LMFDB - Embedded newform 8624.2.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
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:	no (minimal twist has level 1078)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} +1.00000 q^{11} +2.82843 q^{13} +2.00000 q^{15} -5.65685 q^{17} -2.82843 q^{19} +2.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} -6.00000 q^{29} +1.41421 q^{31} +1.41421 q^{33} -2.00000 q^{37} +4.00000 q^{39} +5.65685 q^{41} -4.00000 q^{43} -1.41421 q^{45} -4.24264 q^{47} -8.00000 q^{51} -12.0000 q^{53} +1.41421 q^{55} -4.00000 q^{57} -4.24264 q^{59} -5.65685 q^{61} +4.00000 q^{65} +2.00000 q^{67} +2.82843 q^{69} +10.0000 q^{71} -14.1421 q^{73} -4.24264 q^{75} +8.00000 q^{79} -5.00000 q^{81} +2.82843 q^{83} -8.00000 q^{85} -8.48528 q^{87} -7.07107 q^{89} +2.00000 q^{93} -4.00000 q^{95} -15.5563 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 2 q^{11} + 4 q^{15} + 4 q^{23} - 6 q^{25} - 12 q^{29} - 4 q^{37} + 8 q^{39} - 8 q^{43} - 16 q^{51} - 24 q^{53} - 8 q^{57} + 8 q^{65} + 4 q^{67} + 20 q^{71} + 16 q^{79} - 10 q^{81} - 16 q^{85} + 4 q^{93} - 8 q^{95} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 2 * q^11 + 4 * q^15 + 4 * q^23 - 6 * q^25 - 12 * q^29 - 4 * q^37 + 8 * q^39 - 8 * q^43 - 16 * q^51 - 24 * q^53 - 8 * q^57 + 8 * q^65 + 4 * q^67 + 20 * q^71 + 16 * q^79 - 10 * q^81 - 16 * q^85 + 4 * q^93 - 8 * q^95 - 2 * q^99

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	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	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
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−5.65685	−1.37199	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	−5.65685	−1.37199	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	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
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	1.41421	0.246183
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	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
$47$	−4.24264	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$47$	−4.24264	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−4.24264	−0.552345	−0.276172	−	0.961108i	$-0.589066\pi$
$59$	−4.24264	−0.552345	−0.276172	+	0.961108i	$0.589066\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	2.82843	0.340503
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−14.1421	−1.65521	−0.827606	−	0.561310i	$-0.810298\pi$
$73$	−14.1421	−1.65521	−0.827606	+	0.561310i	$0.810298\pi$
$74$	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
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	−8.48528	−0.909718
$88$	0	0
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−15.5563	−1.57951	−0.789754	−	0.613424i	$-0.789792\pi$
$97$	−15.5563	−1.57951	−0.789754	+	0.613424i	$0.789792\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	2.82843	0.281439	0.140720	−	0.990050i	$-0.455058\pi$
$101$	2.82843	0.281439	0.140720	+	0.990050i	$0.455058\pi$
$102$	0	0
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	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
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	2.82843	0.263752
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	−5.65685	−0.498058
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.00000	−0.688530
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	−8.48528	−0.704664
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	5.65685	0.457330
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	18.3848	1.46726	0.733632	−	0.679546i	$-0.237823\pi$
$157$	18.3848	1.46726	0.733632	+	0.679546i	$0.237823\pi$
$158$	0	0
$159$	−16.9706	−1.34585
$160$	0	0
$161$	0	0
$162$	0	0
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	14.1421	1.07521	0.537603	−	0.843198i	$-0.319330\pi$
