LMFDB - Embedded newform 4023.2.a.f.1.8

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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-0.680129 q^{2} -1.53742 q^{4} -2.78190 q^{5} -0.191226 q^{7} +2.40591 q^{8} +O(q^{10})$	\(q-0.680129 q^{2} -1.53742 q^{4} -2.78190 q^{5} -0.191226 q^{7} +2.40591 q^{8} +1.89205 q^{10} +3.81315 q^{11} +0.780186 q^{13} +0.130058 q^{14} +1.43852 q^{16} +5.90537 q^{17} +4.26424 q^{19} +4.27696 q^{20} -2.59344 q^{22} -5.94776 q^{23} +2.73895 q^{25} -0.530628 q^{26} +0.293995 q^{28} +1.48276 q^{29} -7.36446 q^{31} -5.79019 q^{32} -4.01642 q^{34} +0.531970 q^{35} +6.39900 q^{37} -2.90023 q^{38} -6.69298 q^{40} +3.39169 q^{41} -7.56033 q^{43} -5.86244 q^{44} +4.04524 q^{46} -4.15789 q^{47} -6.96343 q^{49} -1.86284 q^{50} -1.19948 q^{52} +3.46016 q^{53} -10.6078 q^{55} -0.460071 q^{56} -1.00847 q^{58} +11.4225 q^{59} -6.15860 q^{61} +5.00879 q^{62} +1.06104 q^{64} -2.17040 q^{65} -1.97910 q^{67} -9.07906 q^{68} -0.361808 q^{70} -4.38227 q^{71} +1.74168 q^{73} -4.35215 q^{74} -6.55594 q^{76} -0.729173 q^{77} -10.4604 q^{79} -4.00182 q^{80} -2.30679 q^{82} +4.97497 q^{83} -16.4281 q^{85} +5.14200 q^{86} +9.17409 q^{88} +14.1799 q^{89} -0.149192 q^{91} +9.14422 q^{92} +2.82790 q^{94} -11.8627 q^{95} +7.76970 q^{97} +4.73603 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.680129	−0.480924	−0.240462	−	0.970659i	$-0.577299\pi$
$2$	−0.680129	−0.480924	−0.240462	+	0.970659i	$0.577299\pi$
$3$	0	0
$3$	0	0
$4$	−1.53742	−0.768712
$5$	−2.78190	−1.24410	−0.622051	−	0.782977i	$-0.713700\pi$
$5$	−2.78190	−1.24410	−0.622051	+	0.782977i	$0.713700\pi$
$6$	0	0
$7$	−0.191226	−0.0722765	−0.0361382	−	0.999347i	$-0.511506\pi$
$7$	−0.191226	−0.0722765	−0.0361382	+	0.999347i	$0.511506\pi$
$8$	2.40591	0.850616
$9$	0	0
$10$	1.89205	0.598319
$11$	3.81315	1.14971	0.574855	−	0.818256i	$-0.305059\pi$
$11$	3.81315	1.14971	0.574855	+	0.818256i	$0.305059\pi$
$12$	0	0
$13$	0.780186	0.216385	0.108192	−	0.994130i	$-0.465494\pi$
$13$	0.780186	0.216385	0.108192	+	0.994130i	$0.465494\pi$
$14$	0.130058	0.0347595
$15$	0	0
$16$	1.43852	0.359630
$17$	5.90537	1.43226	0.716131	−	0.697966i	$-0.245911\pi$
$17$	5.90537	1.43226	0.716131	+	0.697966i	$0.245911\pi$
$18$	0	0
$19$	4.26424	0.978283	0.489142	−	0.872204i	$-0.337310\pi$
$19$	4.26424	0.978283	0.489142	+	0.872204i	$0.337310\pi$
$20$	4.27696	0.956356
$21$	0	0
$22$	−2.59344	−0.552923
$23$	−5.94776	−1.24019	−0.620096	−	0.784526i	$-0.712907\pi$
$23$	−5.94776	−1.24019	−0.620096	+	0.784526i	$0.712907\pi$
$24$	0	0
$25$	2.73895	0.547790
$26$	−0.530628	−0.104065
$27$	0	0
$28$	0.293995	0.0555598
$29$	1.48276	0.275341	0.137671	−	0.990478i	$-0.456038\pi$
$29$	1.48276	0.275341	0.137671	+	0.990478i	$0.456038\pi$
$30$	0	0
$31$	−7.36446	−1.32270	−0.661348	−	0.750079i	$-0.730016\pi$
$31$	−7.36446	−1.32270	−0.661348	+	0.750079i	$0.730016\pi$
$32$	−5.79019	−1.02357
$33$	0	0
$34$	−4.01642	−0.688810
$35$	0.531970	0.0899194
$36$	0	0
$37$	6.39900	1.05199	0.525995	−	0.850488i	$-0.323693\pi$
$37$	6.39900	1.05199	0.525995	+	0.850488i	$0.323693\pi$
$38$	−2.90023	−0.470480
$39$	0	0
$40$	−6.69298	−1.05825
$41$	3.39169	0.529693	0.264846	−	0.964291i	$-0.414679\pi$
$41$	3.39169	0.529693	0.264846	+	0.964291i	$0.414679\pi$
$42$	0	0
$43$	−7.56033	−1.15294	−0.576470	−	0.817118i	$-0.695570\pi$
$43$	−7.56033	−1.15294	−0.576470	+	0.817118i	$0.695570\pi$
$44$	−5.86244	−0.883795
$45$	0	0
$46$	4.04524	0.596439
$47$	−4.15789	−0.606491	−0.303245	−	0.952913i	$-0.598070\pi$
$47$	−4.15789	−0.606491	−0.303245	+	0.952913i	$0.598070\pi$
$48$	0	0
$49$	−6.96343	−0.994776
$50$	−1.86284	−0.263446
$51$	0	0
$52$	−1.19948	−0.166338
$53$	3.46016	0.475289	0.237645	−	0.971352i	$-0.423625\pi$
$53$	3.46016	0.475289	0.237645	+	0.971352i	$0.423625\pi$
$54$	0	0
$55$	−10.6078	−1.43036
$56$	−0.460071	−0.0614796
$57$	0	0
$58$	−1.00847	−0.132418
$59$	11.4225	1.48708	0.743540	−	0.668691i	$-0.233145\pi$
$59$	11.4225	1.48708	0.743540	+	0.668691i	$0.233145\pi$
$60$	0	0
$61$	−6.15860	−0.788528	−0.394264	−	0.918997i	$-0.629000\pi$
$61$	−6.15860	−0.788528	−0.394264	+	0.918997i	$0.629000\pi$
$62$	5.00879	0.636117
$63$	0	0
$64$	1.06104	0.132630
$65$	−2.17040	−0.269205
$66$	0	0
$67$	−1.97910	−0.241786	−0.120893	−	0.992666i	$-0.538576\pi$
$67$	−1.97910	−0.241786	−0.120893	+	0.992666i	$0.538576\pi$
$68$	−9.07906	−1.10100
$69$	0	0
$70$	−0.361808	−0.0432444
$71$	−4.38227	−0.520080	−0.260040	−	0.965598i	$-0.583736\pi$
$71$	−4.38227	−0.520080	−0.260040	+	0.965598i	$0.583736\pi$
