LMFDB - Embedded newform 8208.2.a.bx.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] 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("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,1,0,-1,0,0,0,-3] 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:	$65.5412099791$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.88980.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 12 $ x^4 - x^3 - 10*x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.10163$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.10163 q^{5} -1.10163 q^{7} -5.34485 q^{11} +4.98967 q^{13} +2.00000 q^{17} -1.00000 q^{19} +2.59840 q^{23} -3.78641 q^{25} +4.43124 q^{29} -8.58807 q^{31} -1.21359 q^{35} -2.18802 q^{37} +5.15683 q^{41} +2.59840 q^{43} -2.74645 q^{47} -5.78641 q^{49} -9.43615 q^{53} -5.88804 q^{55} +3.88804 q^{59} +2.30489 q^{61} +5.49677 q^{65} +5.87771 q^{67} -9.70003 q^{71} -8.71527 q^{73} +5.88804 q^{77} -3.07606 q^{79} +12.2226 q^{83} +2.20326 q^{85} +15.2176 q^{89} -5.49677 q^{91} -1.10163 q^{95} -6.80166 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5} - q^{7} - 3 q^{11} - 3 q^{13} + 8 q^{17} - 4 q^{19} + q^{25} - 3 q^{29} - q^{31} - 21 q^{35} - 3 q^{37} + 8 q^{41} - 3 q^{47} - 7 q^{49} + 7 q^{53} - 4 q^{55} - 4 q^{59} - q^{61} + 15 q^{65}+ \cdots - 10 q^{97}+O(q^{100}) $ 4 * q + q^5 - q^7 - 3 * q^11 - 3 * q^13 + 8 * q^17 - 4 * q^19 + q^25 - 3 * q^29 - q^31 - 21 * q^35 - 3 * q^37 + 8 * q^41 - 3 * q^47 - 7 * q^49 + 7 * q^53 - 4 * q^55 - 4 * q^59 - q^61 + 15 * q^65 - 19 * q^67 - 25 * q^71 - 20 * q^73 + 4 * q^77 + 13 * q^79 - 12 * q^83 + 2 * q^85 + 24 * q^89 - 15 * q^91 - q^95 - 10 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	1.10163	0.492664	0.246332	−	0.969186i	$-0.420775\pi$
$5$	1.10163	0.492664	0.246332	+	0.969186i	$0.420775\pi$
$6$	0	0
$7$	−1.10163	−0.416377	−0.208188	−	0.978089i	$-0.566757\pi$
$7$	−1.10163	−0.416377	−0.208188	+	0.978089i	$0.566757\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.34485	−1.61153	−0.805766	−	0.592234i	$-0.798246\pi$
$11$	−5.34485	−1.61153	−0.805766	+	0.592234i	$0.798246\pi$
$12$	0	0
$13$	4.98967	1.38389	0.691943	−	0.721952i	$-0.256755\pi$
$13$	4.98967	1.38389	0.691943	+	0.721952i	$0.256755\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.59840	0.541803	0.270902	−	0.962607i	$-0.412678\pi$
$23$	2.59840	0.541803	0.270902	+	0.962607i	$0.412678\pi$
$24$	0	0
$25$	−3.78641	−0.757283
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.43124	0.822860	0.411430	−	0.911441i	$-0.365030\pi$
$29$	4.43124	0.822860	0.411430	+	0.911441i	$0.365030\pi$
$30$	0	0
$31$	−8.58807	−1.54246	−0.771231	−	0.636555i	$-0.780359\pi$
$31$	−8.58807	−1.54246	−0.771231	+	0.636555i	$0.780359\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.21359	−0.205134
$36$	0	0
$37$	−2.18802	−0.359708	−0.179854	−	0.983693i	$-0.557562\pi$
$37$	−2.18802	−0.359708	−0.179854	+	0.983693i	$0.557562\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.15683	0.805362	0.402681	−	0.915340i	$-0.368078\pi$
$41$	5.15683	0.805362	0.402681	+	0.915340i	$0.368078\pi$
$42$	0	0
$43$	2.59840	0.396252	0.198126	−	0.980177i	$-0.436514\pi$
$43$	2.59840	0.396252	0.198126	+	0.980177i	$0.436514\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.74645	−0.400611	−0.200306	−	0.979733i	$-0.564193\pi$
$47$	−2.74645	−0.400611	−0.200306	+	0.979733i	$0.564193\pi$
$48$	0	0
$49$	−5.78641	−0.826630
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.43615	−1.29615	−0.648077	−	0.761574i	$-0.724427\pi$
$53$	−9.43615	−1.29615	−0.648077	+	0.761574i	$0.724427\pi$
$54$	0	0
$55$	−5.88804	−0.793943
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.88804	0.506180	0.253090	−	0.967443i	$-0.418553\pi$
$59$	3.88804	0.506180	0.253090	+	0.967443i	$0.418553\pi$
$60$	0	0
$61$	2.30489	0.295111	0.147555	−	0.989054i	$-0.452860\pi$
$61$	2.30489	0.295111	0.147555	+	0.989054i	$0.452860\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.49677	0.681790
$66$	0	0
$67$	5.87771	0.718077	0.359039	−	0.933323i	$-0.383105\pi$
$67$	5.87771	0.718077	0.359039	+	0.933323i	$0.383105\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.70003	−1.15118	−0.575591	−	0.817738i	$-0.695228\pi$
$71$	−9.70003	−1.15118	−0.575591	+	0.817738i	$0.695228\pi$
$72$	0	0
$73$	−8.71527	−1.02005	−0.510023	−	0.860161i	$-0.670363\pi$
$73$	−8.71527	−1.02005	−0.510023	+	0.860161i	$0.670363\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.88804	0.671005
$78$	0	0
