LMFDB - Embedded newform 8512.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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			1.1
Root			$0.772866$ 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)

$f(q)$	$=$	$q-3.40268 q^{3} -1.22713 q^{5} +1.00000 q^{7} +8.57822 q^{9} +O(q^{10})$	$q-3.40268 q^{3} -1.22713 q^{5} +1.00000 q^{7} +8.57822 q^{9} +4.94841 q^{11} +0.454269 q^{13} +4.17554 q^{15} +7.25963 q^{17} -1.00000 q^{19} -3.40268 q^{21} +1.54573 q^{23} -3.49414 q^{25} -18.9809 q^{27} +4.03249 q^{29} -8.35109 q^{31} -16.8378 q^{33} -1.22713 q^{35} -8.03249 q^{37} -1.54573 q^{39} -2.77287 q^{41} -6.03249 q^{43} -10.5266 q^{45} +0.597321 q^{47} +1.00000 q^{49} -24.7022 q^{51} +8.20804 q^{53} -6.07236 q^{55} +3.40268 q^{57} -2.31860 q^{59} -0.143052 q^{61} +8.57822 q^{63} -0.557449 q^{65} -1.25963 q^{67} -5.25963 q^{69} +0.948410 q^{71} -7.89682 q^{73} +11.8894 q^{75} +4.94841 q^{77} -11.5782 q^{79} +38.8512 q^{81} -7.09146 q^{83} -8.90854 q^{85} -13.7213 q^{87} +2.94841 q^{89} +0.454269 q^{91} +28.4161 q^{93} +1.22713 q^{95} -3.85695 q^{97} +42.4486 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 5 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q - q^3 - 5 * q^5 + 3 * q^7 + 6 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	−3.40268	−1.96454	−0.982269	−	0.187478i	$-0.939969\pi$
$3$	−3.40268	−1.96454	−0.982269	+	0.187478i	$0.939969\pi$
$4$	0	0
$5$	−1.22713	−0.548791	−0.274396	−	0.961617i	$-0.588478\pi$
$5$	−1.22713	−0.548791	−0.274396	+	0.961617i	$0.588478\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.57822	2.85941
$10$	0	0
$11$	4.94841	1.49200	0.746001	−	0.665945i	$-0.231971\pi$
$11$	4.94841	1.49200	0.746001	+	0.665945i	$0.231971\pi$
$12$	0	0
$13$	0.454269	0.125992	0.0629958	−	0.998014i	$-0.479935\pi$
$13$	0.454269	0.125992	0.0629958	+	0.998014i	$0.479935\pi$
$14$	0	0
$15$	4.17554	1.07812
$16$	0	0
$17$	7.25963	1.76072	0.880359	−	0.474308i	$-0.157302\pi$
$17$	7.25963	1.76072	0.880359	+	0.474308i	$0.157302\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.40268	−0.742525
$22$	0	0
$23$	1.54573	0.322307	0.161154	−	0.986929i	$-0.448479\pi$
$23$	1.54573	0.322307	0.161154	+	0.986929i	$0.448479\pi$
$24$	0	0
$25$	−3.49414	−0.698828
$26$	0	0
$27$	−18.9809	−3.65288
$28$	0	0
$29$	4.03249	0.748815	0.374407	−	0.927264i	$-0.377846\pi$
$29$	4.03249	0.748815	0.374407	+	0.927264i	$0.377846\pi$
$30$	0	0
$31$	−8.35109	−1.49990	−0.749950	−	0.661495i	$-0.769922\pi$
$31$	−8.35109	−1.49990	−0.749950	+	0.661495i	$0.769922\pi$
$32$	0	0
$33$	−16.8378	−2.93109
$34$	0	0
$35$	−1.22713	−0.207424
$36$	0	0
$37$	−8.03249	−1.32053	−0.660267	−	0.751031i	$-0.729557\pi$
$37$	−8.03249	−1.32053	−0.660267	+	0.751031i	$0.729557\pi$
$38$	0	0
$39$	−1.54573	−0.247515
$40$	0	0
$41$	−2.77287	−0.433049	−0.216524	−	0.976277i	$-0.569472\pi$
$41$	−2.77287	−0.433049	−0.216524	+	0.976277i	$0.569472\pi$
$42$	0	0
$43$	−6.03249	−0.919946	−0.459973	−	0.887933i	$-0.652141\pi$
$43$	−6.03249	−0.919946	−0.459973	+	0.887933i	$0.652141\pi$
$44$	0	0
$45$	−10.5266	−1.56922
$46$	0	0
$47$	0.597321	0.0871282	0.0435641	−	0.999051i	$-0.486129\pi$
$47$	0.597321	0.0871282	0.0435641	+	0.999051i	$0.486129\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−24.7022	−3.45900
$52$	0	0
$53$	8.20804	1.12746	0.563730	−	0.825959i	$-0.309366\pi$
$53$	8.20804	1.12746	0.563730	+	0.825959i	$0.309366\pi$
$54$	0	0
$55$	−6.07236	−0.818797
$56$	0	0
$57$	3.40268	0.450696
$58$	0	0
$59$	−2.31860	−0.301856	−0.150928	−	0.988545i	$-0.548226\pi$
$59$	−2.31860	−0.301856	−0.150928	+	0.988545i	$0.548226\pi$
$60$	0	0
$61$	−0.143052	−0.0183160	−0.00915798	−	0.999958i	$-0.502915\pi$
$61$	−0.143052	−0.0183160	−0.00915798	+	0.999958i	$0.502915\pi$
$62$	0	0
$63$	8.57822	1.08075
$64$	0	0
$65$	−0.557449	−0.0691430
$66$	0	0
$67$	−1.25963	−0.153888	−0.0769439	−	0.997035i	$-0.524516\pi$
$67$	−1.25963	−0.153888	−0.0769439	+	0.997035i	$0.524516\pi$
$68$	0	0
$69$	−5.25963	−0.633185
$70$	0	0
$71$	0.948410	0.112556	0.0562778	−	0.998415i	$-0.482077\pi$
$71$	0.948410	0.112556	0.0562778	+	0.998415i	$0.482077\pi$
$72$	0	0
$73$	−7.89682	−0.924253	−0.462126	−	0.886814i	$-0.652913\pi$
$73$	−7.89682	−0.924253	−0.462126	+	0.886814i	$0.652913\pi$
$74$	0	0
$75$	11.8894	1.37287
$76$	0	0
$77$	4.94841	0.563924
$78$	0	0
$79$	−11.5782	−1.30265	−0.651326	−	0.758798i	$-0.725787\pi$
$79$	−11.5782	−1.30265	−0.651326	+	0.758798i	$0.725787\pi$
$80$	0	0
$81$	38.8512	4.31680
$82$	0	0
$83$	−7.09146	−0.778389	−0.389195	−	0.921156i	$-0.627247\pi$
$83$	−7.09146	−0.778389	−0.389195	+	0.921156i	$0.627247\pi$
