LMFDB - Embedded newform 4624.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4624,2,Mod(1,4624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4624, 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("4624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4624 = 2^{4} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4624.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:	$36.9228258946$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 289)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	4624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.879385 q^{3} -2.34730 q^{5} +1.87939 q^{7} -2.22668 q^{9} +O(q^{10})$	$q-0.879385 q^{3} -2.34730 q^{5} +1.87939 q^{7} -2.22668 q^{9} +5.06418 q^{11} +4.71688 q^{13} +2.06418 q^{15} -0.347296 q^{19} -1.65270 q^{21} +1.77332 q^{23} +0.509800 q^{25} +4.59627 q^{27} +2.22668 q^{29} -1.94356 q^{31} -4.45336 q^{33} -4.41147 q^{35} -6.17024 q^{37} -4.14796 q^{39} +5.17024 q^{41} +1.47565 q^{43} +5.22668 q^{45} +8.53209 q^{47} -3.46791 q^{49} -10.4534 q^{53} -11.8871 q^{55} +0.305407 q^{57} -5.00774 q^{59} -0.184793 q^{61} -4.18479 q^{63} -11.0719 q^{65} +2.44831 q^{67} -1.55943 q^{69} +9.92127 q^{71} +10.9017 q^{73} -0.448311 q^{75} +9.51754 q^{77} -4.43376 q^{79} +2.63816 q^{81} -13.5817 q^{83} -1.95811 q^{87} -6.32770 q^{89} +8.86484 q^{91} +1.70914 q^{93} +0.815207 q^{95} +9.27631 q^{97} -11.2763 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 6 q^{5}+O(q^{10}) $ 3 * q + 3 * q^3 - 6 * q^5	$ 3 q + 3 q^{3} - 6 q^{5} + 6 q^{11} + 6 q^{13} - 3 q^{15} - 6 q^{21} + 12 q^{23} + 3 q^{25} + 9 q^{31} - 3 q^{35} + 3 q^{37} + 3 q^{39} - 6 q^{41} - 15 q^{43} + 9 q^{45} + 21 q^{47} - 15 q^{49} - 18 q^{53} - 6 q^{55} + 3 q^{57} + 9 q^{59} + 3 q^{61} - 9 q^{63} + 9 q^{67} + 21 q^{69} + 21 q^{71} + 21 q^{73} - 3 q^{75} + 6 q^{77} + 3 q^{79} - 9 q^{81} - 9 q^{83} - 9 q^{87} - 15 q^{89} + 3 q^{91} + 21 q^{93} + 6 q^{95} - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^5 + 6 * q^11 + 6 * q^13 - 3 * q^15 - 6 * q^21 + 12 * q^23 + 3 * q^25 + 9 * q^31 - 3 * q^35 + 3 * q^37 + 3 * q^39 - 6 * q^41 - 15 * q^43 + 9 * q^45 + 21 * q^47 - 15 * q^49 - 18 * q^53 - 6 * q^55 + 3 * q^57 + 9 * q^59 + 3 * q^61 - 9 * q^63 + 9 * q^67 + 21 * q^69 + 21 * q^71 + 21 * q^73 - 3 * q^75 + 6 * q^77 + 3 * q^79 - 9 * q^81 - 9 * q^83 - 9 * q^87 - 15 * q^89 + 3 * q^91 + 21 * q^93 + 6 * q^95 - 6 * 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
$3$	−0.879385	−0.507713	−0.253857	−	0.967242i	$-0.581699\pi$
$3$	−0.879385	−0.507713	−0.253857	+	0.967242i	$0.581699\pi$
$4$	0	0
$5$	−2.34730	−1.04974	−0.524871	−	0.851182i	$-0.675887\pi$
$5$	−2.34730	−1.04974	−0.524871	+	0.851182i	$0.675887\pi$
$6$	0	0
$7$	1.87939	0.710341	0.355170	−	0.934802i	$-0.384423\pi$
$7$	1.87939	0.710341	0.355170	+	0.934802i	$0.384423\pi$
$8$	0	0
$9$	−2.22668	−0.742227
$10$	0	0
$11$	5.06418	1.52691	0.763454	−	0.645863i	$-0.223502\pi$
$11$	5.06418	1.52691	0.763454	+	0.645863i	$0.223502\pi$
$12$	0	0
$13$	4.71688	1.30823	0.654114	−	0.756396i	$-0.273042\pi$
$13$	4.71688	1.30823	0.654114	+	0.756396i	$0.273042\pi$
$14$	0	0
$15$	2.06418	0.532968
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.347296	−0.0796752	−0.0398376	−	0.999206i	$-0.512684\pi$
$19$	−0.347296	−0.0796752	−0.0398376	+	0.999206i	$0.512684\pi$
$20$	0	0
$21$	−1.65270	−0.360650
$22$	0	0
$23$	1.77332	0.369762	0.184881	−	0.982761i	$-0.440810\pi$
$23$	1.77332	0.369762	0.184881	+	0.982761i	$0.440810\pi$
$24$	0	0
$25$	0.509800	0.101960
$26$	0	0
$27$	4.59627	0.884552
$28$	0	0
$29$	2.22668	0.413484	0.206742	−	0.978395i	$-0.433714\pi$
$29$	2.22668	0.413484	0.206742	+	0.978395i	$0.433714\pi$
$30$	0	0
$31$	−1.94356	−0.349074	−0.174537	−	0.984651i	$-0.555843\pi$
$31$	−1.94356	−0.349074	−0.174537	+	0.984651i	$0.555843\pi$
$32$	0	0
$33$	−4.45336	−0.775231
$34$	0	0
$35$	−4.41147	−0.745675
$36$	0	0
$37$	−6.17024	−1.01438	−0.507191	−	0.861834i	$-0.669316\pi$
$37$	−6.17024	−1.01438	−0.507191	+	0.861834i	$0.669316\pi$
$38$	0	0
$39$	−4.14796	−0.664205
$40$	0	0
$41$	5.17024	0.807457	0.403728	−	0.914879i	$-0.367714\pi$
$41$	5.17024	0.807457	0.403728	+	0.914879i	$0.367714\pi$
$42$	0	0
$43$	1.47565	0.225035	0.112517	−	0.993650i	$-0.464109\pi$
$43$	1.47565	0.225035	0.112517	+	0.993650i	$0.464109\pi$
$44$	0	0
$45$	5.22668	0.779148
$46$	0	0
$47$	8.53209	1.24453	0.622267	−	0.782805i	$-0.286212\pi$
$47$	8.53209	1.24453	0.622267	+	0.782805i	$0.286212\pi$
$48$	0	0
$49$	−3.46791	−0.495416
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.4534	−1.43588	−0.717940	−	0.696105i	$-0.754915\pi$
$53$	−10.4534	−1.43588	−0.717940	+	0.696105i	$0.754915\pi$
$54$	0	0
$55$	−11.8871	−1.60286
$56$	0	0
$57$	0.305407	0.0404522
$58$	0	0
$59$	−5.00774	−0.651952	−0.325976	−	0.945378i	$-0.605693\pi$
$59$	−5.00774	−0.651952	−0.325976	+	0.945378i	$0.605693\pi$
$60$	0	0
$61$	−0.184793	−0.0236603	−0.0118301	−	0.999930i	$-0.503766\pi$