$79$	−3.07606	−0.346084	−0.173042	−	0.984914i	$-0.555360\pi$
$79$	−3.07606	−0.346084	−0.173042	+	0.984914i	$0.555360\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.2226	1.34160	0.670800	−	0.741638i	$-0.265951\pi$
$83$	12.2226	1.34160	0.670800	+	0.741638i	$0.265951\pi$
$84$	0	0
$85$	2.20326	0.238977
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.2176	1.61307	0.806534	−	0.591188i	$-0.201341\pi$
$89$	15.2176	1.61307	0.806534	+	0.591188i	$0.201341\pi$
$90$	0	0
$91$	−5.49677	−0.576218
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.10163	−0.113025
$96$	0	0
$97$	−6.80166	−0.690604	−0.345302	−	0.938492i	$-0.612223\pi$
$97$	−6.80166	−0.690604	−0.345302	+	0.938492i	$0.612223\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.5674	−1.35001	−0.675004	−	0.737814i	$-0.735858\pi$
$101$	−13.5674	−1.35001	−0.675004	+	0.737814i	$0.735858\pi$
$102$	0	0
$103$	10.8930	1.07331	0.536657	−	0.843800i	$-0.319687\pi$
$103$	10.8930	1.07331	0.536657	+	0.843800i	$0.319687\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.8880	−1.14926	−0.574630	−	0.818413i	$-0.694854\pi$
$107$	−11.8880	−1.14926	−0.574630	+	0.818413i	$0.694854\pi$
$108$	0	0
$109$	−17.7913	−1.70410	−0.852050	−	0.523460i	$-0.824641\pi$
$109$	−17.7913	−1.70410	−0.852050	+	0.523460i	$0.824641\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.55844	0.240677	0.120339	−	0.992733i	$-0.461602\pi$
$113$	2.55844	0.240677	0.120339	+	0.992733i	$0.461602\pi$
$114$	0	0
$115$	2.86247	0.266927
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.20326	−0.201972
$120$	0	0
$121$	17.5674	1.59704
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.67937	−0.865749
$126$	0	0
$127$	15.1761	1.34666	0.673332	−	0.739340i	$-0.264862\pi$
$127$	15.1761	1.34666	0.673332	+	0.739340i	$0.264862\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.47120	−0.740132	−0.370066	−	0.929005i	$-0.620665\pi$
$131$	−8.47120	−0.740132	−0.370066	+	0.929005i	$0.620665\pi$
$132$	0	0
$133$	1.10163	0.0955234
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.36957	0.800496	0.400248	−	0.916407i	$-0.368924\pi$
$137$	9.36957	0.800496	0.400248	+	0.916407i	$0.368924\pi$
$138$	0	0
$139$	−1.59348	−0.135157	−0.0675787	−	0.997714i	$-0.521527\pi$
$139$	−1.59348	−0.135157	−0.0675787	+	0.997714i	$0.521527\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−26.6690	−2.23018
$144$	0	0
$145$	4.88158	0.405393
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.410382	−0.0336198	−0.0168099	−	0.999859i	$-0.505351\pi$
$149$	−0.410382	−0.0336198	−0.0168099	+	0.999859i	$0.505351\pi$
$150$	0	0
$151$	−18.5865	−1.51255	−0.756275	−	0.654254i	$-0.772983\pi$
$151$	−18.5865	−1.51255	−0.756275	+	0.654254i	$0.772983\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.46087	−0.759915
$156$	0	0
$157$	5.87771	0.469093	0.234546	−	0.972105i	$-0.424640\pi$
$157$	5.87771	0.469093	0.234546	+	0.972105i	$0.424640\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.86247	−0.225594
$162$	0	0
$163$	−7.40160	−0.579738	−0.289869	−	0.957066i	$-0.593612\pi$
$163$	−7.40160	−0.579738	−0.289869	+	0.957066i	$0.593612\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.78641	0.215619	0.107810	−	0.994172i	$-0.465616\pi$
$167$	2.78641	0.215619	0.107810	+	0.994172i	$0.465616\pi$
$168$	0	0
$169$	11.8968	0.915140
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.83284	−0.139348	−0.0696740	−	0.997570i	$-0.522196\pi$
$173$	−1.83284	−0.139348	−0.0696740	+	0.997570i	$0.522196\pi$
$174$	0	0
$175$	4.17122	0.315315
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.9744	−1.11924	−0.559621	−	0.828749i	$-0.689053\pi$
$179$	−14.9744	−1.11924	−0.559621	+	0.828749i	$0.689053\pi$
$180$	0	0
$181$	−8.39127	−0.623718	−0.311859	−	0.950128i	$-0.600952\pi$
$181$	−8.39127	−0.623718	−0.311859	+	0.950128i	$0.600952\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.41038	−0.177215
$186$	0	0
$187$	−10.6897	−0.781708
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.2585	−1.03171	−0.515853	−	0.856677i	$-0.672525\pi$
$191$	−14.2585	−1.03171	−0.515853	+	0.856677i	$0.672525\pi$
$192$	0	0
$193$	−23.6755	−1.70420	−0.852100	−	0.523379i	$-0.824671\pi$
$193$	−23.6755	−1.70420	−0.852100	+	0.523379i	$0.824671\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.1761	1.65123	0.825616	−	0.564233i	$-0.190828\pi$
$197$	23.1761	1.65123	0.825616	+	0.564233i	$0.190828\pi$
$198$	0	0
$199$	−18.2625	−1.29460	−0.647298	−	0.762237i	$-0.724101\pi$
