LMFDB - Embedded newform 8016.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.284897.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 $ x^5 - 2x^4 - 6x^3 + 5x^2 + 10x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80442$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.40664 q^{5} -0.548479 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.40664 q^{5} -0.548479 q^{7} +1.00000 q^{9} +0.977163 q^{11} -1.94627 q^{13} +3.40664 q^{15} +1.38380 q^{17} +1.16100 q^{19} +0.548479 q^{21} -4.44417 q^{23} +6.60517 q^{25} -1.00000 q^{27} +3.71999 q^{29} -5.46700 q^{31} -0.977163 q^{33} +1.86847 q^{35} -2.93026 q^{37} +1.94627 q^{39} +6.35002 q^{41} +11.3309 q^{43} -3.40664 q^{45} -10.2154 q^{47} -6.69917 q^{49} -1.38380 q^{51} +3.23512 q^{53} -3.32884 q^{55} -1.16100 q^{57} +12.8823 q^{59} -14.9043 q^{61} -0.548479 q^{63} +6.63022 q^{65} +15.2508 q^{67} +4.44417 q^{69} +11.4987 q^{71} -0.389926 q^{73} -6.60517 q^{75} -0.535953 q^{77} +10.1425 q^{79} +1.00000 q^{81} +5.41931 q^{83} -4.71410 q^{85} -3.71999 q^{87} -17.1536 q^{89} +1.06749 q^{91} +5.46700 q^{93} -3.95512 q^{95} -2.62873 q^{97} +0.977163 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9 + 3 * q^11 - 14 * q^13 - q^15 - 13 * q^17 + 2 * q^19 - 4 * q^21 + 5 * q^23 + 2 * q^25 - 5 * q^27 + 13 * q^29 - 2 * q^31 - 3 * q^33 + 12 * q^35 - 5 * q^37 + 14 * q^39 - 20 * q^41 + 20 * q^43 + q^45 - q^47 - 9 * q^49 + 13 * q^51 - 3 * q^53 + 3 * q^55 - 2 * q^57 + q^59 - 34 * q^61 + 4 * q^63 - 22 * q^65 + 16 * q^67 - 5 * q^69 + 5 * q^71 - 12 * q^73 - 2 * q^75 - 8 * q^77 + 20 * q^79 + 5 * q^81 + 15 * q^83 - 27 * q^85 - 13 * q^87 - 48 * q^89 - 7 * q^91 + 2 * q^93 + 5 * q^95 - 21 * q^97 + 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.40664	−1.52349	−0.761747	−	0.647875i	$-0.775658\pi$
$5$	−3.40664	−1.52349	−0.761747	+	0.647875i	$0.775658\pi$
$6$	0	0
$7$	−0.548479	−0.207306	−0.103653	−	0.994614i	$-0.533053\pi$
$7$	−0.548479	−0.207306	−0.103653	+	0.994614i	$0.533053\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.977163	0.294626	0.147313	−	0.989090i	$-0.452938\pi$
$11$	0.977163	0.294626	0.147313	+	0.989090i	$0.452938\pi$
$12$	0	0
$13$	−1.94627	−0.539797	−0.269899	−	0.962889i	$-0.586990\pi$
$13$	−1.94627	−0.539797	−0.269899	+	0.962889i	$0.586990\pi$
$14$	0	0
$15$	3.40664	0.879590
$16$	0	0
$17$	1.38380	0.335621	0.167810	−	0.985819i	$-0.446330\pi$
$17$	1.38380	0.335621	0.167810	+	0.985819i	$0.446330\pi$
$18$	0	0
$19$	1.16100	0.266352	0.133176	−	0.991092i	$-0.457482\pi$
$19$	1.16100	0.266352	0.133176	+	0.991092i	$0.457482\pi$
$20$	0	0
$21$	0.548479	0.119688
$22$	0	0
$23$	−4.44417	−0.926673	−0.463336	−	0.886182i	$-0.653348\pi$
$23$	−4.44417	−0.926673	−0.463336	+	0.886182i	$0.653348\pi$
$24$	0	0
$25$	6.60517	1.32103
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.71999	0.690785	0.345393	−	0.938458i	$-0.387746\pi$
$29$	3.71999	0.690785	0.345393	+	0.938458i	$0.387746\pi$
$30$	0	0
$31$	−5.46700	−0.981903	−0.490951	−	0.871187i	$-0.663351\pi$
$31$	−5.46700	−0.981903	−0.490951	+	0.871187i	$0.663351\pi$
$32$	0	0
$33$	−0.977163	−0.170102
$34$	0	0
$35$	1.86847	0.315829
$36$	0	0
$37$	−2.93026	−0.481732	−0.240866	−	0.970558i	$-0.577431\pi$
$37$	−2.93026	−0.481732	−0.240866	+	0.970558i	$0.577431\pi$
$38$	0	0
$39$	1.94627	0.311652
$40$	0	0
$41$	6.35002	0.991706	0.495853	−	0.868407i	$-0.334856\pi$
$41$	6.35002	0.991706	0.495853	+	0.868407i	$0.334856\pi$
$42$	0	0
$43$	11.3309	1.72794	0.863970	−	0.503544i	$-0.167971\pi$
$43$	11.3309	1.72794	0.863970	+	0.503544i	$0.167971\pi$
$44$	0	0
$45$	−3.40664	−0.507831
$46$	0	0
$47$	−10.2154	−1.49008	−0.745038	−	0.667022i	$-0.767569\pi$
$47$	−10.2154	−1.49008	−0.745038	+	0.667022i	$0.767569\pi$
$48$	0	0
$49$	−6.69917	−0.957024
$50$	0	0
$51$	−1.38380	−0.193771
$52$	0	0
$53$	3.23512	0.444378	0.222189	−	0.975004i	$-0.428680\pi$
$53$	3.23512	0.444378	0.222189	+	0.975004i	$0.428680\pi$
$54$	0	0
$55$	−3.32884	−0.448860
$56$	0	0
$57$	−1.16100	−0.153779
$58$	0	0
$59$	12.8823	1.67713	0.838566	−	0.544800i	$-0.183395\pi$
$59$	12.8823	1.67713	0.838566	+	0.544800i	$0.183395\pi$
$60$	0	0
$61$	−14.9043	−1.90831	−0.954153	−	0.299321i	$-0.903240\pi$
$61$	−14.9043	−1.90831	−0.954153	+	0.299321i	$0.903240\pi$
$62$	0	0
$63$	−0.548479	−0.0691019
$64$	0	0
$65$	6.63022	0.822378
$66$	0	0
$67$	15.2508	1.86318	0.931591	−	0.363508i	$-0.118421\pi$
$67$	15.2508	1.86318	0.931591	+	0.363508i	$0.118421\pi$
$68$	0	0
$69$	4.44417	0.535015
$70$	0	0
$71$	11.4987	1.36464	0.682322	−	0.731052i	$-0.260970\pi$
$71$	11.4987	1.36464	0.682322	+	0.731052i	$0.260970\pi$
$72$	0	0
