LMFDB - Embedded newform 8712.2.a.ca.1.4

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:	$4$
Coefficient field:	4.4.46224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 15x^{2} + 16x + 16 $ x^4 - 2x^3 - 15x^2 + 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2904)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.39592$ 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)

$f(q)$	$=$	$q+3.39592 q^{5} -0.822116 q^{7} +O(q^{10})$	$q+3.39592 q^{5} -0.822116 q^{7} -4.90993 q^{13} -4.90993 q^{17} +7.61396 q^{19} -2.97197 q^{23} +6.53228 q^{25} -5.34601 q^{29} -4.39592 q^{31} -2.79184 q^{35} +5.97197 q^{37} +2.55417 q^{41} -8.43608 q^{43} -11.7200 q^{47} -6.32412 q^{49} +12.3241 q^{53} +6.97197 q^{59} -1.67001 q^{61} -16.6738 q^{65} -2.57605 q^{67} +9.10833 q^{71} -4.68576 q^{73} -8.68576 q^{79} -5.64423 q^{83} -16.6738 q^{85} -9.21579 q^{89} +4.03654 q^{91} +25.8564 q^{95} +18.5837 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 - 4 * q^7	$ 4 q - 2 q^{5} - 4 q^{7} - 12 q^{13} - 12 q^{17} + 4 q^{23} + 14 q^{25} + 16 q^{29} - 2 q^{31} + 20 q^{35} + 8 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 18 q^{49} + 6 q^{53} + 12 q^{59} - 8 q^{61} - 12 q^{65} - 10 q^{67} + 24 q^{71} - 16 q^{73} - 32 q^{79} - 24 q^{83} - 12 q^{85} - 6 q^{89} - 24 q^{91} + 48 q^{95} + 12 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 4 * q^7 - 12 * q^13 - 12 * q^17 + 4 * q^23 + 14 * q^25 + 16 * q^29 - 2 * q^31 + 20 * q^35 + 8 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 18 * q^49 + 6 * q^53 + 12 * q^59 - 8 * q^61 - 12 * q^65 - 10 * q^67 + 24 * q^71 - 16 * q^73 - 32 * q^79 - 24 * q^83 - 12 * q^85 - 6 * q^89 - 24 * q^91 + 48 * q^95 + 12 * 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
$5$	3.39592	1.51870	0.759351	−	0.650681i	$-0.225517\pi$
$5$	3.39592	1.51870	0.759351	+	0.650681i	$0.225517\pi$
$6$	0	0
$7$	−0.822116	−0.310731	−0.155365	−	0.987857i	$-0.549655\pi$
$7$	−0.822116	−0.310731	−0.155365	+	0.987857i	$0.549655\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.90993	−1.36177	−0.680885	−	0.732390i	$-0.738405\pi$
$13$	−4.90993	−1.36177	−0.680885	+	0.732390i	$0.738405\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.90993	−1.19083	−0.595417	−	0.803417i	$-0.703013\pi$
$17$	−4.90993	−1.19083	−0.595417	+	0.803417i	$0.703013\pi$
$18$	0	0
$19$	7.61396	1.74676	0.873381	−	0.487038i	$-0.161923\pi$
$19$	7.61396	1.74676	0.873381	+	0.487038i	$0.161923\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.97197	−0.619699	−0.309850	−	0.950786i	$-0.600279\pi$
$23$	−2.97197	−0.619699	−0.309850	+	0.950786i	$0.600279\pi$
$24$	0	0
$25$	6.53228	1.30646
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.34601	−0.992729	−0.496365	−	0.868114i	$-0.665332\pi$
$29$	−5.34601	−0.992729	−0.496365	+	0.868114i	$0.665332\pi$
$30$	0	0
$31$	−4.39592	−0.789531	−0.394765	−	0.918782i	$-0.629174\pi$
$31$	−4.39592	−0.789531	−0.394765	+	0.918782i	$0.629174\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.79184	−0.471908
$36$	0	0
$37$	5.97197	0.981786	0.490893	−	0.871220i	$-0.336671\pi$
$37$	5.97197	0.981786	0.490893	+	0.871220i	$0.336671\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.55417	0.398894	0.199447	−	0.979909i	$-0.436085\pi$
$41$	2.55417	0.398894	0.199447	+	0.979909i	$0.436085\pi$
$42$	0	0
$43$	−8.43608	−1.28649	−0.643245	−	0.765661i	$-0.722412\pi$
$43$	−8.43608	−1.28649	−0.643245	+	0.765661i	$0.722412\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.7200	−1.70954	−0.854772	−	0.519003i	$-0.826303\pi$
$47$	−11.7200	−1.70954	−0.854772	+	0.519003i	$0.826303\pi$
$48$	0	0
$49$	−6.32412	−0.903446
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.3241	1.69285	0.846424	−	0.532509i	$-0.178751\pi$
$53$	12.3241	1.69285	0.846424	+	0.532509i	$0.178751\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.97197	0.907674	0.453837	−	0.891085i	$-0.350055\pi$
$59$	6.97197	0.907674	0.453837	+	0.891085i	$0.350055\pi$
$60$	0	0
$61$	−1.67001	−0.213823	−0.106912	−	0.994269i	$-0.534096\pi$
$61$	−1.67001	−0.213823	−0.106912	+	0.994269i	$0.534096\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.6738	−2.06812
$66$	0	0
$67$	−2.57605	−0.314715	−0.157357	−	0.987542i	$-0.550297\pi$
$67$	−2.57605	−0.314715	−0.157357	+	0.987542i	$0.550297\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.10833	1.08096	0.540480	−	0.841357i	$-0.318243\pi$
$71$	9.10833	1.08096	0.540480	+	0.841357i	$0.318243\pi$
