LMFDB - Embedded newform 8020.2.a.d.1.19

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			1.19
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)

$f(q)$	$=$	$q+0.540664 q^{3} +1.00000 q^{5} -1.71282 q^{7} -2.70768 q^{9} +O(q^{10})$	$q+0.540664 q^{3} +1.00000 q^{5} -1.71282 q^{7} -2.70768 q^{9} -1.80842 q^{11} -1.10623 q^{13} +0.540664 q^{15} +2.19624 q^{17} +0.480052 q^{19} -0.926059 q^{21} +7.44588 q^{23} +1.00000 q^{25} -3.08594 q^{27} +7.46233 q^{29} -2.62456 q^{31} -0.977749 q^{33} -1.71282 q^{35} +5.11818 q^{37} -0.598099 q^{39} -9.19794 q^{41} -4.50672 q^{43} -2.70768 q^{45} -3.42999 q^{47} -4.06625 q^{49} +1.18743 q^{51} -6.58123 q^{53} -1.80842 q^{55} +0.259546 q^{57} +2.68514 q^{59} -4.35751 q^{61} +4.63777 q^{63} -1.10623 q^{65} -0.770521 q^{67} +4.02572 q^{69} +11.6268 q^{71} +11.7631 q^{73} +0.540664 q^{75} +3.09751 q^{77} +3.87292 q^{79} +6.45460 q^{81} -3.01510 q^{83} +2.19624 q^{85} +4.03461 q^{87} -7.38341 q^{89} +1.89478 q^{91} -1.41900 q^{93} +0.480052 q^{95} -18.5614 q^{97} +4.89664 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.540664	0.312152	0.156076	−	0.987745i	$-0.450115\pi$
$3$	0.540664	0.312152	0.156076	+	0.987745i	$0.450115\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.71282	−0.647385	−0.323693	−	0.946162i	$-0.604924\pi$
$7$	−1.71282	−0.647385	−0.323693	+	0.946162i	$0.604924\pi$
$8$	0	0
$9$	−2.70768	−0.902561
$10$	0	0
$11$	−1.80842	−0.545260	−0.272630	−	0.962119i	$-0.587894\pi$
$11$	−1.80842	−0.545260	−0.272630	+	0.962119i	$0.587894\pi$
$12$	0	0
$13$	−1.10623	−0.306813	−0.153407	−	0.988163i	$-0.549024\pi$
$13$	−1.10623	−0.306813	−0.153407	+	0.988163i	$0.549024\pi$
$14$	0	0
$15$	0.540664	0.139599
$16$	0	0
$17$	2.19624	0.532667	0.266333	−	0.963881i	$-0.414188\pi$
$17$	2.19624	0.532667	0.266333	+	0.963881i	$0.414188\pi$
$18$	0	0
$19$	0.480052	0.110131	0.0550657	−	0.998483i	$-0.482463\pi$
$19$	0.480052	0.110131	0.0550657	+	0.998483i	$0.482463\pi$
$20$	0	0
$21$	−0.926059	−0.202083
$22$	0	0
$23$	7.44588	1.55257	0.776287	−	0.630380i	$-0.217101\pi$
$23$	7.44588	1.55257	0.776287	+	0.630380i	$0.217101\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.08594	−0.593889
$28$	0	0
$29$	7.46233	1.38572	0.692860	−	0.721073i	$-0.256351\pi$
$29$	7.46233	1.38572	0.692860	+	0.721073i	$0.256351\pi$
$30$	0	0
$31$	−2.62456	−0.471384	−0.235692	−	0.971828i	$-0.575736\pi$
$31$	−2.62456	−0.471384	−0.235692	+	0.971828i	$0.575736\pi$
$32$	0	0
$33$	−0.977749	−0.170204
$34$	0	0
$35$	−1.71282	−0.289519
$36$	0	0
$37$	5.11818	0.841423	0.420712	−	0.907194i	$-0.361780\pi$
$37$	5.11818	0.841423	0.420712	+	0.907194i	$0.361780\pi$
$38$	0	0
$39$	−0.598099	−0.0957725
$40$	0	0
$41$	−9.19794	−1.43648	−0.718238	−	0.695797i	$-0.755051\pi$
$41$	−9.19794	−1.43648	−0.718238	+	0.695797i	$0.755051\pi$
$42$	0	0
$43$	−4.50672	−0.687268	−0.343634	−	0.939104i	$-0.611658\pi$
$43$	−4.50672	−0.687268	−0.343634	+	0.939104i	$0.611658\pi$
$44$	0	0
$45$	−2.70768	−0.403638
$46$	0	0
$47$	−3.42999	−0.500316	−0.250158	−	0.968205i	$-0.580483\pi$
$47$	−3.42999	−0.500316	−0.250158	+	0.968205i	$0.580483\pi$
$48$	0	0
$49$	−4.06625	−0.580893
$50$	0	0
$51$	1.18743	0.166273
$52$	0	0
$53$	−6.58123	−0.904001	−0.452000	−	0.892018i	$-0.649289\pi$
$53$	−6.58123	−0.904001	−0.452000	+	0.892018i	$0.649289\pi$
$54$	0	0
$55$	−1.80842	−0.243848
$56$	0	0
$57$	0.259546	0.0343778
$58$	0	0
$59$	2.68514	0.349576	0.174788	−	0.984606i	$-0.444076\pi$
$59$	2.68514	0.349576	0.174788	+	0.984606i	$0.444076\pi$
$60$	0	0
$61$	−4.35751	−0.557922	−0.278961	−	0.960302i	$-0.589990\pi$
$61$	−4.35751	−0.557922	−0.278961	+	0.960302i	$0.589990\pi$
$62$	0	0
$63$	4.63777	0.584305
$64$	0	0
$65$	−1.10623	−0.137211
$66$	0	0
$67$	−0.770521	−0.0941341	−0.0470671	−	0.998892i	$-0.514987\pi$
$67$	−0.770521	−0.0941341	−0.0470671	+	0.998892i	$0.514987\pi$
$68$	0	0
$69$	4.02572	0.484639
$70$	0	0
$71$	11.6268	1.37984	0.689921	−	0.723884i	$-0.257645\pi$
$71$	11.6268	1.37984	0.689921	+	0.723884i	$0.257645\pi$
$72$	0	0
$73$	11.7631	1.37676	0.688382	−	0.725348i	$-0.258321\pi$
$73$	11.7631	1.37676	0.688382	+	0.725348i	$0.258321\pi$
$74$	0	0
$75$	0.540664	0.0624305
$76$	0	0
$77$	3.09751	0.352993
$78$	0	0
$79$	3.87292	0.435738	0.217869	−	0.975978i	$-0.430089\pi$
$79$	3.87292	0.435738	0.217869	+	0.975978i	$0.430089\pi$
$80$	0	0
$81$	6.45460	0.717177
$82$	0	0
$83$	−3.01510	−0.330950	−0.165475	−	0.986214i	$-0.552916\pi$
$83$	−3.01510	−0.330950	−0.165475	+	0.986214i	$0.552916\pi$
