LMFDB - Embedded newform 8712.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, 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("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.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:	$69.5656702409$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85410 q^{5} -1.85410 q^{7} +O(q^{10})$	$q+2.85410 q^{5} -1.85410 q^{7} +3.23607 q^{13} +4.00000 q^{17} -0.763932 q^{19} -7.23607 q^{23} +3.14590 q^{25} -0.381966 q^{29} -5.85410 q^{31} -5.29180 q^{35} -4.00000 q^{37} -7.23607 q^{41} -9.70820 q^{43} -8.94427 q^{47} -3.56231 q^{49} -11.8541 q^{53} -5.38197 q^{59} +14.1803 q^{61} +9.23607 q^{65} +8.00000 q^{67} -10.9443 q^{71} -14.3820 q^{73} -0.381966 q^{79} -1.61803 q^{83} +11.4164 q^{85} -6.76393 q^{89} -6.00000 q^{91} -2.18034 q^{95} +16.2705 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + 3 q^{7}+O(q^{10}) $ 2 * q - q^5 + 3 * q^7	$ 2 q - q^{5} + 3 q^{7} + 2 q^{13} + 8 q^{17} - 6 q^{19} - 10 q^{23} + 13 q^{25} - 3 q^{29} - 5 q^{31} - 24 q^{35} - 8 q^{37} - 10 q^{41} - 6 q^{43} + 13 q^{49} - 17 q^{53} - 13 q^{59} + 6 q^{61} + 14 q^{65} + 16 q^{67} - 4 q^{71} - 31 q^{73} - 3 q^{79} - q^{83} - 4 q^{85} - 18 q^{89} - 12 q^{91} + 18 q^{95} - q^{97}+O(q^{100}) $ 2 * q - q^5 + 3 * q^7 + 2 * q^13 + 8 * q^17 - 6 * q^19 - 10 * q^23 + 13 * q^25 - 3 * q^29 - 5 * q^31 - 24 * q^35 - 8 * q^37 - 10 * q^41 - 6 * q^43 + 13 * q^49 - 17 * q^53 - 13 * q^59 + 6 * q^61 + 14 * q^65 + 16 * q^67 - 4 * q^71 - 31 * q^73 - 3 * q^79 - q^83 - 4 * q^85 - 18 * q^89 - 12 * q^91 + 18 * q^95 - q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.85410	1.27639	0.638197	−	0.769873i	$-0.279681\pi$
$5$	2.85410	1.27639	0.638197	+	0.769873i	$0.279681\pi$
$6$	0	0
$7$	−1.85410	−0.700785	−0.350392	−	0.936603i	$-0.613952\pi$
$7$	−1.85410	−0.700785	−0.350392	+	0.936603i	$0.613952\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−0.763932	−0.175258	−0.0876290	−	0.996153i	$-0.527929\pi$
$19$	−0.763932	−0.175258	−0.0876290	+	0.996153i	$0.527929\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.23607	−1.50882	−0.754412	−	0.656401i	$-0.772078\pi$
$23$	−7.23607	−1.50882	−0.754412	+	0.656401i	$0.772078\pi$
$24$	0	0
$25$	3.14590	0.629180
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.381966	−0.0709293	−0.0354647	−	0.999371i	$-0.511291\pi$
$29$	−0.381966	−0.0709293	−0.0354647	+	0.999371i	$0.511291\pi$
$30$	0	0
$31$	−5.85410	−1.05143	−0.525714	−	0.850661i	$-0.676202\pi$
$31$	−5.85410	−1.05143	−0.525714	+	0.850661i	$0.676202\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.29180	−0.894477
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.23607	−1.13008	−0.565042	−	0.825062i	$-0.691140\pi$
$41$	−7.23607	−1.13008	−0.565042	+	0.825062i	$0.691140\pi$
$42$	0	0
$43$	−9.70820	−1.48049	−0.740244	−	0.672339i	$-0.765290\pi$
$43$	−9.70820	−1.48049	−0.740244	+	0.672339i	$0.765290\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	−3.56231	−0.508901
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.8541	−1.62829	−0.814143	−	0.580664i	$-0.802793\pi$
$53$	−11.8541	−1.62829	−0.814143	+	0.580664i	$0.802793\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.38197	−0.700672	−0.350336	−	0.936624i	$-0.613933\pi$
$59$	−5.38197	−0.700672	−0.350336	+	0.936624i	$0.613933\pi$
$60$	0	0
$61$	14.1803	1.81561	0.907803	−	0.419396i	$-0.137758\pi$
$61$	14.1803	1.81561	0.907803	+	0.419396i	$0.137758\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.23607	1.14559
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.9443	−1.29885	−0.649423	−	0.760427i	$-0.724990\pi$
$71$	−10.9443	−1.29885	−0.649423	+	0.760427i	$0.724990\pi$
$72$	0	0
$73$	−14.3820	−1.68328	−0.841641	−	0.540038i	$-0.818410\pi$
$73$	−14.3820	−1.68328	−0.841641	+	0.540038i	$0.818410\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.381966	−0.0429745	−0.0214873	−	0.999769i	$-0.506840\pi$
$79$	−0.381966	−0.0429745	−0.0214873	+	0.999769i	$0.506840\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.61803	−0.177602	−0.0888012	−	0.996049i	$-0.528304\pi$
$83$	−1.61803	−0.177602	−0.0888012	+	0.996049i	$0.528304\pi$
$84$	0	0
$85$	11.4164	1.23828
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.76393	−0.716975	−0.358488	−	0.933534i	$-0.616707\pi$
