LMFDB - Embedded newform 8036.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.32542$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32542 q^{3} +1.16439 q^{5} +2.40759 q^{9} +O(q^{10})$	$q-2.32542 q^{3} +1.16439 q^{5} +2.40759 q^{9} +0.577754 q^{11} +2.18295 q^{13} -2.70769 q^{15} +6.65536 q^{17} -2.12850 q^{19} +8.32156 q^{23} -3.64420 q^{25} +1.37760 q^{27} -6.30024 q^{29} -1.38664 q^{31} -1.34352 q^{33} +2.86577 q^{37} -5.07629 q^{39} -1.00000 q^{41} +6.21582 q^{43} +2.80337 q^{45} +12.3492 q^{47} -15.4765 q^{51} +13.0449 q^{53} +0.672730 q^{55} +4.94967 q^{57} -3.27380 q^{59} +5.15551 q^{61} +2.54180 q^{65} -5.35143 q^{67} -19.3511 q^{69} -14.3919 q^{71} +8.30995 q^{73} +8.47431 q^{75} -14.6782 q^{79} -10.4263 q^{81} -11.7691 q^{83} +7.74941 q^{85} +14.6507 q^{87} +6.56350 q^{89} +3.22452 q^{93} -2.47840 q^{95} -2.09480 q^{97} +1.39100 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

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
$3$	−2.32542	−1.34258	−0.671292	−	0.741193i	$-0.734260\pi$
$3$	−2.32542	−1.34258	−0.671292	+	0.741193i	$0.734260\pi$
$4$	0	0
$5$	1.16439	0.520730	0.260365	−	0.965510i	$-0.416157\pi$
$5$	1.16439	0.520730	0.260365	+	0.965510i	$0.416157\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.40759	0.802531
$10$	0	0
$11$	0.577754	0.174199	0.0870997	−	0.996200i	$-0.472240\pi$
$11$	0.577754	0.174199	0.0870997	+	0.996200i	$0.472240\pi$
$12$	0	0
$13$	2.18295	0.605442	0.302721	−	0.953079i	$-0.402105\pi$
$13$	2.18295	0.605442	0.302721	+	0.953079i	$0.402105\pi$
$14$	0	0
$15$	−2.70769	−0.699123
$16$	0	0
$17$	6.65536	1.61416	0.807081	−	0.590441i	$-0.201046\pi$
$17$	6.65536	1.61416	0.807081	+	0.590441i	$0.201046\pi$
$18$	0	0
$19$	−2.12850	−0.488312	−0.244156	−	0.969736i	$-0.578511\pi$
$19$	−2.12850	−0.488312	−0.244156	+	0.969736i	$0.578511\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.32156	1.73516	0.867582	−	0.497294i	$-0.165673\pi$
$23$	8.32156	1.73516	0.867582	+	0.497294i	$0.165673\pi$
$24$	0	0
$25$	−3.64420	−0.728840
$26$	0	0
$27$	1.37760	0.265119
$28$	0	0
$29$	−6.30024	−1.16993	−0.584963	−	0.811060i	$-0.698891\pi$
$29$	−6.30024	−1.16993	−0.584963	+	0.811060i	$0.698891\pi$
$30$	0	0
$31$	−1.38664	−0.249047	−0.124524	−	0.992217i	$-0.539740\pi$
$31$	−1.38664	−0.249047	−0.124524	+	0.992217i	$0.539740\pi$
$32$	0	0
$33$	−1.34352	−0.233877
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.86577	0.471130	0.235565	−	0.971859i	$-0.424306\pi$
$37$	2.86577	0.471130	0.235565	+	0.971859i	$0.424306\pi$
$38$	0	0
$39$	−5.07629	−0.812856
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	6.21582	0.947904	0.473952	−	0.880551i	$-0.342827\pi$
$43$	6.21582	0.947904	0.473952	+	0.880551i	$0.342827\pi$
$44$	0	0
$45$	2.80337	0.417902
$46$	0	0
$47$	12.3492	1.80132	0.900658	−	0.434528i	$-0.143085\pi$
$47$	12.3492	1.80132	0.900658	+	0.434528i	$0.143085\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−15.4765	−2.16715
$52$	0	0
$53$	13.0449	1.79185	0.895925	−	0.444204i	$-0.146514\pi$
$53$	13.0449	1.79185	0.895925	+	0.444204i	$0.146514\pi$
$54$	0	0
$55$	0.672730	0.0907109
$56$	0	0
$57$	4.94967	0.655600
$58$	0	0
$59$	−3.27380	−0.426213	−0.213106	−	0.977029i	$-0.568358\pi$
$59$	−3.27380	−0.426213	−0.213106	+	0.977029i	$0.568358\pi$
$60$	0	0
$61$	5.15551	0.660095	0.330048	−	0.943964i	$-0.392935\pi$
$61$	5.15551	0.660095	0.330048	+	0.943964i	$0.392935\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.54180	0.315272
$66$	0	0
$67$	−5.35143	−0.653781	−0.326890	−	0.945062i	$-0.606001\pi$
$67$	−5.35143	−0.653781	−0.326890	+	0.945062i	$0.606001\pi$
$68$	0	0
$69$	−19.3511	−2.32960
$70$	0	0
$71$	−14.3919	−1.70801	−0.854004	−	0.520267i	$-0.825833\pi$
$71$	−14.3919	−1.70801	−0.854004	+	0.520267i	$0.825833\pi$
$72$	0	0
$73$	8.30995	0.972606	0.486303	−	0.873790i	$-0.338345\pi$
$73$	8.30995	0.972606	0.486303	+	0.873790i	$0.338345\pi$
$74$	0	0
$75$	8.47431	0.978529
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.6782	−1.65143	−0.825716	−	0.564086i	$-0.809229\pi$
$79$	−14.6782	−1.65143	−0.825716	+	0.564086i	$0.809229\pi$
$80$	0	0
$81$	−10.4263	−1.15848
$82$	0	0
$83$	−11.7691	−1.29182	−0.645911	−	0.763413i	$-0.723522\pi$
$83$	−11.7691	−1.29182	−0.645911	+	0.763413i	$0.723522\pi$
$84$	0	0
$85$	7.74941	0.840542
$86$	0	0
$87$	14.6507	1.57072
$88$	0	0
$89$	6.56350	0.695730	0.347865	−	0.937545i	$-0.386907\pi$