$73$	−0.389926	−0.0456373	−0.0228187	−	0.999740i	$-0.507264\pi$
$73$	−0.389926	−0.0456373	−0.0228187	+	0.999740i	$0.507264\pi$
$74$	0	0
$75$	−6.60517	−0.762699
$76$	0	0
$77$	−0.535953	−0.0610776
$78$	0	0
$79$	10.1425	1.14113	0.570563	−	0.821254i	$-0.306725\pi$
$79$	10.1425	1.14113	0.570563	+	0.821254i	$0.306725\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.41931	0.594847	0.297423	−	0.954746i	$-0.403873\pi$
$83$	5.41931	0.594847	0.297423	+	0.954746i	$0.403873\pi$
$84$	0	0
$85$	−4.71410	−0.511316
$86$	0	0
$87$	−3.71999	−0.398825
$88$	0	0
$89$	−17.1536	−1.81828	−0.909142	−	0.416487i	$-0.863261\pi$
$89$	−17.1536	−1.81828	−0.909142	+	0.416487i	$0.863261\pi$
$90$	0	0
$91$	1.06749	0.111903
$92$	0	0
$93$	5.46700	0.566902
$94$	0	0
$95$	−3.95512	−0.405786
$96$	0	0
$97$	−2.62873	−0.266907	−0.133453	−	0.991055i	$-0.542607\pi$
$97$	−2.62873	−0.266907	−0.133453	+	0.991055i	$0.542607\pi$
$98$	0	0
$99$	0.977163	0.0982086
$100$	0	0
$101$	10.7613	1.07079	0.535393	−	0.844603i	$-0.320163\pi$
$101$	10.7613	1.07079	0.535393	+	0.844603i	$0.320163\pi$
$102$	0	0
$103$	−0.545324	−0.0537323	−0.0268662	−	0.999639i	$-0.508553\pi$
$103$	−0.545324	−0.0537323	−0.0268662	+	0.999639i	$0.508553\pi$
$104$	0	0
$105$	−1.86847	−0.182344
$106$	0	0
$107$	12.2316	1.18248	0.591239	−	0.806496i	$-0.298639\pi$
$107$	12.2316	1.18248	0.591239	+	0.806496i	$0.298639\pi$
$108$	0	0
$109$	3.42284	0.327848	0.163924	−	0.986473i	$-0.447585\pi$
$109$	3.42284	0.327848	0.163924	+	0.986473i	$0.447585\pi$
$110$	0	0
$111$	2.93026	0.278128
$112$	0	0
$113$	11.9535	1.12449	0.562243	−	0.826972i	$-0.309938\pi$
$113$	11.9535	1.12449	0.562243	+	0.826972i	$0.309938\pi$
$114$	0	0
$115$	15.1397	1.41178
$116$	0	0
$117$	−1.94627	−0.179932
$118$	0	0
$119$	−0.758985	−0.0695760
$120$	0	0
$121$	−10.0452	−0.913196
$122$	0	0
$123$	−6.35002	−0.572562
$124$	0	0
$125$	−5.46823	−0.489093
$126$	0	0
$127$	3.62801	0.321934	0.160967	−	0.986960i	$-0.448539\pi$
$127$	3.62801	0.321934	0.160967	+	0.986960i	$0.448539\pi$
$128$	0	0
$129$	−11.3309	−0.997626
$130$	0	0
$131$	−6.01253	−0.525317	−0.262658	−	0.964889i	$-0.584599\pi$
$131$	−6.01253	−0.525317	−0.262658	+	0.964889i	$0.584599\pi$
$132$	0	0
$133$	−0.636786	−0.0552163
$134$	0	0
$135$	3.40664	0.293197
$136$	0	0
$137$	−3.87068	−0.330695	−0.165347	−	0.986235i	$-0.552875\pi$
$137$	−3.87068	−0.330695	−0.165347	+	0.986235i	$0.552875\pi$
$138$	0	0
$139$	−5.82501	−0.494071	−0.247035	−	0.969006i	$-0.579456\pi$
$139$	−5.82501	−0.494071	−0.247035	+	0.969006i	$0.579456\pi$
$140$	0	0
$141$	10.2154	0.860295
$142$	0	0
$143$	−1.90182	−0.159038
$144$	0	0
$145$	−12.6727	−1.05241
$146$	0	0
$147$	6.69917	0.552538
$148$	0	0
$149$	14.1095	1.15589	0.577947	−	0.816074i	$-0.303854\pi$
$149$	14.1095	1.15589	0.577947	+	0.816074i	$0.303854\pi$
$150$	0	0
$151$	−20.7002	−1.68456	−0.842278	−	0.539043i	$-0.818786\pi$
$151$	−20.7002	−1.68456	−0.842278	+	0.539043i	$0.818786\pi$
$152$	0	0
$153$	1.38380	0.111874
$154$	0	0
$155$	18.6241	1.49592
$156$	0	0
$157$	6.25155	0.498928	0.249464	−	0.968384i	$-0.419746\pi$
$157$	6.25155	0.498928	0.249464	+	0.968384i	$0.419746\pi$
$158$	0	0
$159$	−3.23512	−0.256562
$160$	0	0
$161$	2.43753	0.192104
$162$	0	0
$163$	−10.3332	−0.809360	−0.404680	−	0.914458i	$-0.632617\pi$
$163$	−10.3332	−0.809360	−0.404680	+	0.914458i	$0.632617\pi$
$164$	0	0
$165$	3.32884	0.259150
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−9.21205	−0.708619
$170$	0	0
$171$	1.16100	0.0887841
$172$	0	0
$173$	−4.48522	−0.341005	−0.170503	−	0.985357i	$-0.554539\pi$
$173$	−4.48522	−0.341005	−0.170503	+	0.985357i	$0.554539\pi$
$174$	0	0
$175$	−3.62280	−0.273858
$176$	0	0
$177$	−12.8823	−0.968292
$178$	0	0
$179$	−20.1276	−1.50441	−0.752205	−	0.658929i	$-0.771010\pi$
$179$	−20.1276	−1.50441	−0.752205	+	0.658929i	$0.771010\pi$
$180$	0	0
$181$	−22.5001	−1.67242	−0.836210	−	0.548409i	$-0.815234\pi$
$181$	−22.5001	−1.67242	−0.836210	+	0.548409i	$0.815234\pi$
$182$	0	0
$183$	14.9043	1.10176
$184$	0	0
$185$	9.98233	0.733916
$186$	0	0
$187$	1.35220	0.0988824
$188$	0	0
$189$	0.548479	0.0398960
$190$	0	0
$191$	10.9408	0.791651	0.395826	−	0.918326i	$-0.370458\pi$
$191$	10.9408	0.791651	0.395826	+	0.918326i	$0.370458\pi$
$192$	0	0
$193$	4.36096	0.313909	0.156954	−	0.987606i	$-0.449832\pi$
$193$	4.36096	0.313909	0.156954	+	0.987606i	$0.449832\pi$
$194$	0	0
$195$	−6.63022	−0.474800
$196$	0	0
$197$	−7.29648	−0.519853	−0.259926	−	0.965628i	$-0.583698\pi$
$197$	−7.29648	−0.519853	−0.259926	+	0.965628i	$0.583698\pi$
$198$	0	0
$199$	22.7900	1.61554	0.807770	−	0.589498i	$-0.200675\pi$
