LMFDB - Embedded newform 8512.2.a.bj.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8512,2,Mod(1,8512)] 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("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-1,0,-5,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.39138$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71871 q^{3} +0.391382 q^{5} +1.00000 q^{7} -0.0460370 q^{9} +3.06406 q^{11} -2.78276 q^{13} +0.672673 q^{15} -6.22018 q^{17} -1.00000 q^{19} +1.71871 q^{21} +4.78276 q^{23} -4.84682 q^{25} -5.23525 q^{27} -7.82880 q^{29} -1.34535 q^{31} +5.26622 q^{33} +0.391382 q^{35} +3.82880 q^{37} -4.78276 q^{39} -4.39138 q^{41} +5.82880 q^{43} -0.0180181 q^{45} +5.71871 q^{47} +1.00000 q^{49} -10.6907 q^{51} -7.15613 q^{53} +1.19922 q^{55} -1.71871 q^{57} -7.17415 q^{59} -8.50147 q^{61} -0.0460370 q^{63} -1.08913 q^{65} +12.2202 q^{67} +8.22018 q^{69} -0.935945 q^{71} -4.12811 q^{73} -8.33028 q^{75} +3.06406 q^{77} -2.95396 q^{79} -8.85977 q^{81} -13.5655 q^{83} -2.43447 q^{85} -13.4554 q^{87} +1.06406 q^{89} -2.78276 q^{91} -2.31226 q^{93} -0.391382 q^{95} +4.50147 q^{97} -0.141060 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 5 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} - 3 q^{19} - q^{21} + 2 q^{23} + 4 q^{25} - 28 q^{27} - 5 q^{29} - 4 q^{31} - 15 q^{33} - 5 q^{35} - 7 q^{37} - 2 q^{39}+ \cdots + 55 q^{99}+O(q^{100}) $ 3 * q - q^3 - 5 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 6 * q^17 - 3 * q^19 - q^21 + 2 * q^23 + 4 * q^25 - 28 * q^27 - 5 * q^29 - 4 * q^31 - 15 * q^33 - 5 * q^35 - 7 * q^37 - 2 * q^39 - 7 * q^41 - q^43 + 11 * q^47 + 3 * q^49 - 32 * q^51 - 3 * q^53 + 16 * q^55 + q^57 - 3 * q^59 - 7 * q^61 + 6 * q^63 - 28 * q^65 + 12 * q^67 - 9 * q^71 + 12 * q^75 + 3 * q^77 - 15 * q^79 + 35 * q^81 - 16 * q^83 - 32 * q^85 - 28 * q^87 - 3 * q^89 + 4 * q^91 + 30 * q^93 + 5 * q^95 - 5 * q^97 + 55 * 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
$3$	1.71871	0.992298	0.496149	−	0.868238i	$-0.334747\pi$
$3$	1.71871	0.992298	0.496149	+	0.868238i	$0.334747\pi$
$4$	0	0
$5$	0.391382	0.175032	0.0875158	−	0.996163i	$-0.472107\pi$
$5$	0.391382	0.175032	0.0875158	+	0.996163i	$0.472107\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.0460370	−0.0153457
$10$	0	0
$11$	3.06406	0.923847	0.461924	−	0.886920i	$-0.347159\pi$
$11$	3.06406	0.923847	0.461924	+	0.886920i	$0.347159\pi$
$12$	0	0
$13$	−2.78276	−0.771800	−0.385900	−	0.922541i	$-0.626109\pi$
$13$	−2.78276	−0.771800	−0.385900	+	0.922541i	$0.626109\pi$
$14$	0	0
$15$	0.672673	0.173683
$16$	0	0
$17$	−6.22018	−1.50862	−0.754308	−	0.656521i	$-0.772028\pi$
$17$	−6.22018	−1.50862	−0.754308	+	0.656521i	$0.772028\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.71871	0.375053
$22$	0	0
$23$	4.78276	0.997275	0.498638	−	0.866811i	$-0.333834\pi$
$23$	4.78276	0.997275	0.498638	+	0.866811i	$0.333834\pi$
$24$	0	0
$25$	−4.84682	−0.969364
$26$	0	0
$27$	−5.23525	−1.00752
$28$	0	0
$29$	−7.82880	−1.45377	−0.726886	−	0.686758i	$-0.759033\pi$
$29$	−7.82880	−1.45377	−0.726886	+	0.686758i	$0.759033\pi$
$30$	0	0
$31$	−1.34535	−0.241631	−0.120816	−	0.992675i	$-0.538551\pi$
$31$	−1.34535	−0.241631	−0.120816	+	0.992675i	$0.538551\pi$
$32$	0	0
$33$	5.26622	0.916731
$34$	0	0
$35$	0.391382	0.0661557
$36$	0	0
$37$	3.82880	0.629451	0.314726	−	0.949183i	$-0.398088\pi$
$37$	3.82880	0.629451	0.314726	+	0.949183i	$0.398088\pi$
$38$	0	0
$39$	−4.78276	−0.765855
$40$	0	0
$41$	−4.39138	−0.685819	−0.342909	−	0.939368i	$-0.611412\pi$
$41$	−4.39138	−0.685819	−0.342909	+	0.939368i	$0.611412\pi$
$42$	0	0
$43$	5.82880	0.888884	0.444442	−	0.895808i	$-0.353402\pi$
$43$	5.82880	0.888884	0.444442	+	0.895808i	$0.353402\pi$
$44$	0	0
$45$	−0.0180181	−0.00268598
$46$	0	0
$47$	5.71871	0.834160	0.417080	−	0.908870i	$-0.363054\pi$
$47$	5.71871	0.834160	0.417080	+	0.908870i	$0.363054\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.6907	−1.49700
$52$	0	0
$53$	−7.15613	−0.982970	−0.491485	−	0.870886i	$-0.663546\pi$
$53$	−7.15613	−0.982970	−0.491485	+	0.870886i	$0.663546\pi$
$54$	0	0
$55$	1.19922	0.161702
$56$	0	0
$57$	−1.71871	−0.227649
$58$	0	0
$59$	−7.17415	−0.933994	−0.466997	−	0.884259i	$-0.654664\pi$
$59$	−7.17415	−0.933994	−0.466997	+	0.884259i	$0.654664\pi$
$60$	0	0
$61$	−8.50147	−1.08850	−0.544251	−	0.838922i	$-0.683186\pi$
$61$	−8.50147	−1.08850	−0.544251	+	0.838922i	$0.683186\pi$
$62$	0	0
$63$	−0.0460370	−0.00580012
$64$	0	0
$65$	−1.08913	−0.135089
$66$	0	0
$67$	12.2202	1.49293	0.746467	−	0.665423i	$-0.231749\pi$
$67$	12.2202	1.49293	0.746467	+	0.665423i	$0.231749\pi$
$68$	0	0
$69$	8.22018	0.989594
$70$	0	0
$71$	−0.935945	−0.111076	−0.0555381	−	0.998457i	$-0.517687\pi$
