LMFDB - Embedded newform 8020.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47047 q^{3} +1.00000 q^{5} -0.726513 q^{7} +3.10322 q^{9} +O(q^{10})$	$q-2.47047 q^{3} +1.00000 q^{5} -0.726513 q^{7} +3.10322 q^{9} +1.45955 q^{11} +3.71653 q^{13} -2.47047 q^{15} -6.16672 q^{17} +2.31425 q^{19} +1.79483 q^{21} +6.33500 q^{23} +1.00000 q^{25} -0.254992 q^{27} +3.60854 q^{29} -2.30751 q^{31} -3.60576 q^{33} -0.726513 q^{35} +0.575787 q^{37} -9.18158 q^{39} +0.674336 q^{41} +5.97471 q^{43} +3.10322 q^{45} -4.22008 q^{47} -6.47218 q^{49} +15.2347 q^{51} +14.4139 q^{53} +1.45955 q^{55} -5.71728 q^{57} -14.6629 q^{59} -1.26703 q^{61} -2.25453 q^{63} +3.71653 q^{65} +7.62513 q^{67} -15.6504 q^{69} +1.13882 q^{71} +7.45595 q^{73} -2.47047 q^{75} -1.06038 q^{77} -1.32427 q^{79} -8.67970 q^{81} -2.58904 q^{83} -6.16672 q^{85} -8.91478 q^{87} +6.09211 q^{89} -2.70011 q^{91} +5.70063 q^{93} +2.31425 q^{95} +7.36737 q^{97} +4.52929 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.47047	−1.42633	−0.713163	−	0.700998i	$-0.752738\pi$
$3$	−2.47047	−1.42633	−0.713163	+	0.700998i	$0.752738\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.726513	−0.274596	−0.137298	−	0.990530i	$-0.543842\pi$
$7$	−0.726513	−0.274596	−0.137298	+	0.990530i	$0.543842\pi$
$8$	0	0
$9$	3.10322	1.03441
$10$	0	0
$11$	1.45955	0.440070	0.220035	−	0.975492i	$-0.429383\pi$
$11$	1.45955	0.440070	0.220035	+	0.975492i	$0.429383\pi$
$12$	0	0
$13$	3.71653	1.03078	0.515390	−	0.856956i	$-0.327647\pi$
$13$	3.71653	1.03078	0.515390	+	0.856956i	$0.327647\pi$
$14$	0	0
$15$	−2.47047	−0.637872
$16$	0	0
$17$	−6.16672	−1.49565	−0.747825	−	0.663896i	$-0.768902\pi$
$17$	−6.16672	−1.49565	−0.747825	+	0.663896i	$0.768902\pi$
$18$	0	0
$19$	2.31425	0.530925	0.265462	−	0.964121i	$-0.414475\pi$
$19$	2.31425	0.530925	0.265462	+	0.964121i	$0.414475\pi$
$20$	0	0
$21$	1.79483	0.391663
$22$	0	0
$23$	6.33500	1.32094	0.660470	−	0.750853i	$-0.270357\pi$
$23$	6.33500	1.32094	0.660470	+	0.750853i	$0.270357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.254992	−0.0490733
$28$	0	0
$29$	3.60854	0.670089	0.335044	−	0.942202i	$-0.391249\pi$
$29$	3.60854	0.670089	0.335044	+	0.942202i	$0.391249\pi$
$30$	0	0
$31$	−2.30751	−0.414441	−0.207221	−	0.978294i	$-0.566442\pi$
$31$	−2.30751	−0.414441	−0.207221	+	0.978294i	$0.566442\pi$
$32$	0	0
$33$	−3.60576	−0.627683
$34$	0	0
$35$	−0.726513	−0.122803
$36$	0	0
$37$	0.575787	0.0946588	0.0473294	−	0.998879i	$-0.484929\pi$
$37$	0.575787	0.0946588	0.0473294	+	0.998879i	$0.484929\pi$
$38$	0	0
$39$	−9.18158	−1.47023
$40$	0	0
$41$	0.674336	0.105314	0.0526568	−	0.998613i	$-0.483231\pi$
$41$	0.674336	0.105314	0.0526568	+	0.998613i	$0.483231\pi$
$42$	0	0
$43$	5.97471	0.911134	0.455567	−	0.890201i	$-0.349437\pi$
$43$	5.97471	0.911134	0.455567	+	0.890201i	$0.349437\pi$
$44$	0	0
$45$	3.10322	0.462600
$46$	0	0
$47$	−4.22008	−0.615562	−0.307781	−	0.951457i	$-0.599586\pi$
$47$	−4.22008	−0.615562	−0.307781	+	0.951457i	$0.599586\pi$
$48$	0	0
$49$	−6.47218	−0.924597
$50$	0	0
$51$	15.2347	2.13328
$52$	0	0
$53$	14.4139	1.97990	0.989951	−	0.141412i	$-0.0451643\pi$
$53$	14.4139	1.97990	0.989951	+	0.141412i	$0.0451643\pi$
$54$	0	0
$55$	1.45955	0.196805
$56$	0	0
$57$	−5.71728	−0.757272
$58$	0	0
$59$	−14.6629	−1.90895	−0.954473	−	0.298297i	$-0.903581\pi$
$59$	−14.6629	−1.90895	−0.954473	+	0.298297i	$0.903581\pi$
$60$	0	0
$61$	−1.26703	−0.162227	−0.0811136	−	0.996705i	$-0.525848\pi$
$61$	−1.26703	−0.162227	−0.0811136	+	0.996705i	$0.525848\pi$
$62$	0	0
$63$	−2.25453	−0.284044
$64$	0	0
$65$	3.71653	0.460979
$66$	0	0
$67$	7.62513	0.931557	0.465779	−	0.884901i	$-0.345774\pi$
$67$	7.62513	0.931557	0.465779	+	0.884901i	$0.345774\pi$
$68$	0	0
$69$	−15.6504	−1.88409
$70$	0	0
$71$	1.13882	0.135153	0.0675766	−	0.997714i	$-0.478473\pi$
$71$	1.13882	0.135153	0.0675766	+	0.997714i	$0.478473\pi$
$72$	0	0
$73$	7.45595	0.872653	0.436326	−	0.899789i	$-0.356279\pi$
$73$	7.45595	0.872653	0.436326	+	0.899789i	$0.356279\pi$
$74$	0	0
$75$	−2.47047	−0.285265
$76$	0	0
$77$	−1.06038	−0.120841
$78$	0	0
$79$	−1.32427	−0.148992	−0.0744961	−	0.997221i	$-0.523735\pi$
$79$	−1.32427	−0.148992	−0.0744961	+	0.997221i	$0.523735\pi$
$80$	0	0
$81$	−8.67970	−0.964411
$82$	0	0
$83$	−2.58904	−0.284184	−0.142092	−	0.989853i	$-0.545383\pi$
$83$	−2.58904	−0.284184	−0.142092	+	0.989853i	$0.545383\pi$
$84$	0	0
$85$	−6.16672	−0.668875
$86$	0	0
$87$	−8.91478	−0.955765
