LMFDB - Embedded newform 4508.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4508,2,Mod(1,4508)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4508, 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("4508.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4508 = 2^{2} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4508.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:	$35.9965612312$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.6963152.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 $ x^5 - 2x^4 - 10x^3 + 10x^2 + 29x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 644)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.76321$ of defining polynomial
Character	$\chi$	$=$	4508.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76321 q^{3} +3.11657 q^{5} +0.108911 q^{9} +O(q^{10})$	$q+1.76321 q^{3} +3.11657 q^{5} +0.108911 q^{9} -5.23940 q^{11} -6.34832 q^{13} +5.49516 q^{15} +0.585105 q^{17} -8.48889 q^{19} -1.00000 q^{23} +4.71298 q^{25} -5.09760 q^{27} -1.59780 q^{29} +5.97640 q^{31} -9.23817 q^{33} +10.0153 q^{37} -11.1934 q^{39} -1.17811 q^{41} -2.96247 q^{43} +0.339428 q^{45} +2.55043 q^{47} +1.03166 q^{51} +8.27107 q^{53} -16.3289 q^{55} -14.9677 q^{57} -14.6004 q^{59} -6.15410 q^{61} -19.7849 q^{65} -8.52015 q^{67} -1.76321 q^{69} +8.38625 q^{71} +0.358396 q^{73} +8.30998 q^{75} -11.0216 q^{79} -9.31487 q^{81} +10.7446 q^{83} +1.82352 q^{85} -2.81726 q^{87} -18.1369 q^{89} +10.5376 q^{93} -26.4562 q^{95} +8.38723 q^{97} -0.570629 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9	$ 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} - 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} - 24 q^{33} - 4 q^{37} + 3 q^{39} - q^{41} - 8 q^{43} + 16 q^{45} - 5 q^{47} - 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} - 20 q^{61} - 12 q^{65} - 12 q^{67} + 3 q^{69} + 9 q^{71} + 9 q^{73} - 35 q^{75} - 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} - 32 q^{89} - 15 q^{93} - 36 q^{95} - 4 q^{97} + 28 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9 + 2 * q^11 - 13 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 - 5 * q^23 + 19 * q^25 - 15 * q^27 + 13 * q^29 + 3 * q^31 - 24 * q^33 - 4 * q^37 + 3 * q^39 - q^41 - 8 * q^43 + 16 * q^45 - 5 * q^47 - 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 12 * q^59 - 20 * q^61 - 12 * q^65 - 12 * q^67 + 3 * q^69 + 9 * q^71 + 9 * q^73 - 35 * q^75 - 8 * q^79 - 11 * q^81 + 28 * q^83 + 16 * q^85 + 15 * q^87 - 32 * q^89 - 15 * q^93 - 36 * q^95 - 4 * q^97 + 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.76321	1.01799	0.508995	−	0.860769i	$-0.330017\pi$
$3$	1.76321	1.01799	0.508995	+	0.860769i	$0.330017\pi$
$4$	0	0
$5$	3.11657	1.39377	0.696885	−	0.717183i	$-0.254569\pi$
$5$	3.11657	1.39377	0.696885	+	0.717183i	$0.254569\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.108911	0.0363037
$10$	0	0
$11$	−5.23940	−1.57974	−0.789870	−	0.613274i	$-0.789852\pi$
$11$	−5.23940	−1.57974	−0.789870	+	0.613274i	$0.789852\pi$
$12$	0	0
$13$	−6.34832	−1.76071	−0.880353	−	0.474319i	$-0.842694\pi$
$13$	−6.34832	−1.76071	−0.880353	+	0.474319i	$0.842694\pi$
$14$	0	0
$15$	5.49516	1.41884
$16$	0	0
$17$	0.585105	0.141909	0.0709544	−	0.997480i	$-0.477396\pi$
$17$	0.585105	0.141909	0.0709544	+	0.997480i	$0.477396\pi$
$18$	0	0
$19$	−8.48889	−1.94748	−0.973742	−	0.227653i	$-0.926895\pi$
$19$	−8.48889	−1.94748	−0.973742	+	0.227653i	$0.926895\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	4.71298	0.942597
$26$	0	0
$27$	−5.09760	−0.981033
$28$	0	0
$29$	−1.59780	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$29$	−1.59780	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$30$	0	0
$31$	5.97640	1.07339	0.536696	−	0.843776i	$-0.319672\pi$
$31$	5.97640	1.07339	0.536696	+	0.843776i	$0.319672\pi$
$32$	0	0
$33$	−9.23817	−1.60816
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0153	1.64651	0.823253	−	0.567674i	$-0.192157\pi$
$37$	10.0153	1.64651	0.823253	+	0.567674i	$0.192157\pi$
$38$	0	0
$39$	−11.1934	−1.79238
$40$	0	0
$41$	−1.17811	−0.183989	−0.0919946	−	0.995760i	$-0.529324\pi$
$41$	−1.17811	−0.183989	−0.0919946	+	0.995760i	$0.529324\pi$
$42$	0	0
$43$	−2.96247	−0.451772	−0.225886	−	0.974154i	$-0.572528\pi$
$43$	−2.96247	−0.451772	−0.225886	+	0.974154i	$0.572528\pi$
$44$	0	0
$45$	0.339428	0.0505990
$46$	0	0
$47$	2.55043	0.372018	0.186009	−	0.982548i	$-0.440445\pi$
$47$	2.55043	0.372018	0.186009	+	0.982548i	$0.440445\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.03166	0.144462
$52$	0	0
$53$	8.27107	1.13612	0.568059	−	0.822988i	$-0.307694\pi$
$53$	8.27107	1.13612	0.568059	+	0.822988i	$0.307694\pi$
$54$	0	0
$55$	−16.3289	−2.20180
$56$	0	0
$57$	−14.9677	−1.98252
$58$	0	0
