LMFDB - Embedded newform 2583.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2583,2,Mod(1,2583)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2583, 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("2583.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2583 = 3^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2583.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:	$20.6253588421$
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:	$\Z[a_1, a_2]$
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	$\chi$	$=$	2583.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03121 q^{2} +2.12582 q^{4} +3.82713 q^{5} +1.00000 q^{7} -0.255573 q^{8} +O(q^{10})$	\(q-2.03121 q^{2} +2.12582 q^{4} +3.82713 q^{5} +1.00000 q^{7} -0.255573 q^{8} -7.77372 q^{10} +5.96294 q^{11} +1.44574 q^{13} -2.03121 q^{14} -3.73252 q^{16} -6.06148 q^{17} -0.0743284 q^{19} +8.13581 q^{20} -12.1120 q^{22} +4.43383 q^{23} +9.64695 q^{25} -2.93660 q^{26} +2.12582 q^{28} +1.92662 q^{29} +1.76471 q^{31} +8.09269 q^{32} +12.3122 q^{34} +3.82713 q^{35} +0.497233 q^{37} +0.150977 q^{38} -0.978111 q^{40} +1.00000 q^{41} +4.10393 q^{43} +12.6762 q^{44} -9.00606 q^{46} +2.92536 q^{47} +1.00000 q^{49} -19.5950 q^{50} +3.07338 q^{52} -3.08431 q^{53} +22.8210 q^{55} -0.255573 q^{56} -3.91337 q^{58} -11.4408 q^{59} +2.94851 q^{61} -3.58450 q^{62} -8.97293 q^{64} +5.53303 q^{65} -1.12488 q^{67} -12.8856 q^{68} -7.77372 q^{70} -5.87671 q^{71} +15.7737 q^{73} -1.00999 q^{74} -0.158009 q^{76} +5.96294 q^{77} -14.5736 q^{79} -14.2849 q^{80} -2.03121 q^{82} +14.4941 q^{83} -23.1981 q^{85} -8.33596 q^{86} -1.52396 q^{88} -0.670099 q^{89} +1.44574 q^{91} +9.42555 q^{92} -5.94203 q^{94} -0.284465 q^{95} -10.5587 q^{97} -2.03121 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8	$ 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8} - 2 q^{11} + 5 q^{13} + q^{14} - q^{16} - 13 q^{17} + 23 q^{20} + q^{22} - 2 q^{23} + 22 q^{25} + 3 q^{28} + 5 q^{29} + 17 q^{31} + 12 q^{32} - 8 q^{34} + 5 q^{35} - 7 q^{37} + 3 q^{38} + 7 q^{40} + 5 q^{41} + q^{43} + 47 q^{44} - 24 q^{46} - 9 q^{47} + 5 q^{49} - 2 q^{50} + 20 q^{52} - 5 q^{53} + 33 q^{55} + 3 q^{56} - 27 q^{58} - 7 q^{59} + 22 q^{61} + 28 q^{62} - 3 q^{64} + 31 q^{65} - 3 q^{67} - 17 q^{68} + 24 q^{71} + 40 q^{73} + 5 q^{74} - 19 q^{76} - 2 q^{77} - 42 q^{79} - 24 q^{80} + q^{82} + 12 q^{83} - 23 q^{85} - 16 q^{86} + 26 q^{88} - 8 q^{89} + 5 q^{91} - 12 q^{92} - 23 q^{94} + 17 q^{95} + 16 q^{97} + q^{98}+O(q^{100}) $ 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8 - 2 * q^11 + 5 * q^13 + q^14 - q^16 - 13 * q^17 + 23 * q^20 + q^22 - 2 * q^23 + 22 * q^25 + 3 * q^28 + 5 * q^29 + 17 * q^31 + 12 * q^32 - 8 * q^34 + 5 * q^35 - 7 * q^37 + 3 * q^38 + 7 * q^40 + 5 * q^41 + q^43 + 47 * q^44 - 24 * q^46 - 9 * q^47 + 5 * q^49 - 2 * q^50 + 20 * q^52 - 5 * q^53 + 33 * q^55 + 3 * q^56 - 27 * q^58 - 7 * q^59 + 22 * q^61 + 28 * q^62 - 3 * q^64 + 31 * q^65 - 3 * q^67 - 17 * q^68 + 24 * q^71 + 40 * q^73 + 5 * q^74 - 19 * q^76 - 2 * q^77 - 42 * q^79 - 24 * q^80 + q^82 + 12 * q^83 - 23 * q^85 - 16 * q^86 + 26 * q^88 - 8 * q^89 + 5 * q^91 - 12 * q^92 - 23 * q^94 + 17 * q^95 + 16 * q^97 + q^98

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 $

$2$	−2.03121	−1.43628	−0.718142	−	0.695897i	$-0.755007\pi$
$2$	−2.03121	−1.43628	−0.718142	+	0.695897i	$0.755007\pi$
$3$	0	0
$3$	0	0
$4$	2.12582	1.06291
$5$	3.82713	1.71155	0.855773	−	0.517351i	$-0.173082\pi$
$5$	3.82713	1.71155	0.855773	+	0.517351i	$0.173082\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.255573	−0.0903586
$9$	0	0
$10$	−7.77372	−2.45827
$11$	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	$0.435747\pi$
$14$	−2.03121	−0.542864
$15$	0	0
$16$	−3.73252	−0.933131
$17$	−6.06148	−1.47012	−0.735062	−	0.677999i	$-0.762847\pi$
$17$	−6.06148	−1.47012	−0.735062	+	0.677999i	$0.762847\pi$
$18$	0	0
$19$	−0.0743284	−0.0170521	−0.00852605	−	0.999964i	$-0.502714\pi$
$19$	−0.0743284	−0.0170521	−0.00852605	+	0.999964i	$0.502714\pi$
$20$	8.13581	1.81922
$21$	0	0
$22$	−12.1120	−2.58229
$23$	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
$26$	−2.93660	−0.575915
$27$	0	0
$28$	2.12582	0.401743
$29$	1.92662	0.357764	0.178882	−	0.983871i	$-0.442752\pi$
$29$	1.92662	0.357764	0.178882	+	0.983871i	$0.442752\pi$
$30$	0	0
$31$	1.76471	0.316951	0.158476	−	0.987363i	$-0.449342\pi$