$89$	−6.76393	−0.716975	−0.358488	+	0.933534i	$0.616707\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.18034	−0.223698
$96$	0	0
$97$	16.2705	1.65202	0.826010	−	0.563655i	$-0.190605\pi$
$97$	16.2705	1.65202	0.826010	+	0.563655i	$0.190605\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.6180	1.15604	0.578019	−	0.816023i	$-0.303826\pi$
$101$	11.6180	1.15604	0.578019	+	0.816023i	$0.303826\pi$
$102$	0	0
$103$	8.61803	0.849160	0.424580	−	0.905390i	$-0.360422\pi$
$103$	8.61803	0.849160	0.424580	+	0.905390i	$0.360422\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.85410	0.372590	0.186295	−	0.982494i	$-0.440352\pi$
$107$	3.85410	0.372590	0.186295	+	0.982494i	$0.440352\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.05573	0.0993145	0.0496573	−	0.998766i	$-0.484187\pi$
$113$	1.05573	0.0993145	0.0496573	+	0.998766i	$0.484187\pi$
$114$	0	0
$115$	−20.6525	−1.92585
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.41641	−0.679861
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.29180	−0.473313
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.09017	0.0952486	0.0476243	−	0.998865i	$-0.484835\pi$
$131$	1.09017	0.0952486	0.0476243	+	0.998865i	$0.484835\pi$
$132$	0	0
$133$	1.41641	0.122818
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.7082	−1.68379	−0.841893	−	0.539645i	$-0.818559\pi$
$137$	−19.7082	−1.68379	−0.841893	+	0.539645i	$0.818559\pi$
$138$	0	0
$139$	10.1803	0.863485	0.431743	−	0.901997i	$-0.357899\pi$
$139$	10.1803	0.863485	0.431743	+	0.901997i	$0.357899\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.09017	−0.0905337
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.5623	−0.947221	−0.473611	−	0.880734i	$-0.657050\pi$
$149$	−11.5623	−0.947221	−0.473611	+	0.880734i	$0.657050\pi$
$150$	0	0
$151$	0.381966	0.0310840	0.0155420	−	0.999879i	$-0.495053\pi$
$151$	0.381966	0.0310840	0.0155420	+	0.999879i	$0.495053\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.7082	−1.34204
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.4164	1.05736
$162$	0	0
$163$	11.5279	0.902932	0.451466	−	0.892288i	$-0.350901\pi$
$163$	11.5279	0.902932	0.451466	+	0.892288i	$0.350901\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.70820	−0.132185	−0.0660924	−	0.997814i	$-0.521053\pi$
$167$	−1.70820	−0.132185	−0.0660924	+	0.997814i	$0.521053\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.09017	−0.0828841	−0.0414420	−	0.999141i	$-0.513195\pi$
$173$	−1.09017	−0.0828841	−0.0414420	+	0.999141i	$0.513195\pi$
$174$	0	0
$175$	−5.83282	−0.440919
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.3820	−0.775985	−0.387992	−	0.921663i	$-0.626831\pi$
$179$	−10.3820	−0.775985	−0.387992	+	0.921663i	$0.626831\pi$
$180$	0	0
$181$	−4.47214	−0.332411	−0.166206	−	0.986091i	$-0.553152\pi$
$181$	−4.47214	−0.332411	−0.166206	+	0.986091i	$0.553152\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.4164	−0.839351
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.9443	0.936615	0.468307	−	0.883566i	$-0.344864\pi$
$191$	12.9443	0.936615	0.468307	+	0.883566i	$0.344864\pi$
$192$	0	0
$193$	−25.5623	−1.84002	−0.920008	−	0.391901i	$-0.871818\pi$
$193$	−25.5623	−1.84002	−0.920008	+	0.391901i	$0.871818\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.0344	1.35615	0.678074	−	0.734994i	$-0.262815\pi$
$197$	19.0344	1.35615	0.678074	+	0.734994i	$0.262815\pi$
$198$	0	0
$199$	19.0344	1.34932	0.674658	−	0.738131i	$-0.264291\pi$
$199$	19.0344	1.34932	0.674658	+	0.738131i	$0.264291\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.708204	0.0497062
$204$	0	0
$205$	−20.6525	−1.44243
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−14.1803	−0.976215	−0.488107	−	0.872784i	$-0.662313\pi$
$211$	−14.1803	−0.976215	−0.488107	+	0.872784i	$0.662313\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−27.7082	−1.88968
$216$	0	0
$217$	10.8541	0.736824
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.9443	0.870726
$222$	0	0
$223$	−13.9098	−0.931471	−0.465736	−	0.884924i	$-0.654210\pi$
$223$	−13.9098	−0.931471	−0.465736	+	0.884924i	$0.654210\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.0344	−1.39610	−0.698052	−	0.716047i	$-0.745950\pi$
$227$	−21.0344	−1.39610	−0.698052	+	0.716047i	$0.745950\pi$
