LMFDB - Embedded newform 8020.2.a.e.1.3

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:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-3.08956 q^{3} -1.00000 q^{5} +4.75064 q^{7} +6.54541 q^{9} +O(q^{10})$	$q-3.08956 q^{3} -1.00000 q^{5} +4.75064 q^{7} +6.54541 q^{9} -1.40093 q^{11} -6.21999 q^{13} +3.08956 q^{15} -4.39587 q^{17} -6.85898 q^{19} -14.6774 q^{21} +4.21440 q^{23} +1.00000 q^{25} -10.9538 q^{27} -6.17522 q^{29} -2.88640 q^{31} +4.32826 q^{33} -4.75064 q^{35} -6.77805 q^{37} +19.2170 q^{39} -4.90908 q^{41} +2.47055 q^{43} -6.54541 q^{45} -4.37622 q^{47} +15.5685 q^{49} +13.5813 q^{51} -0.987349 q^{53} +1.40093 q^{55} +21.1913 q^{57} +1.30307 q^{59} +3.29284 q^{61} +31.0948 q^{63} +6.21999 q^{65} +7.69939 q^{67} -13.0207 q^{69} +6.03743 q^{71} -0.131310 q^{73} -3.08956 q^{75} -6.65530 q^{77} -11.9786 q^{79} +14.2061 q^{81} +10.8246 q^{83} +4.39587 q^{85} +19.0788 q^{87} +9.62028 q^{89} -29.5489 q^{91} +8.91772 q^{93} +6.85898 q^{95} -13.3779 q^{97} -9.16964 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * 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
$3$	−3.08956	−1.78376	−0.891880	−	0.452272i	$-0.850614\pi$
$3$	−3.08956	−1.78376	−0.891880	+	0.452272i	$0.850614\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.75064	1.79557	0.897786	−	0.440432i	$-0.145175\pi$
$7$	4.75064	1.79557	0.897786	+	0.440432i	$0.145175\pi$
$8$	0	0
$9$	6.54541	2.18180
$10$	0	0
$11$	−1.40093	−0.422396	−0.211198	−	0.977443i	$-0.567736\pi$
$11$	−1.40093	−0.422396	−0.211198	+	0.977443i	$0.567736\pi$
$12$	0	0
$13$	−6.21999	−1.72511	−0.862557	−	0.505960i	$-0.831138\pi$
$13$	−6.21999	−1.72511	−0.862557	+	0.505960i	$0.831138\pi$
$14$	0	0
$15$	3.08956	0.797722
$16$	0	0
$17$	−4.39587	−1.06615	−0.533077	−	0.846067i	$-0.678964\pi$
$17$	−4.39587	−1.06615	−0.533077	+	0.846067i	$0.678964\pi$
$18$	0	0
$19$	−6.85898	−1.57356	−0.786779	−	0.617234i	$-0.788253\pi$
$19$	−6.85898	−1.57356	−0.786779	+	0.617234i	$0.788253\pi$
$20$	0	0
$21$	−14.6774	−3.20287
$22$	0	0
$23$	4.21440	0.878763	0.439381	−	0.898301i	$-0.355198\pi$
$23$	4.21440	0.878763	0.439381	+	0.898301i	$0.355198\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−10.9538	−2.10805
$28$	0	0
$29$	−6.17522	−1.14671	−0.573355	−	0.819307i	$-0.694358\pi$
$29$	−6.17522	−1.14671	−0.573355	+	0.819307i	$0.694358\pi$
$30$	0	0
$31$	−2.88640	−0.518413	−0.259206	−	0.965822i	$-0.583461\pi$
$31$	−2.88640	−0.518413	−0.259206	+	0.965822i	$0.583461\pi$
$32$	0	0
$33$	4.32826	0.753453
$34$	0	0
$35$	−4.75064	−0.803004
$36$	0	0
$37$	−6.77805	−1.11430	−0.557152	−	0.830411i	$-0.688106\pi$
$37$	−6.77805	−1.11430	−0.557152	+	0.830411i	$0.688106\pi$
$38$	0	0
$39$	19.2170	3.07719
$40$	0	0
$41$	−4.90908	−0.766670	−0.383335	−	0.923609i	$-0.625224\pi$
$41$	−4.90908	−0.766670	−0.383335	+	0.923609i	$0.625224\pi$
$42$	0	0
$43$	2.47055	0.376755	0.188378	−	0.982097i	$-0.439677\pi$
$43$	2.47055	0.376755	0.188378	+	0.982097i	$0.439677\pi$
$44$	0	0
$45$	−6.54541	−0.975731
$46$	0	0
$47$	−4.37622	−0.638337	−0.319168	−	0.947698i	$-0.603404\pi$
$47$	−4.37622	−0.638337	−0.319168	+	0.947698i	$0.603404\pi$
$48$	0	0
$49$	15.5685	2.22408
$50$	0	0
$51$	13.5813	1.90176
$52$	0	0
$53$	−0.987349	−0.135623	−0.0678114	−	0.997698i	$-0.521602\pi$
$53$	−0.987349	−0.135623	−0.0678114	+	0.997698i	$0.521602\pi$
$54$	0	0
$55$	1.40093	0.188901
$56$	0	0
$57$	21.1913	2.80685
$58$	0	0
$59$	1.30307	0.169645	0.0848225	−	0.996396i	$-0.472968\pi$
$59$	1.30307	0.169645	0.0848225	+	0.996396i	$0.472968\pi$
$60$	0	0
$61$	3.29284	0.421605	0.210802	−	0.977529i	$-0.432392\pi$
$61$	3.29284	0.421605	0.210802	+	0.977529i	$0.432392\pi$
$62$	0	0
$63$	31.0948	3.91758
$64$	0	0
$65$	6.21999	0.771494
$66$	0	0
$67$	7.69939	0.940630	0.470315	−	0.882499i	$-0.344140\pi$
$67$	7.69939	0.940630	0.470315	+	0.882499i	$0.344140\pi$
$68$	0	0
$69$	−13.0207	−1.56750
$70$	0	0
$71$	6.03743	0.716511	0.358255	−	0.933624i	$-0.383372\pi$
$71$	6.03743	0.716511	0.358255	+	0.933624i	$0.383372\pi$
$72$	0	0
$73$	−0.131310	−0.0153687	−0.00768434	−	0.999970i	$-0.502446\pi$
$73$	−0.131310	−0.0153687	−0.00768434	+	0.999970i	$0.502446\pi$
$74$	0	0
$75$	−3.08956	−0.356752
$76$	0	0
$77$	−6.65530	−0.758441
$78$	0	0
$79$	−11.9786	−1.34770	−0.673850	−	0.738869i	$-0.735360\pi$
$79$	−11.9786	−1.34770	−0.673850	+	0.738869i	$0.735360\pi$
$80$	0	0
$81$	14.2061	1.57846
$82$	0	0
$83$	10.8246	1.18816	0.594080	−	0.804406i	$-0.297516\pi$
