LMFDB - Embedded newform 8092.2.a.z.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8092,2,Mod(1,8092)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8092.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8092, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8092 = 2^{2} \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8092.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0,8,0,8,0,20,0,28,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.6149453156$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 12 x^{18} + 240 x^{17} - 224 x^{16} - 2776 x^{15} + 5324 x^{14} + 15280 x^{13} + \cdots + 544 $ x^20 - 8x^19 - 12x^18 + 240x^17 - 224x^16 - 2776x^15 + 5324x^14 + 15280x^13 - 42463x^12 - 36576x^11 + 166532x^10 - 1048x^9 - 328476x^8 + 166744x^7 + 277764x^6 - 258464x^5 - 36367x^4 + 111904x^3 - 40080x^2 + 2688*x + 544
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.20917$ of defining polynomial
Character	$\chi$	$=$	8092.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.20917 q^{3} -3.53097 q^{5} +1.00000 q^{7} -1.53791 q^{9} +4.33450 q^{11} -1.06829 q^{13} +4.26954 q^{15} +0.865933 q^{19} -1.20917 q^{21} -1.81237 q^{23} +7.46774 q^{25} +5.48710 q^{27} +6.30913 q^{29} +6.66297 q^{31} -5.24115 q^{33} -3.53097 q^{35} -7.71962 q^{37} +1.29175 q^{39} -1.33942 q^{41} -3.35689 q^{43} +5.43031 q^{45} -11.4190 q^{47} +1.00000 q^{49} +13.3149 q^{53} -15.3050 q^{55} -1.04706 q^{57} -13.8098 q^{59} -6.86562 q^{61} -1.53791 q^{63} +3.77211 q^{65} -8.07908 q^{67} +2.19147 q^{69} +3.76828 q^{71} -2.16933 q^{73} -9.02977 q^{75} +4.33450 q^{77} -2.94643 q^{79} -2.02111 q^{81} +8.53177 q^{83} -7.62881 q^{87} -9.59243 q^{89} -1.06829 q^{91} -8.05666 q^{93} -3.05758 q^{95} +3.13292 q^{97} -6.66607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) $ 20 * q + 8 * q^3 + 8 * q^5 + 20 * q^7 + 28 * q^9 + 16 * q^11 + 8 * q^13 + 8 * q^15 + 8 * q^19 + 8 * q^21 + 40 * q^23 + 36 * q^25 + 32 * q^27 + 16 * q^29 + 16 * q^31 + 8 * q^33 + 8 * q^35 + 24 * q^37 + 16 * q^39 - 16 * q^41 + 16 * q^43 + 24 * q^45 + 20 * q^49 + 24 * q^53 - 8 * q^55 + 24 * q^57 + 8 * q^61 + 28 * q^63 + 8 * q^65 - 8 * q^67 + 8 * q^69 + 72 * q^71 + 40 * q^75 + 16 * q^77 + 60 * q^81 + 8 * q^83 - 24 * q^89 + 8 * q^91 + 32 * q^93 + 48 * q^95 - 16 * q^97 + 72 * q^99

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
$3$	−1.20917	−0.698114	−0.349057	−	0.937101i	$-0.613498\pi$
$3$	−1.20917	−0.698114	−0.349057	+	0.937101i	$0.613498\pi$
$4$	0	0
$5$	−3.53097	−1.57910	−0.789549	−	0.613688i	$-0.789685\pi$
$5$	−3.53097	−1.57910	−0.789549	+	0.613688i	$0.789685\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.53791	−0.512636
$10$	0	0
$11$	4.33450	1.30690	0.653451	−	0.756969i	$-0.273321\pi$
$11$	4.33450	1.30690	0.653451	+	0.756969i	$0.273321\pi$
$12$	0	0
$13$	−1.06829	−0.296291	−0.148146	−	0.988966i	$-0.547330\pi$
$13$	−1.06829	−0.296291	−0.148146	+	0.988966i	$0.547330\pi$
$14$	0	0
$15$	4.26954	1.10239
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	0.865933	0.198659	0.0993293	−	0.995055i	$-0.468330\pi$
$19$	0.865933	0.198659	0.0993293	+	0.995055i	$0.468330\pi$
$20$	0	0
$21$	−1.20917	−0.263862
$22$	0	0
$23$	−1.81237	−0.377906	−0.188953	−	0.981986i	$-0.560509\pi$
$23$	−1.81237	−0.377906	−0.188953	+	0.981986i	$0.560509\pi$
$24$	0	0
$25$	7.46774	1.49355
$26$	0	0
$27$	5.48710	1.05599
$28$	0	0
$29$	6.30913	1.17158	0.585788	−	0.810464i	$-0.300785\pi$
$29$	6.30913	1.17158	0.585788	+	0.810464i	$0.300785\pi$
$30$	0	0
$31$	6.66297	1.19670	0.598352	−	0.801233i	$-0.295822\pi$
$31$	6.66297	1.19670	0.598352	+	0.801233i	$0.295822\pi$
$32$	0	0
$33$	−5.24115	−0.912367
$34$	0	0
$35$	−3.53097	−0.596843
$36$	0	0
$37$	−7.71962	−1.26910	−0.634549	−	0.772882i	$-0.718814\pi$
$37$	−7.71962	−1.26910	−0.634549	+	0.772882i	$0.718814\pi$
$38$	0	0
$39$	1.29175	0.206845
$40$	0	0
$41$	−1.33942	−0.209183	−0.104591	−	0.994515i	$-0.533353\pi$
$41$	−1.33942	−0.209183	−0.104591	+	0.994515i	$0.533353\pi$
$42$	0	0
$43$	−3.35689	−0.511921	−0.255961	−	0.966687i	$-0.582392\pi$
$43$	−3.35689	−0.511921	−0.255961	+	0.966687i	$0.582392\pi$
$44$	0	0
$45$	5.43031	0.809503
$46$	0	0
$47$	−11.4190	−1.66563	−0.832816	−	0.553551i	$-0.813273\pi$
$47$	−11.4190	−1.66563	−0.832816	+	0.553551i	$0.813273\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.3149	1.82895	0.914473	−	0.404646i	$-0.132605\pi$
$53$	13.3149	1.82895	0.914473	+	0.404646i	$0.132605\pi$
$54$	0	0
$55$	−15.3050	−2.06372
$56$	0	0
$57$	−1.04706	−0.138686
$58$	0	0
$59$	−13.8098	−1.79788	−0.898939	−	0.438075i	$-0.855661\pi$
$59$	−13.8098	−1.79788	−0.898939	+	0.438075i	$0.855661\pi$
$60$	0	0
