LMFDB - Embedded newform 8092.2.a.z.1.9

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)

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")

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);

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			1.9
Root			$0.411306$ 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)

$f(q)$	$=$	$q+0.411306 q^{3} -0.830159 q^{5} +1.00000 q^{7} -2.83083 q^{9} -2.58028 q^{11} -0.648987 q^{13} -0.341449 q^{15} -6.04733 q^{19} +0.411306 q^{21} -2.44561 q^{23} -4.31084 q^{25} -2.39825 q^{27} -8.15224 q^{29} -0.704384 q^{31} -1.06128 q^{33} -0.830159 q^{35} -7.39962 q^{37} -0.266932 q^{39} +7.97071 q^{41} +4.09687 q^{43} +2.35004 q^{45} +6.53831 q^{47} +1.00000 q^{49} +11.8733 q^{53} +2.14204 q^{55} -2.48730 q^{57} -4.49372 q^{59} +10.9567 q^{61} -2.83083 q^{63} +0.538763 q^{65} +9.41150 q^{67} -1.00589 q^{69} +14.3610 q^{71} +12.5276 q^{73} -1.77307 q^{75} -2.58028 q^{77} +5.31955 q^{79} +7.50607 q^{81} +3.59081 q^{83} -3.35306 q^{87} -7.78812 q^{89} -0.648987 q^{91} -0.289717 q^{93} +5.02025 q^{95} +12.7068 q^{97} +7.30432 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.411306	0.237468	0.118734	−	0.992926i	$-0.462116\pi$
$3$	0.411306	0.237468	0.118734	+	0.992926i	$0.462116\pi$
$4$	0	0
$5$	−0.830159	−0.371259	−0.185629	−	0.982620i	$-0.559432\pi$
$5$	−0.830159	−0.371259	−0.185629	+	0.982620i	$0.559432\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.83083	−0.943609
$10$	0	0
$11$	−2.58028	−0.777983	−0.388992	−	0.921241i	$-0.627176\pi$
$11$	−2.58028	−0.777983	−0.388992	+	0.921241i	$0.627176\pi$
$12$	0	0
$13$	−0.648987	−0.179997	−0.0899983	−	0.995942i	$-0.528686\pi$
$13$	−0.648987	−0.179997	−0.0899983	+	0.995942i	$0.528686\pi$
$14$	0	0
$15$	−0.341449	−0.0881619
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−6.04733	−1.38735	−0.693677	−	0.720286i	$-0.744010\pi$
$19$	−6.04733	−1.38735	−0.693677	+	0.720286i	$0.744010\pi$
$20$	0	0
$21$	0.411306	0.0897543
$22$	0	0
$23$	−2.44561	−0.509945	−0.254972	−	0.966948i	$-0.582066\pi$
$23$	−2.44561	−0.509945	−0.254972	+	0.966948i	$0.582066\pi$
$24$	0	0
$25$	−4.31084	−0.862167
$26$	0	0
$27$	−2.39825	−0.461544
$28$	0	0
$29$	−8.15224	−1.51383	−0.756917	−	0.653511i	$-0.773295\pi$
$29$	−8.15224	−1.51383	−0.756917	+	0.653511i	$0.773295\pi$
$30$	0	0
$31$	−0.704384	−0.126511	−0.0632555	−	0.997997i	$-0.520148\pi$
$31$	−0.704384	−0.126511	−0.0632555	+	0.997997i	$0.520148\pi$
$32$	0	0
$33$	−1.06128	−0.184746
$34$	0	0
$35$	−0.830159	−0.140323
$36$	0	0
$37$	−7.39962	−1.21649	−0.608245	−	0.793749i	$-0.708126\pi$
$37$	−7.39962	−1.21649	−0.608245	+	0.793749i	$0.708126\pi$
$38$	0	0
$39$	−0.266932	−0.0427434
$40$	0	0
$41$	7.97071	1.24482	0.622408	−	0.782693i	$-0.286155\pi$
$41$	7.97071	1.24482	0.622408	+	0.782693i	$0.286155\pi$
$42$	0	0
$43$	4.09687	0.624767	0.312384	−	0.949956i	$-0.398873\pi$
$43$	4.09687	0.624767	0.312384	+	0.949956i	$0.398873\pi$
$44$	0	0
$45$	2.35004	0.350323
$46$	0	0
$47$	6.53831	0.953711	0.476855	−	0.878982i	$-0.341777\pi$
$47$	6.53831	0.953711	0.476855	+	0.878982i	$0.341777\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.8733	1.63092	0.815460	−	0.578813i	$-0.196484\pi$
$53$	11.8733	1.63092	0.815460	+	0.578813i	$0.196484\pi$
$54$	0	0
$55$	2.14204	0.288833
$56$	0	0
$57$	−2.48730	−0.329451
$58$	0	0
$59$	−4.49372	−0.585033	−0.292516	−	0.956261i	$-0.594493\pi$
$59$	−4.49372	−0.585033	−0.292516	+	0.956261i	$0.594493\pi$
$60$	0	0
$61$	10.9567	1.40287	0.701433	−	0.712735i	$-0.252544\pi$
$61$	10.9567	1.40287	0.701433	+	0.712735i	$0.252544\pi$
$62$	0	0
$63$	−2.83083	−0.356651
$64$	0	0
$65$	0.538763	0.0668253
$66$	0	0
$67$	9.41150	1.14980	0.574899	−	0.818225i	$-0.305041\pi$
$67$	9.41150	1.14980	0.574899	+	0.818225i	$0.305041\pi$
$68$	0	0
$69$	−1.00589	−0.121095
$70$	0	0
$71$	14.3610	1.70433	0.852166	−	0.523271i	$-0.175288\pi$
$71$	14.3610	1.70433	0.852166	+	0.523271i	$0.175288\pi$
$72$	0	0
$73$	12.5276	1.46624	0.733122	−	0.680097i	$-0.238062\pi$
$73$	12.5276	1.46624	0.733122	+	0.680097i	$0.238062\pi$
$74$	0	0
$75$	−1.77307	−0.204737
$76$	0	0
$77$	−2.58028	−0.294050
$78$	0	0
$79$	5.31955	0.598497	0.299248	−	0.954175i	$-0.403264\pi$
$79$	5.31955	0.598497	0.299248	+	0.954175i	$0.403264\pi$
$80$	0	0
$81$	7.50607	0.834007
$82$	0	0
$83$	3.59081	0.394143	0.197071	−	0.980389i	$-0.436857\pi$
$83$	3.59081	0.394143	0.197071	+	0.980389i	$0.436857\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.35306	−0.359486
