LMFDB - Embedded newform 8712.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$69.5656702409$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.61803 q^{5} +0.618034 q^{7} +O(q^{10})$	$q+1.61803 q^{5} +0.618034 q^{7} -0.854102 q^{13} +4.85410 q^{17} +1.85410 q^{19} +4.00000 q^{23} -2.38197 q^{25} -8.32624 q^{29} -10.0902 q^{31} +1.00000 q^{35} -7.38197 q^{37} -7.38197 q^{41} -10.4721 q^{43} -9.56231 q^{47} -6.61803 q^{49} +1.61803 q^{53} +4.61803 q^{59} -4.85410 q^{61} -1.38197 q^{65} -5.52786 q^{67} -6.09017 q^{71} +9.85410 q^{73} -2.85410 q^{79} +11.6180 q^{83} +7.85410 q^{85} -4.47214 q^{89} -0.527864 q^{91} +3.00000 q^{95} +3.32624 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - q^{7}+O(q^{10}) $ 2 * q + q^5 - q^7	$ 2 q + q^{5} - q^{7} + 5 q^{13} + 3 q^{17} - 3 q^{19} + 8 q^{23} - 7 q^{25} - q^{29} - 9 q^{31} + 2 q^{35} - 17 q^{37} - 17 q^{41} - 12 q^{43} + q^{47} - 11 q^{49} + q^{53} + 7 q^{59} - 3 q^{61} - 5 q^{65} - 20 q^{67} - q^{71} + 13 q^{73} + q^{79} + 21 q^{83} + 9 q^{85} - 10 q^{91} + 6 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q + q^5 - q^7 + 5 * q^13 + 3 * q^17 - 3 * q^19 + 8 * q^23 - 7 * q^25 - q^29 - 9 * q^31 + 2 * q^35 - 17 * q^37 - 17 * q^41 - 12 * q^43 + q^47 - 11 * q^49 + q^53 + 7 * q^59 - 3 * q^61 - 5 * q^65 - 20 * q^67 - q^71 + 13 * q^73 + q^79 + 21 * q^83 + 9 * q^85 - 10 * q^91 + 6 * q^95 - 9 * 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
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.854102	−0.236885	−0.118443	−	0.992961i	$-0.537790\pi$
$13$	−0.854102	−0.236885	−0.118443	+	0.992961i	$0.537790\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.85410	1.17729	0.588646	−	0.808391i	$-0.299661\pi$
$17$	4.85410	1.17729	0.588646	+	0.808391i	$0.299661\pi$
$18$	0	0
$19$	1.85410	0.425360	0.212680	−	0.977122i	$-0.431781\pi$
$19$	1.85410	0.425360	0.212680	+	0.977122i	$0.431781\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.32624	−1.54614	−0.773072	−	0.634319i	$-0.781281\pi$
$29$	−8.32624	−1.54614	−0.773072	+	0.634319i	$0.781281\pi$
$30$	0	0
$31$	−10.0902	−1.81225	−0.906124	−	0.423012i	$-0.860973\pi$
$31$	−10.0902	−1.81225	−0.906124	+	0.423012i	$0.860973\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−7.38197	−1.21359	−0.606794	−	0.794859i	$-0.707545\pi$
$37$	−7.38197	−1.21359	−0.606794	+	0.794859i	$0.707545\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.38197	−1.15287	−0.576435	−	0.817143i	$-0.695556\pi$
$41$	−7.38197	−1.15287	−0.576435	+	0.817143i	$0.695556\pi$
$42$	0	0
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.56231	−1.39481	−0.697403	−	0.716679i	$-0.745661\pi$
$47$	−9.56231	−1.39481	−0.697403	+	0.716679i	$0.745661\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.61803	0.222254	0.111127	−	0.993806i	$-0.464554\pi$
$53$	1.61803	0.222254	0.111127	+	0.993806i	$0.464554\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.61803	0.601217	0.300608	−	0.953748i	$-0.402810\pi$
$59$	4.61803	0.601217	0.300608	+	0.953748i	$0.402810\pi$
$60$	0	0
$61$	−4.85410	−0.621504	−0.310752	−	0.950491i	$-0.600581\pi$
$61$	−4.85410	−0.621504	−0.310752	+	0.950491i	$0.600581\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.38197	−0.171412
$66$	0	0
$67$	−5.52786	−0.675336	−0.337668	−	0.941265i	$-0.609638\pi$
$67$	−5.52786	−0.675336	−0.337668	+	0.941265i	$0.609638\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.09017	−0.722770	−0.361385	−	0.932417i	$-0.617696\pi$
$71$	−6.09017	−0.722770	−0.361385	+	0.932417i	$0.617696\pi$
$72$	0	0
$73$	9.85410	1.15334	0.576668	−	0.816979i	$-0.304353\pi$
$73$	9.85410	1.15334	0.576668	+	0.816979i	$0.304353\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.85410	−0.321112	−0.160556	−	0.987027i	$-0.551329\pi$
$79$	−2.85410	−0.321112	−0.160556	+	0.987027i	$0.551329\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.6180	1.27524	0.637622	−	0.770349i	$-0.279918\pi$
$83$	11.6180	1.27524	0.637622	+	0.770349i	$0.279918\pi$
$84$	0	0
$85$	7.85410	0.851897
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.47214	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$90$	0	0
$91$	−0.527864	−0.0553352
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	3.32624	0.337728	0.168864	−	0.985639i	$-0.445990\pi$
$97$	3.32624	0.337728	0.168864	+	0.985639i	$0.445990\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.09017	−0.805002	−0.402501	−	0.915420i	$-0.631859\pi$
