LMFDB - Embedded newform 820.2.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [820,2,Mod(1,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("820.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 820 = 2^{2} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	820.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.54773296574$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.82405$ of defining polynomial
Character	$\chi$	$=$	820.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.27582 q^{3} +1.00000 q^{5} +1.60298 q^{7} -1.37228 q^{9} +1.27582 q^{11} +0.397017 q^{13} +1.27582 q^{15} +0.724179 q^{17} +6.34755 q^{19} +2.04512 q^{21} +4.37228 q^{23} +1.00000 q^{25} -5.57825 q^{27} +4.69944 q^{29} -5.07172 q^{31} +1.62772 q^{33} +1.60298 q^{35} -3.25109 q^{37} +0.506522 q^{39} +1.00000 q^{41} +1.49348 q^{43} -1.37228 q^{45} -3.32094 q^{47} -4.43045 q^{49} +0.923923 q^{51} +2.35190 q^{53} +1.27582 q^{55} +8.09833 q^{57} +2.69944 q^{59} +6.12989 q^{61} -2.19974 q^{63} +0.397017 q^{65} -12.1052 q^{67} +5.57825 q^{69} +1.93015 q^{71} +13.4837 q^{73} +1.27582 q^{75} +2.04512 q^{77} -15.1318 q^{79} -3.00000 q^{81} +0.569555 q^{83} +0.724179 q^{85} +5.99565 q^{87} -14.0804 q^{89} +0.636411 q^{91} -6.47061 q^{93} +6.34755 q^{95} -2.00682 q^{97} -1.75079 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + q^{7} + 6 q^{9} + 7 q^{13} + 8 q^{17} - 3 q^{19} - 3 q^{21} + 6 q^{23} + 4 q^{25} + 7 q^{29} + 3 q^{31} + 18 q^{33} + q^{35} + 9 q^{37} + 3 q^{39} + 4 q^{41} + 5 q^{43} + 6 q^{45} + 3 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + q^7 + 6 * q^9 + 7 * q^13 + 8 * q^17 - 3 * q^19 - 3 * q^21 + 6 * q^23 + 4 * q^25 + 7 * q^29 + 3 * q^31 + 18 * q^33 + q^35 + 9 * q^37 + 3 * q^39 + 4 * q^41 + 5 * q^43 + 6 * q^45 + 3 * q^47 + 3 * q^49 - 18 * q^51 + 26 * q^53 - 3 * q^57 - q^59 - 8 * q^61 + 18 * q^63 + 7 * q^65 + q^67 + 2 * q^71 + 9 * q^73 - 3 * q^77 + q^79 - 12 * q^81 + 23 * q^83 + 8 * q^85 - 21 * q^87 + 10 * q^89 - 29 * q^91 + 21 * q^93 - 3 * q^95 + 6 * q^97

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
$3$	1.27582	0.736595	0.368298	−	0.929708i	$-0.379941\pi$
$3$	1.27582	0.736595	0.368298	+	0.929708i	$0.379941\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.60298	0.605871	0.302935	−	0.953011i	$-0.402033\pi$
$7$	1.60298	0.605871	0.302935	+	0.953011i	$0.402033\pi$
$8$	0	0
$9$	−1.37228	−0.457427
$10$	0	0
$11$	1.27582	0.384674	0.192337	−	0.981329i	$-0.438393\pi$
$11$	1.27582	0.384674	0.192337	+	0.981329i	$0.438393\pi$
$12$	0	0
$13$	0.397017	0.110113	0.0550563	−	0.998483i	$-0.482466\pi$
$13$	0.397017	0.110113	0.0550563	+	0.998483i	$0.482466\pi$
$14$	0	0
$15$	1.27582	0.329416
$16$	0	0
$17$	0.724179	0.175639	0.0878196	−	0.996136i	$-0.472010\pi$
$17$	0.724179	0.175639	0.0878196	+	0.996136i	$0.472010\pi$
$18$	0	0
$19$	6.34755	1.45623	0.728113	−	0.685457i	$-0.240397\pi$
$19$	6.34755	1.45623	0.728113	+	0.685457i	$0.240397\pi$
$20$	0	0
$21$	2.04512	0.446282
$22$	0	0
$23$	4.37228	0.911684	0.455842	−	0.890061i	$-0.349338\pi$
$23$	4.37228	0.911684	0.455842	+	0.890061i	$0.349338\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.57825	−1.07353
$28$	0	0
$29$	4.69944	0.872665	0.436332	−	0.899786i	$-0.356277\pi$
$29$	4.69944	0.872665	0.436332	+	0.899786i	$0.356277\pi$
$30$	0	0
$31$	−5.07172	−0.910909	−0.455454	−	0.890259i	$-0.650523\pi$
$31$	−5.07172	−0.910909	−0.455454	+	0.890259i	$0.650523\pi$
$32$	0	0
$33$	1.62772	0.283349
$34$	0	0
$35$	1.60298	0.270954
$36$	0	0
$37$	−3.25109	−0.534475	−0.267238	−	0.963631i	$-0.586111\pi$
$37$	−3.25109	−0.534475	−0.267238	+	0.963631i	$0.586111\pi$
$38$	0	0
$39$	0.506522	0.0811085
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	1.49348	0.227753	0.113877	−	0.993495i	$-0.463673\pi$
$43$	1.49348	0.227753	0.113877	+	0.993495i	$0.463673\pi$
$44$	0	0
$45$	−1.37228	−0.204568
$46$	0	0
$47$	−3.32094	−0.484409	−0.242204	−	0.970225i	$-0.577870\pi$
$47$	−3.32094	−0.484409	−0.242204	+	0.970225i	$0.577870\pi$
$48$	0	0
$49$	−4.43045	−0.632921
$50$	0	0
$51$	0.923923	0.129375
$52$	0	0
$53$	2.35190	0.323058	0.161529	−	0.986868i	$-0.448357\pi$
$53$	2.35190	0.323058	0.161529	+	0.986868i	$0.448357\pi$
$54$	0	0
$55$	1.27582	0.172032
$56$	0	0
$57$	8.09833	1.07265
$58$	0	0
$59$	2.69944	0.351438	0.175719	−	0.984440i	$-0.443775\pi$
$59$	2.69944	0.351438	0.175719	+	0.984440i	$0.443775\pi$
$60$	0	0
$61$	6.12989	0.784852	0.392426	−	0.919784i	$-0.371636\pi$
$61$	6.12989	0.784852	0.392426	+	0.919784i	$0.371636\pi$
$62$	0	0
$63$	−2.19974	−0.277142
$64$	0	0
$65$	0.397017	0.0492439
$66$	0	0
$67$	−12.1052	−1.47888	−0.739440	−	0.673223i	$-0.764910\pi$
$67$	−12.1052	−1.47888	−0.739440	+	0.673223i	$0.764910\pi$
$68$	0	0