$61$	−0.184793	−0.0236603	−0.0118301	+	0.999930i	$0.503766\pi$
$62$	0	0
$63$	−4.18479	−0.527234
$64$	0	0
$65$	−11.0719	−1.37330
$66$	0	0
$67$	2.44831	0.299109	0.149554	−	0.988754i	$-0.452216\pi$
$67$	2.44831	0.299109	0.149554	+	0.988754i	$0.452216\pi$
$68$	0	0
$69$	−1.55943	−0.187733
$70$	0	0
$71$	9.92127	1.17744	0.588719	−	0.808338i	$-0.299632\pi$
$71$	9.92127	1.17744	0.588719	+	0.808338i	$0.299632\pi$
$72$	0	0
$73$	10.9017	1.27594	0.637972	−	0.770059i	$-0.279773\pi$
$73$	10.9017	1.27594	0.637972	+	0.770059i	$0.279773\pi$
$74$	0	0
$75$	−0.448311	−0.0517665
$76$	0	0
$77$	9.51754	1.08462
$78$	0	0
$79$	−4.43376	−0.498837	−0.249419	−	0.968396i	$-0.580240\pi$
$79$	−4.43376	−0.498837	−0.249419	+	0.968396i	$0.580240\pi$
$80$	0	0
$81$	2.63816	0.293128
$82$	0	0
$83$	−13.5817	−1.49079	−0.745394	−	0.666625i	$-0.767738\pi$
$83$	−13.5817	−1.49079	−0.745394	+	0.666625i	$0.767738\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.95811	−0.209932
$88$	0	0
$89$	−6.32770	−0.670734	−0.335367	−	0.942087i	$-0.608860\pi$
$89$	−6.32770	−0.670734	−0.335367	+	0.942087i	$0.608860\pi$
$90$	0	0
$91$	8.86484	0.929287
$92$	0	0
$93$	1.70914	0.177230
$94$	0	0
$95$	0.815207	0.0836385
$96$	0	0
$97$	9.27631	0.941867	0.470933	−	0.882169i	$-0.343917\pi$
$97$	9.27631	0.941867	0.470933	+	0.882169i	$0.343917\pi$
$98$	0	0
$99$	−11.2763	−1.13331
$100$	0	0
$101$	−7.04963	−0.701464	−0.350732	−	0.936476i	$-0.614067\pi$
$101$	−7.04963	−0.701464	−0.350732	+	0.936476i	$0.614067\pi$
$102$	0	0
$103$	−5.29860	−0.522087	−0.261043	−	0.965327i	$-0.584067\pi$
$103$	−5.29860	−0.522087	−0.261043	+	0.965327i	$0.584067\pi$
$104$	0	0
$105$	3.87939	0.378589
$106$	0	0
$107$	15.5963	1.50775	0.753874	−	0.657019i	$-0.228183\pi$
$107$	15.5963	1.50775	0.753874	+	0.657019i	$0.228183\pi$
$108$	0	0
$109$	1.87164	0.179271	0.0896355	−	0.995975i	$-0.471430\pi$
$109$	1.87164	0.179271	0.0896355	+	0.995975i	$0.471430\pi$
$110$	0	0
$111$	5.42602	0.515015
$112$	0	0
$113$	−12.1138	−1.13957	−0.569786	−	0.821793i	$-0.692974\pi$
$113$	−12.1138	−1.13957	−0.569786	+	0.821793i	$0.692974\pi$
$114$	0	0
$115$	−4.16250	−0.388155
$116$	0	0
$117$	−10.5030	−0.971002
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.6459	1.33145
$122$	0	0
$123$	−4.54664	−0.409956
$124$	0	0
$125$	10.5398	0.942711
$126$	0	0
$127$	11.5398	1.02399	0.511997	−	0.858987i	$-0.328906\pi$
$127$	11.5398	1.02399	0.511997	+	0.858987i	$0.328906\pi$
$128$	0	0
$129$	−1.29767	−0.114253
$130$	0	0
$131$	19.3746	1.69277	0.846385	−	0.532572i	$-0.178774\pi$
$131$	19.3746	1.69277	0.846385	+	0.532572i	$0.178774\pi$
$132$	0	0
$133$	−0.652704	−0.0565966
$134$	0	0
$135$	−10.7888	−0.928552
$136$	0	0
$137$	−0.448311	−0.0383018	−0.0191509	−	0.999817i	$-0.506096\pi$
$137$	−0.448311	−0.0383018	−0.0191509	+	0.999817i	$0.506096\pi$
$138$	0	0
$139$	−11.6800	−0.990688	−0.495344	−	0.868697i	$-0.664958\pi$
$139$	−11.6800	−0.990688	−0.495344	+	0.868697i	$0.664958\pi$
$140$	0	0
$141$	−7.50299	−0.631866
$142$	0	0
$143$	23.8871	1.99754
$144$	0	0
$145$	−5.22668	−0.434052
$146$	0	0
$147$	3.04963	0.251529
$148$	0	0
$149$	8.46791	0.693718	0.346859	−	0.937917i	$-0.387248\pi$
$149$	8.46791	0.693718	0.346859	+	0.937917i	$0.387248\pi$
$150$	0	0
$151$	13.7665	1.12030	0.560151	−	0.828390i	$-0.310743\pi$
$151$	13.7665	1.12030	0.560151	+	0.828390i	$0.310743\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.56212	0.366438
$156$	0	0
$157$	−17.9786	−1.43485	−0.717426	−	0.696635i	$-0.754680\pi$
$157$	−17.9786	−1.43485	−0.717426	+	0.696635i	$0.754680\pi$
$158$	0	0
$159$	9.19253	0.729015
$160$	0	0
$161$	3.33275	0.262657
$162$	0	0
$163$	7.23442	0.566644	0.283322	−	0.959025i	$-0.408564\pi$
$163$	7.23442	0.566644	0.283322	+	0.959025i	$0.408564\pi$
$164$	0	0
$165$	10.4534	0.813793
$166$	0	0
$167$	5.10607	0.395119	0.197560	−	0.980291i	$-0.436698\pi$
$167$	5.10607	0.395119	0.197560	+	0.980291i	$0.436698\pi$
$168$	0	0
$169$	9.24897	0.711459
$170$	0	0
$171$	0.773318	0.0591371
$172$	0	0
$173$	12.8256	0.975115	0.487558	−	0.873091i	$-0.337888\pi$
$173$	12.8256	0.975115	0.487558	+	0.873091i	$0.337888\pi$
$174$	0	0
$175$	0.958111	0.0724264
$176$	0	0
$177$	4.40373	0.331005
$178$	0	0
$179$	−4.27126	−0.319249	−0.159624	−	0.987178i	$-0.551028\pi$
$179$	−4.27126	−0.319249	−0.159624	+	0.987178i	$0.551028\pi$
$180$	0	0
$181$	−0.462859	−0.0344040	−0.0172020	−	0.999852i	$-0.505476\pi$
$181$	−0.462859	−0.0344040	−0.0172020	+	0.999852i	$0.505476\pi$
$182$	0	0
$183$	0.162504	0.0120126
$184$	0	0
$185$	14.4834	1.06484
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.63816	0.628333
$190$	0	0
$191$	1.43107	0.103549	0.0517745	−	0.998659i	$-0.483512\pi$
$191$	1.43107	0.103549	0.0517745	+	0.998659i	$0.483512\pi$
$192$	0	0
$193$	24.7246	1.77972	0.889859	−	0.456236i	$-0.150803\pi$
$193$	24.7246	1.77972	0.889859	+	0.456236i	$0.150803\pi$
$194$	0	0
$195$	9.73648	0.697244
$196$	0	0
$197$	11.7615	0.837969	0.418985	−	0.907993i	$-0.362386\pi$