$84$	0	0
$85$	2.19624	0.238216
$86$	0	0
$87$	4.03461	0.432555
$88$	0	0
$89$	−7.38341	−0.782640	−0.391320	−	0.920255i	$-0.627981\pi$
$89$	−7.38341	−0.782640	−0.391320	+	0.920255i	$0.627981\pi$
$90$	0	0
$91$	1.89478	0.198626
$92$	0	0
$93$	−1.41900	−0.147144
$94$	0	0
$95$	0.480052	0.0492523
$96$	0	0
$97$	−18.5614	−1.88462	−0.942312	−	0.334737i	$-0.891352\pi$
$97$	−18.5614	−1.88462	−0.942312	+	0.334737i	$0.891352\pi$
$98$	0	0
$99$	4.89664	0.492131
$100$	0	0
$101$	−9.18772	−0.914212	−0.457106	−	0.889412i	$-0.651114\pi$
$101$	−9.18772	−0.914212	−0.457106	+	0.889412i	$0.651114\pi$
$102$	0	0
$103$	2.13149	0.210022	0.105011	−	0.994471i	$-0.466512\pi$
$103$	2.13149	0.210022	0.105011	+	0.994471i	$0.466512\pi$
$104$	0	0
$105$	−0.926059	−0.0903741
$106$	0	0
$107$	13.1015	1.26657	0.633286	−	0.773918i	$-0.281706\pi$
$107$	13.1015	1.26657	0.633286	+	0.773918i	$0.281706\pi$
$108$	0	0
$109$	−5.59275	−0.535688	−0.267844	−	0.963462i	$-0.586311\pi$
$109$	−5.59275	−0.535688	−0.267844	+	0.963462i	$0.586311\pi$
$110$	0	0
$111$	2.76721	0.262652
$112$	0	0
$113$	6.04494	0.568660	0.284330	−	0.958726i	$-0.408229\pi$
$113$	6.04494	0.568660	0.284330	+	0.958726i	$0.408229\pi$
$114$	0	0
$115$	7.44588	0.694332
$116$	0	0
$117$	2.99532	0.276918
$118$	0	0
$119$	−3.76177	−0.344841
$120$	0	0
$121$	−7.72960	−0.702691
$122$	0	0
$123$	−4.97299	−0.448400
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.1119	−1.25223	−0.626113	−	0.779733i	$-0.715355\pi$
$127$	−14.1119	−1.25223	−0.626113	+	0.779733i	$0.715355\pi$
$128$	0	0
$129$	−2.43662	−0.214532
$130$	0	0
$131$	−14.0828	−1.23042	−0.615211	−	0.788362i	$-0.710929\pi$
$131$	−14.0828	−1.23042	−0.615211	+	0.788362i	$0.710929\pi$
$132$	0	0
$133$	−0.822242	−0.0712974
$134$	0	0
$135$	−3.08594	−0.265595
$136$	0	0
$137$	−8.11560	−0.693363	−0.346681	−	0.937983i	$-0.612691\pi$
$137$	−8.11560	−0.693363	−0.346681	+	0.937983i	$0.612691\pi$
$138$	0	0
$139$	−12.2327	−1.03757	−0.518783	−	0.854906i	$-0.673615\pi$
$139$	−12.2327	−1.03757	−0.518783	+	0.854906i	$0.673615\pi$
$140$	0	0
$141$	−1.85447	−0.156175
$142$	0	0
$143$	2.00054	0.167293
$144$	0	0
$145$	7.46233	0.619712
$146$	0	0
$147$	−2.19847	−0.181327
$148$	0	0
$149$	−6.22500	−0.509972	−0.254986	−	0.966945i	$-0.582071\pi$
$149$	−6.22500	−0.509972	−0.254986	+	0.966945i	$0.582071\pi$
$150$	0	0
$151$	−16.2003	−1.31836	−0.659181	−	0.751984i	$-0.729097\pi$
$151$	−16.2003	−1.31836	−0.659181	+	0.751984i	$0.729097\pi$
$152$	0	0
$153$	−5.94673	−0.480764
$154$	0	0
$155$	−2.62456	−0.210809
$156$	0	0
$157$	10.7545	0.858301	0.429150	−	0.903233i	$-0.358813\pi$
$157$	10.7545	0.858301	0.429150	+	0.903233i	$0.358813\pi$
$158$	0	0
$159$	−3.55823	−0.282186
$160$	0	0
$161$	−12.7535	−1.00511
$162$	0	0
$163$	−24.9046	−1.95068	−0.975341	−	0.220703i	$-0.929165\pi$
$163$	−24.9046	−1.95068	−0.975341	+	0.220703i	$0.929165\pi$
$164$	0	0
$165$	−0.977749	−0.0761177
$166$	0	0
$167$	−0.0348718	−0.00269846	−0.00134923	−	0.999999i	$-0.500429\pi$
$167$	−0.0348718	−0.00269846	−0.00134923	+	0.999999i	$0.500429\pi$
$168$	0	0
$169$	−11.7763	−0.905865
$170$	0	0
$171$	−1.29983	−0.0994003
$172$	0	0
$173$	−18.4717	−1.40438	−0.702189	−	0.711990i	$-0.747794\pi$
$173$	−18.4717	−1.40438	−0.702189	+	0.711990i	$0.747794\pi$
$174$	0	0
$175$	−1.71282	−0.129477
$176$	0	0
$177$	1.45176	0.109121
$178$	0	0
$179$	−12.0053	−0.897319	−0.448659	−	0.893703i	$-0.648098\pi$
$179$	−12.0053	−0.897319	−0.448659	+	0.893703i	$0.648098\pi$
$180$	0	0
$181$	−15.9065	−1.18232	−0.591161	−	0.806554i	$-0.701330\pi$
$181$	−15.9065	−1.18232	−0.591161	+	0.806554i	$0.701330\pi$
$182$	0	0
$183$	−2.35595	−0.174157
$184$	0	0
$185$	5.11818	0.376296
$186$	0	0
$187$	−3.97174	−0.290442
$188$	0	0
$189$	5.28565	0.384475
$190$	0	0
$191$	3.62034	0.261959	0.130979	−	0.991385i	$-0.458188\pi$
$191$	3.62034	0.261959	0.130979	+	0.991385i	$0.458188\pi$
$192$	0	0
$193$	−20.9684	−1.50934	−0.754671	−	0.656104i	$-0.772203\pi$
$193$	−20.9684	−1.50934	−0.754671	+	0.656104i	$0.772203\pi$
$194$	0	0
$195$	−0.598099	−0.0428308
$196$	0	0
$197$	13.7325	0.978402	0.489201	−	0.872171i	$-0.337288\pi$
$197$	13.7325	0.978402	0.489201	+	0.872171i	$0.337288\pi$
$198$	0	0
$199$	−4.44364	−0.315001	−0.157500	−	0.987519i	$-0.550344\pi$
$199$	−4.44364	−0.315001	−0.157500	+	0.987519i	$0.550344\pi$
$200$	0	0
$201$	−0.416593	−0.0293842
$202$	0	0
$203$	−12.7816	−0.897094
$204$	0	0
$205$	−9.19794	−0.642412
$206$	0	0
$207$	−20.1611	−1.40129
$208$	0	0
$209$	−0.868137	−0.0600503
