LMFDB - Embedded newform 8020.2.a.f.1.16

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:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-0.435929 q^{3} +1.00000 q^{5} -3.81252 q^{7} -2.80997 q^{9} +O(q^{10})$	$q-0.435929 q^{3} +1.00000 q^{5} -3.81252 q^{7} -2.80997 q^{9} -0.698560 q^{11} -4.30901 q^{13} -0.435929 q^{15} -5.96469 q^{17} +2.02234 q^{19} +1.66199 q^{21} +7.60247 q^{23} +1.00000 q^{25} +2.53273 q^{27} -7.24881 q^{29} -7.12551 q^{31} +0.304523 q^{33} -3.81252 q^{35} -2.65479 q^{37} +1.87842 q^{39} +4.20695 q^{41} +0.0813591 q^{43} -2.80997 q^{45} +6.83468 q^{47} +7.53531 q^{49} +2.60018 q^{51} -8.79165 q^{53} -0.698560 q^{55} -0.881599 q^{57} -9.47024 q^{59} -8.26104 q^{61} +10.7131 q^{63} -4.30901 q^{65} -12.6150 q^{67} -3.31414 q^{69} -7.01822 q^{71} +13.4205 q^{73} -0.435929 q^{75} +2.66327 q^{77} -4.47048 q^{79} +7.32581 q^{81} -13.9679 q^{83} -5.96469 q^{85} +3.15997 q^{87} -4.12463 q^{89} +16.4282 q^{91} +3.10622 q^{93} +2.02234 q^{95} +4.61677 q^{97} +1.96293 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * 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
$3$	−0.435929	−0.251684	−0.125842	−	0.992050i	$-0.540163\pi$
$3$	−0.435929	−0.251684	−0.125842	+	0.992050i	$0.540163\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.81252	−1.44100	−0.720499	−	0.693456i	$-0.756087\pi$
$7$	−3.81252	−1.44100	−0.720499	+	0.693456i	$0.756087\pi$
$8$	0	0
$9$	−2.80997	−0.936655
$10$	0	0
$11$	−0.698560	−0.210624	−0.105312	−	0.994439i	$-0.533584\pi$
$11$	−0.698560	−0.210624	−0.105312	+	0.994439i	$0.533584\pi$
$12$	0	0
$13$	−4.30901	−1.19511	−0.597553	−	0.801830i	$-0.703860\pi$
$13$	−4.30901	−1.19511	−0.597553	+	0.801830i	$0.703860\pi$
$14$	0	0
$15$	−0.435929	−0.112556
$16$	0	0
$17$	−5.96469	−1.44665	−0.723325	−	0.690507i	$-0.757387\pi$
$17$	−5.96469	−1.44665	−0.723325	+	0.690507i	$0.757387\pi$
$18$	0	0
$19$	2.02234	0.463958	0.231979	−	0.972721i	$-0.425480\pi$
$19$	2.02234	0.463958	0.231979	+	0.972721i	$0.425480\pi$
$20$	0	0
$21$	1.66199	0.362676
$22$	0	0
$23$	7.60247	1.58522	0.792612	−	0.609726i	$-0.208720\pi$
$23$	7.60247	1.58522	0.792612	+	0.609726i	$0.208720\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.53273	0.487425
$28$	0	0
$29$	−7.24881	−1.34607	−0.673035	−	0.739610i	$-0.735010\pi$
$29$	−7.24881	−1.34607	−0.673035	+	0.739610i	$0.735010\pi$
$30$	0	0
$31$	−7.12551	−1.27978	−0.639889	−	0.768467i	$-0.721020\pi$
$31$	−7.12551	−1.27978	−0.639889	+	0.768467i	$0.721020\pi$
$32$	0	0
$33$	0.304523	0.0530106
$34$	0	0
$35$	−3.81252	−0.644434
$36$	0	0
$37$	−2.65479	−0.436444	−0.218222	−	0.975899i	$-0.570026\pi$
$37$	−2.65479	−0.436444	−0.218222	+	0.975899i	$0.570026\pi$
$38$	0	0
$39$	1.87842	0.300789
$40$	0	0
$41$	4.20695	0.657015	0.328507	−	0.944501i	$-0.393454\pi$
$41$	4.20695	0.657015	0.328507	+	0.944501i	$0.393454\pi$
$42$	0	0
$43$	0.0813591	0.0124071	0.00620357	−	0.999981i	$-0.498025\pi$
$43$	0.0813591	0.0124071	0.00620357	+	0.999981i	$0.498025\pi$
$44$	0	0
$45$	−2.80997	−0.418885
$46$	0	0
$47$	6.83468	0.996940	0.498470	−	0.866907i	$-0.333895\pi$
$47$	6.83468	0.996940	0.498470	+	0.866907i	$0.333895\pi$
$48$	0	0
$49$	7.53531	1.07647
$50$	0	0
$51$	2.60018	0.364099
$52$	0	0
$53$	−8.79165	−1.20763	−0.603813	−	0.797126i	$-0.706353\pi$
$53$	−8.79165	−1.20763	−0.603813	+	0.797126i	$0.706353\pi$
$54$	0	0
$55$	−0.698560	−0.0941938
$56$	0	0
$57$	−0.881599	−0.116771
$58$	0	0
$59$	−9.47024	−1.23292	−0.616460	−	0.787386i	$-0.711434\pi$
$59$	−9.47024	−1.23292	−0.616460	+	0.787386i	$0.711434\pi$
$60$	0	0
$61$	−8.26104	−1.05772	−0.528859	−	0.848710i	$-0.677380\pi$
$61$	−8.26104	−1.05772	−0.528859	+	0.848710i	$0.677380\pi$
$62$	0	0
$63$	10.7131	1.34972
$64$	0	0
$65$	−4.30901	−0.534468
$66$	0	0
$67$	−12.6150	−1.54117	−0.770586	−	0.637336i	$-0.780036\pi$
$67$	−12.6150	−1.54117	−0.770586	+	0.637336i	$0.780036\pi$
$68$	0	0
$69$	−3.31414	−0.398975
$70$	0	0
$71$	−7.01822	−0.832909	−0.416455	−	0.909157i	$-0.636728\pi$
$71$	−7.01822	−0.832909	−0.416455	+	0.909157i	$0.636728\pi$
$72$	0	0
$73$	13.4205	1.57075	0.785375	−	0.619020i	$-0.212470\pi$
$73$	13.4205	1.57075	0.785375	+	0.619020i	$0.212470\pi$
$74$	0	0
$75$	−0.435929	−0.0503368
$76$	0	0
$77$	2.66327	0.303508
$78$	0	0
$79$	−4.47048	−0.502968	−0.251484	−	0.967861i	$-0.580919\pi$
$79$	−4.47048	−0.502968	−0.251484	+	0.967861i	$0.580919\pi$
$80$	0	0
$81$	7.32581	0.813978
$82$	0	0
$83$	−13.9679	−1.53318	−0.766588	−	0.642139i	$-0.778047\pi$