$72$	0	0
$73$	−4.68576	−0.548426	−0.274213	−	0.961669i	$-0.588417\pi$
$73$	−4.68576	−0.548426	−0.274213	+	0.961669i	$0.588417\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.68576	−0.977224	−0.488612	−	0.872501i	$-0.662497\pi$
$79$	−8.68576	−0.977224	−0.488612	+	0.872501i	$0.662497\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.64423	−0.619535	−0.309768	−	0.950812i	$-0.600251\pi$
$83$	−5.64423	−0.619535	−0.309768	+	0.950812i	$0.600251\pi$
$84$	0	0
$85$	−16.6738	−1.80852
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.21579	−0.976872	−0.488436	−	0.872600i	$-0.662432\pi$
$89$	−9.21579	−0.976872	−0.488436	+	0.872600i	$0.662432\pi$
$90$	0	0
$91$	4.03654	0.423144
$92$	0	0
$93$	0	0
$94$	0	0
$95$	25.8564	2.65281
$96$	0	0
$97$	18.5837	1.88689	0.943444	−	0.331533i	$-0.107566\pi$
$97$	18.5837	1.88689	0.943444	+	0.331533i	$0.107566\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.64423	0.561622	0.280811	−	0.959763i	$-0.409397\pi$
$101$	5.64423	0.561622	0.280811	+	0.959763i	$0.409397\pi$
$102$	0	0
$103$	−10.2158	−1.00659	−0.503296	−	0.864114i	$-0.667879\pi$
$103$	−10.2158	−1.00659	−0.503296	+	0.864114i	$0.667879\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.21941	0.214558	0.107279	−	0.994229i	$-0.465786\pi$
$107$	2.21941	0.214558	0.107279	+	0.994229i	$0.465786\pi$
$108$	0	0
$109$	−6.13159	−0.587300	−0.293650	−	0.955913i	$-0.594870\pi$
$109$	−6.13159	−0.587300	−0.293650	+	0.955913i	$0.594870\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.5043	−1.36445	−0.682223	−	0.731144i	$-0.738987\pi$
$113$	−14.5043	−1.36445	−0.682223	+	0.731144i	$0.738987\pi$
$114$	0	0
$115$	−10.0926	−0.941139
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.03654	0.370029
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.20351	0.465416
$126$	0	0
$127$	−8.68576	−0.770736	−0.385368	−	0.922763i	$-0.625926\pi$
$127$	−8.68576	−0.770736	−0.385368	+	0.922763i	$0.625926\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.43608	−0.737063	−0.368532	−	0.929615i	$-0.620139\pi$
$131$	−8.43608	−0.737063	−0.368532	+	0.929615i	$0.620139\pi$
$132$	0	0
$133$	−6.25956	−0.542773
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.7918	−1.60550	−0.802748	−	0.596319i	$-0.796629\pi$
$137$	−18.7918	−1.60550	−0.802748	+	0.596319i	$0.796629\pi$
$138$	0	0
$139$	−20.0758	−1.70281	−0.851404	−	0.524510i	$-0.824248\pi$
$139$	−20.0758	−1.70281	−0.851404	+	0.524510i	$0.824248\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−18.1546	−1.50766
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.9184	1.14024	0.570122	−	0.821560i	$-0.306896\pi$
$149$	13.9184	1.14024	0.570122	+	0.821560i	$0.306896\pi$
$150$	0	0
$151$	−16.5724	−1.34865	−0.674323	−	0.738437i	$-0.735564\pi$
$151$	−16.5724	−1.34865	−0.674323	+	0.738437i	$0.735564\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−14.9282	−1.19906
$156$	0	0
$157$	22.1160	1.76505	0.882523	−	0.470269i	$-0.155843\pi$
$157$	22.1160	1.76505	0.882523	+	0.470269i	$0.155843\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.44331	0.192560
$162$	0	0
$163$	−14.2158	−1.11347	−0.556733	−	0.830691i	$-0.687945\pi$
$163$	−14.2158	−1.11347	−0.556733	+	0.830691i	$0.687945\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.3643	−0.879394	−0.439697	−	0.898146i	$-0.644914\pi$
$167$	−11.3643	−0.879394	−0.439697	+	0.898146i	$0.644914\pi$
$168$	0	0
$169$	11.1075	0.854420
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.9282	1.13497	0.567485	−	0.823384i	$-0.307916\pi$
$173$	14.9282	1.13497	0.567485	+	0.823384i	$0.307916\pi$
$174$	0	0
$175$	−5.37030	−0.405956
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.0803	1.20190	0.600949	−	0.799287i	$-0.294789\pi$
$179$	16.0803	1.20190	0.600949	+	0.799287i	$0.294789\pi$
$180$	0	0
$181$	−11.8844	−0.883363	−0.441681	−	0.897172i	$-0.645618\pi$
$181$	−11.8844	−0.883363	−0.441681	+	0.897172i	$0.645618\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.2804	1.49104
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.87215	−0.641966	−0.320983	−	0.947085i	$-0.604013\pi$
$191$	−8.87215	−0.641966	−0.320983	+	0.947085i	$0.604013\pi$
$192$	0	0
$193$	−11.3763	−0.818883	−0.409441	−	0.912336i	$-0.634276\pi$
$193$	−11.3763	−0.818883	−0.409441	+	0.912336i	$0.634276\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.9342	1.20651	0.603256	−	0.797548i	$-0.293870\pi$
$197$	16.9342	1.20651	0.603256	+	0.797548i	$0.293870\pi$
$198$	0	0
