LMFDB - Embedded newform 8020.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26002 q^{3} -1.00000 q^{5} -3.39221 q^{7} +2.10771 q^{9} +O(q^{10})$	$q-2.26002 q^{3} -1.00000 q^{5} -3.39221 q^{7} +2.10771 q^{9} -3.74534 q^{11} -4.09762 q^{13} +2.26002 q^{15} +1.81997 q^{17} -0.507197 q^{19} +7.66648 q^{21} +2.07431 q^{23} +1.00000 q^{25} +2.01660 q^{27} -2.08288 q^{29} -3.75361 q^{31} +8.46457 q^{33} +3.39221 q^{35} +5.62876 q^{37} +9.26071 q^{39} -1.53690 q^{41} +6.39486 q^{43} -2.10771 q^{45} +11.8482 q^{47} +4.50709 q^{49} -4.11318 q^{51} -11.1822 q^{53} +3.74534 q^{55} +1.14628 q^{57} +13.8248 q^{59} -3.22244 q^{61} -7.14979 q^{63} +4.09762 q^{65} +2.15862 q^{67} -4.68799 q^{69} +9.69871 q^{71} -5.04323 q^{73} -2.26002 q^{75} +12.7050 q^{77} +0.358510 q^{79} -10.8807 q^{81} +15.6083 q^{83} -1.81997 q^{85} +4.70736 q^{87} -15.3504 q^{89} +13.9000 q^{91} +8.48325 q^{93} +0.507197 q^{95} -6.27268 q^{97} -7.89409 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	−2.26002	−1.30483	−0.652413	−	0.757864i	$-0.726243\pi$
$3$	−2.26002	−1.30483	−0.652413	+	0.757864i	$0.726243\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.39221	−1.28214	−0.641068	−	0.767484i	$-0.721508\pi$
$7$	−3.39221	−1.28214	−0.641068	+	0.767484i	$0.721508\pi$
$8$	0	0
$9$	2.10771	0.702570
$10$	0	0
$11$	−3.74534	−1.12926	−0.564632	−	0.825343i	$-0.690982\pi$
$11$	−3.74534	−1.12926	−0.564632	+	0.825343i	$0.690982\pi$
$12$	0	0
$13$	−4.09762	−1.13647	−0.568237	−	0.822865i	$-0.692374\pi$
$13$	−4.09762	−1.13647	−0.568237	+	0.822865i	$0.692374\pi$
$14$	0	0
$15$	2.26002	0.583536
$16$	0	0
$17$	1.81997	0.441408	0.220704	−	0.975341i	$-0.429165\pi$
$17$	1.81997	0.441408	0.220704	+	0.975341i	$0.429165\pi$
$18$	0	0
$19$	−0.507197	−0.116359	−0.0581795	−	0.998306i	$-0.518530\pi$
$19$	−0.507197	−0.116359	−0.0581795	+	0.998306i	$0.518530\pi$
$20$	0	0
$21$	7.66648	1.67296
$22$	0	0
$23$	2.07431	0.432524	0.216262	−	0.976335i	$-0.430614\pi$
$23$	2.07431	0.432524	0.216262	+	0.976335i	$0.430614\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.01660	0.388095
$28$	0	0
$29$	−2.08288	−0.386781	−0.193391	−	0.981122i	$-0.561948\pi$
$29$	−2.08288	−0.386781	−0.193391	+	0.981122i	$0.561948\pi$
$30$	0	0
$31$	−3.75361	−0.674168	−0.337084	−	0.941474i	$-0.609441\pi$
$31$	−3.75361	−0.674168	−0.337084	+	0.941474i	$0.609441\pi$
$32$	0	0
$33$	8.46457	1.47349
$34$	0	0
$35$	3.39221	0.573388
$36$	0	0
$37$	5.62876	0.925363	0.462681	−	0.886525i	$-0.346887\pi$
$37$	5.62876	0.925363	0.462681	+	0.886525i	$0.346887\pi$
$38$	0	0
$39$	9.26071	1.48290
$40$	0	0
$41$	−1.53690	−0.240024	−0.120012	−	0.992772i	$-0.538293\pi$
$41$	−1.53690	−0.240024	−0.120012	+	0.992772i	$0.538293\pi$
$42$	0	0
$43$	6.39486	0.975207	0.487603	−	0.873065i	$-0.337871\pi$
$43$	6.39486	0.975207	0.487603	+	0.873065i	$0.337871\pi$
$44$	0	0
$45$	−2.10771	−0.314199
$46$	0	0
$47$	11.8482	1.72824	0.864118	−	0.503289i	$-0.167877\pi$
$47$	11.8482	1.72824	0.864118	+	0.503289i	$0.167877\pi$
$48$	0	0
$49$	4.50709	0.643870
$50$	0	0
$51$	−4.11318	−0.575960
$52$	0	0
$53$	−11.1822	−1.53599	−0.767995	−	0.640456i	$-0.778746\pi$
$53$	−11.1822	−1.53599	−0.767995	+	0.640456i	$0.778746\pi$
$54$	0	0
$55$	3.74534	0.505022
$56$	0	0
$57$	1.14628	0.151828
$58$	0	0
$59$	13.8248	1.79983	0.899917	−	0.436061i	$-0.143627\pi$
$59$	13.8248	1.79983	0.899917	+	0.436061i	$0.143627\pi$
$60$	0	0
$61$	−3.22244	−0.412592	−0.206296	−	0.978490i	$-0.566141\pi$
$61$	−3.22244	−0.412592	−0.206296	+	0.978490i	$0.566141\pi$
$62$	0	0
$63$	−7.14979	−0.900789
$64$	0	0
$65$	4.09762	0.508247
$66$	0	0
$67$	2.15862	0.263717	0.131859	−	0.991269i	$-0.457905\pi$
$67$	2.15862	0.263717	0.131859	+	0.991269i	$0.457905\pi$
$68$	0	0
$69$	−4.68799	−0.564368
$70$	0	0
$71$	9.69871	1.15103	0.575513	−	0.817793i	$-0.304803\pi$
$71$	9.69871	1.15103	0.575513	+	0.817793i	$0.304803\pi$
$72$	0	0
$73$	−5.04323	−0.590265	−0.295132	−	0.955456i	$-0.595364\pi$
$73$	−5.04323	−0.590265	−0.295132	+	0.955456i	$0.595364\pi$
$74$	0	0
$75$	−2.26002	−0.260965
$76$	0	0
$77$	12.7050	1.44787
$78$	0	0
$79$	0.358510	0.0403356	0.0201678	−	0.999797i	$-0.493580\pi$
$79$	0.358510	0.0403356	0.0201678	+	0.999797i	$0.493580\pi$
$80$	0	0
$81$	−10.8807	−1.20897
$82$	0	0
$83$	15.6083	1.71323	0.856617	−	0.515952i	$-0.172562\pi$
$83$	15.6083	1.71323	0.856617	+	0.515952i	$0.172562\pi$
$84$	0	0
$85$	−1.81997	−0.197403
$86$	0	0
$87$	4.70736	0.504682
$88$	0	0