$83$	10.8246	1.18816	0.594080	+	0.804406i	$0.297516\pi$
$84$	0	0
$85$	4.39587	0.476799
$86$	0	0
$87$	19.0788	2.04546
$88$	0	0
$89$	9.62028	1.01975	0.509874	−	0.860249i	$-0.329692\pi$
$89$	9.62028	1.01975	0.509874	+	0.860249i	$0.329692\pi$
$90$	0	0
$91$	−29.5489	−3.09756
$92$	0	0
$93$	8.91772	0.924724
$94$	0	0
$95$	6.85898	0.703717
$96$	0	0
$97$	−13.3779	−1.35832	−0.679159	−	0.733991i	$-0.737655\pi$
$97$	−13.3779	−1.35832	−0.679159	+	0.733991i	$0.737655\pi$
$98$	0	0
$99$	−9.16964	−0.921583
$100$	0	0
$101$	18.0389	1.79494	0.897470	−	0.441076i	$-0.145403\pi$
$101$	18.0389	1.79494	0.897470	+	0.441076i	$0.145403\pi$
$102$	0	0
$103$	−2.55939	−0.252184	−0.126092	−	0.992019i	$-0.540244\pi$
$103$	−2.55939	−0.252184	−0.126092	+	0.992019i	$0.540244\pi$
$104$	0	0
$105$	14.6774	1.43237
$106$	0	0
$107$	−16.8747	−1.63134	−0.815671	−	0.578517i	$-0.803632\pi$
$107$	−16.8747	−1.63134	−0.815671	+	0.578517i	$0.803632\pi$
$108$	0	0
$109$	11.0710	1.06041	0.530204	−	0.847870i	$-0.322115\pi$
$109$	11.0710	1.06041	0.530204	+	0.847870i	$0.322115\pi$
$110$	0	0
$111$	20.9412	1.98765
$112$	0	0
$113$	1.99700	0.187862	0.0939309	−	0.995579i	$-0.470057\pi$
$113$	1.99700	0.187862	0.0939309	+	0.995579i	$0.470057\pi$
$114$	0	0
$115$	−4.21440	−0.392995
$116$	0	0
$117$	−40.7123	−3.76386
$118$	0	0
$119$	−20.8832	−1.91436
$120$	0	0
$121$	−9.03740	−0.821582
$122$	0	0
$123$	15.1669	1.36756
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.7019	−1.39332	−0.696659	−	0.717402i	$-0.745331\pi$
$127$	−15.7019	−1.39332	−0.696659	+	0.717402i	$0.745331\pi$
$128$	0	0
$129$	−7.63292	−0.672041
$130$	0	0
$131$	−3.11643	−0.272283	−0.136142	−	0.990689i	$-0.543470\pi$
$131$	−3.11643	−0.272283	−0.136142	+	0.990689i	$0.543470\pi$
$132$	0	0
$133$	−32.5845	−2.82544
$134$	0	0
$135$	10.9538	0.942749
$136$	0	0
$137$	−14.8990	−1.27291	−0.636455	−	0.771314i	$-0.719600\pi$
$137$	−14.8990	−1.27291	−0.636455	+	0.771314i	$0.719600\pi$
$138$	0	0
$139$	12.0631	1.02318	0.511589	−	0.859230i	$-0.329057\pi$
$139$	12.0631	1.02318	0.511589	+	0.859230i	$0.329057\pi$
$140$	0	0
$141$	13.5206	1.13864
$142$	0	0
$143$	8.71375	0.728680
$144$	0	0
$145$	6.17522	0.512825
$146$	0	0
$147$	−48.1000	−3.96722
$148$	0	0
$149$	11.0942	0.908875	0.454437	−	0.890779i	$-0.349840\pi$
$149$	11.0942	0.908875	0.454437	+	0.890779i	$0.349840\pi$
$150$	0	0
$151$	0.328363	0.0267218	0.0133609	−	0.999911i	$-0.495747\pi$
$151$	0.328363	0.0267218	0.0133609	+	0.999911i	$0.495747\pi$
$152$	0	0
$153$	−28.7727	−2.32614
$154$	0	0
$155$	2.88640	0.231841
$156$	0	0
$157$	20.7719	1.65778	0.828890	−	0.559412i	$-0.188973\pi$
$157$	20.7719	1.65778	0.828890	+	0.559412i	$0.188973\pi$
$158$	0	0
$159$	3.05048	0.241919
$160$	0	0
$161$	20.0211	1.57788
$162$	0	0
$163$	−18.2833	−1.43206	−0.716029	−	0.698070i	$-0.754042\pi$
$163$	−18.2833	−1.43206	−0.716029	+	0.698070i	$0.754042\pi$
$164$	0	0
$165$	−4.32826	−0.336954
$166$	0	0
$167$	−14.0115	−1.08424	−0.542122	−	0.840300i	$-0.682379\pi$
$167$	−14.0115	−1.08424	−0.542122	+	0.840300i	$0.682379\pi$
$168$	0	0
$169$	25.6882	1.97602
$170$	0	0
$171$	−44.8948	−3.43319
$172$	0	0
$173$	15.5242	1.18028	0.590142	−	0.807300i	$-0.299072\pi$
$173$	15.5242	1.18028	0.590142	+	0.807300i	$0.299072\pi$
$174$	0	0
$175$	4.75064	0.359114
$176$	0	0
$177$	−4.02591	−0.302606
$178$	0	0
$179$	−6.38806	−0.477466	−0.238733	−	0.971085i	$-0.576732\pi$
$179$	−6.38806	−0.477466	−0.238733	+	0.971085i	$0.576732\pi$
$180$	0	0
$181$	−5.87090	−0.436381	−0.218190	−	0.975906i	$-0.570015\pi$
$181$	−5.87090	−0.436381	−0.218190	+	0.975906i	$0.570015\pi$
$182$	0	0
$183$	−10.1734	−0.752042
$184$	0	0
$185$	6.77805	0.498332
$186$	0	0
$187$	6.15829	0.450339
$188$	0	0
$189$	−52.0373	−3.78516
$190$	0	0
$191$	19.4999	1.41097	0.705483	−	0.708727i	$-0.250730\pi$
$191$	19.4999	1.41097	0.705483	+	0.708727i	$0.250730\pi$
$192$	0	0
$193$	2.36368	0.170141	0.0850706	−	0.996375i	$-0.472888\pi$
$193$	2.36368	0.170141	0.0850706	+	0.996375i	$0.472888\pi$
$194$	0	0
$195$	−19.2170	−1.37616
$196$	0	0
$197$	4.06060	0.289306	0.144653	−	0.989482i	$-0.453793\pi$
$197$	4.06060	0.289306	0.144653	+	0.989482i	$0.453793\pi$
$198$	0	0
$199$	−26.1596	−1.85441	−0.927203	−	0.374559i	$-0.877794\pi$
$199$	−26.1596	−1.85441	−0.927203	+	0.374559i	$0.877794\pi$
$200$	0	0
$201$	−23.7878	−1.67786
$202$	0	0
$203$	−29.3362	−2.05900
$204$	0	0
$205$	4.90908	0.342865
$206$	0	0
$207$	27.5850	1.91729
$208$	0	0
$209$	9.60894	0.664664
