LMFDB - Embedded newform 4592.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4592,2,Mod(1,4592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4592, 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("4592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4592 = 2^{4} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4592.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:	$36.6673046082$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.633117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x^{2} + 6x - 3 $ x^5 - x^4 - 6x^3 + 4x^2 + 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.03121$ of defining polynomial
Character	$\chi$	$=$	4592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.03121 q^{3} -3.82713 q^{5} -1.00000 q^{7} +6.18825 q^{9} +O(q^{10})$	$q-3.03121 q^{3} -3.82713 q^{5} -1.00000 q^{7} +6.18825 q^{9} +5.96294 q^{11} +1.44574 q^{13} +11.6009 q^{15} +6.06148 q^{17} +0.0743284 q^{19} +3.03121 q^{21} +4.43383 q^{23} +9.64695 q^{25} -9.66425 q^{27} -1.92662 q^{29} -1.76471 q^{31} -18.0749 q^{33} +3.82713 q^{35} +0.497233 q^{37} -4.38234 q^{39} -1.00000 q^{41} -4.10393 q^{43} -23.6832 q^{45} +2.92536 q^{47} +1.00000 q^{49} -18.3736 q^{51} +3.08431 q^{53} -22.8210 q^{55} -0.225305 q^{57} -11.4408 q^{59} +2.94851 q^{61} -6.18825 q^{63} -5.53303 q^{65} +1.12488 q^{67} -13.4399 q^{69} -5.87671 q^{71} +15.7737 q^{73} -29.2420 q^{75} -5.96294 q^{77} +14.5736 q^{79} +10.7297 q^{81} +14.4941 q^{83} -23.1981 q^{85} +5.83998 q^{87} +0.670099 q^{89} -1.44574 q^{91} +5.34921 q^{93} -0.284465 q^{95} -10.5587 q^{97} +36.9002 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9	$ 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9} - 2 q^{11} + 5 q^{13} + 5 q^{15} + 13 q^{17} + 4 q^{21} - 2 q^{23} + 22 q^{25} - 10 q^{27} - 5 q^{29} - 17 q^{31} + 3 q^{33} + 5 q^{35} - 7 q^{37} - 5 q^{39} - 5 q^{41} - q^{43} - 23 q^{45} - 9 q^{47} + 5 q^{49} - 5 q^{51} + 5 q^{53} - 33 q^{55} - 3 q^{57} - 7 q^{59} + 22 q^{61} - q^{63} - 31 q^{65} + 3 q^{67} - 22 q^{69} + 24 q^{71} + 40 q^{73} - 24 q^{75} + 2 q^{77} + 42 q^{79} + 9 q^{81} + 12 q^{83} - 23 q^{85} + 32 q^{87} + 8 q^{89} - 5 q^{91} - 11 q^{93} + 17 q^{95} + 16 q^{97} + 45 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9 - 2 * q^11 + 5 * q^13 + 5 * q^15 + 13 * q^17 + 4 * q^21 - 2 * q^23 + 22 * q^25 - 10 * q^27 - 5 * q^29 - 17 * q^31 + 3 * q^33 + 5 * q^35 - 7 * q^37 - 5 * q^39 - 5 * q^41 - q^43 - 23 * q^45 - 9 * q^47 + 5 * q^49 - 5 * q^51 + 5 * q^53 - 33 * q^55 - 3 * q^57 - 7 * q^59 + 22 * q^61 - q^63 - 31 * q^65 + 3 * q^67 - 22 * q^69 + 24 * q^71 + 40 * q^73 - 24 * q^75 + 2 * q^77 + 42 * q^79 + 9 * q^81 + 12 * q^83 - 23 * q^85 + 32 * q^87 + 8 * q^89 - 5 * q^91 - 11 * q^93 + 17 * q^95 + 16 * q^97 + 45 * 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
$3$	−3.03121	−1.75007	−0.875036	−	0.484059i	$-0.839162\pi$
$3$	−3.03121	−1.75007	−0.875036	+	0.484059i	$0.839162\pi$
$4$	0	0
$5$	−3.82713	−1.71155	−0.855773	−	0.517351i	$-0.826918\pi$
$5$	−3.82713	−1.71155	−0.855773	+	0.517351i	$0.826918\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.18825	2.06275
$10$	0	0
$11$	5.96294	1.79789	0.898947	−	0.438057i	$-0.144333\pi$
$11$	5.96294	1.79789	0.898947	+	0.438057i	$0.144333\pi$
$12$	0	0
$13$	1.44574	0.400976	0.200488	−	0.979696i	$-0.435747\pi$
$13$	1.44574	0.400976	0.200488	+	0.979696i	$0.435747\pi$
$14$	0	0
$15$	11.6009	2.99533
$16$	0	0
$17$	6.06148	1.47012	0.735062	−	0.677999i	$-0.237153\pi$
$17$	6.06148	1.47012	0.735062	+	0.677999i	$0.237153\pi$
$18$	0	0
$19$	0.0743284	0.0170521	0.00852605	−	0.999964i	$-0.497286\pi$
$19$	0.0743284	0.0170521	0.00852605	+	0.999964i	$0.497286\pi$
$20$	0	0
$21$	3.03121	0.661465
$22$	0	0
$23$	4.43383	0.924518	0.462259	−	0.886745i	$-0.347039\pi$
$23$	4.43383	0.924518	0.462259	+	0.886745i	$0.347039\pi$
$24$	0	0
$25$	9.64695	1.92939
$26$	0	0
$27$	−9.66425	−1.85989
$28$	0	0
$29$	−1.92662	−0.357764	−0.178882	−	0.983871i	$-0.557248\pi$
$29$	−1.92662	−0.357764	−0.178882	+	0.983871i	$0.557248\pi$
$30$	0	0
$31$	−1.76471	−0.316951	−0.158476	−	0.987363i	$-0.550658\pi$
$31$	−1.76471	−0.316951	−0.158476	+	0.987363i	$0.550658\pi$
$32$	0	0
$33$	−18.0749	−3.14644
$34$	0	0
$35$	3.82713	0.646904
$36$	0	0
$37$	0.497233	0.0817446	0.0408723	−	0.999164i	$-0.486986\pi$
$37$	0.497233	0.0817446	0.0408723	+	0.999164i	$0.486986\pi$
$38$	0	0
$39$	−4.38234	−0.701736
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−4.10393	−0.625844	−0.312922	−	0.949779i	$-0.601308\pi$
$43$	−4.10393	−0.625844	−0.312922	+	0.949779i	$0.601308\pi$
$44$	0	0
$45$	−23.6832	−3.53049
$46$	0	0
$47$	2.92536	0.426708	0.213354	−	0.976975i	$-0.431561\pi$
$47$	2.92536	0.426708	0.213354	+	0.976975i	$0.431561\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−18.3736	−2.57282
$52$	0	0