$89$	−15.3504	−1.62714	−0.813568	−	0.581470i	$-0.802478\pi$
$89$	−15.3504	−1.62714	−0.813568	+	0.581470i	$0.802478\pi$
$90$	0	0
$91$	13.9000	1.45711
$92$	0	0
$93$	8.48325	0.879672
$94$	0	0
$95$	0.507197	0.0520373
$96$	0	0
$97$	−6.27268	−0.636894	−0.318447	−	0.947941i	$-0.603161\pi$
$97$	−6.27268	−0.636894	−0.318447	+	0.947941i	$0.603161\pi$
$98$	0	0
$99$	−7.89409	−0.793386
$100$	0	0
$101$	12.9627	1.28984	0.644919	−	0.764251i	$-0.276891\pi$
$101$	12.9627	1.28984	0.644919	+	0.764251i	$0.276891\pi$
$102$	0	0
$103$	−2.95389	−0.291055	−0.145528	−	0.989354i	$-0.546488\pi$
$103$	−2.95389	−0.291055	−0.145528	+	0.989354i	$0.546488\pi$
$104$	0	0
$105$	−7.66648	−0.748172
$106$	0	0
$107$	4.40788	0.426126	0.213063	−	0.977039i	$-0.431656\pi$
$107$	4.40788	0.426126	0.213063	+	0.977039i	$0.431656\pi$
$108$	0	0
$109$	0.488243	0.0467652	0.0233826	−	0.999727i	$-0.492556\pi$
$109$	0.488243	0.0467652	0.0233826	+	0.999727i	$0.492556\pi$
$110$	0	0
$111$	−12.7211	−1.20744
$112$	0	0
$113$	−9.31385	−0.876173	−0.438087	−	0.898933i	$-0.644344\pi$
$113$	−9.31385	−0.876173	−0.438087	+	0.898933i	$0.644344\pi$
$114$	0	0
$115$	−2.07431	−0.193430
$116$	0	0
$117$	−8.63658	−0.798453
$118$	0	0
$119$	−6.17372	−0.565944
$120$	0	0
$121$	3.02760	0.275236
$122$	0	0
$123$	3.47343	0.313189
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.0353	0.979227	0.489614	−	0.871940i	$-0.337138\pi$
$127$	11.0353	0.979227	0.489614	+	0.871940i	$0.337138\pi$
$128$	0	0
$129$	−14.4525	−1.27247
$130$	0	0
$131$	17.7826	1.55368	0.776838	−	0.629701i	$-0.216822\pi$
$131$	17.7826	1.55368	0.776838	+	0.629701i	$0.216822\pi$
$132$	0	0
$133$	1.72052	0.149188
$134$	0	0
$135$	−2.01660	−0.173561
$136$	0	0
$137$	0.381459	0.0325902	0.0162951	−	0.999867i	$-0.494813\pi$
$137$	0.381459	0.0325902	0.0162951	+	0.999867i	$0.494813\pi$
$138$	0	0
$139$	−18.2364	−1.54679	−0.773396	−	0.633923i	$-0.781444\pi$
$139$	−18.2364	−1.54679	−0.773396	+	0.633923i	$0.781444\pi$
$140$	0	0
$141$	−26.7772	−2.25505
$142$	0	0
$143$	15.3470	1.28338
$144$	0	0
$145$	2.08288	0.172974
$146$	0	0
$147$	−10.1861	−0.840139
$148$	0	0
$149$	−10.2845	−0.842542	−0.421271	−	0.906935i	$-0.638416\pi$
$149$	−10.2845	−0.842542	−0.421271	+	0.906935i	$0.638416\pi$
$150$	0	0
$151$	−12.9619	−1.05482	−0.527412	−	0.849610i	$-0.676837\pi$
$151$	−12.9619	−1.05482	−0.527412	+	0.849610i	$0.676837\pi$
$152$	0	0
$153$	3.83597	0.310120
$154$	0	0
$155$	3.75361	0.301497
$156$	0	0
$157$	22.3779	1.78595	0.892977	−	0.450102i	$-0.148612\pi$
$157$	22.3779	1.78595	0.892977	+	0.450102i	$0.148612\pi$
$158$	0	0
$159$	25.2720	2.00420
$160$	0	0
$161$	−7.03650	−0.554554
$162$	0	0
$163$	5.83282	0.456862	0.228431	−	0.973560i	$-0.426641\pi$
$163$	5.83282	0.456862	0.228431	+	0.973560i	$0.426641\pi$
$164$	0	0
$165$	−8.46457	−0.658966
$166$	0	0
$167$	−16.3904	−1.26833	−0.634165	−	0.773198i	$-0.718656\pi$
$167$	−16.3904	−1.26833	−0.634165	+	0.773198i	$0.718656\pi$
$168$	0	0
$169$	3.79046	0.291574
$170$	0	0
$171$	−1.06902	−0.0817503
$172$	0	0
$173$	−14.8047	−1.12558	−0.562792	−	0.826599i	$-0.690273\pi$
$173$	−14.8047	−1.12558	−0.562792	+	0.826599i	$0.690273\pi$
$174$	0	0
$175$	−3.39221	−0.256427
$176$	0	0
$177$	−31.2444	−2.34847
$178$	0	0
$179$	14.7287	1.10087	0.550436	−	0.834878i	$-0.314462\pi$
$179$	14.7287	1.10087	0.550436	+	0.834878i	$0.314462\pi$
$180$	0	0
$181$	2.65539	0.197373	0.0986867	−	0.995119i	$-0.468536\pi$
$181$	2.65539	0.197373	0.0986867	+	0.995119i	$0.468536\pi$
$182$	0	0
$183$	7.28280	0.538360
$184$	0	0
$185$	−5.62876	−0.413835
$186$	0	0
$187$	−6.81641	−0.498466
$188$	0	0
$189$	−6.84073	−0.497590
$190$	0	0
$191$	−21.3629	−1.54576	−0.772881	−	0.634551i	$-0.781185\pi$
$191$	−21.3629	−1.54576	−0.772881	+	0.634551i	$0.781185\pi$
$192$	0	0
$193$	21.7573	1.56613	0.783064	−	0.621941i	$-0.213656\pi$
$193$	21.7573	1.56613	0.783064	+	0.621941i	$0.213656\pi$
$194$	0	0
$195$	−9.26071	−0.663173
$196$	0	0
$197$	−1.53627	−0.109454	−0.0547272	−	0.998501i	$-0.517429\pi$
$197$	−1.53627	−0.109454	−0.0547272	+	0.998501i	$0.517429\pi$
$198$	0	0
$199$	8.79522	0.623477	0.311738	−	0.950168i	$-0.399089\pi$
$199$	8.79522	0.623477	0.311738	+	0.950168i	$0.399089\pi$
$200$	0	0
$201$	−4.87853	−0.344105
$202$	0	0
$203$	7.06557	0.495906
$204$	0	0
$205$	1.53690	0.107342
$206$	0	0
$207$	4.37204	0.303878
$208$	0	0
$209$	1.89963	0.131400
$210$	0	0
$211$	19.1073	1.31540	0.657700	−	0.753280i	$-0.271530\pi$