$59$	−14.6004	−1.90081	−0.950406	−	0.311012i	$-0.899332\pi$
$59$	−14.6004	−1.90081	−0.950406	+	0.311012i	$0.899332\pi$
$60$	0	0
$61$	−6.15410	−0.787951	−0.393976	−	0.919121i	$-0.628901\pi$
$61$	−6.15410	−0.787951	−0.393976	+	0.919121i	$0.628901\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−19.7849	−2.45402
$66$	0	0
$67$	−8.52015	−1.04090	−0.520451	−	0.853892i	$-0.674236\pi$
$67$	−8.52015	−1.04090	−0.520451	+	0.853892i	$0.674236\pi$
$68$	0	0
$69$	−1.76321	−0.212266
$70$	0	0
$71$	8.38625	0.995265	0.497632	−	0.867388i	$-0.334203\pi$
$71$	8.38625	0.995265	0.497632	+	0.867388i	$0.334203\pi$
$72$	0	0
$73$	0.358396	0.0419471	0.0209735	−	0.999780i	$-0.493323\pi$
$73$	0.358396	0.0419471	0.0209735	+	0.999780i	$0.493323\pi$
$74$	0	0
$75$	8.30998	0.959554
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.0216	−1.24002	−0.620012	−	0.784592i	$-0.712872\pi$
$79$	−11.0216	−1.24002	−0.620012	+	0.784592i	$0.712872\pi$
$80$	0	0
$81$	−9.31487	−1.03499
$82$	0	0
$83$	10.7446	1.17938	0.589689	−	0.807630i	$-0.299250\pi$
$83$	10.7446	1.17938	0.589689	+	0.807630i	$0.299250\pi$
$84$	0	0
$85$	1.82352	0.197788
$86$	0	0
$87$	−2.81726	−0.302042
$88$	0	0
$89$	−18.1369	−1.92251	−0.961255	−	0.275662i	$-0.911103\pi$
$89$	−18.1369	−1.92251	−0.961255	+	0.275662i	$0.911103\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	10.5376	1.09270
$94$	0	0
$95$	−26.4562	−2.71435
$96$	0	0
$97$	8.38723	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$97$	8.38723	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$98$	0	0
$99$	−0.570629	−0.0573503
$100$	0	0
$101$	−6.79708	−0.676335	−0.338168	−	0.941086i	$-0.609807\pi$
$101$	−6.79708	−0.676335	−0.338168	+	0.941086i	$0.609807\pi$
$102$	0	0
$103$	9.75955	0.961637	0.480819	−	0.876820i	$-0.340340\pi$
$103$	9.75955	0.961637	0.480819	+	0.876820i	$0.340340\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.96247	−0.673087	−0.336544	−	0.941668i	$-0.609258\pi$
$107$	−6.96247	−0.673087	−0.336544	+	0.941668i	$0.609258\pi$
$108$	0	0
$109$	−2.03794	−0.195199	−0.0975994	−	0.995226i	$-0.531116\pi$
$109$	−2.03794	−0.195199	−0.0975994	+	0.995226i	$0.531116\pi$
$110$	0	0
$111$	17.6591	1.67613
$112$	0	0
$113$	3.20815	0.301797	0.150898	−	0.988549i	$-0.451783\pi$
$113$	3.20815	0.301797	0.150898	+	0.988549i	$0.451783\pi$
$114$	0	0
$115$	−3.11657	−0.290621
$116$	0	0
$117$	−0.691401	−0.0639201
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.4514	1.49558
$122$	0	0
$123$	−2.07725	−0.187299
$124$	0	0
$125$	−0.894506	−0.0800070
$126$	0	0
$127$	4.83720	0.429232	0.214616	−	0.976698i	$-0.431150\pi$
$127$	4.83720	0.429232	0.214616	+	0.976698i	$0.431150\pi$
$128$	0	0
$129$	−5.22346	−0.459900
$130$	0	0
$131$	−13.9917	−1.22246	−0.611231	−	0.791453i	$-0.709325\pi$
$131$	−13.9917	−1.22246	−0.611231	+	0.791453i	$0.709325\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−15.8870	−1.36734
$136$	0	0
$137$	3.65910	0.312618	0.156309	−	0.987708i	$-0.450040\pi$
$137$	3.65910	0.312618	0.156309	+	0.987708i	$0.450040\pi$
$138$	0	0
$139$	−16.4852	−1.39826	−0.699130	−	0.714995i	$-0.746429\pi$
$139$	−16.4852	−1.39826	−0.699130	+	0.714995i	$0.746429\pi$
$140$	0	0
$141$	4.49694	0.378711
$142$	0	0
$143$	33.2614	2.78146
$144$	0	0
$145$	−4.97965	−0.413537
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.7596	−1.29107	−0.645536	−	0.763729i	$-0.723366\pi$
$149$	−15.7596	−1.29107	−0.645536	+	0.763729i	$0.723366\pi$
$150$	0	0
$151$	6.13049	0.498892	0.249446	−	0.968389i	$-0.419751\pi$
$151$	6.13049	0.498892	0.249446	+	0.968389i	$0.419751\pi$
$152$	0	0
$153$	0.0637243	0.00515181
$154$	0	0
$155$	18.6258	1.49606
$156$	0	0
$157$	−3.65811	−0.291949	−0.145974	−	0.989288i	$-0.546632\pi$
$157$	−3.65811	−0.291949	−0.145974	+	0.989288i	$0.546632\pi$
$158$	0	0
$159$	14.5836	1.15656
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.9381	−1.40502	−0.702509	−	0.711675i	$-0.747937\pi$
$163$	−17.9381	−1.40502	−0.702509	+	0.711675i	$0.747937\pi$
$164$	0	0
$165$	−28.7914	−2.24141
$166$	0	0
$167$	−0.419935	−0.0324956	−0.0162478	−	0.999868i	$-0.505172\pi$
$167$	−0.419935	−0.0324956	−0.0162478	+	0.999868i	$0.505172\pi$
$168$	0	0
$169$	27.3011	2.10009
$170$	0	0
$171$	−0.924533	−0.0707008
$172$	0	0
$173$	−2.48889	−0.189227	−0.0946134	−	0.995514i	$-0.530161\pi$
$173$	−2.48889	−0.189227	−0.0946134	+	0.995514i	$0.530161\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−25.7436	−1.93501
$178$	0	0
$179$	−2.64201	−0.197473	−0.0987365	−	0.995114i	$-0.531480\pi$
$179$	−2.64201	−0.197473	−0.0987365	+	0.995114i	$0.531480\pi$
$180$	0	0
$181$	−1.38950	−0.103281	−0.0516405	−	0.998666i	$-0.516445\pi$
$181$	−1.38950	−0.103281	−0.0516405	+	0.998666i	$0.516445\pi$
$182$	0	0
$183$	−10.8510	−0.802127
$184$	0	0
$185$	31.2134	2.29485
$186$	0	0
