LMFDB - Embedded newform 8512.2.a.i.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-2.61803 q^{3} +2.23607 q^{5} -1.00000 q^{7} +3.85410 q^{9} +O(q^{10})$	$q-2.61803 q^{3} +2.23607 q^{5} -1.00000 q^{7} +3.85410 q^{9} +2.85410 q^{11} +1.76393 q^{13} -5.85410 q^{15} +0.854102 q^{17} +1.00000 q^{19} +2.61803 q^{21} -6.23607 q^{23} -2.23607 q^{27} +4.38197 q^{29} +2.38197 q^{31} -7.47214 q^{33} -2.23607 q^{35} -1.47214 q^{37} -4.61803 q^{39} -10.5623 q^{41} -6.94427 q^{43} +8.61803 q^{45} -7.00000 q^{47} +1.00000 q^{49} -2.23607 q^{51} +1.85410 q^{53} +6.38197 q^{55} -2.61803 q^{57} -10.7082 q^{59} +12.7082 q^{61} -3.85410 q^{63} +3.94427 q^{65} -5.61803 q^{67} +16.3262 q^{69} -3.94427 q^{71} -0.145898 q^{73} -2.85410 q^{77} +4.47214 q^{79} -5.70820 q^{81} -11.3820 q^{83} +1.90983 q^{85} -11.4721 q^{87} +0.763932 q^{89} -1.76393 q^{91} -6.23607 q^{93} +2.23607 q^{95} -3.76393 q^{97} +11.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 2 * q^7 + q^9	$ 2 q - 3 q^{3} - 2 q^{7} + q^{9} - q^{11} + 8 q^{13} - 5 q^{15} - 5 q^{17} + 2 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{29} + 7 q^{31} - 6 q^{33} + 6 q^{37} - 7 q^{39} - q^{41} + 4 q^{43} + 15 q^{45} - 14 q^{47} + 2 q^{49} - 3 q^{53} + 15 q^{55} - 3 q^{57} - 8 q^{59} + 12 q^{61} - q^{63} - 10 q^{65} - 9 q^{67} + 17 q^{69} + 10 q^{71} - 7 q^{73} + q^{77} + 2 q^{81} - 25 q^{83} + 15 q^{85} - 14 q^{87} + 6 q^{89} - 8 q^{91} - 8 q^{93} - 12 q^{97} + 22 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 2 * q^7 + q^9 - q^11 + 8 * q^13 - 5 * q^15 - 5 * q^17 + 2 * q^19 + 3 * q^21 - 8 * q^23 + 11 * q^29 + 7 * q^31 - 6 * q^33 + 6 * q^37 - 7 * q^39 - q^41 + 4 * q^43 + 15 * q^45 - 14 * q^47 + 2 * q^49 - 3 * q^53 + 15 * q^55 - 3 * q^57 - 8 * q^59 + 12 * q^61 - q^63 - 10 * q^65 - 9 * q^67 + 17 * q^69 + 10 * q^71 - 7 * q^73 + q^77 + 2 * q^81 - 25 * q^83 + 15 * q^85 - 14 * q^87 + 6 * q^89 - 8 * q^91 - 8 * q^93 - 12 * q^97 + 22 * 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
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	0	0
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	2.85410	0.860544	0.430272	−	0.902699i	$-0.358418\pi$
$11$	2.85410	0.860544	0.430272	+	0.902699i	$0.358418\pi$
$12$	0	0
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	0	0
$15$	−5.85410	−1.51152
$16$	0	0
$17$	0.854102	0.207150	0.103575	−	0.994622i	$-0.466972\pi$
$17$	0.854102	0.207150	0.103575	+	0.994622i	$0.466972\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.61803	0.571302
$22$	0	0
$23$	−6.23607	−1.30031	−0.650155	−	0.759802i	$-0.725296\pi$
$23$	−6.23607	−1.30031	−0.650155	+	0.759802i	$0.725296\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	4.38197	0.813711	0.406855	−	0.913493i	$-0.366625\pi$
$29$	4.38197	0.813711	0.406855	+	0.913493i	$0.366625\pi$
$30$	0	0
$31$	2.38197	0.427814	0.213907	−	0.976854i	$-0.431381\pi$
$31$	2.38197	0.427814	0.213907	+	0.976854i	$0.431381\pi$
$32$	0	0
$33$	−7.47214	−1.30073
$34$	0	0
$35$	−2.23607	−0.377964
$36$	0	0
$37$	−1.47214	−0.242018	−0.121009	−	0.992651i	$-0.538613\pi$
$37$	−1.47214	−0.242018	−0.121009	+	0.992651i	$0.538613\pi$
$38$	0	0
$39$	−4.61803	−0.739477
$40$	0	0
$41$	−10.5623	−1.64956	−0.824778	−	0.565457i	$-0.808700\pi$
$41$	−10.5623	−1.64956	−0.824778	+	0.565457i	$0.808700\pi$
$42$	0	0
$43$	−6.94427	−1.05899	−0.529496	−	0.848313i	$-0.677619\pi$
$43$	−6.94427	−1.05899	−0.529496	+	0.848313i	$0.677619\pi$
$44$	0	0
$45$	8.61803	1.28470
$46$	0	0
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.23607	−0.313112
$52$	0	0
$53$	1.85410	0.254680	0.127340	−	0.991859i	$-0.459356\pi$
$53$	1.85410	0.254680	0.127340	+	0.991859i	$0.459356\pi$
$54$	0	0
$55$	6.38197	0.860544
$56$	0	0
$57$	−2.61803	−0.346767
$58$	0	0
$59$	−10.7082	−1.39409	−0.697045	−	0.717028i	$-0.745502\pi$
$59$	−10.7082	−1.39409	−0.697045	+	0.717028i	$0.745502\pi$
$60$	0	0
$61$	12.7082	1.62712	0.813559	−	0.581482i	$-0.197527\pi$
$61$	12.7082	1.62712	0.813559	+	0.581482i	$0.197527\pi$
$62$	0	0
$63$	−3.85410	−0.485571
$64$	0	0
$65$	3.94427	0.489227
$66$	0	0
$67$	−5.61803	−0.686352	−0.343176	−	0.939271i	$-0.611503\pi$
$67$	−5.61803	−0.686352	−0.343176	+	0.939271i	$0.611503\pi$
$68$	0	0
$69$	16.3262	1.96545
$70$	0	0
$71$	−3.94427	−0.468099	−0.234049	−	0.972225i	$-0.575198\pi$
$71$	−3.94427	−0.468099	−0.234049	+	0.972225i	$0.575198\pi$
$72$	0	0
$73$	−0.145898	−0.0170761	−0.00853804	−	0.999964i	$-0.502718\pi$
$73$	−0.145898	−0.0170761	−0.00853804	+	0.999964i	$0.502718\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.85410	−0.325255
