LMFDB - Embedded newform 8092.2.a.z.1.13

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			1.13
Root			$1.28083$ 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)

$f(q)$	$=$	$q+1.28083 q^{3} -3.03609 q^{5} +1.00000 q^{7} -1.35947 q^{9} -2.68953 q^{11} -6.64808 q^{13} -3.88873 q^{15} +4.75228 q^{19} +1.28083 q^{21} +7.63946 q^{23} +4.21787 q^{25} -5.58375 q^{27} +0.964919 q^{29} -6.08254 q^{31} -3.44484 q^{33} -3.03609 q^{35} +5.30308 q^{37} -8.51508 q^{39} -11.4695 q^{41} +1.26509 q^{43} +4.12747 q^{45} +0.143498 q^{47} +1.00000 q^{49} -1.46842 q^{53} +8.16566 q^{55} +6.08687 q^{57} -11.7725 q^{59} -12.8689 q^{61} -1.35947 q^{63} +20.1842 q^{65} +5.78961 q^{67} +9.78488 q^{69} +10.3955 q^{71} +7.45558 q^{73} +5.40239 q^{75} -2.68953 q^{77} +0.261371 q^{79} -3.07346 q^{81} +3.13776 q^{83} +1.23590 q^{87} +4.50413 q^{89} -6.64808 q^{91} -7.79071 q^{93} -14.4284 q^{95} -14.2742 q^{97} +3.65632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.28083	0.739489	0.369745	−	0.929133i	$-0.379445\pi$
$3$	1.28083	0.739489	0.369745	+	0.929133i	$0.379445\pi$
$4$	0	0
$5$	−3.03609	−1.35778	−0.678891	−	0.734239i	$-0.737539\pi$
$5$	−3.03609	−1.35778	−0.678891	+	0.734239i	$0.737539\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.35947	−0.453155
$10$	0	0
$11$	−2.68953	−0.810923	−0.405461	−	0.914112i	$-0.632889\pi$
$11$	−2.68953	−0.810923	−0.405461	+	0.914112i	$0.632889\pi$
$12$	0	0
$13$	−6.64808	−1.84385	−0.921923	−	0.387374i	$-0.873382\pi$
$13$	−6.64808	−1.84385	−0.921923	+	0.387374i	$0.873382\pi$
$14$	0	0
$15$	−3.88873	−1.00407
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	4.75228	1.09025	0.545124	−	0.838356i	$-0.316483\pi$
$19$	4.75228	1.09025	0.545124	+	0.838356i	$0.316483\pi$
$20$	0	0
$21$	1.28083	0.279501
$22$	0	0
$23$	7.63946	1.59294	0.796469	−	0.604679i	$-0.206699\pi$
$23$	7.63946	1.59294	0.796469	+	0.604679i	$0.206699\pi$
$24$	0	0
$25$	4.21787	0.843574
$26$	0	0
$27$	−5.58375	−1.07459
$28$	0	0
$29$	0.964919	0.179181	0.0895905	−	0.995979i	$-0.471444\pi$
$29$	0.964919	0.179181	0.0895905	+	0.995979i	$0.471444\pi$
$30$	0	0
$31$	−6.08254	−1.09246	−0.546228	−	0.837637i	$-0.683937\pi$
$31$	−6.08254	−1.09246	−0.546228	+	0.837637i	$0.683937\pi$
$32$	0	0
$33$	−3.44484	−0.599669
$34$	0	0
$35$	−3.03609	−0.513194
$36$	0	0
$37$	5.30308	0.871822	0.435911	−	0.899990i	$-0.356426\pi$
$37$	5.30308	0.871822	0.435911	+	0.899990i	$0.356426\pi$
$38$	0	0
$39$	−8.51508	−1.36350
$40$	0	0
$41$	−11.4695	−1.79124	−0.895619	−	0.444823i	$-0.853267\pi$
$41$	−11.4695	−1.79124	−0.895619	+	0.444823i	$0.853267\pi$
$42$	0	0
$43$	1.26509	0.192925	0.0964626	−	0.995337i	$-0.469247\pi$
$43$	1.26509	0.192925	0.0964626	+	0.995337i	$0.469247\pi$
$44$	0	0
$45$	4.12747	0.615286
$46$	0	0
$47$	0.143498	0.0209314	0.0104657	−	0.999945i	$-0.496669\pi$
$47$	0.143498	0.0209314	0.0104657	+	0.999945i	$0.496669\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.46842	−0.201703	−0.100851	−	0.994902i	$-0.532157\pi$
$53$	−1.46842	−0.201703	−0.100851	+	0.994902i	$0.532157\pi$
$54$	0	0
$55$	8.16566	1.10106
$56$	0	0
$57$	6.08687	0.806226
$58$	0	0
$59$	−11.7725	−1.53265	−0.766324	−	0.642455i	$-0.777916\pi$
$59$	−11.7725	−1.53265	−0.766324	+	0.642455i	$0.777916\pi$
$60$	0	0
$61$	−12.8689	−1.64770	−0.823848	−	0.566810i	$-0.808177\pi$
$61$	−12.8689	−1.64770	−0.823848	+	0.566810i	$0.808177\pi$
$62$	0	0
$63$	−1.35947	−0.171277
$64$	0	0
$65$	20.1842	2.50354
$66$	0	0
$67$	5.78961	0.707314	0.353657	−	0.935375i	$-0.384938\pi$
$67$	5.78961	0.707314	0.353657	+	0.935375i	$0.384938\pi$
$68$	0	0
$69$	9.78488	1.17796
$70$	0	0
$71$	10.3955	1.23371	0.616857	−	0.787075i	$-0.288406\pi$
$71$	10.3955	1.23371	0.616857	+	0.787075i	$0.288406\pi$
$72$	0	0
$73$	7.45558	0.872610	0.436305	−	0.899799i	$-0.356287\pi$
$73$	7.45558	0.872610	0.436305	+	0.899799i	$0.356287\pi$
$74$	0	0
$75$	5.40239	0.623814
$76$	0	0
$77$	−2.68953	−0.306500
$78$	0	0
$79$	0.261371	0.0294065	0.0147033	−	0.999892i	$-0.495320\pi$
$79$	0.261371	0.0294065	0.0147033	+	0.999892i	$0.495320\pi$
$80$	0	0
$81$	−3.07346	−0.341495
$82$	0	0
$83$	3.13776	0.344414	0.172207	−	0.985061i	$-0.444910\pi$
$83$	3.13776	0.344414	0.172207	+	0.985061i	$0.444910\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.23590	0.132502
$88$	0	0
$89$	4.50413	0.477437	0.238718	−	0.971089i	$-0.423273\pi$
$89$	4.50413	0.477437	0.238718	+	0.971089i	$0.423273\pi$
$90$	0	0
$91$	−6.64808	−0.696908
