LMFDB - Embedded newform 8712.2.a.ci.1.6

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:	$6$
Coefficient field:	6.6.62158000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 14x^{4} + 22x^{3} + 38x^{2} - 60x + 11 $ x^6 - 2x^5 - 14x^4 + 22x^3 + 38x^2 - 60*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 792)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.90351$ 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)

$f(q)$	$=$	$q+2.07994 q^{5} +1.39055 q^{7} +O(q^{10})$	$q+2.07994 q^{5} +1.39055 q^{7} -5.79180 q^{13} +7.23340 q^{17} -8.44752 q^{19} -2.19197 q^{23} -0.673854 q^{25} +7.03317 q^{29} -2.55706 q^{31} +2.89226 q^{35} -8.57290 q^{37} -1.07188 q^{41} +7.59881 q^{43} -10.7784 q^{47} -5.06637 q^{49} +0.550965 q^{53} -10.4911 q^{59} +10.5902 q^{61} -12.0466 q^{65} +2.71154 q^{67} +2.01881 q^{71} +0.302349 q^{73} -4.91851 q^{79} -10.6654 q^{83} +15.0450 q^{85} -4.07002 q^{89} -8.05378 q^{91} -17.5703 q^{95} -1.17377 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{5} + q^{7}+O(q^{10}) $ 6 * q - 5 * q^5 + q^7	$ 6 q - 5 q^{5} + q^{7} + 4 q^{13} + 6 q^{17} - 14 q^{19} - 6 q^{23} + 7 q^{25} + 7 q^{29} - 7 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} - 12 q^{43} - 26 q^{47} + 9 q^{49} - 9 q^{53} - 17 q^{59} + 20 q^{61} - 48 q^{65} + 14 q^{67} - 18 q^{71} + 7 q^{73} + 9 q^{79} + 11 q^{83} + 10 q^{85} - 14 q^{89} + 8 q^{91} + 30 q^{95} + 25 q^{97}+O(q^{100}) $ 6 * q - 5 * q^5 + q^7 + 4 * q^13 + 6 * q^17 - 14 * q^19 - 6 * q^23 + 7 * q^25 + 7 * q^29 - 7 * q^31 - 2 * q^35 + 2 * q^37 - 6 * q^41 - 12 * q^43 - 26 * q^47 + 9 * q^49 - 9 * q^53 - 17 * q^59 + 20 * q^61 - 48 * q^65 + 14 * q^67 - 18 * q^71 + 7 * q^73 + 9 * q^79 + 11 * q^83 + 10 * q^85 - 14 * q^89 + 8 * q^91 + 30 * q^95 + 25 * 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
$5$	2.07994	0.930177	0.465088	−	0.885264i	$-0.346023\pi$
$5$	2.07994	0.930177	0.465088	+	0.885264i	$0.346023\pi$
$6$	0	0
$7$	1.39055	0.525578	0.262789	−	0.964853i	$-0.415358\pi$
$7$	1.39055	0.525578	0.262789	+	0.964853i	$0.415358\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.79180	−1.60636	−0.803178	−	0.595739i	$-0.796859\pi$
$13$	−5.79180	−1.60636	−0.803178	+	0.595739i	$0.796859\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.23340	1.75436	0.877179	−	0.480164i	$-0.159423\pi$
$17$	7.23340	1.75436	0.877179	+	0.480164i	$0.159423\pi$
$18$	0	0
$19$	−8.44752	−1.93799	−0.968997	−	0.247073i	$-0.920531\pi$
$19$	−8.44752	−1.93799	−0.968997	+	0.247073i	$0.920531\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.19197	−0.457057	−0.228528	−	0.973537i	$-0.573391\pi$
$23$	−2.19197	−0.457057	−0.228528	+	0.973537i	$0.573391\pi$
$24$	0	0
$25$	−0.673854	−0.134771
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.03317	1.30603	0.653014	−	0.757346i	$-0.273504\pi$
$29$	7.03317	1.30603	0.653014	+	0.757346i	$0.273504\pi$
$30$	0	0
$31$	−2.55706	−0.459261	−0.229631	−	0.973278i	$-0.573752\pi$
$31$	−2.55706	−0.459261	−0.229631	+	0.973278i	$0.573752\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.89226	0.488881
$36$	0	0
$37$	−8.57290	−1.40938	−0.704688	−	0.709517i	$-0.748913\pi$
$37$	−8.57290	−1.40938	−0.704688	+	0.709517i	$0.748913\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.07188	−0.167399	−0.0836995	−	0.996491i	$-0.526674\pi$
$41$	−1.07188	−0.167399	−0.0836995	+	0.996491i	$0.526674\pi$
$42$	0	0
$43$	7.59881	1.15881	0.579404	−	0.815040i	$-0.303285\pi$
$43$	7.59881	1.15881	0.579404	+	0.815040i	$0.303285\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.7784	−1.57220	−0.786098	−	0.618102i	$-0.787902\pi$
$47$	−10.7784	−1.57220	−0.786098	+	0.618102i	$0.787902\pi$
$48$	0	0
$49$	−5.06637	−0.723768
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.550965	0.0756808	0.0378404	−	0.999284i	$-0.487952\pi$
$53$	0.550965	0.0756808	0.0378404	+	0.999284i	$0.487952\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.4911	−1.36582	−0.682912	−	0.730500i	$-0.739287\pi$
$59$	−10.4911	−1.36582	−0.682912	+	0.730500i	$0.739287\pi$
$60$	0	0
$61$	10.5902	1.35594	0.677970	−	0.735090i	$-0.262860\pi$
$61$	10.5902	1.35594	0.677970	+	0.735090i	$0.262860\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.0466	−1.49420
$66$	0	0
$67$	2.71154	0.331267	0.165634	−	0.986187i	$-0.447033\pi$
$67$	2.71154	0.331267	0.165634	+	0.986187i	$0.447033\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.01881	0.239589	0.119795	−	0.992799i	$-0.461776\pi$
$71$	2.01881	0.239589	0.119795	+	0.992799i	$0.461776\pi$