$83$	−13.9679	−1.53318	−0.766588	+	0.642139i	$0.778047\pi$
$84$	0	0
$85$	−5.96469	−0.646962
$86$	0	0
$87$	3.15997	0.338784
$88$	0	0
$89$	−4.12463	−0.437209	−0.218605	−	0.975813i	$-0.570151\pi$
$89$	−4.12463	−0.437209	−0.218605	+	0.975813i	$0.570151\pi$
$90$	0	0
$91$	16.4282	1.72214
$92$	0	0
$93$	3.10622	0.322099
$94$	0	0
$95$	2.02234	0.207488
$96$	0	0
$97$	4.61677	0.468762	0.234381	−	0.972145i	$-0.424694\pi$
$97$	4.61677	0.468762	0.234381	+	0.972145i	$0.424694\pi$
$98$	0	0
$99$	1.96293	0.197282
$100$	0	0
$101$	15.8905	1.58117	0.790583	−	0.612355i	$-0.209778\pi$
$101$	15.8905	1.58117	0.790583	+	0.612355i	$0.209778\pi$
$102$	0	0
$103$	11.1641	1.10003	0.550017	−	0.835153i	$-0.314621\pi$
$103$	11.1641	1.10003	0.550017	+	0.835153i	$0.314621\pi$
$104$	0	0
$105$	1.66199	0.162193
$106$	0	0
$107$	2.21561	0.214191	0.107096	−	0.994249i	$-0.465845\pi$
$107$	2.21561	0.214191	0.107096	+	0.994249i	$0.465845\pi$
$108$	0	0
$109$	−14.2685	−1.36667	−0.683337	−	0.730103i	$-0.739472\pi$
$109$	−14.2685	−1.36667	−0.683337	+	0.730103i	$0.739472\pi$
$110$	0	0
$111$	1.15730	0.109846
$112$	0	0
$113$	7.97026	0.749779	0.374889	−	0.927070i	$-0.377681\pi$
$113$	7.97026	0.749779	0.374889	+	0.927070i	$0.377681\pi$
$114$	0	0
$115$	7.60247	0.708934
$116$	0	0
$117$	12.1082	1.11940
$118$	0	0
$119$	22.7405	2.08462
$120$	0	0
$121$	−10.5120	−0.955638
$122$	0	0
$123$	−1.83393	−0.165360
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.28167	0.291201	0.145601	−	0.989343i	$-0.453489\pi$
$127$	3.28167	0.291201	0.145601	+	0.989343i	$0.453489\pi$
$128$	0	0
$129$	−0.0354668	−0.00312268
$130$	0	0
$131$	9.93413	0.867949	0.433975	−	0.900925i	$-0.357111\pi$
$131$	9.93413	0.867949	0.433975	+	0.900925i	$0.357111\pi$
$132$	0	0
$133$	−7.71023	−0.668562
$134$	0	0
$135$	2.53273	0.217983
$136$	0	0
$137$	−11.2068	−0.957462	−0.478731	−	0.877962i	$-0.658903\pi$
$137$	−11.2068	−0.957462	−0.478731	+	0.877962i	$0.658903\pi$
$138$	0	0
$139$	5.87741	0.498515	0.249257	−	0.968437i	$-0.419813\pi$
$139$	5.87741	0.498515	0.249257	+	0.968437i	$0.419813\pi$
$140$	0	0
$141$	−2.97943	−0.250914
$142$	0	0
$143$	3.01011	0.251718
$144$	0	0
$145$	−7.24881	−0.601981
$146$	0	0
$147$	−3.28486	−0.270931
$148$	0	0
$149$	9.64159	0.789870	0.394935	−	0.918709i	$-0.370767\pi$
$149$	9.64159	0.789870	0.394935	+	0.918709i	$0.370767\pi$
$150$	0	0
$151$	13.1810	1.07266	0.536329	−	0.844009i	$-0.319811\pi$
$151$	13.1810	1.07266	0.536329	+	0.844009i	$0.319811\pi$
$152$	0	0
$153$	16.7606	1.35501
$154$	0	0
$155$	−7.12551	−0.572334
$156$	0	0
$157$	16.6028	1.32505	0.662523	−	0.749041i	$-0.269486\pi$
$157$	16.6028	1.32505	0.662523	+	0.749041i	$0.269486\pi$
$158$	0	0
$159$	3.83253	0.303940
$160$	0	0
$161$	−28.9846	−2.28430
$162$	0	0
$163$	−2.74380	−0.214911	−0.107456	−	0.994210i	$-0.534270\pi$
$163$	−2.74380	−0.214911	−0.107456	+	0.994210i	$0.534270\pi$
$164$	0	0
$165$	0.304523	0.0237071
$166$	0	0
$167$	3.26767	0.252860	0.126430	−	0.991976i	$-0.459648\pi$
$167$	3.26767	0.252860	0.126430	+	0.991976i	$0.459648\pi$
$168$	0	0
$169$	5.56761	0.428278
$170$	0	0
$171$	−5.68272	−0.434568
$172$	0	0
$173$	18.9143	1.43803	0.719014	−	0.694996i	$-0.244594\pi$
$173$	18.9143	1.43803	0.719014	+	0.694996i	$0.244594\pi$
$174$	0	0
$175$	−3.81252	−0.288199
$176$	0	0
$177$	4.12836	0.310306
$178$	0	0
$179$	−23.0509	−1.72290	−0.861452	−	0.507840i	$-0.830444\pi$
$179$	−23.0509	−1.72290	−0.861452	+	0.507840i	$0.830444\pi$
$180$	0	0
$181$	15.6036	1.15981	0.579904	−	0.814685i	$-0.303090\pi$
$181$	15.6036	1.15981	0.579904	+	0.814685i	$0.303090\pi$
$182$	0	0
$183$	3.60123	0.266211
$184$	0	0
$185$	−2.65479	−0.195184
$186$	0	0
$187$	4.16670	0.304699
$188$	0	0
$189$	−9.65610	−0.702378
$190$	0	0
$191$	−8.06410	−0.583498	−0.291749	−	0.956495i	$-0.594237\pi$
$191$	−8.06410	−0.583498	−0.291749	+	0.956495i	$0.594237\pi$
$192$	0	0
$193$	22.9069	1.64888	0.824438	−	0.565952i	$-0.191491\pi$
$193$	22.9069	1.64888	0.824438	+	0.565952i	$0.191491\pi$
$194$	0	0
$195$	1.87842	0.134517
$196$	0	0
$197$	20.1000	1.43207	0.716033	−	0.698067i	$-0.245956\pi$
$197$	20.1000	1.43207	0.716033	+	0.698067i	$0.245956\pi$
$198$	0	0
$199$	−1.93794	−0.137377	−0.0686885	−	0.997638i	$-0.521881\pi$
$199$	−1.93794	−0.137377	−0.0686885	+	0.997638i	$0.521881\pi$
$200$	0	0
$201$	5.49926	0.387888
$202$	0	0
$203$	27.6362	1.93968
$204$	0	0
$205$	4.20695	0.293826
$206$	0	0
$207$	−21.3627	−1.48481
$208$	0	0
$209$	−1.41273	−0.0977205
$210$	0	0