$61$	−6.86562	−0.879053	−0.439527	−	0.898230i	$-0.644854\pi$
$61$	−6.86562	−0.879053	−0.439527	+	0.898230i	$0.644854\pi$
$62$	0	0
$63$	−1.53791	−0.193758
$64$	0	0
$65$	3.77211	0.467873
$66$	0	0
$67$	−8.07908	−0.987016	−0.493508	−	0.869741i	$-0.664286\pi$
$67$	−8.07908	−0.987016	−0.493508	+	0.869741i	$0.664286\pi$
$68$	0	0
$69$	2.19147	0.263821
$70$	0	0
$71$	3.76828	0.447213	0.223607	−	0.974679i	$-0.428217\pi$
$71$	3.76828	0.447213	0.223607	+	0.974679i	$0.428217\pi$
$72$	0	0
$73$	−2.16933	−0.253901	−0.126951	−	0.991909i	$-0.540519\pi$
$73$	−2.16933	−0.253901	−0.126951	+	0.991909i	$0.540519\pi$
$74$	0	0
$75$	−9.02977	−1.04267
$76$	0	0
$77$	4.33450	0.493962
$78$	0	0
$79$	−2.94643	−0.331499	−0.165750	−	0.986168i	$-0.553004\pi$
$79$	−2.94643	−0.331499	−0.165750	+	0.986168i	$0.553004\pi$
$80$	0	0
$81$	−2.02111	−0.224567
$82$	0	0
$83$	8.53177	0.936483	0.468241	−	0.883601i	$-0.344888\pi$
$83$	8.53177	0.936483	0.468241	+	0.883601i	$0.344888\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.62881	−0.817894
$88$	0	0
$89$	−9.59243	−1.01680	−0.508398	−	0.861122i	$-0.669762\pi$
$89$	−9.59243	−1.01680	−0.508398	+	0.861122i	$0.669762\pi$
$90$	0	0
$91$	−1.06829	−0.111988
$92$	0	0
$93$	−8.05666	−0.835436
$94$	0	0
$95$	−3.05758	−0.313701
$96$	0	0
$97$	3.13292	0.318100	0.159050	−	0.987271i	$-0.449157\pi$
$97$	3.13292	0.318100	0.159050	+	0.987271i	$0.449157\pi$
$98$	0	0
$99$	−6.66607	−0.669965
$100$	0	0
$101$	−19.0001	−1.89058	−0.945291	−	0.326228i	$-0.894222\pi$
$101$	−19.0001	−1.89058	−0.945291	+	0.326228i	$0.894222\pi$
$102$	0	0
$103$	−3.36940	−0.331997	−0.165998	−	0.986126i	$-0.553085\pi$
$103$	−3.36940	−0.331997	−0.165998	+	0.986126i	$0.553085\pi$
$104$	0	0
$105$	4.26954	0.416664
$106$	0	0
$107$	−6.83117	−0.660394	−0.330197	−	0.943912i	$-0.607115\pi$
$107$	−6.83117	−0.660394	−0.330197	+	0.943912i	$0.607115\pi$
$108$	0	0
$109$	10.6389	1.01902	0.509510	−	0.860465i	$-0.329827\pi$
$109$	10.6389	1.01902	0.509510	+	0.860465i	$0.329827\pi$
$110$	0	0
$111$	9.33433	0.885976
$112$	0	0
$113$	14.4481	1.35916	0.679580	−	0.733601i	$-0.262162\pi$
$113$	14.4481	1.35916	0.679580	+	0.733601i	$0.262162\pi$
$114$	0	0
$115$	6.39943	0.596750
$116$	0	0
$117$	1.64294	0.151890
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.78790	0.707991
$122$	0	0
$123$	1.61959	0.146033
$124$	0	0
$125$	−8.71352	−0.779361
$126$	0	0
$127$	17.0751	1.51517	0.757586	−	0.652735i	$-0.226379\pi$
$127$	17.0751	1.51517	0.757586	+	0.652735i	$0.226379\pi$
$128$	0	0
$129$	4.05905	0.357379
$130$	0	0
$131$	9.08989	0.794187	0.397093	−	0.917778i	$-0.370019\pi$
$131$	9.08989	0.794187	0.397093	+	0.917778i	$0.370019\pi$
$132$	0	0
$133$	0.865933	0.0750859
$134$	0	0
$135$	−19.3748	−1.66752
$136$	0	0
$137$	−0.472276	−0.0403493	−0.0201746	−	0.999796i	$-0.506422\pi$
$137$	−0.472276	−0.0403493	−0.0201746	+	0.999796i	$0.506422\pi$
$138$	0	0
$139$	7.25396	0.615273	0.307636	−	0.951504i	$-0.400462\pi$
$139$	7.25396	0.615273	0.307636	+	0.951504i	$0.400462\pi$
$140$	0	0
$141$	13.8075	1.16280
$142$	0	0
$143$	−4.63052	−0.387224
$144$	0	0
$145$	−22.2773	−1.85003
$146$	0	0
$147$	−1.20917	−0.0997306
$148$	0	0
$149$	15.2704	1.25100	0.625498	−	0.780226i	$-0.284896\pi$
$149$	15.2704	1.25100	0.625498	+	0.780226i	$0.284896\pi$
$150$	0	0
$151$	−17.4012	−1.41609	−0.708047	−	0.706166i	$-0.750423\pi$
$151$	−17.4012	−1.41609	−0.708047	+	0.706166i	$0.750423\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−23.5267	−1.88971
$156$	0	0
$157$	13.1413	1.04879	0.524396	−	0.851474i	$-0.324291\pi$
$157$	13.1413	1.04879	0.524396	+	0.851474i	$0.324291\pi$
$158$	0	0
$159$	−16.1000	−1.27681
$160$	0	0
$161$	−1.81237	−0.142835
$162$	0	0
$163$	−14.8593	−1.16387	−0.581934	−	0.813236i	$-0.697704\pi$
$163$	−14.8593	−1.16387	−0.581934	+	0.813236i	$0.697704\pi$
$164$	0	0
$165$	18.5063	1.44072
$166$	0	0
$167$	8.44187	0.653251	0.326626	−	0.945154i	$-0.394088\pi$
$167$	8.44187	0.653251	0.326626	+	0.945154i	$0.394088\pi$
$168$	0	0
$169$	−11.8587	−0.912211
$170$	0	0
$171$	−1.33173	−0.101840
$172$	0	0
$173$	12.2065	0.928045	0.464022	−	0.885824i	$-0.346406\pi$
$173$	12.2065	0.928045	0.464022	+	0.885824i	$0.346406\pi$
$174$	0	0
$175$	7.46774	0.564508
$176$	0	0
$177$	16.6983	1.25512
$178$	0	0
$179$	−7.18457	−0.537000	−0.268500	−	0.963280i	$-0.586528\pi$
$179$	−7.18457	−0.537000	−0.268500	+	0.963280i	$0.586528\pi$
$180$	0	0
$181$	11.6459	0.865635	0.432818	−	0.901481i	$-0.357519\pi$
$181$	11.6459	0.865635	0.432818	+	0.901481i	$0.357519\pi$
$182$	0	0
$183$	8.30170	0.613680
$184$	0	0
$185$	27.2578	2.00403
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.48710	0.399128
$190$	0	0
