LMFDB - Embedded newform 8016.2.a.z.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 $ x^8 - 2x^7 - 8x^6 + 15x^5 + 19x^4 - 31x^3 - 13x^2 + 14*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.60046$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.28584 q^{5} +0.120874 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.28584 q^{5} +0.120874 q^{7} +1.00000 q^{9} -5.49363 q^{11} -3.49006 q^{13} +1.28584 q^{15} +4.85075 q^{17} -1.05251 q^{19} +0.120874 q^{21} +7.39997 q^{23} -3.34662 q^{25} +1.00000 q^{27} +0.667361 q^{29} -10.3025 q^{31} -5.49363 q^{33} +0.155425 q^{35} -7.38582 q^{37} -3.49006 q^{39} +6.28456 q^{41} +3.21086 q^{43} +1.28584 q^{45} +11.6838 q^{47} -6.98539 q^{49} +4.85075 q^{51} -12.5451 q^{53} -7.06393 q^{55} -1.05251 q^{57} -1.90048 q^{59} +2.26436 q^{61} +0.120874 q^{63} -4.48766 q^{65} -7.89898 q^{67} +7.39997 q^{69} +6.59825 q^{71} -0.550109 q^{73} -3.34662 q^{75} -0.664039 q^{77} +0.308781 q^{79} +1.00000 q^{81} -12.7914 q^{83} +6.23729 q^{85} +0.667361 q^{87} -0.397213 q^{89} -0.421859 q^{91} -10.3025 q^{93} -1.35335 q^{95} +8.13790 q^{97} -5.49363 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.28584	0.575045	0.287523	−	0.957774i	$-0.407168\pi$
$5$	1.28584	0.575045	0.287523	+	0.957774i	$0.407168\pi$
$6$	0	0
$7$	0.120874	0.0456862	0.0228431	−	0.999739i	$-0.492728\pi$
$7$	0.120874	0.0456862	0.0228431	+	0.999739i	$0.492728\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.49363	−1.65639	−0.828196	−	0.560439i	$-0.810632\pi$
$11$	−5.49363	−1.65639	−0.828196	+	0.560439i	$0.810632\pi$
$12$	0	0
$13$	−3.49006	−0.967969	−0.483985	−	0.875076i	$-0.660811\pi$
$13$	−3.49006	−0.967969	−0.483985	+	0.875076i	$0.660811\pi$
$14$	0	0
$15$	1.28584	0.332003
$16$	0	0
$17$	4.85075	1.17648	0.588240	−	0.808686i	$-0.299821\pi$
$17$	4.85075	1.17648	0.588240	+	0.808686i	$0.299821\pi$
$18$	0	0
$19$	−1.05251	−0.241461	−0.120731	−	0.992685i	$-0.538524\pi$
$19$	−1.05251	−0.241461	−0.120731	+	0.992685i	$0.538524\pi$
$20$	0	0
$21$	0.120874	0.0263769
$22$	0	0
$23$	7.39997	1.54300	0.771501	−	0.636228i	$-0.219506\pi$
$23$	7.39997	1.54300	0.771501	+	0.636228i	$0.219506\pi$
$24$	0	0
$25$	−3.34662	−0.669323
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.667361	0.123926	0.0619629	−	0.998078i	$-0.480264\pi$
$29$	0.667361	0.123926	0.0619629	+	0.998078i	$0.480264\pi$
$30$	0	0
$31$	−10.3025	−1.85039	−0.925194	−	0.379495i	$-0.876098\pi$
$31$	−10.3025	−1.85039	−0.925194	+	0.379495i	$0.876098\pi$
$32$	0	0
$33$	−5.49363	−0.956318
$34$	0	0
$35$	0.155425	0.0262716
$36$	0	0
$37$	−7.38582	−1.21422	−0.607110	−	0.794618i	$-0.707671\pi$
$37$	−7.38582	−1.21422	−0.607110	+	0.794618i	$0.707671\pi$
$38$	0	0
$39$	−3.49006	−0.558857
$40$	0	0
$41$	6.28456	0.981484	0.490742	−	0.871305i	$-0.336726\pi$
$41$	6.28456	0.981484	0.490742	+	0.871305i	$0.336726\pi$
$42$	0	0
$43$	3.21086	0.489651	0.244825	−	0.969567i	$-0.421269\pi$
$43$	3.21086	0.489651	0.244825	+	0.969567i	$0.421269\pi$
$44$	0	0
$45$	1.28584	0.191682
$46$	0	0
$47$	11.6838	1.70425	0.852127	−	0.523334i	$-0.175312\pi$
$47$	11.6838	1.70425	0.852127	+	0.523334i	$0.175312\pi$
$48$	0	0
$49$	−6.98539	−0.997913
$50$	0	0
$51$	4.85075	0.679241
$52$	0	0
$53$	−12.5451	−1.72320	−0.861600	−	0.507589i	$-0.830537\pi$
$53$	−12.5451	−1.72320	−0.861600	+	0.507589i	$0.830537\pi$
$54$	0	0
$55$	−7.06393	−0.952500
$56$	0	0
$57$	−1.05251	−0.139408
$58$	0	0
$59$	−1.90048	−0.247421	−0.123711	−	0.992318i	$-0.539479\pi$
$59$	−1.90048	−0.247421	−0.123711	+	0.992318i	$0.539479\pi$
$60$	0	0
$61$	2.26436	0.289922	0.144961	−	0.989437i	$-0.453694\pi$
$61$	2.26436	0.289922	0.144961	+	0.989437i	$0.453694\pi$
$62$	0	0
$63$	0.120874	0.0152287
$64$	0	0
$65$	−4.48766	−0.556626
$66$	0	0
$67$	−7.89898	−0.965014	−0.482507	−	0.875892i	$-0.660274\pi$
$67$	−7.89898	−0.965014	−0.482507	+	0.875892i	$0.660274\pi$
$68$	0	0
$69$	7.39997	0.890852
$70$	0	0
$71$	6.59825	0.783068	0.391534	−	0.920164i	$-0.371945\pi$
$71$	6.59825	0.783068	0.391534	+	0.920164i	$0.371945\pi$
$72$	0	0
$73$	−0.550109	−0.0643854	−0.0321927	−	0.999482i	$-0.510249\pi$
$73$	−0.550109	−0.0643854	−0.0321927	+	0.999482i	$0.510249\pi$
$74$	0	0
$75$	−3.34662	−0.386434
$76$	0	0
$77$	−0.664039	−0.0756742
$78$	0	0
$79$	0.308781	0.0347406	0.0173703	−	0.999849i	$-0.494471\pi$
$79$	0.308781	0.0347406	0.0173703	+	0.999849i	$0.494471\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.7914	−1.40404	−0.702021	−	0.712157i	$-0.747719\pi$