$211$	19.1073	1.31540	0.657700	+	0.753280i	$0.271530\pi$
$212$	0	0
$213$	−21.9193	−1.50189
$214$	0	0
$215$	−6.39486	−0.436126
$216$	0	0
$217$	12.7330	0.864375
$218$	0	0
$219$	11.3978	0.770193
$220$	0	0
$221$	−7.45754	−0.501648
$222$	0	0
$223$	13.2852	0.889642	0.444821	−	0.895620i	$-0.353267\pi$
$223$	13.2852	0.889642	0.444821	+	0.895620i	$0.353267\pi$
$224$	0	0
$225$	2.10771	0.140514
$226$	0	0
$227$	−7.04735	−0.467749	−0.233874	−	0.972267i	$-0.575140\pi$
$227$	−7.04735	−0.467749	−0.233874	+	0.972267i	$0.575140\pi$
$228$	0	0
$229$	−3.56354	−0.235486	−0.117743	−	0.993044i	$-0.537566\pi$
$229$	−3.56354	−0.235486	−0.117743	+	0.993044i	$0.537566\pi$
$230$	0	0
$231$	−28.7136	−1.88922
$232$	0	0
$233$	29.6161	1.94022	0.970109	−	0.242671i	$-0.0780234\pi$
$233$	29.6161	1.94022	0.970109	+	0.242671i	$0.0780234\pi$
$234$	0	0
$235$	−11.8482	−0.772891
$236$	0	0
$237$	−0.810242	−0.0526309
$238$	0	0
$239$	−10.3630	−0.670328	−0.335164	−	0.942160i	$-0.608792\pi$
$239$	−10.3630	−0.670328	−0.335164	+	0.942160i	$0.608792\pi$
$240$	0	0
$241$	6.38911	0.411559	0.205780	−	0.978598i	$-0.434027\pi$
$241$	6.38911	0.411559	0.205780	+	0.978598i	$0.434027\pi$
$242$	0	0
$243$	18.5408	1.18939
$244$	0	0
$245$	−4.50709	−0.287948
$246$	0	0
$247$	2.07830	0.132239
$248$	0	0
$249$	−35.2752	−2.23547
$250$	0	0
$251$	−18.5430	−1.17042	−0.585212	−	0.810880i	$-0.698989\pi$
$251$	−18.5430	−1.17042	−0.585212	+	0.810880i	$0.698989\pi$
$252$	0	0
$253$	−7.76900	−0.488433
$254$	0	0
$255$	4.11318	0.257577
$256$	0	0
$257$	−20.4171	−1.27359	−0.636793	−	0.771035i	$-0.719739\pi$
$257$	−20.4171	−1.27359	−0.636793	+	0.771035i	$0.719739\pi$
$258$	0	0
$259$	−19.0939	−1.18644
$260$	0	0
$261$	−4.39011	−0.271741
$262$	0	0
$263$	−20.4290	−1.25971	−0.629853	−	0.776714i	$-0.716885\pi$
$263$	−20.4290	−1.25971	−0.629853	+	0.776714i	$0.716885\pi$
$264$	0	0
$265$	11.1822	0.686916
$266$	0	0
$267$	34.6922	2.12313
$268$	0	0
$269$	20.8410	1.27070	0.635348	−	0.772226i	$-0.280856\pi$
$269$	20.8410	1.27070	0.635348	+	0.772226i	$0.280856\pi$
$270$	0	0
$271$	−31.8020	−1.93184	−0.965918	−	0.258847i	$-0.916657\pi$
$271$	−31.8020	−1.93184	−0.965918	+	0.258847i	$0.916657\pi$
$272$	0	0
$273$	−31.4143	−1.90128
$274$	0	0
$275$	−3.74534	−0.225853
$276$	0	0
$277$	12.2380	0.735312	0.367656	−	0.929962i	$-0.380160\pi$
$277$	12.2380	0.735312	0.367656	+	0.929962i	$0.380160\pi$
$278$	0	0
$279$	−7.91152	−0.473650
$280$	0	0
$281$	−9.90590	−0.590936	−0.295468	−	0.955353i	$-0.595476\pi$
$281$	−9.90590	−0.590936	−0.295468	+	0.955353i	$0.595476\pi$
$282$	0	0
$283$	9.95462	0.591741	0.295870	−	0.955228i	$-0.404390\pi$
$283$	9.95462	0.591741	0.295870	+	0.955228i	$0.404390\pi$
$284$	0	0
$285$	−1.14628	−0.0678996
$286$	0	0
$287$	5.21349	0.307743
$288$	0	0
$289$	−13.6877	−0.805159
$290$	0	0
$291$	14.1764	0.831036
$292$	0	0
$293$	16.7691	0.979663	0.489831	−	0.871817i	$-0.337058\pi$
$293$	16.7691	0.979663	0.489831	+	0.871817i	$0.337058\pi$
$294$	0	0
$295$	−13.8248	−0.804910
$296$	0	0
$297$	−7.55286	−0.438261
$298$	0	0
$299$	−8.49973	−0.491552
$300$	0	0
$301$	−21.6927	−1.25035
$302$	0	0
$303$	−29.2960	−1.68301
$304$	0	0
$305$	3.22244	0.184517
$306$	0	0
$307$	−10.7919	−0.615928	−0.307964	−	0.951398i	$-0.599648\pi$
$307$	−10.7919	−0.615928	−0.307964	+	0.951398i	$0.599648\pi$
$308$	0	0
$309$	6.67585	0.379776
$310$	0	0
$311$	20.9120	1.18581	0.592906	−	0.805272i	$-0.297981\pi$
$311$	20.9120	1.18581	0.592906	+	0.805272i	$0.297981\pi$
$312$	0	0
$313$	−17.6297	−0.996490	−0.498245	−	0.867036i	$-0.666022\pi$
$313$	−17.6297	−0.996490	−0.498245	+	0.867036i	$0.666022\pi$
$314$	0	0
$315$	7.14979	0.402845
$316$	0	0
$317$	30.3357	1.70382	0.851910	−	0.523688i	$-0.175444\pi$
$317$	30.3357	1.70382	0.851910	+	0.523688i	$0.175444\pi$
$318$	0	0
$319$	7.80111	0.436778
$320$	0	0
$321$	−9.96191	−0.556020
$322$	0	0
$323$	−0.923083	−0.0513617
$324$	0	0
$325$	−4.09762	−0.227295
$326$	0	0
$327$	−1.10344	−0.0610204
$328$	0	0
$329$	−40.1916	−2.21583
$330$	0	0
$331$	−27.5661	−1.51517	−0.757586	−	0.652736i	$-0.773621\pi$
$331$	−27.5661	−1.51517	−0.757586	+	0.652736i	$0.773621\pi$
$332$	0	0
$333$	11.8638	0.650132
$334$	0	0
$335$	−2.15862	−0.117938
$336$	0	0
$337$	22.0074	1.19882	0.599408	−	0.800443i	$-0.295403\pi$
$337$	22.0074	1.19882	0.599408	+	0.800443i	$0.295403\pi$
$338$	0	0
$339$	21.0495	1.14325
$340$	0	0
$341$	14.0586	0.761314
$342$	0	0
$343$	8.45647	0.456606
$344$	0	0
$345$	4.68799	0.252393
$346$	0	0