$88$	0	0
$89$	−7.78812	−0.825539	−0.412769	−	0.910836i	$-0.635438\pi$
$89$	−7.78812	−0.825539	−0.412769	+	0.910836i	$0.635438\pi$
$90$	0	0
$91$	−0.648987	−0.0680323
$92$	0	0
$93$	−0.289717	−0.0300423
$94$	0	0
$95$	5.02025	0.515067
$96$	0	0
$97$	12.7068	1.29018	0.645088	−	0.764108i	$-0.276821\pi$
$97$	12.7068	1.29018	0.645088	+	0.764108i	$0.276821\pi$
$98$	0	0
$99$	7.30432	0.734112
$100$	0	0
$101$	7.49499	0.745779	0.372890	−	0.927876i	$-0.378367\pi$
$101$	7.49499	0.745779	0.372890	+	0.927876i	$0.378367\pi$
$102$	0	0
$103$	−13.1228	−1.29303	−0.646515	−	0.762901i	$-0.723774\pi$
$103$	−13.1228	−1.29303	−0.646515	+	0.762901i	$0.723774\pi$
$104$	0	0
$105$	−0.341449	−0.0333221
$106$	0	0
$107$	−2.16355	−0.209158	−0.104579	−	0.994517i	$-0.533350\pi$
$107$	−2.16355	−0.209158	−0.104579	+	0.994517i	$0.533350\pi$
$108$	0	0
$109$	−4.14649	−0.397162	−0.198581	−	0.980084i	$-0.563633\pi$
$109$	−4.14649	−0.397162	−0.198581	+	0.980084i	$0.563633\pi$
$110$	0	0
$111$	−3.04351	−0.288877
$112$	0	0
$113$	−11.2656	−1.05978	−0.529891	−	0.848066i	$-0.677767\pi$
$113$	−11.2656	−1.05978	−0.529891	+	0.848066i	$0.677767\pi$
$114$	0	0
$115$	2.03024	0.189321
$116$	0	0
$117$	1.83717	0.169846
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.34216	−0.394742
$122$	0	0
$123$	3.27840	0.295603
$124$	0	0
$125$	7.72948	0.691345
$126$	0	0
$127$	−7.03547	−0.624297	−0.312148	−	0.950033i	$-0.601049\pi$
$127$	−7.03547	−0.624297	−0.312148	+	0.950033i	$0.601049\pi$
$128$	0	0
$129$	1.68507	0.148362
$130$	0	0
$131$	6.48213	0.566346	0.283173	−	0.959069i	$-0.408613\pi$
$131$	6.48213	0.566346	0.283173	+	0.959069i	$0.408613\pi$
$132$	0	0
$133$	−6.04733	−0.524370
$134$	0	0
$135$	1.99093	0.171352
$136$	0	0
$137$	−15.6105	−1.33370	−0.666849	−	0.745193i	$-0.732357\pi$
$137$	−15.6105	−1.33370	−0.666849	+	0.745193i	$0.732357\pi$
$138$	0	0
$139$	9.74681	0.826714	0.413357	−	0.910569i	$-0.364356\pi$
$139$	9.74681	0.826714	0.413357	+	0.910569i	$0.364356\pi$
$140$	0	0
$141$	2.68925	0.226475
$142$	0	0
$143$	1.67457	0.140034
$144$	0	0
$145$	6.76766	0.562024
$146$	0	0
$147$	0.411306	0.0339239
$148$	0	0
$149$	8.41645	0.689502	0.344751	−	0.938694i	$-0.387963\pi$
$149$	8.41645	0.689502	0.344751	+	0.938694i	$0.387963\pi$
$150$	0	0
$151$	−19.1827	−1.56106	−0.780532	−	0.625116i	$-0.785052\pi$
$151$	−19.1827	−1.56106	−0.780532	+	0.625116i	$0.785052\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.584751	0.0469683
$156$	0	0
$157$	20.6310	1.64654	0.823268	−	0.567653i	$-0.192148\pi$
$157$	20.6310	1.64654	0.823268	+	0.567653i	$0.192148\pi$
$158$	0	0
$159$	4.88355	0.387291
$160$	0	0
$161$	−2.44561	−0.192741
$162$	0	0
$163$	20.0232	1.56834	0.784171	−	0.620545i	$-0.213089\pi$
$163$	20.0232	1.56834	0.784171	+	0.620545i	$0.213089\pi$
$164$	0	0
$165$	0.881035	0.0685885
$166$	0	0
$167$	−24.5210	−1.89749	−0.948744	−	0.316045i	$-0.897645\pi$
$167$	−24.5210	−1.89749	−0.948744	+	0.316045i	$0.897645\pi$
$168$	0	0
$169$	−12.5788	−0.967601
$170$	0	0
$171$	17.1190	1.30912
$172$	0	0
$173$	−0.598964	−0.0455384	−0.0227692	−	0.999741i	$-0.507248\pi$
$173$	−0.598964	−0.0455384	−0.0227692	+	0.999741i	$0.507248\pi$
$174$	0	0
$175$	−4.31084	−0.325869
$176$	0	0
$177$	−1.84829	−0.138926
$178$	0	0
$179$	−2.39911	−0.179318	−0.0896588	−	0.995973i	$-0.528578\pi$
$179$	−2.39911	−0.179318	−0.0896588	+	0.995973i	$0.528578\pi$
$180$	0	0
$181$	−6.53938	−0.486068	−0.243034	−	0.970018i	$-0.578143\pi$
$181$	−6.53938	−0.486068	−0.243034	+	0.970018i	$0.578143\pi$
$182$	0	0
$183$	4.50657	0.333135
$184$	0	0
$185$	6.14287	0.451633
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.39825	−0.174447
$190$	0	0
$191$	4.29647	0.310882	0.155441	−	0.987845i	$-0.450320\pi$
$191$	4.29647	0.310882	0.155441	+	0.987845i	$0.450320\pi$
$192$	0	0
$193$	9.07895	0.653517	0.326758	−	0.945108i	$-0.394044\pi$
$193$	9.07895	0.653517	0.326758	+	0.945108i	$0.394044\pi$
$194$	0	0
$195$	0.221596	0.0158688
$196$	0	0
$197$	10.0368	0.715095	0.357547	−	0.933895i	$-0.383613\pi$
$197$	10.0368	0.715095	0.357547	+	0.933895i	$0.383613\pi$
$198$	0	0
$199$	−12.0391	−0.853429	−0.426714	−	0.904386i	$-0.640329\pi$
$199$	−12.0391	−0.853429	−0.426714	+	0.904386i	$0.640329\pi$
$200$	0	0
$201$	3.87100	0.273040
$202$	0	0
$203$	−8.15224	−0.572175
$204$	0	0
$205$	−6.61696	−0.462148
$206$	0	0
$207$	6.92309	0.481188
$208$	0	0
$209$	15.6038	1.07934
$210$	0	0
$211$	21.8913	1.50706	0.753529	−	0.657414i	$-0.228350\pi$
$211$	21.8913	1.50706	0.753529	+	0.657414i	$0.228350\pi$