$31$	1.76471	0.316951	0.158476	+	0.987363i	$0.449342\pi$
$32$	8.09269	1.43060
$33$	0	0
$34$	12.3122	2.11152
$35$	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	$0.486986\pi$
$38$	0.150977	0.0244917
$39$	0	0
$40$	−0.978111	−0.154653
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	4.10393	0.625844	0.312922	−	0.949779i	$-0.398692\pi$
$43$	4.10393	0.625844	0.312922	+	0.949779i	$0.398692\pi$
$44$	12.6762	1.91100
$45$	0	0
$46$	−9.00606	−1.32787
$47$	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
$50$	−19.5950	−2.77115
$51$	0	0
$52$	3.07338	0.426202
$53$	−3.08431	−0.423663	−0.211832	−	0.977306i	$-0.567943\pi$
$53$	−3.08431	−0.423663	−0.211832	+	0.977306i	$0.567943\pi$
$54$	0	0
$55$	22.8210	3.07718
$56$	−0.255573	−0.0341523
$57$	0	0
$58$	−3.91337	−0.513850
$59$	−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	$0.439554\pi$
$62$	−3.58450	−0.455232
$63$	0	0
$64$	−8.97293	−1.12162
$65$	5.53303	0.686288
$66$	0	0
$67$	−1.12488	−0.137426	−0.0687129	−	0.997636i	$-0.521889\pi$
$67$	−1.12488	−0.137426	−0.0687129	+	0.997636i	$0.521889\pi$
$68$	−12.8856	−1.56261
$69$	0	0
$70$	−7.77372	−0.929137
$71$	−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	$0.125657\pi$
$74$	−1.00999	−0.117408
$75$	0	0
$76$	−0.158009	−0.0181249
$77$	5.96294	0.679540
$78$	0	0
$79$	−14.5736	−1.63965	−0.819827	−	0.572611i	$-0.805931\pi$
$79$	−14.5736	−1.63965	−0.819827	+	0.572611i	$0.805931\pi$
$80$	−14.2849	−1.59710
$81$	0	0
$82$	−2.03121	−0.224310
$83$	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
$86$	−8.33596	−0.898890
$87$	0	0
$88$	−1.52396	−0.162455
$89$	−0.670099	−0.0710303	−0.0355152	−	0.999369i	$-0.511307\pi$
$89$	−0.670099	−0.0710303	−0.0355152	+	0.999369i	$0.511307\pi$
$90$	0	0
$91$	1.44574	0.151555
$92$	9.42555	0.982681
$93$	0	0
$94$	−5.94203	−0.612873
$95$	−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	$0.680079\pi$
$98$	−2.03121	−0.205183
$99$	0	0
$100$	20.5077	2.05077
$101$	3.20947	0.319355	0.159677	−	0.987169i	$-0.448955\pi$
$101$	3.20947	0.319355	0.159677	+	0.987169i	$0.448955\pi$
$102$	0	0
$103$	5.79008	0.570513	0.285257	−	0.958451i	$-0.407921\pi$
$103$	5.79008	0.570513	0.285257	+	0.958451i	$0.407921\pi$
$104$	−0.369491	−0.0362316
$105$	0	0
$106$	6.26490	0.608500
$107$	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	$0.722636\pi$
$110$	−46.3542	−4.41970
$111$	0	0
$112$	−3.73252	−0.352690
$113$	−6.01283	−0.565639	−0.282820	−	0.959173i	$-0.591270\pi$
$113$	−6.01283	−0.565639	−0.282820	+	0.959173i	$0.591270\pi$
$114$	0	0
$115$	16.9689	1.58236
$116$	4.09564	0.380271
$117$	0	0
$118$	23.2388	2.13930
$119$	−6.06148	−0.555655
$120$	0	0
$121$	24.5567	2.23242
$122$	−5.98904	−0.542222
$123$	0	0
$124$	3.75146	0.336891
$125$	17.7845	1.59070
$126$	0	0
$127$	−6.27284	−0.556625	−0.278312	−	0.960491i	$-0.589775\pi$
$127$	−6.27284	−0.556625	−0.278312	+	0.960491i	$0.589775\pi$
$128$	2.04053	0.180360
$129$	0	0
$130$	−11.2388	−0.985705
$131$	−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
$134$	2.28487	0.197382
$135$	0	0
$136$	1.54915	0.132838
$137$	−6.46697	−0.552510	−0.276255	−	0.961084i	$-0.589093\pi$
$137$	−6.46697	−0.552510	−0.276255	+	0.961084i	$0.589093\pi$
$138$	0	0
$139$	−8.66331	−0.734812	−0.367406	−	0.930061i	$-0.619754\pi$
$139$	−8.66331	−0.734812	−0.367406	+	0.930061i	$0.619754\pi$
$140$	8.13581	0.687601
$141$	0	0
$142$	11.9368	1.00172
$143$	8.62085	0.720912
$144$	0	0
$145$	7.37342	0.612329
$146$	−32.0398	−2.65163
$147$	0	0
$148$	1.05703	0.0868872
$149$	−8.01997	−0.657022	−0.328511	−	0.944500i	$-0.606547\pi$
$149$	−8.01997	−0.657022	−0.328511	+	0.944500i	$0.606547\pi$
$150$	0	0
$151$	−4.83256	−0.393268	−0.196634	−	0.980477i	$-0.563001\pi$
$151$	−4.83256	−0.393268	−0.196634	+	0.980477i	$0.563001\pi$
$152$	0.0189963	0.00154080
$153$	0	0
$154$	−12.1120	−0.976013
$155$	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	$0.633596\pi$
$158$	29.6020	2.35501
$159$	0	0
$160$	30.9718	2.44854
$161$	4.43383	0.349435
$162$	0	0
$163$	−19.2954	−1.51133	−0.755665	−	0.654958i	$-0.772686\pi$
$163$	−19.2954	−1.51133	−0.755665	+	0.654958i	$0.772686\pi$
$164$	2.12582	0.165999
$165$	0	0
$166$	−29.4405	−2.28503
$167$	−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
$170$	47.1202	3.61396
$171$	0	0
$172$	8.72423	0.665217
$173$	8.79388	0.668587	0.334293	−	0.942469i	$-0.391502\pi$
$173$	8.79388	0.668587	0.334293	+	0.942469i	$0.391502\pi$
$174$	0	0
$175$	9.64695	0.729241