$71$	−0.935945	−0.111076	−0.0555381	+	0.998457i	$0.517687\pi$
$72$	0	0
$73$	−4.12811	−0.483159	−0.241579	−	0.970381i	$-0.577665\pi$
$73$	−4.12811	−0.483159	−0.241579	+	0.970381i	$0.577665\pi$
$74$	0	0
$75$	−8.33028	−0.961897
$76$	0	0
$77$	3.06406	0.349181
$78$	0	0
$79$	−2.95396	−0.332347	−0.166173	−	0.986097i	$-0.553141\pi$
$79$	−2.95396	−0.332347	−0.166173	+	0.986097i	$0.553141\pi$
$80$	0	0
$81$	−8.85977	−0.984419
$82$	0	0
$83$	−13.5655	−1.48901	−0.744505	−	0.667617i	$-0.767315\pi$
$83$	−13.5655	−1.48901	−0.744505	+	0.667617i	$0.767315\pi$
$84$	0	0
$85$	−2.43447	−0.264055
$86$	0	0
$87$	−13.4554	−1.44257
$88$	0	0
$89$	1.06406	0.112790	0.0563948	−	0.998409i	$-0.482039\pi$
$89$	1.06406	0.112790	0.0563948	+	0.998409i	$0.482039\pi$
$90$	0	0
$91$	−2.78276	−0.291713
$92$	0	0
$93$	−2.31226	−0.239770
$94$	0	0
$95$	−0.391382	−0.0401550
$96$	0	0
$97$	4.50147	0.457055	0.228528	−	0.973537i	$-0.426609\pi$
$97$	4.50147	0.457055	0.228528	+	0.973537i	$0.426609\pi$
$98$	0	0
$99$	−0.141060	−0.0141771
$100$	0	0
$101$	−17.1311	−1.70460	−0.852302	−	0.523050i	$-0.824794\pi$
$101$	−17.1311	−1.70460	−0.852302	+	0.523050i	$0.824794\pi$
$102$	0	0
$103$	2.09207	0.206138	0.103069	−	0.994674i	$-0.467134\pi$
$103$	2.09207	0.206138	0.103069	+	0.994674i	$0.467134\pi$
$104$	0	0
$105$	0.672673	0.0656461
$106$	0	0
$107$	−14.0921	−1.36233	−0.681166	−	0.732129i	$-0.738527\pi$
$107$	−14.0921	−1.36233	−0.681166	+	0.732129i	$0.738527\pi$
$108$	0	0
$109$	−15.1561	−1.45169	−0.725847	−	0.687856i	$-0.758552\pi$
$109$	−15.1561	−1.45169	−0.725847	+	0.687856i	$0.758552\pi$
$110$	0	0
$111$	6.58060	0.624603
$112$	0	0
$113$	19.7857	1.86128	0.930642	−	0.365932i	$-0.119250\pi$
$113$	19.7857	1.86128	0.930642	+	0.365932i	$0.119250\pi$
$114$	0	0
$115$	1.87189	0.174555
$116$	0	0
$117$	0.128110	0.0118438
$118$	0	0
$119$	−6.22018	−0.570203
$120$	0	0
$121$	−1.61157	−0.146506
$122$	0	0
$123$	−7.54751	−0.680536
$124$	0	0
$125$	−3.85387	−0.344701
$126$	0	0
$127$	−2.50147	−0.221970	−0.110985	−	0.993822i	$-0.535401\pi$
$127$	−2.50147	−0.221970	−0.110985	+	0.993822i	$0.535401\pi$
$128$	0	0
$129$	10.0180	0.882037
$130$	0	0
$131$	−15.4374	−1.34877	−0.674387	−	0.738378i	$-0.735592\pi$
$131$	−15.4374	−1.34877	−0.674387	+	0.738378i	$0.735592\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−2.04899	−0.176349
$136$	0	0
$137$	13.9209	1.18934	0.594670	−	0.803970i	$-0.297283\pi$
$137$	13.9209	1.18934	0.594670	+	0.803970i	$0.297283\pi$
$138$	0	0
$139$	10.8748	0.922392	0.461196	−	0.887298i	$-0.347421\pi$
$139$	10.8748	0.922392	0.461196	+	0.887298i	$0.347421\pi$
$140$	0	0
$141$	9.82880	0.827734
$142$	0	0
$143$	−8.52654	−0.713025
$144$	0	0
$145$	−3.06406	−0.254456
$146$	0	0
$147$	1.71871	0.141757
$148$	0	0
$149$	1.77982	0.145808	0.0729041	−	0.997339i	$-0.476773\pi$
$149$	1.77982	0.145808	0.0729041	+	0.997339i	$0.476773\pi$
$150$	0	0
$151$	−5.30931	−0.432065	−0.216033	−	0.976386i	$-0.569312\pi$
$151$	−5.30931	−0.432065	−0.216033	+	0.976386i	$0.569312\pi$
$152$	0	0
$153$	0.286359	0.0231507
$154$	0	0
$155$	−0.526544	−0.0422931
$156$	0	0
$157$	−15.3763	−1.22716	−0.613582	−	0.789631i	$-0.710272\pi$
$157$	−15.3763	−1.22716	−0.613582	+	0.789631i	$0.710272\pi$
$158$	0	0
$159$	−12.2993	−0.975399
$160$	0	0
$161$	4.78276	0.376935
$162$	0	0
$163$	17.9389	1.40508	0.702541	−	0.711643i	$-0.252049\pi$
$163$	17.9389	1.40508	0.702541	+	0.711643i	$0.252049\pi$
$164$	0	0
$165$	2.06111	0.160457
$166$	0	0
$167$	13.5295	1.04694	0.523472	−	0.852043i	$-0.324637\pi$
$167$	13.5295	1.04694	0.523472	+	0.852043i	$0.324637\pi$
$168$	0	0
$169$	−5.25622	−0.404325
$170$	0	0
$171$	0.0460370	0.00352054
$172$	0	0
$173$	8.12811	0.617969	0.308984	−	0.951067i	$-0.400011\pi$
$173$	8.12811	0.617969	0.308984	+	0.951067i	$0.400011\pi$
$174$	0	0
$175$	−4.84682	−0.366385
$176$	0	0
$177$	−12.3303	−0.926800
$178$	0	0
$179$	−11.4735	−0.857566	−0.428783	−	0.903407i	$-0.641058\pi$
$179$	−11.4735	−0.857566	−0.428783	+	0.903407i	$0.641058\pi$
$180$	0	0
$181$	−16.3123	−1.21248	−0.606240	−	0.795282i	$-0.707323\pi$
$181$	−16.3123	−1.21248	−0.606240	+	0.795282i	$0.707323\pi$
$182$	0	0
$183$	−14.6116	−1.08012
$184$	0	0
$185$	1.49853	0.110174
$186$	0	0
$187$	−19.0590	−1.39373
$188$	0	0
$189$	−5.23525	−0.380809
$190$	0	0
$191$	15.4735	1.11962	0.559810	−	0.828621i	$-0.310874\pi$
$191$	15.4735	1.11962	0.559810	+	0.828621i	$0.310874\pi$
$192$	0	0
$193$	7.00295	0.504083	0.252042	−	0.967716i	$-0.418898\pi$
$193$	7.00295	0.504083	0.252042	+	0.967716i	$0.418898\pi$
$194$	0	0
$195$	−1.87189	−0.134049
$196$	0	0
$197$	10.4404	0.743845	0.371923	−	0.928264i	$-0.378699\pi$
$197$	10.4404	0.743845	0.371923	+	0.928264i	$0.378699\pi$