$211$	−13.0517	−0.898515	−0.449258	−	0.893402i	$-0.648311\pi$
$211$	−13.0517	−0.898515	−0.449258	+	0.893402i	$0.648311\pi$
$212$	0	0
$213$	3.05944	0.209630
$214$	0	0
$215$	0.0813591	0.00554864
$216$	0	0
$217$	27.1661	1.84416
$218$	0	0
$219$	−5.85039	−0.395333
$220$	0	0
$221$	25.7020	1.72890
$222$	0	0
$223$	29.7571	1.99269	0.996343	−	0.0854479i	$-0.0272321\pi$
$223$	29.7571	1.99269	0.996343	+	0.0854479i	$0.0272321\pi$
$224$	0	0
$225$	−2.80997	−0.187331
$226$	0	0
$227$	−18.7203	−1.24251	−0.621255	−	0.783609i	$-0.713377\pi$
$227$	−18.7203	−1.24251	−0.621255	+	0.783609i	$0.713377\pi$
$228$	0	0
$229$	6.74785	0.445911	0.222955	−	0.974829i	$-0.428430\pi$
$229$	6.74785	0.445911	0.222955	+	0.974829i	$0.428430\pi$
$230$	0	0
$231$	−1.16100	−0.0763881
$232$	0	0
$233$	15.2213	0.997182	0.498591	−	0.866837i	$-0.333851\pi$
$233$	15.2213	0.997182	0.498591	+	0.866837i	$0.333851\pi$
$234$	0	0
$235$	6.83468	0.445845
$236$	0	0
$237$	1.94881	0.126589
$238$	0	0
$239$	26.6226	1.72207	0.861036	−	0.508544i	$-0.169816\pi$
$239$	26.6226	1.72207	0.861036	+	0.508544i	$0.169816\pi$
$240$	0	0
$241$	−13.0046	−0.837699	−0.418850	−	0.908056i	$-0.637567\pi$
$241$	−13.0046	−0.837699	−0.418850	+	0.908056i	$0.637567\pi$
$242$	0	0
$243$	−10.7917	−0.692290
$244$	0	0
$245$	7.53531	0.481413
$246$	0	0
$247$	−8.71431	−0.554478
$248$	0	0
$249$	6.08902	0.385876
$250$	0	0
$251$	−11.7865	−0.743958	−0.371979	−	0.928241i	$-0.621321\pi$
$251$	−11.7865	−0.743958	−0.371979	+	0.928241i	$0.621321\pi$
$252$	0	0
$253$	−5.31078	−0.333886
$254$	0	0
$255$	2.60018	0.162830
$256$	0	0
$257$	25.3079	1.57867	0.789333	−	0.613965i	$-0.210427\pi$
$257$	25.3079	1.57867	0.789333	+	0.613965i	$0.210427\pi$
$258$	0	0
$259$	10.1214	0.628915
$260$	0	0
$261$	20.3689	1.26080
$262$	0	0
$263$	−22.6067	−1.39399	−0.696995	−	0.717076i	$-0.745480\pi$
$263$	−22.6067	−1.39399	−0.696995	+	0.717076i	$0.745480\pi$
$264$	0	0
$265$	−8.79165	−0.540067
$266$	0	0
$267$	1.79804	0.110039
$268$	0	0
$269$	−11.1621	−0.680565	−0.340283	−	0.940323i	$-0.610523\pi$
$269$	−11.1621	−0.680565	−0.340283	+	0.940323i	$0.610523\pi$
$270$	0	0
$271$	−11.2744	−0.684873	−0.342437	−	0.939541i	$-0.611252\pi$
$271$	−11.2744	−0.684873	−0.342437	+	0.939541i	$0.611252\pi$
$272$	0	0
$273$	−7.16153	−0.433436
$274$	0	0
$275$	−0.698560	−0.0421248
$276$	0	0
$277$	8.16707	0.490712	0.245356	−	0.969433i	$-0.421095\pi$
$277$	8.16707	0.490712	0.245356	+	0.969433i	$0.421095\pi$
$278$	0	0
$279$	20.0224	1.19871
$280$	0	0
$281$	19.4147	1.15819	0.579093	−	0.815261i	$-0.303407\pi$
$281$	19.4147	1.15819	0.579093	+	0.815261i	$0.303407\pi$
$282$	0	0
$283$	32.5173	1.93295	0.966475	−	0.256761i	$-0.0826552\pi$
$283$	32.5173	1.93295	0.966475	+	0.256761i	$0.0826552\pi$
$284$	0	0
$285$	−0.881599	−0.0522214
$286$	0	0
$287$	−16.0391	−0.946756
$288$	0	0
$289$	18.5776	1.09280
$290$	0	0
$291$	−2.01258	−0.117980
$292$	0	0
$293$	19.5017	1.13930	0.569651	−	0.821887i	$-0.307078\pi$
$293$	19.5017	1.13930	0.569651	+	0.821887i	$0.307078\pi$
$294$	0	0
$295$	−9.47024	−0.551379
$296$	0	0
$297$	−1.76927	−0.102663
$298$	0	0
$299$	−32.7592	−1.89451
$300$	0	0
$301$	−0.310183	−0.0178787
$302$	0	0
$303$	−6.92714	−0.397954
$304$	0	0
$305$	−8.26104	−0.473026
$306$	0	0
$307$	−16.1698	−0.922861	−0.461430	−	0.887176i	$-0.652664\pi$
$307$	−16.1698	−0.922861	−0.461430	+	0.887176i	$0.652664\pi$
$308$	0	0
$309$	−4.86677	−0.276861
$310$	0	0
$311$	−22.5760	−1.28017	−0.640085	−	0.768304i	$-0.721101\pi$
$311$	−22.5760	−1.28017	−0.640085	+	0.768304i	$0.721101\pi$
$312$	0	0
$313$	22.8074	1.28915	0.644574	−	0.764542i	$-0.277035\pi$
$313$	22.8074	1.28915	0.644574	+	0.764542i	$0.277035\pi$
$314$	0	0
$315$	10.7131	0.603612
$316$	0	0
$317$	−24.7668	−1.39104	−0.695521	−	0.718506i	$-0.744826\pi$
$317$	−24.7668	−1.39104	−0.695521	+	0.718506i	$0.744826\pi$
$318$	0	0
$319$	5.06373	0.283514
$320$	0	0
$321$	−0.965849	−0.0539084
$322$	0	0
$323$	−12.0627	−0.671185
$324$	0	0
$325$	−4.30901	−0.239021
$326$	0	0
$327$	6.22006	0.343970
$328$	0	0
$329$	−26.0573	−1.43659
$330$	0	0
$331$	13.9811	0.768472	0.384236	−	0.923235i	$-0.374465\pi$
$331$	13.9811	0.768472	0.384236	+	0.923235i	$0.374465\pi$
$332$	0	0
$333$	7.45986	0.408798
$334$	0	0
$335$	−12.6150	−0.689233
$336$	0	0
$337$	−0.0268906	−0.00146482	−0.000732411	−	1.00000i	$-0.500233\pi$
$337$	−0.0268906	−0.00146482	−0.000732411		1.00000i	$0.500233\pi$
$338$	0	0
$339$	−3.47447	−0.188707
$340$	0	0
$341$	4.97759	0.269552
$342$	0	0
