LMFDB - Embedded newform 8020.2.a.d.1.8

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:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.59991 q^{3} +1.00000 q^{5} -1.91437 q^{7} -0.440296 q^{9} +O(q^{10})$	$q-1.59991 q^{3} +1.00000 q^{5} -1.91437 q^{7} -0.440296 q^{9} -3.87332 q^{11} +2.13674 q^{13} -1.59991 q^{15} -4.71318 q^{17} +5.63555 q^{19} +3.06281 q^{21} -2.95050 q^{23} +1.00000 q^{25} +5.50416 q^{27} +0.500544 q^{29} +7.18849 q^{31} +6.19695 q^{33} -1.91437 q^{35} -1.27080 q^{37} -3.41859 q^{39} +4.31327 q^{41} -3.34989 q^{43} -0.440296 q^{45} +4.92369 q^{47} -3.33520 q^{49} +7.54065 q^{51} +2.29019 q^{53} -3.87332 q^{55} -9.01636 q^{57} +10.5496 q^{59} -7.22509 q^{61} +0.842889 q^{63} +2.13674 q^{65} +8.21436 q^{67} +4.72053 q^{69} +2.62318 q^{71} +2.35253 q^{73} -1.59991 q^{75} +7.41496 q^{77} +15.6761 q^{79} -7.48525 q^{81} -12.1198 q^{83} -4.71318 q^{85} -0.800823 q^{87} -15.4638 q^{89} -4.09050 q^{91} -11.5009 q^{93} +5.63555 q^{95} +6.13175 q^{97} +1.70541 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * 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
$3$	−1.59991	−0.923707	−0.461853	−	0.886956i	$-0.652815\pi$
$3$	−1.59991	−0.923707	−0.461853	+	0.886956i	$0.652815\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.91437	−0.723563	−0.361781	−	0.932263i	$-0.617831\pi$
$7$	−1.91437	−0.723563	−0.361781	+	0.932263i	$0.617831\pi$
$8$	0	0
$9$	−0.440296	−0.146765
$10$	0	0
$11$	−3.87332	−1.16785	−0.583925	−	0.811808i	$-0.698484\pi$
$11$	−3.87332	−1.16785	−0.583925	+	0.811808i	$0.698484\pi$
$12$	0	0
$13$	2.13674	0.592625	0.296313	−	0.955091i	$-0.404243\pi$
$13$	2.13674	0.592625	0.296313	+	0.955091i	$0.404243\pi$
$14$	0	0
$15$	−1.59991	−0.413094
$16$	0	0
$17$	−4.71318	−1.14311	−0.571557	−	0.820563i	$-0.693660\pi$
$17$	−4.71318	−1.14311	−0.571557	+	0.820563i	$0.693660\pi$
$18$	0	0
$19$	5.63555	1.29288	0.646442	−	0.762963i	$-0.276256\pi$
$19$	5.63555	1.29288	0.646442	+	0.762963i	$0.276256\pi$
$20$	0	0
$21$	3.06281	0.668360
$22$	0	0
$23$	−2.95050	−0.615222	−0.307611	−	0.951512i	$-0.599530\pi$
$23$	−2.95050	−0.615222	−0.307611	+	0.951512i	$0.599530\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.50416	1.05928
$28$	0	0
$29$	0.500544	0.0929486	0.0464743	−	0.998919i	$-0.485201\pi$
$29$	0.500544	0.0929486	0.0464743	+	0.998919i	$0.485201\pi$
$30$	0	0
$31$	7.18849	1.29109	0.645546	−	0.763722i	$-0.276630\pi$
$31$	7.18849	1.29109	0.645546	+	0.763722i	$0.276630\pi$
$32$	0	0
$33$	6.19695	1.07875
$34$	0	0
$35$	−1.91437	−0.323587
$36$	0	0
$37$	−1.27080	−0.208919	−0.104459	−	0.994529i	$-0.533311\pi$
$37$	−1.27080	−0.208919	−0.104459	+	0.994529i	$0.533311\pi$
$38$	0	0
$39$	−3.41859	−0.547412
$40$	0	0
$41$	4.31327	0.673620	0.336810	−	0.941573i	$-0.390652\pi$
$41$	4.31327	0.673620	0.336810	+	0.941573i	$0.390652\pi$
$42$	0	0
$43$	−3.34989	−0.510853	−0.255427	−	0.966828i	$-0.582216\pi$
$43$	−3.34989	−0.510853	−0.255427	+	0.966828i	$0.582216\pi$
$44$	0	0
$45$	−0.440296	−0.0656355
$46$	0	0
$47$	4.92369	0.718193	0.359097	−	0.933300i	$-0.383085\pi$
$47$	4.92369	0.718193	0.359097	+	0.933300i	$0.383085\pi$
$48$	0	0
$49$	−3.33520	−0.476457
$50$	0	0
$51$	7.54065	1.05590
$52$	0	0
$53$	2.29019	0.314582	0.157291	−	0.987552i	$-0.449724\pi$
$53$	2.29019	0.314582	0.157291	+	0.987552i	$0.449724\pi$
$54$	0	0
$55$	−3.87332	−0.522278
$56$	0	0
$57$	−9.01636	−1.19425
$58$	0	0
$59$	10.5496	1.37344	0.686719	−	0.726923i	$-0.259050\pi$
$59$	10.5496	1.37344	0.686719	+	0.726923i	$0.259050\pi$
$60$	0	0
$61$	−7.22509	−0.925078	−0.462539	−	0.886599i	$-0.653061\pi$
$61$	−7.22509	−0.925078	−0.462539	+	0.886599i	$0.653061\pi$
$62$	0	0
$63$	0.842889	0.106194
$64$	0	0
$65$	2.13674	0.265030
$66$	0	0
$67$	8.21436	1.00354	0.501772	−	0.865000i	$-0.332682\pi$
$67$	8.21436	1.00354	0.501772	+	0.865000i	$0.332682\pi$
$68$	0	0
$69$	4.72053	0.568285
$70$	0	0
$71$	2.62318	0.311314	0.155657	−	0.987811i	$-0.450251\pi$
$71$	2.62318	0.311314	0.155657	+	0.987811i	$0.450251\pi$
$72$	0	0
$73$	2.35253	0.275343	0.137671	−	0.990478i	$-0.456038\pi$
$73$	2.35253	0.275343	0.137671	+	0.990478i	$0.456038\pi$
$74$	0	0
$75$	−1.59991	−0.184741
$76$	0	0
$77$	7.41496	0.845013
$78$	0	0
$79$	15.6761	1.76369	0.881847	−	0.471535i	$-0.156300\pi$
$79$	15.6761	1.76369	0.881847	+	0.471535i	$0.156300\pi$
$80$	0	0
$81$	−7.48525	−0.831694
$82$	0	0
$83$	−12.1198	−1.33032	−0.665159	−	0.746701i	$-0.731636\pi$
$83$	−12.1198	−1.33032	−0.665159	+	0.746701i	$0.731636\pi$
