LMFDB - Embedded newform 6724.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6724,2,Mod(1,6724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6724 = 2^{2} \cdot 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6724.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:	$53.6914103191$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 39x^{14} + 594x^{12} - 4428x^{10} + 16529x^{8} - 28236x^{6} + 17856x^{4} - 4032x^{2} + 256 $ x^16 - 39x^14 + 594x^12 - 4428x^10 + 16529x^8 - 28236x^6 + 17856x^4 - 4032*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.66505$ of defining polynomial
Character	$\chi$	$=$	6724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.66505 q^{3} -3.40359 q^{5} +0.711682 q^{7} -0.227602 q^{9} +O(q^{10})$	$q-1.66505 q^{3} -3.40359 q^{5} +0.711682 q^{7} -0.227602 q^{9} +4.55732 q^{11} -0.0240444 q^{13} +5.66715 q^{15} +4.08270 q^{17} +6.90828 q^{19} -1.18499 q^{21} +4.94917 q^{23} +6.58440 q^{25} +5.37412 q^{27} -1.95253 q^{29} +2.58999 q^{31} -7.58817 q^{33} -2.42227 q^{35} -8.36472 q^{37} +0.0400352 q^{39} +2.88868 q^{43} +0.774663 q^{45} -4.22814 q^{47} -6.49351 q^{49} -6.79790 q^{51} -12.2300 q^{53} -15.5112 q^{55} -11.5027 q^{57} +5.56739 q^{59} +13.5779 q^{61} -0.161980 q^{63} +0.0818373 q^{65} +1.85609 q^{67} -8.24062 q^{69} +13.9201 q^{71} -7.18307 q^{73} -10.9634 q^{75} +3.24336 q^{77} +0.414278 q^{79} -8.26539 q^{81} -6.61952 q^{83} -13.8958 q^{85} +3.25107 q^{87} -15.6339 q^{89} -0.0171120 q^{91} -4.31246 q^{93} -23.5129 q^{95} -7.63532 q^{97} -1.03725 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) $ 16 * q + 6 * q^5 + 30 * q^9	$ 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) $ 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

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
$3$	−1.66505	−0.961318	−0.480659	−	0.876908i	$-0.659603\pi$
$3$	−1.66505	−0.961318	−0.480659	+	0.876908i	$0.659603\pi$
$4$	0	0
$5$	−3.40359	−1.52213	−0.761065	−	0.648675i	$-0.775323\pi$
$5$	−3.40359	−1.52213	−0.761065	+	0.648675i	$0.775323\pi$
$6$	0	0
$7$	0.711682	0.268991	0.134495	−	0.990914i	$-0.457059\pi$
$7$	0.711682	0.268991	0.134495	+	0.990914i	$0.457059\pi$
$8$	0	0
$9$	−0.227602	−0.0758673
$10$	0	0
$11$	4.55732	1.37408	0.687042	−	0.726618i	$-0.258909\pi$
$11$	4.55732	1.37408	0.687042	+	0.726618i	$0.258909\pi$
$12$	0	0
$13$	−0.0240444	−0.00666872	−0.00333436	−	0.999994i	$-0.501061\pi$
$13$	−0.0240444	−0.00666872	−0.00333436	+	0.999994i	$0.501061\pi$
$14$	0	0
$15$	5.66715	1.46325
$16$	0	0
$17$	4.08270	0.990200	0.495100	−	0.868836i	$-0.335131\pi$
$17$	4.08270	0.990200	0.495100	+	0.868836i	$0.335131\pi$
$18$	0	0
$19$	6.90828	1.58487	0.792434	−	0.609957i	$-0.208813\pi$
$19$	6.90828	1.58487	0.792434	+	0.609957i	$0.208813\pi$
$20$	0	0
$21$	−1.18499	−0.258586
$22$	0	0
$23$	4.94917	1.03197	0.515986	−	0.856597i	$-0.327425\pi$
$23$	4.94917	1.03197	0.515986	+	0.856597i	$0.327425\pi$
$24$	0	0
$25$	6.58440	1.31688
$26$	0	0
$27$	5.37412	1.03425
$28$	0	0
$29$	−1.95253	−0.362576	−0.181288	−	0.983430i	$-0.558027\pi$
$29$	−1.95253	−0.362576	−0.181288	+	0.983430i	$0.558027\pi$
$30$	0	0
$31$	2.58999	0.465176	0.232588	−	0.972575i	$-0.425281\pi$
$31$	2.58999	0.465176	0.232588	+	0.972575i	$0.425281\pi$
$32$	0	0
$33$	−7.58817	−1.32093
$34$	0	0
$35$	−2.42227	−0.409439
$36$	0	0
$37$	−8.36472	−1.37515	−0.687576	−	0.726112i	$-0.741325\pi$
$37$	−8.36472	−1.37515	−0.687576	+	0.726112i	$0.741325\pi$
$38$	0	0
$39$	0.0400352	0.00641076
$40$	0	0
$41$	0	0
$41$	0	0
$42$	0	0
$43$	2.88868	0.440520	0.220260	−	0.975441i	$-0.429309\pi$
$43$	2.88868	0.440520	0.220260	+	0.975441i	$0.429309\pi$
$44$	0	0
$45$	0.774663	0.115480
$46$	0	0
$47$	−4.22814	−0.616738	−0.308369	−	0.951267i	$-0.599783\pi$
$47$	−4.22814	−0.616738	−0.308369	+	0.951267i	$0.599783\pi$
$48$	0	0
$49$	−6.49351	−0.927644
$50$	0	0
$51$	−6.79790	−0.951897
$52$	0	0
$53$	−12.2300	−1.67992	−0.839959	−	0.542650i	$-0.817421\pi$
$53$	−12.2300	−1.67992	−0.839959	+	0.542650i	$0.817421\pi$
$54$	0	0
$55$	−15.5112	−2.09153
$56$	0	0
$57$	−11.5027	−1.52356
$58$	0	0
$59$	5.56739	0.724812	0.362406	−	0.932020i	$-0.381955\pi$
$59$	5.56739	0.724812	0.362406	+	0.932020i	$0.381955\pi$
$60$	0	0
$61$	13.5779	1.73847	0.869235	−	0.494398i	$-0.164612\pi$
$61$	13.5779	1.73847	0.869235	+	0.494398i	$0.164612\pi$
$62$	0	0
$63$	−0.161980	−0.0204076
$64$	0	0
$65$	0.0818373	0.0101507
$66$	0	0
$67$	1.85609	0.226758	0.113379	−	0.993552i	$-0.463833\pi$
$67$	1.85609	0.226758	0.113379	+	0.993552i	$0.463833\pi$
$68$	0	0
$69$	−8.24062	−0.992054
$70$	0	0
$71$	13.9201	1.65202	0.826009	−	0.563657i	$-0.190606\pi$
$71$	13.9201	1.65202	0.826009	+	0.563657i	$0.190606\pi$
