LMFDB - Embedded newform 8712.2.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8712,2,Mod(1,8712)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,0,-3,0,-4,0,0,0,0,0,0,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-2.61803 q^{5} +0.236068 q^{7} -2.23607 q^{13} -4.85410 q^{17} +2.61803 q^{19} -1.76393 q^{23} +1.85410 q^{25} +8.47214 q^{29} +6.38197 q^{31} -0.618034 q^{35} +8.23607 q^{37} +5.00000 q^{41} +2.52786 q^{43} -6.85410 q^{47} -6.94427 q^{49} -10.5623 q^{53} +5.56231 q^{59} -0.145898 q^{61} +5.85410 q^{65} +15.5623 q^{67} +7.56231 q^{71} -9.70820 q^{73} +9.76393 q^{79} -15.9443 q^{83} +12.7082 q^{85} +5.47214 q^{89} -0.527864 q^{91} -6.85410 q^{95} -0.145898 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} - 4 q^{7} - 3 q^{17} + 3 q^{19} - 8 q^{23} - 3 q^{25} + 8 q^{29} + 15 q^{31} + q^{35} + 12 q^{37} + 10 q^{41} + 14 q^{43} - 7 q^{47} + 4 q^{49} - q^{53} - 9 q^{59} - 7 q^{61} + 5 q^{65}+ \cdots - 7 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 - 4 * q^7 - 3 * q^17 + 3 * q^19 - 8 * q^23 - 3 * q^25 + 8 * q^29 + 15 * q^31 + q^35 + 12 * q^37 + 10 * q^41 + 14 * q^43 - 7 * q^47 + 4 * q^49 - q^53 - 9 * q^59 - 7 * q^61 + 5 * q^65 + 11 * q^67 - 5 * q^71 - 6 * q^73 + 24 * q^79 - 14 * q^83 + 12 * q^85 + 2 * q^89 - 10 * q^91 - 7 * q^95 - 7 * q^97

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
$5$	−2.61803	−1.17082	−0.585410	−	0.810737i	$-0.699067\pi$
$5$	−2.61803	−1.17082	−0.585410	+	0.810737i	$0.699067\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.23607	−0.620174	−0.310087	−	0.950708i	$-0.600358\pi$
$13$	−2.23607	−0.620174	−0.310087	+	0.950708i	$0.600358\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.85410	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$17$	−4.85410	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$18$	0	0
$19$	2.61803	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$19$	2.61803	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.76393	−0.367805	−0.183903	−	0.982944i	$-0.558873\pi$
$23$	−1.76393	−0.367805	−0.183903	+	0.982944i	$0.558873\pi$
$24$	0	0
$25$	1.85410	0.370820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	6.38197	1.14623	0.573117	−	0.819473i	$-0.305734\pi$
$31$	6.38197	1.14623	0.573117	+	0.819473i	$0.305734\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.618034	−0.104467
$36$	0	0
$37$	8.23607	1.35400	0.677001	−	0.735982i	$-0.263279\pi$
$37$	8.23607	1.35400	0.677001	+	0.735982i	$0.263279\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	2.52786	0.385496	0.192748	−	0.981248i	$-0.438260\pi$
$43$	2.52786	0.385496	0.192748	+	0.981248i	$0.438260\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.85410	−0.999774	−0.499887	−	0.866091i	$-0.666625\pi$
$47$	−6.85410	−0.999774	−0.499887	+	0.866091i	$0.666625\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.5623	−1.45084	−0.725422	−	0.688304i	$-0.758355\pi$
$53$	−10.5623	−1.45084	−0.725422	+	0.688304i	$0.758355\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.56231	0.724151	0.362075	−	0.932149i	$-0.382068\pi$
$59$	5.56231	0.724151	0.362075	+	0.932149i	$0.382068\pi$
$60$	0	0
$61$	−0.145898	−0.0186803	−0.00934016	−	0.999956i	$-0.502973\pi$
$61$	−0.145898	−0.0186803	−0.00934016	+	0.999956i	$0.502973\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.85410	0.726112
$66$	0	0
$67$	15.5623	1.90124	0.950619	−	0.310360i	$-0.100450\pi$
$67$	15.5623	1.90124	0.950619	+	0.310360i	$0.100450\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.56231	0.897481	0.448740	−	0.893662i	$-0.351873\pi$
$71$	7.56231	0.897481	0.448740	+	0.893662i	$0.351873\pi$
$72$	0	0
$73$	−9.70820	−1.13626	−0.568130	−	0.822939i	$-0.692333\pi$
$73$	−9.70820	−1.13626	−0.568130	+	0.822939i	$0.692333\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.76393	1.09853	0.549264	−	0.835649i	$-0.314908\pi$
$79$	9.76393	1.09853	0.549264	+	0.835649i	$0.314908\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.9443	−1.75011	−0.875056	−	0.484022i	$-0.839175\pi$
$83$	−15.9443	−1.75011	−0.875056	+	0.484022i	$0.839175\pi$
$84$	0	0
$85$	12.7082	1.37840
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.47214	0.580045	0.290023	−	0.957020i	$-0.406337\pi$
$89$	5.47214	0.580045	0.290023	+	0.957020i	$0.406337\pi$
$90$	0	0