$187$	−3.06560	−0.224179
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.0682	0.945578	0.472789	−	0.881176i	$-0.343247\pi$
$191$	13.0682	0.945578	0.472789	+	0.881176i	$0.343247\pi$
$192$	0	0
$193$	21.5964	1.55454	0.777270	−	0.629167i	$-0.216604\pi$
$193$	21.5964	1.55454	0.777270	+	0.629167i	$0.216604\pi$
$194$	0	0
$195$	−34.8850	−2.49817
$196$	0	0
$197$	10.2410	0.729643	0.364822	−	0.931077i	$-0.381130\pi$
$197$	10.2410	0.729643	0.364822	+	0.931077i	$0.381130\pi$
$198$	0	0
$199$	21.6744	1.53646	0.768229	−	0.640175i	$-0.221138\pi$
$199$	21.6744	1.53646	0.768229	+	0.640175i	$0.221138\pi$
$200$	0	0
$201$	−15.0228	−1.05963
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.67164	−0.256439
$206$	0	0
$207$	−0.108911	−0.00756983
$208$	0	0
$209$	44.4767	3.07652
$210$	0	0
$211$	15.8499	1.09115	0.545577	−	0.838061i	$-0.316311\pi$
$211$	15.8499	1.09115	0.545577	+	0.838061i	$0.316311\pi$
$212$	0	0
$213$	14.7867	1.01317
$214$	0	0
$215$	−9.23273	−0.629667
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.631927	0.0427017
$220$	0	0
$221$	−3.71443	−0.249860
$222$	0	0
$223$	−1.04860	−0.0702197	−0.0351098	−	0.999383i	$-0.511178\pi$
$223$	−1.04860	−0.0702197	−0.0351098	+	0.999383i	$0.511178\pi$
$224$	0	0
$225$	0.513296	0.0342197
$226$	0	0
$227$	−27.9455	−1.85481	−0.927403	−	0.374063i	$-0.877964\pi$
$227$	−27.9455	−1.85481	−0.927403	+	0.374063i	$0.877964\pi$
$228$	0	0
$229$	17.8239	1.17783	0.588917	−	0.808193i	$-0.299554\pi$
$229$	17.8239	1.17783	0.588917	+	0.808193i	$0.299554\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.53528	0.624677	0.312339	−	0.949971i	$-0.398888\pi$
$233$	9.53528	0.624677	0.312339	+	0.949971i	$0.398888\pi$
$234$	0	0
$235$	7.94858	0.518508
$236$	0	0
$237$	−19.4334	−1.26233
$238$	0	0
$239$	13.6190	0.880939	0.440469	−	0.897768i	$-0.354812\pi$
$239$	13.6190	0.880939	0.440469	+	0.897768i	$0.354812\pi$
$240$	0	0
$241$	−16.1115	−1.03783	−0.518917	−	0.854824i	$-0.673665\pi$
$241$	−16.1115	−1.03783	−0.518917	+	0.854824i	$0.673665\pi$
$242$	0	0
$243$	−1.13128	−0.0725718
$244$	0	0
$245$	0	0
$246$	0	0
$247$	53.8901	3.42895
$248$	0	0
$249$	18.9451	1.20060
$250$	0	0
$251$	−17.7822	−1.12240	−0.561201	−	0.827680i	$-0.689660\pi$
$251$	−17.7822	−1.12240	−0.561201	+	0.827680i	$0.689660\pi$
$252$	0	0
$253$	5.23940	0.329399
$254$	0	0
$255$	3.21525	0.201347
$256$	0	0
$257$	11.3733	0.709447	0.354724	−	0.934971i	$-0.384575\pi$
$257$	11.3733	0.709447	0.354724	+	0.934971i	$0.384575\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.174018	−0.0107714
$262$	0	0
$263$	8.22726	0.507315	0.253657	−	0.967294i	$-0.418366\pi$
$263$	8.22726	0.507315	0.253657	+	0.967294i	$0.418366\pi$
$264$	0	0
$265$	25.7773	1.58349
$266$	0	0
$267$	−31.9792	−1.95710
$268$	0	0
$269$	−3.31078	−0.201862	−0.100931	−	0.994893i	$-0.532182\pi$
$269$	−3.31078	−0.201862	−0.100931	+	0.994893i	$0.532182\pi$
$270$	0	0
$271$	−24.3447	−1.47883	−0.739416	−	0.673248i	$-0.764898\pi$
$271$	−24.3447	−1.47883	−0.739416	+	0.673248i	$0.764898\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.6932	−1.48906
$276$	0	0
$277$	−14.1492	−0.850144	−0.425072	−	0.905160i	$-0.639751\pi$
$277$	−14.1492	−0.850144	−0.425072	+	0.905160i	$0.639751\pi$
$278$	0	0
$279$	0.650895	0.0389681
$280$	0	0
$281$	10.2231	0.609856	0.304928	−	0.952375i	$-0.401368\pi$
$281$	10.2231	0.609856	0.304928	+	0.952375i	$0.401368\pi$
$282$	0	0
$283$	−5.80480	−0.345060	−0.172530	−	0.985004i	$-0.555194\pi$
$283$	−5.80480	−0.345060	−0.172530	+	0.985004i	$0.555194\pi$
$284$	0	0
$285$	−46.6478	−2.76318
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.6577	−0.979862
$290$	0	0
$291$	14.7885	0.866914
$292$	0	0
$293$	31.1117	1.81757	0.908783	−	0.417269i	$-0.137013\pi$
$293$	31.1117	1.81757	0.908783	+	0.417269i	$0.137013\pi$
$294$	0	0
$295$	−45.5032	−2.64930
$296$	0	0
$297$	26.7084	1.54978
$298$	0	0
$299$	6.34832	0.367133
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.9847	−0.688502
$304$	0	0
$305$	−19.1797	−1.09822
$306$	0	0
$307$	4.55784	0.260130	0.130065	−	0.991505i	$-0.458481\pi$
$307$	4.55784	0.260130	0.130065	+	0.991505i	$0.458481\pi$
$308$	0	0
$309$	17.2081	0.978937
$310$	0	0
$311$	−1.08872	−0.0617358	−0.0308679	−	0.999523i	$-0.509827\pi$
$311$	−1.08872	−0.0617358	−0.0308679	+	0.999523i	$0.509827\pi$
$312$	0	0
$313$	−16.9086	−0.955731	−0.477866	−	0.878433i	$-0.658589\pi$
$313$	−16.9086	−0.955731	−0.477866	+	0.878433i	$0.658589\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.26439	−0.239512	−0.119756	−	0.992803i	$-0.538211\pi$
$317$	−4.26439	−0.239512	−0.119756	+	0.992803i	$0.538211\pi$
$318$	0	0
$319$	8.37152	0.468715
$320$	0	0
$321$	−12.2763	−0.685196
$322$	0	0
$323$	−4.96689	−0.276365
$324$	0	0
$325$	−29.9195	−1.65964
$326$	0	0
$327$	−3.59331	−0.198710