$92$	0	0
$93$	−7.79071	−0.807859
$94$	0	0
$95$	−14.4284	−1.48032
$96$	0	0
$97$	−14.2742	−1.44933	−0.724664	−	0.689103i	$-0.758005\pi$
$97$	−14.2742	−1.44933	−0.724664	+	0.689103i	$0.758005\pi$
$98$	0	0
$99$	3.65632	0.367474
$100$	0	0
$101$	0.832970	0.0828836	0.0414418	−	0.999141i	$-0.486805\pi$
$101$	0.832970	0.0828836	0.0414418	+	0.999141i	$0.486805\pi$
$102$	0	0
$103$	−19.7300	−1.94405	−0.972027	−	0.234870i	$-0.924534\pi$
$103$	−19.7300	−1.94405	−0.972027	+	0.234870i	$0.924534\pi$
$104$	0	0
$105$	−3.88873	−0.379501
$106$	0	0
$107$	6.30323	0.609356	0.304678	−	0.952455i	$-0.401451\pi$
$107$	6.30323	0.609356	0.304678	+	0.952455i	$0.401451\pi$
$108$	0	0
$109$	17.6944	1.69482	0.847408	−	0.530942i	$-0.178162\pi$
$109$	17.6944	1.69482	0.847408	+	0.530942i	$0.178162\pi$
$110$	0	0
$111$	6.79237	0.644703
$112$	0	0
$113$	5.52861	0.520088	0.260044	−	0.965597i	$-0.416263\pi$
$113$	5.52861	0.520088	0.260044	+	0.965597i	$0.416263\pi$
$114$	0	0
$115$	−23.1941	−2.16286
$116$	0	0
$117$	9.03784	0.835548
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.76645	−0.342404
$122$	0	0
$123$	−14.6905	−1.32460
$124$	0	0
$125$	2.37462	0.212392
$126$	0	0
$127$	−12.0511	−1.06936	−0.534682	−	0.845053i	$-0.679569\pi$
$127$	−12.0511	−1.06936	−0.534682	+	0.845053i	$0.679569\pi$
$128$	0	0
$129$	1.62038	0.142666
$130$	0	0
$131$	15.0374	1.31382	0.656912	−	0.753967i	$-0.271862\pi$
$131$	15.0374	1.31382	0.656912	+	0.753967i	$0.271862\pi$
$132$	0	0
$133$	4.75228	0.412075
$134$	0	0
$135$	16.9528	1.45906
$136$	0	0
$137$	−1.78868	−0.152817	−0.0764085	−	0.997077i	$-0.524345\pi$
$137$	−1.78868	−0.152817	−0.0764085	+	0.997077i	$0.524345\pi$
$138$	0	0
$139$	12.7000	1.07720	0.538598	−	0.842563i	$-0.318954\pi$
$139$	12.7000	1.07720	0.538598	+	0.842563i	$0.318954\pi$
$140$	0	0
$141$	0.183798	0.0154785
$142$	0	0
$143$	17.8802	1.49522
$144$	0	0
$145$	−2.92959	−0.243289
$146$	0	0
$147$	1.28083	0.105641
$148$	0	0
$149$	19.3122	1.58212	0.791060	−	0.611738i	$-0.209529\pi$
$149$	19.3122	1.58212	0.791060	+	0.611738i	$0.209529\pi$
$150$	0	0
$151$	−11.9023	−0.968591	−0.484296	−	0.874904i	$-0.660924\pi$
$151$	−11.9023	−0.968591	−0.484296	+	0.874904i	$0.660924\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.4672	1.48332
$156$	0	0
$157$	14.8113	1.18207	0.591036	−	0.806645i	$-0.298719\pi$
$157$	14.8113	1.18207	0.591036	+	0.806645i	$0.298719\pi$
$158$	0	0
$159$	−1.88080	−0.149157
$160$	0	0
$161$	7.63946	0.602074
$162$	0	0
$163$	0.829745	0.0649907	0.0324953	−	0.999472i	$-0.489655\pi$
$163$	0.829745	0.0649907	0.0324953	+	0.999472i	$0.489655\pi$
$164$	0	0
$165$	10.4588	0.814220
$166$	0	0
$167$	19.0983	1.47787	0.738936	−	0.673775i	$-0.235328\pi$
$167$	19.0983	1.47787	0.738936	+	0.673775i	$0.235328\pi$
$168$	0	0
$169$	31.1970	2.39977
$170$	0	0
$171$	−6.46056	−0.494051
$172$	0	0
$173$	18.4905	1.40581	0.702904	−	0.711285i	$-0.251886\pi$
$173$	18.4905	1.40581	0.702904	+	0.711285i	$0.251886\pi$
$174$	0	0
$175$	4.21787	0.318841
$176$	0	0
$177$	−15.0786	−1.13338
$178$	0	0
$179$	9.33365	0.697630	0.348815	−	0.937192i	$-0.386584\pi$
$179$	9.33365	0.697630	0.348815	+	0.937192i	$0.386584\pi$
$180$	0	0
$181$	11.9356	0.887167	0.443584	−	0.896233i	$-0.353707\pi$
$181$	11.9356	0.887167	0.443584	+	0.896233i	$0.353707\pi$
$182$	0	0
$183$	−16.4829	−1.21845
$184$	0	0
$185$	−16.1007	−1.18374
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.58375	−0.406158
$190$	0	0
$191$	17.5923	1.27293	0.636467	−	0.771304i	$-0.280395\pi$
$191$	17.5923	1.27293	0.636467	+	0.771304i	$0.280395\pi$
$192$	0	0
$193$	10.5138	0.756801	0.378401	−	0.925642i	$-0.376474\pi$
$193$	10.5138	0.756801	0.378401	+	0.925642i	$0.376474\pi$
$194$	0	0
$195$	25.8526	1.85134
$196$	0	0
$197$	21.2138	1.51142	0.755711	−	0.654906i	$-0.227292\pi$
$197$	21.2138	1.51142	0.755711	+	0.654906i	$0.227292\pi$
$198$	0	0
$199$	15.8171	1.12125	0.560623	−	0.828071i	$-0.310562\pi$
$199$	15.8171	1.12125	0.560623	+	0.828071i	$0.310562\pi$
$200$	0	0
$201$	7.41553	0.523051
$202$	0	0
$203$	0.964919	0.0677240
$204$	0	0
$205$	34.8225	2.43211
$206$	0	0
$207$	−10.3856	−0.721848
$208$	0	0
$209$	−12.7814	−0.884106
$210$	0	0
$211$	−7.36095	−0.506748	−0.253374	−	0.967368i	$-0.581540\pi$
$211$	−7.36095	−0.506748	−0.253374	+	0.967368i	$0.581540\pi$
$212$	0	0
$213$	13.3148	0.912318
$214$	0	0
$215$	−3.84095	−0.261950
$216$	0	0
$217$	−6.08254	−0.412909
$218$	0	0
$219$	9.54936	0.645286