$91$	−0.527864	−0.0553352
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.85410	−0.703216
$96$	0	0
$97$	−0.145898	−0.0148137	−0.00740685	−	0.999973i	$-0.502358\pi$
$97$	−0.145898	−0.0148137	−0.00740685	+	0.999973i	$0.502358\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.47214	−0.942513	−0.471256	−	0.881996i	$-0.656199\pi$
$101$	−9.47214	−0.942513	−0.471256	+	0.881996i	$0.656199\pi$
$102$	0	0
$103$	−12.4721	−1.22892	−0.614458	−	0.788950i	$-0.710625\pi$
$103$	−12.4721	−1.22892	−0.614458	+	0.788950i	$0.710625\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.94427	0.574654	0.287327	−	0.957832i	$-0.407233\pi$
$107$	5.94427	0.574654	0.287327	+	0.957832i	$0.407233\pi$
$108$	0	0
$109$	−17.8885	−1.71341	−0.856706	−	0.515805i	$-0.827493\pi$
$109$	−17.8885	−1.71341	−0.856706	+	0.515805i	$0.827493\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.7639	−1.29480	−0.647401	−	0.762150i	$-0.724144\pi$
$113$	−13.7639	−1.29480	−0.647401	+	0.762150i	$0.724144\pi$
$114$	0	0
$115$	4.61803	0.430634
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.14590	−0.105044
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.23607	0.736656
$126$	0	0
$127$	5.23607	0.464626	0.232313	−	0.972641i	$-0.425371\pi$
$127$	5.23607	0.464626	0.232313	+	0.972641i	$0.425371\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.3262	1.16432	0.582159	−	0.813075i	$-0.302208\pi$
$131$	13.3262	1.16432	0.582159	+	0.813075i	$0.302208\pi$
$132$	0	0
$133$	0.618034	0.0535903
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.47214	0.296645	0.148322	−	0.988939i	$-0.452613\pi$
$137$	3.47214	0.296645	0.148322	+	0.988939i	$0.452613\pi$
$138$	0	0
$139$	−16.3820	−1.38950	−0.694750	−	0.719251i	$-0.744485\pi$
$139$	−16.3820	−1.38950	−0.694750	+	0.719251i	$0.744485\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−22.1803	−1.84198
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.47214	0.120602	0.0603010	−	0.998180i	$-0.480794\pi$
$149$	1.47214	0.120602	0.0603010	+	0.998180i	$0.480794\pi$
$150$	0	0
$151$	13.4164	1.09181	0.545906	−	0.837846i	$-0.316186\pi$
$151$	13.4164	1.09181	0.545906	+	0.837846i	$0.316186\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.7082	−1.34204
$156$	0	0
$157$	−6.18034	−0.493245	−0.246622	−	0.969112i	$-0.579321\pi$
$157$	−6.18034	−0.493245	−0.246622	+	0.969112i	$0.579321\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.416408	−0.0328175
$162$	0	0
$163$	−17.6180	−1.37995	−0.689975	−	0.723833i	$-0.742379\pi$
$163$	−17.6180	−1.37995	−0.689975	+	0.723833i	$0.742379\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.67376	−0.593814	−0.296907	−	0.954906i	$-0.595955\pi$
$167$	−7.67376	−0.593814	−0.296907	+	0.954906i	$0.595955\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.0344	−0.914962	−0.457481	−	0.889219i	$-0.651248\pi$
$173$	−12.0344	−0.914962	−0.457481	+	0.889219i	$0.651248\pi$
$174$	0	0
$175$	0.437694	0.0330866
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.41641	−0.330098	−0.165049	−	0.986285i	$-0.552778\pi$
$179$	−4.41641	−0.330098	−0.165049	+	0.986285i	$0.552778\pi$
$180$	0	0
$181$	−20.8885	−1.55263	−0.776317	−	0.630343i	$-0.782914\pi$
$181$	−20.8885	−1.55263	−0.776317	+	0.630343i	$0.782914\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−21.5623	−1.58529
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.2361	−0.740656	−0.370328	−	0.928901i	$-0.620755\pi$
$191$	−10.2361	−0.740656	−0.370328	+	0.928901i	$0.620755\pi$
$192$	0	0
$193$	15.3262	1.10321	0.551603	−	0.834107i	$-0.314016\pi$
$193$	15.3262	1.10321	0.551603	+	0.834107i	$0.314016\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.8541	1.05831	0.529155	−	0.848525i	$-0.322509\pi$
$197$	14.8541	1.05831	0.529155	+	0.848525i	$0.322509\pi$
$198$	0	0
$199$	0.527864	0.0374193	0.0187096	−	0.999825i	$-0.494044\pi$
$199$	0.527864	0.0374193	0.0187096	+	0.999825i	$0.494044\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−13.0902	−0.914257
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.61803	0.386761	0.193381	−	0.981124i	$-0.438055\pi$
$211$	5.61803	0.386761	0.193381	+	0.981124i	$0.438055\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.61803	−0.451346
$216$	0	0
$217$	1.50658	0.102273
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.8541	0.730126
$222$	0	0