$53$	3.08431	0.423663	0.211832	−	0.977306i	$-0.432057\pi$
$53$	3.08431	0.423663	0.211832	+	0.977306i	$0.432057\pi$
$54$	0	0
$55$	−22.8210	−3.07718
$56$	0	0
$57$	−0.225305	−0.0298424
$58$	0	0
$59$	−11.4408	−1.48947	−0.744735	−	0.667360i	$-0.767424\pi$
$59$	−11.4408	−1.48947	−0.744735	+	0.667360i	$0.767424\pi$
$60$	0	0
$61$	2.94851	0.377517	0.188759	−	0.982023i	$-0.439554\pi$
$61$	2.94851	0.377517	0.188759	+	0.982023i	$0.439554\pi$
$62$	0	0
$63$	−6.18825	−0.779646
$64$	0	0
$65$	−5.53303	−0.686288
$66$	0	0
$67$	1.12488	0.137426	0.0687129	−	0.997636i	$-0.478111\pi$
$67$	1.12488	0.137426	0.0687129	+	0.997636i	$0.478111\pi$
$68$	0	0
$69$	−13.4399	−1.61797
$70$	0	0
$71$	−5.87671	−0.697437	−0.348719	−	0.937227i	$-0.613383\pi$
$71$	−5.87671	−0.697437	−0.348719	+	0.937227i	$0.613383\pi$
$72$	0	0
$73$	15.7737	1.84617	0.923087	−	0.384591i	$-0.125657\pi$
$73$	15.7737	1.84617	0.923087	+	0.384591i	$0.125657\pi$
$74$	0	0
$75$	−29.2420	−3.37657
$76$	0	0
$77$	−5.96294	−0.679540
$78$	0	0
$79$	14.5736	1.63965	0.819827	−	0.572611i	$-0.194069\pi$
$79$	14.5736	1.63965	0.819827	+	0.572611i	$0.194069\pi$
$80$	0	0
$81$	10.7297	1.19218
$82$	0	0
$83$	14.4941	1.59093	0.795465	−	0.606000i	$-0.207227\pi$
$83$	14.4941	1.59093	0.795465	+	0.606000i	$0.207227\pi$
$84$	0	0
$85$	−23.1981	−2.51619
$86$	0	0
$87$	5.83998	0.626112
$88$	0	0
$89$	0.670099	0.0710303	0.0355152	−	0.999369i	$-0.488693\pi$
$89$	0.670099	0.0710303	0.0355152	+	0.999369i	$0.488693\pi$
$90$	0	0
$91$	−1.44574	−0.151555
$92$	0	0
$93$	5.34921	0.554687
$94$	0	0
$95$	−0.284465	−0.0291855
$96$	0	0
$97$	−10.5587	−1.07207	−0.536036	−	0.844195i	$-0.680079\pi$
$97$	−10.5587	−1.07207	−0.536036	+	0.844195i	$0.680079\pi$
$98$	0	0
$99$	36.9002	3.70861
$100$	0	0
$101$	−3.20947	−0.319355	−0.159677	−	0.987169i	$-0.551045\pi$
$101$	−3.20947	−0.319355	−0.159677	+	0.987169i	$0.551045\pi$
$102$	0	0
$103$	−5.79008	−0.570513	−0.285257	−	0.958451i	$-0.592079\pi$
$103$	−5.79008	−0.570513	−0.285257	+	0.958451i	$0.592079\pi$
$104$	0	0
$105$	−11.6009	−1.13213
$106$	0	0
$107$	5.90146	0.570516	0.285258	−	0.958451i	$-0.407921\pi$
$107$	5.90146	0.570516	0.285258	+	0.958451i	$0.407921\pi$
$108$	0	0
$109$	−13.4426	−1.28756	−0.643782	−	0.765209i	$-0.722636\pi$
$109$	−13.4426	−1.28756	−0.643782	+	0.765209i	$0.722636\pi$
$110$	0	0
$111$	−1.50722	−0.143059
$112$	0	0
$113$	6.01283	0.565639	0.282820	−	0.959173i	$-0.408730\pi$
$113$	6.01283	0.565639	0.282820	+	0.959173i	$0.408730\pi$
$114$	0	0
$115$	−16.9689	−1.58236
$116$	0	0
$117$	8.94659	0.827112
$118$	0	0
$119$	−6.06148	−0.555655
$120$	0	0
$121$	24.5567	2.23242
$122$	0	0
$123$	3.03121	0.273315
$124$	0	0
$125$	−17.7845	−1.59070
$126$	0	0
$127$	6.27284	0.556625	0.278312	−	0.960491i	$-0.410225\pi$
$127$	6.27284	0.556625	0.278312	+	0.960491i	$0.410225\pi$
$128$	0	0
$129$	12.4399	1.09527
$130$	0	0
$131$	−12.0382	−1.05178	−0.525892	−	0.850551i	$-0.676268\pi$
$131$	−12.0382	−1.05178	−0.525892	+	0.850551i	$0.676268\pi$
$132$	0	0
$133$	−0.0743284	−0.00644509
$134$	0	0
$135$	36.9864	3.18328
$136$	0	0
$137$	6.46697	0.552510	0.276255	−	0.961084i	$-0.410907\pi$
$137$	6.46697	0.552510	0.276255	+	0.961084i	$0.410907\pi$
$138$	0	0
$139$	8.66331	0.734812	0.367406	−	0.930061i	$-0.380246\pi$
$139$	8.66331	0.734812	0.367406	+	0.930061i	$0.380246\pi$
$140$	0	0
$141$	−8.86739	−0.746769
$142$	0	0
$143$	8.62085	0.720912
$144$	0	0
$145$	7.37342	0.612329
$146$	0	0
$147$	−3.03121	−0.250010
$148$	0	0
$149$	8.01997	0.657022	0.328511	−	0.944500i	$-0.393453\pi$
$149$	8.01997	0.657022	0.328511	+	0.944500i	$0.393453\pi$
$150$	0	0
$151$	4.83256	0.393268	0.196634	−	0.980477i	$-0.436999\pi$
$151$	4.83256	0.393268	0.196634	+	0.980477i	$0.436999\pi$
$152$	0	0
$153$	37.5099	3.03250
$154$	0	0
$155$	6.75378	0.542477
$156$	0	0
$157$	−10.2117	−0.814982	−0.407491	−	0.913209i	$-0.633596\pi$
$157$	−10.2117	−0.814982	−0.407491	+	0.913209i	$0.633596\pi$
$158$	0	0
$159$	−9.34921	−0.741441
$160$	0	0
$161$	−4.43383	−0.349435
$162$	0	0
$163$	19.2954	1.51133	0.755665	−	0.654958i	$-0.227314\pi$
$163$	19.2954	1.51133	0.755665	+	0.654958i	$0.227314\pi$
$164$	0	0
$165$	69.1752	5.38528
$166$	0	0
$167$	−2.42397	−0.187572	−0.0937861	−	0.995592i	$-0.529897\pi$
$167$	−2.42397	−0.187572	−0.0937861	+	0.995592i	$0.529897\pi$
$168$	0	0
$169$	−10.9098	−0.839218
$170$	0	0
$171$	0.459963	0.0351742
$172$	0	0
$173$	−8.79388	−0.668587	−0.334293	−	0.942469i	$-0.608498\pi$
$173$	−8.79388	−0.668587	−0.334293	+	0.942469i	$0.608498\pi$
$174$	0	0
$175$	−9.64695	−0.729241
$176$	0	0
$177$	34.6796	2.60668
$178$	0	0
$179$	−14.4831	−1.08252	−0.541259	−	0.840856i	$-0.682052\pi$
$179$	−14.4831	−1.08252	−0.541259	+	0.840856i	$0.682052\pi$
$180$	0	0
$181$	−3.07539	−0.228592	−0.114296	−	0.993447i	$-0.536461\pi$
$181$	−3.07539	−0.228592	−0.114296	+	0.993447i	$0.536461\pi$
$182$	0	0
$183$	−8.93755	−0.660682