$197$	11.7615	0.837969	0.418985	+	0.907993i	$0.362386\pi$
$198$	0	0
$199$	20.5202	1.45464	0.727320	−	0.686298i	$-0.240766\pi$
$199$	20.5202	1.45464	0.727320	+	0.686298i	$0.240766\pi$
$200$	0	0
$201$	−2.15301	−0.151861
$202$	0	0
$203$	4.18479	0.293715
$204$	0	0
$205$	−12.1361	−0.847622
$206$	0	0
$207$	−3.94862	−0.274448
$208$	0	0
$209$	−1.75877	−0.121657
$210$	0	0
$211$	−21.7178	−1.49512	−0.747558	−	0.664197i	$-0.768774\pi$
$211$	−21.7178	−1.49512	−0.747558	+	0.664197i	$0.768774\pi$
$212$	0	0
$213$	−8.72462	−0.597801
$214$	0	0
$215$	−3.46379	−0.236229
$216$	0	0
$217$	−3.65270	−0.247962
$218$	0	0
$219$	−9.58677	−0.647814
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.9513	1.33604	0.668019	−	0.744144i	$-0.267142\pi$
$223$	19.9513	1.33604	0.668019	+	0.744144i	$0.267142\pi$
$224$	0	0
$225$	−1.13516	−0.0756775
$226$	0	0
$227$	4.22163	0.280199	0.140100	−	0.990137i	$-0.455258\pi$
$227$	4.22163	0.280199	0.140100	+	0.990137i	$0.455258\pi$
$228$	0	0
$229$	14.5057	0.958562	0.479281	−	0.877661i	$-0.340897\pi$
$229$	14.5057	0.958562	0.479281	+	0.877661i	$0.340897\pi$
$230$	0	0
$231$	−8.36959	−0.550678
$232$	0	0
$233$	5.12742	0.335909	0.167954	−	0.985795i	$-0.446284\pi$
$233$	5.12742	0.335909	0.167954	+	0.985795i	$0.446284\pi$
$234$	0	0
$235$	−20.0273	−1.30644
$236$	0	0
$237$	3.89899	0.253266
$238$	0	0
$239$	15.8503	1.02527	0.512635	−	0.858607i	$-0.328669\pi$
$239$	15.8503	1.02527	0.512635	+	0.858607i	$0.328669\pi$
$240$	0	0
$241$	−18.1165	−1.16699	−0.583493	−	0.812118i	$-0.698314\pi$
$241$	−18.1165	−1.16699	−0.583493	+	0.812118i	$0.698314\pi$
$242$	0	0
$243$	−16.1088	−1.03338
$244$	0	0
$245$	8.14022	0.520059
$246$	0	0
$247$	−1.63816	−0.104233
$248$	0	0
$249$	11.9436	0.756893
$250$	0	0
$251$	29.6810	1.87345	0.936723	−	0.350070i	$-0.113842\pi$
$251$	29.6810	1.87345	0.936723	+	0.350070i	$0.113842\pi$
$252$	0	0
$253$	8.98040	0.564593
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.39693	0.461408	0.230704	−	0.973024i	$-0.425897\pi$
$257$	7.39693	0.461408	0.230704	+	0.973024i	$0.425897\pi$
$258$	0	0
$259$	−11.5963	−0.720557
$260$	0	0
$261$	−4.95811	−0.306899
$262$	0	0
$263$	23.1908	1.43000	0.715002	−	0.699122i	$-0.246426\pi$
$263$	23.1908	1.43000	0.715002	+	0.699122i	$0.246426\pi$
$264$	0	0
$265$	24.5371	1.50730
$266$	0	0
$267$	5.56448	0.340541
$268$	0	0
$269$	−15.9067	−0.969850	−0.484925	−	0.874556i	$-0.661153\pi$
$269$	−15.9067	−0.969850	−0.484925	+	0.874556i	$0.661153\pi$
$270$	0	0
$271$	−17.0000	−1.03268	−0.516338	−	0.856385i	$-0.672705\pi$
$271$	−17.0000	−1.03268	−0.516338	+	0.856385i	$0.672705\pi$
$272$	0	0
$273$	−7.79561	−0.471812
$274$	0	0
$275$	2.58172	0.155683
$276$	0	0
$277$	16.8057	1.00976	0.504879	−	0.863190i	$-0.331537\pi$
$277$	16.8057	1.00976	0.504879	+	0.863190i	$0.331537\pi$
$278$	0	0
$279$	4.32770	0.259092
$280$	0	0
$281$	28.3209	1.68948	0.844741	−	0.535175i	$-0.179754\pi$
$281$	28.3209	1.68948	0.844741	+	0.535175i	$0.179754\pi$
$282$	0	0
$283$	−32.2344	−1.91614	−0.958069	−	0.286538i	$-0.907495\pi$
$283$	−32.2344	−1.91614	−0.958069	+	0.286538i	$0.907495\pi$
$284$	0	0
$285$	−0.716881	−0.0424644
$286$	0	0
$287$	9.71688	0.573569
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−8.15745	−0.478198
$292$	0	0
$293$	13.9709	0.816189	0.408094	−	0.912940i	$-0.366193\pi$
$293$	13.9709	0.816189	0.408094	+	0.912940i	$0.366193\pi$
$294$	0	0
$295$	11.7547	0.684382
$296$	0	0
$297$	23.2763	1.35063
$298$	0	0
$299$	8.36453	0.483733
$300$	0	0
$301$	2.77332	0.159851
$302$	0	0
$303$	6.19934	0.356143
$304$	0	0
$305$	0.433763	0.0248372
$306$	0	0
$307$	−9.04963	−0.516490	−0.258245	−	0.966080i	$-0.583144\pi$
$307$	−9.04963	−0.516490	−0.258245	+	0.966080i	$0.583144\pi$
$308$	0	0
$309$	4.65951	0.265070
$310$	0	0
$311$	2.34730	0.133103	0.0665515	−	0.997783i	$-0.478800\pi$
$311$	2.34730	0.133103	0.0665515	+	0.997783i	$0.478800\pi$
$312$	0	0
$313$	−15.1284	−0.855105	−0.427553	−	0.903990i	$-0.640624\pi$
$313$	−15.1284	−0.855105	−0.427553	+	0.903990i	$0.640624\pi$
$314$	0	0
$315$	9.82295	0.553460
$316$	0	0
$317$	10.3378	0.580629	0.290314	−	0.956931i	$-0.406240\pi$
$317$	10.3378	0.580629	0.290314	+	0.956931i	$0.406240\pi$
$318$	0	0
$319$	11.2763	0.631352
$320$	0	0
$321$	−13.7151	−0.765504
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.40467	0.133387
$326$	0	0
$327$	−1.64590	−0.0910183
$328$	0	0
$329$	16.0351	0.884043
$330$	0	0
$331$	−18.2567	−1.00348	−0.501740	−	0.865019i	$-0.667307\pi$
$331$	−18.2567	−1.00348	−0.501740	+	0.865019i	$0.667307\pi$
$332$	0	0
$333$	13.7392	0.752902
$334$	0	0
$335$	−5.74691	−0.313987
$336$	0	0
$337$	16.4953	0.898554	0.449277	−	0.893393i	$-0.351682\pi$
$337$	16.4953	0.898554	0.449277	+	0.893393i	$0.351682\pi$
$338$	0	0
$339$	10.6527	0.578575
$340$	0	0
$341$	−9.84255	−0.533004
$342$	0	0
$343$	−19.6732	−1.06225
$344$	0	0
$345$	3.66044	0.197072
$346$	0	0
$347$	−20.5722	−1.10437	−0.552187	−	0.833720i	$-0.686207\pi$