$101$	−8.09017	−0.805002	−0.402501	+	0.915420i	$0.631859\pi$
$102$	0	0
$103$	0.909830	0.0896482	0.0448241	−	0.998995i	$-0.485727\pi$
$103$	0.909830	0.0896482	0.0448241	+	0.998995i	$0.485727\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0902	1.45882	0.729411	−	0.684076i	$-0.239794\pi$
$107$	15.0902	1.45882	0.729411	+	0.684076i	$0.239794\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.14590	−0.201869	−0.100935	−	0.994893i	$-0.532183\pi$
$113$	−2.14590	−0.201869	−0.100935	+	0.994893i	$0.532183\pi$
$114$	0	0
$115$	6.47214	0.603530
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	21.3262	1.89240	0.946199	−	0.323586i	$-0.104888\pi$
$127$	21.3262	1.89240	0.946199	+	0.323586i	$0.104888\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	1.14590	0.0993620
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.56231	0.218913	0.109456	−	0.993992i	$-0.465089\pi$
$137$	2.56231	0.218913	0.109456	+	0.993992i	$0.465089\pi$
$138$	0	0
$139$	−17.2705	−1.46487	−0.732433	−	0.680839i	$-0.761615\pi$
$139$	−17.2705	−1.46487	−0.732433	+	0.680839i	$0.761615\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−13.4721	−1.11880
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.1459	−0.913108	−0.456554	−	0.889696i	$-0.650916\pi$
$149$	−11.1459	−0.913108	−0.456554	+	0.889696i	$0.650916\pi$
$150$	0	0
$151$	8.32624	0.677580	0.338790	−	0.940862i	$-0.389982\pi$
$151$	8.32624	0.677580	0.338790	+	0.940862i	$0.389982\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.3262	−1.31135
$156$	0	0
$157$	−16.3262	−1.30298	−0.651488	−	0.758659i	$-0.725855\pi$
$157$	−16.3262	−1.30298	−0.651488	+	0.758659i	$0.725855\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.47214	0.194832
$162$	0	0
$163$	15.7984	1.23742	0.618712	−	0.785618i	$-0.287655\pi$
$163$	15.7984	1.23742	0.618712	+	0.785618i	$0.287655\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.32624	0.102627	0.0513137	−	0.998683i	$-0.483659\pi$
$167$	1.32624	0.102627	0.0513137	+	0.998683i	$0.483659\pi$
$168$	0	0
$169$	−12.2705	−0.943885
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.14590	0.163150	0.0815748	−	0.996667i	$-0.474005\pi$
$173$	2.14590	0.163150	0.0815748	+	0.996667i	$0.474005\pi$
$174$	0	0
$175$	−1.47214	−0.111283
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.27051	0.692910	0.346455	−	0.938067i	$-0.387385\pi$
$179$	9.27051	0.692910	0.346455	+	0.938067i	$0.387385\pi$
$180$	0	0
$181$	24.2705	1.80401	0.902006	−	0.431723i	$-0.142094\pi$
$181$	24.2705	1.80401	0.902006	+	0.431723i	$0.142094\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.9443	−0.878160
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.8541	−1.58131	−0.790654	−	0.612264i	$-0.790259\pi$
$191$	−21.8541	−1.58131	−0.790654	+	0.612264i	$0.790259\pi$
$192$	0	0
$193$	9.61803	0.692321	0.346161	−	0.938175i	$-0.387485\pi$
$193$	9.61803	0.692321	0.346161	+	0.938175i	$0.387485\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.47214	−0.603615	−0.301807	−	0.953369i	$-0.597590\pi$
$197$	−8.47214	−0.603615	−0.301807	+	0.953369i	$0.597590\pi$
$198$	0	0
$199$	−24.3607	−1.72688	−0.863441	−	0.504449i	$-0.831696\pi$
$199$	−24.3607	−1.72688	−0.863441	+	0.504449i	$0.831696\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.14590	−0.361171
$204$	0	0
$205$	−11.9443	−0.834224
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.90983	−0.131478	−0.0657391	−	0.997837i	$-0.520940\pi$
$211$	−1.90983	−0.131478	−0.0657391	+	0.997837i	$0.520940\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.9443	−1.15559
$216$	0	0
$217$	−6.23607	−0.423332
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.14590	−0.278883
$222$	0	0
$223$	−7.38197	−0.494333	−0.247167	−	0.968973i	$-0.579499\pi$
$223$	−7.38197	−0.494333	−0.247167	+	0.968973i	$0.579499\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.8541	0.919529	0.459765	−	0.888041i	$-0.347934\pi$
$227$	13.8541	0.919529	0.459765	+	0.888041i	$0.347934\pi$
$228$	0	0
$229$	−0.673762	−0.0445235	−0.0222617	−	0.999752i	$-0.507087\pi$
$229$	−0.673762	−0.0445235	−0.0222617	+	0.999752i	$0.507087\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.0344	−0.853915	−0.426957	−	0.904272i	$-0.640415\pi$
$233$	−13.0344	−0.853915	−0.426957	+	0.904272i	$0.640415\pi$
$234$	0	0
$235$	−15.4721	−1.00929
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.5066	0.938353	0.469176	−	0.883105i	$-0.344551\pi$