$343$	−2.04088	−0.110197
$344$	0	0
$345$	−3.31414	−0.178427
$346$	0	0
$347$	−17.9569	−0.963976	−0.481988	−	0.876178i	$-0.660085\pi$
$347$	−17.9569	−0.963976	−0.481988	+	0.876178i	$0.660085\pi$
$348$	0	0
$349$	−13.3645	−0.715384	−0.357692	−	0.933840i	$-0.616436\pi$
$349$	−13.3645	−0.715384	−0.357692	+	0.933840i	$0.616436\pi$
$350$	0	0
$351$	−10.9136	−0.582524
$352$	0	0
$353$	−3.39712	−0.180810	−0.0904051	−	0.995905i	$-0.528816\pi$
$353$	−3.39712	−0.180810	−0.0904051	+	0.995905i	$0.528816\pi$
$354$	0	0
$355$	−7.01822	−0.372488
$356$	0	0
$357$	−9.91325	−0.524665
$358$	0	0
$359$	13.6501	0.720427	0.360213	−	0.932870i	$-0.382704\pi$
$359$	13.6501	0.720427	0.360213	+	0.932870i	$0.382704\pi$
$360$	0	0
$361$	−14.9101	−0.784743
$362$	0	0
$363$	4.58249	0.240518
$364$	0	0
$365$	13.4205	0.702461
$366$	0	0
$367$	−6.41164	−0.334685	−0.167342	−	0.985899i	$-0.553519\pi$
$367$	−6.41164	−0.334685	−0.167342	+	0.985899i	$0.553519\pi$
$368$	0	0
$369$	−11.8214	−0.615396
$370$	0	0
$371$	33.5183	1.74019
$372$	0	0
$373$	−26.2514	−1.35924	−0.679622	−	0.733563i	$-0.737856\pi$
$373$	−26.2514	−1.35924	−0.679622	+	0.733563i	$0.737856\pi$
$374$	0	0
$375$	−0.435929	−0.0225113
$376$	0	0
$377$	31.2352	1.60870
$378$	0	0
$379$	26.5138	1.36192	0.680960	−	0.732320i	$-0.261563\pi$
$379$	26.5138	1.36192	0.680960	+	0.732320i	$0.261563\pi$
$380$	0	0
$381$	−1.43058	−0.0732907
$382$	0	0
$383$	−37.9717	−1.94026	−0.970132	−	0.242576i	$-0.922007\pi$
$383$	−37.9717	−1.94026	−0.970132	+	0.242576i	$0.922007\pi$
$384$	0	0
$385$	2.66327	0.135733
$386$	0	0
$387$	−0.228616	−0.0116212
$388$	0	0
$389$	−6.21787	−0.315259	−0.157629	−	0.987498i	$-0.550385\pi$
$389$	−6.21787	−0.315259	−0.157629	+	0.987498i	$0.550385\pi$
$390$	0	0
$391$	−45.3464	−2.29327
$392$	0	0
$393$	−4.33058	−0.218449
$394$	0	0
$395$	−4.47048	−0.224934
$396$	0	0
$397$	13.4657	0.675823	0.337912	−	0.941178i	$-0.390280\pi$
$397$	13.4657	0.675823	0.337912	+	0.941178i	$0.390280\pi$
$398$	0	0
$399$	3.36111	0.168266
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	30.7039	1.52947
$404$	0	0
$405$	7.32581	0.364022
$406$	0	0
$407$	1.85453	0.0919255
$408$	0	0
$409$	−31.7665	−1.57075	−0.785376	−	0.619019i	$-0.787531\pi$
$409$	−31.7665	−1.57075	−0.785376	+	0.619019i	$0.787531\pi$
$410$	0	0
$411$	4.88537	0.240978
$412$	0	0
$413$	36.1055	1.77664
$414$	0	0
$415$	−13.9679	−0.685658
$416$	0	0
$417$	−2.56213	−0.125468
$418$	0	0
$419$	15.7477	0.769324	0.384662	−	0.923057i	$-0.374318\pi$
$419$	15.7477	0.769324	0.384662	+	0.923057i	$0.374318\pi$
$420$	0	0
$421$	−1.49402	−0.0728138	−0.0364069	−	0.999337i	$-0.511591\pi$
$421$	−1.49402	−0.0728138	−0.0364069	+	0.999337i	$0.511591\pi$
$422$	0	0
$423$	−19.2052	−0.933789
$424$	0	0
$425$	−5.96469	−0.289330
$426$	0	0
$427$	31.4954	1.52417
$428$	0	0
$429$	−1.31219	−0.0633533
$430$	0	0
$431$	−2.27029	−0.109356	−0.0546779	−	0.998504i	$-0.517413\pi$
$431$	−2.27029	−0.109356	−0.0546779	+	0.998504i	$0.517413\pi$
$432$	0	0
$433$	−14.1137	−0.678263	−0.339131	−	0.940739i	$-0.610133\pi$
$433$	−14.1137	−0.678263	−0.339131	+	0.940739i	$0.610133\pi$
$434$	0	0
$435$	3.15997	0.151509
$436$	0	0
$437$	15.3748	0.735477
$438$	0	0
$439$	−9.96558	−0.475631	−0.237816	−	0.971310i	$-0.576431\pi$
$439$	−9.96558	−0.475631	−0.237816	+	0.971310i	$0.576431\pi$
$440$	0	0
$441$	−21.1740	−1.00828
$442$	0	0
$443$	−34.6515	−1.64634	−0.823172	−	0.567792i	$-0.807798\pi$
$443$	−34.6515	−1.64634	−0.823172	+	0.567792i	$0.807798\pi$
$444$	0	0
$445$	−4.12463	−0.195526
$446$	0	0
$447$	−4.20305	−0.198797
$448$	0	0
$449$	−18.2737	−0.862387	−0.431194	−	0.902259i	$-0.641907\pi$
$449$	−18.2737	−0.862387	−0.431194	+	0.902259i	$0.641907\pi$
$450$	0	0
$451$	−2.93881	−0.138383
$452$	0	0
$453$	−5.74600	−0.269971
$454$	0	0
$455$	16.4282	0.770166
$456$	0	0
$457$	21.8077	1.02012	0.510062	−	0.860138i	$-0.329623\pi$
$457$	21.8077	1.02012	0.510062	+	0.860138i	$0.329623\pi$
$458$	0	0
$459$	−15.1070	−0.705133
$460$	0	0
$461$	2.00290	0.0932845	0.0466422	−	0.998912i	$-0.485148\pi$
$461$	2.00290	0.0932845	0.0466422	+	0.998912i	$0.485148\pi$
$462$	0	0
$463$	19.1159	0.888392	0.444196	−	0.895930i	$-0.353489\pi$
$463$	19.1159	0.888392	0.444196	+	0.895930i	$0.353489\pi$
$464$	0	0
$465$	3.10622	0.144047
$466$	0	0
$467$	12.6520	0.585465	0.292732	−	0.956194i	$-0.405436\pi$
$467$	12.6520	0.585465	0.292732	+	0.956194i	$0.405436\pi$
$468$	0	0
$469$	48.0951	2.22082
$470$	0	0
$471$	−7.23764	−0.333493
$472$	0	0