$83$	−12.7914	−1.40404	−0.702021	+	0.712157i	$0.747719\pi$
$84$	0	0
$85$	6.23729	0.676530
$86$	0	0
$87$	0.667361	0.0715486
$88$	0	0
$89$	−0.397213	−0.0421045	−0.0210523	−	0.999778i	$-0.506702\pi$
$89$	−0.397213	−0.0421045	−0.0210523	+	0.999778i	$0.506702\pi$
$90$	0	0
$91$	−0.421859	−0.0442228
$92$	0	0
$93$	−10.3025	−1.06832
$94$	0	0
$95$	−1.35335	−0.138851
$96$	0	0
$97$	8.13790	0.826278	0.413139	−	0.910668i	$-0.364432\pi$
$97$	8.13790	0.826278	0.413139	+	0.910668i	$0.364432\pi$
$98$	0	0
$99$	−5.49363	−0.552131
$100$	0	0
$101$	0.132587	0.0131929	0.00659646	−	0.999978i	$-0.497900\pi$
$101$	0.132587	0.0131929	0.00659646	+	0.999978i	$0.497900\pi$
$102$	0	0
$103$	6.96739	0.686518	0.343259	−	0.939241i	$-0.388469\pi$
$103$	6.96739	0.686518	0.343259	+	0.939241i	$0.388469\pi$
$104$	0	0
$105$	0.155425	0.0151679
$106$	0	0
$107$	−12.8585	−1.24308	−0.621538	−	0.783384i	$-0.713492\pi$
$107$	−12.8585	−1.24308	−0.621538	+	0.783384i	$0.713492\pi$
$108$	0	0
$109$	−6.78250	−0.649646	−0.324823	−	0.945775i	$-0.605305\pi$
$109$	−6.78250	−0.649646	−0.324823	+	0.945775i	$0.605305\pi$
$110$	0	0
$111$	−7.38582	−0.701031
$112$	0	0
$113$	−13.8293	−1.30095	−0.650475	−	0.759527i	$-0.725430\pi$
$113$	−13.8293	−1.30095	−0.650475	+	0.759527i	$0.725430\pi$
$114$	0	0
$115$	9.51518	0.887296
$116$	0	0
$117$	−3.49006	−0.322656
$118$	0	0
$119$	0.586332	0.0537489
$120$	0	0
$121$	19.1800	1.74363
$122$	0	0
$123$	6.28456	0.566660
$124$	0	0
$125$	−10.7324	−0.959936
$126$	0	0
$127$	−12.9151	−1.14603	−0.573014	−	0.819545i	$-0.694226\pi$
$127$	−12.9151	−1.14603	−0.573014	+	0.819545i	$0.694226\pi$
$128$	0	0
$129$	3.21086	0.282700
$130$	0	0
$131$	−1.81550	−0.158621	−0.0793103	−	0.996850i	$-0.525272\pi$
$131$	−1.81550	−0.158621	−0.0793103	+	0.996850i	$0.525272\pi$
$132$	0	0
$133$	−0.127221	−0.0110315
$134$	0	0
$135$	1.28584	0.110668
$136$	0	0
$137$	−6.15600	−0.525943	−0.262971	−	0.964804i	$-0.584702\pi$
$137$	−6.15600	−0.525943	−0.262971	+	0.964804i	$0.584702\pi$
$138$	0	0
$139$	−2.88427	−0.244641	−0.122320	−	0.992491i	$-0.539034\pi$
$139$	−2.88427	−0.244641	−0.122320	+	0.992491i	$0.539034\pi$
$140$	0	0
$141$	11.6838	0.983952
$142$	0	0
$143$	19.1731	1.60334
$144$	0	0
$145$	0.858119	0.0712629
$146$	0	0
$147$	−6.98539	−0.576145
$148$	0	0
$149$	−15.3797	−1.25995	−0.629976	−	0.776615i	$-0.716935\pi$
$149$	−15.3797	−1.25995	−0.629976	+	0.776615i	$0.716935\pi$
$150$	0	0
$151$	−21.5417	−1.75304	−0.876518	−	0.481369i	$-0.840140\pi$
$151$	−21.5417	−1.75304	−0.876518	+	0.481369i	$0.840140\pi$
$152$	0	0
$153$	4.85075	0.392160
$154$	0	0
$155$	−13.2474	−1.06406
$156$	0	0
$157$	21.4579	1.71253	0.856265	−	0.516537i	$-0.172779\pi$
$157$	21.4579	1.71253	0.856265	+	0.516537i	$0.172779\pi$
$158$	0	0
$159$	−12.5451	−0.994890
$160$	0	0
$161$	0.894467	0.0704939
$162$	0	0
$163$	−4.34046	−0.339971	−0.169985	−	0.985447i	$-0.554372\pi$
$163$	−4.34046	−0.339971	−0.169985	+	0.985447i	$0.554372\pi$
$164$	0	0
$165$	−7.06393	−0.549926
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−0.819459	−0.0630353
$170$	0	0
$171$	−1.05251	−0.0804872
$172$	0	0
$173$	−14.3565	−1.09150	−0.545750	−	0.837948i	$-0.683755\pi$
$173$	−14.3565	−1.09150	−0.545750	+	0.837948i	$0.683755\pi$
$174$	0	0
$175$	−0.404520	−0.0305788
$176$	0	0
$177$	−1.90048	−0.142849
$178$	0	0
$179$	−13.2707	−0.991897	−0.495949	−	0.868352i	$-0.665180\pi$
$179$	−13.2707	−0.991897	−0.495949	+	0.868352i	$0.665180\pi$
$180$	0	0
$181$	2.98050	0.221539	0.110770	−	0.993846i	$-0.464668\pi$
$181$	2.98050	0.221539	0.110770	+	0.993846i	$0.464668\pi$
$182$	0	0
$183$	2.26436	0.167386
$184$	0	0
$185$	−9.49698	−0.698232
$186$	0	0
$187$	−26.6482	−1.94871
$188$	0	0
$189$	0.120874	0.00879231
$190$	0	0
$191$	−25.0935	−1.81570	−0.907852	−	0.419292i	$-0.862278\pi$
$191$	−25.0935	−1.81570	−0.907852	+	0.419292i	$0.862278\pi$
$192$	0	0
$193$	25.9678	1.86920	0.934602	−	0.355696i	$-0.115756\pi$
$193$	25.9678	1.86920	0.934602	+	0.355696i	$0.115756\pi$
$194$	0	0
$195$	−4.48766	−0.321368
$196$	0	0
$197$	−9.01491	−0.642285	−0.321143	−	0.947031i	$-0.604067\pi$
$197$	−9.01491	−0.642285	−0.321143	+	0.947031i	$0.604067\pi$
$198$	0	0
$199$	7.13075	0.505485	0.252743	−	0.967534i	$-0.418667\pi$
$199$	7.13075	0.505485	0.252743	+	0.967534i	$0.418667\pi$
$200$	0	0
$201$	−7.89898	−0.557151
$202$	0	0
$203$	0.0806668	0.00566170
$204$	0	0
$205$	8.08095	0.564398
$206$	0	0
$207$	7.39997	0.514334
$208$	0	0
$209$	5.78208	0.399955
$210$	0	0
$211$	−9.22696	−0.635210	−0.317605	−	0.948223i	$-0.602879\pi$
