LMFDB - Embedded newform 8712.2.a.bw.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1436.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 12 $ x^3 - 11*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.76644$ 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)

$f(q)$	$=$	$q-3.76644 q^{5} +0.346835 q^{7} +O(q^{10})$	$q-3.76644 q^{5} +0.346835 q^{7} +0.653165 q^{13} +5.07277 q^{17} -7.18604 q^{19} +2.69367 q^{23} +9.18604 q^{25} +0.233565 q^{29} -7.87971 q^{31} -1.30633 q^{35} +6.53287 q^{37} +6.46010 q^{41} -7.53287 q^{43} +4.00000 q^{47} -6.87971 q^{49} -7.76644 q^{53} -5.30633 q^{59} -5.87971 q^{61} -2.46010 q^{65} -4.34683 q^{67} +16.3721 q^{71} +4.49237 q^{73} -10.4924 q^{79} -11.0657 q^{83} -19.1062 q^{85} +6.92723 q^{89} +0.226540 q^{91} +27.0657 q^{95} -12.6784 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{7}+O(q^{10}) $ 3 * q - q^7	$ 3 q - q^{7} + 4 q^{13} + 8 q^{17} - q^{19} + 4 q^{23} + 7 q^{25} + 12 q^{29} + q^{31} - 8 q^{35} - 3 q^{37} + 4 q^{41} + 12 q^{47} + 4 q^{49} - 12 q^{53} - 20 q^{59} + 7 q^{61} + 8 q^{65} - 11 q^{67} + 8 q^{71} - 3 q^{73} - 15 q^{79} + 12 q^{83} - 6 q^{85} + 28 q^{89} - 26 q^{91} + 36 q^{95} - q^{97}+O(q^{100}) $ 3 * q - q^7 + 4 * q^13 + 8 * q^17 - q^19 + 4 * q^23 + 7 * q^25 + 12 * q^29 + q^31 - 8 * q^35 - 3 * q^37 + 4 * q^41 + 12 * q^47 + 4 * q^49 - 12 * q^53 - 20 * q^59 + 7 * q^61 + 8 * q^65 - 11 * q^67 + 8 * q^71 - 3 * q^73 - 15 * q^79 + 12 * q^83 - 6 * q^85 + 28 * q^89 - 26 * q^91 + 36 * q^95 - q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.76644	−1.68440	−0.842201	−	0.539164i	$-0.818740\pi$
$5$	−3.76644	−1.68440	−0.842201	+	0.539164i	$0.818740\pi$
$6$	0	0
$7$	0.346835	0.131091	0.0655456	−	0.997850i	$-0.479121\pi$
$7$	0.346835	0.131091	0.0655456	+	0.997850i	$0.479121\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.653165	0.181155	0.0905777	−	0.995889i	$-0.471129\pi$
$13$	0.653165	0.181155	0.0905777	+	0.995889i	$0.471129\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.07277	1.23033	0.615163	−	0.788400i	$-0.289090\pi$
$17$	5.07277	1.23033	0.615163	+	0.788400i	$0.289090\pi$
$18$	0	0
$19$	−7.18604	−1.64859	−0.824295	−	0.566161i	$-0.808428\pi$
$19$	−7.18604	−1.64859	−0.824295	+	0.566161i	$0.808428\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.69367	0.561669	0.280834	−	0.959756i	$-0.409389\pi$
$23$	2.69367	0.561669	0.280834	+	0.959756i	$0.409389\pi$
$24$	0	0
$25$	9.18604	1.83721
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.233565	0.0433718	0.0216859	−	0.999765i	$-0.493097\pi$
$29$	0.233565	0.0433718	0.0216859	+	0.999765i	$0.493097\pi$
$30$	0	0
$31$	−7.87971	−1.41524	−0.707618	−	0.706595i	$-0.750230\pi$
$31$	−7.87971	−1.41524	−0.707618	+	0.706595i	$0.750230\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.30633	−0.220810
$36$	0	0
$37$	6.53287	1.07400	0.536999	−	0.843583i	$-0.319558\pi$
$37$	6.53287	1.07400	0.536999	+	0.843583i	$0.319558\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.46010	1.00890	0.504449	−	0.863441i	$-0.331695\pi$
$41$	6.46010	1.00890	0.504449	+	0.863441i	$0.331695\pi$
$42$	0	0
$43$	−7.53287	−1.14875	−0.574376	−	0.818592i	$-0.694755\pi$
$43$	−7.53287	−1.14875	−0.574376	+	0.818592i	$0.694755\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−6.87971	−0.982815
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.76644	−1.06680	−0.533401	−	0.845863i	$-0.679086\pi$
$53$	−7.76644	−1.06680	−0.533401	+	0.845863i	$0.679086\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.30633	−0.690825	−0.345413	−	0.938451i	$-0.612261\pi$
$59$	−5.30633	−0.690825	−0.345413	+	0.938451i	$0.612261\pi$
$60$	0	0
$61$	−5.87971	−0.752819	−0.376410	−	0.926453i	$-0.622841\pi$
$61$	−5.87971	−0.752819	−0.376410	+	0.926453i	$0.622841\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.46010	−0.305138
$66$	0	0
$67$	−4.34683	−0.531050	−0.265525	−	0.964104i	$-0.585545\pi$
$67$	−4.34683	−0.531050	−0.265525	+	0.964104i	$0.585545\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.3721	1.94301	0.971504	−	0.237024i	$-0.0761720\pi$
$71$	16.3721	1.94301	0.971504	+	0.237024i	$0.0761720\pi$
$72$	0	0
$73$	4.49237	0.525792	0.262896	−	0.964824i	$-0.415322\pi$
$73$	4.49237	0.525792	0.262896	+	0.964824i	$0.415322\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.4924	−1.18048	−0.590242	−	0.807227i	$-0.700968\pi$
