698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.d.1.5		5
4.3	odd	2		502.2.a.d.1.1	✓	5
12.11	even	2		4518.2.a.t.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.d.1.1	✓	5	4.3	odd	2
4016.2.a.d.1.5		5	1.1	even	1	trivial
4518.2.a.t.1.1		5	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54179 q^{3} +3.81582 q^{5} -2.11419 q^{7} +3.46068 q^{9} +O(q^{10})\)	\(q+2.54179 q^{3} +3.81582 q^{5} -2.11419 q^{7} +3.46068 q^{9} +0.101432 q^{11} +1.74218 q^{13} +9.69900 q^{15} -1.86048 q^{17} +5.94159 q^{19} -5.37382 q^{21} -6.27187 q^{23} +9.56047 q^{25} +1.17095 q^{27} +4.57076 q^{29} +6.24753 q^{31} +0.257818 q^{33} -8.06736 q^{35} +9.81711 q^{37} +4.42826 q^{39} -1.83614 q^{41} -5.43989 q^{43} +13.2053 q^{45} -2.44200 q^{47} -2.53020 q^{49} -4.72896 q^{51} -1.44282 q^{53} +0.387045 q^{55} +15.1023 q^{57} -8.05923 q^{59} +7.70064 q^{61} -7.31654 q^{63} +6.64785 q^{65} +14.5153 q^{67} -15.9418 q^{69} -11.5924 q^{71} +9.82066 q^{73} +24.3007 q^{75} -0.214446 q^{77} +9.98148 q^{79} -7.40573 q^{81} -2.61542 q^{83} -7.09927 q^{85} +11.6179 q^{87} -4.93696 q^{89} -3.68330 q^{91} +15.8799 q^{93} +22.6720 q^{95} -8.14100 q^{97} +0.351023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9	\( 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9} - 7 q^{11} + 4 q^{13} - 3 q^{15} + 6 q^{17} + 10 q^{19} + 2 q^{21} - 13 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{29} + 13 q^{31} + 6 q^{33} - 13 q^{35} + 16 q^{37} + 16 q^{39} + 6 q^{41} + 8 q^{43} + 26 q^{45} - 29 q^{47} + 4 q^{49} + 7 q^{51} + 25 q^{53} - q^{55} + 19 q^{57} - 11 q^{59} + 11 q^{61} - 15 q^{63} + 7 q^{65} + 6 q^{67} - 12 q^{69} - 15 q^{71} - 9 q^{73} + 25 q^{75} + 12 q^{77} + 29 q^{79} - 7 q^{81} + 9 q^{83} - 7 q^{85} + 11 q^{87} - 9 q^{89} + 34 q^{91} - 8 q^{93} + 13 q^{95} - 6 q^{97} + q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9 - 7 * q^11 + 4 * q^13 - 3 * q^15 + 6 * q^17 + 10 * q^19 + 2 * q^21 - 13 * q^23 + 6 * q^25 - 8 * q^27 + 4 * q^29 + 13 * q^31 + 6 * q^33 - 13 * q^35 + 16 * q^37 + 16 * q^39 + 6 * q^41 + 8 * q^43 + 26 * q^45 - 29 * q^47 + 4 * q^49 + 7 * q^51 + 25 * q^53 - q^55 + 19 * q^57 - 11 * q^59 + 11 * q^61 - 15 * q^63 + 7 * q^65 + 6 * q^67 - 12 * q^69 - 15 * q^71 - 9 * q^73 + 25 * q^75 + 12 * q^77 + 29 * q^79 - 7 * q^81 + 9 * q^83 - 7 * q^85 + 11 * q^87 - 9 * q^89 + 34 * q^91 - 8 * q^93 + 13 * q^95 - 6 * q^97 + q^99

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