$228$	0	0
$229$	−11.5279	−0.761783	−0.380891	−	0.924620i	$-0.624383\pi$
$229$	−11.5279	−0.761783	−0.380891	+	0.924620i	$0.624383\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.8885	0.909869	0.454934	−	0.890525i	$-0.349663\pi$
$233$	13.8885	0.909869	0.454934	+	0.890525i	$0.349663\pi$
$234$	0	0
$235$	−25.5279	−1.66525
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.5279	−0.745676	−0.372838	−	0.927897i	$-0.621615\pi$
$239$	−11.5279	−0.745676	−0.372838	+	0.927897i	$0.621615\pi$
$240$	0	0
$241$	−12.3262	−0.794003	−0.397001	−	0.917818i	$-0.629949\pi$
$241$	−12.3262	−0.794003	−0.397001	+	0.917818i	$0.629949\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.1672	−0.649558
$246$	0	0
$247$	−2.47214	−0.157298
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.5623	1.67660	0.838299	−	0.545211i	$-0.183550\pi$
$251$	26.5623	1.67660	0.838299	+	0.545211i	$0.183550\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.7082	1.47888	0.739439	−	0.673224i	$-0.235091\pi$
$257$	23.7082	1.47888	0.739439	+	0.673224i	$0.235091\pi$
$258$	0	0
$259$	7.41641	0.460833
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.7082	−0.721959	−0.360979	−	0.932574i	$-0.617558\pi$
$263$	−11.7082	−0.721959	−0.360979	+	0.932574i	$0.617558\pi$
$264$	0	0
$265$	−33.8328	−2.07833
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−23.4164	−1.42245	−0.711223	−	0.702967i	$-0.751858\pi$
$271$	−23.4164	−1.42245	−0.711223	+	0.702967i	$0.751858\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.1246	−0.908750	−0.454375	−	0.890811i	$-0.650137\pi$
$277$	−15.1246	−0.908750	−0.454375	+	0.890811i	$0.650137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.6525	1.23202	0.616012	−	0.787737i	$-0.288747\pi$
$281$	20.6525	1.23202	0.616012	+	0.787737i	$0.288747\pi$
$282$	0	0
$283$	26.3607	1.56698	0.783490	−	0.621405i	$-0.213438\pi$
$283$	26.3607	1.56698	0.783490	+	0.621405i	$0.213438\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.4164	0.791946
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.145898	0.00852345	0.00426173	−	0.999991i	$-0.498643\pi$
$293$	0.145898	0.00852345	0.00426173	+	0.999991i	$0.498643\pi$
$294$	0	0
$295$	−15.3607	−0.894333
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−23.4164	−1.35421
$300$	0	0
$301$	18.0000	1.03750
$302$	0	0
$303$	0	0
$304$	0	0
$305$	40.4721	2.31743
$306$	0	0
$307$	1.81966	0.103853	0.0519267	−	0.998651i	$-0.483464\pi$
$307$	1.81966	0.103853	0.0519267	+	0.998651i	$0.483464\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.47214	−0.480411	−0.240205	−	0.970722i	$-0.577215\pi$
$311$	−8.47214	−0.480411	−0.240205	+	0.970722i	$0.577215\pi$
$312$	0	0
$313$	−32.8541	−1.85702	−0.928512	−	0.371303i	$-0.878911\pi$
$313$	−32.8541	−1.85702	−0.928512	+	0.371303i	$0.878911\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.0557	−0.733283	−0.366641	−	0.930362i	$-0.619492\pi$
$317$	−13.0557	−0.733283	−0.366641	+	0.930362i	$0.619492\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.05573	−0.170025
$324$	0	0
$325$	10.1803	0.564704
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.5836	0.914283
$330$	0	0
$331$	27.4164	1.50694	0.753471	−	0.657481i	$-0.228378\pi$
$331$	27.4164	1.50694	0.753471	+	0.657481i	$0.228378\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	22.8328	1.24749
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.5836	1.05741
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.5623	−1.26489	−0.632445	−	0.774605i	$-0.717949\pi$
$347$	−23.5623	−1.26489	−0.632445	+	0.774605i	$0.717949\pi$
$348$	0	0
$349$	21.5967	1.15605	0.578024	−	0.816020i	$-0.303824\pi$
$349$	21.5967	1.15605	0.578024	+	0.816020i	$0.303824\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.23607	0.0657893	0.0328946	−	0.999459i	$-0.489527\pi$
$353$	1.23607	0.0657893	0.0328946	+	0.999459i	$0.489527\pi$
$354$	0	0
$355$	−31.2361	−1.65784
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.7082	−1.56794	−0.783970	−	0.620799i	$-0.786808\pi$
$359$	−29.7082	−1.56794	−0.783970	+	0.620799i	$0.786808\pi$
$360$	0	0
$361$	−18.4164	−0.969285
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−41.0476	−2.14853
$366$	0	0
$367$	13.3262	0.695624	0.347812	−	0.937564i	$-0.386925\pi$