$199$	7.64062	0.541629	0.270814	−	0.962632i	$-0.412707\pi$
$199$	7.64062	0.541629	0.270814	+	0.962632i	$0.412707\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.39504	0.308472
$204$	0	0
$205$	8.67375	0.605801
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−28.7616	−1.98003	−0.990014	−	0.140969i	$-0.954978\pi$
$211$	−28.7616	−1.98003	−0.990014	+	0.140969i	$0.954978\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−28.6482	−1.95379
$216$	0	0
$217$	3.61396	0.245332
$218$	0	0
$219$	0	0
$220$	0	0
$221$	24.1075	1.62164
$222$	0	0
$223$	−23.1317	−1.54901	−0.774507	−	0.632565i	$-0.782002\pi$
$223$	−23.1317	−1.54901	−0.774507	+	0.632565i	$0.782002\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.86364	−0.256439	−0.128219	−	0.991746i	$-0.540926\pi$
$227$	−3.86364	−0.256439	−0.128219	+	0.991746i	$0.540926\pi$
$228$	0	0
$229$	−6.23153	−0.411791	−0.205896	−	0.978574i	$-0.566011\pi$
$229$	−6.23153	−0.411791	−0.205896	+	0.978574i	$0.566011\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0138	1.04910	0.524549	−	0.851381i	$-0.324234\pi$
$233$	16.0138	1.04910	0.524549	+	0.851381i	$0.324234\pi$
$234$	0	0
$235$	−39.8004	−2.59629
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.28397	−0.0830531	−0.0415266	−	0.999137i	$-0.513222\pi$
$239$	−1.28397	−0.0830531	−0.0415266	+	0.999137i	$0.513222\pi$
$240$	0	0
$241$	−9.07180	−0.584366	−0.292183	−	0.956362i	$-0.594382\pi$
$241$	−9.07180	−0.584366	−0.292183	+	0.956362i	$0.594382\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−21.4762	−1.37207
$246$	0	0
$247$	−37.3840	−2.37869
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.0437707	−0.00276278	−0.00138139	−	0.999999i	$-0.500440\pi$
$251$	−0.0437707	−0.00276278	−0.00138139	+	0.999999i	$0.500440\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.2400	−1.44967	−0.724837	−	0.688920i	$-0.758085\pi$
$257$	−23.2400	−1.44967	−0.724837	+	0.688920i	$0.758085\pi$
$258$	0	0
$259$	−4.90966	−0.305071
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.772082	−0.0476086	−0.0238043	−	0.999717i	$-0.507578\pi$
$263$	−0.772082	−0.0476086	−0.0238043	+	0.999717i	$0.507578\pi$
$264$	0	0
$265$	41.8518	2.57093
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.75618	0.350961	0.175480	−	0.984483i	$-0.443852\pi$
$269$	5.75618	0.350961	0.175480	+	0.984483i	$0.443852\pi$
$270$	0	0
$271$	15.0916	0.916747	0.458373	−	0.888760i	$-0.348432\pi$
$271$	15.0916	0.916747	0.458373	+	0.888760i	$0.348432\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.6800	1.84338	0.921692	−	0.387923i	$-0.126807\pi$
$277$	30.6800	1.84338	0.921692	+	0.387923i	$0.126807\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.711534	0.0424466	0.0212233	−	0.999775i	$-0.493244\pi$
$281$	0.711534	0.0424466	0.0212233	+	0.999775i	$0.493244\pi$
$282$	0	0
$283$	22.9782	1.36591	0.682957	−	0.730458i	$-0.260694\pi$
$283$	22.9782	1.36591	0.682957	+	0.730458i	$0.260694\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.09982	−0.123949
$288$	0	0
$289$	7.10746	0.418086
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.6300	−0.854693	−0.427346	−	0.904088i	$-0.640551\pi$
$293$	−14.6300	−0.854693	−0.427346	+	0.904088i	$0.640551\pi$
$294$	0	0
$295$	23.6763	1.37849
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.5922	0.843889
$300$	0	0
$301$	6.93544	0.399752
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.67123	−0.324734
$306$	0	0
$307$	−29.0343	−1.65707	−0.828537	−	0.559934i	$-0.810826\pi$
$307$	−29.0343	−1.65707	−0.828537	+	0.559934i	$0.810826\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.2952	1.49107	0.745533	−	0.666469i	$-0.232195\pi$
$311$	26.2952	1.49107	0.745533	+	0.666469i	$0.232195\pi$
$312$	0	0
$313$	11.3156	0.639596	0.319798	−	0.947486i	$-0.396385\pi$
$313$	11.3156	0.639596	0.319798	+	0.947486i	$0.396385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.7918	1.05546	0.527728	−	0.849414i	$-0.323044\pi$
$317$	18.7918	1.05546	0.527728	+	0.849414i	$0.323044\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−37.3840	−2.08010
$324$	0	0
$325$	−32.0731	−1.77909
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.63524	0.531208
$330$	0	0
$331$	13.1878	0.724865	0.362433	−	0.932010i	$-0.381946\pi$
$331$	13.1878	0.724865	0.362433	+	0.932010i	$0.381946\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.74807	−0.477958
$336$	0	0
$337$	−0.387533	−0.0211103	−0.0105551	−	0.999944i	$-0.503360\pi$