$176$	−22.2568	−1.67767
$177$	0	0
$178$	1.36111	0.102020
$179$	−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	$0.536461\pi$
$182$	−2.93660	−0.217675
$183$	0	0
$184$	−1.13317	−0.0835382
$185$	1.90298	0.139910
$186$	0	0
$187$	−36.1442	−2.64313
$188$	6.21880	0.453552
$189$	0	0
$190$	0.577808	0.0419186
$191$	−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	$0.366618\pi$
$194$	21.4469	1.53980
$195$	0	0
$196$	2.12582	0.151844
$197$	17.8594	1.27243	0.636215	−	0.771512i	$-0.280499\pi$
$197$	17.8594	1.27243	0.636215	+	0.771512i	$0.280499\pi$
$198$	0	0
$199$	13.6330	0.966419	0.483209	−	0.875505i	$-0.339471\pi$
$199$	13.6330	0.966419	0.483209	+	0.875505i	$0.339471\pi$
$200$	−2.46550	−0.174337
$201$	0	0
$202$	−6.51912	−0.458684
$203$	1.92662	0.135222
$204$	0	0
$205$	3.82713	0.267299
$206$	−11.7609	−0.819419
$207$	0	0
$208$	−5.39625	−0.374163
$209$	−0.443216	−0.0306579
$210$	0	0
$211$	22.4460	1.54525	0.772625	−	0.634863i	$-0.218944\pi$
$211$	22.4460	1.54525	0.772625	+	0.634863i	$0.218944\pi$
$212$	−6.55670	−0.450316
$213$	0	0
$214$	−11.9871	−0.819423
$215$	15.7063	1.07116
$216$	0	0
$217$	1.76471	0.119796
$218$	27.3047	1.84931
$219$	0	0
$220$	48.5133	3.27077
$221$	−8.76331	−0.589484
$222$	0	0
$223$	4.08663	0.273661	0.136831	−	0.990594i	$-0.456308\pi$
$223$	4.08663	0.273661	0.136831	+	0.990594i	$0.456308\pi$
$224$	8.09269	0.540716
$225$	0	0
$226$	12.2133	0.812419
$227$	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	$0.224488\pi$
$230$	−34.4674	−2.27271
$231$	0	0
$232$	−0.492390	−0.0323270
$233$	−7.69494	−0.504112	−0.252056	−	0.967713i	$-0.581107\pi$
$233$	−7.69494	−0.504112	−0.252056	+	0.967713i	$0.581107\pi$
$234$	0	0
$235$	11.1957	0.730330
$236$	−24.3212	−1.58317
$237$	0	0
$238$	12.3122	0.798078
$239$	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	$0.537838\pi$
$242$	−49.8798	−3.20640
$243$	0	0
$244$	6.26800	0.401268
$245$	3.82713	0.244507
$246$	0	0
$247$	−0.107459	−0.00683748
$248$	−0.451011	−0.0286393
$249$	0	0
$250$	−36.1241	−2.28469
$251$	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
$254$	12.7415	0.799471
$255$	0	0
$256$	13.8011	0.862568
$257$	4.46344	0.278422	0.139211	−	0.990263i	$-0.455543\pi$
$257$	4.46344	0.278422	0.139211	+	0.990263i	$0.455543\pi$
$258$	0	0
$259$	0.497233	0.0308965
$260$	11.7623	0.729464
$261$	0	0
$262$	24.4522	1.51066
$263$	−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
$266$	0.150977	0.00925698
$267$	0	0
$268$	−2.39129	−0.146071
$269$	−3.43160	−0.209229	−0.104614	−	0.994513i	$-0.533361\pi$
$269$	−3.43160	−0.209229	−0.104614	+	0.994513i	$0.533361\pi$
$270$	0	0
$271$	27.8083	1.68924	0.844618	−	0.535370i	$-0.179828\pi$
$271$	27.8083	1.68924	0.844618	+	0.535370i	$0.179828\pi$
$272$	22.6246	1.37182
$273$	0	0
$274$	13.1358	0.793561
$275$	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	$0.633909\pi$
$278$	17.5970	1.05540
$279$	0	0
$280$	−0.978111	−0.0584533
$281$	−2.37494	−0.141677	−0.0708384	−	0.997488i	$-0.522567\pi$
$281$	−2.37494	−0.141677	−0.0708384	+	0.997488i	$0.522567\pi$
$282$	0	0
$283$	−22.9301	−1.36305	−0.681526	−	0.731794i	$-0.738683\pi$
$283$	−22.9301	−1.36305	−0.681526	+	0.731794i	$0.738683\pi$
$284$	−12.4928	−0.741314
$285$	0	0
$286$	−17.5108	−1.03543
$287$	1.00000	0.0590281
$288$	0	0
$289$	19.7415	1.16127
$290$	−14.9770	−0.879478
$291$	0	0
$292$	33.5321	1.96232
$293$	7.30314	0.426654	0.213327	−	0.976981i	$-0.431570\pi$
$293$	7.30314	0.426654	0.213327	+	0.976981i	$0.431570\pi$
$294$	0	0
$295$	−43.7856	−2.54930
$296$	−0.127079	−0.00738632
$297$	0	0
$298$	16.2903	0.943670
$299$	6.41016	0.370709
$300$	0	0
$301$	4.10393	0.236547
$302$	9.81595	0.564845
$303$	0	0
$304$	0.277433	0.0159118
$305$	11.2843	0.646139
$306$	0	0
$307$	16.2292	0.926247	0.463124	−	0.886294i	$-0.346729\pi$
$307$	16.2292	0.926247	0.463124	+	0.886294i	$0.346729\pi$
$308$	12.6762	0.722291
$309$	0	0
$310$	−13.7184	−0.779150
$311$	−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	$0.490962\pi$
$314$	20.7421	1.17055
$315$	0	0
$316$	−30.9808	−1.74281
$317$	29.9777	1.68372	0.841859	−	0.539697i	$-0.181461\pi$
$317$	29.9777	1.68372	0.841859	+	0.539697i	$0.181461\pi$
$318$	0	0
$319$	11.4883	0.643221
$320$	−34.3406	−1.91970
$321$	0	0
$322$	−9.00606	−0.501888
$323$	0.450540	0.0250687
$324$	0	0
$325$	13.9470	0.773639
$326$	39.1930	2.17070
$327$	0	0
$328$	−0.255573	−0.0141116
$329$	2.92536	0.161280
$330$	0	0