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.9450	0.931380	0.465690	−	0.884948i	$-0.345806\pi$
$331$	16.9450	0.931380	0.465690	+	0.884948i	$0.345806\pi$
$332$	0	0
$333$	1.09078	0.0597742
$334$	0	0
$335$	−26.5536	−1.45078
$336$	0	0
$337$	10.7547	0.585847	0.292924	−	0.956136i	$-0.405372\pi$
$337$	10.7547	0.585847	0.292924	+	0.956136i	$0.405372\pi$
$338$	0	0
$339$	5.65663	0.307226
$340$	0	0
$341$	−31.3128	−1.69568
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.49516	−0.295850
$346$	0	0
$347$	30.2105	1.62178	0.810892	−	0.585195i	$-0.198982\pi$
$347$	30.2105	1.62178	0.810892	+	0.585195i	$0.198982\pi$
$348$	0	0
$349$	20.2309	1.08294	0.541469	−	0.840721i	$-0.317868\pi$
$349$	20.2309	1.08294	0.541469	+	0.840721i	$0.317868\pi$
$350$	0	0
$351$	32.3612	1.72731
$352$	0	0
$353$	−14.0953	−0.750218	−0.375109	−	0.926981i	$-0.622395\pi$
$353$	−14.0953	−0.750218	−0.375109	+	0.926981i	$0.622395\pi$
$354$	0	0
$355$	26.1363	1.38717
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.0086	−0.844903	−0.422452	−	0.906385i	$-0.638830\pi$
$359$	−16.0086	−0.844903	−0.422452	+	0.906385i	$0.638830\pi$
$360$	0	0
$361$	53.0612	2.79270
$362$	0	0
$363$	29.0072	1.52248
$364$	0	0
$365$	1.11696	0.0584646
$366$	0	0
$367$	28.8277	1.50479	0.752397	−	0.658710i	$-0.228898\pi$
$367$	28.8277	1.50479	0.752397	+	0.658710i	$0.228898\pi$
$368$	0	0
$369$	−0.128309	−0.00667948
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−18.6273	−0.964484	−0.482242	−	0.876038i	$-0.660177\pi$
$373$	−18.6273	−0.964484	−0.482242	+	0.876038i	$0.660177\pi$
$374$	0	0
$375$	−1.57720	−0.0814464
$376$	0	0
$377$	10.1433	0.522409
$378$	0	0
$379$	14.8138	0.760936	0.380468	−	0.924794i	$-0.375763\pi$
$379$	14.8138	0.760936	0.380468	+	0.924794i	$0.375763\pi$
$380$	0	0
$381$	8.52901	0.436954
$382$	0	0
$383$	−0.611968	−0.0312701	−0.0156351	−	0.999878i	$-0.504977\pi$
$383$	−0.611968	−0.0312701	−0.0156351	+	0.999878i	$0.504977\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.322645	−0.0164010
$388$	0	0
$389$	11.4514	0.580607	0.290303	−	0.956935i	$-0.406244\pi$
$389$	11.4514	0.580607	0.290303	+	0.956935i	$0.406244\pi$
$390$	0	0
$391$	−0.585105	−0.0295900
$392$	0	0
$393$	−24.6703	−1.24445
$394$	0	0
$395$	−34.3495	−1.72831
$396$	0	0
$397$	−15.9228	−0.799140	−0.399570	−	0.916703i	$-0.630841\pi$
$397$	−15.9228	−0.799140	−0.399570	+	0.916703i	$0.630841\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−35.1157	−1.75359	−0.876797	−	0.480861i	$-0.840324\pi$
$401$	−35.1157	−1.75359	−0.876797	+	0.480861i	$0.840324\pi$
$402$	0	0
$403$	−37.9400	−1.88993
$404$	0	0
$405$	−29.0304	−1.44253
$406$	0	0
$407$	−52.4743	−2.60105
$408$	0	0
$409$	−24.6726	−1.21998	−0.609991	−	0.792408i	$-0.708827\pi$
$409$	−24.6726	−1.21998	−0.609991	+	0.792408i	$0.708827\pi$
$410$	0	0
$411$	6.45176	0.318242
$412$	0	0
$413$	0	0
$414$	0	0
$415$	33.4864	1.64378
$416$	0	0
$417$	−29.0669	−1.42341
$418$	0	0
$419$	−23.0411	−1.12563	−0.562816	−	0.826582i	$-0.690282\pi$
$419$	−23.0411	−1.12563	−0.562816	+	0.826582i	$0.690282\pi$
$420$	0	0
$421$	11.8048	0.575331	0.287665	−	0.957731i	$-0.407121\pi$
$421$	11.8048	0.575331	0.287665	+	0.957731i	$0.407121\pi$
$422$	0	0
$423$	0.277770	0.0135056
$424$	0	0
$425$	2.75759	0.133763
$426$	0	0
$427$	0	0
$428$	0	0
$429$	58.6468	2.83150
$430$	0	0
$431$	−3.56873	−0.171899	−0.0859497	−	0.996299i	$-0.527392\pi$
$431$	−3.56873	−0.171899	−0.0859497	+	0.996299i	$0.527392\pi$
$432$	0	0
$433$	41.0992	1.97510	0.987550	−	0.157305i	$-0.0502806\pi$
$433$	41.0992	1.97510	0.987550	+	0.157305i	$0.0502806\pi$
$434$	0	0
$435$	−8.78017	−0.420977
$436$	0	0
$437$	8.48889	0.406079
$438$	0	0
$439$	−31.6361	−1.50991	−0.754954	−	0.655778i	$-0.772341\pi$
$439$	−31.6361	−1.50991	−0.754954	+	0.655778i	$0.772341\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.30070	0.156821	0.0784106	−	0.996921i	$-0.475015\pi$
$443$	3.30070	0.156821	0.0784106	+	0.996921i	$0.475015\pi$
$444$	0	0
$445$	−56.5249	−2.67954
$446$	0	0
$447$	−27.7874	−1.31430
$448$	0	0
$449$	−19.8820	−0.938289	−0.469145	−	0.883121i	$-0.655438\pi$
$449$	−19.8820	−0.938289	−0.469145	+	0.883121i	$0.655438\pi$
$450$	0	0
$451$	6.17257	0.290655
$452$	0	0
$453$	10.8093	0.507868
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.2707	0.714332	0.357166	−	0.934041i	$-0.383743\pi$
$457$	15.2707	0.714332	0.357166	+	0.934041i	$0.383743\pi$
$458$	0	0
$459$	−2.98263	−0.139217
$460$	0	0
$461$	8.68922	0.404697	0.202349	−	0.979314i	$-0.435143\pi$
$461$	8.68922	0.404697	0.202349	+	0.979314i	$0.435143\pi$
$462$	0	0
$463$	8.57355	0.398447	0.199223	−	0.979954i	$-0.436158\pi$
$463$	8.57355	0.398447	0.199223	+	0.979954i	$0.436158\pi$
$464$	0	0
$465$	32.8413	1.52298
$466$	0	0
$467$	−30.1327	−1.39437	−0.697187	−	0.716889i	$-0.745565\pi$