$84$	0	0
$85$	−8.90854	−0.966267
$86$	0	0
$87$	−13.7213	−1.47108
$88$	0	0
$89$	2.94841	0.312531	0.156265	−	0.987715i	$-0.450054\pi$
$89$	2.94841	0.312531	0.156265	+	0.987715i	$0.450054\pi$
$90$	0	0
$91$	0.454269	0.0476203
$92$	0	0
$93$	28.4161	2.94661
$94$	0	0
$95$	1.22713	0.125901
$96$	0	0
$97$	−3.85695	−0.391614	−0.195807	−	0.980642i	$-0.562733\pi$
$97$	−3.85695	−0.391614	−0.195807	+	0.980642i	$0.562733\pi$
$98$	0	0
$99$	42.4486	4.26624
$100$	0	0
$101$	−4.18292	−0.416217	−0.208108	−	0.978106i	$-0.566731\pi$
$101$	−4.18292	−0.416217	−0.208108	+	0.978106i	$0.566731\pi$
$102$	0	0
$103$	−15.1564	−1.49341	−0.746705	−	0.665156i	$-0.768365\pi$
$103$	−15.1564	−1.49341	−0.746705	+	0.665156i	$0.768365\pi$
$104$	0	0
$105$	4.17554	0.407491
$106$	0	0
$107$	3.15645	0.305145	0.152573	−	0.988292i	$-0.451244\pi$
$107$	3.15645	0.305145	0.152573	+	0.988292i	$0.451244\pi$
$108$	0	0
$109$	0.208036	0.0199263	0.00996314	−	0.999950i	$-0.496829\pi$
$109$	0.208036	0.0199263	0.00996314	+	0.999950i	$0.496829\pi$
$110$	0	0
$111$	27.3320	2.59424
$112$	0	0
$113$	−0.168164	−0.0158196	−0.00790978	−	0.999969i	$-0.502518\pi$
$113$	−0.168164	−0.0158196	−0.00790978	+	0.999969i	$0.502518\pi$
$114$	0	0
$115$	−1.89682	−0.176879
$116$	0	0
$117$	3.89682	0.360261
$118$	0	0
$119$	7.25963	0.665489
$120$	0	0
$121$	13.4868	1.22607
$122$	0	0
$123$	9.43517	0.850741
$124$	0	0
$125$	10.4235	0.932302
$126$	0	0
$127$	5.85695	0.519720	0.259860	−	0.965646i	$-0.416324\pi$
$127$	5.85695	0.519720	0.259860	+	0.965646i	$0.416324\pi$
$128$	0	0
$129$	20.5266	1.80727
$130$	0	0
$131$	−5.19464	−0.453858	−0.226929	−	0.973911i	$-0.572869\pi$
$131$	−5.19464	−0.453858	−0.226929	+	0.973911i	$0.572869\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	23.2921	2.00467
$136$	0	0
$137$	−15.1889	−1.29768	−0.648839	−	0.760925i	$-0.724745\pi$
$137$	−15.1889	−1.29768	−0.648839	+	0.760925i	$0.724745\pi$
$138$	0	0
$139$	−9.61072	−0.815170	−0.407585	−	0.913167i	$-0.633629\pi$
$139$	−9.61072	−0.815170	−0.407585	+	0.913167i	$0.633629\pi$
$140$	0	0
$141$	−2.03249	−0.171167
$142$	0	0
$143$	2.24791	0.187980
$144$	0	0
$145$	−4.94841	−0.410943
$146$	0	0
$147$	−3.40268	−0.280648
$148$	0	0
$149$	15.2596	1.25012	0.625059	−	0.780578i	$-0.285075\pi$
$149$	15.2596	1.25012	0.625059	+	0.780578i	$0.285075\pi$
$150$	0	0
$151$	8.70218	0.708173	0.354087	−	0.935213i	$-0.384792\pi$
$151$	8.70218	0.708173	0.354087	+	0.935213i	$0.384792\pi$
$152$	0	0
$153$	62.2747	5.03461
$154$	0	0
$155$	10.2479	0.823132
$156$	0	0
$157$	13.4677	1.07484	0.537418	−	0.843316i	$-0.319400\pi$
$157$	13.4677	1.07484	0.537418	+	0.843316i	$0.319400\pi$
$158$	0	0
$159$	−27.9293	−2.21494
$160$	0	0
$161$	1.54573	0.121821
$162$	0	0
$163$	−0.662305	−0.0518758	−0.0259379	−	0.999664i	$-0.508257\pi$
$163$	−0.662305	−0.0518758	−0.0259379	+	0.999664i	$0.508257\pi$
$164$	0	0
$165$	20.6623	1.60856
$166$	0	0
$167$	−13.9618	−1.08040	−0.540198	−	0.841538i	$-0.681651\pi$
$167$	−13.9618	−1.08040	−0.540198	+	0.841538i	$0.681651\pi$
$168$	0	0
$169$	−12.7936	−0.984126
$170$	0	0
$171$	−8.57822	−0.655993
$172$	0	0
$173$	11.8968	0.904498	0.452249	−	0.891892i	$-0.350622\pi$
$173$	11.8968	0.904498	0.452249	+	0.891892i	$0.350622\pi$
$174$	0	0
$175$	−3.49414	−0.264132
$176$	0	0
$177$	7.88944	0.593007
$178$	0	0
$179$	−22.2479	−1.66289	−0.831443	−	0.555609i	$-0.812485\pi$
$179$	−22.2479	−1.66289	−0.831443	+	0.555609i	$0.812485\pi$
$180$	0	0
$181$	14.4161	1.07154	0.535769	−	0.844365i	$-0.320022\pi$
$181$	14.4161	1.07154	0.535769	+	0.844365i	$0.320022\pi$
$182$	0	0
$183$	0.486761	0.0359824
$184$	0	0
$185$	9.85695	0.724697
$186$	0	0
$187$	35.9236	2.62699
$188$	0	0
$189$	−18.9809	−1.38066
$190$	0	0
$191$	26.2479	1.89923	0.949616	−	0.313416i	$-0.101473\pi$
$191$	26.2479	1.89923	0.949616	+	0.313416i	$0.101473\pi$
$192$	0	0
$193$	−9.71390	−0.699221	−0.349611	−	0.936895i	$-0.613686\pi$
$193$	−9.71390	−0.699221	−0.349611	+	0.936895i	$0.613686\pi$
$194$	0	0
$195$	1.89682	0.135834
$196$	0	0
$197$	−16.5193	−1.17695	−0.588474	−	0.808516i	$-0.700271\pi$
$197$	−16.5193	−1.17695	−0.588474	+	0.808516i	$0.700271\pi$
$198$	0	0
$199$	−21.7464	−1.54156	−0.770780	−	0.637101i	$-0.780133\pi$
$199$	−21.7464	−1.54156	−0.770780	+	0.637101i	$0.780133\pi$
$200$	0	0
$201$	4.28610	0.302319
$202$	0	0
$203$	4.03249	0.283025
$204$	0	0
$205$	3.40268	0.237653
$206$	0	0
$207$	13.2596	0.921608
$208$	0	0
$209$	−4.94841	−0.342289
$210$	0	0
$211$	4.70218	0.323711	0.161856	−	0.986814i	$-0.448252\pi$
$211$	4.70218	0.323711	0.161856	+	0.986814i	$0.448252\pi$
$212$	0	0
$213$	−3.22713	−0.221120
$214$	0	0