$331$	−25.1471	−1.38221	−0.691106	−	0.722754i	$-0.742876\pi$
$331$	−25.1471	−1.38221	−0.691106	+	0.722754i	$0.742876\pi$
$332$	30.8118	1.69102
$333$	0	0
$334$	4.92359	0.269407
$335$	−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	$0.484249\pi$
$338$	22.1602	1.20536
$339$	0	0
$340$	−49.3150	−2.67448
$341$	10.5229	0.569845
$342$	0	0
$343$	1.00000	0.0539949
$344$	−1.04885	−0.0565504
$345$	0	0
$346$	−17.8622	−0.960280
$347$	−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	$0.295769\pi$
$350$	−19.5950	−1.04740
$351$	0	0
$352$	48.2562	2.57207
$353$	−12.0297	−0.640277	−0.320139	−	0.947371i	$-0.603729\pi$
$353$	−12.0297	−0.640277	−0.320139	+	0.947371i	$0.603729\pi$
$354$	0	0
$355$	−22.4910	−1.19370
$356$	−1.42451	−0.0754990
$357$	0	0
$358$	29.4182	1.55480
$359$	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
$362$	6.24677	0.328323
$363$	0	0
$364$	3.07338	0.161089
$365$	60.3681	3.15981
$366$	0	0
$367$	−34.9839	−1.82615	−0.913073	−	0.407796i	$-0.866298\pi$
$367$	−34.9839	−1.82615	−0.913073	+	0.407796i	$0.866298\pi$
$368$	−16.5494	−0.862697
$369$	0	0
$370$	−3.86535	−0.200950
$371$	−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	$0.275185\pi$
$374$	73.4166	3.79628
$375$	0	0
$376$	−0.747642	−0.0385567
$377$	2.78538	0.143455
$378$	0	0
$379$	16.2692	0.835695	0.417848	−	0.908517i	$-0.362785\pi$
$379$	16.2692	0.835695	0.417848	+	0.908517i	$0.362785\pi$
$380$	−0.604722	−0.0310216
$381$	0	0
$382$	9.97235	0.510230
$383$	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
$386$	−22.9629	−1.16878
$387$	0	0
$388$	−22.4459	−1.13952
$389$	−28.0477	−1.42208	−0.711038	−	0.703154i	$-0.751775\pi$
$389$	−28.0477	−1.42208	−0.711038	+	0.703154i	$0.751775\pi$
$390$	0	0
$391$	−26.8756	−1.35916
$392$	−0.255573	−0.0129084
$393$	0	0
$394$	−36.2762	−1.82757
$395$	−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	$0.395735\pi$
$398$	−27.6915	−1.38805
$399$	0	0
$400$	−36.0075	−1.80037
$401$	1.81684	0.0907285	0.0453643	−	0.998971i	$-0.485555\pi$
$401$	1.81684	0.0907285	0.0453643	+	0.998971i	$0.485555\pi$
$402$	0	0
$403$	2.55131	0.127090
$404$	6.82277	0.339446
$405$	0	0
$406$	−3.91337	−0.194217
$407$	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	$0.836897\pi$
$410$	−7.77372	−0.383917
$411$	0	0
$412$	12.3087	0.606405
$413$	−11.4408	−0.562967
$414$	0	0
$415$	55.4707	2.72295
$416$	11.6999	0.573636
$417$	0	0
$418$	0.900266	0.0440334
$419$	−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	$0.569996\pi$
$422$	−45.5927	−2.21942
$423$	0	0
$424$	0.788266	0.0382816
$425$	−58.4748	−2.83644
$426$	0	0
$427$	2.94851	0.142688
$428$	12.5455	0.606408
$429$	0	0
$430$	−31.9028	−1.53849
$431$	−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	$0.370864\pi$
$434$	−3.58450	−0.172061
$435$	0	0
$436$	−28.5765	−1.36857
$437$	−0.329560	−0.0157650
$438$	0	0
$439$	−25.2770	−1.20640	−0.603202	−	0.797589i	$-0.706109\pi$
$439$	−25.2770	−1.20640	−0.603202	+	0.797589i	$0.706109\pi$
$440$	−5.83242	−0.278050
$441$	0	0
$442$	17.8002	0.846667
$443$	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
$446$	−8.30082	−0.393055
$447$	0	0
$448$	−8.97293	−0.423931
$449$	18.2531	0.861418	0.430709	−	0.902491i	$-0.358264\pi$
$449$	18.2531	0.861418	0.430709	+	0.902491i	$0.358264\pi$
$450$	0	0
$451$	5.96294	0.280784
$452$	−12.7822	−0.601224
$453$	0	0
$454$	−39.6716	−1.86188
$455$	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	$0.267447\pi$
$458$	−46.8105	−2.18731
$459$	0	0
$460$	36.0728	1.68190
$461$	7.30314	0.340141	0.170071	−	0.985432i	$-0.445600\pi$
$461$	7.30314	0.340141	0.170071	+	0.985432i	$0.445600\pi$
$462$	0	0
$463$	−16.5369	−0.768535	−0.384268	−	0.923222i	$-0.625546\pi$
$463$	−16.5369	−0.768535	−0.384268	+	0.923222i	$0.625546\pi$
$464$	−7.19114	−0.333840
$465$	0	0
$466$	15.6301	0.724048
$467$	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
$470$	−22.7409	−1.04896
$471$	0	0
$472$	2.92396	0.134586
$473$	24.4715	1.12520
$474$	0	0
$475$	−0.717043	−0.0329002
$476$	−12.8856	−0.590612
$477$	0	0
$478$	−24.8376	−1.13604
$479$	−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
$482$	7.47911	0.340664
$483$	0	0
$484$	52.2031	2.37287
$485$	−40.4095	−1.83490
$486$	0	0
$487$	−11.6260	−0.526825	−0.263412	−	0.964683i	$-0.584848\pi$
$487$	−11.6260	−0.526825	−0.263412	+	0.964683i	$0.584848\pi$
$488$	−0.753557	−0.0341119
$489$	0	0
$490$	−7.77372	−0.351181
$491$	−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