$89$	6.56350	0.695730	0.347865	+	0.937545i	$0.386907\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.22452	0.334367
$94$	0	0
$95$	−2.47840	−0.254279
$96$	0	0
$97$	−2.09480	−0.212695	−0.106347	−	0.994329i	$-0.533916\pi$
$97$	−2.09480	−0.212695	−0.106347	+	0.994329i	$0.533916\pi$
$98$	0	0
$99$	1.39100	0.139800
$100$	0	0
$101$	4.73209	0.470861	0.235430	−	0.971891i	$-0.424350\pi$
$101$	4.73209	0.470861	0.235430	+	0.971891i	$0.424350\pi$
$102$	0	0
$103$	15.6314	1.54021	0.770106	−	0.637916i	$-0.220204\pi$
$103$	15.6314	1.54021	0.770106	+	0.637916i	$0.220204\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.34990	−0.420520	−0.210260	−	0.977645i	$-0.567431\pi$
$107$	−4.34990	−0.420520	−0.210260	+	0.977645i	$0.567431\pi$
$108$	0	0
$109$	14.7394	1.41178	0.705890	−	0.708321i	$-0.250547\pi$
$109$	14.7394	1.41178	0.705890	+	0.708321i	$0.250547\pi$
$110$	0	0
$111$	−6.66413	−0.632531
$112$	0	0
$113$	−1.29851	−0.122153	−0.0610766	−	0.998133i	$-0.519453\pi$
$113$	−1.29851	−0.122153	−0.0610766	+	0.998133i	$0.519453\pi$
$114$	0	0
$115$	9.68951	0.903552
$116$	0	0
$117$	5.25566	0.485886
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.6662	−0.969655
$122$	0	0
$123$	2.32542	0.209676
$124$	0	0
$125$	−10.0652	−0.900259
$126$	0	0
$127$	−16.6554	−1.47793	−0.738964	−	0.673745i	$-0.764685\pi$
$127$	−16.6554	−1.47793	−0.738964	+	0.673745i	$0.764685\pi$
$128$	0	0
$129$	−14.4544	−1.27264
$130$	0	0
$131$	18.9750	1.65785	0.828926	−	0.559359i	$-0.188953\pi$
$131$	18.9750	1.65785	0.828926	+	0.559359i	$0.188953\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.60406	0.138055
$136$	0	0
$137$	−3.44151	−0.294028	−0.147014	−	0.989134i	$-0.546966\pi$
$137$	−3.44151	−0.294028	−0.147014	+	0.989134i	$0.546966\pi$
$138$	0	0
$139$	−4.88501	−0.414341	−0.207171	−	0.978305i	$-0.566426\pi$
$139$	−4.88501	−0.414341	−0.207171	+	0.978305i	$0.566426\pi$
$140$	0	0
$141$	−28.7171	−2.41842
$142$	0	0
$143$	1.26121	0.105468
$144$	0	0
$145$	−7.33592	−0.609215
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.0258	1.80442	0.902210	−	0.431297i	$-0.141944\pi$
$149$	22.0258	1.80442	0.902210	+	0.431297i	$0.141944\pi$
$150$	0	0
$151$	−9.59867	−0.781128	−0.390564	−	0.920576i	$-0.627720\pi$
$151$	−9.59867	−0.781128	−0.390564	+	0.920576i	$0.627720\pi$
$152$	0	0
$153$	16.0234	1.29541
$154$	0	0
$155$	−1.61458	−0.129686
$156$	0	0
$157$	2.53129	0.202019	0.101010	−	0.994885i	$-0.467793\pi$
$157$	2.53129	0.202019	0.101010	+	0.994885i	$0.467793\pi$
$158$	0	0
$159$	−30.3348	−2.40571
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−18.9641	−1.48539	−0.742693	−	0.669632i	$-0.766452\pi$
$163$	−18.9641	−1.48539	−0.742693	+	0.669632i	$0.766452\pi$
$164$	0	0
$165$	−1.56438	−0.121787
$166$	0	0
$167$	−1.39051	−0.107601	−0.0538004	−	0.998552i	$-0.517133\pi$
$167$	−1.39051	−0.107601	−0.0538004	+	0.998552i	$0.517133\pi$
$168$	0	0
$169$	−8.23472	−0.633440
$170$	0	0
$171$	−5.12457	−0.391885
$172$	0	0
$173$	20.1903	1.53504	0.767518	−	0.641027i	$-0.221491\pi$
$173$	20.1903	1.53504	0.767518	+	0.641027i	$0.221491\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.61298	0.572226
$178$	0	0
$179$	0.927522	0.0693262	0.0346631	−	0.999399i	$-0.488964\pi$
$179$	0.927522	0.0693262	0.0346631	+	0.999399i	$0.488964\pi$
$180$	0	0
$181$	−3.28822	−0.244412	−0.122206	−	0.992505i	$-0.538997\pi$
$181$	−3.28822	−0.244412	−0.122206	+	0.992505i	$0.538997\pi$
$182$	0	0
$183$	−11.9887	−0.886233
$184$	0	0
$185$	3.33687	0.245331
$186$	0	0
$187$	3.84516	0.281186
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.1671	−1.02509	−0.512547	−	0.858659i	$-0.671298\pi$
$191$	−14.1671	−1.02509	−0.512547	+	0.858659i	$0.671298\pi$
$192$	0	0
$193$	−20.1435	−1.44996	−0.724982	−	0.688768i	$-0.758152\pi$
$193$	−20.1435	−1.44996	−0.724982	+	0.688768i	$0.758152\pi$
$194$	0	0
$195$	−5.91076	−0.423279
$196$	0	0
$197$	26.5834	1.89399	0.946996	−	0.321246i	$-0.104102\pi$
$197$	26.5834	1.89399	0.946996	+	0.321246i	$0.104102\pi$
$198$	0	0
$199$	−14.9769	−1.06168	−0.530841	−	0.847471i	$-0.678124\pi$
$199$	−14.9769	−1.06168	−0.530841	+	0.847471i	$0.678124\pi$
$200$	0	0
$201$	12.4443	0.877755
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.16439	−0.0813243
$206$	0	0
$207$	20.0349	1.39252
$208$	0	0
$209$	−1.22975	−0.0850637
$210$	0	0
$211$	8.62894	0.594041	0.297020	−	0.954871i	$-0.404007\pi$
$211$	8.62894	0.594041	0.297020	+	0.954871i	$0.404007\pi$
$212$	0	0
