LMFDB - Embedded newform 6272.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6272,2,Mod(1,6272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6272, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6272.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6272 = 2^{7} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6272.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:	$50.0821721477$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6272.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} -4.24264 q^{13} -2.00000 q^{15} -2.82843 q^{17} -1.41421 q^{19} +2.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} -2.00000 q^{29} -8.48528 q^{31} +2.00000 q^{37} +6.00000 q^{39} +2.82843 q^{41} -4.00000 q^{43} -1.41421 q^{45} +2.82843 q^{47} +4.00000 q^{51} +6.00000 q^{53} +2.00000 q^{57} +9.89949 q^{59} -4.24264 q^{61} -6.00000 q^{65} +12.0000 q^{67} -2.82843 q^{69} -11.3137 q^{73} +4.24264 q^{75} +8.00000 q^{79} -5.00000 q^{81} +7.07107 q^{83} -4.00000 q^{85} +2.82843 q^{87} +16.9706 q^{89} +12.0000 q^{93} -2.00000 q^{95} -14.1421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 4 q^{15} + 4 q^{23} - 6 q^{25} - 4 q^{29} + 4 q^{37} + 12 q^{39} - 8 q^{43} + 8 q^{51} + 12 q^{53} + 4 q^{57} - 12 q^{65} + 24 q^{67} + 16 q^{79} - 10 q^{81} - 8 q^{85} + 24 q^{93} - 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^15 + 4 * q^23 - 6 * q^25 - 4 * q^29 + 4 * q^37 + 12 * q^39 - 8 * q^43 + 8 * q^51 + 12 * q^53 + 4 * q^57 - 12 * q^65 + 24 * q^67 + 16 * q^79 - 10 * q^81 - 8 * q^85 + 24 * q^93 - 4 * q^95

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	−1.41421	−0.324443	−0.162221	−	0.986754i	$-0.551866\pi$
$19$	−1.41421	−0.324443	−0.162221	+	0.986754i	$0.551866\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.41421	−0.210819
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	9.89949	1.28880	0.644402	−	0.764687i	$-0.277106\pi$
$59$	9.89949	1.28880	0.644402	+	0.764687i	$0.277106\pi$
$60$	0	0
$61$	−4.24264	−0.543214	−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264	−0.543214	−0.271607	+	0.962408i	$0.587555\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−11.3137	−1.32417	−0.662085	−	0.749429i	$-0.730328\pi$
$73$	−11.3137	−1.32417	−0.662085	+	0.749429i	$0.730328\pi$
$74$	0	0
$75$	4.24264	0.489898
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	7.07107	0.776151	0.388075	−	0.921628i	$-0.373140\pi$
$83$	7.07107	0.776151	0.388075	+	0.921628i	$0.373140\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	2.82843	0.303239
$88$	0	0
$89$	16.9706	1.79888	0.899438	−	0.437048i	$-0.143976\pi$
$89$	16.9706	1.79888	0.899438	+	0.437048i	$0.143976\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−14.1421	−1.43592	−0.717958	−	0.696086i	$-0.754923\pi$
$97$	−14.1421	−1.43592	−0.717958	+	0.696086i	$0.754923\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.7279	−1.26648	−0.633238	−	0.773957i	$-0.718274\pi$
$101$	−12.7279	−1.26648	−0.633238	+	0.773957i	$0.718274\pi$
$102$	0	0
$103$	14.1421	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$103$	14.1421	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.82843	−0.268462
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	2.82843	0.263752
$116$	0	0
$117$	4.24264	0.392232
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	0	0
$129$	5.65685	0.498058
$130$	0	0
$131$	−4.24264	−0.370681	−0.185341	−	0.982674i	$-0.559339\pi$
$131$	−4.24264	−0.370681	−0.185341	+	0.982674i	$0.559339\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.00000	0.688530
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−4.24264	−0.359856	−0.179928	−	0.983680i	$-0.557586\pi$
$139$	−4.24264	−0.359856	−0.179928	+	0.983680i	$0.557586\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.82843	−0.234888
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	0	0
$153$	2.82843	0.228665
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	1.41421	0.112867	0.0564333	−	0.998406i	$-0.482027\pi$