$199$	−18.2625	−1.29460	−0.647298	+	0.762237i	$0.724101\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.88158	−0.342620
$204$	0	0
$205$	5.68092	0.396773
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.34485	0.369711
$210$	0	0
$211$	−1.56791	−0.107939	−0.0539697	−	0.998543i	$-0.517187\pi$
$211$	−1.56791	−0.107939	−0.0539697	+	0.998543i	$0.517187\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.86247	0.195219
$216$	0	0
$217$	9.46087	0.642246
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.97934	0.671283
$222$	0	0
$223$	5.20712	0.348695	0.174347	−	0.984684i	$-0.444218\pi$
$223$	5.20712	0.348695	0.174347	+	0.984684i	$0.444218\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.486440	0.0322861	0.0161431	−	0.999870i	$-0.494861\pi$
$227$	0.486440	0.0322861	0.0161431	+	0.999870i	$0.494861\pi$
$228$	0	0
$229$	−22.5066	−1.48728	−0.743639	−	0.668581i	$-0.766902\pi$
$229$	−22.5066	−1.48728	−0.743639	+	0.668581i	$0.766902\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.8738	1.23647	0.618233	−	0.785995i	$-0.287849\pi$
$233$	18.8738	1.23647	0.618233	+	0.785995i	$0.287849\pi$
$234$	0	0
$235$	−3.02557	−0.197367
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.8122	0.764066	0.382033	−	0.924149i	$-0.375224\pi$
$239$	11.8122	0.764066	0.382033	+	0.924149i	$0.375224\pi$
$240$	0	0
$241$	−16.7810	−1.08096	−0.540480	−	0.841357i	$-0.681757\pi$
$241$	−16.7810	−1.08096	−0.540480	+	0.841357i	$0.681757\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.37448	−0.407251
$246$	0	0
$247$	−4.98967	−0.317485
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.27777	0.270010	0.135005	−	0.990845i	$-0.456895\pi$
$251$	4.27777	0.270010	0.135005	+	0.990845i	$0.456895\pi$
$252$	0	0
$253$	−13.8880	−0.873134
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.28860	−0.392272	−0.196136	−	0.980577i	$-0.562839\pi$
$257$	−6.28860	−0.392272	−0.196136	+	0.980577i	$0.562839\pi$
$258$	0	0
$259$	2.41038	0.149774
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.2443	−1.06333	−0.531664	−	0.846956i	$-0.678433\pi$
$263$	−17.2443	−1.06333	−0.531664	+	0.846956i	$0.678433\pi$
$264$	0	0
$265$	−10.3951	−0.638568
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.99509	0.182614	0.0913068	−	0.995823i	$-0.470896\pi$
$269$	2.99509	0.182614	0.0913068	+	0.995823i	$0.470896\pi$
$270$	0	0
$271$	−12.1218	−0.736346	−0.368173	−	0.929757i	$-0.620017\pi$
$271$	−12.1218	−0.736346	−0.368173	+	0.929757i	$0.620017\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.2378	1.22039
$276$	0	0
$277$	15.5930	0.936892	0.468446	−	0.883492i	$-0.344814\pi$
$277$	15.5930	0.936892	0.468446	+	0.883492i	$0.344814\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.4011	−1.33634	−0.668169	−	0.744010i	$-0.732922\pi$
$281$	−22.4011	−1.33634	−0.668169	+	0.744010i	$0.732922\pi$
$282$	0	0
$283$	−16.2832	−0.967935	−0.483967	−	0.875086i	$-0.660805\pi$
$283$	−16.2832	−0.967935	−0.483967	+	0.875086i	$0.660805\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.68092	−0.335334
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.09045	−0.531070	−0.265535	−	0.964101i	$-0.585549\pi$
$293$	−9.09045	−0.531070	−0.265535	+	0.964101i	$0.585549\pi$
$294$	0	0
$295$	4.28318	0.249376
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.9651	0.749794
$300$	0	0
$301$	−2.86247	−0.164990
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.53913	0.145390
$306$	0	0
$307$	−26.6033	−1.51833	−0.759166	−	0.650898i	$-0.774393\pi$
$307$	−26.6033	−1.51833	−0.759166	+	0.650898i	$0.774393\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.7307	1.11883	0.559413	−	0.828889i	$-0.311027\pi$
$311$	19.7307	1.11883	0.559413	+	0.828889i	$0.311027\pi$
$312$	0	0
$313$	−16.5979	−0.938169	−0.469084	−	0.883153i	$-0.655416\pi$
$313$	−16.5979	−0.938169	−0.469084	+	0.883153i	$0.655416\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.2946	0.578200	0.289100	−	0.957299i	$-0.406644\pi$
$317$	10.2946	0.578200	0.289100	+	0.957299i	$0.406644\pi$
$318$	0	0
$319$	−23.6843	−1.32607
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	−18.8930	−1.04799
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.02557	0.166805
$330$	0	0
$331$	7.19293	0.395359	0.197680	−	0.980267i	$-0.436659\pi$
$331$	7.19293	0.395359	0.197680	+	0.980267i	$0.436659\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.47506	0.353770
$336$	0	0
$337$	−33.7234	−1.83703	−0.918515	−	0.395386i	$-0.870611\pi$
$337$	−33.7234	−1.83703	−0.918515	+	0.395386i	$0.870611\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	45.9019	2.48573