$72$	0	0
$73$	0.302349	0.0353873	0.0176936	−	0.999843i	$-0.494368\pi$
$73$	0.302349	0.0353873	0.0176936	+	0.999843i	$0.494368\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.91851	−0.553375	−0.276688	−	0.960960i	$-0.589237\pi$
$79$	−4.91851	−0.553375	−0.276688	+	0.960960i	$0.589237\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.6654	−1.17068	−0.585339	−	0.810789i	$-0.699039\pi$
$83$	−10.6654	−1.17068	−0.585339	+	0.810789i	$0.699039\pi$
$84$	0	0
$85$	15.0450	1.63186
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.07002	−0.431421	−0.215710	−	0.976457i	$-0.569207\pi$
$89$	−4.07002	−0.431421	−0.215710	+	0.976457i	$0.569207\pi$
$90$	0	0
$91$	−8.05378	−0.844266
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−17.5703	−1.80268
$96$	0	0
$97$	−1.17377	−0.119178	−0.0595889	−	0.998223i	$-0.518979\pi$
$97$	−1.17377	−0.119178	−0.0595889	+	0.998223i	$0.518979\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.5486	−1.34814	−0.674068	−	0.738670i	$-0.735454\pi$
$101$	−13.5486	−1.34814	−0.674068	+	0.738670i	$0.735454\pi$
$102$	0	0
$103$	9.29833	0.916191	0.458096	−	0.888903i	$-0.348532\pi$
$103$	9.29833	0.916191	0.458096	+	0.888903i	$0.348532\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.0108	1.25781	0.628903	−	0.777484i	$-0.283504\pi$
$107$	13.0108	1.25781	0.628903	+	0.777484i	$0.283504\pi$
$108$	0	0
$109$	−6.97205	−0.667801	−0.333901	−	0.942608i	$-0.608365\pi$
$109$	−6.97205	−0.667801	−0.333901	+	0.942608i	$0.608365\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.27484	0.213999	0.106999	−	0.994259i	$-0.465876\pi$
$113$	2.27484	0.213999	0.106999	+	0.994259i	$0.465876\pi$
$114$	0	0
$115$	−4.55916	−0.425144
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.0584	0.922052
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8013	−1.05554
$126$	0	0
$127$	3.15906	0.280321	0.140161	−	0.990129i	$-0.455238\pi$
$127$	3.15906	0.280321	0.140161	+	0.990129i	$0.455238\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.1519	−1.67331	−0.836655	−	0.547730i	$-0.815492\pi$
$131$	−19.1519	−1.67331	−0.836655	+	0.547730i	$0.815492\pi$
$132$	0	0
$133$	−11.7467	−1.01857
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.61239	−0.821242	−0.410621	−	0.911806i	$-0.634688\pi$
$137$	−9.61239	−0.821242	−0.410621	+	0.911806i	$0.634688\pi$
$138$	0	0
$139$	−14.4601	−1.22649	−0.613244	−	0.789894i	$-0.710136\pi$
$139$	−14.4601	−1.22649	−0.613244	+	0.789894i	$0.710136\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	14.6286	1.21484
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.06471	0.742610	0.371305	−	0.928511i	$-0.378910\pi$
$149$	9.06471	0.742610	0.371305	+	0.928511i	$0.378910\pi$
$150$	0	0
$151$	3.46141	0.281685	0.140843	−	0.990032i	$-0.455019\pi$
$151$	3.46141	0.281685	0.140843	+	0.990032i	$0.455019\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.31852	−0.427194
$156$	0	0
$157$	−17.2040	−1.37303	−0.686514	−	0.727116i	$-0.740860\pi$
$157$	−17.2040	−1.37303	−0.686514	+	0.727116i	$0.740860\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.04804	−0.240219
$162$	0	0
$163$	−1.53727	−0.120408	−0.0602040	−	0.998186i	$-0.519175\pi$
$163$	−1.53727	−0.120408	−0.0602040	+	0.998186i	$0.519175\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.9812	−1.46881	−0.734404	−	0.678712i	$-0.762538\pi$
$167$	−18.9812	−1.46881	−0.734404	+	0.678712i	$0.762538\pi$
$168$	0	0
$169$	20.5449	1.58038
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.9353	1.21154	0.605769	−	0.795641i	$-0.292866\pi$
$173$	15.9353	1.21154	0.605769	+	0.795641i	$0.292866\pi$
$174$	0	0
$175$	−0.937027	−0.0708326
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.63638	−0.496027	−0.248013	−	0.968757i	$-0.579778\pi$
$179$	−6.63638	−0.496027	−0.248013	+	0.968757i	$0.579778\pi$
$180$	0	0
$181$	−9.85656	−0.732632	−0.366316	−	0.930490i	$-0.619381\pi$
$181$	−9.85656	−0.732632	−0.366316	+	0.930490i	$0.619381\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−17.8311	−1.31097
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.25837	−0.452840	−0.226420	−	0.974030i	$-0.572702\pi$
$191$	−6.25837	−0.452840	−0.226420	+	0.974030i	$0.572702\pi$
$192$	0	0
$193$	−14.8818	−1.07121	−0.535607	−	0.844467i	$-0.679917\pi$
$193$	−14.8818	−1.07121	−0.535607	+	0.844467i	$0.679917\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.6099	0.898416	0.449208	−	0.893427i	$-0.351706\pi$
$197$	12.6099	0.898416	0.449208	+	0.893427i	$0.351706\pi$
$198$	0	0