$173$	14.1421	1.07521	0.537603	+	0.843198i	$0.319330\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	12.7279	0.946059	0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279	0.946059	0.473029	+	0.881047i	$0.343160\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	−2.82843	−0.207950
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	5.65685	0.405096
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−21.2132	−1.50376	−0.751882	−	0.659298i	$-0.770854\pi$
$199$	−21.2132	−1.50376	−0.751882	+	0.659298i	$0.770854\pi$
$200$	0	0
$201$	2.82843	0.199502
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	0	0
$213$	14.1421	0.969003
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.0000	−1.35147
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	15.5563	1.04173	0.520865	−	0.853639i	$-0.325609\pi$
$223$	15.5563	1.04173	0.520865	+	0.853639i	$0.325609\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	28.2843	1.87729	0.938647	−	0.344881i	$-0.112081\pi$
$227$	28.2843	1.87729	0.938647	+	0.344881i	$0.112081\pi$
$228$	0	0
$229$	−4.24264	−0.280362	−0.140181	−	0.990126i	$-0.544768\pi$
$229$	−4.24264	−0.280362	−0.140181	+	0.990126i	$0.544768\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	11.3137	0.734904
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−5.65685	−0.364390	−0.182195	−	0.983262i	$-0.558320\pi$
$241$	−5.65685	−0.364390	−0.182195	+	0.983262i	$0.558320\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	24.0416	1.51749	0.758747	−	0.651385i	$-0.225812\pi$
$251$	24.0416	1.51749	0.758747	+	0.651385i	$0.225812\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	−11.3137	−0.708492
$256$	0	0
$257$	−9.89949	−0.617514	−0.308757	−	0.951141i	$-0.599913\pi$
$257$	−9.89949	−0.617514	−0.308757	+	0.951141i	$0.599913\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−16.9706	−1.04249
$266$	0	0
$267$	−10.0000	−0.611990
$268$	0	0
$269$	−24.0416	−1.46584	−0.732922	−	0.680313i	$-0.761844\pi$
$269$	−24.0416	−1.46584	−0.732922	+	0.680313i	$0.761844\pi$
$270$	0	0
$271$	−8.48528	−0.515444	−0.257722	−	0.966219i	$-0.582972\pi$
$271$	−8.48528	−0.515444	−0.257722	+	0.966219i	$0.582972\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−16.9706	−1.00880	−0.504398	−	0.863472i	$-0.668285\pi$
$283$	−16.9706	−1.00880	−0.504398	+	0.863472i	$0.668285\pi$
$284$	0	0
$285$	−5.65685	−0.335083
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	−22.0000	−1.28966
$292$	0	0
$293$	16.9706	0.991431	0.495715	−	0.868485i	$-0.334906\pi$
$293$	16.9706	0.991431	0.495715	+	0.868485i	$0.334906\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.00000	0.229794
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	11.3137	0.645707	0.322854	−	0.946449i	$-0.395358\pi$
$307$	11.3137	0.645707	0.322854	+	0.946449i	$0.395358\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	9.89949	0.561349	0.280674	−	0.959803i	$-0.409442\pi$
$311$	9.89949	0.561349	0.280674	+	0.959803i	$0.409442\pi$
$312$	0	0
$313$	26.8701	1.51879	0.759393	−	0.650633i	$-0.225496\pi$
$313$	26.8701	1.51879	0.759393	+	0.650633i	$0.225496\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	28.2843	1.57867
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−8.48528	−0.470679
$326$	0	0
$327$	14.1421	0.782062
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	2.82843	0.154533
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−25.4558	−1.38257
$340$	0	0
$341$	1.41421	0.0765840
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−5.65685	−0.302804	−0.151402	−	0.988472i	$-0.548379\pi$
$349$	−5.65685	−0.302804	−0.151402	+	0.988472i	$0.548379\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	0	0
$355$	14.1421	0.750587
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	1.41421	0.0742270
$364$	0	0
$365$	−20.0000	−1.04685
$366$	0	0
$367$	−29.6985	−1.55025	−0.775124	−	0.631809i	$-0.782313\pi$
$367$	−29.6985	−1.55025	−0.775124	+	0.631809i	$0.782313\pi$
$368$	0	0
