LMFDB - Embedded newform 8032.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83124 q^{3} +4.45088 q^{5} +3.13811 q^{7} +0.353447 q^{9} +O(q^{10})$	$q-1.83124 q^{3} +4.45088 q^{5} +3.13811 q^{7} +0.353447 q^{9} +4.62858 q^{11} +0.790969 q^{13} -8.15063 q^{15} +2.27963 q^{17} +4.81435 q^{19} -5.74665 q^{21} +3.45456 q^{23} +14.8103 q^{25} +4.84648 q^{27} -5.18070 q^{29} -2.55051 q^{31} -8.47605 q^{33} +13.9674 q^{35} -7.37797 q^{37} -1.44846 q^{39} +0.464049 q^{41} +8.30836 q^{43} +1.57315 q^{45} +9.29454 q^{47} +2.84776 q^{49} -4.17456 q^{51} -9.25686 q^{53} +20.6012 q^{55} -8.81624 q^{57} +2.30647 q^{59} -6.49550 q^{61} +1.10916 q^{63} +3.52051 q^{65} +2.61314 q^{67} -6.32614 q^{69} +3.14739 q^{71} +9.65879 q^{73} -27.1213 q^{75} +14.5250 q^{77} -7.23113 q^{79} -9.93542 q^{81} -4.82000 q^{83} +10.1464 q^{85} +9.48712 q^{87} +13.8922 q^{89} +2.48215 q^{91} +4.67059 q^{93} +21.4281 q^{95} -3.25153 q^{97} +1.63596 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * 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
$3$	−1.83124	−1.05727	−0.528634	−	0.848850i	$-0.677296\pi$
$3$	−1.83124	−1.05727	−0.528634	+	0.848850i	$0.677296\pi$
$4$	0	0
$5$	4.45088	1.99049	0.995246	−	0.0973891i	$-0.0310491\pi$
$5$	4.45088	1.99049	0.995246	+	0.0973891i	$0.0310491\pi$
$6$	0	0
$7$	3.13811	1.18610	0.593048	−	0.805167i	$-0.297925\pi$
$7$	3.13811	1.18610	0.593048	+	0.805167i	$0.297925\pi$
$8$	0	0
$9$	0.353447	0.117816
$10$	0	0
$11$	4.62858	1.39557	0.697785	−	0.716307i	$-0.254169\pi$
$11$	4.62858	1.39557	0.697785	+	0.716307i	$0.254169\pi$
$12$	0	0
$13$	0.790969	0.219375	0.109688	−	0.993966i	$-0.465015\pi$
$13$	0.790969	0.219375	0.109688	+	0.993966i	$0.465015\pi$
$14$	0	0
$15$	−8.15063	−2.10448
$16$	0	0
$17$	2.27963	0.552892	0.276446	−	0.961030i	$-0.410843\pi$
$17$	2.27963	0.552892	0.276446	+	0.961030i	$0.410843\pi$
$18$	0	0
$19$	4.81435	1.10449	0.552244	−	0.833683i	$-0.313772\pi$
$19$	4.81435	1.10449	0.552244	+	0.833683i	$0.313772\pi$
$20$	0	0
$21$	−5.74665	−1.25402
$22$	0	0
$23$	3.45456	0.720326	0.360163	−	0.932889i	$-0.382721\pi$
$23$	3.45456	0.720326	0.360163	+	0.932889i	$0.382721\pi$
$24$	0	0
$25$	14.8103	2.96206
$26$	0	0
$27$	4.84648	0.932705
$28$	0	0
$29$	−5.18070	−0.962032	−0.481016	−	0.876712i	$-0.659732\pi$
$29$	−5.18070	−0.962032	−0.481016	+	0.876712i	$0.659732\pi$
$30$	0	0
$31$	−2.55051	−0.458084	−0.229042	−	0.973417i	$-0.573559\pi$
$31$	−2.55051	−0.458084	−0.229042	+	0.973417i	$0.573559\pi$
$32$	0	0
$33$	−8.47605	−1.47549
$34$	0	0
$35$	13.9674	2.36091
$36$	0	0
$37$	−7.37797	−1.21293	−0.606466	−	0.795110i	$-0.707413\pi$
$37$	−7.37797	−1.21293	−0.606466	+	0.795110i	$0.707413\pi$
$38$	0	0
$39$	−1.44846	−0.231939
$40$	0	0
$41$	0.464049	0.0724722	0.0362361	−	0.999343i	$-0.488463\pi$
$41$	0.464049	0.0724722	0.0362361	+	0.999343i	$0.488463\pi$
$42$	0	0
$43$	8.30836	1.26701	0.633507	−	0.773737i	$-0.281615\pi$
$43$	8.30836	1.26701	0.633507	+	0.773737i	$0.281615\pi$
$44$	0	0
$45$	1.57315	0.234511
$46$	0	0
$47$	9.29454	1.35575	0.677874	−	0.735178i	$-0.262902\pi$
$47$	9.29454	1.35575	0.677874	+	0.735178i	$0.262902\pi$
$48$	0	0
$49$	2.84776	0.406822
$50$	0	0
$51$	−4.17456	−0.584555
$52$	0	0
$53$	−9.25686	−1.27153	−0.635764	−	0.771884i	$-0.719315\pi$
$53$	−9.25686	−1.27153	−0.635764	+	0.771884i	$0.719315\pi$
$54$	0	0
$55$	20.6012	2.77787
$56$	0	0
$57$	−8.81624	−1.16774
$58$	0	0
$59$	2.30647	0.300277	0.150138	−	0.988665i	$-0.452028\pi$
$59$	2.30647	0.300277	0.150138	+	0.988665i	$0.452028\pi$
$60$	0	0
$61$	−6.49550	−0.831663	−0.415832	−	0.909442i	$-0.636509\pi$
$61$	−6.49550	−0.831663	−0.415832	+	0.909442i	$0.636509\pi$
$62$	0	0
$63$	1.10916	0.139741
$64$	0	0
$65$	3.52051	0.436665
$66$	0	0
$67$	2.61314	0.319246	0.159623	−	0.987178i	$-0.448972\pi$
$67$	2.61314	0.319246	0.159623	+	0.987178i	$0.448972\pi$
$68$	0	0
$69$	−6.32614	−0.761578
$70$	0	0
$71$	3.14739	0.373527	0.186763	−	0.982405i	$-0.440200\pi$
$71$	3.14739	0.373527	0.186763	+	0.982405i	$0.440200\pi$
$72$	0	0
$73$	9.65879	1.13048	0.565238	−	0.824928i	$-0.308784\pi$
$73$	9.65879	1.13048	0.565238	+	0.824928i	$0.308784\pi$
$74$	0	0
$75$	−27.1213	−3.13169
$76$	0	0
$77$	14.5250	1.65528
$78$	0	0
$79$	−7.23113	−0.813566	−0.406783	−	0.913525i	$-0.633350\pi$
$79$	−7.23113	−0.813566	−0.406783	+	0.913525i	$0.633350\pi$
$80$	0	0
$81$	−9.93542	−1.10394
$82$	0	0
$83$	−4.82000	−0.529064	−0.264532	−	0.964377i	$-0.585217\pi$