$473$	−0.0568342	−0.00261324
$474$	0	0
$475$	2.02234	0.0927915
$476$	0	0
$477$	24.7042	1.13113
$478$	0	0
$479$	4.03336	0.184289	0.0921444	−	0.995746i	$-0.470628\pi$
$479$	4.03336	0.184289	0.0921444	+	0.995746i	$0.470628\pi$
$480$	0	0
$481$	11.4395	0.521597
$482$	0	0
$483$	12.6352	0.574922
$484$	0	0
$485$	4.61677	0.209637
$486$	0	0
$487$	−31.9530	−1.44793	−0.723965	−	0.689836i	$-0.757682\pi$
$487$	−31.9530	−1.44793	−0.723965	+	0.689836i	$0.757682\pi$
$488$	0	0
$489$	1.19610	0.0540897
$490$	0	0
$491$	−19.4076	−0.875851	−0.437925	−	0.899011i	$-0.644287\pi$
$491$	−19.4076	−0.875851	−0.437925	+	0.899011i	$0.644287\pi$
$492$	0	0
$493$	43.2369	1.94729
$494$	0	0
$495$	1.96293	0.0882271
$496$	0	0
$497$	26.7571	1.20022
$498$	0	0
$499$	−1.08126	−0.0484036	−0.0242018	−	0.999707i	$-0.507704\pi$
$499$	−1.08126	−0.0484036	−0.0242018	+	0.999707i	$0.507704\pi$
$500$	0	0
$501$	−1.42447	−0.0636408
$502$	0	0
$503$	−5.06657	−0.225907	−0.112954	−	0.993600i	$-0.536031\pi$
$503$	−5.06657	−0.225907	−0.112954	+	0.993600i	$0.536031\pi$
$504$	0	0
$505$	15.8905	0.707119
$506$	0	0
$507$	−2.42708	−0.107791
$508$	0	0
$509$	26.0172	1.15319	0.576596	−	0.817030i	$-0.304381\pi$
$509$	26.0172	1.15319	0.576596	+	0.817030i	$0.304381\pi$
$510$	0	0
$511$	−51.1659	−2.26345
$512$	0	0
$513$	5.12206	0.226144
$514$	0	0
$515$	11.1641	0.491950
$516$	0	0
$517$	−4.77443	−0.209979
$518$	0	0
$519$	−8.24530	−0.361928
$520$	0	0
$521$	1.72794	0.0757023	0.0378511	−	0.999283i	$-0.487949\pi$
$521$	1.72794	0.0757023	0.0378511	+	0.999283i	$0.487949\pi$
$522$	0	0
$523$	19.4132	0.848879	0.424439	−	0.905456i	$-0.360471\pi$
$523$	19.4132	0.848879	0.424439	+	0.905456i	$0.360471\pi$
$524$	0	0
$525$	1.66199	0.0725351
$526$	0	0
$527$	42.5015	1.85139
$528$	0	0
$529$	34.7976	1.51294
$530$	0	0
$531$	26.6111	1.15482
$532$	0	0
$533$	−18.1278	−0.785202
$534$	0	0
$535$	2.21561	0.0957892
$536$	0	0
$537$	10.0485	0.433627
$538$	0	0
$539$	−5.26387	−0.226731
$540$	0	0
$541$	42.7037	1.83597	0.917987	−	0.396611i	$-0.129814\pi$
$541$	42.7037	1.83597	0.917987	+	0.396611i	$0.129814\pi$
$542$	0	0
$543$	−6.80207	−0.291905
$544$	0	0
$545$	−14.2685	−0.611196
$546$	0	0
$547$	26.8432	1.14773	0.573865	−	0.818950i	$-0.305443\pi$
$547$	26.8432	1.14773	0.573865	+	0.818950i	$0.305443\pi$
$548$	0	0
$549$	23.2133	0.990717
$550$	0	0
$551$	−14.6596	−0.624520
$552$	0	0
$553$	17.0438	0.724776
$554$	0	0
$555$	1.15730	0.0491246
$556$	0	0
$557$	−27.1504	−1.15040	−0.575200	−	0.818013i	$-0.695076\pi$
$557$	−27.1504	−1.15040	−0.575200	+	0.818013i	$0.695076\pi$
$558$	0	0
$559$	−0.350578	−0.0148279
$560$	0	0
$561$	−1.81638	−0.0766878
$562$	0	0
$563$	6.65248	0.280369	0.140184	−	0.990125i	$-0.455230\pi$
$563$	6.65248	0.280369	0.140184	+	0.990125i	$0.455230\pi$
$564$	0	0
$565$	7.97026	0.335311
$566$	0	0
$567$	−27.9298	−1.17294
$568$	0	0
$569$	40.1239	1.68208	0.841041	−	0.540971i	$-0.181943\pi$
$569$	40.1239	1.68208	0.841041	+	0.540971i	$0.181943\pi$
$570$	0	0
$571$	−14.7881	−0.618864	−0.309432	−	0.950922i	$-0.600139\pi$
$571$	−14.7881	−0.618864	−0.309432	+	0.950922i	$0.600139\pi$
$572$	0	0
$573$	3.51537	0.146857
$574$	0	0
$575$	7.60247	0.317045
$576$	0	0
$577$	−31.4494	−1.30926	−0.654628	−	0.755952i	$-0.727175\pi$
$577$	−31.4494	−1.30926	−0.654628	+	0.755952i	$0.727175\pi$
$578$	0	0
$579$	−9.98579	−0.414995
$580$	0	0
$581$	53.2529	2.20930
$582$	0	0
$583$	6.14149	0.254355
$584$	0	0
$585$	12.1082	0.500612
$586$	0	0
$587$	−24.0481	−0.992573	−0.496286	−	0.868159i	$-0.665303\pi$
$587$	−24.0481	−0.992573	−0.496286	+	0.868159i	$0.665303\pi$
$588$	0	0
$589$	−14.4102	−0.593763
$590$	0	0
$591$	−8.76217	−0.360428
$592$	0	0
$593$	19.3158	0.793203	0.396602	−	0.917991i	$-0.370189\pi$
$593$	19.3158	0.793203	0.396602	+	0.917991i	$0.370189\pi$
$594$	0	0
$595$	22.7405	0.932270
$596$	0	0
$597$	0.844805	0.0345756
$598$	0	0
$599$	0.651692	0.0266274	0.0133137	−	0.999911i	$-0.495762\pi$
$599$	0.651692	0.0266274	0.0133137	+	0.999911i	$0.495762\pi$
$600$	0	0
$601$	−32.0679	−1.30808	−0.654039	−	0.756461i	$-0.726927\pi$
$601$	−32.0679	−1.30808	−0.654039	+	0.756461i	$0.726927\pi$
$602$	0	0
$603$	35.4478	1.44355
$604$	0	0
$605$	−10.5120	−0.427374
$606$	0	0
$607$	−23.0972	−0.937485	−0.468742	−	0.883335i	$-0.655293\pi$
$607$	−23.0972	−0.937485	−0.468742	+	0.883335i	$0.655293\pi$
$608$	0	0
$609$	−12.0474	−0.488187
$610$	0	0
$611$	−29.4507	−1.19145
$612$	0	0
$613$	−32.1971	−1.30043	−0.650214	−	0.759751i	$-0.725321\pi$