$72$	0	0
$73$	1.74168	0.203848	0.101924	−	0.994792i	$-0.467500\pi$
$73$	1.74168	0.203848	0.101924	+	0.994792i	$0.467500\pi$
$74$	−4.35215	−0.505927
$75$	0	0
$76$	−6.55594	−0.752018
$77$	−0.729173	−0.0830970
$78$	0	0
$79$	−10.4604	−1.17689	−0.588445	−	0.808537i	$-0.700260\pi$
$79$	−10.4604	−1.17689	−0.588445	+	0.808537i	$0.700260\pi$
$80$	−4.00182	−0.447417
$81$	0	0
$82$	−2.30679	−0.254742
$83$	4.97497	0.546074	0.273037	−	0.962004i	$-0.411972\pi$
$83$	4.97497	0.546074	0.273037	+	0.962004i	$0.411972\pi$
$84$	0	0
$85$	−16.4281	−1.78188
$86$	5.14200	0.554476
$87$	0	0
$88$	9.17409	0.977961
$89$	14.1799	1.50306	0.751532	−	0.659697i	$-0.229315\pi$
$89$	14.1799	1.50306	0.751532	+	0.659697i	$0.229315\pi$
$90$	0	0
$91$	−0.149192	−0.0156395
$92$	9.14422	0.953351
$93$	0	0
$94$	2.82790	0.291676
$95$	−11.8627	−1.21708
$96$	0	0
$97$	7.76970	0.788894	0.394447	−	0.918919i	$-0.370936\pi$
$97$	7.76970	0.788894	0.394447	+	0.918919i	$0.370936\pi$
$98$	4.73603	0.478412
$99$	0	0
$100$	−4.21093	−0.421093
$101$	12.4047	1.23431	0.617156	−	0.786841i	$-0.288285\pi$
$101$	12.4047	1.23431	0.617156	+	0.786841i	$0.288285\pi$
$102$	0	0
$103$	11.6712	1.15000	0.575001	−	0.818153i	$-0.305002\pi$
$103$	11.6712	1.15000	0.575001	+	0.818153i	$0.305002\pi$
$104$	1.87705	0.184060
$105$	0	0
$106$	−2.35335	−0.228578
$107$	−6.65056	−0.642934	−0.321467	−	0.946921i	$-0.604176\pi$
$107$	−6.65056	−0.642934	−0.321467	+	0.946921i	$0.604176\pi$
$108$	0	0
$109$	−3.24412	−0.310730	−0.155365	−	0.987857i	$-0.549655\pi$
$109$	−3.24412	−0.310730	−0.155365	+	0.987857i	$0.549655\pi$
$110$	7.21468	0.687893
$111$	0	0
$112$	−0.275082	−0.0259928
$113$	10.0431	0.944773	0.472387	−	0.881391i	$-0.343393\pi$
$113$	10.0431	0.944773	0.472387	+	0.881391i	$0.343393\pi$
$114$	0	0
$115$	16.5460	1.54293
$116$	−2.27963	−0.211658
$117$	0	0
$118$	−7.76877	−0.715173
$119$	−1.12926	−0.103519
$120$	0	0
$121$	3.54015	0.321832
$122$	4.18865	0.379222
$123$	0	0
$124$	11.3223	1.01677
$125$	6.29000	0.562595
$126$	0	0
$127$	−8.19875	−0.727521	−0.363761	−	0.931492i	$-0.618507\pi$
$127$	−8.19875	−0.727521	−0.363761	+	0.931492i	$0.618507\pi$
$128$	10.8587	0.959786
$129$	0	0
$130$	1.47615	0.129467
$131$	−6.49287	−0.567285	−0.283642	−	0.958930i	$-0.591543\pi$
$131$	−6.49287	−0.567285	−0.283642	+	0.958930i	$0.591543\pi$
$132$	0	0
$133$	−0.815431	−0.0707069
$134$	1.34604	0.116281
$135$	0	0
$136$	14.2078	1.21831
$137$	0.541055	0.0462254	0.0231127	−	0.999733i	$-0.492642\pi$
$137$	0.541055	0.0462254	0.0231127	+	0.999733i	$0.492642\pi$
$138$	0	0
$139$	18.4109	1.56159	0.780796	−	0.624786i	$-0.214814\pi$
$139$	18.4109	1.56159	0.780796	+	0.624786i	$0.214814\pi$
$140$	−0.817864	−0.0691221
$141$	0	0
$142$	2.98051	0.250119
$143$	2.97497	0.248780
$144$	0	0
$145$	−4.12488	−0.342552
$146$	−1.18457	−0.0980355
$147$	0	0
$148$	−9.83798	−0.808677
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−17.5761	−1.43033	−0.715163	−	0.698958i	$-0.753648\pi$
$151$	−17.5761	−1.43033	−0.715163	+	0.698958i	$0.753648\pi$
$152$	10.2594	0.832143
$153$	0	0
$154$	0.495932	0.0399633
$155$	20.4872	1.64557
$156$	0	0
$157$	12.1084	0.966356	0.483178	−	0.875522i	$-0.339482\pi$
$157$	12.1084	0.966356	0.483178	+	0.875522i	$0.339482\pi$
$158$	7.11444	0.565995
$159$	0	0
$160$	16.1077	1.27343
$161$	1.13736	0.0896368
$162$	0	0
$163$	−10.9066	−0.854274	−0.427137	−	0.904187i	$-0.640478\pi$
$163$	−10.9066	−0.854274	−0.427137	+	0.904187i	$0.640478\pi$
$164$	−5.21447	−0.407181
$165$	0	0
$166$	−3.38362	−0.262620
$167$	−3.85518	−0.298323	−0.149161	−	0.988813i	$-0.547657\pi$
$167$	−3.85518	−0.298323	−0.149161	+	0.988813i	$0.547657\pi$
$168$	0	0
$169$	−12.3913	−0.953178
$170$	11.1733	0.856949
$171$	0	0
$172$	11.6234	0.886279
$173$	−11.5737	−0.879929	−0.439964	−	0.898015i	$-0.645009\pi$
$173$	−11.5737	−0.879929	−0.439964	+	0.898015i	$0.645009\pi$
$174$	0	0
$175$	−0.523758	−0.0395924
$176$	5.48530	0.413470
$177$	0	0
$178$	−9.64415	−0.722859
$179$	11.2812	0.843196	0.421598	−	0.906783i	$-0.361469\pi$
$179$	11.2812	0.843196	0.421598	+	0.906783i	$0.361469\pi$
$180$	0	0
$181$	5.64624	0.419681	0.209841	−	0.977736i	$-0.432705\pi$
$181$	5.64624	0.419681	0.209841	+	0.977736i	$0.432705\pi$
$182$	0.101470	0.00752143
$183$	0	0
$184$	−14.3097	−1.05493
$185$	−17.8014	−1.30878
$186$	0	0
$187$	22.5181	1.64669
$188$	6.39244	0.466217
$189$	0	0
$190$	8.06815	0.585325
$191$	17.8975	1.29502	0.647508	−	0.762059i	$-0.275811\pi$
$191$	17.8975	1.29502	0.647508	+	0.762059i	$0.275811\pi$
$192$	0	0