$347$	27.4502	1.47361	0.736803	−	0.676107i	$-0.236334\pi$
$347$	27.4502	1.47361	0.736803	+	0.676107i	$0.236334\pi$
$348$	0	0
$349$	−18.8851	−1.01090	−0.505449	−	0.862857i	$-0.668673\pi$
$349$	−18.8851	−1.01090	−0.505449	+	0.862857i	$0.668673\pi$
$350$	0	0
$351$	−8.26325	−0.441060
$352$	0	0
$353$	20.3719	1.08429	0.542143	−	0.840286i	$-0.317613\pi$
$353$	20.3719	1.08429	0.542143	+	0.840286i	$0.317613\pi$
$354$	0	0
$355$	−9.69871	−0.514754
$356$	0	0
$357$	13.9528	0.738458
$358$	0	0
$359$	−12.4579	−0.657504	−0.328752	−	0.944416i	$-0.606628\pi$
$359$	−12.4579	−0.657504	−0.328752	+	0.944416i	$0.606628\pi$
$360$	0	0
$361$	−18.7428	−0.986461
$362$	0	0
$363$	−6.84244	−0.359135
$364$	0	0
$365$	5.04323	0.263974
$366$	0	0
$367$	−26.0304	−1.35877	−0.679387	−	0.733780i	$-0.737754\pi$
$367$	−26.0304	−1.35877	−0.679387	+	0.733780i	$0.737754\pi$
$368$	0	0
$369$	−3.23934	−0.168633
$370$	0	0
$371$	37.9323	1.96935
$372$	0	0
$373$	−7.60542	−0.393794	−0.196897	−	0.980424i	$-0.563086\pi$
$373$	−7.60542	−0.393794	−0.196897	+	0.980424i	$0.563086\pi$
$374$	0	0
$375$	2.26002	0.116707
$376$	0	0
$377$	8.53485	0.439567
$378$	0	0
$379$	1.62179	0.0833058	0.0416529	−	0.999132i	$-0.486738\pi$
$379$	1.62179	0.0833058	0.0416529	+	0.999132i	$0.486738\pi$
$380$	0	0
$381$	−24.9401	−1.27772
$382$	0	0
$383$	7.38142	0.377173	0.188586	−	0.982057i	$-0.439609\pi$
$383$	7.38142	0.377173	0.188586	+	0.982057i	$0.439609\pi$
$384$	0	0
$385$	−12.7050	−0.647506
$386$	0	0
$387$	13.4785	0.685151
$388$	0	0
$389$	−8.03121	−0.407199	−0.203599	−	0.979054i	$-0.565264\pi$
$389$	−8.03121	−0.407199	−0.203599	+	0.979054i	$0.565264\pi$
$390$	0	0
$391$	3.77518	0.190919
$392$	0	0
$393$	−40.1892	−2.02728
$394$	0	0
$395$	−0.358510	−0.0180386
$396$	0	0
$397$	29.5537	1.48326	0.741630	−	0.670809i	$-0.234053\pi$
$397$	29.5537	1.48326	0.741630	+	0.670809i	$0.234053\pi$
$398$	0	0
$399$	−3.88841	−0.194664
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	15.3809	0.766175
$404$	0	0
$405$	10.8807	0.540666
$406$	0	0
$407$	−21.0816	−1.04498
$408$	0	0
$409$	9.87886	0.488478	0.244239	−	0.969715i	$-0.421462\pi$
$409$	9.87886	0.488478	0.244239	+	0.969715i	$0.421462\pi$
$410$	0	0
$411$	−0.862106	−0.0425246
$412$	0	0
$413$	−46.8966	−2.30763
$414$	0	0
$415$	−15.6083	−0.766182
$416$	0	0
$417$	41.2147	2.01829
$418$	0	0
$419$	9.54485	0.466297	0.233148	−	0.972441i	$-0.425097\pi$
$419$	9.54485	0.466297	0.233148	+	0.972441i	$0.425097\pi$
$420$	0	0
$421$	−37.6977	−1.83727	−0.918636	−	0.395105i	$-0.870708\pi$
$421$	−37.6977	−1.83727	−0.918636	+	0.395105i	$0.870708\pi$
$422$	0	0
$423$	24.9725	1.21421
$424$	0	0
$425$	1.81997	0.0882815
$426$	0	0
$427$	10.9312	0.528998
$428$	0	0
$429$	−34.6845	−1.67459
$430$	0	0
$431$	−6.48138	−0.312197	−0.156098	−	0.987742i	$-0.549892\pi$
$431$	−6.48138	−0.312197	−0.156098	+	0.987742i	$0.549892\pi$
$432$	0	0
$433$	−6.09537	−0.292925	−0.146462	−	0.989216i	$-0.546789\pi$
$433$	−6.09537	−0.292925	−0.146462	+	0.989216i	$0.546789\pi$
$434$	0	0
$435$	−4.70736	−0.225701
$436$	0	0
$437$	−1.05208	−0.0503280
$438$	0	0
$439$	29.4219	1.40423	0.702114	−	0.712064i	$-0.252239\pi$
$439$	29.4219	1.40423	0.702114	+	0.712064i	$0.252239\pi$
$440$	0	0
$441$	9.49964	0.452364
$442$	0	0
$443$	−13.8208	−0.656648	−0.328324	−	0.944565i	$-0.606484\pi$
$443$	−13.8208	−0.656648	−0.328324	+	0.944565i	$0.606484\pi$
$444$	0	0
$445$	15.3504	0.727677
$446$	0	0
$447$	23.2433	1.09937
$448$	0	0
$449$	13.8881	0.655420	0.327710	−	0.944778i	$-0.393723\pi$
$449$	13.8881	0.655420	0.327710	+	0.944778i	$0.393723\pi$
$450$	0	0
$451$	5.75622	0.271050
$452$	0	0
$453$	29.2942	1.37636
$454$	0	0
$455$	−13.9000	−0.651641
$456$	0	0
$457$	22.2624	1.04139	0.520695	−	0.853743i	$-0.325673\pi$
$457$	22.2624	1.04139	0.520695	+	0.853743i	$0.325673\pi$
$458$	0	0
$459$	3.67015	0.171308
$460$	0	0
$461$	18.8057	0.875867	0.437933	−	0.899007i	$-0.355711\pi$
$461$	18.8057	0.875867	0.437933	+	0.899007i	$0.355711\pi$
$462$	0	0
$463$	5.73689	0.266616	0.133308	−	0.991075i	$-0.457440\pi$
$463$	5.73689	0.266616	0.133308	+	0.991075i	$0.457440\pi$
$464$	0	0
$465$	−8.48325	−0.393401
$466$	0	0
$467$	−25.6640	−1.18759	−0.593793	−	0.804618i	$-0.702370\pi$
$467$	−25.6640	−1.18759	−0.593793	+	0.804618i	$0.702370\pi$
$468$	0	0
$469$	−7.32249	−0.338121
$470$	0	0
$471$	−50.5747	−2.33036
$472$	0	0
$473$	−23.9509	−1.10127
$474$	0	0
$475$	−0.507197	−0.0232718
$476$	0	0
$477$	−23.5688	−1.07914