$199$	20.3905	1.44544	0.722721	−	0.691140i	$-0.242891\pi$
$199$	20.3905	1.44544	0.722721	+	0.691140i	$0.242891\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.77997	0.686420
$204$	0	0
$205$	−2.22944	−0.155711
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.05700	−0.554667	−0.277333	−	0.960774i	$-0.589451\pi$
$211$	−8.05700	−0.554667	−0.277333	+	0.960774i	$0.589451\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.8051	1.07790
$216$	0	0
$217$	−3.55572	−0.241378
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−41.8944	−2.81812
$222$	0	0
$223$	4.42920	0.296601	0.148301	−	0.988942i	$-0.452620\pi$
$223$	4.42920	0.296601	0.148301	+	0.988942i	$0.452620\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.6801	1.43896	0.719479	−	0.694515i	$-0.244381\pi$
$227$	21.6801	1.43896	0.719479	+	0.694515i	$0.244381\pi$
$228$	0	0
$229$	−6.92281	−0.457472	−0.228736	−	0.973488i	$-0.573459\pi$
$229$	−6.92281	−0.457472	−0.228736	+	0.973488i	$0.573459\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.27854	0.280297	0.140148	−	0.990131i	$-0.455242\pi$
$233$	4.27854	0.280297	0.140148	+	0.990131i	$0.455242\pi$
$234$	0	0
$235$	−22.4185	−1.46242
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.3560	0.863926	0.431963	−	0.901891i	$-0.357821\pi$
$239$	13.3560	0.863926	0.431963	+	0.901891i	$0.357821\pi$
$240$	0	0
$241$	10.0451	0.647061	0.323531	−	0.946218i	$-0.395130\pi$
$241$	10.0451	0.647061	0.323531	+	0.946218i	$0.395130\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.5377	−0.673232
$246$	0	0
$247$	48.9263	3.11311
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.6697	−1.30466	−0.652330	−	0.757935i	$-0.726209\pi$
$251$	−20.6697	−1.30466	−0.652330	+	0.757935i	$0.726209\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.4903	−0.779124	−0.389562	−	0.921000i	$-0.627374\pi$
$257$	−12.4903	−0.779124	−0.389562	+	0.921000i	$0.627374\pi$
$258$	0	0
$259$	−11.9210	−0.740737
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.6796	−1.02851	−0.514254	−	0.857638i	$-0.671931\pi$
$263$	−16.6796	−1.02851	−0.514254	+	0.857638i	$0.671931\pi$
$264$	0	0
$265$	1.14597	0.0703966
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.6211	1.37923	0.689615	−	0.724176i	$-0.257780\pi$
$269$	22.6211	1.37923	0.689615	+	0.724176i	$0.257780\pi$
$270$	0	0
$271$	4.39992	0.267276	0.133638	−	0.991030i	$-0.457334\pi$
$271$	4.39992	0.267276	0.133638	+	0.991030i	$0.457334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.8092	0.889800	0.444900	−	0.895580i	$-0.353239\pi$
$277$	14.8092	0.889800	0.444900	+	0.895580i	$0.353239\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	25.6414	1.52964	0.764818	−	0.644246i	$-0.222829\pi$
$281$	25.6414	1.52964	0.764818	+	0.644246i	$0.222829\pi$
$282$	0	0
$283$	−15.4934	−0.920988	−0.460494	−	0.887663i	$-0.652328\pi$
$283$	−15.4934	−0.920988	−0.460494	+	0.887663i	$0.652328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.49050	−0.0879812
$288$	0	0
$289$	35.3221	2.07777
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.0055	−0.642949	−0.321474	−	0.946918i	$-0.604178\pi$
$293$	−11.0055	−0.642949	−0.321474	+	0.946918i	$0.604178\pi$
$294$	0	0
$295$	−21.8208	−1.27046
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.6954	0.734196
$300$	0	0
$301$	10.5665	0.609044
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.0270	1.26126
$306$	0	0
$307$	−9.25608	−0.528272	−0.264136	−	0.964485i	$-0.585087\pi$
$307$	−9.25608	−0.528272	−0.264136	+	0.964485i	$0.585087\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.57301	0.542836	0.271418	−	0.962462i	$-0.412507\pi$
$311$	9.57301	0.542836	0.271418	+	0.962462i	$0.412507\pi$
$312$	0	0
$313$	−5.50665	−0.311254	−0.155627	−	0.987816i	$-0.549740\pi$
$313$	−5.50665	−0.311254	−0.155627	+	0.987816i	$0.549740\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.3576	0.862568	0.431284	−	0.902216i	$-0.358061\pi$
$317$	15.3576	0.862568	0.431284	+	0.902216i	$0.358061\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−61.1043	−3.39993
$324$	0	0
$325$	3.90283	0.216490
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−14.9879	−0.826312
$330$	0	0
$331$	−19.1054	−1.05013	−0.525064	−	0.851063i	$-0.675959\pi$
$331$	−19.1054	−1.05013	−0.525064	+	0.851063i	$0.675959\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.63984	0.308137
$336$	0	0
$337$	19.0067	1.03536	0.517680	−	0.855574i	$-0.326796\pi$
$337$	19.0067	1.03536	0.517680	+	0.855574i	$0.326796\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.7789	−0.905975