$215$	7.40268	0.504859
$216$	0	0
$217$	−8.35109	−0.566909
$218$	0	0
$219$	26.8703	1.81573
$220$	0	0
$221$	3.29782	0.221836
$222$	0	0
$223$	−28.4161	−1.90288	−0.951440	−	0.307833i	$-0.900396\pi$
$223$	−28.4161	−1.90288	−0.951440	+	0.307833i	$0.900396\pi$
$224$	0	0
$225$	−29.9735	−1.99823
$226$	0	0
$227$	−14.5193	−0.963677	−0.481838	−	0.876260i	$-0.660031\pi$
$227$	−14.5193	−0.963677	−0.481838	+	0.876260i	$0.660031\pi$
$228$	0	0
$229$	−17.2271	−1.13840	−0.569201	−	0.822199i	$-0.692747\pi$
$229$	−17.2271	−1.13840	−0.569201	+	0.822199i	$0.692747\pi$
$230$	0	0
$231$	−16.8378	−1.10785
$232$	0	0
$233$	6.31122	0.413462	0.206731	−	0.978398i	$-0.433718\pi$
$233$	6.31122	0.413462	0.206731	+	0.978398i	$0.433718\pi$
$234$	0	0
$235$	−0.732993	−0.0478152
$236$	0	0
$237$	39.3970	2.55911
$238$	0	0
$239$	1.89682	0.122695	0.0613475	−	0.998116i	$-0.480460\pi$
$239$	1.89682	0.122695	0.0613475	+	0.998116i	$0.480460\pi$
$240$	0	0
$241$	−3.61642	−0.232954	−0.116477	−	0.993193i	$-0.537160\pi$
$241$	−3.61642	−0.232954	−0.116477	+	0.993193i	$0.537160\pi$
$242$	0	0
$243$	−75.2556	−4.82765
$244$	0	0
$245$	−1.22713	−0.0783987
$246$	0	0
$247$	−0.454269	−0.0289044
$248$	0	0
$249$	24.1300	1.52917
$250$	0	0
$251$	8.98828	0.567335	0.283668	−	0.958923i	$-0.408449\pi$
$251$	8.98828	0.567335	0.283668	+	0.958923i	$0.408449\pi$
$252$	0	0
$253$	7.64891	0.480883
$254$	0	0
$255$	30.3129	1.89827
$256$	0	0
$257$	−7.68140	−0.479153	−0.239576	−	0.970878i	$-0.577009\pi$
$257$	−7.68140	−0.479153	−0.239576	+	0.970878i	$0.577009\pi$
$258$	0	0
$259$	−8.03249	−0.499115
$260$	0	0
$261$	34.5916	2.14117
$262$	0	0
$263$	−14.2331	−0.877654	−0.438827	−	0.898572i	$-0.644606\pi$
$263$	−14.2331	−0.877654	−0.438827	+	0.898572i	$0.644606\pi$
$264$	0	0
$265$	−10.0724	−0.618740
$266$	0	0
$267$	−10.0325	−0.613979
$268$	0	0
$269$	−16.5193	−1.00720	−0.503598	−	0.863938i	$-0.667991\pi$
$269$	−16.5193	−1.00720	−0.503598	+	0.863938i	$0.667991\pi$
$270$	0	0
$271$	19.9293	1.21062	0.605310	−	0.795990i	$-0.293049\pi$
$271$	19.9293	1.21062	0.605310	+	0.795990i	$0.293049\pi$
$272$	0	0
$273$	−1.54573	−0.0935519
$274$	0	0
$275$	−17.2904	−1.04265
$276$	0	0
$277$	−28.9353	−1.73856	−0.869278	−	0.494324i	$-0.835416\pi$
$277$	−28.9353	−1.73856	−0.869278	+	0.494324i	$0.835416\pi$
$278$	0	0
$279$	−71.6375	−4.28883
$280$	0	0
$281$	−13.6489	−0.814226	−0.407113	−	0.913378i	$-0.633464\pi$
$281$	−13.6489	−0.814226	−0.407113	+	0.913378i	$0.633464\pi$
$282$	0	0
$283$	4.70218	0.279515	0.139758	−	0.990186i	$-0.455368\pi$
$283$	4.70218	0.279515	0.139758	+	0.990186i	$0.455368\pi$
$284$	0	0
$285$	−4.17554	−0.247338
$286$	0	0
$287$	−2.77287	−0.163677
$288$	0	0
$289$	35.7022	2.10013
$290$	0	0
$291$	13.1240	0.769340
$292$	0	0
$293$	22.0650	1.28905	0.644525	−	0.764583i	$-0.277055\pi$
$293$	22.0650	1.28905	0.644525	+	0.764583i	$0.277055\pi$
$294$	0	0
$295$	2.84523	0.165656
$296$	0	0
$297$	−93.9253	−5.45010
$298$	0	0
$299$	0.702178	0.0406080
$300$	0	0
$301$	−6.03249	−0.347707
$302$	0	0
$303$	14.2331	0.817673
$304$	0	0
$305$	0.175544	0.0100516
$306$	0	0
$307$	−10.0473	−0.573427	−0.286713	−	0.958016i	$-0.592563\pi$
$307$	−10.0473	−0.573427	−0.286713	+	0.958016i	$0.592563\pi$
$308$	0	0
$309$	51.5725	2.93386
$310$	0	0
$311$	−10.4941	−0.595068	−0.297534	−	0.954711i	$-0.596164\pi$
$311$	−10.4941	−0.595068	−0.297534	+	0.954711i	$0.596164\pi$
$312$	0	0
$313$	−5.77888	−0.326642	−0.163321	−	0.986573i	$-0.552221\pi$
$313$	−5.77888	−0.326642	−0.163321	+	0.986573i	$0.552221\pi$
$314$	0	0
$315$	−10.5266	−0.593109
$316$	0	0
$317$	−12.9102	−0.725110	−0.362555	−	0.931962i	$-0.618095\pi$
$317$	−12.9102	−0.725110	−0.362555	+	0.931962i	$0.618095\pi$
$318$	0	0
$319$	19.9544	1.11723
$320$	0	0
$321$	−10.7404	−0.599469
$322$	0	0
$323$	−7.25963	−0.403936
$324$	0	0
$325$	−1.58728	−0.0880464
$326$	0	0
$327$	−0.707881	−0.0391459
$328$	0	0
$329$	0.597321	0.0329314
$330$	0	0
$331$	30.8703	1.69679	0.848394	−	0.529366i	$-0.177570\pi$
$331$	30.8703	1.69679	0.848394	+	0.529366i	$0.177570\pi$
$332$	0	0
$333$	−68.9045	−3.77594
$334$	0	0
$335$	1.54573	0.0844523
$336$	0	0
$337$	19.6757	1.07180	0.535902	−	0.844280i	$-0.319972\pi$
$337$	19.6757	1.07180	0.535902	+	0.844280i	$0.319972\pi$
$338$	0	0
$339$	0.572209	0.0310781
$340$	0	0
$341$	−41.3246	−2.23785
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.45427	0.347486
$346$	0	0
$347$	−0.908538	−0.0487729	−0.0243864	−	0.999703i	$-0.507763\pi$
$347$	−0.908538	−0.0487729	−0.0243864	+	0.999703i	$0.507763\pi$
$348$	0	0
$349$	−12.1829	−0.652137	−0.326068	−	0.945346i	$-0.605724\pi$
$349$	−12.1829	−0.652137	−0.326068	+	0.945346i	$0.605724\pi$