$184$	0	0
$185$	−1.90298	−0.139910
$186$	0	0
$187$	36.1442	2.64313
$188$	0	0
$189$	9.66425	0.702971
$190$	0	0
$191$	−4.90956	−0.355243	−0.177622	−	0.984099i	$-0.556840\pi$
$191$	−4.90956	−0.355243	−0.177622	+	0.984099i	$0.556840\pi$
$192$	0	0
$193$	11.3050	0.813754	0.406877	−	0.913483i	$-0.366618\pi$
$193$	11.3050	0.813754	0.406877	+	0.913483i	$0.366618\pi$
$194$	0	0
$195$	16.7718	1.20105
$196$	0	0
$197$	−17.8594	−1.27243	−0.636215	−	0.771512i	$-0.719501\pi$
$197$	−17.8594	−1.27243	−0.636215	+	0.771512i	$0.719501\pi$
$198$	0	0
$199$	−13.6330	−0.966419	−0.483209	−	0.875505i	$-0.660529\pi$
$199$	−13.6330	−0.966419	−0.483209	+	0.875505i	$0.660529\pi$
$200$	0	0
$201$	−3.40974	−0.240505
$202$	0	0
$203$	1.92662	0.135222
$204$	0	0
$205$	3.82713	0.267299
$206$	0	0
$207$	27.4377	1.90705
$208$	0	0
$209$	0.443216	0.0306579
$210$	0	0
$211$	−22.4460	−1.54525	−0.772625	−	0.634863i	$-0.781056\pi$
$211$	−22.4460	−1.54525	−0.772625	+	0.634863i	$0.781056\pi$
$212$	0	0
$213$	17.8136	1.22056
$214$	0	0
$215$	15.7063	1.07116
$216$	0	0
$217$	1.76471	0.119796
$218$	0	0
$219$	−47.8135	−3.23094
$220$	0	0
$221$	8.76331	0.589484
$222$	0	0
$223$	−4.08663	−0.273661	−0.136831	−	0.990594i	$-0.543692\pi$
$223$	−4.08663	−0.273661	−0.136831	+	0.990594i	$0.543692\pi$
$224$	0	0
$225$	59.6977	3.97985
$226$	0	0
$227$	19.5310	1.29632	0.648158	−	0.761506i	$-0.275540\pi$
$227$	19.5310	1.29632	0.648158	+	0.761506i	$0.275540\pi$
$228$	0	0
$229$	23.0456	1.52290	0.761449	−	0.648225i	$-0.224488\pi$
$229$	23.0456	1.52290	0.761449	+	0.648225i	$0.224488\pi$
$230$	0	0
$231$	18.0749	1.18924
$232$	0	0
$233$	7.69494	0.504112	0.252056	−	0.967713i	$-0.418893\pi$
$233$	7.69494	0.504112	0.252056	+	0.967713i	$0.418893\pi$
$234$	0	0
$235$	−11.1957	−0.730330
$236$	0	0
$237$	−44.1756	−2.86951
$238$	0	0
$239$	12.2279	0.790960	0.395480	−	0.918475i	$-0.370578\pi$
$239$	12.2279	0.790960	0.395480	+	0.918475i	$0.370578\pi$
$240$	0	0
$241$	−3.68209	−0.237184	−0.118592	−	0.992943i	$-0.537838\pi$
$241$	−3.68209	−0.237184	−0.118592	+	0.992943i	$0.537838\pi$
$242$	0	0
$243$	−3.53112	−0.226521
$244$	0	0
$245$	−3.82713	−0.244507
$246$	0	0
$247$	0.107459	0.00683748
$248$	0	0
$249$	−43.9346	−2.78424
$250$	0	0
$251$	27.1519	1.71381	0.856907	−	0.515472i	$-0.172383\pi$
$251$	27.1519	1.71381	0.856907	+	0.515472i	$0.172383\pi$
$252$	0	0
$253$	26.4387	1.66219
$254$	0	0
$255$	70.3183	4.40351
$256$	0	0
$257$	−4.46344	−0.278422	−0.139211	−	0.990263i	$-0.544457\pi$
$257$	−4.46344	−0.278422	−0.139211	+	0.990263i	$0.544457\pi$
$258$	0	0
$259$	−0.497233	−0.0308965
$260$	0	0
$261$	−11.9224	−0.737977
$262$	0	0
$263$	−16.6595	−1.02727	−0.513633	−	0.858010i	$-0.671701\pi$
$263$	−16.6595	−1.02727	−0.513633	+	0.858010i	$0.671701\pi$
$264$	0	0
$265$	−11.8041	−0.725119
$266$	0	0
$267$	−2.03121	−0.124308
$268$	0	0
$269$	3.43160	0.209229	0.104614	−	0.994513i	$-0.466639\pi$
$269$	3.43160	0.209229	0.104614	+	0.994513i	$0.466639\pi$
$270$	0	0
$271$	−27.8083	−1.68924	−0.844618	−	0.535370i	$-0.820172\pi$
$271$	−27.8083	−1.68924	−0.844618	+	0.535370i	$0.820172\pi$
$272$	0	0
$273$	4.38234	0.265231
$274$	0	0
$275$	57.5242	3.46884
$276$	0	0
$277$	−13.5938	−0.816774	−0.408387	−	0.912809i	$-0.633909\pi$
$277$	−13.5938	−0.816774	−0.408387	+	0.912809i	$0.633909\pi$
$278$	0	0
$279$	−10.9205	−0.653791
$280$	0	0
$281$	2.37494	0.141677	0.0708384	−	0.997488i	$-0.477433\pi$
$281$	2.37494	0.141677	0.0708384	+	0.997488i	$0.477433\pi$
$282$	0	0
$283$	22.9301	1.36305	0.681526	−	0.731794i	$-0.261317\pi$
$283$	22.9301	1.36305	0.681526	+	0.731794i	$0.261317\pi$
$284$	0	0
$285$	0.862273	0.0510767
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	19.7415	1.16127
$290$	0	0
$291$	32.0056	1.87620
$292$	0	0
$293$	−7.30314	−0.426654	−0.213327	−	0.976981i	$-0.568430\pi$
$293$	−7.30314	−0.426654	−0.213327	+	0.976981i	$0.568430\pi$
$294$	0	0
$295$	43.7856	2.54930
$296$	0	0
$297$	−57.6274	−3.34388
$298$	0	0
$299$	6.41016	0.370709
$300$	0	0
$301$	4.10393	0.236547
$302$	0	0
$303$	9.72860	0.558893
$304$	0	0
$305$	−11.2843	−0.646139
$306$	0	0
$307$	−16.2292	−0.926247	−0.463124	−	0.886294i	$-0.653271\pi$
$307$	−16.2292	−0.926247	−0.463124	+	0.886294i	$0.653271\pi$
$308$	0	0
$309$	17.5509	0.998439
$310$	0	0
$311$	−24.8162	−1.40720	−0.703599	−	0.710598i	$-0.748425\pi$
$311$	−24.8162	−1.40720	−0.703599	+	0.710598i	$0.748425\pi$
$312$	0	0
$313$	1.00459	0.0567828	0.0283914	−	0.999597i	$-0.490962\pi$
$313$	1.00459	0.0567828	0.0283914	+	0.999597i	$0.490962\pi$
$314$	0	0
$315$	23.6832	1.33440
$316$	0	0
$317$	−29.9777	−1.68372	−0.841859	−	0.539697i	$-0.818539\pi$
$317$	−29.9777	−1.68372	−0.841859	+	0.539697i	$0.818539\pi$
$318$	0	0
$319$	−11.4883	−0.643221
$320$	0	0
$321$	−17.8886	−0.998443
$322$	0	0
$323$	0.450540	0.0250687
$324$	0	0
$325$	13.9470	0.773639
$326$	0	0
$327$	40.7473	2.25333