$478$	0	0
$479$	2.81042	0.128411	0.0642056	−	0.997937i	$-0.479549\pi$
$479$	2.81042	0.128411	0.0642056	+	0.997937i	$0.479549\pi$
$480$	0	0
$481$	−23.0645	−1.05165
$482$	0	0
$483$	15.9027	0.723596
$484$	0	0
$485$	6.27268	0.284828
$486$	0	0
$487$	−13.2505	−0.600437	−0.300218	−	0.953870i	$-0.597060\pi$
$487$	−13.2505	−0.600437	−0.300218	+	0.953870i	$0.597060\pi$
$488$	0	0
$489$	−13.1823	−0.596125
$490$	0	0
$491$	6.99717	0.315778	0.157889	−	0.987457i	$-0.449531\pi$
$491$	6.99717	0.315778	0.157889	+	0.987457i	$0.449531\pi$
$492$	0	0
$493$	−3.79078	−0.170728
$494$	0	0
$495$	7.89409	0.354813
$496$	0	0
$497$	−32.9001	−1.47577
$498$	0	0
$499$	34.3136	1.53609	0.768044	−	0.640397i	$-0.221230\pi$
$499$	34.3136	1.53609	0.768044	+	0.640397i	$0.221230\pi$
$500$	0	0
$501$	37.0428	1.65495
$502$	0	0
$503$	−26.0040	−1.15946	−0.579731	−	0.814808i	$-0.696842\pi$
$503$	−26.0040	−1.15946	−0.579731	+	0.814808i	$0.696842\pi$
$504$	0	0
$505$	−12.9627	−0.576833
$506$	0	0
$507$	−8.56654	−0.380453
$508$	0	0
$509$	−12.7628	−0.565701	−0.282850	−	0.959164i	$-0.591280\pi$
$509$	−12.7628	−0.565701	−0.282850	+	0.959164i	$0.591280\pi$
$510$	0	0
$511$	17.1077	0.756799
$512$	0	0
$513$	−1.02281	−0.0451583
$514$	0	0
$515$	2.95389	0.130164
$516$	0	0
$517$	−44.3756	−1.95163
$518$	0	0
$519$	33.4591	1.46869
$520$	0	0
$521$	−40.0435	−1.75434	−0.877168	−	0.480183i	$-0.840570\pi$
$521$	−40.0435	−1.75434	−0.877168	+	0.480183i	$0.840570\pi$
$522$	0	0
$523$	−31.4358	−1.37459	−0.687296	−	0.726377i	$-0.741203\pi$
$523$	−31.4358	−1.37459	−0.687296	+	0.726377i	$0.741203\pi$
$524$	0	0
$525$	7.66648	0.334593
$526$	0	0
$527$	−6.83146	−0.297583
$528$	0	0
$529$	−18.6972	−0.812923
$530$	0	0
$531$	29.1386	1.26451
$532$	0	0
$533$	6.29763	0.272781
$534$	0	0
$535$	−4.40788	−0.190569
$536$	0	0
$537$	−33.2871	−1.43645
$538$	0	0
$539$	−16.8806	−0.727099
$540$	0	0
$541$	−6.01018	−0.258398	−0.129199	−	0.991619i	$-0.541241\pi$
$541$	−6.01018	−0.258398	−0.129199	+	0.991619i	$0.541241\pi$
$542$	0	0
$543$	−6.00124	−0.257538
$544$	0	0
$545$	−0.488243	−0.0209140
$546$	0	0
$547$	−12.0973	−0.517242	−0.258621	−	0.965979i	$-0.583268\pi$
$547$	−12.0973	−0.517242	−0.258621	+	0.965979i	$0.583268\pi$
$548$	0	0
$549$	−6.79197	−0.289874
$550$	0	0
$551$	1.05643	0.0450055
$552$	0	0
$553$	−1.21614	−0.0517156
$554$	0	0
$555$	12.7211	0.539982
$556$	0	0
$557$	−3.38716	−0.143519	−0.0717594	−	0.997422i	$-0.522861\pi$
$557$	−3.38716	−0.143519	−0.0717594	+	0.997422i	$0.522861\pi$
$558$	0	0
$559$	−26.2037	−1.10830
$560$	0	0
$561$	15.4053	0.650411
$562$	0	0
$563$	−16.3650	−0.689703	−0.344852	−	0.938657i	$-0.612071\pi$
$563$	−16.3650	−0.689703	−0.344852	+	0.938657i	$0.612071\pi$
$564$	0	0
$565$	9.31385	0.391837
$566$	0	0
$567$	36.9096	1.55006
$568$	0	0
$569$	−2.67962	−0.112335	−0.0561677	−	0.998421i	$-0.517888\pi$
$569$	−2.67962	−0.112335	−0.0561677	+	0.998421i	$0.517888\pi$
$570$	0	0
$571$	−29.9569	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$571$	−29.9569	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$572$	0	0
$573$	48.2806	2.01695
$574$	0	0
$575$	2.07431	0.0865047
$576$	0	0
$577$	−11.7316	−0.488393	−0.244197	−	0.969726i	$-0.578524\pi$
$577$	−11.7316	−0.488393	−0.244197	+	0.969726i	$0.578524\pi$
$578$	0	0
$579$	−49.1721	−2.04352
$580$	0	0
$581$	−52.9467	−2.19660
$582$	0	0
$583$	41.8811	1.73454
$584$	0	0
$585$	8.63658	0.357079
$586$	0	0
$587$	30.6515	1.26512	0.632561	−	0.774511i	$-0.282004\pi$
$587$	30.6515	1.26512	0.632561	+	0.774511i	$0.282004\pi$
$588$	0	0
$589$	1.90382	0.0784455
$590$	0	0
$591$	3.47200	0.142819
$592$	0	0
$593$	34.6617	1.42338	0.711692	−	0.702491i	$-0.247929\pi$
$593$	34.6617	1.42338	0.711692	+	0.702491i	$0.247929\pi$
$594$	0	0
$595$	6.17372	0.253098
$596$	0	0
$597$	−19.8774	−0.813528
$598$	0	0
$599$	−36.1452	−1.47685	−0.738426	−	0.674334i	$-0.764431\pi$
$599$	−36.1452	−1.47685	−0.738426	+	0.674334i	$0.764431\pi$
$600$	0	0
$601$	12.8712	0.525027	0.262513	−	0.964928i	$-0.415449\pi$
$601$	12.8712	0.525027	0.262513	+	0.964928i	$0.415449\pi$
$602$	0	0
$603$	4.54974	0.185280
$604$	0	0
$605$	−3.02760	−0.123089
$606$	0	0
$607$	−31.3330	−1.27177	−0.635883	−	0.771786i	$-0.719364\pi$
$607$	−31.3330	−1.27177	−0.635883	+	0.771786i	$0.719364\pi$
$608$	0	0
$609$	−15.9684	−0.647071
$610$	0	0
$611$	−48.5494	−1.96410
$612$	0	0
$613$	−12.0796	−0.487889	−0.243944	−	0.969789i	$-0.578441\pi$
$613$	−12.0796	−0.487889	−0.243944	+	0.969789i	$0.578441\pi$