$210$	0	0
$211$	3.75705	0.258646	0.129323	−	0.991603i	$-0.458720\pi$
$211$	3.75705	0.258646	0.129323	+	0.991603i	$0.458720\pi$
$212$	0	0
$213$	6.28617	0.430721
$214$	0	0
$215$	−4.50672	−0.307356
$216$	0	0
$217$	4.49539	0.305167
$218$	0	0
$219$	6.35987	0.429760
$220$	0	0
$221$	−2.42955	−0.163429
$222$	0	0
$223$	−0.131899	−0.00883260	−0.00441630	−	0.999990i	$-0.501406\pi$
$223$	−0.131899	−0.00883260	−0.00441630	+	0.999990i	$0.501406\pi$
$224$	0	0
$225$	−2.70768	−0.180512
$226$	0	0
$227$	18.2033	1.20819	0.604097	−	0.796911i	$-0.293534\pi$
$227$	18.2033	1.20819	0.604097	+	0.796911i	$0.293534\pi$
$228$	0	0
$229$	26.6316	1.75986	0.879932	−	0.475100i	$-0.157588\pi$
$229$	26.6316	1.75986	0.879932	+	0.475100i	$0.157588\pi$
$230$	0	0
$231$	1.67471	0.110188
$232$	0	0
$233$	−13.4792	−0.883055	−0.441527	−	0.897248i	$-0.645563\pi$
$233$	−13.4792	−0.883055	−0.441527	+	0.897248i	$0.645563\pi$
$234$	0	0
$235$	−3.42999	−0.223748
$236$	0	0
$237$	2.09395	0.136017
$238$	0	0
$239$	−6.38590	−0.413070	−0.206535	−	0.978439i	$-0.566219\pi$
$239$	−6.38590	−0.413070	−0.206535	+	0.978439i	$0.566219\pi$
$240$	0	0
$241$	8.14654	0.524765	0.262382	−	0.964964i	$-0.415492\pi$
$241$	8.14654	0.524765	0.262382	+	0.964964i	$0.415492\pi$
$242$	0	0
$243$	12.7476	0.817757
$244$	0	0
$245$	−4.06625	−0.259783
$246$	0	0
$247$	−0.531048	−0.0337898
$248$	0	0
$249$	−1.63016	−0.103307
$250$	0	0
$251$	3.34861	0.211362	0.105681	−	0.994400i	$-0.466298\pi$
$251$	3.34861	0.211362	0.105681	+	0.994400i	$0.466298\pi$
$252$	0	0
$253$	−13.4653	−0.846557
$254$	0	0
$255$	1.18743	0.0743596
$256$	0	0
$257$	9.38777	0.585593	0.292797	−	0.956175i	$-0.405414\pi$
$257$	9.38777	0.585593	0.292797	+	0.956175i	$0.405414\pi$
$258$	0	0
$259$	−8.76652	−0.544725
$260$	0	0
$261$	−20.2056	−1.25070
$262$	0	0
$263$	4.85286	0.299240	0.149620	−	0.988744i	$-0.452195\pi$
$263$	4.85286	0.299240	0.149620	+	0.988744i	$0.452195\pi$
$264$	0	0
$265$	−6.58123	−0.404282
$266$	0	0
$267$	−3.99194	−0.244303
$268$	0	0
$269$	14.9586	0.912042	0.456021	−	0.889969i	$-0.349274\pi$
$269$	14.9586	0.912042	0.456021	+	0.889969i	$0.349274\pi$
$270$	0	0
$271$	28.3951	1.72488	0.862441	−	0.506157i	$-0.168934\pi$
$271$	28.3951	1.72488	0.862441	+	0.506157i	$0.168934\pi$
$272$	0	0
$273$	1.02444	0.0620017
$274$	0	0
$275$	−1.80842	−0.109052
$276$	0	0
$277$	3.71088	0.222965	0.111483	−	0.993766i	$-0.464440\pi$
$277$	3.71088	0.222965	0.111483	+	0.993766i	$0.464440\pi$
$278$	0	0
$279$	7.10646	0.425453
$280$	0	0
$281$	12.6397	0.754022	0.377011	−	0.926209i	$-0.376952\pi$
$281$	12.6397	0.754022	0.377011	+	0.926209i	$0.376952\pi$
$282$	0	0
$283$	−11.3494	−0.674655	−0.337327	−	0.941387i	$-0.609523\pi$
$283$	−11.3494	−0.674655	−0.337327	+	0.941387i	$0.609523\pi$
$284$	0	0
$285$	0.259546	0.0153742
$286$	0	0
$287$	15.7544	0.929954
$288$	0	0
$289$	−12.1765	−0.716266
$290$	0	0
$291$	−10.0355	−0.588289
$292$	0	0
$293$	−9.60848	−0.561334	−0.280667	−	0.959805i	$-0.590556\pi$
$293$	−9.60848	−0.561334	−0.280667	+	0.959805i	$0.590556\pi$
$294$	0	0
$295$	2.68514	0.156335
$296$	0	0
$297$	5.58068	0.323824
$298$	0	0
$299$	−8.23687	−0.476350
$300$	0	0
$301$	7.71920	0.444927
$302$	0	0
$303$	−4.96746	−0.285373
$304$	0	0
$305$	−4.35751	−0.249510
$306$	0	0
$307$	1.81552	0.103617	0.0518087	−	0.998657i	$-0.483501\pi$
$307$	1.81552	0.103617	0.0518087	+	0.998657i	$0.483501\pi$
$308$	0	0
$309$	1.15242	0.0655587
$310$	0	0
$311$	16.6592	0.944654	0.472327	−	0.881423i	$-0.343414\pi$
$311$	16.6592	0.944654	0.472327	+	0.881423i	$0.343414\pi$
$312$	0	0
$313$	−20.5354	−1.16073	−0.580366	−	0.814356i	$-0.697090\pi$
$313$	−20.5354	−1.16073	−0.580366	+	0.814356i	$0.697090\pi$
$314$	0	0
$315$	4.63777	0.261309
$316$	0	0
$317$	−21.0889	−1.18447	−0.592236	−	0.805765i	$-0.701755\pi$
$317$	−21.0889	−1.18447	−0.592236	+	0.805765i	$0.701755\pi$
$318$	0	0
$319$	−13.4951	−0.755578
$320$	0	0
$321$	7.08352	0.395363
$322$	0	0
$323$	1.05431	0.0586633
$324$	0	0
$325$	−1.10623	−0.0613627
$326$	0	0
$327$	−3.02380	−0.167216
$328$	0	0
$329$	5.87496	0.323897
$330$	0	0
$331$	−4.78564	−0.263042	−0.131521	−	0.991313i	$-0.541986\pi$
$331$	−4.78564	−0.263042	−0.131521	+	0.991313i	$0.541986\pi$
$332$	0	0
$333$	−13.8584	−0.759436
$334$	0	0
$335$	−0.770521	−0.0420981
$336$	0	0
$337$	−7.27417	−0.396249	−0.198125	−	0.980177i	$-0.563485\pi$
$337$	−7.27417	−0.396249	−0.198125	+	0.980177i	$0.563485\pi$
$338$	0	0
$339$	3.26828	0.177509
$340$	0	0
$341$	4.74631	0.257027
$342$	0	0
$343$	18.9545	1.02345