$328$	0	0
$329$	−2.92536	−0.161280
$330$	0	0
$331$	25.1471	1.38221	0.691106	−	0.722754i	$-0.257124\pi$
$331$	25.1471	1.38221	0.691106	+	0.722754i	$0.257124\pi$
$332$	0	0
$333$	3.07700	0.168619
$334$	0	0
$335$	−4.30506	−0.235210
$336$	0	0
$337$	1.81608	0.0989285	0.0494642	−	0.998776i	$-0.484249\pi$
$337$	1.81608	0.0989285	0.0494642	+	0.998776i	$0.484249\pi$
$338$	0	0
$339$	−18.2262	−0.989909
$340$	0	0
$341$	−10.5229	−0.569845
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	51.4363	2.76924
$346$	0	0
$347$	−23.5712	−1.26537	−0.632685	−	0.774409i	$-0.718047\pi$
$347$	−23.5712	−1.26537	−0.632685	+	0.774409i	$0.718047\pi$
$348$	0	0
$349$	22.3613	1.19697	0.598486	−	0.801133i	$-0.295769\pi$
$349$	22.3613	1.19697	0.598486	+	0.801133i	$0.295769\pi$
$350$	0	0
$351$	−13.9720	−0.745769
$352$	0	0
$353$	12.0297	0.640277	0.320139	−	0.947371i	$-0.396271\pi$
$353$	12.0297	0.640277	0.320139	+	0.947371i	$0.396271\pi$
$354$	0	0
$355$	22.4910	1.19370
$356$	0	0
$357$	18.3736	0.972436
$358$	0	0
$359$	29.0810	1.53483	0.767417	−	0.641148i	$-0.221541\pi$
$359$	29.0810	1.53483	0.767417	+	0.641148i	$0.221541\pi$
$360$	0	0
$361$	−18.9945	−0.999709
$362$	0	0
$363$	−74.4365	−3.90690
$364$	0	0
$365$	−60.3681	−3.15981
$366$	0	0
$367$	34.9839	1.82615	0.913073	−	0.407796i	$-0.133702\pi$
$367$	34.9839	1.82615	0.913073	+	0.407796i	$0.133702\pi$
$368$	0	0
$369$	−6.18825	−0.322147
$370$	0	0
$371$	−3.08431	−0.160130
$372$	0	0
$373$	25.0688	1.29801	0.649006	−	0.760783i	$-0.275185\pi$
$373$	25.0688	1.29801	0.649006	+	0.760783i	$0.275185\pi$
$374$	0	0
$375$	53.9086	2.78383
$376$	0	0
$377$	−2.78538	−0.143455
$378$	0	0
$379$	−16.2692	−0.835695	−0.417848	−	0.908517i	$-0.637215\pi$
$379$	−16.2692	−0.835695	−0.417848	+	0.908517i	$0.637215\pi$
$380$	0	0
$381$	−19.0143	−0.974133
$382$	0	0
$383$	7.88034	0.402667	0.201333	−	0.979523i	$-0.435473\pi$
$383$	7.88034	0.402667	0.201333	+	0.979523i	$0.435473\pi$
$384$	0	0
$385$	22.8210	1.16306
$386$	0	0
$387$	−25.3962	−1.29096
$388$	0	0
$389$	28.0477	1.42208	0.711038	−	0.703154i	$-0.248225\pi$
$389$	28.0477	1.42208	0.711038	+	0.703154i	$0.248225\pi$
$390$	0	0
$391$	26.8756	1.35916
$392$	0	0
$393$	36.4904	1.84070
$394$	0	0
$395$	−55.7750	−2.80634
$396$	0	0
$397$	12.8210	0.643466	0.321733	−	0.946830i	$-0.395735\pi$
$397$	12.8210	0.643466	0.321733	+	0.946830i	$0.395735\pi$
$398$	0	0
$399$	0.225305	0.0112794
$400$	0	0
$401$	−1.81684	−0.0907285	−0.0453643	−	0.998971i	$-0.514445\pi$
$401$	−1.81684	−0.0907285	−0.0453643	+	0.998971i	$0.514445\pi$
$402$	0	0
$403$	−2.55131	−0.127090
$404$	0	0
$405$	−41.0638	−2.04048
$406$	0	0
$407$	2.96497	0.146968
$408$	0	0
$409$	−35.2528	−1.74314	−0.871569	−	0.490273i	$-0.836897\pi$
$409$	−35.2528	−1.74314	−0.871569	+	0.490273i	$0.836897\pi$
$410$	0	0
$411$	−19.6027	−0.966932
$412$	0	0
$413$	11.4408	0.562967
$414$	0	0
$415$	−55.4707	−2.72295
$416$	0	0
$417$	−26.2603	−1.28597
$418$	0	0
$419$	−16.1804	−0.790464	−0.395232	−	0.918581i	$-0.629336\pi$
$419$	−16.1804	−0.790464	−0.395232	+	0.918581i	$0.629336\pi$
$420$	0	0
$421$	−8.95139	−0.436264	−0.218132	−	0.975919i	$-0.569996\pi$
$421$	−8.95139	−0.436264	−0.218132	+	0.975919i	$0.569996\pi$
$422$	0	0
$423$	18.1028	0.880191
$424$	0	0
$425$	58.4748	2.83644
$426$	0	0
$427$	−2.94851	−0.142688
$428$	0	0
$429$	−26.1316	−1.26165
$430$	0	0
$431$	−12.5790	−0.605907	−0.302954	−	0.953005i	$-0.597973\pi$
$431$	−12.5790	−0.605907	−0.302954	+	0.953005i	$0.597973\pi$
$432$	0	0
$433$	16.4245	0.789309	0.394654	−	0.918830i	$-0.370864\pi$
$433$	16.4245	0.789309	0.394654	+	0.918830i	$0.370864\pi$
$434$	0	0
$435$	−22.3504	−1.07162
$436$	0	0
$437$	0.329560	0.0157650
$438$	0	0
$439$	25.2770	1.20640	0.603202	−	0.797589i	$-0.293891\pi$
$439$	25.2770	1.20640	0.603202	+	0.797589i	$0.293891\pi$
$440$	0	0
$441$	6.18825	0.294678
$442$	0	0
$443$	0.450540	0.0214058	0.0107029	−	0.999943i	$-0.496593\pi$
$443$	0.450540	0.0214058	0.0107029	+	0.999943i	$0.496593\pi$
$444$	0	0
$445$	−2.56456	−0.121572
$446$	0	0
$447$	−24.3102	−1.14983
$448$	0	0
$449$	−18.2531	−0.861418	−0.430709	−	0.902491i	$-0.641736\pi$
$449$	−18.2531	−0.861418	−0.430709	+	0.902491i	$0.641736\pi$
$450$	0	0
$451$	−5.96294	−0.280784
$452$	0	0
$453$	−14.6485	−0.688247
$454$	0	0
$455$	5.53303	0.259393
$456$	0	0
$457$	28.5308	1.33461	0.667307	−	0.744783i	$-0.267447\pi$
$457$	28.5308	1.33461	0.667307	+	0.744783i	$0.267447\pi$
$458$	0	0
$459$	−58.5797	−2.73426
$460$	0	0
$461$	−7.30314	−0.340141	−0.170071	−	0.985432i	$-0.554400\pi$
$461$	−7.30314	−0.340141	−0.170071	+	0.985432i	$0.554400\pi$
$462$	0	0
$463$	16.5369	0.768535	0.384268	−	0.923222i	$-0.374454\pi$
$463$	16.5369	0.768535	0.384268	+	0.923222i	$0.374454\pi$
$464$	0	0
$465$	−20.4721	−0.949373
$466$	0	0
$467$	16.6909	0.772363	0.386181	−	0.922423i	$-0.373794\pi$
$467$	16.6909	0.772363	0.386181	+	0.922423i	$0.373794\pi$