$494$	0.218273	0.00982056
$495$	0	0
$496$	−6.58682	−0.295757
$497$	−5.87671	−0.263606
$498$	0	0
$499$	−23.1920	−1.03821	−0.519107	−	0.854709i	$-0.673736\pi$
$499$	−23.1920	−1.03821	−0.519107	+	0.854709i	$0.673736\pi$
$500$	37.8067	1.69077
$501$	0	0
$502$	−55.1513	−2.46152
$503$	−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
$506$	−53.7026	−2.38737
$507$	0	0
$508$	−13.3350	−0.591643
$509$	−1.33530	−0.0591860	−0.0295930	−	0.999562i	$-0.509421\pi$
$509$	−1.33530	−0.0591860	−0.0295930	+	0.999562i	$0.509421\pi$
$510$	0	0
$511$	15.7737	0.697788
$512$	−32.1140	−1.41925
$513$	0	0
$514$	−9.06619	−0.399893
$515$	22.1594	0.976460
$516$	0	0
$517$	17.4438	0.767175
$518$	−1.00999	−0.0443762
$519$	0	0
$520$	−1.41409	−0.0620120
$521$	−26.5594	−1.16359	−0.581795	−	0.813335i	$-0.697649\pi$
$521$	−26.5594	−1.16359	−0.581795	+	0.813335i	$0.697649\pi$
$522$	0	0
$523$	−9.41507	−0.411692	−0.205846	−	0.978584i	$-0.565995\pi$
$523$	−9.41507	−0.411692	−0.205846	+	0.978584i	$0.565995\pi$
$524$	−25.5911	−1.11795
$525$	0	0
$526$	33.8389	1.47545
$527$	−10.6968	−0.465958
$528$	0	0
$529$	−3.34111	−0.145266
$530$	23.9766	1.04148
$531$	0	0
$532$	−0.158009	−0.00685056
$533$	1.44574	0.0626219
$534$	0	0
$535$	22.5857	0.976464
$536$	0.287488	0.0124176
$537$	0	0
$538$	6.97032	0.300512
$539$	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	$0.559982\pi$
$542$	−56.4846	−2.42622
$543$	0	0
$544$	−49.0537	−2.10316
$545$	−51.4465	−2.20373
$546$	0	0
$547$	−16.6541	−0.712079	−0.356040	−	0.934471i	$-0.615873\pi$
$547$	−16.6541	−0.712079	−0.356040	+	0.934471i	$0.615873\pi$
$548$	−13.7476	−0.587269
$549$	0	0
$550$	−116.844	−4.98224
$551$	−0.143202	−0.00610062
$552$	0	0
$553$	−14.5736	−0.619731
$554$	27.6119	1.17312
$555$	0	0
$556$	−18.4167	−0.781040
$557$	−8.98351	−0.380643	−0.190322	−	0.981722i	$-0.560953\pi$
$557$	−8.98351	−0.380643	−0.190322	+	0.981722i	$0.560953\pi$
$558$	0	0
$559$	5.93321	0.250948
$560$	−14.2849	−0.603646
$561$	0	0
$562$	4.82400	0.203488
$563$	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
$566$	46.5759	1.95773
$567$	0	0
$568$	1.50193	0.0630194
$569$	5.92160	0.248246	0.124123	−	0.992267i	$-0.460388\pi$
$569$	5.92160	0.248246	0.124123	+	0.992267i	$0.460388\pi$
$570$	0	0
$571$	45.8163	1.91735	0.958676	−	0.284502i	$-0.0918281\pi$
$571$	45.8163	1.91735	0.958676	+	0.284502i	$0.0918281\pi$
$572$	18.3264	0.766266
$573$	0	0
$574$	−2.03121	−0.0847812
$575$	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	$0.210220\pi$
$578$	−40.0992	−1.66791
$579$	0	0
$580$	15.6746	0.650852
$581$	14.4941	0.601315
$582$	0	0
$583$	−18.3916	−0.761702
$584$	−4.03133	−0.166818
$585$	0	0
$586$	−14.8342	−0.612796
$587$	−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
$590$	88.9379	3.66151
$591$	0	0
$592$	−1.85593	−0.0762784
$593$	16.7778	0.688980	0.344490	−	0.938790i	$-0.388052\pi$
$593$	16.7778	0.688980	0.344490	+	0.938790i	$0.388052\pi$
$594$	0	0
$595$	−23.1981	−0.951029
$596$	−17.0490	−0.698356
$597$	0	0
$598$	−13.0204	−0.532444
$599$	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	$0.350079\pi$
$602$	−8.33596	−0.339748
$603$	0	0
$604$	−10.2732	−0.418009
$605$	93.9817	3.82090
$606$	0	0
$607$	−2.39107	−0.0970505	−0.0485252	−	0.998822i	$-0.515452\pi$
$607$	−2.39107	−0.0970505	−0.0485252	+	0.998822i	$0.515452\pi$
$608$	−0.601517	−0.0243947
$609$	0	0
$610$	−22.9209	−0.928038
$611$	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	$0.220768\pi$
$614$	−32.9649	−1.33035
$615$	0	0
$616$	−1.52396	−0.0614023
$617$	22.6988	0.913819	0.456910	−	0.889513i	$-0.348956\pi$
$617$	22.6988	0.913819	0.456910	+	0.889513i	$0.348956\pi$
$618$	0	0
$619$	−33.3077	−1.33875	−0.669376	−	0.742924i	$-0.733438\pi$
$619$	−33.3077	−1.33875	−0.669376	+	0.742924i	$0.733438\pi$
$620$	14.3573	0.576605
$621$	0	0
$622$	50.4070	2.02113
$623$	−0.670099	−0.0268469
$624$	0	0
$625$	19.8289	0.793158
$626$	−2.04053	−0.0815562
$627$	0	0
$628$	−21.7083	−0.866254
$629$	−3.01397	−0.120175
$630$	0	0
$631$	−5.52118	−0.219795	−0.109897	−	0.993943i	$-0.535052\pi$
$631$	−5.52118	−0.219795	−0.109897	+	0.993943i	$0.535052\pi$
$632$	3.72461	0.148157
$633$	0	0
$634$	−60.8912	−2.41830
$635$	−24.0070	−0.952689
$636$	0	0
$637$	1.44574	0.0572822
$638$	−23.3352	−0.923848
$639$	0	0
$640$	7.80940	0.308694
$641$	−12.7799	−0.504775	−0.252387	−	0.967626i	$-0.581216\pi$
$641$	−12.7799	−0.504775	−0.252387	+	0.967626i	$0.581216\pi$