$344$	0	0
$345$	4.02572	0.216737
$346$	0	0
$347$	9.03306	0.484920	0.242460	−	0.970161i	$-0.422046\pi$
$347$	9.03306	0.484920	0.242460	+	0.970161i	$0.422046\pi$
$348$	0	0
$349$	−5.77895	−0.309340	−0.154670	−	0.987966i	$-0.549431\pi$
$349$	−5.77895	−0.309340	−0.154670	+	0.987966i	$0.549431\pi$
$350$	0	0
$351$	3.41376	0.182213
$352$	0	0
$353$	−25.8621	−1.37650	−0.688250	−	0.725474i	$-0.741621\pi$
$353$	−25.8621	−1.37650	−0.688250	+	0.725474i	$0.741621\pi$
$354$	0	0
$355$	11.6268	0.617084
$356$	0	0
$357$	−2.03385	−0.107643
$358$	0	0
$359$	−26.3654	−1.39151	−0.695756	−	0.718279i	$-0.744930\pi$
$359$	−26.3654	−1.39151	−0.695756	+	0.718279i	$0.744930\pi$
$360$	0	0
$361$	−18.7696	−0.987871
$362$	0	0
$363$	−4.17911	−0.219347
$364$	0	0
$365$	11.7631	0.615708
$366$	0	0
$367$	−11.8618	−0.619180	−0.309590	−	0.950870i	$-0.600192\pi$
$367$	−11.8618	−0.619180	−0.309590	+	0.950870i	$0.600192\pi$
$368$	0	0
$369$	24.9051	1.29651
$370$	0	0
$371$	11.2725	0.585237
$372$	0	0
$373$	4.82459	0.249808	0.124904	−	0.992169i	$-0.460138\pi$
$373$	4.82459	0.249808	0.124904	+	0.992169i	$0.460138\pi$
$374$	0	0
$375$	0.540664	0.0279197
$376$	0	0
$377$	−8.25506	−0.425157
$378$	0	0
$379$	−28.9084	−1.48492	−0.742462	−	0.669889i	$-0.766342\pi$
$379$	−28.9084	−1.48492	−0.742462	+	0.669889i	$0.766342\pi$
$380$	0	0
$381$	−7.62977	−0.390885
$382$	0	0
$383$	0.806401	0.0412051	0.0206026	−	0.999788i	$-0.493442\pi$
$383$	0.806401	0.0412051	0.0206026	+	0.999788i	$0.493442\pi$
$384$	0	0
$385$	3.09751	0.157863
$386$	0	0
$387$	12.2028	0.620302
$388$	0	0
$389$	−3.48722	−0.176809	−0.0884045	−	0.996085i	$-0.528177\pi$
$389$	−3.48722	−0.176809	−0.0884045	+	0.996085i	$0.528177\pi$
$390$	0	0
$391$	16.3530	0.827004
$392$	0	0
$393$	−7.61407	−0.384079
$394$	0	0
$395$	3.87292	0.194868
$396$	0	0
$397$	18.3095	0.918928	0.459464	−	0.888196i	$-0.348042\pi$
$397$	18.3095	0.918928	0.459464	+	0.888196i	$0.348042\pi$
$398$	0	0
$399$	−0.444556	−0.0222557
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	2.90337	0.144627
$404$	0	0
$405$	6.45460	0.320731
$406$	0	0
$407$	−9.25584	−0.458795
$408$	0	0
$409$	11.6839	0.577733	0.288866	−	0.957369i	$-0.406722\pi$
$409$	11.6839	0.577733	0.288866	+	0.957369i	$0.406722\pi$
$410$	0	0
$411$	−4.38781	−0.216435
$412$	0	0
$413$	−4.59916	−0.226310
$414$	0	0
$415$	−3.01510	−0.148006
$416$	0	0
$417$	−6.61379	−0.323879
$418$	0	0
$419$	2.46746	0.120543	0.0602717	−	0.998182i	$-0.480803\pi$
$419$	2.46746	0.120543	0.0602717	+	0.998182i	$0.480803\pi$
$420$	0	0
$421$	−27.9490	−1.36215	−0.681076	−	0.732213i	$-0.738488\pi$
$421$	−27.9490	−1.36215	−0.681076	+	0.732213i	$0.738488\pi$
$422$	0	0
$423$	9.28734	0.451566
$424$	0	0
$425$	2.19624	0.106533
$426$	0	0
$427$	7.46363	0.361190
$428$	0	0
$429$	1.08162	0.0522210
$430$	0	0
$431$	39.1028	1.88351	0.941757	−	0.336295i	$-0.109174\pi$
$431$	39.1028	1.88351	0.941757	+	0.336295i	$0.109174\pi$
$432$	0	0
$433$	28.4497	1.36721	0.683603	−	0.729854i	$-0.260412\pi$
$433$	28.4497	1.36721	0.683603	+	0.729854i	$0.260412\pi$
$434$	0	0
$435$	4.03461	0.193445
$436$	0	0
$437$	3.57441	0.170987
$438$	0	0
$439$	9.81560	0.468473	0.234237	−	0.972180i	$-0.424741\pi$
$439$	9.81560	0.468473	0.234237	+	0.972180i	$0.424741\pi$
$440$	0	0
$441$	11.0101	0.524291
$442$	0	0
$443$	4.49815	0.213714	0.106857	−	0.994274i	$-0.465921\pi$
$443$	4.49815	0.213714	0.106857	+	0.994274i	$0.465921\pi$
$444$	0	0
$445$	−7.38341	−0.350007
$446$	0	0
$447$	−3.36563	−0.159189
$448$	0	0
$449$	0.668116	0.0315303	0.0157652	−	0.999876i	$-0.494982\pi$
$449$	0.668116	0.0315303	0.0157652	+	0.999876i	$0.494982\pi$
$450$	0	0
$451$	16.6338	0.783254
$452$	0	0
$453$	−8.75891	−0.411530
$454$	0	0
$455$	1.89478	0.0888285
$456$	0	0
$457$	10.6737	0.499295	0.249648	−	0.968337i	$-0.419685\pi$
$457$	10.6737	0.499295	0.249648	+	0.968337i	$0.419685\pi$
$458$	0	0
$459$	−6.77746	−0.316345
$460$	0	0
$461$	4.91610	0.228966	0.114483	−	0.993425i	$-0.463479\pi$
$461$	4.91610	0.228966	0.114483	+	0.993425i	$0.463479\pi$
$462$	0	0
$463$	−16.0776	−0.747190	−0.373595	−	0.927592i	$-0.621875\pi$
$463$	−16.0776	−0.747190	−0.373595	+	0.927592i	$0.621875\pi$
$464$	0	0
$465$	−1.41900	−0.0658046
$466$	0	0
$467$	6.41803	0.296991	0.148495	−	0.988913i	$-0.452557\pi$
$467$	6.41803	0.296991	0.148495	+	0.988913i	$0.452557\pi$
$468$	0	0
$469$	1.31976	0.0609410
$470$	0	0
$471$	5.81455	0.267921
$472$	0	0
$473$	8.15006	0.374740
$474$	0	0
$475$	0.480052	0.0220263
$476$	0	0
$477$	17.8199	0.815916
$478$	0	0