$613$	−32.1971	−1.30043	−0.650214	+	0.759751i	$0.725321\pi$
$614$	0	0
$615$	−1.83393	−0.0739512
$616$	0	0
$617$	21.2211	0.854331	0.427165	−	0.904174i	$-0.359512\pi$
$617$	21.2211	0.854331	0.427165	+	0.904174i	$0.359512\pi$
$618$	0	0
$619$	−17.7816	−0.714704	−0.357352	−	0.933970i	$-0.616320\pi$
$619$	−17.7816	−0.714704	−0.357352	+	0.933970i	$0.616320\pi$
$620$	0	0
$621$	19.2550	0.772678
$622$	0	0
$623$	15.7252	0.630018
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.615850	0.0245947
$628$	0	0
$629$	15.8350	0.631383
$630$	0	0
$631$	34.1871	1.36097	0.680483	−	0.732763i	$-0.261770\pi$
$631$	34.1871	1.36097	0.680483	+	0.732763i	$0.261770\pi$
$632$	0	0
$633$	5.68961	0.226142
$634$	0	0
$635$	3.28167	0.130229
$636$	0	0
$637$	−32.4698	−1.28650
$638$	0	0
$639$	19.7210	0.780149
$640$	0	0
$641$	−6.31553	−0.249449	−0.124724	−	0.992191i	$-0.539805\pi$
$641$	−6.31553	−0.249449	−0.124724	+	0.992191i	$0.539805\pi$
$642$	0	0
$643$	4.96260	0.195706	0.0978529	−	0.995201i	$-0.468803\pi$
$643$	4.96260	0.195706	0.0978529	+	0.995201i	$0.468803\pi$
$644$	0	0
$645$	−0.0354668	−0.00139650
$646$	0	0
$647$	−11.1710	−0.439178	−0.219589	−	0.975592i	$-0.570472\pi$
$647$	−11.1710	−0.439178	−0.219589	+	0.975592i	$0.570472\pi$
$648$	0	0
$649$	6.61553	0.259682
$650$	0	0
$651$	−11.8425	−0.464144
$652$	0	0
$653$	2.54493	0.0995906	0.0497953	−	0.998759i	$-0.484143\pi$
$653$	2.54493	0.0995906	0.0497953	+	0.998759i	$0.484143\pi$
$654$	0	0
$655$	9.93413	0.388159
$656$	0	0
$657$	−37.7112	−1.47125
$658$	0	0
$659$	7.70933	0.300313	0.150156	−	0.988662i	$-0.452022\pi$
$659$	7.70933	0.300313	0.150156	+	0.988662i	$0.452022\pi$
$660$	0	0
$661$	−18.1652	−0.706545	−0.353273	−	0.935520i	$-0.614931\pi$
$661$	−18.1652	−0.706545	−0.353273	+	0.935520i	$0.614931\pi$
$662$	0	0
$663$	−11.2042	−0.435136
$664$	0	0
$665$	−7.71023	−0.298990
$666$	0	0
$667$	−55.1089	−2.13382
$668$	0	0
$669$	−12.9720	−0.501527
$670$	0	0
$671$	5.77084	0.222781
$672$	0	0
$673$	24.2058	0.933065	0.466532	−	0.884504i	$-0.345503\pi$
$673$	24.2058	0.933065	0.466532	+	0.884504i	$0.345503\pi$
$674$	0	0
$675$	2.53273	0.0974849
$676$	0	0
$677$	21.3049	0.818814	0.409407	−	0.912352i	$-0.365736\pi$
$677$	21.3049	0.818814	0.409407	+	0.912352i	$0.365736\pi$
$678$	0	0
$679$	−17.6015	−0.675485
$680$	0	0
$681$	8.16072	0.312719
$682$	0	0
$683$	−12.5461	−0.480062	−0.240031	−	0.970765i	$-0.577158\pi$
$683$	−12.5461	−0.480062	−0.240031	+	0.970765i	$0.577158\pi$
$684$	0	0
$685$	−11.2068	−0.428190
$686$	0	0
$687$	−2.94159	−0.112229
$688$	0	0
$689$	37.8833	1.44324
$690$	0	0
$691$	−25.9450	−0.986994	−0.493497	−	0.869748i	$-0.664282\pi$
$691$	−25.9450	−0.986994	−0.493497	+	0.869748i	$0.664282\pi$
$692$	0	0
$693$	−7.48371	−0.284283
$694$	0	0
$695$	5.87741	0.222943
$696$	0	0
$697$	−25.0932	−0.950471
$698$	0	0
$699$	−6.63541	−0.250974
$700$	0	0
$701$	40.3412	1.52367	0.761834	−	0.647773i	$-0.224299\pi$
$701$	40.3412	1.52367	0.761834	+	0.647773i	$0.224299\pi$
$702$	0	0
$703$	−5.36889	−0.202492
$704$	0	0
$705$	−2.97943	−0.112212
$706$	0	0
$707$	−60.5829	−2.27846
$708$	0	0
$709$	14.6307	0.549468	0.274734	−	0.961520i	$-0.411410\pi$
$709$	14.6307	0.549468	0.274734	+	0.961520i	$0.411410\pi$
$710$	0	0
$711$	12.5619	0.471108
$712$	0	0
$713$	−54.1714	−2.02874
$714$	0	0
$715$	3.01011	0.112572
$716$	0	0
$717$	−11.6056	−0.433418
$718$	0	0
$719$	−4.03361	−0.150428	−0.0752142	−	0.997167i	$-0.523964\pi$
$719$	−4.03361	−0.150428	−0.0752142	+	0.997167i	$0.523964\pi$
$720$	0	0
$721$	−42.5635	−1.58515
$722$	0	0
$723$	5.66908	0.210835
$724$	0	0
$725$	−7.24881	−0.269214
$726$	0	0
$727$	5.72764	0.212426	0.106213	−	0.994343i	$-0.466127\pi$
$727$	5.72764	0.212426	0.106213	+	0.994343i	$0.466127\pi$
$728$	0	0
$729$	−17.2730	−0.639740
$730$	0	0
$731$	−0.485282	−0.0179488
$732$	0	0
$733$	−45.7535	−1.68994	−0.844972	−	0.534810i	$-0.820383\pi$
$733$	−45.7535	−1.68994	−0.844972	+	0.534810i	$0.820383\pi$
$734$	0	0
$735$	−3.28486	−0.121164
$736$	0	0
$737$	8.81236	0.324607
$738$	0	0
$739$	28.0006	1.03002	0.515009	−	0.857185i	$-0.327788\pi$
$739$	28.0006	1.03002	0.515009	+	0.857185i	$0.327788\pi$
$740$	0	0
$741$	3.79882	0.139553
$742$	0	0
$743$	6.94041	0.254619	0.127309	−	0.991863i	$-0.459366\pi$
$743$	6.94041	0.254619	0.127309	+	0.991863i	$0.459366\pi$
$744$	0	0
$745$	9.64159	0.353240
$746$	0	0
$747$	39.2493	1.43606
$748$	0	0
$749$	−8.44706	−0.308649
$750$	0	0
$751$	1.39321	0.0508389	0.0254195	−	0.999677i	$-0.491908\pi$