$213$	33.4673	2.29314
$214$	0	0
$215$	7.23762	0.493602
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−19.3242	−1.30581
$220$	0	0
$221$	14.5283	0.977281
$222$	0	0
$223$	−1.53527	−0.102809	−0.0514045	−	0.998678i	$-0.516370\pi$
$223$	−1.53527	−0.102809	−0.0514045	+	0.998678i	$0.516370\pi$
$224$	0	0
$225$	−8.77375	−0.584917
$226$	0	0
$227$	20.1904	1.34008	0.670042	−	0.742323i	$-0.266276\pi$
$227$	20.1904	1.34008	0.670042	+	0.742323i	$0.266276\pi$
$228$	0	0
$229$	−5.96534	−0.394201	−0.197100	−	0.980383i	$-0.563153\pi$
$229$	−5.96534	−0.394201	−0.197100	+	0.980383i	$0.563153\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.959514	−0.0628599	−0.0314299	−	0.999506i	$-0.510006\pi$
$233$	−0.959514	−0.0628599	−0.0314299	+	0.999506i	$0.510006\pi$
$234$	0	0
$235$	14.3793	0.937999
$236$	0	0
$237$	34.1331	2.21719
$238$	0	0
$239$	6.77258	0.438082	0.219041	−	0.975716i	$-0.429707\pi$
$239$	6.77258	0.438082	0.219041	+	0.975716i	$0.429707\pi$
$240$	0	0
$241$	17.7945	1.14625	0.573123	−	0.819469i	$-0.305732\pi$
$241$	17.7945	1.14625	0.573123	+	0.819469i	$0.305732\pi$
$242$	0	0
$243$	20.1127	1.29023
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.64642	−0.295644
$248$	0	0
$249$	27.3680	1.73438
$250$	0	0
$251$	−21.5340	−1.35922	−0.679608	−	0.733575i	$-0.737850\pi$
$251$	−21.5340	−1.35922	−0.679608	+	0.733575i	$0.737850\pi$
$252$	0	0
$253$	4.80781	0.302265
$254$	0	0
$255$	−18.0207	−1.12850
$256$	0	0
$257$	−22.1592	−1.38225	−0.691127	−	0.722733i	$-0.742885\pi$
$257$	−22.1592	−1.38225	−0.691127	+	0.722733i	$0.742885\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−15.1684	−0.938901
$262$	0	0
$263$	17.7048	1.09173	0.545863	−	0.837874i	$-0.316202\pi$
$263$	17.7048	1.09173	0.545863	+	0.837874i	$0.316202\pi$
$264$	0	0
$265$	15.1893	0.933070
$266$	0	0
$267$	−15.2629	−0.934075
$268$	0	0
$269$	18.4884	1.12726	0.563628	−	0.826029i	$-0.309405\pi$
$269$	18.4884	1.12726	0.563628	+	0.826029i	$0.309405\pi$
$270$	0	0
$271$	−14.1700	−0.860765	−0.430383	−	0.902647i	$-0.641621\pi$
$271$	−14.1700	−0.860765	−0.430383	+	0.902647i	$0.641621\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.10545	−0.126964
$276$	0	0
$277$	15.7359	0.945481	0.472740	−	0.881202i	$-0.343265\pi$
$277$	15.7359	0.945481	0.472740	+	0.881202i	$0.343265\pi$
$278$	0	0
$279$	−3.33846	−0.199868
$280$	0	0
$281$	26.1380	1.55926	0.779632	−	0.626237i	$-0.215406\pi$
$281$	26.1380	1.55926	0.779632	+	0.626237i	$0.215406\pi$
$282$	0	0
$283$	−15.2769	−0.908119	−0.454059	−	0.890971i	$-0.650025\pi$
$283$	−15.2769	−0.908119	−0.454059	+	0.890971i	$0.650025\pi$
$284$	0	0
$285$	5.76333	0.341390
$286$	0	0
$287$	0	0
$288$	0	0
$289$	27.2938	1.60552
$290$	0	0
$291$	4.87130	0.285561
$292$	0	0
$293$	8.49926	0.496532	0.248266	−	0.968692i	$-0.420139\pi$
$293$	8.49926	0.496532	0.248266	+	0.968692i	$0.420139\pi$
$294$	0	0
$295$	−3.81197	−0.221942
$296$	0	0
$297$	0.795913	0.0461836
$298$	0	0
$299$	18.1656	1.05054
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.0041	−0.632170
$304$	0	0
$305$	6.00301	0.343731
$306$	0	0
$307$	−9.03457	−0.515630	−0.257815	−	0.966194i	$-0.583002\pi$
$307$	−9.03457	−0.515630	−0.257815	+	0.966194i	$0.583002\pi$
$308$	0	0
$309$	−36.3497	−2.06786
$310$	0	0
$311$	−24.0195	−1.36202	−0.681010	−	0.732274i	$-0.738459\pi$
$311$	−24.0195	−1.36202	−0.681010	+	0.732274i	$0.738459\pi$
$312$	0	0
$313$	25.9450	1.46650	0.733249	−	0.679960i	$-0.238003\pi$
$313$	25.9450	1.46650	0.733249	+	0.679960i	$0.238003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.0716	−1.74515	−0.872576	−	0.488478i	$-0.837552\pi$
$317$	−31.0716	−1.74515	−0.872576	+	0.488478i	$0.837552\pi$
$318$	0	0
$319$	−3.63999	−0.203800
$320$	0	0
$321$	10.1153	0.564584
$322$	0	0
$323$	−14.1659	−0.788214
$324$	0	0
$325$	−7.95512	−0.441270
$326$	0	0
$327$	−34.2754	−1.89543
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.05839	0.0581746	0.0290873	−	0.999577i	$-0.490740\pi$
$331$	1.05839	0.0581746	0.0290873	+	0.999577i	$0.490740\pi$
$332$	0	0
$333$	6.89961	0.378096
$334$	0	0
$335$	−6.23113	−0.340443
$336$	0	0
$337$	22.0487	1.20107	0.600536	−	0.799598i	$-0.294954\pi$
$337$	22.0487	1.20107	0.600536	+	0.799598i	$0.294954\pi$
$338$	0	0
$339$	3.01958	0.164001
$340$	0	0
$341$	−0.801135	−0.0433839
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−22.5322	−1.21309
$346$	0	0
$347$	−17.4946	−0.939161	−0.469581	−	0.882890i	$-0.655595\pi$