$369$	−5.65685	−0.294484
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	−16.0000	−0.826236
$376$	0	0
$377$	−16.9706	−0.874028
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	−16.9706	−0.869428
$382$	0	0
$383$	−26.8701	−1.37300	−0.686498	−	0.727132i	$-0.740853\pi$
$383$	−26.8701	−1.37300	−0.686498	+	0.727132i	$0.740853\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	0	0
$391$	−11.3137	−0.572159
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	−24.0416	−1.20661	−0.603307	−	0.797509i	$-0.706151\pi$
$397$	−24.0416	−1.20661	−0.603307	+	0.797509i	$0.706151\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−7.07107	−0.351364
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	31.1127	1.53842	0.769212	−	0.638994i	$-0.220649\pi$
$409$	31.1127	1.53842	0.769212	+	0.638994i	$0.220649\pi$
$410$	0	0
$411$	−2.82843	−0.139516
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	12.7279	0.621800	0.310900	−	0.950443i	$-0.399370\pi$
$419$	12.7279	0.621800	0.310900	+	0.950443i	$0.399370\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	4.24264	0.206284
$424$	0	0
$425$	16.9706	0.823193
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	38.1838	1.83499	0.917497	−	0.397742i	$-0.130206\pi$
$433$	38.1838	1.83499	0.917497	+	0.397742i	$0.130206\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	0	0
$437$	−5.65685	−0.270604
$438$	0	0
$439$	25.4558	1.21494	0.607471	−	0.794342i	$-0.292184\pi$
$439$	25.4558	1.21494	0.607471	+	0.794342i	$0.292184\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−14.1421	−0.668900
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	5.65685	0.266371
$452$	0	0
$453$	−22.6274	−1.06313
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	32.0000	1.49363
$460$	0	0
$461$	28.2843	1.31733	0.658665	−	0.752436i	$-0.271121\pi$
$461$	28.2843	1.31733	0.658665	+	0.752436i	$0.271121\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	2.82843	0.131165
$466$	0	0
$467$	−4.24264	−0.196326	−0.0981630	−	0.995170i	$-0.531297\pi$
$467$	−4.24264	−0.196326	−0.0981630	+	0.995170i	$0.531297\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	26.0000	1.19802
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	−25.4558	−1.16311	−0.581554	−	0.813508i	$-0.697555\pi$
$479$	−25.4558	−1.16311	−0.581554	+	0.813508i	$0.697555\pi$
$480$	0	0
$481$	−5.65685	−0.257930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.0000	−0.998969
$486$	0	0
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	0	0
$489$	31.1127	1.40696
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	33.9411	1.52863
$494$	0	0
$495$	−1.41421	−0.0635642
$496$	0	0
$497$	0	0
$498$	0	0
$499$	30.0000	1.34298	0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000	1.34298	0.671492	+	0.741012i	$0.265654\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	−19.7990	−0.882793	−0.441397	−	0.897312i	$-0.645517\pi$
$503$	−19.7990	−0.882793	−0.441397	+	0.897312i	$0.645517\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	−7.07107	−0.314037
$508$	0	0
$509$	4.24264	0.188052	0.0940259	−	0.995570i	$-0.470026\pi$
$509$	4.24264	0.188052	0.0940259	+	0.995570i	$0.470026\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	16.0000	0.706417
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	−4.24264	−0.186591
$518$	0	0
$519$	20.0000	0.877903
$520$	0	0
$521$	−15.5563	−0.681536	−0.340768	−	0.940147i	$-0.610687\pi$
$521$	−15.5563	−0.681536	−0.340768	+	0.940147i	$0.610687\pi$
$522$	0	0
$523$	16.9706	0.742071	0.371035	−	0.928619i	$-0.379003\pi$
$523$	16.9706	0.742071	0.371035	+	0.928619i	$0.379003\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	4.24264	0.184115
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	28.2843	1.22284
$536$	0	0
$537$	−28.2843	−1.22056
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	14.1421	0.605783