$157$	1.41421	0.112867	0.0564333	+	0.998406i	$0.482027\pi$
$158$	0	0
$159$	−8.48528	−0.672927
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.1421	−1.09435	−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421	−1.09435	−0.547176	+	0.837018i	$0.684297\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	1.41421	0.108148
$172$	0	0
$173$	−4.24264	−0.322562	−0.161281	−	0.986909i	$-0.551563\pi$
$173$	−4.24264	−0.322562	−0.161281	+	0.986909i	$0.551563\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.0000	−1.05230
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	15.5563	1.15629	0.578147	−	0.815933i	$-0.303776\pi$
$181$	15.5563	1.15629	0.578147	+	0.815933i	$0.303776\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	2.82843	0.207950
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	8.48528	0.607644
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−19.7990	−1.40351	−0.701757	−	0.712417i	$-0.747601\pi$
$199$	−19.7990	−1.40351	−0.701757	+	0.712417i	$0.747601\pi$
$200$	0	0
$201$	−16.9706	−1.19701
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	16.0000	1.08118
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	4.24264	0.281594	0.140797	−	0.990038i	$-0.455034\pi$
$227$	4.24264	0.281594	0.140797	+	0.990038i	$0.455034\pi$
$228$	0	0
$229$	18.3848	1.21490	0.607450	−	0.794358i	$-0.292192\pi$
$229$	18.3848	1.21490	0.607450	+	0.794358i	$0.292192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	−11.3137	−0.734904
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	−8.48528	−0.546585	−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528	−0.546585	−0.273293	+	0.961931i	$0.588113\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	−18.3848	−1.16044	−0.580218	−	0.814461i	$-0.697033\pi$
$251$	−18.3848	−1.16044	−0.580218	+	0.814461i	$0.697033\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	5.65685	0.354246
$256$	0	0
$257$	−16.9706	−1.05859	−0.529297	−	0.848436i	$-0.677544\pi$
$257$	−16.9706	−1.05859	−0.529297	+	0.848436i	$0.677544\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	8.48528	0.521247
$266$	0	0
$267$	−24.0000	−1.46878
$268$	0	0
$269$	1.41421	0.0862261	0.0431131	−	0.999070i	$-0.486272\pi$
$269$	1.41421	0.0862261	0.0431131	+	0.999070i	$0.486272\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	8.48528	0.508001
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−21.2132	−1.26099	−0.630497	−	0.776192i	$-0.717149\pi$
$283$	−21.2132	−1.26099	−0.630497	+	0.776192i	$0.717149\pi$
$284$	0	0
$285$	2.82843	0.167542
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	20.0000	1.17242
$292$	0	0
$293$	15.5563	0.908812	0.454406	−	0.890795i	$-0.349852\pi$
$293$	15.5563	0.908812	0.454406	+	0.890795i	$0.349852\pi$
$294$	0	0
$295$	14.0000	0.815112
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.48528	−0.490716
$300$	0	0
$301$	0	0
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−15.5563	−0.887848	−0.443924	−	0.896065i	$-0.646414\pi$
$307$	−15.5563	−0.887848	−0.443924	+	0.896065i	$0.646414\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	0	0
$311$	28.2843	1.60385	0.801927	−	0.597422i	$-0.203808\pi$
$311$	28.2843	1.60385	0.801927	+	0.597422i	$0.203808\pi$
$312$	0	0
$313$	14.1421	0.799361	0.399680	−	0.916655i	$-0.369121\pi$
$313$	14.1421	0.799361	0.399680	+	0.916655i	$0.369121\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−22.6274	−1.26294
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	12.7279	0.706018
$326$	0	0
$327$	−14.1421	−0.782062
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	16.9706	0.927201
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	0	0
$339$	−5.65685	−0.307238
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	21.2132	1.13552	0.567758	−	0.823195i	$-0.307811\pi$
$349$	21.2132	1.13552	0.567758	+	0.823195i	$0.307811\pi$
$350$	0	0
$351$	−24.0000	−1.28103
$352$	0	0