$84$	0	0
$85$	−4.71318	−0.511216
$86$	0	0
$87$	−0.800823	−0.0858573
$88$	0	0
$89$	−15.4638	−1.63916	−0.819580	−	0.572965i	$-0.805793\pi$
$89$	−15.4638	−1.63916	−0.819580	+	0.572965i	$0.805793\pi$
$90$	0	0
$91$	−4.09050	−0.428801
$92$	0	0
$93$	−11.5009	−1.19259
$94$	0	0
$95$	5.63555	0.578196
$96$	0	0
$97$	6.13175	0.622585	0.311293	−	0.950314i	$-0.399238\pi$
$97$	6.13175	0.622585	0.311293	+	0.950314i	$0.399238\pi$
$98$	0	0
$99$	1.70541	0.171400
$100$	0	0
$101$	−4.70177	−0.467843	−0.233922	−	0.972255i	$-0.575156\pi$
$101$	−4.70177	−0.467843	−0.233922	+	0.972255i	$0.575156\pi$
$102$	0	0
$103$	9.17004	0.903551	0.451775	−	0.892132i	$-0.350791\pi$
$103$	9.17004	0.903551	0.451775	+	0.892132i	$0.350791\pi$
$104$	0	0
$105$	3.06281	0.298900
$106$	0	0
$107$	−6.45785	−0.624304	−0.312152	−	0.950032i	$-0.601050\pi$
$107$	−6.45785	−0.624304	−0.312152	+	0.950032i	$0.601050\pi$
$108$	0	0
$109$	3.50681	0.335892	0.167946	−	0.985796i	$-0.446287\pi$
$109$	3.50681	0.335892	0.167946	+	0.985796i	$0.446287\pi$
$110$	0	0
$111$	2.03317	0.192980
$112$	0	0
$113$	−4.15356	−0.390734	−0.195367	−	0.980730i	$-0.562590\pi$
$113$	−4.15356	−0.390734	−0.195367	+	0.980730i	$0.562590\pi$
$114$	0	0
$115$	−2.95050	−0.275136
$116$	0	0
$117$	−0.940799	−0.0869769
$118$	0	0
$119$	9.02275	0.827114
$120$	0	0
$121$	4.00261	0.363874
$122$	0	0
$123$	−6.90084	−0.622228
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.850245	−0.0754470	−0.0377235	−	0.999288i	$-0.512011\pi$
$127$	−0.850245	−0.0754470	−0.0377235	+	0.999288i	$0.512011\pi$
$128$	0	0
$129$	5.35951	0.471879
$130$	0	0
$131$	−9.00667	−0.786917	−0.393458	−	0.919342i	$-0.628721\pi$
$131$	−9.00667	−0.786917	−0.393458	+	0.919342i	$0.628721\pi$
$132$	0	0
$133$	−10.7885	−0.935483
$134$	0	0
$135$	5.50416	0.473722
$136$	0	0
$137$	−14.4822	−1.23730	−0.618650	−	0.785667i	$-0.712320\pi$
$137$	−14.4822	−1.23730	−0.618650	+	0.785667i	$0.712320\pi$
$138$	0	0
$139$	−2.69052	−0.228207	−0.114104	−	0.993469i	$-0.536400\pi$
$139$	−2.69052	−0.228207	−0.114104	+	0.993469i	$0.536400\pi$
$140$	0	0
$141$	−7.87744	−0.663400
$142$	0	0
$143$	−8.27628	−0.692097
$144$	0	0
$145$	0.500544	0.0415679
$146$	0	0
$147$	5.33601	0.440107
$148$	0	0
$149$	−20.2802	−1.66142	−0.830709	−	0.556706i	$-0.812065\pi$
$149$	−20.2802	−1.66142	−0.830709	+	0.556706i	$0.812065\pi$
$150$	0	0
$151$	1.39284	0.113348	0.0566740	−	0.998393i	$-0.481950\pi$
$151$	1.39284	0.113348	0.0566740	+	0.998393i	$0.481950\pi$
$152$	0	0
$153$	2.07520	0.167770
$154$	0	0
$155$	7.18849	0.577394
$156$	0	0
$157$	−12.6302	−1.00800	−0.503999	−	0.863704i	$-0.668139\pi$
$157$	−12.6302	−1.00800	−0.503999	+	0.863704i	$0.668139\pi$
$158$	0	0
$159$	−3.66409	−0.290581
$160$	0	0
$161$	5.64834	0.445152
$162$	0	0
$163$	−7.02879	−0.550538	−0.275269	−	0.961367i	$-0.588767\pi$
$163$	−7.02879	−0.550538	−0.275269	+	0.961367i	$0.588767\pi$
$164$	0	0
$165$	6.19695	0.482432
$166$	0	0
$167$	−18.3773	−1.42208	−0.711039	−	0.703153i	$-0.751775\pi$
$167$	−18.3773	−1.42208	−0.711039	+	0.703153i	$0.751775\pi$
$168$	0	0
$169$	−8.43434	−0.648795
$170$	0	0
$171$	−2.48131	−0.189751
$172$	0	0
$173$	−7.68465	−0.584253	−0.292127	−	0.956380i	$-0.594363\pi$
$173$	−7.68465	−0.584253	−0.292127	+	0.956380i	$0.594363\pi$
$174$	0	0
$175$	−1.91437	−0.144713
$176$	0	0
$177$	−16.8783	−1.26865
$178$	0	0
$179$	25.2885	1.89015	0.945076	−	0.326850i	$-0.105987\pi$
$179$	25.2885	1.89015	0.945076	+	0.326850i	$0.105987\pi$
$180$	0	0
$181$	5.16828	0.384155	0.192078	−	0.981380i	$-0.438478\pi$
$181$	5.16828	0.384155	0.192078	+	0.981380i	$0.438478\pi$
$182$	0	0
$183$	11.5595	0.854501
$184$	0	0
$185$	−1.27080	−0.0934313
$186$	0	0
$187$	18.2556	1.33499
$188$	0	0
$189$	−10.5370	−0.766452
$190$	0	0
$191$	3.09060	0.223628	0.111814	−	0.993729i	$-0.464334\pi$
$191$	3.09060	0.223628	0.111814	+	0.993729i	$0.464334\pi$
$192$	0	0
$193$	6.70521	0.482652	0.241326	−	0.970444i	$-0.422418\pi$
$193$	6.70521	0.482652	0.241326	+	0.970444i	$0.422418\pi$
$194$	0	0
$195$	−3.41859	−0.244810
$196$	0	0
$197$	−2.08180	−0.148322	−0.0741611	−	0.997246i	$-0.523628\pi$
$197$	−2.08180	−0.148322	−0.0741611	+	0.997246i	$0.523628\pi$
$198$	0	0
$199$	1.35553	0.0960913	0.0480456	−	0.998845i	$-0.484701\pi$
$199$	1.35553	0.0960913	0.0480456	+	0.998845i	$0.484701\pi$
$200$	0	0
$201$	−13.1422	−0.926980
$202$	0	0
$203$	−0.958224	−0.0672541
$204$	0	0
$205$	4.31327	0.301252
$206$	0	0
$207$	1.29910	0.0902933
$208$	0	0
$209$	−21.8283	−1.50990