$479$	5.58686	0.255270	0.127635	−	0.991821i	$-0.459261\pi$
$479$	5.58686	0.255270	0.127635	+	0.991821i	$0.459261\pi$
$480$	0	0
$481$	−5.66189	−0.258160
$482$	0	0
$483$	−6.89533	−0.313748
$484$	0	0
$485$	−18.5614	−0.842829
$486$	0	0
$487$	11.2794	0.511120	0.255560	−	0.966793i	$-0.417740\pi$
$487$	11.2794	0.511120	0.255560	+	0.966793i	$0.417740\pi$
$488$	0	0
$489$	−13.4650	−0.608910
$490$	0	0
$491$	−24.3590	−1.09931	−0.549653	−	0.835393i	$-0.685240\pi$
$491$	−24.3590	−1.09931	−0.549653	+	0.835393i	$0.685240\pi$
$492$	0	0
$493$	16.3891	0.738127
$494$	0	0
$495$	4.89664	0.220088
$496$	0	0
$497$	−19.9145	−0.893290
$498$	0	0
$499$	24.4438	1.09425	0.547127	−	0.837049i	$-0.315721\pi$
$499$	24.4438	1.09425	0.547127	+	0.837049i	$0.315721\pi$
$500$	0	0
$501$	−0.0188539	−0.000842331 0
$502$	0	0
$503$	−22.1356	−0.986976	−0.493488	−	0.869753i	$-0.664278\pi$
$503$	−22.1356	−0.986976	−0.493488	+	0.869753i	$0.664278\pi$
$504$	0	0
$505$	−9.18772	−0.408848
$506$	0	0
$507$	−6.36699	−0.282768
$508$	0	0
$509$	−17.3224	−0.767800	−0.383900	−	0.923375i	$-0.625419\pi$
$509$	−17.3224	−0.767800	−0.383900	+	0.923375i	$0.625419\pi$
$510$	0	0
$511$	−20.1480	−0.891296
$512$	0	0
$513$	−1.48141	−0.0654058
$514$	0	0
$515$	2.13149	0.0939245
$516$	0	0
$517$	6.20289	0.272803
$518$	0	0
$519$	−9.98698	−0.438380
$520$	0	0
$521$	−3.11144	−0.136315	−0.0681574	−	0.997675i	$-0.521712\pi$
$521$	−3.11144	−0.136315	−0.0681574	+	0.997675i	$0.521712\pi$
$522$	0	0
$523$	4.85347	0.212227	0.106114	−	0.994354i	$-0.466159\pi$
$523$	4.85347	0.212227	0.106114	+	0.994354i	$0.466159\pi$
$524$	0	0
$525$	−0.926059	−0.0404165
$526$	0	0
$527$	−5.76416	−0.251091
$528$	0	0
$529$	32.4411	1.41048
$530$	0	0
$531$	−7.27051	−0.315513
$532$	0	0
$533$	10.1751	0.440730
$534$	0	0
$535$	13.1015	0.566428
$536$	0	0
$537$	−6.49083	−0.280100
$538$	0	0
$539$	7.35350	0.316738
$540$	0	0
$541$	−28.9610	−1.24513	−0.622566	−	0.782567i	$-0.713910\pi$
$541$	−28.9610	−1.24513	−0.622566	+	0.782567i	$0.713910\pi$
$542$	0	0
$543$	−8.60007	−0.369064
$544$	0	0
$545$	−5.59275	−0.239567
$546$	0	0
$547$	−19.0254	−0.813468	−0.406734	−	0.913547i	$-0.633332\pi$
$547$	−19.0254	−0.813468	−0.406734	+	0.913547i	$0.633332\pi$
$548$	0	0
$549$	11.7988	0.503559
$550$	0	0
$551$	3.58230	0.152611
$552$	0	0
$553$	−6.63362	−0.282090
$554$	0	0
$555$	2.76721	0.117462
$556$	0	0
$557$	−24.2283	−1.02659	−0.513293	−	0.858214i	$-0.671575\pi$
$557$	−24.2283	−1.02659	−0.513293	+	0.858214i	$0.671575\pi$
$558$	0	0
$559$	4.98548	0.210863
$560$	0	0
$561$	−2.14737	−0.0906622
$562$	0	0
$563$	−27.5694	−1.16191	−0.580955	−	0.813935i	$-0.697321\pi$
$563$	−27.5694	−1.16191	−0.580955	+	0.813935i	$0.697321\pi$
$564$	0	0
$565$	6.04494	0.254313
$566$	0	0
$567$	−11.0556	−0.464290
$568$	0	0
$569$	−3.89322	−0.163212	−0.0816061	−	0.996665i	$-0.526005\pi$
$569$	−3.89322	−0.163212	−0.0816061	+	0.996665i	$0.526005\pi$
$570$	0	0
$571$	31.2896	1.30943	0.654715	−	0.755876i	$-0.272789\pi$
$571$	31.2896	1.30943	0.654715	+	0.755876i	$0.272789\pi$
$572$	0	0
$573$	1.95739	0.0817711
$574$	0	0
$575$	7.44588	0.310515
$576$	0	0
$577$	−26.4799	−1.10237	−0.551185	−	0.834383i	$-0.685824\pi$
$577$	−26.4799	−1.10237	−0.551185	+	0.834383i	$0.685824\pi$
$578$	0	0
$579$	−11.3369	−0.471144
$580$	0	0
$581$	5.16433	0.214252
$582$	0	0
$583$	11.9016	0.492916
$584$	0	0
$585$	2.99532	0.123841
$586$	0	0
$587$	44.1806	1.82353	0.911764	−	0.410714i	$-0.134721\pi$
$587$	44.1806	1.82353	0.911764	+	0.410714i	$0.134721\pi$
$588$	0	0
$589$	−1.25992	−0.0519142
$590$	0	0
$591$	7.42468	0.305410
$592$	0	0
$593$	8.80898	0.361741	0.180871	−	0.983507i	$-0.442108\pi$
$593$	8.80898	0.361741	0.180871	+	0.983507i	$0.442108\pi$
$594$	0	0
$595$	−3.76177	−0.154217
$596$	0	0
$597$	−2.40251	−0.0983283
$598$	0	0
$599$	12.0040	0.490472	0.245236	−	0.969463i	$-0.421135\pi$
$599$	12.0040	0.490472	0.245236	+	0.969463i	$0.421135\pi$
$600$	0	0
$601$	−8.28873	−0.338105	−0.169052	−	0.985607i	$-0.554071\pi$
$601$	−8.28873	−0.338105	−0.169052	+	0.985607i	$0.554071\pi$
$602$	0	0
$603$	2.08633	0.0849618
$604$	0	0
$605$	−7.72960	−0.314253
$606$	0	0
$607$	−13.0561	−0.529929	−0.264964	−	0.964258i	$-0.585360\pi$
$607$	−13.0561	−0.529929	−0.264964	+	0.964258i	$0.585360\pi$
$608$	0	0
$609$	−6.91056	−0.280030
$610$	0	0
$611$	3.79437	0.153504
$612$	0	0
$613$	−40.6988	−1.64381	−0.821905	−	0.569625i	$-0.807088\pi$
$613$	−40.6988	−1.64381	−0.821905	+	0.569625i	$0.807088\pi$
$614$	0	0
$615$	−4.97299	−0.200530
$616$	0	0