$72$	0	0
$73$	−7.18307	−0.840715	−0.420358	−	0.907359i	$-0.638095\pi$
$73$	−7.18307	−0.840715	−0.420358	+	0.907359i	$0.638095\pi$
$74$	0	0
$75$	−10.9634	−1.26594
$76$	0	0
$77$	3.24336	0.369616
$78$	0	0
$79$	0.414278	0.0466099	0.0233049	−	0.999728i	$-0.492581\pi$
$79$	0.414278	0.0466099	0.0233049	+	0.999728i	$0.492581\pi$
$80$	0	0
$81$	−8.26539	−0.918377
$82$	0	0
$83$	−6.61952	−0.726587	−0.363293	−	0.931675i	$-0.618348\pi$
$83$	−6.61952	−0.726587	−0.363293	+	0.931675i	$0.618348\pi$
$84$	0	0
$85$	−13.8958	−1.50721
$86$	0	0
$87$	3.25107	0.348551
$88$	0	0
$89$	−15.6339	−1.65719	−0.828594	−	0.559850i	$-0.810859\pi$
$89$	−15.6339	−1.65719	−0.828594	+	0.559850i	$0.810859\pi$
$90$	0	0
$91$	−0.0171120	−0.00179382
$92$	0	0
$93$	−4.31246	−0.447182
$94$	0	0
$95$	−23.5129	−2.41238
$96$	0	0
$97$	−7.63532	−0.775250	−0.387625	−	0.921817i	$-0.626704\pi$
$97$	−7.63532	−0.775250	−0.387625	+	0.921817i	$0.626704\pi$
$98$	0	0
$99$	−1.03725	−0.104248
$100$	0	0
$101$	8.98737	0.894277	0.447138	−	0.894465i	$-0.352443\pi$
$101$	8.98737	0.894277	0.447138	+	0.894465i	$0.352443\pi$
$102$	0	0
$103$	−15.8491	−1.56166	−0.780830	−	0.624743i	$-0.785204\pi$
$103$	−15.8491	−1.56166	−0.780830	+	0.624743i	$0.785204\pi$
$104$	0	0
$105$	4.03321	0.393601
$106$	0	0
$107$	9.34074	0.903003	0.451502	−	0.892270i	$-0.350889\pi$
$107$	9.34074	0.903003	0.451502	+	0.892270i	$0.350889\pi$
$108$	0	0
$109$	−4.24041	−0.406158	−0.203079	−	0.979162i	$-0.565095\pi$
$109$	−4.24041	−0.406158	−0.203079	+	0.979162i	$0.565095\pi$
$110$	0	0
$111$	13.9277	1.32196
$112$	0	0
$113$	16.2253	1.52635	0.763175	−	0.646192i	$-0.223639\pi$
$113$	16.2253	1.52635	0.763175	+	0.646192i	$0.223639\pi$
$114$	0	0
$115$	−16.8449	−1.57080
$116$	0	0
$117$	0.00547256	0.000505938 0
$118$	0	0
$119$	2.90558	0.266354
$120$	0	0
$121$	9.76916	0.888105
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.39265	−0.482333
$126$	0	0
$127$	13.7255	1.21794	0.608968	−	0.793194i	$-0.291584\pi$
$127$	13.7255	1.21794	0.608968	+	0.793194i	$0.291584\pi$
$128$	0	0
$129$	−4.80981	−0.423480
$130$	0	0
$131$	9.60192	0.838923	0.419462	−	0.907773i	$-0.362219\pi$
$131$	9.60192	0.838923	0.419462	+	0.907773i	$0.362219\pi$
$132$	0	0
$133$	4.91650	0.426315
$134$	0	0
$135$	−18.2913	−1.57426
$136$	0	0
$137$	−1.67346	−0.142973	−0.0714865	−	0.997442i	$-0.522774\pi$
$137$	−1.67346	−0.142973	−0.0714865	+	0.997442i	$0.522774\pi$
$138$	0	0
$139$	14.2517	1.20881	0.604405	−	0.796677i	$-0.293411\pi$
$139$	14.2517	1.20881	0.604405	+	0.796677i	$0.293411\pi$
$140$	0	0
$141$	7.04008	0.592882
$142$	0	0
$143$	−0.109578	−0.00916338
$144$	0	0
$145$	6.64561	0.551888
$146$	0	0
$147$	10.8120	0.891761
$148$	0	0
$149$	4.85957	0.398112	0.199056	−	0.979988i	$-0.436212\pi$
$149$	4.85957	0.398112	0.199056	+	0.979988i	$0.436212\pi$
$150$	0	0
$151$	10.1872	0.829023	0.414511	−	0.910044i	$-0.363953\pi$
$151$	10.1872	0.829023	0.414511	+	0.910044i	$0.363953\pi$
$152$	0	0
$153$	−0.929230	−0.0751238
$154$	0	0
$155$	−8.81525	−0.708058
$156$	0	0
$157$	−15.3715	−1.22678	−0.613390	−	0.789780i	$-0.710195\pi$
$157$	−15.3715	−1.22678	−0.613390	+	0.789780i	$0.710195\pi$
$158$	0	0
$159$	20.3636	1.61494
$160$	0	0
$161$	3.52224	0.277591
$162$	0	0
$163$	5.13879	0.402501	0.201250	−	0.979540i	$-0.435500\pi$
$163$	5.13879	0.402501	0.201250	+	0.979540i	$0.435500\pi$
$164$	0	0
$165$	25.8270	2.01063
$166$	0	0
$167$	11.9534	0.924984	0.462492	−	0.886623i	$-0.346955\pi$
$167$	11.9534	0.924984	0.462492	+	0.886623i	$0.346955\pi$
$168$	0	0
$169$	−12.9994	−0.999956
$170$	0	0
$171$	−1.57234	−0.120240
$172$	0	0
$173$	6.64287	0.505048	0.252524	−	0.967591i	$-0.418739\pi$
$173$	6.64287	0.505048	0.252524	+	0.967591i	$0.418739\pi$
$174$	0	0
$175$	4.68600	0.354228
$176$	0	0
$177$	−9.26999	−0.696775
$178$	0	0
$179$	14.1385	1.05676	0.528381	−	0.849008i	$-0.322799\pi$
$179$	14.1385	1.05676	0.528381	+	0.849008i	$0.322799\pi$
$180$	0	0
$181$	24.3190	1.80762	0.903808	−	0.427938i	$-0.140760\pi$
$181$	24.3190	1.80762	0.903808	+	0.427938i	$0.140760\pi$
$182$	0	0
$183$	−22.6079	−1.67122
$184$	0	0
$185$	28.4701	2.09316
$186$	0	0
$187$	18.6062	1.36062
$188$	0	0
$189$	3.82467	0.278204
$190$	0	0
$191$	−9.27752	−0.671298	−0.335649	−	0.941987i	$-0.608956\pi$
$191$	−9.27752	−0.671298	−0.335649	+	0.941987i	$0.608956\pi$
$192$	0	0
$193$	10.3852	0.747545	0.373772	−	0.927520i	$-0.378064\pi$
$193$	10.3852	0.747545	0.373772	+	0.927520i	$0.378064\pi$
$194$	0	0
$195$	−0.136263	−0.00975802
$196$	0	0
$197$	−8.43572	−0.601020	−0.300510	−	0.953779i	$-0.597157\pi$
$197$	−8.43572	−0.601020	−0.300510	+	0.953779i	$0.597157\pi$
$198$	0	0