$223$	−21.4721	−1.43788	−0.718940	−	0.695072i	$-0.755373\pi$
$223$	−21.4721	−1.43788	−0.718940	+	0.695072i	$0.755373\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28.5967	−1.89803	−0.949016	−	0.315227i	$-0.897919\pi$
$227$	−28.5967	−1.89803	−0.949016	+	0.315227i	$0.897919\pi$
$228$	0	0
$229$	23.8885	1.57860	0.789300	−	0.614008i	$-0.210444\pi$
$229$	23.8885	1.57860	0.789300	+	0.614008i	$0.210444\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.90983	0.583702	0.291851	−	0.956464i	$-0.405729\pi$
$233$	8.90983	0.583702	0.291851	+	0.956464i	$0.405729\pi$
$234$	0	0
$235$	17.9443	1.17056
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.0344	0.972497	0.486249	−	0.873821i	$-0.338365\pi$
$239$	15.0344	0.972497	0.486249	+	0.873821i	$0.338365\pi$
$240$	0	0
$241$	28.0689	1.80808	0.904038	−	0.427452i	$-0.140589\pi$
$241$	28.0689	1.80808	0.904038	+	0.427452i	$0.140589\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	18.1803	1.16150
$246$	0	0
$247$	−5.85410	−0.372488
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.85410	−0.558866	−0.279433	−	0.960165i	$-0.590146\pi$
$251$	−8.85410	−0.558866	−0.279433	+	0.960165i	$0.590146\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.79837	0.424071	0.212035	−	0.977262i	$-0.431991\pi$
$257$	6.79837	0.424071	0.212035	+	0.977262i	$0.431991\pi$
$258$	0	0
$259$	1.94427	0.120811
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.14590	−0.255647	−0.127824	−	0.991797i	$-0.540799\pi$
$263$	−4.14590	−0.255647	−0.127824	+	0.991797i	$0.540799\pi$
$264$	0	0
$265$	27.6525	1.69868
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.4164	−1.54967	−0.774833	−	0.632166i	$-0.782166\pi$
$269$	−25.4164	−1.54967	−0.774833	+	0.632166i	$0.782166\pi$
$270$	0	0
$271$	15.3820	0.934388	0.467194	−	0.884155i	$-0.345265\pi$
$271$	15.3820	0.934388	0.467194	+	0.884155i	$0.345265\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.85410	−0.531991	−0.265996	−	0.963974i	$-0.585701\pi$
$277$	−8.85410	−0.531991	−0.265996	+	0.963974i	$0.585701\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.5967	1.88490	0.942452	−	0.334342i	$-0.108514\pi$
$281$	31.5967	1.88490	0.942452	+	0.334342i	$0.108514\pi$
$282$	0	0
$283$	−2.29180	−0.136233	−0.0681166	−	0.997677i	$-0.521699\pi$
$283$	−2.29180	−0.136233	−0.0681166	+	0.997677i	$0.521699\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.18034	0.0696733
$288$	0	0
$289$	6.56231	0.386018
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.3607	1.01422	0.507111	−	0.861881i	$-0.330713\pi$
$293$	17.3607	1.01422	0.507111	+	0.861881i	$0.330713\pi$
$294$	0	0
$295$	−14.5623	−0.847850
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.94427	0.228103
$300$	0	0
$301$	0.596748	0.0343960
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.381966	0.0218713
$306$	0	0
$307$	−13.7984	−0.787515	−0.393757	−	0.919214i	$-0.628825\pi$
$307$	−13.7984	−0.787515	−0.393757	+	0.919214i	$0.628825\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.76393	−0.553662	−0.276831	−	0.960919i	$-0.589284\pi$
$311$	−9.76393	−0.553662	−0.276831	+	0.960919i	$0.589284\pi$
$312$	0	0
$313$	−26.8885	−1.51983	−0.759915	−	0.650022i	$-0.774760\pi$
$313$	−26.8885	−1.51983	−0.759915	+	0.650022i	$0.774760\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.9443	−0.895520	−0.447760	−	0.894154i	$-0.647778\pi$
$317$	−15.9443	−0.895520	−0.447760	+	0.894154i	$0.647778\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.7082	−0.707103
$324$	0	0
$325$	−4.14590	−0.229973
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.61803	−0.0892051
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−40.7426	−2.22601
$336$	0	0
$337$	−23.7082	−1.29147	−0.645734	−	0.763562i	$-0.723449\pi$
$337$	−23.7082	−1.29147	−0.645734	+	0.763562i	$0.723449\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	7.76393	0.415594	0.207797	−	0.978172i	$-0.433371\pi$
$349$	7.76393	0.415594	0.207797	+	0.978172i	$0.433371\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.52786	−0.294219	−0.147109	−	0.989120i	$-0.546997\pi$
$353$	−5.52786	−0.294219	−0.147109	+	0.989120i	$0.546997\pi$
$354$	0	0
$355$	−19.7984	−1.05079
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.70820	0.301267	0.150634	−	0.988590i	$-0.451869\pi$
$359$	5.70820	0.301267	0.150634	+	0.988590i	$0.451869\pi$