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	−11.3820	−1.24933	−0.624667	−	0.780892i	$-0.714765\pi$
$83$	−11.3820	−1.24933	−0.624667	+	0.780892i	$0.714765\pi$
$84$	0	0
$85$	1.90983	0.207150
$86$	0	0
$87$	−11.4721	−1.22994
$88$	0	0
$89$	0.763932	0.0809766	0.0404883	−	0.999180i	$-0.487109\pi$
$89$	0.763932	0.0809766	0.0404883	+	0.999180i	$0.487109\pi$
$90$	0	0
$91$	−1.76393	−0.184910
$92$	0	0
$93$	−6.23607	−0.646650
$94$	0	0
$95$	2.23607	0.229416
$96$	0	0
$97$	−3.76393	−0.382169	−0.191085	−	0.981574i	$-0.561201\pi$
$97$	−3.76393	−0.382169	−0.191085	+	0.981574i	$0.561201\pi$
$98$	0	0
$99$	11.0000	1.10554
$100$	0	0
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	0	0
$103$	−8.23607	−0.811524	−0.405762	−	0.913979i	$-0.632994\pi$
$103$	−8.23607	−0.811524	−0.405762	+	0.913979i	$0.632994\pi$
$104$	0	0
$105$	5.85410	0.571302
$106$	0	0
$107$	13.9443	1.34804	0.674022	−	0.738711i	$-0.264565\pi$
$107$	13.9443	1.34804	0.674022	+	0.738711i	$0.264565\pi$
$108$	0	0
$109$	10.2361	0.980437	0.490219	−	0.871599i	$-0.336917\pi$
$109$	10.2361	0.980437	0.490219	+	0.871599i	$0.336917\pi$
$110$	0	0
$111$	3.85410	0.365815
$112$	0	0
$113$	−2.14590	−0.201869	−0.100935	−	0.994893i	$-0.532183\pi$
$113$	−2.14590	−0.201869	−0.100935	+	0.994893i	$0.532183\pi$
$114$	0	0
$115$	−13.9443	−1.30031
$116$	0	0
$117$	6.79837	0.628510
$118$	0	0
$119$	−0.854102	−0.0782954
$120$	0	0
$121$	−2.85410	−0.259464
$122$	0	0
$123$	27.6525	2.49334
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−2.29180	−0.203364	−0.101682	−	0.994817i	$-0.532422\pi$
$127$	−2.29180	−0.203364	−0.101682	+	0.994817i	$0.532422\pi$
$128$	0	0
$129$	18.1803	1.60069
$130$	0	0
$131$	9.79837	0.856088	0.428044	−	0.903758i	$-0.359203\pi$
$131$	9.79837	0.856088	0.428044	+	0.903758i	$0.359203\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−11.4721	−0.973054	−0.486527	−	0.873666i	$-0.661736\pi$
$139$	−11.4721	−0.973054	−0.486527	+	0.873666i	$0.661736\pi$
$140$	0	0
$141$	18.3262	1.54335
$142$	0	0
$143$	5.03444	0.421001
$144$	0	0
$145$	9.79837	0.813711
$146$	0	0
$147$	−2.61803	−0.215932
$148$	0	0
$149$	−12.5279	−1.02632	−0.513161	−	0.858292i	$-0.671526\pi$
$149$	−12.5279	−1.02632	−0.513161	+	0.858292i	$0.671526\pi$
$150$	0	0
$151$	8.85410	0.720537	0.360268	−	0.932849i	$-0.382685\pi$
$151$	8.85410	0.720537	0.360268	+	0.932849i	$0.382685\pi$
$152$	0	0
$153$	3.29180	0.266126
$154$	0	0
$155$	5.32624	0.427814
$156$	0	0
$157$	2.85410	0.227782	0.113891	−	0.993493i	$-0.463669\pi$
$157$	2.85410	0.227782	0.113891	+	0.993493i	$0.463669\pi$
$158$	0	0
$159$	−4.85410	−0.384955
$160$	0	0
$161$	6.23607	0.491471
$162$	0	0
$163$	−7.09017	−0.555345	−0.277672	−	0.960676i	$-0.589563\pi$
$163$	−7.09017	−0.555345	−0.277672	+	0.960676i	$0.589563\pi$
$164$	0	0
$165$	−16.7082	−1.30073
$166$	0	0
$167$	−4.52786	−0.350377	−0.175188	−	0.984535i	$-0.556053\pi$
$167$	−4.52786	−0.350377	−0.175188	+	0.984535i	$0.556053\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	3.85410	0.294731
$172$	0	0
$173$	−13.7082	−1.04222	−0.521108	−	0.853491i	$-0.674481\pi$
$173$	−13.7082	−1.04222	−0.521108	+	0.853491i	$0.674481\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	28.0344	2.10720
$178$	0	0
$179$	0.201626	0.0150702	0.00753512	−	0.999972i	$-0.497601\pi$
$179$	0.201626	0.0150702	0.00753512	+	0.999972i	$0.497601\pi$
$180$	0	0
$181$	20.3820	1.51498	0.757490	−	0.652847i	$-0.226426\pi$
$181$	20.3820	1.51498	0.757490	+	0.652847i	$0.226426\pi$
$182$	0	0
$183$	−33.2705	−2.45943
$184$	0	0
$185$	−3.29180	−0.242018
$186$	0	0
$187$	2.43769	0.178262
$188$	0	0
$189$	2.23607	0.162650
$190$	0	0
$191$	24.0344	1.73907	0.869536	−	0.493870i	$-0.164418\pi$
$191$	24.0344	1.73907	0.869536	+	0.493870i	$0.164418\pi$
$192$	0	0
$193$	6.85410	0.493369	0.246685	−	0.969096i	$-0.420659\pi$
$193$	6.85410	0.493369	0.246685	+	0.969096i	$0.420659\pi$
$194$	0	0
$195$	−10.3262	−0.739477
$196$	0	0
$197$	15.5623	1.10877	0.554384	−	0.832261i	$-0.312954\pi$
$197$	15.5623	1.10877	0.554384	+	0.832261i	$0.312954\pi$
$198$	0	0
$199$	0.0557281	0.00395046	0.00197523	−	0.999998i	$-0.499371\pi$
$199$	0.0557281	0.00395046	0.00197523	+	0.999998i	$0.499371\pi$
$200$	0	0
$201$	14.7082	1.03744
$202$	0	0
$203$	−4.38197	−0.307554
$204$	0	0
$205$	−23.6180	−1.64956
$206$	0	0
$207$	−24.0344	−1.67051
$208$	0	0
$209$	2.85410	0.197422
$210$	0	0
$211$	−21.5623	−1.48441	−0.742205	−	0.670173i	$-0.766220\pi$
$211$	−21.5623	−1.48441	−0.742205	+	0.670173i	$0.766220\pi$
$212$	0	0
$213$	10.3262	0.707542
$214$	0	0
$215$	−15.5279	−1.05899
$216$	0	0
$217$	−2.38197	−0.161698
$218$	0	0
$219$	0.381966	0.0258109
$220$	0	0