$191$	−5.62558	−0.407053	−0.203526	−	0.979069i	$-0.565240\pi$
$191$	−5.62558	−0.407053	−0.203526	+	0.979069i	$0.565240\pi$
$192$	0	0
$193$	21.3297	1.53535	0.767674	−	0.640841i	$-0.221414\pi$
$193$	21.3297	1.53535	0.767674	+	0.640841i	$0.221414\pi$
$194$	0	0
$195$	−4.56112	−0.326629
$196$	0	0
$197$	9.23183	0.657741	0.328870	−	0.944375i	$-0.393332\pi$
$197$	9.23183	0.657741	0.328870	+	0.944375i	$0.393332\pi$
$198$	0	0
$199$	−14.3304	−1.01586	−0.507928	−	0.861399i	$-0.669588\pi$
$199$	−14.3304	−1.01586	−0.507928	+	0.861399i	$0.669588\pi$
$200$	0	0
$201$	9.76897	0.689050
$202$	0	0
$203$	6.30913	0.442814
$204$	0	0
$205$	4.72946	0.330320
$206$	0	0
$207$	2.78726	0.193728
$208$	0	0
$209$	3.75339	0.259627
$210$	0	0
$211$	−2.30044	−0.158369	−0.0791843	−	0.996860i	$-0.525232\pi$
$211$	−2.30044	−0.158369	−0.0791843	+	0.996860i	$0.525232\pi$
$212$	0	0
$213$	−4.55649	−0.312206
$214$	0	0
$215$	11.8531	0.808373
$216$	0	0
$217$	6.66297	0.452312
$218$	0	0
$219$	2.62309	0.177252
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.337611	−0.0226081	−0.0113041	−	0.999936i	$-0.503598\pi$
$223$	−0.337611	−0.0226081	−0.0113041	+	0.999936i	$0.503598\pi$
$224$	0	0
$225$	−11.4847	−0.765647
$226$	0	0
$227$	27.6897	1.83783	0.918914	−	0.394458i	$-0.129068\pi$
$227$	27.6897	1.83783	0.918914	+	0.394458i	$0.129068\pi$
$228$	0	0
$229$	−27.3129	−1.80489	−0.902444	−	0.430807i	$-0.858229\pi$
$229$	−27.3129	−1.80489	−0.902444	+	0.430807i	$0.858229\pi$
$230$	0	0
$231$	−5.24115	−0.344842
$232$	0	0
$233$	−13.5802	−0.889667	−0.444833	−	0.895613i	$-0.646737\pi$
$233$	−13.5802	−0.889667	−0.444833	+	0.895613i	$0.646737\pi$
$234$	0	0
$235$	40.3201	2.63019
$236$	0	0
$237$	3.56274	0.231425
$238$	0	0
$239$	−9.81066	−0.634599	−0.317300	−	0.948325i	$-0.602776\pi$
$239$	−9.81066	−0.634599	−0.317300	+	0.948325i	$0.602776\pi$
$240$	0	0
$241$	−11.0844	−0.714010	−0.357005	−	0.934102i	$-0.616202\pi$
$241$	−11.0844	−0.714010	−0.357005	+	0.934102i	$0.616202\pi$
$242$	0	0
$243$	−14.0174	−0.899219
$244$	0	0
$245$	−3.53097	−0.225585
$246$	0	0
$247$	−0.925071	−0.0588609
$248$	0	0
$249$	−10.3164	−0.653772
$250$	0	0
$251$	5.50834	0.347683	0.173842	−	0.984774i	$-0.444382\pi$
$251$	5.50834	0.347683	0.173842	+	0.984774i	$0.444382\pi$
$252$	0	0
$253$	−7.85573	−0.493886
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.9367	1.05648	0.528242	−	0.849094i	$-0.322851\pi$
$257$	16.9367	1.05648	0.528242	+	0.849094i	$0.322851\pi$
$258$	0	0
$259$	−7.71962	−0.479674
$260$	0	0
$261$	−9.70287	−0.600593
$262$	0	0
$263$	26.7181	1.64751	0.823753	−	0.566948i	$-0.191876\pi$
$263$	26.7181	1.64751	0.823753	+	0.566948i	$0.191876\pi$
$264$	0	0
$265$	−47.0146	−2.88809
$266$	0	0
$267$	11.5989	0.709839
$268$	0	0
$269$	−8.81502	−0.537461	−0.268731	−	0.963215i	$-0.586604\pi$
$269$	−8.81502	−0.537461	−0.268731	+	0.963215i	$0.586604\pi$
$270$	0	0
$271$	3.47254	0.210942	0.105471	−	0.994422i	$-0.466365\pi$
$271$	3.47254	0.210942	0.105471	+	0.994422i	$0.466365\pi$
$272$	0	0
$273$	1.29175	0.0781802
$274$	0	0
$275$	32.3689	1.95192
$276$	0	0
$277$	−10.0420	−0.603366	−0.301683	−	0.953408i	$-0.597548\pi$
$277$	−10.0420	−0.603366	−0.301683	+	0.953408i	$0.597548\pi$
$278$	0	0
$279$	−10.2470	−0.613474
$280$	0	0
$281$	−31.1596	−1.85883	−0.929413	−	0.369041i	$-0.879686\pi$
$281$	−31.1596	−1.85883	−0.929413	+	0.369041i	$0.879686\pi$
$282$	0	0
$283$	2.21090	0.131424	0.0657122	−	0.997839i	$-0.479068\pi$
$283$	2.21090	0.131424	0.0657122	+	0.997839i	$0.479068\pi$
$284$	0	0
$285$	3.69714	0.218999
$286$	0	0
$287$	−1.33942	−0.0790636
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−3.78823	−0.222070
$292$	0	0
$293$	−15.4231	−0.901028	−0.450514	−	0.892769i	$-0.648759\pi$
$293$	−15.4231	−0.901028	−0.450514	+	0.892769i	$0.648759\pi$
$294$	0	0
$295$	48.7618	2.83902
$296$	0	0
$297$	23.7838	1.38008
$298$	0	0
$299$	1.93615	0.111970
$300$	0	0
$301$	−3.35689	−0.193488
$302$	0	0
$303$	22.9744	1.31984
$304$	0	0
$305$	24.2423	1.38811
$306$	0	0
$307$	−2.54658	−0.145341	−0.0726705	−	0.997356i	$-0.523152\pi$
$307$	−2.54658	−0.145341	−0.0726705	+	0.997356i	$0.523152\pi$
$308$	0	0
$309$	4.07417	0.231772
$310$	0	0
$311$	11.0161	0.624667	0.312334	−	0.949972i	$-0.398889\pi$
$311$	11.0161	0.624667	0.312334	+	0.949972i	$0.398889\pi$
$312$	0	0
$313$	2.16259	0.122237	0.0611185	−	0.998131i	$-0.480533\pi$
$313$	2.16259	0.122237	0.0611185	+	0.998131i	$0.480533\pi$
$314$	0	0
$315$	5.43031	0.305963
$316$	0	0
$317$	11.4180	0.641301	0.320650	−	0.947198i	$-0.396099\pi$
$317$	11.4180	0.641301	0.320650	+	0.947198i	$0.396099\pi$
$318$	0	0
$319$	27.3469	1.53113
$320$	0	0
$321$	8.26004	0.461030
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−7.97774	−0.442526
