LMFDB - Embedded newform 8016.2.a.be.1.10

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.83671$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.83671 q^{5} -4.48179 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.83671 q^{5} -4.48179 q^{7} +1.00000 q^{9} -1.10612 q^{11} -5.67402 q^{13} +3.83671 q^{15} -0.581232 q^{17} +4.72213 q^{19} -4.48179 q^{21} +3.33676 q^{23} +9.72035 q^{25} +1.00000 q^{27} +3.38537 q^{29} -2.78132 q^{31} -1.10612 q^{33} -17.1953 q^{35} +4.43128 q^{37} -5.67402 q^{39} +5.90332 q^{41} -10.0030 q^{43} +3.83671 q^{45} +12.4032 q^{47} +13.0864 q^{49} -0.581232 q^{51} +5.10414 q^{53} -4.24388 q^{55} +4.72213 q^{57} +2.77902 q^{59} +9.42616 q^{61} -4.48179 q^{63} -21.7696 q^{65} -9.07361 q^{67} +3.33676 q^{69} -13.8081 q^{71} +9.14053 q^{73} +9.72035 q^{75} +4.95742 q^{77} +17.4215 q^{79} +1.00000 q^{81} -16.9891 q^{83} -2.23002 q^{85} +3.38537 q^{87} -1.50452 q^{89} +25.4298 q^{91} -2.78132 q^{93} +18.1174 q^{95} +0.761888 q^{97} -1.10612 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.83671	1.71583	0.857915	−	0.513792i	$-0.171760\pi$
$5$	3.83671	1.71583	0.857915	+	0.513792i	$0.171760\pi$
$6$	0	0
$7$	−4.48179	−1.69396	−0.846979	−	0.531627i	$-0.821581\pi$
$7$	−4.48179	−1.69396	−0.846979	+	0.531627i	$0.821581\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.10612	−0.333509	−0.166755	−	0.985998i	$-0.553329\pi$
$11$	−1.10612	−0.333509	−0.166755	+	0.985998i	$0.553329\pi$
$12$	0	0
$13$	−5.67402	−1.57369	−0.786845	−	0.617151i	$-0.788287\pi$
$13$	−5.67402	−1.57369	−0.786845	+	0.617151i	$0.788287\pi$
$14$	0	0
$15$	3.83671	0.990635
$16$	0	0
$17$	−0.581232	−0.140969	−0.0704847	−	0.997513i	$-0.522455\pi$
$17$	−0.581232	−0.140969	−0.0704847	+	0.997513i	$0.522455\pi$
$18$	0	0
$19$	4.72213	1.08333	0.541665	−	0.840594i	$-0.317794\pi$
$19$	4.72213	1.08333	0.541665	+	0.840594i	$0.317794\pi$
$20$	0	0
$21$	−4.48179	−0.978007
$22$	0	0
$23$	3.33676	0.695762	0.347881	−	0.937539i	$-0.386901\pi$
$23$	3.33676	0.695762	0.347881	+	0.937539i	$0.386901\pi$
$24$	0	0
$25$	9.72035	1.94407
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.38537	0.628648	0.314324	−	0.949316i	$-0.398222\pi$
$29$	3.38537	0.628648	0.314324	+	0.949316i	$0.398222\pi$
$30$	0	0
$31$	−2.78132	−0.499540	−0.249770	−	0.968305i	$-0.580355\pi$
$31$	−2.78132	−0.499540	−0.249770	+	0.968305i	$0.580355\pi$
$32$	0	0
$33$	−1.10612	−0.192552
$34$	0	0
$35$	−17.1953	−2.90654
$36$	0	0
$37$	4.43128	0.728498	0.364249	−	0.931302i	$-0.381326\pi$
$37$	4.43128	0.728498	0.364249	+	0.931302i	$0.381326\pi$
$38$	0	0
$39$	−5.67402	−0.908570
$40$	0	0
$41$	5.90332	0.921944	0.460972	−	0.887415i	$-0.347501\pi$
$41$	5.90332	0.921944	0.460972	+	0.887415i	$0.347501\pi$
$42$	0	0
$43$	−10.0030	−1.52544	−0.762722	−	0.646727i	$-0.776137\pi$
$43$	−10.0030	−1.52544	−0.762722	+	0.646727i	$0.776137\pi$
$44$	0	0
$45$	3.83671	0.571943
$46$	0	0
$47$	12.4032	1.80920	0.904598	−	0.426266i	$-0.140171\pi$
$47$	12.4032	1.80920	0.904598	+	0.426266i	$0.140171\pi$
$48$	0	0
$49$	13.0864	1.86949
$50$	0	0
$51$	−0.581232	−0.0813887
$52$	0	0
$53$	5.10414	0.701108	0.350554	−	0.936543i	$-0.385993\pi$
$53$	5.10414	0.701108	0.350554	+	0.936543i	$0.385993\pi$
$54$	0	0
$55$	−4.24388	−0.572245
$56$	0	0
$57$	4.72213	0.625461
$58$	0	0
$59$	2.77902	0.361798	0.180899	−	0.983502i	$-0.442099\pi$
$59$	2.77902	0.361798	0.180899	+	0.983502i	$0.442099\pi$
$60$	0	0
$61$	9.42616	1.20690	0.603448	−	0.797402i	$-0.293793\pi$
$61$	9.42616	1.20690	0.603448	+	0.797402i	$0.293793\pi$
$62$	0	0
$63$	−4.48179	−0.564652
$64$	0	0
$65$	−21.7696	−2.70018
$66$	0	0
$67$	−9.07361	−1.10852	−0.554259	−	0.832344i	$-0.686998\pi$
$67$	−9.07361	−1.10852	−0.554259	+	0.832344i	$0.686998\pi$
$68$	0	0
$69$	3.33676	0.401698
$70$	0	0
$71$	−13.8081	−1.63872	−0.819359	−	0.573281i	$-0.805670\pi$
$71$	−13.8081	−1.63872	−0.819359	+	0.573281i	$0.805670\pi$
$72$	0	0
$73$	9.14053	1.06982	0.534909	−	0.844910i	$-0.320346\pi$
$73$	9.14053	1.06982	0.534909	+	0.844910i	$0.320346\pi$
$74$	0	0
$75$	9.72035	1.12241
$76$	0	0
$77$	4.95742	0.564950
$78$	0	0
$79$	17.4215	1.96007	0.980034	−	0.198831i	$-0.0637144\pi$
$79$	17.4215	1.96007	0.980034	+	0.198831i	$0.0637144\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.9891	−1.86479	−0.932397	−	0.361435i	$-0.882287\pi$
$83$	−16.9891	−1.86479	−0.932397	+	0.361435i	$0.882287\pi$
$84$	0	0
$85$	−2.23002	−0.241879
$86$	0	0
$87$	3.38537	0.362950
$88$	0	0