$199$	22.7900	1.61554	0.807770	+	0.589498i	$0.200675\pi$
$200$	0	0
$201$	−15.2508	−1.07571
$202$	0	0
$203$	−2.04034	−0.143204
$204$	0	0
$205$	−21.6322	−1.51086
$206$	0	0
$207$	−4.44417	−0.308891
$208$	0	0
$209$	1.13449	0.0784743
$210$	0	0
$211$	23.3625	1.60834	0.804169	−	0.594401i	$-0.202611\pi$
$211$	23.3625	1.60834	0.804169	+	0.594401i	$0.202611\pi$
$212$	0	0
$213$	−11.4987	−0.787877
$214$	0	0
$215$	−38.6001	−2.63251
$216$	0	0
$217$	2.99854	0.203554
$218$	0	0
$219$	0.389926	0.0263487
$220$	0	0
$221$	−2.69324	−0.181167
$222$	0	0
$223$	−2.93666	−0.196653	−0.0983267	−	0.995154i	$-0.531349\pi$
$223$	−2.93666	−0.196653	−0.0983267	+	0.995154i	$0.531349\pi$
$224$	0	0
$225$	6.60517	0.440345
$226$	0	0
$227$	20.3110	1.34809	0.674045	−	0.738690i	$-0.264555\pi$
$227$	20.3110	1.34809	0.674045	+	0.738690i	$0.264555\pi$
$228$	0	0
$229$	3.90723	0.258197	0.129098	−	0.991632i	$-0.458792\pi$
$229$	3.90723	0.258197	0.129098	+	0.991632i	$0.458792\pi$
$230$	0	0
$231$	0.535953	0.0352631
$232$	0	0
$233$	−26.0582	−1.70713	−0.853564	−	0.520988i	$-0.825564\pi$
$233$	−26.0582	−1.70713	−0.853564	+	0.520988i	$0.825564\pi$
$234$	0	0
$235$	34.8003	2.27012
$236$	0	0
$237$	−10.1425	−0.658829
$238$	0	0
$239$	−22.1406	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$239$	−22.1406	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$240$	0	0
$241$	−19.5932	−1.26211	−0.631053	−	0.775740i	$-0.717377\pi$
$241$	−19.5932	−1.26211	−0.631053	+	0.775740i	$0.717377\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	22.8216	1.45802
$246$	0	0
$247$	−2.25962	−0.143776
$248$	0	0
$249$	−5.41931	−0.343435
$250$	0	0
$251$	−13.5771	−0.856978	−0.428489	−	0.903547i	$-0.640954\pi$
$251$	−13.5771	−0.856978	−0.428489	+	0.903547i	$0.640954\pi$
$252$	0	0
$253$	−4.34268	−0.273022
$254$	0	0
$255$	4.71410	0.295208
$256$	0	0
$257$	−19.8735	−1.23968	−0.619838	−	0.784730i	$-0.712802\pi$
$257$	−19.8735	−1.23968	−0.619838	+	0.784730i	$0.712802\pi$
$258$	0	0
$259$	1.60719	0.0998657
$260$	0	0
$261$	3.71999	0.230262
$262$	0	0
$263$	19.8907	1.22651	0.613257	−	0.789884i	$-0.289859\pi$
$263$	19.8907	1.22651	0.613257	+	0.789884i	$0.289859\pi$
$264$	0	0
$265$	−11.0209	−0.677008
$266$	0	0
$267$	17.1536	1.04979
$268$	0	0
$269$	13.4649	0.820972	0.410486	−	0.911867i	$-0.365359\pi$
$269$	13.4649	0.820972	0.410486	+	0.911867i	$0.365359\pi$
$270$	0	0
$271$	−21.3536	−1.29714	−0.648570	−	0.761155i	$-0.724632\pi$
$271$	−21.3536	−1.29714	−0.648570	+	0.761155i	$0.724632\pi$
$272$	0	0
$273$	−1.06749	−0.0646072
$274$	0	0
$275$	6.45433	0.389211
$276$	0	0
$277$	5.62547	0.338002	0.169001	−	0.985616i	$-0.445946\pi$
$277$	5.62547	0.338002	0.169001	+	0.985616i	$0.445946\pi$
$278$	0	0
$279$	−5.46700	−0.327301
$280$	0	0
$281$	−12.1233	−0.723216	−0.361608	−	0.932330i	$-0.617772\pi$
$281$	−12.1233	−0.723216	−0.361608	+	0.932330i	$0.617772\pi$
$282$	0	0
$283$	2.83994	0.168817	0.0844084	−	0.996431i	$-0.473100\pi$
$283$	2.83994	0.168817	0.0844084	+	0.996431i	$0.473100\pi$
$284$	0	0
$285$	3.95512	0.234281
$286$	0	0
$287$	−3.48285	−0.205586
$288$	0	0
$289$	−15.0851	−0.887359
$290$	0	0
$291$	2.62873	0.154099
$292$	0	0
$293$	−17.8412	−1.04229	−0.521147	−	0.853467i	$-0.674496\pi$
$293$	−17.8412	−1.04229	−0.521147	+	0.853467i	$0.674496\pi$
$294$	0	0
$295$	−43.8853	−2.55510
$296$	0	0
$297$	−0.977163	−0.0567007
$298$	0	0
$299$	8.64953	0.500215
$300$	0	0
$301$	−6.21474	−0.358211
$302$	0	0
$303$	−10.7613	−0.618219
$304$	0	0
$305$	50.7737	2.90729
$306$	0	0
$307$	−17.9879	−1.02662	−0.513311	−	0.858203i	$-0.671581\pi$
$307$	−17.9879	−1.02662	−0.513311	+	0.858203i	$0.671581\pi$
$308$	0	0
$309$	0.545324	0.0310224
$310$	0	0
$311$	8.91959	0.505783	0.252892	−	0.967495i	$-0.418618\pi$
$311$	8.91959	0.505783	0.252892	+	0.967495i	$0.418618\pi$
$312$	0	0
$313$	3.72295	0.210434	0.105217	−	0.994449i	$-0.466446\pi$
$313$	3.72295	0.210434	0.105217	+	0.994449i	$0.466446\pi$
$314$	0	0
$315$	1.86847	0.105276
$316$	0	0
$317$	−17.1341	−0.962348	−0.481174	−	0.876625i	$-0.659790\pi$
$317$	−17.1341	−0.962348	−0.481174	+	0.876625i	$0.659790\pi$
$318$	0	0
$319$	3.63504	0.203523
$320$	0	0
$321$	−12.2316	−0.682704
$322$	0	0
$323$	1.60660	0.0893933
$324$	0	0
$325$	−12.8554	−0.713090
$326$	0	0
$327$	−3.42284	−0.189283
$328$	0	0
$329$	5.60296	0.308901
$330$	0	0
$331$	−36.3810	−1.99968	−0.999839	−	0.0179188i	$-0.994296\pi$
$331$	−36.3810	−1.99968	−0.999839	+	0.0179188i	$0.994296\pi$
$332$	0	0
$333$	−2.93026	−0.160577
$334$	0	0
$335$	−51.9539	−2.83855
$336$	0	0
$337$	15.6702	0.853612	0.426806	−	0.904343i	$-0.359639\pi$
$337$	15.6702	0.853612	0.426806	+	0.904343i	$0.359639\pi$
$338$	0	0
