LMFDB - Embedded newform 4023.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4023,2,Mod(1,4023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4023, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4023 = 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4023.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:	$32.1238167332$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82403 q^{2} +5.97513 q^{4} +0.305598 q^{5} -1.90246 q^{7} -11.2259 q^{8} +O(q^{10})$	\(q-2.82403 q^{2} +5.97513 q^{4} +0.305598 q^{5} -1.90246 q^{7} -11.2259 q^{8} -0.863018 q^{10} -0.144394 q^{11} -1.96848 q^{13} +5.37260 q^{14} +19.7519 q^{16} -3.93402 q^{17} +6.59437 q^{19} +1.82599 q^{20} +0.407772 q^{22} -5.77905 q^{23} -4.90661 q^{25} +5.55905 q^{26} -11.3674 q^{28} +5.85218 q^{29} -2.74653 q^{31} -33.3282 q^{32} +11.1098 q^{34} -0.581389 q^{35} +10.1118 q^{37} -18.6227 q^{38} -3.43061 q^{40} +4.92371 q^{41} +8.01858 q^{43} -0.862772 q^{44} +16.3202 q^{46} +10.1318 q^{47} -3.38064 q^{49} +13.8564 q^{50} -11.7619 q^{52} +1.16398 q^{53} -0.0441265 q^{55} +21.3568 q^{56} -16.5267 q^{58} -5.75048 q^{59} +0.466865 q^{61} +7.75626 q^{62} +54.6158 q^{64} -0.601565 q^{65} -13.2009 q^{67} -23.5063 q^{68} +1.64186 q^{70} +0.0478882 q^{71} +7.71359 q^{73} -28.5559 q^{74} +39.4022 q^{76} +0.274704 q^{77} +15.7615 q^{79} +6.03615 q^{80} -13.9047 q^{82} +4.76473 q^{83} -1.20223 q^{85} -22.6447 q^{86} +1.62095 q^{88} -1.21599 q^{89} +3.74496 q^{91} -34.5305 q^{92} -28.6124 q^{94} +2.01523 q^{95} -5.11338 q^{97} +9.54703 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * 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 $

$2$	−2.82403	−1.99689	−0.998444	−	0.0557587i	$-0.982242\pi$
$2$	−2.82403	−1.99689	−0.998444	+	0.0557587i	$0.982242\pi$
$3$	0	0
$3$	0	0
$4$	5.97513	2.98756
$5$	0.305598	0.136668	0.0683339	−	0.997663i	$-0.478232\pi$
$5$	0.305598	0.136668	0.0683339	+	0.997663i	$0.478232\pi$
$6$	0	0
$7$	−1.90246	−0.719063	−0.359531	−	0.933133i	$-0.617063\pi$
$7$	−1.90246	−0.719063	−0.359531	+	0.933133i	$0.617063\pi$
$8$	−11.2259	−3.96894
$9$	0	0
$10$	−0.863018	−0.272910
$11$	−0.144394	−0.0435364	−0.0217682	−	0.999763i	$-0.506930\pi$
$11$	−0.144394	−0.0435364	−0.0217682	+	0.999763i	$0.506930\pi$
$12$	0	0
$13$	−1.96848	−0.545959	−0.272979	−	0.962020i	$-0.588009\pi$
$13$	−1.96848	−0.545959	−0.272979	+	0.962020i	$0.588009\pi$
$14$	5.37260	1.43589
$15$	0	0
$16$	19.7519	4.93797
$17$	−3.93402	−0.954140	−0.477070	−	0.878865i	$-0.658301\pi$
$17$	−3.93402	−0.954140	−0.477070	+	0.878865i	$0.658301\pi$
$18$	0	0
$19$	6.59437	1.51285	0.756426	−	0.654079i	$-0.226944\pi$
$19$	6.59437	1.51285	0.756426	+	0.654079i	$0.226944\pi$
$20$	1.82599	0.408304
$21$	0	0
$22$	0.407772	0.0869373
$23$	−5.77905	−1.20501	−0.602507	−	0.798114i	$-0.705831\pi$
$23$	−5.77905	−1.20501	−0.602507	+	0.798114i	$0.705831\pi$
$24$	0	0
$25$	−4.90661	−0.981322
$26$	5.55905	1.09022
$27$	0	0
$28$	−11.3674	−2.14825
$29$	5.85218	1.08672	0.543361	−	0.839499i	$-0.317151\pi$
$29$	5.85218	1.08672	0.543361	+	0.839499i	$0.317151\pi$
$30$	0	0
$31$	−2.74653	−0.493291	−0.246645	−	0.969106i	$-0.579328\pi$
$31$	−2.74653	−0.493291	−0.246645	+	0.969106i	$0.579328\pi$
$32$	−33.3282	−5.89164
$33$	0	0
$34$	11.1098	1.90531
$35$	−0.581389	−0.0982727
$36$	0	0
$37$	10.1118	1.66236	0.831182	−	0.556001i	$-0.187665\pi$
$37$	10.1118	1.66236	0.831182	+	0.556001i	$0.187665\pi$
$38$	−18.6227	−3.02100
$39$	0	0
$40$	−3.43061	−0.542427
$41$	4.92371	0.768955	0.384477	−	0.923134i	$-0.374382\pi$
$41$	4.92371	0.768955	0.384477	+	0.923134i	$0.374382\pi$
$42$	0	0
$43$	8.01858	1.22282	0.611411	−	0.791313i	$-0.290602\pi$
$43$	8.01858	1.22282	0.611411	+	0.791313i	$0.290602\pi$
$44$	−0.862772	−0.130068
$45$	0	0
$46$	16.3202	2.40628
$47$	10.1318	1.47787	0.738936	−	0.673776i	$-0.235329\pi$
$47$	10.1318	1.47787	0.738936	+	0.673776i	$0.235329\pi$
$48$	0	0
$49$	−3.38064	−0.482949
$50$	13.8564	1.95959
$51$	0	0
$52$	−11.7619	−1.63109
$53$	1.16398	0.159884	0.0799421	−	0.996800i	$-0.474526\pi$
$53$	1.16398	0.159884	0.0799421	+	0.996800i	$0.474526\pi$
$54$	0	0
$55$	−0.0441265	−0.00595002
$56$	21.3568	2.85392
$57$	0	0
$58$	−16.5267	−2.17006
$59$	−5.75048	−0.748649	−0.374325	−	0.927298i	$-0.622125\pi$
$59$	−5.75048	−0.748649	−0.374325	+	0.927298i	$0.622125\pi$
$60$	0	0
$61$	0.466865	0.0597759	0.0298880	−	0.999553i	$-0.490485\pi$
$61$	0.466865	0.0597759	0.0298880	+	0.999553i	$0.490485\pi$
$62$	7.75626	0.985046
$63$	0	0
$64$	54.6158	6.82698
$65$	−0.601565	−0.0746150
$66$	0	0
$67$	−13.2009	−1.61274	−0.806370	−	0.591411i	$-0.798571\pi$
$67$	−13.2009	−1.61274	−0.806370	+	0.591411i	$0.798571\pi$
$68$	−23.5063	−2.85055
$69$	0	0
$70$	1.64186	0.196240
$71$	0.0478882	0.00568328	0.00284164	−	0.999996i	$-0.499095\pi$
$71$	0.0478882	0.00568328	0.00284164	+	0.999996i	$0.499095\pi$
$72$	0	0
$73$	7.71359	0.902807	0.451403	−	0.892320i	$-0.350924\pi$