$89$	−1.50452	−0.159478	−0.0797391	−	0.996816i	$-0.525409\pi$
$89$	−1.50452	−0.159478	−0.0797391	+	0.996816i	$0.525409\pi$
$90$	0	0
$91$	25.4298	2.66576
$92$	0	0
$93$	−2.78132	−0.288410
$94$	0	0
$95$	18.1174	1.85881
$96$	0	0
$97$	0.761888	0.0773580	0.0386790	−	0.999252i	$-0.487685\pi$
$97$	0.761888	0.0773580	0.0386790	+	0.999252i	$0.487685\pi$
$98$	0	0
$99$	−1.10612	−0.111170
$100$	0	0
$101$	−1.37489	−0.136807	−0.0684034	−	0.997658i	$-0.521790\pi$
$101$	−1.37489	−0.136807	−0.0684034	+	0.997658i	$0.521790\pi$
$102$	0	0
$103$	0.0629727	0.00620488	0.00310244	−	0.999995i	$-0.499012\pi$
$103$	0.0629727	0.00620488	0.00310244	+	0.999995i	$0.499012\pi$
$104$	0	0
$105$	−17.1953	−1.67809
$106$	0	0
$107$	−10.0437	−0.970959	−0.485480	−	0.874248i	$-0.661355\pi$
$107$	−10.0437	−0.970959	−0.485480	+	0.874248i	$0.661355\pi$
$108$	0	0
$109$	1.87839	0.179917	0.0899584	−	0.995946i	$-0.471327\pi$
$109$	1.87839	0.179917	0.0899584	+	0.995946i	$0.471327\pi$
$110$	0	0
$111$	4.43128	0.420598
$112$	0	0
$113$	4.11228	0.386851	0.193425	−	0.981115i	$-0.438040\pi$
$113$	4.11228	0.386851	0.193425	+	0.981115i	$0.438040\pi$
$114$	0	0
$115$	12.8022	1.19381
$116$	0	0
$117$	−5.67402	−0.524563
$118$	0	0
$119$	2.60496	0.238796
$120$	0	0
$121$	−9.77649	−0.888772
$122$	0	0
$123$	5.90332	0.532284
$124$	0	0
$125$	18.1106	1.61986
$126$	0	0
$127$	19.9118	1.76689	0.883445	−	0.468536i	$-0.155218\pi$
$127$	19.9118	1.76689	0.883445	+	0.468536i	$0.155218\pi$
$128$	0	0
$129$	−10.0030	−0.880715
$130$	0	0
$131$	6.42577	0.561422	0.280711	−	0.959792i	$-0.409430\pi$
$131$	6.42577	0.561422	0.280711	+	0.959792i	$0.409430\pi$
$132$	0	0
$133$	−21.1636	−1.83512
$134$	0	0
$135$	3.83671	0.330212
$136$	0	0
$137$	−1.42007	−0.121325	−0.0606625	−	0.998158i	$-0.519321\pi$
$137$	−1.42007	−0.121325	−0.0606625	+	0.998158i	$0.519321\pi$
$138$	0	0
$139$	−9.43659	−0.800401	−0.400200	−	0.916428i	$-0.631059\pi$
$139$	−9.43659	−0.800401	−0.400200	+	0.916428i	$0.631059\pi$
$140$	0	0
$141$	12.4032	1.04454
$142$	0	0
$143$	6.27617	0.524840
$144$	0	0
$145$	12.9887	1.07865
$146$	0	0
$147$	13.0864	1.07935
$148$	0	0
$149$	7.81076	0.639882	0.319941	−	0.947437i	$-0.396337\pi$
$149$	7.81076	0.639882	0.319941	+	0.947437i	$0.396337\pi$
$150$	0	0
$151$	−0.614618	−0.0500169	−0.0250085	−	0.999687i	$-0.507961\pi$
$151$	−0.614618	−0.0500169	−0.0250085	+	0.999687i	$0.507961\pi$
$152$	0	0
$153$	−0.581232	−0.0469898
$154$	0	0
$155$	−10.6711	−0.857126
$156$	0	0
$157$	18.5892	1.48358	0.741792	−	0.670631i	$-0.233976\pi$
$157$	18.5892	1.48358	0.741792	+	0.670631i	$0.233976\pi$
$158$	0	0
$159$	5.10414	0.404785
$160$	0	0
$161$	−14.9547	−1.17859
$162$	0	0
$163$	17.6487	1.38235	0.691175	−	0.722687i	$-0.257093\pi$
$163$	17.6487	1.38235	0.691175	+	0.722687i	$0.257093\pi$
$164$	0	0
$165$	−4.24388	−0.330386
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	19.1945	1.47650
$170$	0	0
$171$	4.72213	0.361110
$172$	0	0
$173$	4.79962	0.364909	0.182454	−	0.983214i	$-0.441596\pi$
$173$	4.79962	0.364909	0.182454	+	0.983214i	$0.441596\pi$
$174$	0	0
$175$	−43.5646	−3.29317
$176$	0	0
$177$	2.77902	0.208884
$178$	0	0
$179$	20.6389	1.54262	0.771310	−	0.636459i	$-0.219602\pi$
$179$	20.6389	1.54262	0.771310	+	0.636459i	$0.219602\pi$
$180$	0	0
$181$	10.9125	0.811123	0.405562	−	0.914068i	$-0.367076\pi$
$181$	10.9125	0.811123	0.405562	+	0.914068i	$0.367076\pi$
$182$	0	0
$183$	9.42616	0.696802
$184$	0	0
$185$	17.0015	1.24998
$186$	0	0
$187$	0.642915	0.0470146
$188$	0	0
$189$	−4.48179	−0.326002
$190$	0	0
$191$	10.4748	0.757931	0.378965	−	0.925411i	$-0.376280\pi$
$191$	10.4748	0.757931	0.378965	+	0.925411i	$0.376280\pi$
$192$	0	0
$193$	−8.30352	−0.597701	−0.298850	−	0.954300i	$-0.596603\pi$
$193$	−8.30352	−0.597701	−0.298850	+	0.954300i	$0.596603\pi$
$194$	0	0
$195$	−21.7696	−1.55895
$196$	0	0
$197$	24.7450	1.76301	0.881503	−	0.472179i	$-0.156533\pi$
$197$	24.7450	1.76301	0.881503	+	0.472179i	$0.156533\pi$
$198$	0	0
$199$	20.9627	1.48601	0.743003	−	0.669288i	$-0.233401\pi$
$199$	20.9627	1.48601	0.743003	+	0.669288i	$0.233401\pi$
$200$	0	0
$201$	−9.07361	−0.640003
$202$	0	0
$203$	−15.1725	−1.06490
$204$	0	0
$205$	22.6493	1.58190
$206$	0	0
$207$	3.33676	0.231921
$208$	0	0
$209$	−5.22326	−0.361301
$210$	0	0
$211$	13.3764	0.920866	0.460433	−	0.887694i	$-0.347694\pi$
$211$	13.3764	0.920866	0.460433	+	0.887694i	$0.347694\pi$
$212$	0	0
$213$	−13.8081	−0.946114
$214$	0	0
$215$	−38.3786	−2.61740
$216$	0	0
$217$	12.4653	0.846200