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	5.65685	0.241429
$550$	0	0
$551$	16.9706	0.722970
$552$	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
$559$	−11.3137	−0.478519
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	14.1421	0.596020	0.298010	−	0.954563i	$-0.403677\pi$
$563$	14.1421	0.596020	0.298010	+	0.954563i	$0.403677\pi$
$564$	0	0
$565$	−25.4558	−1.07094
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−29.6985	−1.23636	−0.618182	−	0.786035i	$-0.712131\pi$
$577$	−29.6985	−1.23636	−0.618182	+	0.786035i	$0.712131\pi$
$578$	0	0
$579$	8.48528	0.352636
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	1.41421	0.0583708	0.0291854	−	0.999574i	$-0.490709\pi$
$587$	1.41421	0.0583708	0.0291854	+	0.999574i	$0.490709\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	25.4558	1.04711
$592$	0	0
$593$	19.7990	0.813047	0.406524	−	0.913640i	$-0.366741\pi$
$593$	19.7990	0.813047	0.406524	+	0.913640i	$0.366741\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−30.0000	−1.22782
$598$	0	0
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	0	0
$601$	45.2548	1.84598	0.922992	−	0.384820i	$-0.125737\pi$
$601$	45.2548	1.84598	0.922992	+	0.384820i	$0.125737\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	1.41421	0.0574960
$606$	0	0
$607$	33.9411	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	11.3137	0.456213
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−12.7279	−0.511578	−0.255789	−	0.966733i	$-0.582335\pi$
$619$	−12.7279	−0.511578	−0.255789	+	0.966733i	$0.582335\pi$
$620$	0	0
$621$	−11.3137	−0.454003
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	11.3137	0.451107
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−39.5980	−1.57388
$634$	0	0
$635$	−16.9706	−0.673456
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	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
$647$	−41.0122	−1.61236	−0.806178	−	0.591673i	$-0.798468\pi$
$647$	−41.0122	−1.61236	−0.806178	+	0.591673i	$0.798468\pi$
$648$	0	0
$649$	−4.24264	−0.166538
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−48.0000	−1.87839	−0.939193	−	0.343391i	$-0.888424\pi$
$653$	−48.0000	−1.87839	−0.939193	+	0.343391i	$0.888424\pi$
$654$	0	0
$655$	−16.0000	−0.625172
$656$	0	0
$657$	14.1421	0.551737
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	38.1838	1.48518	0.742588	−	0.669748i	$-0.233598\pi$
$661$	38.1838	1.48518	0.742588	+	0.669748i	$0.233598\pi$
$662$	0	0
$663$	−22.6274	−0.878776
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	22.0000	0.850569
$670$	0	0
$671$	−5.65685	−0.218380
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	16.9706	0.653197
$676$	0	0
$677$	31.1127	1.19576	0.597879	−	0.801586i	$-0.296010\pi$
$677$	31.1127	1.19576	0.597879	+	0.801586i	$0.296010\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	40.0000	1.53280
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−2.82843	−0.108069
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	−33.9411	−1.29305
$690$	0	0
$691$	32.5269	1.23738	0.618691	−	0.785634i	$-0.287663\pi$
$691$	32.5269	1.23738	0.618691	+	0.785634i	$0.287663\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−32.0000	−1.21209
$698$	0	0
$699$	8.48528	0.320943
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	5.65685	0.213352
$704$	0	0
$705$	−8.48528	−0.319574
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	2.82843	0.105925
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	22.6274	0.845036
$718$	0	0
$719$	15.5563	0.580154	0.290077	−	0.957003i	$-0.406319\pi$
$719$	15.5563	0.580154	0.290077	+	0.957003i	$0.406319\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8.00000	−0.297523
$724$	0	0
$725$	18.0000	0.668503
$726$	0	0
$727$	32.5269	1.20636	0.603178	−	0.797606i	$-0.293901\pi$
$727$	32.5269	1.20636	0.603178	+	0.797606i	$0.293901\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	22.6274	0.836905