$79$	−10.4924	−1.18048	−0.590242	+	0.807227i	$0.700968\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.0657	−1.21462	−0.607311	−	0.794464i	$-0.707752\pi$
$83$	−11.0657	−1.21462	−0.607311	+	0.794464i	$0.707752\pi$
$84$	0	0
$85$	−19.1062	−2.07236
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.92723	0.734285	0.367143	−	0.930165i	$-0.380336\pi$
$89$	6.92723	0.734285	0.367143	+	0.930165i	$0.380336\pi$
$90$	0	0
$91$	0.226540	0.0237479
$92$	0	0
$93$	0	0
$94$	0	0
$95$	27.0657	2.77689
$96$	0	0
$97$	−12.6784	−1.28730	−0.643648	−	0.765321i	$-0.722580\pi$
$97$	−12.6784	−1.28730	−0.643648	+	0.765321i	$0.722580\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	19.4126	1.91278	0.956389	−	0.292096i	$-0.0943527\pi$
$103$	19.4126	1.91278	0.956389	+	0.292096i	$0.0943527\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.37207	−0.809359	−0.404679	−	0.914459i	$-0.632617\pi$
$107$	−8.37207	−0.809359	−0.404679	+	0.914459i	$0.632617\pi$
$108$	0	0
$109$	−6.53287	−0.625736	−0.312868	−	0.949797i	$-0.601290\pi$
$109$	−6.53287	−0.625736	−0.312868	+	0.949797i	$0.601290\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.4601	0.984004	0.492002	−	0.870594i	$-0.336265\pi$
$113$	10.4601	0.984004	0.492002	+	0.870594i	$0.336265\pi$
$114$	0	0
$115$	−10.1455	−0.946076
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.75941	0.161285
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15.7664	−1.41019
$126$	0	0
$127$	−14.4924	−1.28599	−0.642995	−	0.765871i	$-0.722308\pi$
$127$	−14.4924	−1.28599	−0.642995	+	0.765871i	$0.722308\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.3721	1.77992	0.889958	−	0.456042i	$-0.150733\pi$
$131$	20.3721	1.77992	0.889958	+	0.456042i	$0.150733\pi$
$132$	0	0
$133$	−2.49237	−0.216116
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.0657	−1.28715	−0.643577	−	0.765382i	$-0.722550\pi$
$137$	−15.0657	−1.28715	−0.643577	+	0.765382i	$0.722550\pi$
$138$	0	0
$139$	15.5329	1.31748	0.658740	−	0.752370i	$-0.271090\pi$
$139$	15.5329	1.31748	0.658740	+	0.752370i	$0.271090\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.879706	−0.0730556
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.6866	1.69472	0.847358	−	0.531022i	$-0.178192\pi$
$149$	20.6866	1.69472	0.847358	+	0.531022i	$0.178192\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	29.6784	2.38383
$156$	0	0
$157$	3.50763	0.279940	0.139970	−	0.990156i	$-0.455299\pi$
$157$	3.50763	0.279940	0.139970	+	0.990156i	$0.455299\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.934258	0.0736299
$162$	0	0
$163$	−5.04050	−0.394803	−0.197401	−	0.980323i	$-0.563250\pi$
$163$	−5.04050	−0.394803	−0.197401	+	0.980323i	$0.563250\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.08101	−0.315798	−0.157899	−	0.987455i	$-0.550472\pi$
$167$	−4.08101	−0.315798	−0.157899	+	0.987455i	$0.550472\pi$
$168$	0	0
$169$	−12.5734	−0.967183
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.6784	1.34406	0.672032	−	0.740522i	$-0.265422\pi$
$173$	17.6784	1.34406	0.672032	+	0.740522i	$0.265422\pi$
$174$	0	0
$175$	3.18604	0.240842
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.0657	−0.827092	−0.413546	−	0.910483i	$-0.635710\pi$
$179$	−11.0657	−0.827092	−0.413546	+	0.910483i	$0.635710\pi$
$180$	0	0
$181$	11.1455	0.828441	0.414220	−	0.910177i	$-0.364054\pi$
$181$	11.1455	0.828441	0.414220	+	0.910177i	$0.364054\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.6056	−1.80904
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.67840	0.410875	0.205437	−	0.978670i	$-0.434138\pi$
$191$	5.67840	0.410875	0.205437	+	0.978670i	$0.434138\pi$
$192$	0	0
$193$	22.0657	1.58833	0.794163	−	0.607704i	$-0.207909\pi$
$193$	22.0657	1.58833	0.794163	+	0.607704i	$0.207909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.91197	−0.136222	−0.0681110	−	0.997678i	$-0.521697\pi$
$197$	−1.91197	−0.136222	−0.0681110	+	0.997678i	$0.521697\pi$
$198$	0	0
$199$	−0.346835	−0.0245865	−0.0122932	−	0.999924i	$-0.503913\pi$
$199$	−0.346835	−0.0245865	−0.0122932	+	0.999924i	$0.503913\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.0810083	0.00568567
$204$	0	0
$205$	−24.3316	−1.69939
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.71891	0.187177	0.0935886	−	0.995611i	$-0.470166\pi$