$211$	−9.22696	−0.635210	−0.317605	+	0.948223i	$0.602879\pi$
$212$	0	0
$213$	6.59825	0.452104
$214$	0	0
$215$	4.12865	0.281571
$216$	0	0
$217$	−1.24531	−0.0845372
$218$	0	0
$219$	−0.550109	−0.0371729
$220$	0	0
$221$	−16.9294	−1.13880
$222$	0	0
$223$	−14.0521	−0.940999	−0.470500	−	0.882400i	$-0.655926\pi$
$223$	−14.0521	−0.940999	−0.470500	+	0.882400i	$0.655926\pi$
$224$	0	0
$225$	−3.34662	−0.223108
$226$	0	0
$227$	−5.87728	−0.390089	−0.195044	−	0.980794i	$-0.562485\pi$
$227$	−5.87728	−0.390089	−0.195044	+	0.980794i	$0.562485\pi$
$228$	0	0
$229$	−6.58989	−0.435472	−0.217736	−	0.976008i	$-0.569867\pi$
$229$	−6.58989	−0.435472	−0.217736	+	0.976008i	$0.569867\pi$
$230$	0	0
$231$	−0.664039	−0.0436905
$232$	0	0
$233$	21.5949	1.41473	0.707366	−	0.706847i	$-0.249883\pi$
$233$	21.5949	1.41473	0.707366	+	0.706847i	$0.249883\pi$
$234$	0	0
$235$	15.0235	0.980024
$236$	0	0
$237$	0.308781	0.0200575
$238$	0	0
$239$	−13.5344	−0.875470	−0.437735	−	0.899104i	$-0.644219\pi$
$239$	−13.5344	−0.875470	−0.437735	+	0.899104i	$0.644219\pi$
$240$	0	0
$241$	−0.680726	−0.0438494	−0.0219247	−	0.999760i	$-0.506979\pi$
$241$	−0.680726	−0.0438494	−0.0219247	+	0.999760i	$0.506979\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−8.98209	−0.573845
$246$	0	0
$247$	3.67331	0.233727
$248$	0	0
$249$	−12.7914	−0.810624
$250$	0	0
$251$	20.5812	1.29907	0.649537	−	0.760330i	$-0.274963\pi$
$251$	20.5812	1.29907	0.649537	+	0.760330i	$0.274963\pi$
$252$	0	0
$253$	−40.6527	−2.55581
$254$	0	0
$255$	6.23729	0.390595
$256$	0	0
$257$	−10.4824	−0.653875	−0.326938	−	0.945046i	$-0.606017\pi$
$257$	−10.4824	−0.653875	−0.326938	+	0.945046i	$0.606017\pi$
$258$	0	0
$259$	−0.892755	−0.0554731
$260$	0	0
$261$	0.667361	0.0413086
$262$	0	0
$263$	−3.31478	−0.204398	−0.102199	−	0.994764i	$-0.532588\pi$
$263$	−3.31478	−0.204398	−0.102199	+	0.994764i	$0.532588\pi$
$264$	0	0
$265$	−16.1310	−0.990917
$266$	0	0
$267$	−0.397213	−0.0243091
$268$	0	0
$269$	13.2626	0.808638	0.404319	−	0.914618i	$-0.367509\pi$
$269$	13.2626	0.808638	0.404319	+	0.914618i	$0.367509\pi$
$270$	0	0
$271$	11.4876	0.697819	0.348910	−	0.937156i	$-0.386552\pi$
$271$	11.4876	0.697819	0.348910	+	0.937156i	$0.386552\pi$
$272$	0	0
$273$	−0.421859	−0.0255321
$274$	0	0
$275$	18.3851	1.10866
$276$	0	0
$277$	−1.44455	−0.0867943	−0.0433972	−	0.999058i	$-0.513818\pi$
$277$	−1.44455	−0.0867943	−0.0433972	+	0.999058i	$0.513818\pi$
$278$	0	0
$279$	−10.3025	−0.616796
$280$	0	0
$281$	−24.5424	−1.46408	−0.732039	−	0.681262i	$-0.761431\pi$
$281$	−24.5424	−1.46408	−0.732039	+	0.681262i	$0.761431\pi$
$282$	0	0
$283$	−3.57120	−0.212286	−0.106143	−	0.994351i	$-0.533850\pi$
$283$	−3.57120	−0.212286	−0.106143	+	0.994351i	$0.533850\pi$
$284$	0	0
$285$	−1.35335	−0.0801658
$286$	0	0
$287$	0.759642	0.0448403
$288$	0	0
$289$	6.52981	0.384107
$290$	0	0
$291$	8.13790	0.477052
$292$	0	0
$293$	−7.08534	−0.413930	−0.206965	−	0.978348i	$-0.566359\pi$
$293$	−7.08534	−0.413930	−0.206965	+	0.978348i	$0.566359\pi$
$294$	0	0
$295$	−2.44371	−0.142278
$296$	0	0
$297$	−5.49363	−0.318773
$298$	0	0
$299$	−25.8264	−1.49358
$300$	0	0
$301$	0.388110	0.0223703
$302$	0	0
$303$	0.132587	0.00761693
$304$	0	0
$305$	2.91160	0.166718
$306$	0	0
$307$	−2.04358	−0.116633	−0.0583167	−	0.998298i	$-0.518573\pi$
$307$	−2.04358	−0.116633	−0.0583167	+	0.998298i	$0.518573\pi$
$308$	0	0
$309$	6.96739	0.396361
$310$	0	0
$311$	25.5906	1.45111	0.725556	−	0.688163i	$-0.241583\pi$
$311$	25.5906	1.45111	0.725556	+	0.688163i	$0.241583\pi$
$312$	0	0
$313$	12.8093	0.724023	0.362011	−	0.932174i	$-0.382090\pi$
$313$	12.8093	0.724023	0.362011	+	0.932174i	$0.382090\pi$
$314$	0	0
$315$	0.155425	0.00875721
$316$	0	0
$317$	13.3331	0.748861	0.374431	−	0.927255i	$-0.377838\pi$
$317$	13.3331	0.748861	0.374431	+	0.927255i	$0.377838\pi$
$318$	0	0
$319$	−3.66623	−0.205270
$320$	0	0
$321$	−12.8585	−0.717690
$322$	0	0
$323$	−5.10545	−0.284075
$324$	0	0
$325$	11.6799	0.647884
$326$	0	0
$327$	−6.78250	−0.375073
$328$	0	0
$329$	1.41227	0.0778609
$330$	0	0
$331$	32.0159	1.75975	0.879876	−	0.475203i	$-0.157625\pi$
$331$	32.0159	1.75975	0.879876	+	0.475203i	$0.157625\pi$
$332$	0	0
$333$	−7.38582	−0.404740
$334$	0	0
$335$	−10.1568	−0.554927
$336$	0	0
$337$	2.34603	0.127797	0.0638983	−	0.997956i	$-0.479647\pi$
$337$	2.34603	0.127797	0.0638983	+	0.997956i	$0.479647\pi$
$338$	0	0
$339$	−13.8293	−0.751104
$340$	0	0
$341$	56.5982	3.06497
$342$	0	0
$343$	−1.69047	−0.0912770
$344$	0	0
$345$	9.51518	0.512280
$346$	0	0