$367$	13.3262	0.695624	0.347812	+	0.937564i	$0.386925\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	21.9787	1.14108
$372$	0	0
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.23607	−0.0636607
$378$	0	0
$379$	10.1803	0.522929	0.261464	−	0.965213i	$-0.415795\pi$
$379$	10.1803	0.522929	0.261464	+	0.965213i	$0.415795\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.2918	−1.24125	−0.620626	−	0.784106i	$-0.713122\pi$
$383$	−24.2918	−1.24125	−0.620626	+	0.784106i	$0.713122\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.94427	−0.149281	−0.0746403	−	0.997211i	$-0.523781\pi$
$389$	−2.94427	−0.149281	−0.0746403	+	0.997211i	$0.523781\pi$
$390$	0	0
$391$	−28.9443	−1.46377
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.09017	−0.0548524
$396$	0	0
$397$	12.9443	0.649654	0.324827	−	0.945773i	$-0.394694\pi$
$397$	12.9443	0.649654	0.324827	+	0.945773i	$0.394694\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.1803	−0.808008	−0.404004	−	0.914757i	$-0.632382\pi$
$401$	−16.1803	−0.808008	−0.404004	+	0.914757i	$0.632382\pi$
$402$	0	0
$403$	−18.9443	−0.943681
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−20.0902	−0.993395	−0.496697	−	0.867924i	$-0.665454\pi$
$409$	−20.0902	−0.993395	−0.496697	+	0.867924i	$0.665454\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.97871	0.491020
$414$	0	0
$415$	−4.61803	−0.226690
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.7984	0.674095	0.337047	−	0.941488i	$-0.390572\pi$
$419$	13.7984	0.674095	0.337047	+	0.941488i	$0.390572\pi$
$420$	0	0
$421$	7.12461	0.347232	0.173616	−	0.984813i	$-0.444455\pi$
$421$	7.12461	0.347232	0.173616	+	0.984813i	$0.444455\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.5836	0.610394
$426$	0	0
$427$	−26.2918	−1.27235
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.2361	−0.926569	−0.463284	−	0.886210i	$-0.653329\pi$
$431$	−19.2361	−0.926569	−0.463284	+	0.886210i	$0.653329\pi$
$432$	0	0
$433$	1.27051	0.0610568	0.0305284	−	0.999534i	$-0.490281\pi$
$433$	1.27051	0.0610568	0.0305284	+	0.999534i	$0.490281\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.52786	0.264434
$438$	0	0
$439$	6.50658	0.310542	0.155271	−	0.987872i	$-0.450375\pi$
$439$	6.50658	0.310542	0.155271	+	0.987872i	$0.450375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.8541	0.753251	0.376626	−	0.926366i	$-0.377084\pi$
$443$	15.8541	0.753251	0.376626	+	0.926366i	$0.377084\pi$
$444$	0	0
$445$	−19.3050	−0.915142
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.4721	−0.777368	−0.388684	−	0.921371i	$-0.627070\pi$
$449$	−16.4721	−0.777368	−0.388684	+	0.921371i	$0.627070\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.1246	−0.802814
$456$	0	0
$457$	−12.0902	−0.565554	−0.282777	−	0.959186i	$-0.591256\pi$
$457$	−12.0902	−0.565554	−0.282777	+	0.959186i	$0.591256\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−27.3820	−1.27255	−0.636274	−	0.771463i	$-0.719525\pi$
$463$	−27.3820	−1.27255	−0.636274	+	0.771463i	$0.719525\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.20163	0.148154	0.0740768	−	0.997253i	$-0.476399\pi$
$467$	3.20163	0.148154	0.0740768	+	0.997253i	$0.476399\pi$
$468$	0	0
$469$	−14.8328	−0.684916
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.40325	−0.110269
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.1246	1.14797	0.573986	−	0.818865i	$-0.305396\pi$
$479$	25.1246	1.14797	0.573986	+	0.818865i	$0.305396\pi$
$480$	0	0
$481$	−12.9443	−0.590208
$482$	0	0
$483$	0	0
$484$	0	0
$485$	46.4377	2.10863
$486$	0	0
$487$	−25.2705	−1.14512	−0.572558	−	0.819864i	$-0.694049\pi$
$487$	−25.2705	−1.14512	−0.572558	+	0.819864i	$0.694049\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.4721	−1.01415	−0.507077	−	0.861901i	$-0.669274\pi$
$491$	−22.4721	−1.01415	−0.507077	+	0.861901i	$0.669274\pi$
$492$	0	0
$493$	−1.52786	−0.0688115
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.2918	0.910211
$498$	0	0
$499$	27.4164	1.22733	0.613663	−	0.789568i	$-0.289695\pi$
$499$	27.4164	1.22733	0.613663	+	0.789568i	$0.289695\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.3607	0.729487	0.364743	−	0.931108i	$-0.381157\pi$
$503$	16.3607	0.729487	0.364743	+	0.931108i	$0.381157\pi$
$504$	0	0
$505$	33.1591	1.47556