$337$	−0.387533	−0.0211103	−0.0105551	+	0.999944i	$0.503360\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	10.9540	0.591459
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.37151	−0.0736266	−0.0368133	−	0.999322i	$-0.511721\pi$
$347$	−1.37151	−0.0736266	−0.0368133	+	0.999322i	$0.511721\pi$
$348$	0	0
$349$	25.2720	1.35278	0.676389	−	0.736545i	$-0.263544\pi$
$349$	25.2720	1.35278	0.676389	+	0.736545i	$0.263544\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.72816	0.251654	0.125827	−	0.992052i	$-0.459842\pi$
$353$	4.72816	0.251654	0.125827	+	0.992052i	$0.459842\pi$
$354$	0	0
$355$	30.9312	1.64166
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.5765	1.45543	0.727715	−	0.685880i	$-0.240582\pi$
$359$	27.5765	1.45543	0.727715	+	0.685880i	$0.240582\pi$
$360$	0	0
$361$	38.9724	2.05118
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.9125	−0.832896
$366$	0	0
$367$	−12.0365	−0.628302	−0.314151	−	0.949373i	$-0.601720\pi$
$367$	−12.0365	−0.628302	−0.314151	+	0.949373i	$0.601720\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.1319	−0.526020
$372$	0	0
$373$	0.598214	0.0309744	0.0154872	−	0.999880i	$-0.495070\pi$
$373$	0.598214	0.0309744	0.0154872	+	0.999880i	$0.495070\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.2486	1.35187
$378$	0	0
$379$	−0.396800	−0.0203823	−0.0101911	−	0.999948i	$-0.503244\pi$
$379$	−0.396800	−0.0203823	−0.0101911	+	0.999948i	$0.503244\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.09982	−0.107296	−0.0536480	−	0.998560i	$-0.517085\pi$
$383$	−2.09982	−0.107296	−0.0536480	+	0.998560i	$0.517085\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.12320	−0.462565	−0.231282	−	0.972887i	$-0.574292\pi$
$389$	−9.12320	−0.462565	−0.231282	+	0.972887i	$0.574292\pi$
$390$	0	0
$391$	14.5922	0.737959
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−29.4961	−1.48411
$396$	0	0
$397$	−20.1886	−1.01324	−0.506619	−	0.862170i	$-0.669105\pi$
$397$	−20.1886	−1.01324	−0.506619	+	0.862170i	$0.669105\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.65636	−0.382340	−0.191170	−	0.981557i	$-0.561228\pi$
$401$	−7.65636	−0.382340	−0.191170	+	0.981557i	$0.561228\pi$
$402$	0	0
$403$	21.5837	1.07516
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8.95997	−0.443042	−0.221521	−	0.975156i	$-0.571102\pi$
$409$	−8.95997	−0.443042	−0.221521	+	0.975156i	$0.571102\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.73177	−0.282042
$414$	0	0
$415$	−19.1674	−0.940889
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.4121	1.09490	0.547451	−	0.836838i	$-0.315598\pi$
$419$	22.4121	1.09490	0.547451	+	0.836838i	$0.315598\pi$
$420$	0	0
$421$	8.56031	0.417204	0.208602	−	0.978001i	$-0.433109\pi$
$421$	8.56031	0.417204	0.208602	+	0.978001i	$0.433109\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−32.0731	−1.55577
$426$	0	0
$427$	1.37294	0.0664414
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.0758	0.967018	0.483509	−	0.875339i	$-0.339362\pi$
$431$	20.0758	0.967018	0.483509	+	0.875339i	$0.339362\pi$
$432$	0	0
$433$	−11.4316	−0.549367	−0.274683	−	0.961535i	$-0.588573\pi$
$433$	−11.4316	−0.549367	−0.274683	+	0.961535i	$0.588573\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−22.6285	−1.08247
$438$	0	0
$439$	−17.5309	−0.836705	−0.418352	−	0.908285i	$-0.637392\pi$
$439$	−17.5309	−0.836705	−0.418352	+	0.908285i	$0.637392\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.3318	1.44111	0.720553	−	0.693400i	$-0.243888\pi$
$443$	30.3318	1.44111	0.720553	+	0.693400i	$0.243888\pi$
$444$	0	0
$445$	−31.2961	−1.48358
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−19.0722	−0.900073	−0.450036	−	0.893010i	$-0.648589\pi$
$449$	−19.0722	−0.900073	−0.450036	+	0.893010i	$0.648589\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.7078	0.642630
$456$	0	0
$457$	−14.5934	−0.682652	−0.341326	−	0.939945i	$-0.610876\pi$
$457$	−14.5934	−0.682652	−0.341326	+	0.939945i	$0.610876\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.834121	−0.0388489	−0.0194245	−	0.999811i	$-0.506183\pi$
$461$	−0.834121	−0.0388489	−0.0194245	+	0.999811i	$0.506183\pi$
$462$	0	0
$463$	−23.6763	−1.10033	−0.550165	−	0.835056i	$-0.685435\pi$
$463$	−23.6763	−1.10033	−0.550165	+	0.835056i	$0.685435\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.9715	−1.94221	−0.971105	−	0.238654i	$-0.923294\pi$
$467$	−41.9715	−1.94221	−0.971105	+	0.238654i	$0.923294\pi$
$468$	0	0
$469$	2.11782	0.0977916