$347$	−17.4946	−0.939161	−0.469581	+	0.882890i	$0.655595\pi$
$348$	0	0
$349$	−9.53548	−0.510423	−0.255211	−	0.966885i	$-0.582145\pi$
$349$	−9.53548	−0.510423	−0.255211	+	0.966885i	$0.582145\pi$
$350$	0	0
$351$	3.00723	0.160514
$352$	0	0
$353$	5.45842	0.290522	0.145261	−	0.989393i	$-0.453598\pi$
$353$	5.45842	0.290522	0.145261	+	0.989393i	$0.453598\pi$
$354$	0	0
$355$	−16.7578	−0.889411
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.2188	1.48933	0.744665	−	0.667439i	$-0.232609\pi$
$359$	28.2188	1.48933	0.744665	+	0.667439i	$0.232609\pi$
$360$	0	0
$361$	−14.4695	−0.761551
$362$	0	0
$363$	24.8034	1.30184
$364$	0	0
$365$	9.67600	0.506465
$366$	0	0
$367$	12.6548	0.660578	0.330289	−	0.943880i	$-0.392854\pi$
$367$	12.6548	0.660578	0.330289	+	0.943880i	$0.392854\pi$
$368$	0	0
$369$	−2.40759	−0.125334
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0610	0.831605	0.415803	−	0.909455i	$-0.363501\pi$
$373$	16.0610	0.831605	0.415803	+	0.909455i	$0.363501\pi$
$374$	0	0
$375$	23.4058	1.20867
$376$	0	0
$377$	−13.7531	−0.708322
$378$	0	0
$379$	−3.85863	−0.198205	−0.0991023	−	0.995077i	$-0.531597\pi$
$379$	−3.85863	−0.198205	−0.0991023	+	0.995077i	$0.531597\pi$
$380$	0	0
$381$	38.7308	1.98424
$382$	0	0
$383$	9.73227	0.497296	0.248648	−	0.968594i	$-0.420014\pi$
$383$	9.73227	0.497296	0.248648	+	0.968594i	$0.420014\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14.9652	0.760722
$388$	0	0
$389$	4.65923	0.236232	0.118116	−	0.993000i	$-0.462315\pi$
$389$	4.65923	0.236232	0.118116	+	0.993000i	$0.462315\pi$
$390$	0	0
$391$	55.3829	2.80084
$392$	0	0
$393$	−44.1248	−2.22580
$394$	0	0
$395$	−17.0912	−0.859950
$396$	0	0
$397$	23.7699	1.19298	0.596489	−	0.802622i	$-0.296562\pi$
$397$	23.7699	1.19298	0.596489	+	0.802622i	$0.296562\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.1934	−0.808658	−0.404329	−	0.914614i	$-0.632495\pi$
$401$	−16.1934	−0.808658	−0.404329	+	0.914614i	$0.632495\pi$
$402$	0	0
$403$	−3.02696	−0.150784
$404$	0	0
$405$	−12.1402	−0.603253
$406$	0	0
$407$	1.65571	0.0820706
$408$	0	0
$409$	27.8628	1.37773	0.688863	−	0.724891i	$-0.258110\pi$
$409$	27.8628	1.37773	0.688863	+	0.724891i	$0.258110\pi$
$410$	0	0
$411$	8.00297	0.394758
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.7037	−0.672690
$416$	0	0
$417$	11.3597	0.556288
$418$	0	0
$419$	16.5549	0.808759	0.404380	−	0.914591i	$-0.367487\pi$
$419$	16.5549	0.808759	0.404380	+	0.914591i	$0.367487\pi$
$420$	0	0
$421$	−14.7753	−0.720104	−0.360052	−	0.932932i	$-0.617241\pi$
$421$	−14.7753	−0.720104	−0.360052	+	0.932932i	$0.617241\pi$
$422$	0	0
$423$	29.7319	1.44561
$424$	0	0
$425$	−24.2535	−1.17647
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.93285	−0.141599
$430$	0	0
$431$	−11.4352	−0.550815	−0.275407	−	0.961328i	$-0.588813\pi$
$431$	−11.4352	−0.550815	−0.275407	+	0.961328i	$0.588813\pi$
$432$	0	0
$433$	0.174657	0.00839348	0.00419674	−	0.999991i	$-0.498664\pi$
$433$	0.174657	0.00839348	0.00419674	+	0.999991i	$0.498664\pi$
$434$	0	0
$435$	17.0591	0.817922
$436$	0	0
$437$	−17.7125	−0.847302
$438$	0	0
$439$	8.45595	0.403580	0.201790	−	0.979429i	$-0.435324\pi$
$439$	8.45595	0.403580	0.201790	+	0.979429i	$0.435324\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.89323	0.137461	0.0687307	−	0.997635i	$-0.478105\pi$
$443$	2.89323	0.137461	0.0687307	+	0.997635i	$0.478105\pi$
$444$	0	0
$445$	7.64246	0.362287
$446$	0	0
$447$	−51.2192	−2.42258
$448$	0	0
$449$	−8.62766	−0.407164	−0.203582	−	0.979058i	$-0.565258\pi$
$449$	−8.62766	−0.407164	−0.203582	+	0.979058i	$0.565258\pi$
$450$	0	0
$451$	−0.577754	−0.0272054
$452$	0	0
$453$	22.3210	1.04873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.3212	1.18448	0.592238	−	0.805763i	$-0.298245\pi$
$457$	25.3212	1.18448	0.592238	+	0.805763i	$0.298245\pi$
$458$	0	0
$459$	9.16841	0.427945
$460$	0	0
$461$	−13.1693	−0.613354	−0.306677	−	0.951814i	$-0.599217\pi$
$461$	−13.1693	−0.613354	−0.306677	+	0.951814i	$0.599217\pi$
$462$	0	0
$463$	21.7893	1.01264	0.506318	−	0.862347i	$-0.331006\pi$
$463$	21.7893	1.01264	0.506318	+	0.862347i	$0.331006\pi$
$464$	0	0
$465$	3.75459	0.174115
$466$	0	0
$467$	28.9339	1.33890	0.669451	−	0.742856i	$-0.266529\pi$
$467$	28.9339	1.33890	0.669451	+	0.742856i	$0.266529\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.88633	−0.271228
$472$	0	0
$473$	3.59122	0.165124
$474$	0	0
$475$	7.75669	0.355902
$476$	0	0