$239$	14.5066	0.938353	0.469176	+	0.883105i	$0.344551\pi$
$240$	0	0
$241$	−18.9443	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$241$	−18.9443	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.7082	−0.684122
$246$	0	0
$247$	−1.58359	−0.100762
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.38197	−0.529065	−0.264533	−	0.964377i	$-0.585218\pi$
$251$	−8.38197	−0.529065	−0.264533	+	0.964377i	$0.585218\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.5623	−1.09551	−0.547753	−	0.836640i	$-0.684517\pi$
$257$	−17.5623	−1.09551	−0.547753	+	0.836640i	$0.684517\pi$
$258$	0	0
$259$	−4.56231	−0.283488
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.4164	−1.19727	−0.598633	−	0.801023i	$-0.704289\pi$
$263$	−19.4164	−1.19727	−0.598633	+	0.801023i	$0.704289\pi$
$264$	0	0
$265$	2.61803	0.160825
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.0902	0.737151	0.368575	−	0.929598i	$-0.379846\pi$
$269$	12.0902	0.737151	0.368575	+	0.929598i	$0.379846\pi$
$270$	0	0
$271$	28.6180	1.73842	0.869211	−	0.494442i	$-0.164627\pi$
$271$	28.6180	1.73842	0.869211	+	0.494442i	$0.164627\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.61803	0.337555	0.168777	−	0.985654i	$-0.446018\pi$
$277$	5.61803	0.337555	0.168777	+	0.985654i	$0.446018\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.56231	−0.391474	−0.195737	−	0.980656i	$-0.562710\pi$
$281$	−6.56231	−0.391474	−0.195737	+	0.980656i	$0.562710\pi$
$282$	0	0
$283$	10.7984	0.641897	0.320948	−	0.947097i	$-0.395998\pi$
$283$	10.7984	0.641897	0.320948	+	0.947097i	$0.395998\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.56231	−0.269304
$288$	0	0
$289$	6.56231	0.386018
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.56231	0.558636	0.279318	−	0.960199i	$-0.409892\pi$
$293$	9.56231	0.558636	0.279318	+	0.960199i	$0.409892\pi$
$294$	0	0
$295$	7.47214	0.435045
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.41641	−0.197576
$300$	0	0
$301$	−6.47214	−0.373048
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.85410	−0.449725
$306$	0	0
$307$	1.88854	0.107785	0.0538924	−	0.998547i	$-0.482837\pi$
$307$	1.88854	0.107785	0.0538924	+	0.998547i	$0.482837\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.27051	0.0720440	0.0360220	−	0.999351i	$-0.488531\pi$
$311$	1.27051	0.0720440	0.0360220	+	0.999351i	$0.488531\pi$
$312$	0	0
$313$	14.7426	0.833304	0.416652	−	0.909066i	$-0.363203\pi$
$313$	14.7426	0.833304	0.416652	+	0.909066i	$0.363203\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.3262	−0.860807	−0.430404	−	0.902637i	$-0.641629\pi$
$317$	−15.3262	−0.860807	−0.430404	+	0.902637i	$0.641629\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.00000	0.500773
$324$	0	0
$325$	2.03444	0.112851
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.90983	−0.325819
$330$	0	0
$331$	8.94427	0.491622	0.245811	−	0.969318i	$-0.420946\pi$
$331$	8.94427	0.491622	0.245811	+	0.969318i	$0.420946\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.94427	−0.488678
$336$	0	0
$337$	−28.6180	−1.55892	−0.779462	−	0.626450i	$-0.784507\pi$
$337$	−28.6180	−1.55892	−0.779462	+	0.626450i	$0.784507\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−8.41641	−0.454443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.5066	1.58400	0.791998	−	0.610524i	$-0.209041\pi$
$347$	29.5066	1.58400	0.791998	+	0.610524i	$0.209041\pi$
$348$	0	0
$349$	13.2705	0.710354	0.355177	−	0.934799i	$-0.384421\pi$
$349$	13.2705	0.710354	0.355177	+	0.934799i	$0.384421\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.4164	1.35278	0.676389	−	0.736544i	$-0.263544\pi$
$353$	25.4164	1.35278	0.676389	+	0.736544i	$0.263544\pi$
$354$	0	0
$355$	−9.85410	−0.523001
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.5066	−1.18785	−0.593926	−	0.804520i	$-0.702423\pi$
$359$	−22.5066	−1.18785	−0.593926	+	0.804520i	$0.702423\pi$
$360$	0	0
$361$	−15.5623	−0.819069
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.9443	0.834561
$366$	0	0
$367$	−11.6738	−0.609365	−0.304683	−	0.952454i	$-0.598550\pi$
$367$	−11.6738	−0.609365	−0.304683	+	0.952454i	$0.598550\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.00000	0.0519174
$372$	0	0
$373$	−21.4164	−1.10890	−0.554450	−	0.832217i	$-0.687071\pi$
$373$	−21.4164	−1.10890	−0.554450	+	0.832217i	$0.687071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.11146	0.366259
$378$	0	0