$198$	0	0
$199$	6.83175	0.484290	0.242145	−	0.970240i	$-0.422149\pi$
$199$	6.83175	0.484290	0.242145	+	0.970240i	$0.422149\pi$
$200$	0	0
$201$	21.0029	1.48143
$202$	0	0
$203$	−7.82880	−0.549474
$204$	0	0
$205$	−1.71871	−0.120040
$206$	0	0
$207$	−0.220184	−0.0153039
$208$	0	0
$209$	−3.06406	−0.211945
$210$	0	0
$211$	−9.30931	−0.640879	−0.320440	−	0.947269i	$-0.603831\pi$
$211$	−9.30931	−0.640879	−0.320440	+	0.947269i	$0.603831\pi$
$212$	0	0
$213$	−1.60862	−0.110221
$214$	0	0
$215$	2.28129	0.155583
$216$	0	0
$217$	−1.34535	−0.0913280
$218$	0	0
$219$	−7.09502	−0.479437
$220$	0	0
$221$	17.3093	1.16435
$222$	0	0
$223$	2.31226	0.154840	0.0774201	−	0.996999i	$-0.475332\pi$
$223$	2.31226	0.154840	0.0774201	+	0.996999i	$0.475332\pi$
$224$	0	0
$225$	0.223133	0.0148755
$226$	0	0
$227$	12.4404	0.825696	0.412848	−	0.910800i	$-0.364534\pi$
$227$	12.4404	0.825696	0.412848	+	0.910800i	$0.364534\pi$
$228$	0	0
$229$	−15.6086	−1.03145	−0.515723	−	0.856755i	$-0.672477\pi$
$229$	−15.6086	−1.03145	−0.515723	+	0.856755i	$0.672477\pi$
$230$	0	0
$231$	5.26622	0.346492
$232$	0	0
$233$	−5.28424	−0.346182	−0.173091	−	0.984906i	$-0.555375\pi$
$233$	−5.28424	−0.346182	−0.173091	+	0.984906i	$0.555375\pi$
$234$	0	0
$235$	2.23820	0.146004
$236$	0	0
$237$	−5.07700	−0.329787
$238$	0	0
$239$	−1.87189	−0.121082	−0.0605412	−	0.998166i	$-0.519283\pi$
$239$	−1.87189	−0.121082	−0.0605412	+	0.998166i	$0.519283\pi$
$240$	0	0
$241$	−22.4835	−1.44829	−0.724143	−	0.689649i	$-0.757765\pi$
$241$	−22.4835	−1.44829	−0.724143	+	0.689649i	$0.757765\pi$
$242$	0	0
$243$	0.478388	0.0306886
$244$	0	0
$245$	0.391382	0.0250045
$246$	0	0
$247$	2.78276	0.177063
$248$	0	0
$249$	−23.3152	−1.47754
$250$	0	0
$251$	11.6936	0.738096	0.369048	−	0.929410i	$-0.379684\pi$
$251$	11.6936	0.738096	0.369048	+	0.929410i	$0.379684\pi$
$252$	0	0
$253$	14.6547	0.921330
$254$	0	0
$255$	−4.18415	−0.262022
$256$	0	0
$257$	−2.82585	−0.176272	−0.0881359	−	0.996108i	$-0.528091\pi$
$257$	−2.82585	−0.176272	−0.0881359	+	0.996108i	$0.528091\pi$
$258$	0	0
$259$	3.82880	0.237910
$260$	0	0
$261$	0.360415	0.0223091
$262$	0	0
$263$	29.4433	1.81555	0.907776	−	0.419455i	$-0.137779\pi$
$263$	29.4433	1.81555	0.907776	+	0.419455i	$0.137779\pi$
$264$	0	0
$265$	−2.80078	−0.172051
$266$	0	0
$267$	1.82880	0.111921
$268$	0	0
$269$	10.4404	0.636560	0.318280	−	0.947997i	$-0.396895\pi$
$269$	10.4404	0.636560	0.318280	+	0.947997i	$0.396895\pi$
$270$	0	0
$271$	4.29931	0.261164	0.130582	−	0.991437i	$-0.458315\pi$
$271$	4.29931	0.261164	0.130582	+	0.991437i	$0.458315\pi$
$272$	0	0
$273$	−4.78276	−0.289466
$274$	0	0
$275$	−14.8509	−0.895544
$276$	0	0
$277$	28.7526	1.72758	0.863789	−	0.503854i	$-0.168085\pi$
$277$	28.7526	1.72758	0.863789	+	0.503854i	$0.168085\pi$
$278$	0	0
$279$	0.0619357	0.00370799
$280$	0	0
$281$	−20.6547	−1.23215	−0.616077	−	0.787686i	$-0.711279\pi$
$281$	−20.6547	−1.23215	−0.616077	+	0.787686i	$0.711279\pi$
$282$	0	0
$283$	−9.30931	−0.553381	−0.276690	−	0.960959i	$-0.589238\pi$
$283$	−9.30931	−0.553381	−0.276690	+	0.960959i	$0.589238\pi$
$284$	0	0
$285$	−0.672673	−0.0398457
$286$	0	0
$287$	−4.39138	−0.259215
$288$	0	0
$289$	21.6907	1.27592
$290$	0	0
$291$	7.73673	0.453535
$292$	0	0
$293$	−1.65760	−0.0968382	−0.0484191	−	0.998827i	$-0.515418\pi$
$293$	−1.65760	−0.0968382	−0.0484191	+	0.998827i	$0.515418\pi$
$294$	0	0
$295$	−2.80783	−0.163478
$296$	0	0
$297$	−16.0411	−0.930799
$298$	0	0
$299$	−13.3093	−0.769697
$300$	0	0
$301$	5.82880	0.335967
$302$	0	0
$303$	−29.4433	−1.69147
$304$	0	0
$305$	−3.32733	−0.190522
$306$	0	0
$307$	−31.0880	−1.77428	−0.887142	−	0.461496i	$-0.847313\pi$
$307$	−31.0880	−1.77428	−0.887142	+	0.461496i	$0.847313\pi$
$308$	0	0
$309$	3.59567	0.204550
$310$	0	0
$311$	−11.8468	−0.671772	−0.335886	−	0.941903i	$-0.609036\pi$
$311$	−11.8468	−0.671772	−0.335886	+	0.941903i	$0.609036\pi$
$312$	0	0
$313$	34.6606	1.95913	0.979565	−	0.201127i	$-0.0644605\pi$
$313$	34.6606	1.95913	0.979565	+	0.201127i	$0.0644605\pi$
$314$	0	0
$315$	−0.0180181	−0.00101520
$316$	0	0
$317$	16.4654	0.924791	0.462396	−	0.886674i	$-0.346990\pi$
$317$	16.4654	0.924791	0.462396	+	0.886674i	$0.346990\pi$
$318$	0	0
$319$	−23.9879	−1.34306
$320$	0	0
$321$	−24.2202	−1.35184
$322$	0	0
$323$	6.22018	0.346100
$324$	0	0
$325$	13.4876	0.748155
$326$	0	0
$327$	−26.0490	−1.44051
$328$	0	0
$329$	5.71871	0.315283
$330$	0	0
$331$	−3.09502	−0.170118	−0.0850589	−	0.996376i	$-0.527108\pi$
$331$	−3.09502	−0.170118	−0.0850589	+	0.996376i	$0.527108\pi$
$332$	0	0
$333$	−0.176267	−0.00965935
$334$	0	0
$335$	4.78276	0.261310
$336$	0	0
$337$	−24.5324	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$337$	−24.5324	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$338$	0	0