$210$	0	0
$211$	25.2205	1.73625	0.868127	−	0.496342i	$-0.165324\pi$
$211$	25.2205	1.73625	0.868127	+	0.496342i	$0.165324\pi$
$212$	0	0
$213$	−18.6530	−1.27808
$214$	0	0
$215$	−2.47055	−0.168490
$216$	0	0
$217$	−13.7122	−0.930847
$218$	0	0
$219$	0.405691	0.0274141
$220$	0	0
$221$	27.3422	1.83924
$222$	0	0
$223$	6.53564	0.437659	0.218829	−	0.975763i	$-0.429776\pi$
$223$	6.53564	0.437659	0.218829	+	0.975763i	$0.429776\pi$
$224$	0	0
$225$	6.54541	0.436360
$226$	0	0
$227$	−25.5396	−1.69512	−0.847560	−	0.530699i	$-0.821929\pi$
$227$	−25.5396	−1.69512	−0.847560	+	0.530699i	$0.821929\pi$
$228$	0	0
$229$	−24.9512	−1.64882	−0.824410	−	0.565993i	$-0.808493\pi$
$229$	−24.9512	−1.64882	−0.824410	+	0.565993i	$0.808493\pi$
$230$	0	0
$231$	20.5620	1.35288
$232$	0	0
$233$	17.2224	1.12828	0.564140	−	0.825679i	$-0.309208\pi$
$233$	17.2224	1.12828	0.564140	+	0.825679i	$0.309208\pi$
$234$	0	0
$235$	4.37622	0.285473
$236$	0	0
$237$	37.0087	2.40397
$238$	0	0
$239$	21.4161	1.38529	0.692647	−	0.721277i	$-0.256444\pi$
$239$	21.4161	1.38529	0.692647	+	0.721277i	$0.256444\pi$
$240$	0	0
$241$	12.5369	0.807575	0.403787	−	0.914853i	$-0.367694\pi$
$241$	12.5369	0.807575	0.403787	+	0.914853i	$0.367694\pi$
$242$	0	0
$243$	−11.0294	−0.707539
$244$	0	0
$245$	−15.5685	−0.994638
$246$	0	0
$247$	42.6628	2.71457
$248$	0	0
$249$	−33.4434	−2.11939
$250$	0	0
$251$	−10.9838	−0.693292	−0.346646	−	0.937996i	$-0.612680\pi$
$251$	−10.9838	−0.693292	−0.346646	+	0.937996i	$0.612680\pi$
$252$	0	0
$253$	−5.90407	−0.371186
$254$	0	0
$255$	−13.5813	−0.850494
$256$	0	0
$257$	19.8262	1.23673	0.618363	−	0.785893i	$-0.287796\pi$
$257$	19.8262	1.23673	0.618363	+	0.785893i	$0.287796\pi$
$258$	0	0
$259$	−32.2000	−2.00081
$260$	0	0
$261$	−40.4194	−2.50190
$262$	0	0
$263$	21.9113	1.35111	0.675554	−	0.737311i	$-0.263905\pi$
$263$	21.9113	1.35111	0.675554	+	0.737311i	$0.263905\pi$
$264$	0	0
$265$	0.987349	0.0606524
$266$	0	0
$267$	−29.7225	−1.81899
$268$	0	0
$269$	−22.4801	−1.37064	−0.685319	−	0.728243i	$-0.740337\pi$
$269$	−22.4801	−1.37064	−0.685319	+	0.728243i	$0.740337\pi$
$270$	0	0
$271$	19.3606	1.17607	0.588037	−	0.808834i	$-0.299901\pi$
$271$	19.3606	1.17607	0.588037	+	0.808834i	$0.299901\pi$
$272$	0	0
$273$	91.2932	5.52531
$274$	0	0
$275$	−1.40093	−0.0844791
$276$	0	0
$277$	−13.9771	−0.839806	−0.419903	−	0.907569i	$-0.637936\pi$
$277$	−13.9771	−0.839806	−0.419903	+	0.907569i	$0.637936\pi$
$278$	0	0
$279$	−18.8927	−1.13107
$280$	0	0
$281$	23.2128	1.38476	0.692381	−	0.721532i	$-0.256562\pi$
$281$	23.2128	1.38476	0.692381	+	0.721532i	$0.256562\pi$
$282$	0	0
$283$	13.0543	0.775996	0.387998	−	0.921660i	$-0.373167\pi$
$283$	13.0543	0.775996	0.387998	+	0.921660i	$0.373167\pi$
$284$	0	0
$285$	−21.1913	−1.25526
$286$	0	0
$287$	−23.3213	−1.37661
$288$	0	0
$289$	2.32363	0.136684
$290$	0	0
$291$	41.3318	2.42291
$292$	0	0
$293$	15.0649	0.880099	0.440050	−	0.897973i	$-0.354961\pi$
$293$	15.0649	0.880099	0.440050	+	0.897973i	$0.354961\pi$
$294$	0	0
$295$	−1.30307	−0.0758675
$296$	0	0
$297$	15.3454	0.890432
$298$	0	0
$299$	−26.2135	−1.51597
$300$	0	0
$301$	11.7367	0.676491
$302$	0	0
$303$	−55.7324	−3.20174
$304$	0	0
$305$	−3.29284	−0.188547
$306$	0	0
$307$	−30.7413	−1.75450	−0.877249	−	0.480036i	$-0.840624\pi$
$307$	−30.7413	−1.75450	−0.877249	+	0.480036i	$0.840624\pi$
$308$	0	0
$309$	7.90741	0.449837
$310$	0	0
$311$	1.38320	0.0784343	0.0392172	−	0.999231i	$-0.487514\pi$
$311$	1.38320	0.0784343	0.0392172	+	0.999231i	$0.487514\pi$
$312$	0	0
$313$	−11.1696	−0.631342	−0.315671	−	0.948869i	$-0.602230\pi$
$313$	−11.1696	−0.631342	−0.315671	+	0.948869i	$0.602230\pi$
$314$	0	0
$315$	−31.0948	−1.75200
$316$	0	0
$317$	1.78129	0.100047	0.0500235	−	0.998748i	$-0.484070\pi$
$317$	1.78129	0.100047	0.0500235	+	0.998748i	$0.484070\pi$
$318$	0	0
$319$	8.65104	0.484365
$320$	0	0
$321$	52.1355	2.90992
$322$	0	0
$323$	30.1512	1.67766
$324$	0	0
$325$	−6.21999	−0.345023
$326$	0	0
$327$	−34.2045	−1.89151
$328$	0	0
$329$	−20.7898	−1.14618
$330$	0	0
$331$	6.52790	0.358806	0.179403	−	0.983776i	$-0.442583\pi$
$331$	6.52790	0.358806	0.179403	+	0.983776i	$0.442583\pi$
$332$	0	0
$333$	−44.3651	−2.43119
$334$	0	0
$335$	−7.69939	−0.420663
$336$	0	0
$337$	5.96310	0.324831	0.162415	−	0.986722i	$-0.448072\pi$
$337$	5.96310	0.324831	0.162415	+	0.986722i	$0.448072\pi$
$338$	0	0
$339$	−6.16985	−0.335100
$340$	0	0
$341$	4.04364	0.218975
$342$	0	0
$343$	40.7060	2.19792