$73$	7.71359	0.902807	0.451403	+	0.892320i	$0.350924\pi$
$74$	−28.5559	−3.31955
$75$	0	0
$76$	39.4022	4.51974
$77$	0.274704	0.0313054
$78$	0	0
$79$	15.7615	1.77330	0.886651	−	0.462439i	$-0.153026\pi$
$79$	15.7615	1.77330	0.886651	+	0.462439i	$0.153026\pi$
$80$	6.03615	0.674862
$81$	0	0
$82$	−13.9047	−1.53552
$83$	4.76473	0.522997	0.261499	−	0.965204i	$-0.415783\pi$
$83$	4.76473	0.522997	0.261499	+	0.965204i	$0.415783\pi$
$84$	0	0
$85$	−1.20223	−0.130400
$86$	−22.6447	−2.44184
$87$	0	0
$88$	1.62095	0.172793
$89$	−1.21599	−0.128895	−0.0644476	−	0.997921i	$-0.520529\pi$
$89$	−1.21599	−0.128895	−0.0644476	+	0.997921i	$0.520529\pi$
$90$	0	0
$91$	3.74496	0.392579
$92$	−34.5305	−3.60006
$93$	0	0
$94$	−28.6124	−2.95114
$95$	2.01523	0.206758
$96$	0	0
$97$	−5.11338	−0.519185	−0.259593	−	0.965718i	$-0.583588\pi$
$97$	−5.11338	−0.519185	−0.259593	+	0.965718i	$0.583588\pi$
$98$	9.54703	0.964395
$99$	0	0
$100$	−29.3176	−2.93176
$101$	−9.54066	−0.949331	−0.474665	−	0.880166i	$-0.657431\pi$
$101$	−9.54066	−0.949331	−0.474665	+	0.880166i	$0.657431\pi$
$102$	0	0
$103$	−17.4149	−1.71594	−0.857971	−	0.513698i	$-0.828275\pi$
$103$	−17.4149	−1.71594	−0.857971	+	0.513698i	$0.828275\pi$
$104$	22.0979	2.16688
$105$	0	0
$106$	−3.28710	−0.319271
$107$	−16.2024	−1.56634	−0.783171	−	0.621806i	$-0.786399\pi$
$107$	−16.2024	−1.56634	−0.783171	+	0.621806i	$0.786399\pi$
$108$	0	0
$109$	−12.2860	−1.17679	−0.588395	−	0.808574i	$-0.700240\pi$
$109$	−12.2860	−1.17679	−0.588395	+	0.808574i	$0.700240\pi$
$110$	0.124615	0.0118815
$111$	0	0
$112$	−37.5772	−3.55071
$113$	4.14083	0.389536	0.194768	−	0.980849i	$-0.437605\pi$
$113$	4.14083	0.389536	0.194768	+	0.980849i	$0.437605\pi$
$114$	0	0
$115$	−1.76607	−0.164687
$116$	34.9675	3.24665
$117$	0	0
$118$	16.2395	1.49497
$119$	7.48432	0.686086
$120$	0	0
$121$	−10.9792	−0.998105
$122$	−1.31844	−0.119366
$123$	0	0
$124$	−16.4108	−1.47374
$125$	−3.02744	−0.270783
$126$	0	0
$127$	−5.83171	−0.517481	−0.258740	−	0.965947i	$-0.583307\pi$
$127$	−5.83171	−0.517481	−0.258740	+	0.965947i	$0.583307\pi$
$128$	−87.5802	−7.74107
$129$	0	0
$130$	1.69884	0.148998
$131$	−1.59924	−0.139726	−0.0698631	−	0.997557i	$-0.522256\pi$
$131$	−1.59924	−0.139726	−0.0698631	+	0.997557i	$0.522256\pi$
$132$	0	0
$133$	−12.5455	−1.08784
$134$	37.2796	3.22046
$135$	0	0
$136$	44.1628	3.78693
$137$	−7.34100	−0.627184	−0.313592	−	0.949558i	$-0.601532\pi$
$137$	−7.34100	−0.627184	−0.313592	+	0.949558i	$0.601532\pi$
$138$	0	0
$139$	3.65471	0.309988	0.154994	−	0.987915i	$-0.450464\pi$
$139$	3.65471	0.309988	0.154994	+	0.987915i	$0.450464\pi$
$140$	−3.47387	−0.293596
$141$	0	0
$142$	−0.135238	−0.0113489
$143$	0.284237	0.0237691
$144$	0	0
$145$	1.78842	0.148520
$146$	−21.7834	−1.80280
$147$	0	0
$148$	60.4191	4.96642
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−9.82445	−0.799502	−0.399751	−	0.916624i	$-0.630903\pi$
$151$	−9.82445	−0.799502	−0.399751	+	0.916624i	$0.630903\pi$
$152$	−74.0275	−6.00443
$153$	0	0
$154$	−0.775771	−0.0625134
$155$	−0.839334	−0.0674169
$156$	0	0
$157$	−4.50667	−0.359671	−0.179836	−	0.983697i	$-0.557557\pi$
$157$	−4.50667	−0.359671	−0.179836	+	0.983697i	$0.557557\pi$
$158$	−44.5108	−3.54109
$159$	0	0
$160$	−10.1850	−0.805197
$161$	10.9944	0.866481
$162$	0	0
$163$	−0.167178	−0.0130944	−0.00654719	−	0.999979i	$-0.502084\pi$
$163$	−0.167178	−0.0130944	−0.00654719	+	0.999979i	$0.502084\pi$
$164$	29.4198	2.29730
$165$	0	0
$166$	−13.4557	−1.04437
$167$	3.47450	0.268865	0.134432	−	0.990923i	$-0.457079\pi$
$167$	3.47450	0.268865	0.134432	+	0.990923i	$0.457079\pi$
$168$	0	0
$169$	−9.12508	−0.701929
$170$	3.39513	0.260395
$171$	0	0
$172$	47.9120	3.65326
$173$	22.4647	1.70796	0.853982	−	0.520303i	$-0.174181\pi$
$173$	22.4647	1.70796	0.853982	+	0.520303i	$0.174181\pi$
$174$	0	0
$175$	9.33463	0.705632
$176$	−2.85205	−0.214982
$177$	0	0
$178$	3.43400	0.257389
$179$	7.32167	0.547247	0.273624	−	0.961837i	$-0.411778\pi$
$179$	7.32167	0.547247	0.273624	+	0.961837i	$0.411778\pi$
$180$	0	0
$181$	−15.5653	−1.15696	−0.578480	−	0.815697i	$-0.696354\pi$
$181$	−15.5653	−1.15696	−0.578480	+	0.815697i	$0.696354\pi$
$182$	−10.5759	−0.783936
$183$	0	0
$184$	64.8748	4.78263
$185$	3.09014	0.227192
$186$	0	0
$187$	0.568048	0.0415398
$188$	60.5387	4.41524
$189$	0	0
$190$	−5.69106	−0.412873
$191$	15.5571	1.12567	0.562835	−	0.826569i	$-0.309711\pi$
$191$	15.5571	1.12567	0.562835	+	0.826569i	$0.309711\pi$
$192$	0	0
$193$	7.65142	0.550761	0.275381	−	0.961335i	$-0.411196\pi$
$193$	7.65142	0.550761	0.275381	+	0.961335i	$0.411196\pi$
$194$	14.4403	1.03675
$195$	0	0
$196$	−20.1998	−1.44284
$197$	−7.14917	−0.509357	−0.254679	−	0.967026i	$-0.581970\pi$
$197$	−7.14917	−0.509357	−0.254679	+	0.967026i	$0.581970\pi$
$198$	0	0
$199$	−12.0154	−0.851746	−0.425873	−	0.904783i	$-0.640033\pi$