$353$	22.6274	1.20434	0.602168	−	0.798369i	$-0.294304\pi$
$353$	22.6274	1.20434	0.602168	+	0.798369i	$0.294304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	15.5563	0.816497
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	16.9706	0.885856	0.442928	−	0.896557i	$-0.353940\pi$
$367$	16.9706	0.885856	0.442928	+	0.896557i	$0.353940\pi$
$368$	0	0
$369$	−2.82843	−0.147242
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	16.0000	0.826236
$376$	0	0
$377$	8.48528	0.437014
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	14.1421	0.724524
$382$	0	0
$383$	31.1127	1.58978	0.794892	−	0.606751i	$-0.207527\pi$
$383$	31.1127	1.58978	0.794892	+	0.606751i	$0.207527\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	−7.07107	−0.354887	−0.177443	−	0.984131i	$-0.556783\pi$
$397$	−7.07107	−0.354887	−0.177443	+	0.984131i	$0.556783\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	36.0000	1.79329
$404$	0	0
$405$	−7.07107	−0.351364
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8.48528	0.419570	0.209785	−	0.977748i	$-0.432724\pi$
$409$	8.48528	0.419570	0.209785	+	0.977748i	$0.432724\pi$
$410$	0	0
$411$	8.48528	0.418548
$412$	0	0
$413$	0	0
$414$	0	0
$415$	10.0000	0.490881
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	−24.0416	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$419$	−24.0416	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	−2.82843	−0.137523
$424$	0	0
$425$	8.48528	0.411597
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	31.1127	1.49518	0.747590	−	0.664160i	$-0.231211\pi$
$433$	31.1127	1.49518	0.747590	+	0.664160i	$0.231211\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	−2.82843	−0.135302
$438$	0	0
$439$	−11.3137	−0.539974	−0.269987	−	0.962864i	$-0.587019\pi$
$439$	−11.3137	−0.539974	−0.269987	+	0.962864i	$0.587019\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	−25.4558	−1.20402
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−31.1127	−1.46180
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	−16.0000	−0.746816
$460$	0	0
$461$	26.8701	1.25146	0.625732	−	0.780038i	$-0.284800\pi$
$461$	26.8701	1.25146	0.625732	+	0.780038i	$0.284800\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	16.9706	0.786991
$466$	0	0
$467$	38.1838	1.76693	0.883467	−	0.468493i	$-0.155203\pi$
$467$	38.1838	1.76693	0.883467	+	0.468493i	$0.155203\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.24264	0.194666
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−2.82843	−0.129234	−0.0646171	−	0.997910i	$-0.520583\pi$
$479$	−2.82843	−0.129234	−0.0646171	+	0.997910i	$0.520583\pi$
$480$	0	0
$481$	−8.48528	−0.386896
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.0000	−0.908153
$486$	0	0
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	0	0
$489$	−11.3137	−0.511624
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	5.65685	0.254772
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	39.5980	1.76559	0.882793	−	0.469762i	$-0.155660\pi$
$503$	39.5980	1.76559	0.882793	+	0.469762i	$0.155660\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	−7.07107	−0.314037
$508$	0	0
$509$	−18.3848	−0.814891	−0.407445	−	0.913230i	$-0.633580\pi$
$509$	−18.3848	−0.814891	−0.407445	+	0.913230i	$0.633580\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.00000	−0.353209
$514$	0	0
$515$	20.0000	0.881305
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	19.7990	0.867409	0.433705	−	0.901055i	$-0.357206\pi$
$521$	19.7990	0.867409	0.433705	+	0.901055i	$0.357206\pi$
$522$	0	0
$523$	−1.41421	−0.0618392	−0.0309196	−	0.999522i	$-0.509844\pi$
$523$	−1.41421	−0.0618392	−0.0309196	+	0.999522i	$0.509844\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−9.89949	−0.429601
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	22.6274	0.978269
$536$	0	0
$537$	−28.2843	−1.22056
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	14.1421	0.605783
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	4.24264	0.181071