$342$	0	0
$343$	14.0859	0.760566
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.7332	−1.59616	−0.798081	−	0.602550i	$-0.794152\pi$
$347$	−29.7332	−1.59616	−0.798081	+	0.602550i	$0.794152\pi$
$348$	0	0
$349$	8.42667	0.451070	0.225535	−	0.974235i	$-0.427587\pi$
$349$	8.42667	0.451070	0.225535	+	0.974235i	$0.427587\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.1826	0.754864	0.377432	−	0.926037i	$-0.376807\pi$
$353$	14.1826	0.754864	0.377432	+	0.926037i	$0.376807\pi$
$354$	0	0
$355$	−10.6858	−0.567145
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0848	−1.53504	−0.767519	−	0.641026i	$-0.778509\pi$
$359$	−29.0848	−1.53504	−0.767519	+	0.641026i	$0.778509\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.60100	−0.502539
$366$	0	0
$367$	26.4924	1.38289	0.691446	−	0.722428i	$-0.256974\pi$
$367$	26.4924	1.38289	0.691446	+	0.722428i	$0.256974\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.3951	0.539689
$372$	0	0
$373$	−20.7299	−1.07335	−0.536676	−	0.843789i	$-0.680320\pi$
$373$	−20.7299	−1.07335	−0.536676	+	0.843789i	$0.680320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.1104	1.13874
$378$	0	0
$379$	−23.2755	−1.19558	−0.597790	−	0.801653i	$-0.703954\pi$
$379$	−23.2755	−1.19558	−0.597790	+	0.801653i	$0.703954\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	29.3186	1.49811	0.749055	−	0.662508i	$-0.230508\pi$
$383$	29.3186	1.49811	0.749055	+	0.662508i	$0.230508\pi$
$384$	0	0
$385$	6.48644	0.330580
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.6533	1.30067	0.650337	−	0.759646i	$-0.274628\pi$
$389$	25.6533	1.30067	0.650337	+	0.759646i	$0.274628\pi$
$390$	0	0
$391$	5.19679	0.262813
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.38868	−0.170503
$396$	0	0
$397$	2.85214	0.143145	0.0715725	−	0.997435i	$-0.477198\pi$
$397$	2.85214	0.143145	0.0715725	+	0.997435i	$0.477198\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.73593	−0.286439	−0.143219	−	0.989691i	$-0.545745\pi$
$401$	−5.73593	−0.286439	−0.143219	+	0.989691i	$0.545745\pi$
$402$	0	0
$403$	−42.8516	−2.13459
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.6946	0.579680
$408$	0	0
$409$	3.83690	0.189722	0.0948612	−	0.995491i	$-0.469759\pi$
$409$	3.83690	0.189722	0.0948612	+	0.995491i	$0.469759\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.28318	−0.210762
$414$	0	0
$415$	13.4647	0.660958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.3802	0.995640	0.497820	−	0.867280i	$-0.334134\pi$
$419$	20.3802	0.995640	0.497820	+	0.867280i	$0.334134\pi$
$420$	0	0
$421$	4.75979	0.231978	0.115989	−	0.993250i	$-0.462996\pi$
$421$	4.75979	0.231978	0.115989	+	0.993250i	$0.462996\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.57283	−0.367336
$426$	0	0
$427$	−2.53913	−0.122877
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.19138	0.346397	0.173198	−	0.984887i	$-0.444590\pi$
$431$	7.19138	0.346397	0.173198	+	0.984887i	$0.444590\pi$
$432$	0	0
$433$	3.03540	0.145872	0.0729360	−	0.997337i	$-0.476763\pi$
$433$	3.03540	0.145872	0.0729360	+	0.997337i	$0.476763\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.59840	−0.124298
$438$	0	0
$439$	38.1516	1.82088	0.910439	−	0.413644i	$-0.135744\pi$
$439$	38.1516	1.82088	0.910439	+	0.413644i	$0.135744\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.02577	−0.333804	−0.166902	−	0.985973i	$-0.553376\pi$
$443$	−7.02577	−0.333804	−0.166902	+	0.985973i	$0.553376\pi$
$444$	0	0
$445$	16.7642	0.794700
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.3394	1.29023	0.645114	−	0.764086i	$-0.276810\pi$
$449$	27.3394	1.29023	0.645114	+	0.764086i	$0.276810\pi$
$450$	0	0
$451$	−27.5625	−1.29787
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.05540	−0.283882
$456$	0	0
$457$	32.7332	1.53120	0.765598	−	0.643320i	$-0.222443\pi$
$457$	32.7332	1.53120	0.765598	+	0.643320i	$0.222443\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.6489	−0.728841	−0.364421	−	0.931234i	$-0.618733\pi$
$461$	−15.6489	−0.728841	−0.364421	+	0.931234i	$0.618733\pi$
$462$	0	0
$463$	−5.80657	−0.269854	−0.134927	−	0.990856i	$-0.543080\pi$
$463$	−5.80657	−0.269854	−0.134927	+	0.990856i	$0.543080\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.4954	1.17979	0.589894	−	0.807481i	$-0.299170\pi$
$467$	25.4954	1.17979	0.589894	+	0.807481i	$0.299170\pi$
$468$	0	0
$469$	−6.47506	−0.298991
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−13.8880	−0.638573
$474$	0	0