$83$	−4.82000	−0.529064	−0.264532	+	0.964377i	$0.585217\pi$
$84$	0	0
$85$	10.1464	1.10053
$86$	0	0
$87$	9.48712	1.01713
$88$	0	0
$89$	13.8922	1.47257	0.736286	−	0.676670i	$-0.236578\pi$
$89$	13.8922	1.47257	0.736286	+	0.676670i	$0.236578\pi$
$90$	0	0
$91$	2.48215	0.260200
$92$	0	0
$93$	4.67059	0.484318
$94$	0	0
$95$	21.4281	2.19847
$96$	0	0
$97$	−3.25153	−0.330143	−0.165072	−	0.986282i	$-0.552786\pi$
$97$	−3.25153	−0.330143	−0.165072	+	0.986282i	$0.552786\pi$
$98$	0	0
$99$	1.63596	0.164420
$100$	0	0
$101$	−1.48452	−0.147715	−0.0738577	−	0.997269i	$-0.523531\pi$
$101$	−1.48452	−0.147715	−0.0738577	+	0.997269i	$0.523531\pi$
$102$	0	0
$103$	−9.94662	−0.980069	−0.490035	−	0.871703i	$-0.663016\pi$
$103$	−9.94662	−0.980069	−0.490035	+	0.871703i	$0.663016\pi$
$104$	0	0
$105$	−25.5776	−2.49612
$106$	0	0
$107$	−7.70379	−0.744754	−0.372377	−	0.928082i	$-0.621457\pi$
$107$	−7.70379	−0.744754	−0.372377	+	0.928082i	$0.621457\pi$
$108$	0	0
$109$	12.8806	1.23374	0.616871	−	0.787065i	$-0.288400\pi$
$109$	12.8806	1.23374	0.616871	+	0.787065i	$0.288400\pi$
$110$	0	0
$111$	13.5109	1.28239
$112$	0	0
$113$	−15.8378	−1.48989	−0.744946	−	0.667125i	$-0.767525\pi$
$113$	−15.8378	−1.48989	−0.744946	+	0.667125i	$0.767525\pi$
$114$	0	0
$115$	15.3758	1.43380
$116$	0	0
$117$	0.279566	0.0258459
$118$	0	0
$119$	7.15374	0.655782
$120$	0	0
$121$	10.4238	0.947615
$122$	0	0
$123$	−0.849785	−0.0766225
$124$	0	0
$125$	43.6645	3.90547
$126$	0	0
$127$	−20.2362	−1.79567	−0.897837	−	0.440329i	$-0.854862\pi$
$127$	−20.2362	−1.79567	−0.897837	+	0.440329i	$0.854862\pi$
$128$	0	0
$129$	−15.2146	−1.33957
$130$	0	0
$131$	6.45229	0.563739	0.281870	−	0.959453i	$-0.409045\pi$
$131$	6.45229	0.563739	0.281870	+	0.959453i	$0.409045\pi$
$132$	0	0
$133$	15.1080	1.31003
$134$	0	0
$135$	21.5711	1.85654
$136$	0	0
$137$	−16.3431	−1.39629	−0.698144	−	0.715958i	$-0.745990\pi$
$137$	−16.3431	−1.39629	−0.698144	+	0.715958i	$0.745990\pi$
$138$	0	0
$139$	12.7025	1.07742	0.538708	−	0.842493i	$-0.318913\pi$
$139$	12.7025	1.07742	0.538708	+	0.842493i	$0.318913\pi$
$140$	0	0
$141$	−17.0205	−1.43339
$142$	0	0
$143$	3.66106	0.306154
$144$	0	0
$145$	−23.0587	−1.91492
$146$	0	0
$147$	−5.21493	−0.430120
$148$	0	0
$149$	−16.0376	−1.31385	−0.656927	−	0.753954i	$-0.728144\pi$
$149$	−16.0376	−1.31385	−0.656927	+	0.753954i	$0.728144\pi$
$150$	0	0
$151$	−18.2490	−1.48508	−0.742541	−	0.669801i	$-0.766379\pi$
$151$	−18.2490	−1.48508	−0.742541	+	0.669801i	$0.766379\pi$
$152$	0	0
$153$	0.805729	0.0651393
$154$	0	0
$155$	−11.3520	−0.911814
$156$	0	0
$157$	−5.72970	−0.457280	−0.228640	−	0.973511i	$-0.573428\pi$
$157$	−5.72970	−0.457280	−0.228640	+	0.973511i	$0.573428\pi$
$158$	0	0
$159$	16.9515	1.34435
$160$	0	0
$161$	10.8408	0.854376
$162$	0	0
$163$	−7.20756	−0.564539	−0.282270	−	0.959335i	$-0.591087\pi$
$163$	−7.20756	−0.564539	−0.282270	+	0.959335i	$0.591087\pi$
$164$	0	0
$165$	−37.7259	−2.93695
$166$	0	0
$167$	−4.04780	−0.313228	−0.156614	−	0.987660i	$-0.550058\pi$
$167$	−4.04780	−0.313228	−0.156614	+	0.987660i	$0.550058\pi$
$168$	0	0
$169$	−12.3744	−0.951874
$170$	0	0
$171$	1.70162	0.130126
$172$	0	0
$173$	7.04230	0.535416	0.267708	−	0.963500i	$-0.413734\pi$
$173$	7.04230	0.535416	0.267708	+	0.963500i	$0.413734\pi$
$174$	0	0
$175$	46.4764	3.51329
$176$	0	0
$177$	−4.22370	−0.317473
$178$	0	0
$179$	17.1272	1.28015	0.640074	−	0.768313i	$-0.278904\pi$
$179$	17.1272	1.28015	0.640074	+	0.768313i	$0.278904\pi$
$180$	0	0
$181$	23.4748	1.74487	0.872433	−	0.488734i	$-0.162541\pi$
$181$	23.4748	1.74487	0.872433	+	0.488734i	$0.162541\pi$
$182$	0	0
$183$	11.8948	0.879291
$184$	0	0
$185$	−32.8385	−2.41433
$186$	0	0
$187$	10.5515	0.771599
$188$	0	0
$189$	15.2088	1.10628
$190$	0	0
$191$	−9.57707	−0.692972	−0.346486	−	0.938055i	$-0.612625\pi$
$191$	−9.57707	−0.692972	−0.346486	+	0.938055i	$0.612625\pi$
$192$	0	0
$193$	3.10859	0.223761	0.111881	−	0.993722i	$-0.464313\pi$
$193$	3.10859	0.223761	0.111881	+	0.993722i	$0.464313\pi$
$194$	0	0
$195$	−6.44690	−0.461672
$196$	0	0
$197$	−20.8852	−1.48801	−0.744006	−	0.668173i	$-0.767077\pi$
$197$	−20.8852	−1.48801	−0.744006	+	0.668173i	$0.767077\pi$
$198$	0	0
$199$	−14.5655	−1.03252	−0.516262	−	0.856431i	$-0.672677\pi$
$199$	−14.5655	−1.03252	−0.516262	+	0.856431i	$0.672677\pi$
$200$	0	0
$201$	−4.78529	−0.337528
$202$	0	0
$203$	−16.2576	−1.14106
$204$	0	0
$205$	2.06542	0.144255
$206$	0	0
$207$	1.22101	0.0848658
$208$	0	0
$209$	22.2836	1.54139