$339$	−11.9535	−0.649223
$340$	0	0
$341$	−5.34215	−0.289294
$342$	0	0
$343$	7.51371	0.405702
$344$	0	0
$345$	−15.1397	−0.815092
$346$	0	0
$347$	−3.96072	−0.212623	−0.106311	−	0.994333i	$-0.533904\pi$
$347$	−3.96072	−0.212623	−0.106311	+	0.994333i	$0.533904\pi$
$348$	0	0
$349$	−10.7431	−0.575066	−0.287533	−	0.957771i	$-0.592835\pi$
$349$	−10.7431	−0.575066	−0.287533	+	0.957771i	$0.592835\pi$
$350$	0	0
$351$	1.94627	0.103884
$352$	0	0
$353$	14.7105	0.782961	0.391480	−	0.920186i	$-0.371963\pi$
$353$	14.7105	0.782961	0.391480	+	0.920186i	$0.371963\pi$
$354$	0	0
$355$	−39.1719	−2.07903
$356$	0	0
$357$	0.758985	0.0401697
$358$	0	0
$359$	12.3440	0.651490	0.325745	−	0.945458i	$-0.394385\pi$
$359$	12.3440	0.651490	0.325745	+	0.945458i	$0.394385\pi$
$360$	0	0
$361$	−17.6521	−0.929056
$362$	0	0
$363$	10.0452	0.527234
$364$	0	0
$365$	1.32833	0.0695282
$366$	0	0
$367$	−21.3455	−1.11422	−0.557112	−	0.830437i	$-0.688091\pi$
$367$	−21.3455	−1.11422	−0.557112	+	0.830437i	$0.688091\pi$
$368$	0	0
$369$	6.35002	0.330569
$370$	0	0
$371$	−1.77440	−0.0921221
$372$	0	0
$373$	−0.0396748	−0.00205428	−0.00102714	−	0.999999i	$-0.500327\pi$
$373$	−0.0396748	−0.00205428	−0.00102714	+	0.999999i	$0.500327\pi$
$374$	0	0
$375$	5.46823	0.282378
$376$	0	0
$377$	−7.24009	−0.372884
$378$	0	0
$379$	−31.2798	−1.60673	−0.803367	−	0.595485i	$-0.796960\pi$
$379$	−31.2798	−1.60673	−0.803367	+	0.595485i	$0.796960\pi$
$380$	0	0
$381$	−3.62801	−0.185868
$382$	0	0
$383$	18.7930	0.960277	0.480139	−	0.877193i	$-0.340586\pi$
$383$	18.7930	0.960277	0.480139	+	0.877193i	$0.340586\pi$
$384$	0	0
$385$	1.82580	0.0930513
$386$	0	0
$387$	11.3309	0.575980
$388$	0	0
$389$	−34.3125	−1.73971	−0.869856	−	0.493306i	$-0.835788\pi$
$389$	−34.3125	−1.73971	−0.869856	+	0.493306i	$0.835788\pi$
$390$	0	0
$391$	−6.14983	−0.311011
$392$	0	0
$393$	6.01253	0.303292
$394$	0	0
$395$	−34.5520	−1.73850
$396$	0	0
$397$	−16.8147	−0.843903	−0.421952	−	0.906618i	$-0.638655\pi$
$397$	−16.8147	−0.843903	−0.421952	+	0.906618i	$0.638655\pi$
$398$	0	0
$399$	0.636786	0.0318792
$400$	0	0
$401$	37.8464	1.88996	0.944980	−	0.327129i	$-0.106081\pi$
$401$	37.8464	1.88996	0.944980	+	0.327129i	$0.106081\pi$
$402$	0	0
$403$	10.6402	0.530028
$404$	0	0
$405$	−3.40664	−0.169277
$406$	0	0
$407$	−2.86334	−0.141931
$408$	0	0
$409$	−23.8406	−1.17884	−0.589422	−	0.807825i	$-0.700644\pi$
$409$	−23.8406	−1.17884	−0.589422	+	0.807825i	$0.700644\pi$
$410$	0	0
$411$	3.87068	0.190927
$412$	0	0
$413$	−7.06567	−0.347679
$414$	0	0
$415$	−18.4616	−0.906246
$416$	0	0
$417$	5.82501	0.285252
$418$	0	0
$419$	26.1953	1.27972	0.639862	−	0.768490i	$-0.278992\pi$
$419$	26.1953	1.27972	0.639862	+	0.768490i	$0.278992\pi$
$420$	0	0
$421$	−0.289799	−0.0141239	−0.00706196	−	0.999975i	$-0.502248\pi$
$421$	−0.289799	−0.0141239	−0.00706196	+	0.999975i	$0.502248\pi$
$422$	0	0
$423$	−10.2154	−0.496692
$424$	0	0
$425$	9.14023	0.443366
$426$	0	0
$427$	8.17472	0.395602
$428$	0	0
$429$	1.90182	0.0918207
$430$	0	0
$431$	−16.1155	−0.776258	−0.388129	−	0.921605i	$-0.626878\pi$
$431$	−16.1155	−0.776258	−0.388129	+	0.921605i	$0.626878\pi$
$432$	0	0
$433$	12.0680	0.579949	0.289974	−	0.957034i	$-0.406353\pi$
$433$	12.0680	0.579949	0.289974	+	0.957034i	$0.406353\pi$
$434$	0	0
$435$	12.6727	0.607607
$436$	0	0
$437$	−5.15969	−0.246822
$438$	0	0
$439$	10.5088	0.501560	0.250780	−	0.968044i	$-0.419313\pi$
$439$	10.5088	0.501560	0.250780	+	0.968044i	$0.419313\pi$
$440$	0	0
$441$	−6.69917	−0.319008
$442$	0	0
$443$	19.8417	0.942708	0.471354	−	0.881944i	$-0.343765\pi$
$443$	19.8417	0.942708	0.471354	+	0.881944i	$0.343765\pi$
$444$	0	0
$445$	58.4362	2.77014
$446$	0	0
$447$	−14.1095	−0.667356
$448$	0	0
$449$	−16.0031	−0.755234	−0.377617	−	0.925962i	$-0.623256\pi$
$449$	−16.0031	−0.755234	−0.377617	+	0.925962i	$0.623256\pi$
$450$	0	0
$451$	6.20500	0.292182
$452$	0	0
$453$	20.7002	0.972579
$454$	0	0
$455$	−3.63654	−0.170484
$456$	0	0
$457$	23.4461	1.09676	0.548381	−	0.836228i	$-0.315244\pi$
$457$	23.4461	1.09676	0.548381	+	0.836228i	$0.315244\pi$
$458$	0	0
$459$	−1.38380	−0.0645902
$460$	0	0
$461$	24.6649	1.14876	0.574379	−	0.818590i	$-0.305244\pi$
$461$	24.6649	1.14876	0.574379	+	0.818590i	$0.305244\pi$
$462$	0	0
$463$	2.63528	0.122472	0.0612359	−	0.998123i	$-0.480496\pi$
$463$	2.63528	0.122472	0.0612359	+	0.998123i	$0.480496\pi$
$464$	0	0
$465$	−18.6241	−0.863672
$466$	0	0
$467$	11.7325	0.542917	0.271459	−	0.962450i	$-0.412494\pi$
$467$	11.7325	0.542917	0.271459	+	0.962450i	$0.412494\pi$
$468$	0	0
$469$	−8.36475	−0.386248
$470$	0	0
$471$	−6.25155	−0.288056
$472$	0	0
$473$	11.0721	0.509095