$475$	3.78641	0.173733
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.8643	−1.54730	−0.773649	−	0.633614i	$-0.781571\pi$
$479$	−33.8643	−1.54730	−0.773649	+	0.633614i	$0.781571\pi$
$480$	0	0
$481$	−10.9175	−0.497794
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.49290	−0.340235
$486$	0	0
$487$	−15.6745	−0.710277	−0.355139	−	0.934814i	$-0.615566\pi$
$487$	−15.6745	−0.710277	−0.355139	+	0.934814i	$0.615566\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.9634	−1.03632	−0.518162	−	0.855283i	$-0.673384\pi$
$491$	−22.9634	−1.03632	−0.518162	+	0.855283i	$0.673384\pi$
$492$	0	0
$493$	8.86247	0.399146
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.6858	0.479325
$498$	0	0
$499$	−20.7202	−0.927563	−0.463781	−	0.885950i	$-0.653508\pi$
$499$	−20.7202	−0.927563	−0.463781	+	0.885950i	$0.653508\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.1003	1.47587	0.737934	−	0.674873i	$-0.235801\pi$
$503$	33.1003	1.47587	0.737934	+	0.674873i	$0.235801\pi$
$504$	0	0
$505$	−14.9463	−0.665100
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.9802	−0.929931	−0.464965	−	0.885329i	$-0.653933\pi$
$509$	−20.9802	−0.929931	−0.464965	+	0.885329i	$0.653933\pi$
$510$	0	0
$511$	9.60100	0.424723
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	14.6794	0.645598
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.5609	−0.637927	−0.318963	−	0.947767i	$-0.603335\pi$
$521$	−14.5609	−0.637927	−0.318963	+	0.947767i	$0.603335\pi$
$522$	0	0
$523$	−11.1880	−0.489218	−0.244609	−	0.969622i	$-0.578660\pi$
$523$	−11.1880	−0.489218	−0.244609	+	0.969622i	$0.578660\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.1761	−0.748204
$528$	0	0
$529$	−16.2483	−0.706449
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25.7309	1.11453
$534$	0	0
$535$	−13.0962	−0.566199
$536$	0	0
$537$	0	0
$538$	0	0
$539$	30.9275	1.33214
$540$	0	0
$541$	−25.9272	−1.11470	−0.557348	−	0.830279i	$-0.688181\pi$
$541$	−25.9272	−1.11470	−0.557348	+	0.830279i	$0.688181\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.5994	−0.839548
$546$	0	0
$547$	28.4490	1.21639	0.608195	−	0.793787i	$-0.291894\pi$
$547$	28.4490	1.21639	0.608195	+	0.793787i	$0.291894\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.43124	−0.188777
$552$	0	0
$553$	3.38868	0.144101
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.84076	0.416967	0.208483	−	0.978026i	$-0.433147\pi$
$557$	9.84076	0.416967	0.208483	+	0.978026i	$0.433147\pi$
$558$	0	0
$559$	12.9651	0.548367
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.8738	1.04831	0.524154	−	0.851624i	$-0.324382\pi$
$563$	24.8738	1.04831	0.524154	+	0.851624i	$0.324382\pi$
$564$	0	0
$565$	2.81845	0.118573
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.3903	−1.94478	−0.972391	−	0.233359i	$-0.925028\pi$
$569$	−46.3903	−1.94478	−0.972391	+	0.233359i	$0.925028\pi$
$570$	0	0
$571$	−3.04452	−0.127409	−0.0637046	−	0.997969i	$-0.520292\pi$
$571$	−3.04452	−0.127409	−0.0637046	+	0.997969i	$0.520292\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.83861	−0.410298
$576$	0	0
$577$	24.8684	1.03529	0.517643	−	0.855597i	$-0.326809\pi$
$577$	24.8684	1.03529	0.517643	+	0.855597i	$0.326809\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−13.4647	−0.558611
$582$	0	0
$583$	50.4348	2.08880
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.02171	0.124719	0.0623596	−	0.998054i	$-0.480137\pi$
$587$	3.02171	0.124719	0.0623596	+	0.998054i	$0.480137\pi$
$588$	0	0
$589$	8.58807	0.353865
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.2370	0.789967	0.394983	−	0.918688i	$-0.370750\pi$
$593$	19.2370	0.789967	0.394983	+	0.918688i	$0.370750\pi$
$594$	0	0
$595$	−2.42717	−0.0995044
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16.2087	0.662268	0.331134	−	0.943584i	$-0.392569\pi$
$599$	16.2087	0.662268	0.331134	+	0.943584i	$0.392569\pi$
$600$	0	0
$601$	5.49445	0.224123	0.112062	−	0.993701i	$-0.464255\pi$
$601$	5.49445	0.224123	0.112062	+	0.993701i	$0.464255\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.3528	0.786802
$606$	0	0
$607$	−6.47766	−0.262920	−0.131460	−	0.991321i	$-0.541967\pi$
$607$	−6.47766	−0.262920	−0.131460	+	0.991321i	$0.541967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.7039	−0.554400
$612$	0	0
$613$	12.6690	0.511698	0.255849	−	0.966717i	$-0.417645\pi$
$613$	12.6690	0.511698	0.255849	+	0.966717i	$0.417645\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.6912	−0.671964	−0.335982	−	0.941868i	$-0.609068\pi$