$193$	15.0925	1.08638	0.543192	−	0.839609i	$-0.317216\pi$
$193$	15.0925	1.08638	0.543192	+	0.839609i	$0.317216\pi$
$194$	−5.28440	−0.379398
$195$	0	0
$196$	10.7057	0.764696
$197$	12.9601	0.923370	0.461685	−	0.887044i	$-0.347245\pi$
$197$	12.9601	0.923370	0.461685	+	0.887044i	$0.347245\pi$
$198$	0	0
$199$	11.0796	0.785415	0.392707	−	0.919664i	$-0.371539\pi$
$199$	11.0796	0.785415	0.392707	+	0.919664i	$0.371539\pi$
$200$	6.58966	0.465959
$201$	0	0
$202$	−8.43679	−0.593611
$203$	−0.283541	−0.0199007
$204$	0	0
$205$	−9.43533	−0.658992
$206$	−7.93795	−0.553063
$207$	0	0
$208$	1.12231	0.0778185
$209$	16.2602	1.12474
$210$	0	0
$211$	5.19699	0.357776	0.178888	−	0.983869i	$-0.442750\pi$
$211$	5.19699	0.357776	0.178888	+	0.983869i	$0.442750\pi$
$212$	−5.31973	−0.365361
$213$	0	0
$214$	4.52324	0.309203
$215$	21.0321	1.43437
$216$	0	0
$217$	1.40827	0.0955999
$218$	2.20642	0.149437
$219$	0	0
$220$	16.3087	1.09953
$221$	4.60729	0.309920
$222$	0	0
$223$	9.91927	0.664243	0.332122	−	0.943237i	$-0.392236\pi$
$223$	9.91927	0.664243	0.332122	+	0.943237i	$0.392236\pi$
$224$	1.10723	0.0739801
$225$	0	0
$226$	−6.83059	−0.454364
$227$	0.105156	0.00697948	0.00348974	−	0.999994i	$-0.498889\pi$
$227$	0.105156	0.00697948	0.00348974	+	0.999994i	$0.498889\pi$
$228$	0	0
$229$	−20.4017	−1.34818	−0.674090	−	0.738649i	$-0.735464\pi$
$229$	−20.4017	−1.34818	−0.674090	+	0.738649i	$0.735464\pi$
$230$	−11.2534	−0.742031
$231$	0	0
$232$	3.56737	0.234210
$233$	−0.821785	−0.0538369	−0.0269185	−	0.999638i	$-0.508569\pi$
$233$	−0.821785	−0.0538369	−0.0269185	+	0.999638i	$0.508569\pi$
$234$	0	0
$235$	11.5668	0.754536
$236$	−17.5612	−1.14314
$237$	0	0
$238$	0.768042	0.0497847
$239$	−2.82138	−0.182500	−0.0912499	−	0.995828i	$-0.529086\pi$
$239$	−2.82138	−0.182500	−0.0912499	+	0.995828i	$0.529086\pi$
$240$	0	0
$241$	−9.69237	−0.624340	−0.312170	−	0.950026i	$-0.601056\pi$
$241$	−9.69237	−0.624340	−0.312170	+	0.950026i	$0.601056\pi$
$242$	−2.40776	−0.154777
$243$	0	0
$244$	9.46838	0.606151
$245$	19.3716	1.23760
$246$	0	0
$247$	3.32690	0.211686
$248$	−17.7182	−1.12511
$249$	0	0
$250$	−4.27801	−0.270565
$251$	8.17338	0.515899	0.257950	−	0.966158i	$-0.416953\pi$
$251$	8.17338	0.515899	0.257950	+	0.966158i	$0.416953\pi$
$252$	0	0
$253$	−22.6797	−1.42586
$254$	5.57621	0.349882
$255$	0	0
$256$	−9.50742	−0.594214
$257$	17.1335	1.06876	0.534380	−	0.845244i	$-0.320545\pi$
$257$	17.1335	1.06876	0.534380	+	0.845244i	$0.320545\pi$
$258$	0	0
$259$	−1.22365	−0.0760341
$260$	3.33682	0.206941
$261$	0	0
$262$	4.41599	0.272821
$263$	−4.92173	−0.303487	−0.151743	−	0.988420i	$-0.548489\pi$
$263$	−4.92173	−0.303487	−0.151743	+	0.988420i	$0.548489\pi$
$264$	0	0
$265$	−9.62580	−0.591308
$266$	0.554599	0.0340046
$267$	0	0
$268$	3.04272	0.185864
$269$	3.91592	0.238758	0.119379	−	0.992849i	$-0.461910\pi$
$269$	3.91592	0.238758	0.119379	+	0.992849i	$0.461910\pi$
$270$	0	0
$271$	−18.3026	−1.11180	−0.555901	−	0.831249i	$-0.687626\pi$
$271$	−18.3026	−1.11180	−0.555901	+	0.831249i	$0.687626\pi$
$272$	8.49500	0.515085
$273$	0	0
$274$	−0.367987	−0.0222309
$275$	10.4440	0.629800
$276$	0	0
$277$	23.5947	1.41767	0.708833	−	0.705376i	$-0.249222\pi$
$277$	23.5947	1.41767	0.708833	+	0.705376i	$0.249222\pi$
$278$	−12.5218	−0.751007
$279$	0	0
$280$	1.27987	0.0764869
$281$	−0.0794870	−0.00474180	−0.00237090	−	0.999997i	$-0.500755\pi$
$281$	−0.0794870	−0.00474180	−0.00237090	+	0.999997i	$0.500755\pi$
$282$	0	0
$283$	16.3223	0.970260	0.485130	−	0.874442i	$-0.338772\pi$
$283$	16.3223	0.970260	0.485130	+	0.874442i	$0.338772\pi$
$284$	6.73741	0.399792
$285$	0	0
$286$	−2.02337	−0.119644
$287$	−0.648578	−0.0382844
$288$	0	0
$289$	17.8734	1.05138
$290$	2.80545	0.164742
$291$	0	0
$292$	−2.67770	−0.156701
$293$	3.67003	0.214405	0.107203	−	0.994237i	$-0.465811\pi$
$293$	3.67003	0.214405	0.107203	+	0.994237i	$0.465811\pi$
$294$	0	0
$295$	−31.7762	−1.85008
$296$	15.3954	0.894839
$297$	0	0
$298$	−0.680129	−0.0393988
$299$	−4.64036	−0.268359
$300$	0	0
$301$	1.44573	0.0833304
$302$	11.9540	0.687878
$303$	0	0
$304$	6.13420	0.351820
$305$	17.1326	0.981010
$306$	0	0
$307$	−31.5576	−1.80109	−0.900543	−	0.434767i	$-0.856831\pi$
$307$	−31.5576	−1.80109	−0.900543	+	0.434767i	$0.856831\pi$
$308$	1.12105	0.0638776
$309$	0	0
$310$	−13.9339	−0.791394
$311$	−1.98583	−0.112606	−0.0563032	−	0.998414i	$-0.517931\pi$
$311$	−1.98583	−0.112606	−0.0563032	+	0.998414i	$0.517931\pi$
$312$	0	0
$313$	3.49446	0.197519	0.0987594	−	0.995111i	$-0.468513\pi$
$313$	3.49446	0.197519	0.0987594	+	0.995111i	$0.468513\pi$