$210$	0	0
$211$	−12.9361	−0.890558	−0.445279	−	0.895392i	$-0.646895\pi$
$211$	−12.9361	−0.890558	−0.445279	+	0.895392i	$0.646895\pi$
$212$	0	0
$213$	−5.76363	−0.394918
$214$	0	0
$215$	36.9795	2.52198
$216$	0	0
$217$	−8.00378	−0.543332
$218$	0	0
$219$	−17.6876	−1.19522
$220$	0	0
$221$	1.80312	0.121291
$222$	0	0
$223$	8.16183	0.546556	0.273278	−	0.961935i	$-0.411892\pi$
$223$	8.16183	0.546556	0.273278	+	0.961935i	$0.411892\pi$
$224$	0	0
$225$	5.23466	0.348978
$226$	0	0
$227$	−17.6168	−1.16927	−0.584633	−	0.811298i	$-0.698762\pi$
$227$	−17.6168	−1.16927	−0.584633	+	0.811298i	$0.698762\pi$
$228$	0	0
$229$	−8.22037	−0.543218	−0.271609	−	0.962408i	$-0.587556\pi$
$229$	−8.22037	−0.543218	−0.271609	+	0.962408i	$0.587556\pi$
$230$	0	0
$231$	−26.5988	−1.75007
$232$	0	0
$233$	23.0817	1.51213	0.756065	−	0.654496i	$-0.227119\pi$
$233$	23.0817	1.51213	0.756065	+	0.654496i	$0.227119\pi$
$234$	0	0
$235$	41.3688	2.69861
$236$	0	0
$237$	13.2420	0.860158
$238$	0	0
$239$	−6.36858	−0.411949	−0.205975	−	0.978557i	$-0.566036\pi$
$239$	−6.36858	−0.411949	−0.205975	+	0.978557i	$0.566036\pi$
$240$	0	0
$241$	−28.2961	−1.82271	−0.911356	−	0.411619i	$-0.864963\pi$
$241$	−28.2961	−1.82271	−0.911356	+	0.411619i	$0.864963\pi$
$242$	0	0
$243$	3.65472	0.234450
$244$	0	0
$245$	12.6750	0.809777
$246$	0	0
$247$	3.80800	0.242297
$248$	0	0
$249$	8.82658	0.559362
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	15.9897	1.00527
$254$	0	0
$255$	−18.5804	−1.16355
$256$	0	0
$257$	−10.3596	−0.646213	−0.323106	−	0.946363i	$-0.604727\pi$
$257$	−10.3596	−0.646213	−0.323106	+	0.946363i	$0.604727\pi$
$258$	0	0
$259$	−23.1529	−1.43865
$260$	0	0
$261$	−1.83111	−0.113343
$262$	0	0
$263$	−1.89651	−0.116944	−0.0584718	−	0.998289i	$-0.518623\pi$
$263$	−1.89651	−0.116944	−0.0584718	+	0.998289i	$0.518623\pi$
$264$	0	0
$265$	−41.2011	−2.53097
$266$	0	0
$267$	−25.4400	−1.55690
$268$	0	0
$269$	−14.5183	−0.885198	−0.442599	−	0.896720i	$-0.645943\pi$
$269$	−14.5183	−0.885198	−0.442599	+	0.896720i	$0.645943\pi$
$270$	0	0
$271$	−11.0662	−0.672226	−0.336113	−	0.941822i	$-0.609112\pi$
$271$	−11.0662	−0.672226	−0.336113	+	0.941822i	$0.609112\pi$
$272$	0	0
$273$	−4.54542	−0.275101
$274$	0	0
$275$	68.5507	4.13376
$276$	0	0
$277$	1.13487	0.0681877	0.0340939	−	0.999419i	$-0.489145\pi$
$277$	1.13487	0.0681877	0.0340939	+	0.999419i	$0.489145\pi$
$278$	0	0
$279$	−0.901469	−0.0539696
$280$	0	0
$281$	−1.25075	−0.0746137	−0.0373068	−	0.999304i	$-0.511878\pi$
$281$	−1.25075	−0.0746137	−0.0373068	+	0.999304i	$0.511878\pi$
$282$	0	0
$283$	5.59437	0.332551	0.166275	−	0.986079i	$-0.446826\pi$
$283$	5.59437	0.332551	0.166275	+	0.986079i	$0.446826\pi$
$284$	0	0
$285$	−39.2400	−2.32438
$286$	0	0
$287$	1.45624	0.0859589
$288$	0	0
$289$	−11.8033	−0.694311
$290$	0	0
$291$	5.95435	0.349050
$292$	0	0
$293$	−13.2406	−0.773522	−0.386761	−	0.922180i	$-0.626406\pi$
$293$	−13.2406	−0.773522	−0.386761	+	0.922180i	$0.626406\pi$
$294$	0	0
$295$	10.2658	0.597698
$296$	0	0
$297$	22.4323	1.30166
$298$	0	0
$299$	2.73245	0.158022
$300$	0	0
$301$	26.0726	1.50280
$302$	0	0
$303$	2.71852	0.156175
$304$	0	0
$305$	−28.9107	−1.65542
$306$	0	0
$307$	13.1378	0.749813	0.374907	−	0.927063i	$-0.377675\pi$
$307$	13.1378	0.749813	0.374907	+	0.927063i	$0.377675\pi$
$308$	0	0
$309$	18.2147	1.03620
$310$	0	0
$311$	−29.6854	−1.68330	−0.841652	−	0.540020i	$-0.818417\pi$
$311$	−29.6854	−1.68330	−0.841652	+	0.540020i	$0.818417\pi$
$312$	0	0
$313$	−8.01904	−0.453263	−0.226631	−	0.973981i	$-0.572771\pi$
$313$	−8.01904	−0.453263	−0.226631	+	0.973981i	$0.572771\pi$
$314$	0	0
$315$	4.93672	0.278153
$316$	0	0
$317$	−26.9969	−1.51630	−0.758149	−	0.652081i	$-0.773896\pi$
$317$	−26.9969	−1.51630	−0.758149	+	0.652081i	$0.773896\pi$
$318$	0	0
$319$	−23.9793	−1.34258
$320$	0	0
$321$	14.1075	0.787404
$322$	0	0
$323$	10.9749	0.610662
$324$	0	0
$325$	11.7145	0.649803
$326$	0	0
$327$	−23.5876	−1.30440
$328$	0	0
$329$	29.1673	1.60805
$330$	0	0
$331$	8.25536	0.453756	0.226878	−	0.973923i	$-0.427148\pi$
$331$	8.25536	0.453756	0.226878	+	0.973923i	$0.427148\pi$
$332$	0	0
$333$	−2.60772	−0.142902
$334$	0	0
$335$	11.6308	0.635456
$336$	0	0
$337$	−10.2646	−0.559150	−0.279575	−	0.960124i	$-0.590194\pi$
$337$	−10.2646	−0.559150	−0.279575	+	0.960124i	$0.590194\pi$
$338$	0	0
$339$	29.0028	1.57522
$340$	0	0
$341$	−11.8052	−0.639289
$342$	0	0
$343$	−13.0302	−0.703565
$344$	0	0
$345$	−28.1569	−1.51592
$346$	0	0