$69$	5.57825	0.671542
$70$	0	0
$71$	1.93015	0.229066	0.114533	−	0.993419i	$-0.463463\pi$
$71$	1.93015	0.229066	0.114533	+	0.993419i	$0.463463\pi$
$72$	0	0
$73$	13.4837	1.57814	0.789071	−	0.614302i	$-0.210562\pi$
$73$	13.4837	1.57814	0.789071	+	0.614302i	$0.210562\pi$
$74$	0	0
$75$	1.27582	0.147319
$76$	0	0
$77$	2.04512	0.233063
$78$	0	0
$79$	−15.1318	−1.70246	−0.851228	−	0.524796i	$-0.824142\pi$
$79$	−15.1318	−1.70246	−0.851228	+	0.524796i	$0.824142\pi$
$80$	0	0
$81$	−3.00000	−0.333333
$82$	0	0
$83$	0.569555	0.0625167	0.0312584	−	0.999511i	$-0.490049\pi$
$83$	0.569555	0.0625167	0.0312584	+	0.999511i	$0.490049\pi$
$84$	0	0
$85$	0.724179	0.0785483
$86$	0	0
$87$	5.99565	0.642801
$88$	0	0
$89$	−14.0804	−1.49252	−0.746261	−	0.665654i	$-0.768153\pi$
$89$	−14.0804	−1.49252	−0.746261	+	0.665654i	$0.768153\pi$
$90$	0	0
$91$	0.636411	0.0667140
$92$	0	0
$93$	−6.47061	−0.670971
$94$	0	0
$95$	6.34755	0.651244
$96$	0	0
$97$	−2.00682	−0.203762	−0.101881	−	0.994797i	$-0.532486\pi$
$97$	−2.00682	−0.203762	−0.101881	+	0.994797i	$0.532486\pi$
$98$	0	0
$99$	−1.75079	−0.175961
$100$	0	0
$101$	−1.53313	−0.152552	−0.0762760	−	0.997087i	$-0.524303\pi$
$101$	−1.53313	−0.152552	−0.0762760	+	0.997087i	$0.524303\pi$
$102$	0	0
$103$	7.66849	0.755598	0.377799	−	0.925888i	$-0.376681\pi$
$103$	7.66849	0.755598	0.377799	+	0.925888i	$0.376681\pi$
$104$	0	0
$105$	2.04512	0.199583
$106$	0	0
$107$	0.502170	0.0485466	0.0242733	−	0.999705i	$-0.492273\pi$
$107$	0.502170	0.0485466	0.0242733	+	0.999705i	$0.492273\pi$
$108$	0	0
$109$	0.372281	0.0356581	0.0178290	−	0.999841i	$-0.494325\pi$
$109$	0.372281	0.0356581	0.0178290	+	0.999841i	$0.494325\pi$
$110$	0	0
$111$	−4.14780	−0.393692
$112$	0	0
$113$	6.17936	0.581305	0.290653	−	0.956829i	$-0.406128\pi$
$113$	6.17936	0.581305	0.290653	+	0.956829i	$0.406128\pi$
$114$	0	0
$115$	4.37228	0.407717
$116$	0	0
$117$	−0.544819	−0.0503685
$118$	0	0
$119$	1.16085	0.106415
$120$	0	0
$121$	−9.37228	−0.852026
$122$	0	0
$123$	1.27582	0.115037
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.71249	−0.861844	−0.430922	−	0.902389i	$-0.641812\pi$
$127$	−9.71249	−0.861844	−0.430922	+	0.902389i	$0.641812\pi$
$128$	0	0
$129$	1.90541	0.167762
$130$	0	0
$131$	−16.3679	−1.43007	−0.715036	−	0.699087i	$-0.753590\pi$
$131$	−16.3679	−1.43007	−0.715036	+	0.699087i	$0.753590\pi$
$132$	0	0
$133$	10.1750	0.882285
$134$	0	0
$135$	−5.57825	−0.480099
$136$	0	0
$137$	−0.635057	−0.0542566	−0.0271283	−	0.999632i	$-0.508636\pi$
$137$	−0.635057	−0.0542566	−0.0271283	+	0.999632i	$0.508636\pi$
$138$	0	0
$139$	−8.74456	−0.741704	−0.370852	−	0.928692i	$-0.620934\pi$
$139$	−8.74456	−0.741704	−0.370852	+	0.928692i	$0.620934\pi$
$140$	0	0
$141$	−4.23692	−0.356813
$142$	0	0
$143$	0.506522	0.0423575
$144$	0	0
$145$	4.69944	0.390268
$146$	0	0
$147$	−5.65245	−0.466207
$148$	0	0
$149$	9.44401	0.773683	0.386842	−	0.922146i	$-0.373566\pi$
$149$	9.44401	0.773683	0.386842	+	0.922146i	$0.373566\pi$
$150$	0	0
$151$	−2.24921	−0.183039	−0.0915193	−	0.995803i	$-0.529172\pi$
$151$	−2.24921	−0.183039	−0.0915193	+	0.995803i	$0.529172\pi$
$152$	0	0
$153$	−0.993778	−0.0803422
$154$	0	0
$155$	−5.07172	−0.407371
$156$	0	0
$157$	−4.45570	−0.355603	−0.177802	−	0.984066i	$-0.556899\pi$
$157$	−4.45570	−0.355603	−0.177802	+	0.984066i	$0.556899\pi$
$158$	0	0
$159$	3.00060	0.237963
$160$	0	0
$161$	7.00869	0.552362
$162$	0	0
$163$	−5.76308	−0.451399	−0.225699	−	0.974197i	$-0.572467\pi$
$163$	−5.76308	−0.451399	−0.225699	+	0.974197i	$0.572467\pi$
$164$	0	0
$165$	1.62772	0.126718
$166$	0	0
$167$	−14.4953	−1.12168	−0.560842	−	0.827923i	$-0.689522\pi$
$167$	−14.4953	−1.12168	−0.560842	+	0.827923i	$0.689522\pi$
$168$	0	0
$169$	−12.8424	−0.987875
$170$	0	0
$171$	−8.71062	−0.666118
$172$	0	0
$173$	10.5968	0.805657	0.402828	−	0.915276i	$-0.368027\pi$
$173$	10.5968	0.805657	0.402828	+	0.915276i	$0.368027\pi$
$174$	0	0
$175$	1.60298	0.121174
$176$	0	0
$177$	3.44401	0.258867
$178$	0	0
$179$	−8.48179	−0.633959	−0.316979	−	0.948432i	$-0.602669\pi$
$179$	−8.48179	−0.633959	−0.316979	+	0.948432i	$0.602669\pi$
$180$	0	0
$181$	−11.6685	−0.867312	−0.433656	−	0.901079i	$-0.642777\pi$
$181$	−11.6685	−0.867312	−0.433656	+	0.901079i	$0.642777\pi$
$182$	0	0
$183$	7.82064	0.578118
$184$	0	0
$185$	−3.25109	−0.239025
$186$	0	0
$187$	0.923923	0.0675639
$188$	0	0
$189$	−8.94184	−0.650423
$190$	0	0
$191$	4.82798	0.349340	0.174670	−	0.984627i	$-0.444114\pi$
$191$	4.82798	0.349340	0.174670	+	0.984627i	$0.444114\pi$
$192$	0	0
$193$	9.63019	0.693196	0.346598	−	0.938014i	$-0.387337\pi$
$193$	9.63019	0.693196	0.346598	+	0.938014i	$0.387337\pi$
$194$	0	0
$195$	0.506522	0.0362728
$196$	0	0
$197$	24.1255	1.71887	0.859437	−	0.511242i	$-0.170814\pi$