$350$	0	0
$351$	−8.62243	−0.460231
$352$	0	0
$353$	23.9618	1.27536	0.637679	−	0.770302i	$-0.279895\pi$
$353$	23.9618	1.27536	0.637679	+	0.770302i	$0.279895\pi$
$354$	0	0
$355$	−1.16383	−0.0617695
$356$	0	0
$357$	−24.7022	−1.30738
$358$	0	0
$359$	12.7672	0.673825	0.336913	−	0.941536i	$-0.390617\pi$
$359$	12.7672	0.673825	0.336913	+	0.941536i	$0.390617\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−45.8911	−2.40866
$364$	0	0
$365$	9.69046	0.507222
$366$	0	0
$367$	3.68878	0.192553	0.0962765	−	0.995355i	$-0.469307\pi$
$367$	3.68878	0.192553	0.0962765	+	0.995355i	$0.469307\pi$
$368$	0	0
$369$	−23.7863	−1.23826
$370$	0	0
$371$	8.20804	0.426140
$372$	0	0
$373$	−25.0060	−1.29476	−0.647381	−	0.762166i	$-0.724136\pi$
$373$	−25.0060	−1.29476	−0.647381	+	0.762166i	$0.724136\pi$
$374$	0	0
$375$	−35.4677	−1.83154
$376$	0	0
$377$	1.83184	0.0943443
$378$	0	0
$379$	0.286105	0.0146962	0.00734810	−	0.999973i	$-0.497661\pi$
$379$	0.286105	0.0146962	0.00734810	+	0.999973i	$0.497661\pi$
$380$	0	0
$381$	−19.9293	−1.02101
$382$	0	0
$383$	0.973522	0.0497446	0.0248723	−	0.999691i	$-0.492082\pi$
$383$	0.973522	0.0497446	0.0248723	+	0.999691i	$0.492082\pi$
$384$	0	0
$385$	−6.07236	−0.309476
$386$	0	0
$387$	−51.7481	−2.63050
$388$	0	0
$389$	−0.519253	−0.0263272	−0.0131636	−	0.999913i	$-0.504190\pi$
$389$	−0.519253	−0.0263272	−0.0131636	+	0.999913i	$0.504190\pi$
$390$	0	0
$391$	11.2214	0.567492
$392$	0	0
$393$	17.6757	0.891621
$394$	0	0
$395$	14.2080	0.714884
$396$	0	0
$397$	23.2847	1.16863	0.584314	−	0.811528i	$-0.301364\pi$
$397$	23.2847	1.16863	0.584314	+	0.811528i	$0.301364\pi$
$398$	0	0
$399$	3.40268	0.170347
$400$	0	0
$401$	14.6874	0.733455	0.366727	−	0.930328i	$-0.380478\pi$
$401$	14.6874	0.733455	0.366727	+	0.930328i	$0.380478\pi$
$402$	0	0
$403$	−3.79364	−0.188975
$404$	0	0
$405$	−47.6757	−2.36902
$406$	0	0
$407$	−39.7481	−1.97024
$408$	0	0
$409$	−9.46766	−0.468146	−0.234073	−	0.972219i	$-0.575205\pi$
$409$	−9.46766	−0.468146	−0.234073	+	0.972219i	$0.575205\pi$
$410$	0	0
$411$	51.6831	2.54934
$412$	0	0
$413$	−2.31860	−0.114091
$414$	0	0
$415$	8.70218	0.427173
$416$	0	0
$417$	32.7022	1.60143
$418$	0	0
$419$	−33.3896	−1.63119	−0.815594	−	0.578624i	$-0.803590\pi$
$419$	−33.3896	−1.63119	−0.815594	+	0.578624i	$0.803590\pi$
$420$	0	0
$421$	22.9085	1.11649	0.558247	−	0.829675i	$-0.311474\pi$
$421$	22.9085	1.11649	0.558247	+	0.829675i	$0.311474\pi$
$422$	0	0
$423$	5.12395	0.249135
$424$	0	0
$425$	−25.3662	−1.23044
$426$	0	0
$427$	−0.143052	−0.00692279
$428$	0	0
$429$	−7.64891	−0.369293
$430$	0	0
$431$	32.5859	1.56961	0.784804	−	0.619744i	$-0.212764\pi$
$431$	32.5859	1.56961	0.784804	+	0.619744i	$0.212764\pi$
$432$	0	0
$433$	−3.61642	−0.173794	−0.0868970	−	0.996217i	$-0.527695\pi$
$433$	−3.61642	−0.173794	−0.0868970	+	0.996217i	$0.527695\pi$
$434$	0	0
$435$	16.8378	0.807313
$436$	0	0
$437$	−1.54573	−0.0739423
$438$	0	0
$439$	19.4426	0.927942	0.463971	−	0.885850i	$-0.346424\pi$
$439$	19.4426	0.927942	0.463971	+	0.885850i	$0.346424\pi$
$440$	0	0
$441$	8.57822	0.408487
$442$	0	0
$443$	−5.76115	−0.273720	−0.136860	−	0.990590i	$-0.543701\pi$
$443$	−5.76115	−0.273720	−0.136860	+	0.990590i	$0.543701\pi$
$444$	0	0
$445$	−3.61810	−0.171514
$446$	0	0
$447$	−51.9236	−2.45590
$448$	0	0
$449$	1.72866	0.0815803	0.0407902	−	0.999168i	$-0.487012\pi$
$449$	1.72866	0.0815803	0.0407902	+	0.999168i	$0.487012\pi$
$450$	0	0
$451$	−13.7213	−0.646110
$452$	0	0
$453$	−29.6107	−1.39123
$454$	0	0
$455$	−0.557449	−0.0261336
$456$	0	0
$457$	−10.9336	−0.511455	−0.255727	−	0.966749i	$-0.582315\pi$
$457$	−10.9336	−0.511455	−0.255727	+	0.966749i	$0.582315\pi$
$458$	0	0
$459$	−137.794	−6.43169
$460$	0	0
$461$	27.0784	1.26117	0.630583	−	0.776122i	$-0.282816\pi$
$461$	27.0784	1.26117	0.630583	+	0.776122i	$0.282816\pi$
$462$	0	0
$463$	9.32461	0.433351	0.216676	−	0.976244i	$-0.430479\pi$
$463$	9.32461	0.433351	0.216676	+	0.976244i	$0.430479\pi$
$464$	0	0
$465$	−34.8703	−1.61707
$466$	0	0
$467$	26.5842	1.23017	0.615086	−	0.788460i	$-0.289121\pi$
$467$	26.5842	1.23017	0.615086	+	0.788460i	$0.289121\pi$
$468$	0	0
$469$	−1.25963	−0.0581641
$470$	0	0
$471$	−45.8261	−2.11156
$472$	0	0
$473$	−29.8512	−1.37256
$474$	0	0
$475$	3.49414	0.160322
$476$	0	0
$477$	70.4104	3.22387
$478$	0	0
$479$	−36.9028	−1.68613	−0.843067	−	0.537809i	$-0.819252\pi$
$479$	−36.9028	−1.68613	−0.843067	+	0.537809i	$0.819252\pi$
$480$	0	0
$481$	−3.64891	−0.166376
$482$	0	0
$483$	−5.25963	−0.239321
$484$	0	0
$485$	4.73299	0.214914
$486$	0	0
$487$	−34.4338	−1.56034	−0.780172	−	0.625565i	$-0.784869\pi$