$474$	0	0
$475$	7.66862	0.351861
$476$	0	0
$477$	3.23512	0.148126
$478$	0	0
$479$	−26.1101	−1.19300	−0.596501	−	0.802612i	$-0.703443\pi$
$479$	−26.1101	−1.19300	−0.596501	+	0.802612i	$0.703443\pi$
$480$	0	0
$481$	5.70307	0.260038
$482$	0	0
$483$	−2.43753	−0.110912
$484$	0	0
$485$	8.95512	0.406631
$486$	0	0
$487$	−29.1754	−1.32206	−0.661032	−	0.750358i	$-0.729881\pi$
$487$	−29.1754	−1.32206	−0.661032	+	0.750358i	$0.729881\pi$
$488$	0	0
$489$	10.3332	0.467284
$490$	0	0
$491$	−8.31324	−0.375171	−0.187586	−	0.982248i	$-0.560066\pi$
$491$	−8.31324	−0.375171	−0.187586	+	0.982248i	$0.560066\pi$
$492$	0	0
$493$	5.14772	0.231842
$494$	0	0
$495$	−3.32884	−0.149620
$496$	0	0
$497$	−6.30679	−0.282898
$498$	0	0
$499$	−33.0939	−1.48148	−0.740742	−	0.671789i	$-0.765526\pi$
$499$	−33.0939	−1.48148	−0.740742	+	0.671789i	$0.765526\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	22.4327	1.00022	0.500112	−	0.865961i	$-0.333292\pi$
$503$	22.4327	1.00022	0.500112	+	0.865961i	$0.333292\pi$
$504$	0	0
$505$	−36.6597	−1.63134
$506$	0	0
$507$	9.21205	0.409121
$508$	0	0
$509$	7.47758	0.331438	0.165719	−	0.986173i	$-0.447006\pi$
$509$	7.47758	0.331438	0.165719	+	0.986173i	$0.447006\pi$
$510$	0	0
$511$	0.213866	0.00946088
$512$	0	0
$513$	−1.16100	−0.0512595
$514$	0	0
$515$	1.85772	0.0818609
$516$	0	0
$517$	−9.98215	−0.439014
$518$	0	0
$519$	4.48522	0.196879
$520$	0	0
$521$	−4.51154	−0.197654	−0.0988271	−	0.995105i	$-0.531509\pi$
$521$	−4.51154	−0.197654	−0.0988271	+	0.995105i	$0.531509\pi$
$522$	0	0
$523$	−27.9098	−1.22041	−0.610206	−	0.792243i	$-0.708913\pi$
$523$	−27.9098	−1.22041	−0.610206	+	0.792243i	$0.708913\pi$
$524$	0	0
$525$	3.62280	0.158112
$526$	0	0
$527$	−7.56524	−0.329547
$528$	0	0
$529$	−3.24938	−0.141277
$530$	0	0
$531$	12.8823	0.559044
$532$	0	0
$533$	−12.3588	−0.535320
$534$	0	0
$535$	−41.6688	−1.80150
$536$	0	0
$537$	20.1276	0.868572
$538$	0	0
$539$	−6.54618	−0.281964
$540$	0	0
$541$	−28.7584	−1.23642	−0.618211	−	0.786012i	$-0.712142\pi$
$541$	−28.7584	−1.23642	−0.618211	+	0.786012i	$0.712142\pi$
$542$	0	0
$543$	22.5001	0.965572
$544$	0	0
$545$	−11.6604	−0.499475
$546$	0	0
$547$	22.8889	0.978659	0.489330	−	0.872099i	$-0.337242\pi$
$547$	22.8889	0.978659	0.489330	+	0.872099i	$0.337242\pi$
$548$	0	0
$549$	−14.9043	−0.636102
$550$	0	0
$551$	4.31892	0.183992
$552$	0	0
$553$	−5.56298	−0.236562
$554$	0	0
$555$	−9.98233	−0.423727
$556$	0	0
$557$	41.8344	1.77258	0.886291	−	0.463129i	$-0.153273\pi$
$557$	41.8344	1.77258	0.886291	+	0.463129i	$0.153273\pi$
$558$	0	0
$559$	−22.0529	−0.932737
$560$	0	0
$561$	−1.35220	−0.0570898
$562$	0	0
$563$	−33.9234	−1.42970	−0.714851	−	0.699277i	$-0.753505\pi$
$563$	−33.9234	−1.42970	−0.714851	+	0.699277i	$0.753505\pi$
$564$	0	0
$565$	−40.7211	−1.71315
$566$	0	0
$567$	−0.548479	−0.0230340
$568$	0	0
$569$	−26.0510	−1.09212	−0.546058	−	0.837747i	$-0.683872\pi$
$569$	−26.0510	−1.09212	−0.546058	+	0.837747i	$0.683872\pi$
$570$	0	0
$571$	3.91990	0.164043	0.0820214	−	0.996631i	$-0.473862\pi$
$571$	3.91990	0.164043	0.0820214	+	0.996631i	$0.473862\pi$
$572$	0	0
$573$	−10.9408	−0.457060
$574$	0	0
$575$	−29.3545	−1.22417
$576$	0	0
$577$	−30.8384	−1.28382	−0.641910	−	0.766780i	$-0.721858\pi$
$577$	−30.8384	−1.28382	−0.641910	+	0.766780i	$0.721858\pi$
$578$	0	0
$579$	−4.36096	−0.181235
$580$	0	0
$581$	−2.97238	−0.123315
$582$	0	0
$583$	3.16124	0.130925
$584$	0	0
$585$	6.63022	0.274126
$586$	0	0
$587$	−33.0392	−1.36367	−0.681837	−	0.731504i	$-0.738819\pi$
$587$	−33.0392	−1.36367	−0.681837	+	0.731504i	$0.738819\pi$
$588$	0	0
$589$	−6.34721	−0.261532
$590$	0	0
$591$	7.29648	0.300137
$592$	0	0
$593$	−39.8895	−1.63807	−0.819033	−	0.573747i	$-0.805489\pi$
$593$	−39.8895	−1.63807	−0.819033	+	0.573747i	$0.805489\pi$
$594$	0	0
$595$	2.58559	0.105999
$596$	0	0
$597$	−22.7900	−0.932732
$598$	0	0
$599$	−7.18551	−0.293592	−0.146796	−	0.989167i	$-0.546896\pi$
$599$	−7.18551	−0.293592	−0.146796	+	0.989167i	$0.546896\pi$
$600$	0	0
$601$	−12.6866	−0.517499	−0.258749	−	0.965944i	$-0.583310\pi$
$601$	−12.6866	−0.517499	−0.258749	+	0.965944i	$0.583310\pi$
$602$	0	0
$603$	15.2508	0.621061
$604$	0	0
$605$	34.2202	1.39125
$606$	0	0
$607$	25.5231	1.03595	0.517975	−	0.855396i	$-0.326686\pi$
$607$	25.5231	1.03595	0.517975	+	0.855396i	$0.326686\pi$
$608$	0	0
$609$	2.04034	0.0826786
$610$	0	0
$611$	19.8820	0.804339
$612$	0	0
$613$	8.66709	0.350060	0.175030	−	0.984563i	$-0.443998\pi$
$613$	8.66709	0.350060	0.175030	+	0.984563i	$0.443998\pi$
$614$	0	0
$615$	21.6322	0.872294
$616$	0	0
$617$	41.3383	1.66422	0.832109	−	0.554613i	$-0.187134\pi$