$379$	−14.2705	−0.733027	−0.366513	−	0.930413i	$-0.619449\pi$
$379$	−14.2705	−0.733027	−0.366513	+	0.930413i	$0.619449\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.0344	1.58579	0.792893	−	0.609361i	$-0.208574\pi$
$383$	31.0344	1.58579	0.792893	+	0.609361i	$0.208574\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.7426	0.798184	0.399092	−	0.916911i	$-0.369325\pi$
$389$	15.7426	0.798184	0.399092	+	0.916911i	$0.369325\pi$
$390$	0	0
$391$	19.4164	0.981930
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.61803	−0.232359
$396$	0	0
$397$	−29.4164	−1.47637	−0.738184	−	0.674600i	$-0.764316\pi$
$397$	−29.4164	−1.47637	−0.738184	+	0.674600i	$0.764316\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.09017	0.204253	0.102127	−	0.994771i	$-0.467435\pi$
$401$	4.09017	0.204253	0.102127	+	0.994771i	$0.467435\pi$
$402$	0	0
$403$	8.61803	0.429295
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	18.5623	0.917847	0.458923	−	0.888476i	$-0.348235\pi$
$409$	18.5623	0.917847	0.458923	+	0.888476i	$0.348235\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.85410	0.140441
$414$	0	0
$415$	18.7984	0.922776
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.9443	−1.02319	−0.511597	−	0.859225i	$-0.670946\pi$
$419$	−20.9443	−1.02319	−0.511597	+	0.859225i	$0.670946\pi$
$420$	0	0
$421$	3.09017	0.150606	0.0753028	−	0.997161i	$-0.476008\pi$
$421$	3.09017	0.150606	0.0753028	+	0.997161i	$0.476008\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11.5623	−0.560854
$426$	0	0
$427$	−3.00000	−0.145180
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.1591	1.74172	0.870860	−	0.491531i	$-0.163562\pi$
$431$	36.1591	1.74172	0.870860	+	0.491531i	$0.163562\pi$
$432$	0	0
$433$	−26.7984	−1.28785	−0.643924	−	0.765090i	$-0.722695\pi$
$433$	−26.7984	−1.28785	−0.643924	+	0.765090i	$0.722695\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.41641	0.354775
$438$	0	0
$439$	−16.9443	−0.808706	−0.404353	−	0.914603i	$-0.632503\pi$
$439$	−16.9443	−0.808706	−0.404353	+	0.914603i	$0.632503\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.7984	0.893138	0.446569	−	0.894749i	$-0.352646\pi$
$443$	18.7984	0.893138	0.446569	+	0.894749i	$0.352646\pi$
$444$	0	0
$445$	−7.23607	−0.343023
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.4508	0.776364	0.388182	−	0.921583i	$-0.373103\pi$
$449$	16.4508	0.776364	0.388182	+	0.921583i	$0.373103\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.854102	−0.0400409
$456$	0	0
$457$	10.5623	0.494084	0.247042	−	0.969005i	$-0.420541\pi$
$457$	10.5623	0.494084	0.247042	+	0.969005i	$0.420541\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.3607	0.668844	0.334422	−	0.942424i	$-0.391459\pi$
$461$	14.3607	0.668844	0.334422	+	0.942424i	$0.391459\pi$
$462$	0	0
$463$	−11.4164	−0.530565	−0.265283	−	0.964171i	$-0.585465\pi$
$463$	−11.4164	−0.530565	−0.265283	+	0.964171i	$0.585465\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.7984	1.28636	0.643178	−	0.765717i	$-0.277616\pi$
$467$	27.7984	1.28636	0.643178	+	0.765717i	$0.277616\pi$
$468$	0	0
$469$	−3.41641	−0.157755
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.41641	−0.202639
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.9787	1.09562	0.547808	−	0.836604i	$-0.315463\pi$
$479$	23.9787	1.09562	0.547808	+	0.836604i	$0.315463\pi$
$480$	0	0
$481$	6.30495	0.287481
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.38197	0.244382
$486$	0	0
$487$	−3.96556	−0.179697	−0.0898483	−	0.995955i	$-0.528638\pi$
$487$	−3.96556	−0.179697	−0.0898483	+	0.995955i	$0.528638\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.5066	−0.835190	−0.417595	−	0.908633i	$-0.637127\pi$
$491$	−18.5066	−0.835190	−0.417595	+	0.908633i	$0.637127\pi$
$492$	0	0
$493$	−40.4164	−1.82026
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.76393	−0.168835
$498$	0	0
$499$	−23.0902	−1.03366	−0.516829	−	0.856089i	$-0.672888\pi$
$499$	−23.0902	−1.03366	−0.516829	+	0.856089i	$0.672888\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.09017	0.137784	0.0688919	−	0.997624i	$-0.478054\pi$
$503$	3.09017	0.137784	0.0688919	+	0.997624i	$0.478054\pi$
$504$	0	0
$505$	−13.0902	−0.582505
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.5066	−0.820290	−0.410145	−	0.912020i	$-0.634522\pi$
$509$	−18.5066	−0.820290	−0.410145	+	0.912020i	$0.634522\pi$
$510$	0	0
$511$	6.09017	0.269413