$88$	0	0
$89$	6.09211	0.645762	0.322881	−	0.946440i	$-0.395349\pi$
$89$	6.09211	0.645762	0.322881	+	0.946440i	$0.395349\pi$
$90$	0	0
$91$	−2.70011	−0.283048
$92$	0	0
$93$	5.70063	0.591128
$94$	0	0
$95$	2.31425	0.237437
$96$	0	0
$97$	7.36737	0.748043	0.374022	−	0.927420i	$-0.377979\pi$
$97$	7.36737	0.748043	0.374022	+	0.927420i	$0.377979\pi$
$98$	0	0
$99$	4.52929	0.455210
$100$	0	0
$101$	4.87752	0.485331	0.242666	−	0.970110i	$-0.421978\pi$
$101$	4.87752	0.485331	0.242666	+	0.970110i	$0.421978\pi$
$102$	0	0
$103$	−15.7618	−1.55305	−0.776527	−	0.630084i	$-0.783021\pi$
$103$	−15.7618	−1.55305	−0.776527	+	0.630084i	$0.783021\pi$
$104$	0	0
$105$	1.79483	0.175157
$106$	0	0
$107$	2.22827	0.215415	0.107708	−	0.994183i	$-0.465649\pi$
$107$	2.22827	0.215415	0.107708	+	0.994183i	$0.465649\pi$
$108$	0	0
$109$	16.0544	1.53773	0.768864	−	0.639412i	$-0.220822\pi$
$109$	16.0544	1.53773	0.768864	+	0.639412i	$0.220822\pi$
$110$	0	0
$111$	−1.42246	−0.135014
$112$	0	0
$113$	3.06250	0.288096	0.144048	−	0.989571i	$-0.453988\pi$
$113$	3.06250	0.288096	0.144048	+	0.989571i	$0.453988\pi$
$114$	0	0
$115$	6.33500	0.590742
$116$	0	0
$117$	11.5332	1.06624
$118$	0	0
$119$	4.48020	0.410700
$120$	0	0
$121$	−8.86973	−0.806339
$122$	0	0
$123$	−1.66593	−0.150211
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.63777	−0.234065	−0.117032	−	0.993128i	$-0.537338\pi$
$127$	−2.63777	−0.234065	−0.117032	+	0.993128i	$0.537338\pi$
$128$	0	0
$129$	−14.7603	−1.29957
$130$	0	0
$131$	−12.5589	−1.09727	−0.548636	−	0.836061i	$-0.684853\pi$
$131$	−12.5589	−1.09727	−0.548636	+	0.836061i	$0.684853\pi$
$132$	0	0
$133$	−1.68133	−0.145790
$134$	0	0
$135$	−0.254992	−0.0219463
$136$	0	0
$137$	6.98876	0.597090	0.298545	−	0.954396i	$-0.403499\pi$
$137$	6.98876	0.597090	0.298545	+	0.954396i	$0.403499\pi$
$138$	0	0
$139$	−13.3797	−1.13485	−0.567427	−	0.823424i	$-0.692061\pi$
$139$	−13.3797	−1.13485	−0.567427	+	0.823424i	$0.692061\pi$
$140$	0	0
$141$	10.4256	0.877992
$142$	0	0
$143$	5.42445	0.453615
$144$	0	0
$145$	3.60854	0.299673
$146$	0	0
$147$	15.9893	1.31878
$148$	0	0
$149$	5.00899	0.410353	0.205176	−	0.978725i	$-0.434223\pi$
$149$	5.00899	0.410353	0.205176	+	0.978725i	$0.434223\pi$
$150$	0	0
$151$	−24.4395	−1.98886	−0.994431	−	0.105390i	$-0.966391\pi$
$151$	−24.4395	−1.98886	−0.994431	+	0.105390i	$0.966391\pi$
$152$	0	0
$153$	−19.1367	−1.54711
$154$	0	0
$155$	−2.30751	−0.185344
$156$	0	0
$157$	−8.32288	−0.664238	−0.332119	−	0.943237i	$-0.607764\pi$
$157$	−8.32288	−0.664238	−0.332119	+	0.943237i	$0.607764\pi$
$158$	0	0
$159$	−35.6091	−2.82398
$160$	0	0
$161$	−4.60246	−0.362725
$162$	0	0
$163$	−10.0051	−0.783656	−0.391828	−	0.920038i	$-0.628157\pi$
$163$	−10.0051	−0.783656	−0.391828	+	0.920038i	$0.628157\pi$
$164$	0	0
$165$	−3.60576	−0.280708
$166$	0	0
$167$	3.70550	0.286740	0.143370	−	0.989669i	$-0.454206\pi$
$167$	3.70550	0.286740	0.143370	+	0.989669i	$0.454206\pi$
$168$	0	0
$169$	0.812607	0.0625082
$170$	0	0
$171$	7.18161	0.549191
$172$	0	0
$173$	−6.19393	−0.470915	−0.235458	−	0.971885i	$-0.575659\pi$
$173$	−6.19393	−0.470915	−0.235458	+	0.971885i	$0.575659\pi$
$174$	0	0
$175$	−0.726513	−0.0549192
$176$	0	0
$177$	36.2242	2.72278
$178$	0	0
$179$	17.0201	1.27214	0.636070	−	0.771631i	$-0.280559\pi$
$179$	17.0201	1.27214	0.636070	+	0.771631i	$0.280559\pi$
$180$	0	0
$181$	10.6866	0.794329	0.397165	−	0.917747i	$-0.369994\pi$
$181$	10.6866	0.794329	0.397165	+	0.917747i	$0.369994\pi$
$182$	0	0
$183$	3.13017	0.231389
$184$	0	0
$185$	0.575787	0.0423327
$186$	0	0
$187$	−9.00061	−0.658190
$188$	0	0
$189$	0.185255	0.0134753
$190$	0	0
$191$	−6.17850	−0.447060	−0.223530	−	0.974697i	$-0.571758\pi$
$191$	−6.17850	−0.447060	−0.223530	+	0.974697i	$0.571758\pi$
$192$	0	0
$193$	14.2624	1.02663	0.513315	−	0.858200i	$-0.328417\pi$
$193$	14.2624	1.02663	0.513315	+	0.858200i	$0.328417\pi$
$194$	0	0
$195$	−9.18158	−0.657506
$196$	0	0
$197$	25.8069	1.83867	0.919334	−	0.393478i	$-0.128728\pi$
$197$	25.8069	1.83867	0.919334	+	0.393478i	$0.128728\pi$
$198$	0	0
$199$	−20.5786	−1.45878	−0.729390	−	0.684098i	$-0.760196\pi$
$199$	−20.5786	−1.45878	−0.729390	+	0.684098i	$0.760196\pi$
$200$	0	0
$201$	−18.8376	−1.32870
$202$	0	0
$203$	−2.62165	−0.184004
$204$	0	0
$205$	0.674336	0.0470977
$206$	0	0
$207$	19.6589	1.36639
$208$	0	0
$209$	3.37775	0.233644
$210$	0	0
$211$	18.9388	1.30380	0.651899	−	0.758306i	$-0.273973\pi$
$211$	18.9388	1.30380	0.651899	+	0.758306i	$0.273973\pi$