$199$	0.259985	0.0184298	0.00921491	−	0.999958i	$-0.497067\pi$
$199$	0.259985	0.0184298	0.00921491	+	0.999958i	$0.497067\pi$
$200$	0	0
$201$	−3.09049	−0.217986
$202$	0	0
$203$	−1.38958	−0.0975295
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.12644	−0.0782930
$208$	0	0
$209$	31.4833	2.17774
$210$	0	0
$211$	−14.4387	−0.994003	−0.497001	−	0.867750i	$-0.665566\pi$
$211$	−14.4387	−0.994003	−0.497001	+	0.867750i	$0.665566\pi$
$212$	0	0
$213$	−23.1778	−1.58811
$214$	0	0
$215$	−9.83188	−0.670529
$216$	0	0
$217$	1.84325	0.125128
$218$	0	0
$219$	11.9602	0.808195
$220$	0	0
$221$	−0.0981661	−0.00660337
$222$	0	0
$223$	−14.8117	−0.991864	−0.495932	−	0.868361i	$-0.665173\pi$
$223$	−14.8117	−0.991864	−0.495932	+	0.868361i	$0.665173\pi$
$224$	0	0
$225$	−1.49862	−0.0999082
$226$	0	0
$227$	−6.15836	−0.408744	−0.204372	−	0.978893i	$-0.565515\pi$
$227$	−6.15836	−0.408744	−0.204372	+	0.978893i	$0.565515\pi$
$228$	0	0
$229$	−24.7514	−1.63562	−0.817808	−	0.575491i	$-0.804811\pi$
$229$	−24.7514	−1.63562	−0.817808	+	0.575491i	$0.804811\pi$
$230$	0	0
$231$	−5.40037	−0.355318
$232$	0	0
$233$	22.8007	1.49372	0.746861	−	0.664980i	$-0.231560\pi$
$233$	22.8007	1.49372	0.746861	+	0.664980i	$0.231560\pi$
$234$	0	0
$235$	14.3909	0.938756
$236$	0	0
$237$	−0.689794	−0.0448069
$238$	0	0
$239$	−22.6174	−1.46300	−0.731499	−	0.681842i	$-0.761179\pi$
$239$	−22.6174	−1.46300	−0.731499	+	0.681842i	$0.761179\pi$
$240$	0	0
$241$	−22.5387	−1.45184	−0.725922	−	0.687777i	$-0.758587\pi$
$241$	−22.5387	−1.45184	−0.725922	+	0.687777i	$0.758587\pi$
$242$	0	0
$243$	−2.36007	−0.151398
$244$	0	0
$245$	22.1012	1.41200
$246$	0	0
$247$	−0.166106	−0.0105690
$248$	0	0
$249$	11.0218	0.698481
$250$	0	0
$251$	19.9165	1.25712	0.628558	−	0.777763i	$-0.283645\pi$
$251$	19.9165	1.25712	0.628558	+	0.777763i	$0.283645\pi$
$252$	0	0
$253$	22.5549	1.41802
$254$	0	0
$255$	23.1373	1.44891
$256$	0	0
$257$	−6.83205	−0.426171	−0.213086	−	0.977034i	$-0.568351\pi$
$257$	−6.83205	−0.426171	−0.213086	+	0.977034i	$0.568351\pi$
$258$	0	0
$259$	−5.95302	−0.369903
$260$	0	0
$261$	0.444400	0.0275077
$262$	0	0
$263$	5.54618	0.341992	0.170996	−	0.985272i	$-0.445301\pi$
$263$	5.54618	0.341992	0.170996	+	0.985272i	$0.445301\pi$
$264$	0	0
$265$	41.6258	2.55705
$266$	0	0
$267$	26.0312	1.59308
$268$	0	0
$269$	−1.77034	−0.107940	−0.0539699	−	0.998543i	$-0.517188\pi$
$269$	−1.77034	−0.107940	−0.0539699	+	0.998543i	$0.517188\pi$
$270$	0	0
$271$	−16.9917	−1.03217	−0.516085	−	0.856537i	$-0.672611\pi$
$271$	−16.9917	−1.03217	−0.516085	+	0.856537i	$0.672611\pi$
$272$	0	0
$273$	0.0284923	0.00172444
$274$	0	0
$275$	30.0072	1.80950
$276$	0	0
$277$	−32.0696	−1.92687	−0.963437	−	0.267934i	$-0.913659\pi$
$277$	−32.0696	−1.92687	−0.963437	+	0.267934i	$0.913659\pi$
$278$	0	0
$279$	−0.589486	−0.0352916
$280$	0	0
$281$	−1.56010	−0.0930678	−0.0465339	−	0.998917i	$-0.514818\pi$
$281$	−1.56010	−0.0930678	−0.0465339	+	0.998917i	$0.514818\pi$
$282$	0	0
$283$	30.4079	1.80756	0.903780	−	0.427998i	$-0.140781\pi$
$283$	30.4079	1.80756	0.903780	+	0.427998i	$0.140781\pi$
$284$	0	0
$285$	39.1503	2.31906
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−0.331578	−0.0195046
$290$	0	0
$291$	12.7132	0.745262
$292$	0	0
$293$	−0.680027	−0.0397276	−0.0198638	−	0.999803i	$-0.506323\pi$
$293$	−0.680027	−0.0397276	−0.0198638	+	0.999803i	$0.506323\pi$
$294$	0	0
$295$	−18.9491	−1.10326
$296$	0	0
$297$	24.4916	1.42115
$298$	0	0
$299$	−0.119000	−0.00688194
$300$	0	0
$301$	2.05582	0.118496
$302$	0	0
$303$	−14.9644	−0.859685
$304$	0	0
$305$	−46.2135	−2.64618
$306$	0	0
$307$	28.1502	1.60661	0.803307	−	0.595565i	$-0.203072\pi$
$307$	28.1502	1.60661	0.803307	+	0.595565i	$0.203072\pi$
$308$	0	0
$309$	26.3896	1.50125
$310$	0	0
$311$	10.6335	0.602969	0.301485	−	0.953471i	$-0.402518\pi$
$311$	10.6335	0.602969	0.301485	+	0.953471i	$0.402518\pi$
$312$	0	0
$313$	22.4117	1.26679	0.633393	−	0.773830i	$-0.281662\pi$
$313$	22.4117	1.26679	0.633393	+	0.773830i	$0.281662\pi$
$314$	0	0
$315$	0.551314	0.0310630
$316$	0	0
$317$	−10.1093	−0.567794	−0.283897	−	0.958855i	$-0.591627\pi$
$317$	−10.1093	−0.567794	−0.283897	+	0.958855i	$0.591627\pi$
$318$	0	0
$319$	−8.89831	−0.498210
$320$	0	0
$321$	−15.5528	−0.868073
$322$	0	0
$323$	28.2044	1.56934
$324$	0	0
$325$	−0.158318	−0.00878191
$326$	0	0
$327$	7.06051	0.390447
$328$	0	0
$329$	−3.00909	−0.165897
$330$	0	0
$331$	−1.78846	−0.0983025	−0.0491513	−	0.998791i	$-0.515652\pi$
$331$	−1.78846	−0.0983025	−0.0491513	+	0.998791i	$0.515652\pi$
$332$	0	0
$333$	1.90383	0.104329
$334$	0	0
$335$	−6.31737	−0.345155
$336$	0	0
$337$	19.8296	1.08019	0.540094	−	0.841605i	$-0.318389\pi$