$199$	−12.0154	−0.851746	−0.425873	+	0.904783i	$0.640033\pi$
$200$	55.0809	3.89481
$201$	0	0
$202$	26.9431	1.89571
$203$	−11.1335	−0.781421
$204$	0	0
$205$	1.50468	0.105091
$206$	49.1802	3.42655
$207$	0	0
$208$	−38.8813	−2.69593
$209$	−0.952187	−0.0658641
$210$	0	0
$211$	12.1655	0.837504	0.418752	−	0.908101i	$-0.362468\pi$
$211$	12.1655	0.837504	0.418752	+	0.908101i	$0.362468\pi$
$212$	6.95490	0.477665
$213$	0	0
$214$	45.7559	3.12781
$215$	2.45046	0.167120
$216$	0	0
$217$	5.22516	0.354707
$218$	34.6961	2.34992
$219$	0	0
$220$	−0.263662	−0.0177761
$221$	7.74405	0.520921
$222$	0	0
$223$	−8.80907	−0.589899	−0.294949	−	0.955513i	$-0.595303\pi$
$223$	−8.80907	−0.589899	−0.294949	+	0.955513i	$0.595303\pi$
$224$	63.4055	4.23646
$225$	0	0
$226$	−11.6938	−0.777860
$227$	−13.2382	−0.878650	−0.439325	−	0.898328i	$-0.644782\pi$
$227$	−13.2382	−0.878650	−0.439325	+	0.898328i	$0.644782\pi$
$228$	0	0
$229$	−11.9360	−0.788750	−0.394375	−	0.918950i	$-0.629039\pi$
$229$	−11.9360	−0.788750	−0.394375	+	0.918950i	$0.629039\pi$
$230$	4.98742	0.328861
$231$	0	0
$232$	−65.6958	−4.31314
$233$	14.9461	0.979153	0.489577	−	0.871960i	$-0.337151\pi$
$233$	14.9461	0.979153	0.489577	+	0.871960i	$0.337151\pi$
$234$	0	0
$235$	3.09626	0.201977
$236$	−34.3599	−2.23664
$237$	0	0
$238$	−21.1359	−1.37004
$239$	−24.8134	−1.60505	−0.802523	−	0.596622i	$-0.796509\pi$
$239$	−24.8134	−1.60505	−0.802523	+	0.596622i	$0.796509\pi$
$240$	0	0
$241$	4.00608	0.258054	0.129027	−	0.991641i	$-0.458815\pi$
$241$	4.00608	0.258054	0.129027	+	0.991641i	$0.458815\pi$
$242$	31.0054	1.99310
$243$	0	0
$244$	2.78958	0.178584
$245$	−1.03312	−0.0660036
$246$	0	0
$247$	−12.9809	−0.825955
$248$	30.8321	1.95784
$249$	0	0
$250$	8.54958	0.540723
$251$	17.9532	1.13319	0.566597	−	0.823995i	$-0.308260\pi$
$251$	17.9532	1.13319	0.566597	+	0.823995i	$0.308260\pi$
$252$	0	0
$253$	0.834459	0.0524620
$254$	16.4689	1.03335
$255$	0	0
$256$	138.097	8.63107
$257$	28.9880	1.80822	0.904111	−	0.427297i	$-0.140534\pi$
$257$	28.9880	1.80822	0.904111	+	0.427297i	$0.140534\pi$
$258$	0	0
$259$	−19.2372	−1.19534
$260$	−3.59443	−0.222917
$261$	0	0
$262$	4.51629	0.279017
$263$	14.6374	0.902580	0.451290	−	0.892377i	$-0.350964\pi$
$263$	14.6374	0.902580	0.451290	+	0.892377i	$0.350964\pi$
$264$	0	0
$265$	0.355709	0.0218510
$266$	35.4289	2.17229
$267$	0	0
$268$	−78.8768	−4.81817
$269$	−25.5360	−1.55696	−0.778478	−	0.627672i	$-0.784008\pi$
$269$	−25.5360	−1.55696	−0.778478	+	0.627672i	$0.784008\pi$
$270$	0	0
$271$	−25.7071	−1.56159	−0.780796	−	0.624785i	$-0.785186\pi$
$271$	−25.7071	−1.56159	−0.780796	+	0.624785i	$0.785186\pi$
$272$	−77.7043	−4.71152
$273$	0	0
$274$	20.7312	1.25242
$275$	0.708484	0.0427232
$276$	0	0
$277$	−24.3071	−1.46047	−0.730237	−	0.683194i	$-0.760590\pi$
$277$	−24.3071	−1.46047	−0.730237	+	0.683194i	$0.760590\pi$
$278$	−10.3210	−0.619012
$279$	0	0
$280$	6.52660	0.390039
$281$	5.10736	0.304679	0.152340	−	0.988328i	$-0.451319\pi$
$281$	5.10736	0.304679	0.152340	+	0.988328i	$0.451319\pi$
$282$	0	0
$283$	−10.2565	−0.609686	−0.304843	−	0.952403i	$-0.598604\pi$
$283$	−10.2565	−0.609686	−0.304843	+	0.952403i	$0.598604\pi$
$284$	0.286138	0.0169792
$285$	0	0
$286$	−0.802693	−0.0474642
$287$	−9.36717	−0.552927
$288$	0	0
$289$	−1.52350	−0.0896175
$290$	−5.05054	−0.296578
$291$	0	0
$292$	46.0897	2.69719
$293$	−25.8821	−1.51205	−0.756024	−	0.654544i	$-0.772861\pi$
$293$	−25.8821	−1.51205	−0.756024	+	0.654544i	$0.772861\pi$
$294$	0	0
$295$	−1.75734	−0.102316
$296$	−113.513	−6.59783
$297$	0	0
$298$	2.82403	0.163591
$299$	11.3760	0.657888
$300$	0	0
$301$	−15.2550	−0.879285
$302$	27.7445	1.59652
$303$	0	0
$304$	130.251	7.47043
$305$	0.142673	0.00816944
$306$	0	0
$307$	−27.3157	−1.55899	−0.779496	−	0.626407i	$-0.784525\pi$
$307$	−27.3157	−1.55899	−0.779496	+	0.626407i	$0.784525\pi$
$308$	1.64139	0.0935268
$309$	0	0
$310$	2.37030	0.134624
$311$	−15.4557	−0.876415	−0.438207	−	0.898874i	$-0.644386\pi$
$311$	−15.4557	−0.876415	−0.438207	+	0.898874i	$0.644386\pi$
$312$	0	0
$313$	−13.9659	−0.789402	−0.394701	−	0.918810i	$-0.629152\pi$
$313$	−13.9659	−0.789402	−0.394701	+	0.918810i	$0.629152\pi$
$314$	12.7270	0.718224
$315$	0	0
$316$	94.1767	5.29785
$317$	−8.32127	−0.467369	−0.233684	−	0.972312i	$-0.575078\pi$
$317$	−8.32127	−0.467369	−0.233684	+	0.972312i	$0.575078\pi$
$318$	0	0
$319$	−0.845019	−0.0473120
$320$	16.6905	0.933027
$321$	0	0
$322$	−31.0485	−1.73027
$323$	−25.9424	−1.44347
$324$	0	0
$325$	9.65858	0.535761
$326$	0.472114	0.0261480
$327$	0	0
$328$	−55.2730	−3.05194
$329$	−19.2753	−1.06268
$330$	0	0
$331$	−7.02220	−0.385975	−0.192988	−	0.981201i	$-0.561818\pi$
$331$	−7.02220	−0.385975	−0.192988	+	0.981201i	$0.561818\pi$
$332$	28.4699	1.56249
$333$	0	0
$334$	−9.81207	−0.536892
$335$	−4.03416	−0.220410
$336$	0	0
$337$	7.04771	0.383913	0.191957	−	0.981403i	$-0.438517\pi$
$337$	7.04771	0.383913	0.191957	+	0.981403i	$0.438517\pi$