$314$	−8.23528	−0.464744
$315$	0	0
$316$	16.0821	0.904689
$317$	4.19203	0.235448	0.117724	−	0.993046i	$-0.462440\pi$
$317$	4.19203	0.235448	0.117724	+	0.993046i	$0.462440\pi$
$318$	0	0
$319$	5.65398	0.316562
$320$	−2.95170	−0.165005
$321$	0	0
$322$	−0.773554	−0.0431085
$323$	25.1819	1.40116
$324$	0	0
$325$	2.13689	0.118534
$326$	7.41793	0.410841
$327$	0	0
$328$	8.16009	0.450565
$329$	0.795095	0.0438350
$330$	0	0
$331$	23.9087	1.31414	0.657071	−	0.753829i	$-0.271795\pi$
$331$	23.9087	1.31414	0.657071	+	0.753829i	$0.271795\pi$
$332$	−7.64864	−0.419773
$333$	0	0
$334$	2.62202	0.143471
$335$	5.50565	0.300806
$336$	0	0
$337$	26.8723	1.46383	0.731914	−	0.681397i	$-0.238627\pi$
$337$	26.8723	1.46383	0.731914	+	0.681397i	$0.238627\pi$
$338$	8.42769	0.458406
$339$	0	0
$340$	25.2570	1.36975
$341$	−28.0818	−1.52072
$342$	0	0
$343$	2.67017	0.144175
$344$	−18.1894	−0.980709
$345$	0	0
$346$	7.87158	0.423179
$347$	0.410569	0.0220405	0.0110203	−	0.999939i	$-0.496492\pi$
$347$	0.410569	0.0220405	0.0110203	+	0.999939i	$0.496492\pi$
$348$	0	0
$349$	−1.47715	−0.0790699	−0.0395349	−	0.999218i	$-0.512588\pi$
$349$	−1.47715	−0.0790699	−0.0395349	+	0.999218i	$0.512588\pi$
$350$	0.356223	0.0190409
$351$	0	0
$352$	−22.0789	−1.17681
$353$	−10.7493	−0.572127	−0.286063	−	0.958211i	$-0.592347\pi$
$353$	−10.7493	−0.572127	−0.286063	+	0.958211i	$0.592347\pi$
$354$	0	0
$355$	12.1910	0.647032
$356$	−21.8005	−1.15542
$357$	0	0
$358$	−7.67267	−0.405513
$359$	−10.4334	−0.550652	−0.275326	−	0.961351i	$-0.588786\pi$
$359$	−10.4334	−0.550652	−0.275326	+	0.961351i	$0.588786\pi$
$360$	0	0
$361$	−0.816281	−0.0429622
$362$	−3.84017	−0.201835
$363$	0	0
$364$	0.229371	0.0120223
$365$	−4.84518	−0.253608
$366$	0	0
$367$	9.56209	0.499137	0.249569	−	0.968357i	$-0.419711\pi$
$367$	9.56209	0.499137	0.249569	+	0.968357i	$0.419711\pi$
$368$	−8.55597	−0.446011
$369$	0	0
$370$	12.1072	0.629425
$371$	−0.661671	−0.0343522
$372$	0	0
$373$	−6.48317	−0.335686	−0.167843	−	0.985814i	$-0.553680\pi$
$373$	−6.48317	−0.335686	−0.167843	+	0.985814i	$0.553680\pi$
$374$	−15.3152	−0.791931
$375$	0	0
$376$	−10.0035	−0.515891
$377$	1.15683	0.0595796
$378$	0	0
$379$	1.48772	0.0764191	0.0382096	−	0.999270i	$-0.487835\pi$
$379$	1.48772	0.0764191	0.0382096	+	0.999270i	$0.487835\pi$
$380$	18.2380	0.935587
$381$	0	0
$382$	−12.1726	−0.622804
$383$	20.7269	1.05909	0.529546	−	0.848281i	$-0.322362\pi$
$383$	20.7269	1.05909	0.529546	+	0.848281i	$0.322362\pi$
$384$	0	0
$385$	2.02848	0.103381
$386$	−10.2649	−0.522468
$387$	0	0
$388$	−11.9453	−0.606432
$389$	9.32445	0.472768	0.236384	−	0.971660i	$-0.424038\pi$
$389$	9.32445	0.472768	0.236384	+	0.971660i	$0.424038\pi$
$390$	0	0
$391$	−35.1237	−1.77628
$392$	−16.7534	−0.846173
$393$	0	0
$394$	−8.81455	−0.444071
$395$	29.0998	1.46417
$396$	0	0
$397$	−35.6763	−1.79054	−0.895272	−	0.445520i	$-0.853019\pi$
$397$	−35.6763	−1.79054	−0.895272	+	0.445520i	$0.853019\pi$
$398$	−7.53559	−0.377725
$399$	0	0
$400$	3.94004	0.197002
$401$	−23.4743	−1.17225	−0.586126	−	0.810220i	$-0.699348\pi$
$401$	−23.4743	−1.17225	−0.586126	+	0.810220i	$0.699348\pi$
$402$	0	0
$403$	−5.74565	−0.286211
$404$	−19.0713	−0.948831
$405$	0	0
$406$	0.192845	0.00957072
$407$	24.4004	1.20948
$408$	0	0
$409$	30.0070	1.48375	0.741876	−	0.670537i	$-0.233936\pi$
$409$	30.0070	1.48375	0.741876	+	0.670537i	$0.233936\pi$
$410$	6.41725	0.316925
$411$	0	0
$412$	−17.9436	−0.884020
$413$	−2.18427	−0.107481
$414$	0	0
$415$	−13.8398	−0.679371
$416$	−4.51743	−0.221485
$417$	0	0
$418$	−11.0590	−0.540915
$419$	−25.5850	−1.24991	−0.624954	−	0.780662i	$-0.714882\pi$
$419$	−25.5850	−1.24991	−0.624954	+	0.780662i	$0.714882\pi$
$420$	0	0
$421$	−8.42486	−0.410603	−0.205301	−	0.978699i	$-0.565817\pi$
$421$	−8.42486	−0.410603	−0.205301	+	0.978699i	$0.565817\pi$
$422$	−3.53463	−0.172063
$423$	0	0
$424$	8.32481	0.404289
$425$	16.1745	0.784580
$426$	0	0
$427$	1.17768	0.0569921
$428$	10.2247	0.494231
$429$	0	0
$430$	−14.3045	−0.689825
$431$	−7.45195	−0.358948	−0.179474	−	0.983763i	$-0.557440\pi$
$431$	−7.45195	−0.358948	−0.179474	+	0.983763i	$0.557440\pi$
$432$	0	0
$433$	−30.2392	−1.45320	−0.726602	−	0.687059i	$-0.758902\pi$
$433$	−30.2392	−1.45320	−0.726602	+	0.687059i	$0.758902\pi$
$434$	−0.957809	−0.0459763
$435$	0	0
$436$	4.98758	0.238862
$437$	−25.3626	−1.21326
$438$	0	0
$439$	26.5916	1.26915	0.634575	−	0.772862i	$-0.281175\pi$
$439$	26.5916	1.26915	0.634575	+	0.772862i	$0.281175\pi$
$440$	−25.5214	−1.21668
$441$	0	0