$210$	0	0
$211$	−14.7358	−1.01446	−0.507228	−	0.861812i	$-0.669330\pi$
$211$	−14.7358	−1.01446	−0.507228	+	0.861812i	$0.669330\pi$
$212$	0	0
$213$	−4.19684	−0.287563
$214$	0	0
$215$	−3.34989	−0.228461
$216$	0	0
$217$	−13.7614	−0.934185
$218$	0	0
$219$	−3.76383	−0.254336
$220$	0	0
$221$	−10.0708	−0.677438
$222$	0	0
$223$	−9.59268	−0.642373	−0.321187	−	0.947016i	$-0.604082\pi$
$223$	−9.59268	−0.642373	−0.321187	+	0.947016i	$0.604082\pi$
$224$	0	0
$225$	−0.440296	−0.0293531
$226$	0	0
$227$	−29.1940	−1.93768	−0.968838	−	0.247697i	$-0.920326\pi$
$227$	−29.1940	−1.93768	−0.968838	+	0.247697i	$0.920326\pi$
$228$	0	0
$229$	5.14763	0.340165	0.170083	−	0.985430i	$-0.445597\pi$
$229$	5.14763	0.340165	0.170083	+	0.985430i	$0.445597\pi$
$230$	0	0
$231$	−11.8632	−0.780544
$232$	0	0
$233$	26.2347	1.71869	0.859347	−	0.511392i	$-0.170870\pi$
$233$	26.2347	1.71869	0.859347	+	0.511392i	$0.170870\pi$
$234$	0	0
$235$	4.92369	0.321186
$236$	0	0
$237$	−25.0802	−1.62914
$238$	0	0
$239$	6.33765	0.409948	0.204974	−	0.978767i	$-0.434289\pi$
$239$	6.33765	0.409948	0.204974	+	0.978767i	$0.434289\pi$
$240$	0	0
$241$	16.6622	1.07330	0.536652	−	0.843804i	$-0.319689\pi$
$241$	16.6622	1.07330	0.536652	+	0.843804i	$0.319689\pi$
$242$	0	0
$243$	−4.53676	−0.291033
$244$	0	0
$245$	−3.33520	−0.213078
$246$	0	0
$247$	12.0417	0.766196
$248$	0	0
$249$	19.3905	1.22882
$250$	0	0
$251$	7.53553	0.475639	0.237819	−	0.971309i	$-0.423567\pi$
$251$	7.53553	0.475639	0.237819	+	0.971309i	$0.423567\pi$
$252$	0	0
$253$	11.4282	0.718487
$254$	0	0
$255$	7.54065	0.472214
$256$	0	0
$257$	12.7392	0.794647	0.397324	−	0.917679i	$-0.369939\pi$
$257$	12.7392	0.794647	0.397324	+	0.917679i	$0.369939\pi$
$258$	0	0
$259$	2.43278	0.151166
$260$	0	0
$261$	−0.220388	−0.0136416
$262$	0	0
$263$	9.91760	0.611545	0.305773	−	0.952105i	$-0.401085\pi$
$263$	9.91760	0.611545	0.305773	+	0.952105i	$0.401085\pi$
$264$	0	0
$265$	2.29019	0.140685
$266$	0	0
$267$	24.7407	1.51410
$268$	0	0
$269$	6.27707	0.382720	0.191360	−	0.981520i	$-0.438710\pi$
$269$	6.27707	0.382720	0.191360	+	0.981520i	$0.438710\pi$
$270$	0	0
$271$	25.6697	1.55932	0.779661	−	0.626202i	$-0.215391\pi$
$271$	25.6697	1.55932	0.779661	+	0.626202i	$0.215391\pi$
$272$	0	0
$273$	6.54443	0.396087
$274$	0	0
$275$	−3.87332	−0.233570
$276$	0	0
$277$	6.55884	0.394083	0.197041	−	0.980395i	$-0.436867\pi$
$277$	6.55884	0.394083	0.197041	+	0.980395i	$0.436867\pi$
$278$	0	0
$279$	−3.16507	−0.189488
$280$	0	0
$281$	−0.649878	−0.0387685	−0.0193842	−	0.999812i	$-0.506171\pi$
$281$	−0.649878	−0.0387685	−0.0193842	+	0.999812i	$0.506171\pi$
$282$	0	0
$283$	8.38543	0.498462	0.249231	−	0.968444i	$-0.419822\pi$
$283$	8.38543	0.498462	0.249231	+	0.968444i	$0.419822\pi$
$284$	0	0
$285$	−9.01636	−0.534083
$286$	0	0
$287$	−8.25719	−0.487406
$288$	0	0
$289$	5.21405	0.306709
$290$	0	0
$291$	−9.81024	−0.575086
$292$	0	0
$293$	−12.7668	−0.745847	−0.372924	−	0.927862i	$-0.621645\pi$
$293$	−12.7668	−0.745847	−0.372924	+	0.927862i	$0.621645\pi$
$294$	0	0
$295$	10.5496	0.614220
$296$	0	0
$297$	−21.3194	−1.23707
$298$	0	0
$299$	−6.30446	−0.364596
$300$	0	0
$301$	6.41292	0.369634
$302$	0	0
$303$	7.52239	0.432150
$304$	0	0
$305$	−7.22509	−0.413707
$306$	0	0
$307$	1.22475	0.0698999	0.0349500	−	0.999389i	$-0.488873\pi$
$307$	1.22475	0.0698999	0.0349500	+	0.999389i	$0.488873\pi$
$308$	0	0
$309$	−14.6712	−0.834616
$310$	0	0
$311$	−29.9291	−1.69712	−0.848560	−	0.529098i	$-0.822530\pi$
$311$	−29.9291	−1.69712	−0.848560	+	0.529098i	$0.822530\pi$
$312$	0	0
$313$	−16.2074	−0.916096	−0.458048	−	0.888927i	$-0.651451\pi$
$313$	−16.2074	−0.916096	−0.458048	+	0.888927i	$0.651451\pi$
$314$	0	0
$315$	0.842889	0.0474914
$316$	0	0
$317$	14.2822	0.802168	0.401084	−	0.916041i	$-0.368633\pi$
$317$	14.2822	0.802168	0.401084	+	0.916041i	$0.368633\pi$
$318$	0	0
$319$	−1.93877	−0.108550
$320$	0	0
$321$	10.3320	0.576674
$322$	0	0
$323$	−26.5614	−1.47791
$324$	0	0
$325$	2.13674	0.118525
$326$	0	0
$327$	−5.61058	−0.310266
$328$	0	0
$329$	−9.42574	−0.519658
$330$	0	0
$331$	−8.13613	−0.447202	−0.223601	−	0.974681i	$-0.571781\pi$
$331$	−8.13613	−0.447202	−0.223601	+	0.974681i	$0.571781\pi$
$332$	0	0
$333$	0.559530	0.0306621
$334$	0	0
$335$	8.21436	0.448798
$336$	0	0
$337$	−19.5691	−1.06600	−0.532999	−	0.846116i	$-0.678935\pi$
$337$	−19.5691	−1.06600	−0.532999	+	0.846116i	$0.678935\pi$
$338$	0	0
$339$	6.64531	0.360923
$340$	0	0
$341$	−27.8433	−1.50780
$342$	0	0