$617$	−16.6912	−0.671964	−0.335982	+	0.941868i	$0.609068\pi$
$618$	0	0
$619$	−35.2354	−1.41623	−0.708115	−	0.706097i	$-0.750454\pi$
$619$	−35.2354	−1.41623	−0.708115	+	0.706097i	$0.750454\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.7642	−0.671644
$624$	0	0
$625$	8.26899	0.330760
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.37603	−0.174484
$630$	0	0
$631$	−34.8555	−1.38758	−0.693788	−	0.720179i	$-0.744059\pi$
$631$	−34.8555	−1.38758	−0.693788	+	0.720179i	$0.744059\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.7185	0.663453
$636$	0	0
$637$	−28.8723	−1.14396
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−47.0628	−1.85887	−0.929435	−	0.368986i	$-0.879705\pi$
$641$	−47.0628	−1.85887	−0.929435	+	0.368986i	$0.879705\pi$
$642$	0	0
$643$	−29.8979	−1.17906	−0.589528	−	0.807748i	$-0.700686\pi$
$643$	−29.8979	−1.17906	−0.589528	+	0.807748i	$0.700686\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.84703	−0.269185	−0.134592	−	0.990901i	$-0.542972\pi$
$647$	−6.84703	−0.269185	−0.134592	+	0.990901i	$0.542972\pi$
$648$	0	0
$649$	−20.7810	−0.815726
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.82336	−0.306152	−0.153076	−	0.988214i	$-0.548918\pi$
$653$	−7.82336	−0.306152	−0.153076	+	0.988214i	$0.548918\pi$
$654$	0	0
$655$	−9.33212	−0.364636
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.4978	0.642664	0.321332	−	0.946967i	$-0.395870\pi$
$659$	16.4978	0.642664	0.321332	+	0.946967i	$0.395870\pi$
$660$	0	0
$661$	33.7593	1.31308	0.656542	−	0.754289i	$-0.272018\pi$
$661$	33.7593	1.31308	0.656542	+	0.754289i	$0.272018\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.21359	0.0470609
$666$	0	0
$667$	11.5141	0.445828
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.3193	−0.475580
$672$	0	0
$673$	−36.2517	−1.39740	−0.698700	−	0.715415i	$-0.746238\pi$
$673$	−36.2517	−1.39740	−0.698700	+	0.715415i	$0.746238\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.34570	0.167019	0.0835095	−	0.996507i	$-0.473387\pi$
$677$	4.34570	0.167019	0.0835095	+	0.996507i	$0.473387\pi$
$678$	0	0
$679$	7.49290	0.287551
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−41.6533	−1.59382	−0.796910	−	0.604099i	$-0.793533\pi$
$683$	−41.6533	−1.59382	−0.796910	+	0.604099i	$0.793533\pi$
$684$	0	0
$685$	10.3218	0.394375
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−47.0833	−1.79373
$690$	0	0
$691$	0.598397	0.0227641	0.0113821	−	0.999935i	$-0.496377\pi$
$691$	0.598397	0.0227641	0.0113821	+	0.999935i	$0.496377\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.75543	−0.0665872
$696$	0	0
$697$	10.3137	0.390658
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.9773	−1.39661	−0.698307	−	0.715799i	$-0.746063\pi$
$701$	−36.9773	−1.39661	−0.698307	+	0.715799i	$0.746063\pi$
$702$	0	0
$703$	2.18802	0.0825226
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.9463	0.562112
$708$	0	0
$709$	13.7456	0.516227	0.258113	−	0.966115i	$-0.416899\pi$
$709$	13.7456	0.516227	0.258113	+	0.966115i	$0.416899\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−22.3152	−0.835711
$714$	0	0
$715$	−29.3794	−1.09873
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.6154	−0.843412	−0.421706	−	0.906733i	$-0.638568\pi$
$719$	−22.6154	−0.843412	−0.421706	+	0.906733i	$0.638568\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−16.7785	−0.623137
$726$	0	0
$727$	24.9995	0.927180	0.463590	−	0.886050i	$-0.346561\pi$
$727$	24.9995	0.927180	0.463590	+	0.886050i	$0.346561\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.19679	0.192210
$732$	0	0
$733$	−42.1707	−1.55761	−0.778806	−	0.627265i	$-0.784174\pi$
$733$	−42.1707	−1.55761	−0.778806	+	0.627265i	$0.784174\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.4155	−1.15720
$738$	0	0
$739$	34.4436	1.26703	0.633514	−	0.773731i	$-0.281612\pi$
$739$	34.4436	1.26703	0.633514	+	0.773731i	$0.281612\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.9527	1.53910	0.769548	−	0.638589i	$-0.220482\pi$
$743$	41.9527	1.53910	0.769548	+	0.638589i	$0.220482\pi$
$744$	0	0
$745$	−0.452089	−0.0165633
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.0962	0.478525
$750$	0	0
$751$	52.2083	1.90511	0.952554	−	0.304371i	$-0.0984462\pi$
$751$	52.2083	1.90511	0.952554	+	0.304371i	$0.0984462\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−20.4755	−0.745178
$756$	0	0
$757$	−16.1929	−0.588542	−0.294271	−	0.955722i	$-0.595077\pi$