$197$	24.1255	1.71887	0.859437	+	0.511242i	$0.170814\pi$
$198$	0	0
$199$	5.04699	0.357772	0.178886	−	0.983870i	$-0.442751\pi$
$199$	5.04699	0.357772	0.178886	+	0.983870i	$0.442751\pi$
$200$	0	0
$201$	−15.4440	−1.08934
$202$	0	0
$203$	7.53313	0.528722
$204$	0	0
$205$	1.00000	0.0698430
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	8.09833	0.560173
$210$	0	0
$211$	−18.9759	−1.30635	−0.653176	−	0.757206i	$-0.726564\pi$
$211$	−18.9759	−1.30635	−0.653176	+	0.757206i	$0.726564\pi$
$212$	0	0
$213$	2.46252	0.168729
$214$	0	0
$215$	1.49348	0.101854
$216$	0	0
$217$	−8.12989	−0.551893
$218$	0	0
$219$	17.2027	1.16245
$220$	0	0
$221$	0.287511	0.0193401
$222$	0	0
$223$	−24.4032	−1.63416	−0.817080	−	0.576524i	$-0.804409\pi$
$223$	−24.4032	−1.63416	−0.817080	+	0.576524i	$0.804409\pi$
$224$	0	0
$225$	−1.37228	−0.0914854
$226$	0	0
$227$	−23.2536	−1.54339	−0.771696	−	0.635991i	$-0.780591\pi$
$227$	−23.2536	−1.54339	−0.771696	+	0.635991i	$0.780591\pi$
$228$	0	0
$229$	−10.0587	−0.664696	−0.332348	−	0.943157i	$-0.607841\pi$
$229$	−10.0587	−0.664696	−0.332348	+	0.943157i	$0.607841\pi$
$230$	0	0
$231$	2.60921	0.171673
$232$	0	0
$233$	3.74269	0.245192	0.122596	−	0.992457i	$-0.460878\pi$
$233$	3.74269	0.245192	0.122596	+	0.992457i	$0.460878\pi$
$234$	0	0
$235$	−3.32094	−0.216634
$236$	0	0
$237$	−19.3054	−1.25402
$238$	0	0
$239$	10.5845	0.684652	0.342326	−	0.939581i	$-0.388785\pi$
$239$	10.5845	0.684652	0.342326	+	0.939581i	$0.388785\pi$
$240$	0	0
$241$	−12.0848	−0.778448	−0.389224	−	0.921143i	$-0.627257\pi$
$241$	−12.0848	−0.778448	−0.389224	+	0.921143i	$0.627257\pi$
$242$	0	0
$243$	12.9073	0.828002
$244$	0	0
$245$	−4.43045	−0.283051
$246$	0	0
$247$	2.52008	0.160349
$248$	0	0
$249$	0.726650	0.0460495
$250$	0	0
$251$	−5.28264	−0.333437	−0.166719	−	0.986004i	$-0.553317\pi$
$251$	−5.28264	−0.333437	−0.166719	+	0.986004i	$0.553317\pi$
$252$	0	0
$253$	5.57825	0.350701
$254$	0	0
$255$	0.923923	0.0578583
$256$	0	0
$257$	14.2943	0.891656	0.445828	−	0.895119i	$-0.352909\pi$
$257$	14.2943	0.891656	0.445828	+	0.895119i	$0.352909\pi$
$258$	0	0
$259$	−5.21143	−0.323823
$260$	0	0
$261$	−6.44896	−0.399181
$262$	0	0
$263$	9.71113	0.598814	0.299407	−	0.954125i	$-0.403211\pi$
$263$	9.71113	0.598814	0.299407	+	0.954125i	$0.403211\pi$
$264$	0	0
$265$	2.35190	0.144476
$266$	0	0
$267$	−17.9641	−1.09938
$268$	0	0
$269$	−12.6005	−0.768266	−0.384133	−	0.923278i	$-0.625500\pi$
$269$	−12.6005	−0.768266	−0.384133	+	0.923278i	$0.625500\pi$
$270$	0	0
$271$	−0.319070	−0.0193821	−0.00969105	−	0.999953i	$-0.503085\pi$
$271$	−0.319070	−0.0193821	−0.00969105	+	0.999953i	$0.503085\pi$
$272$	0	0
$273$	0.811947	0.0491413
$274$	0	0
$275$	1.27582	0.0769349
$276$	0	0
$277$	4.87071	0.292653	0.146326	−	0.989236i	$-0.453255\pi$
$277$	4.87071	0.292653	0.146326	+	0.989236i	$0.453255\pi$
$278$	0	0
$279$	6.95983	0.416674
$280$	0	0
$281$	−11.5288	−0.687749	−0.343875	−	0.939016i	$-0.611739\pi$
$281$	−11.5288	−0.687749	−0.343875	+	0.939016i	$0.611739\pi$
$282$	0	0
$283$	−16.8690	−1.00276	−0.501378	−	0.865228i	$-0.667174\pi$
$283$	−16.8690	−1.00276	−0.501378	+	0.865228i	$0.667174\pi$
$284$	0	0
$285$	8.09833	0.479704
$286$	0	0
$287$	1.60298	0.0946211
$288$	0	0
$289$	−16.4756	−0.969151
$290$	0	0
$291$	−2.56035	−0.150090
$292$	0	0
$293$	10.3530	0.604830	0.302415	−	0.953176i	$-0.402207\pi$
$293$	10.3530	0.604830	0.302415	+	0.953176i	$0.402207\pi$
$294$	0	0
$295$	2.69944	0.157168
$296$	0	0
$297$	−7.11684	−0.412961
$298$	0	0
$299$	1.73587	0.100388
$300$	0	0
$301$	2.39402	0.137989
$302$	0	0
$303$	−1.95600	−0.112369
$304$	0	0
$305$	6.12989	0.350996
$306$	0	0
$307$	6.76630	0.386173	0.193087	−	0.981182i	$-0.438150\pi$
$307$	6.76630	0.386173	0.193087	+	0.981182i	$0.438150\pi$
$308$	0	0
$309$	9.78361	0.556570
$310$	0	0
$311$	−12.0557	−0.683615	−0.341808	−	0.939770i	$-0.611039\pi$
$311$	−12.0557	−0.683615	−0.341808	+	0.939770i	$0.611039\pi$
$312$	0	0
$313$	17.5095	0.989696	0.494848	−	0.868980i	$-0.335224\pi$
$313$	17.5095	0.989696	0.494848	+	0.868980i	$0.335224\pi$
$314$	0	0
$315$	−2.19974	−0.123942
$316$	0	0
$317$	27.2671	1.53147	0.765737	−	0.643154i	$-0.222374\pi$
$317$	27.2671	1.53147	0.765737	+	0.643154i	$0.222374\pi$
$318$	0	0
$319$	5.99565	0.335692
$320$	0	0
$321$	0.640679	0.0357592
$322$	0	0
$323$	4.59676	0.255771
$324$	0	0
$325$	0.397017	0.0220225
$326$	0	0
$327$	0.474964	0.0262656
$328$	0	0
$329$	−5.32341	−0.293489
$330$	0	0
$331$	−16.4502	−0.904186	−0.452093	−	0.891971i	$-0.649323\pi$
$331$	−16.4502	−0.904186	−0.452093	+	0.891971i	$0.649323\pi$
$332$	0	0
$333$	4.46140	0.244483
$334$	0	0
$335$	−12.1052	−0.661375
$336$	0	0
$337$	30.0815	1.63865	0.819323	−	0.573333i	$-0.194350\pi$
$337$	30.0815	1.63865	0.819323	+	0.573333i	$0.194350\pi$