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.4727	1.47481	0.737406	−	0.675450i	$-0.236051\pi$
$347$	27.4727	1.47481	0.737406	+	0.675450i	$0.236051\pi$
$348$	0	0
$349$	3.31972	0.177700	0.0888502	−	0.996045i	$-0.471681\pi$
$349$	3.31972	0.177700	0.0888502	+	0.996045i	$0.471681\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.1782	0.594956	0.297478	−	0.954729i	$-0.403855\pi$
$353$	11.1782	0.594956	0.297478	+	0.954729i	$0.403855\pi$
$354$	0	0
$355$	4.19901	0.222860
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.590367	−0.0311584	−0.0155792	−	0.999879i	$-0.504959\pi$
$359$	−0.590367	−0.0311584	−0.0155792	+	0.999879i	$0.504959\pi$
$360$	0	0
$361$	52.3606	2.75582
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.628867	0.0329164
$366$	0	0
$367$	−30.6794	−1.60145	−0.800726	−	0.599030i	$-0.795553\pi$
$367$	−30.6794	−1.60145	−0.800726	+	0.599030i	$0.795553\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.766144	0.0397762
$372$	0	0
$373$	−8.85036	−0.458254	−0.229127	−	0.973396i	$-0.573587\pi$
$373$	−8.85036	−0.458254	−0.229127	+	0.973396i	$0.573587\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−40.7347	−2.09795
$378$	0	0
$379$	−22.6542	−1.16367	−0.581834	−	0.813308i	$-0.697665\pi$
$379$	−22.6542	−1.16367	−0.581834	+	0.813308i	$0.697665\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.26423	0.320087	0.160044	−	0.987110i	$-0.448837\pi$
$383$	6.26423	0.320087	0.160044	+	0.987110i	$0.448837\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−31.6929	−1.60689	−0.803447	−	0.595377i	$-0.797003\pi$
$389$	−31.6929	−1.60689	−0.803447	+	0.595377i	$0.797003\pi$
$390$	0	0
$391$	−15.8554	−0.801841
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.2302	−0.514737
$396$	0	0
$397$	−3.24531	−0.162877	−0.0814386	−	0.996678i	$-0.525951\pi$
$397$	−3.24531	−0.162877	−0.0814386	+	0.996678i	$0.525951\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.8348	1.04044	0.520220	−	0.854032i	$-0.325850\pi$
$401$	20.8348	1.04044	0.520220	+	0.854032i	$0.325850\pi$
$402$	0	0
$403$	14.8100	0.737737
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	13.6722	0.676046	0.338023	−	0.941138i	$-0.390242\pi$
$409$	13.6722	0.676046	0.338023	+	0.941138i	$0.390242\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.5884	−0.717848
$414$	0	0
$415$	−22.1834	−1.08894
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.05402	0.0514924	0.0257462	−	0.999669i	$-0.491804\pi$
$419$	1.05402	0.0514924	0.0257462	+	0.999669i	$0.491804\pi$
$420$	0	0
$421$	19.7281	0.961487	0.480744	−	0.876861i	$-0.340367\pi$
$421$	19.7281	0.961487	0.480744	+	0.876861i	$0.340367\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.87426	−0.236436
$426$	0	0
$427$	14.7262	0.712653
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.80184	−0.327633	−0.163817	−	0.986491i	$-0.552381\pi$
$431$	−6.80184	−0.327633	−0.163817	+	0.986491i	$0.552381\pi$
$432$	0	0
$433$	−4.36860	−0.209942	−0.104971	−	0.994475i	$-0.533475\pi$
$433$	−4.36860	−0.209942	−0.104971	+	0.994475i	$0.533475\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.5167	0.885773
$438$	0	0
$439$	−0.269843	−0.0128789	−0.00643944	−	0.999979i	$-0.502050\pi$
$439$	−0.269843	−0.0128789	−0.00643944	+	0.999979i	$0.502050\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.6718	−1.26721	−0.633607	−	0.773655i	$-0.718426\pi$
$443$	−26.6718	−1.26721	−0.633607	+	0.773655i	$0.718426\pi$
$444$	0	0
$445$	−8.46538	−0.401298
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.49052	−0.164728	−0.0823639	−	0.996602i	$-0.526247\pi$
$449$	−3.49052	−0.164728	−0.0823639	+	0.996602i	$0.526247\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16.7514	−0.785317
$456$	0	0
$457$	30.5942	1.43113	0.715567	−	0.698544i	$-0.246168\pi$
$457$	30.5942	1.43113	0.715567	+	0.698544i	$0.246168\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−28.8689	−1.34456	−0.672280	−	0.740297i	$-0.734685\pi$
$461$	−28.8689	−1.34456	−0.672280	+	0.740297i	$0.734685\pi$
$462$	0	0
$463$	−7.46908	−0.347118	−0.173559	−	0.984824i	$-0.555527\pi$
$463$	−7.46908	−0.347118	−0.173559	+	0.984824i	$0.555527\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.52200	−0.301802	−0.150901	−	0.988549i	$-0.548218\pi$
$467$	−6.52200	−0.301802	−0.150901	+	0.988549i	$0.548218\pi$
$468$	0	0
$469$	3.77053	0.174107
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.69239	0.261185
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.9726	−0.866882	−0.433441	−	0.901182i	$-0.642701\pi$