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.9787	1.01851	0.509257	−	0.860615i	$-0.329920\pi$
$509$	22.9787	1.01851	0.509257	+	0.860615i	$0.329920\pi$
$510$	0	0
$511$	26.6656	1.17962
$512$	0	0
$513$	0	0
$514$	0	0
$515$	24.5967	1.08386
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.8328	0.474594	0.237297	−	0.971437i	$-0.423739\pi$
$521$	10.8328	0.474594	0.237297	+	0.971437i	$0.423739\pi$
$522$	0	0
$523$	−9.34752	−0.408739	−0.204369	−	0.978894i	$-0.565514\pi$
$523$	−9.34752	−0.408739	−0.204369	+	0.978894i	$0.565514\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.4164	−1.02003
$528$	0	0
$529$	29.3607	1.27655
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23.4164	−1.01428
$534$	0	0
$535$	11.0000	0.475571
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.5279	−1.01154	−0.505771	−	0.862668i	$-0.668792\pi$
$541$	−23.5279	−1.01154	−0.505771	+	0.862668i	$0.668792\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	42.6525	1.82703
$546$	0	0
$547$	30.7639	1.31537	0.657685	−	0.753293i	$-0.271536\pi$
$547$	30.7639	1.31537	0.657685	+	0.753293i	$0.271536\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.291796	0.0124309
$552$	0	0
$553$	0.708204	0.0301159
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.7426	1.42972	0.714861	−	0.699267i	$-0.246490\pi$
$557$	33.7426	1.42972	0.714861	+	0.699267i	$0.246490\pi$
$558$	0	0
$559$	−31.4164	−1.32877
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.7771	1.84498	0.922492	−	0.386016i	$-0.126149\pi$
$563$	43.7771	1.84498	0.922492	+	0.386016i	$0.126149\pi$
$564$	0	0
$565$	3.01316	0.126764
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.4721	−1.02593	−0.512963	−	0.858411i	$-0.671452\pi$
$569$	−24.4721	−1.02593	−0.512963	+	0.858411i	$0.671452\pi$
$570$	0	0
$571$	37.5967	1.57337	0.786687	−	0.617351i	$-0.211794\pi$
$571$	37.5967	1.57337	0.786687	+	0.617351i	$0.211794\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−22.7639	−0.949322
$576$	0	0
$577$	−10.7984	−0.449542	−0.224771	−	0.974412i	$-0.572163\pi$
$577$	−10.7984	−0.449542	−0.224771	+	0.974412i	$0.572163\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.00000	0.124461
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.1459	−1.28553	−0.642764	−	0.766064i	$-0.722213\pi$
$587$	−31.1459	−1.28553	−0.642764	+	0.766064i	$0.722213\pi$
$588$	0	0
$589$	4.47214	0.184271
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.9443	−0.942208	−0.471104	−	0.882078i	$-0.656144\pi$
$593$	−22.9443	−0.942208	−0.471104	+	0.882078i	$0.656144\pi$
$594$	0	0
$595$	−21.1672	−0.867770
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.41641	0.303026	0.151513	−	0.988455i	$-0.451585\pi$
$599$	7.41641	0.303026	0.151513	+	0.988455i	$0.451585\pi$
$600$	0	0
$601$	44.0344	1.79620	0.898101	−	0.439789i	$-0.144947\pi$
$601$	44.0344	1.79620	0.898101	+	0.439789i	$0.144947\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.0557	0.611093	0.305547	−	0.952177i	$-0.401161\pi$
$607$	15.0557	0.611093	0.305547	+	0.952177i	$0.401161\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−28.9443	−1.17096
$612$	0	0
$613$	19.8197	0.800509	0.400254	−	0.916404i	$-0.368922\pi$
$613$	19.8197	0.800509	0.400254	+	0.916404i	$0.368922\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.2361	−0.613381	−0.306691	−	0.951809i	$-0.599222\pi$
$617$	−15.2361	−0.613381	−0.306691	+	0.951809i	$0.599222\pi$
$618$	0	0
$619$	7.05573	0.283594	0.141797	−	0.989896i	$-0.454712\pi$
$619$	7.05573	0.283594	0.141797	+	0.989896i	$0.454712\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.5410	0.502445
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	7.79837	0.310448	0.155224	−	0.987879i	$-0.450390\pi$
$631$	7.79837	0.310448	0.155224	+	0.987879i	$0.450390\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	34.2492	1.35914
$636$	0	0
$637$	−11.5279	−0.456751
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20.0689	−0.792673	−0.396337	−	0.918105i	$-0.629719\pi$
$641$	−20.0689	−0.792673	−0.396337	+	0.918105i	$0.629719\pi$
$642$	0	0
$643$	−29.7771	−1.17429	−0.587147	−	0.809480i	$-0.699749\pi$
$643$	−29.7771	−1.17429	−0.587147	+	0.809480i	$0.699749\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.7639	1.28808	0.644042	−	0.764991i	$-0.277256\pi$