$338$	25.7695	1.40167
$339$	0	0
$340$	−7.18348	−0.389579
$341$	0.396582	0.0214761
$342$	0	0
$343$	19.7488	1.06633
$344$	−90.0155	−4.85331
$345$	0	0
$346$	−63.4411	−3.41061
$347$	5.49587	0.295034	0.147517	−	0.989060i	$-0.452872\pi$
$347$	5.49587	0.295034	0.147517	+	0.989060i	$0.452872\pi$
$348$	0	0
$349$	19.3544	1.03602	0.518008	−	0.855376i	$-0.326674\pi$
$349$	19.3544	1.03602	0.518008	+	0.855376i	$0.326674\pi$
$350$	−26.3613	−1.40907
$351$	0	0
$352$	4.81238	0.256501
$353$	11.9287	0.634903	0.317451	−	0.948275i	$-0.397173\pi$
$353$	11.9287	0.634903	0.317451	+	0.948275i	$0.397173\pi$
$354$	0	0
$355$	0.0146346	0.000776722 0
$356$	−7.26573	−0.385083
$357$	0	0
$358$	−20.6766	−1.09279
$359$	7.48775	0.395188	0.197594	−	0.980284i	$-0.436687\pi$
$359$	7.48775	0.395188	0.197594	+	0.980284i	$0.436687\pi$
$360$	0	0
$361$	24.4857	1.28872
$362$	43.9568	2.31032
$363$	0	0
$364$	22.3766	1.17285
$365$	2.35726	0.123385
$366$	0	0
$367$	26.3400	1.37494	0.687469	−	0.726213i	$-0.258722\pi$
$367$	26.3400	1.37494	0.687469	+	0.726213i	$0.258722\pi$
$368$	−114.147	−5.95033
$369$	0	0
$370$	−8.72663	−0.453676
$371$	−2.21442	−0.114967
$372$	0	0
$373$	7.24016	0.374881	0.187441	−	0.982276i	$-0.439981\pi$
$373$	7.24016	0.374881	0.187441	+	0.982276i	$0.439981\pi$
$374$	−1.60418	−0.0829503
$375$	0	0
$376$	−113.738	−5.86559
$377$	−11.5199	−0.593306
$378$	0	0
$379$	0.791218	0.0406422	0.0203211	−	0.999794i	$-0.493531\pi$
$379$	0.791218	0.0406422	0.0203211	+	0.999794i	$0.493531\pi$
$380$	12.0413	0.617703
$381$	0	0
$382$	−43.9336	−2.24784
$383$	35.5707	1.81758	0.908789	−	0.417256i	$-0.137008\pi$
$383$	35.5707	1.81758	0.908789	+	0.417256i	$0.137008\pi$
$384$	0	0
$385$	0.0839490	0.00427844
$386$	−21.6078	−1.09981
$387$	0	0
$388$	−30.5531	−1.55110
$389$	−25.7964	−1.30793	−0.653965	−	0.756525i	$-0.726896\pi$
$389$	−25.7964	−1.30793	−0.653965	+	0.756525i	$0.726896\pi$
$390$	0	0
$391$	22.7349	1.14975
$392$	37.9507	1.91680
$393$	0	0
$394$	20.1895	1.01713
$395$	4.81668	0.242353
$396$	0	0
$397$	−13.0463	−0.654777	−0.327389	−	0.944890i	$-0.606169\pi$
$397$	−13.0463	−0.654777	−0.327389	+	0.944890i	$0.606169\pi$
$398$	33.9317	1.70084
$399$	0	0
$400$	−96.9148	−4.84574
$401$	−23.8128	−1.18915	−0.594577	−	0.804039i	$-0.702681\pi$
$401$	−23.8128	−1.18915	−0.594577	+	0.804039i	$0.702681\pi$
$402$	0	0
$403$	5.40649	0.269316
$404$	−57.0066	−2.83619
$405$	0	0
$406$	31.4414	1.56041
$407$	−1.46008	−0.0723733
$408$	0	0
$409$	−14.6864	−0.726197	−0.363098	−	0.931751i	$-0.618281\pi$
$409$	−14.6864	−0.726197	−0.363098	+	0.931751i	$0.618281\pi$
$410$	−4.24925	−0.209856
$411$	0	0
$412$	−104.056	−5.12649
$413$	10.9401	0.538326
$414$	0	0
$415$	1.45610	0.0714769
$416$	65.6059	3.21659
$417$	0	0
$418$	2.68900	0.131523
$419$	0.0220058	0.00107506	0.000537528	−	1.00000i	$-0.499829\pi$
$419$	0.0220058	0.00107506	0.000537528		1.00000i	$0.499829\pi$
$420$	0	0
$421$	6.89954	0.336263	0.168131	−	0.985765i	$-0.446227\pi$
$421$	6.89954	0.336263	0.168131	+	0.985765i	$0.446227\pi$
$422$	−34.3556	−1.67240
$423$	0	0
$424$	−13.0666	−0.634572
$425$	19.3027	0.936318
$426$	0	0
$427$	−0.888192	−0.0429826
$428$	−96.8113	−4.67955
$429$	0	0
$430$	−6.92018	−0.333721
$431$	18.0430	0.869103	0.434551	−	0.900647i	$-0.356907\pi$
$431$	18.0430	0.869103	0.434551	+	0.900647i	$0.356907\pi$
$432$	0	0
$433$	35.7910	1.72001	0.860004	−	0.510288i	$-0.170461\pi$
$433$	35.7910	1.72001	0.860004	+	0.510288i	$0.170461\pi$
$434$	−14.7560	−0.708310
$435$	0	0
$436$	−73.4107	−3.51573
$437$	−38.1092	−1.82301
$438$	0	0
$439$	18.6094	0.888179	0.444090	−	0.895982i	$-0.353527\pi$
$439$	18.6094	0.888179	0.444090	+	0.895982i	$0.353527\pi$
$440$	0.495359	0.0236153
$441$	0	0
$442$	−21.8694	−1.04022
$443$	−14.1496	−0.672266	−0.336133	−	0.941815i	$-0.609119\pi$
$443$	−14.1496	−0.672266	−0.336133	+	0.941815i	$0.609119\pi$
$444$	0	0
$445$	−0.371606	−0.0176158
$446$	24.8771	1.17796
$447$	0	0
$448$	−103.904	−4.90902
$449$	−35.4467	−1.67283	−0.836416	−	0.548095i	$-0.815353\pi$
$449$	−35.4467	−1.67283	−0.836416	+	0.548095i	$0.815353\pi$
$450$	0	0
$451$	−0.710954	−0.0334775
$452$	24.7420	1.16376
$453$	0	0
$454$	37.3850	1.75457
$455$	1.14445	0.0536528
$456$	0	0
$457$	−6.05588	−0.283282	−0.141641	−	0.989918i	$-0.545238\pi$
$457$	−6.05588	−0.283282	−0.141641	+	0.989918i	$0.545238\pi$
$458$	33.7075	1.57505
$459$	0	0
$460$	−10.5525	−0.492012
$461$	5.76689	0.268591	0.134295	−	0.990941i	$-0.457123\pi$
$461$	5.76689	0.268591	0.134295	+	0.990941i	$0.457123\pi$
$462$	0	0
$463$	−4.57734	−0.212727	−0.106364	−	0.994327i	$-0.533921\pi$
$463$	−4.57734	−0.212727	−0.106364	+	0.994327i	$0.533921\pi$
$464$	115.592	5.36621
$465$	0	0
$466$	−42.2083	−1.95526
$467$	11.7163	0.542165	0.271083	−	0.962556i	$-0.412618\pi$
$467$	11.7163	0.542165	0.271083	+	0.962556i	$0.412618\pi$
$468$	0	0
$469$	25.1141	1.15966
$470$	−8.74391	−0.403326
$471$	0	0