$347$	−14.4364	−0.774989	−0.387495	−	0.921872i	$-0.626659\pi$
$347$	−14.4364	−0.774989	−0.387495	+	0.921872i	$0.626659\pi$
$348$	0	0
$349$	9.50535	0.508810	0.254405	−	0.967098i	$-0.418120\pi$
$349$	9.50535	0.508810	0.254405	+	0.967098i	$0.418120\pi$
$350$	0	0
$351$	3.83341	0.204613
$352$	0	0
$353$	−27.0174	−1.43799	−0.718996	−	0.695014i	$-0.755398\pi$
$353$	−27.0174	−1.43799	−0.718996	+	0.695014i	$0.755398\pi$
$354$	0	0
$355$	14.0086	0.743502
$356$	0	0
$357$	−13.1002	−0.693338
$358$	0	0
$359$	−6.71012	−0.354147	−0.177073	−	0.984198i	$-0.556663\pi$
$359$	−6.71012	−0.354147	−0.177073	+	0.984198i	$0.556663\pi$
$360$	0	0
$361$	4.17795	0.219892
$362$	0	0
$363$	−19.0884	−1.00188
$364$	0	0
$365$	42.9901	2.25020
$366$	0	0
$367$	26.9444	1.40649	0.703243	−	0.710950i	$-0.251735\pi$
$367$	26.9444	1.40649	0.703243	+	0.710950i	$0.251735\pi$
$368$	0	0
$369$	0.164017	0.00853837
$370$	0	0
$371$	−29.0491	−1.50815
$372$	0	0
$373$	−18.3503	−0.950144	−0.475072	−	0.879947i	$-0.657578\pi$
$373$	−18.3503	−0.950144	−0.475072	+	0.879947i	$0.657578\pi$
$374$	0	0
$375$	−79.9602	−4.12913
$376$	0	0
$377$	−4.09778	−0.211046
$378$	0	0
$379$	24.2058	1.24337	0.621684	−	0.783268i	$-0.286449\pi$
$379$	24.2058	1.24337	0.621684	+	0.783268i	$0.286449\pi$
$380$	0	0
$381$	37.0574	1.89851
$382$	0	0
$383$	0.140721	0.00719049	0.00359525	−	0.999994i	$-0.498856\pi$
$383$	0.140721	0.00719049	0.00359525	+	0.999994i	$0.498856\pi$
$384$	0	0
$385$	64.6490	3.29482
$386$	0	0
$387$	2.93657	0.149274
$388$	0	0
$389$	13.9382	0.706694	0.353347	−	0.935492i	$-0.385044\pi$
$389$	13.9382	0.706694	0.353347	+	0.935492i	$0.385044\pi$
$390$	0	0
$391$	7.87513	0.398262
$392$	0	0
$393$	−11.8157	−0.596023
$394$	0	0
$395$	−32.1849	−1.61940
$396$	0	0
$397$	16.4624	0.826225	0.413112	−	0.910680i	$-0.364442\pi$
$397$	16.4624	0.826225	0.413112	+	0.910680i	$0.364442\pi$
$398$	0	0
$399$	−27.6664	−1.38505
$400$	0	0
$401$	15.5398	0.776020	0.388010	−	0.921655i	$-0.373163\pi$
$401$	15.5398	0.776020	0.388010	+	0.921655i	$0.373163\pi$
$402$	0	0
$403$	−2.01737	−0.100492
$404$	0	0
$405$	−44.2213	−2.19738
$406$	0	0
$407$	−34.1495	−1.69273
$408$	0	0
$409$	14.4038	0.712221	0.356110	−	0.934444i	$-0.384103\pi$
$409$	14.4038	0.712221	0.356110	+	0.934444i	$0.384103\pi$
$410$	0	0
$411$	29.9282	1.47625
$412$	0	0
$413$	7.23796	0.356157
$414$	0	0
$415$	−21.4532	−1.05310
$416$	0	0
$417$	−23.2614	−1.13912
$418$	0	0
$419$	−22.2146	−1.08525	−0.542627	−	0.839974i	$-0.682570\pi$
$419$	−22.2146	−1.08525	−0.542627	+	0.839974i	$0.682570\pi$
$420$	0	0
$421$	−8.45992	−0.412311	−0.206156	−	0.978519i	$-0.566095\pi$
$421$	−8.45992	−0.412311	−0.206156	+	0.978519i	$0.566095\pi$
$422$	0	0
$423$	3.28513	0.159728
$424$	0	0
$425$	33.7620	1.63770
$426$	0	0
$427$	−20.3836	−0.986432
$428$	0	0
$429$	−6.70430	−0.323686
$430$	0	0
$431$	−6.07701	−0.292719	−0.146360	−	0.989231i	$-0.546756\pi$
$431$	−6.07701	−0.292719	−0.146360	+	0.989231i	$0.546756\pi$
$432$	0	0
$433$	12.2468	0.588545	0.294273	−	0.955722i	$-0.404923\pi$
$433$	12.2468	0.588545	0.294273	+	0.955722i	$0.404923\pi$
$434$	0	0
$435$	42.2260	2.02458
$436$	0	0
$437$	16.6315	0.795591
$438$	0	0
$439$	12.0878	0.576920	0.288460	−	0.957492i	$-0.406857\pi$
$439$	12.0878	0.576920	0.288460	+	0.957492i	$0.406857\pi$
$440$	0	0
$441$	1.00653	0.0479301
$442$	0	0
$443$	18.3841	0.873454	0.436727	−	0.899594i	$-0.356138\pi$
$443$	18.3841	0.873454	0.436727	+	0.899594i	$0.356138\pi$
$444$	0	0
$445$	61.8326	2.93115
$446$	0	0
$447$	29.3688	1.38910
$448$	0	0
$449$	0.881148	0.0415839	0.0207920	−	0.999784i	$-0.493381\pi$
$449$	0.881148	0.0415839	0.0207920	+	0.999784i	$0.493381\pi$
$450$	0	0
$451$	2.14789	0.101140
$452$	0	0
$453$	33.4183	1.57013
$454$	0	0
$455$	11.0477	0.517926
$456$	0	0
$457$	−18.8421	−0.881395	−0.440698	−	0.897656i	$-0.645269\pi$
$457$	−18.8421	−0.881395	−0.440698	+	0.897656i	$0.645269\pi$
$458$	0	0
$459$	11.0482	0.515685
$460$	0	0
$461$	19.8734	0.925596	0.462798	−	0.886464i	$-0.346846\pi$
$461$	19.8734	0.925596	0.462798	+	0.886464i	$0.346846\pi$
$462$	0	0
$463$	41.1868	1.91411	0.957057	−	0.289901i	$-0.0936222\pi$
$463$	41.1868	1.91411	0.957057	+	0.289901i	$0.0936222\pi$
$464$	0	0
$465$	20.7882	0.964031
$466$	0	0
$467$	26.3226	1.21806	0.609032	−	0.793146i	$-0.291558\pi$
$467$	26.3226	1.21806	0.609032	+	0.793146i	$0.291558\pi$
$468$	0	0
$469$	8.20032	0.378656
$470$	0	0
$471$	10.4925	0.483468
$472$	0	0
$473$	38.4559	1.76821
$474$	0	0