$617$	41.3383	1.66422	0.832109	+	0.554613i	$0.187134\pi$
$618$	0	0
$619$	5.59969	0.225071	0.112535	−	0.993648i	$-0.464103\pi$
$619$	5.59969	0.225071	0.112535	+	0.993648i	$0.464103\pi$
$620$	0	0
$621$	4.44417	0.178338
$622$	0	0
$623$	9.40842	0.376940
$624$	0	0
$625$	−14.3976	−0.575903
$626$	0	0
$627$	−1.13449	−0.0453071
$628$	0	0
$629$	−4.05489	−0.161679
$630$	0	0
$631$	−0.900479	−0.0358475	−0.0179237	−	0.999839i	$-0.505706\pi$
$631$	−0.900479	−0.0358475	−0.0179237	+	0.999839i	$0.505706\pi$
$632$	0	0
$633$	−23.3625	−0.928574
$634$	0	0
$635$	−12.3593	−0.490464
$636$	0	0
$637$	13.0384	0.516599
$638$	0	0
$639$	11.4987	0.454881
$640$	0	0
$641$	17.5091	0.691567	0.345783	−	0.938314i	$-0.387613\pi$
$641$	17.5091	0.691567	0.345783	+	0.938314i	$0.387613\pi$
$642$	0	0
$643$	3.47894	0.137196	0.0685980	−	0.997644i	$-0.478147\pi$
$643$	3.47894	0.137196	0.0685980	+	0.997644i	$0.478147\pi$
$644$	0	0
$645$	38.6001	1.51988
$646$	0	0
$647$	−9.60160	−0.377478	−0.188739	−	0.982027i	$-0.560440\pi$
$647$	−9.60160	−0.377478	−0.188739	+	0.982027i	$0.560440\pi$
$648$	0	0
$649$	12.5881	0.494126
$650$	0	0
$651$	−2.99854	−0.117522
$652$	0	0
$653$	15.8659	0.620879	0.310440	−	0.950593i	$-0.399524\pi$
$653$	15.8659	0.620879	0.310440	+	0.950593i	$0.399524\pi$
$654$	0	0
$655$	20.4825	0.800317
$656$	0	0
$657$	−0.389926	−0.0152124
$658$	0	0
$659$	24.2593	0.945008	0.472504	−	0.881328i	$-0.343350\pi$
$659$	24.2593	0.945008	0.472504	+	0.881328i	$0.343350\pi$
$660$	0	0
$661$	23.6372	0.919381	0.459690	−	0.888079i	$-0.347960\pi$
$661$	23.6372	0.919381	0.459690	+	0.888079i	$0.347960\pi$
$662$	0	0
$663$	2.69324	0.104597
$664$	0	0
$665$	2.16930	0.0841218
$666$	0	0
$667$	−16.5323	−0.640132
$668$	0	0
$669$	2.93666	0.113538
$670$	0	0
$671$	−14.5640	−0.562236
$672$	0	0
$673$	37.6912	1.45289	0.726445	−	0.687224i	$-0.241171\pi$
$673$	37.6912	1.45289	0.726445	+	0.687224i	$0.241171\pi$
$674$	0	0
$675$	−6.60517	−0.254233
$676$	0	0
$677$	−37.7588	−1.45119	−0.725594	−	0.688123i	$-0.758435\pi$
$677$	−37.7588	−1.45119	−0.725594	+	0.688123i	$0.758435\pi$
$678$	0	0
$679$	1.44180	0.0553313
$680$	0	0
$681$	−20.3110	−0.778320
$682$	0	0
$683$	−24.5037	−0.937608	−0.468804	−	0.883302i	$-0.655315\pi$
$683$	−24.5037	−0.937608	−0.468804	+	0.883302i	$0.655315\pi$
$684$	0	0
$685$	13.1860	0.503811
$686$	0	0
$687$	−3.90723	−0.149070
$688$	0	0
$689$	−6.29641	−0.239874
$690$	0	0
$691$	34.7858	1.32331	0.661657	−	0.749807i	$-0.269854\pi$
$691$	34.7858	1.32331	0.661657	+	0.749807i	$0.269854\pi$
$692$	0	0
$693$	−0.535953	−0.0203592
$694$	0	0
$695$	19.8437	0.752714
$696$	0	0
$697$	8.78715	0.332837
$698$	0	0
$699$	26.0582	0.985611
$700$	0	0
$701$	−14.1262	−0.533540	−0.266770	−	0.963760i	$-0.585956\pi$
$701$	−14.1262	−0.533540	−0.266770	+	0.963760i	$0.585956\pi$
$702$	0	0
$703$	−3.40204	−0.128310
$704$	0	0
$705$	−34.8003	−1.31065
$706$	0	0
$707$	−5.90233	−0.221980
$708$	0	0
$709$	−31.3280	−1.17655	−0.588275	−	0.808661i	$-0.700193\pi$
$709$	−31.3280	−1.17655	−0.588275	+	0.808661i	$0.700193\pi$
$710$	0	0
$711$	10.1425	0.380375
$712$	0	0
$713$	24.2963	0.909903
$714$	0	0
$715$	6.47881	0.242294
$716$	0	0
$717$	22.1406	0.826858
$718$	0	0
$719$	−13.1522	−0.490494	−0.245247	−	0.969461i	$-0.578869\pi$
$719$	−13.1522	−0.490494	−0.245247	+	0.969461i	$0.578869\pi$
$720$	0	0
$721$	0.299099	0.0111390
$722$	0	0
$723$	19.5932	0.728677
$724$	0	0
$725$	24.5712	0.912551
$726$	0	0
$727$	43.9706	1.63078	0.815390	−	0.578913i	$-0.196523\pi$
$727$	43.9706	1.63078	0.815390	+	0.578913i	$0.196523\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.6796	0.579932
$732$	0	0
$733$	−29.8792	−1.10361	−0.551806	−	0.833972i	$-0.686061\pi$
$733$	−29.8792	−1.10361	−0.551806	+	0.833972i	$0.686061\pi$
$734$	0	0
$735$	−22.8216	−0.841789
$736$	0	0
$737$	14.9025	0.548941
$738$	0	0
$739$	−42.1071	−1.54894	−0.774468	−	0.632613i	$-0.781982\pi$
$739$	−42.1071	−1.54894	−0.774468	+	0.632613i	$0.781982\pi$
$740$	0	0
$741$	2.25962	0.0830093
$742$	0	0
$743$	−5.40968	−0.198462	−0.0992310	−	0.995064i	$-0.531638\pi$
$743$	−5.40968	−0.198462	−0.0992310	+	0.995064i	$0.531638\pi$
$744$	0	0
$745$	−48.0659	−1.76100
$746$	0	0
$747$	5.41931	0.198282
$748$	0	0
$749$	−6.70880	−0.245134
$750$	0	0
$751$	−24.2028	−0.883173	−0.441587	−	0.897219i	$-0.645584\pi$
$751$	−24.2028	−0.883173	−0.441587	+	0.897219i	$0.645584\pi$
$752$	0	0
$753$	13.5771	0.494776
$754$	0	0
$755$	70.5180	2.56641
$756$	0	0
$757$	11.1230	0.404273	0.202137	−	0.979357i	$-0.435212\pi$
$757$	11.1230	0.404273	0.202137	+	0.979357i	$0.435212\pi$
$758$	0	0
$759$	4.34268	0.157629
$760$	0	0