$344$	0	0
$345$	13.0207	0.701009
$346$	0	0
$347$	−19.8171	−1.06384	−0.531919	−	0.846795i	$-0.678529\pi$
$347$	−19.8171	−1.06384	−0.531919	+	0.846795i	$0.678529\pi$
$348$	0	0
$349$	2.12644	0.113826	0.0569129	−	0.998379i	$-0.481874\pi$
$349$	2.12644	0.113826	0.0569129	+	0.998379i	$0.481874\pi$
$350$	0	0
$351$	68.1322	3.63663
$352$	0	0
$353$	−21.3077	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$353$	−21.3077	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$354$	0	0
$355$	−6.03743	−0.320433
$356$	0	0
$357$	64.5198	3.41475
$358$	0	0
$359$	−6.84327	−0.361174	−0.180587	−	0.983559i	$-0.557800\pi$
$359$	−6.84327	−0.361174	−0.180587	+	0.983559i	$0.557800\pi$
$360$	0	0
$361$	28.0457	1.47609
$362$	0	0
$363$	27.9216	1.46551
$364$	0	0
$365$	0.131310	0.00687309
$366$	0	0
$367$	−17.1419	−0.894798	−0.447399	−	0.894335i	$-0.647650\pi$
$367$	−17.1419	−0.894798	−0.447399	+	0.894335i	$0.647650\pi$
$368$	0	0
$369$	−32.1319	−1.67272
$370$	0	0
$371$	−4.69054	−0.243520
$372$	0	0
$373$	25.6913	1.33024	0.665122	−	0.746735i	$-0.268380\pi$
$373$	25.6913	1.33024	0.665122	+	0.746735i	$0.268380\pi$
$374$	0	0
$375$	3.08956	0.159544
$376$	0	0
$377$	38.4098	1.97821
$378$	0	0
$379$	30.3419	1.55856	0.779280	−	0.626676i	$-0.215585\pi$
$379$	30.3419	1.55856	0.779280	+	0.626676i	$0.215585\pi$
$380$	0	0
$381$	48.5120	2.48535
$382$	0	0
$383$	8.03909	0.410779	0.205389	−	0.978680i	$-0.434154\pi$
$383$	8.03909	0.410779	0.205389	+	0.978680i	$0.434154\pi$
$384$	0	0
$385$	6.65530	0.339185
$386$	0	0
$387$	16.1708	0.822006
$388$	0	0
$389$	0.930577	0.0471821	0.0235911	−	0.999722i	$-0.492490\pi$
$389$	0.930577	0.0471821	0.0235911	+	0.999722i	$0.492490\pi$
$390$	0	0
$391$	−18.5259	−0.936897
$392$	0	0
$393$	9.62840	0.485689
$394$	0	0
$395$	11.9786	0.602709
$396$	0	0
$397$	−31.7592	−1.59395	−0.796975	−	0.604012i	$-0.793568\pi$
$397$	−31.7592	−1.59395	−0.796975	+	0.604012i	$0.793568\pi$
$398$	0	0
$399$	100.672	5.03990
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	17.9534	0.894321
$404$	0	0
$405$	−14.2061	−0.705908
$406$	0	0
$407$	9.49555	0.470677
$408$	0	0
$409$	31.7625	1.57056	0.785278	−	0.619143i	$-0.212520\pi$
$409$	31.7625	1.57056	0.785278	+	0.619143i	$0.212520\pi$
$410$	0	0
$411$	46.0315	2.27057
$412$	0	0
$413$	6.19040	0.304610
$414$	0	0
$415$	−10.8246	−0.531361
$416$	0	0
$417$	−37.2697	−1.82511
$418$	0	0
$419$	24.8382	1.21343	0.606713	−	0.794921i	$-0.292488\pi$
$419$	24.8382	1.21343	0.606713	+	0.794921i	$0.292488\pi$
$420$	0	0
$421$	−7.42620	−0.361931	−0.180966	−	0.983489i	$-0.557922\pi$
$421$	−7.42620	−0.361931	−0.180966	+	0.983489i	$0.557922\pi$
$422$	0	0
$423$	−28.6441	−1.39272
$424$	0	0
$425$	−4.39587	−0.213231
$426$	0	0
$427$	15.6431	0.757022
$428$	0	0
$429$	−26.9217	−1.29979
$430$	0	0
$431$	−19.0212	−0.916217	−0.458108	−	0.888896i	$-0.651473\pi$
$431$	−19.0212	−0.916217	−0.458108	+	0.888896i	$0.651473\pi$
$432$	0	0
$433$	−5.14268	−0.247141	−0.123571	−	0.992336i	$-0.539435\pi$
$433$	−5.14268	−0.247141	−0.123571	+	0.992336i	$0.539435\pi$
$434$	0	0
$435$	−19.0788	−0.914756
$436$	0	0
$437$	−28.9065	−1.38279
$438$	0	0
$439$	14.4145	0.687967	0.343983	−	0.938976i	$-0.388224\pi$
$439$	14.4145	0.687967	0.343983	+	0.938976i	$0.388224\pi$
$440$	0	0
$441$	101.902	4.85250
$442$	0	0
$443$	18.3212	0.870466	0.435233	−	0.900318i	$-0.356666\pi$
$443$	18.3212	0.870466	0.435233	+	0.900318i	$0.356666\pi$
$444$	0	0
$445$	−9.62028	−0.456045
$446$	0	0
$447$	−34.2763	−1.62122
$448$	0	0
$449$	31.9639	1.50847	0.754234	−	0.656605i	$-0.228008\pi$
$449$	31.9639	1.50847	0.754234	+	0.656605i	$0.228008\pi$
$450$	0	0
$451$	6.87727	0.323838
$452$	0	0
$453$	−1.01450	−0.0476653
$454$	0	0
$455$	29.5489	1.38527
$456$	0	0
$457$	−8.53394	−0.399201	−0.199601	−	0.979877i	$-0.563964\pi$
$457$	−8.53394	−0.399201	−0.199601	+	0.979877i	$0.563964\pi$
$458$	0	0
$459$	48.1512	2.24751
$460$	0	0
$461$	7.04045	0.327907	0.163953	−	0.986468i	$-0.447575\pi$
$461$	7.04045	0.327907	0.163953	+	0.986468i	$0.447575\pi$
$462$	0	0
$463$	−22.0094	−1.02286	−0.511431	−	0.859324i	$-0.670885\pi$
$463$	−22.0094	−1.02286	−0.511431	+	0.859324i	$0.670885\pi$
$464$	0	0
$465$	−8.91772	−0.413549
$466$	0	0
$467$	21.4219	0.991285	0.495643	−	0.868527i	$-0.334933\pi$
$467$	21.4219	0.991285	0.495643	+	0.868527i	$0.334933\pi$
$468$	0	0
$469$	36.5770	1.68897
$470$	0	0
$471$	−64.1762	−2.95708
$472$	0	0
$473$	−3.46106	−0.159140
$474$	0	0
$475$	−6.85898	−0.314712