$475$	71.3020	3.27156
$476$	0	0
$477$	−3.27181	−0.149806
$478$	0	0
$479$	9.02674	0.412442	0.206221	−	0.978505i	$-0.433883\pi$
$479$	9.02674	0.412442	0.206221	+	0.978505i	$0.433883\pi$
$480$	0	0
$481$	−5.83575	−0.266087
$482$	0	0
$483$	−19.8522	−0.903304
$484$	0	0
$485$	−14.4722	−0.657148
$486$	0	0
$487$	−18.5615	−0.841103	−0.420551	−	0.907269i	$-0.638163\pi$
$487$	−18.5615	−0.841103	−0.420551	+	0.907269i	$0.638163\pi$
$488$	0	0
$489$	13.1988	0.596869
$490$	0	0
$491$	29.4693	1.32993	0.664965	−	0.746875i	$-0.268447\pi$
$491$	29.4693	1.32993	0.664965	+	0.746875i	$0.268447\pi$
$492$	0	0
$493$	−11.8101	−0.531900
$494$	0	0
$495$	7.28145	0.327277
$496$	0	0
$497$	9.87687	0.443038
$498$	0	0
$499$	18.4316	0.825112	0.412556	−	0.910932i	$-0.364636\pi$
$499$	18.4316	0.825112	0.412556	+	0.910932i	$0.364636\pi$
$500$	0	0
$501$	7.41249	0.331166
$502$	0	0
$503$	1.36669	0.0609377	0.0304689	−	0.999536i	$-0.490300\pi$
$503$	1.36669	0.0609377	0.0304689	+	0.999536i	$0.490300\pi$
$504$	0	0
$505$	−6.60742	−0.294026
$506$	0	0
$507$	22.6605	1.00639
$508$	0	0
$509$	33.2521	1.47387	0.736936	−	0.675963i	$-0.236272\pi$
$509$	33.2521	1.47387	0.736936	+	0.675963i	$0.236272\pi$
$510$	0	0
$511$	30.3104	1.34085
$512$	0	0
$513$	23.3326	1.03016
$514$	0	0
$515$	−44.2712	−1.95082
$516$	0	0
$517$	43.0205	1.89204
$518$	0	0
$519$	−12.8962	−0.566079
$520$	0	0
$521$	11.3748	0.498341	0.249171	−	0.968460i	$-0.419842\pi$
$521$	11.3748	0.498341	0.249171	+	0.968460i	$0.419842\pi$
$522$	0	0
$523$	23.0957	1.00991	0.504953	−	0.863147i	$-0.331510\pi$
$523$	23.0957	1.00991	0.504953	+	0.863147i	$0.331510\pi$
$524$	0	0
$525$	−85.1096	−3.71449
$526$	0	0
$527$	−5.81421	−0.253271
$528$	0	0
$529$	−11.0660	−0.481130
$530$	0	0
$531$	0.815215	0.0353773
$532$	0	0
$533$	0.367048	0.0158986
$534$	0	0
$535$	−34.2886	−1.48243
$536$	0	0
$537$	−31.3641	−1.35346
$538$	0	0
$539$	13.1811	0.567749
$540$	0	0
$541$	−36.3006	−1.56069	−0.780343	−	0.625351i	$-0.784956\pi$
$541$	−36.3006	−1.56069	−0.780343	+	0.625351i	$0.784956\pi$
$542$	0	0
$543$	−42.9880	−1.84479
$544$	0	0
$545$	57.3301	2.45575
$546$	0	0
$547$	41.5715	1.77747	0.888734	−	0.458423i	$-0.151585\pi$
$547$	41.5715	1.77747	0.888734	+	0.458423i	$0.151585\pi$
$548$	0	0
$549$	−2.29582	−0.0979830
$550$	0	0
$551$	−24.9417	−1.06255
$552$	0	0
$553$	−22.6921	−0.964967
$554$	0	0
$555$	60.1352	2.55260
$556$	0	0
$557$	22.7368	0.963391	0.481696	−	0.876339i	$-0.340021\pi$
$557$	22.7368	0.963391	0.481696	+	0.876339i	$0.340021\pi$
$558$	0	0
$559$	6.57166	0.277952
$560$	0	0
$561$	−19.3223	−0.815787
$562$	0	0
$563$	−42.5706	−1.79414	−0.897068	−	0.441892i	$-0.854308\pi$
$563$	−42.5706	−1.79414	−0.897068	+	0.441892i	$0.854308\pi$
$564$	0	0
$565$	−70.4920	−2.96562
$566$	0	0
$567$	−31.1785	−1.30937
$568$	0	0
$569$	11.3517	0.475890	0.237945	−	0.971279i	$-0.423526\pi$
$569$	11.3517	0.475890	0.237945	+	0.971279i	$0.423526\pi$
$570$	0	0
$571$	−13.6448	−0.571016	−0.285508	−	0.958376i	$-0.592162\pi$
$571$	−13.6448	−0.571016	−0.285508	+	0.958376i	$0.592162\pi$
$572$	0	0
$573$	17.5379	0.732657
$574$	0	0
$575$	51.1632	2.13365
$576$	0	0
$577$	37.8838	1.57712	0.788561	−	0.614956i	$-0.210826\pi$
$577$	37.8838	1.57712	0.788561	+	0.614956i	$0.210826\pi$
$578$	0	0
$579$	−5.69258	−0.236576
$580$	0	0
$581$	−15.1257	−0.627520
$582$	0	0
$583$	−42.8461	−1.77450
$584$	0	0
$585$	1.24431	0.0514460
$586$	0	0
$587$	−14.5455	−0.600359	−0.300179	−	0.953883i	$-0.597047\pi$
$587$	−14.5455	−0.600359	−0.300179	+	0.953883i	$0.597047\pi$
$588$	0	0
$589$	−12.2790	−0.505948
$590$	0	0
$591$	38.2459	1.57323
$592$	0	0
$593$	7.64237	0.313835	0.156917	−	0.987612i	$-0.449844\pi$
$593$	7.64237	0.313835	0.156917	+	0.987612i	$0.449844\pi$
$594$	0	0
$595$	31.8404	1.30533
$596$	0	0
$597$	26.6730	1.09165
$598$	0	0
$599$	15.9795	0.652904	0.326452	−	0.945214i	$-0.394147\pi$
$599$	15.9795	0.652904	0.326452	+	0.945214i	$0.394147\pi$
$600$	0	0
$601$	−0.00926781	−0.000378042 0	−0.000189021	−	1.00000i	$-0.500060\pi$
$601$	−0.00926781	−0.000378042 0	−0.000189021		1.00000i	$0.500060\pi$
$602$	0	0
$603$	0.923607	0.0376122
$604$	0	0
$605$	46.3949	1.88622
$606$	0	0
$607$	25.3143	1.02748	0.513738	−	0.857947i	$-0.328260\pi$
$607$	25.3143	1.02748	0.513738	+	0.857947i	$0.328260\pi$
$608$	0	0
$609$	29.7717	1.20641
$610$	0	0
$611$	7.35169	0.297418
$612$	0	0
$613$	45.3976	1.83359	0.916795	−	0.399358i	$-0.130767\pi$
$613$	45.3976	1.83359	0.916795	+	0.399358i	$0.130767\pi$