$467$	−30.1327	−1.39437	−0.697187	+	0.716889i	$0.745565\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.45001	−0.297201
$472$	0	0
$473$	15.5216	0.713683
$474$	0	0
$475$	−40.0080	−1.83569
$476$	0	0
$477$	0.900810	0.0412452
$478$	0	0
$479$	−4.74988	−0.217027	−0.108514	−	0.994095i	$-0.534609\pi$
$479$	−4.74988	−0.217027	−0.108514	+	0.994095i	$0.534609\pi$
$480$	0	0
$481$	−63.5803	−2.89901
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26.1394	1.18693
$486$	0	0
$487$	−18.9728	−0.859741	−0.429870	−	0.902891i	$-0.641441\pi$
$487$	−18.9728	−0.859741	−0.429870	+	0.902891i	$0.641441\pi$
$488$	0	0
$489$	−31.6286	−1.43029
$490$	0	0
$491$	6.18017	0.278907	0.139453	−	0.990229i	$-0.455465\pi$
$491$	6.18017	0.278907	0.139453	+	0.990229i	$0.455465\pi$
$492$	0	0
$493$	−0.934881	−0.0421049
$494$	0	0
$495$	−1.77840	−0.0799332
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.9756	−0.938997	−0.469498	−	0.882933i	$-0.655565\pi$
$499$	−20.9756	−0.938997	−0.469498	+	0.882933i	$0.655565\pi$
$500$	0	0
$501$	−0.740435	−0.0330802
$502$	0	0
$503$	3.42597	0.152756	0.0763782	−	0.997079i	$-0.475664\pi$
$503$	3.42597	0.152756	0.0763782	+	0.997079i	$0.475664\pi$
$504$	0	0
$505$	−21.1836	−0.942656
$506$	0	0
$507$	48.1376	2.13787
$508$	0	0
$509$	35.6250	1.57905	0.789526	−	0.613718i	$-0.210327\pi$
$509$	35.6250	1.57905	0.789526	+	0.613718i	$0.210327\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	43.2729	1.91055
$514$	0	0
$515$	30.4163	1.34030
$516$	0	0
$517$	−13.3627	−0.587692
$518$	0	0
$519$	−4.38844	−0.192631
$520$	0	0
$521$	28.6482	1.25510	0.627551	−	0.778576i	$-0.284058\pi$
$521$	28.6482	1.25510	0.627551	+	0.778576i	$0.284058\pi$
$522$	0	0
$523$	14.7717	0.645921	0.322961	−	0.946412i	$-0.395322\pi$
$523$	14.7717	0.645921	0.322961	+	0.946412i	$0.395322\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.49682	0.152324
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.59015	−0.0690064
$532$	0	0
$533$	7.47899	0.323951
$534$	0	0
$535$	−21.6990	−0.938129
$536$	0	0
$537$	−4.65842	−0.201025
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.3001	−0.528824	−0.264412	−	0.964410i	$-0.585178\pi$
$541$	−12.3001	−0.528824	−0.264412	+	0.964410i	$0.585178\pi$
$542$	0	0
$543$	−2.44999	−0.105139
$544$	0	0
$545$	−6.35136	−0.272062
$546$	0	0
$547$	−20.4059	−0.872495	−0.436247	−	0.899827i	$-0.643693\pi$
$547$	−20.4059	−0.872495	−0.436247	+	0.899827i	$0.643693\pi$
$548$	0	0
$549$	−0.670249	−0.0286055
$550$	0	0
$551$	13.5635	0.577827
$552$	0	0
$553$	0	0
$554$	0	0
$555$	55.0357	2.33614
$556$	0	0
$557$	13.9455	0.590889	0.295444	−	0.955360i	$-0.404532\pi$
$557$	13.9455	0.590889	0.295444	+	0.955360i	$0.404532\pi$
$558$	0	0
$559$	18.8067	0.795438
$560$	0	0
$561$	−5.40530	−0.228212
$562$	0	0
$563$	22.3868	0.943492	0.471746	−	0.881734i	$-0.343624\pi$
$563$	22.3868	0.943492	0.471746	+	0.881734i	$0.343624\pi$
$564$	0	0
$565$	9.99840	0.420636
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.0581	−0.966645	−0.483322	−	0.875442i	$-0.660570\pi$
$569$	−23.0581	−0.966645	−0.483322	+	0.875442i	$0.660570\pi$
$570$	0	0
$571$	0.606257	0.0253711	0.0126855	−	0.999920i	$-0.495962\pi$
$571$	0.606257	0.0253711	0.0126855	+	0.999920i	$0.495962\pi$
$572$	0	0
$573$	23.0419	0.962589
$574$	0	0
$575$	−4.71298	−0.196545
$576$	0	0
$577$	−5.57828	−0.232227	−0.116113	−	0.993236i	$-0.537044\pi$
$577$	−5.57828	−0.232227	−0.116113	+	0.993236i	$0.537044\pi$
$578$	0	0
$579$	38.0789	1.58251
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−43.3355	−1.79477
$584$	0	0
$585$	−2.15480	−0.0890899
$586$	0	0
$587$	15.5804	0.643071	0.321536	−	0.946898i	$-0.395801\pi$
$587$	15.5804	0.643071	0.321536	+	0.946898i	$0.395801\pi$
$588$	0	0
$589$	−50.7330	−2.09042
$590$	0	0
$591$	18.0571	0.742769
$592$	0	0
$593$	12.4788	0.512443	0.256222	−	0.966618i	$-0.417522\pi$
$593$	12.4788	0.512443	0.256222	+	0.966618i	$0.417522\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	38.2165	1.56410
$598$	0	0
$599$	−25.6897	−1.04965	−0.524827	−	0.851209i	$-0.675870\pi$
$599$	−25.6897	−1.04965	−0.524827	+	0.851209i	$0.675870\pi$
$600$	0	0
$601$	−14.1712	−0.578055	−0.289028	−	0.957321i	$-0.593332\pi$
$601$	−14.1712	−0.578055	−0.289028	+	0.957321i	$0.593332\pi$
$602$	0	0
$603$	−0.927938	−0.0377885
$604$	0	0
$605$	51.2717	2.08449
$606$	0	0
$607$	38.6010	1.56677	0.783383	−	0.621539i	$-0.213492\pi$
$607$	38.6010	1.56677	0.783383	+	0.621539i	$0.213492\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.1909	−0.655015
$612$	0	0
$613$	−2.40423	−0.0971058	−0.0485529	−	0.998821i	$-0.515461\pi$
$613$	−2.40423	−0.0971058	−0.0485529	+	0.998821i	$0.515461\pi$
$614$	0	0
$615$	−6.47388	−0.261052
$616$	0	0
$617$	24.5675	0.989051	0.494526	−	0.869163i	$-0.335342\pi$
$617$	24.5675	0.989051	0.494526	+	0.869163i	$0.335342\pi$