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	49.7365	2.28207
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.47638	−0.341605	−0.170802	−	0.985305i	$-0.554636\pi$
$479$	−7.47638	−0.341605	−0.170802	+	0.985305i	$0.554636\pi$
$480$	0	0
$481$	−29.3220	−1.33697
$482$	0	0
$483$	0	0
$484$	0	0
$485$	63.1087	2.86562
$486$	0	0
$487$	−7.40355	−0.335487	−0.167744	−	0.985831i	$-0.553648\pi$
$487$	−7.40355	−0.335487	−0.167744	+	0.985831i	$0.553648\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.7398	1.43240	0.716199	−	0.697896i	$-0.245880\pi$
$491$	31.7398	1.43240	0.716199	+	0.697896i	$0.245880\pi$
$492$	0	0
$493$	26.2486	1.18218
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.48811	−0.335888
$498$	0	0
$499$	−8.63210	−0.386426	−0.193213	−	0.981157i	$-0.561891\pi$
$499$	−8.63210	−0.386426	−0.193213	+	0.981157i	$0.561891\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.0961	−0.762275	−0.381138	−	0.924518i	$-0.624468\pi$
$503$	−17.0961	−0.762275	−0.381138	+	0.924518i	$0.624468\pi$
$504$	0	0
$505$	19.1674	0.852937
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.8564	0.525526	0.262763	−	0.964860i	$-0.415366\pi$
$509$	11.8564	0.525526	0.262763	+	0.964860i	$0.415366\pi$
$510$	0	0
$511$	3.85224	0.170413
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−34.6920	−1.52871
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.7115	0.644524	0.322262	−	0.946651i	$-0.395557\pi$
$521$	14.7115	0.644524	0.322262	+	0.946651i	$0.395557\pi$
$522$	0	0
$523$	4.54666	0.198811	0.0994057	−	0.995047i	$-0.468306\pi$
$523$	4.54666	0.198811	0.0994057	+	0.995047i	$0.468306\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.5837	0.940200
$528$	0	0
$529$	−14.1674	−0.615973
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.5408	−0.543202
$534$	0	0
$535$	7.53693	0.325850
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−19.3670	−0.832653	−0.416327	−	0.909215i	$-0.636683\pi$
$541$	−19.3670	−0.832653	−0.416327	+	0.909215i	$0.636683\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−20.8224	−0.891933
$546$	0	0
$547$	2.44331	0.104468	0.0522342	−	0.998635i	$-0.483366\pi$
$547$	2.44331	0.104468	0.0522342	+	0.998635i	$0.483366\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−40.7043	−1.73406
$552$	0	0
$553$	7.14070	0.303654
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.87215	0.206440	0.103220	−	0.994659i	$-0.467085\pi$
$557$	4.87215	0.206440	0.103220	+	0.994659i	$0.467085\pi$
$558$	0	0
$559$	41.4206	1.75190
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.2082	−0.556657	−0.278329	−	0.960486i	$-0.589780\pi$
$563$	−13.2082	−0.556657	−0.278329	+	0.960486i	$0.589780\pi$
$564$	0	0
$565$	−49.2553	−2.07219
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0956	1.26167	0.630836	−	0.775916i	$-0.282712\pi$
$569$	30.0956	1.26167	0.630836	+	0.775916i	$0.282712\pi$
$570$	0	0
$571$	−39.0052	−1.63232	−0.816159	−	0.577827i	$-0.803901\pi$
$571$	−39.0052	−1.63232	−0.816159	+	0.577827i	$0.803901\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−19.4138	−0.809610
$576$	0	0
$577$	7.30421	0.304078	0.152039	−	0.988374i	$-0.451416\pi$
$577$	7.30421	0.304078	0.152039	+	0.988374i	$0.451416\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.64022	0.192509
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.3122	1.33367	0.666834	−	0.745206i	$-0.267649\pi$
$587$	32.3122	1.33367	0.666834	+	0.745206i	$0.267649\pi$
$588$	0	0
$589$	−33.4704	−1.37912
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.42632	0.140702	0.0703510	−	0.997522i	$-0.477588\pi$
$593$	3.42632	0.140702	0.0703510	+	0.997522i	$0.477588\pi$
$594$	0	0
$595$	13.7078	0.561964
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28.6725	−1.17153	−0.585763	−	0.810482i	$-0.699205\pi$
$599$	−28.6725	−1.17153	−0.585763	+	0.810482i	$0.699205\pi$
$600$	0	0
$601$	33.3112	1.35879	0.679397	−	0.733771i	$-0.262241\pi$
$601$	33.3112	1.35879	0.679397	+	0.733771i	$0.262241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.3795	−1.27366	−0.636828	−	0.771006i	$-0.719754\pi$
$607$	−31.3795	−1.27366	−0.636828	+	0.771006i	$0.719754\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	57.5447	2.32801
$612$	0	0
$613$	18.9162	0.764018	0.382009	−	0.924159i	$-0.375232\pi$
$613$	18.9162	0.764018	0.382009	+	0.924159i	$0.375232\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.855528	0.0344422	0.0172211	−	0.999852i	$-0.494518\pi$