$617$	19.9198	0.801940	0.400970	−	0.916091i	$-0.368673\pi$
$617$	19.9198	0.801940	0.400970	+	0.916091i	$0.368673\pi$
$618$	0	0
$619$	8.87180	0.356588	0.178294	−	0.983977i	$-0.442942\pi$
$619$	8.87180	0.356588	0.178294	+	0.983977i	$0.442942\pi$
$620$	0	0
$621$	−22.9775	−0.922056
$622$	0	0
$623$	12.6465	0.506669
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.469370	−0.0187448
$628$	0	0
$629$	11.2408	0.448198
$630$	0	0
$631$	30.9735	1.23303	0.616517	−	0.787342i	$-0.288543\pi$
$631$	30.9735	1.23303	0.616517	+	0.787342i	$0.288543\pi$
$632$	0	0
$633$	2.03130	0.0807370
$634$	0	0
$635$	−14.1119	−0.560012
$636$	0	0
$637$	4.49821	0.178226
$638$	0	0
$639$	−31.4816	−1.24539
$640$	0	0
$641$	16.3968	0.647633	0.323817	−	0.946120i	$-0.395034\pi$
$641$	16.3968	0.647633	0.323817	+	0.946120i	$0.395034\pi$
$642$	0	0
$643$	17.6789	0.697188	0.348594	−	0.937274i	$-0.386659\pi$
$643$	17.6789	0.697188	0.348594	+	0.937274i	$0.386659\pi$
$644$	0	0
$645$	−2.43662	−0.0959418
$646$	0	0
$647$	8.86192	0.348398	0.174199	−	0.984710i	$-0.444266\pi$
$647$	8.86192	0.348398	0.174199	+	0.984710i	$0.444266\pi$
$648$	0	0
$649$	−4.85587	−0.190610
$650$	0	0
$651$	2.43049	0.0952586
$652$	0	0
$653$	−11.3151	−0.442794	−0.221397	−	0.975184i	$-0.571062\pi$
$653$	−11.3151	−0.442794	−0.221397	+	0.975184i	$0.571062\pi$
$654$	0	0
$655$	−14.0828	−0.550262
$656$	0	0
$657$	−31.8507	−1.24261
$658$	0	0
$659$	6.60837	0.257426	0.128713	−	0.991682i	$-0.458915\pi$
$659$	6.60837	0.257426	0.128713	+	0.991682i	$0.458915\pi$
$660$	0	0
$661$	49.5721	1.92813	0.964066	−	0.265663i	$-0.0855910\pi$
$661$	49.5721	1.92813	0.964066	+	0.265663i	$0.0855910\pi$
$662$	0	0
$663$	−1.31357	−0.0510148
$664$	0	0
$665$	−0.822242	−0.0318852
$666$	0	0
$667$	55.5636	2.15143
$668$	0	0
$669$	−0.0713129	−0.00275712
$670$	0	0
$671$	7.88023	0.304213
$672$	0	0
$673$	−18.0971	−0.697593	−0.348797	−	0.937198i	$-0.613410\pi$
$673$	−18.0971	−0.697593	−0.348797	+	0.937198i	$0.613410\pi$
$674$	0	0
$675$	−3.08594	−0.118778
$676$	0	0
$677$	−36.5973	−1.40655	−0.703274	−	0.710919i	$-0.748279\pi$
$677$	−36.5973	−1.40655	−0.703274	+	0.710919i	$0.748279\pi$
$678$	0	0
$679$	31.7923	1.22008
$680$	0	0
$681$	9.84185	0.377141
$682$	0	0
$683$	−7.96922	−0.304934	−0.152467	−	0.988309i	$-0.548722\pi$
$683$	−7.96922	−0.304934	−0.152467	+	0.988309i	$0.548722\pi$
$684$	0	0
$685$	−8.11560	−0.310081
$686$	0	0
$687$	14.3987	0.549345
$688$	0	0
$689$	7.28036	0.277360
$690$	0	0
$691$	−10.9134	−0.415164	−0.207582	−	0.978218i	$-0.566559\pi$
$691$	−10.9134	−0.415164	−0.207582	+	0.978218i	$0.566559\pi$
$692$	0	0
$693$	−8.38706	−0.318598
$694$	0	0
$695$	−12.2327	−0.464014
$696$	0	0
$697$	−20.2009	−0.765164
$698$	0	0
$699$	−7.28774	−0.275647
$700$	0	0
$701$	−2.35912	−0.0891026	−0.0445513	−	0.999007i	$-0.514186\pi$
$701$	−2.35912	−0.0891026	−0.0445513	+	0.999007i	$0.514186\pi$
$702$	0	0
$703$	2.45699	0.0926671
$704$	0	0
$705$	−1.85447	−0.0698435
$706$	0	0
$707$	15.7369	0.591847
$708$	0	0
$709$	26.4771	0.994367	0.497184	−	0.867645i	$-0.334368\pi$
$709$	26.4771	0.994367	0.497184	+	0.867645i	$0.334368\pi$
$710$	0	0
$711$	−10.4867	−0.393280
$712$	0	0
$713$	−19.5421	−0.731858
$714$	0	0
$715$	2.00054	0.0748158
$716$	0	0
$717$	−3.45262	−0.128941
$718$	0	0
$719$	−25.4734	−0.949999	−0.474999	−	0.879986i	$-0.657552\pi$
$719$	−25.4734	−0.949999	−0.474999	+	0.879986i	$0.657552\pi$
$720$	0	0
$721$	−3.65085	−0.135965
$722$	0	0
$723$	4.40454	0.163806
$724$	0	0
$725$	7.46233	0.277144
$726$	0	0
$727$	15.6029	0.578681	0.289340	−	0.957226i	$-0.406564\pi$
$727$	15.6029	0.578681	0.289340	+	0.957226i	$0.406564\pi$
$728$	0	0
$729$	−12.4716	−0.461912
$730$	0	0
$731$	−9.89785	−0.366085
$732$	0	0
$733$	−18.2341	−0.673492	−0.336746	−	0.941595i	$-0.609326\pi$
$733$	−18.2341	−0.673492	−0.336746	+	0.941595i	$0.609326\pi$
$734$	0	0
$735$	−2.19847	−0.0810919
$736$	0	0
$737$	1.39343	0.0513276
$738$	0	0
$739$	10.9196	0.401684	0.200842	−	0.979624i	$-0.435632\pi$
$739$	10.9196	0.401684	0.200842	+	0.979624i	$0.435632\pi$
$740$	0	0
$741$	−0.287119	−0.0105476
$742$	0	0
$743$	−22.7178	−0.833435	−0.416718	−	0.909036i	$-0.636820\pi$
$743$	−22.7178	−0.833435	−0.416718	+	0.909036i	$0.636820\pi$
$744$	0	0
$745$	−6.22500	−0.228066
$746$	0	0
$747$	8.16394	0.298703
$748$	0	0
$749$	−22.4406	−0.819960
$750$	0	0
$751$	−24.0597	−0.877951	−0.438975	−	0.898499i	$-0.644658\pi$
$751$	−24.0597	−0.877951	−0.438975	+	0.898499i	$0.644658\pi$
$752$	0	0
$753$	1.81047	0.0659772
$754$	0	0