$360$	0	0
$361$	−12.1459	−0.639258
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.4164	1.33036
$366$	0	0
$367$	−5.67376	−0.296168	−0.148084	−	0.988975i	$-0.547311\pi$
$367$	−5.67376	−0.296168	−0.148084	+	0.988975i	$0.547311\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.49342	−0.129452
$372$	0	0
$373$	15.9443	0.825563	0.412782	−	0.910830i	$-0.364557\pi$
$373$	15.9443	0.825563	0.412782	+	0.910830i	$0.364557\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.9443	−0.975680
$378$	0	0
$379$	−17.1803	−0.882495	−0.441247	−	0.897386i	$-0.645464\pi$
$379$	−17.1803	−0.882495	−0.441247	+	0.897386i	$0.645464\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.7082	1.67131	0.835656	−	0.549254i	$-0.185088\pi$
$383$	32.7082	1.67131	0.835656	+	0.549254i	$0.185088\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.2016	−0.517243	−0.258621	−	0.965979i	$-0.583268\pi$
$389$	−10.2016	−0.517243	−0.258621	+	0.965979i	$0.583268\pi$
$390$	0	0
$391$	8.56231	0.433014
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−25.5623	−1.28618
$396$	0	0
$397$	−34.7082	−1.74195	−0.870977	−	0.491323i	$-0.836513\pi$
$397$	−34.7082	−1.74195	−0.870977	+	0.491323i	$0.836513\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.4508	−1.12114	−0.560571	−	0.828106i	$-0.689418\pi$
$401$	−22.4508	−1.12114	−0.560571	+	0.828106i	$0.689418\pi$
$402$	0	0
$403$	−14.2705	−0.710865
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.88854	−0.0933825	−0.0466912	−	0.998909i	$-0.514868\pi$
$409$	−1.88854	−0.0933825	−0.0466912	+	0.998909i	$0.514868\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.31308	0.0646126
$414$	0	0
$415$	41.7426	2.04907
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−35.4508	−1.73189	−0.865944	−	0.500142i	$-0.833281\pi$
$419$	−35.4508	−1.73189	−0.865944	+	0.500142i	$0.833281\pi$
$420$	0	0
$421$	−19.7426	−0.962198	−0.481099	−	0.876666i	$-0.659762\pi$
$421$	−19.7426	−0.962198	−0.481099	+	0.876666i	$0.659762\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9.00000	−0.436564
$426$	0	0
$427$	−0.0344419	−0.00166676
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.9787	−0.576994	−0.288497	−	0.957481i	$-0.593156\pi$
$431$	−11.9787	−0.576994	−0.288497	+	0.957481i	$0.593156\pi$
$432$	0	0
$433$	−13.0557	−0.627418	−0.313709	−	0.949519i	$-0.601572\pi$
$433$	−13.0557	−0.627418	−0.313709	+	0.949519i	$0.601572\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.61803	−0.220911
$438$	0	0
$439$	20.5279	0.979741	0.489871	−	0.871795i	$-0.337044\pi$
$439$	20.5279	0.979741	0.489871	+	0.871795i	$0.337044\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.81966	−0.371523	−0.185762	−	0.982595i	$-0.559475\pi$
$443$	−7.81966	−0.371523	−0.185762	+	0.982595i	$0.559475\pi$
$444$	0	0
$445$	−14.3262	−0.679129
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.3050	1.28860	0.644300	−	0.764773i	$-0.277149\pi$
$449$	27.3050	1.28860	0.644300	+	0.764773i	$0.277149\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.38197	0.0647876
$456$	0	0
$457$	−40.7426	−1.90586	−0.952930	−	0.303189i	$-0.901948\pi$
$457$	−40.7426	−1.90586	−0.952930	+	0.303189i	$0.901948\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.9098	0.508121	0.254061	−	0.967188i	$-0.418234\pi$
$461$	10.9098	0.508121	0.254061	+	0.967188i	$0.418234\pi$
$462$	0	0
$463$	24.4508	1.13633	0.568164	−	0.822916i	$-0.307654\pi$
$463$	24.4508	1.13633	0.568164	+	0.822916i	$0.307654\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.4164	0.944759	0.472379	−	0.881395i	$-0.343395\pi$
$467$	20.4164	0.944759	0.472379	+	0.881395i	$0.343395\pi$
$468$	0	0
$469$	3.67376	0.169639
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.85410	0.222721
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.20163	−0.374742	−0.187371	−	0.982289i	$-0.559997\pi$
$479$	−8.20163	−0.374742	−0.187371	+	0.982289i	$0.559997\pi$
$480$	0	0
$481$	−18.4164	−0.839716
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.381966	0.0173442
$486$	0	0
$487$	−8.05573	−0.365040	−0.182520	−	0.983202i	$-0.558425\pi$
$487$	−8.05573	−0.365040	−0.182520	+	0.983202i	$0.558425\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.0902	−1.71899	−0.859493	−	0.511148i	$-0.829220\pi$
$491$	−38.0902	−1.71899	−0.859493	+	0.511148i	$0.829220\pi$