$343$	19.7854	1.06831
$344$	0	0
$345$	4.72053	0.254145
$346$	0	0
$347$	−4.59553	−0.246701	−0.123350	−	0.992363i	$-0.539364\pi$
$347$	−4.59553	−0.246701	−0.123350	+	0.992363i	$0.539364\pi$
$348$	0	0
$349$	−20.1304	−1.07756	−0.538778	−	0.842448i	$-0.681114\pi$
$349$	−20.1304	−1.07756	−0.538778	+	0.842448i	$0.681114\pi$
$350$	0	0
$351$	11.7610	0.627753
$352$	0	0
$353$	−16.3829	−0.871971	−0.435986	−	0.899954i	$-0.643600\pi$
$353$	−16.3829	−0.871971	−0.435986	+	0.899954i	$0.643600\pi$
$354$	0	0
$355$	2.62318	0.139224
$356$	0	0
$357$	−14.4356	−0.764011
$358$	0	0
$359$	−6.01438	−0.317427	−0.158713	−	0.987325i	$-0.550735\pi$
$359$	−6.01438	−0.317427	−0.158713	+	0.987325i	$0.550735\pi$
$360$	0	0
$361$	12.7595	0.671551
$362$	0	0
$363$	−6.40381	−0.336113
$364$	0	0
$365$	2.35253	0.123137
$366$	0	0
$367$	5.41180	0.282494	0.141247	−	0.989974i	$-0.454889\pi$
$367$	5.41180	0.282494	0.141247	+	0.989974i	$0.454889\pi$
$368$	0	0
$369$	−1.89912	−0.0988642
$370$	0	0
$371$	−4.38426	−0.227620
$372$	0	0
$373$	−31.3338	−1.62240	−0.811202	−	0.584766i	$-0.801186\pi$
$373$	−31.3338	−1.62240	−0.811202	+	0.584766i	$0.801186\pi$
$374$	0	0
$375$	−1.59991	−0.0826189
$376$	0	0
$377$	1.06953	0.0550837
$378$	0	0
$379$	27.6405	1.41980	0.709898	−	0.704305i	$-0.248741\pi$
$379$	27.6405	1.41980	0.709898	+	0.704305i	$0.248741\pi$
$380$	0	0
$381$	1.36031	0.0696909
$382$	0	0
$383$	−16.4770	−0.841938	−0.420969	−	0.907075i	$-0.638310\pi$
$383$	−16.4770	−0.841938	−0.420969	+	0.907075i	$0.638310\pi$
$384$	0	0
$385$	7.41496	0.377901
$386$	0	0
$387$	1.47494	0.0749756
$388$	0	0
$389$	−1.74949	−0.0887026	−0.0443513	−	0.999016i	$-0.514122\pi$
$389$	−1.74949	−0.0887026	−0.0443513	+	0.999016i	$0.514122\pi$
$390$	0	0
$391$	13.9062	0.703269
$392$	0	0
$393$	14.4098	0.726881
$394$	0	0
$395$	15.6761	0.788748
$396$	0	0
$397$	−4.80924	−0.241369	−0.120684	−	0.992691i	$-0.538509\pi$
$397$	−4.80924	−0.241369	−0.120684	+	0.992691i	$0.538509\pi$
$398$	0	0
$399$	17.2606	0.864112
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	15.3599	0.765133
$404$	0	0
$405$	−7.48525	−0.371945
$406$	0	0
$407$	4.92223	0.243986
$408$	0	0
$409$	−30.5776	−1.51197	−0.755983	−	0.654591i	$-0.772841\pi$
$409$	−30.5776	−1.51197	−0.755983	+	0.654591i	$0.772841\pi$
$410$	0	0
$411$	23.1702	1.14290
$412$	0	0
$413$	−20.1957	−0.993768
$414$	0	0
$415$	−12.1198	−0.594937
$416$	0	0
$417$	4.30459	0.210797
$418$	0	0
$419$	−14.0329	−0.685551	−0.342775	−	0.939417i	$-0.611367\pi$
$419$	−14.0329	−0.685551	−0.342775	+	0.939417i	$0.611367\pi$
$420$	0	0
$421$	17.7891	0.866986	0.433493	−	0.901157i	$-0.357281\pi$
$421$	17.7891	0.866986	0.433493	+	0.901157i	$0.357281\pi$
$422$	0	0
$423$	−2.16788	−0.105406
$424$	0	0
$425$	−4.71318	−0.228623
$426$	0	0
$427$	13.8315	0.669352
$428$	0	0
$429$	13.2413	0.639295
$430$	0	0
$431$	−14.2846	−0.688067	−0.344033	−	0.938957i	$-0.611793\pi$
$431$	−14.2846	−0.688067	−0.344033	+	0.938957i	$0.611793\pi$
$432$	0	0
$433$	−10.9236	−0.524957	−0.262479	−	0.964938i	$-0.584540\pi$
$433$	−10.9236	−0.524957	−0.262479	+	0.964938i	$0.584540\pi$
$434$	0	0
$435$	−0.800823	−0.0383965
$436$	0	0
$437$	−16.6277	−0.795411
$438$	0	0
$439$	−16.9942	−0.811088	−0.405544	−	0.914076i	$-0.632918\pi$
$439$	−16.9942	−0.811088	−0.405544	+	0.914076i	$0.632918\pi$
$440$	0	0
$441$	1.46848	0.0699275
$442$	0	0
$443$	−34.5455	−1.64131	−0.820653	−	0.571426i	$-0.806390\pi$
$443$	−34.5455	−1.64131	−0.820653	+	0.571426i	$0.806390\pi$
$444$	0	0
$445$	−15.4638	−0.733055
$446$	0	0
$447$	32.4464	1.53466
$448$	0	0
$449$	0.850890	0.0401560	0.0200780	−	0.999798i	$-0.493609\pi$
$449$	0.850890	0.0401560	0.0200780	+	0.999798i	$0.493609\pi$
$450$	0	0
$451$	−16.7067	−0.786687
$452$	0	0
$453$	−2.22842	−0.104700
$454$	0	0
$455$	−4.09050	−0.191766
$456$	0	0
$457$	−18.5187	−0.866267	−0.433134	−	0.901330i	$-0.642592\pi$
$457$	−18.5187	−0.866267	−0.433134	+	0.901330i	$0.642592\pi$
$458$	0	0
$459$	−25.9421	−1.21087
$460$	0	0
$461$	12.7430	0.593499	0.296749	−	0.954955i	$-0.404097\pi$
$461$	12.7430	0.593499	0.296749	+	0.954955i	$0.404097\pi$
$462$	0	0
$463$	−18.0268	−0.837778	−0.418889	−	0.908037i	$-0.637580\pi$
$463$	−18.0268	−0.837778	−0.418889	+	0.908037i	$0.637580\pi$
$464$	0	0
$465$	−11.5009	−0.533342
$466$	0	0
$467$	40.6494	1.88103	0.940515	−	0.339751i	$-0.110343\pi$
$467$	40.6494	1.88103	0.940515	+	0.339751i	$0.110343\pi$
$468$	0	0
$469$	−15.7253	−0.726127
$470$	0	0
$471$	20.2071	0.931095
$472$	0	0
$473$	12.9752	0.596600