$347$	−20.5722	−1.10437	−0.552187	+	0.833720i	$0.686207\pi$
$348$	0	0
$349$	6.42427	0.343883	0.171942	−	0.985107i	$-0.444996\pi$
$349$	6.42427	0.343883	0.171942	+	0.985107i	$0.444996\pi$
$350$	0	0
$351$	21.6800	1.15720
$352$	0	0
$353$	27.1685	1.44603	0.723016	−	0.690831i	$-0.242755\pi$
$353$	27.1685	1.44603	0.723016	+	0.690831i	$0.242755\pi$
$354$	0	0
$355$	−23.2882	−1.23601
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.8949	−0.891677	−0.445838	−	0.895113i	$-0.647094\pi$
$359$	−16.8949	−0.891677	−0.445838	+	0.895113i	$0.647094\pi$
$360$	0	0
$361$	−18.8794	−0.993652
$362$	0	0
$363$	−12.8794	−0.675992
$364$	0	0
$365$	−25.5895	−1.33941
$366$	0	0
$367$	1.48751	0.0776475	0.0388237	−	0.999246i	$-0.487639\pi$
$367$	1.48751	0.0776475	0.0388237	+	0.999246i	$0.487639\pi$
$368$	0	0
$369$	−11.5125	−0.599316
$370$	0	0
$371$	−19.6459	−1.01996
$372$	0	0
$373$	−24.0496	−1.24524	−0.622621	−	0.782523i	$-0.713932\pi$
$373$	−24.0496	−1.24524	−0.622621	+	0.782523i	$0.713932\pi$
$374$	0	0
$375$	−9.26857	−0.478627
$376$	0	0
$377$	10.5030	0.540932
$378$	0	0
$379$	20.1976	1.03748	0.518740	−	0.854932i	$-0.326401\pi$
$379$	20.1976	1.03748	0.518740	+	0.854932i	$0.326401\pi$
$380$	0	0
$381$	−10.1480	−0.519896
$382$	0	0
$383$	8.52528	0.435622	0.217811	−	0.975991i	$-0.430108\pi$
$383$	8.52528	0.435622	0.217811	+	0.975991i	$0.430108\pi$
$384$	0	0
$385$	−22.3405	−1.13858
$386$	0	0
$387$	−3.28581	−0.167027
$388$	0	0
$389$	−12.9162	−0.654878	−0.327439	−	0.944872i	$-0.606186\pi$
$389$	−12.9162	−0.654878	−0.327439	+	0.944872i	$0.606186\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−17.0378	−0.859442
$394$	0	0
$395$	10.4074	0.523651
$396$	0	0
$397$	−24.2249	−1.21581	−0.607907	−	0.794008i	$-0.707991\pi$
$397$	−24.2249	−1.21581	−0.607907	+	0.794008i	$0.707991\pi$
$398$	0	0
$399$	0.573978	0.0287348
$400$	0	0
$401$	−25.5371	−1.27526	−0.637632	−	0.770341i	$-0.720086\pi$
$401$	−25.5371	−1.27526	−0.637632	+	0.770341i	$0.720086\pi$
$402$	0	0
$403$	−9.16756	−0.456669
$404$	0	0
$405$	−6.19253	−0.307709
$406$	0	0
$407$	−31.2472	−1.54887
$408$	0	0
$409$	10.3523	0.511891	0.255945	−	0.966691i	$-0.417613\pi$
$409$	10.3523	0.511891	0.255945	+	0.966691i	$0.417613\pi$
$410$	0	0
$411$	0.394238	0.0194463
$412$	0	0
$413$	−9.41147	−0.463108
$414$	0	0
$415$	31.8803	1.56494
$416$	0	0
$417$	10.2713	0.502986
$418$	0	0
$419$	1.31315	0.0641515	0.0320757	−	0.999485i	$-0.489788\pi$
$419$	1.31315	0.0641515	0.0320757	+	0.999485i	$0.489788\pi$
$420$	0	0
$421$	8.01548	0.390651	0.195325	−	0.980739i	$-0.437424\pi$
$421$	8.01548	0.390651	0.195325	+	0.980739i	$0.437424\pi$
$422$	0	0
$423$	−18.9982	−0.923726
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.347296	−0.0168068
$428$	0	0
$429$	−21.0060	−1.01418
$430$	0	0
$431$	14.7270	0.709374	0.354687	−	0.934985i	$-0.384587\pi$
$431$	14.7270	0.709374	0.354687	+	0.934985i	$0.384587\pi$
$432$	0	0
$433$	−8.24123	−0.396048	−0.198024	−	0.980197i	$-0.563452\pi$
$433$	−8.24123	−0.396048	−0.198024	+	0.980197i	$0.563452\pi$
$434$	0	0
$435$	4.59627	0.220374
$436$	0	0
$437$	−0.615867	−0.0294609
$438$	0	0
$439$	39.9564	1.90701	0.953506	−	0.301373i	$-0.0974448\pi$
$439$	39.9564	1.90701	0.953506	+	0.301373i	$0.0974448\pi$
$440$	0	0
$441$	7.72193	0.367711
$442$	0	0
$443$	−13.9463	−0.662606	−0.331303	−	0.943524i	$-0.607488\pi$
$443$	−13.9463	−0.662606	−0.331303	+	0.943524i	$0.607488\pi$
$444$	0	0
$445$	14.8530	0.704099
$446$	0	0
$447$	−7.44656	−0.352210
$448$	0	0
$449$	−10.2317	−0.482865	−0.241433	−	0.970418i	$-0.577617\pi$
$449$	−10.2317	−0.482865	−0.241433	+	0.970418i	$0.577617\pi$
$450$	0	0
$451$	26.1830	1.23291
$452$	0	0
$453$	−12.1061	−0.568793
$454$	0	0
$455$	−20.8084	−0.975513
$456$	0	0
$457$	−11.7706	−0.550607	−0.275303	−	0.961357i	$-0.588778\pi$
$457$	−11.7706	−0.550607	−0.275303	+	0.961357i	$0.588778\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.5003	0.908220	0.454110	−	0.890946i	$-0.349957\pi$
$461$	19.5003	0.908220	0.454110	+	0.890946i	$0.349957\pi$
$462$	0	0
$463$	1.43107	0.0665077	0.0332538	−	0.999447i	$-0.489413\pi$
$463$	1.43107	0.0665077	0.0332538	+	0.999447i	$0.489413\pi$
$464$	0	0
$465$	−4.01186	−0.186046
$466$	0	0
$467$	10.6895	0.494653	0.247326	−	0.968932i	$-0.420448\pi$
$467$	10.6895	0.494653	0.247326	+	0.968932i	$0.420448\pi$
$468$	0	0
$469$	4.60132	0.212469
$470$	0	0
$471$	15.8102	0.728493
$472$	0	0
$473$	7.47296	0.343607
$474$	0	0
$475$	−0.177052	−0.00812369
$476$	0	0
$477$	23.2763	1.06575
$478$	0	0
$479$	38.8675	1.77590	0.887951	−	0.459938i	$-0.152128\pi$
$479$	38.8675	1.77590	0.887951	+	0.459938i	$0.152128\pi$
$480$	0	0
$481$	−29.1043	−1.32704
$482$	0	0
$483$	−2.93077	−0.133355
$484$	0	0
$485$	−21.7743	−0.988718
$486$	0	0
$487$	37.0137	1.67725	0.838626	−	0.544708i	$-0.183359\pi$
$487$	37.0137	1.67725	0.838626	+	0.544708i	$0.183359\pi$
$488$	0	0
$489$	−6.36184	−0.287693
$490$	0	0
$491$	25.4175	1.14707	0.573537	−	0.819180i	$-0.305571\pi$