$550$	0	0
$551$	2.82843	0.120495
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.2132	−0.894030	−0.447015	−	0.894526i	$-0.647513\pi$
$563$	−21.2132	−0.894030	−0.447015	+	0.894526i	$0.647513\pi$
$564$	0	0
$565$	5.65685	0.237986
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.00000	−0.335377	−0.167689	−	0.985840i	$-0.553630\pi$
$569$	−8.00000	−0.335377	−0.167689	+	0.985840i	$0.553630\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−33.9411	−1.41791
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−11.3137	−0.470996	−0.235498	−	0.971875i	$-0.575672\pi$
$577$	−11.3137	−0.470996	−0.235498	+	0.971875i	$0.575672\pi$
$578$	0	0
$579$	22.6274	0.940363
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	−9.89949	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$587$	−9.89949	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	2.82843	0.116346
$592$	0	0
$593$	45.2548	1.85839	0.929197	−	0.369586i	$-0.120500\pi$
$593$	45.2548	1.85839	0.929197	+	0.369586i	$0.120500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.0000	1.14596
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	16.9706	0.692244	0.346122	−	0.938190i	$-0.387498\pi$
$601$	16.9706	0.692244	0.346122	+	0.938190i	$0.387498\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	−15.5563	−0.632456
$606$	0	0
$607$	5.65685	0.229605	0.114802	−	0.993388i	$-0.463377\pi$
$607$	5.65685	0.229605	0.114802	+	0.993388i	$0.463377\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	0	0
$615$	−5.65685	−0.228106
$616$	0	0
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	0	0
$619$	−26.8701	−1.08000	−0.539999	−	0.841665i	$-0.681576\pi$
$619$	−26.8701	−1.08000	−0.539999	+	0.841665i	$0.681576\pi$
$620$	0	0
$621$	11.3137	0.454003
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−16.9706	−0.674519
$634$	0	0
$635$	−14.1421	−0.561214
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	12.7279	0.501940	0.250970	−	0.967995i	$-0.419250\pi$
$643$	12.7279	0.501940	0.250970	+	0.967995i	$0.419250\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−31.1127	−1.22317	−0.611583	−	0.791180i	$-0.709467\pi$
$647$	−31.1127	−1.22317	−0.611583	+	0.791180i	$0.709467\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	11.3137	0.441390
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	9.89949	0.385046	0.192523	−	0.981292i	$-0.438333\pi$
$661$	9.89949	0.385046	0.192523	+	0.981292i	$0.438333\pi$
$662$	0	0
$663$	−16.9706	−0.659082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.00000	−0.154881
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	0	0
$675$	−16.9706	−0.653197
$676$	0	0
$677$	4.24264	0.163058	0.0815290	−	0.996671i	$-0.474020\pi$
$677$	4.24264	0.163058	0.0815290	+	0.996671i	$0.474020\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	−8.48528	−0.324206
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	−25.4558	−0.969790
$690$	0	0
$691$	−26.8701	−1.02219	−0.511093	−	0.859526i	$-0.670759\pi$
$691$	−26.8701	−1.02219	−0.511093	+	0.859526i	$0.670759\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.00000	−0.227593
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	25.4558	0.962828
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	−2.82843	−0.106676
$704$	0	0
$705$	−5.65685	−0.213049
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−16.9706	−0.635553
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.1127	1.16193
$718$	0	0
$719$	14.1421	0.527413	0.263706	−	0.964603i	$-0.415055\pi$
$719$	14.1421	0.527413	0.263706	+	0.964603i	$0.415055\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	12.0000	0.446285
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	36.7696	1.36371	0.681854	−	0.731489i	$-0.261174\pi$
$727$	36.7696	1.36371	0.681854	+	0.731489i	$0.261174\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	11.3137	0.418453
$732$	0	0
$733$	35.3553	1.30588	0.652940	−	0.757410i	$-0.273536\pi$