$474$	0	0
$475$	5.63555	0.258577
$476$	0	0
$477$	−1.00836	−0.0461697
$478$	0	0
$479$	14.0579	0.642322	0.321161	−	0.947025i	$-0.395927\pi$
$479$	14.0579	0.642322	0.321161	+	0.947025i	$0.395927\pi$
$480$	0	0
$481$	−2.71538	−0.123811
$482$	0	0
$483$	−9.03682	−0.411190
$484$	0	0
$485$	6.13175	0.278429
$486$	0	0
$487$	23.6034	1.06957	0.534785	−	0.844988i	$-0.320392\pi$
$487$	23.6034	1.06957	0.534785	+	0.844988i	$0.320392\pi$
$488$	0	0
$489$	11.2454	0.508535
$490$	0	0
$491$	20.9890	0.947221	0.473610	−	0.880735i	$-0.342951\pi$
$491$	20.9890	0.947221	0.473610	+	0.880735i	$0.342951\pi$
$492$	0	0
$493$	−2.35915	−0.106251
$494$	0	0
$495$	1.70541	0.0766524
$496$	0	0
$497$	−5.02172	−0.225255
$498$	0	0
$499$	2.76962	0.123985	0.0619927	−	0.998077i	$-0.480254\pi$
$499$	2.76962	0.123985	0.0619927	+	0.998077i	$0.480254\pi$
$500$	0	0
$501$	29.4020	1.31358
$502$	0	0
$503$	−25.5717	−1.14018	−0.570092	−	0.821581i	$-0.693092\pi$
$503$	−25.5717	−1.14018	−0.570092	+	0.821581i	$0.693092\pi$
$504$	0	0
$505$	−4.70177	−0.209226
$506$	0	0
$507$	13.4942	0.599297
$508$	0	0
$509$	−32.9620	−1.46101	−0.730507	−	0.682905i	$-0.760716\pi$
$509$	−32.9620	−1.46101	−0.730507	+	0.682905i	$0.760716\pi$
$510$	0	0
$511$	−4.50361	−0.199228
$512$	0	0
$513$	31.0190	1.36952
$514$	0	0
$515$	9.17004	0.404080
$516$	0	0
$517$	−19.0710	−0.838742
$518$	0	0
$519$	12.2947	0.539679
$520$	0	0
$521$	−0.467503	−0.0204817	−0.0102408	−	0.999948i	$-0.503260\pi$
$521$	−0.467503	−0.0204817	−0.0102408	+	0.999948i	$0.503260\pi$
$522$	0	0
$523$	−31.7220	−1.38711	−0.693553	−	0.720405i	$-0.743956\pi$
$523$	−31.7220	−1.38711	−0.693553	+	0.720405i	$0.743956\pi$
$524$	0	0
$525$	3.06281	0.133672
$526$	0	0
$527$	−33.8806	−1.47586
$528$	0	0
$529$	−14.2945	−0.621502
$530$	0	0
$531$	−4.64494	−0.201573
$532$	0	0
$533$	9.21635	0.399204
$534$	0	0
$535$	−6.45785	−0.279197
$536$	0	0
$537$	−40.4593	−1.74595
$538$	0	0
$539$	12.9183	0.556431
$540$	0	0
$541$	−20.5425	−0.883190	−0.441595	−	0.897214i	$-0.645587\pi$
$541$	−20.5425	−0.883190	−0.441595	+	0.897214i	$0.645587\pi$
$542$	0	0
$543$	−8.26876	−0.354847
$544$	0	0
$545$	3.50681	0.150215
$546$	0	0
$547$	−44.3226	−1.89510	−0.947549	−	0.319611i	$-0.896448\pi$
$547$	−44.3226	−1.89510	−0.947549	+	0.319611i	$0.896448\pi$
$548$	0	0
$549$	3.18118	0.135769
$550$	0	0
$551$	2.82084	0.120172
$552$	0	0
$553$	−30.0097	−1.27614
$554$	0	0
$555$	2.03317	0.0863032
$556$	0	0
$557$	37.7558	1.59977	0.799883	−	0.600156i	$-0.204895\pi$
$557$	37.7558	1.59977	0.799883	+	0.600156i	$0.204895\pi$
$558$	0	0
$559$	−7.15784	−0.302745
$560$	0	0
$561$	−29.2073	−1.23314
$562$	0	0
$563$	−24.5221	−1.03348	−0.516742	−	0.856141i	$-0.672855\pi$
$563$	−24.5221	−1.03348	−0.516742	+	0.856141i	$0.672855\pi$
$564$	0	0
$565$	−4.15356	−0.174741
$566$	0	0
$567$	14.3295	0.601783
$568$	0	0
$569$	−25.7613	−1.07997	−0.539986	−	0.841674i	$-0.681570\pi$
$569$	−25.7613	−1.07997	−0.539986	+	0.841674i	$0.681570\pi$
$570$	0	0
$571$	11.7622	0.492235	0.246117	−	0.969240i	$-0.420845\pi$
$571$	11.7622	0.492235	0.246117	+	0.969240i	$0.420845\pi$
$572$	0	0
$573$	−4.94468	−0.206567
$574$	0	0
$575$	−2.95050	−0.123044
$576$	0	0
$577$	15.8914	0.661566	0.330783	−	0.943707i	$-0.392687\pi$
$577$	15.8914	0.661566	0.330783	+	0.943707i	$0.392687\pi$
$578$	0	0
$579$	−10.7277	−0.445829
$580$	0	0
$581$	23.2017	0.962569
$582$	0	0
$583$	−8.87064	−0.367384
$584$	0	0
$585$	−0.940799	−0.0388973
$586$	0	0
$587$	22.0912	0.911803	0.455901	−	0.890030i	$-0.349317\pi$
$587$	22.0912	0.911803	0.455901	+	0.890030i	$0.349317\pi$
$588$	0	0
$589$	40.5111	1.66923
$590$	0	0
$591$	3.33069	0.137006
$592$	0	0
$593$	−33.7376	−1.38544	−0.692718	−	0.721208i	$-0.743587\pi$
$593$	−33.7376	−1.38544	−0.692718	+	0.721208i	$0.743587\pi$
$594$	0	0
$595$	9.02275	0.369897
$596$	0	0
$597$	−2.16873	−0.0887602
$598$	0	0
$599$	−36.4465	−1.48916	−0.744581	−	0.667532i	$-0.767351\pi$
$599$	−36.4465	−1.48916	−0.744581	+	0.667532i	$0.767351\pi$
$600$	0	0
$601$	4.52989	0.184778	0.0923891	−	0.995723i	$-0.470550\pi$
$601$	4.52989	0.184778	0.0923891	+	0.995723i	$0.470550\pi$
$602$	0	0
$603$	−3.61675	−0.147286
$604$	0	0
$605$	4.00261	0.162729
$606$	0	0
$607$	−17.1292	−0.695252	−0.347626	−	0.937633i	$-0.613012\pi$
$607$	−17.1292	−0.695252	−0.347626	+	0.937633i	$0.613012\pi$
$608$	0	0
$609$	1.53307	0.0621231
$610$	0	0
$611$	10.5206	0.425619
$612$	0	0
$613$	11.2847	0.455787	0.227893	−	0.973686i	$-0.426816\pi$
$613$	11.2847	0.455787	0.227893	+	0.973686i	$0.426816\pi$