$442$	−3.13355	−0.149048
$443$	35.4520	1.68438	0.842189	−	0.539183i	$-0.181267\pi$
$443$	35.4520	1.68438	0.842189	+	0.539183i	$0.181267\pi$
$444$	0	0
$445$	−39.4469	−1.86996
$446$	−6.74638	−0.319451
$447$	0	0
$448$	−0.202897	−0.00958600
$449$	20.1447	0.950687	0.475344	−	0.879800i	$-0.342324\pi$
$449$	20.1447	0.950687	0.475344	+	0.879800i	$0.342324\pi$
$450$	0	0
$451$	12.9330	0.608993
$452$	−15.4405	−0.726259
$453$	0	0
$454$	−0.0715200	−0.00335660
$455$	0.415036	0.0194572
$456$	0	0
$457$	−19.0897	−0.892981	−0.446490	−	0.894788i	$-0.647326\pi$
$457$	−19.0897	−0.892981	−0.446490	+	0.894788i	$0.647326\pi$
$458$	13.8758	0.648372
$459$	0	0
$460$	−25.4383	−1.18607
$461$	22.2929	1.03828	0.519141	−	0.854689i	$-0.326252\pi$
$461$	22.2929	1.03828	0.519141	+	0.854689i	$0.326252\pi$
$462$	0	0
$463$	26.0080	1.20869	0.604346	−	0.796722i	$-0.293434\pi$
$463$	26.0080	1.20869	0.604346	+	0.796722i	$0.293434\pi$
$464$	2.13298	0.0990210
$465$	0	0
$466$	0.558920	0.0258915
$467$	17.3724	0.803899	0.401949	−	0.915662i	$-0.368333\pi$
$467$	17.3724	0.803899	0.401949	+	0.915662i	$0.368333\pi$
$468$	0	0
$469$	0.378455	0.0174754
$470$	−7.86694	−0.362875
$471$	0	0
$472$	27.4814	1.26493
$473$	−28.8287	−1.32555
$474$	0	0
$475$	11.6795	0.535894
$476$	1.73615	0.0795762
$477$	0	0
$478$	1.91890	0.0877686
$479$	5.15886	0.235714	0.117857	−	0.993031i	$-0.462397\pi$
$479$	5.15886	0.235714	0.117857	+	0.993031i	$0.462397\pi$
$480$	0	0
$481$	4.99242	0.227635
$482$	6.59207	0.300260
$483$	0	0
$484$	−5.44271	−0.247396
$485$	−21.6145	−0.981465
$486$	0	0
$487$	9.77281	0.442848	0.221424	−	0.975178i	$-0.428929\pi$
$487$	9.77281	0.442848	0.221424	+	0.975178i	$0.428929\pi$
$488$	−14.8170	−0.670735
$489$	0	0
$490$	−13.1752	−0.595193
$491$	−36.1198	−1.63007	−0.815033	−	0.579415i	$-0.803281\pi$
$491$	−36.1198	−1.63007	−0.815033	+	0.579415i	$0.803281\pi$
$492$	0	0
$493$	8.75623	0.394361
$494$	−2.26272	−0.101805
$495$	0	0
$496$	−10.5939	−0.475682
$497$	0.838002	0.0375895
$498$	0	0
$499$	30.7383	1.37604	0.688018	−	0.725694i	$-0.258481\pi$
$499$	30.7383	1.37604	0.688018	+	0.725694i	$0.258481\pi$
$500$	−9.67040	−0.432473
$501$	0	0
$502$	−5.55896	−0.248108
$503$	−0.319080	−0.0142271	−0.00711353	−	0.999975i	$-0.502264\pi$
$503$	−0.319080	−0.0142271	−0.00711353	+	0.999975i	$0.502264\pi$
$504$	0	0
$505$	−34.5086	−1.53561
$506$	15.4251	0.685731
$507$	0	0
$508$	12.6050	0.559254
$509$	35.1070	1.55609	0.778046	−	0.628207i	$-0.216211\pi$
$509$	35.1070	1.55609	0.778046	+	0.628207i	$0.216211\pi$
$510$	0	0
$511$	−0.333054	−0.0147334
$512$	−15.2512	−0.674015
$513$	0	0
$514$	−11.6530	−0.513993
$515$	−32.4682	−1.43072
$516$	0	0
$517$	−15.8547	−0.697288
$518$	0.832243	0.0365666
$519$	0	0
$520$	−5.22177	−0.228990
$521$	−29.4763	−1.29138	−0.645690	−	0.763599i	$-0.723430\pi$
$521$	−29.4763	−1.29138	−0.645690	+	0.763599i	$0.723430\pi$
$522$	0	0
$523$	18.4653	0.807433	0.403716	−	0.914884i	$-0.367718\pi$
$523$	18.4653	0.807433	0.403716	+	0.914884i	$0.367718\pi$
$524$	9.98229	0.436079
$525$	0	0
$526$	3.34741	0.145954
$527$	−43.4899	−1.89445
$528$	0	0
$529$	12.3758	0.538078
$530$	6.54679	0.284374
$531$	0	0
$532$	1.25366	0.0543532
$533$	2.64615	0.114617
$534$	0	0
$535$	18.5012	0.799876
$536$	−4.76153	−0.205667
$537$	0	0
$538$	−2.66333	−0.114824
$539$	−26.5526	−1.14370
$540$	0	0
$541$	9.79337	0.421050	0.210525	−	0.977588i	$-0.432483\pi$
$541$	9.79337	0.421050	0.210525	+	0.977588i	$0.432483\pi$
$542$	12.4481	0.534692
$543$	0	0
$544$	−34.1932	−1.46602
$545$	9.02479	0.386580
$546$	0	0
$547$	40.6173	1.73667	0.868335	−	0.495979i	$-0.165191\pi$
$547$	40.6173	1.73667	0.868335	+	0.495979i	$0.165191\pi$
$548$	−0.831830	−0.0355340
$549$	0	0
$550$	−7.10330	−0.302886
$551$	6.32283	0.269361
$552$	0	0
$553$	2.00030	0.0850615
$554$	−16.0474	−0.681790
$555$	0	0
$556$	−28.3054	−1.20042
$557$	24.1514	1.02333	0.511664	−	0.859185i	$-0.329029\pi$
$557$	24.1514	1.02333	0.511664	+	0.859185i	$0.329029\pi$
$558$	0	0
$559$	−5.89847	−0.249479
$560$	0.765250	0.0323377
$561$	0	0
$562$	0.0540614	0.00228044
$563$	45.0142	1.89712	0.948562	−	0.316591i	$-0.102538\pi$
$563$	45.0142	1.89712	0.948562	+	0.316591i	$0.102538\pi$
$564$	0	0
$565$	−27.9388	−1.17539
$566$	−11.1013	−0.466621
$567$	0	0
$568$	−10.5433	−0.442388
$569$	44.8797	1.88146	0.940728	−	0.339162i	$-0.110144\pi$
$569$	44.8797	1.88146	0.940728	+	0.339162i	$0.110144\pi$
$570$	0	0
$571$	41.5202	1.73757	0.868783	−	0.495193i	$-0.164903\pi$
$571$	41.5202	1.73757	0.868783	+	0.495193i	$0.164903\pi$
$572$	−4.57379	−0.191240