$755$	−16.2003	−0.589589
$756$	0	0
$757$	−20.0187	−0.727590	−0.363795	−	0.931479i	$-0.618519\pi$
$757$	−20.0187	−0.727590	−0.363795	+	0.931479i	$0.618519\pi$
$758$	0	0
$759$	−7.28020	−0.264255
$760$	0	0
$761$	−31.4399	−1.13970	−0.569848	−	0.821750i	$-0.692998\pi$
$761$	−31.4399	−1.13970	−0.569848	+	0.821750i	$0.692998\pi$
$762$	0	0
$763$	9.57937	0.346797
$764$	0	0
$765$	−5.94673	−0.215004
$766$	0	0
$767$	−2.97039	−0.107254
$768$	0	0
$769$	32.1784	1.16038	0.580191	−	0.814480i	$-0.302978\pi$
$769$	32.1784	1.16038	0.580191	+	0.814480i	$0.302978\pi$
$770$	0	0
$771$	5.07563	0.182794
$772$	0	0
$773$	22.7452	0.818087	0.409044	−	0.912515i	$-0.365862\pi$
$773$	22.7452	0.818087	0.409044	+	0.912515i	$0.365862\pi$
$774$	0	0
$775$	−2.62456	−0.0942768
$776$	0	0
$777$	−4.73974	−0.170037
$778$	0	0
$779$	−4.41549	−0.158201
$780$	0	0
$781$	−21.0261	−0.752374
$782$	0	0
$783$	−23.0283	−0.822963
$784$	0	0
$785$	10.7545	0.383844
$786$	0	0
$787$	−26.3472	−0.939176	−0.469588	−	0.882886i	$-0.655598\pi$
$787$	−26.3472	−0.939176	−0.469588	+	0.882886i	$0.655598\pi$
$788$	0	0
$789$	2.62376	0.0934085
$790$	0	0
$791$	−10.3539	−0.368142
$792$	0	0
$793$	4.82041	0.171178
$794$	0	0
$795$	−3.55823	−0.126197
$796$	0	0
$797$	−39.8082	−1.41008	−0.705039	−	0.709168i	$-0.749071\pi$
$797$	−39.8082	−1.41008	−0.705039	+	0.709168i	$0.749071\pi$
$798$	0	0
$799$	−7.53310	−0.266502
$800$	0	0
$801$	19.9919	0.706380
$802$	0	0
$803$	−21.2726	−0.750695
$804$	0	0
$805$	−12.7535	−0.449500
$806$	0	0
$807$	8.08757	0.284696
$808$	0	0
$809$	5.20233	0.182904	0.0914521	−	0.995809i	$-0.470849\pi$
$809$	5.20233	0.182904	0.0914521	+	0.995809i	$0.470849\pi$
$810$	0	0
$811$	−37.2838	−1.30921	−0.654605	−	0.755971i	$-0.727165\pi$
$811$	−37.2838	−1.30921	−0.654605	+	0.755971i	$0.727165\pi$
$812$	0	0
$813$	15.3522	0.538426
$814$	0	0
$815$	−24.9046	−0.872372
$816$	0	0
$817$	−2.16346	−0.0756898
$818$	0	0
$819$	−5.13045	−0.179272
$820$	0	0
$821$	−4.21473	−0.147095	−0.0735476	−	0.997292i	$-0.523432\pi$
$821$	−4.21473	−0.147095	−0.0735476	+	0.997292i	$0.523432\pi$
$822$	0	0
$823$	6.79217	0.236760	0.118380	−	0.992968i	$-0.462230\pi$
$823$	6.79217	0.236760	0.118380	+	0.992968i	$0.462230\pi$
$824$	0	0
$825$	−0.977749	−0.0340409
$826$	0	0
$827$	44.0552	1.53195	0.765976	−	0.642870i	$-0.222256\pi$
$827$	44.0552	1.53195	0.765976	+	0.642870i	$0.222256\pi$
$828$	0	0
$829$	18.5777	0.645229	0.322614	−	0.946531i	$-0.395438\pi$
$829$	18.5777	0.645229	0.322614	+	0.946531i	$0.395438\pi$
$830$	0	0
$831$	2.00634	0.0695992
$832$	0	0
$833$	−8.93046	−0.309422
$834$	0	0
$835$	−0.0348718	−0.00120679
$836$	0	0
$837$	8.09921	0.279950
$838$	0	0
$839$	35.3484	1.22036	0.610181	−	0.792262i	$-0.291097\pi$
$839$	35.3484	1.22036	0.610181	+	0.792262i	$0.291097\pi$
$840$	0	0
$841$	26.6863	0.920218
$842$	0	0
$843$	6.83383	0.235370
$844$	0	0
$845$	−11.7763	−0.405115
$846$	0	0
$847$	13.2394	0.454912
$848$	0	0
$849$	−6.13623	−0.210595
$850$	0	0
$851$	38.1093	1.30637
$852$	0	0
$853$	−9.31295	−0.318869	−0.159435	−	0.987208i	$-0.550967\pi$
$853$	−9.31295	−0.318869	−0.159435	+	0.987208i	$0.550967\pi$
$854$	0	0
$855$	−1.29983	−0.0444532
$856$	0	0
$857$	29.2450	0.998992	0.499496	−	0.866316i	$-0.333519\pi$
$857$	29.2450	0.998992	0.499496	+	0.866316i	$0.333519\pi$
$858$	0	0
$859$	57.3204	1.95575	0.977874	−	0.209195i	$-0.0670843\pi$
$859$	57.3204	1.95575	0.977874	+	0.209195i	$0.0670843\pi$
$860$	0	0
$861$	8.51784	0.290287
$862$	0	0
$863$	−39.2157	−1.33492	−0.667459	−	0.744647i	$-0.732618\pi$
$863$	−39.2157	−1.33492	−0.667459	+	0.744647i	$0.732618\pi$
$864$	0	0
$865$	−18.4717	−0.628057
$866$	0	0
$867$	−6.58340	−0.223584
$868$	0	0
$869$	−7.00389	−0.237591
$870$	0	0
$871$	0.852375	0.0288816
$872$	0	0
$873$	50.2583	1.70099
$874$	0	0
$875$	−1.71282	−0.0579039
$876$	0	0
$877$	50.7803	1.71473	0.857364	−	0.514710i	$-0.172100\pi$
$877$	50.7803	1.71473	0.857364	+	0.514710i	$0.172100\pi$
$878$	0	0
$879$	−5.19496	−0.175222
$880$	0	0
$881$	26.8086	0.903206	0.451603	−	0.892219i	$-0.350852\pi$
$881$	26.8086	0.903206	0.451603	+	0.892219i	$0.350852\pi$
$882$	0	0
$883$	−27.6148	−0.929310	−0.464655	−	0.885492i	$-0.653822\pi$
$883$	−27.6148	−0.929310	−0.464655	+	0.885492i	$0.653822\pi$
$884$	0	0
$885$	1.45176	0.0488003
$886$	0	0
$887$	8.31821	0.279298	0.139649	−	0.990201i	$-0.455403\pi$
$887$	8.31821	0.279298	0.139649	+	0.990201i	$0.455403\pi$
$888$	0	0
$889$	24.1711	0.810672
$890$	0	0
$891$	−11.6726	−0.391048
$892$	0	0
$893$	−1.64657	−0.0551005