$732$	0	0
$733$	−16.9706	−0.626822	−0.313411	−	0.949618i	$-0.601472\pi$
$733$	−16.9706	−0.626822	−0.313411	+	0.949618i	$0.601472\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.00000	0.0736709
$738$	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
$741$	−11.3137	−0.415619
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−14.1421	−0.518128
$746$	0	0
$747$	−2.82843	−0.103487
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	34.0000	1.23903
$754$	0	0
$755$	−22.6274	−0.823496
$756$	0	0
$757$	−44.0000	−1.59921	−0.799604	−	0.600528i	$-0.794957\pi$
$757$	−44.0000	−1.59921	−0.799604	+	0.600528i	$0.794957\pi$
$758$	0	0
$759$	2.82843	0.102665
$760$	0	0
$761$	25.4558	0.922774	0.461387	−	0.887199i	$-0.347352\pi$
$761$	25.4558	0.922774	0.461387	+	0.887199i	$0.347352\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	8.48528	0.305987	0.152994	−	0.988227i	$-0.451109\pi$
$769$	8.48528	0.305987	0.152994	+	0.988227i	$0.451109\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−1.41421	−0.0508657	−0.0254329	−	0.999677i	$-0.508096\pi$
$773$	−1.41421	−0.0508657	−0.0254329	+	0.999677i	$0.508096\pi$
$774$	0	0
$775$	−4.24264	−0.152400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	33.9411	1.21296
$784$	0	0
$785$	26.0000	0.927980
$786$	0	0
$787$	−42.4264	−1.51234	−0.756169	−	0.654376i	$-0.772931\pi$
$787$	−42.4264	−1.51234	−0.756169	+	0.654376i	$0.772931\pi$
$788$	0	0
$789$	−33.9411	−1.20834
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−16.0000	−0.568177
$794$	0	0
$795$	−24.0000	−0.851192
$796$	0	0
$797$	−18.3848	−0.651222	−0.325611	−	0.945504i	$-0.605570\pi$
$797$	−18.3848	−0.651222	−0.325611	+	0.945504i	$0.605570\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	7.07107	0.249844
$802$	0	0
$803$	−14.1421	−0.499065
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−34.0000	−1.19686
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	0	0
$815$	31.1127	1.08983
$816$	0	0
$817$	11.3137	0.395817
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	−4.24264	−0.147710
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	−35.3553	−1.22794	−0.613971	−	0.789329i	$-0.710429\pi$
$829$	−35.3553	−1.22794	−0.613971	+	0.789329i	$0.710429\pi$
$830$	0	0
$831$	−14.1421	−0.490585
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	43.8406	1.51355	0.756773	−	0.653678i	$-0.226775\pi$
$839$	43.8406	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	25.4558	0.876746
$844$	0	0
$845$	−7.07107	−0.243252
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−2.82843	−0.0968435	−0.0484218	−	0.998827i	$-0.515419\pi$
$853$	−2.82843	−0.0968435	−0.0484218	+	0.998827i	$0.515419\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−8.48528	−0.289852	−0.144926	−	0.989443i	$-0.546294\pi$
$857$	−8.48528	−0.289852	−0.144926	+	0.989443i	$0.546294\pi$
$858$	0	0
$859$	−38.1838	−1.30281	−0.651407	−	0.758729i	$-0.725821\pi$
$859$	−38.1838	−1.30281	−0.651407	+	0.758729i	$0.725821\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	21.2132	0.720438
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	5.65685	0.191675
$872$	0	0
$873$	15.5563	0.526503
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	4.24264	0.142938	0.0714691	−	0.997443i	$-0.477231\pi$
$881$	4.24264	0.142938	0.0714691	+	0.997443i	$0.477231\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	−8.48528	−0.285230
$886$	0	0
$887$	−48.0833	−1.61448	−0.807239	−	0.590225i	$-0.799039\pi$
$887$	−48.0833	−1.61448	−0.807239	+	0.590225i	$0.799039\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	−28.2843	−0.945439
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	−8.48528	−0.283000
$900$	0	0
$901$	67.8823	2.26149
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	0	0
$909$	−2.82843	−0.0938130
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	2.82843	0.0936073
$914$	0	0