$573$	0	0
$574$	0.441117	0.0184119
$575$	−16.2906	−0.679366
$576$	0	0
$577$	−17.6664	−0.735462	−0.367731	−	0.929932i	$-0.619865\pi$
$577$	−17.6664	−0.735462	−0.367731	+	0.929932i	$0.619865\pi$
$578$	−12.1562	−0.505632
$579$	0	0
$580$	6.34169	0.263324
$581$	−0.951341	−0.0394683
$582$	0	0
$583$	13.1941	0.546444
$584$	4.19032	0.173397
$585$	0	0
$586$	−2.49609	−0.103113
$587$	−44.0011	−1.81612	−0.908060	−	0.418839i	$-0.862437\pi$
$587$	−44.0011	−1.81612	−0.908060	+	0.418839i	$0.862437\pi$
$588$	0	0
$589$	−31.4038	−1.29397
$590$	21.6119	0.889748
$591$	0	0
$592$	9.20510	0.378327
$593$	−35.0980	−1.44130	−0.720652	−	0.693297i	$-0.756157\pi$
$593$	−35.0980	−1.44130	−0.720652	+	0.693297i	$0.756157\pi$
$594$	0	0
$595$	3.14148	0.128788
$596$	−1.53742	−0.0629753
$597$	0	0
$598$	3.15604	0.129060
$599$	6.49449	0.265357	0.132679	−	0.991159i	$-0.457642\pi$
$599$	6.49449	0.265357	0.132679	+	0.991159i	$0.457642\pi$
$600$	0	0
$601$	13.4774	0.549754	0.274877	−	0.961479i	$-0.411363\pi$
$601$	13.4774	0.549754	0.274877	+	0.961479i	$0.411363\pi$
$602$	−0.983283	−0.0400756
$603$	0	0
$604$	27.0220	1.09951
$605$	−9.84833	−0.400391
$606$	0	0
$607$	42.2726	1.71579	0.857897	−	0.513822i	$-0.171771\pi$
$607$	42.2726	1.71579	0.857897	+	0.513822i	$0.171771\pi$
$608$	−24.6908	−1.00134
$609$	0	0
$610$	−11.6524	−0.471791
$611$	−3.24393	−0.131235
$612$	0	0
$613$	30.0980	1.21565	0.607824	−	0.794072i	$-0.292043\pi$
$613$	30.0980	1.21565	0.607824	+	0.794072i	$0.292043\pi$
$614$	21.4632	0.866186
$615$	0	0
$616$	−1.75432	−0.0706836
$617$	−10.8695	−0.437590	−0.218795	−	0.975771i	$-0.570213\pi$
$617$	−10.8695	−0.437590	−0.218795	+	0.975771i	$0.570213\pi$
$618$	0	0
$619$	−9.12340	−0.366701	−0.183350	−	0.983048i	$-0.558694\pi$
$619$	−9.12340	−0.366701	−0.183350	+	0.983048i	$0.558694\pi$
$620$	−31.4975	−1.26497
$621$	0	0
$622$	1.35062	0.0541551
$623$	−2.71155	−0.108636
$624$	0	0
$625$	−31.1929	−1.24772
$626$	−2.37669	−0.0949915
$627$	0	0
$628$	−18.6157	−0.742849
$629$	37.7885	1.50673
$630$	0	0
$631$	34.6128	1.37791	0.688957	−	0.724802i	$-0.258069\pi$
$631$	34.6128	1.37791	0.688957	+	0.724802i	$0.258069\pi$
$632$	−25.1668	−1.00108
$633$	0	0
$634$	−2.85112	−0.113232
$635$	22.8081	0.905111
$636$	0	0
$637$	−5.43278	−0.215254
$638$	−3.84544	−0.152242
$639$	0	0
$640$	−30.2079	−1.19407
$641$	18.8891	0.746075	0.373038	−	0.927816i	$-0.378316\pi$
$641$	18.8891	0.746075	0.373038	+	0.927816i	$0.378316\pi$
$642$	0	0
$643$	21.1718	0.834935	0.417467	−	0.908692i	$-0.362918\pi$
$643$	21.1718	0.834935	0.417467	+	0.908692i	$0.362918\pi$
$644$	−1.74861	−0.0689049
$645$	0	0
$646$	−17.1269	−0.673851
$647$	4.01183	0.157721	0.0788606	−	0.996886i	$-0.474872\pi$
$647$	4.01183	0.157721	0.0788606	+	0.996886i	$0.474872\pi$
$648$	0	0
$649$	43.5557	1.70971
$650$	−1.45336	−0.0570056
$651$	0	0
$652$	16.7681	0.656691
$653$	28.0625	1.09817	0.549085	−	0.835767i	$-0.314976\pi$
$653$	28.0625	1.09817	0.549085	+	0.835767i	$0.314976\pi$
$654$	0	0
$655$	18.0625	0.705760
$656$	4.87902	0.190494
$657$	0	0
$658$	−0.540768	−0.0210813
$659$	−19.3247	−0.752784	−0.376392	−	0.926460i	$-0.622835\pi$
$659$	−19.3247	−0.752784	−0.376392	+	0.926460i	$0.622835\pi$
$660$	0	0
$661$	−4.39981	−0.171133	−0.0855664	−	0.996332i	$-0.527270\pi$
$661$	−4.39981	−0.171133	−0.0855664	+	0.996332i	$0.527270\pi$
$662$	−16.2610	−0.632002
$663$	0	0
$664$	11.9693	0.464499
$665$	2.26845	0.0879666
$666$	0	0
$667$	−8.81908	−0.341476
$668$	5.92705	0.229324
$669$	0	0
$670$	−3.74456	−0.144665
$671$	−23.4837	−0.906578
$672$	0	0
$673$	29.6144	1.14155	0.570775	−	0.821106i	$-0.306643\pi$
$673$	29.6144	1.14155	0.570775	+	0.821106i	$0.306643\pi$
$674$	−18.2766	−0.703990
$675$	0	0
$676$	19.0507	0.732719
$677$	−0.870348	−0.0334502	−0.0167251	−	0.999860i	$-0.505324\pi$
$677$	−0.870348	−0.0334502	−0.0167251	+	0.999860i	$0.505324\pi$
$678$	0	0
$679$	−1.48577	−0.0570185
$680$	−39.5245	−1.51570
$681$	0	0
$682$	19.0993	0.731349
$683$	−9.15518	−0.350313	−0.175157	−	0.984541i	$-0.556043\pi$
$683$	−9.15518	−0.350313	−0.175157	+	0.984541i	$0.556043\pi$
$684$	0	0
$685$	−1.50516	−0.0575091
$686$	−1.81606	−0.0693374
$687$	0	0
$688$	−10.8757	−0.414632
$689$	2.69957	0.102845
$690$	0	0
$691$	−0.972064	−0.0369791	−0.0184895	−	0.999829i	$-0.505886\pi$
$691$	−0.972064	−0.0369791	−0.0184895	+	0.999829i	$0.505886\pi$
$692$	17.7936	0.676412
$693$	0	0
$694$	−0.279240	−0.0105998
$695$	−51.2172	−1.94278
$696$	0	0
$697$	20.0292	0.758659
$698$	1.00465	0.0380266
$699$	0	0
$700$	0.805238	0.0304351