$337$	19.8296	1.08019	0.540094	+	0.841605i	$0.318389\pi$
$338$	0	0
$339$	−27.0160	−1.46731
$340$	0	0
$341$	11.8034	0.639190
$342$	0	0
$343$	−9.60309	−0.518518
$344$	0	0
$345$	28.0477	1.51004
$346$	0	0
$347$	−35.7301	−1.91809	−0.959046	−	0.283249i	$-0.908588\pi$
$347$	−35.7301	−1.91809	−0.959046	+	0.283249i	$0.908588\pi$
$348$	0	0
$349$	3.72595	0.199446	0.0997228	−	0.995015i	$-0.468204\pi$
$349$	3.72595	0.199446	0.0997228	+	0.995015i	$0.468204\pi$
$350$	0	0
$351$	−0.129218	−0.00689713
$352$	0	0
$353$	19.0289	1.01280	0.506402	−	0.862297i	$-0.330975\pi$
$353$	19.0289	1.01280	0.506402	+	0.862297i	$0.330975\pi$
$354$	0	0
$355$	−47.3784	−2.51459
$356$	0	0
$357$	−4.83795	−0.256051
$358$	0	0
$359$	0.326475	0.0172307	0.00861535	−	0.999963i	$-0.497258\pi$
$359$	0.326475	0.0172307	0.00861535	+	0.999963i	$0.497258\pi$
$360$	0	0
$361$	28.7244	1.51181
$362$	0	0
$363$	−16.2662	−0.853752
$364$	0	0
$365$	24.4482	1.27968
$366$	0	0
$367$	−3.98260	−0.207890	−0.103945	−	0.994583i	$-0.533147\pi$
$367$	−3.98260	−0.207890	−0.103945	+	0.994583i	$0.533147\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.70386	−0.451882
$372$	0	0
$373$	6.32698	0.327599	0.163799	−	0.986494i	$-0.447625\pi$
$373$	6.32698	0.327599	0.163799	+	0.986494i	$0.447625\pi$
$374$	0	0
$375$	8.97904	0.463676
$376$	0	0
$377$	0.0469475	0.00241792
$378$	0	0
$379$	10.8108	0.555316	0.277658	−	0.960680i	$-0.410442\pi$
$379$	10.8108	0.555316	0.277658	+	0.960680i	$0.410442\pi$
$380$	0	0
$381$	−22.8536	−1.17082
$382$	0	0
$383$	7.37101	0.376641	0.188321	−	0.982108i	$-0.439696\pi$
$383$	7.37101	0.376641	0.188321	+	0.982108i	$0.439696\pi$
$384$	0	0
$385$	−11.0391	−0.562603
$386$	0	0
$387$	−0.657470	−0.0334211
$388$	0	0
$389$	21.8862	1.10967	0.554836	−	0.831960i	$-0.312781\pi$
$389$	21.8862	1.10967	0.554836	+	0.831960i	$0.312781\pi$
$390$	0	0
$391$	20.2060	1.02186
$392$	0	0
$393$	−15.9877	−0.806472
$394$	0	0
$395$	−1.41003	−0.0709463
$396$	0	0
$397$	13.6193	0.683533	0.341767	−	0.939785i	$-0.388975\pi$
$397$	13.6193	0.683533	0.341767	+	0.939785i	$0.388975\pi$
$398$	0	0
$399$	−8.18623	−0.409824
$400$	0	0
$401$	−11.0671	−0.552663	−0.276331	−	0.961062i	$-0.589119\pi$
$401$	−11.0671	−0.552663	−0.276331	+	0.961062i	$0.589119\pi$
$402$	0	0
$403$	−0.0622747	−0.00310213
$404$	0	0
$405$	28.1320	1.39789
$406$	0	0
$407$	−38.1207	−1.88957
$408$	0	0
$409$	−19.5993	−0.969122	−0.484561	−	0.874757i	$-0.661021\pi$
$409$	−19.5993	−0.969122	−0.484561	+	0.874757i	$0.661021\pi$
$410$	0	0
$411$	2.78639	0.137443
$412$	0	0
$413$	3.96221	0.194968
$414$	0	0
$415$	22.5301	1.10596
$416$	0	0
$417$	−23.7297	−1.16205
$418$	0	0
$419$	23.3775	1.14207	0.571034	−	0.820927i	$-0.306543\pi$
$419$	23.3775	1.14207	0.571034	+	0.820927i	$0.306543\pi$
$420$	0	0
$421$	−3.95752	−0.192878	−0.0964388	−	0.995339i	$-0.530745\pi$
$421$	−3.95752	−0.192878	−0.0964388	+	0.995339i	$0.530745\pi$
$422$	0	0
$423$	0.962334	0.0467903
$424$	0	0
$425$	26.8821	1.30397
$426$	0	0
$427$	9.66314	0.467632
$428$	0	0
$429$	0.182453	0.00880892
$430$	0	0
$431$	−14.8963	−0.717531	−0.358765	−	0.933428i	$-0.616802\pi$
$431$	−14.8963	−0.717531	−0.358765	+	0.933428i	$0.616802\pi$
$432$	0	0
$433$	−12.0171	−0.577507	−0.288753	−	0.957404i	$-0.593241\pi$
$433$	−12.0171	−0.577507	−0.288753	+	0.957404i	$0.593241\pi$
$434$	0	0
$435$	−11.0653	−0.530540
$436$	0	0
$437$	34.1903	1.63554
$438$	0	0
$439$	26.9201	1.28483	0.642413	−	0.766359i	$-0.277933\pi$
$439$	26.9201	1.28483	0.642413	+	0.766359i	$0.277933\pi$
$440$	0	0
$441$	1.47794	0.0703779
$442$	0	0
$443$	−9.80762	−0.465974	−0.232987	−	0.972480i	$-0.574850\pi$
$443$	−9.80762	−0.465974	−0.232987	+	0.972480i	$0.574850\pi$
$444$	0	0
$445$	53.2112	2.52245
$446$	0	0
$447$	−8.09144	−0.382712
$448$	0	0
$449$	10.5868	0.499624	0.249812	−	0.968294i	$-0.419631\pi$
$449$	10.5868	0.499624	0.249812	+	0.968294i	$0.419631\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−16.9622	−0.796955
$454$	0	0
$455$	0.0582421	0.00273043
$456$	0	0
$457$	−33.9855	−1.58978	−0.794888	−	0.606756i	$-0.792470\pi$
$457$	−33.9855	−1.58978	−0.794888	+	0.606756i	$0.792470\pi$
$458$	0	0
$459$	21.9409	1.02411
$460$	0	0
$461$	33.7435	1.57159	0.785796	−	0.618486i	$-0.212253\pi$
$461$	33.7435	1.57159	0.785796	+	0.618486i	$0.212253\pi$
$462$	0	0
$463$	24.3348	1.13094	0.565468	−	0.824770i	$-0.308696\pi$
$463$	24.3348	1.13094	0.565468	+	0.824770i	$0.308696\pi$
$464$	0	0
$465$	14.6778	0.680669
$466$	0	0
$467$	16.6668	0.771245	0.385623	−	0.922657i	$-0.373987\pi$
$467$	16.6668	0.771245	0.385623	+	0.922657i	$0.373987\pi$
$468$	0	0
$469$	1.32095	0.0609957
$470$	0	0
$471$	25.5944	1.17933
$472$	0	0
$473$	13.1647	0.605311
$474$	0	0