$647$	32.7639	1.28808	0.644042	+	0.764991i	$0.277256\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.03444	−0.275279	−0.137639	−	0.990482i	$-0.543952\pi$
$653$	−7.03444	−0.275279	−0.137639	+	0.990482i	$0.543952\pi$
$654$	0	0
$655$	3.11146	0.121575
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.90983	−0.152305	−0.0761527	−	0.997096i	$-0.524264\pi$
$659$	−3.90983	−0.152305	−0.0761527	+	0.997096i	$0.524264\pi$
$660$	0	0
$661$	−32.0689	−1.24734	−0.623668	−	0.781690i	$-0.714358\pi$
$661$	−32.0689	−1.24734	−0.623668	+	0.781690i	$0.714358\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.04257	0.156764
$666$	0	0
$667$	2.76393	0.107020
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.03444	0.0398748	0.0199374	−	0.999801i	$-0.493653\pi$
$673$	1.03444	0.0398748	0.0199374	+	0.999801i	$0.493653\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.506578	−0.0194694	−0.00973468	−	0.999953i	$-0.503099\pi$
$677$	−0.506578	−0.0194694	−0.00973468	+	0.999953i	$0.503099\pi$
$678$	0	0
$679$	−30.1672	−1.15771
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.437694	0.0167479	0.00837395	−	0.999965i	$-0.497334\pi$
$683$	0.437694	0.0167479	0.00837395	+	0.999965i	$0.497334\pi$
$684$	0	0
$685$	−56.2492	−2.14917
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−38.3607	−1.46143
$690$	0	0
$691$	32.6525	1.24216	0.621079	−	0.783748i	$-0.286694\pi$
$691$	32.6525	1.24216	0.621079	+	0.783748i	$0.286694\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	29.0557	1.10215
$696$	0	0
$697$	−28.9443	−1.09634
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.111456	−0.00420964	−0.00210482	−	0.999998i	$-0.500670\pi$
$701$	−0.111456	−0.00420964	−0.00210482	+	0.999998i	$0.500670\pi$
$702$	0	0
$703$	3.05573	0.115249
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.5410	−0.810133
$708$	0	0
$709$	3.23607	0.121533	0.0607665	−	0.998152i	$-0.480645\pi$
$709$	3.23607	0.121533	0.0607665	+	0.998152i	$0.480645\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	42.3607	1.58642
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.0557	0.785246	0.392623	−	0.919699i	$-0.371568\pi$
$719$	21.0557	0.785246	0.392623	+	0.919699i	$0.371568\pi$
$720$	0	0
$721$	−15.9787	−0.595078
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.20163	−0.0446273
$726$	0	0
$727$	−6.47214	−0.240038	−0.120019	−	0.992772i	$-0.538296\pi$
$727$	−6.47214	−0.240038	−0.120019	+	0.992772i	$0.538296\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−38.8328	−1.43628
$732$	0	0
$733$	10.7639	0.397575	0.198787	−	0.980043i	$-0.436300\pi$
$733$	10.7639	0.397575	0.198787	+	0.980043i	$0.436300\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	4.36068	0.160410	0.0802051	−	0.996778i	$-0.474442\pi$
$739$	4.36068	0.160410	0.0802051	+	0.996778i	$0.474442\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.88854	−0.142657	−0.0713284	−	0.997453i	$-0.522724\pi$
$743$	−3.88854	−0.142657	−0.0713284	+	0.997453i	$0.522724\pi$
$744$	0	0
$745$	−33.0000	−1.20903
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.14590	−0.261105
$750$	0	0
$751$	−18.8328	−0.687219	−0.343610	−	0.939113i	$-0.611650\pi$
$751$	−18.8328	−0.687219	−0.343610	+	0.939113i	$0.611650\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.09017	0.0396753
$756$	0	0
$757$	16.6525	0.605245	0.302622	−	0.953111i	$-0.402138\pi$
$757$	16.6525	0.605245	0.302622	+	0.953111i	$0.402138\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.70820	−0.0619223	−0.0309612	−	0.999521i	$-0.509857\pi$
$761$	−1.70820	−0.0619223	−0.0309612	+	0.999521i	$0.509857\pi$
$762$	0	0
$763$	−27.7082	−1.00310
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17.4164	−0.628870
$768$	0	0
$769$	29.6180	1.06805	0.534027	−	0.845468i	$-0.320678\pi$
$769$	29.6180	1.06805	0.534027	+	0.845468i	$0.320678\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	47.6180	1.71270	0.856351	−	0.516394i	$-0.172726\pi$
$773$	47.6180	1.71270	0.856351	+	0.516394i	$0.172726\pi$
$774$	0	0
$775$	−18.4164	−0.661537
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.52786	0.198056
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.70820	−0.203735
$786$	0	0
$787$	−48.1803	−1.71744	−0.858722	−	0.512442i	$-0.828741\pi$
$787$	−48.1803	−1.71744	−0.858722	+	0.512442i	$0.828741\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.95743	−0.0695981