$476$	0	0
$477$	−6.46260	−0.295902
$478$	0	0
$479$	−1.13613	−0.0519109	−0.0259555	−	0.999663i	$-0.508263\pi$
$479$	−1.13613	−0.0519109	−0.0259555	+	0.999663i	$0.508263\pi$
$480$	0	0
$481$	42.1594	1.92230
$482$	0	0
$483$	−61.8564	−2.81456
$484$	0	0
$485$	13.3779	0.607458
$486$	0	0
$487$	33.0421	1.49728	0.748641	−	0.662975i	$-0.230707\pi$
$487$	33.0421	1.49728	0.748641	+	0.662975i	$0.230707\pi$
$488$	0	0
$489$	56.4874	2.55445
$490$	0	0
$491$	−8.30134	−0.374634	−0.187317	−	0.982299i	$-0.559979\pi$
$491$	−8.30134	−0.374634	−0.187317	+	0.982299i	$0.559979\pi$
$492$	0	0
$493$	27.1455	1.22257
$494$	0	0
$495$	9.16964	0.412145
$496$	0	0
$497$	28.6816	1.28655
$498$	0	0
$499$	−15.2730	−0.683715	−0.341857	−	0.939752i	$-0.611056\pi$
$499$	−15.2730	−0.683715	−0.341857	+	0.939752i	$0.611056\pi$
$500$	0	0
$501$	43.2895	1.93403
$502$	0	0
$503$	−28.2153	−1.25806	−0.629029	−	0.777382i	$-0.716547\pi$
$503$	−28.2153	−1.25806	−0.629029	+	0.777382i	$0.716547\pi$
$504$	0	0
$505$	−18.0389	−0.802721
$506$	0	0
$507$	−79.3654	−3.52474
$508$	0	0
$509$	34.2129	1.51646	0.758230	−	0.651988i	$-0.226065\pi$
$509$	34.2129	1.51646	0.758230	+	0.651988i	$0.226065\pi$
$510$	0	0
$511$	−0.623807	−0.0275956
$512$	0	0
$513$	75.1317	3.31714
$514$	0	0
$515$	2.55939	0.112780
$516$	0	0
$517$	6.13076	0.269631
$518$	0	0
$519$	−47.9630	−2.10534
$520$	0	0
$521$	7.38881	0.323709	0.161855	−	0.986815i	$-0.448252\pi$
$521$	7.38881	0.323709	0.161855	+	0.986815i	$0.448252\pi$
$522$	0	0
$523$	6.11089	0.267210	0.133605	−	0.991035i	$-0.457345\pi$
$523$	6.11089	0.267210	0.133605	+	0.991035i	$0.457345\pi$
$524$	0	0
$525$	−14.6774	−0.640574
$526$	0	0
$527$	12.6882	0.552708
$528$	0	0
$529$	−5.23884	−0.227776
$530$	0	0
$531$	8.52910	0.370132
$532$	0	0
$533$	30.5344	1.32259
$534$	0	0
$535$	16.8747	0.729558
$536$	0	0
$537$	19.7363	0.851685
$538$	0	0
$539$	−21.8104	−0.939440
$540$	0	0
$541$	32.7202	1.40675	0.703376	−	0.710818i	$-0.251675\pi$
$541$	32.7202	1.40675	0.703376	+	0.710818i	$0.251675\pi$
$542$	0	0
$543$	18.1385	0.778399
$544$	0	0
$545$	−11.0710	−0.474229
$546$	0	0
$547$	7.12204	0.304516	0.152258	−	0.988341i	$-0.451345\pi$
$547$	7.12204	0.304516	0.152258	+	0.988341i	$0.451345\pi$
$548$	0	0
$549$	21.5530	0.919858
$550$	0	0
$551$	42.3558	1.80442
$552$	0	0
$553$	−56.9060	−2.41989
$554$	0	0
$555$	−20.9412	−0.888905
$556$	0	0
$557$	−32.7771	−1.38881	−0.694404	−	0.719585i	$-0.744332\pi$
$557$	−32.7771	−1.38881	−0.694404	+	0.719585i	$0.744332\pi$
$558$	0	0
$559$	−15.3668	−0.649946
$560$	0	0
$561$	−19.0264	−0.803296
$562$	0	0
$563$	−35.8120	−1.50929	−0.754647	−	0.656131i	$-0.772192\pi$
$563$	−35.8120	−1.50929	−0.754647	+	0.656131i	$0.772192\pi$
$564$	0	0
$565$	−1.99700	−0.0840143
$566$	0	0
$567$	67.4881	2.83423
$568$	0	0
$569$	−43.0834	−1.80615	−0.903074	−	0.429484i	$-0.858695\pi$
$569$	−43.0834	−1.80615	−0.903074	+	0.429484i	$0.858695\pi$
$570$	0	0
$571$	−13.6770	−0.572366	−0.286183	−	0.958175i	$-0.592387\pi$
$571$	−13.6770	−0.572366	−0.286183	+	0.958175i	$0.592387\pi$
$572$	0	0
$573$	−60.2463	−2.51683
$574$	0	0
$575$	4.21440	0.175753
$576$	0	0
$577$	5.54653	0.230905	0.115452	−	0.993313i	$-0.463168\pi$
$577$	5.54653	0.230905	0.115452	+	0.993313i	$0.463168\pi$
$578$	0	0
$579$	−7.30274	−0.303491
$580$	0	0
$581$	51.4239	2.13342
$582$	0	0
$583$	1.38320	0.0572865
$584$	0	0
$585$	40.7123	1.68325
$586$	0	0
$587$	1.74578	0.0720560	0.0360280	−	0.999351i	$-0.488529\pi$
$587$	1.74578	0.0720560	0.0360280	+	0.999351i	$0.488529\pi$
$588$	0	0
$589$	19.7978	0.815753
$590$	0	0
$591$	−12.5455	−0.516052
$592$	0	0
$593$	−16.7013	−0.685838	−0.342919	−	0.939365i	$-0.611416\pi$
$593$	−16.7013	−0.685838	−0.342919	+	0.939365i	$0.611416\pi$
$594$	0	0
$595$	20.8832	0.856126
$596$	0	0
$597$	80.8218	3.30782
$598$	0	0
$599$	−14.4092	−0.588744	−0.294372	−	0.955691i	$-0.595111\pi$
$599$	−14.4092	−0.588744	−0.294372	+	0.955691i	$0.595111\pi$
$600$	0	0
$601$	−7.12117	−0.290479	−0.145239	−	0.989397i	$-0.546395\pi$
$601$	−7.12117	−0.290479	−0.145239	+	0.989397i	$0.546395\pi$
$602$	0	0
$603$	50.3956	2.05227
$604$	0	0
$605$	9.03740	0.367423
$606$	0	0
$607$	−30.7388	−1.24765	−0.623825	−	0.781564i	$-0.714422\pi$
$607$	−30.7388	−1.24765	−0.623825	+	0.781564i	$0.714422\pi$
$608$	0	0
$609$	90.6362	3.67276
$610$	0	0
$611$	27.2200	1.10120
$612$	0	0
$613$	45.6708	1.84463	0.922314	−	0.386442i	$-0.126296\pi$
$613$	45.6708	1.84463	0.922314	+	0.386442i	$0.126296\pi$
$614$	0	0