$733$	35.3553	1.30588	0.652940	+	0.757410i	$0.273536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	−8.48528	−0.311715
$742$	0	0
$743$	2.00000	0.0733729	0.0366864	−	0.999327i	$-0.488320\pi$
$743$	2.00000	0.0733729	0.0366864	+	0.999327i	$0.488320\pi$
$744$	0	0
$745$	25.4558	0.932630
$746$	0	0
$747$	−7.07107	−0.258717
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.0000	−0.656829	−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000	−0.656829	−0.328415	+	0.944534i	$0.606514\pi$
$752$	0	0
$753$	26.0000	0.947493
$754$	0	0
$755$	31.1127	1.13231
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.7990	−0.717713	−0.358856	−	0.933393i	$-0.616833\pi$
$761$	−19.7990	−0.717713	−0.358856	+	0.933393i	$0.616833\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	−42.0000	−1.51653
$768$	0	0
$769$	19.7990	0.713970	0.356985	−	0.934110i	$-0.383805\pi$
$769$	19.7990	0.713970	0.356985	+	0.934110i	$0.383805\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	41.0122	1.47511	0.737553	−	0.675289i	$-0.235981\pi$
$773$	41.0122	1.47511	0.737553	+	0.675289i	$0.235981\pi$
$774$	0	0
$775$	25.4558	0.914401
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.3137	−0.404319
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	49.4975	1.76439	0.882197	−	0.470880i	$-0.156064\pi$
$787$	49.4975	1.76439	0.882197	+	0.470880i	$0.156064\pi$
$788$	0	0
$789$	−33.9411	−1.20834
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	46.6690	1.65310	0.826551	−	0.562861i	$-0.190300\pi$
$797$	46.6690	1.65310	0.826551	+	0.562861i	$0.190300\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−16.9706	−0.599625
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−15.5563	−0.546257	−0.273129	−	0.961978i	$-0.588058\pi$
$811$	−15.5563	−0.546257	−0.273129	+	0.961978i	$0.588058\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	0	0
$815$	11.3137	0.396302
$816$	0	0
$817$	5.65685	0.197908
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−26.8701	−0.933236	−0.466618	−	0.884459i	$-0.654528\pi$
$829$	−26.8701	−0.933236	−0.466618	+	0.884459i	$0.654528\pi$
$830$	0	0
$831$	−36.7696	−1.27552
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−20.0000	−0.692129
$836$	0	0
$837$	−48.0000	−1.65912
$838$	0	0
$839$	−36.7696	−1.26943	−0.634713	−	0.772748i	$-0.718882\pi$
$839$	−36.7696	−1.26943	−0.634713	+	0.772748i	$0.718882\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−14.1421	−0.487081
$844$	0	0
$845$	7.07107	0.243252
$846$	0	0
$847$	0	0
$848$	0	0
$849$	30.0000	1.02960
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−49.4975	−1.69476	−0.847381	−	0.530986i	$-0.821822\pi$
$853$	−49.4975	−1.69476	−0.847381	+	0.530986i	$0.821822\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−31.1127	−1.06279	−0.531395	−	0.847124i	$-0.678332\pi$
$857$	−31.1127	−1.06279	−0.531395	+	0.847124i	$0.678332\pi$
$858$	0	0
$859$	−12.7279	−0.434271	−0.217136	−	0.976141i	$-0.569671\pi$
$859$	−12.7279	−0.434271	−0.217136	+	0.976141i	$0.569671\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	12.7279	0.432263
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−50.9117	−1.72508
$872$	0	0
$873$	14.1421	0.478639
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	−22.0000	−0.742042
$880$	0	0
$881$	−5.65685	−0.190584	−0.0952921	−	0.995449i	$-0.530379\pi$
$881$	−5.65685	−0.190584	−0.0952921	+	0.995449i	$0.530379\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	−19.7990	−0.665536
$886$	0	0
$887$	2.82843	0.0949693	0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843	0.0949693	0.0474846	+	0.998872i	$0.484879\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.00000	−0.133855
$894$	0	0
$895$	28.2843	0.945439
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	16.9706	0.566000
$900$	0	0
$901$	−16.9706	−0.565371
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	12.7279	0.422159
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	8.48528	0.280515