$212$	0	0
$213$	5.90675	0.404724
$214$	0	0
$215$	−3.40106	−0.231950
$216$	0	0
$217$	−0.704384	−0.0478167
$218$	0	0
$219$	5.15268	0.348186
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.57622	−0.574306	−0.287153	−	0.957885i	$-0.592709\pi$
$223$	−8.57622	−0.574306	−0.287153	+	0.957885i	$0.592709\pi$
$224$	0	0
$225$	12.2032	0.813549
$226$	0	0
$227$	15.8544	1.05230	0.526148	−	0.850393i	$-0.323636\pi$
$227$	15.8544	1.05230	0.526148	+	0.850393i	$0.323636\pi$
$228$	0	0
$229$	−19.4863	−1.28769	−0.643845	−	0.765156i	$-0.722662\pi$
$229$	−19.4863	−1.28769	−0.643845	+	0.765156i	$0.722662\pi$
$230$	0	0
$231$	−1.06128	−0.0698273
$232$	0	0
$233$	−15.8467	−1.03815	−0.519074	−	0.854729i	$-0.673723\pi$
$233$	−15.8467	−1.03815	−0.519074	+	0.854729i	$0.673723\pi$
$234$	0	0
$235$	−5.42784	−0.354073
$236$	0	0
$237$	2.18796	0.142124
$238$	0	0
$239$	−9.86627	−0.638196	−0.319098	−	0.947722i	$-0.603380\pi$
$239$	−9.86627	−0.638196	−0.319098	+	0.947722i	$0.603380\pi$
$240$	0	0
$241$	17.4717	1.12545	0.562724	−	0.826645i	$-0.309753\pi$
$241$	17.4717	1.12545	0.562724	+	0.826645i	$0.309753\pi$
$242$	0	0
$243$	10.2821	0.659594
$244$	0	0
$245$	−0.830159	−0.0530369
$246$	0	0
$247$	3.92464	0.249719
$248$	0	0
$249$	1.47692	0.0935961
$250$	0	0
$251$	24.0492	1.51797	0.758986	−	0.651107i	$-0.225695\pi$
$251$	24.0492	1.51797	0.758986	+	0.651107i	$0.225695\pi$
$252$	0	0
$253$	6.31035	0.396728
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.8241	−0.799945	−0.399973	−	0.916527i	$-0.630980\pi$
$257$	−12.8241	−0.799945	−0.399973	+	0.916527i	$0.630980\pi$
$258$	0	0
$259$	−7.39962	−0.459790
$260$	0	0
$261$	23.0776	1.42847
$262$	0	0
$263$	14.5779	0.898914	0.449457	−	0.893302i	$-0.351618\pi$
$263$	14.5779	0.898914	0.449457	+	0.893302i	$0.351618\pi$
$264$	0	0
$265$	−9.85671	−0.605493
$266$	0	0
$267$	−3.20330	−0.196039
$268$	0	0
$269$	7.33879	0.447454	0.223727	−	0.974652i	$-0.428178\pi$
$269$	7.33879	0.447454	0.223727	+	0.974652i	$0.428178\pi$
$270$	0	0
$271$	11.0167	0.669218	0.334609	−	0.942357i	$-0.391396\pi$
$271$	11.0167	0.669218	0.334609	+	0.942357i	$0.391396\pi$
$272$	0	0
$273$	−0.266932	−0.0161555
$274$	0	0
$275$	11.1232	0.670752
$276$	0	0
$277$	−10.0398	−0.603232	−0.301616	−	0.953429i	$-0.597526\pi$
$277$	−10.0398	−0.603232	−0.301616	+	0.953429i	$0.597526\pi$
$278$	0	0
$279$	1.99399	0.119377
$280$	0	0
$281$	6.22839	0.371555	0.185777	−	0.982592i	$-0.440520\pi$
$281$	6.22839	0.371555	0.185777	+	0.982592i	$0.440520\pi$
$282$	0	0
$283$	6.88123	0.409047	0.204523	−	0.978862i	$-0.434436\pi$
$283$	6.88123	0.409047	0.204523	+	0.978862i	$0.434436\pi$
$284$	0	0
$285$	2.06486	0.122312
$286$	0	0
$287$	7.97071	0.470496
$288$	0	0
$289$	0	0
$290$	0	0
$291$	5.22636	0.306375
$292$	0	0
$293$	20.8767	1.21963	0.609814	−	0.792544i	$-0.291244\pi$
$293$	20.8767	1.21963	0.609814	+	0.792544i	$0.291244\pi$
$294$	0	0
$295$	3.73050	0.217198
$296$	0	0
$297$	6.18816	0.359074
$298$	0	0
$299$	1.58717	0.0917883
$300$	0	0
$301$	4.09687	0.236140
$302$	0	0
$303$	3.08273	0.177098
$304$	0	0
$305$	−9.09584	−0.520826
$306$	0	0
$307$	−20.9303	−1.19455	−0.597277	−	0.802035i	$-0.703751\pi$
$307$	−20.9303	−1.19455	−0.597277	+	0.802035i	$0.703751\pi$
$308$	0	0
$309$	−5.39750	−0.307053
$310$	0	0
$311$	8.31252	0.471360	0.235680	−	0.971831i	$-0.424268\pi$
$311$	8.31252	0.471360	0.235680	+	0.971831i	$0.424268\pi$
$312$	0	0
$313$	4.48015	0.253233	0.126617	−	0.991952i	$-0.459588\pi$
$313$	4.48015	0.253233	0.126617	+	0.991952i	$0.459588\pi$
$314$	0	0
$315$	2.35004	0.132410
$316$	0	0
$317$	−0.553633	−0.0310951	−0.0155476	−	0.999879i	$-0.504949\pi$
$317$	−0.553633	−0.0310951	−0.0155476	+	0.999879i	$0.504949\pi$
$318$	0	0
$319$	21.0351	1.17774
$320$	0	0
$321$	−0.889881	−0.0496683
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.79768	0.155187
$326$	0	0
$327$	−1.70548	−0.0943131
$328$	0	0
$329$	6.53831	0.360469
$330$	0	0
$331$	−33.1492	−1.82204	−0.911022	−	0.412357i	$-0.864706\pi$
$331$	−33.1492	−1.82204	−0.911022	+	0.412357i	$0.864706\pi$
$332$	0	0
$333$	20.9471	1.14789
$334$	0	0
$335$	−7.81304	−0.426872
$336$	0	0
$337$	2.00018	0.108957	0.0544783	−	0.998515i	$-0.482650\pi$
$337$	2.00018	0.108957	0.0544783	+	0.998515i	$0.482650\pi$
$338$	0	0
$339$	−4.63362	−0.251664
$340$	0	0
$341$	1.81751	0.0984235
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.835051	0.0449577
$346$	0	0
$347$	−16.1307	−0.865941	−0.432971	−	0.901408i	$-0.642535\pi$
$347$	−16.1307	−0.865941	−0.432971	+	0.901408i	$0.642535\pi$