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.65182	−0.110614	−0.0553071	−	0.998469i	$-0.517614\pi$
$223$	−1.65182	−0.110614	−0.0553071	+	0.998469i	$0.517614\pi$
$224$	0	0
$225$	−5.73405	−0.382270
$226$	0	0
$227$	−10.9425	−0.726278	−0.363139	−	0.931735i	$-0.618295\pi$
$227$	−10.9425	−0.726278	−0.363139	+	0.931735i	$0.618295\pi$
$228$	0	0
$229$	9.98702	0.659961	0.329981	−	0.943988i	$-0.392958\pi$
$229$	9.98702	0.659961	0.329981	+	0.943988i	$0.392958\pi$
$230$	0	0
$231$	−3.44484	−0.226654
$232$	0	0
$233$	−6.17981	−0.404853	−0.202426	−	0.979297i	$-0.564883\pi$
$233$	−6.17981	−0.404853	−0.202426	+	0.979297i	$0.564883\pi$
$234$	0	0
$235$	−0.435675	−0.0284203
$236$	0	0
$237$	0.334773	0.0217458
$238$	0	0
$239$	−7.79655	−0.504317	−0.252159	−	0.967686i	$-0.581140\pi$
$239$	−7.79655	−0.504317	−0.252159	+	0.967686i	$0.581140\pi$
$240$	0	0
$241$	8.65492	0.557512	0.278756	−	0.960362i	$-0.410078\pi$
$241$	8.65492	0.557512	0.278756	+	0.960362i	$0.410078\pi$
$242$	0	0
$243$	12.8147	0.822061
$244$	0	0
$245$	−3.03609	−0.193969
$246$	0	0
$247$	−31.5935	−2.01025
$248$	0	0
$249$	4.01895	0.254691
$250$	0	0
$251$	5.05922	0.319335	0.159668	−	0.987171i	$-0.448958\pi$
$251$	5.05922	0.319335	0.159668	+	0.987171i	$0.448958\pi$
$252$	0	0
$253$	−20.5465	−1.29175
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.75949	−0.546402	−0.273201	−	0.961957i	$-0.588082\pi$
$257$	−8.75949	−0.546402	−0.273201	+	0.961957i	$0.588082\pi$
$258$	0	0
$259$	5.30308	0.329518
$260$	0	0
$261$	−1.31177	−0.0811968
$262$	0	0
$263$	−10.0878	−0.622038	−0.311019	−	0.950404i	$-0.600670\pi$
$263$	−10.0878	−0.622038	−0.311019	+	0.950404i	$0.600670\pi$
$264$	0	0
$265$	4.45825	0.273868
$266$	0	0
$267$	5.76904	0.353060
$268$	0	0
$269$	13.8464	0.844230	0.422115	−	0.906542i	$-0.361288\pi$
$269$	13.8464	0.844230	0.422115	+	0.906542i	$0.361288\pi$
$270$	0	0
$271$	−8.06767	−0.490076	−0.245038	−	0.969514i	$-0.578800\pi$
$271$	−8.06767	−0.490076	−0.245038	+	0.969514i	$0.578800\pi$
$272$	0	0
$273$	−8.51508	−0.515356
$274$	0	0
$275$	−11.3441	−0.684073
$276$	0	0
$277$	5.33110	0.320315	0.160157	−	0.987091i	$-0.448800\pi$
$277$	5.33110	0.320315	0.160157	+	0.987091i	$0.448800\pi$
$278$	0	0
$279$	8.26900	0.495052
$280$	0	0
$281$	21.5681	1.28664	0.643322	−	0.765596i	$-0.277556\pi$
$281$	21.5681	1.28664	0.643322	+	0.765596i	$0.277556\pi$
$282$	0	0
$283$	12.3467	0.733933	0.366966	−	0.930234i	$-0.380396\pi$
$283$	12.3467	0.733933	0.366966	+	0.930234i	$0.380396\pi$
$284$	0	0
$285$	−18.4803	−1.09468
$286$	0	0
$287$	−11.4695	−0.677024
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−18.2829	−1.07176
$292$	0	0
$293$	17.6846	1.03314	0.516572	−	0.856244i	$-0.327208\pi$
$293$	17.6846	1.03314	0.516572	+	0.856244i	$0.327208\pi$
$294$	0	0
$295$	35.7424	2.08100
$296$	0	0
$297$	15.0176	0.871412
$298$	0	0
$299$	−50.7877	−2.93713
$300$	0	0
$301$	1.26509	0.0729188
$302$	0	0
$303$	1.06690	0.0612916
$304$	0	0
$305$	39.0713	2.23721
$306$	0	0
$307$	−5.45438	−0.311298	−0.155649	−	0.987812i	$-0.549747\pi$
$307$	−5.45438	−0.311298	−0.155649	+	0.987812i	$0.549747\pi$
$308$	0	0
$309$	−25.2708	−1.43761
$310$	0	0
$311$	3.31549	0.188004	0.0940020	−	0.995572i	$-0.470034\pi$
$311$	3.31549	0.188004	0.0940020	+	0.995572i	$0.470034\pi$
$312$	0	0
$313$	−17.7228	−1.00175	−0.500876	−	0.865519i	$-0.666989\pi$
$313$	−17.7228	−1.00175	−0.500876	+	0.865519i	$0.666989\pi$
$314$	0	0
$315$	4.12747	0.232556
$316$	0	0
$317$	−6.35079	−0.356696	−0.178348	−	0.983967i	$-0.557075\pi$
$317$	−6.35079	−0.356696	−0.178348	+	0.983967i	$0.557075\pi$
$318$	0	0
$319$	−2.59518	−0.145302
$320$	0	0
$321$	8.07339	0.450613
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−28.0407	−1.55542
$326$	0	0
$327$	22.6636	1.25330
$328$	0	0
$329$	0.143498	0.00791132
$330$	0	0
$331$	7.42992	0.408385	0.204193	−	0.978931i	$-0.434543\pi$
$331$	7.42992	0.408385	0.204193	+	0.978931i	$0.434543\pi$
$332$	0	0
$333$	−7.20936	−0.395071
$334$	0	0
$335$	−17.5778	−0.960378
$336$	0	0
$337$	−13.9526	−0.760046	−0.380023	−	0.924977i	$-0.624084\pi$
$337$	−13.9526	−0.760046	−0.380023	+	0.924977i	$0.624084\pi$
$338$	0	0
$339$	7.08123	0.384600
$340$	0	0
$341$	16.3591	0.885897
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−29.7078	−1.59941
$346$	0	0
$347$	−26.3572	−1.41493	−0.707466	−	0.706748i	$-0.750162\pi$
$347$	−26.3572	−1.41493	−0.707466	+	0.706748i	$0.750162\pi$
$348$	0	0
$349$	14.0123	0.750063	0.375031	−	0.927012i	$-0.377632\pi$