$491$	25.4175	1.14707	0.573537	+	0.819180i	$0.305571\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	26.4688	1.18969
$496$	0	0
$497$	18.6459	0.836383
$498$	0	0
$499$	−21.8239	−0.976971	−0.488486	−	0.872572i	$-0.662451\pi$
$499$	−21.8239	−0.976971	−0.488486	+	0.872572i	$0.662451\pi$
$500$	0	0
$501$	−4.49020	−0.200607
$502$	0	0
$503$	33.4371	1.49088	0.745442	−	0.666570i	$-0.232238\pi$
$503$	33.4371	1.49088	0.745442	+	0.666570i	$0.232238\pi$
$504$	0	0
$505$	16.5476	0.736357
$506$	0	0
$507$	−8.13341	−0.361217
$508$	0	0
$509$	−19.1530	−0.848942	−0.424471	−	0.905441i	$-0.639540\pi$
$509$	−19.1530	−0.848942	−0.424471	+	0.905441i	$0.639540\pi$
$510$	0	0
$511$	20.4884	0.906355
$512$	0	0
$513$	−1.59627	−0.0704769
$514$	0	0
$515$	12.4374	0.548057
$516$	0	0
$517$	43.2080	1.90029
$518$	0	0
$519$	−11.2787	−0.495079
$520$	0	0
$521$	35.5330	1.55673	0.778365	−	0.627812i	$-0.216049\pi$
$521$	35.5330	1.55673	0.778365	+	0.627812i	$0.216049\pi$
$522$	0	0
$523$	11.8307	0.517320	0.258660	−	0.965968i	$-0.416719\pi$
$523$	11.8307	0.517320	0.258660	+	0.965968i	$0.416719\pi$
$524$	0	0
$525$	−0.842549	−0.0367718
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.8553	−0.863276
$530$	0	0
$531$	11.1506	0.483897
$532$	0	0
$533$	24.3874	1.05634
$534$	0	0
$535$	−36.6091	−1.58275
$536$	0	0
$537$	3.75608	0.162087
$538$	0	0
$539$	−17.5621	−0.756454
$540$	0	0
$541$	−5.55850	−0.238978	−0.119489	−	0.992835i	$-0.538126\pi$
$541$	−5.55850	−0.238978	−0.119489	+	0.992835i	$0.538126\pi$
$542$	0	0
$543$	0.407031	0.0174674
$544$	0	0
$545$	−4.39330	−0.188188
$546$	0	0
$547$	5.02465	0.214839	0.107419	−	0.994214i	$-0.465741\pi$
$547$	5.02465	0.214839	0.107419	+	0.994214i	$0.465741\pi$
$548$	0	0
$549$	0.411474	0.0175613
$550$	0	0
$551$	−0.773318	−0.0329445
$552$	0	0
$553$	−8.33275	−0.354345
$554$	0	0
$555$	−12.7365	−0.540634
$556$	0	0
$557$	3.86659	0.163833	0.0819164	−	0.996639i	$-0.473896\pi$
$557$	3.86659	0.163833	0.0819164	+	0.996639i	$0.473896\pi$
$558$	0	0
$559$	6.96048	0.294397
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.8411	−1.21551	−0.607754	−	0.794125i	$-0.707929\pi$
$563$	−28.8411	−1.21551	−0.607754	+	0.794125i	$0.707929\pi$
$564$	0	0
$565$	28.4347	1.19626
$566$	0	0
$567$	4.95811	0.208221
$568$	0	0
$569$	2.16157	0.0906177	0.0453089	−	0.998973i	$-0.485573\pi$
$569$	2.16157	0.0906177	0.0453089	+	0.998973i	$0.485573\pi$
$570$	0	0
$571$	−5.71925	−0.239343	−0.119671	−	0.992814i	$-0.538184\pi$
$571$	−5.71925	−0.239343	−0.119671	+	0.992814i	$0.538184\pi$
$572$	0	0
$573$	−1.25847	−0.0525732
$574$	0	0
$575$	0.904038	0.0377010
$576$	0	0
$577$	−10.8007	−0.449637	−0.224819	−	0.974401i	$-0.572179\pi$
$577$	−10.8007	−0.449637	−0.224819	+	0.974401i	$0.572179\pi$
$578$	0	0
$579$	−21.7425	−0.903586
$580$	0	0
$581$	−25.5253	−1.05897
$582$	0	0
$583$	−52.9377	−2.19246
$584$	0	0
$585$	24.6536	1.01930
$586$	0	0
$587$	−20.8188	−0.859285	−0.429643	−	0.902999i	$-0.641360\pi$
$587$	−20.8188	−0.859285	−0.429643	+	0.902999i	$0.641360\pi$
$588$	0	0
$589$	0.674992	0.0278126
$590$	0	0
$591$	−10.3429	−0.425448
$592$	0	0
$593$	20.6313	0.847228	0.423614	−	0.905843i	$-0.360761\pi$
$593$	20.6313	0.847228	0.423614	+	0.905843i	$0.360761\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−18.0452	−0.738540
$598$	0	0
$599$	−31.8212	−1.30018	−0.650089	−	0.759858i	$-0.725269\pi$
$599$	−31.8212	−1.30018	−0.650089	+	0.759858i	$0.725269\pi$
$600$	0	0
$601$	−48.1174	−1.96275	−0.981375	−	0.192100i	$-0.938470\pi$
$601$	−48.1174	−1.96275	−0.981375	+	0.192100i	$0.938470\pi$
$602$	0	0
$603$	−5.45161	−0.222007
$604$	0	0
$605$	−34.3783	−1.39768
$606$	0	0
$607$	−15.4757	−0.628137	−0.314069	−	0.949400i	$-0.601692\pi$
$607$	−15.4757	−0.628137	−0.314069	+	0.949400i	$0.601692\pi$
$608$	0	0
$609$	−3.68004	−0.149123
$610$	0	0
$611$	40.2449	1.62813
$612$	0	0
$613$	−5.04963	−0.203953	−0.101976	−	0.994787i	$-0.532517\pi$
$613$	−5.04963	−0.203953	−0.101976	+	0.994787i	$0.532517\pi$
$614$	0	0
$615$	10.6723	0.430349
$616$	0	0
$617$	27.6064	1.11139	0.555695	−	0.831386i	$-0.312452\pi$
$617$	27.6064	1.11139	0.555695	+	0.831386i	$0.312452\pi$
$618$	0	0
$619$	14.8331	0.596191	0.298095	−	0.954536i	$-0.403649\pi$
$619$	14.8331	0.596191	0.298095	+	0.954536i	$0.403649\pi$
$620$	0	0
$621$	8.15064	0.327074
$622$	0	0
$623$	−11.8922	−0.476450
$624$	0	0
$625$	−27.2891	−1.09156
$626$	0	0
$627$	1.54664	0.0617667
$628$	0	0
$629$	0	0
$630$	0	0
$631$	34.0506	1.35553	0.677766	−	0.735278i	$-0.262948\pi$
$631$	34.0506	1.35553	0.677766	+	0.735278i	$0.262948\pi$
$632$	0	0
$633$	19.0983	0.759090
$634$	0	0
$635$	−27.0874	−1.07493
$636$	0	0
$637$	−16.3577	−0.648117
$638$	0	0
$639$	−22.0915	−0.873927
$640$	0	0
$641$	15.0232	0.593382	0.296691	−	0.954974i	$-0.404117\pi$
$641$	15.0232	0.593382	0.296691	+	0.954974i	$0.404117\pi$
$642$	0	0
$643$	−17.7980	−0.701883	−0.350942	−	0.936397i	$-0.614138\pi$
$643$	−17.7980	−0.701883	−0.350942	+	0.936397i	$0.614138\pi$
$644$	0	0