$477$	31.4067	1.43802
$478$	0	0
$479$	7.12399	0.325503	0.162752	−	0.986667i	$-0.447963\pi$
$479$	7.12399	0.325503	0.162752	+	0.986667i	$0.447963\pi$
$480$	0	0
$481$	6.25584	0.285242
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.43916	−0.110757
$486$	0	0
$487$	33.1983	1.50436	0.752179	−	0.658959i	$-0.229003\pi$
$487$	33.1983	1.50436	0.752179	+	0.658959i	$0.229003\pi$
$488$	0	0
$489$	44.0997	1.99426
$490$	0	0
$491$	−0.0506885	−0.00228754	−0.00114377	−	0.999999i	$-0.500364\pi$
$491$	−0.0506885	−0.00228754	−0.00114377	+	0.999999i	$0.500364\pi$
$492$	0	0
$493$	−41.9303	−1.88845
$494$	0	0
$495$	1.61966	0.0727983
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−27.6020	−1.23563	−0.617817	−	0.786322i	$-0.711983\pi$
$499$	−27.6020	−1.23563	−0.617817	+	0.786322i	$0.711983\pi$
$500$	0	0
$501$	3.23352	0.144463
$502$	0	0
$503$	31.7270	1.41464	0.707318	−	0.706895i	$-0.249905\pi$
$503$	31.7270	1.41464	0.707318	+	0.706895i	$0.249905\pi$
$504$	0	0
$505$	5.50999	0.245191
$506$	0	0
$507$	19.1492	0.850446
$508$	0	0
$509$	−5.24465	−0.232465	−0.116233	−	0.993222i	$-0.537082\pi$
$509$	−5.24465	−0.232465	−0.116233	+	0.993222i	$0.537082\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.93222	−0.129461
$514$	0	0
$515$	18.2010	0.802034
$516$	0	0
$517$	7.13481	0.313788
$518$	0	0
$519$	−46.9509	−2.06092
$520$	0	0
$521$	−7.11443	−0.311689	−0.155844	−	0.987782i	$-0.549810\pi$
$521$	−7.11443	−0.311689	−0.155844	+	0.987782i	$0.549810\pi$
$522$	0	0
$523$	−7.45011	−0.325770	−0.162885	−	0.986645i	$-0.552080\pi$
$523$	−7.45011	−0.325770	−0.162885	+	0.986645i	$0.552080\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.22856	−0.402003
$528$	0	0
$529$	46.2483	2.01080
$530$	0	0
$531$	−7.88198	−0.342049
$532$	0	0
$533$	−2.18295	−0.0945541
$534$	0	0
$535$	−5.06496	−0.218977
$536$	0	0
$537$	−2.15688	−0.0930762
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.23267	−0.0959899	−0.0479949	−	0.998848i	$-0.515283\pi$
$541$	−2.23267	−0.0959899	−0.0479949	+	0.998848i	$0.515283\pi$
$542$	0	0
$543$	7.64651	0.328143
$544$	0	0
$545$	17.1624	0.735156
$546$	0	0
$547$	20.7822	0.888582	0.444291	−	0.895882i	$-0.353456\pi$
$547$	20.7822	0.888582	0.444291	+	0.895882i	$0.353456\pi$
$548$	0	0
$549$	12.4124	0.529747
$550$	0	0
$551$	13.4101	0.571288
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.75963	−0.329378
$556$	0	0
$557$	−0.0133890	−0.000567310 0	−0.000283655	−	1.00000i	$-0.500090\pi$
$557$	−0.0133890	−0.000567310 0	−0.000283655		1.00000i	$0.500090\pi$
$558$	0	0
$559$	13.5688	0.573901
$560$	0	0
$561$	−8.94163	−0.377516
$562$	0	0
$563$	−17.7441	−0.747824	−0.373912	−	0.927464i	$-0.621984\pi$
$563$	−17.7441	−0.747824	−0.373912	+	0.927464i	$0.621984\pi$
$564$	0	0
$565$	−1.51196	−0.0636088
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.4381	1.44372	0.721859	−	0.692040i	$-0.243288\pi$
$569$	34.4381	1.44372	0.721859	+	0.692040i	$0.243288\pi$
$570$	0	0
$571$	−26.8148	−1.12216	−0.561081	−	0.827761i	$-0.689615\pi$
$571$	−26.8148	−1.12216	−0.561081	+	0.827761i	$0.689615\pi$
$572$	0	0
$573$	32.9444	1.37627
$574$	0	0
$575$	−30.3254	−1.26466
$576$	0	0
$577$	−2.11828	−0.0881853	−0.0440926	−	0.999027i	$-0.514040\pi$
$577$	−2.11828	−0.0881853	−0.0440926	+	0.999027i	$0.514040\pi$
$578$	0	0
$579$	46.8422	1.94670
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.53673	0.312140
$584$	0	0
$585$	6.11962	0.253015
$586$	0	0
$587$	32.6369	1.34707	0.673534	−	0.739156i	$-0.264775\pi$
$587$	32.6369	1.34707	0.673534	+	0.739156i	$0.264775\pi$
$588$	0	0
$589$	2.95146	0.121613
$590$	0	0
$591$	−61.8177	−2.54284
$592$	0	0
$593$	−10.8698	−0.446371	−0.223186	−	0.974776i	$-0.571646\pi$
$593$	−10.8698	−0.446371	−0.223186	+	0.974776i	$0.571646\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	34.8276	1.42540
$598$	0	0
$599$	13.0873	0.534731	0.267366	−	0.963595i	$-0.413847\pi$
$599$	13.0873	0.534731	0.267366	+	0.963595i	$0.413847\pi$
$600$	0	0
$601$	−6.48319	−0.264455	−0.132227	−	0.991219i	$-0.542213\pi$
$601$	−6.48319	−0.264455	−0.132227	+	0.991219i	$0.542213\pi$
$602$	0	0
$603$	−12.8841	−0.524679
$604$	0	0
$605$	−12.4196	−0.504928
$606$	0	0
$607$	29.4411	1.19498	0.597488	−	0.801878i	$-0.296166\pi$
$607$	29.4411	1.19498	0.597488	+	0.801878i	$0.296166\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.9577	1.09059
$612$	0	0
$613$	−0.185150	−0.00747814	−0.00373907	−	0.999993i	$-0.501190\pi$