$218$	0	0
$219$	9.14053	0.617660
$220$	0	0
$221$	3.29792	0.221842
$222$	0	0
$223$	7.31317	0.489726	0.244863	−	0.969558i	$-0.421257\pi$
$223$	7.31317	0.489726	0.244863	+	0.969558i	$0.421257\pi$
$224$	0	0
$225$	9.72035	0.648024
$226$	0	0
$227$	−2.35302	−0.156175	−0.0780875	−	0.996947i	$-0.524881\pi$
$227$	−2.35302	−0.156175	−0.0780875	+	0.996947i	$0.524881\pi$
$228$	0	0
$229$	16.3540	1.08070	0.540350	−	0.841440i	$-0.318292\pi$
$229$	16.3540	1.08070	0.540350	+	0.841440i	$0.318292\pi$
$230$	0	0
$231$	4.95742	0.326174
$232$	0	0
$233$	−24.5380	−1.60753	−0.803767	−	0.594944i	$-0.797174\pi$
$233$	−24.5380	−1.60753	−0.803767	+	0.594944i	$0.797174\pi$
$234$	0	0
$235$	47.5876	3.10427
$236$	0	0
$237$	17.4215	1.13165
$238$	0	0
$239$	16.5560	1.07092	0.535460	−	0.844561i	$-0.320138\pi$
$239$	16.5560	1.07092	0.535460	+	0.844561i	$0.320138\pi$
$240$	0	0
$241$	13.4248	0.864765	0.432383	−	0.901690i	$-0.357673\pi$
$241$	13.4248	0.864765	0.432383	+	0.901690i	$0.357673\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	50.2089	3.20773
$246$	0	0
$247$	−26.7935	−1.70483
$248$	0	0
$249$	−16.9891	−1.07664
$250$	0	0
$251$	−7.08588	−0.447257	−0.223628	−	0.974674i	$-0.571790\pi$
$251$	−7.08588	−0.447257	−0.223628	+	0.974674i	$0.571790\pi$
$252$	0	0
$253$	−3.69087	−0.232043
$254$	0	0
$255$	−2.23002	−0.139649
$256$	0	0
$257$	9.32571	0.581722	0.290861	−	0.956765i	$-0.406058\pi$
$257$	9.32571	0.581722	0.290861	+	0.956765i	$0.406058\pi$
$258$	0	0
$259$	−19.8601	−1.23404
$260$	0	0
$261$	3.38537	0.209549
$262$	0	0
$263$	−1.03494	−0.0638170	−0.0319085	−	0.999491i	$-0.510159\pi$
$263$	−1.03494	−0.0638170	−0.0319085	+	0.999491i	$0.510159\pi$
$264$	0	0
$265$	19.5831	1.20298
$266$	0	0
$267$	−1.50452	−0.0920748
$268$	0	0
$269$	−13.2297	−0.806630	−0.403315	−	0.915061i	$-0.632142\pi$
$269$	−13.2297	−0.806630	−0.403315	+	0.915061i	$0.632142\pi$
$270$	0	0
$271$	−15.6279	−0.949330	−0.474665	−	0.880166i	$-0.657431\pi$
$271$	−15.6279	−0.949330	−0.474665	+	0.880166i	$0.657431\pi$
$272$	0	0
$273$	25.4298	1.53908
$274$	0	0
$275$	−10.7519	−0.648365
$276$	0	0
$277$	−0.591123	−0.0355171	−0.0177586	−	0.999842i	$-0.505653\pi$
$277$	−0.591123	−0.0355171	−0.0177586	+	0.999842i	$0.505653\pi$
$278$	0	0
$279$	−2.78132	−0.166513
$280$	0	0
$281$	−4.64950	−0.277366	−0.138683	−	0.990337i	$-0.544287\pi$
$281$	−4.64950	−0.277366	−0.138683	+	0.990337i	$0.544287\pi$
$282$	0	0
$283$	23.7408	1.41124	0.705621	−	0.708590i	$-0.250668\pi$
$283$	23.7408	1.41124	0.705621	+	0.708590i	$0.250668\pi$
$284$	0	0
$285$	18.1174	1.07319
$286$	0	0
$287$	−26.4574	−1.56173
$288$	0	0
$289$	−16.6622	−0.980128
$290$	0	0
$291$	0.761888	0.0446627
$292$	0	0
$293$	−27.7554	−1.62149	−0.810743	−	0.585402i	$-0.800937\pi$
$293$	−27.7554	−1.62149	−0.810743	+	0.585402i	$0.800937\pi$
$294$	0	0
$295$	10.6623	0.620783
$296$	0	0
$297$	−1.10612	−0.0641839
$298$	0	0
$299$	−18.9328	−1.09491
$300$	0	0
$301$	44.8314	2.58404
$302$	0	0
$303$	−1.37489	−0.0789854
$304$	0	0
$305$	36.1654	2.07083
$306$	0	0
$307$	−1.47952	−0.0844405	−0.0422202	−	0.999108i	$-0.513443\pi$
$307$	−1.47952	−0.0844405	−0.0422202	+	0.999108i	$0.513443\pi$
$308$	0	0
$309$	0.0629727	0.00358239
$310$	0	0
$311$	−19.6599	−1.11481	−0.557404	−	0.830241i	$-0.688203\pi$
$311$	−19.6599	−1.11481	−0.557404	+	0.830241i	$0.688203\pi$
$312$	0	0
$313$	−13.0409	−0.737116	−0.368558	−	0.929605i	$-0.620148\pi$
$313$	−13.0409	−0.737116	−0.368558	+	0.929605i	$0.620148\pi$
$314$	0	0
$315$	−17.1953	−0.968847
$316$	0	0
$317$	13.2143	0.742191	0.371096	−	0.928595i	$-0.378982\pi$
$317$	13.2143	0.742191	0.371096	+	0.928595i	$0.378982\pi$
$318$	0	0
$319$	−3.74464	−0.209660
$320$	0	0
$321$	−10.0437	−0.560584
$322$	0	0
$323$	−2.74465	−0.152717
$324$	0	0
$325$	−55.1535	−3.05936
$326$	0	0
$327$	1.87839	0.103875
$328$	0	0
$329$	−55.5886	−3.06470
$330$	0	0
$331$	27.4467	1.50860	0.754302	−	0.656527i	$-0.227975\pi$
$331$	27.4467	1.50860	0.754302	+	0.656527i	$0.227975\pi$
$332$	0	0
$333$	4.43128	0.242833
$334$	0	0
$335$	−34.8128	−1.90203
$336$	0	0
$337$	−28.0032	−1.52543	−0.762716	−	0.646733i	$-0.776135\pi$
$337$	−28.0032	−1.52543	−0.762716	+	0.646733i	$0.776135\pi$
$338$	0	0
$339$	4.11228	0.223348
$340$	0	0
$341$	3.07649	0.166601
$342$	0	0
$343$	−27.2782	−1.47288
$344$	0	0
$345$	12.8022	0.689246
$346$	0	0
$347$	−31.7281	−1.70325	−0.851627	−	0.524148i	$-0.824384\pi$
$347$	−31.7281	−1.70325	−0.851627	+	0.524148i	$0.824384\pi$
$348$	0	0