$894$	0	0
$895$	−12.0053	−0.401293
$896$	0	0
$897$	−4.45337	−0.148694
$898$	0	0
$899$	−19.5853	−0.653206
$900$	0	0
$901$	−14.4540	−0.481531
$902$	0	0
$903$	4.17349	0.138885
$904$	0	0
$905$	−15.9065	−0.528750
$906$	0	0
$907$	−29.5911	−0.982555	−0.491277	−	0.871003i	$-0.663470\pi$
$907$	−29.5911	−0.982555	−0.491277	+	0.871003i	$0.663470\pi$
$908$	0	0
$909$	24.8774	0.825132
$910$	0	0
$911$	−23.6659	−0.784088	−0.392044	−	0.919947i	$-0.628232\pi$
$911$	−23.6659	−0.784088	−0.392044	+	0.919947i	$0.628232\pi$
$912$	0	0
$913$	5.45258	0.180454
$914$	0	0
$915$	−2.35595	−0.0778852
$916$	0	0
$917$	24.1213	0.796557
$918$	0	0
$919$	26.0953	0.860803	0.430402	−	0.902638i	$-0.358372\pi$
$919$	26.0953	0.860803	0.430402	+	0.902638i	$0.358372\pi$
$920$	0	0
$921$	0.981587	0.0323444
$922$	0	0
$923$	−12.8619	−0.423354
$924$	0	0
$925$	5.11818	0.168285
$926$	0	0
$927$	−5.77139	−0.189557
$928$	0	0
$929$	7.29609	0.239377	0.119688	−	0.992811i	$-0.461810\pi$
$929$	7.29609	0.239377	0.119688	+	0.992811i	$0.461810\pi$
$930$	0	0
$931$	−1.95201	−0.0639745
$932$	0	0
$933$	9.00700	0.294876
$934$	0	0
$935$	−3.97174	−0.129890
$936$	0	0
$937$	−10.7786	−0.352122	−0.176061	−	0.984379i	$-0.556336\pi$
$937$	−10.7786	−0.352122	−0.176061	+	0.984379i	$0.556336\pi$
$938$	0	0
$939$	−11.1028	−0.362325
$940$	0	0
$941$	−24.6832	−0.804651	−0.402325	−	0.915497i	$-0.631798\pi$
$941$	−24.6832	−0.804651	−0.402325	+	0.915497i	$0.631798\pi$
$942$	0	0
$943$	−68.4868	−2.23024
$944$	0	0
$945$	5.28565	0.171942
$946$	0	0
$947$	30.2435	0.982781	0.491391	−	0.870939i	$-0.336489\pi$
$947$	30.2435	0.982781	0.491391	+	0.870939i	$0.336489\pi$
$948$	0	0
$949$	−13.0127	−0.422410
$950$	0	0
$951$	−11.4020	−0.369735
$952$	0	0
$953$	21.3041	0.690109	0.345054	−	0.938583i	$-0.387860\pi$
$953$	21.3041	0.690109	0.345054	+	0.938583i	$0.387860\pi$
$954$	0	0
$955$	3.62034	0.117152
$956$	0	0
$957$	−7.29628	−0.235855
$958$	0	0
$959$	13.9006	0.448873
$960$	0	0
$961$	−24.1117	−0.777797
$962$	0	0
$963$	−35.4748	−1.14316
$964$	0	0
$965$	−20.9684	−0.674998
$966$	0	0
$967$	−13.8637	−0.445827	−0.222913	−	0.974838i	$-0.571557\pi$
$967$	−13.8637	−0.445827	−0.222913	+	0.974838i	$0.571557\pi$
$968$	0	0
$969$	0.570027	0.0183119
$970$	0	0
$971$	15.8671	0.509201	0.254600	−	0.967046i	$-0.418056\pi$
$971$	15.8671	0.509201	0.254600	+	0.967046i	$0.418056\pi$
$972$	0	0
$973$	20.9525	0.671705
$974$	0	0
$975$	−0.598099	−0.0191545
$976$	0	0
$977$	−5.32861	−0.170477	−0.0852386	−	0.996361i	$-0.527165\pi$
$977$	−5.32861	−0.170477	−0.0852386	+	0.996361i	$0.527165\pi$
$978$	0	0
$979$	13.3523	0.426743
$980$	0	0
$981$	15.1434	0.483491
$982$	0	0
$983$	−29.0567	−0.926764	−0.463382	−	0.886159i	$-0.653364\pi$
$983$	−29.0567	−0.926764	−0.463382	+	0.886159i	$0.653364\pi$
$984$	0	0
$985$	13.7325	0.437555
$986$	0	0
$987$	3.17638	0.101105
$988$	0	0
$989$	−33.5565	−1.06703
$990$	0	0
$991$	25.8381	0.820776	0.410388	−	0.911911i	$-0.365393\pi$
$991$	25.8381	0.820776	0.410388	+	0.911911i	$0.365393\pi$
$992$	0	0
$993$	−2.58742	−0.0821093
$994$	0	0
$995$	−4.44364	−0.140873
$996$	0	0
$997$	−0.226115	−0.00716113	−0.00358056	−	0.999994i	$-0.501140\pi$
$997$	−0.226115	−0.00716113	−0.00358056	+	0.999994i	$0.501140\pi$
$998$	0	0
$999$	−15.7944	−0.499712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.19	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.19	✓	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)

\(f(q)\)	\(=\)	\(q+0.540664 q^{3} +1.00000 q^{5} -1.71282 q^{7} -2.70768 q^{9} +O(q^{10})\)	\(q+0.540664 q^{3} +1.00000 q^{5} -1.71282 q^{7} -2.70768 q^{9} -1.80842 q^{11} -1.10623 q^{13} +0.540664 q^{15} +2.19624 q^{17} +0.480052 q^{19} -0.926059 q^{21} +7.44588 q^{23} +1.00000 q^{25} -3.08594 q^{27} +7.46233 q^{29} -2.62456 q^{31} -0.977749 q^{33} -1.71282 q^{35} +5.11818 q^{37} -0.598099 q^{39} -9.19794 q^{41} -4.50672 q^{43} -2.70768 q^{45} -3.42999 q^{47} -4.06625 q^{49} +1.18743 q^{51} -6.58123 q^{53} -1.80842 q^{55} +0.259546 q^{57} +2.68514 q^{59} -4.35751 q^{61} +4.63777 q^{63} -1.10623 q^{65} -0.770521 q^{67} +4.02572 q^{69} +11.6268 q^{71} +11.7631 q^{73} +0.540664 q^{75} +3.09751 q^{77} +3.87292 q^{79} +6.45460 q^{81} -3.01510 q^{83} +2.19624 q^{85} +4.03461 q^{87} -7.38341 q^{89} +1.89478 q^{91} -1.41900 q^{93} +0.480052 q^{95} -18.5614 q^{97} +4.89664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 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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