$617$	0.855528	0.0344422	0.0172211	+	0.999852i	$0.494518\pi$
$618$	0	0
$619$	−25.9805	−1.04424	−0.522122	−	0.852871i	$-0.674859\pi$
$619$	−25.9805	−1.04424	−0.522122	+	0.852871i	$0.674859\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.57645	0.303544
$624$	0	0
$625$	−14.9907	−0.599628
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.3220	−1.16914
$630$	0	0
$631$	11.9635	0.476258	0.238129	−	0.971234i	$-0.423466\pi$
$631$	11.9635	0.476258	0.238129	+	0.971234i	$0.423466\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.4961	−1.17052
$636$	0	0
$637$	31.0510	1.23029
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.2000	−0.402878	−0.201439	−	0.979501i	$-0.564562\pi$
$641$	−10.2000	−0.402878	−0.201439	+	0.979501i	$0.564562\pi$
$642$	0	0
$643$	38.0527	1.50065	0.750326	−	0.661068i	$-0.229897\pi$
$643$	38.0527	1.50065	0.750326	+	0.661068i	$0.229897\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.4333	−1.27509	−0.637543	−	0.770415i	$-0.720049\pi$
$647$	−32.4333	−1.27509	−0.637543	+	0.770415i	$0.720049\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−44.0080	−1.72217	−0.861084	−	0.508463i	$-0.830214\pi$
$653$	−44.0080	−1.72217	−0.861084	+	0.508463i	$0.830214\pi$
$654$	0	0
$655$	−28.6482	−1.11938
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.4446	−1.14700	−0.573499	−	0.819206i	$-0.694414\pi$
$659$	−29.4446	−1.14700	−0.573499	+	0.819206i	$0.694414\pi$
$660$	0	0
$661$	5.03654	0.195899	0.0979493	−	0.995191i	$-0.468772\pi$
$661$	5.03654	0.195899	0.0979493	+	0.995191i	$0.468772\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.2570	−0.824310
$666$	0	0
$667$	15.8882	0.615194
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−38.2550	−1.47462	−0.737310	−	0.675554i	$-0.763904\pi$
$673$	−38.2550	−1.47462	−0.737310	+	0.675554i	$0.763904\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.7018	−0.449736	−0.224868	−	0.974389i	$-0.572195\pi$
$677$	−11.7018	−0.449736	−0.224868	+	0.974389i	$0.572195\pi$
$678$	0	0
$679$	−15.2780	−0.586314
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.17290	0.236199	0.118100	−	0.993002i	$-0.462320\pi$
$683$	6.17290	0.236199	0.118100	+	0.993002i	$0.462320\pi$
$684$	0	0
$685$	−63.8156	−2.43827
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−60.5106	−2.30527
$690$	0	0
$691$	45.6559	1.73683	0.868416	−	0.495837i	$-0.165139\pi$
$691$	45.6559	1.73683	0.868416	+	0.495837i	$0.165139\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−68.1759	−2.58606
$696$	0	0
$697$	−12.5408	−0.475016
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.6255	1.08117	0.540585	−	0.841290i	$-0.318203\pi$
$701$	28.6255	1.08117	0.540585	+	0.841290i	$0.318203\pi$
$702$	0	0
$703$	45.4704	1.71495
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.64022	−0.174513
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13.0646	0.489272
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12.1954	−0.454812	−0.227406	−	0.973800i	$-0.573024\pi$
$719$	−12.1954	−0.454812	−0.227406	+	0.973800i	$0.573024\pi$
$720$	0	0
$721$	8.39857	0.312779
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−34.9216	−1.29696
$726$	0	0
$727$	−43.9244	−1.62907	−0.814534	−	0.580116i	$-0.803007\pi$
$727$	−43.9244	−1.62907	−0.814534	+	0.580116i	$0.803007\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	41.4206	1.53200
$732$	0	0
$733$	−44.9860	−1.66160	−0.830798	−	0.556574i	$-0.812115\pi$
$733$	−44.9860	−1.66160	−0.830798	+	0.556574i	$0.812115\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	49.0585	1.80465	0.902324	−	0.431059i	$-0.141860\pi$
$739$	49.0585	1.80465	0.902324	+	0.431059i	$0.141860\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.9525	−0.842044	−0.421022	−	0.907050i	$-0.638328\pi$
$743$	−22.9525	−0.842044	−0.421022	+	0.907050i	$0.638328\pi$
$744$	0	0
$745$	47.2659	1.73169
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.82461	−0.0666698
$750$	0	0
$751$	25.3118	0.923642	0.461821	−	0.886973i	$-0.347196\pi$
$751$	25.3118	0.923642	0.461821	+	0.886973i	$0.347196\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−56.2787	−2.04819
$756$	0	0
$757$	17.5802	0.638964	0.319482	−	0.947592i	$-0.396491\pi$
$757$	17.5802	0.638964	0.319482	+	0.947592i	$0.396491\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.471121	0.0170781	0.00853906	−	0.999964i	$-0.497282\pi$
$761$	0.471121	0.0170781	0.00853906	+	0.999964i	$0.497282\pi$
$762$	0	0
$763$	5.04088	0.182492