$339$	34.0059	1.84695
$340$	0	0
$341$	−4.12221	−0.223230
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.21724	0.173210
$346$	0	0
$347$	5.56553	0.298773	0.149387	−	0.988779i	$-0.452270\pi$
$347$	5.56553	0.298773	0.149387	+	0.988779i	$0.452270\pi$
$348$	0	0
$349$	−25.1311	−1.34523	−0.672617	−	0.739990i	$-0.734830\pi$
$349$	−25.1311	−1.34523	−0.672617	+	0.739990i	$0.734830\pi$
$350$	0	0
$351$	14.5685	0.777608
$352$	0	0
$353$	−3.52949	−0.187856	−0.0939280	−	0.995579i	$-0.529942\pi$
$353$	−3.52949	−0.187856	−0.0939280	+	0.995579i	$0.529942\pi$
$354$	0	0
$355$	−0.366312	−0.0194418
$356$	0	0
$357$	−10.6907	−0.565811
$358$	0	0
$359$	−24.9669	−1.31770	−0.658852	−	0.752273i	$-0.728958\pi$
$359$	−24.9669	−1.31770	−0.658852	+	0.752273i	$0.728958\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.76981	−0.145378
$364$	0	0
$365$	−1.61567	−0.0845680
$366$	0	0
$367$	15.2842	0.797831	0.398915	−	0.916988i	$-0.369387\pi$
$367$	15.2842	0.797831	0.398915	+	0.916988i	$0.369387\pi$
$368$	0	0
$369$	0.202166	0.0105243
$370$	0	0
$371$	−7.15613	−0.371528
$372$	0	0
$373$	17.0519	0.882916	0.441458	−	0.897282i	$-0.354461\pi$
$373$	17.0519	0.882916	0.441458	+	0.897282i	$0.354461\pi$
$374$	0	0
$375$	−6.62369	−0.342046
$376$	0	0
$377$	21.7857	1.12202
$378$	0	0
$379$	17.0029	0.873383	0.436691	−	0.899611i	$-0.356150\pi$
$379$	17.0029	0.873383	0.436691	+	0.899611i	$0.356150\pi$
$380$	0	0
$381$	−4.29931	−0.220260
$382$	0	0
$383$	−29.2231	−1.49323	−0.746616	−	0.665255i	$-0.768323\pi$
$383$	−29.2231	−1.49323	−0.746616	+	0.665255i	$0.768323\pi$
$384$	0	0
$385$	1.19922	0.0611178
$386$	0	0
$387$	−0.268341	−0.0136405
$388$	0	0
$389$	26.4404	1.34058	0.670290	−	0.742099i	$-0.266170\pi$
$389$	26.4404	1.34058	0.670290	+	0.742099i	$0.266170\pi$
$390$	0	0
$391$	−29.7497	−1.50451
$392$	0	0
$393$	−26.5324	−1.33838
$394$	0	0
$395$	−1.15613	−0.0581712
$396$	0	0
$397$	−18.5074	−0.928858	−0.464429	−	0.885610i	$-0.653740\pi$
$397$	−18.5074	−0.928858	−0.464429	+	0.885610i	$0.653740\pi$
$398$	0	0
$399$	−1.71871	−0.0860431
$400$	0	0
$401$	−32.2261	−1.60929	−0.804647	−	0.593754i	$-0.797645\pi$
$401$	−32.2261	−1.60929	−0.804647	+	0.593754i	$0.797645\pi$
$402$	0	0
$403$	3.74378	0.186491
$404$	0	0
$405$	−3.46756	−0.172304
$406$	0	0
$407$	11.7317	0.581517
$408$	0	0
$409$	19.3763	0.958097	0.479049	−	0.877788i	$-0.340982\pi$
$409$	19.3763	0.958097	0.479049	+	0.877788i	$0.340982\pi$
$410$	0	0
$411$	23.9259	1.18018
$412$	0	0
$413$	−7.17415	−0.353017
$414$	0	0
$415$	−5.30931	−0.260624
$416$	0	0
$417$	18.6907	0.915287
$418$	0	0
$419$	27.5354	1.34519	0.672596	−	0.740010i	$-0.265179\pi$
$419$	27.5354	1.34519	0.672596	+	0.740010i	$0.265179\pi$
$420$	0	0
$421$	16.4345	0.800967	0.400484	−	0.916304i	$-0.368842\pi$
$421$	16.4345	0.800967	0.400484	+	0.916304i	$0.368842\pi$
$422$	0	0
$423$	−0.263272	−0.0128007
$424$	0	0
$425$	30.1481	1.46240
$426$	0	0
$427$	−8.50147	−0.411415
$428$	0	0
$429$	−14.6547	−0.707533
$430$	0	0
$431$	−40.9979	−1.97480	−0.987399	−	0.158249i	$-0.949415\pi$
$431$	−40.9979	−1.97480	−0.987399	+	0.158249i	$0.949415\pi$
$432$	0	0
$433$	−22.4835	−1.08049	−0.540243	−	0.841509i	$-0.681668\pi$
$433$	−22.4835	−1.08049	−0.540243	+	0.841509i	$0.681668\pi$
$434$	0	0
$435$	−5.26622	−0.252496
$436$	0	0
$437$	−4.78276	−0.228791
$438$	0	0
$439$	18.9109	0.902567	0.451283	−	0.892381i	$-0.350966\pi$
$439$	18.9109	0.902567	0.451283	+	0.892381i	$0.350966\pi$
$440$	0	0
$441$	−0.0460370	−0.00219224
$442$	0	0
$443$	−10.0850	−0.479154	−0.239577	−	0.970877i	$-0.577009\pi$
$443$	−10.0850	−0.479154	−0.239577	+	0.970877i	$0.577009\pi$
$444$	0	0
$445$	0.416452	0.0197417
$446$	0	0
$447$	3.05899	0.144685
$448$	0	0
$449$	17.9138	0.845406	0.422703	−	0.906268i	$-0.361081\pi$
$449$	17.9138	0.845406	0.422703	+	0.906268i	$0.361081\pi$
$450$	0	0
$451$	−13.4554	−0.633592
$452$	0	0
$453$	−9.12516	−0.428737
$454$	0	0
$455$	−1.08913	−0.0510590
$456$	0	0
$457$	23.8527	1.11578	0.557892	−	0.829914i	$-0.311611\pi$
$457$	23.8527	1.11578	0.557892	+	0.829914i	$0.311611\pi$
$458$	0	0
$459$	32.5642	1.51997
$460$	0	0
$461$	−22.2512	−1.03634	−0.518170	−	0.855278i	$-0.673386\pi$
$461$	−22.2512	−1.03634	−0.518170	+	0.855278i	$0.673386\pi$
$462$	0	0
$463$	−27.8778	−1.29559	−0.647795	−	0.761814i	$-0.724309\pi$
$463$	−27.8778	−1.29559	−0.647795	+	0.761814i	$0.724309\pi$
$464$	0	0
$465$	−0.904977	−0.0419673
$466$	0	0
$467$	−24.0980	−1.11512	−0.557561	−	0.830136i	$-0.688263\pi$
$467$	−24.0980	−1.11512	−0.557561	+	0.830136i	$0.688263\pi$
$468$	0	0
$469$	12.2202	0.564276
$470$	0	0
$471$	−26.4274	−1.21771
$472$	0	0
$473$	17.8598	0.821193
$474$	0	0
$475$	4.84682	0.222387