$221$	1.50658	0.101343
$222$	0	0
$223$	27.9443	1.87129	0.935643	−	0.352947i	$-0.114820\pi$
$223$	27.9443	1.87129	0.935643	+	0.352947i	$0.114820\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.7984	−1.71230	−0.856149	−	0.516729i	$-0.827150\pi$
$227$	−25.7984	−1.71230	−0.856149	+	0.516729i	$0.827150\pi$
$228$	0	0
$229$	−28.8328	−1.90533	−0.952663	−	0.304028i	$-0.901668\pi$
$229$	−28.8328	−1.90533	−0.952663	+	0.304028i	$0.901668\pi$
$230$	0	0
$231$	7.47214	0.491630
$232$	0	0
$233$	−13.7984	−0.903962	−0.451981	−	0.892028i	$-0.649282\pi$
$233$	−13.7984	−0.903962	−0.451981	+	0.892028i	$0.649282\pi$
$234$	0	0
$235$	−15.6525	−1.02105
$236$	0	0
$237$	−11.7082	−0.760530
$238$	0	0
$239$	−22.7082	−1.46887	−0.734436	−	0.678678i	$-0.762553\pi$
$239$	−22.7082	−1.46887	−0.734436	+	0.678678i	$0.762553\pi$
$240$	0	0
$241$	15.1246	0.974262	0.487131	−	0.873329i	$-0.338043\pi$
$241$	15.1246	0.974262	0.487131	+	0.873329i	$0.338043\pi$
$242$	0	0
$243$	21.6525	1.38901
$244$	0	0
$245$	2.23607	0.142857
$246$	0	0
$247$	1.76393	0.112236
$248$	0	0
$249$	29.7984	1.88840
$250$	0	0
$251$	27.9787	1.76600	0.883000	−	0.469372i	$-0.155520\pi$
$251$	27.9787	1.76600	0.883000	+	0.469372i	$0.155520\pi$
$252$	0	0
$253$	−17.7984	−1.11897
$254$	0	0
$255$	−5.00000	−0.313112
$256$	0	0
$257$	19.0902	1.19081	0.595406	−	0.803425i	$-0.296991\pi$
$257$	19.0902	1.19081	0.595406	+	0.803425i	$0.296991\pi$
$258$	0	0
$259$	1.47214	0.0914741
$260$	0	0
$261$	16.8885	1.04537
$262$	0	0
$263$	−11.6738	−0.719835	−0.359918	−	0.932984i	$-0.617195\pi$
$263$	−11.6738	−0.719835	−0.359918	+	0.932984i	$0.617195\pi$
$264$	0	0
$265$	4.14590	0.254680
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	15.8541	0.966642	0.483321	−	0.875443i	$-0.339430\pi$
$269$	15.8541	0.966642	0.483321	+	0.875443i	$0.339430\pi$
$270$	0	0
$271$	17.0344	1.03477	0.517384	−	0.855753i	$-0.326906\pi$
$271$	17.0344	1.03477	0.517384	+	0.855753i	$0.326906\pi$
$272$	0	0
$273$	4.61803	0.279496
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−20.6525	−1.24089	−0.620444	−	0.784251i	$-0.713047\pi$
$277$	−20.6525	−1.24089	−0.620444	+	0.784251i	$0.713047\pi$
$278$	0	0
$279$	9.18034	0.549613
$280$	0	0
$281$	−7.76393	−0.463157	−0.231579	−	0.972816i	$-0.574389\pi$
$281$	−7.76393	−0.463157	−0.231579	+	0.972816i	$0.574389\pi$
$282$	0	0
$283$	18.7984	1.11745	0.558724	−	0.829354i	$-0.311291\pi$
$283$	18.7984	1.11745	0.558724	+	0.829354i	$0.311291\pi$
$284$	0	0
$285$	−5.85410	−0.346767
$286$	0	0
$287$	10.5623	0.623473
$288$	0	0
$289$	−16.2705	−0.957089
$290$	0	0
$291$	9.85410	0.577658
$292$	0	0
$293$	23.0000	1.34367	0.671837	−	0.740699i	$-0.265505\pi$
$293$	23.0000	1.34367	0.671837	+	0.740699i	$0.265505\pi$
$294$	0	0
$295$	−23.9443	−1.39409
$296$	0	0
$297$	−6.38197	−0.370319
$298$	0	0
$299$	−11.0000	−0.636146
$300$	0	0
$301$	6.94427	0.400261
$302$	0	0
$303$	2.61803	0.150402
$304$	0	0
$305$	28.4164	1.62712
$306$	0	0
$307$	12.2705	0.700315	0.350157	−	0.936691i	$-0.386128\pi$
$307$	12.2705	0.700315	0.350157	+	0.936691i	$0.386128\pi$
$308$	0	0
$309$	21.5623	1.22664
$310$	0	0
$311$	25.5066	1.44635	0.723173	−	0.690667i	$-0.242683\pi$
$311$	25.5066	1.44635	0.723173	+	0.690667i	$0.242683\pi$
$312$	0	0
$313$	−21.5967	−1.22072	−0.610360	−	0.792124i	$-0.708975\pi$
$313$	−21.5967	−1.22072	−0.610360	+	0.792124i	$0.708975\pi$
$314$	0	0
$315$	−8.61803	−0.485571
$316$	0	0
$317$	−10.3607	−0.581914	−0.290957	−	0.956736i	$-0.593974\pi$
$317$	−10.3607	−0.581914	−0.290957	+	0.956736i	$0.593974\pi$
$318$	0	0
$319$	12.5066	0.700234
$320$	0	0
$321$	−36.5066	−2.03760
$322$	0	0
$323$	0.854102	0.0475235
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−26.7984	−1.48195
$328$	0	0
$329$	7.00000	0.385922
$330$	0	0
$331$	−31.3820	−1.72491	−0.862454	−	0.506135i	$-0.831074\pi$
$331$	−31.3820	−1.72491	−0.862454	+	0.506135i	$0.831074\pi$
$332$	0	0
$333$	−5.67376	−0.310920
$334$	0	0
$335$	−12.5623	−0.686352
$336$	0	0
$337$	−27.1459	−1.47873	−0.739366	−	0.673304i	$-0.764874\pi$
$337$	−27.1459	−1.47873	−0.739366	+	0.673304i	$0.764874\pi$
$338$	0	0
$339$	5.61803	0.305130
$340$	0	0
$341$	6.79837	0.368153
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	36.5066	1.96545
$346$	0	0
$347$	14.0344	0.753408	0.376704	−	0.926334i	$-0.377057\pi$
$347$	14.0344	0.753408	0.376704	+	0.926334i	$0.377057\pi$
$348$	0	0
$349$	8.03444	0.430074	0.215037	−	0.976606i	$-0.431013\pi$
$349$	8.03444	0.430074	0.215037	+	0.976606i	$0.431013\pi$
$350$	0	0
$351$	−3.94427	−0.210530
$352$	0	0
$353$	−27.9787	−1.48916	−0.744578	−	0.667535i	$-0.767349\pi$