$642$	0	0
$643$	19.0687	0.751998	0.375999	−	0.926620i	$-0.377300\pi$
$643$	19.0687	0.751998	0.375999	+	0.926620i	$0.377300\pi$
$644$	9.42555	0.371419
$645$	0	0
$646$	−0.915143	−0.0360058
$647$	−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
$650$	−28.3293	−1.11116
$651$	0	0
$652$	−41.0185	−1.60641
$653$	36.4523	1.42649	0.713244	−	0.700916i	$-0.247225\pi$
$653$	36.4523	1.42649	0.713244	+	0.700916i	$0.247225\pi$
$654$	0	0
$655$	−46.0719	−1.80018
$656$	−3.73252	−0.145731
$657$	0	0
$658$	−5.94203	−0.231644
$659$	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	$0.0375550\pi$
$662$	51.0792	1.98525
$663$	0	0
$664$	−3.70428	−0.143754
$665$	−0.284465	−0.0110311
$666$	0	0
$667$	8.54230	0.330759
$668$	−5.15293	−0.199373
$669$	0	0
$670$	8.74449	0.337829
$671$	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	$0.195229\pi$
$674$	−3.68885	−0.142089
$675$	0	0
$676$	−23.1924	−0.892015
$677$	−15.1486	−0.582209	−0.291105	−	0.956691i	$-0.594023\pi$
$677$	−15.1486	−0.582209	−0.291105	+	0.956691i	$0.594023\pi$
$678$	0	0
$679$	−10.5587	−0.405205
$680$	5.92880	0.227359
$681$	0	0
$682$	−21.3742	−0.818459
$683$	−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
$686$	−2.03121	−0.0775520
$687$	0	0
$688$	−15.3180	−0.583994
$689$	−4.45911	−0.169879
$690$	0	0
$691$	24.8701	0.946102	0.473051	−	0.881035i	$-0.343153\pi$
$691$	24.8701	0.946102	0.473051	+	0.881035i	$0.343153\pi$
$692$	18.6942	0.710648
$693$	0	0
$694$	47.8782	1.81743
$695$	−33.1556	−1.25767
$696$	0	0
$697$	−6.06148	−0.229595
$698$	−45.4205	−1.71919
$699$	0	0
$700$	20.5077	0.775119
$701$	−38.5821	−1.45723	−0.728613	−	0.684926i	$-0.759835\pi$
$701$	−38.5821	−1.45723	−0.728613	+	0.684926i	$0.759835\pi$
$702$	0	0
$703$	−0.0369585	−0.00139392
$704$	−53.5050	−2.01655
$705$	0	0
$706$	24.4349	0.919620
$707$	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	$0.419464\pi$
$710$	45.6839	1.71449
$711$	0	0
$712$	0.171259	0.00641820
$713$	7.82443	0.293027
$714$	0	0
$715$	32.9932	1.23387
$716$	−30.7885	−1.15062
$717$	0	0
$718$	−59.0696	−2.20446
$719$	−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
$722$	38.5818	1.43587
$723$	0	0
$724$	−6.53774	−0.242973
$725$	18.5860	0.690266
$726$	0	0
$727$	19.0336	0.705916	0.352958	−	0.935639i	$-0.385176\pi$
$727$	19.0336	0.705916	0.352958	+	0.935639i	$0.385176\pi$
$728$	−0.369491	−0.0136943
$729$	0	0
$730$	−122.620	−4.53839
$731$	−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	$0.113352\pi$
$734$	71.0598	2.62286
$735$	0	0
$736$	35.8817	1.32262
$737$	−6.70758	−0.247077
$738$	0	0
$739$	−6.70443	−0.246626	−0.123313	−	0.992368i	$-0.539352\pi$
$739$	−6.70443	−0.246626	−0.123313	+	0.992368i	$0.539352\pi$
$740$	4.04539	0.148712
$741$	0	0
$742$	6.26490	0.229992
$743$	−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
$746$	−50.9200	−1.86432
$747$	0	0
$748$	−76.8363	−2.80941
$749$	5.90146	0.215635
$750$	0	0
$751$	−38.8738	−1.41852	−0.709262	−	0.704945i	$-0.750972\pi$
$751$	−38.8738	−1.41852	−0.709262	+	0.704945i	$0.750972\pi$
$752$	−10.9190	−0.398174
$753$	0	0
$754$	−5.65770	−0.206041
$755$	−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	$0.477426\pi$
$758$	−33.0463	−1.20030
$759$	0	0
$760$	0.0727014	0.00263716
$761$	−13.9575	−0.505959	−0.252979	−	0.967472i	$-0.581410\pi$
$761$	−13.9575	−0.505959	−0.252979	+	0.967472i	$0.581410\pi$
$762$	0	0
$763$	−13.4426	−0.486653
$764$	−10.4369	−0.377592
$765$	0	0
$766$	−16.0067	−0.578344
$767$	−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	$0.445754\pi$
$770$	−46.3542	−1.67049
$771$	0	0
$772$	24.0325	0.864948
$773$	−6.38151	−0.229527	−0.114764	−	0.993393i	$-0.536611\pi$
$773$	−6.38151	−0.229527	−0.114764	+	0.993393i	$0.536611\pi$
$774$	0	0
$775$	17.0241	0.611523
$776$	2.69851	0.0968709
$777$	0	0
$778$	56.9709	2.04250
$779$	−0.0743284	−0.00266309
$780$	0	0
$781$	−35.0425	−1.25392
$782$	54.5900	1.95214
$783$	0	0
$784$	−3.73252	−0.133304
$785$	−39.0815	−1.39488
$786$	0	0
$787$	−31.4514	−1.12112	−0.560561	−	0.828113i	$-0.689414\pi$
$787$	−31.4514	−1.12112	−0.560561	+	0.828113i	$0.689414\pi$
$788$	37.9659	1.35248
$789$	0	0
$790$	113.291	4.03071
$791$	−6.01283	−0.213792
$792$	0	0
$793$	4.26277	0.151375
$794$	−26.0421	−0.924200
$795$	0	0
$796$	28.9814	1.02722
$797$	12.7701	0.452339	0.226170	−	0.974088i	$-0.427380\pi$
$797$	12.7701	0.452339	0.226170	+	0.974088i	$0.427380\pi$
$798$	0	0
$799$	−17.7320	−0.627313