$472$	64.5542	2.97135
$473$	−1.15783	−0.0532372
$474$	0	0
$475$	−32.3560	−1.48460
$476$	44.7197	2.04973
$477$	0	0
$478$	70.0737	3.20510
$479$	−40.8765	−1.86769	−0.933847	−	0.357674i	$-0.883570\pi$
$479$	−40.8765	−1.86769	−0.933847	+	0.357674i	$0.883570\pi$
$480$	0	0
$481$	−19.9048	−0.907582
$482$	−11.3133	−0.515306
$483$	0	0
$484$	−65.6018	−2.98190
$485$	−1.56264	−0.0709559
$486$	0	0
$487$	−33.0418	−1.49727	−0.748633	−	0.662985i	$-0.769289\pi$
$487$	−33.0418	−1.49727	−0.748633	+	0.662985i	$0.769289\pi$
$488$	−5.24096	−0.237247
$489$	0	0
$490$	2.91756	0.131802
$491$	−21.8794	−0.987402	−0.493701	−	0.869632i	$-0.664356\pi$
$491$	−21.8794	−0.987402	−0.493701	+	0.869632i	$0.664356\pi$
$492$	0	0
$493$	−23.0226	−1.03689
$494$	36.6584	1.64934
$495$	0	0
$496$	−54.2491	−2.43586
$497$	−0.0911054	−0.00408664
$498$	0	0
$499$	−20.9035	−0.935770	−0.467885	−	0.883789i	$-0.654984\pi$
$499$	−20.9035	−0.935770	−0.467885	+	0.883789i	$0.654984\pi$
$500$	−18.0894	−0.808981
$501$	0	0
$502$	−50.7002	−2.26286
$503$	−22.2752	−0.993200	−0.496600	−	0.867979i	$-0.665418\pi$
$503$	−22.2752	−0.993200	−0.496600	+	0.867979i	$0.665418\pi$
$504$	0	0
$505$	−2.91561	−0.129743
$506$	−2.35653	−0.104761
$507$	0	0
$508$	−34.8452	−1.54601
$509$	−12.5564	−0.556554	−0.278277	−	0.960501i	$-0.589763\pi$
$509$	−12.5564	−0.556554	−0.278277	+	0.960501i	$0.589763\pi$
$510$	0	0
$511$	−14.6748	−0.649175
$512$	−214.830	−9.49423
$513$	0	0
$514$	−81.8629	−3.61082
$515$	−5.32197	−0.234514
$516$	0	0
$517$	−1.46297	−0.0643412
$518$	54.3265	2.38697
$519$	0	0
$520$	6.75309	0.296143
$521$	−26.1303	−1.14479	−0.572395	−	0.819978i	$-0.693986\pi$
$521$	−26.1303	−1.14479	−0.572395	+	0.819978i	$0.693986\pi$
$522$	0	0
$523$	5.59842	0.244802	0.122401	−	0.992481i	$-0.460941\pi$
$523$	5.59842	0.244802	0.122401	+	0.992481i	$0.460941\pi$
$524$	−9.55565	−0.417441
$525$	0	0
$526$	−41.3364	−1.80235
$527$	10.8049	0.470668
$528$	0	0
$529$	10.3974	0.452059
$530$	−1.00453	−0.0436341
$531$	0	0
$532$	−74.9612	−3.24998
$533$	−9.69225	−0.419818
$534$	0	0
$535$	−4.95142	−0.214069
$536$	148.191	6.40088
$537$	0	0
$538$	72.1143	3.10907
$539$	0.488144	0.0210259
$540$	0	0
$541$	12.1646	0.522997	0.261499	−	0.965204i	$-0.415783\pi$
$541$	12.1646	0.522997	0.261499	+	0.965204i	$0.415783\pi$
$542$	72.5974	3.11833
$543$	0	0
$544$	131.114	5.62145
$545$	−3.75459	−0.160829
$546$	0	0
$547$	17.3506	0.741859	0.370930	−	0.928661i	$-0.379039\pi$
$547$	17.3506	0.741859	0.370930	+	0.928661i	$0.379039\pi$
$548$	−43.8634	−1.87375
$549$	0	0
$550$	−2.00078	−0.0853135
$551$	38.5915	1.64405
$552$	0	0
$553$	−29.9855	−1.27512
$554$	68.6440	2.91640
$555$	0	0
$556$	21.8373	0.926109
$557$	26.4254	1.11968	0.559839	−	0.828601i	$-0.310863\pi$
$557$	26.4254	1.11968	0.559839	+	0.828601i	$0.310863\pi$
$558$	0	0
$559$	−15.7844	−0.667610
$560$	−11.4835	−0.485268
$561$	0	0
$562$	−14.4233	−0.608411
$563$	2.62323	0.110556	0.0552780	−	0.998471i	$-0.482396\pi$
$563$	2.62323	0.110556	0.0552780	+	0.998471i	$0.482396\pi$
$564$	0	0
$565$	1.26543	0.0532370
$566$	28.9647	1.21748
$567$	0	0
$568$	−0.537587	−0.0225566
$569$	−20.7256	−0.868863	−0.434431	−	0.900705i	$-0.643051\pi$
$569$	−20.7256	−0.868863	−0.434431	+	0.900705i	$0.643051\pi$
$570$	0	0
$571$	8.06008	0.337304	0.168652	−	0.985676i	$-0.446059\pi$
$571$	8.06008	0.337304	0.168652	+	0.985676i	$0.446059\pi$
$572$	1.69835	0.0710116
$573$	0	0
$574$	26.4531	1.10413
$575$	28.3555	1.18251
$576$	0	0
$577$	−7.00109	−0.291459	−0.145730	−	0.989324i	$-0.546553\pi$
$577$	−7.00109	−0.291459	−0.145730	+	0.989324i	$0.546553\pi$
$578$	4.30240	0.178956
$579$	0	0
$580$	10.6860	0.443713
$581$	−9.06472	−0.376068
$582$	0	0
$583$	−0.168071	−0.00696078
$584$	−86.5917	−3.58319
$585$	0	0
$586$	73.0917	3.01939
$587$	−46.3413	−1.91271	−0.956356	−	0.292205i	$-0.905611\pi$
$587$	−46.3413	−1.91271	−0.956356	+	0.292205i	$0.905611\pi$
$588$	0	0
$589$	−18.1116	−0.746276
$590$	4.96277	0.204314
$591$	0	0
$592$	199.726	8.20871
$593$	−1.90983	−0.0784275	−0.0392137	−	0.999231i	$-0.512485\pi$
$593$	−1.90983	−0.0784275	−0.0392137	+	0.999231i	$0.512485\pi$
$594$	0	0
$595$	2.28720	0.0937659
$596$	−5.97513	−0.244751
$597$	0	0
$598$	−32.1260	−1.31373
$599$	41.6216	1.70061	0.850306	−	0.526288i	$-0.176417\pi$
$599$	41.6216	1.70061	0.850306	+	0.526288i	$0.176417\pi$
$600$	0	0
$601$	2.36787	0.0965874	0.0482937	−	0.998833i	$-0.484622\pi$
$601$	2.36787	0.0965874	0.0482937	+	0.998833i	$0.484622\pi$
$602$	43.0806	1.75583
$603$	0	0
$604$	−58.7023	−2.38856
$605$	−3.35521	−0.136409
$606$	0	0
$607$	28.3205	1.14949	0.574746	−	0.818332i	$-0.305101\pi$
$607$	28.3205	1.14949	0.574746	+	0.818332i	$0.305101\pi$
$608$	−219.778	−8.91318
$609$	0	0
$610$	−0.402913	−0.0163135
$611$	−19.9442	−0.806857
$612$	0	0
$613$	−12.3322	−0.498094	−0.249047	−	0.968491i	$-0.580117\pi$
$613$	−12.3322	−0.498094	−0.249047	+	0.968491i	$0.580117\pi$