$353$	−27.9787	−1.48916	−0.744578	+	0.667535i	$0.767349\pi$
$354$	0	0
$355$	−8.81966	−0.468099
$356$	0	0
$357$	2.23607	0.118345
$358$	0	0
$359$	−28.3820	−1.49794	−0.748971	−	0.662602i	$-0.769452\pi$
$359$	−28.3820	−1.49794	−0.748971	+	0.662602i	$0.769452\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.47214	0.392185
$364$	0	0
$365$	−0.326238	−0.0170761
$366$	0	0
$367$	0.819660	0.0427859	0.0213930	−	0.999771i	$-0.493190\pi$
$367$	0.819660	0.0427859	0.0213930	+	0.999771i	$0.493190\pi$
$368$	0	0
$369$	−40.7082	−2.11918
$370$	0	0
$371$	−1.85410	−0.0962602
$372$	0	0
$373$	−8.90983	−0.461334	−0.230667	−	0.973033i	$-0.574091\pi$
$373$	−8.90983	−0.461334	−0.230667	+	0.973033i	$0.574091\pi$
$374$	0	0
$375$	29.2705	1.51152
$376$	0	0
$377$	7.72949	0.398089
$378$	0	0
$379$	9.65248	0.495814	0.247907	−	0.968784i	$-0.420257\pi$
$379$	9.65248	0.495814	0.247907	+	0.968784i	$0.420257\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	8.23607	0.420843	0.210422	−	0.977611i	$-0.432516\pi$
$383$	8.23607	0.420843	0.210422	+	0.977611i	$0.432516\pi$
$384$	0	0
$385$	−6.38197	−0.325255
$386$	0	0
$387$	−26.7639	−1.36049
$388$	0	0
$389$	11.3262	0.574263	0.287132	−	0.957891i	$-0.407298\pi$
$389$	11.3262	0.574263	0.287132	+	0.957891i	$0.407298\pi$
$390$	0	0
$391$	−5.32624	−0.269359
$392$	0	0
$393$	−25.6525	−1.29400
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	−23.2361	−1.16618	−0.583092	−	0.812406i	$-0.698157\pi$
$397$	−23.2361	−1.16618	−0.583092	+	0.812406i	$0.698157\pi$
$398$	0	0
$399$	2.61803	0.131066
$400$	0	0
$401$	−37.5066	−1.87299	−0.936495	−	0.350682i	$-0.885950\pi$
$401$	−37.5066	−1.87299	−0.936495	+	0.350682i	$0.885950\pi$
$402$	0	0
$403$	4.20163	0.209298
$404$	0	0
$405$	−12.7639	−0.634245
$406$	0	0
$407$	−4.20163	−0.208267
$408$	0	0
$409$	−2.14590	−0.106108	−0.0530539	−	0.998592i	$-0.516896\pi$
$409$	−2.14590	−0.106108	−0.0530539	+	0.998592i	$0.516896\pi$
$410$	0	0
$411$	−5.23607	−0.258276
$412$	0	0
$413$	10.7082	0.526916
$414$	0	0
$415$	−25.4508	−1.24933
$416$	0	0
$417$	30.0344	1.47079
$418$	0	0
$419$	34.3050	1.67591	0.837953	−	0.545742i	$-0.183752\pi$
$419$	34.3050	1.67591	0.837953	+	0.545742i	$0.183752\pi$
$420$	0	0
$421$	−3.94427	−0.192232	−0.0961160	−	0.995370i	$-0.530642\pi$
$421$	−3.94427	−0.192232	−0.0961160	+	0.995370i	$0.530642\pi$
$422$	0	0
$423$	−26.9787	−1.31175
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.7082	−0.614993
$428$	0	0
$429$	−13.1803	−0.636353
$430$	0	0
$431$	7.88854	0.379978	0.189989	−	0.981786i	$-0.439155\pi$
$431$	7.88854	0.379978	0.189989	+	0.981786i	$0.439155\pi$
$432$	0	0
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	0	0
$435$	−25.6525	−1.22994
$436$	0	0
$437$	−6.23607	−0.298312
$438$	0	0
$439$	−9.88854	−0.471954	−0.235977	−	0.971759i	$-0.575829\pi$
$439$	−9.88854	−0.471954	−0.235977	+	0.971759i	$0.575829\pi$
$440$	0	0
$441$	3.85410	0.183529
$442$	0	0
$443$	−22.2148	−1.05546	−0.527728	−	0.849413i	$-0.676956\pi$
$443$	−22.2148	−1.05546	−0.527728	+	0.849413i	$0.676956\pi$
$444$	0	0
$445$	1.70820	0.0809766
$446$	0	0
$447$	32.7984	1.55131
$448$	0	0
$449$	12.6180	0.595482	0.297741	−	0.954647i	$-0.403767\pi$
$449$	12.6180	0.595482	0.297741	+	0.954647i	$0.403767\pi$
$450$	0	0
$451$	−30.1459	−1.41951
$452$	0	0
$453$	−23.1803	−1.08911
$454$	0	0
$455$	−3.94427	−0.184910
$456$	0	0
$457$	−23.9230	−1.11907	−0.559535	−	0.828807i	$-0.689020\pi$
$457$	−23.9230	−1.11907	−0.559535	+	0.828807i	$0.689020\pi$
$458$	0	0
$459$	−1.90983	−0.0891432
$460$	0	0
$461$	−30.9787	−1.44282	−0.721411	−	0.692507i	$-0.756506\pi$
$461$	−30.9787	−1.44282	−0.721411	+	0.692507i	$0.756506\pi$
$462$	0	0
$463$	−21.3607	−0.992715	−0.496357	−	0.868118i	$-0.665329\pi$
$463$	−21.3607	−0.992715	−0.496357	+	0.868118i	$0.665329\pi$
$464$	0	0
$465$	−13.9443	−0.646650
$466$	0	0
$467$	−39.0902	−1.80888	−0.904439	−	0.426604i	$-0.859710\pi$
$467$	−39.0902	−1.80888	−0.904439	+	0.426604i	$0.859710\pi$
$468$	0	0
$469$	5.61803	0.259417
$470$	0	0
$471$	−7.47214	−0.344298
$472$	0	0
$473$	−19.8197	−0.911309
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.14590	0.327188
$478$	0	0
$479$	6.74265	0.308079	0.154040	−	0.988065i	$-0.450772\pi$
$479$	6.74265	0.308079	0.154040	+	0.988065i	$0.450772\pi$
$480$	0	0
$481$	−2.59675	−0.118402
$482$	0	0
$483$	−16.3262	−0.742870
$484$	0	0
$485$	−8.41641	−0.382169
$486$	0	0
$487$	26.8885	1.21844	0.609218	−	0.793003i	$-0.291483\pi$
$487$	26.8885	1.21844	0.609218	+	0.793003i	$0.291483\pi$
$488$	0	0
$489$	18.5623	0.839416
$490$	0	0
$491$	0.944272	0.0426144	0.0213072	−	0.999773i	$-0.493217\pi$
$491$	0.944272	0.0426144	0.0213072	+	0.999773i	$0.493217\pi$