$487$	−34.4338	−1.56034	−0.780172	+	0.625565i	$0.784869\pi$
$488$	0	0
$489$	2.25361	0.101912
$490$	0	0
$491$	33.0385	1.49101	0.745503	−	0.666502i	$-0.232209\pi$
$491$	33.0385	1.49101	0.745503	+	0.666502i	$0.232209\pi$
$492$	0	0
$493$	29.2744	1.31845
$494$	0	0
$495$	−52.0901	−2.34128
$496$	0	0
$497$	0.948410	0.0425420
$498$	0	0
$499$	38.7347	1.73400	0.867001	−	0.498306i	$-0.166045\pi$
$499$	38.7347	1.73400	0.867001	+	0.498306i	$0.166045\pi$
$500$	0	0
$501$	47.5075	2.12248
$502$	0	0
$503$	10.5250	0.469285	0.234642	−	0.972082i	$-0.424608\pi$
$503$	10.5250	0.469285	0.234642	+	0.972082i	$0.424608\pi$
$504$	0	0
$505$	5.13301	0.228416
$506$	0	0
$507$	43.5326	1.93335
$508$	0	0
$509$	14.9883	0.664344	0.332172	−	0.943219i	$-0.392219\pi$
$509$	14.9883	0.664344	0.332172	+	0.943219i	$0.392219\pi$
$510$	0	0
$511$	−7.89682	−0.349335
$512$	0	0
$513$	18.9809	0.838027
$514$	0	0
$515$	18.5990	0.819570
$516$	0	0
$517$	2.95579	0.129995
$518$	0	0
$519$	−40.4811	−1.77692
$520$	0	0
$521$	34.1300	1.49526	0.747631	−	0.664115i	$-0.231191\pi$
$521$	34.1300	1.49526	0.747631	+	0.664115i	$0.231191\pi$
$522$	0	0
$523$	−17.4807	−0.764380	−0.382190	−	0.924084i	$-0.624830\pi$
$523$	−17.4807	−0.764380	−0.382190	+	0.924084i	$0.624830\pi$
$524$	0	0
$525$	11.8894	0.518898
$526$	0	0
$527$	−60.6258	−2.64090
$528$	0	0
$529$	−20.6107	−0.896118
$530$	0	0
$531$	−19.8894	−0.863128
$532$	0	0
$533$	−1.25963	−0.0545605
$534$	0	0
$535$	−3.87338	−0.167461
$536$	0	0
$537$	75.7025	3.26680
$538$	0	0
$539$	4.94841	0.213143
$540$	0	0
$541$	−28.8054	−1.23844	−0.619220	−	0.785218i	$-0.712551\pi$
$541$	−28.8054	−1.23844	−0.619220	+	0.785218i	$0.712551\pi$
$542$	0	0
$543$	−49.0533	−2.10508
$544$	0	0
$545$	−0.255289	−0.0109354
$546$	0	0
$547$	25.4693	1.08899	0.544495	−	0.838764i	$-0.316721\pi$
$547$	25.4693	1.08899	0.544495	+	0.838764i	$0.316721\pi$
$548$	0	0
$549$	−1.22713	−0.0523728
$550$	0	0
$551$	−4.03249	−0.171790
$552$	0	0
$553$	−11.5782	−0.492356
$554$	0	0
$555$	−33.5400	−1.42369
$556$	0	0
$557$	−30.4308	−1.28940	−0.644698	−	0.764437i	$-0.723017\pi$
$557$	−30.4308	−1.28940	−0.644698	+	0.764437i	$0.723017\pi$
$558$	0	0
$559$	−2.74037	−0.115905
$560$	0	0
$561$	−122.237	−5.16083
$562$	0	0
$563$	−5.84219	−0.246219	−0.123109	−	0.992393i	$-0.539287\pi$
$563$	−5.84219	−0.246219	−0.123109	+	0.992393i	$0.539287\pi$
$564$	0	0
$565$	0.206360	0.00868164
$566$	0	0
$567$	38.8512	1.63160
$568$	0	0
$569$	−10.9883	−0.460653	−0.230326	−	0.973113i	$-0.573979\pi$
$569$	−10.9883	−0.460653	−0.230326	+	0.973113i	$0.573979\pi$
$570$	0	0
$571$	0.662305	0.0277166	0.0138583	−	0.999904i	$-0.495589\pi$
$571$	0.662305	0.0277166	0.0138583	+	0.999904i	$0.495589\pi$
$572$	0	0
$573$	−89.3132	−3.73111
$574$	0	0
$575$	−5.40100	−0.225237
$576$	0	0
$577$	−39.9766	−1.66425	−0.832123	−	0.554591i	$-0.812875\pi$
$577$	−39.9766	−1.66425	−0.832123	+	0.554591i	$0.812875\pi$
$578$	0	0
$579$	33.0533	1.37365
$580$	0	0
$581$	−7.09146	−0.294203
$582$	0	0
$583$	40.6167	1.68217
$584$	0	0
$585$	−4.78192	−0.197708
$586$	0	0
$587$	−24.6372	−1.01689	−0.508443	−	0.861096i	$-0.669779\pi$
$587$	−24.6372	−1.01689	−0.508443	+	0.861096i	$0.669779\pi$
$588$	0	0
$589$	8.35109	0.344101
$590$	0	0
$591$	56.2097	2.31216
$592$	0	0
$593$	16.8054	0.690113	0.345057	−	0.938582i	$-0.387860\pi$
$593$	16.8054	0.690113	0.345057	+	0.938582i	$0.387860\pi$
$594$	0	0
$595$	−8.90854	−0.365214
$596$	0	0
$597$	73.9960	3.02845
$598$	0	0
$599$	4.03987	0.165065	0.0825324	−	0.996588i	$-0.473699\pi$
$599$	4.03987	0.165065	0.0825324	+	0.996588i	$0.473699\pi$
$600$	0	0
$601$	−16.5193	−0.673834	−0.336917	−	0.941534i	$-0.609384\pi$
$601$	−16.5193	−0.673834	−0.336917	+	0.941534i	$0.609384\pi$
$602$	0	0
$603$	−10.8054	−0.440028
$604$	0	0
$605$	−16.5501	−0.672856
$606$	0	0
$607$	24.6224	0.999394	0.499697	−	0.866200i	$-0.333445\pi$
$607$	24.6224	0.999394	0.499697	+	0.866200i	$0.333445\pi$
$608$	0	0
$609$	−13.7213	−0.556014
$610$	0	0
$611$	0.271344	0.0109774
$612$	0	0
$613$	7.12966	0.287964	0.143982	−	0.989580i	$-0.454009\pi$
$613$	7.12966	0.287964	0.143982	+	0.989580i	$0.454009\pi$
$614$	0	0
$615$	−11.5782	−0.466879
$616$	0	0
$617$	−32.1625	−1.29481	−0.647406	−	0.762145i	$-0.724146\pi$
$617$	−32.1625	−1.29481	−0.647406	+	0.762145i	$0.724146\pi$
$618$	0	0
$619$	−22.9501	−0.922442	−0.461221	−	0.887285i	$-0.652588\pi$
$619$	−22.9501	−0.922442	−0.461221	+	0.887285i	$0.652588\pi$
$620$	0	0
$621$	−29.3394	−1.17735
$622$	0	0
$623$	2.94841	0.118126
$624$	0	0
$625$	4.67973	0.187189
$626$	0	0
$627$	16.8378	0.672439
$628$	0	0
$629$	−58.3129	−2.32509