$468$	0	0
$469$	−1.12488	−0.0519420
$470$	0	0
$471$	30.9538	1.42628
$472$	0	0
$473$	−24.4715	−1.12520
$474$	0	0
$475$	0.717043	0.0329002
$476$	0	0
$477$	19.0865	0.873911
$478$	0	0
$479$	−7.64133	−0.349141	−0.174571	−	0.984645i	$-0.555854\pi$
$479$	−7.64133	−0.349141	−0.174571	+	0.984645i	$0.555854\pi$
$480$	0	0
$481$	0.718869	0.0327776
$482$	0	0
$483$	13.4399	0.611536
$484$	0	0
$485$	40.4095	1.83490
$486$	0	0
$487$	11.6260	0.526825	0.263412	−	0.964683i	$-0.415152\pi$
$487$	11.6260	0.526825	0.263412	+	0.964683i	$0.415152\pi$
$488$	0	0
$489$	−58.4884	−2.64494
$490$	0	0
$491$	−25.9788	−1.17241	−0.586204	−	0.810164i	$-0.699378\pi$
$491$	−25.9788	−1.17241	−0.586204	+	0.810164i	$0.699378\pi$
$492$	0	0
$493$	−11.6781	−0.525957
$494$	0	0
$495$	−141.222	−6.34745
$496$	0	0
$497$	5.87671	0.263606
$498$	0	0
$499$	23.1920	1.03821	0.519107	−	0.854709i	$-0.326264\pi$
$499$	23.1920	1.03821	0.519107	+	0.854709i	$0.326264\pi$
$500$	0	0
$501$	7.34756	0.328265
$502$	0	0
$503$	−10.6434	−0.474564	−0.237282	−	0.971441i	$-0.576257\pi$
$503$	−10.6434	−0.474564	−0.237282	+	0.971441i	$0.576257\pi$
$504$	0	0
$505$	12.2831	0.546590
$506$	0	0
$507$	33.0700	1.46869
$508$	0	0
$509$	1.33530	0.0591860	0.0295930	−	0.999562i	$-0.490579\pi$
$509$	1.33530	0.0591860	0.0295930	+	0.999562i	$0.490579\pi$
$510$	0	0
$511$	−15.7737	−0.697788
$512$	0	0
$513$	−0.718329	−0.0317150
$514$	0	0
$515$	22.1594	0.976460
$516$	0	0
$517$	17.4438	0.767175
$518$	0	0
$519$	26.6561	1.17007
$520$	0	0
$521$	26.5594	1.16359	0.581795	−	0.813335i	$-0.302351\pi$
$521$	26.5594	1.16359	0.581795	+	0.813335i	$0.302351\pi$
$522$	0	0
$523$	9.41507	0.411692	0.205846	−	0.978584i	$-0.434005\pi$
$523$	9.41507	0.411692	0.205846	+	0.978584i	$0.434005\pi$
$524$	0	0
$525$	29.2420	1.27622
$526$	0	0
$527$	−10.6968	−0.465958
$528$	0	0
$529$	−3.34111	−0.145266
$530$	0	0
$531$	−70.7987	−3.07240
$532$	0	0
$533$	−1.44574	−0.0626219
$534$	0	0
$535$	−22.5857	−0.976464
$536$	0	0
$537$	43.9013	1.89448
$538$	0	0
$539$	5.96294	0.256842
$540$	0	0
$541$	−8.71416	−0.374651	−0.187325	−	0.982298i	$-0.559982\pi$
$541$	−8.71416	−0.374651	−0.187325	+	0.982298i	$0.559982\pi$
$542$	0	0
$543$	9.32217	0.400052
$544$	0	0
$545$	51.4465	2.20373
$546$	0	0
$547$	16.6541	0.712079	0.356040	−	0.934471i	$-0.384127\pi$
$547$	16.6541	0.712079	0.356040	+	0.934471i	$0.384127\pi$
$548$	0	0
$549$	18.2461	0.778724
$550$	0	0
$551$	−0.143202	−0.00610062
$552$	0	0
$553$	−14.5736	−0.619731
$554$	0	0
$555$	5.76833	0.244852
$556$	0	0
$557$	8.98351	0.380643	0.190322	−	0.981722i	$-0.439047\pi$
$557$	8.98351	0.380643	0.190322	+	0.981722i	$0.439047\pi$
$558$	0	0
$559$	−5.93321	−0.250948
$560$	0	0
$561$	−109.561	−4.62566
$562$	0	0
$563$	6.35931	0.268013	0.134007	−	0.990980i	$-0.457216\pi$
$563$	6.35931	0.268013	0.134007	+	0.990980i	$0.457216\pi$
$564$	0	0
$565$	−23.0119	−0.968118
$566$	0	0
$567$	−10.7297	−0.450603
$568$	0	0
$569$	−5.92160	−0.248246	−0.124123	−	0.992267i	$-0.539612\pi$
$569$	−5.92160	−0.248246	−0.124123	+	0.992267i	$0.539612\pi$
$570$	0	0
$571$	−45.8163	−1.91735	−0.958676	−	0.284502i	$-0.908172\pi$
$571$	−45.8163	−1.91735	−0.958676	+	0.284502i	$0.908172\pi$
$572$	0	0
$573$	14.8819	0.621701
$574$	0	0
$575$	42.7730	1.78376
$576$	0	0
$577$	37.9400	1.57946	0.789731	−	0.613453i	$-0.210220\pi$
$577$	37.9400	1.57946	0.789731	+	0.613453i	$0.210220\pi$
$578$	0	0
$579$	−34.2679	−1.42413
$580$	0	0
$581$	−14.4941	−0.601315
$582$	0	0
$583$	18.3916	0.761702
$584$	0	0
$585$	−34.2398	−1.41564
$586$	0	0
$587$	−5.81661	−0.240077	−0.120039	−	0.992769i	$-0.538302\pi$
$587$	−5.81661	−0.240077	−0.120039	+	0.992769i	$0.538302\pi$
$588$	0	0
$589$	−0.131168	−0.00540469
$590$	0	0
$591$	54.1357	2.22684
$592$	0	0
$593$	−16.7778	−0.688980	−0.344490	−	0.938790i	$-0.611948\pi$
$593$	−16.7778	−0.688980	−0.344490	+	0.938790i	$0.611948\pi$
$594$	0	0
$595$	23.1981	0.951029
$596$	0	0
$597$	41.3245	1.69130
$598$	0	0
$599$	13.8344	0.565260	0.282630	−	0.959229i	$-0.408793\pi$
$599$	13.8344	0.565260	0.282630	+	0.959229i	$0.408793\pi$
$600$	0	0
$601$	22.2486	0.907539	0.453769	−	0.891119i	$-0.350079\pi$
$601$	22.2486	0.907539	0.453769	+	0.891119i	$0.350079\pi$
$602$	0	0
$603$	6.96102	0.283475
$604$	0	0
$605$	−93.9817	−3.82090
$606$	0	0
$607$	2.39107	0.0970505	0.0485252	−	0.998822i	$-0.484548\pi$
$607$	2.39107	0.0970505	0.0485252	+	0.998822i	$0.484548\pi$
$608$	0	0
$609$	−5.83998	−0.236648
$610$	0	0
$611$	4.22931	0.171099
$612$	0	0
$613$	38.0777	1.53795	0.768973	−	0.639282i	$-0.220768\pi$
$613$	38.0777	1.53795	0.768973	+	0.639282i	$0.220768\pi$
$614$	0	0
$615$	−11.6009	−0.467792
$616$	0	0
$617$	−22.6988	−0.913819	−0.456910	−	0.889513i	$-0.651044\pi$
$617$	−22.6988	−0.913819	−0.456910	+	0.889513i	$0.651044\pi$
$618$	0	0
$619$	33.3077	1.33875	0.669376	−	0.742924i	$-0.266562\pi$