$212$	0	0
$213$	−2.81342	−0.192772
$214$	0	0
$215$	5.97471	0.407472
$216$	0	0
$217$	1.67644	0.113804
$218$	0	0
$219$	−18.4197	−1.24469
$220$	0	0
$221$	−22.9188	−1.54169
$222$	0	0
$223$	24.3883	1.63316	0.816580	−	0.577233i	$-0.195867\pi$
$223$	24.3883	1.63316	0.816580	+	0.577233i	$0.195867\pi$
$224$	0	0
$225$	3.10322	0.206881
$226$	0	0
$227$	17.9988	1.19462	0.597312	−	0.802009i	$-0.296235\pi$
$227$	17.9988	1.19462	0.597312	+	0.802009i	$0.296235\pi$
$228$	0	0
$229$	3.95818	0.261564	0.130782	−	0.991411i	$-0.458251\pi$
$229$	3.95818	0.261564	0.130782	+	0.991411i	$0.458251\pi$
$230$	0	0
$231$	2.61963	0.172359
$232$	0	0
$233$	4.17047	0.273217	0.136608	−	0.990625i	$-0.456380\pi$
$233$	4.17047	0.273217	0.136608	+	0.990625i	$0.456380\pi$
$234$	0	0
$235$	−4.22008	−0.275288
$236$	0	0
$237$	3.27157	0.212511
$238$	0	0
$239$	−5.54470	−0.358657	−0.179328	−	0.983789i	$-0.557392\pi$
$239$	−5.54470	−0.358657	−0.179328	+	0.983789i	$0.557392\pi$
$240$	0	0
$241$	15.1780	0.977700	0.488850	−	0.872368i	$-0.337417\pi$
$241$	15.1780	0.977700	0.488850	+	0.872368i	$0.337417\pi$
$242$	0	0
$243$	22.2079	1.42464
$244$	0	0
$245$	−6.47218	−0.413492
$246$	0	0
$247$	8.60097	0.547267
$248$	0	0
$249$	6.39613	0.405338
$250$	0	0
$251$	−27.7844	−1.75373	−0.876867	−	0.480732i	$-0.840371\pi$
$251$	−27.7844	−1.75373	−0.876867	+	0.480732i	$0.840371\pi$
$252$	0	0
$253$	9.24623	0.581305
$254$	0	0
$255$	15.2347	0.954034
$256$	0	0
$257$	−15.2813	−0.953223	−0.476611	−	0.879114i	$-0.658135\pi$
$257$	−15.2813	−0.953223	−0.476611	+	0.879114i	$0.658135\pi$
$258$	0	0
$259$	−0.418317	−0.0259929
$260$	0	0
$261$	11.1981	0.693143
$262$	0	0
$263$	−29.9217	−1.84505	−0.922524	−	0.385939i	$-0.873878\pi$
$263$	−29.9217	−1.84505	−0.922524	+	0.385939i	$0.873878\pi$
$264$	0	0
$265$	14.4139	0.885439
$266$	0	0
$267$	−15.0504	−0.921067
$268$	0	0
$269$	−3.28426	−0.200245	−0.100123	−	0.994975i	$-0.531923\pi$
$269$	−3.28426	−0.200245	−0.100123	+	0.994975i	$0.531923\pi$
$270$	0	0
$271$	12.8398	0.779964	0.389982	−	0.920823i	$-0.372481\pi$
$271$	12.8398	0.779964	0.389982	+	0.920823i	$0.372481\pi$
$272$	0	0
$273$	6.67053	0.403719
$274$	0	0
$275$	1.45955	0.0880139
$276$	0	0
$277$	−21.1556	−1.27112	−0.635558	−	0.772053i	$-0.719230\pi$
$277$	−21.1556	−1.27112	−0.635558	+	0.772053i	$0.719230\pi$
$278$	0	0
$279$	−7.16071	−0.428700
$280$	0	0
$281$	−13.6501	−0.814294	−0.407147	−	0.913363i	$-0.633476\pi$
$281$	−13.6501	−0.814294	−0.407147	+	0.913363i	$0.633476\pi$
$282$	0	0
$283$	−16.3614	−0.972586	−0.486293	−	0.873796i	$-0.661651\pi$
$283$	−16.3614	−0.972586	−0.486293	+	0.873796i	$0.661651\pi$
$284$	0	0
$285$	−5.71728	−0.338662
$286$	0	0
$287$	−0.489914	−0.0289187
$288$	0	0
$289$	21.0285	1.23697
$290$	0	0
$291$	−18.2009	−1.06695
$292$	0	0
$293$	2.31367	0.135166	0.0675830	−	0.997714i	$-0.478471\pi$
$293$	2.31367	0.135166	0.0675830	+	0.997714i	$0.478471\pi$
$294$	0	0
$295$	−14.6629	−0.853707
$296$	0	0
$297$	−0.372173	−0.0215957
$298$	0	0
$299$	23.5442	1.36160
$300$	0	0
$301$	−4.34070	−0.250194
$302$	0	0
$303$	−12.0498	−0.692240
$304$	0	0
$305$	−1.26703	−0.0725502
$306$	0	0
$307$	−25.7601	−1.47021	−0.735105	−	0.677954i	$-0.762867\pi$
$307$	−25.7601	−1.47021	−0.735105	+	0.677954i	$0.762867\pi$
$308$	0	0
$309$	38.9390	2.21516
$310$	0	0
$311$	−5.35859	−0.303858	−0.151929	−	0.988391i	$-0.548548\pi$
$311$	−5.35859	−0.303858	−0.151929	+	0.988391i	$0.548548\pi$
$312$	0	0
$313$	32.6618	1.84615	0.923077	−	0.384616i	$-0.125666\pi$
$313$	32.6618	1.84615	0.923077	+	0.384616i	$0.125666\pi$
$314$	0	0
$315$	−2.25453	−0.127028
$316$	0	0
$317$	26.4637	1.48635	0.743175	−	0.669097i	$-0.233319\pi$
$317$	26.4637	1.48635	0.743175	+	0.669097i	$0.233319\pi$
$318$	0	0
$319$	5.26683	0.294886
$320$	0	0
$321$	−5.50487	−0.307252
$322$	0	0
$323$	−14.2713	−0.794077
$324$	0	0
$325$	3.71653	0.206156
$326$	0	0
$327$	−39.6618	−2.19330
$328$	0	0
$329$	3.06594	0.169031
$330$	0	0
$331$	27.1069	1.48993	0.744966	−	0.667102i	$-0.232466\pi$
$331$	27.1069	1.48993	0.744966	+	0.667102i	$0.232466\pi$
$332$	0	0
$333$	1.78679	0.0979156
$334$	0	0
$335$	7.62513	0.416605
$336$	0	0
$337$	−3.87037	−0.210833	−0.105416	−	0.994428i	$-0.533618\pi$
$337$	−3.87037	−0.210833	−0.105416	+	0.994428i	$0.533618\pi$
$338$	0	0
$339$	−7.56581	−0.410918
$340$	0	0
$341$	−3.36792	−0.182383
$342$	0	0
$343$	9.78771	0.528487
$344$	0	0
$345$	−15.6504	−0.842591
$346$	0	0
$347$	20.3819	1.09416	0.547079	−	0.837081i	$-0.315740\pi$