$614$	77.1404	3.11313
$615$	0	0
$616$	−3.08379	−0.124249
$617$	−12.5472	−0.505131	−0.252566	−	0.967580i	$-0.581274\pi$
$617$	−12.5472	−0.505131	−0.252566	+	0.967580i	$0.581274\pi$
$618$	0	0
$619$	−18.0160	−0.724124	−0.362062	−	0.932154i	$-0.617927\pi$
$619$	−18.0160	−0.724124	−0.362062	+	0.932154i	$0.617927\pi$
$620$	−5.01513	−0.201412
$621$	0	0
$622$	43.6474	1.75010
$623$	2.31338	0.0926837
$624$	0	0
$625$	23.6079	0.944315
$626$	39.4402	1.57635
$627$	0	0
$628$	−26.9279	−1.07454
$629$	−39.7799	−1.58613
$630$	0	0
$631$	−12.6681	−0.504310	−0.252155	−	0.967687i	$-0.581139\pi$
$631$	−12.6681	−0.504310	−0.252155	+	0.967687i	$0.581139\pi$
$632$	−176.936	−7.03814
$633$	0	0
$634$	23.4995	0.933284
$635$	−1.78216	−0.0707229
$636$	0	0
$637$	6.65474	0.263670
$638$	2.38636	0.0944767
$639$	0	0
$640$	−26.7644	−1.05795
$641$	−29.2160	−1.15396	−0.576982	−	0.816757i	$-0.695770\pi$
$641$	−29.2160	−1.15396	−0.576982	+	0.816757i	$0.695770\pi$
$642$	0	0
$643$	−28.2228	−1.11300	−0.556498	−	0.830849i	$-0.687855\pi$
$643$	−28.2228	−1.11300	−0.556498	+	0.830849i	$0.687855\pi$
$644$	65.6930	2.58867
$645$	0	0
$646$	73.2620	2.88245
$647$	13.0708	0.513865	0.256933	−	0.966429i	$-0.417288\pi$
$647$	13.0708	0.513865	0.256933	+	0.966429i	$0.417288\pi$
$648$	0	0
$649$	0.830335	0.0325935
$650$	−27.2761	−1.06986
$651$	0	0
$652$	−0.998908	−0.0391203
$653$	−5.07155	−0.198465	−0.0992326	−	0.995064i	$-0.531639\pi$
$653$	−5.07155	−0.198465	−0.0992326	+	0.995064i	$0.531639\pi$
$654$	0	0
$655$	−0.488725	−0.0190961
$656$	97.2527	3.79708
$657$	0	0
$658$	54.4340	2.12206
$659$	29.5096	1.14953	0.574765	−	0.818319i	$-0.305093\pi$
$659$	29.5096	1.14953	0.574765	+	0.818319i	$0.305093\pi$
$660$	0	0
$661$	−30.0972	−1.17065	−0.585324	−	0.810800i	$-0.699032\pi$
$661$	−30.0972	−1.17065	−0.585324	+	0.810800i	$0.699032\pi$
$662$	19.8309	0.770749
$663$	0	0
$664$	−53.4883	−2.07575
$665$	−3.83390	−0.148672
$666$	0	0
$667$	−33.8200	−1.30952
$668$	20.7606	0.803250
$669$	0	0
$670$	11.3926	0.440134
$671$	−0.0674124	−0.00260243
$672$	0	0
$673$	10.9238	0.421080	0.210540	−	0.977585i	$-0.432478\pi$
$673$	10.9238	0.421080	0.210540	+	0.977585i	$0.432478\pi$
$674$	−19.9029	−0.766632
$675$	0	0
$676$	−54.5235	−2.09706
$677$	21.4703	0.825170	0.412585	−	0.910919i	$-0.364626\pi$
$677$	21.4703	0.825170	0.412585	+	0.910919i	$0.364626\pi$
$678$	0	0
$679$	9.72801	0.373327
$680$	13.4961	0.517551
$681$	0	0
$682$	−1.11996	−0.0428854
$683$	−32.7242	−1.25216	−0.626078	−	0.779760i	$-0.715341\pi$
$683$	−32.7242	−1.25216	−0.626078	+	0.779760i	$0.715341\pi$
$684$	0	0
$685$	−2.24340	−0.0857158
$686$	−55.7710	−2.12935
$687$	0	0
$688$	158.382	6.03826
$689$	−2.29127	−0.0872903
$690$	0	0
$691$	−31.3362	−1.19209	−0.596043	−	0.802953i	$-0.703261\pi$
$691$	−31.3362	−1.19209	−0.596043	+	0.802953i	$0.703261\pi$
$692$	134.230	5.10265
$693$	0	0
$694$	−15.5205	−0.589150
$695$	1.11687	0.0423654
$696$	0	0
$697$	−19.3700	−0.733690
$698$	−54.6572	−2.06881
$699$	0	0
$700$	55.7756	2.10812
$701$	−43.0030	−1.62420	−0.812101	−	0.583518i	$-0.801676\pi$
$701$	−43.0030	−1.62420	−0.812101	+	0.583518i	$0.801676\pi$
$702$	0	0
$703$	66.6807	2.51491
$704$	−7.88619	−0.297222
$705$	0	0
$706$	−33.6871	−1.26783
$707$	18.1507	0.682628
$708$	0	0
$709$	−23.8564	−0.895947	−0.447974	−	0.894047i	$-0.647854\pi$
$709$	−23.8564	−0.895947	−0.447974	+	0.894047i	$0.647854\pi$
$710$	−0.0413284	−0.00155103
$711$	0	0
$712$	13.6506	0.511578
$713$	15.8723	0.594422
$714$	0	0
$715$	0.0868623	0.00324847
$716$	43.7479	1.63494
$717$	0	0
$718$	−21.1456	−0.789147
$719$	39.1369	1.45956	0.729780	−	0.683682i	$-0.239623\pi$
$719$	39.1369	1.45956	0.729780	+	0.683682i	$0.239623\pi$
$720$	0	0
$721$	33.1312	1.23387
$722$	−69.1484	−2.57344
$723$	0	0
$724$	−93.0047	−3.45649
$725$	−28.7144	−1.06642
$726$	0	0
$727$	−32.4688	−1.20420	−0.602101	−	0.798420i	$-0.705669\pi$
$727$	−32.4688	−1.20420	−0.602101	+	0.798420i	$0.705669\pi$
$728$	−42.0404	−1.55812
$729$	0	0
$730$	−6.65696	−0.246385
$731$	−31.5452	−1.16674
$732$	0	0
$733$	−10.3993	−0.384108	−0.192054	−	0.981384i	$-0.561515\pi$
$733$	−10.3993	−0.384108	−0.192054	+	0.981384i	$0.561515\pi$
$734$	−74.3850	−2.74560
$735$	0	0
$736$	192.605	7.09951
$737$	1.90612	0.0702129
$738$	0	0
$739$	52.2968	1.92377	0.961885	−	0.273454i	$-0.0881662\pi$
$739$	52.2968	1.92377	0.961885	+	0.273454i	$0.0881662\pi$
$740$	18.4640	0.678749
$741$	0	0
$742$	6.25357	0.229576
$743$	31.6582	1.16143	0.580714	−	0.814108i	$-0.302773\pi$
$743$	31.6582	1.16143	0.580714	+	0.814108i	$0.302773\pi$
$744$	0	0
$745$	−0.305598	−0.0111963
$746$	−20.4464	−0.748596
$747$	0	0
$748$	3.39416	0.124103
$749$	30.8244	1.12630
$750$	0	0
$751$	−33.0043	−1.20434	−0.602172	−	0.798367i	$-0.705698\pi$
$751$	−33.0043	−1.20434	−0.602172	+	0.798367i	$0.705698\pi$
$752$	200.122	7.29769
$753$	0	0
$754$	32.5326	1.18477
$755$	−3.00234	−0.109266
$756$	0	0