$618$	0	0
$619$	−31.0657	−1.24864	−0.624318	−	0.781171i	$-0.714623\pi$
$619$	−31.0657	−1.24864	−0.624318	+	0.781171i	$0.714623\pi$
$620$	0	0
$621$	5.09760	0.204560
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.3527	−1.05411
$626$	0	0
$627$	78.4218	3.13187
$628$	0	0
$629$	5.86001	0.233654
$630$	0	0
$631$	−15.9461	−0.634805	−0.317402	−	0.948291i	$-0.602811\pi$
$631$	−15.9461	−0.634805	−0.317402	+	0.948291i	$0.602811\pi$
$632$	0	0
$633$	27.9468	1.11078
$634$	0	0
$635$	15.0755	0.598252
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.913355	0.0361317
$640$	0	0
$641$	−34.1109	−1.34730	−0.673650	−	0.739051i	$-0.735274\pi$
$641$	−34.1109	−1.34730	−0.673650	+	0.739051i	$0.735274\pi$
$642$	0	0
$643$	−22.7215	−0.896050	−0.448025	−	0.894021i	$-0.647872\pi$
$643$	−22.7215	−0.896050	−0.448025	+	0.894021i	$0.647872\pi$
$644$	0	0
$645$	−16.2792	−0.640995
$646$	0	0
$647$	4.53740	0.178384	0.0891918	−	0.996014i	$-0.471572\pi$
$647$	4.53740	0.178384	0.0891918	+	0.996014i	$0.471572\pi$
$648$	0	0
$649$	76.4975	3.00279
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−38.9525	−1.52433	−0.762164	−	0.647384i	$-0.775863\pi$
$653$	−38.9525	−1.52433	−0.762164	+	0.647384i	$0.775863\pi$
$654$	0	0
$655$	−43.6061	−1.70383
$656$	0	0
$657$	0.0390332	0.00152283
$658$	0	0
$659$	−14.8983	−0.580357	−0.290179	−	0.956973i	$-0.593715\pi$
$659$	−14.8983	−0.580357	−0.290179	+	0.956973i	$0.593715\pi$
$660$	0	0
$661$	−17.2118	−0.669461	−0.334731	−	0.942314i	$-0.608645\pi$
$661$	−17.2118	−0.669461	−0.334731	+	0.942314i	$0.608645\pi$
$662$	0	0
$663$	−6.54932	−0.254355
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.59780	0.0618671
$668$	0	0
$669$	−1.84891	−0.0714829
$670$	0	0
$671$	32.2438	1.24476
$672$	0	0
$673$	19.7685	0.762018	0.381009	−	0.924571i	$-0.375577\pi$
$673$	19.7685	0.762018	0.381009	+	0.924571i	$0.375577\pi$
$674$	0	0
$675$	−24.0249	−0.924719
$676$	0	0
$677$	36.0117	1.38404	0.692020	−	0.721878i	$-0.256721\pi$
$677$	36.0117	1.38404	0.692020	+	0.721878i	$0.256721\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−49.2738	−1.88817
$682$	0	0
$683$	11.0796	0.423949	0.211975	−	0.977275i	$-0.432011\pi$
$683$	11.0796	0.423949	0.211975	+	0.977275i	$0.432011\pi$
$684$	0	0
$685$	11.4038	0.435718
$686$	0	0
$687$	31.4272	1.19902
$688$	0	0
$689$	−52.5073	−2.00037
$690$	0	0
$691$	16.5328	0.628936	0.314468	−	0.949268i	$-0.398174\pi$
$691$	16.5328	0.628936	0.314468	+	0.949268i	$0.398174\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−51.3773	−1.94885
$696$	0	0
$697$	−0.689315	−0.0261097
$698$	0	0
$699$	16.8127	0.635915
$700$	0	0
$701$	17.2642	0.652058	0.326029	−	0.945360i	$-0.394289\pi$
$701$	17.2642	0.652058	0.326029	+	0.945360i	$0.394289\pi$
$702$	0	0
$703$	−85.0189	−3.20655
$704$	0	0
$705$	14.0150	0.527836
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−3.98260	−0.149570	−0.0747849	−	0.997200i	$-0.523827\pi$
$709$	−3.98260	−0.149570	−0.0747849	+	0.997200i	$0.523827\pi$
$710$	0	0
$711$	−1.20037	−0.0450174
$712$	0	0
$713$	−5.97640	−0.223818
$714$	0	0
$715$	103.661	3.87671
$716$	0	0
$717$	24.0131	0.896787
$718$	0	0
$719$	7.33439	0.273527	0.136763	−	0.990604i	$-0.456330\pi$
$719$	7.33439	0.273527	0.136763	+	0.990604i	$0.456330\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−28.4080	−1.05651
$724$	0	0
$725$	−7.53041	−0.279672
$726$	0	0
$727$	−4.06498	−0.150762	−0.0753809	−	0.997155i	$-0.524017\pi$
$727$	−4.06498	−0.150762	−0.0753809	+	0.997155i	$0.524017\pi$
$728$	0	0
$729$	25.9499	0.961108
$730$	0	0
$731$	−1.73335	−0.0641104
$732$	0	0
$733$	−17.2247	−0.636207	−0.318104	−	0.948056i	$-0.603046\pi$
$733$	−17.2247	−0.636207	−0.318104	+	0.948056i	$0.603046\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	44.6405	1.64435
$738$	0	0
$739$	13.0829	0.481262	0.240631	−	0.970617i	$-0.422646\pi$
$739$	13.0829	0.481262	0.240631	+	0.970617i	$0.422646\pi$
$740$	0	0
$741$	95.0197	3.49063
$742$	0	0
$743$	−9.58553	−0.351659	−0.175830	−	0.984421i	$-0.556261\pi$
$743$	−9.58553	−0.351659	−0.175830	+	0.984421i	$0.556261\pi$
$744$	0	0
$745$	−49.1157	−1.79946
$746$	0	0
$747$	1.17021	0.0428157
$748$	0	0
$749$	0	0
$750$	0	0
$751$	39.3151	1.43463	0.717314	−	0.696750i	$-0.245371\pi$
$751$	39.3151	1.43463	0.717314	+	0.696750i	$0.245371\pi$
$752$	0	0
$753$	−31.3537	−1.14259
$754$	0	0
$755$	19.1061	0.695342
$756$	0	0
$757$	−30.7971	−1.11934	−0.559670	−	0.828716i	$-0.689072\pi$
$757$	−30.7971	−1.11934	−0.559670	+	0.828716i	$0.689072\pi$
$758$	0	0
$759$	9.23817	0.335324
$760$	0	0
$761$	−19.0651	−0.691110	−0.345555	−	0.938399i	$-0.612309\pi$
$761$	−19.0651	−0.691110	−0.345555	+	0.938399i	$0.612309\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.198601	0.00718044
$766$	0	0
$767$	92.6880	3.34677