$630$	0	0
$631$	−3.66367	−0.145848	−0.0729242	−	0.997337i	$-0.523233\pi$
$631$	−3.66367	−0.145848	−0.0729242	+	0.997337i	$0.523233\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	0	0
$635$	−7.18726	−0.285218
$636$	0	0
$637$	0.454269	0.0179988
$638$	0	0
$639$	8.13567	0.321842
$640$	0	0
$641$	−34.5460	−1.36449	−0.682243	−	0.731125i	$-0.738996\pi$
$641$	−34.5460	−1.36449	−0.682243	+	0.731125i	$0.738996\pi$
$642$	0	0
$643$	28.7022	1.13190	0.565952	−	0.824438i	$-0.308509\pi$
$643$	28.7022	1.13190	0.565952	+	0.824438i	$0.308509\pi$
$644$	0	0
$645$	−25.1889	−0.991813
$646$	0	0
$647$	12.2007	0.479657	0.239829	−	0.970815i	$-0.422909\pi$
$647$	12.2007	0.479657	0.239829	+	0.970815i	$0.422909\pi$
$648$	0	0
$649$	−11.4734	−0.450369
$650$	0	0
$651$	28.4161	1.11371
$652$	0	0
$653$	−33.5223	−1.31183	−0.655914	−	0.754835i	$-0.727717\pi$
$653$	−33.5223	−1.31183	−0.655914	+	0.754835i	$0.727717\pi$
$654$	0	0
$655$	6.37452	0.249073
$656$	0	0
$657$	−67.7407	−2.64282
$658$	0	0
$659$	−33.7407	−1.31435	−0.657175	−	0.753738i	$-0.728249\pi$
$659$	−33.7407	−1.31435	−0.657175	+	0.753738i	$0.728249\pi$
$660$	0	0
$661$	−15.2449	−0.592957	−0.296478	−	0.955040i	$-0.595812\pi$
$661$	−15.2449	−0.592957	−0.296478	+	0.955040i	$0.595812\pi$
$662$	0	0
$663$	−11.2214	−0.435804
$664$	0	0
$665$	1.22713	0.0475862
$666$	0	0
$667$	6.23315	0.241348
$668$	0	0
$669$	96.6908	3.73828
$670$	0	0
$671$	−0.707881	−0.0273275
$672$	0	0
$673$	−46.4308	−1.78978	−0.894889	−	0.446290i	$-0.852745\pi$
$673$	−46.4308	−1.78978	−0.894889	+	0.446290i	$0.852745\pi$
$674$	0	0
$675$	66.3219	2.55273
$676$	0	0
$677$	−44.3129	−1.70308	−0.851541	−	0.524287i	$-0.824332\pi$
$677$	−44.3129	−1.70308	−0.851541	+	0.524287i	$0.824332\pi$
$678$	0	0
$679$	−3.85695	−0.148016
$680$	0	0
$681$	49.4044	1.89318
$682$	0	0
$683$	27.7789	1.06293	0.531465	−	0.847080i	$-0.321642\pi$
$683$	27.7789	1.06293	0.531465	+	0.847080i	$0.321642\pi$
$684$	0	0
$685$	18.6389	0.712155
$686$	0	0
$687$	58.6184	2.23643
$688$	0	0
$689$	3.72866	0.142050
$690$	0	0
$691$	−19.2864	−0.733690	−0.366845	−	0.930282i	$-0.619562\pi$
$691$	−19.2864	−0.733690	−0.366845	+	0.930282i	$0.619562\pi$
$692$	0	0
$693$	42.4486	1.61249
$694$	0	0
$695$	11.7936	0.447358
$696$	0	0
$697$	−20.1300	−0.762477
$698$	0	0
$699$	−21.4750	−0.812261
$700$	0	0
$701$	−40.0268	−1.51179	−0.755895	−	0.654692i	$-0.772798\pi$
$701$	−40.0268	−1.51179	−0.755895	+	0.654692i	$0.772798\pi$
$702$	0	0
$703$	8.03249	0.302951
$704$	0	0
$705$	2.49414	0.0939348
$706$	0	0
$707$	−4.18292	−0.157315
$708$	0	0
$709$	17.8586	0.670695	0.335347	−	0.942095i	$-0.391146\pi$
$709$	17.8586	0.670695	0.335347	+	0.942095i	$0.391146\pi$
$710$	0	0
$711$	−99.3206	−3.72481
$712$	0	0
$713$	−12.9085	−0.483429
$714$	0	0
$715$	−2.75849	−0.103162
$716$	0	0
$717$	−6.45427	−0.241039
$718$	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
$721$	−15.1564	−0.564456
$722$	0	0
$723$	12.3055	0.457647
$724$	0	0
$725$	−14.0901	−0.523293
$726$	0	0
$727$	17.8717	0.662825	0.331412	−	0.943486i	$-0.392475\pi$
$727$	17.8717	0.662825	0.331412	+	0.943486i	$0.392475\pi$
$728$	0	0
$729$	139.517	5.16729
$730$	0	0
$731$	−43.7936	−1.61977
$732$	0	0
$733$	2.43654	0.0899955	0.0449978	−	0.998987i	$-0.485672\pi$
$733$	2.43654	0.0899955	0.0449978	+	0.998987i	$0.485672\pi$
$734$	0	0
$735$	4.17554	0.154017
$736$	0	0
$737$	−6.23315	−0.229601
$738$	0	0
$739$	−17.8261	−0.655745	−0.327872	−	0.944722i	$-0.606332\pi$
$739$	−17.8261	−0.655745	−0.327872	+	0.944722i	$0.606332\pi$
$740$	0	0
$741$	1.54573	0.0567839
$742$	0	0
$743$	−28.4218	−1.04269	−0.521347	−	0.853345i	$-0.674570\pi$
$743$	−28.4218	−1.04269	−0.521347	+	0.853345i	$0.674570\pi$
$744$	0	0
$745$	−18.7256	−0.686053
$746$	0	0
$747$	−60.8321	−2.22573
$748$	0	0
$749$	3.15645	0.115334
$750$	0	0
$751$	−21.1240	−0.770824	−0.385412	−	0.922745i	$-0.625941\pi$
$751$	−21.1240	−0.770824	−0.385412	+	0.922745i	$0.625941\pi$
$752$	0	0
$753$	−30.5842	−1.11455
$754$	0	0
$755$	−10.6787	−0.388639
$756$	0	0
$757$	−6.92330	−0.251632	−0.125816	−	0.992054i	$-0.540155\pi$
$757$	−6.92330	−0.251632	−0.125816	+	0.992054i	$0.540155\pi$
$758$	0	0
$759$	−26.0268	−0.944713
$760$	0	0
$761$	−13.4278	−0.486757	−0.243379	−	0.969931i	$-0.578256\pi$
$761$	−13.4278	−0.486757	−0.243379	+	0.969931i	$0.578256\pi$
$762$	0	0
$763$	0.208036	0.00753143
$764$	0	0
$765$	−76.4194	−2.76295
$766$	0	0
$767$	−1.05327	−0.0380312
$768$	0	0
$769$	38.2861	1.38063	0.690316	−	0.723508i	$-0.257471\pi$
$769$	38.2861	1.38063	0.690316	+	0.723508i	$0.257471\pi$
$770$	0	0
$771$	26.1373	0.941314
$772$	0	0