$613$	−0.185150	−0.00747814	−0.00373907	+	0.999993i	$0.501190\pi$
$614$	0	0
$615$	2.70769	0.109185
$616$	0	0
$617$	−37.6266	−1.51479	−0.757395	−	0.652957i	$-0.773528\pi$
$617$	−37.6266	−1.51479	−0.757395	+	0.652957i	$0.773528\pi$
$618$	0	0
$619$	34.9704	1.40558	0.702790	−	0.711397i	$-0.251937\pi$
$619$	34.9704	1.40558	0.702790	+	0.711397i	$0.251937\pi$
$620$	0	0
$621$	11.4638	0.460025
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.50122	0.260049
$626$	0	0
$627$	2.85969	0.114205
$628$	0	0
$629$	19.0727	0.760480
$630$	0	0
$631$	39.4821	1.57176	0.785878	−	0.618381i	$-0.212211\pi$
$631$	39.4821	1.57176	0.785878	+	0.618381i	$0.212211\pi$
$632$	0	0
$633$	−20.0659	−0.797550
$634$	0	0
$635$	−19.3933	−0.769601
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−34.6499	−1.37073
$640$	0	0
$641$	26.1071	1.03117	0.515583	−	0.856839i	$-0.327575\pi$
$641$	26.1071	1.03117	0.515583	+	0.856839i	$0.327575\pi$
$642$	0	0
$643$	13.0439	0.514403	0.257201	−	0.966358i	$-0.417200\pi$
$643$	13.0439	0.514403	0.257201	+	0.966358i	$0.417200\pi$
$644$	0	0
$645$	−16.8305	−0.662702
$646$	0	0
$647$	25.3524	0.996704	0.498352	−	0.866975i	$-0.333939\pi$
$647$	25.3524	0.996704	0.498352	+	0.866975i	$0.333939\pi$
$648$	0	0
$649$	−1.89145	−0.0742461
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.7755	1.28260	0.641301	−	0.767289i	$-0.278395\pi$
$653$	32.7755	1.28260	0.641301	+	0.767289i	$0.278395\pi$
$654$	0	0
$655$	22.0942	0.863293
$656$	0	0
$657$	20.0070	0.780546
$658$	0	0
$659$	18.6120	0.725019	0.362509	−	0.931980i	$-0.381920\pi$
$659$	18.6120	0.725019	0.362509	+	0.931980i	$0.381920\pi$
$660$	0	0
$661$	−21.6989	−0.843991	−0.421995	−	0.906598i	$-0.638670\pi$
$661$	−21.6989	−0.843991	−0.421995	+	0.906598i	$0.638670\pi$
$662$	0	0
$663$	−33.7845	−1.31208
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−52.4278	−2.03001
$668$	0	0
$669$	3.57014	0.138030
$670$	0	0
$671$	2.97862	0.114988
$672$	0	0
$673$	−4.48404	−0.172847	−0.0864235	−	0.996258i	$-0.527544\pi$
$673$	−4.48404	−0.172847	−0.0864235	+	0.996258i	$0.527544\pi$
$674$	0	0
$675$	−5.02024	−0.193229
$676$	0	0
$677$	−24.7715	−0.952047	−0.476023	−	0.879433i	$-0.657922\pi$
$677$	−24.7715	−0.952047	−0.476023	+	0.879433i	$0.657922\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−46.9512	−1.79917
$682$	0	0
$683$	34.2572	1.31082	0.655408	−	0.755275i	$-0.272497\pi$
$683$	34.2572	1.31082	0.655408	+	0.755275i	$0.272497\pi$
$684$	0	0
$685$	−4.00725	−0.153109
$686$	0	0
$687$	13.8719	0.529248
$688$	0	0
$689$	28.4763	1.08486
$690$	0	0
$691$	50.4922	1.92081	0.960407	−	0.278600i	$-0.0898703\pi$
$691$	50.4922	1.92081	0.960407	+	0.278600i	$0.0898703\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.68804	−0.215760
$696$	0	0
$697$	−6.65536	−0.252090
$698$	0	0
$699$	2.23128	0.0843946
$700$	0	0
$701$	35.6345	1.34590	0.672948	−	0.739690i	$-0.265028\pi$
$701$	35.6345	1.34590	0.672948	+	0.739690i	$0.265028\pi$
$702$	0	0
$703$	−6.09980	−0.230058
$704$	0	0
$705$	−33.4379	−1.25934
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.63467	0.174059	0.0870294	−	0.996206i	$-0.472263\pi$
$709$	4.63467	0.174059	0.0870294	+	0.996206i	$0.472263\pi$
$710$	0	0
$711$	−35.3392	−1.32533
$712$	0	0
$713$	−11.5390	−0.432138
$714$	0	0
$715$	1.46854	0.0549202
$716$	0	0
$717$	−15.7491	−0.588161
$718$	0	0
$719$	−38.0114	−1.41759	−0.708794	−	0.705416i	$-0.750760\pi$
$719$	−38.0114	−1.41759	−0.708794	+	0.705416i	$0.750760\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−41.3798	−1.53893
$724$	0	0
$725$	22.9593	0.852689
$726$	0	0
$727$	−8.03179	−0.297882	−0.148941	−	0.988846i	$-0.547587\pi$
$727$	−8.03179	−0.297882	−0.148941	+	0.988846i	$0.547587\pi$
$728$	0	0
$729$	−15.4917	−0.573768
$730$	0	0
$731$	41.3685	1.53007
$732$	0	0
$733$	−17.0042	−0.628063	−0.314032	−	0.949413i	$-0.601680\pi$
$733$	−17.0042	−0.628063	−0.314032	+	0.949413i	$0.601680\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.09181	−0.113888
$738$	0	0
$739$	50.4482	1.85577	0.927883	−	0.372871i	$-0.121626\pi$
$739$	50.4482	1.85577	0.927883	+	0.372871i	$0.121626\pi$
$740$	0	0
$741$	10.8049	0.396927
$742$	0	0
$743$	49.1625	1.80360	0.901799	−	0.432156i	$-0.142247\pi$
$743$	49.1625	1.80360	0.901799	+	0.432156i	$0.142247\pi$
$744$	0	0
$745$	25.6465	0.939615
$746$	0	0
$747$	−28.3351	−1.03673
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−0.812005	−0.0296305	−0.0148152	−	0.999890i	$-0.504716\pi$