$326$	0	0
$327$	−12.8642	−0.711393
$328$	0	0
$329$	−11.4190	−0.629549
$330$	0	0
$331$	0.846845	0.0465468	0.0232734	−	0.999729i	$-0.492591\pi$
$331$	0.846845	0.0465468	0.0232734	+	0.999729i	$0.492591\pi$
$332$	0	0
$333$	11.8721	0.650586
$334$	0	0
$335$	28.5270	1.55860
$336$	0	0
$337$	4.47432	0.243732	0.121866	−	0.992547i	$-0.461112\pi$
$337$	4.47432	0.243732	0.121866	+	0.992547i	$0.461112\pi$
$338$	0	0
$339$	−17.4702	−0.948849
$340$	0	0
$341$	28.8806	1.56397
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.73800	−0.416600
$346$	0	0
$347$	11.3776	0.610782	0.305391	−	0.952227i	$-0.401213\pi$
$347$	11.3776	0.610782	0.305391	+	0.952227i	$0.401213\pi$
$348$	0	0
$349$	25.4509	1.36236	0.681179	−	0.732117i	$-0.261468\pi$
$349$	25.4509	1.36236	0.681179	+	0.732117i	$0.261468\pi$
$350$	0	0
$351$	−5.86184	−0.312882
$352$	0	0
$353$	28.8137	1.53360	0.766800	−	0.641887i	$-0.221848\pi$
$353$	28.8137	1.53360	0.766800	+	0.641887i	$0.221848\pi$
$354$	0	0
$355$	−13.3057	−0.706193
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.4745	0.658380	0.329190	−	0.944264i	$-0.393224\pi$
$359$	12.4745	0.658380	0.329190	+	0.944264i	$0.393224\pi$
$360$	0	0
$361$	−18.2502	−0.960535
$362$	0	0
$363$	−9.41690	−0.494259
$364$	0	0
$365$	7.65984	0.400935
$366$	0	0
$367$	−31.8914	−1.66472	−0.832360	−	0.554235i	$-0.813011\pi$
$367$	−31.8914	−1.66472	−0.832360	+	0.554235i	$0.813011\pi$
$368$	0	0
$369$	2.05991	0.107235
$370$	0	0
$371$	13.3149	0.691277
$372$	0	0
$373$	32.3338	1.67418	0.837090	−	0.547066i	$-0.184255\pi$
$373$	32.3338	1.67418	0.837090	+	0.547066i	$0.184255\pi$
$374$	0	0
$375$	10.5361	0.544083
$376$	0	0
$377$	−6.74000	−0.347128
$378$	0	0
$379$	27.4866	1.41189	0.705947	−	0.708265i	$-0.250522\pi$
$379$	27.4866	1.41189	0.705947	+	0.708265i	$0.250522\pi$
$380$	0	0
$381$	−20.6467	−1.05776
$382$	0	0
$383$	24.5806	1.25601	0.628004	−	0.778210i	$-0.283872\pi$
$383$	24.5806	1.25601	0.628004	+	0.778210i	$0.283872\pi$
$384$	0	0
$385$	−15.3050	−0.780015
$386$	0	0
$387$	5.16259	0.262429
$388$	0	0
$389$	31.1352	1.57862	0.789308	−	0.613998i	$-0.210440\pi$
$389$	31.1352	1.57862	0.789308	+	0.613998i	$0.210440\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−10.9912	−0.554433
$394$	0	0
$395$	10.4038	0.523470
$396$	0	0
$397$	−4.65419	−0.233587	−0.116793	−	0.993156i	$-0.537262\pi$
$397$	−4.65419	−0.233587	−0.116793	+	0.993156i	$0.537262\pi$
$398$	0	0
$399$	−1.04706	−0.0524186
$400$	0	0
$401$	−13.9586	−0.697058	−0.348529	−	0.937298i	$-0.613319\pi$
$401$	−13.9586	−0.697058	−0.348529	+	0.937298i	$0.613319\pi$
$402$	0	0
$403$	−7.11801	−0.354573
$404$	0	0
$405$	7.13647	0.354614
$406$	0	0
$407$	−33.4607	−1.65859
$408$	0	0
$409$	18.9180	0.935433	0.467717	−	0.883878i	$-0.345077\pi$
$409$	18.9180	0.935433	0.467717	+	0.883878i	$0.345077\pi$
$410$	0	0
$411$	0.571062	0.0281684
$412$	0	0
$413$	−13.8098	−0.679534
$414$	0	0
$415$	−30.1254	−1.47880
$416$	0	0
$417$	−8.77126	−0.429531
$418$	0	0
$419$	−38.7131	−1.89126	−0.945629	−	0.325247i	$-0.894553\pi$
$419$	−38.7131	−1.89126	−0.945629	+	0.325247i	$0.894553\pi$
$420$	0	0
$421$	26.4823	1.29067	0.645333	−	0.763901i	$-0.276718\pi$
$421$	26.4823	1.29067	0.645333	+	0.763901i	$0.276718\pi$
$422$	0	0
$423$	17.5614	0.853863
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.86562	−0.332251
$428$	0	0
$429$	5.59908	0.270326
$430$	0	0
$431$	31.4453	1.51467	0.757334	−	0.653028i	$-0.226501\pi$
$431$	31.4453	1.51467	0.757334	+	0.653028i	$0.226501\pi$
$432$	0	0
$433$	−16.7690	−0.805868	−0.402934	−	0.915229i	$-0.632010\pi$
$433$	−16.7690	−0.805868	−0.402934	+	0.915229i	$0.632010\pi$
$434$	0	0
$435$	26.9371	1.29153
$436$	0	0
$437$	−1.56939	−0.0750743
$438$	0	0
$439$	8.77883	0.418991	0.209495	−	0.977810i	$-0.432818\pi$
$439$	8.77883	0.418991	0.209495	+	0.977810i	$0.432818\pi$
$440$	0	0
$441$	−1.53791	−0.0732338
$442$	0	0
$443$	35.9940	1.71013	0.855063	−	0.518524i	$-0.173518\pi$
$443$	35.9940	1.71013	0.855063	+	0.518524i	$0.173518\pi$
$444$	0	0
$445$	33.8706	1.60562
$446$	0	0
$447$	−18.4644	−0.873338
$448$	0	0
$449$	8.83589	0.416991	0.208496	−	0.978023i	$-0.433143\pi$
$449$	8.83589	0.416991	0.208496	+	0.978023i	$0.433143\pi$
$450$	0	0
$451$	−5.80573	−0.273381
$452$	0	0
$453$	21.0411	0.988595
$454$	0	0
$455$	3.77211	0.176839
$456$	0	0
$457$	15.8172	0.739898	0.369949	−	0.929052i	$-0.379375\pi$
$457$	15.8172	0.739898	0.369949	+	0.929052i	$0.379375\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.8173	0.969557	0.484779	−	0.874637i	$-0.338900\pi$
$461$	20.8173	0.969557	0.484779	+	0.874637i	$0.338900\pi$
$462$	0	0
$463$	39.9133	1.85493	0.927464	−	0.373914i	$-0.121984\pi$