$773$	24.9353	0.896861	0.448431	−	0.893818i	$-0.351983\pi$
$773$	24.9353	0.896861	0.448431	+	0.893818i	$0.351983\pi$
$774$	0	0
$775$	29.1799	1.04817
$776$	0	0
$777$	27.3320	0.980530
$778$	0	0
$779$	2.77287	0.0993482
$780$	0	0
$781$	4.69312	0.167933
$782$	0	0
$783$	−76.5403	−2.73533
$784$	0	0
$785$	−16.5266	−0.589861
$786$	0	0
$787$	31.9293	1.13816	0.569079	−	0.822283i	$-0.307300\pi$
$787$	31.9293	1.13816	0.569079	+	0.822283i	$0.307300\pi$
$788$	0	0
$789$	48.4308	1.72418
$790$	0	0
$791$	−0.168164	−0.00597923
$792$	0	0
$793$	−0.0649842	−0.00230766
$794$	0	0
$795$	34.2730	1.21554
$796$	0	0
$797$	−41.3012	−1.46296	−0.731481	−	0.681861i	$-0.761171\pi$
$797$	−41.3012	−1.46296	−0.731481	+	0.681861i	$0.761171\pi$
$798$	0	0
$799$	4.33633	0.153408
$800$	0	0
$801$	25.2921	0.893653
$802$	0	0
$803$	−39.0767	−1.37899
$804$	0	0
$805$	−1.89682	−0.0668541
$806$	0	0
$807$	56.2097	1.97868
$808$	0	0
$809$	26.3454	0.926254	0.463127	−	0.886292i	$-0.346727\pi$
$809$	26.3454	0.926254	0.463127	+	0.886292i	$0.346727\pi$
$810$	0	0
$811$	15.0664	0.529051	0.264526	−	0.964379i	$-0.414785\pi$
$811$	15.0664	0.529051	0.264526	+	0.964379i	$0.414785\pi$
$812$	0	0
$813$	−67.8130	−2.37831
$814$	0	0
$815$	0.812738	0.0284690
$816$	0	0
$817$	6.03249	0.211050
$818$	0	0
$819$	3.89682	0.136166
$820$	0	0
$821$	50.1300	1.74955	0.874774	−	0.484531i	$-0.161010\pi$
$821$	50.1300	1.74955	0.874774	+	0.484531i	$0.161010\pi$
$822$	0	0
$823$	22.5340	0.785486	0.392743	−	0.919648i	$-0.371526\pi$
$823$	22.5340	0.785486	0.392743	+	0.919648i	$0.371526\pi$
$824$	0	0
$825$	58.8338	2.04833
$826$	0	0
$827$	−33.3394	−1.15932	−0.579662	−	0.814857i	$-0.696815\pi$
$827$	−33.3394	−1.15932	−0.579662	+	0.814857i	$0.696815\pi$
$828$	0	0
$829$	−42.2747	−1.46826	−0.734130	−	0.679008i	$-0.762410\pi$
$829$	−42.2747	−1.46826	−0.734130	+	0.679008i	$0.762410\pi$
$830$	0	0
$831$	98.4576	3.41546
$832$	0	0
$833$	7.25963	0.251531
$834$	0	0
$835$	17.1330	0.592912
$836$	0	0
$837$	158.511	5.47895
$838$	0	0
$839$	19.4426	0.671231	0.335616	−	0.941999i	$-0.391056\pi$
$839$	19.4426	0.671231	0.335616	+	0.941999i	$0.391056\pi$
$840$	0	0
$841$	−12.7390	−0.439276
$842$	0	0
$843$	46.4429	1.59958
$844$	0	0
$845$	15.6995	0.540080
$846$	0	0
$847$	13.4868	0.463411
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−12.4161	−0.425618
$852$	0	0
$853$	−9.08576	−0.311090	−0.155545	−	0.987829i	$-0.549713\pi$
$853$	−9.08576	−0.311090	−0.155545	+	0.987829i	$0.549713\pi$
$854$	0	0
$855$	10.5266	0.360003
$856$	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
$859$	24.7672	0.845045	0.422522	−	0.906353i	$-0.361145\pi$
$859$	24.7672	0.845045	0.422522	+	0.906353i	$0.361145\pi$
$860$	0	0
$861$	9.43517	0.321550
$862$	0	0
$863$	19.0060	0.646972	0.323486	−	0.946233i	$-0.395145\pi$
$863$	19.0060	0.646972	0.323486	+	0.946233i	$0.395145\pi$
$864$	0	0
$865$	−14.5990	−0.496381
$866$	0	0
$867$	−121.483	−4.12578
$868$	0	0
$869$	−57.2938	−1.94356
$870$	0	0
$871$	−0.572209	−0.0193886
$872$	0	0
$873$	−33.0858	−1.11978
$874$	0	0
$875$	10.4235	0.352377
$876$	0	0
$877$	−16.4941	−0.556968	−0.278484	−	0.960441i	$-0.589832\pi$
$877$	−16.4941	−0.556968	−0.278484	+	0.960441i	$0.589832\pi$
$878$	0	0
$879$	−75.0801	−2.53239
$880$	0	0
$881$	−38.5608	−1.29915	−0.649573	−	0.760299i	$-0.725052\pi$
$881$	−38.5608	−1.29915	−0.649573	+	0.760299i	$0.725052\pi$
$882$	0	0
$883$	56.1698	1.89027	0.945133	−	0.326686i	$-0.105932\pi$
$883$	56.1698	1.89027	0.945133	+	0.326686i	$0.105932\pi$
$884$	0	0
$885$	−9.68140	−0.325437
$886$	0	0
$887$	17.7554	0.596169	0.298085	−	0.954539i	$-0.403652\pi$
$887$	17.7554	0.596169	0.298085	+	0.954539i	$0.403652\pi$
$888$	0	0
$889$	5.85695	0.196436
$890$	0	0
$891$	192.252	6.44068
$892$	0	0
$893$	−0.597321	−0.0199886
$894$	0	0
$895$	27.3012	0.912578
$896$	0	0
$897$	−2.38928	−0.0797759
$898$	0	0
$899$	−33.6757	−1.12315
$900$	0	0
$901$	59.5873	1.98514
$902$	0	0
$903$	20.5266	0.683084
$904$	0	0
$905$	−17.6905	−0.588051
$906$	0	0
$907$	23.7139	0.787407	0.393703	−	0.919237i	$-0.371194\pi$
$907$	23.7139	0.787407	0.393703	+	0.919237i	$0.371194\pi$
$908$	0	0
$909$	−35.8821	−1.19013
$910$	0	0
$911$	39.5018	1.30875	0.654377	−	0.756168i	$-0.272931\pi$
$911$	39.5018	1.30875	0.654377	+	0.756168i	$0.272931\pi$
$912$	0	0
$913$	−35.0915	−1.16136
$914$	0	0
$915$	−0.597321	−0.0197468
$916$	0	0
$917$	−5.19464	−0.171542
$918$	0	0
$919$	33.7052	1.11183	0.555916	−	0.831238i	$-0.312367\pi$
$919$	33.7052	1.11183	0.555916	+	0.831238i	$0.312367\pi$
$920$	0	0
$921$	34.1876	1.12652
$922$	0	0
$923$	0.430833	0.0141810
$924$	0	0
$925$	28.0667	0.922826
$926$	0	0
$927$	−130.015	−4.27027