$349$	14.0123	0.750063	0.375031	+	0.927012i	$0.377632\pi$
$350$	0	0
$351$	37.1212	1.98138
$352$	0	0
$353$	6.36885	0.338980	0.169490	−	0.985532i	$-0.445788\pi$
$353$	6.36885	0.338980	0.169490	+	0.985532i	$0.445788\pi$
$354$	0	0
$355$	−31.5616	−1.67512
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.3702	−0.811210	−0.405605	−	0.914049i	$-0.632939\pi$
$359$	−15.3702	−0.811210	−0.405605	+	0.914049i	$0.632939\pi$
$360$	0	0
$361$	3.58413	0.188639
$362$	0	0
$363$	−4.82419	−0.253204
$364$	0	0
$365$	−22.6358	−1.18481
$366$	0	0
$367$	26.4916	1.38285	0.691425	−	0.722448i	$-0.256983\pi$
$367$	26.4916	1.38285	0.691425	+	0.722448i	$0.256983\pi$
$368$	0	0
$369$	15.5924	0.811709
$370$	0	0
$371$	−1.46842	−0.0762364
$372$	0	0
$373$	−10.1950	−0.527879	−0.263940	−	0.964539i	$-0.585022\pi$
$373$	−10.1950	−0.527879	−0.263940	+	0.964539i	$0.585022\pi$
$374$	0	0
$375$	3.04149	0.157062
$376$	0	0
$377$	−6.41486	−0.330382
$378$	0	0
$379$	8.92051	0.458216	0.229108	−	0.973401i	$-0.426419\pi$
$379$	8.92051	0.458216	0.229108	+	0.973401i	$0.426419\pi$
$380$	0	0
$381$	−15.4355	−0.790784
$382$	0	0
$383$	−3.95980	−0.202336	−0.101168	−	0.994869i	$-0.532258\pi$
$383$	−3.95980	−0.202336	−0.101168	+	0.994869i	$0.532258\pi$
$384$	0	0
$385$	8.16566	0.416160
$386$	0	0
$387$	−1.71985	−0.0874250
$388$	0	0
$389$	−19.5342	−0.990424	−0.495212	−	0.868772i	$-0.664910\pi$
$389$	−19.5342	−0.990424	−0.495212	+	0.868772i	$0.664910\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	19.2604	0.971559
$394$	0	0
$395$	−0.793547	−0.0399277
$396$	0	0
$397$	−20.6895	−1.03837	−0.519187	−	0.854660i	$-0.673765\pi$
$397$	−20.6895	−1.03837	−0.519187	+	0.854660i	$0.673765\pi$
$398$	0	0
$399$	6.08687	0.304725
$400$	0	0
$401$	10.9484	0.546737	0.273369	−	0.961909i	$-0.411862\pi$
$401$	10.9484	0.546737	0.273369	+	0.961909i	$0.411862\pi$
$402$	0	0
$403$	40.4372	2.01432
$404$	0	0
$405$	9.33130	0.463676
$406$	0	0
$407$	−14.2628	−0.706980
$408$	0	0
$409$	−3.76332	−0.186084	−0.0930421	−	0.995662i	$-0.529659\pi$
$409$	−3.76332	−0.186084	−0.0930421	+	0.995662i	$0.529659\pi$
$410$	0	0
$411$	−2.29100	−0.113007
$412$	0	0
$413$	−11.7725	−0.579286
$414$	0	0
$415$	−9.52654	−0.467640
$416$	0	0
$417$	16.2665	0.796575
$418$	0	0
$419$	−12.1123	−0.591724	−0.295862	−	0.955231i	$-0.595607\pi$
$419$	−12.1123	−0.591724	−0.295862	+	0.955231i	$0.595607\pi$
$420$	0	0
$421$	−2.86973	−0.139862	−0.0699310	−	0.997552i	$-0.522278\pi$
$421$	−2.86973	−0.139862	−0.0699310	+	0.997552i	$0.522278\pi$
$422$	0	0
$423$	−0.195081	−0.00948517
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.8689	−0.622771
$428$	0	0
$429$	22.9015	1.10570
$430$	0	0
$431$	−30.7960	−1.48339	−0.741695	−	0.670738i	$-0.765978\pi$
$431$	−30.7960	−1.48339	−0.741695	+	0.670738i	$0.765978\pi$
$432$	0	0
$433$	−38.8474	−1.86689	−0.933443	−	0.358725i	$-0.883211\pi$
$433$	−38.8474	−1.86689	−0.933443	+	0.358725i	$0.883211\pi$
$434$	0	0
$435$	−3.75231	−0.179910
$436$	0	0
$437$	36.3048	1.73670
$438$	0	0
$439$	23.4721	1.12026	0.560131	−	0.828404i	$-0.310751\pi$
$439$	23.4721	1.12026	0.560131	+	0.828404i	$0.310751\pi$
$440$	0	0
$441$	−1.35947	−0.0647365
$442$	0	0
$443$	−20.2489	−0.962053	−0.481027	−	0.876706i	$-0.659736\pi$
$443$	−20.2489	−0.962053	−0.481027	+	0.876706i	$0.659736\pi$
$444$	0	0
$445$	−13.6750	−0.648256
$446$	0	0
$447$	24.7358	1.16996
$448$	0	0
$449$	5.91885	0.279328	0.139664	−	0.990199i	$-0.455398\pi$
$449$	5.91885	0.279328	0.139664	+	0.990199i	$0.455398\pi$
$450$	0	0
$451$	30.8476	1.45256
$452$	0	0
$453$	−15.2448	−0.716263
$454$	0	0
$455$	20.1842	0.946250
$456$	0	0
$457$	−23.3723	−1.09331	−0.546655	−	0.837358i	$-0.684099\pi$
$457$	−23.3723	−1.09331	−0.546655	+	0.837358i	$0.684099\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.4332	−1.37084	−0.685421	−	0.728147i	$-0.740382\pi$
$461$	−29.4332	−1.37084	−0.685421	+	0.728147i	$0.740382\pi$
$462$	0	0
$463$	−37.8397	−1.75856	−0.879280	−	0.476305i	$-0.841976\pi$
$463$	−37.8397	−1.75856	−0.879280	+	0.476305i	$0.841976\pi$
$464$	0	0
$465$	23.6533	1.09690
$466$	0	0
$467$	13.7935	0.638287	0.319144	−	0.947706i	$-0.396605\pi$
$467$	13.7935	0.638287	0.319144	+	0.947706i	$0.396605\pi$
$468$	0	0
$469$	5.78961	0.267339
$470$	0	0
$471$	18.9708	0.874130
$472$	0	0
$473$	−3.40251	−0.156447
$474$	0	0
$475$	20.0445	0.919704
$476$	0	0
$477$	1.99626	0.0914026
$478$	0	0
$479$	1.64548	0.0751840	0.0375920	−	0.999293i	$-0.488031\pi$
$479$	1.64548	0.0751840	0.0375920	+	0.999293i	$0.488031\pi$