$645$	3.04601	0.119936
$646$	0	0
$647$	46.4971	1.82799	0.913995	−	0.405725i	$-0.132981\pi$
$647$	46.4971	1.82799	0.913995	+	0.405725i	$0.132981\pi$
$648$	0	0
$649$	−25.3601	−0.995471
$650$	0	0
$651$	3.21213	0.125893
$652$	0	0
$653$	−30.5303	−1.19474	−0.597372	−	0.801964i	$-0.703788\pi$
$653$	−30.5303	−1.19474	−0.597372	+	0.801964i	$0.703788\pi$
$654$	0	0
$655$	−45.4780	−1.77697
$656$	0	0
$657$	−24.2746	−0.947041
$658$	0	0
$659$	9.83481	0.383110	0.191555	−	0.981482i	$-0.438647\pi$
$659$	9.83481	0.383110	0.191555	+	0.981482i	$0.438647\pi$
$660$	0	0
$661$	12.4361	0.483709	0.241855	−	0.970312i	$-0.422244\pi$
$661$	12.4361	0.483709	0.241855	+	0.970312i	$0.422244\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.53209	0.0594119
$666$	0	0
$667$	3.94862	0.152891
$668$	0	0
$669$	−17.5449	−0.678324
$670$	0	0
$671$	−0.935822	−0.0361270
$672$	0	0
$673$	12.6117	0.486147	0.243074	−	0.970008i	$-0.421844\pi$
$673$	12.6117	0.486147	0.243074	+	0.970008i	$0.421844\pi$
$674$	0	0
$675$	2.34318	0.0901889
$676$	0	0
$677$	4.54900	0.174832	0.0874162	−	0.996172i	$-0.472139\pi$
$677$	4.54900	0.174832	0.0874162	+	0.996172i	$0.472139\pi$
$678$	0	0
$679$	17.4338	0.669046
$680$	0	0
$681$	−3.71244	−0.142261
$682$	0	0
$683$	4.86753	0.186251	0.0931253	−	0.995654i	$-0.470314\pi$
$683$	4.86753	0.186251	0.0931253	+	0.995654i	$0.470314\pi$
$684$	0	0
$685$	1.05232	0.0402070
$686$	0	0
$687$	−12.7561	−0.486675
$688$	0	0
$689$	−49.3073	−1.87846
$690$	0	0
$691$	5.51661	0.209862	0.104931	−	0.994480i	$-0.466538\pi$
$691$	5.51661	0.209862	0.104931	+	0.994480i	$0.466538\pi$
$692$	0	0
$693$	−21.1925	−0.805038
$694$	0	0
$695$	27.4165	1.03997
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−4.50898	−0.170545
$700$	0	0
$701$	22.3233	0.843138	0.421569	−	0.906796i	$-0.361480\pi$
$701$	22.3233	0.843138	0.421569	+	0.906796i	$0.361480\pi$
$702$	0	0
$703$	2.14290	0.0808211
$704$	0	0
$705$	17.6117	0.663297
$706$	0	0
$707$	−13.2490	−0.498279
$708$	0	0
$709$	−34.7766	−1.30606	−0.653032	−	0.757331i	$-0.726503\pi$
$709$	−34.7766	−1.30606	−0.653032	+	0.757331i	$0.726503\pi$
$710$	0	0
$711$	9.87258	0.370251
$712$	0	0
$713$	−3.44656	−0.129075
$714$	0	0
$715$	−56.0702	−2.09691
$716$	0	0
$717$	−13.9385	−0.520543
$718$	0	0
$719$	−4.24990	−0.158495	−0.0792473	−	0.996855i	$-0.525252\pi$
$719$	−4.24990	−0.158495	−0.0792473	+	0.996855i	$0.525252\pi$
$720$	0	0
$721$	−9.95811	−0.370859
$722$	0	0
$723$	15.9314	0.592494
$724$	0	0
$725$	1.13516	0.0421589
$726$	0	0
$727$	−29.1644	−1.08165	−0.540823	−	0.841136i	$-0.681887\pi$
$727$	−29.1644	−1.08165	−0.540823	+	0.841136i	$0.681887\pi$
$728$	0	0
$729$	6.25133	0.231531
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−0.502993	−0.0185785	−0.00928924	−	0.999957i	$-0.502957\pi$
$733$	−0.502993	−0.0185785	−0.00928924	+	0.999957i	$0.502957\pi$
$734$	0	0
$735$	−7.15839	−0.264041
$736$	0	0
$737$	12.3987	0.456711
$738$	0	0
$739$	−14.9581	−0.550243	−0.275122	−	0.961409i	$-0.588718\pi$
$739$	−14.9581	−0.550243	−0.275122	+	0.961409i	$0.588718\pi$
$740$	0	0
$741$	1.44057	0.0529207
$742$	0	0
$743$	−23.3756	−0.857567	−0.428783	−	0.903407i	$-0.641058\pi$
$743$	−23.3756	−0.857567	−0.428783	+	0.903407i	$0.641058\pi$
$744$	0	0
$745$	−19.8767	−0.728226
$746$	0	0
$747$	30.2422	1.10650
$748$	0	0
$749$	29.3114	1.07102
$750$	0	0
$751$	−17.0291	−0.621401	−0.310700	−	0.950508i	$-0.600564\pi$
$751$	−17.0291	−0.621401	−0.310700	+	0.950508i	$0.600564\pi$
$752$	0	0
$753$	−26.1010	−0.951174
$754$	0	0
$755$	−32.3141	−1.17603
$756$	0	0
$757$	16.3746	0.595146	0.297573	−	0.954699i	$-0.403823\pi$
$757$	16.3746	0.595146	0.297573	+	0.954699i	$0.403823\pi$
$758$	0	0
$759$	−7.89723	−0.286651
$760$	0	0
$761$	−18.8803	−0.684411	−0.342206	−	0.939625i	$-0.611174\pi$
$761$	−18.8803	−0.684411	−0.342206	+	0.939625i	$0.611174\pi$
$762$	0	0
$763$	3.51754	0.127344
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.6209	−0.852902
$768$	0	0
$769$	−27.4766	−0.990831	−0.495416	−	0.868656i	$-0.664984\pi$
$769$	−27.4766	−0.990831	−0.495416	+	0.868656i	$0.664984\pi$
$770$	0	0
$771$	−6.50475	−0.234263
$772$	0	0
$773$	9.90436	0.356235	0.178118	−	0.984009i	$-0.442999\pi$
$773$	9.90436	0.356235	0.178118	+	0.984009i	$0.442999\pi$
$774$	0	0
$775$	−0.990829	−0.0355916
$776$	0	0
$777$	10.1976	0.365836
$778$	0	0
$779$	−1.79561	−0.0643343
$780$	0	0
$781$	50.2431	1.79784
$782$	0	0
$783$	10.2344	0.365748
$784$	0	0
$785$	42.2012	1.50623
$786$	0	0
$787$	50.8444	1.81241	0.906204	−	0.422841i	$-0.138967\pi$
$787$	50.8444	1.81241	0.906204	+	0.422841i	$0.138967\pi$
$788$	0	0
$789$	−20.3936	−0.726032
$790$	0	0
$791$	−22.7665	−0.809484
$792$	0	0
$793$	−0.871644	−0.0309530
$794$	0	0
$795$	−21.5776	−0.765279
$796$	0	0
$797$	5.38650	0.190800	0.0953998	−	0.995439i	$-0.469587\pi$
$797$	5.38650	0.190800	0.0953998	+	0.995439i	$0.469587\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	14.0898	0.497837
$802$	0	0
$803$	55.2080	1.94825
$804$	0	0