$347$	20.3819	1.09416	0.547079	+	0.837081i	$0.315740\pi$
$348$	0	0
$349$	−11.8958	−0.636766	−0.318383	−	0.947962i	$-0.603140\pi$
$349$	−11.8958	−0.636766	−0.318383	+	0.947962i	$0.603140\pi$
$350$	0	0
$351$	−0.947687	−0.0505838
$352$	0	0
$353$	26.0907	1.38867	0.694333	−	0.719654i	$-0.255700\pi$
$353$	26.0907	1.38867	0.694333	+	0.719654i	$0.255700\pi$
$354$	0	0
$355$	1.13882	0.0604423
$356$	0	0
$357$	−11.0682	−0.585791
$358$	0	0
$359$	13.5500	0.715141	0.357571	−	0.933886i	$-0.383605\pi$
$359$	13.5500	0.715141	0.357571	+	0.933886i	$0.383605\pi$
$360$	0	0
$361$	−13.6443	−0.718119
$362$	0	0
$363$	21.9124	1.15010
$364$	0	0
$365$	7.45595	0.390262
$366$	0	0
$367$	−21.5521	−1.12501	−0.562505	−	0.826794i	$-0.690162\pi$
$367$	−21.5521	−1.12501	−0.562505	+	0.826794i	$0.690162\pi$
$368$	0	0
$369$	2.09261	0.108937
$370$	0	0
$371$	−10.4719	−0.543673
$372$	0	0
$373$	12.0797	0.625461	0.312730	−	0.949842i	$-0.398756\pi$
$373$	12.0797	0.625461	0.312730	+	0.949842i	$0.398756\pi$
$374$	0	0
$375$	−2.47047	−0.127574
$376$	0	0
$377$	13.4112	0.690714
$378$	0	0
$379$	27.8380	1.42994	0.714972	−	0.699153i	$-0.246439\pi$
$379$	27.8380	1.42994	0.714972	+	0.699153i	$0.246439\pi$
$380$	0	0
$381$	6.51654	0.333852
$382$	0	0
$383$	10.0629	0.514188	0.257094	−	0.966386i	$-0.417235\pi$
$383$	10.0629	0.514188	0.257094	+	0.966386i	$0.417235\pi$
$384$	0	0
$385$	−1.06038	−0.0540419
$386$	0	0
$387$	18.5408	0.942482
$388$	0	0
$389$	7.07206	0.358568	0.179284	−	0.983797i	$-0.442622\pi$
$389$	7.07206	0.358568	0.179284	+	0.983797i	$0.442622\pi$
$390$	0	0
$391$	−39.0662	−1.97566
$392$	0	0
$393$	31.0263	1.56507
$394$	0	0
$395$	−1.32427	−0.0666313
$396$	0	0
$397$	−0.764152	−0.0383517	−0.0191758	−	0.999816i	$-0.506104\pi$
$397$	−0.764152	−0.0383517	−0.0191758	+	0.999816i	$0.506104\pi$
$398$	0	0
$399$	4.15367	0.207944
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−8.57594	−0.427198
$404$	0	0
$405$	−8.67970	−0.431298
$406$	0	0
$407$	0.840388	0.0416565
$408$	0	0
$409$	5.82301	0.287929	0.143965	−	0.989583i	$-0.454015\pi$
$409$	5.82301	0.287929	0.143965	+	0.989583i	$0.454015\pi$
$410$	0	0
$411$	−17.2655	−0.851645
$412$	0	0
$413$	10.6528	0.524189
$414$	0	0
$415$	−2.58904	−0.127091
$416$	0	0
$417$	33.0542	1.61867
$418$	0	0
$419$	11.7846	0.575718	0.287859	−	0.957673i	$-0.407057\pi$
$419$	11.7846	0.575718	0.287859	+	0.957673i	$0.407057\pi$
$420$	0	0
$421$	−13.5483	−0.660301	−0.330151	−	0.943928i	$-0.607100\pi$
$421$	−13.5483	−0.660301	−0.330151	+	0.943928i	$0.607100\pi$
$422$	0	0
$423$	−13.0958	−0.636741
$424$	0	0
$425$	−6.16672	−0.299130
$426$	0	0
$427$	0.920517	0.0445469
$428$	0	0
$429$	−13.4009	−0.647003
$430$	0	0
$431$	39.2624	1.89120	0.945601	−	0.325328i	$-0.105475\pi$
$431$	39.2624	1.89120	0.945601	+	0.325328i	$0.105475\pi$
$432$	0	0
$433$	16.7474	0.804828	0.402414	−	0.915458i	$-0.368171\pi$
$433$	16.7474	0.804828	0.402414	+	0.915458i	$0.368171\pi$
$434$	0	0
$435$	−8.91478	−0.427431
$436$	0	0
$437$	14.6608	0.701319
$438$	0	0
$439$	39.3978	1.88035	0.940177	−	0.340688i	$-0.110660\pi$
$439$	39.3978	1.88035	0.940177	+	0.340688i	$0.110660\pi$
$440$	0	0
$441$	−20.0846	−0.956408
$442$	0	0
$443$	18.8812	0.897075	0.448538	−	0.893764i	$-0.351945\pi$
$443$	18.8812	0.897075	0.448538	+	0.893764i	$0.351945\pi$
$444$	0	0
$445$	6.09211	0.288794
$446$	0	0
$447$	−12.3746	−0.585296
$448$	0	0
$449$	21.5278	1.01596	0.507979	−	0.861369i	$-0.330393\pi$
$449$	21.5278	1.01596	0.507979	+	0.861369i	$0.330393\pi$
$450$	0	0
$451$	0.984224	0.0463453
$452$	0	0
$453$	60.3771	2.83677
$454$	0	0
$455$	−2.70011	−0.126583
$456$	0	0
$457$	22.6308	1.05862	0.529311	−	0.848428i	$-0.322450\pi$
$457$	22.6308	1.05862	0.529311	+	0.848428i	$0.322450\pi$
$458$	0	0
$459$	1.57247	0.0733965
$460$	0	0
$461$	22.1289	1.03065	0.515323	−	0.856996i	$-0.327672\pi$
$461$	22.1289	1.03065	0.515323	+	0.856996i	$0.327672\pi$
$462$	0	0
$463$	−5.78716	−0.268952	−0.134476	−	0.990917i	$-0.542935\pi$
$463$	−5.78716	−0.268952	−0.134476	+	0.990917i	$0.542935\pi$
$464$	0	0
$465$	5.70063	0.264361
$466$	0	0
$467$	0.556011	0.0257291	0.0128646	−	0.999917i	$-0.495905\pi$
$467$	0.556011	0.0257291	0.0128646	+	0.999917i	$0.495905\pi$
$468$	0	0
$469$	−5.53975	−0.255802
$470$	0	0
$471$	20.5614	0.947420
$472$	0	0
$473$	8.72036	0.400963
$474$	0	0
$475$	2.31425	0.106185
$476$	0	0
$477$	44.7295	2.04802
$478$	0	0
$479$	21.6638	0.989845	0.494923	−	0.868937i	$-0.335196\pi$