$492$	0	0
$493$	3.74265	0.168560
$494$	0	0
$495$	24.5967	1.10554
$496$	0	0
$497$	3.94427	0.176925
$498$	0	0
$499$	−13.3820	−0.599059	−0.299530	−	0.954087i	$-0.596830\pi$
$499$	−13.3820	−0.599059	−0.299530	+	0.954087i	$0.596830\pi$
$500$	0	0
$501$	11.8541	0.529602
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−2.23607	−0.0995037
$506$	0	0
$507$	25.8885	1.14975
$508$	0	0
$509$	−23.5279	−1.04285	−0.521427	−	0.853296i	$-0.674600\pi$
$509$	−23.5279	−1.04285	−0.521427	+	0.853296i	$0.674600\pi$
$510$	0	0
$511$	0.145898	0.00645415
$512$	0	0
$513$	−2.23607	−0.0987248
$514$	0	0
$515$	−18.4164	−0.811524
$516$	0	0
$517$	−19.9787	−0.878663
$518$	0	0
$519$	35.8885	1.57533
$520$	0	0
$521$	−14.8885	−0.652279	−0.326139	−	0.945322i	$-0.605748\pi$
$521$	−14.8885	−0.652279	−0.326139	+	0.945322i	$0.605748\pi$
$522$	0	0
$523$	13.9443	0.609740	0.304870	−	0.952394i	$-0.401387\pi$
$523$	13.9443	0.609740	0.304870	+	0.952394i	$0.401387\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.03444	0.0886217
$528$	0	0
$529$	15.8885	0.690806
$530$	0	0
$531$	−41.2705	−1.79099
$532$	0	0
$533$	−18.6312	−0.807006
$534$	0	0
$535$	31.1803	1.34804
$536$	0	0
$537$	−0.527864	−0.0227790
$538$	0	0
$539$	2.85410	0.122935
$540$	0	0
$541$	−14.9443	−0.642504	−0.321252	−	0.946994i	$-0.604104\pi$
$541$	−14.9443	−0.642504	−0.321252	+	0.946994i	$0.604104\pi$
$542$	0	0
$543$	−53.3607	−2.28993
$544$	0	0
$545$	22.8885	0.980437
$546$	0	0
$547$	−24.0344	−1.02764	−0.513819	−	0.857898i	$-0.671770\pi$
$547$	−24.0344	−1.02764	−0.513819	+	0.857898i	$0.671770\pi$
$548$	0	0
$549$	48.9787	2.09036
$550$	0	0
$551$	4.38197	0.186678
$552$	0	0
$553$	−4.47214	−0.190175
$554$	0	0
$555$	8.61803	0.365815
$556$	0	0
$557$	−5.90983	−0.250408	−0.125204	−	0.992131i	$-0.539958\pi$
$557$	−5.90983	−0.250408	−0.125204	+	0.992131i	$0.539958\pi$
$558$	0	0
$559$	−12.2492	−0.518087
$560$	0	0
$561$	−6.38197	−0.269447
$562$	0	0
$563$	13.7639	0.580081	0.290040	−	0.957014i	$-0.406331\pi$
$563$	13.7639	0.580081	0.290040	+	0.957014i	$0.406331\pi$
$564$	0	0
$565$	−4.79837	−0.201869
$566$	0	0
$567$	5.70820	0.239722
$568$	0	0
$569$	17.3607	0.727798	0.363899	−	0.931438i	$-0.381445\pi$
$569$	17.3607	0.727798	0.363899	+	0.931438i	$0.381445\pi$
$570$	0	0
$571$	17.0000	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$572$	0	0
$573$	−62.9230	−2.62865
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.2148	0.591769	0.295885	−	0.955224i	$-0.404386\pi$
$577$	14.2148	0.591769	0.295885	+	0.955224i	$0.404386\pi$
$578$	0	0
$579$	−17.9443	−0.745738
$580$	0	0
$581$	11.3820	0.472204
$582$	0	0
$583$	5.29180	0.219164
$584$	0	0
$585$	15.2016	0.628510
$586$	0	0
$587$	−8.47214	−0.349682	−0.174841	−	0.984597i	$-0.555941\pi$
$587$	−8.47214	−0.349682	−0.174841	+	0.984597i	$0.555941\pi$
$588$	0	0
$589$	2.38197	0.0981472
$590$	0	0
$591$	−40.7426	−1.67593
$592$	0	0
$593$	−28.7771	−1.18173	−0.590867	−	0.806769i	$-0.701214\pi$
$593$	−28.7771	−1.18173	−0.590867	+	0.806769i	$0.701214\pi$
$594$	0	0
$595$	−1.90983	−0.0782954
$596$	0	0
$597$	−0.145898	−0.00597121
$598$	0	0
$599$	−32.6869	−1.33555	−0.667776	−	0.744363i	$-0.732753\pi$
$599$	−32.6869	−1.33555	−0.667776	+	0.744363i	$0.732753\pi$
$600$	0	0
$601$	9.74265	0.397411	0.198705	−	0.980059i	$-0.436326\pi$
$601$	9.74265	0.397411	0.198705	+	0.980059i	$0.436326\pi$
$602$	0	0
$603$	−21.6525	−0.881757
$604$	0	0
$605$	−6.38197	−0.259464
$606$	0	0
$607$	−3.18034	−0.129086	−0.0645430	−	0.997915i	$-0.520559\pi$
$607$	−3.18034	−0.129086	−0.0645430	+	0.997915i	$0.520559\pi$
$608$	0	0
$609$	11.4721	0.464874
$610$	0	0
$611$	−12.3475	−0.499527
$612$	0	0
$613$	−14.6180	−0.590417	−0.295208	−	0.955433i	$-0.595389\pi$
$613$	−14.6180	−0.590417	−0.295208	+	0.955433i	$0.595389\pi$
$614$	0	0
$615$	61.8328	2.49334
$616$	0	0
$617$	−30.2705	−1.21864	−0.609322	−	0.792923i	$-0.708558\pi$
$617$	−30.2705	−1.21864	−0.609322	+	0.792923i	$0.708558\pi$
$618$	0	0
$619$	−30.6869	−1.23341	−0.616706	−	0.787194i	$-0.711533\pi$
$619$	−30.6869	−1.23341	−0.616706	+	0.787194i	$0.711533\pi$
$620$	0	0
$621$	13.9443	0.559564
$622$	0	0
$623$	−0.763932	−0.0306063
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	−7.47214	−0.298408
$628$	0	0
$629$	−1.25735	−0.0501340
$630$	0	0
$631$	−28.0557	−1.11688	−0.558440	−	0.829545i	$-0.688600\pi$
$631$	−28.0557	−1.11688	−0.558440	+	0.829545i	$0.688600\pi$
$632$	0	0
$633$	56.4508	2.24372
$634$	0	0
$635$	−5.12461	−0.203364
$636$	0	0
$637$	1.76393	0.0698895
$638$	0	0
$639$	−15.2016	−0.601367
$640$	0	0
$641$	−8.38197	−0.331068	−0.165534	−	0.986204i	$-0.552935\pi$