$492$	0	0
$493$	−41.1246	−1.85216
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.78522	0.0800780
$498$	0	0
$499$	13.0902	0.585996	0.292998	−	0.956113i	$-0.405347\pi$
$499$	13.0902	0.585996	0.292998	+	0.956113i	$0.405347\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.23607	0.0551135	0.0275568	−	0.999620i	$-0.491227\pi$
$503$	1.23607	0.0551135	0.0275568	+	0.999620i	$0.491227\pi$
$504$	0	0
$505$	24.7984	1.10351
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.14590	0.405385	0.202692	−	0.979242i	$-0.435031\pi$
$509$	9.14590	0.405385	0.202692	+	0.979242i	$0.435031\pi$
$510$	0	0
$511$	−2.29180	−0.101383
$512$	0	0
$513$	0	0
$514$	0	0
$515$	32.6525	1.43884
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	0	0
$523$	−26.5623	−1.16149	−0.580744	−	0.814086i	$-0.697238\pi$
$523$	−26.5623	−1.16149	−0.580744	+	0.814086i	$0.697238\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−30.9787	−1.34945
$528$	0	0
$529$	−19.8885	−0.864719
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.1803	−0.484274
$534$	0	0
$535$	−15.5623	−0.672817
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0344	0.947335	0.473667	−	0.880704i	$-0.342930\pi$
$541$	22.0344	0.947335	0.473667	+	0.880704i	$0.342930\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	46.8328	2.00610
$546$	0	0
$547$	−22.8541	−0.977171	−0.488585	−	0.872516i	$-0.662487\pi$
$547$	−22.8541	−0.977171	−0.488585	+	0.872516i	$0.662487\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22.1803	0.944914
$552$	0	0
$553$	2.30495	0.0980165
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.67376	−0.0709196	−0.0354598	−	0.999371i	$-0.511290\pi$
$557$	−1.67376	−0.0709196	−0.0354598	+	0.999371i	$0.511290\pi$
$558$	0	0
$559$	−5.65248	−0.239074
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.7639	−1.08582	−0.542910	−	0.839791i	$-0.682678\pi$
$563$	−25.7639	−1.08582	−0.542910	+	0.839791i	$0.682678\pi$
$564$	0	0
$565$	36.0344	1.51598
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.2918	0.850676	0.425338	−	0.905034i	$-0.360155\pi$
$569$	20.2918	0.850676	0.425338	+	0.905034i	$0.360155\pi$
$570$	0	0
$571$	−29.8541	−1.24936	−0.624678	−	0.780883i	$-0.714770\pi$
$571$	−29.8541	−1.24936	−0.624678	+	0.780883i	$0.714770\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.27051	−0.136390
$576$	0	0
$577$	−5.81966	−0.242276	−0.121138	−	0.992636i	$-0.538654\pi$
$577$	−5.81966	−0.242276	−0.121138	+	0.992636i	$0.538654\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.76393	−0.156154
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.708204	−0.0292307	−0.0146154	−	0.999893i	$-0.504652\pi$
$587$	−0.708204	−0.0292307	−0.0146154	+	0.999893i	$0.504652\pi$
$588$	0	0
$589$	16.7082	0.688450
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.4377	0.757145	0.378573	−	0.925572i	$-0.376415\pi$
$593$	18.4377	0.757145	0.378573	+	0.925572i	$0.376415\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.7082	1.13213	0.566063	−	0.824362i	$-0.308466\pi$
$599$	27.7082	1.13213	0.566063	+	0.824362i	$0.308466\pi$
$600$	0	0
$601$	3.94427	0.160890	0.0804451	−	0.996759i	$-0.474366\pi$
$601$	3.94427	0.160890	0.0804451	+	0.996759i	$0.474366\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.3820	0.421391	0.210696	−	0.977552i	$-0.432427\pi$
$607$	10.3820	0.421391	0.210696	+	0.977552i	$0.432427\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15.3262	0.620033
$612$	0	0
$613$	28.1803	1.13819	0.569097	−	0.822271i	$-0.307293\pi$
$613$	28.1803	1.13819	0.569097	+	0.822271i	$0.307293\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.4721	0.864436	0.432218	−	0.901769i	$-0.357731\pi$
$617$	21.4721	0.864436	0.432218	+	0.901769i	$0.357731\pi$
$618$	0	0
$619$	−25.4721	−1.02381	−0.511906	−	0.859042i	$-0.671060\pi$
$619$	−25.4721	−1.02381	−0.511906	+	0.859042i	$0.671060\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.29180	0.0517547
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−39.9787	−1.59406
$630$	0	0
$631$	−11.5066	−0.458070	−0.229035	−	0.973418i	$-0.573557\pi$
$631$	−11.5066	−0.458070	−0.229035	+	0.973418i	$0.573557\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.7082	−0.543993
$636$	0	0
$637$	15.5279	0.615236
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.79837	0.268520	0.134260	−	0.990946i	$-0.457134\pi$