$479$	−18.9726	−0.866882	−0.433441	+	0.901182i	$0.642701\pi$
$480$	0	0
$481$	49.6525	2.26396
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.44136	−0.110857
$486$	0	0
$487$	20.6822	0.937199	0.468599	−	0.883411i	$-0.344759\pi$
$487$	20.6822	0.937199	0.468599	+	0.883411i	$0.344759\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.2184	−0.731925	−0.365963	−	0.930630i	$-0.619260\pi$
$491$	−16.2184	−0.731925	−0.365963	+	0.930630i	$0.619260\pi$
$492$	0	0
$493$	50.8738	2.29124
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.80726	0.125923
$498$	0	0
$499$	26.8935	1.20392	0.601958	−	0.798527i	$-0.294387\pi$
$499$	26.8935	1.20392	0.601958	+	0.798527i	$0.294387\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.561440	−0.0250334	−0.0125167	−	0.999922i	$-0.503984\pi$
$503$	−0.561440	−0.0250334	−0.0125167	+	0.999922i	$0.503984\pi$
$504$	0	0
$505$	−28.1802	−1.25400
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.50788	−0.332781	−0.166391	−	0.986060i	$-0.553211\pi$
$509$	−7.50788	−0.332781	−0.166391	+	0.986060i	$0.553211\pi$
$510$	0	0
$511$	0.420431	0.0185988
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.3399	0.852220
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.60817	−0.158077	−0.0790384	−	0.996872i	$-0.525185\pi$
$521$	−3.60817	−0.158077	−0.0790384	+	0.996872i	$0.525185\pi$
$522$	0	0
$523$	−15.3913	−0.673013	−0.336506	−	0.941681i	$-0.609245\pi$
$523$	−15.3913	−0.673013	−0.336506	+	0.941681i	$0.609245\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.4962	−0.805708
$528$	0	0
$529$	−18.1953	−0.791099
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.20809	0.268902
$534$	0	0
$535$	27.0618	1.16998
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.85837	0.337858	0.168929	−	0.985628i	$-0.445969\pi$
$541$	7.85837	0.337858	0.168929	+	0.985628i	$0.445969\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.5014	−0.621173
$546$	0	0
$547$	−34.7441	−1.48555	−0.742775	−	0.669541i	$-0.766491\pi$
$547$	−34.7441	−1.48555	−0.742775	+	0.669541i	$0.766491\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−59.4129	−2.53107
$552$	0	0
$553$	−6.83942	−0.290842
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.1420	0.599214	0.299607	−	0.954063i	$-0.403144\pi$
$557$	14.1420	0.599214	0.299607	+	0.954063i	$0.403144\pi$
$558$	0	0
$559$	−44.0108	−1.86146
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.6519	−0.870373	−0.435187	−	0.900340i	$-0.643318\pi$
$563$	−20.6519	−0.870373	−0.435187	+	0.900340i	$0.643318\pi$
$564$	0	0
$565$	4.73152	0.199057
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.0737	0.799611	0.399805	−	0.916600i	$-0.369078\pi$
$569$	19.0737	0.799611	0.399805	+	0.916600i	$0.369078\pi$
$570$	0	0
$571$	−3.33635	−0.139622	−0.0698110	−	0.997560i	$-0.522240\pi$
$571$	−3.33635	−0.139622	−0.0698110	+	0.997560i	$0.522240\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.47707	0.0615979
$576$	0	0
$577$	−5.54116	−0.230682	−0.115341	−	0.993326i	$-0.536796\pi$
$577$	−5.54116	−0.230682	−0.115341	+	0.993326i	$0.536796\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.8307	−0.615283
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.5147	0.764183	0.382092	−	0.924124i	$-0.375204\pi$
$587$	18.5147	0.764183	0.382092	+	0.924124i	$0.375204\pi$
$588$	0	0
$589$	21.6008	0.890045
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−39.4582	−1.62035	−0.810177	−	0.586186i	$-0.800629\pi$
$593$	−39.4582	−1.62035	−0.810177	+	0.586186i	$0.800629\pi$
$594$	0	0
$595$	20.9209	0.857672
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.3132	−1.52457	−0.762287	−	0.647239i	$-0.775924\pi$
$599$	−37.3132	−1.52457	−0.762287	+	0.647239i	$0.775924\pi$
$600$	0	0
$601$	−33.5132	−1.36703	−0.683516	−	0.729935i	$-0.739550\pi$
$601$	−33.5132	−1.36703	−0.683516	+	0.729935i	$0.739550\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.5608	0.631591	0.315796	−	0.948827i	$-0.397729\pi$
$607$	15.5608	0.631591	0.315796	+	0.948827i	$0.397729\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	62.4265	2.52551
$612$	0	0
$613$	26.9534	1.08864	0.544319	−	0.838878i	$-0.316788\pi$
$613$	26.9534	1.08864	0.544319	+	0.838878i	$0.316788\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.1595	−0.570040	−0.285020	−	0.958522i	$-0.592000\pi$
$617$	−14.1595	−0.570040	−0.285020	+	0.958522i	$0.592000\pi$
$618$	0	0
$619$	−22.7923	−0.916098	−0.458049	−	0.888927i	$-0.651452\pi$