$614$	0	0
$615$	−6.90084	−0.278269
$616$	0	0
$617$	3.02369	0.121729	0.0608646	−	0.998146i	$-0.480614\pi$
$617$	3.02369	0.121729	0.0608646	+	0.998146i	$0.480614\pi$
$618$	0	0
$619$	−39.8050	−1.59990	−0.799950	−	0.600067i	$-0.795141\pi$
$619$	−39.8050	−1.59990	−0.799950	+	0.600067i	$0.795141\pi$
$620$	0	0
$621$	−16.2400	−0.651689
$622$	0	0
$623$	29.6034	1.18603
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	34.9233	1.39470
$628$	0	0
$629$	5.98952	0.238818
$630$	0	0
$631$	46.0056	1.83145	0.915727	−	0.401801i	$-0.131616\pi$
$631$	46.0056	1.83145	0.915727	+	0.401801i	$0.131616\pi$
$632$	0	0
$633$	23.5759	0.937059
$634$	0	0
$635$	−0.850245	−0.0337409
$636$	0	0
$637$	−7.12646	−0.282361
$638$	0	0
$639$	−1.15498	−0.0456901
$640$	0	0
$641$	−6.79772	−0.268494	−0.134247	−	0.990948i	$-0.542862\pi$
$641$	−6.79772	−0.268494	−0.134247	+	0.990948i	$0.542862\pi$
$642$	0	0
$643$	−22.1346	−0.872904	−0.436452	−	0.899727i	$-0.643765\pi$
$643$	−22.1346	−0.872904	−0.436452	+	0.899727i	$0.643765\pi$
$644$	0	0
$645$	5.35951	0.211031
$646$	0	0
$647$	29.8985	1.17543	0.587715	−	0.809068i	$-0.300027\pi$
$647$	29.8985	1.17543	0.587715	+	0.809068i	$0.300027\pi$
$648$	0	0
$649$	−40.8619	−1.60397
$650$	0	0
$651$	22.0170	0.862913
$652$	0	0
$653$	−21.0918	−0.825385	−0.412693	−	0.910870i	$-0.635412\pi$
$653$	−21.0918	−0.825385	−0.412693	+	0.910870i	$0.635412\pi$
$654$	0	0
$655$	−9.00667	−0.351920
$656$	0	0
$657$	−1.03581	−0.0404108
$658$	0	0
$659$	−35.8445	−1.39630	−0.698152	−	0.715949i	$-0.745994\pi$
$659$	−35.8445	−1.39630	−0.698152	+	0.715949i	$0.745994\pi$
$660$	0	0
$661$	6.19738	0.241050	0.120525	−	0.992710i	$-0.461542\pi$
$661$	6.19738	0.241050	0.120525	+	0.992710i	$0.461542\pi$
$662$	0	0
$663$	16.1124	0.625754
$664$	0	0
$665$	−10.7885	−0.418361
$666$	0	0
$667$	−1.47685	−0.0571840
$668$	0	0
$669$	15.3474	0.593365
$670$	0	0
$671$	27.9851	1.08035
$672$	0	0
$673$	−21.5452	−0.830506	−0.415253	−	0.909706i	$-0.636307\pi$
$673$	−21.5452	−0.830506	−0.415253	+	0.909706i	$0.636307\pi$
$674$	0	0
$675$	5.50416	0.211855
$676$	0	0
$677$	38.3387	1.47348	0.736739	−	0.676177i	$-0.236365\pi$
$677$	38.3387	1.47348	0.736739	+	0.676177i	$0.236365\pi$
$678$	0	0
$679$	−11.7384	−0.450479
$680$	0	0
$681$	46.7077	1.78984
$682$	0	0
$683$	49.3083	1.88673	0.943365	−	0.331757i	$-0.107641\pi$
$683$	49.3083	1.88673	0.943365	+	0.331757i	$0.107641\pi$
$684$	0	0
$685$	−14.4822	−0.553337
$686$	0	0
$687$	−8.23573	−0.314213
$688$	0	0
$689$	4.89354	0.186429
$690$	0	0
$691$	−9.43604	−0.358964	−0.179482	−	0.983761i	$-0.557442\pi$
$691$	−9.43604	−0.358964	−0.179482	+	0.983761i	$0.557442\pi$
$692$	0	0
$693$	−3.26478	−0.124019
$694$	0	0
$695$	−2.69052	−0.102057
$696$	0	0
$697$	−20.3292	−0.770024
$698$	0	0
$699$	−41.9732	−1.58757
$700$	0	0
$701$	50.5938	1.91090	0.955451	−	0.295150i	$-0.0953695\pi$
$701$	50.5938	1.91090	0.955451	+	0.295150i	$0.0953695\pi$
$702$	0	0
$703$	−7.16168	−0.270108
$704$	0	0
$705$	−7.87744	−0.296682
$706$	0	0
$707$	9.00091	0.338514
$708$	0	0
$709$	19.3822	0.727912	0.363956	−	0.931416i	$-0.381426\pi$
$709$	19.3822	0.727912	0.363956	+	0.931416i	$0.381426\pi$
$710$	0	0
$711$	−6.90211	−0.258849
$712$	0	0
$713$	−21.2097	−0.794308
$714$	0	0
$715$	−8.27628	−0.309515
$716$	0	0
$717$	−10.1396	−0.378672
$718$	0	0
$719$	0.0462493	0.00172481	0.000862404	−	1.00000i	$-0.499725\pi$
$719$	0.0462493	0.00172481	0.000862404		1.00000i	$0.499725\pi$
$720$	0	0
$721$	−17.5548	−0.653775
$722$	0	0
$723$	−26.6579	−0.991418
$724$	0	0
$725$	0.500544	0.0185897
$726$	0	0
$727$	−28.5914	−1.06040	−0.530198	−	0.847874i	$-0.677882\pi$
$727$	−28.5914	−1.06040	−0.530198	+	0.847874i	$0.677882\pi$
$728$	0	0
$729$	29.7141	1.10052
$730$	0	0
$731$	15.7886	0.583963
$732$	0	0
$733$	−6.72962	−0.248564	−0.124282	−	0.992247i	$-0.539663\pi$
$733$	−6.72962	−0.248564	−0.124282	+	0.992247i	$0.539663\pi$
$734$	0	0
$735$	5.33601	0.196822
$736$	0	0
$737$	−31.8168	−1.17199
$738$	0	0
$739$	−26.7192	−0.982881	−0.491440	−	0.870911i	$-0.663529\pi$
$739$	−26.7192	−0.982881	−0.491440	+	0.870911i	$0.663529\pi$
$740$	0	0
$741$	−19.2656	−0.707741
$742$	0	0
$743$	−9.46945	−0.347400	−0.173700	−	0.984799i	$-0.555572\pi$
$743$	−9.46945	−0.347400	−0.173700	+	0.984799i	$0.555572\pi$
$744$	0	0
$745$	−20.2802	−0.743009
$746$	0	0
$747$	5.33630	0.195245
$748$	0	0
$749$	12.3627	0.451723
$750$	0	0
$751$	30.8197	1.12463	0.562314	−	0.826924i	$-0.309911\pi$
$751$	30.8197	1.12463	0.562314	+	0.826924i	$0.309911\pi$