$757$	6.50821	0.236545	0.118272	−	0.992981i	$-0.462264\pi$
$757$	6.50821	0.236545	0.118272	+	0.992981i	$0.462264\pi$
$758$	−2.23442	−0.0811579
$759$	0	0
$760$	−22.6227	−0.820612
$761$	−35.1736	−1.27504	−0.637521	−	0.770433i	$-0.720040\pi$
$761$	−35.1736	−1.27504	−0.637521	+	0.770433i	$0.720040\pi$
$762$	0	0
$763$	23.3737	0.846185
$764$	92.9555	3.36301
$765$	0	0
$766$	−100.453	−3.62950
$767$	11.3197	0.408732
$768$	0	0
$769$	3.68124	0.132749	0.0663744	−	0.997795i	$-0.478857\pi$
$769$	3.68124	0.132749	0.0663744	+	0.997795i	$0.478857\pi$
$770$	−0.237074	−0.00854356
$771$	0	0
$772$	45.7182	1.64543
$773$	28.0880	1.01025	0.505127	−	0.863045i	$-0.331445\pi$
$773$	28.0880	1.01025	0.505127	+	0.863045i	$0.331445\pi$
$774$	0	0
$775$	13.4761	0.484077
$776$	57.4021	2.06062
$777$	0	0
$778$	72.8497	2.61179
$779$	32.4688	1.16332
$780$	0	0
$781$	−0.00691476	−0.000247430 0
$782$	−64.2039	−2.29593
$783$	0	0
$784$	−66.7741	−2.38479
$785$	−1.37723	−0.0491555
$786$	0	0
$787$	−41.2225	−1.46942	−0.734712	−	0.678380i	$-0.762683\pi$
$787$	−41.2225	−1.46942	−0.734712	+	0.678380i	$0.762683\pi$
$788$	−42.7172	−1.52174
$789$	0	0
$790$	−13.6024	−0.483952
$791$	−7.87776	−0.280101
$792$	0	0
$793$	−0.919016	−0.0326352
$794$	36.8432	1.30752
$795$	0	0
$796$	−71.7933	−2.54464
$797$	−37.4192	−1.32546	−0.662729	−	0.748860i	$-0.730602\pi$
$797$	−37.4192	−1.32546	−0.662729	+	0.748860i	$0.730602\pi$
$798$	0	0
$799$	−39.8586	−1.41010
$800$	163.528	5.78160
$801$	0	0
$802$	67.2480	2.37461
$803$	−1.11379	−0.0393050
$804$	0	0
$805$	3.35987	0.118420
$806$	−15.2681	−0.537795
$807$	0	0
$808$	107.102	3.76784
$809$	10.6689	0.375097	0.187549	−	0.982255i	$-0.439946\pi$
$809$	10.6689	0.375097	0.187549	+	0.982255i	$0.439946\pi$
$810$	0	0
$811$	42.3528	1.48721	0.743604	−	0.668620i	$-0.233115\pi$
$811$	42.3528	1.48721	0.743604	+	0.668620i	$0.233115\pi$
$812$	−66.5243	−2.33455
$813$	0	0
$814$	4.12330	0.144521
$815$	−0.0510893	−0.00178958
$816$	0	0
$817$	52.8775	1.84995
$818$	41.4749	1.45013
$819$	0	0
$820$	8.99065	0.313967
$821$	−52.7448	−1.84081	−0.920403	−	0.390970i	$-0.872140\pi$
$821$	−52.7448	−1.84081	−0.920403	+	0.390970i	$0.872140\pi$
$822$	0	0
$823$	45.5255	1.58692	0.793460	−	0.608623i	$-0.208278\pi$
$823$	45.5255	1.58692	0.793460	+	0.608623i	$0.208278\pi$
$824$	195.498	6.81048
$825$	0	0
$826$	−30.8950	−1.07498
$827$	−26.5967	−0.924857	−0.462428	−	0.886657i	$-0.653022\pi$
$827$	−26.5967	−0.924857	−0.462428	+	0.886657i	$0.653022\pi$
$828$	0	0
$829$	−16.1426	−0.560656	−0.280328	−	0.959904i	$-0.590443\pi$
$829$	−16.1426	−0.560656	−0.280328	+	0.959904i	$0.590443\pi$
$830$	−4.11205	−0.142731
$831$	0	0
$832$	−107.510	−3.72725
$833$	13.2995	0.460801
$834$	0	0
$835$	1.06180	0.0367451
$836$	−5.68944	−0.196773
$837$	0	0
$838$	−0.0621451	−0.00214677
$839$	47.7051	1.64696	0.823481	−	0.567344i	$-0.192029\pi$
$839$	47.7051	1.64696	0.823481	+	0.567344i	$0.192029\pi$
$840$	0	0
$841$	5.24801	0.180966
$842$	−19.4845	−0.671479
$843$	0	0
$844$	72.6901	2.50210
$845$	−2.78861	−0.0959311
$846$	0	0
$847$	20.8874	0.717700
$848$	22.9907	0.789504
$849$	0	0
$850$	−54.5113	−1.86972
$851$	−58.4363	−2.00317
$852$	0	0
$853$	17.4238	0.596580	0.298290	−	0.954475i	$-0.403584\pi$
$853$	17.4238	0.596580	0.298290	+	0.954475i	$0.403584\pi$
$854$	2.50828	0.0858315
$855$	0	0
$856$	181.886	6.21673
$857$	−39.2056	−1.33924	−0.669619	−	0.742704i	$-0.733543\pi$
$857$	−39.2056	−1.33924	−0.669619	+	0.742704i	$0.733543\pi$
$858$	0	0
$859$	−0.358150	−0.0122199	−0.00610995	−	0.999981i	$-0.501945\pi$
$859$	−0.358150	−0.0122199	−0.00610995	+	0.999981i	$0.501945\pi$
$860$	14.6418	0.499283
$861$	0	0
$862$	−50.9540	−1.73550
$863$	−48.7339	−1.65892	−0.829460	−	0.558566i	$-0.811352\pi$
$863$	−48.7339	−1.65892	−0.829460	+	0.558566i	$0.811352\pi$
$864$	0	0
$865$	6.86519	0.233424
$866$	−101.075	−3.43466
$867$	0	0
$868$	31.2210	1.05971
$869$	−2.27586	−0.0772032
$870$	0	0
$871$	25.9857	0.880490
$872$	137.921	4.67061
$873$	0	0
$874$	107.621	3.64035
$875$	5.75959	0.194710
$876$	0	0
$877$	6.75342	0.228047	0.114023	−	0.993478i	$-0.463626\pi$
$877$	6.75342	0.228047	0.114023	+	0.993478i	$0.463626\pi$
$878$	−52.5535	−1.77360
$879$	0	0
$880$	−0.871583	−0.0293811
$881$	−43.6676	−1.47120	−0.735600	−	0.677416i	$-0.763100\pi$
$881$	−43.6676	−1.47120	−0.735600	+	0.677416i	$0.763100\pi$
$882$	0	0
$883$	−0.814635	−0.0274147	−0.0137073	−	0.999906i	$-0.504363\pi$
$883$	−0.814635	−0.0274147	−0.0137073	+	0.999906i	$0.504363\pi$
$884$	46.2717	1.55628
$885$	0	0
$886$	39.9587	1.34244
$887$	26.6048	0.893303	0.446652	−	0.894708i	$-0.352616\pi$
$887$	26.6048	0.893303	0.446652	+	0.894708i	$0.352616\pi$
$888$	0	0
$889$	11.0946	0.372101
$890$	1.04943	0.0351768
$891$	0	0
$892$	−52.6353	−1.76236
$893$	66.8127	2.23580
$894$	0	0
$895$	2.23749	0.0747910
$896$	166.618	5.56631
$897$	0	0
$898$	100.102	3.34046
$899$	−16.0732	−0.536070
$900$	0	0