$805$	−7.82295	−0.275723
$806$	0	0
$807$	13.9881	0.492406
$808$	0	0
$809$	−48.6819	−1.71156	−0.855782	−	0.517336i	$-0.826924\pi$
$809$	−48.6819	−1.71156	−0.855782	+	0.517336i	$0.826924\pi$
$810$	0	0
$811$	−15.9281	−0.559311	−0.279655	−	0.960100i	$-0.590220\pi$
$811$	−15.9281	−0.559311	−0.279655	+	0.960100i	$0.590220\pi$
$812$	0	0
$813$	14.9495	0.524304
$814$	0	0
$815$	−16.9813	−0.594830
$816$	0	0
$817$	−0.512489	−0.0179297
$818$	0	0
$819$	−19.7392	−0.689742
$820$	0	0
$821$	17.1010	0.596830	0.298415	−	0.954436i	$-0.403542\pi$
$821$	17.1010	0.596830	0.298415	+	0.954436i	$0.403542\pi$
$822$	0	0
$823$	−5.62267	−0.195994	−0.0979971	−	0.995187i	$-0.531244\pi$
$823$	−5.62267	−0.195994	−0.0979971	+	0.995187i	$0.531244\pi$
$824$	0	0
$825$	−2.27033	−0.0790426
$826$	0	0
$827$	17.8607	0.621078	0.310539	−	0.950561i	$-0.399490\pi$
$827$	17.8607	0.621078	0.310539	+	0.950561i	$0.399490\pi$
$828$	0	0
$829$	35.8161	1.24395	0.621973	−	0.783039i	$-0.286331\pi$
$829$	35.8161	1.24395	0.621973	+	0.783039i	$0.286331\pi$
$830$	0	0
$831$	−14.7787	−0.512667
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11.9855	−0.414774
$836$	0	0
$837$	−8.93313	−0.308774
$838$	0	0
$839$	35.7733	1.23503	0.617516	−	0.786558i	$-0.288139\pi$
$839$	35.7733	1.23503	0.617516	+	0.786558i	$0.288139\pi$
$840$	0	0
$841$	−24.0419	−0.829031
$842$	0	0
$843$	−24.9050	−0.857773
$844$	0	0
$845$	−21.7101	−0.746849
$846$	0	0
$847$	27.5253	0.945780
$848$	0	0
$849$	28.3465	0.972849
$850$	0	0
$851$	−10.9418	−0.375080
$852$	0	0
$853$	−11.0669	−0.378922	−0.189461	−	0.981888i	$-0.560674\pi$
$853$	−11.0669	−0.378922	−0.189461	+	0.981888i	$0.560674\pi$
$854$	0	0
$855$	−1.81521	−0.0620788
$856$	0	0
$857$	−27.0746	−0.924851	−0.462425	−	0.886658i	$-0.653021\pi$
$857$	−27.0746	−0.924851	−0.462425	+	0.886658i	$0.653021\pi$
$858$	0	0
$859$	51.7279	1.76493	0.882467	−	0.470374i	$-0.155881\pi$
$859$	51.7279	1.76493	0.882467	+	0.470374i	$0.155881\pi$
$860$	0	0
$861$	−8.54488	−0.291209
$862$	0	0
$863$	−45.4712	−1.54786	−0.773929	−	0.633272i	$-0.781711\pi$
$863$	−45.4712	−1.54786	−0.773929	+	0.633272i	$0.781711\pi$
$864$	0	0
$865$	−30.1056	−1.02362
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−22.4534	−0.761678
$870$	0	0
$871$	11.5484	0.391302
$872$	0	0
$873$	−20.6554	−0.699079
$874$	0	0
$875$	19.8084	0.669646
$876$	0	0
$877$	−37.4834	−1.26572	−0.632862	−	0.774265i	$-0.718120\pi$
$877$	−37.4834	−1.26572	−0.632862	+	0.774265i	$0.718120\pi$
$878$	0	0
$879$	−12.2858	−0.414390
$880$	0	0
$881$	18.1958	0.613033	0.306517	−	0.951865i	$-0.400837\pi$
$881$	18.1958	0.613033	0.306517	+	0.951865i	$0.400837\pi$
$882$	0	0
$883$	−0.397860	−0.0133891	−0.00669453	−	0.999978i	$-0.502131\pi$
$883$	−0.397860	−0.0133891	−0.00669453	+	0.999978i	$0.502131\pi$
$884$	0	0
$885$	−10.3369	−0.347470
$886$	0	0
$887$	−23.7834	−0.798569	−0.399285	−	0.916827i	$-0.630741\pi$
$887$	−23.7834	−0.798569	−0.399285	+	0.916827i	$0.630741\pi$
$888$	0	0
$889$	21.6878	0.727385
$890$	0	0
$891$	13.3601	0.447580
$892$	0	0
$893$	−2.96316	−0.0991585
$894$	0	0
$895$	10.0259	0.335129
$896$	0	0
$897$	−7.35565	−0.245598
$898$	0	0
$899$	−4.32770	−0.144337
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−2.43882	−0.0811587
$904$	0	0
$905$	1.08647	0.0361154
$906$	0	0
$907$	−37.5212	−1.24587	−0.622935	−	0.782274i	$-0.714060\pi$
$907$	−37.5212	−1.24587	−0.622935	+	0.782274i	$0.714060\pi$
$908$	0	0
$909$	15.6973	0.520646
$910$	0	0
$911$	−14.7648	−0.489178	−0.244589	−	0.969627i	$-0.578653\pi$
$911$	−14.7648	−0.489178	−0.244589	+	0.969627i	$0.578653\pi$
$912$	0	0
$913$	−68.7802	−2.27629
$914$	0	0
$915$	−0.381445	−0.0126102
$916$	0	0
$917$	36.4124	1.20244
$918$	0	0
$919$	48.5476	1.60144	0.800718	−	0.599041i	$-0.204451\pi$
$919$	48.5476	1.60144	0.800718	+	0.599041i	$0.204451\pi$
$920$	0	0
$921$	7.95811	0.262229
$922$	0	0
$923$	46.7975	1.54036
$924$	0	0
$925$	−3.14559	−0.103426
$926$	0	0
$927$	11.7983	0.387507
$928$	0	0
$929$	−11.2635	−0.369544	−0.184772	−	0.982781i	$-0.559155\pi$
$929$	−11.2635	−0.369544	−0.184772	+	0.982781i	$0.559155\pi$
$930$	0	0
$931$	1.20439	0.0394724
$932$	0	0
$933$	−2.06418	−0.0675781
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.39775	0.0783310	0.0391655	−	0.999233i	$-0.487530\pi$
$937$	2.39775	0.0783310	0.0391655	+	0.999233i	$0.487530\pi$
$938$	0	0
$939$	13.3037	0.434148
$940$	0	0
$941$	34.3797	1.12075	0.560373	−	0.828240i	$-0.310658\pi$
$941$	34.3797	1.12075	0.560373	+	0.828240i	$0.310658\pi$
$942$	0	0
$943$	9.16849	0.298567
$944$	0	0
$945$	−20.2763	−0.659588
$946$	0	0
$947$	−20.5294	−0.667116	−0.333558	−	0.942730i	$-0.608249\pi$
$947$	−20.5294	−0.667116	−0.333558	+	0.942730i	$0.608249\pi$
$948$	0	0
$949$	51.4219	1.66923
$950$	0	0
$951$	−9.09091	−0.294793
$952$	0	0
$953$	16.5517	0.536162	0.268081	−	0.963396i	$-0.413611\pi$
$953$	16.5517	0.536162	0.268081	+	0.963396i	$0.413611\pi$
$954$	0	0
$955$	−3.35916	−0.108700
$956$	0	0
$957$	−9.91622	−0.320546