$347$	−14.7593	−0.792321	−0.396161	−	0.918181i	$-0.629658\pi$
$347$	−14.7593	−0.792321	−0.396161	+	0.918181i	$0.629658\pi$
$348$	0	0
$349$	32.0582	1.71604	0.858019	−	0.513618i	$-0.171695\pi$
$349$	32.0582	1.71604	0.858019	+	0.513618i	$0.171695\pi$
$350$	0	0
$351$	−3.49006	−0.186286
$352$	0	0
$353$	−4.46611	−0.237707	−0.118853	−	0.992912i	$-0.537922\pi$
$353$	−4.46611	−0.237707	−0.118853	+	0.992912i	$0.537922\pi$
$354$	0	0
$355$	8.48429	0.450299
$356$	0	0
$357$	0.586332	0.0310320
$358$	0	0
$359$	17.0582	0.900296	0.450148	−	0.892954i	$-0.351371\pi$
$359$	17.0582	0.900296	0.450148	+	0.892954i	$0.351371\pi$
$360$	0	0
$361$	−17.8922	−0.941696
$362$	0	0
$363$	19.1800	1.00669
$364$	0	0
$365$	−0.707353	−0.0370245
$366$	0	0
$367$	12.2661	0.640284	0.320142	−	0.947370i	$-0.396269\pi$
$367$	12.2661	0.640284	0.320142	+	0.947370i	$0.396269\pi$
$368$	0	0
$369$	6.28456	0.327161
$370$	0	0
$371$	−1.51638	−0.0787264
$372$	0	0
$373$	23.5400	1.21885	0.609427	−	0.792842i	$-0.291399\pi$
$373$	23.5400	1.21885	0.609427	+	0.792842i	$0.291399\pi$
$374$	0	0
$375$	−10.7324	−0.554219
$376$	0	0
$377$	−2.32913	−0.119956
$378$	0	0
$379$	9.01330	0.462982	0.231491	−	0.972837i	$-0.425640\pi$
$379$	9.01330	0.462982	0.231491	+	0.972837i	$0.425640\pi$
$380$	0	0
$381$	−12.9151	−0.661660
$382$	0	0
$383$	−9.43055	−0.481878	−0.240939	−	0.970540i	$-0.577455\pi$
$383$	−9.43055	−0.481878	−0.240939	+	0.970540i	$0.577455\pi$
$384$	0	0
$385$	−0.853848	−0.0435161
$386$	0	0
$387$	3.21086	0.163217
$388$	0	0
$389$	30.2853	1.53553	0.767764	−	0.640733i	$-0.221369\pi$
$389$	30.2853	1.53553	0.767764	+	0.640733i	$0.221369\pi$
$390$	0	0
$391$	35.8955	1.81531
$392$	0	0
$393$	−1.81550	−0.0915796
$394$	0	0
$395$	0.397043	0.0199774
$396$	0	0
$397$	−24.6678	−1.23804	−0.619022	−	0.785374i	$-0.712471\pi$
$397$	−24.6678	−1.23804	−0.619022	+	0.785374i	$0.712471\pi$
$398$	0	0
$399$	−0.127221	−0.00636902
$400$	0	0
$401$	−8.69148	−0.434032	−0.217016	−	0.976168i	$-0.569632\pi$
$401$	−8.69148	−0.434032	−0.217016	+	0.976168i	$0.569632\pi$
$402$	0	0
$403$	35.9565	1.79112
$404$	0	0
$405$	1.28584	0.0638939
$406$	0	0
$407$	40.5749	2.01122
$408$	0	0
$409$	−15.3317	−0.758102	−0.379051	−	0.925376i	$-0.623749\pi$
$409$	−15.3317	−0.758102	−0.379051	+	0.925376i	$0.623749\pi$
$410$	0	0
$411$	−6.15600	−0.303653
$412$	0	0
$413$	−0.229719	−0.0113037
$414$	0	0
$415$	−16.4477	−0.807387
$416$	0	0
$417$	−2.88427	−0.141243
$418$	0	0
$419$	−18.3214	−0.895057	−0.447528	−	0.894270i	$-0.647696\pi$
$419$	−18.3214	−0.895057	−0.447528	+	0.894270i	$0.647696\pi$
$420$	0	0
$421$	5.96176	0.290558	0.145279	−	0.989391i	$-0.453592\pi$
$421$	5.96176	0.290558	0.145279	+	0.989391i	$0.453592\pi$
$422$	0	0
$423$	11.6838	0.568085
$424$	0	0
$425$	−16.2336	−0.787446
$426$	0	0
$427$	0.273703	0.0132454
$428$	0	0
$429$	19.1731	0.925687
$430$	0	0
$431$	36.5741	1.76171	0.880856	−	0.473384i	$-0.156968\pi$
$431$	36.5741	1.76171	0.880856	+	0.473384i	$0.156968\pi$
$432$	0	0
$433$	2.22728	0.107036	0.0535180	−	0.998567i	$-0.482957\pi$
$433$	2.22728	0.107036	0.0535180	+	0.998567i	$0.482957\pi$
$434$	0	0
$435$	0.858119	0.0411437
$436$	0	0
$437$	−7.78852	−0.372575
$438$	0	0
$439$	34.6751	1.65495	0.827477	−	0.561500i	$-0.189776\pi$
$439$	34.6751	1.65495	0.827477	+	0.561500i	$0.189776\pi$
$440$	0	0
$441$	−6.98539	−0.332638
$442$	0	0
$443$	23.1363	1.09924	0.549619	−	0.835415i	$-0.314773\pi$
$443$	23.1363	1.09924	0.549619	+	0.835415i	$0.314773\pi$
$444$	0	0
$445$	−0.510753	−0.0242120
$446$	0	0
$447$	−15.3797	−0.727433
$448$	0	0
$449$	−37.5542	−1.77229	−0.886147	−	0.463404i	$-0.846628\pi$
$449$	−37.5542	−1.77229	−0.886147	+	0.463404i	$0.846628\pi$
$450$	0	0
$451$	−34.5251	−1.62572
$452$	0	0
$453$	−21.5417	−1.01212
$454$	0	0
$455$	−0.542443	−0.0254301
$456$	0	0
$457$	−37.9985	−1.77749	−0.888747	−	0.458398i	$-0.848423\pi$
$457$	−37.9985	−1.77749	−0.888747	+	0.458398i	$0.848423\pi$
$458$	0	0
$459$	4.85075	0.226414
$460$	0	0
$461$	31.9901	1.48993	0.744964	−	0.667105i	$-0.232467\pi$
$461$	31.9901	1.48993	0.744964	+	0.667105i	$0.232467\pi$
$462$	0	0
$463$	−8.45887	−0.393117	−0.196559	−	0.980492i	$-0.562977\pi$
$463$	−8.45887	−0.393117	−0.196559	+	0.980492i	$0.562977\pi$
$464$	0	0
$465$	−13.2474	−0.614333
$466$	0	0
$467$	18.7519	0.867736	0.433868	−	0.900977i	$-0.357148\pi$
$467$	18.7519	0.867736	0.433868	+	0.900977i	$0.357148\pi$
$468$	0	0
$469$	−0.954784	−0.0440878
$470$	0	0
$471$	21.4579	0.988730
$472$	0	0
$473$	−17.6393	−0.811054
$474$	0	0
$475$	3.52233	0.161616