$349$	−22.0742	−1.18160	−0.590802	−	0.806817i	$-0.701189\pi$
$349$	−22.0742	−1.18160	−0.590802	+	0.806817i	$0.701189\pi$
$350$	0	0
$351$	−5.67402	−0.302857
$352$	0	0
$353$	24.7084	1.31509	0.657547	−	0.753413i	$-0.271594\pi$
$353$	24.7084	1.31509	0.657547	+	0.753413i	$0.271594\pi$
$354$	0	0
$355$	−52.9776	−2.81176
$356$	0	0
$357$	2.60496	0.137869
$358$	0	0
$359$	26.1424	1.37975	0.689873	−	0.723931i	$-0.257666\pi$
$359$	26.1424	1.37975	0.689873	+	0.723931i	$0.257666\pi$
$360$	0	0
$361$	3.29851	0.173606
$362$	0	0
$363$	−9.77649	−0.513133
$364$	0	0
$365$	35.0696	1.83563
$366$	0	0
$367$	−10.7397	−0.560607	−0.280303	−	0.959911i	$-0.590435\pi$
$367$	−10.7397	−0.560607	−0.280303	+	0.959911i	$0.590435\pi$
$368$	0	0
$369$	5.90332	0.307315
$370$	0	0
$371$	−22.8757	−1.18765
$372$	0	0
$373$	−32.2357	−1.66910	−0.834551	−	0.550930i	$-0.814273\pi$
$373$	−32.2357	−1.66910	−0.834551	+	0.550930i	$0.814273\pi$
$374$	0	0
$375$	18.1106	0.935229
$376$	0	0
$377$	−19.2087	−0.989296
$378$	0	0
$379$	25.1380	1.29125	0.645626	−	0.763654i	$-0.276597\pi$
$379$	25.1380	1.29125	0.645626	+	0.763654i	$0.276597\pi$
$380$	0	0
$381$	19.9118	1.02011
$382$	0	0
$383$	−25.2766	−1.29157	−0.645787	−	0.763518i	$-0.723470\pi$
$383$	−25.2766	−1.29157	−0.645787	+	0.763518i	$0.723470\pi$
$384$	0	0
$385$	19.0202	0.969358
$386$	0	0
$387$	−10.0030	−0.508481
$388$	0	0
$389$	−2.01440	−0.102134	−0.0510670	−	0.998695i	$-0.516262\pi$
$389$	−2.01440	−0.102134	−0.0510670	+	0.998695i	$0.516262\pi$
$390$	0	0
$391$	−1.93943	−0.0980812
$392$	0	0
$393$	6.42577	0.324137
$394$	0	0
$395$	66.8411	3.36314
$396$	0	0
$397$	−29.6705	−1.48912	−0.744561	−	0.667555i	$-0.767341\pi$
$397$	−29.6705	−1.48912	−0.744561	+	0.667555i	$0.767341\pi$
$398$	0	0
$399$	−21.1636	−1.05950
$400$	0	0
$401$	−11.0918	−0.553900	−0.276950	−	0.960884i	$-0.589324\pi$
$401$	−11.0918	−0.553900	−0.276950	+	0.960884i	$0.589324\pi$
$402$	0	0
$403$	15.7813	0.786121
$404$	0	0
$405$	3.83671	0.190648
$406$	0	0
$407$	−4.90155	−0.242961
$408$	0	0
$409$	21.5517	1.06566	0.532831	−	0.846221i	$-0.321128\pi$
$409$	21.5517	1.06566	0.532831	+	0.846221i	$0.321128\pi$
$410$	0	0
$411$	−1.42007	−0.0700470
$412$	0	0
$413$	−12.4550	−0.612870
$414$	0	0
$415$	−65.1822	−3.19967
$416$	0	0
$417$	−9.43659	−0.462112
$418$	0	0
$419$	−2.34797	−0.114706	−0.0573530	−	0.998354i	$-0.518266\pi$
$419$	−2.34797	−0.114706	−0.0573530	+	0.998354i	$0.518266\pi$
$420$	0	0
$421$	−24.1533	−1.17716	−0.588581	−	0.808439i	$-0.700313\pi$
$421$	−24.1533	−1.17716	−0.588581	+	0.808439i	$0.700313\pi$
$422$	0	0
$423$	12.4032	0.603065
$424$	0	0
$425$	−5.64978	−0.274055
$426$	0	0
$427$	−42.2461	−2.04443
$428$	0	0
$429$	6.27617	0.303016
$430$	0	0
$431$	14.9384	0.719558	0.359779	−	0.933038i	$-0.382852\pi$
$431$	14.9384	0.719558	0.359779	+	0.933038i	$0.382852\pi$
$432$	0	0
$433$	−10.2775	−0.493907	−0.246954	−	0.969027i	$-0.579430\pi$
$433$	−10.2775	−0.493907	−0.246954	+	0.969027i	$0.579430\pi$
$434$	0	0
$435$	12.9887	0.622760
$436$	0	0
$437$	15.7566	0.753741
$438$	0	0
$439$	−2.86030	−0.136514	−0.0682572	−	0.997668i	$-0.521744\pi$
$439$	−2.86030	−0.136514	−0.0682572	+	0.997668i	$0.521744\pi$
$440$	0	0
$441$	13.0864	0.623164
$442$	0	0
$443$	25.8556	1.22844	0.614219	−	0.789136i	$-0.289471\pi$
$443$	25.8556	1.22844	0.614219	+	0.789136i	$0.289471\pi$
$444$	0	0
$445$	−5.77239	−0.273638
$446$	0	0
$447$	7.81076	0.369436
$448$	0	0
$449$	−7.46562	−0.352325	−0.176162	−	0.984361i	$-0.556368\pi$
$449$	−7.46562	−0.352325	−0.176162	+	0.984361i	$0.556368\pi$
$450$	0	0
$451$	−6.52981	−0.307477
$452$	0	0
$453$	−0.614618	−0.0288773
$454$	0	0
$455$	97.5667	4.57400
$456$	0	0
$457$	40.6342	1.90079	0.950393	−	0.311052i	$-0.100681\pi$
$457$	40.6342	1.90079	0.950393	+	0.311052i	$0.100681\pi$
$458$	0	0
$459$	−0.581232	−0.0271296
$460$	0	0
$461$	0.791635	0.0368701	0.0184351	−	0.999830i	$-0.494132\pi$
$461$	0.791635	0.0368701	0.0184351	+	0.999830i	$0.494132\pi$
$462$	0	0
$463$	35.1065	1.63154	0.815769	−	0.578377i	$-0.196314\pi$
$463$	35.1065	1.63154	0.815769	+	0.578377i	$0.196314\pi$
$464$	0	0
$465$	−10.6711	−0.494862
$466$	0	0
$467$	37.1756	1.72028	0.860141	−	0.510056i	$-0.170375\pi$
$467$	37.1756	1.72028	0.860141	+	0.510056i	$0.170375\pi$
$468$	0	0
$469$	40.6660	1.87778
$470$	0	0
$471$	18.5892	0.856547
$472$	0	0
$473$	11.0646	0.508749
$474$	0	0
$475$	45.9008	2.10607
$476$	0	0
$477$	5.10414	0.233703
$478$	0	0
$479$	−19.1213	−0.873675	−0.436838	−	0.899540i	$-0.643902\pi$