$701$	0.638668	0.0241222	0.0120611	−	0.999927i	$-0.496161\pi$
$701$	0.638668	0.0241222	0.0120611	+	0.999927i	$0.496161\pi$
$702$	0	0
$703$	27.2869	1.02914
$704$	4.04590	0.152485
$705$	0	0
$706$	7.31091	0.275150
$707$	−2.37209	−0.0892118
$708$	0	0
$709$	44.9592	1.68848	0.844239	−	0.535967i	$-0.180053\pi$
$709$	44.9592	1.68848	0.844239	+	0.535967i	$0.180053\pi$
$710$	−8.29147	−0.311173
$711$	0	0
$712$	34.1154	1.27853
$713$	43.8020	1.64040
$714$	0	0
$715$	−8.27606	−0.309507
$716$	−17.3440	−0.648175
$717$	0	0
$718$	7.09604	0.264822
$719$	12.9293	0.482182	0.241091	−	0.970503i	$-0.422495\pi$
$719$	12.9293	0.482182	0.241091	+	0.970503i	$0.422495\pi$
$720$	0	0
$721$	−2.23184	−0.0831181
$722$	0.555177	0.0206615
$723$	0	0
$724$	−8.68066	−0.322614
$725$	4.06120	0.150829
$726$	0	0
$727$	−35.8730	−1.33046	−0.665228	−	0.746641i	$-0.731666\pi$
$727$	−35.8730	−1.33046	−0.665228	+	0.746641i	$0.731666\pi$
$728$	−0.358941	−0.0133032
$729$	0	0
$730$	3.29535	0.121966
$731$	−44.6466	−1.65131
$732$	0	0
$733$	28.7202	1.06080	0.530402	−	0.847746i	$-0.322041\pi$
$733$	28.7202	1.06080	0.530402	+	0.847746i	$0.322041\pi$
$734$	−6.50346	−0.240047
$735$	0	0
$736$	34.4386	1.26943
$737$	−7.54662	−0.277983
$738$	0	0
$739$	23.9463	0.880877	0.440439	−	0.897783i	$-0.354823\pi$
$739$	23.9463	0.880877	0.440439	+	0.897783i	$0.354823\pi$
$740$	27.3683	1.00608
$741$	0	0
$742$	0.450022	0.0165208
$743$	−42.0678	−1.54332	−0.771658	−	0.636037i	$-0.780572\pi$
$743$	−42.0678	−1.54332	−0.771658	+	0.636037i	$0.780572\pi$
$744$	0	0
$745$	−2.78190	−0.101921
$746$	4.40940	0.161439
$747$	0	0
$748$	−34.6199	−1.26583
$749$	1.27176	0.0464690
$750$	0	0
$751$	34.5818	1.26191	0.630953	−	0.775821i	$-0.282664\pi$
$751$	34.5818	1.26191	0.630953	+	0.775821i	$0.282664\pi$
$752$	−5.98121	−0.218112
$753$	0	0
$754$	−0.786792	−0.0286533
$755$	48.8950	1.77947
$756$	0	0
$757$	−31.7885	−1.15537	−0.577686	−	0.816259i	$-0.696044\pi$
$757$	−31.7885	−1.15537	−0.577686	+	0.816259i	$0.696044\pi$
$758$	−1.01184	−0.0367518
$759$	0	0
$760$	−28.5405	−1.03527
$761$	−1.76940	−0.0641408	−0.0320704	−	0.999486i	$-0.510210\pi$
$761$	−1.76940	−0.0641408	−0.0320704	+	0.999486i	$0.510210\pi$
$762$	0	0
$763$	0.620358	0.0224585
$764$	−27.5160	−0.995495
$765$	0	0
$766$	−14.0969	−0.509343
$767$	8.91167	0.321782
$768$	0	0
$769$	39.0821	1.40934	0.704669	−	0.709537i	$-0.251096\pi$
$769$	39.0821	1.40934	0.704669	+	0.709537i	$0.251096\pi$
$770$	−1.37963	−0.0497185
$771$	0	0
$772$	−23.2036	−0.835116
$773$	−40.5756	−1.45940	−0.729701	−	0.683767i	$-0.760341\pi$
$773$	−40.5756	−1.45940	−0.729701	+	0.683767i	$0.760341\pi$
$774$	0	0
$775$	−20.1709	−0.724561
$776$	18.6932	0.671046
$777$	0	0
$778$	−6.34183	−0.227366
$779$	14.4630	0.518190
$780$	0	0
$781$	−16.7103	−0.597941
$782$	23.8887	0.854257
$783$	0	0
$784$	−10.0170	−0.357752
$785$	−33.6843	−1.20225
$786$	0	0
$787$	21.1600	0.754272	0.377136	−	0.926158i	$-0.376909\pi$
$787$	21.1600	0.754272	0.377136	+	0.926158i	$0.376909\pi$
$788$	−19.9252	−0.709805
$789$	0	0
$790$	−19.7916	−0.704155
$791$	−1.92049	−0.0682849
$792$	0	0
$793$	−4.80486	−0.170625
$794$	24.2645	0.861116
$795$	0	0
$796$	−17.0341	−0.603758
$797$	44.5341	1.57748	0.788740	−	0.614728i	$-0.210734\pi$
$797$	44.5341	1.57748	0.788740	+	0.614728i	$0.210734\pi$
$798$	0	0
$799$	−24.5539	−0.868654
$800$	−15.8591	−0.560702
$801$	0	0
$802$	15.9656	0.563764
$803$	6.64130	0.234366
$804$	0	0
$805$	−3.16403	−0.111517
$806$	3.90779	0.137646
$807$	0	0
$808$	29.8445	1.04993
$809$	18.8305	0.662046	0.331023	−	0.943623i	$-0.392606\pi$
$809$	18.8305	0.662046	0.331023	+	0.943623i	$0.392606\pi$
$810$	0	0
$811$	−20.0810	−0.705139	−0.352569	−	0.935786i	$-0.614692\pi$
$811$	−20.0810	−0.705139	−0.352569	+	0.935786i	$0.614692\pi$
$812$	0.435923	0.0152979
$813$	0	0
$814$	−16.5954	−0.581669
$815$	30.3412	1.06280
$816$	0	0
$817$	−32.2390	−1.12790
$818$	−20.4087	−0.713572
$819$	0	0
$820$	14.5061	0.506575
$821$	−26.4011	−0.921404	−0.460702	−	0.887555i	$-0.652402\pi$
$821$	−26.4011	−0.921404	−0.460702	+	0.887555i	$0.652402\pi$
$822$	0	0
$823$	53.5151	1.86542	0.932709	−	0.360630i	$-0.117438\pi$
$823$	53.5151	1.86542	0.932709	+	0.360630i	$0.117438\pi$
$824$	28.0799	0.978210
$825$	0	0
$826$	1.48559	0.0516902
$827$	2.88421	0.100294	0.0501469	−	0.998742i	$-0.484031\pi$
$827$	2.88421	0.100294	0.0501469	+	0.998742i	$0.484031\pi$
$828$	0	0
$829$	20.0460	0.696228	0.348114	−	0.937452i	$-0.386822\pi$
$829$	20.0460	0.696228	0.348114	+	0.937452i	$0.386822\pi$
$830$	9.41289	0.326726
$831$	0	0