$615$	−15.1669	−0.611589
$616$	0	0
$617$	19.1780	0.772077	0.386039	−	0.922483i	$-0.373843\pi$
$617$	19.1780	0.772077	0.386039	+	0.922483i	$0.373843\pi$
$618$	0	0
$619$	7.98508	0.320948	0.160474	−	0.987040i	$-0.448698\pi$
$619$	7.98508	0.320948	0.160474	+	0.987040i	$0.448698\pi$
$620$	0	0
$621$	−46.1635	−1.85248
$622$	0	0
$623$	45.7025	1.83103
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.6874	−1.18560
$628$	0	0
$629$	29.7954	1.18802
$630$	0	0
$631$	−37.1383	−1.47845	−0.739227	−	0.673457i	$-0.764809\pi$
$631$	−37.1383	−1.47845	−0.739227	+	0.673457i	$0.764809\pi$
$632$	0	0
$633$	−77.9205	−3.09706
$634$	0	0
$635$	15.7019	0.623111
$636$	0	0
$637$	−96.8361	−3.83679
$638$	0	0
$639$	39.5174	1.56328
$640$	0	0
$641$	45.3022	1.78933	0.894666	−	0.446737i	$-0.147414\pi$
$641$	45.3022	1.78933	0.894666	+	0.446737i	$0.147414\pi$
$642$	0	0
$643$	13.6406	0.537931	0.268965	−	0.963150i	$-0.413318\pi$
$643$	13.6406	0.537931	0.268965	+	0.963150i	$0.413318\pi$
$644$	0	0
$645$	7.63292	0.300546
$646$	0	0
$647$	29.2098	1.14836	0.574178	−	0.818731i	$-0.305322\pi$
$647$	29.2098	1.14836	0.574178	+	0.818731i	$0.305322\pi$
$648$	0	0
$649$	−1.82550	−0.0716573
$650$	0	0
$651$	42.3648	1.66041
$652$	0	0
$653$	49.8480	1.95070	0.975352	−	0.220655i	$-0.0708196\pi$
$653$	49.8480	1.95070	0.975352	+	0.220655i	$0.0708196\pi$
$654$	0	0
$655$	3.11643	0.121769
$656$	0	0
$657$	−0.859478	−0.0335314
$658$	0	0
$659$	32.1563	1.25263	0.626315	−	0.779570i	$-0.284562\pi$
$659$	32.1563	1.25263	0.626315	+	0.779570i	$0.284562\pi$
$660$	0	0
$661$	0.687534	0.0267420	0.0133710	−	0.999911i	$-0.495744\pi$
$661$	0.687534	0.0267420	0.0133710	+	0.999911i	$0.495744\pi$
$662$	0	0
$663$	−84.4755	−3.28076
$664$	0	0
$665$	32.5845	1.26357
$666$	0	0
$667$	−26.0249	−1.00769
$668$	0	0
$669$	−20.1923	−0.780679
$670$	0	0
$671$	−4.61303	−0.178084
$672$	0	0
$673$	−12.2781	−0.473287	−0.236644	−	0.971597i	$-0.576047\pi$
$673$	−12.2781	−0.473287	−0.236644	+	0.971597i	$0.576047\pi$
$674$	0	0
$675$	−10.9538	−0.421610
$676$	0	0
$677$	14.8891	0.572234	0.286117	−	0.958195i	$-0.407635\pi$
$677$	14.8891	0.572234	0.286117	+	0.958195i	$0.407635\pi$
$678$	0	0
$679$	−63.5535	−2.43896
$680$	0	0
$681$	78.9061	3.02369
$682$	0	0
$683$	36.1913	1.38482	0.692410	−	0.721504i	$-0.256549\pi$
$683$	36.1913	1.38482	0.692410	+	0.721504i	$0.256549\pi$
$684$	0	0
$685$	14.8990	0.569262
$686$	0	0
$687$	77.0883	2.94110
$688$	0	0
$689$	6.14130	0.233965
$690$	0	0
$691$	−17.6870	−0.672845	−0.336422	−	0.941711i	$-0.609217\pi$
$691$	−17.6870	−0.672845	−0.336422	+	0.941711i	$0.609217\pi$
$692$	0	0
$693$	−43.5616	−1.65477
$694$	0	0
$695$	−12.0631	−0.457579
$696$	0	0
$697$	21.5797	0.817388
$698$	0	0
$699$	−53.2098	−2.01258
$700$	0	0
$701$	9.59054	0.362230	0.181115	−	0.983462i	$-0.442029\pi$
$701$	9.59054	0.362230	0.181115	+	0.983462i	$0.442029\pi$
$702$	0	0
$703$	46.4905	1.75342
$704$	0	0
$705$	−13.5206	−0.509215
$706$	0	0
$707$	85.6963	3.22294
$708$	0	0
$709$	35.0589	1.31667	0.658333	−	0.752727i	$-0.271262\pi$
$709$	35.0589	1.31667	0.658333	+	0.752727i	$0.271262\pi$
$710$	0	0
$711$	−78.4049	−2.94041
$712$	0	0
$713$	−12.1644	−0.455562
$714$	0	0
$715$	−8.71375	−0.325876
$716$	0	0
$717$	−66.1665	−2.47103
$718$	0	0
$719$	28.8878	1.07733	0.538666	−	0.842519i	$-0.318928\pi$
$719$	28.8878	1.07733	0.538666	+	0.842519i	$0.318928\pi$
$720$	0	0
$721$	−12.1587	−0.452815
$722$	0	0
$723$	−38.7336	−1.44052
$724$	0	0
$725$	−6.17522	−0.229342
$726$	0	0
$727$	−14.1916	−0.526338	−0.263169	−	0.964750i	$-0.584768\pi$
$727$	−14.1916	−0.526338	−0.263169	+	0.964750i	$0.584768\pi$
$728$	0	0
$729$	−8.54219	−0.316378
$730$	0	0
$731$	−10.8602	−0.401679
$732$	0	0
$733$	8.18639	0.302371	0.151186	−	0.988505i	$-0.451691\pi$
$733$	8.18639	0.302371	0.151186	+	0.988505i	$0.451691\pi$
$734$	0	0
$735$	48.1000	1.77420
$736$	0	0
$737$	−10.7863	−0.397318
$738$	0	0
$739$	15.5716	0.572812	0.286406	−	0.958108i	$-0.407539\pi$
$739$	15.5716	0.572812	0.286406	+	0.958108i	$0.407539\pi$
$740$	0	0
$741$	−131.809	−4.84214
$742$	0	0
$743$	45.1118	1.65499	0.827497	−	0.561471i	$-0.189764\pi$
$743$	45.1118	1.65499	0.827497	+	0.561471i	$0.189764\pi$
$744$	0	0
$745$	−11.0942	−0.406461
$746$	0	0
$747$	70.8517	2.59233
$748$	0	0
$749$	−80.1657	−2.92919
$750$	0	0
$751$	−30.3068	−1.10591	−0.552955	−	0.833211i	$-0.686500\pi$
$751$	−30.3068	−1.10591	−0.552955	+	0.833211i	$0.686500\pi$
$752$	0	0
$753$	33.9352	1.23667
$754$	0	0