$479$	−19.1213	−0.873675	−0.436838	+	0.899540i	$0.643902\pi$
$480$	0	0
$481$	−25.1432	−1.14643
$482$	0	0
$483$	−14.9547	−0.680460
$484$	0	0
$485$	2.92315	0.132733
$486$	0	0
$487$	9.51757	0.431282	0.215641	−	0.976473i	$-0.430816\pi$
$487$	9.51757	0.431282	0.215641	+	0.976473i	$0.430816\pi$
$488$	0	0
$489$	17.6487	0.798100
$490$	0	0
$491$	−2.94043	−0.132700	−0.0663499	−	0.997796i	$-0.521135\pi$
$491$	−2.94043	−0.132700	−0.0663499	+	0.997796i	$0.521135\pi$
$492$	0	0
$493$	−1.96769	−0.0886201
$494$	0	0
$495$	−4.24388	−0.190748
$496$	0	0
$497$	61.8849	2.77592
$498$	0	0
$499$	−28.1976	−1.26230	−0.631149	−	0.775662i	$-0.717416\pi$
$499$	−28.1976	−1.26230	−0.631149	+	0.775662i	$0.717416\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−4.54162	−0.202501	−0.101250	−	0.994861i	$-0.532284\pi$
$503$	−4.54162	−0.202501	−0.101250	+	0.994861i	$0.532284\pi$
$504$	0	0
$505$	−5.27506	−0.234737
$506$	0	0
$507$	19.1945	0.852457
$508$	0	0
$509$	23.0856	1.02325	0.511625	−	0.859209i	$-0.329044\pi$
$509$	23.0856	1.02325	0.511625	+	0.859209i	$0.329044\pi$
$510$	0	0
$511$	−40.9659	−1.81223
$512$	0	0
$513$	4.72213	0.208487
$514$	0	0
$515$	0.241608	0.0106465
$516$	0	0
$517$	−13.7195	−0.603383
$518$	0	0
$519$	4.79962	0.210680
$520$	0	0
$521$	7.64748	0.335042	0.167521	−	0.985868i	$-0.446424\pi$
$521$	7.64748	0.335042	0.167521	+	0.985868i	$0.446424\pi$
$522$	0	0
$523$	−31.1172	−1.36066	−0.680329	−	0.732906i	$-0.738163\pi$
$523$	−31.1172	−1.36066	−0.680329	+	0.732906i	$0.738163\pi$
$524$	0	0
$525$	−43.5646	−1.90131
$526$	0	0
$527$	1.61659	0.0704199
$528$	0	0
$529$	−11.8660	−0.515915
$530$	0	0
$531$	2.77902	0.120599
$532$	0	0
$533$	−33.4955	−1.45085
$534$	0	0
$535$	−38.5347	−1.66600
$536$	0	0
$537$	20.6389	0.890633
$538$	0	0
$539$	−14.4752	−0.623493
$540$	0	0
$541$	−25.3270	−1.08889	−0.544447	−	0.838795i	$-0.683260\pi$
$541$	−25.3270	−1.08889	−0.544447	+	0.838795i	$0.683260\pi$
$542$	0	0
$543$	10.9125	0.468302
$544$	0	0
$545$	7.20683	0.308707
$546$	0	0
$547$	27.8497	1.19077	0.595383	−	0.803442i	$-0.297000\pi$
$547$	27.8497	1.19077	0.595383	+	0.803442i	$0.297000\pi$
$548$	0	0
$549$	9.42616	0.402299
$550$	0	0
$551$	15.9862	0.681034
$552$	0	0
$553$	−78.0793	−3.32027
$554$	0	0
$555$	17.0015	0.721675
$556$	0	0
$557$	9.21977	0.390654	0.195327	−	0.980738i	$-0.437423\pi$
$557$	9.21977	0.390654	0.195327	+	0.980738i	$0.437423\pi$
$558$	0	0
$559$	56.7572	2.40057
$560$	0	0
$561$	0.642915	0.0271439
$562$	0	0
$563$	16.8905	0.711851	0.355926	−	0.934514i	$-0.384166\pi$
$563$	16.8905	0.711851	0.355926	+	0.934514i	$0.384166\pi$
$564$	0	0
$565$	15.7776	0.663770
$566$	0	0
$567$	−4.48179	−0.188217
$568$	0	0
$569$	−38.5370	−1.61556	−0.807778	−	0.589487i	$-0.799330\pi$
$569$	−38.5370	−1.61556	−0.807778	+	0.589487i	$0.799330\pi$
$570$	0	0
$571$	−13.6738	−0.572232	−0.286116	−	0.958195i	$-0.592364\pi$
$571$	−13.6738	−0.572232	−0.286116	+	0.958195i	$0.592364\pi$
$572$	0	0
$573$	10.4748	0.437592
$574$	0	0
$575$	32.4345	1.35261
$576$	0	0
$577$	−0.888543	−0.0369905	−0.0184953	−	0.999829i	$-0.505888\pi$
$577$	−0.888543	−0.0369905	−0.0184953	+	0.999829i	$0.505888\pi$
$578$	0	0
$579$	−8.30352	−0.345083
$580$	0	0
$581$	76.1415	3.15888
$582$	0	0
$583$	−5.64582	−0.233826
$584$	0	0
$585$	−21.7696	−0.900061
$586$	0	0
$587$	−24.2849	−1.00234	−0.501172	−	0.865348i	$-0.667098\pi$
$587$	−24.2849	−1.00234	−0.501172	+	0.865348i	$0.667098\pi$
$588$	0	0
$589$	−13.1338	−0.541167
$590$	0	0
$591$	24.7450	1.01787
$592$	0	0
$593$	−6.05351	−0.248588	−0.124294	−	0.992245i	$-0.539667\pi$
$593$	−6.05351	−0.248588	−0.124294	+	0.992245i	$0.539667\pi$
$594$	0	0
$595$	9.99448	0.409734
$596$	0	0
$597$	20.9627	0.857946
$598$	0	0
$599$	−29.9256	−1.22273	−0.611364	−	0.791349i	$-0.709379\pi$
$599$	−29.9256	−1.22273	−0.611364	+	0.791349i	$0.709379\pi$
$600$	0	0
$601$	−45.2604	−1.84621	−0.923105	−	0.384548i	$-0.874357\pi$
$601$	−45.2604	−1.84621	−0.923105	+	0.384548i	$0.874357\pi$
$602$	0	0
$603$	−9.07361	−0.369506
$604$	0	0
$605$	−37.5096	−1.52498
$606$	0	0
$607$	−3.72389	−0.151148	−0.0755740	−	0.997140i	$-0.524079\pi$
$607$	−3.72389	−0.151148	−0.0755740	+	0.997140i	$0.524079\pi$
$608$	0	0
$609$	−15.1725	−0.614822
$610$	0	0
$611$	−70.3761	−2.84711
$612$	0	0
$613$	−35.3721	−1.42867	−0.714333	−	0.699806i	$-0.753270\pi$
$613$	−35.3721	−1.42867	−0.714333	+	0.699806i	$0.753270\pi$
$614$	0	0
$615$	22.6493	0.913309
$616$	0	0
$617$	−7.32100	−0.294732	−0.147366	−	0.989082i	$-0.547080\pi$