$619$	33.3077	1.33875	0.669376	+	0.742924i	$0.266562\pi$
$620$	0	0
$621$	−42.8497	−1.71950
$622$	0	0
$623$	−0.670099	−0.0268469
$624$	0	0
$625$	19.8289	0.793158
$626$	0	0
$627$	−1.34348	−0.0536535
$628$	0	0
$629$	3.01397	0.120175
$630$	0	0
$631$	5.52118	0.219795	0.109897	−	0.993943i	$-0.464948\pi$
$631$	5.52118	0.219795	0.109897	+	0.993943i	$0.464948\pi$
$632$	0	0
$633$	68.0387	2.70430
$634$	0	0
$635$	−24.0070	−0.952689
$636$	0	0
$637$	1.44574	0.0572822
$638$	0	0
$639$	−36.3665	−1.43864
$640$	0	0
$641$	12.7799	0.504775	0.252387	−	0.967626i	$-0.418784\pi$
$641$	12.7799	0.504775	0.252387	+	0.967626i	$0.418784\pi$
$642$	0	0
$643$	−19.0687	−0.751998	−0.375999	−	0.926620i	$-0.622700\pi$
$643$	−19.0687	−0.751998	−0.375999	+	0.926620i	$0.622700\pi$
$644$	0	0
$645$	−47.6091	−1.87461
$646$	0	0
$647$	−20.9933	−0.825331	−0.412665	−	0.910883i	$-0.635402\pi$
$647$	−20.9933	−0.825331	−0.412665	+	0.910883i	$0.635402\pi$
$648$	0	0
$649$	−68.2210	−2.67791
$650$	0	0
$651$	−5.34921	−0.209652
$652$	0	0
$653$	−36.4523	−1.42649	−0.713244	−	0.700916i	$-0.752775\pi$
$653$	−36.4523	−1.42649	−0.713244	+	0.700916i	$0.752775\pi$
$654$	0	0
$655$	46.0719	1.80018
$656$	0	0
$657$	97.6117	3.80819
$658$	0	0
$659$	20.9058	0.814373	0.407187	−	0.913345i	$-0.366510\pi$
$659$	20.9058	0.814373	0.407187	+	0.913345i	$0.366510\pi$
$660$	0	0
$661$	51.0624	1.98610	0.993048	−	0.117709i	$-0.0375550\pi$
$661$	51.0624	1.98610	0.993048	+	0.117709i	$0.0375550\pi$
$662$	0	0
$663$	−26.5635	−1.03164
$664$	0	0
$665$	0.284465	0.0110311
$666$	0	0
$667$	−8.54230	−0.330759
$668$	0	0
$669$	12.3875	0.478927
$670$	0	0
$671$	17.5818	0.678737
$672$	0	0
$673$	42.4278	1.63547	0.817736	−	0.575594i	$-0.195229\pi$
$673$	42.4278	1.63547	0.817736	+	0.575594i	$0.195229\pi$
$674$	0	0
$675$	−93.2306	−3.58845
$676$	0	0
$677$	15.1486	0.582209	0.291105	−	0.956691i	$-0.405977\pi$
$677$	15.1486	0.582209	0.291105	+	0.956691i	$0.405977\pi$
$678$	0	0
$679$	10.5587	0.405205
$680$	0	0
$681$	−59.2025	−2.26865
$682$	0	0
$683$	−8.69113	−0.332557	−0.166278	−	0.986079i	$-0.553175\pi$
$683$	−8.69113	−0.332557	−0.166278	+	0.986079i	$0.553175\pi$
$684$	0	0
$685$	−24.7499	−0.945647
$686$	0	0
$687$	−69.8562	−2.66518
$688$	0	0
$689$	4.45911	0.169879
$690$	0	0
$691$	−24.8701	−0.946102	−0.473051	−	0.881035i	$-0.656847\pi$
$691$	−24.8701	−0.946102	−0.473051	+	0.881035i	$0.656847\pi$
$692$	0	0
$693$	−36.9002	−1.40172
$694$	0	0
$695$	−33.1556	−1.25767
$696$	0	0
$697$	−6.06148	−0.229595
$698$	0	0
$699$	−23.3250	−0.882233
$700$	0	0
$701$	38.5821	1.45723	0.728613	−	0.684926i	$-0.240165\pi$
$701$	38.5821	1.45723	0.728613	+	0.684926i	$0.240165\pi$
$702$	0	0
$703$	0.0369585	0.00139392
$704$	0	0
$705$	33.9367	1.27813
$706$	0	0
$707$	3.20947	0.120705
$708$	0	0
$709$	13.3306	0.500642	0.250321	−	0.968163i	$-0.419464\pi$
$709$	13.3306	0.500642	0.250321	+	0.968163i	$0.419464\pi$
$710$	0	0
$711$	90.1848	3.38220
$712$	0	0
$713$	−7.82443	−0.293027
$714$	0	0
$715$	−32.9932	−1.23387
$716$	0	0
$717$	−37.0655	−1.38424
$718$	0	0
$719$	−19.0012	−0.708625	−0.354312	−	0.935127i	$-0.615285\pi$
$719$	−19.0012	−0.708625	−0.354312	+	0.935127i	$0.615285\pi$
$720$	0	0
$721$	5.79008	0.215634
$722$	0	0
$723$	11.1612	0.415090
$724$	0	0
$725$	−18.5860	−0.690266
$726$	0	0
$727$	−19.0336	−0.705916	−0.352958	−	0.935639i	$-0.614824\pi$
$727$	−19.0336	−0.705916	−0.352958	+	0.935639i	$0.614824\pi$
$728$	0	0
$729$	−21.4854	−0.795756
$730$	0	0
$731$	−24.8759	−0.920069
$732$	0	0
$733$	50.7508	1.87452	0.937261	−	0.348628i	$-0.113352\pi$
$733$	50.7508	1.87452	0.937261	+	0.348628i	$0.113352\pi$
$734$	0	0
$735$	11.6009	0.427904
$736$	0	0
$737$	6.70758	0.247077
$738$	0	0
$739$	6.70443	0.246626	0.123313	−	0.992368i	$-0.460648\pi$
$739$	6.70443	0.246626	0.123313	+	0.992368i	$0.460648\pi$
$740$	0	0
$741$	−0.325732	−0.0119661
$742$	0	0
$743$	−30.7597	−1.12846	−0.564232	−	0.825617i	$-0.690827\pi$
$743$	−30.7597	−1.12846	−0.564232	+	0.825617i	$0.690827\pi$
$744$	0	0
$745$	−30.6935	−1.12452
$746$	0	0
$747$	89.6928	3.28169
$748$	0	0
$749$	−5.90146	−0.215635
$750$	0	0
$751$	38.8738	1.41852	0.709262	−	0.704945i	$-0.249028\pi$
$751$	38.8738	1.41852	0.709262	+	0.704945i	$0.249028\pi$
$752$	0	0
$753$	−82.3032	−2.99930
$754$	0	0
$755$	−18.4948	−0.673096
$756$	0	0
$757$	3.89914	0.141717	0.0708583	−	0.997486i	$-0.477426\pi$
$757$	3.89914	0.141717	0.0708583	+	0.997486i	$0.477426\pi$
$758$	0	0
$759$	−80.1413	−2.90894
$760$	0	0
$761$	13.9575	0.505959	0.252979	−	0.967472i	$-0.418590\pi$
$761$	13.9575	0.505959	0.252979	+	0.967472i	$0.418590\pi$
$762$	0	0
$763$	13.4426	0.486653
$764$	0	0
$765$	−143.556	−5.19026
$766$	0	0
$767$	−16.5405	−0.597241
$768$	0	0
$769$	9.40607	0.339192	0.169596	−	0.985514i	$-0.445754\pi$
$769$	9.40607	0.339192	0.169596	+	0.985514i	$0.445754\pi$
$770$	0	0
$771$	13.5296	0.487258
$772$	0	0