$463$	39.9133	1.85493	0.927464	+	0.373914i	$0.121984\pi$
$464$	0	0
$465$	28.4478	1.31924
$466$	0	0
$467$	3.99404	0.184822	0.0924110	−	0.995721i	$-0.470543\pi$
$467$	3.99404	0.184822	0.0924110	+	0.995721i	$0.470543\pi$
$468$	0	0
$469$	−8.07908	−0.373057
$470$	0	0
$471$	−15.8901	−0.732177
$472$	0	0
$473$	−14.5505	−0.669031
$474$	0	0
$475$	6.46657	0.296706
$476$	0	0
$477$	−20.4772	−0.937585
$478$	0	0
$479$	−1.54405	−0.0705494	−0.0352747	−	0.999378i	$-0.511231\pi$
$479$	−1.54405	−0.0705494	−0.0352747	+	0.999378i	$0.511231\pi$
$480$	0	0
$481$	8.24683	0.376023
$482$	0	0
$483$	2.19147	0.0997151
$484$	0	0
$485$	−11.0623	−0.502311
$486$	0	0
$487$	−2.20153	−0.0997608	−0.0498804	−	0.998755i	$-0.515884\pi$
$487$	−2.20153	−0.0997608	−0.0498804	+	0.998755i	$0.515884\pi$
$488$	0	0
$489$	17.9674	0.812514
$490$	0	0
$491$	38.0842	1.71871	0.859357	−	0.511376i	$-0.170864\pi$
$491$	38.0842	1.71871	0.859357	+	0.511376i	$0.170864\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	23.5377	1.05794
$496$	0	0
$497$	3.76828	0.169031
$498$	0	0
$499$	36.2687	1.62361	0.811806	−	0.583927i	$-0.198485\pi$
$499$	36.2687	1.62361	0.811806	+	0.583927i	$0.198485\pi$
$500$	0	0
$501$	−10.2076	−0.456044
$502$	0	0
$503$	14.6396	0.652746	0.326373	−	0.945241i	$-0.394174\pi$
$503$	14.6396	0.652746	0.326373	+	0.945241i	$0.394174\pi$
$504$	0	0
$505$	67.0888	2.98541
$506$	0	0
$507$	14.3392	0.636828
$508$	0	0
$509$	−8.33470	−0.369429	−0.184715	−	0.982792i	$-0.559136\pi$
$509$	−8.33470	−0.369429	−0.184715	+	0.982792i	$0.559136\pi$
$510$	0	0
$511$	−2.16933	−0.0959656
$512$	0	0
$513$	4.75146	0.209782
$514$	0	0
$515$	11.8972	0.524255
$516$	0	0
$517$	−49.4956	−2.17682
$518$	0	0
$519$	−14.7598	−0.647881
$520$	0	0
$521$	−8.21783	−0.360030	−0.180015	−	0.983664i	$-0.557615\pi$
$521$	−8.21783	−0.360030	−0.180015	+	0.983664i	$0.557615\pi$
$522$	0	0
$523$	−9.82413	−0.429579	−0.214790	−	0.976660i	$-0.568907\pi$
$523$	−9.82413	−0.429579	−0.214790	+	0.976660i	$0.568907\pi$
$524$	0	0
$525$	−9.02977	−0.394091
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.7153	−0.857187
$530$	0	0
$531$	21.2382	0.921657
$532$	0	0
$533$	1.43090	0.0619790
$534$	0	0
$535$	24.1206	1.04283
$536$	0	0
$537$	8.68736	0.374887
$538$	0	0
$539$	4.33450	0.186700
$540$	0	0
$541$	17.6235	0.757693	0.378847	−	0.925459i	$-0.376321\pi$
$541$	17.6235	0.757693	0.378847	+	0.925459i	$0.376321\pi$
$542$	0	0
$543$	−14.0819	−0.604312
$544$	0	0
$545$	−37.5656	−1.60913
$546$	0	0
$547$	−19.5130	−0.834314	−0.417157	−	0.908834i	$-0.636973\pi$
$547$	−19.5130	−0.834314	−0.417157	+	0.908834i	$0.636973\pi$
$548$	0	0
$549$	10.5587	0.450635
$550$	0	0
$551$	5.46329	0.232744
$552$	0	0
$553$	−2.94643	−0.125295
$554$	0	0
$555$	−32.9592	−1.39904
$556$	0	0
$557$	16.4920	0.698787	0.349393	−	0.936976i	$-0.386388\pi$
$557$	16.4920	0.698787	0.349393	+	0.936976i	$0.386388\pi$
$558$	0	0
$559$	3.58615	0.151678
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.6951	−0.914339	−0.457169	−	0.889380i	$-0.651137\pi$
$563$	−21.6951	−0.914339	−0.457169	+	0.889380i	$0.651137\pi$
$564$	0	0
$565$	−51.0157	−2.14625
$566$	0	0
$567$	−2.02111	−0.0848785
$568$	0	0
$569$	−9.31653	−0.390569	−0.195285	−	0.980747i	$-0.562563\pi$
$569$	−9.31653	−0.390569	−0.195285	+	0.980747i	$0.562563\pi$
$570$	0	0
$571$	13.4541	0.563036	0.281518	−	0.959556i	$-0.409162\pi$
$571$	13.4541	0.563036	0.281518	+	0.959556i	$0.409162\pi$
$572$	0	0
$573$	6.80228	0.284169
$574$	0	0
$575$	−13.5343	−0.564421
$576$	0	0
$577$	−5.13162	−0.213632	−0.106816	−	0.994279i	$-0.534066\pi$
$577$	−5.13162	−0.213632	−0.106816	+	0.994279i	$0.534066\pi$
$578$	0	0
$579$	−25.7913	−1.07185
$580$	0	0
$581$	8.53177	0.353957
$582$	0	0
$583$	57.7136	2.39025
$584$	0	0
$585$	−5.80117	−0.239849
$586$	0	0
$587$	−2.24329	−0.0925905	−0.0462953	−	0.998928i	$-0.514742\pi$
$587$	−2.24329	−0.0925905	−0.0462953	+	0.998928i	$0.514742\pi$
$588$	0	0
$589$	5.76968	0.237736
$590$	0	0
$591$	−11.1628	−0.459178
$592$	0	0
$593$	21.9578	0.901698	0.450849	−	0.892600i	$-0.351121\pi$
$593$	21.9578	0.901698	0.450849	+	0.892600i	$0.351121\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.3279	0.709184
$598$	0	0
$599$	−21.9301	−0.896039	−0.448020	−	0.894024i	$-0.647871\pi$
$599$	−21.9301	−0.896039	−0.448020	+	0.894024i	$0.647871\pi$
$600$	0	0
$601$	−22.5628	−0.920355	−0.460178	−	0.887827i	$-0.652214\pi$
$601$	−22.5628	−0.920355	−0.460178	+	0.887827i	$0.652214\pi$
$602$	0	0
$603$	12.4249	0.505981
$604$	0	0
$605$	−27.4988	−1.11799
$606$	0	0
$607$	−34.5481	−1.40227	−0.701133	−	0.713031i	$-0.747322\pi$
$607$	−34.5481	−1.40227	−0.701133	+	0.713031i	$0.747322\pi$
$608$	0	0
$609$	−7.62881	−0.309135
$610$	0	0
$611$	12.1988	0.493512