$751$	1.39321	0.0508389	0.0254195	+	0.999677i	$0.491908\pi$
$752$	0	0
$753$	5.13808	0.187242
$754$	0	0
$755$	13.1810	0.479708
$756$	0	0
$757$	−2.24105	−0.0814522	−0.0407261	−	0.999170i	$-0.512967\pi$
$757$	−2.24105	−0.0814522	−0.0407261	+	0.999170i	$0.512967\pi$
$758$	0	0
$759$	2.31512	0.0840337
$760$	0	0
$761$	−35.3361	−1.28093	−0.640467	−	0.767986i	$-0.721259\pi$
$761$	−35.3361	−1.28093	−0.640467	+	0.767986i	$0.721259\pi$
$762$	0	0
$763$	54.3990	1.96937
$764$	0	0
$765$	16.7606	0.605980
$766$	0	0
$767$	40.8074	1.47347
$768$	0	0
$769$	−32.2391	−1.16257	−0.581285	−	0.813700i	$-0.697450\pi$
$769$	−32.2391	−1.16257	−0.581285	+	0.813700i	$0.697450\pi$
$770$	0	0
$771$	−11.0325	−0.397325
$772$	0	0
$773$	−25.9701	−0.934078	−0.467039	−	0.884237i	$-0.654679\pi$
$773$	−25.9701	−0.934078	−0.467039	+	0.884237i	$0.654679\pi$
$774$	0	0
$775$	−7.12551	−0.255956
$776$	0	0
$777$	−4.41223	−0.158288
$778$	0	0
$779$	8.50790	0.304827
$780$	0	0
$781$	4.90265	0.175430
$782$	0	0
$783$	−18.3593	−0.656108
$784$	0	0
$785$	16.6028	0.592579
$786$	0	0
$787$	20.7503	0.739668	0.369834	−	0.929098i	$-0.379415\pi$
$787$	20.7503	0.739668	0.369834	+	0.929098i	$0.379415\pi$
$788$	0	0
$789$	9.85493	0.350845
$790$	0	0
$791$	−30.3868	−1.08043
$792$	0	0
$793$	35.5970	1.26409
$794$	0	0
$795$	3.83253	0.135926
$796$	0	0
$797$	4.99957	0.177094	0.0885470	−	0.996072i	$-0.471778\pi$
$797$	4.99957	0.177094	0.0885470	+	0.996072i	$0.471778\pi$
$798$	0	0
$799$	−40.7668	−1.44222
$800$	0	0
$801$	11.5901	0.409515
$802$	0	0
$803$	−9.37503	−0.330838
$804$	0	0
$805$	−28.9846	−1.02157
$806$	0	0
$807$	4.86588	0.171287
$808$	0	0
$809$	1.51554	0.0532835	0.0266418	−	0.999645i	$-0.491519\pi$
$809$	1.51554	0.0532835	0.0266418	+	0.999645i	$0.491519\pi$
$810$	0	0
$811$	−9.84140	−0.345578	−0.172789	−	0.984959i	$-0.555278\pi$
$811$	−9.84140	−0.345578	−0.172789	+	0.984959i	$0.555278\pi$
$812$	0	0
$813$	4.91485	0.172371
$814$	0	0
$815$	−2.74380	−0.0961112
$816$	0	0
$817$	0.164536	0.00575639
$818$	0	0
$819$	−46.1627	−1.61306
$820$	0	0
$821$	31.0414	1.08335	0.541677	−	0.840587i	$-0.317790\pi$
$821$	31.0414	1.08335	0.541677	+	0.840587i	$0.317790\pi$
$822$	0	0
$823$	47.4898	1.65539	0.827695	−	0.561179i	$-0.189652\pi$
$823$	47.4898	1.65539	0.827695	+	0.561179i	$0.189652\pi$
$824$	0	0
$825$	0.304523	0.0106021
$826$	0	0
$827$	4.32160	0.150277	0.0751384	−	0.997173i	$-0.476060\pi$
$827$	4.32160	0.150277	0.0751384	+	0.997173i	$0.476060\pi$
$828$	0	0
$829$	−6.02225	−0.209161	−0.104581	−	0.994516i	$-0.533350\pi$
$829$	−6.02225	−0.209161	−0.104581	+	0.994516i	$0.533350\pi$
$830$	0	0
$831$	−3.56026	−0.123504
$832$	0	0
$833$	−44.9458	−1.55728
$834$	0	0
$835$	3.26767	0.113083
$836$	0	0
$837$	−18.0470	−0.623796
$838$	0	0
$839$	−25.3071	−0.873697	−0.436849	−	0.899535i	$-0.643905\pi$
$839$	−25.3071	−0.873697	−0.436849	+	0.899535i	$0.643905\pi$
$840$	0	0
$841$	23.5453	0.811905
$842$	0	0
$843$	−8.46345	−0.291497
$844$	0	0
$845$	5.56761	0.191532
$846$	0	0
$847$	40.0773	1.37707
$848$	0	0
$849$	−14.1752	−0.486492
$850$	0	0
$851$	−20.1829	−0.691862
$852$	0	0
$853$	6.09421	0.208662	0.104331	−	0.994543i	$-0.466730\pi$
$853$	6.09421	0.208662	0.104331	+	0.994543i	$0.466730\pi$
$854$	0	0
$855$	−5.68272	−0.194345
$856$	0	0
$857$	−9.68807	−0.330938	−0.165469	−	0.986215i	$-0.552914\pi$
$857$	−9.68807	−0.330938	−0.165469	+	0.986215i	$0.552914\pi$
$858$	0	0
$859$	36.1387	1.23304	0.616519	−	0.787340i	$-0.288542\pi$
$859$	36.1387	1.23304	0.616519	+	0.787340i	$0.288542\pi$
$860$	0	0
$861$	6.99190	0.238283
$862$	0	0
$863$	−36.0187	−1.22609	−0.613046	−	0.790047i	$-0.710056\pi$
$863$	−36.0187	−1.22609	−0.613046	+	0.790047i	$0.710056\pi$
$864$	0	0
$865$	18.9143	0.643106
$866$	0	0
$867$	−8.09851	−0.275040
$868$	0	0
$869$	3.12290	0.105937
$870$	0	0
$871$	54.3584	1.84186
$872$	0	0
$873$	−12.9730	−0.439068
$874$	0	0
$875$	−3.81252	−0.128887
$876$	0	0
$877$	17.9603	0.606477	0.303239	−	0.952915i	$-0.401932\pi$
$877$	17.9603	0.606477	0.303239	+	0.952915i	$0.401932\pi$
$878$	0	0
$879$	−8.50136	−0.286744
$880$	0	0
$881$	5.20188	0.175256	0.0876278	−	0.996153i	$-0.472071\pi$
$881$	5.20188	0.175256	0.0876278	+	0.996153i	$0.472071\pi$
$882$	0	0
$883$	40.3984	1.35951	0.679757	−	0.733437i	$-0.262085\pi$
$883$	40.3984	1.35951	0.679757	+	0.733437i	$0.262085\pi$
$884$	0	0
$885$	4.12836	0.138773
$886$	0	0
$887$	−31.7579	−1.06633	−0.533164	−	0.846012i	$-0.678997\pi$
$887$	−31.7579	−1.06633	−0.533164	+	0.846012i	$0.678997\pi$
$888$	0	0