$614$	0	0
$615$	−3.47343	−0.140062
$616$	0	0
$617$	−43.2988	−1.74314	−0.871572	−	0.490267i	$-0.836899\pi$
$617$	−43.2988	−1.74314	−0.871572	+	0.490267i	$0.836899\pi$
$618$	0	0
$619$	16.7755	0.674266	0.337133	−	0.941457i	$-0.390543\pi$
$619$	16.7755	0.674266	0.337133	+	0.941457i	$0.390543\pi$
$620$	0	0
$621$	4.18305	0.167860
$622$	0	0
$623$	52.0717	2.08621
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.29320	−0.171454
$628$	0	0
$629$	10.2442	0.408462
$630$	0	0
$631$	−41.2859	−1.64357	−0.821783	−	0.569801i	$-0.807020\pi$
$631$	−41.2859	−1.64357	−0.821783	+	0.569801i	$0.807020\pi$
$632$	0	0
$633$	−43.1829	−1.71637
$634$	0	0
$635$	−11.0353	−0.437924
$636$	0	0
$637$	−18.4683	−0.731742
$638$	0	0
$639$	20.4421	0.808676
$640$	0	0
$641$	36.9689	1.46018	0.730092	−	0.683349i	$-0.239477\pi$
$641$	36.9689	1.46018	0.730092	+	0.683349i	$0.239477\pi$
$642$	0	0
$643$	−6.65748	−0.262545	−0.131273	−	0.991346i	$-0.541906\pi$
$643$	−6.65748	−0.262545	−0.131273	+	0.991346i	$0.541906\pi$
$644$	0	0
$645$	14.4525	0.569068
$646$	0	0
$647$	16.6223	0.653491	0.326745	−	0.945112i	$-0.394048\pi$
$647$	16.6223	0.653491	0.326745	+	0.945112i	$0.394048\pi$
$648$	0	0
$649$	−51.7786	−2.03249
$650$	0	0
$651$	−28.7770	−1.12786
$652$	0	0
$653$	−38.0685	−1.48974	−0.744869	−	0.667211i	$-0.767488\pi$
$653$	−38.0685	−1.48974	−0.744869	+	0.667211i	$0.767488\pi$
$654$	0	0
$655$	−17.7826	−0.694825
$656$	0	0
$657$	−10.6297	−0.414702
$658$	0	0
$659$	23.6347	0.920678	0.460339	−	0.887743i	$-0.347728\pi$
$659$	23.6347	0.920678	0.460339	+	0.887743i	$0.347728\pi$
$660$	0	0
$661$	27.6608	1.07588	0.537940	−	0.842983i	$-0.319203\pi$
$661$	27.6608	1.07588	0.537940	+	0.842983i	$0.319203\pi$
$662$	0	0
$663$	16.8542	0.654564
$664$	0	0
$665$	−1.72052	−0.0667189
$666$	0	0
$667$	−4.32054	−0.167292
$668$	0	0
$669$	−30.0248	−1.16083
$670$	0	0
$671$	12.0692	0.465925
$672$	0	0
$673$	−3.49481	−0.134715	−0.0673575	−	0.997729i	$-0.521457\pi$
$673$	−3.49481	−0.134715	−0.0673575	+	0.997729i	$0.521457\pi$
$674$	0	0
$675$	2.01660	0.0776189
$676$	0	0
$677$	−37.6498	−1.44700	−0.723500	−	0.690325i	$-0.757468\pi$
$677$	−37.6498	−1.44700	−0.723500	+	0.690325i	$0.757468\pi$
$678$	0	0
$679$	21.2783	0.816585
$680$	0	0
$681$	15.9272	0.610331
$682$	0	0
$683$	−26.4630	−1.01258	−0.506288	−	0.862364i	$-0.668983\pi$
$683$	−26.4630	−1.01258	−0.506288	+	0.862364i	$0.668983\pi$
$684$	0	0
$685$	−0.381459	−0.0145748
$686$	0	0
$687$	8.05369	0.307268
$688$	0	0
$689$	45.8203	1.74561
$690$	0	0
$691$	0.435246	0.0165575	0.00827877	−	0.999966i	$-0.497365\pi$
$691$	0.435246	0.0165575	0.00827877	+	0.999966i	$0.497365\pi$
$692$	0	0
$693$	26.7784	1.01723
$694$	0	0
$695$	18.2364	0.691747
$696$	0	0
$697$	−2.79711	−0.105948
$698$	0	0
$699$	−66.9332	−2.53165
$700$	0	0
$701$	−49.6967	−1.87702	−0.938509	−	0.345254i	$-0.887793\pi$
$701$	−49.6967	−1.87702	−0.938509	+	0.345254i	$0.887793\pi$
$702$	0	0
$703$	−2.85489	−0.107674
$704$	0	0
$705$	26.7772	1.00849
$706$	0	0
$707$	−43.9722	−1.65375
$708$	0	0
$709$	−26.3514	−0.989646	−0.494823	−	0.868994i	$-0.664767\pi$
$709$	−26.3514	−0.989646	−0.494823	+	0.868994i	$0.664767\pi$
$710$	0	0
$711$	0.755636	0.0283386
$712$	0	0
$713$	−7.78615	−0.291594
$714$	0	0
$715$	−15.3470	−0.573945
$716$	0	0
$717$	23.4207	0.874661
$718$	0	0
$719$	44.6579	1.66546	0.832730	−	0.553679i	$-0.186777\pi$
$719$	44.6579	1.66546	0.832730	+	0.553679i	$0.186777\pi$
$720$	0	0
$721$	10.0202	0.373172
$722$	0	0
$723$	−14.4396	−0.537013
$724$	0	0
$725$	−2.08288	−0.0773563
$726$	0	0
$727$	8.39329	0.311290	0.155645	−	0.987813i	$-0.450254\pi$
$727$	8.39329	0.311290	0.155645	+	0.987813i	$0.450254\pi$
$728$	0	0
$729$	−9.26064	−0.342987
$730$	0	0
$731$	11.6385	0.430464
$732$	0	0
$733$	10.7144	0.395746	0.197873	−	0.980228i	$-0.436597\pi$
$733$	10.7144	0.395746	0.197873	+	0.980228i	$0.436597\pi$
$734$	0	0
$735$	10.1861	0.375721
$736$	0	0
$737$	−8.08477	−0.297806
$738$	0	0
$739$	31.6523	1.16435	0.582173	−	0.813065i	$-0.302202\pi$
$739$	31.6523	1.16435	0.582173	+	0.813065i	$0.302202\pi$
$740$	0	0
$741$	−4.69700	−0.172549
$742$	0	0
$743$	37.7808	1.38604	0.693021	−	0.720918i	$-0.256279\pi$
$743$	37.7808	1.38604	0.693021	+	0.720918i	$0.256279\pi$
$744$	0	0
$745$	10.2845	0.376796
$746$	0	0
$747$	32.8978	1.20367
$748$	0	0
$749$	−14.9524	−0.546351
$750$	0	0
$751$	40.0582	1.46174	0.730871	−	0.682515i	$-0.239114\pi$
$751$	40.0582	1.46174	0.730871	+	0.682515i	$0.239114\pi$