$773$	6.38151	0.229527	0.114764	−	0.993393i	$-0.463389\pi$
$773$	6.38151	0.229527	0.114764	+	0.993393i	$0.463389\pi$
$774$	0	0
$775$	−17.0241	−0.611523
$776$	0	0
$777$	1.50722	0.0540711
$778$	0	0
$779$	−0.0743284	−0.00266309
$780$	0	0
$781$	−35.0425	−1.25392
$782$	0	0
$783$	18.6193	0.665400
$784$	0	0
$785$	39.0815	1.39488
$786$	0	0
$787$	31.4514	1.12112	0.560561	−	0.828113i	$-0.310586\pi$
$787$	31.4514	1.12112	0.560561	+	0.828113i	$0.310586\pi$
$788$	0	0
$789$	50.4984	1.79779
$790$	0	0
$791$	−6.01283	−0.213792
$792$	0	0
$793$	4.26277	0.151375
$794$	0	0
$795$	35.7807	1.26901
$796$	0	0
$797$	−12.7701	−0.452339	−0.226170	−	0.974088i	$-0.572620\pi$
$797$	−12.7701	−0.452339	−0.226170	+	0.974088i	$0.572620\pi$
$798$	0	0
$799$	17.7320	0.627313
$800$	0	0
$801$	4.14674	0.146518
$802$	0	0
$803$	94.0578	3.31923
$804$	0	0
$805$	16.9689	0.598074
$806$	0	0
$807$	−10.4019	−0.366165
$808$	0	0
$809$	37.8116	1.32939	0.664693	−	0.747117i	$-0.268563\pi$
$809$	37.8116	1.32939	0.664693	+	0.747117i	$0.268563\pi$
$810$	0	0
$811$	29.0566	1.02031	0.510157	−	0.860081i	$-0.329587\pi$
$811$	29.0566	1.02031	0.510157	+	0.860081i	$0.329587\pi$
$812$	0	0
$813$	84.2929	2.95628
$814$	0	0
$815$	−73.8460	−2.58671
$816$	0	0
$817$	−0.305039	−0.0106720
$818$	0	0
$819$	−8.94659	−0.312619
$820$	0	0
$821$	−22.1598	−0.773381	−0.386691	−	0.922209i	$-0.626382\pi$
$821$	−22.1598	−0.773381	−0.386691	+	0.922209i	$0.626382\pi$
$822$	0	0
$823$	44.5895	1.55429	0.777147	−	0.629319i	$-0.216666\pi$
$823$	44.5895	1.55429	0.777147	+	0.629319i	$0.216666\pi$
$824$	0	0
$825$	−174.368	−6.07072
$826$	0	0
$827$	−17.7093	−0.615814	−0.307907	−	0.951416i	$-0.599629\pi$
$827$	−17.7093	−0.615814	−0.307907	+	0.951416i	$0.599629\pi$
$828$	0	0
$829$	29.8270	1.03594	0.517968	−	0.855400i	$-0.326689\pi$
$829$	29.8270	1.03594	0.517968	+	0.855400i	$0.326689\pi$
$830$	0	0
$831$	41.2058	1.42941
$832$	0	0
$833$	6.06148	0.210018
$834$	0	0
$835$	9.27685	0.321039
$836$	0	0
$837$	17.0546	0.589493
$838$	0	0
$839$	−31.3123	−1.08102	−0.540511	−	0.841337i	$-0.681769\pi$
$839$	−31.3123	−1.08102	−0.540511	+	0.841337i	$0.681769\pi$
$840$	0	0
$841$	−25.2881	−0.872005
$842$	0	0
$843$	−7.19894	−0.247945
$844$	0	0
$845$	41.7534	1.43636
$846$	0	0
$847$	−24.5567	−0.843777
$848$	0	0
$849$	−69.5060	−2.38544
$850$	0	0
$851$	2.20465	0.0755744
$852$	0	0
$853$	33.0949	1.13315	0.566574	−	0.824011i	$-0.308269\pi$
$853$	33.0949	1.13315	0.566574	+	0.824011i	$0.308269\pi$
$854$	0	0
$855$	−1.76034	−0.0602023
$856$	0	0
$857$	−20.0025	−0.683272	−0.341636	−	0.939832i	$-0.610981\pi$
$857$	−20.0025	−0.683272	−0.341636	+	0.939832i	$0.610981\pi$
$858$	0	0
$859$	37.3721	1.27512	0.637559	−	0.770401i	$-0.279944\pi$
$859$	37.3721	1.27512	0.637559	+	0.770401i	$0.279944\pi$
$860$	0	0
$861$	−3.03121	−0.103303
$862$	0	0
$863$	−46.4668	−1.58175	−0.790875	−	0.611978i	$-0.790374\pi$
$863$	−46.4668	−1.58175	−0.790875	+	0.611978i	$0.790374\pi$
$864$	0	0
$865$	33.6554	1.14432
$866$	0	0
$867$	−59.8408	−2.03230
$868$	0	0
$869$	86.9013	2.94793
$870$	0	0
$871$	1.62628	0.0551044
$872$	0	0
$873$	−65.3397	−2.21142
$874$	0	0
$875$	17.7845	0.601226
$876$	0	0
$877$	−10.5546	−0.356404	−0.178202	−	0.983994i	$-0.557028\pi$
$877$	−10.5546	−0.356404	−0.178202	+	0.983994i	$0.557028\pi$
$878$	0	0
$879$	22.1374	0.746675
$880$	0	0
$881$	−25.0214	−0.842993	−0.421497	−	0.906830i	$-0.638495\pi$
$881$	−25.0214	−0.842993	−0.421497	+	0.906830i	$0.638495\pi$
$882$	0	0
$883$	−14.4626	−0.486704	−0.243352	−	0.969938i	$-0.578247\pi$
$883$	−14.4626	−0.486704	−0.243352	+	0.969938i	$0.578247\pi$
$884$	0	0
$885$	−132.723	−4.46145
$886$	0	0
$887$	5.65795	0.189975	0.0949877	−	0.995478i	$-0.469719\pi$
$887$	5.65795	0.189975	0.0949877	+	0.995478i	$0.469719\pi$
$888$	0	0
$889$	−6.27284	−0.210384
$890$	0	0
$891$	63.9803	2.14342
$892$	0	0
$893$	0.217437	0.00727626
$894$	0	0
$895$	55.4288	1.85278
$896$	0	0
$897$	−19.4306	−0.648768
$898$	0	0
$899$	3.39992	0.113394
$900$	0	0
$901$	18.6955	0.622838
$902$	0	0
$903$	−12.4399	−0.413974
$904$	0	0
$905$	11.7699	0.391246
$906$	0	0
$907$	45.9050	1.52425	0.762126	−	0.647429i	$-0.224156\pi$
$907$	45.9050	1.52425	0.762126	+	0.647429i	$0.224156\pi$
$908$	0	0
$909$	−19.8610	−0.658748
$910$	0	0
$911$	−42.5495	−1.40973	−0.704864	−	0.709342i	$-0.748992\pi$
$911$	−42.5495	−1.40973	−0.704864	+	0.709342i	$0.748992\pi$
$912$	0	0
$913$	86.4272	2.86032
$914$	0	0
$915$	34.2052	1.13079
$916$	0	0
$917$	12.0382	0.397537
$918$	0	0
$919$	25.1999	0.831269	0.415634	−	0.909532i	$-0.363560\pi$
$919$	25.1999	0.831269	0.415634	+	0.909532i	$0.363560\pi$
$920$	0	0
$921$	49.1940	1.62100
$922$	0	0
$923$	−8.49618	−0.279655
$924$	0	0
$925$	4.79678	0.157717
$926$	0	0
$927$	−35.8304	−1.17683
$928$	0	0
$929$	37.0410	1.21528	0.607639	−	0.794214i	$-0.292117\pi$
$929$	37.0410	1.21528	0.607639	+	0.794214i	$0.292117\pi$
$930$	0	0