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.47214	0.0648701
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.03444	−0.176752	−0.0883761	−	0.996087i	$-0.528168\pi$
$521$	−4.03444	−0.176752	−0.0883761	+	0.996087i	$0.528168\pi$
$522$	0	0
$523$	−25.5066	−1.11532	−0.557662	−	0.830068i	$-0.688302\pi$
$523$	−25.5066	−1.11532	−0.557662	+	0.830068i	$0.688302\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−48.9787	−2.13355
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.30495	0.273098
$534$	0	0
$535$	24.4164	1.05561
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−36.2705	−1.55939	−0.779696	−	0.626159i	$-0.784626\pi$
$541$	−36.2705	−1.55939	−0.779696	+	0.626159i	$0.784626\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.76393	0.204064
$546$	0	0
$547$	34.7984	1.48787	0.743936	−	0.668251i	$-0.232957\pi$
$547$	34.7984	1.48787	0.743936	+	0.668251i	$0.232957\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.4377	−0.657668
$552$	0	0
$553$	−1.76393	−0.0750100
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.1459	1.10784	0.553919	−	0.832571i	$-0.313132\pi$
$557$	26.1459	1.10784	0.553919	+	0.832571i	$0.313132\pi$
$558$	0	0
$559$	8.94427	0.378302
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.56231	−0.192278	−0.0961391	−	0.995368i	$-0.530649\pi$
$563$	−4.56231	−0.192278	−0.0961391	+	0.995368i	$0.530649\pi$
$564$	0	0
$565$	−3.47214	−0.146074
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.20163	0.0503748	0.0251874	−	0.999683i	$-0.491982\pi$
$569$	1.20163	0.0503748	0.0251874	+	0.999683i	$0.491982\pi$
$570$	0	0
$571$	−22.8328	−0.955524	−0.477762	−	0.878489i	$-0.658552\pi$
$571$	−22.8328	−0.955524	−0.477762	+	0.878489i	$0.658552\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.52786	−0.397339
$576$	0	0
$577$	−21.6180	−0.899971	−0.449985	−	0.893036i	$-0.648571\pi$
$577$	−21.6180	−0.899971	−0.449985	+	0.893036i	$0.648571\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.18034	0.297891
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.0902	0.457740	0.228870	−	0.973457i	$-0.426497\pi$
$587$	11.0902	0.457740	0.228870	+	0.973457i	$0.426497\pi$
$588$	0	0
$589$	−18.7082	−0.770858
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.47214	−0.347909	−0.173954	−	0.984754i	$-0.555655\pi$
$593$	−8.47214	−0.347909	−0.173954	+	0.984754i	$0.555655\pi$
$594$	0	0
$595$	4.85410	0.198999
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.9098	0.568340	0.284170	−	0.958774i	$-0.408282\pi$
$599$	13.9098	0.568340	0.284170	+	0.958774i	$0.408282\pi$
$600$	0	0
$601$	37.8541	1.54410	0.772051	−	0.635561i	$-0.219231\pi$
$601$	37.8541	1.54410	0.772051	+	0.635561i	$0.219231\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	31.6180	1.28334	0.641668	−	0.766982i	$-0.278243\pi$
$607$	31.6180	1.28334	0.641668	+	0.766982i	$0.278243\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.16718	0.330409
$612$	0	0
$613$	−4.97871	−0.201088	−0.100544	−	0.994933i	$-0.532058\pi$
$613$	−4.97871	−0.201088	−0.100544	+	0.994933i	$0.532058\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.41641	−0.0570224	−0.0285112	−	0.999593i	$-0.509077\pi$
$617$	−1.41641	−0.0570224	−0.0285112	+	0.999593i	$0.509077\pi$
$618$	0	0
$619$	−13.8541	−0.556843	−0.278422	−	0.960459i	$-0.589811\pi$
$619$	−13.8541	−0.556843	−0.278422	+	0.960459i	$0.589811\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.76393	−0.110735
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−35.8328	−1.42875
$630$	0	0
$631$	0.909830	0.0362198	0.0181099	−	0.999836i	$-0.494235\pi$
$631$	0.909830	0.0362198	0.0181099	+	0.999836i	$0.494235\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	34.5066	1.36935
$636$	0	0
$637$	5.65248	0.223959
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−33.5623	−1.32563	−0.662816	−	0.748783i	$-0.730639\pi$
$641$	−33.5623	−1.32563	−0.662816	+	0.748783i	$0.730639\pi$
$642$	0	0
$643$	−24.3820	−0.961531	−0.480765	−	0.876849i	$-0.659641\pi$
$643$	−24.3820	−0.961531	−0.480765	+	0.876849i	$0.659641\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.5623	1.28016	0.640078	−	0.768310i	$-0.278902\pi$
$647$	32.5623	1.28016	0.640078	+	0.768310i	$0.278902\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.9230	−1.01444	−0.507222	−	0.861815i	$-0.669328\pi$
$653$	−25.9230	−1.01444	−0.507222	+	0.861815i	$0.669328\pi$
$654$	0	0
$655$	12.9443	0.505775
$656$	0	0
$657$	0	0