$612$	0	0
$613$	37.5011	1.51466	0.757328	−	0.653035i	$-0.226504\pi$
$613$	37.5011	1.51466	0.757328	+	0.653035i	$0.226504\pi$
$614$	0	0
$615$	−5.71871	−0.230601
$616$	0	0
$617$	−5.04568	−0.203132	−0.101566	−	0.994829i	$-0.532385\pi$
$617$	−5.04568	−0.203132	−0.101566	+	0.994829i	$0.532385\pi$
$618$	0	0
$619$	24.1657	0.971300	0.485650	−	0.874153i	$-0.338583\pi$
$619$	24.1657	0.971300	0.485650	+	0.874153i	$0.338583\pi$
$620$	0	0
$621$	−9.94467	−0.399066
$622$	0	0
$623$	−9.59243	−0.384312
$624$	0	0
$625$	−6.57153	−0.262861
$626$	0	0
$627$	−4.53848	−0.181250
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.767437	0.0305512	0.0152756	−	0.999883i	$-0.495137\pi$
$631$	0.767437	0.0305512	0.0152756	+	0.999883i	$0.495137\pi$
$632$	0	0
$633$	2.78162	0.110559
$634$	0	0
$635$	−60.2917	−2.39260
$636$	0	0
$637$	−1.06829	−0.0423273
$638$	0	0
$639$	−5.79528	−0.229258
$640$	0	0
$641$	−45.6602	−1.80347	−0.901735	−	0.432290i	$-0.857706\pi$
$641$	−45.6602	−1.80347	−0.901735	+	0.432290i	$0.857706\pi$
$642$	0	0
$643$	44.3325	1.74830	0.874152	−	0.485652i	$-0.161418\pi$
$643$	44.3325	1.74830	0.874152	+	0.485652i	$0.161418\pi$
$644$	0	0
$645$	−14.3324	−0.564337
$646$	0	0
$647$	8.61462	0.338676	0.169338	−	0.985558i	$-0.445837\pi$
$647$	8.61462	0.338676	0.169338	+	0.985558i	$0.445837\pi$
$648$	0	0
$649$	−59.8584	−2.34965
$650$	0	0
$651$	−8.05666	−0.315765
$652$	0	0
$653$	1.61420	0.0631687	0.0315843	−	0.999501i	$-0.489945\pi$
$653$	1.61420	0.0631687	0.0315843	+	0.999501i	$0.489945\pi$
$654$	0	0
$655$	−32.0961	−1.25410
$656$	0	0
$657$	3.33624	0.130159
$658$	0	0
$659$	−23.3794	−0.910732	−0.455366	−	0.890304i	$-0.650492\pi$
$659$	−23.3794	−0.910732	−0.455366	+	0.890304i	$0.650492\pi$
$660$	0	0
$661$	−41.7893	−1.62542	−0.812708	−	0.582671i	$-0.802008\pi$
$661$	−41.7893	−1.62542	−0.812708	+	0.582671i	$0.802008\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.05758	−0.118568
$666$	0	0
$667$	−11.4345	−0.442745
$668$	0	0
$669$	0.408229	0.0157831
$670$	0	0
$671$	−29.7591	−1.14884
$672$	0	0
$673$	2.85786	0.110162	0.0550812	−	0.998482i	$-0.482458\pi$
$673$	2.85786	0.110162	0.0550812	+	0.998482i	$0.482458\pi$
$674$	0	0
$675$	40.9763	1.57718
$676$	0	0
$677$	42.2233	1.62277	0.811386	−	0.584511i	$-0.198713\pi$
$677$	42.2233	1.62277	0.811386	+	0.584511i	$0.198713\pi$
$678$	0	0
$679$	3.13292	0.120231
$680$	0	0
$681$	−33.4815	−1.28301
$682$	0	0
$683$	6.97143	0.266754	0.133377	−	0.991065i	$-0.457418\pi$
$683$	6.97143	0.266754	0.133377	+	0.991065i	$0.457418\pi$
$684$	0	0
$685$	1.66759	0.0637154
$686$	0	0
$687$	33.0259	1.26002
$688$	0	0
$689$	−14.2243	−0.541901
$690$	0	0
$691$	9.68902	0.368588	0.184294	−	0.982871i	$-0.441000\pi$
$691$	9.68902	0.368588	0.184294	+	0.982871i	$0.441000\pi$
$692$	0	0
$693$	−6.66607	−0.253223
$694$	0	0
$695$	−25.6135	−0.971575
$696$	0	0
$697$	0	0
$698$	0	0
$699$	16.4207	0.621089
$700$	0	0
$701$	47.7392	1.80308	0.901542	−	0.432692i	$-0.142436\pi$
$701$	47.7392	1.80308	0.901542	+	0.432692i	$0.142436\pi$
$702$	0	0
$703$	−6.68468	−0.252117
$704$	0	0
$705$	−48.7538	−1.83618
$706$	0	0
$707$	−19.0001	−0.714573
$708$	0	0
$709$	2.16659	0.0813680	0.0406840	−	0.999172i	$-0.487046\pi$
$709$	2.16659	0.0813680	0.0406840	+	0.999172i	$0.487046\pi$
$710$	0	0
$711$	4.53135	0.169939
$712$	0	0
$713$	−12.0758	−0.452242
$714$	0	0
$715$	16.3502	0.611464
$716$	0	0
$717$	11.8628	0.443023
$718$	0	0
$719$	21.2309	0.791778	0.395889	−	0.918298i	$-0.370437\pi$
$719$	21.2309	0.791778	0.395889	+	0.918298i	$0.370437\pi$
$720$	0	0
$721$	−3.36940	−0.125483
$722$	0	0
$723$	13.4029	0.498461
$724$	0	0
$725$	47.1150	1.74981
$726$	0	0
$727$	−35.0863	−1.30128	−0.650639	−	0.759387i	$-0.725499\pi$
$727$	−35.0863	−1.30128	−0.650639	+	0.759387i	$0.725499\pi$
$728$	0	0
$729$	23.0128	0.852325
$730$	0	0
$731$	0	0
$732$	0	0
$733$	48.1243	1.77751	0.888756	−	0.458380i	$-0.151570\pi$
$733$	48.1243	1.77751	0.888756	+	0.458380i	$0.151570\pi$
$734$	0	0
$735$	4.26954	0.157484
$736$	0	0
$737$	−35.0188	−1.28993
$738$	0	0
$739$	−9.08895	−0.334343	−0.167171	−	0.985928i	$-0.553463\pi$
$739$	−9.08895	−0.334343	−0.167171	+	0.985928i	$0.553463\pi$
$740$	0	0
$741$	1.11857	0.0410916
$742$	0	0
$743$	−15.8471	−0.581373	−0.290686	−	0.956818i	$-0.593884\pi$
$743$	−15.8471	−0.581373	−0.290686	+	0.956818i	$0.593884\pi$
$744$	0	0
$745$	−53.9192	−1.97544
$746$	0	0
$747$	−13.1211	−0.480075
$748$	0	0
$749$	−6.83117	−0.249605
$750$	0	0
$751$	21.7164	0.792443	0.396222	−	0.918155i	$-0.370321\pi$
$751$	21.7164	0.792443	0.396222	+	0.918155i	$0.370321\pi$
$752$	0	0
$753$	−6.66052	−0.242723
$754$	0	0
$755$	61.4433	2.23615
$756$	0	0
$757$	−33.4660	−1.21634	−0.608172	−	0.793805i	$-0.708097\pi$