$479$	21.6638	0.989845	0.494923	+	0.868937i	$0.335196\pi$
$480$	0	0
$481$	2.13993	0.0975724
$482$	0	0
$483$	11.3702	0.517364
$484$	0	0
$485$	7.36737	0.334535
$486$	0	0
$487$	−20.4708	−0.927621	−0.463811	−	0.885934i	$-0.653518\pi$
$487$	−20.4708	−0.927621	−0.463811	+	0.885934i	$0.653518\pi$
$488$	0	0
$489$	24.7172	1.11775
$490$	0	0
$491$	12.1429	0.548002	0.274001	−	0.961729i	$-0.411653\pi$
$491$	12.1429	0.548002	0.274001	+	0.961729i	$0.411653\pi$
$492$	0	0
$493$	−22.2528	−1.00222
$494$	0	0
$495$	4.52929	0.203576
$496$	0	0
$497$	−0.827368	−0.0371125
$498$	0	0
$499$	8.50320	0.380655	0.190328	−	0.981721i	$-0.439045\pi$
$499$	8.50320	0.380655	0.190328	+	0.981721i	$0.439045\pi$
$500$	0	0
$501$	−9.15433	−0.408985
$502$	0	0
$503$	−15.6010	−0.695615	−0.347807	−	0.937566i	$-0.613074\pi$
$503$	−15.6010	−0.695615	−0.347807	+	0.937566i	$0.613074\pi$
$504$	0	0
$505$	4.87752	0.217047
$506$	0	0
$507$	−2.00752	−0.0891571
$508$	0	0
$509$	19.9556	0.884515	0.442257	−	0.896888i	$-0.354178\pi$
$509$	19.9556	0.884515	0.442257	+	0.896888i	$0.354178\pi$
$510$	0	0
$511$	−5.41684	−0.239627
$512$	0	0
$513$	−0.590115	−0.0260542
$514$	0	0
$515$	−15.7618	−0.694547
$516$	0	0
$517$	−6.15940	−0.270890
$518$	0	0
$519$	15.3019	0.671679
$520$	0	0
$521$	−14.4373	−0.632510	−0.316255	−	0.948674i	$-0.602425\pi$
$521$	−14.4373	−0.632510	−0.316255	+	0.948674i	$0.602425\pi$
$522$	0	0
$523$	18.7908	0.821664	0.410832	−	0.911711i	$-0.365238\pi$
$523$	18.7908	0.821664	0.410832	+	0.911711i	$0.365238\pi$
$524$	0	0
$525$	1.79483	0.0783327
$526$	0	0
$527$	14.2298	0.619859
$528$	0	0
$529$	17.1322	0.744880
$530$	0	0
$531$	−45.5021	−1.97462
$532$	0	0
$533$	2.50619	0.108555
$534$	0	0
$535$	2.22827	0.0963366
$536$	0	0
$537$	−42.0476	−1.81449
$538$	0	0
$539$	−9.44644	−0.406887
$540$	0	0
$541$	37.8477	1.62720	0.813601	−	0.581424i	$-0.197504\pi$
$541$	37.8477	1.62720	0.813601	+	0.581424i	$0.197504\pi$
$542$	0	0
$543$	−26.4009	−1.13297
$544$	0	0
$545$	16.0544	0.687693
$546$	0	0
$547$	30.3659	1.29835	0.649176	−	0.760638i	$-0.275114\pi$
$547$	30.3659	1.29835	0.649176	+	0.760638i	$0.275114\pi$
$548$	0	0
$549$	−3.93188	−0.167809
$550$	0	0
$551$	8.35105	0.355767
$552$	0	0
$553$	0.962100	0.0409127
$554$	0	0
$555$	−1.42246	−0.0603802
$556$	0	0
$557$	14.8269	0.628238	0.314119	−	0.949384i	$-0.398291\pi$
$557$	14.8269	0.628238	0.314119	+	0.949384i	$0.398291\pi$
$558$	0	0
$559$	22.2052	0.939179
$560$	0	0
$561$	22.2357	0.938794
$562$	0	0
$563$	−4.26621	−0.179799	−0.0898997	−	0.995951i	$-0.528655\pi$
$563$	−4.26621	−0.179799	−0.0898997	+	0.995951i	$0.528655\pi$
$564$	0	0
$565$	3.06250	0.128840
$566$	0	0
$567$	6.30591	0.264823
$568$	0	0
$569$	23.8430	0.999549	0.499775	−	0.866155i	$-0.333416\pi$
$569$	23.8430	0.999549	0.499775	+	0.866155i	$0.333416\pi$
$570$	0	0
$571$	−28.9759	−1.21260	−0.606302	−	0.795234i	$-0.707348\pi$
$571$	−28.9759	−1.21260	−0.606302	+	0.795234i	$0.707348\pi$
$572$	0	0
$573$	15.2638	0.637654
$574$	0	0
$575$	6.33500	0.264188
$576$	0	0
$577$	40.4129	1.68241	0.841206	−	0.540715i	$-0.181846\pi$
$577$	40.4129	1.68241	0.841206	+	0.540715i	$0.181846\pi$
$578$	0	0
$579$	−35.2348	−1.46431
$580$	0	0
$581$	1.88097	0.0780357
$582$	0	0
$583$	21.0378	0.871295
$584$	0	0
$585$	11.5332	0.476839
$586$	0	0
$587$	28.1088	1.16018	0.580088	−	0.814554i	$-0.303018\pi$
$587$	28.1088	1.16018	0.580088	+	0.814554i	$0.303018\pi$
$588$	0	0
$589$	−5.34015	−0.220037
$590$	0	0
$591$	−63.7552	−2.62254
$592$	0	0
$593$	−2.56359	−0.105274	−0.0526369	−	0.998614i	$-0.516763\pi$
$593$	−2.56359	−0.105274	−0.0526369	+	0.998614i	$0.516763\pi$
$594$	0	0
$595$	4.48020	0.183670
$596$	0	0
$597$	50.8388	2.08069
$598$	0	0
$599$	−34.8711	−1.42480	−0.712398	−	0.701776i	$-0.752391\pi$
$599$	−34.8711	−1.42480	−0.712398	+	0.701776i	$0.752391\pi$
$600$	0	0
$601$	−2.38421	−0.0972538	−0.0486269	−	0.998817i	$-0.515485\pi$
$601$	−2.38421	−0.0972538	−0.0486269	+	0.998817i	$0.515485\pi$
$602$	0	0
$603$	23.6624	0.963608
$604$	0	0
$605$	−8.86973	−0.360606
$606$	0	0
$607$	−0.0153986	−0.000625011 0	−0.000312506	−	1.00000i	$-0.500099\pi$
$607$	−0.0153986	−0.000625011 0	−0.000312506		1.00000i	$0.500099\pi$
$608$	0	0
$609$	6.47670	0.262449
$610$	0	0
$611$	−15.6841	−0.634509
$612$	0	0
$613$	−1.36715	−0.0552186	−0.0276093	−	0.999619i	$-0.508789\pi$
$613$	−1.36715	−0.0552186	−0.0276093	+	0.999619i	$0.508789\pi$
$614$	0	0
$615$	−1.66593	−0.0671766
$616$	0	0
$617$	18.8726	0.759781	0.379890	−	0.925032i	$-0.375962\pi$