$480$	0	0
$481$	−35.2553	−1.60750
$482$	0	0
$483$	9.78488	0.445227
$484$	0	0
$485$	43.3379	1.96787
$486$	0	0
$487$	9.26931	0.420032	0.210016	−	0.977698i	$-0.432648\pi$
$487$	9.26931	0.420032	0.210016	+	0.977698i	$0.432648\pi$
$488$	0	0
$489$	1.06277	0.0480599
$490$	0	0
$491$	4.57593	0.206509	0.103254	−	0.994655i	$-0.467074\pi$
$491$	4.57593	0.206509	0.103254	+	0.994655i	$0.467074\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−11.1009	−0.498950
$496$	0	0
$497$	10.3955	0.466300
$498$	0	0
$499$	−29.8152	−1.33471	−0.667355	−	0.744740i	$-0.732574\pi$
$499$	−29.8152	−1.33471	−0.667355	+	0.744740i	$0.732574\pi$
$500$	0	0
$501$	24.4618	1.09287
$502$	0	0
$503$	18.4376	0.822090	0.411045	−	0.911615i	$-0.365164\pi$
$503$	18.4376	0.822090	0.411045	+	0.911615i	$0.365164\pi$
$504$	0	0
$505$	−2.52898	−0.112538
$506$	0	0
$507$	39.9581	1.77460
$508$	0	0
$509$	−15.7727	−0.699113	−0.349556	−	0.936915i	$-0.613668\pi$
$509$	−15.7727	−0.699113	−0.349556	+	0.936915i	$0.613668\pi$
$510$	0	0
$511$	7.45558	0.329815
$512$	0	0
$513$	−26.5355	−1.17157
$514$	0	0
$515$	59.9021	2.63960
$516$	0	0
$517$	−0.385943	−0.0169737
$518$	0	0
$519$	23.6833	1.03958
$520$	0	0
$521$	23.0475	1.00973	0.504865	−	0.863198i	$-0.331542\pi$
$521$	23.0475	1.00973	0.504865	+	0.863198i	$0.331542\pi$
$522$	0	0
$523$	−31.2824	−1.36788	−0.683941	−	0.729537i	$-0.739735\pi$
$523$	−31.2824	−1.36788	−0.683941	+	0.729537i	$0.739735\pi$
$524$	0	0
$525$	5.40239	0.235780
$526$	0	0
$527$	0	0
$528$	0	0
$529$	35.3614	1.53745
$530$	0	0
$531$	16.0043	0.694527
$532$	0	0
$533$	76.2503	3.30277
$534$	0	0
$535$	−19.1372	−0.827374
$536$	0	0
$537$	11.9549	0.515890
$538$	0	0
$539$	−2.68953	−0.115846
$540$	0	0
$541$	15.7721	0.678094	0.339047	−	0.940769i	$-0.389895\pi$
$541$	15.7721	0.678094	0.339047	+	0.940769i	$0.389895\pi$
$542$	0	0
$543$	15.2875	0.656051
$544$	0	0
$545$	−53.7219	−2.30119
$546$	0	0
$547$	33.2705	1.42254	0.711271	−	0.702917i	$-0.248120\pi$
$547$	33.2705	1.42254	0.711271	+	0.702917i	$0.248120\pi$
$548$	0	0
$549$	17.4949	0.746662
$550$	0	0
$551$	4.58556	0.195352
$552$	0	0
$553$	0.261371	0.0111146
$554$	0	0
$555$	−20.6223	−0.875367
$556$	0	0
$557$	15.8429	0.671286	0.335643	−	0.941989i	$-0.391046\pi$
$557$	15.8429	0.671286	0.335643	+	0.941989i	$0.391046\pi$
$558$	0	0
$559$	−8.41045	−0.355724
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.1038	1.56374	0.781869	−	0.623442i	$-0.214266\pi$
$563$	37.1038	1.56374	0.781869	+	0.623442i	$0.214266\pi$
$564$	0	0
$565$	−16.7854	−0.706167
$566$	0	0
$567$	−3.07346	−0.129073
$568$	0	0
$569$	22.8488	0.957872	0.478936	−	0.877850i	$-0.341023\pi$
$569$	22.8488	0.957872	0.478936	+	0.877850i	$0.341023\pi$
$570$	0	0
$571$	14.9854	0.627117	0.313559	−	0.949569i	$-0.398479\pi$
$571$	14.9854	0.627117	0.313559	+	0.949569i	$0.398479\pi$
$572$	0	0
$573$	22.5328	0.941322
$574$	0	0
$575$	32.2223	1.34376
$576$	0	0
$577$	24.7542	1.03053	0.515266	−	0.857030i	$-0.327693\pi$
$577$	24.7542	1.03053	0.515266	+	0.857030i	$0.327693\pi$
$578$	0	0
$579$	13.4665	0.559647
$580$	0	0
$581$	3.13776	0.130176
$582$	0	0
$583$	3.94935	0.163565
$584$	0	0
$585$	−27.4397	−1.13449
$586$	0	0
$587$	−22.7196	−0.937738	−0.468869	−	0.883268i	$-0.655338\pi$
$587$	−22.7196	−0.937738	−0.468869	+	0.883268i	$0.655338\pi$
$588$	0	0
$589$	−28.9059	−1.19105
$590$	0	0
$591$	27.1714	1.11768
$592$	0	0
$593$	8.42261	0.345875	0.172938	−	0.984933i	$-0.444674\pi$
$593$	8.42261	0.345875	0.172938	+	0.984933i	$0.444674\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.2591	0.829150
$598$	0	0
$599$	−9.55705	−0.390490	−0.195245	−	0.980754i	$-0.562550\pi$
$599$	−9.55705	−0.390490	−0.195245	+	0.980754i	$0.562550\pi$
$600$	0	0
$601$	−37.3815	−1.52482	−0.762411	−	0.647093i	$-0.775985\pi$
$601$	−37.3815	−1.52482	−0.762411	+	0.647093i	$0.775985\pi$
$602$	0	0
$603$	−7.87078	−0.320523
$604$	0	0
$605$	11.4353	0.464911
$606$	0	0
$607$	−40.7855	−1.65543	−0.827716	−	0.561147i	$-0.810360\pi$
$607$	−40.7855	−1.65543	−0.827716	+	0.561147i	$0.810360\pi$
$608$	0	0
$609$	1.23590	0.0500812
$610$	0	0
$611$	−0.953989	−0.0385943
$612$	0	0
$613$	20.5567	0.830276	0.415138	−	0.909758i	$-0.363733\pi$
$613$	20.5567	0.830276	0.415138	+	0.909758i	$0.363733\pi$
$614$	0	0
$615$	44.6019	1.79852
$616$	0	0
$617$	13.9092	0.559962	0.279981	−	0.960005i	$-0.409672\pi$
$617$	13.9092	0.559962	0.279981	+	0.960005i	$0.409672\pi$
$618$	0	0