$614$	0	0
$615$	−3.78229	−0.152517
$616$	0	0
$617$	3.69842	0.148893	0.0744463	−	0.997225i	$-0.476281\pi$
$617$	3.69842	0.148893	0.0744463	+	0.997225i	$0.476281\pi$
$618$	0	0
$619$	−39.7902	−1.59930	−0.799651	−	0.600465i	$-0.794982\pi$
$619$	−39.7902	−1.59930	−0.799651	+	0.600465i	$0.794982\pi$
$620$	0	0
$621$	16.7425	0.671852
$622$	0	0
$623$	43.5954	1.74661
$624$	0	0
$625$	120.294	4.81175
$626$	0	0
$627$	−40.8067	−1.62966
$628$	0	0
$629$	−16.8191	−0.670620
$630$	0	0
$631$	−1.92262	−0.0765382	−0.0382691	−	0.999267i	$-0.512184\pi$
$631$	−1.92262	−0.0765382	−0.0382691	+	0.999267i	$0.512184\pi$
$632$	0	0
$633$	23.6891	0.941558
$634$	0	0
$635$	−90.0689	−3.57427
$636$	0	0
$637$	2.25249	0.0892468
$638$	0	0
$639$	1.11244	0.0440073
$640$	0	0
$641$	3.10437	0.122615	0.0613077	−	0.998119i	$-0.480473\pi$
$641$	3.10437	0.122615	0.0613077	+	0.998119i	$0.480473\pi$
$642$	0	0
$643$	−17.6644	−0.696614	−0.348307	−	0.937380i	$-0.613243\pi$
$643$	−17.6644	−0.696614	−0.348307	+	0.937380i	$0.613243\pi$
$644$	0	0
$645$	−67.7184	−2.66641
$646$	0	0
$647$	−11.5894	−0.455626	−0.227813	−	0.973705i	$-0.573157\pi$
$647$	−11.5894	−0.455626	−0.227813	+	0.973705i	$0.573157\pi$
$648$	0	0
$649$	10.6757	0.419057
$650$	0	0
$651$	14.6569	0.574447
$652$	0	0
$653$	3.06370	0.119892	0.0599459	−	0.998202i	$-0.480907\pi$
$653$	3.06370	0.119892	0.0599459	+	0.998202i	$0.480907\pi$
$654$	0	0
$655$	28.7183	1.12212
$656$	0	0
$657$	3.41387	0.133188
$658$	0	0
$659$	21.0135	0.818569	0.409285	−	0.912407i	$-0.365778\pi$
$659$	21.0135	0.818569	0.409285	+	0.912407i	$0.365778\pi$
$660$	0	0
$661$	−46.3360	−1.80226	−0.901130	−	0.433549i	$-0.857261\pi$
$661$	−46.3360	−1.80226	−0.901130	+	0.433549i	$0.857261\pi$
$662$	0	0
$663$	−3.30194	−0.128237
$664$	0	0
$665$	67.2437	2.60760
$666$	0	0
$667$	−17.8971	−0.692977
$668$	0	0
$669$	−14.9463	−0.577857
$670$	0	0
$671$	−30.0649	−1.16064
$672$	0	0
$673$	1.76558	0.0680580	0.0340290	−	0.999421i	$-0.489166\pi$
$673$	1.76558	0.0680580	0.0340290	+	0.999421i	$0.489166\pi$
$674$	0	0
$675$	71.7778	2.76273
$676$	0	0
$677$	37.1878	1.42924	0.714621	−	0.699512i	$-0.246599\pi$
$677$	37.1878	1.42924	0.714621	+	0.699512i	$0.246599\pi$
$678$	0	0
$679$	−10.2037	−0.391581
$680$	0	0
$681$	32.2606	1.23623
$682$	0	0
$683$	41.6976	1.59552	0.797758	−	0.602978i	$-0.206019\pi$
$683$	41.6976	1.59552	0.797758	+	0.602978i	$0.206019\pi$
$684$	0	0
$685$	−72.7412	−2.77930
$686$	0	0
$687$	15.0535	0.574327
$688$	0	0
$689$	−7.32189	−0.278942
$690$	0	0
$691$	35.7510	1.36003	0.680015	−	0.733198i	$-0.261973\pi$
$691$	35.7510	1.36003	0.680015	+	0.733198i	$0.261973\pi$
$692$	0	0
$693$	5.13383	0.195018
$694$	0	0
$695$	56.5375	2.14459
$696$	0	0
$697$	1.05786	0.0400693
$698$	0	0
$699$	−42.2681	−1.59873
$700$	0	0
$701$	7.83620	0.295969	0.147985	−	0.988990i	$-0.452721\pi$
$701$	7.83620	0.295969	0.147985	+	0.988990i	$0.452721\pi$
$702$	0	0
$703$	−35.5201	−1.33967
$704$	0	0
$705$	−75.7564	−2.85315
$706$	0	0
$707$	−4.65860	−0.175205
$708$	0	0
$709$	−10.4499	−0.392453	−0.196227	−	0.980559i	$-0.562869\pi$
$709$	−10.4499	−0.392453	−0.196227	+	0.980559i	$0.562869\pi$
$710$	0	0
$711$	−2.55582	−0.0958509
$712$	0	0
$713$	−8.81088	−0.329970
$714$	0	0
$715$	16.2949	0.609397
$716$	0	0
$717$	11.6624	0.435541
$718$	0	0
$719$	−42.7845	−1.59559	−0.797796	−	0.602928i	$-0.794001\pi$
$719$	−42.7845	−1.59559	−0.797796	+	0.602928i	$0.794001\pi$
$720$	0	0
$721$	−31.2136	−1.16246
$722$	0	0
$723$	51.8170	1.92709
$724$	0	0
$725$	−76.7278	−2.84960
$726$	0	0
$727$	0.640295	0.0237472	0.0118736	−	0.999930i	$-0.496220\pi$
$727$	0.640295	0.0237472	0.0118736	+	0.999930i	$0.496220\pi$
$728$	0	0
$729$	23.1136	0.856058
$730$	0	0
$731$	18.9400	0.700521
$732$	0	0
$733$	26.6556	0.984547	0.492273	−	0.870441i	$-0.336166\pi$
$733$	26.6556	0.984547	0.492273	+	0.870441i	$0.336166\pi$
$734$	0	0
$735$	−23.2110	−0.856151
$736$	0	0
$737$	12.0951	0.445530
$738$	0	0
$739$	−37.3307	−1.37323	−0.686616	−	0.727021i	$-0.740904\pi$
$739$	−37.3307	−1.37323	−0.686616	+	0.727021i	$0.740904\pi$
$740$	0	0
$741$	−6.97337	−0.256173
$742$	0	0
$743$	−36.2770	−1.33087	−0.665437	−	0.746454i	$-0.731755\pi$
$743$	−36.2770	−1.33087	−0.665437	+	0.746454i	$0.731755\pi$
$744$	0	0
$745$	−71.3815	−2.61522
$746$	0	0
$747$	−1.70362	−0.0623320
$748$	0	0
$749$	−24.1754	−0.883349
$750$	0	0
$751$	6.93874	0.253198	0.126599	−	0.991954i	$-0.459594\pi$
$751$	6.93874	0.253198	0.126599	+	0.991954i	$0.459594\pi$