$751$	−0.812005	−0.0296305	−0.0148152	+	0.999890i	$0.504716\pi$
$752$	0	0
$753$	50.0758	1.82486
$754$	0	0
$755$	−11.1766	−0.406757
$756$	0	0
$757$	−6.57037	−0.238804	−0.119402	−	0.992846i	$-0.538098\pi$
$757$	−6.57037	−0.238804	−0.119402	+	0.992846i	$0.538098\pi$
$758$	0	0
$759$	−11.1802	−0.405816
$760$	0	0
$761$	20.4663	0.741904	0.370952	−	0.928652i	$-0.379031\pi$
$761$	20.4663	0.741904	0.370952	+	0.928652i	$0.379031\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	18.6574	0.674561
$766$	0	0
$767$	−7.14655	−0.258047
$768$	0	0
$769$	−12.6132	−0.454844	−0.227422	−	0.973796i	$-0.573030\pi$
$769$	−12.6132	−0.454844	−0.227422	+	0.973796i	$0.573030\pi$
$770$	0	0
$771$	51.5295	1.85579
$772$	0	0
$773$	−46.2197	−1.66241	−0.831203	−	0.555969i	$-0.812347\pi$
$773$	−46.2197	−1.66241	−0.831203	+	0.555969i	$0.812347\pi$
$774$	0	0
$775$	5.05318	0.181516
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.12850	0.0762615
$780$	0	0
$781$	−8.31500	−0.297534
$782$	0	0
$783$	−8.67920	−0.310169
$784$	0	0
$785$	2.94741	0.105197
$786$	0	0
$787$	−42.1247	−1.50158	−0.750791	−	0.660539i	$-0.770328\pi$
$787$	−42.1247	−1.50158	−0.750791	+	0.660539i	$0.770328\pi$
$788$	0	0
$789$	−41.1712	−1.46573
$790$	0	0
$791$	0	0
$792$	0	0
$793$	11.2542	0.399649
$794$	0	0
$795$	−35.3215	−1.25272
$796$	0	0
$797$	−20.2562	−0.717510	−0.358755	−	0.933432i	$-0.616799\pi$
$797$	−20.2562	−0.717510	−0.358755	+	0.933432i	$0.616799\pi$
$798$	0	0
$799$	82.1884	2.90762
$800$	0	0
$801$	15.8022	0.558344
$802$	0	0
$803$	4.80111	0.169427
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−42.9933	−1.51344
$808$	0	0
$809$	−43.4001	−1.52587	−0.762934	−	0.646477i	$-0.776242\pi$
$809$	−43.4001	−1.52587	−0.762934	+	0.646477i	$0.776242\pi$
$810$	0	0
$811$	−3.14538	−0.110449	−0.0552246	−	0.998474i	$-0.517587\pi$
$811$	−3.14538	−0.110449	−0.0552246	+	0.998474i	$0.517587\pi$
$812$	0	0
$813$	32.9512	1.15565
$814$	0	0
$815$	−22.0816	−0.773485
$816$	0	0
$817$	−13.2304	−0.462873
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.9800	1.39531	0.697656	−	0.716432i	$-0.254226\pi$
$821$	39.9800	1.39531	0.697656	+	0.716432i	$0.254226\pi$
$822$	0	0
$823$	−38.4780	−1.34126	−0.670630	−	0.741792i	$-0.733976\pi$
$823$	−38.4780	−1.34126	−0.670630	+	0.741792i	$0.733976\pi$
$824$	0	0
$825$	4.89607	0.170459
$826$	0	0
$827$	23.8196	0.828289	0.414145	−	0.910211i	$-0.364081\pi$
$827$	23.8196	0.828289	0.414145	+	0.910211i	$0.364081\pi$
$828$	0	0
$829$	25.5478	0.887311	0.443655	−	0.896197i	$-0.353681\pi$
$829$	25.5478	0.887311	0.443655	+	0.896197i	$0.353681\pi$
$830$	0	0
$831$	−36.5927	−1.26939
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.61909	−0.0560310
$836$	0	0
$837$	−1.91023	−0.0660271
$838$	0	0
$839$	−35.2200	−1.21593	−0.607966	−	0.793963i	$-0.708014\pi$
$839$	−35.2200	−1.21593	−0.607966	+	0.793963i	$0.708014\pi$
$840$	0	0
$841$	10.6930	0.368725
$842$	0	0
$843$	−60.7820	−2.09344
$844$	0	0
$845$	−9.58841	−0.329851
$846$	0	0
$847$	0	0
$848$	0	0
$849$	35.5253	1.21923
$850$	0	0
$851$	23.8477	0.817488
$852$	0	0
$853$	−23.5636	−0.806803	−0.403401	−	0.915023i	$-0.632172\pi$
$853$	−23.5636	−0.806803	−0.403401	+	0.915023i	$0.632172\pi$
$854$	0	0
$855$	−5.96698	−0.204066
$856$	0	0
$857$	−13.1481	−0.449131	−0.224565	−	0.974459i	$-0.572096\pi$
$857$	−13.1481	−0.449131	−0.224565	+	0.974459i	$0.572096\pi$
$858$	0	0
$859$	41.6355	1.42058	0.710292	−	0.703907i	$-0.248563\pi$
$859$	41.6355	1.42058	0.710292	+	0.703907i	$0.248563\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.6615	−1.24797	−0.623986	−	0.781435i	$-0.714488\pi$
$863$	−36.6615	−1.24797	−0.623986	+	0.781435i	$0.714488\pi$
$864$	0	0
$865$	23.5093	0.799339
$866$	0	0
$867$	−63.4696	−2.15554
$868$	0	0
$869$	−8.48042	−0.287679
$870$	0	0
$871$	−11.6819	−0.395826
$872$	0	0
$873$	−5.04343	−0.170694
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.3610	0.788844	0.394422	−	0.918929i	$-0.370945\pi$
$877$	23.3610	0.788844	0.394422	+	0.918929i	$0.370945\pi$
$878$	0	0
$879$	−19.7644	−0.666636
$880$	0	0
$881$	34.3795	1.15827	0.579137	−	0.815230i	$-0.303389\pi$
$881$	34.3795	1.15827	0.579137	+	0.815230i	$0.303389\pi$
$882$	0	0
$883$	6.66499	0.224295	0.112147	−	0.993692i	$-0.464227\pi$
$883$	6.66499	0.224295	0.112147	+	0.993692i	$0.464227\pi$
$884$	0	0
$885$	8.86445	0.297975
$886$	0	0
$887$	12.1283	0.407229	0.203615	−	0.979051i	$-0.434731\pi$