$641$	−8.38197	−0.331068	−0.165534	+	0.986204i	$0.552935\pi$
$642$	0	0
$643$	6.23607	0.245927	0.122963	−	0.992411i	$-0.460760\pi$
$643$	6.23607	0.245927	0.122963	+	0.992411i	$0.460760\pi$
$644$	0	0
$645$	40.6525	1.60069
$646$	0	0
$647$	−23.3607	−0.918403	−0.459202	−	0.888332i	$-0.651864\pi$
$647$	−23.3607	−0.918403	−0.459202	+	0.888332i	$0.651864\pi$
$648$	0	0
$649$	−30.5623	−1.19968
$650$	0	0
$651$	6.23607	0.244411
$652$	0	0
$653$	24.5967	0.962545	0.481273	−	0.876571i	$-0.340175\pi$
$653$	24.5967	0.962545	0.481273	+	0.876571i	$0.340175\pi$
$654$	0	0
$655$	21.9098	0.856088
$656$	0	0
$657$	−0.562306	−0.0219376
$658$	0	0
$659$	−18.1591	−0.707376	−0.353688	−	0.935363i	$-0.615073\pi$
$659$	−18.1591	−0.707376	−0.353688	+	0.935363i	$0.615073\pi$
$660$	0	0
$661$	−39.3050	−1.52879	−0.764393	−	0.644751i	$-0.776961\pi$
$661$	−39.3050	−1.52879	−0.764393	+	0.644751i	$0.776961\pi$
$662$	0	0
$663$	−3.94427	−0.153183
$664$	0	0
$665$	−2.23607	−0.0867110
$666$	0	0
$667$	−27.3262	−1.05808
$668$	0	0
$669$	−73.1591	−2.82849
$670$	0	0
$671$	36.2705	1.40021
$672$	0	0
$673$	−9.56231	−0.368600	−0.184300	−	0.982870i	$-0.559002\pi$
$673$	−9.56231	−0.368600	−0.184300	+	0.982870i	$0.559002\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.0344	−0.539387	−0.269694	−	0.962946i	$-0.586922\pi$
$677$	−14.0344	−0.539387	−0.269694	+	0.962946i	$0.586922\pi$
$678$	0	0
$679$	3.76393	0.144446
$680$	0	0
$681$	67.5410	2.58818
$682$	0	0
$683$	38.7771	1.48376	0.741882	−	0.670530i	$-0.233933\pi$
$683$	38.7771	1.48376	0.741882	+	0.670530i	$0.233933\pi$
$684$	0	0
$685$	4.47214	0.170872
$686$	0	0
$687$	75.4853	2.87994
$688$	0	0
$689$	3.27051	0.124597
$690$	0	0
$691$	−5.81966	−0.221390	−0.110695	−	0.993854i	$-0.535308\pi$
$691$	−5.81966	−0.221390	−0.110695	+	0.993854i	$0.535308\pi$
$692$	0	0
$693$	−11.0000	−0.417855
$694$	0	0
$695$	−25.6525	−0.973054
$696$	0	0
$697$	−9.02129	−0.341706
$698$	0	0
$699$	36.1246	1.36636
$700$	0	0
$701$	−10.8328	−0.409150	−0.204575	−	0.978851i	$-0.565581\pi$
$701$	−10.8328	−0.409150	−0.204575	+	0.978851i	$0.565581\pi$
$702$	0	0
$703$	−1.47214	−0.0555227
$704$	0	0
$705$	40.9787	1.54335
$706$	0	0
$707$	1.00000	0.0376089
$708$	0	0
$709$	50.7082	1.90439	0.952193	−	0.305496i	$-0.0988223\pi$
$709$	50.7082	1.90439	0.952193	+	0.305496i	$0.0988223\pi$
$710$	0	0
$711$	17.2361	0.646403
$712$	0	0
$713$	−14.8541	−0.556290
$714$	0	0
$715$	11.2574	0.421001
$716$	0	0
$717$	59.4508	2.22023
$718$	0	0
$719$	−33.3050	−1.24207	−0.621033	−	0.783785i	$-0.713287\pi$
$719$	−33.3050	−1.24207	−0.621033	+	0.783785i	$0.713287\pi$
$720$	0	0
$721$	8.23607	0.306727
$722$	0	0
$723$	−39.5967	−1.47262
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.5410	1.54067	0.770336	−	0.637639i	$-0.220089\pi$
$727$	41.5410	1.54067	0.770336	+	0.637639i	$0.220089\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	−5.93112	−0.219370
$732$	0	0
$733$	−12.3607	−0.456552	−0.228276	−	0.973596i	$-0.573309\pi$
$733$	−12.3607	−0.456552	−0.228276	+	0.973596i	$0.573309\pi$
$734$	0	0
$735$	−5.85410	−0.215932
$736$	0	0
$737$	−16.0344	−0.590636
$738$	0	0
$739$	40.5967	1.49338	0.746688	−	0.665175i	$-0.231643\pi$
$739$	40.5967	1.49338	0.746688	+	0.665175i	$0.231643\pi$
$740$	0	0
$741$	−4.61803	−0.169648
$742$	0	0
$743$	−9.11146	−0.334267	−0.167133	−	0.985934i	$-0.553451\pi$
$743$	−9.11146	−0.334267	−0.167133	+	0.985934i	$0.553451\pi$
$744$	0	0
$745$	−28.0132	−1.02632
$746$	0	0
$747$	−43.8673	−1.60502
$748$	0	0
$749$	−13.9443	−0.509513
$750$	0	0
$751$	35.0902	1.28046	0.640229	−	0.768184i	$-0.278839\pi$
$751$	35.0902	1.28046	0.640229	+	0.768184i	$0.278839\pi$
$752$	0	0
$753$	−73.2492	−2.66935
$754$	0	0
$755$	19.7984	0.720537
$756$	0	0
$757$	−5.63932	−0.204965	−0.102482	−	0.994735i	$-0.532678\pi$
$757$	−5.63932	−0.204965	−0.102482	+	0.994735i	$0.532678\pi$
$758$	0	0
$759$	46.5967	1.69135
$760$	0	0
$761$	25.7639	0.933942	0.466971	−	0.884273i	$-0.345345\pi$
$761$	25.7639	0.933942	0.466971	+	0.884273i	$0.345345\pi$
$762$	0	0
$763$	−10.2361	−0.370571
$764$	0	0
$765$	7.36068	0.266126
$766$	0	0
$767$	−18.8885	−0.682026
$768$	0	0
$769$	−11.0557	−0.398680	−0.199340	−	0.979930i	$-0.563880\pi$
$769$	−11.0557	−0.398680	−0.199340	+	0.979930i	$0.563880\pi$
$770$	0	0
$771$	−49.9787	−1.79994
$772$	0	0
$773$	−13.2361	−0.476068	−0.238034	−	0.971257i	$-0.576503\pi$
$773$	−13.2361	−0.476068	−0.238034	+	0.971257i	$0.576503\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.85410	−0.138265
$778$	0	0
$779$	−10.5623	−0.378434
$780$	0	0
$781$	−11.2574	−0.402820
$782$	0	0
$783$	−9.79837	−0.350165
$784$	0	0