$931$	0.0743284	0.00243602
$932$	0	0
$933$	75.2231	2.46270
$934$	0	0
$935$	−138.329	−4.52384
$936$	0	0
$937$	24.4307	0.798115	0.399058	−	0.916926i	$-0.369337\pi$
$937$	24.4307	0.798115	0.399058	+	0.916926i	$0.369337\pi$
$938$	0	0
$939$	−3.04512	−0.0993739
$940$	0	0
$941$	−42.4153	−1.38270	−0.691350	−	0.722520i	$-0.742984\pi$
$941$	−42.4153	−1.38270	−0.691350	+	0.722520i	$0.742984\pi$
$942$	0	0
$943$	−4.43383	−0.144386
$944$	0	0
$945$	−36.9864	−1.20317
$946$	0	0
$947$	10.0783	0.327501	0.163750	−	0.986502i	$-0.447641\pi$
$947$	10.0783	0.327501	0.163750	+	0.986502i	$0.447641\pi$
$948$	0	0
$949$	22.8047	0.740271
$950$	0	0
$951$	90.8689	2.94663
$952$	0	0
$953$	−50.1799	−1.62549	−0.812743	−	0.582623i	$-0.802026\pi$
$953$	−50.1799	−1.62549	−0.812743	+	0.582623i	$0.802026\pi$
$954$	0	0
$955$	18.7895	0.608015
$956$	0	0
$957$	34.8235	1.12568
$958$	0	0
$959$	−6.46697	−0.208829
$960$	0	0
$961$	−27.8858	−0.899542
$962$	0	0
$963$	36.5197	1.17683
$964$	0	0
$965$	−43.2659	−1.39278
$966$	0	0
$967$	−33.4107	−1.07441	−0.537207	−	0.843450i	$-0.680521\pi$
$967$	−33.4107	−1.07441	−0.537207	+	0.843450i	$0.680521\pi$
$968$	0	0
$969$	−1.36568	−0.0438721
$970$	0	0
$971$	57.1446	1.83386	0.916929	−	0.399051i	$-0.130660\pi$
$971$	57.1446	1.83386	0.916929	+	0.399051i	$0.130660\pi$
$972$	0	0
$973$	−8.66331	−0.277733
$974$	0	0
$975$	−42.2762	−1.35392
$976$	0	0
$977$	−5.17315	−0.165504	−0.0827519	−	0.996570i	$-0.526371\pi$
$977$	−5.17315	−0.165504	−0.0827519	+	0.996570i	$0.526371\pi$
$978$	0	0
$979$	3.99576	0.127705
$980$	0	0
$981$	−83.1859	−2.65592
$982$	0	0
$983$	−53.4989	−1.70635	−0.853175	−	0.521624i	$-0.825326\pi$
$983$	−53.4989	−1.70635	−0.853175	+	0.521624i	$0.825326\pi$
$984$	0	0
$985$	68.3503	2.17782
$986$	0	0
$987$	8.86739	0.282252
$988$	0	0
$989$	−18.1962	−0.578604
$990$	0	0
$991$	−16.6924	−0.530252	−0.265126	−	0.964214i	$-0.585413\pi$
$991$	−16.6924	−0.530252	−0.265126	+	0.964214i	$0.585413\pi$
$992$	0	0
$993$	−76.2263	−2.41897
$994$	0	0
$995$	52.1754	1.65407
$996$	0	0
$997$	0.818514	0.0259226	0.0129613	−	0.999916i	$-0.495874\pi$
$997$	0.818514	0.0259226	0.0129613	+	0.999916i	$0.495874\pi$
$998$	0	0
$999$	−4.80538	−0.152036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4592.2.a.bb.1.1		5
4.3	odd	2		287.2.a.e.1.5	✓	5
12.11	even	2		2583.2.a.r.1.1		5
20.19	odd	2		7175.2.a.n.1.1		5
28.27	even	2		2009.2.a.n.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.e.1.5	✓	5	4.3	odd	2
2009.2.a.n.1.5		5	28.27	even	2
2583.2.a.r.1.1		5	12.11	even	2
4592.2.a.bb.1.1		5	1.1	even	1	trivial
7175.2.a.n.1.1		5	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:	\( N \)	\(=\)	\( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4592.a (trivial)

\(f(q)\)	\(=\)	\(q-3.03121 q^{3} -3.82713 q^{5} -1.00000 q^{7} +6.18825 q^{9} +O(q^{10})\)	\(q-3.03121 q^{3} -3.82713 q^{5} -1.00000 q^{7} +6.18825 q^{9} +5.96294 q^{11} +1.44574 q^{13} +11.6009 q^{15} +6.06148 q^{17} +0.0743284 q^{19} +3.03121 q^{21} +4.43383 q^{23} +9.64695 q^{25} -9.66425 q^{27} -1.92662 q^{29} -1.76471 q^{31} -18.0749 q^{33} +3.82713 q^{35} +0.497233 q^{37} -4.38234 q^{39} -1.00000 q^{41} -4.10393 q^{43} -23.6832 q^{45} +2.92536 q^{47} +1.00000 q^{49} -18.3736 q^{51} +3.08431 q^{53} -22.8210 q^{55} -0.225305 q^{57} -11.4408 q^{59} +2.94851 q^{61} -6.18825 q^{63} -5.53303 q^{65} +1.12488 q^{67} -13.4399 q^{69} -5.87671 q^{71} +15.7737 q^{73} -29.2420 q^{75} -5.96294 q^{77} +14.5736 q^{79} +10.7297 q^{81} +14.4941 q^{83} -23.1981 q^{85} +5.83998 q^{87} +0.670099 q^{89} -1.44574 q^{91} +5.34921 q^{93} -0.284465 q^{95} -10.5587 q^{97} +36.9002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9	\( 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9} - 2 q^{11} + 5 q^{13} + 5 q^{15} + 13 q^{17} + 4 q^{21} - 2 q^{23} + 22 q^{25} - 10 q^{27} - 5 q^{29} - 17 q^{31} + 3 q^{33} + 5 q^{35} - 7 q^{37} - 5 q^{39} - 5 q^{41} - q^{43} - 23 q^{45} - 9 q^{47} + 5 q^{49} - 5 q^{51} + 5 q^{53} - 33 q^{55} - 3 q^{57} - 7 q^{59} + 22 q^{61} - q^{63} - 31 q^{65} + 3 q^{67} - 22 q^{69} + 24 q^{71} + 40 q^{73} - 24 q^{75} + 2 q^{77} + 42 q^{79} + 9 q^{81} + 12 q^{83} - 23 q^{85} + 32 q^{87} + 8 q^{89} - 5 q^{91} - 11 q^{93} + 17 q^{95} + 16 q^{97} + 45 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9 - 2 * q^11 + 5 * q^13 + 5 * q^15 + 13 * q^17 + 4 * q^21 - 2 * q^23 + 22 * q^25 - 10 * q^27 - 5 * q^29 - 17 * q^31 + 3 * q^33 + 5 * q^35 - 7 * q^37 - 5 * q^39 - 5 * q^41 - q^43 - 23 * q^45 - 9 * q^47 + 5 * q^49 - 5 * q^51 + 5 * q^53 - 33 * q^55 - 3 * q^57 - 7 * q^59 + 22 * q^61 - q^63 - 31 * q^65 + 3 * q^67 - 22 * q^69 + 24 * q^71 + 40 * q^73 - 24 * q^75 + 2 * q^77 + 42 * q^79 + 9 * q^81 + 12 * q^83 - 23 * q^85 + 32 * q^87 + 8 * q^89 - 5 * q^91 - 11 * q^93 + 17 * q^95 + 16 * q^97 + 45 * q^99

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