$338$	0	0
$339$	7.88376	0.428187
$340$	0	0
$341$	−6.47061	−0.350403
$342$	0	0
$343$	−18.3228	−0.989339
$344$	0	0
$345$	5.57825	0.300323
$346$	0	0
$347$	−4.97578	−0.267114	−0.133557	−	0.991041i	$-0.542640\pi$
$347$	−4.97578	−0.267114	−0.133557	+	0.991041i	$0.542640\pi$
$348$	0	0
$349$	21.1478	1.13202	0.566008	−	0.824400i	$-0.308487\pi$
$349$	21.1478	1.13202	0.566008	+	0.824400i	$0.308487\pi$
$350$	0	0
$351$	−2.21466	−0.118210
$352$	0	0
$353$	1.14729	0.0610639	0.0305319	−	0.999534i	$-0.490280\pi$
$353$	1.14729	0.0610639	0.0305319	+	0.999534i	$0.490280\pi$
$354$	0	0
$355$	1.93015	0.102441
$356$	0	0
$357$	1.48103	0.0783846
$358$	0	0
$359$	31.7353	1.67493	0.837464	−	0.546493i	$-0.184038\pi$
$359$	31.7353	1.67493	0.837464	+	0.546493i	$0.184038\pi$
$360$	0	0
$361$	21.2913	1.12060
$362$	0	0
$363$	−11.9574	−0.627598
$364$	0	0
$365$	13.4837	0.705767
$366$	0	0
$367$	28.5114	1.48828	0.744141	−	0.668023i	$-0.232859\pi$
$367$	28.5114	1.48828	0.744141	+	0.668023i	$0.232859\pi$
$368$	0	0
$369$	−1.37228	−0.0714381
$370$	0	0
$371$	3.77005	0.195731
$372$	0	0
$373$	−12.7581	−0.660591	−0.330295	−	0.943878i	$-0.607148\pi$
$373$	−12.7581	−0.660591	−0.330295	+	0.943878i	$0.607148\pi$
$374$	0	0
$375$	1.27582	0.0658831
$376$	0	0
$377$	1.86576	0.0960915
$378$	0	0
$379$	2.64076	0.135647	0.0678235	−	0.997697i	$-0.478395\pi$
$379$	2.64076	0.135647	0.0678235	+	0.997697i	$0.478395\pi$
$380$	0	0
$381$	−12.3914	−0.634830
$382$	0	0
$383$	36.4334	1.86166	0.930831	−	0.365451i	$-0.119085\pi$
$383$	36.4334	1.86166	0.930831	+	0.365451i	$0.119085\pi$
$384$	0	0
$385$	2.04512	0.104229
$386$	0	0
$387$	−2.04947	−0.104180
$388$	0	0
$389$	10.2913	0.521791	0.260896	−	0.965367i	$-0.415982\pi$
$389$	10.2913	0.521791	0.260896	+	0.965367i	$0.415982\pi$
$390$	0	0
$391$	3.16632	0.160127
$392$	0	0
$393$	−20.8825	−1.05339
$394$	0	0
$395$	−15.1318	−0.761361
$396$	0	0
$397$	3.83181	0.192313	0.0961566	−	0.995366i	$-0.469345\pi$
$397$	3.83181	0.192313	0.0961566	+	0.995366i	$0.469345\pi$
$398$	0	0
$399$	12.9815	0.649887
$400$	0	0
$401$	25.0810	1.25249	0.626243	−	0.779628i	$-0.284592\pi$
$401$	25.0810	1.25249	0.626243	+	0.779628i	$0.284592\pi$
$402$	0	0
$403$	−2.01356	−0.100303
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−4.14780	−0.205599
$408$	0	0
$409$	−32.3445	−1.59933	−0.799667	−	0.600443i	$-0.794991\pi$
$409$	−32.3445	−1.59933	−0.799667	+	0.600443i	$0.794991\pi$
$410$	0	0
$411$	−0.810219	−0.0399652
$412$	0	0
$413$	4.32716	0.212926
$414$	0	0
$415$	0.569555	0.0279583
$416$	0	0
$417$	−11.1565	−0.546336
$418$	0	0
$419$	4.53808	0.221700	0.110850	−	0.993837i	$-0.464643\pi$
$419$	4.53808	0.221700	0.110850	+	0.993837i	$0.464643\pi$
$420$	0	0
$421$	35.6598	1.73795	0.868976	−	0.494855i	$-0.164779\pi$
$421$	35.6598	1.73795	0.868976	+	0.494855i	$0.164779\pi$
$422$	0	0
$423$	4.55726	0.221582
$424$	0	0
$425$	0.724179	0.0351279
$426$	0	0
$427$	9.82611	0.475519
$428$	0	0
$429$	0.646232	0.0312004
$430$	0	0
$431$	3.52504	0.169795	0.0848975	−	0.996390i	$-0.472944\pi$
$431$	3.52504	0.169795	0.0848975	+	0.996390i	$0.472944\pi$
$432$	0	0
$433$	6.81577	0.327545	0.163773	−	0.986498i	$-0.447634\pi$
$433$	6.81577	0.327545	0.163773	+	0.986498i	$0.447634\pi$
$434$	0	0
$435$	5.99565	0.287469
$436$	0	0
$437$	27.7533	1.32762
$438$	0	0
$439$	4.52317	0.215879	0.107939	−	0.994157i	$-0.465575\pi$
$439$	4.52317	0.215879	0.107939	+	0.994157i	$0.465575\pi$
$440$	0	0
$441$	6.07982	0.289515
$442$	0	0
$443$	22.7979	1.08316	0.541579	−	0.840650i	$-0.317827\pi$
$443$	22.7979	1.08316	0.541579	+	0.840650i	$0.317827\pi$
$444$	0	0
$445$	−14.0804	−0.667476
$446$	0	0
$447$	12.0489	0.569891
$448$	0	0
$449$	−25.3092	−1.19442	−0.597208	−	0.802086i	$-0.703723\pi$
$449$	−25.3092	−1.19442	−0.597208	+	0.802086i	$0.703723\pi$
$450$	0	0
$451$	1.27582	0.0600761
$452$	0	0
$453$	−2.86960	−0.134825
$454$	0	0
$455$	0.636411	0.0298354
$456$	0	0
$457$	−5.37962	−0.251648	−0.125824	−	0.992053i	$-0.540157\pi$
$457$	−5.37962	−0.251648	−0.125824	+	0.992053i	$0.540157\pi$
$458$	0	0
$459$	−4.03965	−0.188555
$460$	0	0
$461$	−4.33646	−0.201969	−0.100984	−	0.994888i	$-0.532199\pi$
$461$	−4.33646	−0.201969	−0.100984	+	0.994888i	$0.532199\pi$
$462$	0	0
$463$	8.21219	0.381653	0.190826	−	0.981624i	$-0.438883\pi$
$463$	8.21219	0.381653	0.190826	+	0.981624i	$0.438883\pi$
$464$	0	0
$465$	−6.47061	−0.300067
$466$	0	0
$467$	−9.18805	−0.425172	−0.212586	−	0.977142i	$-0.568189\pi$
$467$	−9.18805	−0.425172	−0.212586	+	0.977142i	$0.568189\pi$
$468$	0	0
$469$	−19.4044	−0.896010
$470$	0	0
$471$	−5.68467	−0.261936
$472$	0	0
$473$	1.90541	0.0876108
$474$	0	0
$475$	6.34755	0.291245
$476$	0	0
$477$	−3.22747	−0.147775
$478$	0	0
$479$	30.3160	1.38517	0.692586	−	0.721335i	$-0.256471\pi$