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−34.2319	−1.23604
$768$	0	0
$769$	−26.7283	−0.963846	−0.481923	−	0.876213i	$-0.660062\pi$
$769$	−26.7283	−0.963846	−0.481923	+	0.876213i	$0.660062\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.5047	1.24105	0.620523	−	0.784188i	$-0.286920\pi$
$773$	34.5047	1.24105	0.620523	+	0.784188i	$0.286920\pi$
$774$	0	0
$775$	−28.7154	−1.03149
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.4473	0.696773
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	75.1041	2.68058
$786$	0	0
$787$	−7.70028	−0.274486	−0.137243	−	0.990537i	$-0.543824\pi$
$787$	−7.70028	−0.274486	−0.137243	+	0.990537i	$0.543824\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.9242	0.423975
$792$	0	0
$793$	8.19965	0.291178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.43206	0.263257	0.131629	−	0.991299i	$-0.457979\pi$
$797$	7.43206	0.263257	0.131629	+	0.991299i	$0.457979\pi$
$798$	0	0
$799$	57.5447	2.03578
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	8.29728	0.292441
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−46.1686	−1.62320	−0.811602	−	0.584211i	$-0.801404\pi$
$809$	−46.1686	−1.62320	−0.811602	+	0.584211i	$0.801404\pi$
$810$	0	0
$811$	1.90918	0.0670403	0.0335202	−	0.999438i	$-0.489328\pi$
$811$	1.90918	0.0670403	0.0335202	+	0.999438i	$0.489328\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−48.2757	−1.69102
$816$	0	0
$817$	−64.2319	−2.24719
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.2279	0.531458	0.265729	−	0.964048i	$-0.414387\pi$
$821$	15.2279	0.531458	0.265729	+	0.964048i	$0.414387\pi$
$822$	0	0
$823$	−32.4325	−1.13052	−0.565262	−	0.824911i	$-0.691225\pi$
$823$	−32.4325	−1.13052	−0.565262	+	0.824911i	$0.691225\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.0040	1.21721	0.608604	−	0.793474i	$-0.291730\pi$
$827$	35.0040	1.21721	0.608604	+	0.793474i	$0.291730\pi$
$828$	0	0
$829$	36.5642	1.26993	0.634963	−	0.772542i	$-0.281015\pi$
$829$	36.5642	1.26993	0.634963	+	0.772542i	$0.281015\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.0510	1.07585
$834$	0	0
$835$	−38.5922	−1.33554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.4558	1.32764	0.663821	−	0.747891i	$-0.268933\pi$
$839$	38.4558	1.32764	0.663821	+	0.747891i	$0.268933\pi$
$840$	0	0
$841$	−0.420178	−0.0144889
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.7200	1.29761
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−17.7486	−0.608412
$852$	0	0
$853$	−32.2002	−1.10251	−0.551256	−	0.834336i	$-0.685851\pi$
$853$	−32.2002	−1.10251	−0.551256	+	0.834336i	$0.685851\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.2404	1.34043	0.670214	−	0.742168i	$-0.266202\pi$
$857$	39.2404	1.34043	0.670214	+	0.742168i	$0.266202\pi$
$858$	0	0
$859$	4.43246	0.151234	0.0756168	−	0.997137i	$-0.475907\pi$
$859$	4.43246	0.151234	0.0756168	+	0.997137i	$0.475907\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.52417	−0.0859237	−0.0429619	−	0.999077i	$-0.513679\pi$
$863$	−2.52417	−0.0859237	−0.0429619	+	0.999077i	$0.513679\pi$
$864$	0	0
$865$	50.6950	1.72368
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	12.6482	0.428570
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.27789	−0.144619
$876$	0	0
$877$	26.7676	0.903876	0.451938	−	0.892049i	$-0.350733\pi$
$877$	26.7676	0.903876	0.451938	+	0.892049i	$0.350733\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.0148659	−0.000500844 0	−0.000250422	−	1.00000i	$-0.500080\pi$
$881$	−0.0148659	−0.000500844 0	−0.000250422		1.00000i	$0.500080\pi$
$882$	0	0
$883$	23.7629	0.799687	0.399843	−	0.916584i	$-0.369065\pi$
$883$	23.7629	0.799687	0.399843	+	0.916584i	$0.369065\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.7963	−0.429659	−0.214829	−	0.976652i	$-0.568920\pi$
$887$	−12.7963	−0.429659	−0.214829	+	0.976652i	$0.568920\pi$
$888$	0	0
$889$	7.14070	0.239491
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−89.2360	−2.98617
$894$	0	0
$895$	54.6075	1.82533
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.5006	0.783790
$900$	0	0
$901$	−60.5106	−2.01590
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−40.3586	−1.34157
$906$	0	0
$907$	11.0076	0.365502	0.182751	−	0.983159i	$-0.441500\pi$
$907$	11.0076	0.365502	0.182751	+	0.983159i	$0.441500\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.3475	0.839800	0.419900	−	0.907570i	$-0.362065\pi$
$911$	25.3475	0.839800	0.419900	+	0.907570i	$0.362065\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.93544	0.229028