$617$	−7.32100	−0.294732	−0.147366	+	0.989082i	$0.547080\pi$
$618$	0	0
$619$	15.2901	0.614559	0.307280	−	0.951619i	$-0.400581\pi$
$619$	15.2901	0.614559	0.307280	+	0.951619i	$0.400581\pi$
$620$	0	0
$621$	3.33676	0.133899
$622$	0	0
$623$	6.74292	0.270149
$624$	0	0
$625$	20.8835	0.835341
$626$	0	0
$627$	−5.22326	−0.208597
$628$	0	0
$629$	−2.57560	−0.102696
$630$	0	0
$631$	13.2764	0.528524	0.264262	−	0.964451i	$-0.414872\pi$
$631$	13.2764	0.528524	0.264262	+	0.964451i	$0.414872\pi$
$632$	0	0
$633$	13.3764	0.531662
$634$	0	0
$635$	76.3959	3.03168
$636$	0	0
$637$	−74.2527	−2.94200
$638$	0	0
$639$	−13.8081	−0.546239
$640$	0	0
$641$	6.17423	0.243867	0.121934	−	0.992538i	$-0.461090\pi$
$641$	6.17423	0.243867	0.121934	+	0.992538i	$0.461090\pi$
$642$	0	0
$643$	−0.865911	−0.0341482	−0.0170741	−	0.999854i	$-0.505435\pi$
$643$	−0.865911	−0.0341482	−0.0170741	+	0.999854i	$0.505435\pi$
$644$	0	0
$645$	−38.3786	−1.51116
$646$	0	0
$647$	−34.3420	−1.35012	−0.675062	−	0.737761i	$-0.735883\pi$
$647$	−34.3420	−1.35012	−0.675062	+	0.737761i	$0.735883\pi$
$648$	0	0
$649$	−3.07394	−0.120663
$650$	0	0
$651$	12.4653	0.488554
$652$	0	0
$653$	17.1698	0.671908	0.335954	−	0.941878i	$-0.390941\pi$
$653$	17.1698	0.671908	0.335954	+	0.941878i	$0.390941\pi$
$654$	0	0
$655$	24.6538	0.963305
$656$	0	0
$657$	9.14053	0.356606
$658$	0	0
$659$	−36.8878	−1.43695	−0.718473	−	0.695555i	$-0.755159\pi$
$659$	−36.8878	−1.43695	−0.718473	+	0.695555i	$0.755159\pi$
$660$	0	0
$661$	26.5260	1.03174	0.515871	−	0.856666i	$-0.327468\pi$
$661$	26.5260	1.03174	0.515871	+	0.856666i	$0.327468\pi$
$662$	0	0
$663$	3.29792	0.128081
$664$	0	0
$665$	−81.1986	−3.14875
$666$	0	0
$667$	11.2962	0.437389
$668$	0	0
$669$	7.31317	0.282744
$670$	0	0
$671$	−10.4265	−0.402511
$672$	0	0
$673$	−27.2730	−1.05130	−0.525648	−	0.850702i	$-0.676177\pi$
$673$	−27.2730	−1.05130	−0.525648	+	0.850702i	$0.676177\pi$
$674$	0	0
$675$	9.72035	0.374137
$676$	0	0
$677$	12.4893	0.480004	0.240002	−	0.970772i	$-0.422852\pi$
$677$	12.4893	0.480004	0.240002	+	0.970772i	$0.422852\pi$
$678$	0	0
$679$	−3.41462	−0.131041
$680$	0	0
$681$	−2.35302	−0.0901677
$682$	0	0
$683$	29.6433	1.13427	0.567134	−	0.823626i	$-0.308052\pi$
$683$	29.6433	1.13427	0.567134	+	0.823626i	$0.308052\pi$
$684$	0	0
$685$	−5.44841	−0.208173
$686$	0	0
$687$	16.3540	0.623943
$688$	0	0
$689$	−28.9610	−1.10333
$690$	0	0
$691$	−1.78092	−0.0677493	−0.0338747	−	0.999426i	$-0.510785\pi$
$691$	−1.78092	−0.0677493	−0.0338747	+	0.999426i	$0.510785\pi$
$692$	0	0
$693$	4.95742	0.188317
$694$	0	0
$695$	−36.2055	−1.37335
$696$	0	0
$697$	−3.43120	−0.129966
$698$	0	0
$699$	−24.5380	−0.928111
$700$	0	0
$701$	19.8886	0.751182	0.375591	−	0.926785i	$-0.377440\pi$
$701$	19.8886	0.751182	0.375591	+	0.926785i	$0.377440\pi$
$702$	0	0
$703$	20.9251	0.789204
$704$	0	0
$705$	47.5876	1.79225
$706$	0	0
$707$	6.16197	0.231745
$708$	0	0
$709$	−41.6852	−1.56552	−0.782760	−	0.622323i	$-0.786189\pi$
$709$	−41.6852	−1.56552	−0.782760	+	0.622323i	$0.786189\pi$
$710$	0	0
$711$	17.4215	0.653356
$712$	0	0
$713$	−9.28060	−0.347561
$714$	0	0
$715$	24.0799	0.900536
$716$	0	0
$717$	16.5560	0.618296
$718$	0	0
$719$	−16.0390	−0.598153	−0.299076	−	0.954229i	$-0.596679\pi$
$719$	−16.0390	−0.598153	−0.299076	+	0.954229i	$0.596679\pi$
$720$	0	0
$721$	−0.282230	−0.0105108
$722$	0	0
$723$	13.4248	0.499272
$724$	0	0
$725$	32.9070	1.22214
$726$	0	0
$727$	−4.28104	−0.158775	−0.0793874	−	0.996844i	$-0.525296\pi$
$727$	−4.28104	−0.158775	−0.0793874	+	0.996844i	$0.525296\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.81406	0.215041
$732$	0	0
$733$	−14.9604	−0.552574	−0.276287	−	0.961075i	$-0.589104\pi$
$733$	−14.9604	−0.552574	−0.276287	+	0.961075i	$0.589104\pi$
$734$	0	0
$735$	50.2089	1.85198
$736$	0	0
$737$	10.0365	0.369701
$738$	0	0
$739$	−7.78573	−0.286403	−0.143201	−	0.989694i	$-0.545740\pi$
$739$	−7.78573	−0.286403	−0.143201	+	0.989694i	$0.545740\pi$
$740$	0	0
$741$	−26.7935	−0.984282
$742$	0	0
$743$	4.23167	0.155245	0.0776224	−	0.996983i	$-0.475267\pi$
$743$	4.23167	0.155245	0.0776224	+	0.996983i	$0.475267\pi$
$744$	0	0
$745$	29.9676	1.09793
$746$	0	0
$747$	−16.9891	−0.621598
$748$	0	0
$749$	45.0137	1.64476
$750$	0	0
$751$	49.4751	1.80537	0.902686	−	0.430301i	$-0.141592\pi$
$751$	49.4751	1.80537	0.902686	+	0.430301i	$0.141592\pi$
$752$	0	0
$753$	−7.08588	−0.258224
$754$	0	0
$755$	−2.35811	−0.0858205
$756$	0	0
$757$	−13.6590	−0.496444	−0.248222	−	0.968703i	$-0.579846\pi$