$348$	0	0
$349$	−12.3185	−0.659393	−0.329697	−	0.944087i	$-0.606946\pi$
$349$	−12.3185	−0.659393	−0.329697	+	0.944087i	$0.606946\pi$
$350$	0	0
$351$	1.55644	0.0830764
$352$	0	0
$353$	−10.5411	−0.561048	−0.280524	−	0.959847i	$-0.590508\pi$
$353$	−10.5411	−0.561048	−0.280524	+	0.959847i	$0.590508\pi$
$354$	0	0
$355$	−11.9219	−0.632748
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.14847	0.166170	0.0830849	−	0.996542i	$-0.473523\pi$
$359$	3.14847	0.166170	0.0830849	+	0.996542i	$0.473523\pi$
$360$	0	0
$361$	17.5702	0.924750
$362$	0	0
$363$	−1.78596	−0.0937385
$364$	0	0
$365$	−10.3999	−0.544356
$366$	0	0
$367$	21.2023	1.10675	0.553377	−	0.832931i	$-0.313339\pi$
$367$	21.2023	1.10675	0.553377	+	0.832931i	$0.313339\pi$
$368$	0	0
$369$	−22.5637	−1.17462
$370$	0	0
$371$	11.8733	0.616430
$372$	0	0
$373$	−2.58465	−0.133828	−0.0669142	−	0.997759i	$-0.521315\pi$
$373$	−2.58465	−0.133828	−0.0669142	+	0.997759i	$0.521315\pi$
$374$	0	0
$375$	3.17918	0.164172
$376$	0	0
$377$	5.29070	0.272485
$378$	0	0
$379$	14.4376	0.741611	0.370805	−	0.928711i	$-0.379082\pi$
$379$	14.4376	0.741611	0.370805	+	0.928711i	$0.379082\pi$
$380$	0	0
$381$	−2.89373	−0.148250
$382$	0	0
$383$	−29.0751	−1.48567	−0.742835	−	0.669475i	$-0.766519\pi$
$383$	−29.0751	−1.48567	−0.742835	+	0.669475i	$0.766519\pi$
$384$	0	0
$385$	2.14204	0.109169
$386$	0	0
$387$	−11.5975	−0.589536
$388$	0	0
$389$	24.2825	1.23117	0.615587	−	0.788069i	$-0.288919\pi$
$389$	24.2825	1.23117	0.615587	+	0.788069i	$0.288919\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.66614	0.134489
$394$	0	0
$395$	−4.41608	−0.222197
$396$	0	0
$397$	−29.3485	−1.47296	−0.736480	−	0.676460i	$-0.763513\pi$
$397$	−29.3485	−1.47296	−0.736480	+	0.676460i	$0.763513\pi$
$398$	0	0
$399$	−2.48730	−0.124521
$400$	0	0
$401$	34.2340	1.70957	0.854783	−	0.518985i	$-0.173690\pi$
$401$	34.2340	1.70957	0.854783	+	0.518985i	$0.173690\pi$
$402$	0	0
$403$	0.457136	0.0227716
$404$	0	0
$405$	−6.23123	−0.309632
$406$	0	0
$407$	19.0931	0.946409
$408$	0	0
$409$	−30.4230	−1.50432	−0.752161	−	0.658979i	$-0.770988\pi$
$409$	−30.4230	−1.50432	−0.752161	+	0.658979i	$0.770988\pi$
$410$	0	0
$411$	−6.42070	−0.316710
$412$	0	0
$413$	−4.49372	−0.221122
$414$	0	0
$415$	−2.98095	−0.146329
$416$	0	0
$417$	4.00892	0.196318
$418$	0	0
$419$	−6.24873	−0.305270	−0.152635	−	0.988283i	$-0.548776\pi$
$419$	−6.24873	−0.305270	−0.152635	+	0.988283i	$0.548776\pi$
$420$	0	0
$421$	11.1927	0.545500	0.272750	−	0.962085i	$-0.412067\pi$
$421$	11.1927	0.545500	0.272750	+	0.962085i	$0.412067\pi$
$422$	0	0
$423$	−18.5088	−0.899930
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.9567	0.530234
$428$	0	0
$429$	0.688760	0.0332536
$430$	0	0
$431$	17.4889	0.842413	0.421207	−	0.906965i	$-0.361607\pi$
$431$	17.4889	0.842413	0.421207	+	0.906965i	$0.361607\pi$
$432$	0	0
$433$	−11.0024	−0.528742	−0.264371	−	0.964421i	$-0.585164\pi$
$433$	−11.0024	−0.528742	−0.264371	+	0.964421i	$0.585164\pi$
$434$	0	0
$435$	2.78358	0.133462
$436$	0	0
$437$	14.7894	0.707473
$438$	0	0
$439$	−11.9799	−0.571771	−0.285886	−	0.958264i	$-0.592288\pi$
$439$	−11.9799	−0.571771	−0.285886	+	0.958264i	$0.592288\pi$
$440$	0	0
$441$	−2.83083	−0.134801
$442$	0	0
$443$	8.15044	0.387239	0.193619	−	0.981077i	$-0.437977\pi$
$443$	8.15044	0.387239	0.193619	+	0.981077i	$0.437977\pi$
$444$	0	0
$445$	6.46538	0.306488
$446$	0	0
$447$	3.46174	0.163734
$448$	0	0
$449$	1.55566	0.0734159	0.0367080	−	0.999326i	$-0.488313\pi$
$449$	1.55566	0.0734159	0.0367080	+	0.999326i	$0.488313\pi$
$450$	0	0
$451$	−20.5666	−0.968446
$452$	0	0
$453$	−7.88995	−0.370702
$454$	0	0
$455$	0.538763	0.0252576
$456$	0	0
$457$	13.0862	0.612146	0.306073	−	0.952008i	$-0.400985\pi$
$457$	13.0862	0.612146	0.306073	+	0.952008i	$0.400985\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.6025	1.19243	0.596214	−	0.802826i	$-0.296671\pi$
$461$	25.6025	1.19243	0.596214	+	0.802826i	$0.296671\pi$
$462$	0	0
$463$	−33.4214	−1.55323	−0.776613	−	0.629978i	$-0.783064\pi$
$463$	−33.4214	−1.55323	−0.776613	+	0.629978i	$0.783064\pi$
$464$	0	0
$465$	0.240511	0.0111535
$466$	0	0
$467$	−37.4711	−1.73396	−0.866978	−	0.498346i	$-0.833941\pi$
$467$	−37.4711	−1.73396	−0.866978	+	0.498346i	$0.833941\pi$
$468$	0	0
$469$	9.41150	0.434583
$470$	0	0
$471$	8.48567	0.390999
$472$	0	0
$473$	−10.5711	−0.486059
$474$	0	0
$475$	26.0691	1.19613
$476$	0	0
$477$	−33.6112	−1.53895
$478$	0	0
$479$	−6.31911	−0.288727	−0.144364	−	0.989525i	$-0.546114\pi$