$476$	0	0
$477$	0.329447	0.0150843
$478$	0	0
$479$	8.92382	0.407740	0.203870	−	0.978998i	$-0.434648\pi$
$479$	8.92382	0.407740	0.203870	+	0.978998i	$0.434648\pi$
$480$	0	0
$481$	−10.6547	−0.485810
$482$	0	0
$483$	8.22018	0.374031
$484$	0	0
$485$	1.76180	0.0799991
$486$	0	0
$487$	41.0578	1.86051	0.930254	−	0.366916i	$-0.119586\pi$
$487$	41.0578	1.86051	0.930254	+	0.366916i	$0.119586\pi$
$488$	0	0
$489$	30.8318	1.39426
$490$	0	0
$491$	−20.8807	−0.942334	−0.471167	−	0.882044i	$-0.656167\pi$
$491$	−20.8807	−0.942334	−0.471167	+	0.882044i	$0.656167\pi$
$492$	0	0
$493$	48.6966	2.19318
$494$	0	0
$495$	−0.0552084	−0.00248143
$496$	0	0
$497$	−0.935945	−0.0419829
$498$	0	0
$499$	12.8619	0.575777	0.287889	−	0.957664i	$-0.407047\pi$
$499$	12.8619	0.575777	0.287889	+	0.957664i	$0.407047\pi$
$500$	0	0
$501$	23.2533	1.03888
$502$	0	0
$503$	22.9179	1.02186	0.510930	−	0.859622i	$-0.329301\pi$
$503$	22.9179	1.02186	0.510930	+	0.859622i	$0.329301\pi$
$504$	0	0
$505$	−6.70479	−0.298359
$506$	0	0
$507$	−9.03392	−0.401210
$508$	0	0
$509$	17.6936	0.784257	0.392128	−	0.919910i	$-0.371739\pi$
$509$	17.6936	0.784257	0.392128	+	0.919910i	$0.371739\pi$
$510$	0	0
$511$	−4.12811	−0.182617
$512$	0	0
$513$	5.23525	0.231142
$514$	0	0
$515$	0.818801	0.0360807
$516$	0	0
$517$	17.5224	0.770636
$518$	0	0
$519$	13.9699	0.613209
$520$	0	0
$521$	−13.3152	−0.583350	−0.291675	−	0.956518i	$-0.594213\pi$
$521$	−13.3152	−0.583350	−0.291675	+	0.956518i	$0.594213\pi$
$522$	0	0
$523$	−44.4404	−1.94324	−0.971621	−	0.236544i	$-0.923985\pi$
$523$	−44.4404	−1.94324	−0.971621	+	0.236544i	$0.923985\pi$
$524$	0	0
$525$	−8.33028	−0.363563
$526$	0	0
$527$	8.36830	0.364529
$528$	0	0
$529$	−0.125161	−0.00544179
$530$	0	0
$531$	0.330276	0.0143328
$532$	0	0
$533$	12.2202	0.529315
$534$	0	0
$535$	−5.51539	−0.238451
$536$	0	0
$537$	−19.7195	−0.850961
$538$	0	0
$539$	3.06406	0.131978
$540$	0	0
$541$	−18.5626	−0.798068	−0.399034	−	0.916936i	$-0.630654\pi$
$541$	−18.5626	−0.798068	−0.399034	+	0.916936i	$0.630654\pi$
$542$	0	0
$543$	−28.0360	−1.20314
$544$	0	0
$545$	−5.93184	−0.254092
$546$	0	0
$547$	−26.2762	−1.12349	−0.561745	−	0.827310i	$-0.689870\pi$
$547$	−26.2762	−1.12349	−0.561745	+	0.827310i	$0.689870\pi$
$548$	0	0
$549$	0.391382	0.0167038
$550$	0	0
$551$	7.82880	0.333518
$552$	0	0
$553$	−2.95396	−0.125615
$554$	0	0
$555$	2.57553	0.109325
$556$	0	0
$557$	−32.6045	−1.38150	−0.690749	−	0.723095i	$-0.742719\pi$
$557$	−32.6045	−1.38150	−0.690749	+	0.723095i	$0.742719\pi$
$558$	0	0
$559$	−16.2202	−0.686041
$560$	0	0
$561$	−32.7569	−1.38300
$562$	0	0
$563$	35.4182	1.49270	0.746351	−	0.665553i	$-0.231804\pi$
$563$	35.4182	1.49270	0.746351	+	0.665553i	$0.231804\pi$
$564$	0	0
$565$	7.74378	0.325783
$566$	0	0
$567$	−8.85977	−0.372075
$568$	0	0
$569$	−13.6936	−0.574067	−0.287034	−	0.957921i	$-0.592669\pi$
$569$	−13.6936	−0.574067	−0.287034	+	0.957921i	$0.592669\pi$
$570$	0	0
$571$	−17.9389	−0.750719	−0.375360	−	0.926879i	$-0.622481\pi$
$571$	−17.9389	−0.750719	−0.375360	+	0.926879i	$0.622481\pi$
$572$	0	0
$573$	26.5944	1.11100
$574$	0	0
$575$	−23.1812	−0.966723
$576$	0	0
$577$	−45.3873	−1.88950	−0.944749	−	0.327796i	$-0.893694\pi$
$577$	−45.3873	−1.88950	−0.944749	+	0.327796i	$0.893694\pi$
$578$	0	0
$579$	12.0360	0.500201
$580$	0	0
$581$	−13.5655	−0.562793
$582$	0	0
$583$	−21.9268	−0.908114
$584$	0	0
$585$	0.0501401	0.00207304
$586$	0	0
$587$	−34.3483	−1.41771	−0.708853	−	0.705356i	$-0.750787\pi$
$587$	−34.3483	−1.41771	−0.708853	+	0.705356i	$0.750787\pi$
$588$	0	0
$589$	1.34535	0.0554340
$590$	0	0
$591$	17.9440	0.738116
$592$	0	0
$593$	6.56258	0.269493	0.134746	−	0.990880i	$-0.456978\pi$
$593$	6.56258	0.269493	0.134746	+	0.990880i	$0.456978\pi$
$594$	0	0
$595$	−2.43447	−0.0998036
$596$	0	0
$597$	11.7418	0.480560
$598$	0	0
$599$	8.62958	0.352595	0.176298	−	0.984337i	$-0.443588\pi$
$599$	8.62958	0.352595	0.176298	+	0.984337i	$0.443588\pi$
$600$	0	0
$601$	10.4404	0.425872	0.212936	−	0.977066i	$-0.431697\pi$
$601$	10.4404	0.425872	0.212936	+	0.977066i	$0.431697\pi$
$602$	0	0
$603$	−0.562581	−0.0229101
$604$	0	0
$605$	−0.630739	−0.0256432
$606$	0	0
$607$	1.43152	0.0581037	0.0290518	−	0.999578i	$-0.490751\pi$
$607$	1.43152	0.0581037	0.0290518	+	0.999578i	$0.490751\pi$
$608$	0	0
$609$	−13.4554	−0.545242
$610$	0	0
$611$	−15.9138	−0.643804
$612$	0	0
$613$	41.0950	1.65981	0.829906	−	0.557903i	$-0.188394\pi$
$613$	41.0950	1.65981	0.829906	+	0.557903i	$0.188394\pi$
$614$	0	0
$615$	−2.95396	−0.119115
$616$	0	0
$617$	27.1440	1.09278	0.546388	−	0.837532i	$-0.316002\pi$
$617$	27.1440	1.09278	0.546388	+	0.837532i	$0.316002\pi$
$618$	0	0
$619$	1.83585	0.0737892	0.0368946	−	0.999319i	$-0.488253\pi$