$475$	45.4869	2.08708
$476$	0	0
$477$	2.78357	0.127451
$478$	0	0
$479$	−16.8764	−0.771102	−0.385551	−	0.922686i	$-0.625989\pi$
$479$	−16.8764	−0.771102	−0.385551	+	0.922686i	$0.625989\pi$
$480$	0	0
$481$	0.201125	0.00917050
$482$	0	0
$483$	−5.86470	−0.266853
$484$	0	0
$485$	25.9875	1.18003
$486$	0	0
$487$	−28.5559	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$487$	−28.5559	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$488$	0	0
$489$	−8.55634	−0.386931
$490$	0	0
$491$	−22.2214	−1.00284	−0.501420	−	0.865204i	$-0.667189\pi$
$491$	−22.2214	−1.00284	−0.501420	+	0.865204i	$0.667189\pi$
$492$	0	0
$493$	−7.97159	−0.359023
$494$	0	0
$495$	3.53039	0.158679
$496$	0	0
$497$	9.90672	0.444377
$498$	0	0
$499$	21.5838	0.966224	0.483112	−	0.875559i	$-0.339506\pi$
$499$	21.5838	0.966224	0.483112	+	0.875559i	$0.339506\pi$
$500$	0	0
$501$	−19.9031	−0.889204
$502$	0	0
$503$	1.07649	0.0479982	0.0239991	−	0.999712i	$-0.492360\pi$
$503$	1.07649	0.0479982	0.0239991	+	0.999712i	$0.492360\pi$
$504$	0	0
$505$	−30.5893	−1.36121
$506$	0	0
$507$	21.6447	0.961275
$508$	0	0
$509$	−13.1937	−0.584802	−0.292401	−	0.956296i	$-0.594454\pi$
$509$	−13.1937	−0.584802	−0.292401	+	0.956296i	$0.594454\pi$
$510$	0	0
$511$	−5.11207	−0.226144
$512$	0	0
$513$	37.1260	1.63915
$514$	0	0
$515$	53.9439	2.37705
$516$	0	0
$517$	−19.2690	−0.847450
$518$	0	0
$519$	−11.0607	−0.485512
$520$	0	0
$521$	0.977316	0.0428170	0.0214085	−	0.999771i	$-0.493185\pi$
$521$	0.977316	0.0428170	0.0214085	+	0.999771i	$0.493185\pi$
$522$	0	0
$523$	41.2522	1.80383	0.901915	−	0.431913i	$-0.142161\pi$
$523$	41.2522	1.80383	0.901915	+	0.431913i	$0.142161\pi$
$524$	0	0
$525$	−7.80244	−0.340526
$526$	0	0
$527$	10.5741	0.460617
$528$	0	0
$529$	1.49427	0.0649681
$530$	0	0
$531$	−1.26715	−0.0549896
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−31.7920	−1.37449
$536$	0	0
$537$	−23.5413	−1.01588
$538$	0	0
$539$	−29.5930	−1.27466
$540$	0	0
$541$	4.78508	0.205726	0.102863	−	0.994696i	$-0.467200\pi$
$541$	4.78508	0.205726	0.102863	+	0.994696i	$0.467200\pi$
$542$	0	0
$543$	−40.4924	−1.73769
$544$	0	0
$545$	14.4326	0.618225
$546$	0	0
$547$	−28.7197	−1.22797	−0.613983	−	0.789319i	$-0.710434\pi$
$547$	−28.7197	−1.22797	−0.613983	+	0.789319i	$0.710434\pi$
$548$	0	0
$549$	−3.09035	−0.131893
$550$	0	0
$551$	−13.4886	−0.574635
$552$	0	0
$553$	0.294834	0.0125376
$554$	0	0
$555$	−47.4041	−2.01219
$556$	0	0
$557$	5.96535	0.252760	0.126380	−	0.991982i	$-0.459664\pi$
$557$	5.96535	0.252760	0.126380	+	0.991982i	$0.459664\pi$
$558$	0	0
$559$	−0.0694567	−0.00293771
$560$	0	0
$561$	−30.9802	−1.30799
$562$	0	0
$563$	22.6732	0.955561	0.477780	−	0.878479i	$-0.341442\pi$
$563$	22.6732	0.955561	0.477780	+	0.878479i	$0.341442\pi$
$564$	0	0
$565$	−55.2243	−2.32330
$566$	0	0
$567$	−5.88233	−0.247035
$568$	0	0
$569$	33.0068	1.38372	0.691858	−	0.722033i	$-0.256792\pi$
$569$	33.0068	1.38372	0.691858	+	0.722033i	$0.256792\pi$
$570$	0	0
$571$	47.3216	1.98035	0.990173	−	0.139848i	$-0.0446614\pi$
$571$	47.3216	1.98035	0.990173	+	0.139848i	$0.0446614\pi$
$572$	0	0
$573$	15.4476	0.645331
$574$	0	0
$575$	32.5873	1.35898
$576$	0	0
$577$	2.14793	0.0894195	0.0447097	−	0.999000i	$-0.485764\pi$
$577$	2.14793	0.0894195	0.0447097	+	0.999000i	$0.485764\pi$
$578$	0	0
$579$	−17.2919	−0.718629
$580$	0	0
$581$	−4.71099	−0.195445
$582$	0	0
$583$	−55.7359	−2.30835
$584$	0	0
$585$	−0.0186263	−0.000770104 0
$586$	0	0
$587$	4.85855	0.200534	0.100267	−	0.994961i	$-0.468030\pi$
$587$	4.85855	0.200534	0.100267	+	0.994961i	$0.468030\pi$
$588$	0	0
$589$	17.8924	0.737242
$590$	0	0
$591$	14.0459	0.577772
$592$	0	0
$593$	4.98676	0.204782	0.102391	−	0.994744i	$-0.467351\pi$
$593$	4.98676	0.204782	0.102391	+	0.994744i	$0.467351\pi$
$594$	0	0
$595$	−9.88941	−0.405426
$596$	0	0
$597$	−0.432888	−0.0177169
$598$	0	0
$599$	−16.9065	−0.690783	−0.345391	−	0.938459i	$-0.612254\pi$
$599$	−16.9065	−0.690783	−0.345391	+	0.938459i	$0.612254\pi$
$600$	0	0
$601$	20.8010	0.848489	0.424245	−	0.905548i	$-0.360540\pi$
$601$	20.8010	0.848489	0.424245	+	0.905548i	$0.360540\pi$
$602$	0	0
$603$	−0.422450	−0.0172035
$604$	0	0
$605$	−33.2502	−1.35181
$606$	0	0
$607$	−7.84719	−0.318508	−0.159254	−	0.987238i	$-0.550909\pi$
$607$	−7.84719	−0.318508	−0.159254	+	0.987238i	$0.550909\pi$
$608$	0	0
$609$	2.31373	0.0937569
$610$	0	0
$611$	0.101663	0.00411285
$612$	0	0
$613$	11.3549	0.458619	0.229309	−	0.973354i	$-0.426353\pi$
$613$	11.3549	0.458619	0.229309	+	0.973354i	$0.426353\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.58188	0.264976	0.132488	−	0.991185i	$-0.457703\pi$
$617$	6.58188	0.264976	0.132488	+	0.991185i	$0.457703\pi$