$757$	−16.1929	−0.588542	−0.294271	+	0.955722i	$0.595077\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.4529	1.24892	0.624458	−	0.781059i	$-0.285320\pi$
$761$	34.4529	1.24892	0.624458	+	0.781059i	$0.285320\pi$
$762$	0	0
$763$	19.5994	0.709548
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.4001	0.700495
$768$	0	0
$769$	37.6251	1.35680	0.678398	−	0.734694i	$-0.262674\pi$
$769$	37.6251	1.35680	0.678398	+	0.734694i	$0.262674\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.1976	0.510654	0.255327	−	0.966855i	$-0.417817\pi$
$773$	14.1976	0.510654	0.255327	+	0.966855i	$0.417817\pi$
$774$	0	0
$775$	32.5180	1.16808
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.15683	−0.184763
$780$	0	0
$781$	51.8452	1.85517
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.47506	0.231105
$786$	0	0
$787$	−4.15493	−0.148107	−0.0740536	−	0.997254i	$-0.523594\pi$
$787$	−4.15493	−0.148107	−0.0740536	+	0.997254i	$0.523594\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.81845	−0.100212
$792$	0	0
$793$	11.5006	0.408399
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.4856	−0.761059	−0.380529	−	0.924769i	$-0.624258\pi$
$797$	−21.4856	−0.761059	−0.380529	+	0.924769i	$0.624258\pi$
$798$	0	0
$799$	−5.49290	−0.194325
$800$	0	0
$801$	0	0
$802$	0	0
$803$	46.5818	1.64384
$804$	0	0
$805$	−3.15338	−0.111142
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−15.3711	−0.540420	−0.270210	−	0.962801i	$-0.587093\pi$
$809$	−15.3711	−0.540420	−0.270210	+	0.962801i	$0.587093\pi$
$810$	0	0
$811$	−37.2229	−1.30707	−0.653537	−	0.756895i	$-0.726715\pi$
$811$	−37.2229	−1.30707	−0.653537	+	0.756895i	$0.726715\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.15382	−0.285616
$816$	0	0
$817$	−2.59840	−0.0909064
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.3164	−1.47685	−0.738426	−	0.674335i	$-0.764431\pi$
$821$	−42.3164	−1.47685	−0.738426	+	0.674335i	$0.764431\pi$
$822$	0	0
$823$	18.6082	0.648642	0.324321	−	0.945947i	$-0.394864\pi$
$823$	18.6082	0.648642	0.324321	+	0.945947i	$0.394864\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.8501	1.49004	0.745022	−	0.667040i	$-0.232439\pi$
$827$	42.8501	1.49004	0.745022	+	0.667040i	$0.232439\pi$
$828$	0	0
$829$	−57.1310	−1.98424	−0.992120	−	0.125290i	$-0.960014\pi$
$829$	−57.1310	−1.98424	−0.992120	+	0.125290i	$0.960014\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11.5728	−0.400975
$834$	0	0
$835$	3.06959	0.106228
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46.0614	1.59021	0.795107	−	0.606469i	$-0.207414\pi$
$839$	46.0614	1.59021	0.795107	+	0.606469i	$0.207414\pi$
$840$	0	0
$841$	−9.36415	−0.322902
$842$	0	0
$843$	0	0
$844$	0	0
$845$	13.1059	0.450856
$846$	0	0
$847$	−19.3528	−0.664969
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.68533	−0.194891
$852$	0	0
$853$	−22.5886	−0.773420	−0.386710	−	0.922201i	$-0.626388\pi$
$853$	−22.5886	−0.773420	−0.386710	+	0.922201i	$0.626388\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.3307	−0.899438	−0.449719	−	0.893170i	$-0.648476\pi$
$857$	−26.3307	−0.899438	−0.449719	+	0.893170i	$0.648476\pi$
$858$	0	0
$859$	41.2246	1.40656	0.703282	−	0.710911i	$-0.251717\pi$
$859$	41.2246	1.40656	0.703282	+	0.710911i	$0.251717\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.1692	−1.06101	−0.530506	−	0.847681i	$-0.677998\pi$
$863$	−31.1692	−1.06101	−0.530506	+	0.847681i	$0.677998\pi$
$864$	0	0
$865$	−2.01911	−0.0686517
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.4411	0.557725
$870$	0	0
$871$	29.3279	0.993737
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.6631	0.360478
$876$	0	0
$877$	−36.7908	−1.24234	−0.621169	−	0.783677i	$-0.713342\pi$
$877$	−36.7908	−1.24234	−0.621169	+	0.783677i	$0.713342\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.5783	1.03021	0.515104	−	0.857127i	$-0.327753\pi$
$881$	30.5783	1.03021	0.515104	+	0.857127i	$0.327753\pi$
$882$	0	0
$883$	−18.2641	−0.614635	−0.307318	−	0.951607i	$-0.599431\pi$
$883$	−18.2641	−0.614635	−0.307318	+	0.951607i	$0.599431\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.8746	1.30528	0.652641	−	0.757667i	$-0.273661\pi$
$887$	38.8746	1.30528	0.652641	+	0.757667i	$0.273661\pi$
$888$	0	0
$889$	−16.7185	−0.560720
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.74645	0.0919065
$894$	0	0
$895$	−16.4963	−0.551409
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38.0558	−1.26923
$900$	0	0
$901$	−18.8723	−0.628727
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.24407	−0.307283
$906$	0	0
$907$	−12.7962	−0.424892	−0.212446	−	0.977173i	$-0.568143\pi$