$800$	78.0698	2.76018
$801$	0	0
$802$	−3.69038	−0.130312
$803$	94.0578	3.31923
$804$	0	0
$805$	16.9689	0.598074
$806$	−5.18225	−0.182537
$807$	0	0
$808$	−0.820254	−0.0288564
$809$	−37.8116	−1.32939	−0.664693	−	0.747117i	$-0.731437\pi$
$809$	−37.8116	−1.32939	−0.664693	+	0.747117i	$0.731437\pi$
$810$	0	0
$811$	−29.0566	−1.02031	−0.510157	−	0.860081i	$-0.670413\pi$
$811$	−29.0566	−1.02031	−0.510157	+	0.860081i	$0.670413\pi$
$812$	4.09564	0.143729
$813$	0	0
$814$	−6.02248	−0.211088
$815$	−73.8460	−2.58671
$816$	0	0
$817$	−0.305039	−0.0106720
$818$	71.6059	2.50364
$819$	0	0
$820$	8.13581	0.284115
$821$	22.1598	0.773381	0.386691	−	0.922209i	$-0.373618\pi$
$821$	22.1598	0.773381	0.386691	+	0.922209i	$0.373618\pi$
$822$	0	0
$823$	−44.5895	−1.55429	−0.777147	−	0.629319i	$-0.783334\pi$
$823$	−44.5895	−1.55429	−0.777147	+	0.629319i	$0.783334\pi$
$824$	−1.47978	−0.0515507
$825$	0	0
$826$	23.2388	0.808580
$827$	−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	$0.326689\pi$
$830$	−112.673	−3.91093
$831$	0	0
$832$	−12.9725	−0.449741
$833$	−6.06148	−0.210018
$834$	0	0
$835$	−9.27685	−0.321039
$836$	−0.942199	−0.0325866
$837$	0	0
$838$	32.8658	1.13533
$839$	−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
$842$	18.1822	0.626599
$843$	0	0
$844$	47.7163	1.64246
$845$	−41.7534	−1.43636
$846$	0	0
$847$	24.5567	0.843777
$848$	11.5123	0.395333
$849$	0	0
$850$	118.775	4.07394
$851$	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	$0.308269\pi$
$854$	−5.98904	−0.204941
$855$	0	0
$856$	−1.50825	−0.0515510
$857$	20.0025	0.683272	0.341636	−	0.939832i	$-0.389019\pi$
$857$	20.0025	0.683272	0.341636	+	0.939832i	$0.389019\pi$
$858$	0	0
$859$	−37.3721	−1.27512	−0.637559	−	0.770401i	$-0.720056\pi$
$859$	−37.3721	−1.27512	−0.637559	+	0.770401i	$0.720056\pi$
$860$	33.3888	1.13855
$861$	0	0
$862$	25.5505	0.870255
$863$	−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
$866$	−33.3615	−1.13367
$867$	0	0
$868$	3.75146	0.127333
$869$	−86.9013	−2.94793
$870$	0	0
$871$	−1.62628	−0.0551044
$872$	3.43555	0.116342
$873$	0	0
$874$	0.669406	0.0226430
$875$	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	$0.557028\pi$
$878$	51.3429	1.73274
$879$	0	0
$880$	−85.1798	−2.87141
$881$	25.0214	0.842993	0.421497	−	0.906830i	$-0.361505\pi$
$881$	25.0214	0.842993	0.421497	+	0.906830i	$0.361505\pi$
$882$	0	0
$883$	14.4626	0.486704	0.243352	−	0.969938i	$-0.421753\pi$
$883$	14.4626	0.486704	0.243352	+	0.969938i	$0.421753\pi$
$884$	−18.6293	−0.626570
$885$	0	0
$886$	−0.915143	−0.0307448
$887$	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
$890$	5.20916	0.174612
$891$	0	0
$892$	8.68746	0.290878
$893$	−0.217437	−0.00727626
$894$	0	0
$895$	−55.4288	−1.85278
$896$	2.04053	0.0681695
$897$	0	0
$898$	−37.0759	−1.23724
$899$	3.39992	0.113394
$900$	0	0
$901$	18.6955	0.622838
$902$	−12.1120	−0.403285
$903$	0	0
$904$	1.53671	0.0511104
$905$	−11.7699	−0.391246
$906$	0	0
$907$	−45.9050	−1.52425	−0.762126	−	0.647429i	$-0.775844\pi$
$907$	−45.9050	−1.52425	−0.762126	+	0.647429i	$0.775844\pi$
$908$	41.5194	1.37787
$909$	0	0
$910$	−11.2388	−0.372562
$911$	−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
$914$	−57.9521	−1.91688
$915$	0	0
$916$	48.9909	1.61870
$917$	−12.0382	−0.397537
$918$	0	0
$919$	−25.1999	−0.831269	−0.415634	−	0.909532i	$-0.636440\pi$
$919$	−25.1999	−0.831269	−0.415634	+	0.909532i	$0.636440\pi$
$920$	−4.33678	−0.142979
$921$	0	0
$922$	−14.8342	−0.488539
$923$	−8.49618	−0.279655
$924$	0	0
$925$	4.79678	0.157717
$926$	33.5900	1.10384
$927$	0	0
$928$	15.5915	0.511816
$929$	−37.0410	−1.21528	−0.607639	−	0.794214i	$-0.707883\pi$
$929$	−37.0410	−1.21528	−0.607639	+	0.794214i	$0.707883\pi$
$930$	0	0
$931$	−0.0743284	−0.00243602
$932$	−16.3581	−0.535827
$933$	0	0
$934$	−33.9027	−1.10933
$935$	−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	$0.369337\pi$
$938$	2.28487	0.0746035
$939$	0	0
$940$	23.8002	0.776276
$941$	42.4153	1.38270	0.691350	−	0.722520i	$-0.257016\pi$
$941$	42.4153	1.38270	0.691350	+	0.722520i	$0.257016\pi$
$942$	0	0
$943$	4.43383	0.144386
$944$	42.7032	1.38987
$945$	0	0
$946$	−49.7068	−1.61611
$947$	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
$950$	1.45647	0.0472540
$951$	0	0
$952$	1.54915	0.0502082
$953$	50.1799	1.62549	0.812743	−	0.582623i	$-0.197974\pi$
$953$	50.1799	1.62549	0.812743	+	0.582623i	$0.197974\pi$
$954$	0	0
$955$	−18.7895	−0.608015