$619$	1.83585	0.0737892	0.0368946	+	0.999319i	$0.488253\pi$
$620$	0	0
$621$	−25.0390	−1.00478
$622$	0	0
$623$	1.06406	0.0426305
$624$	0	0
$625$	22.7258	0.909030
$626$	0	0
$627$	−5.26622	−0.210313
$628$	0	0
$629$	−23.8159	−0.949600
$630$	0	0
$631$	−43.5714	−1.73455	−0.867276	−	0.497828i	$-0.834131\pi$
$631$	−43.5714	−1.73455	−0.867276	+	0.497828i	$0.834131\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	0	0
$635$	−0.979033	−0.0388517
$636$	0	0
$637$	−2.78276	−0.110257
$638$	0	0
$639$	0.0430881	0.00170454
$640$	0	0
$641$	43.6275	1.72318	0.861591	−	0.507604i	$-0.169469\pi$
$641$	43.6275	1.72318	0.861591	+	0.507604i	$0.169469\pi$
$642$	0	0
$643$	14.6907	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$643$	14.6907	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$644$	0	0
$645$	3.92088	0.154384
$646$	0	0
$647$	−19.6145	−0.771126	−0.385563	−	0.922681i	$-0.625993\pi$
$647$	−19.6145	−0.771126	−0.385563	+	0.922681i	$0.625993\pi$
$648$	0	0
$649$	−21.9820	−0.862868
$650$	0	0
$651$	−2.31226	−0.0906245
$652$	0	0
$653$	−42.1700	−1.65024	−0.825121	−	0.564957i	$-0.808893\pi$
$653$	−42.1700	−1.65024	−0.825121	+	0.564957i	$0.808893\pi$
$654$	0	0
$655$	−6.04193	−0.236078
$656$	0	0
$657$	0.190046	0.00741439
$658$	0	0
$659$	34.1900	1.33186	0.665928	−	0.746016i	$-0.268036\pi$
$659$	34.1900	1.33186	0.665928	+	0.746016i	$0.268036\pi$
$660$	0	0
$661$	31.1370	1.21109	0.605544	−	0.795812i	$-0.292956\pi$
$661$	31.1370	1.21109	0.605544	+	0.795812i	$0.292956\pi$
$662$	0	0
$663$	29.7497	1.15538
$664$	0	0
$665$	−0.391382	−0.0151772
$666$	0	0
$667$	−37.4433	−1.44981
$668$	0	0
$669$	3.97410	0.153648
$670$	0	0
$671$	−26.0490	−1.00561
$672$	0	0
$673$	−48.6045	−1.87357	−0.936783	−	0.349910i	$-0.886212\pi$
$673$	−48.6045	−1.87357	−0.936783	+	0.349910i	$0.886212\pi$
$674$	0	0
$675$	25.3743	0.976658
$676$	0	0
$677$	−9.81585	−0.377254	−0.188627	−	0.982049i	$-0.560404\pi$
$677$	−9.81585	−0.377254	−0.188627	+	0.982049i	$0.560404\pi$
$678$	0	0
$679$	4.50147	0.172751
$680$	0	0
$681$	21.3814	0.819336
$682$	0	0
$683$	−12.6606	−0.484443	−0.242221	−	0.970221i	$-0.577876\pi$
$683$	−12.6606	−0.484443	−0.242221	+	0.970221i	$0.577876\pi$
$684$	0	0
$685$	5.44839	0.208172
$686$	0	0
$687$	−26.8267	−1.02350
$688$	0	0
$689$	19.9138	0.758656
$690$	0	0
$691$	45.4073	1.72737	0.863687	−	0.504028i	$-0.168149\pi$
$691$	45.4073	1.72737	0.863687	+	0.504028i	$0.168149\pi$
$692$	0	0
$693$	−0.141060	−0.00535842
$694$	0	0
$695$	4.25622	0.161448
$696$	0	0
$697$	27.3152	1.03464
$698$	0	0
$699$	−9.08207	−0.343516
$700$	0	0
$701$	11.1871	0.422531	0.211265	−	0.977429i	$-0.432242\pi$
$701$	11.1871	0.422531	0.211265	+	0.977429i	$0.432242\pi$
$702$	0	0
$703$	−3.82880	−0.144406
$704$	0	0
$705$	3.84682	0.144880
$706$	0	0
$707$	−17.1311	−0.644280
$708$	0	0
$709$	−13.4014	−0.503300	−0.251650	−	0.967818i	$-0.580973\pi$
$709$	−13.4014	−0.503300	−0.251650	+	0.967818i	$0.580973\pi$
$710$	0	0
$711$	0.135992	0.00510008
$712$	0	0
$713$	−6.43447	−0.240973
$714$	0	0
$715$	−3.33714	−0.124802
$716$	0	0
$717$	−3.21724	−0.120150
$718$	0	0
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	2.09207	0.0779129
$722$	0	0
$723$	−38.6425	−1.43713
$724$	0	0
$725$	37.9448	1.40923
$726$	0	0
$727$	42.4153	1.57310	0.786548	−	0.617529i	$-0.211866\pi$
$727$	42.4153	1.57310	0.786548	+	0.617529i	$0.211866\pi$
$728$	0	0
$729$	27.4015	1.01487
$730$	0	0
$731$	−36.2562	−1.34098
$732$	0	0
$733$	43.9628	1.62380	0.811902	−	0.583794i	$-0.198432\pi$
$733$	43.9628	1.62380	0.811902	+	0.583794i	$0.198432\pi$
$734$	0	0
$735$	0.672673	0.0248119
$736$	0	0
$737$	37.4433	1.37924
$738$	0	0
$739$	1.57258	0.0578483	0.0289242	−	0.999582i	$-0.490792\pi$
$739$	1.57258	0.0578483	0.0289242	+	0.999582i	$0.490792\pi$
$740$	0	0
$741$	4.78276	0.175699
$742$	0	0
$743$	−37.0460	−1.35909	−0.679544	−	0.733635i	$-0.737822\pi$
$743$	−37.0460	−1.35909	−0.679544	+	0.733635i	$0.737822\pi$
$744$	0	0
$745$	0.696589	0.0255210
$746$	0	0
$747$	0.624516	0.0228499
$748$	0	0
$749$	−14.0921	−0.514913
$750$	0	0
$751$	−15.7367	−0.574241	−0.287121	−	0.957894i	$-0.592698\pi$
$751$	−15.7367	−0.574241	−0.287121	+	0.957894i	$0.592698\pi$
$752$	0	0
$753$	20.0980	0.732411
$754$	0	0
$755$	−2.07797	−0.0756251
$756$	0	0
$757$	−33.3512	−1.21217	−0.606086	−	0.795399i	$-0.707261\pi$
$757$	−33.3512	−1.21217	−0.606086	+	0.795399i	$0.707261\pi$
$758$	0	0
$759$	25.1871	0.914234
$760$	0	0
$761$	20.0059	0.725213	0.362607	−	0.931942i	$-0.381887\pi$
$761$	20.0059	0.725213	0.362607	+	0.931942i	$0.381887\pi$
$762$	0	0
$763$	−15.1561	−0.548689
$764$	0	0
$765$	0.112076	0.00405211
$766$	0	0
$767$	19.9640	0.720857
$768$	0	0
$769$	55.0029	1.98346	0.991729	−	0.128353i	$-0.0409691\pi$