$619$	−22.7923	−0.916098	−0.458049	+	0.888927i	$0.651452\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.65956	−0.226745
$624$	0	0
$625$	−21.1767	−0.847066
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−62.0112	−2.47255
$630$	0	0
$631$	3.60859	0.143656	0.0718278	−	0.997417i	$-0.477117\pi$
$631$	3.60859	0.143656	0.0718278	+	0.997417i	$0.477117\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.57065	0.260748
$636$	0	0
$637$	29.3434	1.16263
$638$	0	0
$639$	0	0
$640$	0	0
$641$	49.4830	1.95446	0.977230	−	0.212183i	$-0.0680572\pi$
$641$	49.4830	1.95446	0.977230	+	0.212183i	$0.0680572\pi$
$642$	0	0
$643$	47.4709	1.87207	0.936034	−	0.351908i	$-0.114467\pi$
$643$	47.4709	1.87207	0.936034	+	0.351908i	$0.114467\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.6658	−1.00903	−0.504514	−	0.863404i	$-0.668328\pi$
$647$	−25.6658	−1.00903	−0.504514	+	0.863404i	$0.668328\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.11555	0.0436547	0.0218273	−	0.999762i	$-0.493052\pi$
$653$	1.11555	0.0436547	0.0218273	+	0.999762i	$0.493052\pi$
$654$	0	0
$655$	−39.8348	−1.55647
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.5988	1.03614	0.518071	−	0.855338i	$-0.326650\pi$
$659$	26.5988	1.03614	0.518071	+	0.855338i	$0.326650\pi$
$660$	0	0
$661$	6.45103	0.250916	0.125458	−	0.992099i	$-0.459960\pi$
$661$	6.45103	0.250916	0.125458	+	0.992099i	$0.459960\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.4324	−0.947448
$666$	0	0
$667$	−15.4165	−0.596929
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.48012	0.249790	0.124895	−	0.992170i	$-0.460141\pi$
$673$	6.48012	0.249790	0.124895	+	0.992170i	$0.460141\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.48486	0.210800	0.105400	−	0.994430i	$-0.466388\pi$
$677$	5.48486	0.210800	0.105400	+	0.994430i	$0.466388\pi$
$678$	0	0
$679$	−1.63218	−0.0626373
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.9800	−0.458402	−0.229201	−	0.973379i	$-0.573611\pi$
$683$	−11.9800	−0.458402	−0.229201	+	0.973379i	$0.573611\pi$
$684$	0	0
$685$	−19.9932	−0.763901
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.19108	−0.121570
$690$	0	0
$691$	−24.9522	−0.949226	−0.474613	−	0.880195i	$-0.657412\pi$
$691$	−24.9522	−0.949226	−0.474613	+	0.880195i	$0.657412\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−30.0761	−1.14085
$696$	0	0
$697$	−7.75331	−0.293678
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−16.1665	−0.610600	−0.305300	−	0.952256i	$-0.598757\pi$
$701$	−16.1665	−0.610600	−0.305300	+	0.952256i	$0.598757\pi$
$702$	0	0
$703$	72.4197	2.73136
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.8400	−0.708550
$708$	0	0
$709$	−30.3518	−1.13989	−0.569943	−	0.821684i	$-0.693035\pi$
$709$	−30.3518	−1.13989	−0.569943	+	0.821684i	$0.693035\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.60499	0.209908
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	37.6654	1.40468	0.702341	−	0.711840i	$-0.252138\pi$
$719$	37.6654	1.40468	0.702341	+	0.711840i	$0.252138\pi$
$720$	0	0
$721$	12.9298	0.481530
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.73933	−0.176014
$726$	0	0
$727$	47.4997	1.76167	0.880833	−	0.473428i	$-0.156984\pi$
$727$	47.4997	1.76167	0.880833	+	0.473428i	$0.156984\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	54.9653	2.03296
$732$	0	0
$733$	−16.1572	−0.596781	−0.298391	−	0.954444i	$-0.596450\pi$
$733$	−16.1572	−0.596781	−0.298391	+	0.954444i	$0.596450\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−50.4917	−1.85737	−0.928684	−	0.370871i	$-0.879059\pi$
$739$	−50.4917	−1.85737	−0.928684	+	0.370871i	$0.879059\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.2859	−0.780902	−0.390451	−	0.920624i	$-0.627681\pi$
$743$	−21.2859	−0.780902	−0.390451	+	0.920624i	$0.627681\pi$
$744$	0	0
$745$	18.8540	0.690759
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.0922	0.661075
$750$	0	0
$751$	4.79656	0.175029	0.0875145	−	0.996163i	$-0.472108\pi$
$751$	4.79656	0.175029	0.0875145	+	0.996163i	$0.472108\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.19952	0.262017
$756$	0	0
$757$	−28.2400	−1.02640	−0.513200	−	0.858269i	$-0.671540\pi$
$757$	−28.2400	−1.02640	−0.513200	+	0.858269i	$0.671540\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9.78628	−0.354752	−0.177376	−	0.984143i	$-0.556761\pi$
$761$	−9.78628	−0.354752	−0.177376	+	0.984143i	$0.556761\pi$
$762$	0	0
$763$	−9.69498	−0.350982
$764$	0	0
$765$	0	0
$766$	0	0
$767$	60.7623	2.19400
$768$	0	0