$476$	0	0
$477$	−12.5451	−0.574400
$478$	0	0
$479$	23.1095	1.05590	0.527951	−	0.849275i	$-0.322961\pi$
$479$	23.1095	1.05590	0.527951	+	0.849275i	$0.322961\pi$
$480$	0	0
$481$	25.7770	1.17533
$482$	0	0
$483$	0.894467	0.0406997
$484$	0	0
$485$	10.4640	0.475147
$486$	0	0
$487$	−28.4422	−1.28884	−0.644420	−	0.764672i	$-0.722901\pi$
$487$	−28.4422	−1.28884	−0.644420	+	0.764672i	$0.722901\pi$
$488$	0	0
$489$	−4.34046	−0.196282
$490$	0	0
$491$	−20.8355	−0.940291	−0.470146	−	0.882589i	$-0.655799\pi$
$491$	−20.8355	−0.940291	−0.470146	+	0.882589i	$0.655799\pi$
$492$	0	0
$493$	3.23720	0.145796
$494$	0	0
$495$	−7.06393	−0.317500
$496$	0	0
$497$	0.797559	0.0357754
$498$	0	0
$499$	1.66112	0.0743618	0.0371809	−	0.999309i	$-0.488162\pi$
$499$	1.66112	0.0743618	0.0371809	+	0.999309i	$0.488162\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−31.4205	−1.40097	−0.700486	−	0.713666i	$-0.747033\pi$
$503$	−31.4205	−1.40097	−0.700486	+	0.713666i	$0.747033\pi$
$504$	0	0
$505$	0.170486	0.00758652
$506$	0	0
$507$	−0.819459	−0.0363935
$508$	0	0
$509$	37.9012	1.67994	0.839970	−	0.542633i	$-0.182572\pi$
$509$	37.9012	1.67994	0.839970	+	0.542633i	$0.182572\pi$
$510$	0	0
$511$	−0.0664941	−0.00294153
$512$	0	0
$513$	−1.05251	−0.0464693
$514$	0	0
$515$	8.95895	0.394779
$516$	0	0
$517$	−64.1864	−2.82291
$518$	0	0
$519$	−14.3565	−0.630178
$520$	0	0
$521$	−18.4349	−0.807648	−0.403824	−	0.914837i	$-0.632319\pi$
$521$	−18.4349	−0.807648	−0.403824	+	0.914837i	$0.632319\pi$
$522$	0	0
$523$	−16.4271	−0.718306	−0.359153	−	0.933279i	$-0.616934\pi$
$523$	−16.4271	−0.718306	−0.359153	+	0.933279i	$0.616934\pi$
$524$	0	0
$525$	−0.404520	−0.0176547
$526$	0	0
$527$	−49.9750	−2.17695
$528$	0	0
$529$	31.7596	1.38085
$530$	0	0
$531$	−1.90048	−0.0824737
$532$	0	0
$533$	−21.9335	−0.950047
$534$	0	0
$535$	−16.5339	−0.714825
$536$	0	0
$537$	−13.2707	−0.572672
$538$	0	0
$539$	38.3751	1.65293
$540$	0	0
$541$	−25.4813	−1.09553	−0.547764	−	0.836633i	$-0.684521\pi$
$541$	−25.4813	−1.09553	−0.547764	+	0.836633i	$0.684521\pi$
$542$	0	0
$543$	2.98050	0.127906
$544$	0	0
$545$	−8.72122	−0.373576
$546$	0	0
$547$	−21.0203	−0.898763	−0.449381	−	0.893340i	$-0.648356\pi$
$547$	−21.0203	−0.898763	−0.449381	+	0.893340i	$0.648356\pi$
$548$	0	0
$549$	2.26436	0.0966405
$550$	0	0
$551$	−0.702401	−0.0299233
$552$	0	0
$553$	0.0373237	0.00158716
$554$	0	0
$555$	−9.49698	−0.403124
$556$	0	0
$557$	−19.5030	−0.826369	−0.413184	−	0.910647i	$-0.635583\pi$
$557$	−19.5030	−0.826369	−0.413184	+	0.910647i	$0.635583\pi$
$558$	0	0
$559$	−11.2061	−0.473967
$560$	0	0
$561$	−26.6482	−1.12509
$562$	0	0
$563$	−3.37436	−0.142212	−0.0711061	−	0.997469i	$-0.522653\pi$
$563$	−3.37436	−0.142212	−0.0711061	+	0.997469i	$0.522653\pi$
$564$	0	0
$565$	−17.7823	−0.748105
$566$	0	0
$567$	0.120874	0.00507624
$568$	0	0
$569$	7.56267	0.317044	0.158522	−	0.987355i	$-0.449327\pi$
$569$	7.56267	0.317044	0.158522	+	0.987355i	$0.449327\pi$
$570$	0	0
$571$	−10.2362	−0.428371	−0.214186	−	0.976793i	$-0.568710\pi$
$571$	−10.2362	−0.428371	−0.214186	+	0.976793i	$0.568710\pi$
$572$	0	0
$573$	−25.0935	−1.04830
$574$	0	0
$575$	−24.7649	−1.03277
$576$	0	0
$577$	33.5713	1.39759	0.698795	−	0.715322i	$-0.253720\pi$
$577$	33.5713	1.39759	0.698795	+	0.715322i	$0.253720\pi$
$578$	0	0
$579$	25.9678	1.07919
$580$	0	0
$581$	−1.54615	−0.0641453
$582$	0	0
$583$	68.9180	2.85429
$584$	0	0
$585$	−4.48766	−0.185542
$586$	0	0
$587$	8.74690	0.361023	0.180512	−	0.983573i	$-0.442225\pi$
$587$	8.74690	0.361023	0.180512	+	0.983573i	$0.442225\pi$
$588$	0	0
$589$	10.8435	0.446797
$590$	0	0
$591$	−9.01491	−0.370824
$592$	0	0
$593$	−3.87828	−0.159262	−0.0796309	−	0.996824i	$-0.525374\pi$
$593$	−3.87828	−0.159262	−0.0796309	+	0.996824i	$0.525374\pi$
$594$	0	0
$595$	0.753929	0.0309081
$596$	0	0
$597$	7.13075	0.291842
$598$	0	0
$599$	−28.8381	−1.17829	−0.589146	−	0.808026i	$-0.700536\pi$
$599$	−28.8381	−1.17829	−0.589146	+	0.808026i	$0.700536\pi$
$600$	0	0
$601$	−46.7913	−1.90866	−0.954329	−	0.298757i	$-0.903428\pi$
$601$	−46.7913	−1.90866	−0.954329	+	0.298757i	$0.903428\pi$
$602$	0	0
$603$	−7.89898	−0.321671
$604$	0	0
$605$	24.6624	1.00267
$606$	0	0
$607$	−11.9106	−0.483438	−0.241719	−	0.970346i	$-0.577711\pi$
$607$	−11.9106	−0.483438	−0.241719	+	0.970346i	$0.577711\pi$
$608$	0	0
$609$	0.0806668	0.00326878
$610$	0	0
$611$	−40.7771	−1.64967
$612$	0	0
$613$	14.1608	0.571948	0.285974	−	0.958237i	$-0.407683\pi$
$613$	14.1608	0.571948	0.285974	+	0.958237i	$0.407683\pi$