$211$	2.71891	0.187177	0.0935886	+	0.995611i	$0.470166\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	28.3721	1.93496
$216$	0	0
$217$	−2.73296	−0.185525
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.31335	0.222880
$222$	0	0
$223$	−2.20130	−0.147410	−0.0737051	−	0.997280i	$-0.523482\pi$
$223$	−2.20130	−0.147410	−0.0737051	+	0.997280i	$0.523482\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.6937	1.50623	0.753116	−	0.657888i	$-0.228550\pi$
$227$	22.6937	1.50623	0.753116	+	0.657888i	$0.228550\pi$
$228$	0	0
$229$	18.3316	1.21138	0.605692	−	0.795699i	$-0.292896\pi$
$229$	18.3316	1.21138	0.605692	+	0.795699i	$0.292896\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.07277	−0.594377	−0.297188	−	0.954819i	$-0.596049\pi$
$233$	−9.07277	−0.594377	−0.297188	+	0.954819i	$0.596049\pi$
$234$	0	0
$235$	−15.0657	−0.982781
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.61266	0.427737	0.213869	−	0.976862i	$-0.431394\pi$
$239$	6.61266	0.427737	0.213869	+	0.976862i	$0.431394\pi$
$240$	0	0
$241$	3.38734	0.218198	0.109099	−	0.994031i	$-0.465203\pi$
$241$	3.38734	0.218198	0.109099	+	0.994031i	$0.465203\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	25.9120	1.65545
$246$	0	0
$247$	−4.69367	−0.298651
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.1468	−1.71349	−0.856744	−	0.515742i	$-0.827516\pi$
$251$	−27.1468	−1.71349	−0.856744	+	0.515742i	$0.827516\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.6056	−1.78437	−0.892185	−	0.451670i	$-0.850829\pi$
$257$	−28.6056	−1.78437	−0.892185	+	0.451670i	$0.850829\pi$
$258$	0	0
$259$	2.26583	0.140792
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.67840	−0.596796	−0.298398	−	0.954442i	$-0.596452\pi$
$263$	−9.67840	−0.596796	−0.298398	+	0.954442i	$0.596452\pi$
$264$	0	0
$265$	29.2518	1.79692
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.29931	−0.201162	−0.100581	−	0.994929i	$-0.532070\pi$
$269$	−3.29931	−0.201162	−0.100581	+	0.994929i	$0.532070\pi$
$270$	0	0
$271$	4.46713	0.271359	0.135679	−	0.990753i	$-0.456678\pi$
$271$	4.46713	0.271359	0.135679	+	0.990753i	$0.456678\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.1455	1.39068	0.695340	−	0.718681i	$-0.255254\pi$
$277$	23.1455	1.39068	0.695340	+	0.718681i	$0.255254\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.38734	0.560002	0.280001	−	0.960000i	$-0.409665\pi$
$281$	9.38734	0.560002	0.280001	+	0.960000i	$0.409665\pi$
$282$	0	0
$283$	15.4126	0.916183	0.458091	−	0.888905i	$-0.348533\pi$
$283$	15.4126	0.916183	0.458091	+	0.888905i	$0.348533\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.24059	0.132258
$288$	0	0
$289$	8.73296	0.513703
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.7007	0.741982	0.370991	−	0.928636i	$-0.379018\pi$
$293$	12.7007	0.741982	0.370991	+	0.928636i	$0.379018\pi$
$294$	0	0
$295$	19.9860	1.16363
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.75941	0.101749
$300$	0	0
$301$	−2.61266	−0.150591
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.1455	1.26805
$306$	0	0
$307$	22.0252	1.25705	0.628523	−	0.777791i	$-0.283660\pi$
$307$	22.0252	1.25705	0.628523	+	0.777791i	$0.283660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.0657	1.53476	0.767379	−	0.641194i	$-0.221561\pi$
$311$	27.0657	1.53476	0.767379	+	0.641194i	$0.221561\pi$
$312$	0	0
$313$	31.3328	1.77103	0.885517	−	0.464607i	$-0.153804\pi$
$313$	31.3328	1.77103	0.885517	+	0.464607i	$0.153804\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.3873	−1.20123	−0.600616	−	0.799537i	$-0.705078\pi$
$317$	−21.3873	−1.20123	−0.600616	+	0.799537i	$0.705078\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−36.4531	−2.02830
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.38734	0.0764865
$330$	0	0
$331$	−28.5734	−1.57053	−0.785267	−	0.619157i	$-0.787475\pi$
$331$	−28.5734	−1.57053	−0.785267	+	0.619157i	$0.787475\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.3721	0.894502
$336$	0	0
$337$	−19.0000	−1.03500	−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000	−1.03500	−0.517498	+	0.855684i	$0.673136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4.81396	−0.259930
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.7441	−0.898873	−0.449436	−	0.893312i	$-0.648375\pi$
$347$	−16.7441	−0.898873	−0.449436	+	0.893312i	$0.648375\pi$