$769$	55.0029	1.98346	0.991729	+	0.128353i	$0.0409691\pi$
$770$	0	0
$771$	−4.85682	−0.174914
$772$	0	0
$773$	−32.7526	−1.17803	−0.589015	−	0.808122i	$-0.700484\pi$
$773$	−32.7526	−1.17803	−0.589015	+	0.808122i	$0.700484\pi$
$774$	0	0
$775$	6.52065	0.234229
$776$	0	0
$777$	6.58060	0.236078
$778$	0	0
$779$	4.39138	0.157338
$780$	0	0
$781$	−2.86779	−0.102617
$782$	0	0
$783$	40.9858	1.46471
$784$	0	0
$785$	−6.01802	−0.214792
$786$	0	0
$787$	16.2993	0.581008	0.290504	−	0.956874i	$-0.406177\pi$
$787$	16.2993	0.581008	0.290504	+	0.956874i	$0.406177\pi$
$788$	0	0
$789$	50.6045	1.80157
$790$	0	0
$791$	19.7857	0.703499
$792$	0	0
$793$	23.6576	0.840106
$794$	0	0
$795$	−4.81373	−0.170726
$796$	0	0
$797$	−9.50949	−0.336843	−0.168422	−	0.985715i	$-0.553867\pi$
$797$	−9.50949	−0.336843	−0.168422	+	0.985715i	$0.553867\pi$
$798$	0	0
$799$	−35.5714	−1.25843
$800$	0	0
$801$	−0.0489859	−0.00173083
$802$	0	0
$803$	−12.6488	−0.446365
$804$	0	0
$805$	1.87189	0.0659754
$806$	0	0
$807$	17.9440	0.631657
$808$	0	0
$809$	−20.0129	−0.703618	−0.351809	−	0.936072i	$-0.614433\pi$
$809$	−20.0129	−0.703618	−0.351809	+	0.936072i	$0.614433\pi$
$810$	0	0
$811$	49.8527	1.75057	0.875283	−	0.483611i	$-0.160675\pi$
$811$	49.8527	1.75057	0.875283	+	0.483611i	$0.160675\pi$
$812$	0	0
$813$	7.38926	0.259153
$814$	0	0
$815$	7.02097	0.245934
$816$	0	0
$817$	−5.82880	−0.203924
$818$	0	0
$819$	0.128110	0.00447653
$820$	0	0
$821$	2.68479	0.0936999	0.0468500	−	0.998902i	$-0.485082\pi$
$821$	2.68479	0.0936999	0.0468500	+	0.998902i	$0.485082\pi$
$822$	0	0
$823$	28.4764	0.992625	0.496313	−	0.868144i	$-0.334687\pi$
$823$	28.4764	0.992625	0.496313	+	0.868144i	$0.334687\pi$
$824$	0	0
$825$	−25.5244	−0.888646
$826$	0	0
$827$	−29.0390	−1.00978	−0.504892	−	0.863182i	$-0.668468\pi$
$827$	−29.0390	−1.00978	−0.504892	+	0.863182i	$0.668468\pi$
$828$	0	0
$829$	19.7136	0.684683	0.342342	−	0.939576i	$-0.388780\pi$
$829$	19.7136	0.684683	0.342342	+	0.939576i	$0.388780\pi$
$830$	0	0
$831$	49.4174	1.71427
$832$	0	0
$833$	−6.22018	−0.215517
$834$	0	0
$835$	5.29521	0.183248
$836$	0	0
$837$	7.04322	0.243449
$838$	0	0
$839$	18.9109	0.652876	0.326438	−	0.945219i	$-0.394152\pi$
$839$	18.9109	0.652876	0.326438	+	0.945219i	$0.394152\pi$
$840$	0	0
$841$	32.2901	1.11345
$842$	0	0
$843$	−35.4994	−1.22266
$844$	0	0
$845$	−2.05719	−0.0707696
$846$	0	0
$847$	−1.61157	−0.0553741
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	18.3123	0.627736
$852$	0	0
$853$	23.7928	0.814649	0.407324	−	0.913284i	$-0.366462\pi$
$853$	23.7928	0.814649	0.407324	+	0.913284i	$0.366462\pi$
$854$	0	0
$855$	0.0180181	0.000616205 0
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−12.9669	−0.442425	−0.221213	−	0.975226i	$-0.571001\pi$
$859$	−12.9669	−0.442425	−0.221213	+	0.975226i	$0.571001\pi$
$860$	0	0
$861$	−7.54751	−0.257219
$862$	0	0
$863$	−23.0519	−0.784697	−0.392348	−	0.919817i	$-0.628337\pi$
$863$	−23.0519	−0.784697	−0.392348	+	0.919817i	$0.628337\pi$
$864$	0	0
$865$	3.18120	0.108164
$866$	0	0
$867$	37.2800	1.26610
$868$	0	0
$869$	−9.05111	−0.307038
$870$	0	0
$871$	−34.0059	−1.15225
$872$	0	0
$873$	−0.207234	−0.00701382
$874$	0	0
$875$	−3.85387	−0.130285
$876$	0	0
$877$	−17.8468	−0.602644	−0.301322	−	0.953522i	$-0.597428\pi$
$877$	−17.8468	−0.602644	−0.301322	+	0.953522i	$0.597428\pi$
$878$	0	0
$879$	−2.84894	−0.0960923
$880$	0	0
$881$	6.71069	0.226089	0.113044	−	0.993590i	$-0.463940\pi$
$881$	6.71069	0.226089	0.113044	+	0.993590i	$0.463940\pi$
$882$	0	0
$883$	13.3144	0.448064	0.224032	−	0.974582i	$-0.428078\pi$
$883$	13.3144	0.448064	0.224032	+	0.974582i	$0.428078\pi$
$884$	0	0
$885$	−4.82585	−0.162219
$886$	0	0
$887$	−17.2733	−0.579980	−0.289990	−	0.957030i	$-0.593652\pi$
$887$	−17.2733	−0.579980	−0.289990	+	0.957030i	$0.593652\pi$
$888$	0	0
$889$	−2.50147	−0.0838968
$890$	0	0
$891$	−27.1468	−0.909453
$892$	0	0
$893$	−5.71871	−0.191369
$894$	0	0
$895$	−4.49051	−0.150101
$896$	0	0
$897$	−22.8748	−0.763769
$898$	0	0
$899$	10.5324	0.351277
$900$	0	0
$901$	44.5124	1.48292
$902$	0	0
$903$	10.0180	0.333379
$904$	0	0
$905$	−6.38433	−0.212222
$906$	0	0
$907$	6.99705	0.232333	0.116167	−	0.993230i	$-0.462939\pi$
$907$	6.99705	0.232333	0.116167	+	0.993230i	$0.462939\pi$
$908$	0	0
$909$	0.788663	0.0261583
$910$	0	0
$911$	−24.1050	−0.798635	−0.399318	−	0.916813i	$-0.630753\pi$
$911$	−24.1050	−0.798635	−0.399318	+	0.916813i	$0.630753\pi$
$912$	0	0
$913$	−41.5655	−1.37562
$914$	0	0
$915$	−5.71871	−0.189055
$916$	0	0
$917$	−15.4374	−0.509789
$918$	0	0
$919$	55.3011	1.82422	0.912108	−	0.409951i	$-0.134454\pi$
$919$	55.3011	1.82422	0.912108	+	0.409951i	$0.134454\pi$
$920$	0	0
$921$	−53.4312	−1.76062