$761$	46.2131	1.67522	0.837612	−	0.546266i	$-0.183951\pi$
$761$	46.2131	1.67522	0.837612	+	0.546266i	$0.183951\pi$
$762$	0	0
$763$	−1.87736	−0.0679648
$764$	0	0
$765$	−4.71410	−0.170439
$766$	0	0
$767$	−25.0724	−0.905311
$768$	0	0
$769$	−2.85940	−0.103113	−0.0515563	−	0.998670i	$-0.516418\pi$
$769$	−2.85940	−0.103113	−0.0515563	+	0.998670i	$0.516418\pi$
$770$	0	0
$771$	19.8735	0.715727
$772$	0	0
$773$	−15.3191	−0.550991	−0.275496	−	0.961302i	$-0.588842\pi$
$773$	−15.3191	−0.550991	−0.275496	+	0.961302i	$0.588842\pi$
$774$	0	0
$775$	−36.1105	−1.29713
$776$	0	0
$777$	−1.60719	−0.0576575
$778$	0	0
$779$	7.37239	0.264143
$780$	0	0
$781$	11.2361	0.402059
$782$	0	0
$783$	−3.71999	−0.132942
$784$	0	0
$785$	−21.2968	−0.760114
$786$	0	0
$787$	22.1474	0.789471	0.394735	−	0.918795i	$-0.370836\pi$
$787$	22.1474	0.789471	0.394735	+	0.918795i	$0.370836\pi$
$788$	0	0
$789$	−19.8907	−0.708128
$790$	0	0
$791$	−6.55622	−0.233112
$792$	0	0
$793$	29.0078	1.03010
$794$	0	0
$795$	11.0209	0.390871
$796$	0	0
$797$	−3.24008	−0.114770	−0.0573848	−	0.998352i	$-0.518276\pi$
$797$	−3.24008	−0.114770	−0.0573848	+	0.998352i	$0.518276\pi$
$798$	0	0
$799$	−14.1361	−0.500100
$800$	0	0
$801$	−17.1536	−0.606094
$802$	0	0
$803$	−0.381021	−0.0134459
$804$	0	0
$805$	−8.30379	−0.292670
$806$	0	0
$807$	−13.4649	−0.473989
$808$	0	0
$809$	10.7561	0.378163	0.189081	−	0.981961i	$-0.439449\pi$
$809$	10.7561	0.378163	0.189081	+	0.981961i	$0.439449\pi$
$810$	0	0
$811$	56.3522	1.97879	0.989396	−	0.145241i	$-0.0463958\pi$
$811$	56.3522	1.97879	0.989396	+	0.145241i	$0.0463958\pi$
$812$	0	0
$813$	21.3536	0.748904
$814$	0	0
$815$	35.2015	1.23306
$816$	0	0
$817$	13.1552	0.460241
$818$	0	0
$819$	1.06749	0.0373010
$820$	0	0
$821$	−6.76637	−0.236148	−0.118074	−	0.993005i	$-0.537672\pi$
$821$	−6.76637	−0.236148	−0.118074	+	0.993005i	$0.537672\pi$
$822$	0	0
$823$	−28.4265	−0.990886	−0.495443	−	0.868640i	$-0.664994\pi$
$823$	−28.4265	−0.990886	−0.495443	+	0.868640i	$0.664994\pi$
$824$	0	0
$825$	−6.45433	−0.224711
$826$	0	0
$827$	−13.3921	−0.465689	−0.232844	−	0.972514i	$-0.574803\pi$
$827$	−13.3921	−0.465689	−0.232844	+	0.972514i	$0.574803\pi$
$828$	0	0
$829$	23.5846	0.819126	0.409563	−	0.912282i	$-0.365681\pi$
$829$	23.5846	0.819126	0.409563	+	0.912282i	$0.365681\pi$
$830$	0	0
$831$	−5.62547	−0.195145
$832$	0	0
$833$	−9.27031	−0.321197
$834$	0	0
$835$	3.40664	0.117892
$836$	0	0
$837$	5.46700	0.188967
$838$	0	0
$839$	24.8349	0.857394	0.428697	−	0.903448i	$-0.358973\pi$
$839$	24.8349	0.857394	0.428697	+	0.903448i	$0.358973\pi$
$840$	0	0
$841$	−15.1617	−0.522816
$842$	0	0
$843$	12.1233	0.417549
$844$	0	0
$845$	31.3821	1.07958
$846$	0	0
$847$	5.50956	0.189311
$848$	0	0
$849$	−2.83994	−0.0974664
$850$	0	0
$851$	13.0226	0.446408
$852$	0	0
$853$	−15.2493	−0.522127	−0.261063	−	0.965322i	$-0.584073\pi$
$853$	−15.2493	−0.522127	−0.261063	+	0.965322i	$0.584073\pi$
$854$	0	0
$855$	−3.95512	−0.135262
$856$	0	0
$857$	3.00641	0.102697	0.0513485	−	0.998681i	$-0.483648\pi$
$857$	3.00641	0.102697	0.0513485	+	0.998681i	$0.483648\pi$
$858$	0	0
$859$	−26.7316	−0.912070	−0.456035	−	0.889962i	$-0.650731\pi$
$859$	−26.7316	−0.912070	−0.456035	+	0.889962i	$0.650731\pi$
$860$	0	0
$861$	3.48285	0.118695
$862$	0	0
$863$	−16.1526	−0.549839	−0.274920	−	0.961467i	$-0.588651\pi$
$863$	−16.1526	−0.549839	−0.274920	+	0.961467i	$0.588651\pi$
$864$	0	0
$865$	15.2795	0.519519
$866$	0	0
$867$	15.0851	0.512317
$868$	0	0
$869$	9.91092	0.336205
$870$	0	0
$871$	−29.6821	−1.00574
$872$	0	0
$873$	−2.62873	−0.0889689
$874$	0	0
$875$	2.99921	0.101392
$876$	0	0
$877$	52.9473	1.78790	0.893952	−	0.448162i	$-0.147921\pi$
$877$	52.9473	1.78790	0.893952	+	0.448162i	$0.147921\pi$
$878$	0	0
$879$	17.8412	0.601769
$880$	0	0
$881$	−42.3111	−1.42550	−0.712749	−	0.701420i	$-0.752550\pi$
$881$	−42.3111	−1.42550	−0.712749	+	0.701420i	$0.752550\pi$
$882$	0	0
$883$	−21.9912	−0.740063	−0.370032	−	0.929019i	$-0.620653\pi$
$883$	−21.9912	−0.740063	−0.370032	+	0.929019i	$0.620653\pi$
$884$	0	0
$885$	43.8853	1.47519
$886$	0	0
$887$	−32.9683	−1.10697	−0.553483	−	0.832860i	$-0.686702\pi$
$887$	−32.9683	−1.10697	−0.553483	+	0.832860i	$0.686702\pi$
$888$	0	0
$889$	−1.98989	−0.0667386
$890$	0	0
$891$	0.977163	0.0327362
$892$	0	0
$893$	−11.8602	−0.396885
$894$	0	0
$895$	68.5676	2.29196
$896$	0	0
$897$	−8.64953	−0.288800
$898$	0	0
$899$	−20.3372	−0.678284
$900$	0	0
$901$	4.47676	0.149143
$902$	0	0
$903$	6.21474	0.206813
$904$	0	0
$905$	76.6497	2.54792
$906$	0	0
$907$	20.7799	0.689985	0.344992	−	0.938606i	$-0.387882\pi$
$907$	20.7799	0.689985	0.344992	+	0.938606i	$0.387882\pi$