$641$	6.79837	0.268520	0.134260	+	0.990946i	$0.457134\pi$
$642$	0	0
$643$	−26.0902	−1.02890	−0.514448	−	0.857522i	$-0.672003\pi$
$643$	−26.0902	−1.02890	−0.514448	+	0.857522i	$0.672003\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.3951	−1.31290	−0.656449	−	0.754371i	$-0.727942\pi$
$647$	−33.3951	−1.31290	−0.656449	+	0.754371i	$0.727942\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.7426	−0.968255	−0.484127	−	0.874998i	$-0.660863\pi$
$653$	−24.7426	−0.968255	−0.484127	+	0.874998i	$0.660863\pi$
$654$	0	0
$655$	−34.8885	−1.36321
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21.1246	−0.822898	−0.411449	−	0.911433i	$-0.634977\pi$
$659$	−21.1246	−0.822898	−0.411449	+	0.911433i	$0.634977\pi$
$660$	0	0
$661$	−14.8541	−0.577758	−0.288879	−	0.957366i	$-0.593282\pi$
$661$	−14.8541	−0.577758	−0.288879	+	0.957366i	$0.593282\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.61803	−0.0627447
$666$	0	0
$667$	−14.9443	−0.578645
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−17.4721	−0.673501	−0.336751	−	0.941594i	$-0.609328\pi$
$673$	−17.4721	−0.673501	−0.336751	+	0.941594i	$0.609328\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.4164	0.592501	0.296250	−	0.955110i	$-0.404264\pi$
$677$	15.4164	0.592501	0.296250	+	0.955110i	$0.404264\pi$
$678$	0	0
$679$	−0.0344419	−0.00132176
$680$	0	0
$681$	0	0
$682$	0	0
$683$	50.4853	1.93177	0.965883	−	0.258979i	$-0.0833860\pi$
$683$	50.4853	1.93177	0.965883	+	0.258979i	$0.0833860\pi$
$684$	0	0
$685$	−9.09017	−0.347318
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23.6180	0.899775
$690$	0	0
$691$	−35.5967	−1.35416	−0.677082	−	0.735908i	$-0.736756\pi$
$691$	−35.5967	−1.35416	−0.677082	+	0.735908i	$0.736756\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42.8885	1.62686
$696$	0	0
$697$	−24.2705	−0.919311
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.9230	1.20571	0.602857	−	0.797849i	$-0.294029\pi$
$701$	31.9230	1.20571	0.602857	+	0.797849i	$0.294029\pi$
$702$	0	0
$703$	21.5623	0.813238
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.23607	−0.0840960
$708$	0	0
$709$	5.32624	0.200031	0.100016	−	0.994986i	$-0.468111\pi$
$709$	5.32624	0.200031	0.100016	+	0.994986i	$0.468111\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.2574	−0.421591
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.81966	0.328918	0.164459	−	0.986384i	$-0.447412\pi$
$719$	8.81966	0.328918	0.164459	+	0.986384i	$0.447412\pi$
$720$	0	0
$721$	−2.94427	−0.109650
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.7082	0.583388
$726$	0	0
$727$	10.7426	0.398423	0.199211	−	0.979957i	$-0.436162\pi$
$727$	10.7426	0.398423	0.199211	+	0.979957i	$0.436162\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.2705	−0.453841
$732$	0	0
$733$	−22.0689	−0.815133	−0.407566	−	0.913176i	$-0.633623\pi$
$733$	−22.0689	−0.815133	−0.407566	+	0.913176i	$0.633623\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−49.0689	−1.80503	−0.902514	−	0.430660i	$-0.858281\pi$
$739$	−49.0689	−1.80503	−0.902514	+	0.430660i	$0.858281\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.94427	0.364820	0.182410	−	0.983223i	$-0.441610\pi$
$743$	9.94427	0.364820	0.182410	+	0.983223i	$0.441610\pi$
$744$	0	0
$745$	−3.85410	−0.141203
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.40325	0.0512737
$750$	0	0
$751$	−7.09017	−0.258724	−0.129362	−	0.991597i	$-0.541293\pi$
$751$	−7.09017	−0.258724	−0.129362	+	0.991597i	$0.541293\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−35.1246	−1.27832
$756$	0	0
$757$	−17.4721	−0.635036	−0.317518	−	0.948252i	$-0.602849\pi$
$757$	−17.4721	−0.635036	−0.317518	+	0.948252i	$0.602849\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.94427	0.287980	0.143990	−	0.989579i	$-0.454007\pi$
$761$	7.94427	0.287980	0.143990	+	0.989579i	$0.454007\pi$
$762$	0	0
$763$	−4.22291	−0.152880
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.4377	−0.449099
$768$	0	0
$769$	6.43769	0.232149	0.116075	−	0.993240i	$-0.462969\pi$
$769$	6.43769	0.232149	0.116075	+	0.993240i	$0.462969\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	45.8328	1.64849	0.824246	−	0.566232i	$-0.191599\pi$
$773$	45.8328	1.64849	0.824246	+	0.566232i	$0.191599\pi$
$774$	0	0
$775$	11.8328	0.425047