$752$	0	0
$753$	−12.0562	−0.439351
$754$	0	0
$755$	1.39284	0.0506908
$756$	0	0
$757$	−18.8132	−0.683779	−0.341890	−	0.939740i	$-0.611067\pi$
$757$	−18.8132	−0.683779	−0.341890	+	0.939740i	$0.611067\pi$
$758$	0	0
$759$	−18.2841	−0.663672
$760$	0	0
$761$	6.59754	0.239161	0.119580	−	0.992825i	$-0.461845\pi$
$761$	6.59754	0.239161	0.119580	+	0.992825i	$0.461845\pi$
$762$	0	0
$763$	−6.71333	−0.243039
$764$	0	0
$765$	2.07520	0.0750288
$766$	0	0
$767$	22.5417	0.813933
$768$	0	0
$769$	−14.1560	−0.510477	−0.255239	−	0.966878i	$-0.582154\pi$
$769$	−14.1560	−0.510477	−0.255239	+	0.966878i	$0.582154\pi$
$770$	0	0
$771$	−20.3815	−0.734021
$772$	0	0
$773$	−3.54033	−0.127337	−0.0636685	−	0.997971i	$-0.520280\pi$
$773$	−3.54033	−0.127337	−0.0636685	+	0.997971i	$0.520280\pi$
$774$	0	0
$775$	7.18849	0.258218
$776$	0	0
$777$	−3.89223	−0.139633
$778$	0	0
$779$	24.3077	0.870913
$780$	0	0
$781$	−10.1604	−0.363568
$782$	0	0
$783$	2.75507	0.0984582
$784$	0	0
$785$	−12.6302	−0.450791
$786$	0	0
$787$	−47.3299	−1.68713	−0.843565	−	0.537027i	$-0.819547\pi$
$787$	−47.3299	−1.68713	−0.843565	+	0.537027i	$0.819547\pi$
$788$	0	0
$789$	−15.8672	−0.564888
$790$	0	0
$791$	7.95143	0.282720
$792$	0	0
$793$	−15.4381	−0.548224
$794$	0	0
$795$	−3.66409	−0.129952
$796$	0	0
$797$	−24.7688	−0.877355	−0.438677	−	0.898645i	$-0.644553\pi$
$797$	−24.7688	−0.877355	−0.438677	+	0.898645i	$0.644553\pi$
$798$	0	0
$799$	−23.2062	−0.820977
$800$	0	0
$801$	6.80866	0.240572
$802$	0	0
$803$	−9.11211	−0.321559
$804$	0	0
$805$	5.64834	0.199078
$806$	0	0
$807$	−10.0427	−0.353521
$808$	0	0
$809$	33.8143	1.18885	0.594424	−	0.804152i	$-0.297380\pi$
$809$	33.8143	1.18885	0.594424	+	0.804152i	$0.297380\pi$
$810$	0	0
$811$	40.2140	1.41210	0.706051	−	0.708161i	$-0.250475\pi$
$811$	40.2140	1.41210	0.706051	+	0.708161i	$0.250475\pi$
$812$	0	0
$813$	−41.0691	−1.44036
$814$	0	0
$815$	−7.02879	−0.246208
$816$	0	0
$817$	−18.8785	−0.660474
$818$	0	0
$819$	1.80103	0.0629332
$820$	0	0
$821$	10.8471	0.378568	0.189284	−	0.981922i	$-0.439383\pi$
$821$	10.8471	0.378568	0.189284	+	0.981922i	$0.439383\pi$
$822$	0	0
$823$	−43.6015	−1.51985	−0.759927	−	0.650009i	$-0.774765\pi$
$823$	−43.6015	−1.51985	−0.759927	+	0.650009i	$0.774765\pi$
$824$	0	0
$825$	6.19695	0.215750
$826$	0	0
$827$	−19.5133	−0.678545	−0.339273	−	0.940688i	$-0.610181\pi$
$827$	−19.5133	−0.678545	−0.339273	+	0.940688i	$0.610181\pi$
$828$	0	0
$829$	56.6154	1.96634	0.983168	−	0.182704i	$-0.0584851\pi$
$829$	56.6154	1.96634	0.983168	+	0.182704i	$0.0584851\pi$
$830$	0	0
$831$	−10.4935	−0.364017
$832$	0	0
$833$	15.7194	0.544645
$834$	0	0
$835$	−18.3773	−0.635973
$836$	0	0
$837$	39.5666	1.36762
$838$	0	0
$839$	−17.9781	−0.620672	−0.310336	−	0.950627i	$-0.600442\pi$
$839$	−17.9781	−0.620672	−0.310336	+	0.950627i	$0.600442\pi$
$840$	0	0
$841$	−28.7495	−0.991361
$842$	0	0
$843$	1.03975	0.0358107
$844$	0	0
$845$	−8.43434	−0.290150
$846$	0	0
$847$	−7.66247	−0.263286
$848$	0	0
$849$	−13.4159	−0.460433
$850$	0	0
$851$	3.74951	0.128531
$852$	0	0
$853$	36.9642	1.26563	0.632814	−	0.774304i	$-0.281900\pi$
$853$	36.9642	1.26563	0.632814	+	0.774304i	$0.281900\pi$
$854$	0	0
$855$	−2.48131	−0.0848591
$856$	0	0
$857$	35.2422	1.20385	0.601926	−	0.798552i	$-0.294400\pi$
$857$	35.2422	1.20385	0.601926	+	0.798552i	$0.294400\pi$
$858$	0	0
$859$	14.7194	0.502221	0.251110	−	0.967958i	$-0.419204\pi$
$859$	14.7194	0.502221	0.251110	+	0.967958i	$0.419204\pi$
$860$	0	0
$861$	13.2107	0.450221
$862$	0	0
$863$	−7.88349	−0.268357	−0.134179	−	0.990957i	$-0.542840\pi$
$863$	−7.88349	−0.268357	−0.134179	+	0.990957i	$0.542840\pi$
$864$	0	0
$865$	−7.68465	−0.261286
$866$	0	0
$867$	−8.34199	−0.283309
$868$	0	0
$869$	−60.7184	−2.05973
$870$	0	0
$871$	17.5520	0.594725
$872$	0	0
$873$	−2.69979	−0.0913740
$874$	0	0
$875$	−1.91437	−0.0647174
$876$	0	0
$877$	12.2639	0.414123	0.207061	−	0.978328i	$-0.433610\pi$
$877$	12.2639	0.414123	0.207061	+	0.978328i	$0.433610\pi$
$878$	0	0
$879$	20.4258	0.688944
$880$	0	0
$881$	−10.0539	−0.338726	−0.169363	−	0.985554i	$-0.554171\pi$
$881$	−10.0539	−0.338726	−0.169363	+	0.985554i	$0.554171\pi$
$882$	0	0
$883$	−36.1676	−1.21714	−0.608569	−	0.793501i	$-0.708256\pi$
$883$	−36.1676	−1.21714	−0.608569	+	0.793501i	$0.708256\pi$
$884$	0	0
$885$	−16.8783	−0.567359
$886$	0	0
$887$	−48.3684	−1.62405	−0.812025	−	0.583622i	$-0.801635\pi$
$887$	−48.3684	−1.62405	−0.812025	+	0.583622i	$0.801635\pi$
$888$	0	0