$908$	0	0
$909$	10.7613	0.356929
$910$	0	0
$911$	15.6556	0.518694	0.259347	−	0.965784i	$-0.416493\pi$
$911$	15.6556	0.518694	0.259347	+	0.965784i	$0.416493\pi$
$912$	0	0
$913$	5.29555	0.175257
$914$	0	0
$915$	−50.7737	−1.67853
$916$	0	0
$917$	3.29774	0.108901
$918$	0	0
$919$	−12.9838	−0.428296	−0.214148	−	0.976801i	$-0.568697\pi$
$919$	−12.9838	−0.428296	−0.214148	+	0.976801i	$0.568697\pi$
$920$	0	0
$921$	17.9879	0.592720
$922$	0	0
$923$	−22.3795	−0.736631
$924$	0	0
$925$	−19.3549	−0.636384
$926$	0	0
$927$	−0.545324	−0.0179108
$928$	0	0
$929$	1.96973	0.0646246	0.0323123	−	0.999478i	$-0.489713\pi$
$929$	1.96973	0.0646246	0.0323123	+	0.999478i	$0.489713\pi$
$930$	0	0
$931$	−7.77776	−0.254906
$932$	0	0
$933$	−8.91959	−0.292014
$934$	0	0
$935$	−4.60644	−0.150647
$936$	0	0
$937$	−43.8351	−1.43203	−0.716016	−	0.698084i	$-0.754036\pi$
$937$	−43.8351	−1.43203	−0.716016	+	0.698084i	$0.754036\pi$
$938$	0	0
$939$	−3.72295	−0.121494
$940$	0	0
$941$	−5.35649	−0.174617	−0.0873083	−	0.996181i	$-0.527827\pi$
$941$	−5.35649	−0.174617	−0.0873083	+	0.996181i	$0.527827\pi$
$942$	0	0
$943$	−28.2205	−0.918987
$944$	0	0
$945$	−1.86847	−0.0607813
$946$	0	0
$947$	36.7424	1.19397	0.596985	−	0.802253i	$-0.296365\pi$
$947$	36.7424	1.19397	0.596985	+	0.802253i	$0.296365\pi$
$948$	0	0
$949$	0.758899	0.0246349
$950$	0	0
$951$	17.1341	0.555612
$952$	0	0
$953$	21.0794	0.682828	0.341414	−	0.939913i	$-0.389094\pi$
$953$	21.0794	0.682828	0.341414	+	0.939913i	$0.389094\pi$
$954$	0	0
$955$	−37.2715	−1.20608
$956$	0	0
$957$	−3.63504	−0.117504
$958$	0	0
$959$	2.12299	0.0685549
$960$	0	0
$961$	−1.11186	−0.0358666
$962$	0	0
$963$	12.2316	0.394159
$964$	0	0
$965$	−14.8562	−0.478238
$966$	0	0
$967$	2.90443	0.0934001	0.0467001	−	0.998909i	$-0.485129\pi$
$967$	2.90443	0.0934001	0.0467001	+	0.998909i	$0.485129\pi$
$968$	0	0
$969$	−1.60660	−0.0516113
$970$	0	0
$971$	−34.3856	−1.10349	−0.551743	−	0.834014i	$-0.686037\pi$
$971$	−34.3856	−1.10349	−0.551743	+	0.834014i	$0.686037\pi$
$972$	0	0
$973$	3.19490	0.102424
$974$	0	0
$975$	12.8554	0.411703
$976$	0	0
$977$	18.2981	0.585407	0.292704	−	0.956203i	$-0.405445\pi$
$977$	18.2981	0.585407	0.292704	+	0.956203i	$0.405445\pi$
$978$	0	0
$979$	−16.7619	−0.535713
$980$	0	0
$981$	3.42284	0.109283
$982$	0	0
$983$	28.5885	0.911831	0.455915	−	0.890023i	$-0.349312\pi$
$983$	28.5885	0.911831	0.455915	+	0.890023i	$0.349312\pi$
$984$	0	0
$985$	24.8565	0.791992
$986$	0	0
$987$	−5.60296	−0.178344
$988$	0	0
$989$	−50.3562	−1.60123
$990$	0	0
$991$	−3.21688	−0.102188	−0.0510938	−	0.998694i	$-0.516271\pi$
$991$	−3.21688	−0.102188	−0.0510938	+	0.998694i	$0.516271\pi$
$992$	0	0
$993$	36.3810	1.15452
$994$	0	0
$995$	−77.6372	−2.46126
$996$	0	0
$997$	−37.7255	−1.19478	−0.597390	−	0.801951i	$-0.703796\pi$
$997$	−37.7255	−1.19478	−0.597390	+	0.801951i	$0.703796\pi$
$998$	0	0
$999$	2.93026	0.0927094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.s.1.1		5
4.3	odd	2		4008.2.a.f.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.f.1.1	✓	5	4.3	odd	2
8016.2.a.s.1.1		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.40664 q^{5} -0.548479 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.40664 q^{5} -0.548479 q^{7} +1.00000 q^{9} +0.977163 q^{11} -1.94627 q^{13} +3.40664 q^{15} +1.38380 q^{17} +1.16100 q^{19} +0.548479 q^{21} -4.44417 q^{23} +6.60517 q^{25} -1.00000 q^{27} +3.71999 q^{29} -5.46700 q^{31} -0.977163 q^{33} +1.86847 q^{35} -2.93026 q^{37} +1.94627 q^{39} +6.35002 q^{41} +11.3309 q^{43} -3.40664 q^{45} -10.2154 q^{47} -6.69917 q^{49} -1.38380 q^{51} +3.23512 q^{53} -3.32884 q^{55} -1.16100 q^{57} +12.8823 q^{59} -14.9043 q^{61} -0.548479 q^{63} +6.63022 q^{65} +15.2508 q^{67} +4.44417 q^{69} +11.4987 q^{71} -0.389926 q^{73} -6.60517 q^{75} -0.535953 q^{77} +10.1425 q^{79} +1.00000 q^{81} +5.41931 q^{83} -4.71410 q^{85} -3.71999 q^{87} -17.1536 q^{89} +1.06749 q^{91} +5.46700 q^{93} -3.95512 q^{95} -2.62873 q^{97} +0.977163 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9 + 3 * q^11 - 14 * q^13 - q^15 - 13 * q^17 + 2 * q^19 - 4 * q^21 + 5 * q^23 + 2 * q^25 - 5 * q^27 + 13 * q^29 - 2 * q^31 - 3 * q^33 + 12 * q^35 - 5 * q^37 + 14 * q^39 - 20 * q^41 + 20 * q^43 + q^45 - q^47 - 9 * q^49 + 13 * q^51 - 3 * q^53 + 3 * q^55 - 2 * q^57 + q^59 - 34 * q^61 + 4 * q^63 - 22 * q^65 + 16 * q^67 - 5 * q^69 + 5 * q^71 - 12 * q^73 - 2 * q^75 - 8 * q^77 + 20 * q^79 + 5 * q^81 + 15 * q^83 - 27 * q^85 - 13 * q^87 - 48 * q^89 - 7 * q^91 + 2 * q^93 + 5 * q^95 - 21 * q^97 + 3 * q^99

Introduction

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