$614$	0	0
$615$	8.08095	0.325855
$616$	0	0
$617$	−40.5999	−1.63449	−0.817246	−	0.576289i	$-0.804500\pi$
$617$	−40.5999	−1.63449	−0.817246	+	0.576289i	$0.804500\pi$
$618$	0	0
$619$	13.4795	0.541785	0.270893	−	0.962610i	$-0.412681\pi$
$619$	13.4795	0.541785	0.270893	+	0.962610i	$0.412681\pi$
$620$	0	0
$621$	7.39997	0.296951
$622$	0	0
$623$	−0.0480129	−0.00192360
$624$	0	0
$625$	2.93291	0.117316
$626$	0	0
$627$	5.78208	0.230914
$628$	0	0
$629$	−35.8268	−1.42851
$630$	0	0
$631$	−11.8321	−0.471028	−0.235514	−	0.971871i	$-0.575677\pi$
$631$	−11.8321	−0.471028	−0.235514	+	0.971871i	$0.575677\pi$
$632$	0	0
$633$	−9.22696	−0.366739
$634$	0	0
$635$	−16.6067	−0.659018
$636$	0	0
$637$	24.3795	0.965949
$638$	0	0
$639$	6.59825	0.261023
$640$	0	0
$641$	−8.03953	−0.317542	−0.158771	−	0.987315i	$-0.550753\pi$
$641$	−8.03953	−0.317542	−0.158771	+	0.987315i	$0.550753\pi$
$642$	0	0
$643$	−48.1186	−1.89761	−0.948806	−	0.315859i	$-0.897707\pi$
$643$	−48.1186	−1.89761	−0.948806	+	0.315859i	$0.897707\pi$
$644$	0	0
$645$	4.12865	0.162565
$646$	0	0
$647$	7.96647	0.313194	0.156597	−	0.987663i	$-0.449948\pi$
$647$	7.96647	0.313194	0.156597	+	0.987663i	$0.449948\pi$
$648$	0	0
$649$	10.4405	0.409826
$650$	0	0
$651$	−1.24531	−0.0488076
$652$	0	0
$653$	40.5566	1.58710	0.793551	−	0.608503i	$-0.208230\pi$
$653$	40.5566	1.58710	0.793551	+	0.608503i	$0.208230\pi$
$654$	0	0
$655$	−2.33444	−0.0912140
$656$	0	0
$657$	−0.550109	−0.0214618
$658$	0	0
$659$	37.3255	1.45399	0.726997	−	0.686641i	$-0.240915\pi$
$659$	37.3255	1.45399	0.726997	+	0.686641i	$0.240915\pi$
$660$	0	0
$661$	45.1593	1.75649	0.878246	−	0.478209i	$-0.158714\pi$
$661$	45.1593	1.75649	0.878246	+	0.478209i	$0.158714\pi$
$662$	0	0
$663$	−16.9294	−0.657485
$664$	0	0
$665$	−0.163586	−0.00634359
$666$	0	0
$667$	4.93845	0.191218
$668$	0	0
$669$	−14.0521	−0.543286
$670$	0	0
$671$	−12.4396	−0.480224
$672$	0	0
$673$	−16.3233	−0.629218	−0.314609	−	0.949221i	$-0.601873\pi$
$673$	−16.3233	−0.629218	−0.314609	+	0.949221i	$0.601873\pi$
$674$	0	0
$675$	−3.34662	−0.128811
$676$	0	0
$677$	−24.3569	−0.936114	−0.468057	−	0.883698i	$-0.655046\pi$
$677$	−24.3569	−0.936114	−0.468057	+	0.883698i	$0.655046\pi$
$678$	0	0
$679$	0.983663	0.0377495
$680$	0	0
$681$	−5.87728	−0.225218
$682$	0	0
$683$	11.9635	0.457771	0.228885	−	0.973453i	$-0.426492\pi$
$683$	11.9635	0.457771	0.228885	+	0.973453i	$0.426492\pi$
$684$	0	0
$685$	−7.91563	−0.302441
$686$	0	0
$687$	−6.58989	−0.251420
$688$	0	0
$689$	43.7831	1.66800
$690$	0	0
$691$	14.3688	0.546616	0.273308	−	0.961927i	$-0.411882\pi$
$691$	14.3688	0.546616	0.273308	+	0.961927i	$0.411882\pi$
$692$	0	0
$693$	−0.664039	−0.0252247
$694$	0	0
$695$	−3.70871	−0.140680
$696$	0	0
$697$	30.4849	1.15470
$698$	0	0
$699$	21.5949	0.816796
$700$	0	0
$701$	6.26375	0.236579	0.118289	−	0.992979i	$-0.462259\pi$
$701$	6.26375	0.236579	0.118289	+	0.992979i	$0.462259\pi$
$702$	0	0
$703$	7.77362	0.293188
$704$	0	0
$705$	15.0235	0.565817
$706$	0	0
$707$	0.0160264	0.000602734 0
$708$	0	0
$709$	13.4610	0.505539	0.252769	−	0.967527i	$-0.418659\pi$
$709$	13.4610	0.505539	0.252769	+	0.967527i	$0.418659\pi$
$710$	0	0
$711$	0.308781	0.0115802
$712$	0	0
$713$	−76.2384	−2.85515
$714$	0	0
$715$	24.6536	0.921991
$716$	0	0
$717$	−13.5344	−0.505453
$718$	0	0
$719$	−22.9156	−0.854609	−0.427305	−	0.904108i	$-0.640537\pi$
$719$	−22.9156	−0.854609	−0.427305	+	0.904108i	$0.640537\pi$
$720$	0	0
$721$	0.842179	0.0313644
$722$	0	0
$723$	−0.680726	−0.0253165
$724$	0	0
$725$	−2.23340	−0.0829464
$726$	0	0
$727$	−2.59130	−0.0961061	−0.0480530	−	0.998845i	$-0.515302\pi$
$727$	−2.59130	−0.0961061	−0.0480530	+	0.998845i	$0.515302\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.5751	0.576065
$732$	0	0
$733$	3.25837	0.120351	0.0601753	−	0.998188i	$-0.480834\pi$
$733$	3.25837	0.120351	0.0601753	+	0.998188i	$0.480834\pi$
$734$	0	0
$735$	−8.98209	−0.331310
$736$	0	0
$737$	43.3941	1.59844
$738$	0	0
$739$	28.0467	1.03171	0.515856	−	0.856675i	$-0.327474\pi$
$739$	28.0467	1.03171	0.515856	+	0.856675i	$0.327474\pi$
$740$	0	0
$741$	3.67331	0.134943
$742$	0	0
$743$	34.4338	1.26326	0.631628	−	0.775272i	$-0.282387\pi$
$743$	34.4338	1.26326	0.631628	+	0.775272i	$0.282387\pi$
$744$	0	0
$745$	−19.7758	−0.724529
$746$	0	0
$747$	−12.7914	−0.468014
$748$	0	0
$749$	−1.55426	−0.0567914
$750$	0	0
$751$	33.5730	1.22510	0.612549	−	0.790433i	$-0.290144\pi$
$751$	33.5730	1.22510	0.612549	+	0.790433i	$0.290144\pi$
$752$	0	0
$753$	20.5812	0.750021
$754$	0	0