$757$	−33.4660	−1.21634	−0.608172	+	0.793805i	$0.708097\pi$
$758$	0	0
$759$	9.49891	0.344789
$760$	0	0
$761$	−41.0963	−1.48974	−0.744870	−	0.667209i	$-0.767489\pi$
$761$	−41.0963	−1.48974	−0.744870	+	0.667209i	$0.767489\pi$
$762$	0	0
$763$	10.6389	0.385153
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.7529	0.532695
$768$	0	0
$769$	−33.5980	−1.21157	−0.605787	−	0.795627i	$-0.707142\pi$
$769$	−33.5980	−1.21157	−0.605787	+	0.795627i	$0.707142\pi$
$770$	0	0
$771$	−20.4794	−0.737546
$772$	0	0
$773$	10.2680	0.369314	0.184657	−	0.982803i	$-0.440883\pi$
$773$	10.2680	0.369314	0.184657	+	0.982803i	$0.440883\pi$
$774$	0	0
$775$	49.7573	1.78734
$776$	0	0
$777$	9.33433	0.334867
$778$	0	0
$779$	−1.15985	−0.0415559
$780$	0	0
$781$	16.3336	0.584463
$782$	0	0
$783$	34.6188	1.23718
$784$	0	0
$785$	−46.4016	−1.65614
$786$	0	0
$787$	14.4117	0.513722	0.256861	−	0.966448i	$-0.417312\pi$
$787$	14.4117	0.513722	0.256861	+	0.966448i	$0.417312\pi$
$788$	0	0
$789$	−32.3067	−1.15015
$790$	0	0
$791$	14.4481	0.513714
$792$	0	0
$793$	7.33450	0.260456
$794$	0	0
$795$	56.8486	2.01621
$796$	0	0
$797$	9.69924	0.343565	0.171782	−	0.985135i	$-0.445047\pi$
$797$	9.69924	0.343565	0.171782	+	0.985135i	$0.445047\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	14.7523	0.521246
$802$	0	0
$803$	−9.40297	−0.331824
$804$	0	0
$805$	6.39943	0.225550
$806$	0	0
$807$	10.6588	0.375209
$808$	0	0
$809$	56.1783	1.97513	0.987563	−	0.157226i	$-0.0502551\pi$
$809$	56.1783	1.97513	0.987563	+	0.157226i	$0.0502551\pi$
$810$	0	0
$811$	−15.0603	−0.528838	−0.264419	−	0.964408i	$-0.585180\pi$
$811$	−15.0603	−0.528838	−0.264419	+	0.964408i	$0.585180\pi$
$812$	0	0
$813$	−4.19889	−0.147262
$814$	0	0
$815$	52.4677	1.83786
$816$	0	0
$817$	−2.90684	−0.101698
$818$	0	0
$819$	1.64294	0.0574089
$820$	0	0
$821$	−47.2701	−1.64974	−0.824870	−	0.565323i	$-0.808752\pi$
$821$	−47.2701	−1.64974	−0.824870	+	0.565323i	$0.808752\pi$
$822$	0	0
$823$	−46.6576	−1.62638	−0.813190	−	0.581998i	$-0.802271\pi$
$823$	−46.6576	−1.62638	−0.813190	+	0.581998i	$0.802271\pi$
$824$	0	0
$825$	−39.1395	−1.36266
$826$	0	0
$827$	28.7773	1.00068	0.500342	−	0.865828i	$-0.333208\pi$
$827$	28.7773	1.00068	0.500342	+	0.865828i	$0.333208\pi$
$828$	0	0
$829$	4.59550	0.159608	0.0798042	−	0.996811i	$-0.474570\pi$
$829$	4.59550	0.159608	0.0798042	+	0.996811i	$0.474570\pi$
$830$	0	0
$831$	12.1425	0.421218
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−29.8080	−1.03155
$836$	0	0
$837$	36.5604	1.26371
$838$	0	0
$839$	16.1546	0.557717	0.278859	−	0.960332i	$-0.410044\pi$
$839$	16.1546	0.557717	0.278859	+	0.960332i	$0.410044\pi$
$840$	0	0
$841$	10.8051	0.372591
$842$	0	0
$843$	37.6772	1.29767
$844$	0	0
$845$	41.8729	1.44047
$846$	0	0
$847$	7.78790	0.267596
$848$	0	0
$849$	−2.67335	−0.0917493
$850$	0	0
$851$	13.9908	0.479600
$852$	0	0
$853$	−23.1870	−0.793908	−0.396954	−	0.917838i	$-0.629933\pi$
$853$	−23.1870	−0.793908	−0.396954	+	0.917838i	$0.629933\pi$
$854$	0	0
$855$	4.70229	0.160815
$856$	0	0
$857$	24.3105	0.830430	0.415215	−	0.909723i	$-0.363706\pi$
$857$	24.3105	0.830430	0.415215	+	0.909723i	$0.363706\pi$
$858$	0	0
$859$	−1.83430	−0.0625856	−0.0312928	−	0.999510i	$-0.509962\pi$
$859$	−1.83430	−0.0625856	−0.0312928	+	0.999510i	$0.509962\pi$
$860$	0	0
$861$	1.61959	0.0551954
$862$	0	0
$863$	25.5721	0.870485	0.435243	−	0.900313i	$-0.356663\pi$
$863$	25.5721	0.870485	0.435243	+	0.900313i	$0.356663\pi$
$864$	0	0
$865$	−43.1008	−1.46547
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.7713	−0.433237
$870$	0	0
$871$	8.63083	0.292444
$872$	0	0
$873$	−4.81815	−0.163070
$874$	0	0
$875$	−8.71352	−0.294571
$876$	0	0
$877$	1.16799	0.0394404	0.0197202	−	0.999806i	$-0.493722\pi$
$877$	1.16799	0.0394404	0.0197202	+	0.999806i	$0.493722\pi$
$878$	0	0
$879$	18.6492	0.629021
$880$	0	0
$881$	−40.2931	−1.35751	−0.678755	−	0.734365i	$-0.737480\pi$
$881$	−40.2931	−1.35751	−0.678755	+	0.734365i	$0.737480\pi$
$882$	0	0
$883$	−44.5385	−1.49884	−0.749421	−	0.662094i	$-0.769668\pi$
$883$	−44.5385	−1.49884	−0.749421	+	0.662094i	$0.769668\pi$
$884$	0	0
$885$	−58.9613	−1.98196
$886$	0	0
$887$	−1.06222	−0.0356659	−0.0178329	−	0.999841i	$-0.505677\pi$
$887$	−1.06222	−0.0356659	−0.0178329	+	0.999841i	$0.505677\pi$
$888$	0	0
$889$	17.0751	0.572681
$890$	0	0
$891$	−8.76049	−0.293488
$892$	0	0
$893$	−9.88808	−0.330892
$894$	0	0
$895$	25.3685	0.847975
$896$	0	0
$897$	−2.34113	−0.0781680
$898$	0	0
$899$	42.0375	1.40203
$900$	0	0
$901$	0	0
$902$	0	0
$903$	4.05905	0.135077
$904$	0	0
$905$	−41.1214	−1.36692
$906$	0	0
$907$	−32.8756	−1.09162	−0.545808	−	0.837910i	$-0.683777\pi$