$479$	30.3160	1.38517	0.692586	+	0.721335i	$0.256471\pi$
$480$	0	0
$481$	−1.29074	−0.0588525
$482$	0	0
$483$	8.94184	0.406868
$484$	0	0
$485$	−2.00682	−0.0911251
$486$	0	0
$487$	7.77125	0.352149	0.176075	−	0.984377i	$-0.443660\pi$
$487$	7.77125	0.352149	0.176075	+	0.984377i	$0.443660\pi$
$488$	0	0
$489$	−7.35265	−0.332498
$490$	0	0
$491$	−2.31360	−0.104411	−0.0522057	−	0.998636i	$-0.516625\pi$
$491$	−2.31360	−0.104411	−0.0522057	+	0.998636i	$0.516625\pi$
$492$	0	0
$493$	3.40324	0.153274
$494$	0	0
$495$	−1.75079	−0.0786919
$496$	0	0
$497$	3.09399	0.138784
$498$	0	0
$499$	2.40623	0.107717	0.0538587	−	0.998549i	$-0.482848\pi$
$499$	2.40623	0.107717	0.0538587	+	0.998549i	$0.482848\pi$
$500$	0	0
$501$	−18.4935	−0.826227
$502$	0	0
$503$	26.8442	1.19693	0.598463	−	0.801151i	$-0.295778\pi$
$503$	26.8442	1.19693	0.598463	+	0.801151i	$0.295778\pi$
$504$	0	0
$505$	−1.53313	−0.0682233
$506$	0	0
$507$	−16.3846	−0.727664
$508$	0	0
$509$	19.7657	0.876099	0.438050	−	0.898951i	$-0.355669\pi$
$509$	19.7657	0.876099	0.438050	+	0.898951i	$0.355669\pi$
$510$	0	0
$511$	21.6141	0.956150
$512$	0	0
$513$	−35.4082	−1.56331
$514$	0	0
$515$	7.66849	0.337914
$516$	0	0
$517$	−4.23692	−0.186340
$518$	0	0
$519$	13.5196	0.593443
$520$	0	0
$521$	31.6228	1.38542	0.692711	−	0.721215i	$-0.256416\pi$
$521$	31.6228	1.38542	0.692711	+	0.721215i	$0.256416\pi$
$522$	0	0
$523$	23.0549	1.00812	0.504061	−	0.863668i	$-0.331839\pi$
$523$	23.0549	1.00812	0.504061	+	0.863668i	$0.331839\pi$
$524$	0	0
$525$	2.04512	0.0892563
$526$	0	0
$527$	−3.67284	−0.159991
$528$	0	0
$529$	−3.88316	−0.168833
$530$	0	0
$531$	−3.70440	−0.160757
$532$	0	0
$533$	0.397017	0.0171967
$534$	0	0
$535$	0.502170	0.0217107
$536$	0	0
$537$	−10.8212	−0.466971
$538$	0	0
$539$	−5.65245	−0.243468
$540$	0	0
$541$	−5.87011	−0.252376	−0.126188	−	0.992006i	$-0.540274\pi$
$541$	−5.87011	−0.252376	−0.126188	+	0.992006i	$0.540274\pi$
$542$	0	0
$543$	−14.8869	−0.638858
$544$	0	0
$545$	0.372281	0.0159468
$546$	0	0
$547$	−39.8002	−1.70174	−0.850868	−	0.525380i	$-0.823923\pi$
$547$	−39.8002	−1.70174	−0.850868	+	0.525380i	$0.823923\pi$
$548$	0	0
$549$	−8.41193	−0.359013
$550$	0	0
$551$	29.8299	1.27080
$552$	0	0
$553$	−24.2560	−1.03147
$554$	0	0
$555$	−4.14780	−0.176064
$556$	0	0
$557$	17.4916	0.741143	0.370571	−	0.928804i	$-0.379162\pi$
$557$	17.4916	0.741143	0.370571	+	0.928804i	$0.379162\pi$
$558$	0	0
$559$	0.592936	0.0250785
$560$	0	0
$561$	1.17876	0.0497673
$562$	0	0
$563$	−23.4726	−0.989251	−0.494625	−	0.869106i	$-0.664695\pi$
$563$	−23.4726	−0.989251	−0.494625	+	0.869106i	$0.664695\pi$
$564$	0	0
$565$	6.17936	0.259968
$566$	0	0
$567$	−4.80895	−0.201957
$568$	0	0
$569$	1.49175	0.0625374	0.0312687	−	0.999511i	$-0.490045\pi$
$569$	1.49175	0.0625374	0.0312687	+	0.999511i	$0.490045\pi$
$570$	0	0
$571$	0.970396	0.0406098	0.0203049	−	0.999794i	$-0.493536\pi$
$571$	0.970396	0.0406098	0.0203049	+	0.999794i	$0.493536\pi$
$572$	0	0
$573$	6.15963	0.257322
$574$	0	0
$575$	4.37228	0.182337
$576$	0	0
$577$	−10.9498	−0.455845	−0.227923	−	0.973679i	$-0.573193\pi$
$577$	−10.9498	−0.455845	−0.227923	+	0.973679i	$0.573193\pi$
$578$	0	0
$579$	12.2864	0.510605
$580$	0	0
$581$	0.912986	0.0378771
$582$	0	0
$583$	3.00060	0.124272
$584$	0	0
$585$	−0.544819	−0.0225255
$586$	0	0
$587$	−3.56768	−0.147254	−0.0736270	−	0.997286i	$-0.523457\pi$
$587$	−3.56768	−0.147254	−0.0736270	+	0.997286i	$0.523457\pi$
$588$	0	0
$589$	−32.1930	−1.32649
$590$	0	0
$591$	30.7799	1.26611
$592$	0	0
$593$	30.3965	1.24823	0.624117	−	0.781331i	$-0.285459\pi$
$593$	30.3965	1.24823	0.624117	+	0.781331i	$0.285459\pi$
$594$	0	0
$595$	1.16085	0.0475901
$596$	0	0
$597$	6.43905	0.263533
$598$	0	0
$599$	−24.9809	−1.02069	−0.510345	−	0.859969i	$-0.670482\pi$
$599$	−24.9809	−1.02069	−0.510345	+	0.859969i	$0.670482\pi$
$600$	0	0
$601$	−19.7173	−0.804287	−0.402144	−	0.915577i	$-0.631735\pi$
$601$	−19.7173	−0.804287	−0.402144	+	0.915577i	$0.631735\pi$
$602$	0	0
$603$	16.6117	0.676480
$604$	0	0
$605$	−9.37228	−0.381037
$606$	0	0
$607$	33.0674	1.34216	0.671082	−	0.741383i	$-0.265830\pi$
$607$	33.0674	1.34216	0.671082	+	0.741383i	$0.265830\pi$
$608$	0	0
$609$	9.61092	0.389454
$610$	0	0
$611$	−1.31847	−0.0533396
$612$	0	0
$613$	−40.1473	−1.62153	−0.810767	−	0.585370i	$-0.800949\pi$
$613$	−40.1473	−1.62153	−0.810767	+	0.585370i	$0.800949\pi$
$614$	0	0
$615$	1.27582	0.0514461
$616$	0	0
$617$	−13.6191	−0.548284	−0.274142	−	0.961689i	$-0.588394\pi$
$617$	−13.6191	−0.548284	−0.274142	+	0.961689i	$0.588394\pi$
$618$	0	0
$619$	−17.3908	−0.698995	−0.349498	−	0.936937i	$-0.613648\pi$
$619$	−17.3908	−0.698995	−0.349498	+	0.936937i	$0.613648\pi$
$620$	0	0
$621$	−24.3897	−0.978724
$622$	0	0
$623$	−22.5707	−0.904275