$956$	25.9944	0.840720
$957$	0	0
$958$	15.5212	0.501466
$959$	−6.46697	−0.208829
$960$	0	0
$961$	−27.8858	−0.899542
$962$	−1.46017	−0.0470779
$963$	0	0
$964$	−7.82748	−0.252106
$965$	43.2659	1.39278
$966$	0	0
$967$	33.4107	1.07441	0.537207	−	0.843450i	$-0.319479\pi$
$967$	33.4107	1.07441	0.537207	+	0.843450i	$0.319479\pi$
$968$	−6.27601	−0.201719
$969$	0	0
$970$	82.0803	2.63544
$971$	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
$974$	23.6149	0.756670
$975$	0	0
$976$	−11.0054	−0.352273
$977$	5.17315	0.165504	0.0827519	−	0.996570i	$-0.473629\pi$
$977$	5.17315	0.165504	0.0827519	+	0.996570i	$0.473629\pi$
$978$	0	0
$979$	−3.99576	−0.127705
$980$	8.13581	0.259889
$981$	0	0
$982$	52.7685	1.68391
$983$	−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
$986$	23.7208	0.755424
$987$	0	0
$988$	−0.228440	−0.00726764
$989$	18.1962	0.578604
$990$	0	0
$991$	16.6924	0.530252	0.265126	−	0.964214i	$-0.414587\pi$
$991$	16.6924	0.530252	0.265126	+	0.964214i	$0.414587\pi$
$992$	14.2813	0.453430
$993$	0	0
$994$	11.9368	0.378614
$995$	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	$0.495874\pi$
$998$	47.1078	1.49117
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2583.2.a.r.1.1		5
3.2	odd	2		287.2.a.e.1.5	✓	5
12.11	even	2		4592.2.a.bb.1.1		5
15.14	odd	2		7175.2.a.n.1.1		5
21.20	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	3.2	odd	2
2009.2.a.n.1.5		5	21.20	even	2
2583.2.a.r.1.1		5	1.1	even	1	trivial
4592.2.a.bb.1.1		5	12.11	even	2
7175.2.a.n.1.1		5	15.14	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:	\( N \)	\(=\)	\( 2583 = 3^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2583.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03121 q^{2} +2.12582 q^{4} +3.82713 q^{5} +1.00000 q^{7} -0.255573 q^{8} +O(q^{10})\)	\(q-2.03121 q^{2} +2.12582 q^{4} +3.82713 q^{5} +1.00000 q^{7} -0.255573 q^{8} -7.77372 q^{10} +5.96294 q^{11} +1.44574 q^{13} -2.03121 q^{14} -3.73252 q^{16} -6.06148 q^{17} -0.0743284 q^{19} +8.13581 q^{20} -12.1120 q^{22} +4.43383 q^{23} +9.64695 q^{25} -2.93660 q^{26} +2.12582 q^{28} +1.92662 q^{29} +1.76471 q^{31} +8.09269 q^{32} +12.3122 q^{34} +3.82713 q^{35} +0.497233 q^{37} +0.150977 q^{38} -0.978111 q^{40} +1.00000 q^{41} +4.10393 q^{43} +12.6762 q^{44} -9.00606 q^{46} +2.92536 q^{47} +1.00000 q^{49} -19.5950 q^{50} +3.07338 q^{52} -3.08431 q^{53} +22.8210 q^{55} -0.255573 q^{56} -3.91337 q^{58} -11.4408 q^{59} +2.94851 q^{61} -3.58450 q^{62} -8.97293 q^{64} +5.53303 q^{65} -1.12488 q^{67} -12.8856 q^{68} -7.77372 q^{70} -5.87671 q^{71} +15.7737 q^{73} -1.00999 q^{74} -0.158009 q^{76} +5.96294 q^{77} -14.5736 q^{79} -14.2849 q^{80} -2.03121 q^{82} +14.4941 q^{83} -23.1981 q^{85} -8.33596 q^{86} -1.52396 q^{88} -0.670099 q^{89} +1.44574 q^{91} +9.42555 q^{92} -5.94203 q^{94} -0.284465 q^{95} -10.5587 q^{97} -2.03121 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8	\( 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8} - 2 q^{11} + 5 q^{13} + q^{14} - q^{16} - 13 q^{17} + 23 q^{20} + q^{22} - 2 q^{23} + 22 q^{25} + 3 q^{28} + 5 q^{29} + 17 q^{31} + 12 q^{32} - 8 q^{34} + 5 q^{35} - 7 q^{37} + 3 q^{38} + 7 q^{40} + 5 q^{41} + q^{43} + 47 q^{44} - 24 q^{46} - 9 q^{47} + 5 q^{49} - 2 q^{50} + 20 q^{52} - 5 q^{53} + 33 q^{55} + 3 q^{56} - 27 q^{58} - 7 q^{59} + 22 q^{61} + 28 q^{62} - 3 q^{64} + 31 q^{65} - 3 q^{67} - 17 q^{68} + 24 q^{71} + 40 q^{73} + 5 q^{74} - 19 q^{76} - 2 q^{77} - 42 q^{79} - 24 q^{80} + q^{82} + 12 q^{83} - 23 q^{85} - 16 q^{86} + 26 q^{88} - 8 q^{89} + 5 q^{91} - 12 q^{92} - 23 q^{94} + 17 q^{95} + 16 q^{97} + q^{98}+O(q^{100}) \) 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8 - 2 * q^11 + 5 * q^13 + q^14 - q^16 - 13 * q^17 + 23 * q^20 + q^22 - 2 * q^23 + 22 * q^25 + 3 * q^28 + 5 * q^29 + 17 * q^31 + 12 * q^32 - 8 * q^34 + 5 * q^35 - 7 * q^37 + 3 * q^38 + 7 * q^40 + 5 * q^41 + q^43 + 47 * q^44 - 24 * q^46 - 9 * q^47 + 5 * q^49 - 2 * q^50 + 20 * q^52 - 5 * q^53 + 33 * q^55 + 3 * q^56 - 27 * q^58 - 7 * q^59 + 22 * q^61 + 28 * q^62 - 3 * q^64 + 31 * q^65 - 3 * q^67 - 17 * q^68 + 24 * q^71 + 40 * q^73 + 5 * q^74 - 19 * q^76 - 2 * q^77 - 42 * q^79 - 24 * q^80 + q^82 + 12 * q^83 - 23 * q^85 - 16 * q^86 + 26 * q^88 - 8 * q^89 + 5 * q^91 - 12 * q^92 - 23 * q^94 + 17 * q^95 + 16 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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