$901$	−4.57910	−0.152552
$902$	2.00775	0.0668509
$903$	0	0
$904$	−46.4844	−1.54605
$905$	−4.75673	−0.158119
$906$	0	0
$907$	50.0070	1.66046	0.830228	−	0.557424i	$-0.188210\pi$
$907$	50.0070	1.66046	0.830228	+	0.557424i	$0.188210\pi$
$908$	−79.1000	−2.62502
$909$	0	0
$910$	−3.23197	−0.107139
$911$	8.64611	0.286458	0.143229	−	0.989690i	$-0.454251\pi$
$911$	8.64611	0.286458	0.143229	+	0.989690i	$0.454251\pi$
$912$	0	0
$913$	−0.687998	−0.0227694
$914$	17.1020	0.565683
$915$	0	0
$916$	−71.3189	−2.35644
$917$	3.04249	0.100472
$918$	0	0
$919$	4.88285	0.161070	0.0805352	−	0.996752i	$-0.474337\pi$
$919$	4.88285	0.161070	0.0805352	+	0.996752i	$0.474337\pi$
$920$	19.8256	0.653632
$921$	0	0
$922$	−16.2858	−0.536346
$923$	−0.0942671	−0.00310284
$924$	0	0
$925$	−49.6145	−1.63131
$926$	12.9265	0.424793
$927$	0	0
$928$	−195.042	−6.40258
$929$	13.6723	0.448574	0.224287	−	0.974523i	$-0.427995\pi$
$929$	13.6723	0.448574	0.224287	+	0.974523i	$0.427995\pi$
$930$	0	0
$931$	−22.2932	−0.730631
$932$	89.3050	2.92528
$933$	0	0
$934$	−33.0871	−1.08264
$935$	0.173595	0.00567715
$936$	0	0
$937$	16.9631	0.554162	0.277081	−	0.960847i	$-0.410633\pi$
$937$	16.9631	0.554162	0.277081	+	0.960847i	$0.410633\pi$
$938$	−70.9229	−2.31571
$939$	0	0
$940$	18.5005	0.603420
$941$	26.7050	0.870559	0.435279	−	0.900295i	$-0.356650\pi$
$941$	26.7050	0.870559	0.435279	+	0.900295i	$0.356650\pi$
$942$	0	0
$943$	−28.4544	−0.926602
$944$	−113.583	−3.69681
$945$	0	0
$946$	3.26975	0.106309
$947$	37.7256	1.22592	0.612958	−	0.790116i	$-0.289980\pi$
$947$	37.7256	1.22592	0.612958	+	0.790116i	$0.289980\pi$
$948$	0	0
$949$	−15.1841	−0.492895
$950$	91.3742	2.96457
$951$	0	0
$952$	−84.0179	−2.72304
$953$	−48.2534	−1.56308	−0.781540	−	0.623855i	$-0.785566\pi$
$953$	−48.2534	−1.56308	−0.781540	+	0.623855i	$0.785566\pi$
$954$	0	0
$955$	4.75422	0.153843
$956$	−148.263	−4.79518
$957$	0	0
$958$	115.436	3.72958
$959$	13.9660	0.450985
$960$	0	0
$961$	−23.4566	−0.756664
$962$	56.2118	1.81234
$963$	0	0
$964$	23.9369	0.770954
$965$	2.33826	0.0752713
$966$	0	0
$967$	32.7649	1.05365	0.526825	−	0.849974i	$-0.323382\pi$
$967$	32.7649	1.05365	0.526825	+	0.849974i	$0.323382\pi$
$968$	123.250	3.96142
$969$	0	0
$970$	4.41294	0.141691
$971$	29.2210	0.937746	0.468873	−	0.883265i	$-0.344660\pi$
$971$	29.2210	0.937746	0.468873	+	0.883265i	$0.344660\pi$
$972$	0	0
$973$	−6.95294	−0.222901
$974$	93.3108	2.98987
$975$	0	0
$976$	9.22147	0.295172
$977$	−0.304920	−0.00975525	−0.00487763	−	0.999988i	$-0.501553\pi$
$977$	−0.304920	−0.00975525	−0.00487763	+	0.999988i	$0.501553\pi$
$978$	0	0
$979$	0.175582	0.00561163
$980$	−6.17302	−0.197190
$981$	0	0
$982$	61.7879	1.97173
$983$	16.3625	0.521881	0.260941	−	0.965355i	$-0.415967\pi$
$983$	16.3625	0.521881	0.260941	+	0.965355i	$0.415967\pi$
$984$	0	0
$985$	−2.18478	−0.0696127
$986$	65.0164	2.07054
$987$	0	0
$988$	−77.5626	−2.46759
$989$	−46.3397	−1.47352
$990$	0	0
$991$	13.2445	0.420725	0.210362	−	0.977623i	$-0.432536\pi$
$991$	13.2445	0.420725	0.210362	+	0.977623i	$0.432536\pi$
$992$	91.5366	2.90629
$993$	0	0
$994$	0.257284	0.00816056
$995$	−3.67187	−0.116406
$996$	0	0
$997$	−22.8151	−0.722562	−0.361281	−	0.932457i	$-0.617660\pi$
$997$	−22.8151	−0.722562	−0.361281	+	0.932457i	$0.617660\pi$
$998$	59.0321	1.86863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.1	✓	25
3.2	odd	2		4023.2.a.f.1.25	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.1	✓	25	1.1	even	1	trivial
4023.2.a.f.1.25	yes	25	3.2	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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82403 q^{2} +5.97513 q^{4} +0.305598 q^{5} -1.90246 q^{7} -11.2259 q^{8} +O(q^{10})\)	\(q-2.82403 q^{2} +5.97513 q^{4} +0.305598 q^{5} -1.90246 q^{7} -11.2259 q^{8} -0.863018 q^{10} -0.144394 q^{11} -1.96848 q^{13} +5.37260 q^{14} +19.7519 q^{16} -3.93402 q^{17} +6.59437 q^{19} +1.82599 q^{20} +0.407772 q^{22} -5.77905 q^{23} -4.90661 q^{25} +5.55905 q^{26} -11.3674 q^{28} +5.85218 q^{29} -2.74653 q^{31} -33.3282 q^{32} +11.1098 q^{34} -0.581389 q^{35} +10.1118 q^{37} -18.6227 q^{38} -3.43061 q^{40} +4.92371 q^{41} +8.01858 q^{43} -0.862772 q^{44} +16.3202 q^{46} +10.1318 q^{47} -3.38064 q^{49} +13.8564 q^{50} -11.7619 q^{52} +1.16398 q^{53} -0.0441265 q^{55} +21.3568 q^{56} -16.5267 q^{58} -5.75048 q^{59} +0.466865 q^{61} +7.75626 q^{62} +54.6158 q^{64} -0.601565 q^{65} -13.2009 q^{67} -23.5063 q^{68} +1.64186 q^{70} +0.0478882 q^{71} +7.71359 q^{73} -28.5559 q^{74} +39.4022 q^{76} +0.274704 q^{77} +15.7615 q^{79} +6.03615 q^{80} -13.9047 q^{82} +4.76473 q^{83} -1.20223 q^{85} -22.6447 q^{86} +1.62095 q^{88} -1.21599 q^{89} +3.74496 q^{91} -34.5305 q^{92} -28.6124 q^{94} +2.01523 q^{95} -5.11338 q^{97} +9.54703 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

Introduction

L-functions

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