$755$	−27.6991	−1.00807
$756$	0	0
$757$	−5.17670	−0.188150	−0.0940751	−	0.995565i	$-0.529989\pi$
$757$	−5.17670	−0.188150	−0.0940751	+	0.995565i	$0.529989\pi$
$758$	0	0
$759$	−40.6527	−1.47560
$760$	0	0
$761$	54.7529	1.98479	0.992395	−	0.123090i	$-0.0392805\pi$
$761$	54.7529	1.98479	0.992395	+	0.123090i	$0.0392805\pi$
$762$	0	0
$763$	−0.819831	−0.0296799
$764$	0	0
$765$	6.23729	0.225510
$766$	0	0
$767$	6.63279	0.239496
$768$	0	0
$769$	−3.44405	−0.124196	−0.0620978	−	0.998070i	$-0.519779\pi$
$769$	−3.44405	−0.124196	−0.0620978	+	0.998070i	$0.519779\pi$
$770$	0	0
$771$	−10.4824	−0.377515
$772$	0	0
$773$	−12.3892	−0.445608	−0.222804	−	0.974863i	$-0.571521\pi$
$773$	−12.3892	−0.445608	−0.222804	+	0.974863i	$0.571521\pi$
$774$	0	0
$775$	34.4786	1.23851
$776$	0	0
$777$	−0.892755	−0.0320274
$778$	0	0
$779$	−6.61454	−0.236991
$780$	0	0
$781$	−36.2483	−1.29707
$782$	0	0
$783$	0.667361	0.0238495
$784$	0	0
$785$	27.5915	0.984782
$786$	0	0
$787$	−45.0571	−1.60611	−0.803056	−	0.595903i	$-0.796794\pi$
$787$	−45.0571	−1.60611	−0.803056	+	0.595903i	$0.796794\pi$
$788$	0	0
$789$	−3.31478	−0.118009
$790$	0	0
$791$	−1.67161	−0.0594355
$792$	0	0
$793$	−7.90276	−0.280635
$794$	0	0
$795$	−16.1310	−0.572106
$796$	0	0
$797$	40.7782	1.44444	0.722218	−	0.691665i	$-0.243123\pi$
$797$	40.7782	1.44444	0.722218	+	0.691665i	$0.243123\pi$
$798$	0	0
$799$	56.6752	2.00502
$800$	0	0
$801$	−0.397213	−0.0140348
$802$	0	0
$803$	3.02210	0.106647
$804$	0	0
$805$	1.15014	0.0405372
$806$	0	0
$807$	13.2626	0.466867
$808$	0	0
$809$	−13.5270	−0.475582	−0.237791	−	0.971316i	$-0.576423\pi$
$809$	−13.5270	−0.475582	−0.237791	+	0.971316i	$0.576423\pi$
$810$	0	0
$811$	19.9196	0.699472	0.349736	−	0.936848i	$-0.386271\pi$
$811$	19.9196	0.699472	0.349736	+	0.936848i	$0.386271\pi$
$812$	0	0
$813$	11.4876	0.402886
$814$	0	0
$815$	−5.58113	−0.195499
$816$	0	0
$817$	−3.37945	−0.118232
$818$	0	0
$819$	−0.421859	−0.0147409
$820$	0	0
$821$	−3.50620	−0.122367	−0.0611836	−	0.998127i	$-0.519488\pi$
$821$	−3.50620	−0.122367	−0.0611836	+	0.998127i	$0.519488\pi$
$822$	0	0
$823$	−28.9034	−1.00751	−0.503754	−	0.863847i	$-0.668048\pi$
$823$	−28.9034	−1.00751	−0.503754	+	0.863847i	$0.668048\pi$
$824$	0	0
$825$	18.3851	0.640086
$826$	0	0
$827$	−37.4569	−1.30250	−0.651251	−	0.758862i	$-0.725756\pi$
$827$	−37.4569	−1.30250	−0.651251	+	0.758862i	$0.725756\pi$
$828$	0	0
$829$	10.1413	0.352220	0.176110	−	0.984370i	$-0.443649\pi$
$829$	10.1413	0.352220	0.176110	+	0.984370i	$0.443649\pi$
$830$	0	0
$831$	−1.44455	−0.0501107
$832$	0	0
$833$	−33.8844	−1.17403
$834$	0	0
$835$	−1.28584	−0.0444983
$836$	0	0
$837$	−10.3025	−0.356107
$838$	0	0
$839$	−17.3392	−0.598616	−0.299308	−	0.954156i	$-0.596756\pi$
$839$	−17.3392	−0.598616	−0.299308	+	0.954156i	$0.596756\pi$
$840$	0	0
$841$	−28.5546	−0.984642
$842$	0	0
$843$	−24.5424	−0.845286
$844$	0	0
$845$	−1.05369	−0.0362482
$846$	0	0
$847$	2.31836	0.0796600
$848$	0	0
$849$	−3.57120	−0.122563
$850$	0	0
$851$	−54.6548	−1.87354
$852$	0	0
$853$	43.1242	1.47655	0.738273	−	0.674502i	$-0.235642\pi$
$853$	43.1242	1.47655	0.738273	+	0.674502i	$0.235642\pi$
$854$	0	0
$855$	−1.35335	−0.0462838
$856$	0	0
$857$	25.3962	0.867518	0.433759	−	0.901029i	$-0.357187\pi$
$857$	25.3962	0.867518	0.433759	+	0.901029i	$0.357187\pi$
$858$	0	0
$859$	−35.6397	−1.21601	−0.608005	−	0.793933i	$-0.708030\pi$
$859$	−35.6397	−1.21601	−0.608005	+	0.793933i	$0.708030\pi$
$860$	0	0
$861$	0.759642	0.0258885
$862$	0	0
$863$	−37.9455	−1.29168	−0.645840	−	0.763473i	$-0.723493\pi$
$863$	−37.9455	−1.29168	−0.645840	+	0.763473i	$0.723493\pi$
$864$	0	0
$865$	−18.4601	−0.627662
$866$	0	0
$867$	6.52981	0.221764
$868$	0	0
$869$	−1.69633	−0.0575440
$870$	0	0
$871$	27.5679	0.934104
$872$	0	0
$873$	8.13790	0.275426
$874$	0	0
$875$	−1.29727	−0.0438558
$876$	0	0
$877$	8.17854	0.276170	0.138085	−	0.990420i	$-0.455905\pi$
$877$	8.17854	0.276170	0.138085	+	0.990420i	$0.455905\pi$
$878$	0	0
$879$	−7.08534	−0.238983
$880$	0	0
$881$	35.3123	1.18970	0.594851	−	0.803836i	$-0.297211\pi$
$881$	35.3123	1.18970	0.594851	+	0.803836i	$0.297211\pi$
$882$	0	0
$883$	8.60384	0.289542	0.144771	−	0.989465i	$-0.453755\pi$
$883$	8.60384	0.289542	0.144771	+	0.989465i	$0.453755\pi$
$884$	0	0
$885$	−2.44371	−0.0821444
$886$	0	0
$887$	26.5619	0.891862	0.445931	−	0.895067i	$-0.352873\pi$
$887$	26.5619	0.891862	0.445931	+	0.895067i	$0.352873\pi$
$888$	0	0
$889$	−1.56110	−0.0523577
$890$	0	0
$891$	−5.49363	−0.184044
$892$	0	0
$893$	−12.2973	−0.411512