$619$	20.9992	0.844028	0.422014	−	0.906589i	$-0.361323\pi$
$619$	20.9992	0.844028	0.422014	+	0.906589i	$0.361323\pi$
$620$	0	0
$621$	−42.6568	−1.71176
$622$	0	0
$623$	4.50413	0.180454
$624$	0	0
$625$	−28.2989	−1.13196
$626$	0	0
$627$	−16.3708	−0.653787
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−28.1343	−1.12001	−0.560004	−	0.828490i	$-0.689201\pi$
$631$	−28.1343	−1.12001	−0.560004	+	0.828490i	$0.689201\pi$
$632$	0	0
$633$	−9.42814	−0.374735
$634$	0	0
$635$	36.5884	1.45196
$636$	0	0
$637$	−6.64808	−0.263406
$638$	0	0
$639$	−14.1323	−0.559064
$640$	0	0
$641$	−36.8497	−1.45547	−0.727737	−	0.685856i	$-0.759428\pi$
$641$	−36.8497	−1.45547	−0.727737	+	0.685856i	$0.759428\pi$
$642$	0	0
$643$	22.6853	0.894621	0.447311	−	0.894379i	$-0.352382\pi$
$643$	22.6853	0.894621	0.447311	+	0.894379i	$0.352382\pi$
$644$	0	0
$645$	−4.91961	−0.193710
$646$	0	0
$647$	−17.5195	−0.688761	−0.344380	−	0.938830i	$-0.611911\pi$
$647$	−17.5195	−0.688761	−0.344380	+	0.938830i	$0.611911\pi$
$648$	0	0
$649$	31.6624	1.24286
$650$	0	0
$651$	−7.79071	−0.305342
$652$	0	0
$653$	−5.58294	−0.218477	−0.109239	−	0.994016i	$-0.534841\pi$
$653$	−5.58294	−0.218477	−0.109239	+	0.994016i	$0.534841\pi$
$654$	0	0
$655$	−45.6550	−1.78389
$656$	0	0
$657$	−10.1356	−0.395428
$658$	0	0
$659$	−29.3526	−1.14342	−0.571708	−	0.820457i	$-0.693719\pi$
$659$	−29.3526	−1.14342	−0.571708	+	0.820457i	$0.693719\pi$
$660$	0	0
$661$	37.3445	1.45253	0.726267	−	0.687412i	$-0.241254\pi$
$661$	37.3445	1.45253	0.726267	+	0.687412i	$0.241254\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−14.4284	−0.559508
$666$	0	0
$667$	7.37146	0.285424
$668$	0	0
$669$	−2.11571	−0.0817980
$670$	0	0
$671$	34.6113	1.33615
$672$	0	0
$673$	−13.5430	−0.522044	−0.261022	−	0.965333i	$-0.584059\pi$
$673$	−13.5430	−0.522044	−0.261022	+	0.965333i	$0.584059\pi$
$674$	0	0
$675$	−23.5515	−0.906499
$676$	0	0
$677$	−6.97444	−0.268050	−0.134025	−	0.990978i	$-0.542790\pi$
$677$	−6.97444	−0.268050	−0.134025	+	0.990978i	$0.542790\pi$
$678$	0	0
$679$	−14.2742	−0.547794
$680$	0	0
$681$	−14.0155	−0.537075
$682$	0	0
$683$	−0.201732	−0.00771906	−0.00385953	−	0.999993i	$-0.501229\pi$
$683$	−0.201732	−0.00771906	−0.00385953	+	0.999993i	$0.501229\pi$
$684$	0	0
$685$	5.43059	0.207492
$686$	0	0
$687$	12.7917	0.488034
$688$	0	0
$689$	9.76215	0.371908
$690$	0	0
$691$	−51.9674	−1.97693	−0.988466	−	0.151444i	$-0.951608\pi$
$691$	−51.9674	−1.97693	−0.988466	+	0.151444i	$0.951608\pi$
$692$	0	0
$693$	3.65632	0.138892
$694$	0	0
$695$	−38.5583	−1.46260
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−7.91531	−0.299384
$700$	0	0
$701$	33.7022	1.27291	0.636457	−	0.771312i	$-0.280399\pi$
$701$	33.7022	1.27291	0.636457	+	0.771312i	$0.280399\pi$
$702$	0	0
$703$	25.2017	0.950501
$704$	0	0
$705$	−0.558027	−0.0210165
$706$	0	0
$707$	0.832970	0.0313271
$708$	0	0
$709$	46.6896	1.75347	0.876733	−	0.480977i	$-0.159718\pi$
$709$	46.6896	1.75347	0.876733	+	0.480977i	$0.159718\pi$
$710$	0	0
$711$	−0.355325	−0.0133257
$712$	0	0
$713$	−46.4673	−1.74021
$714$	0	0
$715$	−54.2859	−2.03018
$716$	0	0
$717$	−9.98609	−0.372937
$718$	0	0
$719$	21.7426	0.810861	0.405430	−	0.914126i	$-0.367122\pi$
$719$	21.7426	0.810861	0.405430	+	0.914126i	$0.367122\pi$
$720$	0	0
$721$	−19.7300	−0.734783
$722$	0	0
$723$	11.0855	0.412275
$724$	0	0
$725$	4.06990	0.151152
$726$	0	0
$727$	−14.2664	−0.529112	−0.264556	−	0.964370i	$-0.585225\pi$
$727$	−14.2664	−0.529112	−0.264556	+	0.964370i	$0.585225\pi$
$728$	0	0
$729$	25.6338	0.949401
$730$	0	0
$731$	0	0
$732$	0	0
$733$	8.28950	0.306180	0.153090	−	0.988212i	$-0.451078\pi$
$733$	8.28950	0.306180	0.153090	+	0.988212i	$0.451078\pi$
$734$	0	0
$735$	−3.88873	−0.143438
$736$	0	0
$737$	−15.5713	−0.573577
$738$	0	0
$739$	−6.78017	−0.249413	−0.124706	−	0.992194i	$-0.539799\pi$
$739$	−6.78017	−0.249413	−0.124706	+	0.992194i	$0.539799\pi$
$740$	0	0
$741$	−40.4660	−1.48656
$742$	0	0
$743$	21.2017	0.777816	0.388908	−	0.921277i	$-0.372852\pi$
$743$	21.2017	0.777816	0.388908	+	0.921277i	$0.372852\pi$
$744$	0	0
$745$	−58.6338	−2.14818
$746$	0	0
$747$	−4.26568	−0.156073
$748$	0	0
$749$	6.30323	0.230315
$750$	0	0
$751$	26.1690	0.954921	0.477461	−	0.878653i	$-0.341557\pi$
$751$	26.1690	0.954921	0.477461	+	0.878653i	$0.341557\pi$
$752$	0	0
$753$	6.48002	0.236145
$754$	0	0
$755$	36.1364	1.31514
$756$	0	0
$757$	41.6338	1.51320	0.756602	−	0.653875i	$-0.226858\pi$
$757$	41.6338	1.51320	0.756602	+	0.653875i	$0.226858\pi$