$928$	0	0
$929$	2.43083	0.0797530	0.0398765	−	0.999205i	$-0.487304\pi$
$929$	2.43083	0.0797530	0.0398765	+	0.999205i	$0.487304\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	35.7082	1.16903
$934$	0	0
$935$	−44.0831	−1.44167
$936$	0	0
$937$	−49.7025	−1.62371	−0.811855	−	0.583859i	$-0.801542\pi$
$937$	−49.7025	−1.62371	−0.811855	+	0.583859i	$0.801542\pi$
$938$	0	0
$939$	19.6637	0.641700
$940$	0	0
$941$	11.6757	0.380617	0.190308	−	0.981724i	$-0.439051\pi$
$941$	11.6757	0.380617	0.190308	+	0.981724i	$0.439051\pi$
$942$	0	0
$943$	−4.28610	−0.139575
$944$	0	0
$945$	23.2921	0.757693
$946$	0	0
$947$	−36.9484	−1.20066	−0.600331	−	0.799752i	$-0.704964\pi$
$947$	−36.9484	−1.20066	−0.600331	+	0.799752i	$0.704964\pi$
$948$	0	0
$949$	−3.58728	−0.116448
$950$	0	0
$951$	43.9293	1.42451
$952$	0	0
$953$	49.4278	1.60112	0.800562	−	0.599250i	$-0.204535\pi$
$953$	49.4278	1.60112	0.800562	+	0.599250i	$0.204535\pi$
$954$	0	0
$955$	−32.2097	−1.04228
$956$	0	0
$957$	−67.8985	−2.19485
$958$	0	0
$959$	−15.1889	−0.490476
$960$	0	0
$961$	38.7407	1.24970
$962$	0	0
$963$	27.0767	0.872535
$964$	0	0
$965$	11.9203	0.383727
$966$	0	0
$967$	−58.1065	−1.86858	−0.934290	−	0.356514i	$-0.883965\pi$
$967$	−58.1065	−1.86858	−0.934290	+	0.356514i	$0.883965\pi$
$968$	0	0
$969$	24.7022	0.793548
$970$	0	0
$971$	17.6653	0.566908	0.283454	−	0.958986i	$-0.408520\pi$
$971$	17.6653	0.566908	0.283454	+	0.958986i	$0.408520\pi$
$972$	0	0
$973$	−9.61072	−0.308105
$974$	0	0
$975$	5.40100	0.172971
$976$	0	0
$977$	10.6224	0.339842	0.169921	−	0.985458i	$-0.445649\pi$
$977$	10.6224	0.339842	0.169921	+	0.985458i	$0.445649\pi$
$978$	0	0
$979$	14.5899	0.466297
$980$	0	0
$981$	1.78458	0.0569774
$982$	0	0
$983$	4.90854	0.156558	0.0782790	−	0.996931i	$-0.475057\pi$
$983$	4.90854	0.156558	0.0782790	+	0.996931i	$0.475057\pi$
$984$	0	0
$985$	20.2713	0.645899
$986$	0	0
$987$	−2.03249	−0.0646949
$988$	0	0
$989$	−9.32461	−0.296505
$990$	0	0
$991$	4.61208	0.146508	0.0732538	−	0.997313i	$-0.476662\pi$
$991$	4.61208	0.146508	0.0732538	+	0.997313i	$0.476662\pi$
$992$	0	0
$993$	−105.042	−3.33340
$994$	0	0
$995$	26.6857	0.845995
$996$	0	0
$997$	−5.73436	−0.181609	−0.0908045	−	0.995869i	$-0.528944\pi$
$997$	−5.73436	−0.181609	−0.0908045	+	0.995869i	$0.528944\pi$
$998$	0	0
$999$	152.464	4.82375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bj.1.1		3
4.3	odd	2		8512.2.a.bm.1.3		3
8.3	odd	2		266.2.a.d.1.1	✓	3
8.5	even	2		2128.2.a.s.1.3		3
24.11	even	2		2394.2.a.ba.1.2		3
40.19	odd	2		6650.2.a.cd.1.3		3
56.27	even	2		1862.2.a.r.1.3		3
152.75	even	2		5054.2.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.a.d.1.1	✓	3	8.3	odd	2
1862.2.a.r.1.3		3	56.27	even	2
2128.2.a.s.1.3		3	8.5	even	2
2394.2.a.ba.1.2		3	24.11	even	2
5054.2.a.r.1.3		3	152.75	even	2
6650.2.a.cd.1.3		3	40.19	odd	2
8512.2.a.bj.1.1		3	1.1	even	1	trivial
8512.2.a.bm.1.3		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.40268 q^{3} -1.22713 q^{5} +1.00000 q^{7} +8.57822 q^{9} +O(q^{10})\)	\(q-3.40268 q^{3} -1.22713 q^{5} +1.00000 q^{7} +8.57822 q^{9} +4.94841 q^{11} +0.454269 q^{13} +4.17554 q^{15} +7.25963 q^{17} -1.00000 q^{19} -3.40268 q^{21} +1.54573 q^{23} -3.49414 q^{25} -18.9809 q^{27} +4.03249 q^{29} -8.35109 q^{31} -16.8378 q^{33} -1.22713 q^{35} -8.03249 q^{37} -1.54573 q^{39} -2.77287 q^{41} -6.03249 q^{43} -10.5266 q^{45} +0.597321 q^{47} +1.00000 q^{49} -24.7022 q^{51} +8.20804 q^{53} -6.07236 q^{55} +3.40268 q^{57} -2.31860 q^{59} -0.143052 q^{61} +8.57822 q^{63} -0.557449 q^{65} -1.25963 q^{67} -5.25963 q^{69} +0.948410 q^{71} -7.89682 q^{73} +11.8894 q^{75} +4.94841 q^{77} -11.5782 q^{79} +38.8512 q^{81} -7.09146 q^{83} -8.90854 q^{85} -13.7213 q^{87} +2.94841 q^{89} +0.454269 q^{91} +28.4161 q^{93} +1.22713 q^{95} -3.85695 q^{97} +42.4486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 5 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q - q^3 - 5 * q^5 + 3 * q^7 + 6 * q^9	\( 3 q - q^{3} - 5 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} - 3 q^{19} - q^{21} + 2 q^{23} + 4 q^{25} - 28 q^{27} - 5 q^{29} - 4 q^{31} - 15 q^{33} - 5 q^{35} - 7 q^{37} - 2 q^{39} - 7 q^{41} - q^{43} + 11 q^{47} + 3 q^{49} - 32 q^{51} - 3 q^{53} + 16 q^{55} + q^{57} - 3 q^{59} - 7 q^{61} + 6 q^{63} - 28 q^{65} + 12 q^{67} - 9 q^{71} + 12 q^{75} + 3 q^{77} - 15 q^{79} + 35 q^{81} - 16 q^{83} - 32 q^{85} - 28 q^{87} - 3 q^{89} + 4 q^{91} + 30 q^{93} + 5 q^{95} - 5 q^{97} + 55 q^{99}+O(q^{100}) \) 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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