$958$	0	0
$959$	−0.842549	−0.0272073
$960$	0	0
$961$	−27.2226	−0.878147
$962$	0	0
$963$	−34.7279	−1.11909
$964$	0	0
$965$	−58.0360	−1.86825
$966$	0	0
$967$	−3.92665	−0.126273	−0.0631363	−	0.998005i	$-0.520110\pi$
$967$	−3.92665	−0.126273	−0.0631363	+	0.998005i	$0.520110\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.25402	0.136518	0.0682590	−	0.997668i	$-0.478256\pi$
$971$	4.25402	0.136518	0.0682590	+	0.997668i	$0.478256\pi$
$972$	0	0
$973$	−21.9513	−0.703726
$974$	0	0
$975$	−2.11463	−0.0677223
$976$	0	0
$977$	−48.5431	−1.55303	−0.776516	−	0.630097i	$-0.783015\pi$
$977$	−48.5431	−1.55303	−0.776516	+	0.630097i	$0.783015\pi$
$978$	0	0
$979$	−32.0446	−1.02415
$980$	0	0
$981$	−4.16756	−0.133060
$982$	0	0
$983$	35.0597	1.11823	0.559116	−	0.829089i	$-0.311141\pi$
$983$	35.0597	1.11823	0.559116	+	0.829089i	$0.311141\pi$
$984$	0	0
$985$	−27.6076	−0.879652
$986$	0	0
$987$	−14.1010	−0.448840
$988$	0	0
$989$	2.61680	0.0832094
$990$	0	0
$991$	−53.1995	−1.68994	−0.844968	−	0.534817i	$-0.820381\pi$
$991$	−53.1995	−1.68994	−0.844968	+	0.534817i	$0.820381\pi$
$992$	0	0
$993$	16.0547	0.509480
$994$	0	0
$995$	−48.1671	−1.52700
$996$	0	0
$997$	22.5794	0.715095	0.357548	−	0.933895i	$-0.383613\pi$
$997$	22.5794	0.715095	0.357548	+	0.933895i	$0.383613\pi$
$998$	0	0
$999$	−28.3601	−0.897274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4624.2.a.bg.1.1		3
4.3	odd	2		289.2.a.d.1.2	✓	3
12.11	even	2		2601.2.a.x.1.2		3
17.16	even	2		4624.2.a.bd.1.3		3
20.19	odd	2		7225.2.a.t.1.2		3
68.3	even	16		289.2.d.f.179.4		24
68.7	even	16		289.2.d.f.134.3		24
68.11	even	16		289.2.d.f.155.3		24
68.15	odd	8		289.2.c.d.38.3		12
68.19	odd	8		289.2.c.d.38.4		12
68.23	even	16		289.2.d.f.155.4		24
68.27	even	16		289.2.d.f.134.4		24
68.31	even	16		289.2.d.f.179.3		24
68.39	even	16		289.2.d.f.110.3		24
68.43	odd	8		289.2.c.d.251.4		12
68.47	odd	4		289.2.b.d.288.4		6
68.55	odd	4		289.2.b.d.288.3		6
68.59	odd	8		289.2.c.d.251.3		12
68.63	even	16		289.2.d.f.110.4		24
68.67	odd	2		289.2.a.e.1.2	yes	3
204.203	even	2		2601.2.a.w.1.2		3
340.339	odd	2		7225.2.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.a.d.1.2	✓	3	4.3	odd	2
289.2.a.e.1.2	yes	3	68.67	odd	2
289.2.b.d.288.3		6	68.55	odd	4
289.2.b.d.288.4		6	68.47	odd	4
289.2.c.d.38.3		12	68.15	odd	8
289.2.c.d.38.4		12	68.19	odd	8
289.2.c.d.251.3		12	68.59	odd	8
289.2.c.d.251.4		12	68.43	odd	8
289.2.d.f.110.3		24	68.39	even	16
289.2.d.f.110.4		24	68.63	even	16
289.2.d.f.134.3		24	68.7	even	16
289.2.d.f.134.4		24	68.27	even	16
289.2.d.f.155.3		24	68.11	even	16
289.2.d.f.155.4		24	68.23	even	16
289.2.d.f.179.3		24	68.31	even	16
289.2.d.f.179.4		24	68.3	even	16
2601.2.a.w.1.2		3	204.203	even	2
2601.2.a.x.1.2		3	12.11	even	2
4624.2.a.bd.1.3		3	17.16	even	2
4624.2.a.bg.1.1		3	1.1	even	1	trivial
7225.2.a.s.1.2		3	340.339	odd	2
7225.2.a.t.1.2		3	20.19	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:	\( N \)	\(=\)	\( 4624 = 2^{4} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4624.a (trivial)

\(f(q)\)	\(=\)	\(q-0.879385 q^{3} -2.34730 q^{5} +1.87939 q^{7} -2.22668 q^{9} +O(q^{10})\)	\(q-0.879385 q^{3} -2.34730 q^{5} +1.87939 q^{7} -2.22668 q^{9} +5.06418 q^{11} +4.71688 q^{13} +2.06418 q^{15} -0.347296 q^{19} -1.65270 q^{21} +1.77332 q^{23} +0.509800 q^{25} +4.59627 q^{27} +2.22668 q^{29} -1.94356 q^{31} -4.45336 q^{33} -4.41147 q^{35} -6.17024 q^{37} -4.14796 q^{39} +5.17024 q^{41} +1.47565 q^{43} +5.22668 q^{45} +8.53209 q^{47} -3.46791 q^{49} -10.4534 q^{53} -11.8871 q^{55} +0.305407 q^{57} -5.00774 q^{59} -0.184793 q^{61} -4.18479 q^{63} -11.0719 q^{65} +2.44831 q^{67} -1.55943 q^{69} +9.92127 q^{71} +10.9017 q^{73} -0.448311 q^{75} +9.51754 q^{77} -4.43376 q^{79} +2.63816 q^{81} -13.5817 q^{83} -1.95811 q^{87} -6.32770 q^{89} +8.86484 q^{91} +1.70914 q^{93} +0.815207 q^{95} +9.27631 q^{97} -11.2763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{5}+O(q^{10}) \) 3 * q + 3 * q^3 - 6 * q^5	\( 3 q + 3 q^{3} - 6 q^{5} + 6 q^{11} + 6 q^{13} - 3 q^{15} - 6 q^{21} + 12 q^{23} + 3 q^{25} + 9 q^{31} - 3 q^{35} + 3 q^{37} + 3 q^{39} - 6 q^{41} - 15 q^{43} + 9 q^{45} + 21 q^{47} - 15 q^{49} - 18 q^{53} - 6 q^{55} + 3 q^{57} + 9 q^{59} + 3 q^{61} - 9 q^{63} + 9 q^{67} + 21 q^{69} + 21 q^{71} + 21 q^{73} - 3 q^{75} + 6 q^{77} + 3 q^{79} - 9 q^{81} - 9 q^{83} - 9 q^{87} - 15 q^{89} + 3 q^{91} + 21 q^{93} + 6 q^{95} - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^5 + 6 * q^11 + 6 * q^13 - 3 * q^15 - 6 * q^21 + 12 * q^23 + 3 * q^25 + 9 * q^31 - 3 * q^35 + 3 * q^37 + 3 * q^39 - 6 * q^41 - 15 * q^43 + 9 * q^45 + 21 * q^47 - 15 * q^49 - 18 * q^53 - 6 * q^55 + 3 * q^57 + 9 * q^59 + 3 * q^61 - 9 * q^63 + 9 * q^67 + 21 * q^69 + 21 * q^71 + 21 * q^73 - 3 * q^75 + 6 * q^77 + 3 * q^79 - 9 * q^81 - 9 * q^83 - 9 * q^87 - 15 * q^89 + 3 * q^91 + 21 * q^93 + 6 * q^95 - 6 * q^97

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