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.3320	0.412621
$628$	0	0
$629$	−2.35437	−0.0938748
$630$	0	0
$631$	32.4571	1.29209	0.646047	−	0.763297i	$-0.276421\pi$
$631$	32.4571	1.29209	0.646047	+	0.763297i	$0.276421\pi$
$632$	0	0
$633$	−24.2098	−0.962253
$634$	0	0
$635$	−9.71249	−0.385428
$636$	0	0
$637$	−1.75896	−0.0696926
$638$	0	0
$639$	−2.64870	−0.104781
$640$	0	0
$641$	33.0810	1.30662	0.653311	−	0.757090i	$-0.273379\pi$
$641$	33.0810	1.30662	0.653311	+	0.757090i	$0.273379\pi$
$642$	0	0
$643$	5.00734	0.197470	0.0987351	−	0.995114i	$-0.468520\pi$
$643$	5.00734	0.197470	0.0987351	+	0.995114i	$0.468520\pi$
$644$	0	0
$645$	1.90541	0.0750254
$646$	0	0
$647$	−3.70004	−0.145464	−0.0727319	−	0.997352i	$-0.523172\pi$
$647$	−3.70004	−0.145464	−0.0727319	+	0.997352i	$0.523172\pi$
$648$	0	0
$649$	3.44401	0.135189
$650$	0	0
$651$	−10.3723	−0.406522
$652$	0	0
$653$	−31.0154	−1.21373	−0.606864	−	0.794806i	$-0.707573\pi$
$653$	−31.0154	−1.21373	−0.606864	+	0.794806i	$0.707573\pi$
$654$	0	0
$655$	−16.3679	−0.639548
$656$	0	0
$657$	−18.5034	−0.721885
$658$	0	0
$659$	1.09758	0.0427555	0.0213778	−	0.999771i	$-0.493195\pi$
$659$	1.09758	0.0427555	0.0213778	+	0.999771i	$0.493195\pi$
$660$	0	0
$661$	−8.00929	−0.311525	−0.155763	−	0.987795i	$-0.549784\pi$
$661$	−8.00929	−0.311525	−0.155763	+	0.987795i	$0.549784\pi$
$662$	0	0
$663$	0.366813	0.0142458
$664$	0	0
$665$	10.1750	0.394570
$666$	0	0
$667$	20.5473	0.795594
$668$	0	0
$669$	−31.1341	−1.20372
$670$	0	0
$671$	7.82064	0.301912
$672$	0	0
$673$	−13.5312	−0.521591	−0.260796	−	0.965394i	$-0.583985\pi$
$673$	−13.5312	−0.521591	−0.260796	+	0.965394i	$0.583985\pi$
$674$	0	0
$675$	−5.57825	−0.214707
$676$	0	0
$677$	−17.7228	−0.681143	−0.340572	−	0.940219i	$-0.610621\pi$
$677$	−17.7228	−0.681143	−0.340572	+	0.940219i	$0.610621\pi$
$678$	0	0
$679$	−3.21690	−0.123453
$680$	0	0
$681$	−29.6674	−1.13686
$682$	0	0
$683$	−19.3203	−0.739272	−0.369636	−	0.929177i	$-0.620518\pi$
$683$	−19.3203	−0.739272	−0.369636	+	0.929177i	$0.620518\pi$
$684$	0	0
$685$	−0.635057	−0.0242643
$686$	0	0
$687$	−12.8331	−0.489612
$688$	0	0
$689$	0.933743	0.0355728
$690$	0	0
$691$	3.97526	0.151226	0.0756131	−	0.997137i	$-0.475909\pi$
$691$	3.97526	0.151226	0.0756131	+	0.997137i	$0.475909\pi$
$692$	0	0
$693$	−2.80648	−0.106609
$694$	0	0
$695$	−8.74456	−0.331700
$696$	0	0
$697$	0.724179	0.0274302
$698$	0	0
$699$	4.77500	0.180607
$700$	0	0
$701$	−36.2102	−1.36764	−0.683821	−	0.729650i	$-0.739683\pi$
$701$	−36.2102	−1.36764	−0.683821	+	0.729650i	$0.739683\pi$
$702$	0	0
$703$	−20.6364	−0.778317
$704$	0	0
$705$	−4.23692	−0.159572
$706$	0	0
$707$	−2.45758	−0.0924268
$708$	0	0
$709$	−39.6587	−1.48941	−0.744706	−	0.667392i	$-0.767410\pi$
$709$	−39.6587	−1.48941	−0.744706	+	0.667392i	$0.767410\pi$
$710$	0	0
$711$	20.7650	0.778749
$712$	0	0
$713$	−22.1750	−0.830461
$714$	0	0
$715$	0.506522	0.0189429
$716$	0	0
$717$	13.5039	0.504312
$718$	0	0
$719$	−6.07368	−0.226510	−0.113255	−	0.993566i	$-0.536128\pi$
$719$	−6.07368	−0.226510	−0.113255	+	0.993566i	$0.536128\pi$
$720$	0	0
$721$	12.2925	0.457795
$722$	0	0
$723$	−15.4180	−0.573402
$724$	0	0
$725$	4.69944	0.174533
$726$	0	0
$727$	−7.49969	−0.278148	−0.139074	−	0.990282i	$-0.544413\pi$
$727$	−7.49969	−0.278148	−0.139074	+	0.990282i	$0.544413\pi$
$728$	0	0
$729$	25.4674	0.943236
$730$	0	0
$731$	1.08155	0.0400024
$732$	0	0
$733$	−37.2331	−1.37524	−0.687618	−	0.726073i	$-0.741343\pi$
$733$	−37.2331	−1.37524	−0.687618	+	0.726073i	$0.741343\pi$
$734$	0	0
$735$	−5.65245	−0.208494
$736$	0	0
$737$	−15.4440	−0.568887
$738$	0	0
$739$	18.3211	0.673952	0.336976	−	0.941513i	$-0.390596\pi$
$739$	18.3211	0.673952	0.336976	+	0.941513i	$0.390596\pi$
$740$	0	0
$741$	3.21517	0.118112
$742$	0	0
$743$	23.3810	0.857765	0.428882	−	0.903360i	$-0.358907\pi$
$743$	23.3810	0.857765	0.428882	+	0.903360i	$0.358907\pi$
$744$	0	0
$745$	9.44401	0.346002
$746$	0	0
$747$	−0.781589	−0.0285969
$748$	0	0
$749$	0.804970	0.0294130
$750$	0	0
$751$	25.1997	0.919551	0.459776	−	0.888035i	$-0.347930\pi$
$751$	25.1997	0.919551	0.459776	+	0.888035i	$0.347930\pi$
$752$	0	0
$753$	−6.73971	−0.245609
$754$	0	0
$755$	−2.24921	−0.0818573
$756$	0	0
$757$	47.8448	1.73895	0.869475	−	0.493976i	$-0.164457\pi$
$757$	47.8448	1.73895	0.869475	+	0.493976i	$0.164457\pi$
$758$	0	0
$759$	7.11684	0.258325
$760$	0	0
$761$	26.1256	0.947053	0.473527	−	0.880780i	$-0.342981\pi$
$761$	26.1256	0.947053	0.473527	+	0.880780i	$0.342981\pi$
$762$	0	0
$763$	0.596761	0.0216042
$764$	0	0
$765$	−0.993778	−0.0359301
$766$	0	0
$767$	1.07172	0.0386977
$768$	0	0
$769$	6.25978	0.225733	0.112867	−	0.993610i	$-0.463997\pi$
$769$	6.25978	0.225733	0.112867	+	0.993610i	$0.463997\pi$
$770$	0	0