$792$	0	0
$793$	45.8885	1.62955
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.5623	−1.25968	−0.629841	−	0.776724i	$-0.716880\pi$
$797$	−35.5623	−1.25968	−0.629841	+	0.776724i	$0.716880\pi$
$798$	0	0
$799$	−35.7771	−1.26570
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	38.2918	1.34961
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.0000	1.40633	0.703163	−	0.711029i	$-0.251771\pi$
$809$	40.0000	1.40633	0.703163	+	0.711029i	$0.251771\pi$
$810$	0	0
$811$	6.06888	0.213107	0.106554	−	0.994307i	$-0.466018\pi$
$811$	6.06888	0.213107	0.106554	+	0.994307i	$0.466018\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	32.9017	1.15250
$816$	0	0
$817$	7.41641	0.259467
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.8541	−1.39092	−0.695459	−	0.718566i	$-0.744799\pi$
$821$	−39.8541	−1.39092	−0.695459	+	0.718566i	$0.744799\pi$
$822$	0	0
$823$	−50.2705	−1.75232	−0.876160	−	0.482021i	$-0.839903\pi$
$823$	−50.2705	−1.75232	−0.876160	+	0.482021i	$0.839903\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.6738	−0.475483	−0.237742	−	0.971328i	$-0.576407\pi$
$827$	−13.6738	−0.475483	−0.237742	+	0.971328i	$0.576407\pi$
$828$	0	0
$829$	31.8885	1.10753	0.553767	−	0.832671i	$-0.313190\pi$
$829$	31.8885	1.10753	0.553767	+	0.832671i	$0.313190\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.2492	−0.493706
$834$	0	0
$835$	−4.87539	−0.168720
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.18034	0.0752737	0.0376368	−	0.999291i	$-0.488017\pi$
$839$	2.18034	0.0752737	0.0376368	+	0.999291i	$0.488017\pi$
$840$	0	0
$841$	−28.8541	−0.994969
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.21478	−0.248196
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	28.9443	0.992197
$852$	0	0
$853$	18.4721	0.632474	0.316237	−	0.948680i	$-0.397581\pi$
$853$	18.4721	0.632474	0.316237	+	0.948680i	$0.397581\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.7639	0.572645	0.286323	−	0.958133i	$-0.407567\pi$
$857$	16.7639	0.572645	0.286323	+	0.958133i	$0.407567\pi$
$858$	0	0
$859$	−39.0132	−1.33111	−0.665556	−	0.746348i	$-0.731805\pi$
$859$	−39.0132	−1.33111	−0.665556	+	0.746348i	$0.731805\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.70820	0.262390	0.131195	−	0.991357i	$-0.458119\pi$
$863$	7.70820	0.262390	0.131195	+	0.991357i	$0.458119\pi$
$864$	0	0
$865$	−3.11146	−0.105793
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	25.8885	0.877200
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.81153	0.331690
$876$	0	0
$877$	−4.83282	−0.163193	−0.0815963	−	0.996665i	$-0.526002\pi$
$877$	−4.83282	−0.163193	−0.0815963	+	0.996665i	$0.526002\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.0689	0.541374	0.270687	−	0.962667i	$-0.412749\pi$
$881$	16.0689	0.541374	0.270687	+	0.962667i	$0.412749\pi$
$882$	0	0
$883$	21.1246	0.710900	0.355450	−	0.934695i	$-0.384328\pi$
$883$	21.1246	0.710900	0.355450	+	0.934695i	$0.384328\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.29180	0.0769510	0.0384755	−	0.999260i	$-0.487750\pi$
$887$	2.29180	0.0769510	0.0384755	+	0.999260i	$0.487750\pi$
$888$	0	0
$889$	−22.2492	−0.746215
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.83282	0.228651
$894$	0	0
$895$	−29.6312	−0.990461
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.23607	0.0745770
$900$	0	0
$901$	−47.4164	−1.57967
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.7639	−0.424287
$906$	0	0
$907$	11.2361	0.373088	0.186544	−	0.982447i	$-0.440271\pi$
$907$	11.2361	0.373088	0.186544	+	0.982447i	$0.440271\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.7082	−0.652962	−0.326481	−	0.945204i	$-0.605863\pi$
$911$	−19.7082	−0.652962	−0.326481	+	0.945204i	$0.605863\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.02129	−0.0667488
$918$	0	0
$919$	14.2705	0.470741	0.235370	−	0.971906i	$-0.424370\pi$
$919$	14.2705	0.470741	0.235370	+	0.971906i	$0.424370\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−35.4164	−1.16575
$924$	0	0
$925$	−12.5836	−0.413746
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−37.9574	−1.24534	−0.622671	−	0.782483i	$-0.713953\pi$
$929$	−37.9574	−1.24534	−0.622671	+	0.782483i	$0.713953\pi$
$930$	0	0