$889$	−12.5114	−0.419620
$890$	0	0
$891$	−5.11752	−0.171443
$892$	0	0
$893$	13.8221	0.462538
$894$	0	0
$895$	−23.0509	−0.770506
$896$	0	0
$897$	14.2807	0.476818
$898$	0	0
$899$	51.6514	1.72267
$900$	0	0
$901$	52.4395	1.74701
$902$	0	0
$903$	0.135218	0.00449977
$904$	0	0
$905$	15.6036	0.518682
$906$	0	0
$907$	−8.37632	−0.278131	−0.139066	−	0.990283i	$-0.544410\pi$
$907$	−8.37632	−0.278131	−0.139066	+	0.990283i	$0.544410\pi$
$908$	0	0
$909$	−44.6518	−1.48101
$910$	0	0
$911$	51.3757	1.70215	0.851077	−	0.525041i	$-0.175950\pi$
$911$	51.3757	1.70215	0.851077	+	0.525041i	$0.175950\pi$
$912$	0	0
$913$	9.75742	0.322924
$914$	0	0
$915$	3.60123	0.119053
$916$	0	0
$917$	−37.8741	−1.25071
$918$	0	0
$919$	51.0499	1.68398	0.841990	−	0.539493i	$-0.181384\pi$
$919$	51.0499	1.68398	0.841990	+	0.539493i	$0.181384\pi$
$920$	0	0
$921$	7.04890	0.232269
$922$	0	0
$923$	30.2416	0.995415
$924$	0	0
$925$	−2.65479	−0.0872889
$926$	0	0
$927$	−31.3708	−1.03035
$928$	0	0
$929$	9.56270	0.313742	0.156871	−	0.987619i	$-0.449859\pi$
$929$	9.56270	0.313742	0.156871	+	0.987619i	$0.449859\pi$
$930$	0	0
$931$	15.2390	0.499438
$932$	0	0
$933$	9.84155	0.322198
$934$	0	0
$935$	4.16670	0.136266
$936$	0	0
$937$	45.5096	1.48673	0.743367	−	0.668884i	$-0.233228\pi$
$937$	45.5096	1.48673	0.743367	+	0.668884i	$0.233228\pi$
$938$	0	0
$939$	−9.94239	−0.324458
$940$	0	0
$941$	−57.3863	−1.87074	−0.935369	−	0.353673i	$-0.884933\pi$
$941$	−57.3863	−1.87074	−0.935369	+	0.353673i	$0.884933\pi$
$942$	0	0
$943$	31.9832	1.04152
$944$	0	0
$945$	−9.65610	−0.314113
$946$	0	0
$947$	−33.7817	−1.09776	−0.548879	−	0.835902i	$-0.684945\pi$
$947$	−33.7817	−1.09776	−0.548879	+	0.835902i	$0.684945\pi$
$948$	0	0
$949$	−57.8291	−1.87721
$950$	0	0
$951$	10.7966	0.350103
$952$	0	0
$953$	11.7447	0.380448	0.190224	−	0.981741i	$-0.439079\pi$
$953$	11.7447	0.380448	0.190224	+	0.981741i	$0.439079\pi$
$954$	0	0
$955$	−8.06410	−0.260948
$956$	0	0
$957$	−2.20743	−0.0713560
$958$	0	0
$959$	42.7262	1.37970
$960$	0	0
$961$	19.7728	0.637833
$962$	0	0
$963$	−6.22579	−0.200623
$964$	0	0
$965$	22.9069	0.737400
$966$	0	0
$967$	34.3829	1.10568	0.552840	−	0.833288i	$-0.313544\pi$
$967$	34.3829	1.10568	0.552840	+	0.833288i	$0.313544\pi$
$968$	0	0
$969$	5.25847	0.168926
$970$	0	0
$971$	6.16778	0.197933	0.0989667	−	0.995091i	$-0.468446\pi$
$971$	6.16778	0.197933	0.0989667	+	0.995091i	$0.468446\pi$
$972$	0	0
$973$	−22.4077	−0.718359
$974$	0	0
$975$	1.87842	0.0601577
$976$	0	0
$977$	−59.6143	−1.90723	−0.953615	−	0.301028i	$-0.902670\pi$
$977$	−59.6143	−1.90723	−0.953615	+	0.301028i	$0.902670\pi$
$978$	0	0
$979$	2.88130	0.0920867
$980$	0	0
$981$	40.0940	1.28010
$982$	0	0
$983$	−44.2770	−1.41222	−0.706109	−	0.708103i	$-0.749551\pi$
$983$	−44.2770	−1.41222	−0.706109	+	0.708103i	$0.749551\pi$
$984$	0	0
$985$	20.1000	0.640439
$986$	0	0
$987$	11.3592	0.361566
$988$	0	0
$989$	0.618530	0.0196681
$990$	0	0
$991$	11.8852	0.377546	0.188773	−	0.982021i	$-0.439549\pi$
$991$	11.8852	0.377546	0.188773	+	0.982021i	$0.439549\pi$
$992$	0	0
$993$	−6.09478	−0.193412
$994$	0	0
$995$	−1.93794	−0.0614369
$996$	0	0
$997$	19.1444	0.606311	0.303155	−	0.952941i	$-0.401960\pi$
$997$	19.1444	0.606311	0.303155	+	0.952941i	$0.401960\pi$
$998$	0	0
$999$	−6.72387	−0.212734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.16	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.16	✓	37	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)

\(f(q)\)	\(=\)	\(q-0.435929 q^{3} +1.00000 q^{5} -3.81252 q^{7} -2.80997 q^{9} +O(q^{10})\)	\(q-0.435929 q^{3} +1.00000 q^{5} -3.81252 q^{7} -2.80997 q^{9} -0.698560 q^{11} -4.30901 q^{13} -0.435929 q^{15} -5.96469 q^{17} +2.02234 q^{19} +1.66199 q^{21} +7.60247 q^{23} +1.00000 q^{25} +2.53273 q^{27} -7.24881 q^{29} -7.12551 q^{31} +0.304523 q^{33} -3.81252 q^{35} -2.65479 q^{37} +1.87842 q^{39} +4.20695 q^{41} +0.0813591 q^{43} -2.80997 q^{45} +6.83468 q^{47} +7.53531 q^{49} +2.60018 q^{51} -8.79165 q^{53} -0.698560 q^{55} -0.881599 q^{57} -9.47024 q^{59} -8.26104 q^{61} +10.7131 q^{63} -4.30901 q^{65} -12.6150 q^{67} -3.31414 q^{69} -7.01822 q^{71} +13.4205 q^{73} -0.435929 q^{75} +2.66327 q^{77} -4.47048 q^{79} +7.32581 q^{81} -13.9679 q^{83} -5.96469 q^{85} +3.15997 q^{87} -4.12463 q^{89} +16.4282 q^{91} +3.10622 q^{93} +2.02234 q^{95} +4.61677 q^{97} +1.96293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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