$348$	0	0
$349$	23.2770	1.24599	0.622995	−	0.782226i	$-0.285916\pi$
$349$	23.2770	1.24599	0.622995	+	0.782226i	$0.285916\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.0728	0.908692	0.454346	−	0.890825i	$-0.349873\pi$
$353$	17.0728	0.908692	0.454346	+	0.890825i	$0.349873\pi$
$354$	0	0
$355$	−61.6644	−3.27280
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.9847	1.63531	0.817656	−	0.575707i	$-0.195273\pi$
$359$	30.9847	1.63531	0.817656	+	0.575707i	$0.195273\pi$
$360$	0	0
$361$	32.6391	1.71785
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.9202	−0.885644
$366$	0	0
$367$	−28.2770	−1.47605	−0.738024	−	0.674774i	$-0.764241\pi$
$367$	−28.2770	−1.47605	−0.738024	+	0.674774i	$0.764241\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.69367	−0.139848
$372$	0	0
$373$	18.8644	0.976764	0.488382	−	0.872630i	$-0.337587\pi$
$373$	18.8644	0.976764	0.488382	+	0.872630i	$0.337587\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.152556	0.00785705
$378$	0	0
$379$	17.2113	0.884084	0.442042	−	0.896994i	$-0.354254\pi$
$379$	17.2113	0.884084	0.442042	+	0.896994i	$0.354254\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.8252	1.06412	0.532058	−	0.846708i	$-0.321419\pi$
$383$	20.8252	1.06412	0.532058	+	0.846708i	$0.321419\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.62090	−0.0821831	−0.0410915	−	0.999155i	$-0.513084\pi$
$389$	−1.62090	−0.0821831	−0.0410915	+	0.999155i	$0.513084\pi$
$390$	0	0
$391$	13.6644	0.691036
$392$	0	0
$393$	0	0
$394$	0	0
$395$	39.5188	1.98841
$396$	0	0
$397$	2.07979	0.104382	0.0521908	−	0.998637i	$-0.483380\pi$
$397$	2.07979	0.104382	0.0521908	+	0.998637i	$0.483380\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.1385	1.00567	0.502835	−	0.864383i	$-0.332290\pi$
$401$	20.1385	1.00567	0.502835	+	0.864383i	$0.332290\pi$
$402$	0	0
$403$	−5.14675	−0.256378
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	34.8404	1.72275	0.861374	−	0.507971i	$-0.169604\pi$
$409$	34.8404	1.72275	0.861374	+	0.507971i	$0.169604\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.84042	−0.0905611
$414$	0	0
$415$	41.6784	2.04591
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.7594	1.64926	0.824628	−	0.565676i	$-0.191385\pi$
$419$	33.7594	1.64926	0.824628	+	0.565676i	$0.191385\pi$
$420$	0	0
$421$	19.4278	0.946855	0.473428	−	0.880833i	$-0.343017\pi$
$421$	19.4278	0.946855	0.473428	+	0.880833i	$0.343017\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	46.5986	2.26036
$426$	0	0
$427$	−2.03929	−0.0986880
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.0505	−1.44748	−0.723740	−	0.690073i	$-0.757579\pi$
$431$	−30.0505	−1.44748	−0.723740	+	0.690073i	$0.757579\pi$
$432$	0	0
$433$	−17.4531	−0.838742	−0.419371	−	0.907815i	$-0.637749\pi$
$433$	−17.4531	−0.838742	−0.419371	+	0.907815i	$0.637749\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−19.3568	−0.925962
$438$	0	0
$439$	−21.7987	−1.04040	−0.520198	−	0.854046i	$-0.674142\pi$
$439$	−21.7987	−1.04040	−0.520198	+	0.854046i	$0.674142\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.62793	0.362414	0.181207	−	0.983445i	$-0.442000\pi$
$443$	7.62793	0.362414	0.181207	+	0.983445i	$0.442000\pi$
$444$	0	0
$445$	−26.0910	−1.23683
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.7512	0.507379	0.253690	−	0.967286i	$-0.418356\pi$
$449$	10.7512	0.507379	0.253690	+	0.967286i	$0.418356\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.853250	−0.0400010
$456$	0	0
$457$	−15.1708	−0.709659	−0.354829	−	0.934931i	$-0.615461\pi$
$457$	−15.1708	−0.709659	−0.354829	+	0.934931i	$0.615461\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.0575	0.934171	0.467085	−	0.884212i	$-0.345304\pi$
$461$	20.0575	0.934171	0.467085	+	0.884212i	$0.345304\pi$
$462$	0	0
$463$	14.3075	0.664928	0.332464	−	0.943116i	$-0.392120\pi$
$463$	14.3075	0.664928	0.332464	+	0.943116i	$0.392120\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.1468	−0.515810	−0.257905	−	0.966170i	$-0.583032\pi$
$467$	−11.1468	−0.515810	−0.257905	+	0.966170i	$0.583032\pi$
$468$	0	0
$469$	−1.50763	−0.0696160
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−66.0112	−3.02880
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.8252	1.68258	0.841292	−	0.540581i	$-0.181795\pi$
$479$	36.8252	1.68258	0.841292	+	0.540581i	$0.181795\pi$