$931$	2.72136	0.0891890
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.2148	−0.856400	−0.428200	−	0.903684i	$-0.640852\pi$
$937$	−26.2148	−0.856400	−0.428200	+	0.903684i	$0.640852\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.8885	−0.387555	−0.193778	−	0.981045i	$-0.562074\pi$
$941$	−11.8885	−0.387555	−0.193778	+	0.981045i	$0.562074\pi$
$942$	0	0
$943$	52.3607	1.70510
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−58.9787	−1.91655	−0.958275	−	0.285847i	$-0.907725\pi$
$947$	−58.9787	−1.91655	−0.958275	+	0.285847i	$0.907725\pi$
$948$	0	0
$949$	−46.5410	−1.51079
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.3607	0.789120	0.394560	−	0.918870i	$-0.370897\pi$
$953$	24.3607	0.789120	0.394560	+	0.918870i	$0.370897\pi$
$954$	0	0
$955$	36.9443	1.19549
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.5410	1.17997
$960$	0	0
$961$	3.27051	0.105500
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−72.9574	−2.34858
$966$	0	0
$967$	−51.9787	−1.67152	−0.835761	−	0.549093i	$-0.814973\pi$
$967$	−51.9787	−1.67152	−0.835761	+	0.549093i	$0.814973\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.52786	0.177398	0.0886988	−	0.996058i	$-0.471729\pi$
$971$	5.52786	0.177398	0.0886988	+	0.996058i	$0.471729\pi$
$972$	0	0
$973$	−18.8754	−0.605117
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.0689	0.706046	0.353023	−	0.935615i	$-0.385154\pi$
$977$	22.0689	0.706046	0.353023	+	0.935615i	$0.385154\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.12461	−0.163450	−0.0817249	−	0.996655i	$-0.526043\pi$
$983$	−5.12461	−0.163450	−0.0817249	+	0.996655i	$0.526043\pi$
$984$	0	0
$985$	54.3262	1.73098
$986$	0	0
$987$	0	0
$988$	0	0
$989$	70.2492	2.23380
$990$	0	0
$991$	−41.6869	−1.32423	−0.662114	−	0.749403i	$-0.730341\pi$
$991$	−41.6869	−1.32423	−0.662114	+	0.749403i	$0.730341\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	54.3262	1.72226
$996$	0	0
$997$	−3.12461	−0.0989574	−0.0494787	−	0.998775i	$-0.515756\pi$
$997$	−3.12461	−0.0989574	−0.0494787	+	0.998775i	$0.515756\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bg.1.2		2
3.2	odd	2		2904.2.a.w.1.1		2
11.3	even	5		792.2.r.e.361.1		4
11.4	even	5		792.2.r.e.577.1		4
11.10	odd	2		8712.2.a.bf.1.2		2
12.11	even	2		5808.2.a.br.1.1		2
33.14	odd	10		264.2.q.a.97.1	yes	4
33.26	odd	10		264.2.q.a.49.1	✓	4
33.32	even	2		2904.2.a.v.1.1		2
132.47	even	10		528.2.y.e.97.1		4
132.59	even	10		528.2.y.e.49.1		4
132.131	odd	2		5808.2.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.q.a.49.1	✓	4	33.26	odd	10
264.2.q.a.97.1	yes	4	33.14	odd	10
528.2.y.e.49.1		4	132.59	even	10
528.2.y.e.97.1		4	132.47	even	10
792.2.r.e.361.1		4	11.3	even	5
792.2.r.e.577.1		4	11.4	even	5
2904.2.a.v.1.1		2	33.32	even	2
2904.2.a.w.1.1		2	3.2	odd	2
5808.2.a.br.1.1		2	12.11	even	2
5808.2.a.bs.1.1		2	132.131	odd	2
8712.2.a.bf.1.2		2	11.10	odd	2
8712.2.a.bg.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q+2.85410 q^{5} -1.85410 q^{7} +O(q^{10})\)	\(q+2.85410 q^{5} -1.85410 q^{7} +3.23607 q^{13} +4.00000 q^{17} -0.763932 q^{19} -7.23607 q^{23} +3.14590 q^{25} -0.381966 q^{29} -5.85410 q^{31} -5.29180 q^{35} -4.00000 q^{37} -7.23607 q^{41} -9.70820 q^{43} -8.94427 q^{47} -3.56231 q^{49} -11.8541 q^{53} -5.38197 q^{59} +14.1803 q^{61} +9.23607 q^{65} +8.00000 q^{67} -10.9443 q^{71} -14.3820 q^{73} -0.381966 q^{79} -1.61803 q^{83} +11.4164 q^{85} -6.76393 q^{89} -6.00000 q^{91} -2.18034 q^{95} +16.2705 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + 3 q^{7}+O(q^{10}) \) 2 * q - q^5 + 3 * q^7	\( 2 q - q^{5} + 3 q^{7} + 2 q^{13} + 8 q^{17} - 6 q^{19} - 10 q^{23} + 13 q^{25} - 3 q^{29} - 5 q^{31} - 24 q^{35} - 8 q^{37} - 10 q^{41} - 6 q^{43} + 13 q^{49} - 17 q^{53} - 13 q^{59} + 6 q^{61} + 14 q^{65} + 16 q^{67} - 4 q^{71} - 31 q^{73} - 3 q^{79} - q^{83} - 4 q^{85} - 18 q^{89} - 12 q^{91} + 18 q^{95} - q^{97}+O(q^{100}) \) 2 * q - q^5 + 3 * q^7 + 2 * q^13 + 8 * q^17 - 6 * q^19 - 10 * q^23 + 13 * q^25 - 3 * q^29 - 5 * q^31 - 24 * q^35 - 8 * q^37 - 10 * q^41 - 6 * q^43 + 13 * q^49 - 17 * q^53 - 13 * q^59 + 6 * q^61 + 14 * q^65 + 16 * q^67 - 4 * q^71 - 31 * q^73 - 3 * q^79 - q^83 - 4 * q^85 - 18 * q^89 - 12 * q^91 + 18 * q^95 - q^97

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