$758$	0	0
$759$	−26.3167	−0.955235
$760$	0	0
$761$	−40.0049	−1.45018	−0.725089	−	0.688655i	$-0.758201\pi$
$761$	−40.0049	−1.45018	−0.725089	+	0.688655i	$0.758201\pi$
$762$	0	0
$763$	17.6944	0.640580
$764$	0	0
$765$	0	0
$766$	0	0
$767$	78.2644	2.82596
$768$	0	0
$769$	−20.7081	−0.746752	−0.373376	−	0.927680i	$-0.621800\pi$
$769$	−20.7081	−0.746752	−0.373376	+	0.927680i	$0.621800\pi$
$770$	0	0
$771$	−11.2194	−0.404058
$772$	0	0
$773$	2.58332	0.0929155	0.0464578	−	0.998920i	$-0.485207\pi$
$773$	2.58332	0.0929155	0.0464578	+	0.998920i	$0.485207\pi$
$774$	0	0
$775$	−25.6553	−0.921567
$776$	0	0
$777$	6.79237	0.243675
$778$	0	0
$779$	−54.5063	−1.95289
$780$	0	0
$781$	−27.9589	−1.00045
$782$	0	0
$783$	−5.38787	−0.192547
$784$	0	0
$785$	−44.9686	−1.60500
$786$	0	0
$787$	−8.28463	−0.295315	−0.147658	−	0.989039i	$-0.547173\pi$
$787$	−8.28463	−0.295315	−0.147658	+	0.989039i	$0.547173\pi$
$788$	0	0
$789$	−12.9207	−0.459990
$790$	0	0
$791$	5.52861	0.196575
$792$	0	0
$793$	85.5536	3.03810
$794$	0	0
$795$	5.71028	0.202523
$796$	0	0
$797$	−2.62926	−0.0931330	−0.0465665	−	0.998915i	$-0.514828\pi$
$797$	−2.62926	−0.0931330	−0.0465665	+	0.998915i	$0.514828\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.12321	−0.216353
$802$	0	0
$803$	−20.0520	−0.707619
$804$	0	0
$805$	−23.1941	−0.817486
$806$	0	0
$807$	17.7349	0.624300
$808$	0	0
$809$	−13.9032	−0.488810	−0.244405	−	0.969673i	$-0.578593\pi$
$809$	−13.9032	−0.488810	−0.244405	+	0.969673i	$0.578593\pi$
$810$	0	0
$811$	11.5950	0.407157	0.203578	−	0.979059i	$-0.434743\pi$
$811$	11.5950	0.407157	0.203578	+	0.979059i	$0.434743\pi$
$812$	0	0
$813$	−10.3333	−0.362406
$814$	0	0
$815$	−2.51919	−0.0882432
$816$	0	0
$817$	6.01208	0.210336
$818$	0	0
$819$	9.03784	0.315808
$820$	0	0
$821$	25.8949	0.903737	0.451868	−	0.892085i	$-0.350758\pi$
$821$	25.8949	0.903737	0.451868	+	0.892085i	$0.350758\pi$
$822$	0	0
$823$	−33.1231	−1.15460	−0.577299	−	0.816532i	$-0.695893\pi$
$823$	−33.1231	−1.15460	−0.577299	+	0.816532i	$0.695893\pi$
$824$	0	0
$825$	−14.5299	−0.505865
$826$	0	0
$827$	5.01985	0.174557	0.0872786	−	0.996184i	$-0.472183\pi$
$827$	5.01985	0.174557	0.0872786	+	0.996184i	$0.472183\pi$
$828$	0	0
$829$	1.77854	0.0617712	0.0308856	−	0.999523i	$-0.490167\pi$
$829$	1.77854	0.0617712	0.0308856	+	0.999523i	$0.490167\pi$
$830$	0	0
$831$	6.82825	0.236869
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−57.9843	−2.00663
$836$	0	0
$837$	33.9634	1.17395
$838$	0	0
$839$	35.1621	1.21393	0.606965	−	0.794729i	$-0.292387\pi$
$839$	35.1621	1.21393	0.606965	+	0.794729i	$0.292387\pi$
$840$	0	0
$841$	−28.0689	−0.967894
$842$	0	0
$843$	27.6251	0.951460
$844$	0	0
$845$	−94.7169	−3.25836
$846$	0	0
$847$	−3.76645	−0.129417
$848$	0	0
$849$	15.8140	0.542736
$850$	0	0
$851$	40.5127	1.38876
$852$	0	0
$853$	12.9251	0.442548	0.221274	−	0.975212i	$-0.428978\pi$
$853$	12.9251	0.442548	0.221274	+	0.975212i	$0.428978\pi$
$854$	0	0
$855$	19.6149	0.670814
$856$	0	0
$857$	−29.3537	−1.00270	−0.501352	−	0.865243i	$-0.667164\pi$
$857$	−29.3537	−1.00270	−0.501352	+	0.865243i	$0.667164\pi$
$858$	0	0
$859$	−26.1494	−0.892206	−0.446103	−	0.894982i	$-0.647188\pi$
$859$	−26.1494	−0.892206	−0.446103	+	0.894982i	$0.647188\pi$
$860$	0	0
$861$	−14.6905	−0.500652
$862$	0	0
$863$	−10.2288	−0.348192	−0.174096	−	0.984729i	$-0.555700\pi$
$863$	−10.2288	−0.348192	−0.174096	+	0.984729i	$0.555700\pi$
$864$	0	0
$865$	−56.1390	−1.90878
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.702964	−0.0238464
$870$	0	0
$871$	−38.4898	−1.30418
$872$	0	0
$873$	19.4053	0.656770
$874$	0	0
$875$	2.37462	0.0802768
$876$	0	0
$877$	−40.7864	−1.37726	−0.688629	−	0.725114i	$-0.741787\pi$
$877$	−40.7864	−1.37726	−0.688629	+	0.725114i	$0.741787\pi$
$878$	0	0
$879$	22.6510	0.763999
$880$	0	0
$881$	22.5511	0.759766	0.379883	−	0.925035i	$-0.375964\pi$
$881$	22.5511	0.759766	0.379883	+	0.925035i	$0.375964\pi$
$882$	0	0
$883$	19.9441	0.671172	0.335586	−	0.942010i	$-0.391066\pi$
$883$	19.9441	0.671172	0.335586	+	0.942010i	$0.391066\pi$
$884$	0	0
$885$	45.7800	1.53888
$886$	0	0
$887$	20.9331	0.702865	0.351432	−	0.936213i	$-0.385695\pi$
$887$	20.9331	0.702865	0.351432	+	0.936213i	$0.385695\pi$
$888$	0	0
$889$	−12.0511	−0.404182
$890$	0	0
$891$	8.26614	0.276926
$892$	0	0
$893$	0.681944	0.0228204
$894$	0	0
$895$	−28.3379	−0.947230
$896$	0	0
$897$	−65.0506	−2.17198
$898$	0	0
$899$	−5.86915	−0.195747
$900$	0	0
$901$	0	0