$617$	18.8726	0.759781	0.379890	+	0.925032i	$0.375962\pi$
$618$	0	0
$619$	24.1507	0.970698	0.485349	−	0.874320i	$-0.338693\pi$
$619$	24.1507	0.970698	0.485349	+	0.874320i	$0.338693\pi$
$620$	0	0
$621$	−1.61538	−0.0648229
$622$	0	0
$623$	−4.42600	−0.177324
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.34463	−0.333252
$628$	0	0
$629$	−3.55072	−0.141576
$630$	0	0
$631$	30.5122	1.21467	0.607336	−	0.794445i	$-0.292238\pi$
$631$	30.5122	1.21467	0.607336	+	0.794445i	$0.292238\pi$
$632$	0	0
$633$	−46.7876	−1.85964
$634$	0	0
$635$	−2.63777	−0.104677
$636$	0	0
$637$	−24.0541	−0.953056
$638$	0	0
$639$	3.53401	0.139803
$640$	0	0
$641$	−32.4463	−1.28155	−0.640776	−	0.767728i	$-0.721387\pi$
$641$	−32.4463	−1.28155	−0.640776	+	0.767728i	$0.721387\pi$
$642$	0	0
$643$	−0.110690	−0.00436518	−0.00218259	−	0.999998i	$-0.500695\pi$
$643$	−0.110690	−0.00436518	−0.00218259	+	0.999998i	$0.500695\pi$
$644$	0	0
$645$	−14.7603	−0.581187
$646$	0	0
$647$	8.45331	0.332334	0.166167	−	0.986098i	$-0.446861\pi$
$647$	8.45331	0.332334	0.166167	+	0.986098i	$0.446861\pi$
$648$	0	0
$649$	−21.4012	−0.840069
$650$	0	0
$651$	−4.14158	−0.162322
$652$	0	0
$653$	−37.5478	−1.46936	−0.734680	−	0.678414i	$-0.762668\pi$
$653$	−37.5478	−1.46936	−0.734680	+	0.678414i	$0.762668\pi$
$654$	0	0
$655$	−12.5589	−0.490715
$656$	0	0
$657$	23.1374	0.902677
$658$	0	0
$659$	15.8692	0.618175	0.309087	−	0.951034i	$-0.399976\pi$
$659$	15.8692	0.618175	0.309087	+	0.951034i	$0.399976\pi$
$660$	0	0
$661$	−40.9566	−1.59303	−0.796514	−	0.604621i	$-0.793325\pi$
$661$	−40.9566	−1.59303	−0.796514	+	0.604621i	$0.793325\pi$
$662$	0	0
$663$	56.6202	2.19895
$664$	0	0
$665$	−1.68133	−0.0651992
$666$	0	0
$667$	22.8601	0.885146
$668$	0	0
$669$	−60.2505	−2.32942
$670$	0	0
$671$	−1.84929	−0.0713912
$672$	0	0
$673$	−43.3073	−1.66938	−0.834688	−	0.550724i	$-0.814352\pi$
$673$	−43.3073	−1.66938	−0.834688	+	0.550724i	$0.814352\pi$
$674$	0	0
$675$	−0.254992	−0.00981466
$676$	0	0
$677$	−19.5692	−0.752107	−0.376053	−	0.926598i	$-0.622719\pi$
$677$	−19.5692	−0.752107	−0.376053	+	0.926598i	$0.622719\pi$
$678$	0	0
$679$	−5.35249	−0.205410
$680$	0	0
$681$	−44.4655	−1.70392
$682$	0	0
$683$	12.7377	0.487395	0.243697	−	0.969851i	$-0.421640\pi$
$683$	12.7377	0.487395	0.243697	+	0.969851i	$0.421640\pi$
$684$	0	0
$685$	6.98876	0.267027
$686$	0	0
$687$	−9.77855	−0.373075
$688$	0	0
$689$	53.5697	2.04084
$690$	0	0
$691$	−9.22621	−0.350982	−0.175491	−	0.984481i	$-0.556151\pi$
$691$	−9.22621	−0.350982	−0.175491	+	0.984481i	$0.556151\pi$
$692$	0	0
$693$	−3.29059	−0.124999
$694$	0	0
$695$	−13.3797	−0.507522
$696$	0	0
$697$	−4.15844	−0.157512
$698$	0	0
$699$	−10.3030	−0.389696
$700$	0	0
$701$	−45.1740	−1.70620	−0.853098	−	0.521750i	$-0.825279\pi$
$701$	−45.1740	−1.70620	−0.853098	+	0.521750i	$0.825279\pi$
$702$	0	0
$703$	1.33251	0.0502567
$704$	0	0
$705$	10.4256	0.392650
$706$	0	0
$707$	−3.54358	−0.133270
$708$	0	0
$709$	7.10059	0.266668	0.133334	−	0.991071i	$-0.457432\pi$
$709$	7.10059	0.266668	0.133334	+	0.991071i	$0.457432\pi$
$710$	0	0
$711$	−4.10950	−0.154118
$712$	0	0
$713$	−14.6181	−0.547452
$714$	0	0
$715$	5.42445	0.202863
$716$	0	0
$717$	13.6980	0.511561
$718$	0	0
$719$	22.2807	0.830928	0.415464	−	0.909610i	$-0.363619\pi$
$719$	22.2807	0.830928	0.415464	+	0.909610i	$0.363619\pi$
$720$	0	0
$721$	11.4511	0.426463
$722$	0	0
$723$	−37.4967	−1.39452
$724$	0	0
$725$	3.60854	0.134018
$726$	0	0
$727$	42.3064	1.56906	0.784529	−	0.620092i	$-0.212905\pi$
$727$	42.3064	1.56906	0.784529	+	0.620092i	$0.212905\pi$
$728$	0	0
$729$	−28.8248	−1.06759
$730$	0	0
$731$	−36.8444	−1.36274
$732$	0	0
$733$	−47.1373	−1.74106	−0.870528	−	0.492118i	$-0.836223\pi$
$733$	−47.1373	−1.74106	−0.870528	+	0.492118i	$0.836223\pi$
$734$	0	0
$735$	15.9893	0.589775
$736$	0	0
$737$	11.1292	0.409950
$738$	0	0
$739$	−10.3658	−0.381312	−0.190656	−	0.981657i	$-0.561062\pi$
$739$	−10.3658	−0.381312	−0.190656	+	0.981657i	$0.561062\pi$
$740$	0	0
$741$	−21.2484	−0.780581
$742$	0	0
$743$	6.11245	0.224244	0.112122	−	0.993694i	$-0.464235\pi$
$743$	6.11245	0.224244	0.112122	+	0.993694i	$0.464235\pi$
$744$	0	0
$745$	5.00899	0.183515
$746$	0	0
$747$	−8.03434	−0.293961
$748$	0	0
$749$	−1.61887	−0.0591521
$750$	0	0
$751$	−3.98955	−0.145581	−0.0727903	−	0.997347i	$-0.523190\pi$
$751$	−3.98955	−0.145581	−0.0727903	+	0.997347i	$0.523190\pi$
$752$	0	0
$753$	68.6404	2.50140
$754$	0	0