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	−10.9443	−0.425683	−0.212841	−	0.977087i	$-0.568272\pi$
$661$	−10.9443	−0.425683	−0.212841	+	0.977087i	$0.568272\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.85410	0.0718990
$666$	0	0
$667$	−33.3050	−1.28957
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−34.7426	−1.33923	−0.669615	−	0.742708i	$-0.733541\pi$
$673$	−34.7426	−1.33923	−0.669615	+	0.742708i	$0.733541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.67376	−0.179627	−0.0898136	−	0.995959i	$-0.528627\pi$
$677$	−4.67376	−0.179627	−0.0898136	+	0.995959i	$0.528627\pi$
$678$	0	0
$679$	2.05573	0.0788916
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.9443	−0.648355	−0.324177	−	0.945996i	$-0.605087\pi$
$683$	−16.9443	−0.648355	−0.324177	+	0.945996i	$0.605087\pi$
$684$	0	0
$685$	4.14590	0.158407
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.38197	−0.0526487
$690$	0	0
$691$	3.43769	0.130776	0.0653880	−	0.997860i	$-0.479171\pi$
$691$	3.43769	0.130776	0.0653880	+	0.997860i	$0.479171\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−27.9443	−1.05999
$696$	0	0
$697$	−35.8328	−1.35726
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−49.2705	−1.86092	−0.930461	−	0.366392i	$-0.880593\pi$
$701$	−49.2705	−1.86092	−0.930461	+	0.366392i	$0.880593\pi$
$702$	0	0
$703$	−13.6869	−0.516212
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.00000	−0.188044
$708$	0	0
$709$	22.7426	0.854118	0.427059	−	0.904224i	$-0.359550\pi$
$709$	22.7426	0.854118	0.427059	+	0.904224i	$0.359550\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−40.3607	−1.51152
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.0902	0.413594	0.206797	−	0.978384i	$-0.433696\pi$
$719$	11.0902	0.413594	0.206797	+	0.978384i	$0.433696\pi$
$720$	0	0
$721$	0.562306	0.0209414
$722$	0	0
$723$	0	0
$724$	0	0
$725$	19.8328	0.736572
$726$	0	0
$727$	−11.0557	−0.410034	−0.205017	−	0.978758i	$-0.565725\pi$
$727$	−11.0557	−0.410034	−0.205017	+	0.978758i	$0.565725\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−50.8328	−1.88012
$732$	0	0
$733$	−16.0344	−0.592246	−0.296123	−	0.955150i	$-0.595694\pi$
$733$	−16.0344	−0.592246	−0.296123	+	0.955150i	$0.595694\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−14.8541	−0.546417	−0.273208	−	0.961955i	$-0.588085\pi$
$739$	−14.8541	−0.546417	−0.273208	+	0.961955i	$0.588085\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.6869	−0.502124	−0.251062	−	0.967971i	$-0.580780\pi$
$743$	−13.6869	−0.502124	−0.251062	+	0.967971i	$0.580780\pi$
$744$	0	0
$745$	−18.0344	−0.660731
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.32624	0.340773
$750$	0	0
$751$	−12.3262	−0.449791	−0.224895	−	0.974383i	$-0.572204\pi$
$751$	−12.3262	−0.449791	−0.224895	+	0.974383i	$0.572204\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.4721	0.490301
$756$	0	0
$757$	42.1591	1.53230	0.766148	−	0.642664i	$-0.222171\pi$
$757$	42.1591	1.53230	0.766148	+	0.642664i	$0.222171\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.5066	−1.14211	−0.571056	−	0.820911i	$-0.693466\pi$
$761$	−31.5066	−1.14211	−0.571056	+	0.820911i	$0.693466\pi$
$762$	0	0
$763$	1.81966	0.0658761
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.94427	−0.142419
$768$	0	0
$769$	44.2492	1.59567	0.797834	−	0.602877i	$-0.205979\pi$
$769$	44.2492	1.59567	0.797834	+	0.602877i	$0.205979\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.6312	1.35350	0.676750	−	0.736213i	$-0.263388\pi$
$773$	37.6312	1.35350	0.676750	+	0.736213i	$0.263388\pi$
$774$	0	0
$775$	24.0344	0.863343
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.6869	−0.490385
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−26.4164	−0.942842
$786$	0	0
$787$	3.43769	0.122541	0.0612703	−	0.998121i	$-0.480485\pi$
$787$	3.43769	0.122541	0.0612703	+	0.998121i	$0.480485\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.32624	−0.0471556
$792$	0	0
$793$	4.14590	0.147225
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.4508	1.57453	0.787265	−	0.616615i	$-0.211496\pi$
$797$	44.4508	1.57453	0.787265	+	0.616615i	$0.211496\pi$
$798$	0	0
$799$	−46.4164	−1.64209
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.5066	−1.38898	−0.694489	−	0.719504i	$-0.744369\pi$
$809$	−39.5066	−1.38898	−0.694489	+	0.719504i	$0.744369\pi$
$810$	0	0
$811$	−0.257354	−0.00903693	−0.00451846	−	0.999990i	$-0.501438\pi$