$618$	0	0
$619$	−34.8449	−1.40053	−0.700267	−	0.713881i	$-0.746936\pi$
$619$	−34.8449	−1.40053	−0.700267	+	0.713881i	$0.746936\pi$
$620$	0	0
$621$	26.5974	1.06732
$622$	0	0
$623$	−11.1264	−0.445768
$624$	0	0
$625$	−14.5677	−0.582706
$626$	0	0
$627$	−52.4212	−2.09350
$628$	0	0
$629$	−34.1506	−1.36167
$630$	0	0
$631$	−12.5028	−0.497730	−0.248865	−	0.968538i	$-0.580058\pi$
$631$	−12.5028	−0.497730	−0.248865	+	0.968538i	$0.580058\pi$
$632$	0	0
$633$	24.0412	0.955553
$634$	0	0
$635$	−46.7158	−1.85386
$636$	0	0
$637$	0.156133	0.00618620
$638$	0	0
$639$	−3.16825	−0.125334
$640$	0	0
$641$	0.588696	0.0232521	0.0116261	−	0.999932i	$-0.496299\pi$
$641$	0.588696	0.0232521	0.0116261	+	0.999932i	$0.496299\pi$
$642$	0	0
$643$	−21.9740	−0.866570	−0.433285	−	0.901257i	$-0.642646\pi$
$643$	−21.9740	−0.866570	−0.433285	+	0.901257i	$0.642646\pi$
$644$	0	0
$645$	16.3706	0.644592
$646$	0	0
$647$	21.2194	0.834221	0.417110	−	0.908856i	$-0.363043\pi$
$647$	21.2194	0.834221	0.417110	+	0.908856i	$0.363043\pi$
$648$	0	0
$649$	25.3724	0.995952
$650$	0	0
$651$	−3.06910	−0.120288
$652$	0	0
$653$	−28.3160	−1.10809	−0.554046	−	0.832486i	$-0.686917\pi$
$653$	−28.3160	−1.10809	−0.554046	+	0.832486i	$0.686917\pi$
$654$	0	0
$655$	−32.6810	−1.27695
$656$	0	0
$657$	1.63488	0.0637828
$658$	0	0
$659$	8.66565	0.337566	0.168783	−	0.985653i	$-0.446016\pi$
$659$	8.66565	0.337566	0.168783	+	0.985653i	$0.446016\pi$
$660$	0	0
$661$	−6.74466	−0.262337	−0.131169	−	0.991360i	$-0.541873\pi$
$661$	−6.74466	−0.262337	−0.131169	+	0.991360i	$0.541873\pi$
$662$	0	0
$663$	0.163452	0.00634794
$664$	0	0
$665$	−16.7337	−0.648907
$666$	0	0
$667$	−9.66341	−0.374169
$668$	0	0
$669$	24.6622	0.953497
$670$	0	0
$671$	61.8788	2.38880
$672$	0	0
$673$	2.11024	0.0813438	0.0406719	−	0.999173i	$-0.487050\pi$
$673$	2.11024	0.0813438	0.0406719	+	0.999173i	$0.487050\pi$
$674$	0	0
$675$	35.3854	1.36198
$676$	0	0
$677$	−15.2869	−0.587524	−0.293762	−	0.955879i	$-0.594907\pi$
$677$	−15.2869	−0.587524	−0.293762	+	0.955879i	$0.594907\pi$
$678$	0	0
$679$	−5.43393	−0.208535
$680$	0	0
$681$	10.2540	0.392933
$682$	0	0
$683$	27.1347	1.03828	0.519139	−	0.854690i	$-0.326253\pi$
$683$	27.1347	1.03828	0.519139	+	0.854690i	$0.326253\pi$
$684$	0	0
$685$	5.69575	0.217623
$686$	0	0
$687$	41.2123	1.57235
$688$	0	0
$689$	0.294063	0.0112029
$690$	0	0
$691$	−2.93674	−0.111719	−0.0558595	−	0.998439i	$-0.517790\pi$
$691$	−2.93674	−0.111719	−0.0558595	+	0.998439i	$0.517790\pi$
$692$	0	0
$693$	−0.738196	−0.0280417
$694$	0	0
$695$	−48.5067	−1.83997
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−37.9643	−1.43594
$700$	0	0
$701$	8.67049	0.327480	0.163740	−	0.986504i	$-0.447644\pi$
$701$	8.67049	0.327480	0.163740	+	0.986504i	$0.447644\pi$
$702$	0	0
$703$	−57.7859	−2.17944
$704$	0	0
$705$	−23.9615	−0.902443
$706$	0	0
$707$	6.39615	0.240552
$708$	0	0
$709$	−1.36075	−0.0511039	−0.0255519	−	0.999673i	$-0.508134\pi$
$709$	−1.36075	−0.0511039	−0.0255519	+	0.999673i	$0.508134\pi$
$710$	0	0
$711$	−0.0942904	−0.00353617
$712$	0	0
$713$	12.8183	0.480049
$714$	0	0
$715$	0.372958	0.0139479
$716$	0	0
$717$	37.6592	1.40641
$718$	0	0
$719$	27.0076	1.00722	0.503608	−	0.863933i	$-0.332006\pi$
$719$	27.0076	1.00722	0.503608	+	0.863933i	$0.332006\pi$
$720$	0	0
$721$	−11.2795	−0.420072
$722$	0	0
$723$	37.5281	1.39568
$724$	0	0
$725$	−12.8563	−0.477469
$726$	0	0
$727$	−28.5929	−1.06045	−0.530227	−	0.847856i	$-0.677893\pi$
$727$	−28.5929	−1.06045	−0.530227	+	0.847856i	$0.677893\pi$
$728$	0	0
$729$	28.7258	1.06392
$730$	0	0
$731$	11.7936	0.436203
$732$	0	0
$733$	18.1786	0.671441	0.335720	−	0.941962i	$-0.391020\pi$
$733$	18.1786	0.671441	0.335720	+	0.941962i	$0.391020\pi$
$734$	0	0
$735$	−36.7997	−1.35738
$736$	0	0
$737$	8.45881	0.311584
$738$	0	0
$739$	−15.7856	−0.580683	−0.290341	−	0.956923i	$-0.593769\pi$
$739$	−15.7856	−0.580683	−0.290341	+	0.956923i	$0.593769\pi$
$740$	0	0
$741$	0.276575	0.0101602
$742$	0	0
$743$	27.8154	1.02045	0.510224	−	0.860041i	$-0.329562\pi$
$743$	27.8154	1.02045	0.510224	+	0.860041i	$0.329562\pi$
$744$	0	0
$745$	−16.5400	−0.605978
$746$	0	0
$747$	1.50662	0.0551242
$748$	0	0
$749$	6.64764	0.242899
$750$	0	0
$751$	−7.54128	−0.275185	−0.137593	−	0.990489i	$-0.543936\pi$
$751$	−7.54128	−0.275185	−0.137593	+	0.990489i	$0.543936\pi$
$752$	0	0
$753$	−33.1620	−1.20849
$754$	0	0
$755$	−34.6730	−1.26188
$756$	0	0
$757$	43.2442	1.57174	0.785868	−	0.618394i	$-0.212216\pi$
$757$	43.2442	1.57174	0.785868	+	0.618394i	$0.212216\pi$
$758$	0	0
$759$	−37.5551	−1.36317
$760$	0	0
$761$	39.3062	1.42485	0.712425	−	0.701748i	$-0.247597\pi$
$761$	39.3062	1.42485	0.712425	+	0.701748i	$0.247597\pi$