$755$	−24.4395	−0.889446
$756$	0	0
$757$	2.89107	0.105078	0.0525389	−	0.998619i	$-0.483269\pi$
$757$	2.89107	0.105078	0.0525389	+	0.998619i	$0.483269\pi$
$758$	0	0
$759$	−22.8425	−0.829131
$760$	0	0
$761$	38.2107	1.38514	0.692569	−	0.721352i	$-0.256479\pi$
$761$	38.2107	1.38514	0.692569	+	0.721352i	$0.256479\pi$
$762$	0	0
$763$	−11.6637	−0.422254
$764$	0	0
$765$	−19.1367	−0.691888
$766$	0	0
$767$	−54.4951	−1.96770
$768$	0	0
$769$	43.4090	1.56537	0.782684	−	0.622420i	$-0.213850\pi$
$769$	43.4090	1.56537	0.782684	+	0.622420i	$0.213850\pi$
$770$	0	0
$771$	37.7520	1.35961
$772$	0	0
$773$	47.6466	1.71373	0.856864	−	0.515543i	$-0.172410\pi$
$773$	47.6466	1.71373	0.856864	+	0.515543i	$0.172410\pi$
$774$	0	0
$775$	−2.30751	−0.0828883
$776$	0	0
$777$	1.03344	0.0370744
$778$	0	0
$779$	1.56058	0.0559136
$780$	0	0
$781$	1.66216	0.0594768
$782$	0	0
$783$	−0.920150	−0.0328835
$784$	0	0
$785$	−8.32288	−0.297056
$786$	0	0
$787$	44.4071	1.58294	0.791470	−	0.611208i	$-0.209316\pi$
$787$	44.4071	1.58294	0.791470	+	0.611208i	$0.209316\pi$
$788$	0	0
$789$	73.9205	2.63164
$790$	0	0
$791$	−2.22494	−0.0791099
$792$	0	0
$793$	−4.70897	−0.167221
$794$	0	0
$795$	−35.6091	−1.26292
$796$	0	0
$797$	10.9489	0.387829	0.193914	−	0.981018i	$-0.437882\pi$
$797$	10.9489	0.387829	0.193914	+	0.981018i	$0.437882\pi$
$798$	0	0
$799$	26.0241	0.920665
$800$	0	0
$801$	18.9051	0.667980
$802$	0	0
$803$	10.8823	0.384028
$804$	0	0
$805$	−4.60246	−0.162215
$806$	0	0
$807$	8.11367	0.285615
$808$	0	0
$809$	−22.7150	−0.798617	−0.399309	−	0.916817i	$-0.630750\pi$
$809$	−22.7150	−0.798617	−0.399309	+	0.916817i	$0.630750\pi$
$810$	0	0
$811$	−6.03580	−0.211946	−0.105973	−	0.994369i	$-0.533796\pi$
$811$	−6.03580	−0.211946	−0.105973	+	0.994369i	$0.533796\pi$
$812$	0	0
$813$	−31.7204	−1.11248
$814$	0	0
$815$	−10.0051	−0.350462
$816$	0	0
$817$	13.8269	0.483744
$818$	0	0
$819$	−8.37902	−0.292787
$820$	0	0
$821$	−2.57926	−0.0900167	−0.0450083	−	0.998987i	$-0.514331\pi$
$821$	−2.57926	−0.0900167	−0.0450083	+	0.998987i	$0.514331\pi$
$822$	0	0
$823$	9.82892	0.342615	0.171307	−	0.985218i	$-0.445201\pi$
$823$	9.82892	0.342615	0.171307	+	0.985218i	$0.445201\pi$
$824$	0	0
$825$	−3.60576	−0.125537
$826$	0	0
$827$	−27.4158	−0.953342	−0.476671	−	0.879082i	$-0.658157\pi$
$827$	−27.4158	−0.953342	−0.476671	+	0.879082i	$0.658157\pi$
$828$	0	0
$829$	−47.2280	−1.64030	−0.820148	−	0.572151i	$-0.806109\pi$
$829$	−47.2280	−1.64030	−0.820148	+	0.572151i	$0.806109\pi$
$830$	0	0
$831$	52.2642	1.81302
$832$	0	0
$833$	39.9121	1.38287
$834$	0	0
$835$	3.70550	0.128234
$836$	0	0
$837$	0.588398	0.0203380
$838$	0	0
$839$	45.1716	1.55950	0.779749	−	0.626092i	$-0.215347\pi$
$839$	45.1716	1.55950	0.779749	+	0.626092i	$0.215347\pi$
$840$	0	0
$841$	−15.9785	−0.550981
$842$	0	0
$843$	33.7221	1.16145
$844$	0	0
$845$	0.812607	0.0279545
$846$	0	0
$847$	6.44397	0.221417
$848$	0	0
$849$	40.4204	1.38722
$850$	0	0
$851$	3.64761	0.125038
$852$	0	0
$853$	−45.6370	−1.56258	−0.781290	−	0.624168i	$-0.785438\pi$
$853$	−45.6370	−1.56258	−0.781290	+	0.624168i	$0.785438\pi$
$854$	0	0
$855$	7.18161	0.245606
$856$	0	0
$857$	−22.5351	−0.769783	−0.384891	−	0.922962i	$-0.625761\pi$
$857$	−22.5351	−0.769783	−0.384891	+	0.922962i	$0.625761\pi$
$858$	0	0
$859$	−24.2935	−0.828882	−0.414441	−	0.910076i	$-0.636023\pi$
$859$	−24.2935	−0.828882	−0.414441	+	0.910076i	$0.636023\pi$
$860$	0	0
$861$	1.21032	0.0412475
$862$	0	0
$863$	40.8615	1.39094	0.695471	−	0.718554i	$-0.255196\pi$
$863$	40.8615	1.39094	0.695471	+	0.718554i	$0.255196\pi$
$864$	0	0
$865$	−6.19393	−0.210600
$866$	0	0
$867$	−51.9501	−1.76432
$868$	0	0
$869$	−1.93284	−0.0655670
$870$	0	0
$871$	28.3390	0.960231
$872$	0	0
$873$	22.8625	0.773780
$874$	0	0
$875$	−0.726513	−0.0245606
$876$	0	0
$877$	1.61456	0.0545198	0.0272599	−	0.999628i	$-0.491322\pi$
$877$	1.61456	0.0545198	0.0272599	+	0.999628i	$0.491322\pi$
$878$	0	0
$879$	−5.71585	−0.192791
$880$	0	0
$881$	−39.7743	−1.34003	−0.670015	−	0.742348i	$-0.733712\pi$
$881$	−39.7743	−1.34003	−0.670015	+	0.742348i	$0.733712\pi$
$882$	0	0
$883$	−33.0006	−1.11056	−0.555279	−	0.831664i	$-0.687389\pi$
$883$	−33.0006	−1.11056	−0.555279	+	0.831664i	$0.687389\pi$
$884$	0	0
$885$	36.2242	1.21766
$886$	0	0
$887$	−40.2071	−1.35002	−0.675012	−	0.737807i	$-0.735861\pi$
$887$	−40.2071	−1.35002	−0.675012	+	0.737807i	$0.735861\pi$
$888$	0	0
$889$	1.91638	0.0642732
$890$	0	0
$891$	−12.6684	−0.424408