$887$	12.1283	0.407229	0.203615	+	0.979051i	$0.434731\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.02383	−0.201806
$892$	0	0
$893$	−26.2853	−0.879604
$894$	0	0
$895$	1.07999	0.0361002
$896$	0	0
$897$	−42.2426	−1.41044
$898$	0	0
$899$	8.73614	0.291367
$900$	0	0
$901$	86.8183	2.89234
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.82876	−0.127272
$906$	0	0
$907$	8.69194	0.288611	0.144306	−	0.989533i	$-0.453905\pi$
$907$	8.69194	0.288611	0.144306	+	0.989533i	$0.453905\pi$
$908$	0	0
$909$	11.3929	0.377880
$910$	0	0
$911$	−10.6870	−0.354076	−0.177038	−	0.984204i	$-0.556652\pi$
$911$	−10.6870	−0.354076	−0.177038	+	0.984204i	$0.556652\pi$
$912$	0	0
$913$	−6.79963	−0.225035
$914$	0	0
$915$	−13.9595	−0.461488
$916$	0	0
$917$	0	0
$918$	0	0
$919$	37.0944	1.22363	0.611815	−	0.791001i	$-0.290440\pi$
$919$	37.0944	1.22363	0.611815	+	0.791001i	$0.290440\pi$
$920$	0	0
$921$	21.0092	0.692276
$922$	0	0
$923$	−31.4169	−1.03410
$924$	0	0
$925$	−10.4435	−0.343379
$926$	0	0
$927$	37.6341	1.23607
$928$	0	0
$929$	−13.7333	−0.450574	−0.225287	−	0.974292i	$-0.572332\pi$
$929$	−13.7333	−0.450574	−0.225287	+	0.974292i	$0.572332\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	55.8555	1.82863
$934$	0	0
$935$	4.47726	0.146422
$936$	0	0
$937$	−14.5987	−0.476920	−0.238460	−	0.971152i	$-0.576643\pi$
$937$	−14.5987	−0.476920	−0.238460	+	0.971152i	$0.576643\pi$
$938$	0	0
$939$	−60.3331	−1.96890
$940$	0	0
$941$	−10.5934	−0.345334	−0.172667	−	0.984980i	$-0.555238\pi$
$941$	−10.5934	−0.345334	−0.172667	+	0.984980i	$0.555238\pi$
$942$	0	0
$943$	−8.32156	−0.270987
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.15610	0.265038	0.132519	−	0.991180i	$-0.457693\pi$
$947$	8.15610	0.265038	0.132519	+	0.991180i	$0.457693\pi$
$948$	0	0
$949$	18.1402	0.588856
$950$	0	0
$951$	72.2545	2.34301
$952$	0	0
$953$	−58.9994	−1.91118	−0.955589	−	0.294703i	$-0.904779\pi$
$953$	−58.9994	−1.91118	−0.955589	+	0.294703i	$0.904779\pi$
$954$	0	0
$955$	−16.4960	−0.533797
$956$	0	0
$957$	8.46452	0.273619
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.0772	−0.937975
$962$	0	0
$963$	−10.4728	−0.337480
$964$	0	0
$965$	−23.4549	−0.755039
$966$	0	0
$967$	−22.9767	−0.738880	−0.369440	−	0.929255i	$-0.620450\pi$
$967$	−22.9767	−0.738880	−0.369440	+	0.929255i	$0.620450\pi$
$968$	0	0
$969$	32.9418	1.05824
$970$	0	0
$971$	−53.3603	−1.71241	−0.856206	−	0.516634i	$-0.827185\pi$
$971$	−53.3603	−1.71241	−0.856206	+	0.516634i	$0.827185\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	18.4990	0.592443
$976$	0	0
$977$	14.6210	0.467769	0.233884	−	0.972264i	$-0.424856\pi$
$977$	14.6210	0.467769	0.233884	+	0.972264i	$0.424856\pi$
$978$	0	0
$979$	3.79209	0.121196
$980$	0	0
$981$	35.4865	1.13300
$982$	0	0
$983$	21.9944	0.701512	0.350756	−	0.936467i	$-0.385925\pi$
$983$	21.9944	0.701512	0.350756	+	0.936467i	$0.385925\pi$
$984$	0	0
$985$	30.9534	0.986258
$986$	0	0
$987$	0	0
$988$	0	0
$989$	51.7253	1.64477
$990$	0	0
$991$	−28.0746	−0.891820	−0.445910	−	0.895078i	$-0.647120\pi$
$991$	−28.0746	−0.891820	−0.445910	+	0.895078i	$0.647120\pi$
$992$	0	0
$993$	−2.46122	−0.0781043
$994$	0	0
$995$	−17.4389	−0.552850
$996$	0	0
$997$	28.3418	0.897593	0.448797	−	0.893634i	$-0.351853\pi$
$997$	28.3418	0.897593	0.448797	+	0.893634i	$0.351853\pi$
$998$	0	0
$999$	3.94788	0.124905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.3	yes	20
7.6	odd	2		8036.2.a.s.1.18	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.18	✓	20	7.6	odd	2
8036.2.a.t.1.3	yes	20	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-2.32542 q^{3} +1.16439 q^{5} +2.40759 q^{9} +O(q^{10})\)	\(q-2.32542 q^{3} +1.16439 q^{5} +2.40759 q^{9} +0.577754 q^{11} +2.18295 q^{13} -2.70769 q^{15} +6.65536 q^{17} -2.12850 q^{19} +8.32156 q^{23} -3.64420 q^{25} +1.37760 q^{27} -6.30024 q^{29} -1.38664 q^{31} -1.34352 q^{33} +2.86577 q^{37} -5.07629 q^{39} -1.00000 q^{41} +6.21582 q^{43} +2.80337 q^{45} +12.3492 q^{47} -15.4765 q^{51} +13.0449 q^{53} +0.672730 q^{55} +4.94967 q^{57} -3.27380 q^{59} +5.15551 q^{61} +2.54180 q^{65} -5.35143 q^{67} -19.3511 q^{69} -14.3919 q^{71} +8.30995 q^{73} +8.47431 q^{75} -14.6782 q^{79} -10.4263 q^{81} -11.7691 q^{83} +7.74941 q^{85} +14.6507 q^{87} +6.56350 q^{89} +3.22452 q^{93} -2.47840 q^{95} -2.09480 q^{97} +1.39100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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