$768$	0	0
$769$	−14.0861	−0.507959	−0.253980	−	0.967210i	$-0.581740\pi$
$769$	−14.0861	−0.507959	−0.253980	+	0.967210i	$0.581740\pi$
$770$	0	0
$771$	20.0535	0.722210
$772$	0	0
$773$	5.46538	0.196576	0.0982879	−	0.995158i	$-0.468663\pi$
$773$	5.46538	0.196576	0.0982879	+	0.995158i	$0.468663\pi$
$774$	0	0
$775$	28.1667	1.01178
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.0008	0.358316
$780$	0	0
$781$	−43.9390	−1.57226
$782$	0	0
$783$	8.14494	0.291077
$784$	0	0
$785$	−11.4007	−0.406910
$786$	0	0
$787$	−36.3921	−1.29724	−0.648618	−	0.761114i	$-0.724653\pi$
$787$	−36.3921	−1.29724	−0.648618	+	0.761114i	$0.724653\pi$
$788$	0	0
$789$	14.5064	0.516441
$790$	0	0
$791$	0	0
$792$	0	0
$793$	39.0682	1.38735
$794$	0	0
$795$	45.4509	1.61198
$796$	0	0
$797$	−43.5810	−1.54372	−0.771858	−	0.635794i	$-0.780673\pi$
$797$	−43.5810	−1.54372	−0.771858	+	0.635794i	$0.780673\pi$
$798$	0	0
$799$	1.49227	0.0527927
$800$	0	0
$801$	−1.97531	−0.0697941
$802$	0	0
$803$	−1.87778	−0.0662654
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.83761	−0.205494
$808$	0	0
$809$	5.11013	0.179663	0.0898313	−	0.995957i	$-0.471367\pi$
$809$	5.11013	0.179663	0.0898313	+	0.995957i	$0.471367\pi$
$810$	0	0
$811$	2.06953	0.0726712	0.0363356	−	0.999340i	$-0.488431\pi$
$811$	2.06953	0.0726712	0.0363356	+	0.999340i	$0.488431\pi$
$812$	0	0
$813$	−42.9248	−1.50544
$814$	0	0
$815$	−55.9052	−1.95827
$816$	0	0
$817$	25.1481	0.879819
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.87359	0.100289	0.0501446	−	0.998742i	$-0.484032\pi$
$821$	2.87359	0.100289	0.0501446	+	0.998742i	$0.484032\pi$
$822$	0	0
$823$	3.81222	0.132886	0.0664428	−	0.997790i	$-0.478835\pi$
$823$	3.81222	0.132886	0.0664428	+	0.997790i	$0.478835\pi$
$824$	0	0
$825$	−43.5394	−1.51585
$826$	0	0
$827$	−6.87210	−0.238966	−0.119483	−	0.992836i	$-0.538124\pi$
$827$	−6.87210	−0.238966	−0.119483	+	0.992836i	$0.538124\pi$
$828$	0	0
$829$	−5.53374	−0.192195	−0.0960973	−	0.995372i	$-0.530636\pi$
$829$	−5.53374	−0.192195	−0.0960973	+	0.995372i	$0.530636\pi$
$830$	0	0
$831$	−24.9480	−0.865438
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.30876	−0.0452914
$836$	0	0
$837$	−30.4653	−1.05303
$838$	0	0
$839$	−1.16338	−0.0401642	−0.0200821	−	0.999798i	$-0.506393\pi$
$839$	−1.16338	−0.0401642	−0.0200821	+	0.999798i	$0.506393\pi$
$840$	0	0
$841$	−26.4470	−0.911967
$842$	0	0
$843$	18.0254	0.620827
$844$	0	0
$845$	85.0857	2.92704
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−10.2351	−0.351267
$850$	0	0
$851$	−10.0153	−0.343320
$852$	0	0
$853$	−5.30694	−0.181706	−0.0908531	−	0.995864i	$-0.528959\pi$
$853$	−5.30694	−0.181706	−0.0908531	+	0.995864i	$0.528959\pi$
$854$	0	0
$855$	−2.88137	−0.0985407
$856$	0	0
$857$	1.84412	0.0629938	0.0314969	−	0.999504i	$-0.489973\pi$
$857$	1.84412	0.0629938	0.0314969	+	0.999504i	$0.489973\pi$
$858$	0	0
$859$	−45.1077	−1.53905	−0.769527	−	0.638615i	$-0.779508\pi$
$859$	−45.1077	−1.53905	−0.769527	+	0.638615i	$0.779508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.75173	0.0936701	0.0468351	−	0.998903i	$-0.485086\pi$
$863$	2.75173	0.0936701	0.0468351	+	0.998903i	$0.485086\pi$
$864$	0	0
$865$	−7.75679	−0.263739
$866$	0	0
$867$	−29.3709	−0.997490
$868$	0	0
$869$	57.7465	1.95892
$870$	0	0
$871$	54.0886	1.83272
$872$	0	0
$873$	0.913461	0.0309160
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.9340	1.11210	0.556052	−	0.831148i	$-0.312316\pi$
$877$	32.9340	1.11210	0.556052	+	0.831148i	$0.312316\pi$
$878$	0	0
$879$	54.8565	1.85026
$880$	0	0
$881$	−40.6201	−1.36853	−0.684264	−	0.729235i	$-0.739876\pi$
$881$	−40.6201	−1.36853	−0.684264	+	0.729235i	$0.739876\pi$
$882$	0	0
$883$	0.817246	0.0275025	0.0137513	−	0.999905i	$-0.495623\pi$
$883$	0.817246	0.0275025	0.0137513	+	0.999905i	$0.495623\pi$
$884$	0	0
$885$	−80.2316	−2.69696
$886$	0	0
$887$	−4.29164	−0.144099	−0.0720496	−	0.997401i	$-0.522954\pi$
$887$	−4.29164	−0.144099	−0.0720496	+	0.997401i	$0.522954\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	48.8044	1.63501
$892$	0	0
$893$	−21.6503	−0.724500
$894$	0	0
$895$	−8.23399	−0.275232
$896$	0	0
$897$	11.1934	0.373737
$898$	0	0
$899$	−9.54909	−0.318480
$900$	0	0
$901$	4.83944	0.161225
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.33048	−0.143950
$906$	0	0
$907$	−43.7845	−1.45384	−0.726920	−	0.686723i	$-0.759049\pi$
$907$	−43.7845	−1.45384	−0.726920	+	0.686723i	$0.759049\pi$
$908$	0	0
$909$	−0.740277	−0.0245534
$910$	0	0
$911$	−48.8009	−1.61684	−0.808422	−	0.588603i	$-0.799678\pi$
$911$	−48.8009	−1.61684	−0.808422	+	0.588603i	$0.799678\pi$
$912$	0	0
$913$	−56.2955	−1.86311
$914$	0	0
$915$	−33.8178	−1.11798
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10.5677	0.348596	0.174298	−	0.984693i	$-0.444234\pi$