$922$	0	0
$923$	2.60451	0.0857286
$924$	0	0
$925$	−18.5575	−0.610167
$926$	0	0
$927$	−0.0963128	−0.00316333
$928$	0	0
$929$	4.60451	0.151069	0.0755346	−	0.997143i	$-0.475934\pi$
$929$	4.60451	0.151069	0.0755346	+	0.997143i	$0.475934\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−20.3612	−0.666597
$934$	0	0
$935$	−7.45935	−0.243947
$936$	0	0
$937$	45.7195	1.49359	0.746796	−	0.665053i	$-0.231591\pi$
$937$	45.7195	1.49359	0.746796	+	0.665053i	$0.231591\pi$
$938$	0	0
$939$	59.5714	1.94404
$940$	0	0
$941$	−32.5324	−1.06053	−0.530264	−	0.847833i	$-0.677907\pi$
$941$	−32.5324	−1.06053	−0.530264	+	0.847833i	$0.677907\pi$
$942$	0	0
$943$	−21.0029	−0.683950
$944$	0	0
$945$	−2.04899	−0.0666535
$946$	0	0
$947$	−35.0641	−1.13943	−0.569714	−	0.821843i	$-0.692946\pi$
$947$	−35.0641	−1.13943	−0.569714	+	0.821843i	$0.692946\pi$
$948$	0	0
$949$	11.4876	0.372902
$950$	0	0
$951$	28.2993	0.917668
$952$	0	0
$953$	15.9941	0.518100	0.259050	−	0.965864i	$-0.416591\pi$
$953$	15.9941	0.518100	0.259050	+	0.965864i	$0.416591\pi$
$954$	0	0
$955$	6.05604	0.195969
$956$	0	0
$957$	−41.2282	−1.33272
$958$	0	0
$959$	13.9209	0.449529
$960$	0	0
$961$	−29.1900	−0.941614
$962$	0	0
$963$	0.648757	0.0209059
$964$	0	0
$965$	2.74083	0.0882305
$966$	0	0
$967$	−16.0721	−0.516843	−0.258422	−	0.966032i	$-0.583202\pi$
$967$	−16.0721	−0.516843	−0.258422	+	0.966032i	$0.583202\pi$
$968$	0	0
$969$	10.6907	0.343434
$970$	0	0
$971$	34.6715	1.11266	0.556331	−	0.830961i	$-0.312209\pi$
$971$	34.6715	1.11266	0.556331	+	0.830961i	$0.312209\pi$
$972$	0	0
$973$	10.8748	0.348631
$974$	0	0
$975$	23.1812	0.742393
$976$	0	0
$977$	−12.5685	−0.402101	−0.201051	−	0.979581i	$-0.564436\pi$
$977$	−12.5685	−0.402101	−0.201051	+	0.979581i	$0.564436\pi$
$978$	0	0
$979$	3.26032	0.104200
$980$	0	0
$981$	0.697743	0.0222772
$982$	0	0
$983$	−1.56553	−0.0499326	−0.0249663	−	0.999688i	$-0.507948\pi$
$983$	−1.56553	−0.0499326	−0.0249663	+	0.999688i	$0.507948\pi$
$984$	0	0
$985$	4.08618	0.130196
$986$	0	0
$987$	9.82880	0.312854
$988$	0	0
$989$	27.8778	0.886462
$990$	0	0
$991$	42.6355	1.35436	0.677180	−	0.735817i	$-0.263202\pi$
$991$	42.6355	1.35436	0.677180	+	0.735817i	$0.263202\pi$
$992$	0	0
$993$	−5.31945	−0.168808
$994$	0	0
$995$	2.67383	0.0847660
$996$	0	0
$997$	−61.2721	−1.94051	−0.970254	−	0.242090i	$-0.922167\pi$
$997$	−61.2721	−1.94051	−0.970254	+	0.242090i	$0.922167\pi$
$998$	0	0
$999$	−20.0447	−0.634188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bj.1.3		3
4.3	odd	2		8512.2.a.bm.1.1		3
8.3	odd	2		266.2.a.d.1.3	✓	3
8.5	even	2		2128.2.a.s.1.1		3
24.11	even	2		2394.2.a.ba.1.3		3
40.19	odd	2		6650.2.a.cd.1.1		3
56.27	even	2		1862.2.a.r.1.1		3
152.75	even	2		5054.2.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.a.d.1.3	✓	3	8.3	odd	2
1862.2.a.r.1.1		3	56.27	even	2
2128.2.a.s.1.1		3	8.5	even	2
2394.2.a.ba.1.3		3	24.11	even	2
5054.2.a.r.1.1		3	152.75	even	2
6650.2.a.cd.1.1		3	40.19	odd	2
8512.2.a.bj.1.3		3	1.1	even	1	trivial
8512.2.a.bm.1.1		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71871 q^{3} +0.391382 q^{5} +1.00000 q^{7} -0.0460370 q^{9} +3.06406 q^{11} -2.78276 q^{13} +0.672673 q^{15} -6.22018 q^{17} -1.00000 q^{19} +1.71871 q^{21} +4.78276 q^{23} -4.84682 q^{25} -5.23525 q^{27} -7.82880 q^{29} -1.34535 q^{31} +5.26622 q^{33} +0.391382 q^{35} +3.82880 q^{37} -4.78276 q^{39} -4.39138 q^{41} +5.82880 q^{43} -0.0180181 q^{45} +5.71871 q^{47} +1.00000 q^{49} -10.6907 q^{51} -7.15613 q^{53} +1.19922 q^{55} -1.71871 q^{57} -7.17415 q^{59} -8.50147 q^{61} -0.0460370 q^{63} -1.08913 q^{65} +12.2202 q^{67} +8.22018 q^{69} -0.935945 q^{71} -4.12811 q^{73} -8.33028 q^{75} +3.06406 q^{77} -2.95396 q^{79} -8.85977 q^{81} -13.5655 q^{83} -2.43447 q^{85} -13.4554 q^{87} +1.06406 q^{89} -2.78276 q^{91} -2.31226 q^{93} -0.391382 q^{95} +4.50147 q^{97} -0.141060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 5 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} - 3 q^{19} - q^{21} + 2 q^{23} + 4 q^{25} - 28 q^{27} - 5 q^{29} - 4 q^{31} - 15 q^{33} - 5 q^{35} - 7 q^{37} - 2 q^{39}+ \cdots + 55 q^{99}+O(q^{100}) \) 3 * q - q^3 - 5 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 6 * q^17 - 3 * q^19 - q^21 + 2 * q^23 + 4 * q^25 - 28 * q^27 - 5 * q^29 - 4 * q^31 - 15 * q^33 - 5 * q^35 - 7 * q^37 - 2 * q^39 - 7 * q^41 - q^43 + 11 * q^47 + 3 * q^49 - 32 * q^51 - 3 * q^53 + 16 * q^55 + q^57 - 3 * q^59 - 7 * q^61 + 6 * q^63 - 28 * q^65 + 12 * q^67 - 9 * q^71 + 12 * q^75 + 3 * q^77 - 15 * q^79 + 35 * q^81 - 16 * q^83 - 32 * q^85 - 28 * q^87 - 3 * q^89 + 4 * q^91 + 30 * q^93 + 5 * q^95 - 5 * q^97 + 55 * q^99

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