$479$	−6.31911	−0.288727	−0.144364	+	0.989525i	$0.546114\pi$
$480$	0	0
$481$	4.80226	0.218964
$482$	0	0
$483$	−1.00589	−0.0457697
$484$	0	0
$485$	−10.5486	−0.478989
$486$	0	0
$487$	1.43736	0.0651329	0.0325664	−	0.999470i	$-0.489632\pi$
$487$	1.43736	0.0651329	0.0325664	+	0.999470i	$0.489632\pi$
$488$	0	0
$489$	8.23568	0.372430
$490$	0	0
$491$	−13.7106	−0.618749	−0.309375	−	0.950940i	$-0.600120\pi$
$491$	−13.7106	−0.618749	−0.309375	+	0.950940i	$0.600120\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−6.06375	−0.272545
$496$	0	0
$497$	14.3610	0.644177
$498$	0	0
$499$	27.3393	1.22388	0.611938	−	0.790906i	$-0.290390\pi$
$499$	27.3393	1.22388	0.611938	+	0.790906i	$0.290390\pi$
$500$	0	0
$501$	−10.0856	−0.450592
$502$	0	0
$503$	19.7853	0.882184	0.441092	−	0.897462i	$-0.354591\pi$
$503$	19.7853	0.882184	0.441092	+	0.897462i	$0.354591\pi$
$504$	0	0
$505$	−6.22203	−0.276877
$506$	0	0
$507$	−5.17374	−0.229774
$508$	0	0
$509$	20.8928	0.926056	0.463028	−	0.886344i	$-0.346763\pi$
$509$	20.8928	0.926056	0.463028	+	0.886344i	$0.346763\pi$
$510$	0	0
$511$	12.5276	0.554188
$512$	0	0
$513$	14.5030	0.640325
$514$	0	0
$515$	10.8940	0.480049
$516$	0	0
$517$	−16.8707	−0.741971
$518$	0	0
$519$	−0.246357	−0.0108139
$520$	0	0
$521$	13.5525	0.593747	0.296873	−	0.954917i	$-0.404056\pi$
$521$	13.5525	0.593747	0.296873	+	0.954917i	$0.404056\pi$
$522$	0	0
$523$	27.2512	1.19161	0.595806	−	0.803128i	$-0.296833\pi$
$523$	27.2512	1.19161	0.595806	+	0.803128i	$0.296833\pi$
$524$	0	0
$525$	−1.77307	−0.0773832
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−17.0190	−0.739957
$530$	0	0
$531$	12.7209	0.552042
$532$	0	0
$533$	−5.17289	−0.224063
$534$	0	0
$535$	1.79609	0.0776518
$536$	0	0
$537$	−0.986767	−0.0425821
$538$	0	0
$539$	−2.58028	−0.111140
$540$	0	0
$541$	4.47470	0.192383	0.0961913	−	0.995363i	$-0.469334\pi$
$541$	4.47470	0.192383	0.0961913	+	0.995363i	$0.469334\pi$
$542$	0	0
$543$	−2.68969	−0.115425
$544$	0	0
$545$	3.44225	0.147450
$546$	0	0
$547$	−5.30223	−0.226707	−0.113353	−	0.993555i	$-0.536159\pi$
$547$	−5.30223	−0.226707	−0.113353	+	0.993555i	$0.536159\pi$
$548$	0	0
$549$	−31.0166	−1.32376
$550$	0	0
$551$	49.2993	2.10022
$552$	0	0
$553$	5.31955	0.226210
$554$	0	0
$555$	2.52660	0.107248
$556$	0	0
$557$	2.15891	0.0914760	0.0457380	−	0.998953i	$-0.485436\pi$
$557$	2.15891	0.0914760	0.0457380	+	0.998953i	$0.485436\pi$
$558$	0	0
$559$	−2.65882	−0.112456
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.55509	0.276264	0.138132	−	0.990414i	$-0.455890\pi$
$563$	6.55509	0.276264	0.138132	+	0.990414i	$0.455890\pi$
$564$	0	0
$565$	9.35227	0.393453
$566$	0	0
$567$	7.50607	0.315225
$568$	0	0
$569$	44.9389	1.88394	0.941969	−	0.335700i	$-0.108973\pi$
$569$	44.9389	1.88394	0.941969	+	0.335700i	$0.108973\pi$
$570$	0	0
$571$	−37.3323	−1.56231	−0.781154	−	0.624338i	$-0.785369\pi$
$571$	−37.3323	−1.56231	−0.781154	+	0.624338i	$0.785369\pi$
$572$	0	0
$573$	1.76716	0.0738243
$574$	0	0
$575$	10.5426	0.439657
$576$	0	0
$577$	13.3042	0.553860	0.276930	−	0.960890i	$-0.410683\pi$
$577$	13.3042	0.553860	0.276930	+	0.960890i	$0.410683\pi$
$578$	0	0
$579$	3.73422	0.155189
$580$	0	0
$581$	3.59081	0.148972
$582$	0	0
$583$	−30.6364	−1.26883
$584$	0	0
$585$	−1.52514	−0.0630570
$586$	0	0
$587$	17.0629	0.704261	0.352130	−	0.935951i	$-0.385457\pi$
$587$	17.0629	0.704261	0.352130	+	0.935951i	$0.385457\pi$
$588$	0	0
$589$	4.25964	0.175516
$590$	0	0
$591$	4.12821	0.169812
$592$	0	0
$593$	7.90337	0.324553	0.162276	−	0.986745i	$-0.448116\pi$
$593$	7.90337	0.324553	0.162276	+	0.986745i	$0.448116\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.95175	−0.202662
$598$	0	0
$599$	36.6728	1.49841	0.749205	−	0.662338i	$-0.230436\pi$
$599$	36.6728	1.49841	0.749205	+	0.662338i	$0.230436\pi$
$600$	0	0
$601$	−5.25010	−0.214156	−0.107078	−	0.994251i	$-0.534149\pi$
$601$	−5.25010	−0.214156	−0.107078	+	0.994251i	$0.534149\pi$
$602$	0	0
$603$	−26.6423	−1.08496
$604$	0	0
$605$	3.60469	0.146551
$606$	0	0
$607$	−9.28740	−0.376964	−0.188482	−	0.982077i	$-0.560357\pi$
$607$	−9.28740	−0.376964	−0.188482	+	0.982077i	$0.560357\pi$
$608$	0	0
$609$	−3.35306	−0.135873
$610$	0	0
$611$	−4.24328	−0.171665
$612$	0	0
$613$	3.96358	0.160087	0.0800437	−	0.996791i	$-0.474494\pi$
$613$	3.96358	0.160087	0.0800437	+	0.996791i	$0.474494\pi$
$614$	0	0
$615$	−2.72159	−0.109745
$616$	0	0
$617$	19.2531	0.775099	0.387549	−	0.921849i	$-0.373322\pi$