$919$	10.5677	0.348596	0.174298	+	0.984693i	$0.444234\pi$
$920$	0	0
$921$	8.03644	0.264810
$922$	0	0
$923$	−53.2386	−1.75237
$924$	0	0
$925$	47.2020	1.55199
$926$	0	0
$927$	1.06292	0.0349109
$928$	0	0
$929$	−5.51933	−0.181083	−0.0905417	−	0.995893i	$-0.528860\pi$
$929$	−5.51933	−0.181083	−0.0905417	+	0.995893i	$0.528860\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.91965	−0.0628465
$934$	0	0
$935$	−9.55415	−0.312454
$936$	0	0
$937$	51.4761	1.68165	0.840826	−	0.541305i	$-0.182070\pi$
$937$	51.4761	1.68165	0.840826	+	0.541305i	$0.182070\pi$
$938$	0	0
$939$	−29.8134	−0.972925
$940$	0	0
$941$	−23.0595	−0.751718	−0.375859	−	0.926677i	$-0.622652\pi$
$941$	−23.0595	−0.751718	−0.375859	+	0.926677i	$0.622652\pi$
$942$	0	0
$943$	1.17811	0.0383644
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.0207	0.878054	0.439027	−	0.898474i	$-0.355323\pi$
$947$	27.0207	0.878054	0.439027	+	0.898474i	$0.355323\pi$
$948$	0	0
$949$	−2.27521	−0.0738564
$950$	0	0
$951$	−7.51902	−0.243821
$952$	0	0
$953$	29.6466	0.960348	0.480174	−	0.877173i	$-0.340574\pi$
$953$	29.6466	0.960348	0.480174	+	0.877173i	$0.340574\pi$
$954$	0	0
$955$	40.7278	1.31792
$956$	0	0
$957$	14.7608	0.477147
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.71731	0.152171
$962$	0	0
$963$	−0.758289	−0.0244355
$964$	0	0
$965$	67.3065	2.16667
$966$	0	0
$967$	−41.6275	−1.33865	−0.669324	−	0.742970i	$-0.733416\pi$
$967$	−41.6275	−1.33865	−0.669324	+	0.742970i	$0.733416\pi$
$968$	0	0
$969$	−8.75767	−0.281337
$970$	0	0
$971$	−37.4436	−1.20162	−0.600811	−	0.799391i	$-0.705155\pi$
$971$	−37.4436	−1.20162	−0.600811	+	0.799391i	$0.705155\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−52.7544	−1.68949
$976$	0	0
$977$	25.8568	0.827231	0.413616	−	0.910452i	$-0.364266\pi$
$977$	25.8568	0.827231	0.413616	+	0.910452i	$0.364266\pi$
$978$	0	0
$979$	95.0266	3.03706
$980$	0	0
$981$	−0.221953	−0.00708643
$982$	0	0
$983$	−33.8201	−1.07869	−0.539347	−	0.842084i	$-0.681329\pi$
$983$	−33.8201	−1.07869	−0.539347	+	0.842084i	$0.681329\pi$
$984$	0	0
$985$	31.9168	1.01696
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.96247	0.0942010
$990$	0	0
$991$	−26.9278	−0.855390	−0.427695	−	0.903923i	$-0.640674\pi$
$991$	−26.9278	−0.855390	−0.427695	+	0.903923i	$0.640674\pi$
$992$	0	0
$993$	29.8776	0.948136
$994$	0	0
$995$	67.5497	2.14147
$996$	0	0
$997$	−4.87456	−0.154379	−0.0771894	−	0.997016i	$-0.524595\pi$
$997$	−4.87456	−0.154379	−0.0771894	+	0.997016i	$0.524595\pi$
$998$	0	0
$999$	−51.0540	−1.61528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4508.2.a.f.1.4		5
7.6	odd	2		644.2.a.d.1.2	✓	5
21.20	even	2		5796.2.a.t.1.5		5
28.27	even	2		2576.2.a.bb.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.d.1.2	✓	5	7.6	odd	2
2576.2.a.bb.1.4		5	28.27	even	2
4508.2.a.f.1.4		5	1.1	even	1	trivial
5796.2.a.t.1.5		5	21.20	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4508.a (trivial)

\(f(q)\)	\(=\)	\(q+1.76321 q^{3} +3.11657 q^{5} +0.108911 q^{9} +O(q^{10})\)	\(q+1.76321 q^{3} +3.11657 q^{5} +0.108911 q^{9} -5.23940 q^{11} -6.34832 q^{13} +5.49516 q^{15} +0.585105 q^{17} -8.48889 q^{19} -1.00000 q^{23} +4.71298 q^{25} -5.09760 q^{27} -1.59780 q^{29} +5.97640 q^{31} -9.23817 q^{33} +10.0153 q^{37} -11.1934 q^{39} -1.17811 q^{41} -2.96247 q^{43} +0.339428 q^{45} +2.55043 q^{47} +1.03166 q^{51} +8.27107 q^{53} -16.3289 q^{55} -14.9677 q^{57} -14.6004 q^{59} -6.15410 q^{61} -19.7849 q^{65} -8.52015 q^{67} -1.76321 q^{69} +8.38625 q^{71} +0.358396 q^{73} +8.30998 q^{75} -11.0216 q^{79} -9.31487 q^{81} +10.7446 q^{83} +1.82352 q^{85} -2.81726 q^{87} -18.1369 q^{89} +10.5376 q^{93} -26.4562 q^{95} +8.38723 q^{97} -0.570629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9	\( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} - 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} - 24 q^{33} - 4 q^{37} + 3 q^{39} - q^{41} - 8 q^{43} + 16 q^{45} - 5 q^{47} - 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} - 20 q^{61} - 12 q^{65} - 12 q^{67} + 3 q^{69} + 9 q^{71} + 9 q^{73} - 35 q^{75} - 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} - 32 q^{89} - 15 q^{93} - 36 q^{95} - 4 q^{97} + 28 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9 + 2 * q^11 - 13 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 - 5 * q^23 + 19 * q^25 - 15 * q^27 + 13 * q^29 + 3 * q^31 - 24 * q^33 - 4 * q^37 + 3 * q^39 - q^41 - 8 * q^43 + 16 * q^45 - 5 * q^47 - 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 12 * q^59 - 20 * q^61 - 12 * q^65 - 12 * q^67 + 3 * q^69 + 9 * q^71 + 9 * q^73 - 35 * q^75 - 8 * q^79 - 11 * q^81 + 28 * q^83 + 16 * q^85 + 15 * q^87 - 32 * q^89 - 15 * q^93 - 36 * q^95 - 4 * q^97 + 28 * q^99

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