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.0902	0.469004
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16.1803	0.577501
$786$	0	0
$787$	31.0132	1.10550	0.552750	−	0.833347i	$-0.313579\pi$
$787$	31.0132	1.10550	0.552750	+	0.833347i	$0.313579\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.24922	−0.115529
$792$	0	0
$793$	0.326238	0.0115850
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.70820	−0.273039	−0.136519	−	0.990637i	$-0.543592\pi$
$797$	−7.70820	−0.273039	−0.136519	+	0.990637i	$0.543592\pi$
$798$	0	0
$799$	33.2705	1.17703
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1.09017	0.0384234
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.2705	−0.818148	−0.409074	−	0.912501i	$-0.634148\pi$
$809$	−23.2705	−0.818148	−0.409074	+	0.912501i	$0.634148\pi$
$810$	0	0
$811$	−15.3262	−0.538177	−0.269089	−	0.963115i	$-0.586722\pi$
$811$	−15.3262	−0.538177	−0.269089	+	0.963115i	$0.586722\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	46.1246	1.61567
$816$	0	0
$817$	6.61803	0.231536
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31.0000	−1.08191	−0.540954	−	0.841052i	$-0.681937\pi$
$821$	−31.0000	−1.08191	−0.540954	+	0.841052i	$0.681937\pi$
$822$	0	0
$823$	25.5410	0.890304	0.445152	−	0.895455i	$-0.353150\pi$
$823$	25.5410	0.890304	0.445152	+	0.895455i	$0.353150\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.2918	−0.357881	−0.178940	−	0.983860i	$-0.557267\pi$
$827$	−10.2918	−0.357881	−0.178940	+	0.983860i	$0.557267\pi$
$828$	0	0
$829$	27.2148	0.945208	0.472604	−	0.881275i	$-0.343314\pi$
$829$	27.2148	0.945208	0.472604	+	0.881275i	$0.343314\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	33.7082	1.16792
$834$	0	0
$835$	20.0902	0.695249
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.5279	−1.19203	−0.596017	−	0.802972i	$-0.703251\pi$
$839$	−34.5279	−1.19203	−0.596017	+	0.802972i	$0.703251\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20.9443	0.720505
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.5279	−0.498009
$852$	0	0
$853$	37.4721	1.28302	0.641511	−	0.767114i	$-0.278308\pi$
$853$	37.4721	1.28302	0.641511	+	0.767114i	$0.278308\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.8885	0.440264	0.220132	−	0.975470i	$-0.429351\pi$
$857$	12.8885	0.440264	0.220132	+	0.975470i	$0.429351\pi$
$858$	0	0
$859$	−11.6525	−0.397577	−0.198789	−	0.980042i	$-0.563701\pi$
$859$	−11.6525	−0.397577	−0.198789	+	0.980042i	$0.563701\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.3050	1.61028	0.805140	−	0.593085i	$-0.202090\pi$
$863$	47.3050	1.61028	0.805140	+	0.593085i	$0.202090\pi$
$864$	0	0
$865$	31.5066	1.07126
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−34.7984	−1.17910
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.94427	0.0657284
$876$	0	0
$877$	41.8328	1.41259	0.706297	−	0.707916i	$-0.250364\pi$
$877$	41.8328	1.41259	0.706297	+	0.707916i	$0.250364\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.14590	−0.0386063	−0.0193031	−	0.999814i	$-0.506145\pi$
$881$	−1.14590	−0.0386063	−0.0193031	+	0.999814i	$0.506145\pi$
$882$	0	0
$883$	−37.7771	−1.27130	−0.635650	−	0.771977i	$-0.719268\pi$
$883$	−37.7771	−1.27130	−0.635650	+	0.771977i	$0.719268\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.18034	0.173939	0.0869694	−	0.996211i	$-0.472282\pi$
$887$	5.18034	0.173939	0.0869694	+	0.996211i	$0.472282\pi$
$888$	0	0
$889$	1.23607	0.0414564
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17.9443	−0.600482
$894$	0	0
$895$	11.5623	0.386485
$896$	0	0
$897$	0	0
$898$	0	0
$899$	54.0689	1.80330
$900$	0	0
$901$	51.2705	1.70807
$902$	0	0
$903$	0	0
$904$	0	0
$905$	54.6869	1.81785
$906$	0	0
$907$	25.5623	0.848782	0.424391	−	0.905479i	$-0.360488\pi$
$907$	25.5623	0.848782	0.424391	+	0.905479i	$0.360488\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.708204	−0.0234638	−0.0117319	−	0.999931i	$-0.503734\pi$
$911$	−0.708204	−0.0234638	−0.0117319	+	0.999931i	$0.503734\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.14590	0.103887
$918$	0	0
$919$	−32.8885	−1.08489	−0.542446	−	0.840090i	$-0.682502\pi$
$919$	−32.8885	−1.08489	−0.542446	+	0.840090i	$0.682502\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16.9098	−0.556594
$924$	0	0
$925$	15.2705	0.502091