$769$	−46.4358	−1.67452	−0.837259	−	0.546807i	$-0.815843\pi$
$769$	−46.4358	−1.67452	−0.837259	+	0.546807i	$0.815843\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−48.8843	−1.75825	−0.879123	−	0.476595i	$-0.841871\pi$
$773$	−48.8843	−1.75825	−0.879123	+	0.476595i	$0.841871\pi$
$774$	0	0
$775$	1.72308	0.0618950
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.05470	0.324418
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−35.7833	−1.27716
$786$	0	0
$787$	−3.29963	−0.117619	−0.0588096	−	0.998269i	$-0.518730\pi$
$787$	−3.29963	−0.117619	−0.0588096	+	0.998269i	$0.518730\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.16327	0.112473
$792$	0	0
$793$	−61.3365	−2.17812
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.7072	−1.08770	−0.543852	−	0.839181i	$-0.683035\pi$
$797$	−30.7072	−1.08770	−0.543852	+	0.839181i	$0.683035\pi$
$798$	0	0
$799$	−77.9647	−2.75819
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−6.33974	−0.223446
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.0909	0.741518	0.370759	−	0.928729i	$-0.379098\pi$
$809$	21.0909	0.741518	0.370759	+	0.928729i	$0.379098\pi$
$810$	0	0
$811$	−45.2483	−1.58888	−0.794441	−	0.607342i	$-0.792236\pi$
$811$	−45.2483	−1.58888	−0.794441	+	0.607342i	$0.792236\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.19742	−0.112001
$816$	0	0
$817$	−64.1911	−2.24576
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.29830	−0.254713	−0.127356	−	0.991857i	$-0.540649\pi$
$821$	−7.29830	−0.254713	−0.127356	+	0.991857i	$0.540649\pi$
$822$	0	0
$823$	−30.8711	−1.07610	−0.538050	−	0.842913i	$-0.680839\pi$
$823$	−30.8711	−1.07610	−0.538050	+	0.842913i	$0.680839\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.0220	1.18306	0.591530	−	0.806283i	$-0.298524\pi$
$827$	34.0220	1.18306	0.591530	+	0.806283i	$0.298524\pi$
$828$	0	0
$829$	43.5040	1.51096	0.755478	−	0.655174i	$-0.227405\pi$
$829$	43.5040	1.51096	0.755478	+	0.655174i	$0.227405\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−36.6471	−1.26975
$834$	0	0
$835$	−39.4797	−1.36625
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.8583	1.13439	0.567197	−	0.823582i	$-0.308028\pi$
$839$	32.8583	1.13439	0.567197	+	0.823582i	$0.308028\pi$
$840$	0	0
$841$	20.4655	0.705708
$842$	0	0
$843$	0	0
$844$	0	0
$845$	42.7322	1.47003
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.7915	0.644165
$852$	0	0
$853$	−11.5423	−0.395201	−0.197600	−	0.980283i	$-0.563315\pi$
$853$	−11.5423	−0.395201	−0.197600	+	0.980283i	$0.563315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.54480	0.291885	0.145942	−	0.989293i	$-0.453379\pi$
$857$	8.54480	0.291885	0.145942	+	0.989293i	$0.453379\pi$
$858$	0	0
$859$	55.4781	1.89289	0.946443	−	0.322869i	$-0.104647\pi$
$859$	55.4781	1.89289	0.946443	+	0.322869i	$0.104647\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.751023	−0.0255651	−0.0127826	−	0.999918i	$-0.504069\pi$
$863$	−0.751023	−0.0255651	−0.0127826	+	0.999918i	$0.504069\pi$
$864$	0	0
$865$	33.1444	1.12694
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−15.7047	−0.532133
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−16.4102	−0.554768
$876$	0	0
$877$	−6.93104	−0.234045	−0.117022	−	0.993129i	$-0.537335\pi$
$877$	−6.93104	−0.234045	−0.117022	+	0.993129i	$0.537335\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.45456	−0.183769	−0.0918844	−	0.995770i	$-0.529289\pi$
$881$	−5.45456	−0.183769	−0.0918844	+	0.995770i	$0.529289\pi$
$882$	0	0
$883$	−45.1936	−1.52088	−0.760442	−	0.649406i	$-0.775018\pi$
$883$	−45.1936	−1.52088	−0.760442	+	0.649406i	$0.775018\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.7926	1.33611	0.668053	−	0.744114i	$-0.267128\pi$
$887$	39.7926	1.33611	0.668053	+	0.744114i	$0.267128\pi$
$888$	0	0
$889$	4.39283	0.147331
$890$	0	0
$891$	0	0
$892$	0	0
$893$	91.0510	3.04691
$894$	0	0
$895$	−13.8033	−0.461393
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.9842	−0.599808
$900$	0	0
$901$	3.98535	0.132771
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20.5010	−0.681478
$906$	0	0
$907$	6.25231	0.207605	0.103802	−	0.994598i	$-0.466899\pi$
$907$	6.25231	0.207605	0.103802	+	0.994598i	$0.466899\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.0042	−1.55732	−0.778659	−	0.627447i	$-0.784100\pi$
$911$	−47.0042	−1.55732	−0.778659	+	0.627447i	$0.784100\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26.6317	−0.879455
$918$	0	0
$919$	36.7422	1.21201	0.606007	−	0.795460i	$-0.292770\pi$