$892$	0	0
$893$	−9.76631	−0.326817
$894$	0	0
$895$	17.0201	0.568918
$896$	0	0
$897$	−58.1653	−1.94208
$898$	0	0
$899$	−8.32674	−0.277712
$900$	0	0
$901$	−88.8865	−2.96124
$902$	0	0
$903$	10.7236	0.356858
$904$	0	0
$905$	10.6866	0.355235
$906$	0	0
$907$	−1.02603	−0.0340686	−0.0170343	−	0.999855i	$-0.505422\pi$
$907$	−1.02603	−0.0340686	−0.0170343	+	0.999855i	$0.505422\pi$
$908$	0	0
$909$	15.1360	0.502029
$910$	0	0
$911$	−5.41039	−0.179254	−0.0896271	−	0.995975i	$-0.528568\pi$
$911$	−5.41039	−0.179254	−0.0896271	+	0.995975i	$0.528568\pi$
$912$	0	0
$913$	−3.77882	−0.125061
$914$	0	0
$915$	3.13017	0.103480
$916$	0	0
$917$	9.12417	0.301307
$918$	0	0
$919$	35.3384	1.16571	0.582853	−	0.812578i	$-0.301936\pi$
$919$	35.3384	1.16571	0.582853	+	0.812578i	$0.301936\pi$
$920$	0	0
$921$	63.6396	2.09700
$922$	0	0
$923$	4.23246	0.139313
$924$	0	0
$925$	0.575787	0.0189318
$926$	0	0
$927$	−48.9122	−1.60649
$928$	0	0
$929$	12.1573	0.398867	0.199433	−	0.979911i	$-0.436090\pi$
$929$	12.1573	0.398867	0.199433	+	0.979911i	$0.436090\pi$
$930$	0	0
$931$	−14.9782	−0.490891
$932$	0	0
$933$	13.2382	0.433400
$934$	0	0
$935$	−9.00061	−0.294352
$936$	0	0
$937$	16.6996	0.545554	0.272777	−	0.962077i	$-0.412058\pi$
$937$	16.6996	0.545554	0.272777	+	0.962077i	$0.412058\pi$
$938$	0	0
$939$	−80.6899	−2.63322
$940$	0	0
$941$	42.3099	1.37926	0.689632	−	0.724160i	$-0.257772\pi$
$941$	42.3099	1.37926	0.689632	+	0.724160i	$0.257772\pi$
$942$	0	0
$943$	4.27192	0.139113
$944$	0	0
$945$	0.185255	0.00602635
$946$	0	0
$947$	3.40261	0.110570	0.0552849	−	0.998471i	$-0.482393\pi$
$947$	3.40261	0.110570	0.0552849	+	0.998471i	$0.482393\pi$
$948$	0	0
$949$	27.7103	0.899513
$950$	0	0
$951$	−65.3778	−2.12002
$952$	0	0
$953$	−30.8392	−0.998979	−0.499489	−	0.866320i	$-0.666479\pi$
$953$	−30.8392	−0.998979	−0.499489	+	0.866320i	$0.666479\pi$
$954$	0	0
$955$	−6.17850	−0.199931
$956$	0	0
$957$	−13.0115	−0.420603
$958$	0	0
$959$	−5.07742	−0.163959
$960$	0	0
$961$	−25.6754	−0.828238
$962$	0	0
$963$	6.91481	0.222827
$964$	0	0
$965$	14.2624	0.459123
$966$	0	0
$967$	46.8323	1.50602	0.753012	−	0.658006i	$-0.228600\pi$
$967$	46.8323	1.50602	0.753012	+	0.658006i	$0.228600\pi$
$968$	0	0
$969$	35.2568	1.13261
$970$	0	0
$971$	16.7271	0.536797	0.268398	−	0.963308i	$-0.413506\pi$
$971$	16.7271	0.536797	0.268398	+	0.963308i	$0.413506\pi$
$972$	0	0
$973$	9.72055	0.311626
$974$	0	0
$975$	−9.18158	−0.294046
$976$	0	0
$977$	2.36237	0.0755789	0.0377895	−	0.999286i	$-0.487968\pi$
$977$	2.36237	0.0755789	0.0377895	+	0.999286i	$0.487968\pi$
$978$	0	0
$979$	8.89171	0.284180
$980$	0	0
$981$	49.8201	1.59063
$982$	0	0
$983$	12.5751	0.401082	0.200541	−	0.979685i	$-0.435730\pi$
$983$	12.5751	0.401082	0.200541	+	0.979685i	$0.435730\pi$
$984$	0	0
$985$	25.8069	0.822277
$986$	0	0
$987$	−7.57432	−0.241093
$988$	0	0
$989$	37.8498	1.20355
$990$	0	0
$991$	46.7263	1.48431	0.742155	−	0.670228i	$-0.233804\pi$
$991$	46.7263	1.48431	0.742155	+	0.670228i	$0.233804\pi$
$992$	0	0
$993$	−66.9669	−2.12513
$994$	0	0
$995$	−20.5786	−0.652386
$996$	0	0
$997$	−43.3761	−1.37373	−0.686867	−	0.726783i	$-0.741015\pi$
$997$	−43.3761	−1.37373	−0.686867	+	0.726783i	$0.741015\pi$
$998$	0	0
$999$	−0.146821	−0.00464522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.5	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-2.47047 q^{3} +1.00000 q^{5} -0.726513 q^{7} +3.10322 q^{9} +O(q^{10})\)	\(q-2.47047 q^{3} +1.00000 q^{5} -0.726513 q^{7} +3.10322 q^{9} +1.45955 q^{11} +3.71653 q^{13} -2.47047 q^{15} -6.16672 q^{17} +2.31425 q^{19} +1.79483 q^{21} +6.33500 q^{23} +1.00000 q^{25} -0.254992 q^{27} +3.60854 q^{29} -2.30751 q^{31} -3.60576 q^{33} -0.726513 q^{35} +0.575787 q^{37} -9.18158 q^{39} +0.674336 q^{41} +5.97471 q^{43} +3.10322 q^{45} -4.22008 q^{47} -6.47218 q^{49} +15.2347 q^{51} +14.4139 q^{53} +1.45955 q^{55} -5.71728 q^{57} -14.6629 q^{59} -1.26703 q^{61} -2.25453 q^{63} +3.71653 q^{65} +7.62513 q^{67} -15.6504 q^{69} +1.13882 q^{71} +7.45595 q^{73} -2.47047 q^{75} -1.06038 q^{77} -1.32427 q^{79} -8.67970 q^{81} -2.58904 q^{83} -6.16672 q^{85} -8.91478 q^{87} +6.09211 q^{89} -2.70011 q^{91} +5.70063 q^{93} +2.31425 q^{95} +7.36737 q^{97} +4.52929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 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

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Database

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