$785$	6.38197	0.227782
$786$	0	0
$787$	52.7214	1.87931	0.939657	−	0.342119i	$-0.111144\pi$
$787$	52.7214	1.87931	0.939657	+	0.342119i	$0.111144\pi$
$788$	0	0
$789$	30.5623	1.08805
$790$	0	0
$791$	2.14590	0.0762994
$792$	0	0
$793$	22.4164	0.796030
$794$	0	0
$795$	−10.8541	−0.384955
$796$	0	0
$797$	−2.20163	−0.0779856	−0.0389928	−	0.999239i	$-0.512415\pi$
$797$	−2.20163	−0.0779856	−0.0389928	+	0.999239i	$0.512415\pi$
$798$	0	0
$799$	−5.97871	−0.211512
$800$	0	0
$801$	2.94427	0.104031
$802$	0	0
$803$	−0.416408	−0.0146947
$804$	0	0
$805$	13.9443	0.491471
$806$	0	0
$807$	−41.5066	−1.46110
$808$	0	0
$809$	−17.7639	−0.624547	−0.312273	−	0.949992i	$-0.601090\pi$
$809$	−17.7639	−0.624547	−0.312273	+	0.949992i	$0.601090\pi$
$810$	0	0
$811$	−48.2492	−1.69426	−0.847130	−	0.531386i	$-0.821671\pi$
$811$	−48.2492	−1.69426	−0.847130	+	0.531386i	$0.821671\pi$
$812$	0	0
$813$	−44.5967	−1.56408
$814$	0	0
$815$	−15.8541	−0.555345
$816$	0	0
$817$	−6.94427	−0.242949
$818$	0	0
$819$	−6.79837	−0.237554
$820$	0	0
$821$	38.8328	1.35527	0.677637	−	0.735396i	$-0.263004\pi$
$821$	38.8328	1.35527	0.677637	+	0.735396i	$0.263004\pi$
$822$	0	0
$823$	−35.7082	−1.24471	−0.622355	−	0.782735i	$-0.713824\pi$
$823$	−35.7082	−1.24471	−0.622355	+	0.782735i	$0.713824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.3050	1.54063	0.770317	−	0.637661i	$-0.220098\pi$
$827$	44.3050	1.54063	0.770317	+	0.637661i	$0.220098\pi$
$828$	0	0
$829$	12.5836	0.437046	0.218523	−	0.975832i	$-0.429876\pi$
$829$	12.5836	0.437046	0.218523	+	0.975832i	$0.429876\pi$
$830$	0	0
$831$	54.0689	1.87563
$832$	0	0
$833$	0.854102	0.0295929
$834$	0	0
$835$	−10.1246	−0.350377
$836$	0	0
$837$	−5.32624	−0.184102
$838$	0	0
$839$	−10.4721	−0.361538	−0.180769	−	0.983526i	$-0.557859\pi$
$839$	−10.4721	−0.361538	−0.180769	+	0.983526i	$0.557859\pi$
$840$	0	0
$841$	−9.79837	−0.337875
$842$	0	0
$843$	20.3262	0.700073
$844$	0	0
$845$	−22.1115	−0.760657
$846$	0	0
$847$	2.85410	0.0980681
$848$	0	0
$849$	−49.2148	−1.68905
$850$	0	0
$851$	9.18034	0.314698
$852$	0	0
$853$	51.0476	1.74784	0.873918	−	0.486073i	$-0.161571\pi$
$853$	51.0476	1.74784	0.873918	+	0.486073i	$0.161571\pi$
$854$	0	0
$855$	8.61803	0.294731
$856$	0	0
$857$	25.1459	0.858968	0.429484	−	0.903075i	$-0.358696\pi$
$857$	25.1459	0.858968	0.429484	+	0.903075i	$0.358696\pi$
$858$	0	0
$859$	37.2148	1.26975	0.634876	−	0.772614i	$-0.281051\pi$
$859$	37.2148	1.26975	0.634876	+	0.772614i	$0.281051\pi$
$860$	0	0
$861$	−27.6525	−0.942394
$862$	0	0
$863$	8.03444	0.273496	0.136748	−	0.990606i	$-0.456335\pi$
$863$	8.03444	0.273496	0.136748	+	0.990606i	$0.456335\pi$
$864$	0	0
$865$	−30.6525	−1.04222
$866$	0	0
$867$	42.5967	1.44666
$868$	0	0
$869$	12.7639	0.432987
$870$	0	0
$871$	−9.90983	−0.335782
$872$	0	0
$873$	−14.5066	−0.490973
$874$	0	0
$875$	11.1803	0.377964
$876$	0	0
$877$	−29.7082	−1.00317	−0.501587	−	0.865107i	$-0.667250\pi$
$877$	−29.7082	−1.00317	−0.501587	+	0.865107i	$0.667250\pi$
$878$	0	0
$879$	−60.2148	−2.03099
$880$	0	0
$881$	13.2705	0.447095	0.223547	−	0.974693i	$-0.428236\pi$
$881$	13.2705	0.447095	0.223547	+	0.974693i	$0.428236\pi$
$882$	0	0
$883$	−4.12461	−0.138804	−0.0694021	−	0.997589i	$-0.522109\pi$
$883$	−4.12461	−0.138804	−0.0694021	+	0.997589i	$0.522109\pi$
$884$	0	0
$885$	62.6869	2.10720
$886$	0	0
$887$	−21.4721	−0.720964	−0.360482	−	0.932766i	$-0.617388\pi$
$887$	−21.4721	−0.720964	−0.360482	+	0.932766i	$0.617388\pi$
$888$	0	0
$889$	2.29180	0.0768644
$890$	0	0
$891$	−16.2918	−0.545796
$892$	0	0
$893$	−7.00000	−0.234246
$894$	0	0
$895$	0.450850	0.0150702
$896$	0	0
$897$	28.7984	0.961550
$898$	0	0
$899$	10.4377	0.348117
$900$	0	0
$901$	1.58359	0.0527571
$902$	0	0
$903$	−18.1803	−0.605004
$904$	0	0
$905$	45.5755	1.51498
$906$	0	0
$907$	−41.1803	−1.36737	−0.683685	−	0.729777i	$-0.739624\pi$
$907$	−41.1803	−1.36737	−0.683685	+	0.729777i	$0.739624\pi$
$908$	0	0
$909$	−3.85410	−0.127832
$910$	0	0
$911$	−32.4721	−1.07585	−0.537925	−	0.842993i	$-0.680792\pi$
$911$	−32.4721	−1.07585	−0.537925	+	0.842993i	$0.680792\pi$
$912$	0	0
$913$	−32.4853	−1.07511
$914$	0	0
$915$	−74.3951	−2.45943
$916$	0	0
$917$	−9.79837	−0.323571
$918$	0	0
$919$	−56.1935	−1.85365	−0.926826	−	0.375491i	$-0.877474\pi$
$919$	−56.1935	−1.85365	−0.926826	+	0.375491i	$0.877474\pi$
$920$	0	0
$921$	−32.1246	−1.05854
$922$	0	0
$923$	−6.95743	−0.229007
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−31.7426	−1.04257
$928$	0	0
$929$	19.7426	0.647735	0.323868	−	0.946102i	$-0.395017\pi$
$929$	19.7426	0.647735	0.323868	+	0.946102i	$0.395017\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−66.7771	−2.18618