$752$	0	0
$753$	41.9076	1.52720
$754$	0	0
$755$	12.9619	0.471731
$756$	0	0
$757$	−20.9954	−0.763092	−0.381546	−	0.924350i	$-0.624608\pi$
$757$	−20.9954	−0.763092	−0.381546	+	0.924350i	$0.624608\pi$
$758$	0	0
$759$	17.5581	0.637320
$760$	0	0
$761$	−47.3283	−1.71565	−0.857825	−	0.513942i	$-0.828185\pi$
$761$	−47.3283	−1.71565	−0.857825	+	0.513942i	$0.828185\pi$
$762$	0	0
$763$	−1.65622	−0.0599593
$764$	0	0
$765$	−3.83597	−0.138690
$766$	0	0
$767$	−56.6487	−2.04547
$768$	0	0
$769$	21.0421	0.758797	0.379399	−	0.925233i	$-0.376131\pi$
$769$	21.0421	0.758797	0.379399	+	0.925233i	$0.376131\pi$
$770$	0	0
$771$	46.1432	1.66181
$772$	0	0
$773$	6.72545	0.241898	0.120949	−	0.992659i	$-0.461406\pi$
$773$	6.72545	0.241898	0.120949	+	0.992659i	$0.461406\pi$
$774$	0	0
$775$	−3.75361	−0.134834
$776$	0	0
$777$	43.1528	1.54810
$778$	0	0
$779$	0.779511	0.0279289
$780$	0	0
$781$	−36.3250	−1.29981
$782$	0	0
$783$	−4.20034	−0.150108
$784$	0	0
$785$	−22.3779	−0.798703
$786$	0	0
$787$	33.0061	1.17654	0.588270	−	0.808665i	$-0.299809\pi$
$787$	33.0061	1.17654	0.588270	+	0.808665i	$0.299809\pi$
$788$	0	0
$789$	46.1701	1.64370
$790$	0	0
$791$	31.5945	1.12337
$792$	0	0
$793$	13.2043	0.468900
$794$	0	0
$795$	−25.2720	−0.896305
$796$	0	0
$797$	−26.2697	−0.930522	−0.465261	−	0.885173i	$-0.654040\pi$
$797$	−26.2697	−0.930522	−0.465261	+	0.885173i	$0.654040\pi$
$798$	0	0
$799$	21.5634	0.762857
$800$	0	0
$801$	−32.3541	−1.14318
$802$	0	0
$803$	18.8886	0.666565
$804$	0	0
$805$	7.03650	0.248004
$806$	0	0
$807$	−47.1011	−1.65804
$808$	0	0
$809$	50.6705	1.78148	0.890740	−	0.454513i	$-0.150187\pi$
$809$	50.6705	1.78148	0.890740	+	0.454513i	$0.150187\pi$
$810$	0	0
$811$	16.6755	0.585554	0.292777	−	0.956181i	$-0.405421\pi$
$811$	16.6755	0.585554	0.292777	+	0.956181i	$0.405421\pi$
$812$	0	0
$813$	71.8734	2.52071
$814$	0	0
$815$	−5.83282	−0.204315
$816$	0	0
$817$	−3.24345	−0.113474
$818$	0	0
$819$	29.2971	1.02372
$820$	0	0
$821$	−20.5920	−0.718667	−0.359334	−	0.933209i	$-0.616996\pi$
$821$	−20.5920	−0.718667	−0.359334	+	0.933209i	$0.616996\pi$
$822$	0	0
$823$	−13.3359	−0.464859	−0.232430	−	0.972613i	$-0.574668\pi$
$823$	−13.3359	−0.464859	−0.232430	+	0.972613i	$0.574668\pi$
$824$	0	0
$825$	8.46457	0.294698
$826$	0	0
$827$	−51.7704	−1.80023	−0.900116	−	0.435650i	$-0.856518\pi$
$827$	−51.7704	−1.80023	−0.900116	+	0.435650i	$0.856518\pi$
$828$	0	0
$829$	40.9639	1.42273	0.711367	−	0.702821i	$-0.248076\pi$
$829$	40.9639	1.42273	0.711367	+	0.702821i	$0.248076\pi$
$830$	0	0
$831$	−27.6583	−0.959454
$832$	0	0
$833$	8.20277	0.284209
$834$	0	0
$835$	16.3904	0.567214
$836$	0	0
$837$	−7.56953	−0.261641
$838$	0	0
$839$	−4.44499	−0.153458	−0.0767291	−	0.997052i	$-0.524448\pi$
$839$	−4.44499	−0.153458	−0.0767291	+	0.997052i	$0.524448\pi$
$840$	0	0
$841$	−24.6616	−0.850400
$842$	0	0
$843$	22.3876	0.771069
$844$	0	0
$845$	−3.79046	−0.130396
$846$	0	0
$847$	−10.2702	−0.352890
$848$	0	0
$849$	−22.4977	−0.772118
$850$	0	0
$851$	11.6758	0.400241
$852$	0	0
$853$	17.0743	0.584613	0.292307	−	0.956325i	$-0.405577\pi$
$853$	17.0743	0.584613	0.292307	+	0.956325i	$0.405577\pi$
$854$	0	0
$855$	1.06902	0.0365598
$856$	0	0
$857$	4.15338	0.141877	0.0709384	−	0.997481i	$-0.477401\pi$
$857$	4.15338	0.141877	0.0709384	+	0.997481i	$0.477401\pi$
$858$	0	0
$859$	−15.0953	−0.515046	−0.257523	−	0.966272i	$-0.582906\pi$
$859$	−15.0953	−0.515046	−0.257523	+	0.966272i	$0.582906\pi$
$860$	0	0
$861$	−11.7826	−0.401550
$862$	0	0
$863$	10.9329	0.372159	0.186080	−	0.982535i	$-0.440422\pi$
$863$	10.9329	0.372159	0.186080	+	0.982535i	$0.440422\pi$
$864$	0	0
$865$	14.8047	0.503376
$866$	0	0
$867$	30.9346	1.05059
$868$	0	0
$869$	−1.34274	−0.0455495
$870$	0	0
$871$	−8.84520	−0.299708
$872$	0	0
$873$	−13.2210	−0.447463
$874$	0	0
$875$	3.39221	0.114678
$876$	0	0
$877$	37.9563	1.28169	0.640847	−	0.767669i	$-0.278583\pi$
$877$	37.9563	1.28169	0.640847	+	0.767669i	$0.278583\pi$
$878$	0	0
$879$	−37.8986	−1.27829
$880$	0	0
$881$	29.4011	0.990549	0.495274	−	0.868737i	$-0.335068\pi$
$881$	29.4011	0.990549	0.495274	+	0.868737i	$0.335068\pi$
$882$	0	0
$883$	29.5819	0.995509	0.497754	−	0.867318i	$-0.334158\pi$
$883$	29.5819	0.995509	0.497754	+	0.867318i	$0.334158\pi$
$884$	0	0
$885$	31.2444	1.05027
$886$	0	0
$887$	−15.7308	−0.528189	−0.264095	−	0.964497i	$-0.585073\pi$
$887$	−15.7308	−0.528189	−0.264095	+	0.964497i	$0.585073\pi$
$888$	0	0