$811$	−0.257354	−0.00903693	−0.00451846	+	0.999990i	$0.501438\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	25.5623	0.895409
$816$	0	0
$817$	−19.4164	−0.679294
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.4377	−0.922682	−0.461341	−	0.887223i	$-0.652632\pi$
$821$	−26.4377	−0.922682	−0.461341	+	0.887223i	$0.652632\pi$
$822$	0	0
$823$	29.3262	1.02225	0.511124	−	0.859507i	$-0.329229\pi$
$823$	29.3262	1.02225	0.511124	+	0.859507i	$0.329229\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.3262	1.01977	0.509887	−	0.860242i	$-0.329687\pi$
$827$	29.3262	1.01977	0.509887	+	0.860242i	$0.329687\pi$
$828$	0	0
$829$	12.0344	0.417973	0.208987	−	0.977918i	$-0.432983\pi$
$829$	12.0344	0.417973	0.208987	+	0.977918i	$0.432983\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−32.1246	−1.11305
$834$	0	0
$835$	2.14590	0.0742619
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−48.0344	−1.65833	−0.829167	−	0.559002i	$-0.811185\pi$
$839$	−48.0344	−1.65833	−0.829167	+	0.559002i	$0.811185\pi$
$840$	0	0
$841$	40.3262	1.39056
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−19.8541	−0.683002
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−29.5279	−1.01220
$852$	0	0
$853$	44.0902	1.50962	0.754809	−	0.655944i	$-0.227729\pi$
$853$	44.0902	1.50962	0.754809	+	0.655944i	$0.227729\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.3607	0.627189	0.313594	−	0.949557i	$-0.398467\pi$
$857$	18.3607	0.627189	0.313594	+	0.949557i	$0.398467\pi$
$858$	0	0
$859$	−21.8885	−0.746827	−0.373414	−	0.927665i	$-0.621813\pi$
$859$	−21.8885	−0.746827	−0.373414	+	0.927665i	$0.621813\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.5066	−0.868254	−0.434127	−	0.900852i	$-0.642943\pi$
$863$	−25.5066	−0.868254	−0.434127	+	0.900852i	$0.642943\pi$
$864$	0	0
$865$	3.47214	0.118056
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.72136	0.159977
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.38197	−0.249556
$876$	0	0
$877$	−48.0344	−1.62201	−0.811004	−	0.585041i	$-0.801079\pi$
$877$	−48.0344	−1.62201	−0.811004	+	0.585041i	$0.801079\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40.4721	−1.36354	−0.681770	−	0.731566i	$-0.738790\pi$
$881$	−40.4721	−1.36354	−0.681770	+	0.731566i	$0.738790\pi$
$882$	0	0
$883$	6.14590	0.206826	0.103413	−	0.994639i	$-0.467024\pi$
$883$	6.14590	0.206826	0.103413	+	0.994639i	$0.467024\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.5755	1.56385	0.781925	−	0.623372i	$-0.214238\pi$
$887$	46.5755	1.56385	0.781925	+	0.623372i	$0.214238\pi$
$888$	0	0
$889$	13.1803	0.442054
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17.7295	−0.593295
$894$	0	0
$895$	15.0000	0.501395
$896$	0	0
$897$	0	0
$898$	0	0
$899$	84.0132	2.80200
$900$	0	0
$901$	7.85410	0.261658
$902$	0	0
$903$	0	0
$904$	0	0
$905$	39.2705	1.30540
$906$	0	0
$907$	−6.27051	−0.208209	−0.104104	−	0.994566i	$-0.533198\pi$
$907$	−6.27051	−0.208209	−0.104104	+	0.994566i	$0.533198\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.0212862	−0.000705244 0	−0.000352622	−	1.00000i	$-0.500112\pi$
$911$	−0.0212862	−0.000705244 0	−0.000352622		1.00000i	$0.500112\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.94427	0.163274
$918$	0	0
$919$	16.3820	0.540391	0.270196	−	0.962805i	$-0.412912\pi$
$919$	16.3820	0.540391	0.270196	+	0.962805i	$0.412912\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.20163	0.171214
$924$	0	0
$925$	17.5836	0.578145
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.5623	1.13395	0.566976	−	0.823734i	$-0.308113\pi$
$929$	34.5623	1.13395	0.566976	+	0.823734i	$0.308113\pi$
$930$	0	0
$931$	−12.2705	−0.402150
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.5623	0.606404	0.303202	−	0.952926i	$-0.401944\pi$
$937$	18.5623	0.606404	0.303202	+	0.952926i	$0.401944\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.27051	0.139215	0.0696073	−	0.997574i	$-0.477825\pi$
$941$	4.27051	0.139215	0.0696073	+	0.997574i	$0.477825\pi$
$942$	0	0
$943$	−29.5279	−0.961560
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.8328	1.13191	0.565957	−	0.824435i	$-0.308507\pi$
$947$	34.8328	1.13191	0.565957	+	0.824435i	$0.308507\pi$
$948$	0	0
$949$	−8.41641	−0.273208
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.9787	0.549994	0.274997	−	0.961445i	$-0.411323\pi$
$953$	16.9787	0.549994	0.274997	+	0.961445i	$0.411323\pi$
$954$	0	0
$955$	−35.3607	−1.14424