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19.9443	−0.654350	−0.327175	−	0.944964i	$-0.606097\pi$
$929$	−19.9443	−0.654350	−0.327175	+	0.944964i	$0.606097\pi$
$930$	0	0
$931$	−18.1803	−0.595837
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.70820	0.284485	0.142242	−	0.989832i	$-0.454569\pi$
$937$	8.70820	0.284485	0.142242	+	0.989832i	$0.454569\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−45.2148	−1.47396	−0.736980	−	0.675915i	$-0.763749\pi$
$941$	−45.2148	−1.47396	−0.736980	+	0.675915i	$0.763749\pi$
$942$	0	0
$943$	−8.81966	−0.287208
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.2705	−1.34111	−0.670556	−	0.741859i	$-0.733944\pi$
$947$	−41.2705	−1.34111	−0.670556	+	0.741859i	$0.733944\pi$
$948$	0	0
$949$	21.7082	0.704678
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−12.6525	−0.409854	−0.204927	−	0.978777i	$-0.565696\pi$
$953$	−12.6525	−0.409854	−0.204927	+	0.978777i	$0.565696\pi$
$954$	0	0
$955$	26.7984	0.867175
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.819660	0.0264682
$960$	0	0
$961$	9.72949	0.313855
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−40.1246	−1.29166
$966$	0	0
$967$	−24.6180	−0.791663	−0.395831	−	0.918323i	$-0.629544\pi$
$967$	−24.6180	−0.791663	−0.395831	+	0.918323i	$0.629544\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.3050	−0.876258	−0.438129	−	0.898912i	$-0.644359\pi$
$971$	−27.3050	−0.876258	−0.438129	+	0.898912i	$0.644359\pi$
$972$	0	0
$973$	−3.86726	−0.123979
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.4164	−0.653179	−0.326589	−	0.945166i	$-0.605899\pi$
$977$	−20.4164	−0.653179	−0.326589	+	0.945166i	$0.605899\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.7771	−1.07732	−0.538661	−	0.842523i	$-0.681070\pi$
$983$	−33.7771	−1.07732	−0.538661	+	0.842523i	$0.681070\pi$
$984$	0	0
$985$	−38.8885	−1.23909
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.45898	−0.141787
$990$	0	0
$991$	−14.3262	−0.455088	−0.227544	−	0.973768i	$-0.573070\pi$
$991$	−14.3262	−0.455088	−0.227544	+	0.973768i	$0.573070\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.38197	−0.0438113
$996$	0	0
$997$	50.7984	1.60880	0.804400	−	0.594088i	$-0.202487\pi$
$997$	50.7984	1.60880	0.804400	+	0.594088i	$0.202487\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.ba.1.1		2
3.2	odd	2		2904.2.a.z.1.2		2
11.3	even	5		792.2.r.d.361.1		4
11.4	even	5		792.2.r.d.577.1		4
11.10	odd	2		8712.2.a.bc.1.1		2
12.11	even	2		5808.2.a.bw.1.2		2
33.14	odd	10		264.2.q.b.97.1	yes	4
33.26	odd	10		264.2.q.b.49.1	✓	4
33.32	even	2		2904.2.a.ba.1.2		2
132.47	even	10		528.2.y.h.97.1		4
132.59	even	10		528.2.y.h.49.1		4
132.131	odd	2		5808.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.q.b.49.1	✓	4	33.26	odd	10
264.2.q.b.97.1	yes	4	33.14	odd	10
528.2.y.h.49.1		4	132.59	even	10
528.2.y.h.97.1		4	132.47	even	10
792.2.r.d.361.1		4	11.3	even	5
792.2.r.d.577.1		4	11.4	even	5
2904.2.a.z.1.2		2	3.2	odd	2
2904.2.a.ba.1.2		2	33.32	even	2
5808.2.a.bv.1.2		2	132.131	odd	2
5808.2.a.bw.1.2		2	12.11	even	2
8712.2.a.ba.1.1		2	1.1	even	1	trivial
8712.2.a.bc.1.1		2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.61803 q^{5} +0.236068 q^{7} -2.23607 q^{13} -4.85410 q^{17} +2.61803 q^{19} -1.76393 q^{23} +1.85410 q^{25} +8.47214 q^{29} +6.38197 q^{31} -0.618034 q^{35} +8.23607 q^{37} +5.00000 q^{41} +2.52786 q^{43} -6.85410 q^{47} -6.94427 q^{49} -10.5623 q^{53} +5.56231 q^{59} -0.145898 q^{61} +5.85410 q^{65} +15.5623 q^{67} +7.56231 q^{71} -9.70820 q^{73} +9.76393 q^{79} -15.9443 q^{83} +12.7082 q^{85} +5.47214 q^{89} -0.527864 q^{91} -6.85410 q^{95} -0.145898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} - 4 q^{7} - 3 q^{17} + 3 q^{19} - 8 q^{23} - 3 q^{25} + 8 q^{29} + 15 q^{31} + q^{35} + 12 q^{37} + 10 q^{41} + 14 q^{43} - 7 q^{47} + 4 q^{49} - q^{53} - 9 q^{59} - 7 q^{61} + 5 q^{65}+ \cdots - 7 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 - 4 * q^7 - 3 * q^17 + 3 * q^19 - 8 * q^23 - 3 * q^25 + 8 * q^29 + 15 * q^31 + q^35 + 12 * q^37 + 10 * q^41 + 14 * q^43 - 7 * q^47 + 4 * q^49 - q^53 - 9 * q^59 - 7 * q^61 + 5 * q^65 + 11 * q^67 - 5 * q^71 - 6 * q^73 + 24 * q^79 - 14 * q^83 + 12 * q^85 + 2 * q^89 - 10 * q^91 - 7 * q^95 - 7 * q^97

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