$889$	1.62768	0.0545906
$890$	0	0
$891$	28.9928	0.971294
$892$	0	0
$893$	27.7477	0.928541
$894$	0	0
$895$	25.2885	0.845302
$896$	0	0
$897$	10.0865	0.336780
$898$	0	0
$899$	3.59815	0.120005
$900$	0	0
$901$	−10.7941	−0.359603
$902$	0	0
$903$	−10.2601	−0.341434
$904$	0	0
$905$	5.16828	0.171799
$906$	0	0
$907$	−47.6824	−1.58327	−0.791635	−	0.610995i	$-0.790770\pi$
$907$	−47.6824	−1.58327	−0.791635	+	0.610995i	$0.790770\pi$
$908$	0	0
$909$	2.07017	0.0686633
$910$	0	0
$911$	51.4808	1.70563	0.852817	−	0.522209i	$-0.174892\pi$
$911$	51.4808	1.70563	0.852817	+	0.522209i	$0.174892\pi$
$912$	0	0
$913$	46.9438	1.55361
$914$	0	0
$915$	11.5595	0.382144
$916$	0	0
$917$	17.2421	0.569384
$918$	0	0
$919$	−30.2678	−0.998444	−0.499222	−	0.866474i	$-0.666381\pi$
$919$	−30.2678	−0.998444	−0.499222	+	0.866474i	$0.666381\pi$
$920$	0	0
$921$	−1.95948	−0.0645671
$922$	0	0
$923$	5.60505	0.184492
$924$	0	0
$925$	−1.27080	−0.0417838
$926$	0	0
$927$	−4.03753	−0.132610
$928$	0	0
$929$	−10.8773	−0.356871	−0.178436	−	0.983952i	$-0.557104\pi$
$929$	−10.8773	−0.356871	−0.178436	+	0.983952i	$0.557104\pi$
$930$	0	0
$931$	−18.7957	−0.616004
$932$	0	0
$933$	47.8837	1.56764
$934$	0	0
$935$	18.2556	0.597024
$936$	0	0
$937$	−12.5990	−0.411590	−0.205795	−	0.978595i	$-0.565978\pi$
$937$	−12.5990	−0.411590	−0.205795	+	0.978595i	$0.565978\pi$
$938$	0	0
$939$	25.9303	0.846205
$940$	0	0
$941$	6.64142	0.216504	0.108252	−	0.994123i	$-0.465475\pi$
$941$	6.64142	0.216504	0.108252	+	0.994123i	$0.465475\pi$
$942$	0	0
$943$	−12.7263	−0.414426
$944$	0	0
$945$	−10.5370	−0.342768
$946$	0	0
$947$	−21.3470	−0.693686	−0.346843	−	0.937923i	$-0.612746\pi$
$947$	−21.3470	−0.693686	−0.346843	+	0.937923i	$0.612746\pi$
$948$	0	0
$949$	5.02675	0.163175
$950$	0	0
$951$	−22.8502	−0.740968
$952$	0	0
$953$	−31.0729	−1.00655	−0.503276	−	0.864126i	$-0.667872\pi$
$953$	−31.0729	−1.00655	−0.503276	+	0.864126i	$0.667872\pi$
$954$	0	0
$955$	3.09060	0.100010
$956$	0	0
$957$	3.10185	0.100268
$958$	0	0
$959$	27.7243	0.895264
$960$	0	0
$961$	20.6744	0.666917
$962$	0	0
$963$	2.84337	0.0916263
$964$	0	0
$965$	6.70521	0.215848
$966$	0	0
$967$	24.6906	0.793996	0.396998	−	0.917819i	$-0.370052\pi$
$967$	24.6906	0.793996	0.396998	+	0.917819i	$0.370052\pi$
$968$	0	0
$969$	42.4957	1.36516
$970$	0	0
$971$	8.85188	0.284070	0.142035	−	0.989862i	$-0.454635\pi$
$971$	8.85188	0.284070	0.142035	+	0.989862i	$0.454635\pi$
$972$	0	0
$973$	5.15065	0.165122
$974$	0	0
$975$	−3.41859	−0.109482
$976$	0	0
$977$	−37.9982	−1.21567	−0.607835	−	0.794064i	$-0.707962\pi$
$977$	−37.9982	−1.21567	−0.607835	+	0.794064i	$0.707962\pi$
$978$	0	0
$979$	59.8963	1.91429
$980$	0	0
$981$	−1.54404	−0.0492973
$982$	0	0
$983$	20.1166	0.641621	0.320811	−	0.947143i	$-0.396045\pi$
$983$	20.1166	0.641621	0.320811	+	0.947143i	$0.396045\pi$
$984$	0	0
$985$	−2.08180	−0.0663317
$986$	0	0
$987$	15.0803	0.480012
$988$	0	0
$989$	9.88385	0.314288
$990$	0	0
$991$	14.2224	0.451789	0.225894	−	0.974152i	$-0.427470\pi$
$991$	14.2224	0.451789	0.225894	+	0.974152i	$0.427470\pi$
$992$	0	0
$993$	13.0171	0.413084
$994$	0	0
$995$	1.35553	0.0429733
$996$	0	0
$997$	−41.8093	−1.32411	−0.662057	−	0.749454i	$-0.730316\pi$
$997$	−41.8093	−1.32411	−0.662057	+	0.749454i	$0.730316\pi$
$998$	0	0
$999$	−6.99470	−0.221303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.8	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.8	✓	29	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)

\(f(q)\)	\(=\)	\(q-1.59991 q^{3} +1.00000 q^{5} -1.91437 q^{7} -0.440296 q^{9} +O(q^{10})\)	\(q-1.59991 q^{3} +1.00000 q^{5} -1.91437 q^{7} -0.440296 q^{9} -3.87332 q^{11} +2.13674 q^{13} -1.59991 q^{15} -4.71318 q^{17} +5.63555 q^{19} +3.06281 q^{21} -2.95050 q^{23} +1.00000 q^{25} +5.50416 q^{27} +0.500544 q^{29} +7.18849 q^{31} +6.19695 q^{33} -1.91437 q^{35} -1.27080 q^{37} -3.41859 q^{39} +4.31327 q^{41} -3.34989 q^{43} -0.440296 q^{45} +4.92369 q^{47} -3.33520 q^{49} +7.54065 q^{51} +2.29019 q^{53} -3.87332 q^{55} -9.01636 q^{57} +10.5496 q^{59} -7.22509 q^{61} +0.842889 q^{63} +2.13674 q^{65} +8.21436 q^{67} +4.72053 q^{69} +2.62318 q^{71} +2.35253 q^{73} -1.59991 q^{75} +7.41496 q^{77} +15.6761 q^{79} -7.48525 q^{81} -12.1198 q^{83} -4.71318 q^{85} -0.800823 q^{87} -15.4638 q^{89} -4.09050 q^{91} -11.5009 q^{93} +5.63555 q^{95} +6.13175 q^{97} +1.70541 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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