$752$	0	0
$753$	1.83124	0.0667342
$754$	0	0
$755$	−81.2240	−2.95604
$756$	0	0
$757$	−53.9566	−1.96109	−0.980543	−	0.196305i	$-0.937106\pi$
$757$	−53.9566	−1.96109	−0.980543	+	0.196305i	$0.937106\pi$
$758$	0	0
$759$	−29.2811	−1.06284
$760$	0	0
$761$	−8.07668	−0.292779	−0.146390	−	0.989227i	$-0.546765\pi$
$761$	−8.07668	−0.292779	−0.146390	+	0.989227i	$0.546765\pi$
$762$	0	0
$763$	40.4209	1.46333
$764$	0	0
$765$	3.58620	0.129659
$766$	0	0
$767$	1.82434	0.0658733
$768$	0	0
$769$	40.1797	1.44892	0.724458	−	0.689319i	$-0.242090\pi$
$769$	40.1797	1.44892	0.724458	+	0.689319i	$0.242090\pi$
$770$	0	0
$771$	18.9709	0.683220
$772$	0	0
$773$	39.1086	1.40664	0.703319	−	0.710874i	$-0.251701\pi$
$773$	39.1086	1.40664	0.703319	+	0.710874i	$0.251701\pi$
$774$	0	0
$775$	−37.7738	−1.35687
$776$	0	0
$777$	42.3986	1.52104
$778$	0	0
$779$	2.23409	0.0800446
$780$	0	0
$781$	14.5680	0.521282
$782$	0	0
$783$	−25.1082	−0.897293
$784$	0	0
$785$	−25.5022	−0.910213
$786$	0	0
$787$	47.2327	1.68366	0.841832	−	0.539740i	$-0.181477\pi$
$787$	47.2327	1.68366	0.841832	+	0.539740i	$0.181477\pi$
$788$	0	0
$789$	3.47296	0.123641
$790$	0	0
$791$	−49.7007	−1.76715
$792$	0	0
$793$	−5.13774	−0.182446
$794$	0	0
$795$	75.4493	2.67591
$796$	0	0
$797$	−49.5646	−1.75567	−0.877835	−	0.478964i	$-0.841013\pi$
$797$	−49.5646	−1.75567	−0.877835	+	0.478964i	$0.841013\pi$
$798$	0	0
$799$	21.1881	0.749582
$800$	0	0
$801$	4.91017	0.173492
$802$	0	0
$803$	44.7065	1.57766
$804$	0	0
$805$	48.2511	1.70063
$806$	0	0
$807$	26.5866	0.935891
$808$	0	0
$809$	−35.1152	−1.23459	−0.617293	−	0.786733i	$-0.711771\pi$
$809$	−35.1152	−1.23459	−0.617293	+	0.786733i	$0.711771\pi$
$810$	0	0
$811$	40.9708	1.43868	0.719340	−	0.694659i	$-0.244445\pi$
$811$	40.9708	1.43868	0.719340	+	0.694659i	$0.244445\pi$
$812$	0	0
$813$	20.2650	0.710723
$814$	0	0
$815$	−32.0799	−1.12371
$816$	0	0
$817$	39.9994	1.39940
$818$	0	0
$819$	0.877309	0.0306557
$820$	0	0
$821$	−9.27997	−0.323873	−0.161937	−	0.986801i	$-0.551774\pi$
$821$	−9.27997	−0.323873	−0.161937	+	0.986801i	$0.551774\pi$
$822$	0	0
$823$	7.25473	0.252884	0.126442	−	0.991974i	$-0.459644\pi$
$823$	7.25473	0.252884	0.126442	+	0.991974i	$0.459644\pi$
$824$	0	0
$825$	−125.533	−4.37050
$826$	0	0
$827$	−10.4529	−0.363484	−0.181742	−	0.983346i	$-0.558174\pi$
$827$	−10.4529	−0.363484	−0.181742	+	0.983346i	$0.558174\pi$
$828$	0	0
$829$	−6.43336	−0.223440	−0.111720	−	0.993740i	$-0.535636\pi$
$829$	−6.43336	−0.223440	−0.111720	+	0.993740i	$0.535636\pi$
$830$	0	0
$831$	−2.07822	−0.0720927
$832$	0	0
$833$	6.49183	0.224929
$834$	0	0
$835$	−18.0162	−0.623478
$836$	0	0
$837$	−12.3610	−0.427258
$838$	0	0
$839$	52.2469	1.80376	0.901882	−	0.431983i	$-0.142186\pi$
$839$	52.2469	1.80376	0.901882	+	0.431983i	$0.142186\pi$
$840$	0	0
$841$	−2.16032	−0.0744937
$842$	0	0
$843$	2.29043	0.0788867
$844$	0	0
$845$	−55.0768	−1.89470
$846$	0	0
$847$	32.7109	1.12396
$848$	0	0
$849$	−10.2446	−0.351595
$850$	0	0
$851$	−25.4877	−0.873706
$852$	0	0
$853$	24.0375	0.823027	0.411514	−	0.911404i	$-0.365000\pi$
$853$	24.0375	0.823027	0.411514	+	0.911404i	$0.365000\pi$
$854$	0	0
$855$	7.57369	0.259015
$856$	0	0
$857$	−54.8808	−1.87469	−0.937346	−	0.348399i	$-0.886725\pi$
$857$	−54.8808	−1.87469	−0.937346	+	0.348399i	$0.886725\pi$
$858$	0	0
$859$	−10.3844	−0.354312	−0.177156	−	0.984183i	$-0.556690\pi$
$859$	−10.3844	−0.354312	−0.177156	+	0.984183i	$0.556690\pi$
$860$	0	0
$861$	−2.66672	−0.0908816
$862$	0	0
$863$	35.9218	1.22279	0.611395	−	0.791325i	$-0.290609\pi$
$863$	35.9218	1.22279	0.611395	+	0.791325i	$0.290609\pi$
$864$	0	0
$865$	31.3444	1.06574
$866$	0	0
$867$	21.6147	0.734073
$868$	0	0
$869$	−33.4699	−1.13539
$870$	0	0
$871$	2.06691	0.0700346
$872$	0	0
$873$	−1.14925	−0.0388961
$874$	0	0
$875$	137.024	4.63226
$876$	0	0
$877$	32.2344	1.08848	0.544239	−	0.838931i	$-0.316819\pi$
$877$	32.2344	1.08848	0.544239	+	0.838931i	$0.316819\pi$
$878$	0	0
$879$	24.2467	0.817820
$880$	0	0
$881$	0.528573	0.0178081	0.00890404	−	0.999960i	$-0.497166\pi$
$881$	0.528573	0.0178081	0.00890404	+	0.999960i	$0.497166\pi$
$882$	0	0
$883$	36.2015	1.21828	0.609139	−	0.793063i	$-0.291515\pi$
$883$	36.2015	1.21828	0.609139	+	0.793063i	$0.291515\pi$
$884$	0	0
$885$	−18.7992	−0.631927
$886$	0	0
$887$	−17.7690	−0.596624	−0.298312	−	0.954468i	$-0.596424\pi$
$887$	−17.7690	−0.596624	−0.298312	+	0.954468i	$0.596424\pi$
$888$	0	0
$889$	−63.5035	−2.12984
$890$	0	0