$480$	0	0
$481$	4.26704	0.194560
$482$	0	0
$483$	0	0
$484$	0	0
$485$	47.7524	2.16832
$486$	0	0
$487$	−39.8239	−1.80459	−0.902297	−	0.431114i	$-0.858121\pi$
$487$	−39.8239	−1.80459	−0.902297	+	0.431114i	$0.858121\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.1315	−0.457227	−0.228614	−	0.973517i	$-0.573419\pi$
$491$	−10.1315	−0.457227	−0.228614	+	0.973517i	$0.573419\pi$
$492$	0	0
$493$	1.18482	0.0533615
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.67840	0.254711
$498$	0	0
$499$	31.0100	1.38820	0.694098	−	0.719880i	$-0.255803\pi$
$499$	31.0100	1.38820	0.694098	+	0.719880i	$0.255803\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.6937	1.01186	0.505930	−	0.862574i	$-0.331149\pi$
$503$	22.6937	1.01186	0.505930	+	0.862574i	$0.331149\pi$
$504$	0	0
$505$	15.0657	0.670417
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.9695	−1.32837	−0.664187	−	0.747567i	$-0.731222\pi$
$509$	−29.9695	−1.32837	−0.664187	+	0.747567i	$0.731222\pi$
$510$	0	0
$511$	1.55811	0.0689267
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−73.1162	−3.22189
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.7441	0.558331	0.279166	−	0.960243i	$-0.409942\pi$
$521$	12.7441	0.558331	0.279166	+	0.960243i	$0.409942\pi$
$522$	0	0
$523$	−3.18604	−0.139316	−0.0696578	−	0.997571i	$-0.522191\pi$
$523$	−3.18604	−0.139316	−0.0696578	+	0.997571i	$0.522191\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−39.9719	−1.74120
$528$	0	0
$529$	−15.7441	−0.684528
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.21952	0.182768
$534$	0	0
$535$	31.5329	1.36328
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.7747	0.549226	0.274613	−	0.961555i	$-0.411450\pi$
$541$	12.7747	0.549226	0.274613	+	0.961555i	$0.411450\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	24.6056	1.05399
$546$	0	0
$547$	17.6784	0.755874	0.377937	−	0.925831i	$-0.376634\pi$
$547$	17.6784	0.755874	0.377937	+	0.925831i	$0.376634\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.67840	−0.0715024
$552$	0	0
$553$	−3.63912	−0.154751
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.1315	1.78517	0.892584	−	0.450881i	$-0.148890\pi$
$557$	42.1315	1.78517	0.892584	+	0.450881i	$0.148890\pi$
$558$	0	0
$559$	−4.92021	−0.208103
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.4378	1.32495	0.662473	−	0.749086i	$-0.269507\pi$
$563$	31.4378	1.32495	0.662473	+	0.749086i	$0.269507\pi$
$564$	0	0
$565$	−39.3973	−1.65746
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.6784	−1.24418	−0.622092	−	0.782944i	$-0.713717\pi$
$569$	−29.6784	−1.24418	−0.622092	+	0.782944i	$0.713717\pi$
$570$	0	0
$571$	18.4278	0.771181	0.385591	−	0.922670i	$-0.373998\pi$
$571$	18.4278	0.771181	0.385591	+	0.922670i	$0.373998\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.7441	1.03190
$576$	0	0
$577$	13.7747	0.573447	0.286724	−	0.958013i	$-0.407434\pi$
$577$	13.7747	0.573447	0.286724	+	0.958013i	$0.407434\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.83798	−0.159226
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.3568	−0.964039	−0.482019	−	0.876161i	$-0.660096\pi$
$587$	−23.3568	−0.964039	−0.482019	+	0.876161i	$0.660096\pi$
$588$	0	0
$589$	56.6239	2.33315
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.0728	1.02962	0.514808	−	0.857306i	$-0.327863\pi$
$593$	25.0728	1.02962	0.514808	+	0.857306i	$0.327863\pi$
$594$	0	0
$595$	−6.62671	−0.271669
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.7441	−0.847583	−0.423791	−	0.905760i	$-0.639301\pi$
$599$	−20.7441	−0.847583	−0.423791	+	0.905760i	$0.639301\pi$
$600$	0	0
$601$	−3.61266	−0.147364	−0.0736818	−	0.997282i	$-0.523475\pi$
$601$	−3.61266	−0.147364	−0.0736818	+	0.997282i	$0.523475\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.3568	0.948024	0.474012	−	0.880519i	$-0.342805\pi$
$607$	23.3568	0.948024	0.474012	+	0.880519i	$0.342805\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.61266	0.105697
$612$	0	0
$613$	−29.9202	−1.20847	−0.604233	−	0.796808i	$-0.706520\pi$
$613$	−29.9202	−1.20847	−0.604233	+	0.796808i	$0.706520\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00702	0.0807997	0.0403999	−	0.999184i	$-0.487137\pi$
$617$	2.00702	0.0807997	0.0403999	+	0.999184i	$0.487137\pi$
$618$	0	0
$619$	13.6644	0.549217	0.274608	−	0.961556i	$-0.411452\pi$