$902$	0	0
$903$	1.62038	0.0539227
$904$	0	0
$905$	−36.2377	−1.20458
$906$	0	0
$907$	20.2828	0.673479	0.336740	−	0.941598i	$-0.390676\pi$
$907$	20.2828	0.673479	0.336740	+	0.941598i	$0.390676\pi$
$908$	0	0
$909$	−1.13239	−0.0375591
$910$	0	0
$911$	8.66274	0.287009	0.143505	−	0.989650i	$-0.454163\pi$
$911$	8.66274	0.287009	0.143505	+	0.989650i	$0.454163\pi$
$912$	0	0
$913$	−8.43910	−0.279293
$914$	0	0
$915$	50.0438	1.65440
$916$	0	0
$917$	15.0374	0.496579
$918$	0	0
$919$	−11.4527	−0.377790	−0.188895	−	0.981997i	$-0.560491\pi$
$919$	−11.4527	−0.377790	−0.188895	+	0.981997i	$0.560491\pi$
$920$	0	0
$921$	−6.98615	−0.230202
$922$	0	0
$923$	−69.1098	−2.27478
$924$	0	0
$925$	22.3677	0.735446
$926$	0	0
$927$	26.8222	0.880958
$928$	0	0
$929$	−13.7812	−0.452147	−0.226074	−	0.974110i	$-0.572589\pi$
$929$	−13.7812	−0.452147	−0.226074	+	0.974110i	$0.572589\pi$
$930$	0	0
$931$	4.75228	0.155750
$932$	0	0
$933$	4.24659	0.139027
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41.7236	−1.36305	−0.681526	−	0.731794i	$-0.738683\pi$
$937$	−41.7236	−1.36305	−0.681526	+	0.731794i	$0.738683\pi$
$938$	0	0
$939$	−22.7000	−0.740785
$940$	0	0
$941$	54.1171	1.76417	0.882084	−	0.471093i	$-0.156140\pi$
$941$	54.1171	1.76417	0.882084	+	0.471093i	$0.156140\pi$
$942$	0	0
$943$	−87.6209	−2.85333
$944$	0	0
$945$	16.9528	0.551474
$946$	0	0
$947$	−4.60920	−0.149779	−0.0748894	−	0.997192i	$-0.523860\pi$
$947$	−4.60920	−0.149779	−0.0748894	+	0.997192i	$0.523860\pi$
$948$	0	0
$949$	−49.5653	−1.60896
$950$	0	0
$951$	−8.13430	−0.263773
$952$	0	0
$953$	40.6740	1.31756	0.658780	−	0.752335i	$-0.271073\pi$
$953$	40.6740	1.31756	0.658780	+	0.752335i	$0.271073\pi$
$954$	0	0
$955$	−53.4119	−1.72837
$956$	0	0
$957$	−3.32399	−0.107449
$958$	0	0
$959$	−1.78868	−0.0577594
$960$	0	0
$961$	5.99723	0.193459
$962$	0	0
$963$	−8.56903	−0.276133
$964$	0	0
$965$	−31.9210	−1.02757
$966$	0	0
$967$	42.6607	1.37188	0.685938	−	0.727660i	$-0.259392\pi$
$967$	42.6607	1.37188	0.685938	+	0.727660i	$0.259392\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.2834	0.586741	0.293371	−	0.955999i	$-0.405223\pi$
$971$	18.2834	0.586741	0.293371	+	0.955999i	$0.405223\pi$
$972$	0	0
$973$	12.7000	0.407142
$974$	0	0
$975$	−35.9155	−1.15022
$976$	0	0
$977$	42.5565	1.36150	0.680751	−	0.732515i	$-0.261654\pi$
$977$	42.5565	1.36150	0.680751	+	0.732515i	$0.261654\pi$
$978$	0	0
$979$	−12.1140	−0.387164
$980$	0	0
$981$	−24.0549	−0.768015
$982$	0	0
$983$	24.0781	0.767971	0.383985	−	0.923339i	$-0.374551\pi$
$983$	24.0781	0.767971	0.383985	+	0.923339i	$0.374551\pi$
$984$	0	0
$985$	−64.4071	−2.05218
$986$	0	0
$987$	0.183798	0.00585034
$988$	0	0
$989$	9.66464	0.307318
$990$	0	0
$991$	−38.1108	−1.21063	−0.605315	−	0.795986i	$-0.706953\pi$
$991$	−38.1108	−1.21063	−0.605315	+	0.795986i	$0.706953\pi$
$992$	0	0
$993$	9.51649	0.301997
$994$	0	0
$995$	−48.0223	−1.52241
$996$	0	0
$997$	4.40175	0.139405	0.0697024	−	0.997568i	$-0.477795\pi$
$997$	4.40175	0.139405	0.0697024	+	0.997568i	$0.477795\pi$
$998$	0	0
$999$	−29.6111	−0.936854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.z.1.13		20
17.11	odd	16		476.2.u.a.393.4	yes	40
17.14	odd	16		476.2.u.a.281.4	✓	40
17.16	even	2		8092.2.a.y.1.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.u.a.281.4	✓	40	17.14	odd	16
476.2.u.a.393.4	yes	40	17.11	odd	16
8092.2.a.y.1.8		20	17.16	even	2
8092.2.a.z.1.13		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)

\(f(q)\)	\(=\)	\(q+1.28083 q^{3} -3.03609 q^{5} +1.00000 q^{7} -1.35947 q^{9} -2.68953 q^{11} -6.64808 q^{13} -3.88873 q^{15} +4.75228 q^{19} +1.28083 q^{21} +7.63946 q^{23} +4.21787 q^{25} -5.58375 q^{27} +0.964919 q^{29} -6.08254 q^{31} -3.44484 q^{33} -3.03609 q^{35} +5.30308 q^{37} -8.51508 q^{39} -11.4695 q^{41} +1.26509 q^{43} +4.12747 q^{45} +0.143498 q^{47} +1.00000 q^{49} -1.46842 q^{53} +8.16566 q^{55} +6.08687 q^{57} -11.7725 q^{59} -12.8689 q^{61} -1.35947 q^{63} +20.1842 q^{65} +5.78961 q^{67} +9.78488 q^{69} +10.3955 q^{71} +7.45558 q^{73} +5.40239 q^{75} -2.68953 q^{77} +0.261371 q^{79} -3.07346 q^{81} +3.13776 q^{83} +1.23590 q^{87} +4.50413 q^{89} -6.64808 q^{91} -7.79071 q^{93} -14.4284 q^{95} -14.2742 q^{97} +3.65632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 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

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Database

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