$891$	−45.9869	−1.54062
$892$	0	0
$893$	44.7471	1.49741
$894$	0	0
$895$	76.2311	2.54813
$896$	0	0
$897$	−5.00378	−0.167071
$898$	0	0
$899$	13.2134	0.440692
$900$	0	0
$901$	−21.1022	−0.703017
$902$	0	0
$903$	−47.7452	−1.58886
$904$	0	0
$905$	104.483	3.47314
$906$	0	0
$907$	−23.0725	−0.766109	−0.383054	−	0.923726i	$-0.625128\pi$
$907$	−23.0725	−0.766109	−0.383054	+	0.923726i	$0.625128\pi$
$908$	0	0
$909$	−0.524700	−0.0174032
$910$	0	0
$911$	4.47965	0.148417	0.0742087	−	0.997243i	$-0.476357\pi$
$911$	4.47965	0.148417	0.0742087	+	0.997243i	$0.476357\pi$
$912$	0	0
$913$	−22.3098	−0.738345
$914$	0	0
$915$	52.9424	1.75022
$916$	0	0
$917$	20.2480	0.668648
$918$	0	0
$919$	39.8745	1.31534	0.657669	−	0.753307i	$-0.271543\pi$
$919$	39.8745	1.31534	0.657669	+	0.753307i	$0.271543\pi$
$920$	0	0
$921$	−24.0585	−0.792754
$922$	0	0
$923$	2.48949	0.0819425
$924$	0	0
$925$	−109.270	−3.59278
$926$	0	0
$927$	−3.51560	−0.115468
$928$	0	0
$929$	−24.0062	−0.787619	−0.393809	−	0.919192i	$-0.628843\pi$
$929$	−24.0062	−0.787619	−0.393809	+	0.919192i	$0.628843\pi$
$930$	0	0
$931$	13.7101	0.449330
$932$	0	0
$933$	54.3611	1.77970
$934$	0	0
$935$	46.9632	1.53586
$936$	0	0
$937$	51.2395	1.67392	0.836962	−	0.547262i	$-0.184330\pi$
$937$	51.2395	1.67392	0.836962	+	0.547262i	$0.184330\pi$
$938$	0	0
$939$	14.6848	0.479220
$940$	0	0
$941$	8.31201	0.270964	0.135482	−	0.990780i	$-0.456742\pi$
$941$	8.31201	0.270964	0.135482	+	0.990780i	$0.456742\pi$
$942$	0	0
$943$	1.60309	0.0522036
$944$	0	0
$945$	67.6925	2.20204
$946$	0	0
$947$	41.1004	1.33558	0.667792	−	0.744348i	$-0.267240\pi$
$947$	41.1004	1.33558	0.667792	+	0.744348i	$0.267240\pi$
$948$	0	0
$949$	7.63980	0.247998
$950$	0	0
$951$	49.4379	1.60313
$952$	0	0
$953$	−38.7539	−1.25536	−0.627682	−	0.778470i	$-0.715996\pi$
$953$	−38.7539	−1.25536	−0.627682	+	0.778470i	$0.715996\pi$
$954$	0	0
$955$	−42.6263	−1.37936
$956$	0	0
$957$	43.9119	1.41947
$958$	0	0
$959$	−51.2866	−1.65613
$960$	0	0
$961$	−24.4949	−0.790159
$962$	0	0
$963$	−2.72288	−0.0877437
$964$	0	0
$965$	13.8359	0.445395
$966$	0	0
$967$	−22.5151	−0.724036	−0.362018	−	0.932171i	$-0.617912\pi$
$967$	−22.5151	−0.724036	−0.362018	+	0.932171i	$0.617912\pi$
$968$	0	0
$969$	−20.0978	−0.645633
$970$	0	0
$971$	54.1975	1.73928	0.869640	−	0.493686i	$-0.164351\pi$
$971$	54.1975	1.73928	0.869640	+	0.493686i	$0.164351\pi$
$972$	0	0
$973$	39.8620	1.27792
$974$	0	0
$975$	−21.4521	−0.687016
$976$	0	0
$977$	−13.6396	−0.436370	−0.218185	−	0.975907i	$-0.570014\pi$
$977$	−13.6396	−0.436370	−0.218185	+	0.975907i	$0.570014\pi$
$978$	0	0
$979$	64.3013	2.05508
$980$	0	0
$981$	4.55263	0.145354
$982$	0	0
$983$	56.6625	1.80725	0.903626	−	0.428322i	$-0.140895\pi$
$983$	56.6625	1.80725	0.903626	+	0.428322i	$0.140895\pi$
$984$	0	0
$985$	−92.9577	−2.96188
$986$	0	0
$987$	−53.4124	−1.70014
$988$	0	0
$989$	28.7018	0.912663
$990$	0	0
$991$	−32.6957	−1.03861	−0.519307	−	0.854588i	$-0.673810\pi$
$991$	−32.6957	−1.03861	−0.519307	+	0.854588i	$0.673810\pi$
$992$	0	0
$993$	−15.1176	−0.479741
$994$	0	0
$995$	−64.8294	−2.05523
$996$	0	0
$997$	58.0684	1.83905	0.919523	−	0.393036i	$-0.128575\pi$
$997$	58.0684	1.83905	0.919523	+	0.393036i	$0.128575\pi$
$998$	0	0
$999$	−35.7572	−1.13131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.6	yes	30
4.3	odd	2		8032.2.a.g.1.25	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.25	✓	30	4.3	odd	2
8032.2.a.j.1.6	yes	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83124 q^{3} +4.45088 q^{5} +3.13811 q^{7} +0.353447 q^{9} +O(q^{10})\)	\(q-1.83124 q^{3} +4.45088 q^{5} +3.13811 q^{7} +0.353447 q^{9} +4.62858 q^{11} +0.790969 q^{13} -8.15063 q^{15} +2.27963 q^{17} +4.81435 q^{19} -5.74665 q^{21} +3.45456 q^{23} +14.8103 q^{25} +4.84648 q^{27} -5.18070 q^{29} -2.55051 q^{31} -8.47605 q^{33} +13.9674 q^{35} -7.37797 q^{37} -1.44846 q^{39} +0.464049 q^{41} +8.30836 q^{43} +1.57315 q^{45} +9.29454 q^{47} +2.84776 q^{49} -4.17456 q^{51} -9.25686 q^{53} +20.6012 q^{55} -8.81624 q^{57} +2.30647 q^{59} -6.49550 q^{61} +1.10916 q^{63} +3.52051 q^{65} +2.61314 q^{67} -6.32614 q^{69} +3.14739 q^{71} +9.65879 q^{73} -27.1213 q^{75} +14.5250 q^{77} -7.23113 q^{79} -9.93542 q^{81} -4.82000 q^{83} +10.1464 q^{85} +9.48712 q^{87} +13.8922 q^{89} +2.48215 q^{91} +4.67059 q^{93} +21.4281 q^{95} -3.25153 q^{97} +1.63596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

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