$915$	−11.3137	−0.374020
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	28.2843	0.930988
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−4.24264	−0.139347
$928$	0	0
$929$	−15.5563	−0.510387	−0.255194	−	0.966890i	$-0.582139\pi$
$929$	−15.5563	−0.510387	−0.255194	+	0.966890i	$0.582139\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	14.0000	0.458339
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−25.4558	−0.831606	−0.415803	−	0.909455i	$-0.636499\pi$
$937$	−25.4558	−0.831606	−0.415803	+	0.909455i	$0.636499\pi$
$938$	0	0
$939$	38.0000	1.24008
$940$	0	0
$941$	22.6274	0.737633	0.368816	−	0.929502i	$-0.379763\pi$
$941$	22.6274	0.737633	0.368816	+	0.929502i	$0.379763\pi$
$942$	0	0
$943$	11.3137	0.368425
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	−36.7696	−1.19233
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.48528	−0.274290
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	−20.0000	−0.644491
$964$	0	0
$965$	8.48528	0.273151
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	0	0
$969$	22.6274	0.726897
$970$	0	0
$971$	−26.8701	−0.862301	−0.431151	−	0.902280i	$-0.641892\pi$
$971$	−26.8701	−0.862301	−0.431151	+	0.902280i	$0.641892\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	−7.07107	−0.225992
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−57.9828	−1.84936	−0.924681	−	0.380742i	$-0.875669\pi$
$983$	−57.9828	−1.84936	−0.924681	+	0.380742i	$0.875669\pi$
$984$	0	0
$985$	25.4558	0.811091
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−50.0000	−1.58830	−0.794151	−	0.607720i	$-0.792084\pi$
$991$	−50.0000	−1.58830	−0.794151	+	0.607720i	$0.792084\pi$
$992$	0	0
$993$	−39.5980	−1.25660
$994$	0	0
$995$	−30.0000	−0.951064
$996$	0	0
$997$	59.3970	1.88112	0.940560	−	0.339626i	$-0.110301\pi$
$997$	59.3970	1.88112	0.940560	+	0.339626i	$0.110301\pi$
$998$	0	0
$999$	11.3137	0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bw.1.2		2
4.3	odd	2		1078.2.a.q.1.1	✓	2
7.6	odd	2	inner	8624.2.a.bw.1.1		2
12.11	even	2		9702.2.a.dq.1.1		2
28.3	even	6		1078.2.e.s.177.1		4
28.11	odd	6		1078.2.e.s.177.2		4
28.19	even	6		1078.2.e.s.67.1		4
28.23	odd	6		1078.2.e.s.67.2		4
28.27	even	2		1078.2.a.q.1.2	yes	2
84.83	odd	2		9702.2.a.dq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.q.1.1	✓	2	4.3	odd	2
1078.2.a.q.1.2	yes	2	28.27	even	2
1078.2.e.s.67.1		4	28.19	even	6
1078.2.e.s.67.2		4	28.23	odd	6
1078.2.e.s.177.1		4	28.3	even	6
1078.2.e.s.177.2		4	28.11	odd	6
8624.2.a.bw.1.1		2	7.6	odd	2	inner
8624.2.a.bw.1.2		2	1.1	even	1	trivial
9702.2.a.dq.1.1		2	12.11	even	2
9702.2.a.dq.1.2		2	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} +1.00000 q^{11} +2.82843 q^{13} +2.00000 q^{15} -5.65685 q^{17} -2.82843 q^{19} +2.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} -6.00000 q^{29} +1.41421 q^{31} +1.41421 q^{33} -2.00000 q^{37} +4.00000 q^{39} +5.65685 q^{41} -4.00000 q^{43} -1.41421 q^{45} -4.24264 q^{47} -8.00000 q^{51} -12.0000 q^{53} +1.41421 q^{55} -4.00000 q^{57} -4.24264 q^{59} -5.65685 q^{61} +4.00000 q^{65} +2.00000 q^{67} +2.82843 q^{69} +10.0000 q^{71} -14.1421 q^{73} -4.24264 q^{75} +8.00000 q^{79} -5.00000 q^{81} +2.82843 q^{83} -8.00000 q^{85} -8.48528 q^{87} -7.07107 q^{89} +2.00000 q^{93} -4.00000 q^{95} -15.5563 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 2 q^{11} + 4 q^{15} + 4 q^{23} - 6 q^{25} - 12 q^{29} - 4 q^{37} + 8 q^{39} - 8 q^{43} - 16 q^{51} - 24 q^{53} - 8 q^{57} + 8 q^{65} + 4 q^{67} + 20 q^{71} + 16 q^{79} - 10 q^{81} - 16 q^{85} + 4 q^{93} - 8 q^{95} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 2 * q^11 + 4 * q^15 + 4 * q^23 - 6 * q^25 - 12 * q^29 - 4 * q^37 + 8 * q^39 - 8 * q^43 - 16 * q^51 - 24 * q^53 - 8 * q^57 + 8 * q^65 + 4 * q^67 + 20 * q^71 + 16 * q^79 - 10 * q^81 - 16 * q^85 + 4 * q^93 - 8 * q^95 - 2 * q^99

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