$617$	19.2531	0.775099	0.387549	+	0.921849i	$0.373322\pi$
$618$	0	0
$619$	12.7211	0.511306	0.255653	−	0.966769i	$-0.417710\pi$
$619$	12.7211	0.511306	0.255653	+	0.966769i	$0.417710\pi$
$620$	0	0
$621$	5.86519	0.235362
$622$	0	0
$623$	−7.78812	−0.312024
$624$	0	0
$625$	15.1375	0.605499
$626$	0	0
$627$	6.41794	0.256308
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.8286	−1.26708	−0.633539	−	0.773711i	$-0.718398\pi$
$631$	−31.8286	−1.26708	−0.633539	+	0.773711i	$0.718398\pi$
$632$	0	0
$633$	9.00402	0.357878
$634$	0	0
$635$	5.84056	0.231776
$636$	0	0
$637$	−0.648987	−0.0257138
$638$	0	0
$639$	−40.6534	−1.60822
$640$	0	0
$641$	−26.1190	−1.03164	−0.515819	−	0.856698i	$-0.672512\pi$
$641$	−26.1190	−1.03164	−0.515819	+	0.856698i	$0.672512\pi$
$642$	0	0
$643$	40.2971	1.58916	0.794581	−	0.607158i	$-0.207691\pi$
$643$	40.2971	1.58916	0.794581	+	0.607158i	$0.207691\pi$
$644$	0	0
$645$	−1.39888	−0.0550807
$646$	0	0
$647$	−31.0279	−1.21983	−0.609917	−	0.792465i	$-0.708797\pi$
$647$	−31.0279	−1.21983	−0.609917	+	0.792465i	$0.708797\pi$
$648$	0	0
$649$	11.5951	0.455146
$650$	0	0
$651$	−0.289717	−0.0113549
$652$	0	0
$653$	−9.42264	−0.368736	−0.184368	−	0.982857i	$-0.559024\pi$
$653$	−9.42264	−0.368736	−0.184368	+	0.982857i	$0.559024\pi$
$654$	0	0
$655$	−5.38120	−0.210261
$656$	0	0
$657$	−35.4635	−1.38356
$658$	0	0
$659$	9.30384	0.362426	0.181213	−	0.983444i	$-0.441998\pi$
$659$	9.30384	0.362426	0.181213	+	0.983444i	$0.441998\pi$
$660$	0	0
$661$	15.5049	0.603071	0.301536	−	0.953455i	$-0.402501\pi$
$661$	15.5049	0.603071	0.301536	+	0.953455i	$0.402501\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.02025	0.194677
$666$	0	0
$667$	19.9372	0.771971
$668$	0	0
$669$	−3.52745	−0.136379
$670$	0	0
$671$	−28.2714	−1.09141
$672$	0	0
$673$	−24.0223	−0.925990	−0.462995	−	0.886361i	$-0.653225\pi$
$673$	−24.0223	−0.925990	−0.462995	+	0.886361i	$0.653225\pi$
$674$	0	0
$675$	10.3385	0.397928
$676$	0	0
$677$	12.8826	0.495119	0.247560	−	0.968873i	$-0.420371\pi$
$677$	12.8826	0.495119	0.247560	+	0.968873i	$0.420371\pi$
$678$	0	0
$679$	12.7068	0.487641
$680$	0	0
$681$	6.52103	0.249886
$682$	0	0
$683$	35.4977	1.35828	0.679142	−	0.734007i	$-0.262352\pi$
$683$	35.4977	1.35828	0.679142	+	0.734007i	$0.262352\pi$
$684$	0	0
$685$	12.9592	0.495147
$686$	0	0
$687$	−8.01483	−0.305785
$688$	0	0
$689$	−7.70560	−0.293560
$690$	0	0
$691$	−41.3441	−1.57281	−0.786403	−	0.617714i	$-0.788059\pi$
$691$	−41.3441	−1.57281	−0.786403	+	0.617714i	$0.788059\pi$
$692$	0	0
$693$	7.30432	0.277468
$694$	0	0
$695$	−8.09141	−0.306925
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−6.51782	−0.246527
$700$	0	0
$701$	−8.10139	−0.305985	−0.152993	−	0.988227i	$-0.548891\pi$
$701$	−8.10139	−0.305985	−0.152993	+	0.988227i	$0.548891\pi$
$702$	0	0
$703$	44.7480	1.68770
$704$	0	0
$705$	−2.23250	−0.0840809
$706$	0	0
$707$	7.49499	0.281878
$708$	0	0
$709$	−31.1598	−1.17023	−0.585115	−	0.810950i	$-0.698951\pi$
$709$	−31.1598	−1.17023	−0.585115	+	0.810950i	$0.698951\pi$
$710$	0	0
$711$	−15.0587	−0.564747
$712$	0	0
$713$	1.72265	0.0645136
$714$	0	0
$715$	−1.39016	−0.0519890
$716$	0	0
$717$	−4.05806	−0.151551
$718$	0	0
$719$	48.6941	1.81598	0.907992	−	0.418987i	$-0.137615\pi$
$719$	48.6941	1.81598	0.907992	+	0.418987i	$0.137615\pi$
$720$	0	0
$721$	−13.1228	−0.488720
$722$	0	0
$723$	7.18619	0.267257
$724$	0	0
$725$	35.1430	1.30518
$726$	0	0
$727$	−30.5854	−1.13435	−0.567176	−	0.823597i	$-0.691964\pi$
$727$	−30.5854	−1.13435	−0.567176	+	0.823597i	$0.691964\pi$
$728$	0	0
$729$	−18.2891	−0.677375
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−45.4998	−1.68057	−0.840286	−	0.542143i	$-0.817613\pi$
$733$	−45.4998	−1.68057	−0.840286	+	0.542143i	$0.817613\pi$
$734$	0	0
$735$	−0.341449	−0.0125946
$736$	0	0
$737$	−24.2843	−0.894523
$738$	0	0
$739$	51.2964	1.88697	0.943484	−	0.331419i	$-0.107527\pi$
$739$	51.2964	1.88697	0.943484	+	0.331419i	$0.107527\pi$
$740$	0	0
$741$	1.61423	0.0593002
$742$	0	0
$743$	44.4995	1.63253	0.816264	−	0.577680i	$-0.196042\pi$
$743$	44.4995	1.63253	0.816264	+	0.577680i	$0.196042\pi$
$744$	0	0
$745$	−6.98700	−0.255984
$746$	0	0
$747$	−10.1650	−0.371917
$748$	0	0
$749$	−2.16355	−0.0790544
$750$	0	0
$751$	12.5724	0.458774	0.229387	−	0.973335i	$-0.426328\pi$
$751$	12.5724	0.458774	0.229387	+	0.973335i	$0.426328\pi$
$752$	0	0
$753$	9.89158	0.360469
$754$	0	0
$755$	15.9247	0.579558
$756$	0	0
$757$	−11.6350	−0.422883	−0.211441	−	0.977391i	$-0.567816\pi$