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.58359	0.0511369
$960$	0	0
$961$	70.8115	2.28424
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.5623	0.500968
$966$	0	0
$967$	29.3050	0.942384	0.471192	−	0.882031i	$-0.343824\pi$
$967$	29.3050	0.942384	0.471192	+	0.882031i	$0.343824\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.8541	0.572965	0.286483	−	0.958085i	$-0.407514\pi$
$971$	17.8541	0.572965	0.286483	+	0.958085i	$0.407514\pi$
$972$	0	0
$973$	−10.6738	−0.342185
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−51.6869	−1.65361	−0.826805	−	0.562488i	$-0.809844\pi$
$977$	−51.6869	−1.65361	−0.826805	+	0.562488i	$0.809844\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.09017	0.0985611	0.0492806	−	0.998785i	$-0.484307\pi$
$983$	3.09017	0.0985611	0.0492806	+	0.998785i	$0.484307\pi$
$984$	0	0
$985$	−13.7082	−0.436780
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41.8885	−1.33198
$990$	0	0
$991$	44.9443	1.42770	0.713851	−	0.700298i	$-0.246949\pi$
$991$	44.9443	1.42770	0.713851	+	0.700298i	$0.246949\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−39.4164	−1.24958
$996$	0	0
$997$	49.6312	1.57184	0.785918	−	0.618331i	$-0.212191\pi$
$997$	49.6312	1.57184	0.785918	+	0.618331i	$0.212191\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bm.1.2		2
3.2	odd	2		968.2.a.h.1.2		2
11.2	odd	10		792.2.r.b.433.1		4
11.6	odd	10		792.2.r.b.289.1		4
11.10	odd	2		8712.2.a.bp.1.2		2
12.11	even	2		1936.2.a.u.1.1		2
24.5	odd	2		7744.2.a.cq.1.1		2
24.11	even	2		7744.2.a.cc.1.2		2
33.2	even	10		88.2.i.a.81.1	yes	4
33.5	odd	10		968.2.i.k.729.1		4
33.8	even	10		968.2.i.d.9.1		4
33.14	odd	10		968.2.i.c.9.1		4
33.17	even	10		88.2.i.a.25.1	✓	4
33.20	odd	10		968.2.i.k.81.1		4
33.26	odd	10		968.2.i.c.753.1		4
33.29	even	10		968.2.i.d.753.1		4
33.32	even	2		968.2.a.i.1.2		2
132.35	odd	10		176.2.m.a.81.1		4
132.83	odd	10		176.2.m.a.113.1		4
132.131	odd	2		1936.2.a.t.1.1		2
264.35	odd	10		704.2.m.g.257.1		4
264.83	odd	10		704.2.m.g.641.1		4
264.101	even	10		704.2.m.b.257.1		4
264.131	odd	2		7744.2.a.cb.1.2		2
264.149	even	10		704.2.m.b.641.1		4
264.197	even	2		7744.2.a.cr.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.i.a.25.1	✓	4	33.17	even	10
88.2.i.a.81.1	yes	4	33.2	even	10
176.2.m.a.81.1		4	132.35	odd	10
176.2.m.a.113.1		4	132.83	odd	10
704.2.m.b.257.1		4	264.101	even	10
704.2.m.b.641.1		4	264.149	even	10
704.2.m.g.257.1		4	264.35	odd	10
704.2.m.g.641.1		4	264.83	odd	10
792.2.r.b.289.1		4	11.6	odd	10
792.2.r.b.433.1		4	11.2	odd	10
968.2.a.h.1.2		2	3.2	odd	2
968.2.a.i.1.2		2	33.32	even	2
968.2.i.c.9.1		4	33.14	odd	10
968.2.i.c.753.1		4	33.26	odd	10
968.2.i.d.9.1		4	33.8	even	10
968.2.i.d.753.1		4	33.29	even	10
968.2.i.k.81.1		4	33.20	odd	10
968.2.i.k.729.1		4	33.5	odd	10
1936.2.a.t.1.1		2	132.131	odd	2
1936.2.a.u.1.1		2	12.11	even	2
7744.2.a.cb.1.2		2	264.131	odd	2
7744.2.a.cc.1.2		2	24.11	even	2
7744.2.a.cq.1.1		2	24.5	odd	2
7744.2.a.cr.1.1		2	264.197	even	2
8712.2.a.bm.1.2		2	1.1	even	1	trivial
8712.2.a.bp.1.2		2	11.10	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.61803 q^{5} +0.618034 q^{7} +O(q^{10})\)	\(q+1.61803 q^{5} +0.618034 q^{7} -0.854102 q^{13} +4.85410 q^{17} +1.85410 q^{19} +4.00000 q^{23} -2.38197 q^{25} -8.32624 q^{29} -10.0902 q^{31} +1.00000 q^{35} -7.38197 q^{37} -7.38197 q^{41} -10.4721 q^{43} -9.56231 q^{47} -6.61803 q^{49} +1.61803 q^{53} +4.61803 q^{59} -4.85410 q^{61} -1.38197 q^{65} -5.52786 q^{67} -6.09017 q^{71} +9.85410 q^{73} -2.85410 q^{79} +11.6180 q^{83} +7.85410 q^{85} -4.47214 q^{89} -0.527864 q^{91} +3.00000 q^{95} +3.32624 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - q^{7}+O(q^{10}) \) 2 * q + q^5 - q^7	\( 2 q + q^{5} - q^{7} + 5 q^{13} + 3 q^{17} - 3 q^{19} + 8 q^{23} - 7 q^{25} - q^{29} - 9 q^{31} + 2 q^{35} - 17 q^{37} - 17 q^{41} - 12 q^{43} + q^{47} - 11 q^{49} + q^{53} + 7 q^{59} - 3 q^{61} - 5 q^{65} - 20 q^{67} - q^{71} + 13 q^{73} + q^{79} + 21 q^{83} + 9 q^{85} - 10 q^{91} + 6 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q + q^5 - q^7 + 5 * q^13 + 3 * q^17 - 3 * q^19 + 8 * q^23 - 7 * q^25 - q^29 - 9 * q^31 + 2 * q^35 - 17 * q^37 - 17 * q^41 - 12 * q^43 + q^47 - 11 * q^49 + q^53 + 7 * q^59 - 3 * q^61 - 5 * q^65 - 20 * q^67 - q^71 + 13 * q^73 + q^79 + 21 * q^83 + 9 * q^85 - 10 * q^91 + 6 * q^95 - 9 * q^97

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