$755$	−0.328363	−0.0119504
$756$	0	0
$757$	−32.3799	−1.17687	−0.588434	−	0.808545i	$-0.700255\pi$
$757$	−32.3799	−1.17687	−0.588434	+	0.808545i	$0.700255\pi$
$758$	0	0
$759$	18.2410	0.662106
$760$	0	0
$761$	4.78285	0.173378	0.0866890	−	0.996235i	$-0.472371\pi$
$761$	4.78285	0.173378	0.0866890	+	0.996235i	$0.472371\pi$
$762$	0	0
$763$	52.5942	1.90404
$764$	0	0
$765$	28.7727	1.04028
$766$	0	0
$767$	−8.10506	−0.292657
$768$	0	0
$769$	27.2334	0.982062	0.491031	−	0.871142i	$-0.336620\pi$
$769$	27.2334	0.982062	0.491031	+	0.871142i	$0.336620\pi$
$770$	0	0
$771$	−61.2544	−2.20602
$772$	0	0
$773$	−17.7932	−0.639978	−0.319989	−	0.947421i	$-0.603679\pi$
$773$	−17.7932	−0.639978	−0.319989	+	0.947421i	$0.603679\pi$
$774$	0	0
$775$	−2.88640	−0.103683
$776$	0	0
$777$	99.4841	3.56897
$778$	0	0
$779$	33.6713	1.20640
$780$	0	0
$781$	−8.45800	−0.302651
$782$	0	0
$783$	67.6419	2.41732
$784$	0	0
$785$	−20.7719	−0.741382
$786$	0	0
$787$	−28.1435	−1.00321	−0.501604	−	0.865098i	$-0.667256\pi$
$787$	−28.1435	−1.00321	−0.501604	+	0.865098i	$0.667256\pi$
$788$	0	0
$789$	−67.6963	−2.41005
$790$	0	0
$791$	9.48701	0.337319
$792$	0	0
$793$	−20.4814	−0.727316
$794$	0	0
$795$	−3.05048	−0.108189
$796$	0	0
$797$	37.7468	1.33706	0.668530	−	0.743685i	$-0.266924\pi$
$797$	37.7468	1.33706	0.668530	+	0.743685i	$0.266924\pi$
$798$	0	0
$799$	19.2373	0.680566
$800$	0	0
$801$	62.9686	2.22489
$802$	0	0
$803$	0.183956	0.00649166
$804$	0	0
$805$	−20.0211	−0.705650
$806$	0	0
$807$	69.4538	2.44489
$808$	0	0
$809$	5.65916	0.198966	0.0994828	−	0.995039i	$-0.468281\pi$
$809$	5.65916	0.198966	0.0994828	+	0.995039i	$0.468281\pi$
$810$	0	0
$811$	−33.9019	−1.19046	−0.595228	−	0.803557i	$-0.702938\pi$
$811$	−33.9019	−1.19046	−0.595228	+	0.803557i	$0.702938\pi$
$812$	0	0
$813$	−59.8159	−2.09783
$814$	0	0
$815$	18.2833	0.640436
$816$	0	0
$817$	−16.9455	−0.592847
$818$	0	0
$819$	−193.409	−6.75827
$820$	0	0
$821$	35.0512	1.22329	0.611647	−	0.791131i	$-0.290507\pi$
$821$	35.0512	1.22329	0.611647	+	0.791131i	$0.290507\pi$
$822$	0	0
$823$	−45.0034	−1.56872	−0.784359	−	0.620307i	$-0.787008\pi$
$823$	−45.0034	−1.56872	−0.784359	+	0.620307i	$0.787008\pi$
$824$	0	0
$825$	4.32826	0.150691
$826$	0	0
$827$	−19.5178	−0.678700	−0.339350	−	0.940660i	$-0.610207\pi$
$827$	−19.5178	−0.678700	−0.339350	+	0.940660i	$0.610207\pi$
$828$	0	0
$829$	−11.6796	−0.405650	−0.202825	−	0.979215i	$-0.565012\pi$
$829$	−11.6796	−0.405650	−0.202825	+	0.979215i	$0.565012\pi$
$830$	0	0
$831$	43.1833	1.49801
$832$	0	0
$833$	−68.4372	−2.37121
$834$	0	0
$835$	14.0115	0.484889
$836$	0	0
$837$	31.6169	1.09284
$838$	0	0
$839$	17.9822	0.620815	0.310408	−	0.950604i	$-0.399534\pi$
$839$	17.9822	0.620815	0.310408	+	0.950604i	$0.399534\pi$
$840$	0	0
$841$	9.13340	0.314945
$842$	0	0
$843$	−71.7176	−2.47008
$844$	0	0
$845$	−25.6882	−0.883702
$846$	0	0
$847$	−42.9334	−1.47521
$848$	0	0
$849$	−40.3320	−1.38419
$850$	0	0
$851$	−28.5654	−0.979209
$852$	0	0
$853$	−23.1696	−0.793312	−0.396656	−	0.917967i	$-0.629829\pi$
$853$	−23.1696	−0.793312	−0.396656	+	0.917967i	$0.629829\pi$
$854$	0	0
$855$	44.8948	1.53537
$856$	0	0
$857$	−35.2612	−1.20450	−0.602251	−	0.798307i	$-0.705729\pi$
$857$	−35.2612	−1.20450	−0.602251	+	0.798307i	$0.705729\pi$
$858$	0	0
$859$	17.2125	0.587284	0.293642	−	0.955915i	$-0.405133\pi$
$859$	17.2125	0.587284	0.293642	+	0.955915i	$0.405133\pi$
$860$	0	0
$861$	72.0525	2.45554
$862$	0	0
$863$	17.3176	0.589499	0.294749	−	0.955575i	$-0.404764\pi$
$863$	17.3176	0.589499	0.294749	+	0.955575i	$0.404764\pi$
$864$	0	0
$865$	−15.5242	−0.527839
$866$	0	0
$867$	−7.17901	−0.243812
$868$	0	0
$869$	16.7812	0.569262
$870$	0	0
$871$	−47.8901	−1.62269
$872$	0	0
$873$	−87.5637	−2.96358
$874$	0	0
$875$	−4.75064	−0.160601
$876$	0	0
$877$	16.0442	0.541775	0.270887	−	0.962611i	$-0.412683\pi$
$877$	16.0442	0.541775	0.270887	+	0.962611i	$0.412683\pi$
$878$	0	0
$879$	−46.5439	−1.56989
$880$	0	0
$881$	−24.7098	−0.832494	−0.416247	−	0.909252i	$-0.636655\pi$
$881$	−24.7098	−0.832494	−0.416247	+	0.909252i	$0.636655\pi$
$882$	0	0
$883$	−21.8153	−0.734142	−0.367071	−	0.930193i	$-0.619639\pi$
$883$	−21.8153	−0.734142	−0.367071	+	0.930193i	$0.619639\pi$
$884$	0	0
$885$	4.02591	0.135329
$886$	0	0
$887$	−41.7369	−1.40139	−0.700693	−	0.713463i	$-0.747126\pi$
$887$	−41.7369	−1.40139	−0.700693	+	0.713463i	$0.747126\pi$
$888$	0	0
$889$	−74.5940	−2.50180
$890$	0	0
$891$	−19.9017	−0.666734
$892$	0	0