$757$	−13.6590	−0.496444	−0.248222	+	0.968703i	$0.579846\pi$
$758$	0	0
$759$	−3.69087	−0.133970
$760$	0	0
$761$	7.48874	0.271467	0.135733	−	0.990745i	$-0.456661\pi$
$761$	7.48874	0.271467	0.135733	+	0.990745i	$0.456661\pi$
$762$	0	0
$763$	−8.41853	−0.304771
$764$	0	0
$765$	−2.23002	−0.0806265
$766$	0	0
$767$	−15.7682	−0.569357
$768$	0	0
$769$	51.7046	1.86452	0.932258	−	0.361795i	$-0.117836\pi$
$769$	51.7046	1.86452	0.932258	+	0.361795i	$0.117836\pi$
$770$	0	0
$771$	9.32571	0.335857
$772$	0	0
$773$	17.9350	0.645078	0.322539	−	0.946556i	$-0.395464\pi$
$773$	17.9350	0.645078	0.322539	+	0.946556i	$0.395464\pi$
$774$	0	0
$775$	−27.0354	−0.971142
$776$	0	0
$777$	−19.8601	−0.712476
$778$	0	0
$779$	27.8762	0.998770
$780$	0	0
$781$	15.2735	0.546527
$782$	0	0
$783$	3.38537	0.120983
$784$	0	0
$785$	71.3216	2.54558
$786$	0	0
$787$	−10.1192	−0.360711	−0.180356	−	0.983601i	$-0.557725\pi$
$787$	−10.1192	−0.360711	−0.180356	+	0.983601i	$0.557725\pi$
$788$	0	0
$789$	−1.03494	−0.0368448
$790$	0	0
$791$	−18.4304	−0.655309
$792$	0	0
$793$	−53.4842	−1.89928
$794$	0	0
$795$	19.5831	0.694542
$796$	0	0
$797$	2.51921	0.0892351	0.0446175	−	0.999004i	$-0.485793\pi$
$797$	2.51921	0.0892351	0.0446175	+	0.999004i	$0.485793\pi$
$798$	0	0
$799$	−7.20915	−0.255041
$800$	0	0
$801$	−1.50452	−0.0531594
$802$	0	0
$803$	−10.1106	−0.356794
$804$	0	0
$805$	−57.3767	−2.02226
$806$	0	0
$807$	−13.2297	−0.465708
$808$	0	0
$809$	3.10619	0.109208	0.0546039	−	0.998508i	$-0.482610\pi$
$809$	3.10619	0.109208	0.0546039	+	0.998508i	$0.482610\pi$
$810$	0	0
$811$	21.4811	0.754303	0.377152	−	0.926152i	$-0.376904\pi$
$811$	21.4811	0.754303	0.377152	+	0.926152i	$0.376904\pi$
$812$	0	0
$813$	−15.6279	−0.548096
$814$	0	0
$815$	67.7128	2.37188
$816$	0	0
$817$	−47.2355	−1.65256
$818$	0	0
$819$	25.4298	0.888588
$820$	0	0
$821$	15.2130	0.530936	0.265468	−	0.964120i	$-0.414473\pi$
$821$	15.2130	0.530936	0.265468	+	0.964120i	$0.414473\pi$
$822$	0	0
$823$	−11.0861	−0.386439	−0.193219	−	0.981156i	$-0.561893\pi$
$823$	−11.0861	−0.386439	−0.193219	+	0.981156i	$0.561893\pi$
$824$	0	0
$825$	−10.7519	−0.374334
$826$	0	0
$827$	−1.19279	−0.0414773	−0.0207387	−	0.999785i	$-0.506602\pi$
$827$	−1.19279	−0.0414773	−0.0207387	+	0.999785i	$0.506602\pi$
$828$	0	0
$829$	35.5477	1.23462	0.617311	−	0.786719i	$-0.288222\pi$
$829$	35.5477	1.23462	0.617311	+	0.786719i	$0.288222\pi$
$830$	0	0
$831$	−0.591123	−0.0205058
$832$	0	0
$833$	−7.60626	−0.263541
$834$	0	0
$835$	−3.83671	−0.132775
$836$	0	0
$837$	−2.78132	−0.0961366
$838$	0	0
$839$	6.12432	0.211435	0.105718	−	0.994396i	$-0.466286\pi$
$839$	6.12432	0.211435	0.105718	+	0.994396i	$0.466286\pi$
$840$	0	0
$841$	−17.5393	−0.604802
$842$	0	0
$843$	−4.64950	−0.160137
$844$	0	0
$845$	73.6437	2.53342
$846$	0	0
$847$	43.8162	1.50554
$848$	0	0
$849$	23.7408	0.814781
$850$	0	0
$851$	14.7861	0.506861
$852$	0	0
$853$	−16.7670	−0.574091	−0.287046	−	0.957917i	$-0.592673\pi$
$853$	−16.7670	−0.574091	−0.287046	+	0.957917i	$0.592673\pi$
$854$	0	0
$855$	18.1174	0.619604
$856$	0	0
$857$	−11.0064	−0.375972	−0.187986	−	0.982172i	$-0.560196\pi$
$857$	−11.0064	−0.375972	−0.187986	+	0.982172i	$0.560196\pi$
$858$	0	0
$859$	−30.9158	−1.05483	−0.527416	−	0.849607i	$-0.676839\pi$
$859$	−30.9158	−1.05483	−0.527416	+	0.849607i	$0.676839\pi$
$860$	0	0
$861$	−26.4574	−0.901667
$862$	0	0
$863$	18.7791	0.639247	0.319624	−	0.947545i	$-0.396444\pi$
$863$	18.7791	0.639247	0.319624	+	0.947545i	$0.396444\pi$
$864$	0	0
$865$	18.4148	0.626121
$866$	0	0
$867$	−16.6622	−0.565877
$868$	0	0
$869$	−19.2703	−0.653700
$870$	0	0
$871$	51.4838	1.74446
$872$	0	0
$873$	0.761888	0.0257860
$874$	0	0
$875$	−81.1681	−2.74398
$876$	0	0
$877$	−8.89364	−0.300317	−0.150158	−	0.988662i	$-0.547978\pi$
$877$	−8.89364	−0.300317	−0.150158	+	0.988662i	$0.547978\pi$
$878$	0	0
$879$	−27.7554	−0.936166
$880$	0	0
$881$	14.0530	0.473457	0.236728	−	0.971576i	$-0.423925\pi$
$881$	14.0530	0.473457	0.236728	+	0.971576i	$0.423925\pi$
$882$	0	0
$883$	−31.0318	−1.04430	−0.522151	−	0.852853i	$-0.674870\pi$
$883$	−31.0318	−1.04430	−0.522151	+	0.852853i	$0.674870\pi$
$884$	0	0
$885$	10.6623	0.358409
$886$	0	0
$887$	17.0309	0.571842	0.285921	−	0.958253i	$-0.407700\pi$
$887$	17.0309	0.571842	0.285921	+	0.958253i	$0.407700\pi$
$888$	0	0
$889$	−89.2406	−2.99304
$890$	0	0
$891$	−1.10612	−0.0370566
$892$	0	0
$893$	58.5696	1.95996
$894$	0	0
$895$	79.1854	2.64687
$896$	0	0
$897$	−18.9328	−0.632149