$907$	−32.8756	−1.09162	−0.545808	+	0.837910i	$0.683777\pi$
$908$	0	0
$909$	29.2205	0.969181
$910$	0	0
$911$	27.1282	0.898799	0.449399	−	0.893331i	$-0.351638\pi$
$911$	27.1282	0.898799	0.449399	+	0.893331i	$0.351638\pi$
$912$	0	0
$913$	36.9810	1.22389
$914$	0	0
$915$	−29.3131	−0.969060
$916$	0	0
$917$	9.08989	0.300174
$918$	0	0
$919$	−13.3575	−0.440624	−0.220312	−	0.975429i	$-0.570708\pi$
$919$	−13.3575	−0.440624	−0.220312	+	0.975429i	$0.570708\pi$
$920$	0	0
$921$	3.07925	0.101465
$922$	0	0
$923$	−4.02564	−0.132505
$924$	0	0
$925$	−57.6482	−1.89546
$926$	0	0
$927$	5.18183	0.170194
$928$	0	0
$929$	−6.61737	−0.217109	−0.108554	−	0.994090i	$-0.534622\pi$
$929$	−6.61737	−0.217109	−0.108554	+	0.994090i	$0.534622\pi$
$930$	0	0
$931$	0.865933	0.0283798
$932$	0	0
$933$	−13.3204	−0.436089
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.8317	1.30125	0.650623	−	0.759401i	$-0.274508\pi$
$937$	39.8317	1.30125	0.650623	+	0.759401i	$0.274508\pi$
$938$	0	0
$939$	−2.61494	−0.0853354
$940$	0	0
$941$	11.1026	0.361934	0.180967	−	0.983489i	$-0.442077\pi$
$941$	11.1026	0.361934	0.180967	+	0.983489i	$0.442077\pi$
$942$	0	0
$943$	2.42753	0.0790513
$944$	0	0
$945$	−19.3748	−0.630262
$946$	0	0
$947$	27.4531	0.892106	0.446053	−	0.895007i	$-0.352829\pi$
$947$	27.4531	0.892106	0.446053	+	0.895007i	$0.352829\pi$
$948$	0	0
$949$	2.31748	0.0752287
$950$	0	0
$951$	−13.8063	−0.447701
$952$	0	0
$953$	11.7203	0.379658	0.189829	−	0.981817i	$-0.439207\pi$
$953$	11.7203	0.379658	0.189829	+	0.981817i	$0.439207\pi$
$954$	0	0
$955$	19.8638	0.642776
$956$	0	0
$957$	−33.0671	−1.06891
$958$	0	0
$959$	−0.472276	−0.0152506
$960$	0	0
$961$	13.3951	0.432101
$962$	0	0
$963$	10.5057	0.338542
$964$	0	0
$965$	−75.3146	−2.42446
$966$	0	0
$967$	−16.4982	−0.530547	−0.265273	−	0.964173i	$-0.585462\pi$
$967$	−16.4982	−0.530547	−0.265273	+	0.964173i	$0.585462\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	59.0436	1.89480	0.947400	−	0.320051i	$-0.103700\pi$
$971$	59.0436	1.89480	0.947400	+	0.320051i	$0.103700\pi$
$972$	0	0
$973$	7.25396	0.232551
$974$	0	0
$975$	9.64644	0.308933
$976$	0	0
$977$	−11.9321	−0.381742	−0.190871	−	0.981615i	$-0.561131\pi$
$977$	−11.9321	−0.381742	−0.190871	+	0.981615i	$0.561131\pi$
$978$	0	0
$979$	−41.5784	−1.32885
$980$	0	0
$981$	−16.3616	−0.522387
$982$	0	0
$983$	37.7789	1.20496	0.602479	−	0.798135i	$-0.294180\pi$
$983$	37.7789	1.20496	0.602479	+	0.798135i	$0.294180\pi$
$984$	0	0
$985$	−32.5973	−1.03864
$986$	0	0
$987$	13.8075	0.439497
$988$	0	0
$989$	6.08394	0.193458
$990$	0	0
$991$	43.8544	1.39308	0.696540	−	0.717517i	$-0.254722\pi$
$991$	43.8544	1.39308	0.696540	+	0.717517i	$0.254722\pi$
$992$	0	0
$993$	−1.02398	−0.0324950
$994$	0	0
$995$	50.6003	1.60414
$996$	0	0
$997$	−49.7475	−1.57552	−0.787759	−	0.615984i	$-0.788759\pi$
$997$	−49.7475	−1.57552	−0.787759	+	0.615984i	$0.788759\pi$
$998$	0	0
$999$	−42.3584	−1.34016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.z.1.6		20
17.3	odd	16		476.2.u.a.281.5	✓	40
17.6	odd	16		476.2.u.a.393.5	yes	40
17.16	even	2		8092.2.a.y.1.15		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.u.a.281.5	✓	40	17.3	odd	16
476.2.u.a.393.5	yes	40	17.6	odd	16
8092.2.a.y.1.15		20	17.16	even	2
8092.2.a.z.1.6		20	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8092.a (trivial)

\(f(q)\)	\(=\)	\(q-1.20917 q^{3} -3.53097 q^{5} +1.00000 q^{7} -1.53791 q^{9} +4.33450 q^{11} -1.06829 q^{13} +4.26954 q^{15} +0.865933 q^{19} -1.20917 q^{21} -1.81237 q^{23} +7.46774 q^{25} +5.48710 q^{27} +6.30913 q^{29} +6.66297 q^{31} -5.24115 q^{33} -3.53097 q^{35} -7.71962 q^{37} +1.29175 q^{39} -1.33942 q^{41} -3.35689 q^{43} +5.43031 q^{45} -11.4190 q^{47} +1.00000 q^{49} +13.3149 q^{53} -15.3050 q^{55} -1.04706 q^{57} -13.8098 q^{59} -6.86562 q^{61} -1.53791 q^{63} +3.77211 q^{65} -8.07908 q^{67} +2.19147 q^{69} +3.76828 q^{71} -2.16933 q^{73} -9.02977 q^{75} +4.33450 q^{77} -2.94643 q^{79} -2.02111 q^{81} +8.53177 q^{83} -7.62881 q^{87} -9.59243 q^{89} -1.06829 q^{91} -8.05666 q^{93} -3.05758 q^{95} +3.13292 q^{97} -6.66607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 20 * q + 8 * q^3 + 8 * q^5 + 20 * q^7 + 28 * q^9 + 16 * q^11 + 8 * q^13 + 8 * q^15 + 8 * q^19 + 8 * q^21 + 40 * q^23 + 36 * q^25 + 32 * q^27 + 16 * q^29 + 16 * q^31 + 8 * q^33 + 8 * q^35 + 24 * q^37 + 16 * q^39 - 16 * q^41 + 16 * q^43 + 24 * q^45 + 20 * q^49 + 24 * q^53 - 8 * q^55 + 24 * q^57 + 8 * q^61 + 28 * q^63 + 8 * q^65 - 8 * q^67 + 8 * q^69 + 72 * q^71 + 40 * q^75 + 16 * q^77 + 60 * q^81 + 8 * q^83 - 24 * q^89 + 8 * q^91 + 32 * q^93 + 48 * q^95 - 16 * q^97 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more