$619$	13.6644	0.549217	0.274608	+	0.961556i	$0.411452\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.40261	0.0962583
$624$	0	0
$625$	13.4531	0.538123
$626$	0	0
$627$	0	0
$628$	0	0
$629$	33.1397	1.32137
$630$	0	0
$631$	22.1455	0.881600	0.440800	−	0.897605i	$-0.354695\pi$
$631$	22.1455	0.881600	0.440800	+	0.897605i	$0.354695\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	54.5846	2.16612
$636$	0	0
$637$	−4.49358	−0.178042
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.8474	0.941917	0.470959	−	0.882155i	$-0.343908\pi$
$641$	23.8474	0.941917	0.470959	+	0.882155i	$0.343908\pi$
$642$	0	0
$643$	−14.9454	−0.589391	−0.294695	−	0.955591i	$-0.595218\pi$
$643$	−14.9454	−0.589391	−0.294695	+	0.955591i	$0.595218\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.6784	0.695010	0.347505	−	0.937678i	$-0.387029\pi$
$647$	17.6784	0.695010	0.347505	+	0.937678i	$0.387029\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.22532	−0.361015	−0.180507	−	0.983574i	$-0.557774\pi$
$653$	−9.22532	−0.361015	−0.180507	+	0.983574i	$0.557774\pi$
$654$	0	0
$655$	−76.7301	−2.99809
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.3873	−0.677315	−0.338657	−	0.940910i	$-0.609973\pi$
$659$	−17.3873	−0.677315	−0.338657	+	0.940910i	$0.609973\pi$
$660$	0	0
$661$	26.8239	1.04333	0.521665	−	0.853150i	$-0.325311\pi$
$661$	26.8239	1.04333	0.521665	+	0.853150i	$0.325311\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.38734	0.364025
$666$	0	0
$667$	0.629146	0.0243606
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	10.1203	0.390109	0.195054	−	0.980792i	$-0.437512\pi$
$673$	10.1203	0.390109	0.195054	+	0.980792i	$0.437512\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.1538	−0.505541	−0.252770	−	0.967526i	$-0.581342\pi$
$677$	−13.1538	−0.505541	−0.252770	+	0.967526i	$0.581342\pi$
$678$	0	0
$679$	−4.39731	−0.168753
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.6631	−1.09676	−0.548382	−	0.836228i	$-0.684756\pi$
$683$	−28.6631	−1.09676	−0.548382	+	0.836228i	$0.684756\pi$
$684$	0	0
$685$	56.7441	2.16808
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.07277	−0.193257
$690$	0	0
$691$	−31.4126	−1.19499	−0.597495	−	0.801872i	$-0.703837\pi$
$691$	−31.4126	−1.19499	−0.597495	+	0.801872i	$0.703837\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−58.5036	−2.21917
$696$	0	0
$697$	32.7706	1.24128
$698$	0	0
$699$	0	0
$700$	0	0
$701$	46.3650	1.75118	0.875592	−	0.483052i	$-0.160472\pi$
$701$	46.3650	1.75118	0.875592	+	0.483052i	$0.160472\pi$
$702$	0	0
$703$	−46.9454	−1.77058
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.38734	−0.0521762
$708$	0	0
$709$	45.3568	1.70341	0.851705	−	0.524021i	$-0.175569\pi$
$709$	45.3568	1.70341	0.851705	+	0.524021i	$0.175569\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.2253	−0.794895
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.3721	−1.35645	−0.678225	−	0.734855i	$-0.737250\pi$
$719$	−36.3721	−1.35645	−0.678225	+	0.734855i	$0.737250\pi$
$720$	0	0
$721$	6.73296	0.250748
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.14553	0.0796831
$726$	0	0
$727$	1.69245	0.0627695	0.0313848	−	0.999507i	$-0.490008\pi$
$727$	1.69245	0.0627695	0.0313848	+	0.999507i	$0.490008\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−38.2125	−1.41334
$732$	0	0
$733$	−16.2811	−0.601356	−0.300678	−	0.953726i	$-0.597213\pi$
$733$	−16.2811	−0.601356	−0.300678	+	0.953726i	$0.597213\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−22.8950	−0.842205	−0.421103	−	0.907013i	$-0.638357\pi$
$739$	−22.8950	−0.842205	−0.421103	+	0.907013i	$0.638357\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.8252	1.05749	0.528746	−	0.848780i	$-0.322662\pi$
$743$	28.8252	1.05749	0.528746	+	0.848780i	$0.322662\pi$
$744$	0	0
$745$	−77.9149	−2.85458
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.90373	−0.106100
$750$	0	0
$751$	−4.05577	−0.147997	−0.0739986	−	0.997258i	$-0.523576\pi$
$751$	−4.05577	−0.147997	−0.0739986	+	0.997258i	$0.523576\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−45.1972	−1.64490
$756$	0	0
$757$	−44.9554	−1.63393	−0.816966	−	0.576685i	$-0.804346\pi$
$757$	−44.9554	−1.63393	−0.816966	+	0.576685i	$0.804346\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.2042	−1.27615	−0.638077	−	0.769973i	$-0.720270\pi$