$889$	−37.4342	−1.25550
$890$	0	0
$891$	40.7519	1.36524
$892$	0	0
$893$	−6.00937	−0.201096
$894$	0	0
$895$	−14.7287	−0.492325
$896$	0	0
$897$	19.2096	0.641390
$898$	0	0
$899$	7.81833	0.260756
$900$	0	0
$901$	−20.3512	−0.677998
$902$	0	0
$903$	49.0260	1.63148
$904$	0	0
$905$	−2.65539	−0.0882681
$906$	0	0
$907$	21.8268	0.724746	0.362373	−	0.932033i	$-0.381967\pi$
$907$	21.8268	0.724746	0.362373	+	0.932033i	$0.381967\pi$
$908$	0	0
$909$	27.3216	0.906201
$910$	0	0
$911$	−30.3478	−1.00547	−0.502734	−	0.864441i	$-0.667673\pi$
$911$	−30.3478	−1.00547	−0.502734	+	0.864441i	$0.667673\pi$
$912$	0	0
$913$	−58.4585	−1.93469
$914$	0	0
$915$	−7.28280	−0.240762
$916$	0	0
$917$	−60.3224	−1.99202
$918$	0	0
$919$	−37.7901	−1.24658	−0.623290	−	0.781991i	$-0.714204\pi$
$919$	−37.7901	−1.24658	−0.623290	+	0.781991i	$0.714204\pi$
$920$	0	0
$921$	24.3900	0.803679
$922$	0	0
$923$	−39.7416	−1.30811
$924$	0	0
$925$	5.62876	0.185073
$926$	0	0
$927$	−6.22593	−0.204486
$928$	0	0
$929$	45.8919	1.50567	0.752833	−	0.658212i	$-0.228687\pi$
$929$	45.8919	1.50567	0.752833	+	0.658212i	$0.228687\pi$
$930$	0	0
$931$	−2.28598	−0.0749201
$932$	0	0
$933$	−47.2617	−1.54728
$934$	0	0
$935$	6.81641	0.222921
$936$	0	0
$937$	6.88964	0.225075	0.112537	−	0.993647i	$-0.464102\pi$
$937$	6.88964	0.225075	0.112537	+	0.993647i	$0.464102\pi$
$938$	0	0
$939$	39.8436	1.30025
$940$	0	0
$941$	−27.3078	−0.890210	−0.445105	−	0.895478i	$-0.646834\pi$
$941$	−27.3078	−0.890210	−0.445105	+	0.895478i	$0.646834\pi$
$942$	0	0
$943$	−3.18801	−0.103816
$944$	0	0
$945$	6.84073	0.222529
$946$	0	0
$947$	−39.4271	−1.28121	−0.640604	−	0.767872i	$-0.721316\pi$
$947$	−39.4271	−1.28121	−0.640604	+	0.767872i	$0.721316\pi$
$948$	0	0
$949$	20.6652	0.670821
$950$	0	0
$951$	−68.5593	−2.22319
$952$	0	0
$953$	38.0324	1.23199	0.615995	−	0.787750i	$-0.288754\pi$
$953$	38.0324	1.23199	0.615995	+	0.787750i	$0.288754\pi$
$954$	0	0
$955$	21.3629	0.691286
$956$	0	0
$957$	−17.6307	−0.569919
$958$	0	0
$959$	−1.29399	−0.0417851
$960$	0	0
$961$	−16.9104	−0.545497
$962$	0	0
$963$	9.29052	0.299383
$964$	0	0
$965$	−21.7573	−0.700394
$966$	0	0
$967$	25.7662	0.828587	0.414293	−	0.910143i	$-0.364029\pi$
$967$	25.7662	0.828587	0.414293	+	0.910143i	$0.364029\pi$
$968$	0	0
$969$	2.08619	0.0670181
$970$	0	0
$971$	5.29562	0.169944	0.0849722	−	0.996383i	$-0.472920\pi$
$971$	5.29562	0.169944	0.0849722	+	0.996383i	$0.472920\pi$
$972$	0	0
$973$	61.8618	1.98320
$974$	0	0
$975$	9.26071	0.296580
$976$	0	0
$977$	−35.1477	−1.12447	−0.562237	−	0.826976i	$-0.690059\pi$
$977$	−35.1477	−1.12447	−0.562237	+	0.826976i	$0.690059\pi$
$978$	0	0
$979$	57.4924	1.83747
$980$	0	0
$981$	1.02907	0.0328558
$982$	0	0
$983$	13.7793	0.439492	0.219746	−	0.975557i	$-0.429477\pi$
$983$	13.7793	0.439492	0.219746	+	0.975557i	$0.429477\pi$
$984$	0	0
$985$	1.53627	0.0489495
$986$	0	0
$987$	90.8339	2.89128
$988$	0	0
$989$	13.2649	0.421800
$990$	0	0
$991$	−4.03898	−0.128302	−0.0641512	−	0.997940i	$-0.520434\pi$
$991$	−4.03898	−0.128302	−0.0641512	+	0.997940i	$0.520434\pi$
$992$	0	0
$993$	62.3001	1.97703
$994$	0	0
$995$	−8.79522	−0.278827
$996$	0	0
$997$	52.7065	1.66923	0.834616	−	0.550833i	$-0.185690\pi$
$997$	52.7065	1.66923	0.834616	+	0.550833i	$0.185690\pi$
$998$	0	0
$999$	11.3510	0.359128

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.5	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.5	✓	28	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26002 q^{3} -1.00000 q^{5} -3.39221 q^{7} +2.10771 q^{9} +O(q^{10})\)	\(q-2.26002 q^{3} -1.00000 q^{5} -3.39221 q^{7} +2.10771 q^{9} -3.74534 q^{11} -4.09762 q^{13} +2.26002 q^{15} +1.81997 q^{17} -0.507197 q^{19} +7.66648 q^{21} +2.07431 q^{23} +1.00000 q^{25} +2.01660 q^{27} -2.08288 q^{29} -3.75361 q^{31} +8.46457 q^{33} +3.39221 q^{35} +5.62876 q^{37} +9.26071 q^{39} -1.53690 q^{41} +6.39486 q^{43} -2.10771 q^{45} +11.8482 q^{47} +4.50709 q^{49} -4.11318 q^{51} -11.1822 q^{53} +3.74534 q^{55} +1.14628 q^{57} +13.8248 q^{59} -3.22244 q^{61} -7.14979 q^{63} +4.09762 q^{65} +2.15862 q^{67} -4.68799 q^{69} +9.69871 q^{71} -5.04323 q^{73} -2.26002 q^{75} +12.7050 q^{77} +0.358510 q^{79} -10.8807 q^{81} +15.6083 q^{83} -1.81997 q^{85} +4.70736 q^{87} -15.3504 q^{89} +13.9000 q^{91} +8.48325 q^{93} +0.507197 q^{95} -6.27268 q^{97} -7.89409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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