$757$	−11.6350	−0.422883	−0.211441	+	0.977391i	$0.567816\pi$
$758$	0	0
$759$	2.59548	0.0942101
$760$	0	0
$761$	−20.4844	−0.742558	−0.371279	−	0.928521i	$-0.621081\pi$
$761$	−20.4844	−0.742558	−0.371279	+	0.928521i	$0.621081\pi$
$762$	0	0
$763$	−4.14649	−0.150113
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.91637	0.105304
$768$	0	0
$769$	−11.8655	−0.427880	−0.213940	−	0.976847i	$-0.568630\pi$
$769$	−11.8655	−0.427880	−0.213940	+	0.976847i	$0.568630\pi$
$770$	0	0
$771$	−5.27463	−0.189961
$772$	0	0
$773$	−4.86427	−0.174956	−0.0874778	−	0.996166i	$-0.527881\pi$
$773$	−4.86427	−0.174956	−0.0874778	+	0.996166i	$0.527881\pi$
$774$	0	0
$775$	3.03648	0.109074
$776$	0	0
$777$	−3.04351	−0.109185
$778$	0	0
$779$	−48.2015	−1.72700
$780$	0	0
$781$	−37.0553	−1.32594
$782$	0	0
$783$	19.5511	0.698701
$784$	0	0
$785$	−17.1270	−0.611291
$786$	0	0
$787$	11.1663	0.398037	0.199018	−	0.979996i	$-0.436225\pi$
$787$	11.1663	0.398037	0.199018	+	0.979996i	$0.436225\pi$
$788$	0	0
$789$	5.99599	0.213463
$790$	0	0
$791$	−11.2656	−0.400560
$792$	0	0
$793$	−7.11078	−0.252511
$794$	0	0
$795$	−4.05412	−0.143785
$796$	0	0
$797$	13.7439	0.486834	0.243417	−	0.969922i	$-0.421732\pi$
$797$	13.7439	0.486834	0.243417	+	0.969922i	$0.421732\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	22.0468	0.778986
$802$	0	0
$803$	−32.3247	−1.14071
$804$	0	0
$805$	2.03024	0.0715567
$806$	0	0
$807$	3.01849	0.106256
$808$	0	0
$809$	31.2368	1.09823	0.549114	−	0.835748i	$-0.314965\pi$
$809$	31.2368	1.09823	0.549114	+	0.835748i	$0.314965\pi$
$810$	0	0
$811$	35.6728	1.25264	0.626321	−	0.779565i	$-0.284560\pi$
$811$	35.6728	1.25264	0.626321	+	0.779565i	$0.284560\pi$
$812$	0	0
$813$	4.53124	0.158918
$814$	0	0
$815$	−16.6225	−0.582260
$816$	0	0
$817$	−24.7752	−0.866773
$818$	0	0
$819$	1.83717	0.0641959
$820$	0	0
$821$	10.4805	0.365774	0.182887	−	0.983134i	$-0.441456\pi$
$821$	10.4805	0.365774	0.182887	+	0.983134i	$0.441456\pi$
$822$	0	0
$823$	20.5994	0.718052	0.359026	−	0.933328i	$-0.383109\pi$
$823$	20.5994	0.718052	0.359026	+	0.933328i	$0.383109\pi$
$824$	0	0
$825$	4.57502	0.159282
$826$	0	0
$827$	26.7409	0.929871	0.464936	−	0.885344i	$-0.346077\pi$
$827$	26.7409	0.929871	0.464936	+	0.885344i	$0.346077\pi$
$828$	0	0
$829$	44.5408	1.54697	0.773483	−	0.633817i	$-0.218513\pi$
$829$	44.5408	1.54697	0.773483	+	0.633817i	$0.218513\pi$
$830$	0	0
$831$	−4.12942	−0.143248
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.3563	0.704459
$836$	0	0
$837$	1.68929	0.0583904
$838$	0	0
$839$	4.39922	0.151878	0.0759391	−	0.997112i	$-0.475805\pi$
$839$	4.39922	0.151878	0.0759391	+	0.997112i	$0.475805\pi$
$840$	0	0
$841$	37.4590	1.29169
$842$	0	0
$843$	2.56178	0.0882322
$844$	0	0
$845$	10.4424	0.359230
$846$	0	0
$847$	−4.34216	−0.149199
$848$	0	0
$849$	2.83029	0.0971354
$850$	0	0
$851$	18.0966	0.620343
$852$	0	0
$853$	−18.4937	−0.633213	−0.316606	−	0.948557i	$-0.602543\pi$
$853$	−18.4937	−0.633213	−0.316606	+	0.948557i	$0.602543\pi$
$854$	0	0
$855$	−14.2115	−0.486022
$856$	0	0
$857$	−28.5034	−0.973659	−0.486830	−	0.873497i	$-0.661847\pi$
$857$	−28.5034	−0.973659	−0.486830	+	0.873497i	$0.661847\pi$
$858$	0	0
$859$	53.7580	1.83420	0.917100	−	0.398658i	$-0.130524\pi$
$859$	53.7580	1.83420	0.917100	+	0.398658i	$0.130524\pi$
$860$	0	0
$861$	3.27840	0.111728
$862$	0	0
$863$	−6.24562	−0.212603	−0.106302	−	0.994334i	$-0.533901\pi$
$863$	−6.24562	−0.212603	−0.106302	+	0.994334i	$0.533901\pi$
$864$	0	0
$865$	0.497235	0.0169065
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.7259	−0.465620
$870$	0	0
$871$	−6.10794	−0.206960
$872$	0	0
$873$	−35.9706	−1.21742
$874$	0	0
$875$	7.72948	0.261304
$876$	0	0
$877$	−3.50585	−0.118384	−0.0591921	−	0.998247i	$-0.518852\pi$
$877$	−3.50585	−0.118384	−0.0591921	+	0.998247i	$0.518852\pi$
$878$	0	0
$879$	8.58670	0.289622
$880$	0	0
$881$	16.2054	0.545975	0.272988	−	0.962018i	$-0.411988\pi$
$881$	16.2054	0.545975	0.272988	+	0.962018i	$0.411988\pi$
$882$	0	0
$883$	−13.3564	−0.449477	−0.224739	−	0.974419i	$-0.572153\pi$
$883$	−13.3564	−0.449477	−0.224739	+	0.974419i	$0.572153\pi$
$884$	0	0
$885$	1.53438	0.0515776
$886$	0	0
$887$	50.3882	1.69187	0.845935	−	0.533287i	$-0.179043\pi$
$887$	50.3882	1.69187	0.845935	+	0.533287i	$0.179043\pi$
$888$	0	0
$889$	−7.03547	−0.235962
$890$	0	0
$891$	−19.3677	−0.648844
$892$	0	0
$893$	−39.5394	−1.32313
$894$	0	0
$895$	1.99164	0.0665732
$896$	0	0
$897$	0.652812	0.0217967
$898$	0	0
$899$	5.74231	0.191517
$900$	0	0
$901$	0	0