$762$	0	0
$763$	−3.01783	−0.109253
$764$	0	0
$765$	3.16272	0.114348
$766$	0	0
$767$	−0.133865	−0.00483357
$768$	0	0
$769$	23.8652	0.860600	0.430300	−	0.902686i	$-0.358408\pi$
$769$	23.8652	0.860600	0.430300	+	0.902686i	$0.358408\pi$
$770$	0	0
$771$	11.3757	0.409686
$772$	0	0
$773$	−23.1050	−0.831030	−0.415515	−	0.909586i	$-0.636399\pi$
$773$	−23.1050	−0.831030	−0.415515	+	0.909586i	$0.636399\pi$
$774$	0	0
$775$	17.0535	0.612580
$776$	0	0
$777$	9.91210	0.355594
$778$	0	0
$779$	0	0
$780$	0	0
$781$	63.4385	2.27001
$782$	0	0
$783$	−10.4931	−0.374994
$784$	0	0
$785$	52.3183	1.86732
$786$	0	0
$787$	20.9478	0.746710	0.373355	−	0.927689i	$-0.378207\pi$
$787$	20.9478	0.746710	0.373355	+	0.927689i	$0.378207\pi$
$788$	0	0
$789$	−9.23468	−0.328763
$790$	0	0
$791$	11.5473	0.410574
$792$	0	0
$793$	−0.326472	−0.0115934
$794$	0	0
$795$	−69.3091	−2.45814
$796$	0	0
$797$	33.9615	1.20298	0.601489	−	0.798881i	$-0.294574\pi$
$797$	33.9615	1.20298	0.601489	+	0.798881i	$0.294574\pi$
$798$	0	0
$799$	−17.2622	−0.610694
$800$	0	0
$801$	3.55830	0.125726
$802$	0	0
$803$	−32.7356	−1.15521
$804$	0	0
$805$	−11.9882	−0.422530
$806$	0	0
$807$	2.94772	0.103765
$808$	0	0
$809$	6.25783	0.220013	0.110007	−	0.993931i	$-0.464913\pi$
$809$	6.25783	0.220013	0.110007	+	0.993931i	$0.464913\pi$
$810$	0	0
$811$	33.8680	1.18927	0.594633	−	0.803997i	$-0.297297\pi$
$811$	33.8680	1.18927	0.594633	+	0.803997i	$0.297297\pi$
$812$	0	0
$813$	28.2920	0.992244
$814$	0	0
$815$	−17.4903	−0.612658
$816$	0	0
$817$	19.9558	0.698167
$818$	0	0
$819$	0.00389472	0.000136093 0
$820$	0	0
$821$	−38.0107	−1.32658	−0.663292	−	0.748361i	$-0.730841\pi$
$821$	−38.0107	−1.32658	−0.663292	+	0.748361i	$0.730841\pi$
$822$	0	0
$823$	−12.3387	−0.430101	−0.215050	−	0.976603i	$-0.568992\pi$
$823$	−12.3387	−0.430101	−0.215050	+	0.976603i	$0.568992\pi$
$824$	0	0
$825$	−49.9636	−1.73951
$826$	0	0
$827$	39.9202	1.38816	0.694080	−	0.719898i	$-0.255811\pi$
$827$	39.9202	1.38816	0.694080	+	0.719898i	$0.255811\pi$
$828$	0	0
$829$	12.9343	0.449227	0.224614	−	0.974448i	$-0.427888\pi$
$829$	12.9343	0.449227	0.224614	+	0.974448i	$0.427888\pi$
$830$	0	0
$831$	53.3975	1.85234
$832$	0	0
$833$	−26.5110	−0.918553
$834$	0	0
$835$	−40.6845	−1.40795
$836$	0	0
$837$	13.9189	0.481108
$838$	0	0
$839$	−20.3297	−0.701860	−0.350930	−	0.936402i	$-0.614135\pi$
$839$	−20.3297	−0.701860	−0.350930	+	0.936402i	$0.614135\pi$
$840$	0	0
$841$	−25.1876	−0.868539
$842$	0	0
$843$	2.59765	0.0894678
$844$	0	0
$845$	44.2447	1.52206
$846$	0	0
$847$	6.95254	0.238892
$848$	0	0
$849$	−50.6307	−1.73764
$850$	0	0
$851$	−41.3984	−1.41912
$852$	0	0
$853$	39.5033	1.35257	0.676284	−	0.736641i	$-0.263589\pi$
$853$	39.5033	1.35257	0.676284	+	0.736641i	$0.263589\pi$
$854$	0	0
$855$	5.35159	0.183021
$856$	0	0
$857$	−11.6287	−0.397229	−0.198615	−	0.980078i	$-0.563644\pi$
$857$	−11.6287	−0.397229	−0.198615	+	0.980078i	$0.563644\pi$
$858$	0	0
$859$	3.37820	0.115263	0.0576314	−	0.998338i	$-0.481645\pi$
$859$	3.37820	0.115263	0.0576314	+	0.998338i	$0.481645\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.5178	1.17500	0.587500	−	0.809224i	$-0.300112\pi$
$863$	34.5178	1.17500	0.587500	+	0.809224i	$0.300112\pi$
$864$	0	0
$865$	−22.6096	−0.768749
$866$	0	0
$867$	0.552095	0.0187501
$868$	0	0
$869$	1.88799	0.0640458
$870$	0	0
$871$	−0.0446287	−0.00151218
$872$	0	0
$873$	1.73782	0.0588161
$874$	0	0
$875$	−3.83785	−0.129743
$876$	0	0
$877$	10.1304	0.342080	0.171040	−	0.985264i	$-0.445287\pi$
$877$	10.1304	0.342080	0.171040	+	0.985264i	$0.445287\pi$
$878$	0	0
$879$	1.13228	0.0381909
$880$	0	0
$881$	41.5062	1.39838	0.699190	−	0.714936i	$-0.253544\pi$
$881$	41.5062	1.39838	0.699190	+	0.714936i	$0.253544\pi$
$882$	0	0
$883$	−8.83275	−0.297246	−0.148623	−	0.988894i	$-0.547484\pi$
$883$	−8.83275	−0.297246	−0.148623	+	0.988894i	$0.547484\pi$
$884$	0	0
$885$	31.5512	1.06058
$886$	0	0
$887$	−4.53003	−0.152103	−0.0760517	−	0.997104i	$-0.524231\pi$
$887$	−4.53003	−0.152103	−0.0760517	+	0.997104i	$0.524231\pi$
$888$	0	0
$889$	9.76816	0.327614
$890$	0	0
$891$	−37.6680	−1.26193
$892$	0	0
$893$	−29.2092	−0.977449
$894$	0	0
$895$	−48.1216	−1.60853
$896$	0	0
$897$	0.198141	0.00661573
$898$	0	0
$899$	−5.05703	−0.168661
$900$	0	0
$901$	−49.9313	−1.66345
$902$	0	0
$903$	−3.42305	−0.113912
$904$	0	0
$905$	−82.7718	−2.75143
$906$	0	0
$907$	−6.59984	−0.219144	−0.109572	−	0.993979i	$-0.534948\pi$
$907$	−6.59984	−0.219144	−0.109572	+	0.993979i	$0.534948\pi$
$908$	0	0
$909$	−2.04554	−0.0678464
$910$	0	0
$911$	7.45685	0.247056	0.123528	−	0.992341i	$-0.460579\pi$
$911$	7.45685	0.247056	0.123528	+	0.992341i	$0.460579\pi$