$894$	0	0
$895$	−17.0640	−0.570386
$896$	0	0
$897$	−25.8264	−0.862318
$898$	0	0
$899$	−6.87550	−0.229311
$900$	0	0
$901$	−60.8531	−2.02731
$902$	0	0
$903$	0.388110	0.0129155
$904$	0	0
$905$	3.83245	0.127395
$906$	0	0
$907$	11.4872	0.381427	0.190714	−	0.981646i	$-0.438920\pi$
$907$	11.4872	0.381427	0.190714	+	0.981646i	$0.438920\pi$
$908$	0	0
$909$	0.132587	0.00439764
$910$	0	0
$911$	27.1535	0.899634	0.449817	−	0.893121i	$-0.351489\pi$
$911$	27.1535	0.899634	0.449817	+	0.893121i	$0.351489\pi$
$912$	0	0
$913$	70.2713	2.32564
$914$	0	0
$915$	2.91160	0.0962547
$916$	0	0
$917$	−0.219447	−0.00724677
$918$	0	0
$919$	−47.6921	−1.57322	−0.786609	−	0.617451i	$-0.788165\pi$
$919$	−47.6921	−1.57322	−0.786609	+	0.617451i	$0.788165\pi$
$920$	0	0
$921$	−2.04358	−0.0673384
$922$	0	0
$923$	−23.0283	−0.757986
$924$	0	0
$925$	24.7175	0.812706
$926$	0	0
$927$	6.96739	0.228839
$928$	0	0
$929$	−9.64215	−0.316349	−0.158174	−	0.987411i	$-0.550561\pi$
$929$	−9.64215	−0.316349	−0.158174	+	0.987411i	$0.550561\pi$
$930$	0	0
$931$	7.35217	0.240958
$932$	0	0
$933$	25.5906	0.837800
$934$	0	0
$935$	−34.2654	−1.12060
$936$	0	0
$937$	−38.8134	−1.26798	−0.633989	−	0.773342i	$-0.718583\pi$
$937$	−38.8134	−1.26798	−0.633989	+	0.773342i	$0.718583\pi$
$938$	0	0
$939$	12.8093	0.418015
$940$	0	0
$941$	−25.3020	−0.824822	−0.412411	−	0.910998i	$-0.635313\pi$
$941$	−25.3020	−0.824822	−0.412411	+	0.910998i	$0.635313\pi$
$942$	0	0
$943$	46.5056	1.51443
$944$	0	0
$945$	0.155425	0.00505598
$946$	0	0
$947$	−58.1583	−1.88989	−0.944945	−	0.327228i	$-0.893885\pi$
$947$	−58.1583	−1.88989	−0.944945	+	0.327228i	$0.893885\pi$
$948$	0	0
$949$	1.91992	0.0623231
$950$	0	0
$951$	13.3331	0.432355
$952$	0	0
$953$	−36.7947	−1.19190	−0.595948	−	0.803023i	$-0.703224\pi$
$953$	−36.7947	−1.19190	−0.595948	+	0.803023i	$0.703224\pi$
$954$	0	0
$955$	−32.2662	−1.04411
$956$	0	0
$957$	−3.66623	−0.118512
$958$	0	0
$959$	−0.744102	−0.0240283
$960$	0	0
$961$	75.1420	2.42394
$962$	0	0
$963$	−12.8585	−0.414359
$964$	0	0
$965$	33.3904	1.07488
$966$	0	0
$967$	57.7727	1.85785	0.928923	−	0.370273i	$-0.120736\pi$
$967$	57.7727	1.85785	0.928923	+	0.370273i	$0.120736\pi$
$968$	0	0
$969$	−5.10545	−0.164011
$970$	0	0
$971$	13.2652	0.425702	0.212851	−	0.977085i	$-0.431725\pi$
$971$	13.2652	0.425702	0.212851	+	0.977085i	$0.431725\pi$
$972$	0	0
$973$	−0.348635	−0.0111767
$974$	0	0
$975$	11.6799	0.374056
$976$	0	0
$977$	−4.72472	−0.151157	−0.0755785	−	0.997140i	$-0.524080\pi$
$977$	−4.72472	−0.151157	−0.0755785	+	0.997140i	$0.524080\pi$
$978$	0	0
$979$	2.18214	0.0697416
$980$	0	0
$981$	−6.78250	−0.216549
$982$	0	0
$983$	38.9354	1.24185	0.620923	−	0.783871i	$-0.286758\pi$
$983$	38.9354	1.24185	0.620923	+	0.783871i	$0.286758\pi$
$984$	0	0
$985$	−11.5917	−0.369343
$986$	0	0
$987$	1.41227	0.0449530
$988$	0	0
$989$	23.7603	0.755532
$990$	0	0
$991$	−38.1471	−1.21178	−0.605892	−	0.795547i	$-0.707184\pi$
$991$	−38.1471	−1.21178	−0.605892	+	0.795547i	$0.707184\pi$
$992$	0	0
$993$	32.0159	1.01599
$994$	0	0
$995$	9.16900	0.290677
$996$	0	0
$997$	6.51907	0.206461	0.103231	−	0.994657i	$-0.467082\pi$
$997$	6.51907	0.206461	0.103231	+	0.994657i	$0.467082\pi$
$998$	0	0
$999$	−7.38582	−0.233677

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.z.1.6		8
4.3	odd	2		501.2.a.d.1.7	✓	8
12.11	even	2		1503.2.a.f.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.d.1.7	✓	8	4.3	odd	2
1503.2.a.f.1.2		8	12.11	even	2
8016.2.a.z.1.6		8	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.28584 q^{5} +0.120874 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.28584 q^{5} +0.120874 q^{7} +1.00000 q^{9} -5.49363 q^{11} -3.49006 q^{13} +1.28584 q^{15} +4.85075 q^{17} -1.05251 q^{19} +0.120874 q^{21} +7.39997 q^{23} -3.34662 q^{25} +1.00000 q^{27} +0.667361 q^{29} -10.3025 q^{31} -5.49363 q^{33} +0.155425 q^{35} -7.38582 q^{37} -3.49006 q^{39} +6.28456 q^{41} +3.21086 q^{43} +1.28584 q^{45} +11.6838 q^{47} -6.98539 q^{49} +4.85075 q^{51} -12.5451 q^{53} -7.06393 q^{55} -1.05251 q^{57} -1.90048 q^{59} +2.26436 q^{61} +0.120874 q^{63} -4.48766 q^{65} -7.89898 q^{67} +7.39997 q^{69} +6.59825 q^{71} -0.550109 q^{73} -3.34662 q^{75} -0.664039 q^{77} +0.308781 q^{79} +1.00000 q^{81} -12.7914 q^{83} +6.23729 q^{85} +0.667361 q^{87} -0.397213 q^{89} -0.421859 q^{91} -10.3025 q^{93} -1.35335 q^{95} +8.13790 q^{97} -5.49363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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