$919$	36.7422	1.21201	0.606007	+	0.795460i	$0.292770\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.6926	−0.384866
$924$	0	0
$925$	5.77688	0.189943
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.1704	0.793005	0.396503	−	0.918034i	$-0.370224\pi$
$929$	24.1704	0.793005	0.396503	+	0.918034i	$0.370224\pi$
$930$	0	0
$931$	42.7983	1.40266
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.8459	0.811681	0.405840	−	0.913944i	$-0.366979\pi$
$937$	24.8459	0.811681	0.405840	+	0.913944i	$0.366979\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.36285	0.174824	0.0874120	−	0.996172i	$-0.472140\pi$
$941$	5.36285	0.174824	0.0874120	+	0.996172i	$0.472140\pi$
$942$	0	0
$943$	2.34952	0.0765108
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.39581	−0.0453576	−0.0226788	−	0.999743i	$-0.507220\pi$
$947$	−1.39581	−0.0453576	−0.0226788	+	0.999743i	$0.507220\pi$
$948$	0	0
$949$	−1.75114	−0.0568445
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25.7702	−0.834780	−0.417390	−	0.908728i	$-0.637055\pi$
$953$	−25.7702	−0.834780	−0.417390	+	0.908728i	$0.637055\pi$
$954$	0	0
$955$	−13.0170	−0.421221
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.3665	−0.431627
$960$	0	0
$961$	−24.4615	−0.789079
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−30.9532	−0.996420
$966$	0	0
$967$	37.4436	1.20410	0.602052	−	0.798457i	$-0.294350\pi$
$967$	37.4436	1.20410	0.602052	+	0.798457i	$0.294350\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.1236	−1.25554	−0.627768	−	0.778400i	$-0.716031\pi$
$971$	−39.1236	−1.25554	−0.627768	+	0.778400i	$0.716031\pi$
$972$	0	0
$973$	−20.1075	−0.644615
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.3777	0.843896	0.421948	−	0.906620i	$-0.361347\pi$
$977$	26.3777	0.843896	0.421948	+	0.906620i	$0.361347\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.7779	−1.20493	−0.602463	−	0.798147i	$-0.705814\pi$
$983$	−37.7779	−1.20493	−0.602463	+	0.798147i	$0.705814\pi$
$984$	0	0
$985$	26.2278	0.835686
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.6564	−0.529641
$990$	0	0
$991$	−28.8482	−0.916393	−0.458197	−	0.888851i	$-0.651504\pi$
$991$	−28.8482	−0.916393	−0.458197	+	0.888851i	$0.651504\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	42.4109	1.34452
$996$	0	0
$997$	17.0341	0.539474	0.269737	−	0.962934i	$-0.413063\pi$
$997$	17.0341	0.539474	0.269737	+	0.962934i	$0.413063\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.ci.1.6		6
3.2	odd	2		8712.2.a.ck.1.1		6
11.2	odd	10		792.2.r.i.433.1	yes	12
11.6	odd	10		792.2.r.i.289.1	yes	12
11.10	odd	2		8712.2.a.ch.1.6		6
33.2	even	10		792.2.r.h.433.3	yes	12
33.17	even	10		792.2.r.h.289.3	✓	12
33.32	even	2		8712.2.a.cj.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.r.h.289.3	✓	12	33.17	even	10
792.2.r.h.433.3	yes	12	33.2	even	10
792.2.r.i.289.1	yes	12	11.6	odd	10
792.2.r.i.433.1	yes	12	11.2	odd	10
8712.2.a.ch.1.6		6	11.10	odd	2
8712.2.a.ci.1.6		6	1.1	even	1	trivial
8712.2.a.cj.1.1		6	33.32	even	2
8712.2.a.ck.1.1		6	3.2	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:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q+2.07994 q^{5} +1.39055 q^{7} +O(q^{10})\)	\(q+2.07994 q^{5} +1.39055 q^{7} -5.79180 q^{13} +7.23340 q^{17} -8.44752 q^{19} -2.19197 q^{23} -0.673854 q^{25} +7.03317 q^{29} -2.55706 q^{31} +2.89226 q^{35} -8.57290 q^{37} -1.07188 q^{41} +7.59881 q^{43} -10.7784 q^{47} -5.06637 q^{49} +0.550965 q^{53} -10.4911 q^{59} +10.5902 q^{61} -12.0466 q^{65} +2.71154 q^{67} +2.01881 q^{71} +0.302349 q^{73} -4.91851 q^{79} -10.6654 q^{83} +15.0450 q^{85} -4.07002 q^{89} -8.05378 q^{91} -17.5703 q^{95} -1.17377 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{5} + q^{7}+O(q^{10}) \) 6 * q - 5 * q^5 + q^7	\( 6 q - 5 q^{5} + q^{7} + 4 q^{13} + 6 q^{17} - 14 q^{19} - 6 q^{23} + 7 q^{25} + 7 q^{29} - 7 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} - 12 q^{43} - 26 q^{47} + 9 q^{49} - 9 q^{53} - 17 q^{59} + 20 q^{61} - 48 q^{65} + 14 q^{67} - 18 q^{71} + 7 q^{73} + 9 q^{79} + 11 q^{83} + 10 q^{85} - 14 q^{89} + 8 q^{91} + 30 q^{95} + 25 q^{97}+O(q^{100}) \) 6 * q - 5 * q^5 + q^7 + 4 * q^13 + 6 * q^17 - 14 * q^19 - 6 * q^23 + 7 * q^25 + 7 * q^29 - 7 * q^31 - 2 * q^35 + 2 * q^37 - 6 * q^41 - 12 * q^43 - 26 * q^47 + 9 * q^49 - 9 * q^53 - 17 * q^59 + 20 * q^61 - 48 * q^65 + 14 * q^67 - 18 * q^71 + 7 * q^73 + 9 * q^79 + 11 * q^83 + 10 * q^85 - 14 * q^89 + 8 * q^91 + 30 * q^95 + 25 * q^97

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