$893$	30.0164	1.00446
$894$	0	0
$895$	6.38806	0.213529
$896$	0	0
$897$	80.9883	2.70412
$898$	0	0
$899$	17.8242	0.594469
$900$	0	0
$901$	4.34025	0.144595
$902$	0	0
$903$	−36.2612	−1.20670
$904$	0	0
$905$	5.87090	0.195155
$906$	0	0
$907$	32.0134	1.06299	0.531494	−	0.847062i	$-0.321631\pi$
$907$	32.0134	1.06299	0.531494	+	0.847062i	$0.321631\pi$
$908$	0	0
$909$	118.072	3.91620
$910$	0	0
$911$	11.2822	0.373796	0.186898	−	0.982379i	$-0.440157\pi$
$911$	11.2822	0.373796	0.186898	+	0.982379i	$0.440157\pi$
$912$	0	0
$913$	−15.1645	−0.501873
$914$	0	0
$915$	10.1734	0.336323
$916$	0	0
$917$	−14.8050	−0.488904
$918$	0	0
$919$	1.59447	0.0525968	0.0262984	−	0.999654i	$-0.491628\pi$
$919$	1.59447	0.0525968	0.0262984	+	0.999654i	$0.491628\pi$
$920$	0	0
$921$	94.9772	3.12960
$922$	0	0
$923$	−37.5527	−1.23606
$924$	0	0
$925$	−6.77805	−0.222861
$926$	0	0
$927$	−16.7523	−0.550216
$928$	0	0
$929$	3.96461	0.130075	0.0650373	−	0.997883i	$-0.479283\pi$
$929$	3.96461	0.130075	0.0650373	+	0.997883i	$0.479283\pi$
$930$	0	0
$931$	−106.784	−3.49972
$932$	0	0
$933$	−4.27350	−0.139908
$934$	0	0
$935$	−6.15829	−0.201398
$936$	0	0
$937$	9.70408	0.317018	0.158509	−	0.987358i	$-0.449331\pi$
$937$	9.70408	0.317018	0.158509	+	0.987358i	$0.449331\pi$
$938$	0	0
$939$	34.5091	1.12616
$940$	0	0
$941$	−8.56911	−0.279345	−0.139672	−	0.990198i	$-0.544605\pi$
$941$	−8.56911	−0.279345	−0.139672	+	0.990198i	$0.544605\pi$
$942$	0	0
$943$	−20.6888	−0.673721
$944$	0	0
$945$	52.0373	1.69277
$946$	0	0
$947$	44.9246	1.45985	0.729927	−	0.683525i	$-0.239554\pi$
$947$	44.9246	1.45985	0.729927	+	0.683525i	$0.239554\pi$
$948$	0	0
$949$	0.816747	0.0265127
$950$	0	0
$951$	−5.50340	−0.178460
$952$	0	0
$953$	27.7440	0.898715	0.449357	−	0.893352i	$-0.351653\pi$
$953$	27.7440	0.898715	0.449357	+	0.893352i	$0.351653\pi$
$954$	0	0
$955$	−19.4999	−0.631003
$956$	0	0
$957$	−26.7279	−0.863992
$958$	0	0
$959$	−70.7798	−2.28560
$960$	0	0
$961$	−22.6687	−0.731248
$962$	0	0
$963$	−110.452	−3.55926
$964$	0	0
$965$	−2.36368	−0.0760895
$966$	0	0
$967$	4.32776	0.139171	0.0695857	−	0.997576i	$-0.477832\pi$
$967$	4.32776	0.139171	0.0695857	+	0.997576i	$0.477832\pi$
$968$	0	0
$969$	−93.1540	−2.99254
$970$	0	0
$971$	−10.7264	−0.344228	−0.172114	−	0.985077i	$-0.555060\pi$
$971$	−10.7264	−0.344228	−0.172114	+	0.985077i	$0.555060\pi$
$972$	0	0
$973$	57.3074	1.83719
$974$	0	0
$975$	19.2170	0.615438
$976$	0	0
$977$	14.7639	0.472337	0.236169	−	0.971712i	$-0.424108\pi$
$977$	14.7639	0.472337	0.236169	+	0.971712i	$0.424108\pi$
$978$	0	0
$979$	−13.4773	−0.430737
$980$	0	0
$981$	72.4641	2.31360
$982$	0	0
$983$	−0.788408	−0.0251463	−0.0125732	−	0.999921i	$-0.504002\pi$
$983$	−0.788408	−0.0251463	−0.0125732	+	0.999921i	$0.504002\pi$
$984$	0	0
$985$	−4.06060	−0.129381
$986$	0	0
$987$	64.2315	2.04451
$988$	0	0
$989$	10.4119	0.331079
$990$	0	0
$991$	−56.6090	−1.79824	−0.899122	−	0.437698i	$-0.855794\pi$
$991$	−56.6090	−1.79824	−0.899122	+	0.437698i	$0.855794\pi$
$992$	0	0
$993$	−20.1684	−0.640024
$994$	0	0
$995$	26.1596	0.829316
$996$	0	0
$997$	−35.2265	−1.11563	−0.557817	−	0.829964i	$-0.688361\pi$
$997$	−35.2265	−1.11563	−0.557817	+	0.829964i	$0.688361\pi$
$998$	0	0
$999$	74.2451	2.34901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.3	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.3	✓	35	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)

\(f(q)\)	\(=\)	\(q-3.08956 q^{3} -1.00000 q^{5} +4.75064 q^{7} +6.54541 q^{9} +O(q^{10})\)	\(q-3.08956 q^{3} -1.00000 q^{5} +4.75064 q^{7} +6.54541 q^{9} -1.40093 q^{11} -6.21999 q^{13} +3.08956 q^{15} -4.39587 q^{17} -6.85898 q^{19} -14.6774 q^{21} +4.21440 q^{23} +1.00000 q^{25} -10.9538 q^{27} -6.17522 q^{29} -2.88640 q^{31} +4.32826 q^{33} -4.75064 q^{35} -6.77805 q^{37} +19.2170 q^{39} -4.90908 q^{41} +2.47055 q^{43} -6.54541 q^{45} -4.37622 q^{47} +15.5685 q^{49} +13.5813 q^{51} -0.987349 q^{53} +1.40093 q^{55} +21.1913 q^{57} +1.30307 q^{59} +3.29284 q^{61} +31.0948 q^{63} +6.21999 q^{65} +7.69939 q^{67} -13.0207 q^{69} +6.03743 q^{71} -0.131310 q^{73} -3.08956 q^{75} -6.65530 q^{77} -11.9786 q^{79} +14.2061 q^{81} +10.8246 q^{83} +4.39587 q^{85} +19.0788 q^{87} +9.62028 q^{89} -29.5489 q^{91} +8.91772 q^{93} +6.85898 q^{95} -13.3779 q^{97} -9.16964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Introduction

L-functions

Modular forms

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