$771$	18.2370	0.656790
$772$	0	0
$773$	−4.61776	−0.166089	−0.0830446	−	0.996546i	$-0.526464\pi$
$773$	−4.61776	−0.166089	−0.0830446	+	0.996546i	$0.526464\pi$
$774$	0	0
$775$	−5.07172	−0.182182
$776$	0	0
$777$	−6.64886	−0.238526
$778$	0	0
$779$	6.34755	0.227424
$780$	0	0
$781$	2.46252	0.0881159
$782$	0	0
$783$	−26.2147	−0.936835
$784$	0	0
$785$	−4.45570	−0.159031
$786$	0	0
$787$	0.110775	0.00394872	0.00197436	−	0.999998i	$-0.499372\pi$
$787$	0.110775	0.00394872	0.00197436	+	0.999998i	$0.499372\pi$
$788$	0	0
$789$	12.3897	0.441084
$790$	0	0
$791$	9.90541	0.352196
$792$	0	0
$793$	2.43367	0.0864222
$794$	0	0
$795$	3.00060	0.106420
$796$	0	0
$797$	−35.1973	−1.24675	−0.623377	−	0.781921i	$-0.714240\pi$
$797$	−35.1973	−1.24675	−0.623377	+	0.781921i	$0.714240\pi$
$798$	0	0
$799$	−2.40496	−0.0850812
$800$	0	0
$801$	19.3223	0.682720
$802$	0	0
$803$	17.2027	0.607071
$804$	0	0
$805$	7.00869	0.247024
$806$	0	0
$807$	−16.0760	−0.565901
$808$	0	0
$809$	4.09842	0.144093	0.0720463	−	0.997401i	$-0.477047\pi$
$809$	4.09842	0.144093	0.0720463	+	0.997401i	$0.477047\pi$
$810$	0	0
$811$	14.9213	0.523958	0.261979	−	0.965074i	$-0.415625\pi$
$811$	14.9213	0.523958	0.261979	+	0.965074i	$0.415625\pi$
$812$	0	0
$813$	−0.407076	−0.0142768
$814$	0	0
$815$	−5.76308	−0.201872
$816$	0	0
$817$	9.47992	0.331660
$818$	0	0
$819$	−0.873336	−0.0305168
$820$	0	0
$821$	7.31959	0.255455	0.127728	−	0.991809i	$-0.459232\pi$
$821$	7.31959	0.255455	0.127728	+	0.991809i	$0.459232\pi$
$822$	0	0
$823$	21.2981	0.742404	0.371202	−	0.928552i	$-0.378946\pi$
$823$	21.2981	0.742404	0.371202	+	0.928552i	$0.378946\pi$
$824$	0	0
$825$	1.62772	0.0566699
$826$	0	0
$827$	−18.8231	−0.654544	−0.327272	−	0.944930i	$-0.606129\pi$
$827$	−18.8231	−0.654544	−0.327272	+	0.944930i	$0.606129\pi$
$828$	0	0
$829$	−0.214659	−0.00745541	−0.00372771	−	0.999993i	$-0.501187\pi$
$829$	−0.214659	−0.00745541	−0.00372771	+	0.999993i	$0.501187\pi$
$830$	0	0
$831$	6.21415	0.215567
$832$	0	0
$833$	−3.20844	−0.111166
$834$	0	0
$835$	−14.4953	−0.501632
$836$	0	0
$837$	28.2913	0.977892
$838$	0	0
$839$	34.0432	1.17530	0.587652	−	0.809114i	$-0.300053\pi$
$839$	34.0432	1.17530	0.587652	+	0.809114i	$0.300053\pi$
$840$	0	0
$841$	−6.91523	−0.238456
$842$	0	0
$843$	−14.7087	−0.506593
$844$	0	0
$845$	−12.8424	−0.441791
$846$	0	0
$847$	−15.0236	−0.516217
$848$	0	0
$849$	−21.5218	−0.738626
$850$	0	0
$851$	−14.2147	−0.487272
$852$	0	0
$853$	−1.99018	−0.0681424	−0.0340712	−	0.999419i	$-0.510847\pi$
$853$	−1.99018	−0.0681424	−0.0340712	+	0.999419i	$0.510847\pi$
$854$	0	0
$855$	−8.71062	−0.297897
$856$	0	0
$857$	−45.3037	−1.54754	−0.773772	−	0.633464i	$-0.781632\pi$
$857$	−45.3037	−1.54754	−0.773772	+	0.633464i	$0.781632\pi$
$858$	0	0
$859$	5.61790	0.191680	0.0958401	−	0.995397i	$-0.469446\pi$
$859$	5.61790	0.191680	0.0958401	+	0.995397i	$0.469446\pi$
$860$	0	0
$861$	2.04512	0.0696975
$862$	0	0
$863$	−49.6228	−1.68918	−0.844589	−	0.535415i	$-0.820155\pi$
$863$	−49.6228	−1.68918	−0.844589	+	0.535415i	$0.820155\pi$
$864$	0	0
$865$	10.5968	0.360301
$866$	0	0
$867$	−21.0199	−0.713872
$868$	0	0
$869$	−19.3054	−0.654891
$870$	0	0
$871$	−4.80595	−0.162843
$872$	0	0
$873$	2.75393	0.0932062
$874$	0	0
$875$	1.60298	0.0541907
$876$	0	0
$877$	15.8325	0.534625	0.267312	−	0.963610i	$-0.413864\pi$
$877$	15.8325	0.534625	0.267312	+	0.963610i	$0.413864\pi$
$878$	0	0
$879$	13.2086	0.445515
$880$	0	0
$881$	51.6103	1.73880	0.869398	−	0.494113i	$-0.164507\pi$
$881$	51.6103	1.73880	0.869398	+	0.494113i	$0.164507\pi$
$882$	0	0
$883$	−12.7910	−0.430453	−0.215226	−	0.976564i	$-0.569049\pi$
$883$	−12.7910	−0.430453	−0.215226	+	0.976564i	$0.569049\pi$
$884$	0	0
$885$	3.44401	0.115769
$886$	0	0
$887$	16.7106	0.561088	0.280544	−	0.959841i	$-0.409485\pi$
$887$	16.7106	0.561088	0.280544	+	0.959841i	$0.409485\pi$
$888$	0	0
$889$	−15.5690	−0.522166
$890$	0	0
$891$	−3.82746	−0.128225
$892$	0	0
$893$	−21.0798	−0.705409
$894$	0	0
$895$	−8.48179	−0.283515
$896$	0	0
$897$	2.21466	0.0739453
$898$	0	0
$899$	−23.8343	−0.794918
$900$	0	0
$901$	1.70320	0.0567417
$902$	0	0
$903$	3.05434	0.101642
$904$	0	0
$905$	−11.6685	−0.387874
$906$	0	0
$907$	41.2098	1.36835	0.684174	−	0.729319i	$-0.260163\pi$
$907$	41.2098	1.36835	0.684174	+	0.729319i	$0.260163\pi$
$908$	0	0
$909$	2.10388	0.0697814
$910$	0	0
$911$	42.1967	1.39804	0.699020	−	0.715102i	$-0.253620\pi$
$911$	42.1967	1.39804	0.699020	+	0.715102i	$0.253620\pi$
$912$	0	0
$913$	0.726650	0.0240486
$914$	0	0
$915$	7.82064	0.258542
$916$	0	0
$917$	−26.2375	−0.866439
$918$	0	0
$919$	16.3915	0.540707	0.270354	−	0.962761i	$-0.412859\pi$
$919$	16.3915	0.540707	0.270354	+	0.962761i	$0.412859\pi$
$920$	0	0
$921$	8.63259	0.284453
$922$	0	0