$898$	0	0
$899$	−9.41581	−0.314035
$900$	0	0
$901$	−2.96669	−0.0988348
$902$	0	0
$903$	44.8314	1.49189
$904$	0	0
$905$	41.8683	1.39175
$906$	0	0
$907$	−56.3207	−1.87010	−0.935049	−	0.354520i	$-0.884644\pi$
$907$	−56.3207	−1.87010	−0.935049	+	0.354520i	$0.884644\pi$
$908$	0	0
$909$	−1.37489	−0.0456023
$910$	0	0
$911$	57.9696	1.92062	0.960309	−	0.278937i	$-0.0899821\pi$
$911$	57.9696	1.92062	0.960309	+	0.278937i	$0.0899821\pi$
$912$	0	0
$913$	18.7920	0.621926
$914$	0	0
$915$	36.1654	1.19559
$916$	0	0
$917$	−28.7990	−0.951026
$918$	0	0
$919$	7.93192	0.261650	0.130825	−	0.991405i	$-0.458237\pi$
$919$	7.93192	0.261650	0.130825	+	0.991405i	$0.458237\pi$
$920$	0	0
$921$	−1.47952	−0.0487517
$922$	0	0
$923$	78.3473	2.57883
$924$	0	0
$925$	43.0736	1.41625
$926$	0	0
$927$	0.0629727	0.00206829
$928$	0	0
$929$	11.1314	0.365209	0.182604	−	0.983186i	$-0.441547\pi$
$929$	11.1314	0.365209	0.182604	+	0.983186i	$0.441547\pi$
$930$	0	0
$931$	61.7959	2.02528
$932$	0	0
$933$	−19.6599	−0.643635
$934$	0	0
$935$	2.46668	0.0806690
$936$	0	0
$937$	4.29223	0.140221	0.0701105	−	0.997539i	$-0.477665\pi$
$937$	4.29223	0.140221	0.0701105	+	0.997539i	$0.477665\pi$
$938$	0	0
$939$	−13.0409	−0.425574
$940$	0	0
$941$	−39.4909	−1.28737	−0.643683	−	0.765292i	$-0.722595\pi$
$941$	−39.4909	−1.28737	−0.643683	+	0.765292i	$0.722595\pi$
$942$	0	0
$943$	19.6979	0.641453
$944$	0	0
$945$	−17.1953	−0.559364
$946$	0	0
$947$	−39.0910	−1.27029	−0.635144	−	0.772394i	$-0.719059\pi$
$947$	−39.0910	−1.27029	−0.635144	+	0.772394i	$0.719059\pi$
$948$	0	0
$949$	−51.8635	−1.68356
$950$	0	0
$951$	13.2143	0.428504
$952$	0	0
$953$	33.7855	1.09442	0.547210	−	0.836995i	$-0.315690\pi$
$953$	33.7855	1.09442	0.547210	+	0.836995i	$0.315690\pi$
$954$	0	0
$955$	40.1888	1.30048
$956$	0	0
$957$	−3.74464	−0.121047
$958$	0	0
$959$	6.36447	0.205519
$960$	0	0
$961$	−23.2642	−0.750459
$962$	0	0
$963$	−10.0437	−0.323653
$964$	0	0
$965$	−31.8582	−1.02555
$966$	0	0
$967$	5.47656	0.176114	0.0880572	−	0.996115i	$-0.471934\pi$
$967$	5.47656	0.176114	0.0880572	+	0.996115i	$0.471934\pi$
$968$	0	0
$969$	−2.74465	−0.0881709
$970$	0	0
$971$	−23.8330	−0.764836	−0.382418	−	0.923989i	$-0.624909\pi$
$971$	−23.8330	−0.764836	−0.382418	+	0.923989i	$0.624909\pi$
$972$	0	0
$973$	42.2928	1.35584
$974$	0	0
$975$	−55.1535	−1.76632
$976$	0	0
$977$	−31.9054	−1.02075	−0.510373	−	0.859953i	$-0.670493\pi$
$977$	−31.9054	−1.02075	−0.510373	+	0.859953i	$0.670493\pi$
$978$	0	0
$979$	1.66418	0.0531875
$980$	0	0
$981$	1.87839	0.0599723
$982$	0	0
$983$	−23.5991	−0.752696	−0.376348	−	0.926478i	$-0.622820\pi$
$983$	−23.5991	−0.752696	−0.376348	+	0.926478i	$0.622820\pi$
$984$	0	0
$985$	94.9393	3.02502
$986$	0	0
$987$	−55.5886	−1.76941
$988$	0	0
$989$	−33.3776	−1.06135
$990$	0	0
$991$	35.7794	1.13657	0.568285	−	0.822832i	$-0.307607\pi$
$991$	35.7794	1.13657	0.568285	+	0.822832i	$0.307607\pi$
$992$	0	0
$993$	27.4467	0.870993
$994$	0	0
$995$	80.4278	2.54973
$996$	0	0
$997$	−35.1383	−1.11284	−0.556420	−	0.830901i	$-0.687825\pi$
$997$	−35.1383	−1.11284	−0.556420	+	0.830901i	$0.687825\pi$
$998$	0	0
$999$	4.43128	0.140199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.be.1.10		11
4.3	odd	2		4008.2.a.k.1.10	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.10	✓	11	4.3	odd	2
8016.2.a.be.1.10		11	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.83671 q^{5} -4.48179 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.83671 q^{5} -4.48179 q^{7} +1.00000 q^{9} -1.10612 q^{11} -5.67402 q^{13} +3.83671 q^{15} -0.581232 q^{17} +4.72213 q^{19} -4.48179 q^{21} +3.33676 q^{23} +9.72035 q^{25} +1.00000 q^{27} +3.38537 q^{29} -2.78132 q^{31} -1.10612 q^{33} -17.1953 q^{35} +4.43128 q^{37} -5.67402 q^{39} +5.90332 q^{41} -10.0030 q^{43} +3.83671 q^{45} +12.4032 q^{47} +13.0864 q^{49} -0.581232 q^{51} +5.10414 q^{53} -4.24388 q^{55} +4.72213 q^{57} +2.77902 q^{59} +9.42616 q^{61} -4.48179 q^{63} -21.7696 q^{65} -9.07361 q^{67} +3.33676 q^{69} -13.8081 q^{71} +9.14053 q^{73} +9.72035 q^{75} +4.95742 q^{77} +17.4215 q^{79} +1.00000 q^{81} -16.9891 q^{83} -2.23002 q^{85} +3.38537 q^{87} -1.50452 q^{89} +25.4298 q^{91} -2.78132 q^{93} +18.1174 q^{95} +0.761888 q^{97} -1.10612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + q^99

Introduction

L-functions

Modular forms

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Database

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