$934$	0	0
$935$	5.45085	0.178262
$936$	0	0
$937$	39.3262	1.28473	0.642366	−	0.766398i	$-0.277953\pi$
$937$	39.3262	1.28473	0.642366	+	0.766398i	$0.277953\pi$
$938$	0	0
$939$	56.5410	1.84515
$940$	0	0
$941$	41.0557	1.33838	0.669189	−	0.743092i	$-0.266642\pi$
$941$	41.0557	1.33838	0.669189	+	0.743092i	$0.266642\pi$
$942$	0	0
$943$	65.8673	2.14493
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	20.4377	0.664136	0.332068	−	0.943255i	$-0.392254\pi$
$947$	20.4377	0.664136	0.332068	+	0.943255i	$0.392254\pi$
$948$	0	0
$949$	−0.257354	−0.00835407
$950$	0	0
$951$	27.1246	0.879576
$952$	0	0
$953$	49.2837	1.59645	0.798227	−	0.602356i	$-0.205771\pi$
$953$	49.2837	1.59645	0.798227	+	0.602356i	$0.205771\pi$
$954$	0	0
$955$	53.7426	1.73907
$956$	0	0
$957$	−32.7426	−1.05842
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−25.3262	−0.816975
$962$	0	0
$963$	53.7426	1.73183
$964$	0	0
$965$	15.3262	0.493369
$966$	0	0
$967$	−39.1459	−1.25885	−0.629424	−	0.777062i	$-0.716709\pi$
$967$	−39.1459	−1.25885	−0.629424	+	0.777062i	$0.716709\pi$
$968$	0	0
$969$	−2.23607	−0.0718329
$970$	0	0
$971$	−27.7639	−0.890987	−0.445493	−	0.895285i	$-0.646972\pi$
$971$	−27.7639	−0.890987	−0.445493	+	0.895285i	$0.646972\pi$
$972$	0	0
$973$	11.4721	0.367780
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−59.7082	−1.91023	−0.955117	−	0.296228i	$-0.904271\pi$
$977$	−59.7082	−1.91023	−0.955117	+	0.296228i	$0.904271\pi$
$978$	0	0
$979$	2.18034	0.0696840
$980$	0	0
$981$	39.4508	1.25957
$982$	0	0
$983$	−1.18034	−0.0376470	−0.0188235	−	0.999823i	$-0.505992\pi$
$983$	−1.18034	−0.0376470	−0.0188235	+	0.999823i	$0.505992\pi$
$984$	0	0
$985$	34.7984	1.10877
$986$	0	0
$987$	−18.3262	−0.583331
$988$	0	0
$989$	43.3050	1.37702
$990$	0	0
$991$	17.5967	0.558979	0.279490	−	0.960149i	$-0.409835\pi$
$991$	17.5967	0.558979	0.279490	+	0.960149i	$0.409835\pi$
$992$	0	0
$993$	82.1591	2.60724
$994$	0	0
$995$	0.124612	0.00395046
$996$	0	0
$997$	−20.5066	−0.649450	−0.324725	−	0.945809i	$-0.605272\pi$
$997$	−20.5066	−0.649450	−0.324725	+	0.945809i	$0.605272\pi$
$998$	0	0
$999$	3.29180	0.104148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.i.1.1		2
4.3	odd	2		8512.2.a.bf.1.2		2
8.3	odd	2		1064.2.a.a.1.1	✓	2
8.5	even	2		2128.2.a.n.1.2		2
24.11	even	2		9576.2.a.bm.1.2		2
56.27	even	2		7448.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.a.1.1	✓	2	8.3	odd	2
2128.2.a.n.1.2		2	8.5	even	2
7448.2.a.bc.1.2		2	56.27	even	2
8512.2.a.i.1.1		2	1.1	even	1	trivial
8512.2.a.bf.1.2		2	4.3	odd	2
9576.2.a.bm.1.2		2	24.11	even	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)

\(f(q)\)	\(=\)	\(q-2.61803 q^{3} +2.23607 q^{5} -1.00000 q^{7} +3.85410 q^{9} +O(q^{10})\)	\(q-2.61803 q^{3} +2.23607 q^{5} -1.00000 q^{7} +3.85410 q^{9} +2.85410 q^{11} +1.76393 q^{13} -5.85410 q^{15} +0.854102 q^{17} +1.00000 q^{19} +2.61803 q^{21} -6.23607 q^{23} -2.23607 q^{27} +4.38197 q^{29} +2.38197 q^{31} -7.47214 q^{33} -2.23607 q^{35} -1.47214 q^{37} -4.61803 q^{39} -10.5623 q^{41} -6.94427 q^{43} +8.61803 q^{45} -7.00000 q^{47} +1.00000 q^{49} -2.23607 q^{51} +1.85410 q^{53} +6.38197 q^{55} -2.61803 q^{57} -10.7082 q^{59} +12.7082 q^{61} -3.85410 q^{63} +3.94427 q^{65} -5.61803 q^{67} +16.3262 q^{69} -3.94427 q^{71} -0.145898 q^{73} -2.85410 q^{77} +4.47214 q^{79} -5.70820 q^{81} -11.3820 q^{83} +1.90983 q^{85} -11.4721 q^{87} +0.763932 q^{89} -1.76393 q^{91} -6.23607 q^{93} +2.23607 q^{95} -3.76393 q^{97} +11.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 2 * q^7 + q^9	\( 2 q - 3 q^{3} - 2 q^{7} + q^{9} - q^{11} + 8 q^{13} - 5 q^{15} - 5 q^{17} + 2 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{29} + 7 q^{31} - 6 q^{33} + 6 q^{37} - 7 q^{39} - q^{41} + 4 q^{43} + 15 q^{45} - 14 q^{47} + 2 q^{49} - 3 q^{53} + 15 q^{55} - 3 q^{57} - 8 q^{59} + 12 q^{61} - q^{63} - 10 q^{65} - 9 q^{67} + 17 q^{69} + 10 q^{71} - 7 q^{73} + q^{77} + 2 q^{81} - 25 q^{83} + 15 q^{85} - 14 q^{87} + 6 q^{89} - 8 q^{91} - 8 q^{93} - 12 q^{97} + 22 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 2 * q^7 + q^9 - q^11 + 8 * q^13 - 5 * q^15 - 5 * q^17 + 2 * q^19 + 3 * q^21 - 8 * q^23 + 11 * q^29 + 7 * q^31 - 6 * q^33 + 6 * q^37 - 7 * q^39 - q^41 + 4 * q^43 + 15 * q^45 - 14 * q^47 + 2 * q^49 - 3 * q^53 + 15 * q^55 - 3 * q^57 - 8 * q^59 + 12 * q^61 - q^63 - 10 * q^65 - 9 * q^67 + 17 * q^69 + 10 * q^71 - 7 * q^73 + q^77 + 2 * q^81 - 25 * q^83 + 15 * q^85 - 14 * q^87 + 6 * q^89 - 8 * q^91 - 8 * q^93 - 12 * q^97 + 22 * q^99

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