$761$	−35.2042	−1.27615	−0.638077	+	0.769973i	$0.720270\pi$
$762$	0	0
$763$	−2.26583	−0.0820284
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.46591	−0.125147
$768$	0	0
$769$	−10.6784	−0.385073	−0.192537	−	0.981290i	$-0.561671\pi$
$769$	−10.6784	−0.385073	−0.192537	+	0.981290i	$0.561671\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.7441	1.17773	0.588863	−	0.808233i	$-0.299576\pi$
$773$	32.7441	1.17773	0.588863	+	0.808233i	$0.299576\pi$
$774$	0	0
$775$	−72.3833	−2.60008
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−46.4225	−1.66326
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.2113	−0.471531
$786$	0	0
$787$	27.5188	0.980940	0.490470	−	0.871458i	$-0.336825\pi$
$787$	27.5188	0.980940	0.490470	+	0.871458i	$0.336825\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.62793	0.128994
$792$	0	0
$793$	−3.84042	−0.136377
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.3568	1.39409	0.697045	−	0.717028i	$-0.254498\pi$
$797$	39.3568	1.39409	0.697045	+	0.717028i	$0.254498\pi$
$798$	0	0
$799$	20.2911	0.717846
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3.51882	−0.124022
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−50.8756	−1.78869	−0.894346	−	0.447376i	$-0.852359\pi$
$809$	−50.8756	−1.78869	−0.894346	+	0.447376i	$0.852359\pi$
$810$	0	0
$811$	−26.9454	−0.946183	−0.473091	−	0.881013i	$-0.656862\pi$
$811$	−26.9454	−0.946183	−0.473091	+	0.881013i	$0.656862\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.9847	0.665006
$816$	0	0
$817$	54.1315	1.89382
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.6127	0.370384	0.185192	−	0.982702i	$-0.440709\pi$
$821$	10.6127	0.370384	0.185192	+	0.982702i	$0.440709\pi$
$822$	0	0
$823$	1.28109	0.0446561	0.0223280	−	0.999751i	$-0.492892\pi$
$823$	1.28109	0.0446561	0.0223280	+	0.999751i	$0.492892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.6631	1.13581	0.567904	−	0.823095i	$-0.307754\pi$
$827$	32.6631	1.13581	0.567904	+	0.823095i	$0.307754\pi$
$828$	0	0
$829$	12.6949	0.440912	0.220456	−	0.975397i	$-0.429246\pi$
$829$	12.6949	0.440912	0.220456	+	0.975397i	$0.429246\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−34.8991	−1.20918
$834$	0	0
$835$	15.3709	0.531930
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.8404	0.754015	0.377008	−	0.926210i	$-0.376953\pi$
$839$	21.8404	0.754015	0.377008	+	0.926210i	$0.376953\pi$
$840$	0	0
$841$	−28.9454	−0.998119
$842$	0	0
$843$	0	0
$844$	0	0
$845$	47.3568	1.62912
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	17.5974	0.603231
$852$	0	0
$853$	−24.9860	−0.855503	−0.427751	−	0.903896i	$-0.640694\pi$
$853$	−24.9860	−0.855503	−0.427751	+	0.903896i	$0.640694\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.453081	0.0154769	0.00773847	−	0.999970i	$-0.497537\pi$
$857$	0.453081	0.0154769	0.00773847	+	0.999970i	$0.497537\pi$
$858$	0	0
$859$	−51.4126	−1.75417	−0.877087	−	0.480331i	$-0.840517\pi$
$859$	−51.4126	−1.75417	−0.877087	+	0.480331i	$0.840517\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.9847	−0.782409	−0.391205	−	0.920304i	$-0.627942\pi$
$863$	−22.9847	−0.782409	−0.391205	+	0.920304i	$0.627942\pi$
$864$	0	0
$865$	−66.5846	−2.26394
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2.83920	−0.0962027
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.46835	−0.184864
$876$	0	0
$877$	−36.0517	−1.21738	−0.608690	−	0.793408i	$-0.708305\pi$
$877$	−36.0517	−1.21738	−0.608690	+	0.793408i	$0.708305\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.8029	1.40837	0.704187	−	0.710014i	$-0.251312\pi$
$881$	41.8029	1.40837	0.704187	+	0.710014i	$0.251312\pi$
$882$	0	0
$883$	11.2365	0.378139	0.189069	−	0.981964i	$-0.439453\pi$
$883$	11.2365	0.378139	0.189069	+	0.981964i	$0.439453\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.75941	0.327689	0.163844	−	0.986486i	$-0.447610\pi$
$887$	9.75941	0.327689	0.163844	+	0.986486i	$0.447610\pi$
$888$	0	0
$889$	−5.02646	−0.168582
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−28.7441	−0.961886
$894$	0	0
$895$	41.6784	1.39316
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.84042	−0.0613814
$900$	0	0
$901$	−39.3973	−1.31251
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−41.9789	−1.39543
$906$	0	0
$907$	8.63790	0.286817	0.143408	−	0.989664i	$-0.454194\pi$