$912$	0	0
$913$	−30.1673	−0.998391
$914$	0	0
$915$	76.9479	2.54382
$916$	0	0
$917$	6.83351	0.225662
$918$	0	0
$919$	4.69285	0.154803	0.0774014	−	0.997000i	$-0.475338\pi$
$919$	4.69285	0.154803	0.0774014	+	0.997000i	$0.475338\pi$
$920$	0	0
$921$	−46.8715	−1.54447
$922$	0	0
$923$	−0.334702	−0.0110168
$924$	0	0
$925$	−55.0767	−1.81091
$926$	0	0
$927$	3.60729	0.118479
$928$	0	0
$929$	−6.14293	−0.201543	−0.100771	−	0.994910i	$-0.532131\pi$
$929$	−6.14293	−0.201543	−0.100771	+	0.994910i	$0.532131\pi$
$930$	0	0
$931$	−44.8590	−1.47019
$932$	0	0
$933$	−17.7053	−0.579645
$934$	0	0
$935$	−63.3277	−2.07104
$936$	0	0
$937$	−22.3585	−0.730420	−0.365210	−	0.930925i	$-0.619003\pi$
$937$	−22.3585	−0.730420	−0.365210	+	0.930925i	$0.619003\pi$
$938$	0	0
$939$	−37.3167	−1.21778
$940$	0	0
$941$	12.2659	0.399858	0.199929	−	0.979810i	$-0.435929\pi$
$941$	12.2659	0.399858	0.199929	+	0.979810i	$0.435929\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−13.0176	−0.423462
$946$	0	0
$947$	−14.5922	−0.474184	−0.237092	−	0.971487i	$-0.576194\pi$
$947$	−14.5922	−0.474184	−0.237092	+	0.971487i	$0.576194\pi$
$948$	0	0
$949$	0.172713	0.00560649
$950$	0	0
$951$	16.8325	0.545831
$952$	0	0
$953$	−13.2497	−0.429200	−0.214600	−	0.976702i	$-0.568845\pi$
$953$	−13.2497	−0.429200	−0.214600	+	0.976702i	$0.568845\pi$
$954$	0	0
$955$	31.5769	1.02180
$956$	0	0
$957$	14.8161	0.478938
$958$	0	0
$959$	−1.19097	−0.0384584
$960$	0	0
$961$	−24.2920	−0.783612
$962$	0	0
$963$	−2.12597	−0.0685084
$964$	0	0
$965$	−35.3470	−1.13786
$966$	0	0
$967$	−10.0047	−0.321729	−0.160864	−	0.986977i	$-0.551428\pi$
$967$	−10.0047	−0.321729	−0.160864	+	0.986977i	$0.551428\pi$
$968$	0	0
$969$	−46.9618	−1.50863
$970$	0	0
$971$	36.4920	1.17108	0.585542	−	0.810642i	$-0.300882\pi$
$971$	36.4920	1.17108	0.585542	+	0.810642i	$0.300882\pi$
$972$	0	0
$973$	10.1427	0.325158
$974$	0	0
$975$	0.263608	0.00844221
$976$	0	0
$977$	48.0072	1.53589	0.767943	−	0.640518i	$-0.221280\pi$
$977$	48.0072	1.53589	0.767943	+	0.640518i	$0.221280\pi$
$978$	0	0
$979$	−71.2486	−2.27711
$980$	0	0
$981$	0.965127	0.0308141
$982$	0	0
$983$	54.5159	1.73879	0.869394	−	0.494119i	$-0.164509\pi$
$983$	54.5159	1.73879	0.869394	+	0.494119i	$0.164509\pi$
$984$	0	0
$985$	28.7117	0.914831
$986$	0	0
$987$	5.01030	0.159480
$988$	0	0
$989$	14.2966	0.454605
$990$	0	0
$991$	−47.5725	−1.51119	−0.755595	−	0.655039i	$-0.772652\pi$
$991$	−47.5725	−1.51119	−0.755595	+	0.655039i	$0.772652\pi$
$992$	0	0
$993$	2.97787	0.0945000
$994$	0	0
$995$	−0.884880	−0.0280526
$996$	0	0
$997$	−35.5951	−1.12731	−0.563654	−	0.826011i	$-0.690605\pi$
$997$	−35.5951	−1.12731	−0.563654	+	0.826011i	$0.690605\pi$
$998$	0	0
$999$	−44.9531	−1.42225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6724.2.a.j.1.5		16
41.5	even	20		164.2.k.a.25.2	✓	16
41.33	even	20		164.2.k.a.105.3	yes	16
41.40	even	2	inner	6724.2.a.j.1.12		16
123.5	odd	20		1476.2.bb.b.1009.4		16
123.74	odd	20		1476.2.bb.b.433.4		16
164.87	odd	20		656.2.be.e.353.3		16
164.115	odd	20		656.2.be.e.433.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.k.a.25.2	✓	16	41.5	even	20
164.2.k.a.105.3	yes	16	41.33	even	20
656.2.be.e.353.3		16	164.87	odd	20
656.2.be.e.433.2		16	164.115	odd	20
1476.2.bb.b.433.4		16	123.74	odd	20
1476.2.bb.b.1009.4		16	123.5	odd	20
6724.2.a.j.1.5		16	1.1	even	1	trivial
6724.2.a.j.1.12		16	41.40	even	2	inner

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:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)

\(f(q)\)	\(=\)	\(q-1.66505 q^{3} -3.40359 q^{5} +0.711682 q^{7} -0.227602 q^{9} +O(q^{10})\)	\(q-1.66505 q^{3} -3.40359 q^{5} +0.711682 q^{7} -0.227602 q^{9} +4.55732 q^{11} -0.0240444 q^{13} +5.66715 q^{15} +4.08270 q^{17} +6.90828 q^{19} -1.18499 q^{21} +4.94917 q^{23} +6.58440 q^{25} +5.37412 q^{27} -1.95253 q^{29} +2.58999 q^{31} -7.58817 q^{33} -2.42227 q^{35} -8.36472 q^{37} +0.0400352 q^{39} +2.88868 q^{43} +0.774663 q^{45} -4.22814 q^{47} -6.49351 q^{49} -6.79790 q^{51} -12.2300 q^{53} -15.5112 q^{55} -11.5027 q^{57} +5.56739 q^{59} +13.5779 q^{61} -0.161980 q^{63} +0.0818373 q^{65} +1.85609 q^{67} -8.24062 q^{69} +13.9201 q^{71} -7.18307 q^{73} -10.9634 q^{75} +3.24336 q^{77} +0.414278 q^{79} -8.26539 q^{81} -6.61952 q^{83} -13.8958 q^{85} +3.25107 q^{87} -15.6339 q^{89} -0.0171120 q^{91} -4.31246 q^{93} -23.5129 q^{95} -7.63532 q^{97} -1.03725 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) \) 16 * q + 6 * q^5 + 30 * q^9	\( 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) \) 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

Introduction

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