$902$	0	0
$903$	1.68507	0.0560756
$904$	0	0
$905$	5.42873	0.180457
$906$	0	0
$907$	17.0911	0.567502	0.283751	−	0.958898i	$-0.408421\pi$
$907$	17.0911	0.567502	0.283751	+	0.958898i	$0.408421\pi$
$908$	0	0
$909$	−21.2170	−0.703724
$910$	0	0
$911$	39.3835	1.30484	0.652418	−	0.757860i	$-0.273755\pi$
$911$	39.3835	1.30484	0.652418	+	0.757860i	$0.273755\pi$
$912$	0	0
$913$	−9.26529	−0.306636
$914$	0	0
$915$	−3.74117	−0.123679
$916$	0	0
$917$	6.48213	0.214059
$918$	0	0
$919$	26.5781	0.876731	0.438366	−	0.898797i	$-0.355558\pi$
$919$	26.5781	0.876731	0.438366	+	0.898797i	$0.355558\pi$
$920$	0	0
$921$	−8.60875	−0.283668
$922$	0	0
$923$	−9.32008	−0.306774
$924$	0	0
$925$	31.8986	1.04882
$926$	0	0
$927$	37.1485	1.22012
$928$	0	0
$929$	−42.3940	−1.39090	−0.695450	−	0.718574i	$-0.744795\pi$
$929$	−42.3940	−1.39090	−0.695450	+	0.718574i	$0.744795\pi$
$930$	0	0
$931$	−6.04733	−0.198193
$932$	0	0
$933$	3.41899	0.111933
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.45274	−0.0801276	−0.0400638	−	0.999197i	$-0.512756\pi$
$937$	−2.45274	−0.0801276	−0.0400638	+	0.999197i	$0.512756\pi$
$938$	0	0
$939$	1.84271	0.0601347
$940$	0	0
$941$	−23.6834	−0.772058	−0.386029	−	0.922487i	$-0.626153\pi$
$941$	−23.6834	−0.772058	−0.386029	+	0.922487i	$0.626153\pi$
$942$	0	0
$943$	−19.4932	−0.634787
$944$	0	0
$945$	1.99093	0.0647650
$946$	0	0
$947$	15.1939	0.493734	0.246867	−	0.969049i	$-0.420599\pi$
$947$	15.1939	0.493734	0.246867	+	0.969049i	$0.420599\pi$
$948$	0	0
$949$	−8.13025	−0.263919
$950$	0	0
$951$	−0.227712	−0.00738408
$952$	0	0
$953$	−29.4565	−0.954191	−0.477095	−	0.878852i	$-0.658310\pi$
$953$	−29.4565	−0.954191	−0.477095	+	0.878852i	$0.658310\pi$
$954$	0	0
$955$	−3.56675	−0.115417
$956$	0	0
$957$	8.65184	0.279674
$958$	0	0
$959$	−15.6105	−0.504090
$960$	0	0
$961$	−30.5038	−0.983995
$962$	0	0
$963$	6.12464	0.197364
$964$	0	0
$965$	−7.53697	−0.242624
$966$	0	0
$967$	5.05212	0.162465	0.0812326	−	0.996695i	$-0.474114\pi$
$967$	5.05212	0.162465	0.0812326	+	0.996695i	$0.474114\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−60.6866	−1.94753	−0.973763	−	0.227566i	$-0.926923\pi$
$971$	−60.6866	−1.94753	−0.973763	+	0.227566i	$0.926923\pi$
$972$	0	0
$973$	9.74681	0.312468
$974$	0	0
$975$	1.15070	0.0368519
$976$	0	0
$977$	−8.68190	−0.277759	−0.138879	−	0.990309i	$-0.544350\pi$
$977$	−8.68190	−0.277759	−0.138879	+	0.990309i	$0.544350\pi$
$978$	0	0
$979$	20.0955	0.642255
$980$	0	0
$981$	11.7380	0.374766
$982$	0	0
$983$	26.7276	0.852477	0.426239	−	0.904611i	$-0.359838\pi$
$983$	26.7276	0.852477	0.426239	+	0.904611i	$0.359838\pi$
$984$	0	0
$985$	−8.33217	−0.265485
$986$	0	0
$987$	2.68925	0.0855997
$988$	0	0
$989$	−10.0193	−0.318597
$990$	0	0
$991$	−43.0587	−1.36780	−0.683902	−	0.729574i	$-0.739718\pi$
$991$	−43.0587	−1.36780	−0.683902	+	0.729574i	$0.739718\pi$
$992$	0	0
$993$	−13.6345	−0.432677
$994$	0	0
$995$	9.99436	0.316843
$996$	0	0
$997$	−28.4278	−0.900318	−0.450159	−	0.892948i	$-0.648633\pi$
$997$	−28.4278	−0.900318	−0.450159	+	0.892948i	$0.648633\pi$
$998$	0	0
$999$	17.7462	0.561464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.z.1.9		20
17.10	odd	16		476.2.u.a.253.5	✓	40
17.12	odd	16		476.2.u.a.365.5	yes	40
17.16	even	2		8092.2.a.y.1.12		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.u.a.253.5	✓	40	17.10	odd	16
476.2.u.a.365.5	yes	40	17.12	odd	16
8092.2.a.y.1.12		20	17.16	even	2
8092.2.a.z.1.9		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)

\(f(q)\)	\(=\)	\(q+0.411306 q^{3} -0.830159 q^{5} +1.00000 q^{7} -2.83083 q^{9} -2.58028 q^{11} -0.648987 q^{13} -0.341449 q^{15} -6.04733 q^{19} +0.411306 q^{21} -2.44561 q^{23} -4.31084 q^{25} -2.39825 q^{27} -8.15224 q^{29} -0.704384 q^{31} -1.06128 q^{33} -0.830159 q^{35} -7.39962 q^{37} -0.266932 q^{39} +7.97071 q^{41} +4.09687 q^{43} +2.35004 q^{45} +6.53831 q^{47} +1.00000 q^{49} +11.8733 q^{53} +2.14204 q^{55} -2.48730 q^{57} -4.49372 q^{59} +10.9567 q^{61} -2.83083 q^{63} +0.538763 q^{65} +9.41150 q^{67} -1.00589 q^{69} +14.3610 q^{71} +12.5276 q^{73} -1.77307 q^{75} -2.58028 q^{77} +5.31955 q^{79} +7.50607 q^{81} +3.59081 q^{83} -3.35306 q^{87} -7.78812 q^{89} -0.648987 q^{91} -0.289717 q^{93} +5.02025 q^{95} +12.7068 q^{97} +7.30432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 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

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Database

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