$907$	−12.7962	−0.424892	−0.212446	+	0.977173i	$0.568143\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15.4059	0.510419	0.255209	−	0.966886i	$-0.417856\pi$
$911$	15.4059	0.510419	0.255209	+	0.966886i	$0.417856\pi$
$912$	0	0
$913$	−65.3278	−2.16203
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.33212	0.308174
$918$	0	0
$919$	−2.18415	−0.0720485	−0.0360242	−	0.999351i	$-0.511469\pi$
$919$	−2.18415	−0.0720485	−0.0360242	+	0.999351i	$0.511469\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−48.3999	−1.59310
$924$	0	0
$925$	8.28473	0.272400
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.05435	0.231446	0.115723	−	0.993282i	$-0.463082\pi$
$929$	7.05435	0.231446	0.115723	+	0.993282i	$0.463082\pi$
$930$	0	0
$931$	5.78641	0.189642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.7761	−0.385119
$936$	0	0
$937$	50.8511	1.66123	0.830617	−	0.556844i	$-0.187988\pi$
$937$	50.8511	1.66123	0.830617	+	0.556844i	$0.187988\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39.3375	−1.28237	−0.641183	−	0.767388i	$-0.721556\pi$
$941$	−39.3375	−1.28237	−0.641183	+	0.767388i	$0.721556\pi$
$942$	0	0
$943$	13.3995	0.436348
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.3105	−0.887472	−0.443736	−	0.896158i	$-0.646347\pi$
$947$	−27.3105	−0.887472	−0.443736	+	0.896158i	$0.646347\pi$
$948$	0	0
$949$	−43.4863	−1.41163
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.9432	−0.646025	−0.323013	−	0.946395i	$-0.604696\pi$
$953$	−19.9432	−0.646025	−0.323013	+	0.946395i	$0.604696\pi$
$954$	0	0
$955$	−15.7075	−0.508284
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.3218	−0.333308
$960$	0	0
$961$	42.7549	1.37919
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.0816	−0.839597
$966$	0	0
$967$	6.20481	0.199533	0.0997666	−	0.995011i	$-0.468190\pi$
$967$	6.20481	0.199533	0.0997666	+	0.995011i	$0.468190\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.4072	−1.52137	−0.760684	−	0.649122i	$-0.775136\pi$
$971$	−47.4072	−1.52137	−0.760684	+	0.649122i	$0.775136\pi$
$972$	0	0
$973$	1.75543	0.0562764
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.3502	1.61084	0.805422	−	0.592702i	$-0.201939\pi$
$977$	50.3502	1.61084	0.805422	+	0.592702i	$0.201939\pi$
$978$	0	0
$979$	−81.3360	−2.59951
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.9245	0.571703	0.285851	−	0.958274i	$-0.407724\pi$
$983$	17.9245	0.571703	0.285851	+	0.958274i	$0.407724\pi$
$984$	0	0
$985$	25.5315	0.813502
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.75167	0.214691
$990$	0	0
$991$	4.66092	0.148059	0.0740295	−	0.997256i	$-0.476414\pi$
$991$	4.66092	0.148059	0.0740295	+	0.997256i	$0.476414\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20.1185	−0.637800
$996$	0	0
$997$	2.63816	0.0835514	0.0417757	−	0.999127i	$-0.486699\pi$
$997$	2.63816	0.0835514	0.0417757	+	0.999127i	$0.486699\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.bx.1.3		4
3.2	odd	2		8208.2.a.bs.1.2		4
4.3	odd	2		4104.2.a.l.1.3	yes	4
12.11	even	2		4104.2.a.k.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.k.1.2	✓	4	12.11	even	2
4104.2.a.l.1.3	yes	4	4.3	odd	2
8208.2.a.bs.1.2		4	3.2	odd	2
8208.2.a.bx.1.3		4	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:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q+1.10163 q^{5} -1.10163 q^{7} -5.34485 q^{11} +4.98967 q^{13} +2.00000 q^{17} -1.00000 q^{19} +2.59840 q^{23} -3.78641 q^{25} +4.43124 q^{29} -8.58807 q^{31} -1.21359 q^{35} -2.18802 q^{37} +5.15683 q^{41} +2.59840 q^{43} -2.74645 q^{47} -5.78641 q^{49} -9.43615 q^{53} -5.88804 q^{55} +3.88804 q^{59} +2.30489 q^{61} +5.49677 q^{65} +5.87771 q^{67} -9.70003 q^{71} -8.71527 q^{73} +5.88804 q^{77} -3.07606 q^{79} +12.2226 q^{83} +2.20326 q^{85} +15.2176 q^{89} -5.49677 q^{91} -1.10163 q^{95} -6.80166 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} - q^{7} - 3 q^{11} - 3 q^{13} + 8 q^{17} - 4 q^{19} + q^{25} - 3 q^{29} - q^{31} - 21 q^{35} - 3 q^{37} + 8 q^{41} - 3 q^{47} - 7 q^{49} + 7 q^{53} - 4 q^{55} - 4 q^{59} - q^{61} + 15 q^{65}+ \cdots - 10 q^{97}+O(q^{100}) \) 4 * q + q^5 - q^7 - 3 * q^11 - 3 * q^13 + 8 * q^17 - 4 * q^19 + q^25 - 3 * q^29 - q^31 - 21 * q^35 - 3 * q^37 + 8 * q^41 - 3 * q^47 - 7 * q^49 + 7 * q^53 - 4 * q^55 - 4 * q^59 - q^61 + 15 * q^65 - 19 * q^67 - 25 * q^71 - 20 * q^73 + 4 * q^77 + 13 * q^79 - 12 * q^83 + 2 * q^85 + 24 * q^89 - 15 * q^91 - q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more