$918$	0	0
$919$	25.4215	0.838579	0.419290	−	0.907852i	$-0.362279\pi$
$919$	25.4215	0.838579	0.419290	+	0.907852i	$0.362279\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−44.7213	−1.47202
$924$	0	0
$925$	39.0106	1.28266
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.2243	0.794774	0.397387	−	0.917651i	$-0.369917\pi$
$929$	24.2243	0.794774	0.397387	+	0.917651i	$0.369917\pi$
$930$	0	0
$931$	−48.1516	−1.57811
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.2792	0.368475	0.184238	−	0.982882i	$-0.441018\pi$
$937$	11.2792	0.368475	0.184238	+	0.982882i	$0.441018\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.7130	1.06641	0.533207	−	0.845985i	$-0.320987\pi$
$941$	32.7130	1.06641	0.533207	+	0.845985i	$0.320987\pi$
$942$	0	0
$943$	−7.59092	−0.247194
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.6445	−0.898325	−0.449162	−	0.893450i	$-0.648278\pi$
$947$	−27.6445	−0.898325	−0.449162	+	0.893450i	$0.648278\pi$
$948$	0	0
$949$	23.0068	0.746831
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.94198	−0.0953000	−0.0476500	−	0.998864i	$-0.515173\pi$
$953$	−2.94198	−0.0953000	−0.0476500	+	0.998864i	$0.515173\pi$
$954$	0	0
$955$	−30.1291	−0.974956
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.4491	0.498877
$960$	0	0
$961$	−11.6759	−0.376641
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−38.6330	−1.24364
$966$	0	0
$967$	−3.58970	−0.115437	−0.0577185	−	0.998333i	$-0.518383\pi$
$967$	−3.58970	−0.115437	−0.0577185	+	0.998333i	$0.518383\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.9002	−1.08791	−0.543954	−	0.839115i	$-0.683073\pi$
$971$	−33.9002	−1.08791	−0.543954	+	0.839115i	$0.683073\pi$
$972$	0	0
$973$	16.5047	0.529115
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.4885	−0.623493	−0.311746	−	0.950165i	$-0.600914\pi$
$977$	−19.4885	−0.623493	−0.311746	+	0.950165i	$0.600914\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	19.2477	0.613906	0.306953	−	0.951725i	$-0.400691\pi$
$983$	19.2477	0.613906	0.306953	+	0.951725i	$0.400691\pi$
$984$	0	0
$985$	57.5072	1.83233
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25.0718	0.797237
$990$	0	0
$991$	−13.6712	−0.434281	−0.217140	−	0.976140i	$-0.569673\pi$
$991$	−13.6712	−0.434281	−0.217140	+	0.976140i	$0.569673\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.9469	0.822573
$996$	0	0
$997$	−16.2973	−0.516139	−0.258070	−	0.966126i	$-0.583086\pi$
$997$	−16.2973	−0.516139	−0.258070	+	0.966126i	$0.583086\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.ca.1.4		4
3.2	odd	2		2904.2.a.bc.1.1	✓	4
11.10	odd	2		8712.2.a.cb.1.4		4
12.11	even	2		5808.2.a.cn.1.1		4
33.32	even	2		2904.2.a.bd.1.1	yes	4
132.131	odd	2		5808.2.a.cm.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2904.2.a.bc.1.1	✓	4	3.2	odd	2
2904.2.a.bd.1.1	yes	4	33.32	even	2
5808.2.a.cm.1.1		4	132.131	odd	2
5808.2.a.cn.1.1		4	12.11	even	2
8712.2.a.ca.1.4		4	1.1	even	1	trivial
8712.2.a.cb.1.4		4	11.10	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)

\(f(q)\)	\(=\)	\(q+3.39592 q^{5} -0.822116 q^{7} +O(q^{10})\)	\(q+3.39592 q^{5} -0.822116 q^{7} -4.90993 q^{13} -4.90993 q^{17} +7.61396 q^{19} -2.97197 q^{23} +6.53228 q^{25} -5.34601 q^{29} -4.39592 q^{31} -2.79184 q^{35} +5.97197 q^{37} +2.55417 q^{41} -8.43608 q^{43} -11.7200 q^{47} -6.32412 q^{49} +12.3241 q^{53} +6.97197 q^{59} -1.67001 q^{61} -16.6738 q^{65} -2.57605 q^{67} +9.10833 q^{71} -4.68576 q^{73} -8.68576 q^{79} -5.64423 q^{83} -16.6738 q^{85} -9.21579 q^{89} +4.03654 q^{91} +25.8564 q^{95} +18.5837 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 - 4 * q^7	\( 4 q - 2 q^{5} - 4 q^{7} - 12 q^{13} - 12 q^{17} + 4 q^{23} + 14 q^{25} + 16 q^{29} - 2 q^{31} + 20 q^{35} + 8 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 18 q^{49} + 6 q^{53} + 12 q^{59} - 8 q^{61} - 12 q^{65} - 10 q^{67} + 24 q^{71} - 16 q^{73} - 32 q^{79} - 24 q^{83} - 12 q^{85} - 6 q^{89} - 24 q^{91} + 48 q^{95} + 12 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 4 * q^7 - 12 * q^13 - 12 * q^17 + 4 * q^23 + 14 * q^25 + 16 * q^29 - 2 * q^31 + 20 * q^35 + 8 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 18 * q^49 + 6 * q^53 + 12 * q^59 - 8 * q^61 - 12 * q^65 - 10 * q^67 + 24 * q^71 - 16 * q^73 - 32 * q^79 - 24 * q^83 - 12 * q^85 - 6 * q^89 - 24 * q^91 + 48 * q^95 + 12 * q^97

Introduction

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