$907$	8.63790	0.286817	0.143408	+	0.989664i	$0.454194\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.6631	0.817126	0.408563	−	0.912730i	$-0.366030\pi$
$911$	24.6631	0.817126	0.408563	+	0.912730i	$0.366030\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.06574	0.233331
$918$	0	0
$919$	−23.1215	−0.762708	−0.381354	−	0.924429i	$-0.624542\pi$
$919$	−23.1215	−0.762708	−0.381354	+	0.924429i	$0.624542\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.6937	0.351986
$924$	0	0
$925$	60.0112	1.97316
$926$	0	0
$927$	0	0
$928$	0	0
$929$	38.7371	1.27092	0.635462	−	0.772132i	$-0.280810\pi$
$929$	38.7371	1.27092	0.635462	+	0.772132i	$0.280810\pi$
$930$	0	0
$931$	49.4378	1.62026
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.0000	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.5246	−0.408291	−0.204146	−	0.978941i	$-0.565442\pi$
$941$	−12.5246	−0.408291	−0.204146	+	0.978941i	$0.565442\pi$
$942$	0	0
$943$	17.4014	0.566667
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.0810	−0.392580	−0.196290	−	0.980546i	$-0.562889\pi$
$947$	−12.0810	−0.392580	−0.196290	+	0.980546i	$0.562889\pi$
$948$	0	0
$949$	2.93426	0.0952500
$950$	0	0
$951$	0	0
$952$	0	0
$953$	29.3498	0.950733	0.475366	−	0.879788i	$-0.342316\pi$
$953$	29.3498	0.950733	0.475366	+	0.879788i	$0.342316\pi$
$954$	0	0
$955$	−21.3873	−0.692078
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.22532	−0.168734
$960$	0	0
$961$	31.0898	1.00290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−83.1092	−2.67538
$966$	0	0
$967$	3.41258	0.109741	0.0548705	−	0.998493i	$-0.482525\pi$
$967$	3.41258	0.109741	0.0548705	+	0.998493i	$0.482525\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.4378	−1.52235	−0.761176	−	0.648545i	$-0.775378\pi$
$971$	−47.4378	−1.52235	−0.761176	+	0.648545i	$0.775378\pi$
$972$	0	0
$973$	5.38734	0.172710
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.8310	0.698434	0.349217	−	0.937042i	$-0.386448\pi$
$977$	21.8310	0.698434	0.349217	+	0.937042i	$0.386448\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.6784	1.45691	0.728457	−	0.685091i	$-0.240238\pi$
$983$	45.6784	1.45691	0.728457	+	0.685091i	$0.240238\pi$
$984$	0	0
$985$	7.20130	0.229453
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.2911	−0.645218
$990$	0	0
$991$	−22.4225	−0.712276	−0.356138	−	0.934433i	$-0.615907\pi$
$991$	−22.4225	−0.712276	−0.356138	+	0.934433i	$0.615907\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.30633	0.0414135
$996$	0	0
$997$	48.6644	1.54122	0.770608	−	0.637310i	$-0.219953\pi$
$997$	48.6644	1.54122	0.770608	+	0.637310i	$0.219953\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bw.1.1	yes	3
3.2	odd	2		8712.2.a.bv.1.3	✓	3
11.10	odd	2		8712.2.a.bx.1.1	yes	3
33.32	even	2		8712.2.a.by.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8712.2.a.bv.1.3	✓	3	3.2	odd	2
8712.2.a.bw.1.1	yes	3	1.1	even	1	trivial
8712.2.a.bx.1.1	yes	3	11.10	odd	2
8712.2.a.by.1.3	yes	3	33.32	even	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)

\(f(q)\)	\(=\)	\(q-3.76644 q^{5} +0.346835 q^{7} +O(q^{10})\)	\(q-3.76644 q^{5} +0.346835 q^{7} +0.653165 q^{13} +5.07277 q^{17} -7.18604 q^{19} +2.69367 q^{23} +9.18604 q^{25} +0.233565 q^{29} -7.87971 q^{31} -1.30633 q^{35} +6.53287 q^{37} +6.46010 q^{41} -7.53287 q^{43} +4.00000 q^{47} -6.87971 q^{49} -7.76644 q^{53} -5.30633 q^{59} -5.87971 q^{61} -2.46010 q^{65} -4.34683 q^{67} +16.3721 q^{71} +4.49237 q^{73} -10.4924 q^{79} -11.0657 q^{83} -19.1062 q^{85} +6.92723 q^{89} +0.226540 q^{91} +27.0657 q^{95} -12.6784 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{7}+O(q^{10}) \) 3 * q - q^7	\( 3 q - q^{7} + 4 q^{13} + 8 q^{17} - q^{19} + 4 q^{23} + 7 q^{25} + 12 q^{29} + q^{31} - 8 q^{35} - 3 q^{37} + 4 q^{41} + 12 q^{47} + 4 q^{49} - 12 q^{53} - 20 q^{59} + 7 q^{61} + 8 q^{65} - 11 q^{67} + 8 q^{71} - 3 q^{73} - 15 q^{79} + 12 q^{83} - 6 q^{85} + 28 q^{89} - 26 q^{91} + 36 q^{95} - q^{97}+O(q^{100}) \) 3 * q - q^7 + 4 * q^13 + 8 * q^17 - q^19 + 4 * q^23 + 7 * q^25 + 12 * q^29 + q^31 - 8 * q^35 - 3 * q^37 + 4 * q^41 + 12 * q^47 + 4 * q^49 - 12 * q^53 - 20 * q^59 + 7 * q^61 + 8 * q^65 - 11 * q^67 + 8 * q^71 - 3 * q^73 - 15 * q^79 + 12 * q^83 - 6 * q^85 + 28 * q^89 - 26 * q^91 + 36 * q^95 - q^97

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