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	22.0000	0.724925
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	−14.1421	−0.464489
$928$	0	0
$929$	2.82843	0.0927977	0.0463988	−	0.998923i	$-0.485225\pi$
$929$	2.82843	0.0927977	0.0463988	+	0.998923i	$0.485225\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−40.0000	−1.30954
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39.5980	−1.29361	−0.646805	−	0.762656i	$-0.723895\pi$
$937$	−39.5980	−1.29361	−0.646805	+	0.762656i	$0.723895\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	−57.9828	−1.89018	−0.945092	−	0.326805i	$-0.894028\pi$
$941$	−57.9828	−1.89018	−0.945092	+	0.326805i	$0.894028\pi$
$942$	0	0
$943$	5.65685	0.184213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	48.0000	1.55815
$950$	0	0
$951$	−31.1127	−1.00890
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	33.9411	1.09831
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−16.0000	−0.515593
$964$	0	0
$965$	−22.6274	−0.728402
$966$	0	0
$967$	42.0000	1.35063	0.675314	−	0.737530i	$-0.264008\pi$
$967$	42.0000	1.35063	0.675314	+	0.737530i	$0.264008\pi$
$968$	0	0
$969$	−5.65685	−0.181724
$970$	0	0
$971$	43.8406	1.40691	0.703456	−	0.710739i	$-0.251639\pi$
$971$	43.8406	1.40691	0.703456	+	0.710739i	$0.251639\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−18.0000	−0.576461
$976$	0	0
$977$	14.0000	0.447900	0.223950	−	0.974601i	$-0.428105\pi$
$977$	14.0000	0.447900	0.223950	+	0.974601i	$0.428105\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	19.7990	0.631490	0.315745	−	0.948844i	$-0.397746\pi$
$983$	19.7990	0.631490	0.315745	+	0.948844i	$0.397746\pi$
$984$	0	0
$985$	−2.82843	−0.0901212
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−28.2843	−0.897574
$994$	0	0
$995$	−28.0000	−0.887660
$996$	0	0
$997$	−49.4975	−1.56760	−0.783800	−	0.621013i	$-0.786721\pi$
$997$	−49.4975	−1.56760	−0.783800	+	0.621013i	$0.786721\pi$
$998$	0	0
$999$	11.3137	0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6272.2.a.m.1.1	✓	2
4.3	odd	2		6272.2.a.o.1.2	yes	2
7.6	odd	2	inner	6272.2.a.m.1.2	yes	2
8.3	odd	2		6272.2.a.p.1.1	yes	2
8.5	even	2		6272.2.a.n.1.2	yes	2
28.27	even	2		6272.2.a.o.1.1	yes	2
56.13	odd	2		6272.2.a.n.1.1	yes	2
56.27	even	2		6272.2.a.p.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6272.2.a.m.1.1	✓	2	1.1	even	1	trivial
6272.2.a.m.1.2	yes	2	7.6	odd	2	inner
6272.2.a.n.1.1	yes	2	56.13	odd	2
6272.2.a.n.1.2	yes	2	8.5	even	2
6272.2.a.o.1.1	yes	2	28.27	even	2
6272.2.a.o.1.2	yes	2	4.3	odd	2
6272.2.a.p.1.1	yes	2	8.3	odd	2
6272.2.a.p.1.2	yes	2	56.27	even	2

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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:	\( N \)	\(=\)	\( 6272 = 2^{7} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6272.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} -4.24264 q^{13} -2.00000 q^{15} -2.82843 q^{17} -1.41421 q^{19} +2.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} -2.00000 q^{29} -8.48528 q^{31} +2.00000 q^{37} +6.00000 q^{39} +2.82843 q^{41} -4.00000 q^{43} -1.41421 q^{45} +2.82843 q^{47} +4.00000 q^{51} +6.00000 q^{53} +2.00000 q^{57} +9.89949 q^{59} -4.24264 q^{61} -6.00000 q^{65} +12.0000 q^{67} -2.82843 q^{69} -11.3137 q^{73} +4.24264 q^{75} +8.00000 q^{79} -5.00000 q^{81} +7.07107 q^{83} -4.00000 q^{85} +2.82843 q^{87} +16.9706 q^{89} +12.0000 q^{93} -2.00000 q^{95} -14.1421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 4 q^{15} + 4 q^{23} - 6 q^{25} - 4 q^{29} + 4 q^{37} + 12 q^{39} - 8 q^{43} + 8 q^{51} + 12 q^{53} + 4 q^{57} - 12 q^{65} + 24 q^{67} + 16 q^{79} - 10 q^{81} - 8 q^{85} + 24 q^{93} - 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^15 + 4 * q^23 - 6 * q^25 - 4 * q^29 + 4 * q^37 + 12 * q^39 - 8 * q^43 + 8 * q^51 + 12 * q^53 + 4 * q^57 - 12 * q^65 + 24 * q^67 + 16 * q^79 - 10 * q^81 - 8 * q^85 + 24 * q^93 - 4 * q^95

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