$923$	0.766300	0.0252231
$924$	0	0
$925$	−3.25109	−0.106895
$926$	0	0
$927$	−10.5233	−0.345631
$928$	0	0
$929$	52.3744	1.71835	0.859174	−	0.511684i	$-0.170978\pi$
$929$	52.3744	1.71835	0.859174	+	0.511684i	$0.170978\pi$
$930$	0	0
$931$	−28.1225	−0.921676
$932$	0	0
$933$	−15.3809	−0.503548
$934$	0	0
$935$	0.923923	0.0302155
$936$	0	0
$937$	15.1361	0.494475	0.247238	−	0.968955i	$-0.420477\pi$
$937$	15.1361	0.494475	0.247238	+	0.968955i	$0.420477\pi$
$938$	0	0
$939$	22.3390	0.729006
$940$	0	0
$941$	−29.4588	−0.960328	−0.480164	−	0.877179i	$-0.659423\pi$
$941$	−29.4588	−0.960328	−0.480164	+	0.877179i	$0.659423\pi$
$942$	0	0
$943$	4.37228	0.142381
$944$	0	0
$945$	−8.94184	−0.290878
$946$	0	0
$947$	−53.1207	−1.72619	−0.863095	−	0.505042i	$-0.831477\pi$
$947$	−53.1207	−1.72619	−0.863095	+	0.505042i	$0.831477\pi$
$948$	0	0
$949$	5.35324	0.173774
$950$	0	0
$951$	34.7880	1.12808
$952$	0	0
$953$	−7.73587	−0.250589	−0.125295	−	0.992120i	$-0.539988\pi$
$953$	−7.73587	−0.250589	−0.125295	+	0.992120i	$0.539988\pi$
$954$	0	0
$955$	4.82798	0.156230
$956$	0	0
$957$	7.64937	0.247269
$958$	0	0
$959$	−1.01799	−0.0328725
$960$	0	0
$961$	−5.27761	−0.170245
$962$	0	0
$963$	−0.689119	−0.0222065
$964$	0	0
$965$	9.63019	0.310007
$966$	0	0
$967$	17.8301	0.573377	0.286688	−	0.958024i	$-0.407446\pi$
$967$	17.8301	0.573377	0.286688	+	0.958024i	$0.407446\pi$
$968$	0	0
$969$	5.86464	0.188399
$970$	0	0
$971$	−24.1090	−0.773694	−0.386847	−	0.922144i	$-0.626436\pi$
$971$	−24.1090	−0.773694	−0.386847	+	0.922144i	$0.626436\pi$
$972$	0	0
$973$	−14.0174	−0.449377
$974$	0	0
$975$	0.506522	0.0162217
$976$	0	0
$977$	41.9856	1.34324	0.671619	−	0.740897i	$-0.265599\pi$
$977$	41.9856	1.34324	0.671619	+	0.740897i	$0.265599\pi$
$978$	0	0
$979$	−17.9641	−0.574135
$980$	0	0
$981$	−0.510875	−0.0163110
$982$	0	0
$983$	−45.6216	−1.45510	−0.727552	−	0.686052i	$-0.759342\pi$
$983$	−45.6216	−1.45510	−0.727552	+	0.686052i	$0.759342\pi$
$984$	0	0
$985$	24.1255	0.768704
$986$	0	0
$987$	−6.79172	−0.216183
$988$	0	0
$989$	6.52990	0.207639
$990$	0	0
$991$	−44.1182	−1.40146	−0.700730	−	0.713426i	$-0.747142\pi$
$991$	−44.1182	−1.40146	−0.700730	+	0.713426i	$0.747142\pi$
$992$	0	0
$993$	−20.9875	−0.666020
$994$	0	0
$995$	5.04699	0.160000
$996$	0	0
$997$	31.9611	1.01222	0.506109	−	0.862470i	$-0.331083\pi$
$997$	31.9611	1.01222	0.506109	+	0.862470i	$0.331083\pi$
$998$	0	0
$999$	18.1354	0.573777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	820.2.a.d.1.3	✓	4
3.2	odd	2		7380.2.a.t.1.3		4
4.3	odd	2		3280.2.a.be.1.2		4
5.2	odd	4		4100.2.d.e.1149.3		8
5.3	odd	4		4100.2.d.e.1149.6		8
5.4	even	2		4100.2.a.d.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.a.d.1.3	✓	4	1.1	even	1	trivial
3280.2.a.be.1.2		4	4.3	odd	2
4100.2.a.d.1.2		4	5.4	even	2
4100.2.d.e.1149.3		8	5.2	odd	4
4100.2.d.e.1149.6		8	5.3	odd	4
7380.2.a.t.1.3		4	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	820.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27582 q^{3} +1.00000 q^{5} +1.60298 q^{7} -1.37228 q^{9} +1.27582 q^{11} +0.397017 q^{13} +1.27582 q^{15} +0.724179 q^{17} +6.34755 q^{19} +2.04512 q^{21} +4.37228 q^{23} +1.00000 q^{25} -5.57825 q^{27} +4.69944 q^{29} -5.07172 q^{31} +1.62772 q^{33} +1.60298 q^{35} -3.25109 q^{37} +0.506522 q^{39} +1.00000 q^{41} +1.49348 q^{43} -1.37228 q^{45} -3.32094 q^{47} -4.43045 q^{49} +0.923923 q^{51} +2.35190 q^{53} +1.27582 q^{55} +8.09833 q^{57} +2.69944 q^{59} +6.12989 q^{61} -2.19974 q^{63} +0.397017 q^{65} -12.1052 q^{67} +5.57825 q^{69} +1.93015 q^{71} +13.4837 q^{73} +1.27582 q^{75} +2.04512 q^{77} -15.1318 q^{79} -3.00000 q^{81} +0.569555 q^{83} +0.724179 q^{85} +5.99565 q^{87} -14.0804 q^{89} +0.636411 q^{91} -6.47061 q^{93} +6.34755 q^{95} -2.00682 q^{97} -1.75079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + q^{7} + 6 q^{9} + 7 q^{13} + 8 q^{17} - 3 q^{19} - 3 q^{21} + 6 q^{23} + 4 q^{25} + 7 q^{29} + 3 q^{31} + 18 q^{33} + q^{35} + 9 q^{37} + 3 q^{39} + 4 q^{41} + 5 q^{43} + 6 q^{45} + 3 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + q^7 + 6 * q^9 + 7 * q^13 + 8 * q^17 - 3 * q^19 - 3 * q^21 + 6 * q^23 + 4 * q^25 + 7 * q^29 + 3 * q^31 + 18 * q^33 + q^35 + 9 * q^37 + 3 * q^39 + 4 * q^41 + 5 * q^43 + 6 * q^45 + 3 * q^47 + 3 * q^49 - 18 * q^51 + 26 * q^53 - 3 * q^57 - q^59 - 8 * q^61 + 18 * q^63 + 7 * q^65 + q^67 + 2 * q^71 + 9 * q^73 - 3 * q^77 + q^79 - 12 * q^81 + 23 * q^83 + 8 * q^85 - 21 * q^87 + 10 * q^89 - 29 * q^91 + 21 * q^93 - 3 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

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Representations

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Database

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