$832$	0.827806	0.0286990
$833$	−41.1216	−1.42478
$834$	0	0
$835$	10.7247	0.371144
$836$	−24.9988	−0.864602
$837$	0	0
$838$	17.4011	0.601111
$839$	54.0057	1.86448	0.932241	−	0.361837i	$-0.117850\pi$
$839$	54.0057	1.86448	0.932241	+	0.361837i	$0.117850\pi$
$840$	0	0
$841$	−26.8014	−0.924187
$842$	5.72999	0.197469
$843$	0	0
$844$	−7.98998	−0.275027
$845$	34.4713	1.18585
$846$	0	0
$847$	−0.676967	−0.0232609
$848$	4.97751	0.170928
$849$	0	0
$850$	−11.0008	−0.377323
$851$	−38.0597	−1.30467
$852$	0	0
$853$	−32.8420	−1.12449	−0.562244	−	0.826971i	$-0.690062\pi$
$853$	−32.8420	−1.12449	−0.562244	+	0.826971i	$0.690062\pi$
$854$	−0.800976	−0.0274088
$855$	0	0
$856$	−16.0006	−0.546890
$857$	9.66710	0.330222	0.165111	−	0.986275i	$-0.447202\pi$
$857$	9.66710	0.330222	0.165111	+	0.986275i	$0.447202\pi$
$858$	0	0
$859$	−51.9637	−1.77298	−0.886490	−	0.462748i	$-0.846863\pi$
$859$	−51.9637	−1.77298	−0.886490	+	0.462748i	$0.846863\pi$
$860$	−32.3352	−1.10262
$861$	0	0
$862$	5.06829	0.172627
$863$	−31.4070	−1.06911	−0.534553	−	0.845135i	$-0.679520\pi$
$863$	−31.4070	−1.06911	−0.534553	+	0.845135i	$0.679520\pi$
$864$	0	0
$865$	32.1967	1.09472
$866$	20.5666	0.698881
$867$	0	0
$868$	−2.16512	−0.0734888
$869$	−39.8872	−1.35308
$870$	0	0
$871$	−1.54407	−0.0523187
$872$	−7.80504	−0.264312
$873$	0	0
$874$	17.2499	0.583486
$875$	−1.20281	−0.0406624
$876$	0	0
$877$	−28.1367	−0.950110	−0.475055	−	0.879956i	$-0.657572\pi$
$877$	−28.1367	−0.950110	−0.475055	+	0.879956i	$0.657572\pi$
$878$	−18.0857	−0.610364
$879$	0	0
$880$	−15.2596	−0.514399
$881$	−50.9182	−1.71548	−0.857738	−	0.514086i	$-0.828131\pi$
$881$	−50.9182	−1.71548	−0.857738	+	0.514086i	$0.828131\pi$
$882$	0	0
$883$	−19.1744	−0.645269	−0.322635	−	0.946524i	$-0.604569\pi$
$883$	−19.1744	−0.645269	−0.322635	+	0.946524i	$0.604569\pi$
$884$	−7.08336	−0.238239
$885$	0	0
$886$	−24.1120	−0.810058
$887$	−55.3183	−1.85741	−0.928703	−	0.370825i	$-0.879075\pi$
$887$	−55.3183	−1.85741	−0.928703	+	0.370825i	$0.879075\pi$
$888$	0	0
$889$	1.56781	0.0525827
$890$	26.8290	0.899311
$891$	0	0
$892$	−15.2501	−0.510612
$893$	−17.7302	−0.593320
$894$	0	0
$895$	−31.3831	−1.04902
$896$	−2.07647	−0.0693700
$897$	0	0
$898$	−13.7010	−0.457208
$899$	−10.9197	−0.364193
$900$	0	0
$901$	20.4335	0.680739
$902$	−8.79614	−0.292879
$903$	0	0
$904$	24.1627	0.803639
$905$	−15.7072	−0.522127
$906$	0	0
$907$	−4.38057	−0.145454	−0.0727272	−	0.997352i	$-0.523170\pi$
$907$	−4.38057	−0.145454	−0.0727272	+	0.997352i	$0.523170\pi$
$908$	−0.161670	−0.00536521
$909$	0	0
$910$	−0.282278	−0.00935743
$911$	2.83687	0.0939897	0.0469948	−	0.998895i	$-0.485036\pi$
$911$	2.83687	0.0939897	0.0469948	+	0.998895i	$0.485036\pi$
$912$	0	0
$913$	18.9703	0.627826
$914$	12.9835	0.429456
$915$	0	0
$916$	31.3660	1.03636
$917$	1.24160	0.0410013
$918$	0	0
$919$	−32.6972	−1.07858	−0.539290	−	0.842120i	$-0.681307\pi$
$919$	−32.6972	−1.07858	−0.539290	+	0.842120i	$0.681307\pi$
$920$	39.8082	1.31244
$921$	0	0
$922$	−15.1620	−0.499335
$923$	−3.41899	−0.112537
$924$	0	0
$925$	17.5266	0.576270
$926$	−17.6888	−0.581289
$927$	0	0
$928$	−8.58545	−0.281831
$929$	−6.93912	−0.227665	−0.113833	−	0.993500i	$-0.536313\pi$
$929$	−6.93912	−0.227665	−0.113833	+	0.993500i	$0.536313\pi$
$930$	0	0
$931$	−29.6937	−0.973173
$932$	1.26343	0.0413851
$933$	0	0
$934$	−11.8155	−0.386614
$935$	−62.6430	−2.04865
$936$	0	0
$937$	15.6266	0.510500	0.255250	−	0.966875i	$-0.417842\pi$
$937$	15.6266	0.510500	0.255250	+	0.966875i	$0.417842\pi$
$938$	−0.257398	−0.00840435
$939$	0	0
$940$	−17.7831	−0.580021
$941$	21.3769	0.696867	0.348434	−	0.937333i	$-0.386714\pi$
$941$	21.3769	0.696867	0.348434	+	0.937333i	$0.386714\pi$
$942$	0	0
$943$	−20.1729	−0.656921
$944$	16.4315	0.534799
$945$	0	0
$946$	19.6072	0.637487
$947$	33.7861	1.09790	0.548950	−	0.835855i	$-0.315028\pi$
$947$	33.7861	1.09790	0.548950	+	0.835855i	$0.315028\pi$
$948$	0	0
$949$	1.35884	0.0441097
$950$	−7.94360	−0.257724
$951$	0	0
$952$	−2.71689	−0.0880549
$953$	49.4995	1.60345	0.801724	−	0.597695i	$-0.203917\pi$
$953$	49.4995	1.60345	0.801724	+	0.597695i	$0.203917\pi$
$954$	0	0
$955$	−49.7889	−1.61113
$956$	4.33766	0.140290
$957$	0	0
$958$	−3.50869	−0.113361
$959$	−0.103464	−0.00334101
$960$	0	0
$961$	23.2353	0.749527
$962$	−3.39549	−0.109475
$963$	0	0
$964$	14.9013	0.479938
$965$	−41.9858	−1.35157
$966$	0	0
$967$	−57.1688	−1.83842	−0.919212	−	0.393763i	$-0.871173\pi$
$967$	−57.1688	−1.83842	−0.919212	+	0.393763i	$0.871173\pi$
$968$	8.51726	0.273755
$969$	0	0
$970$	14.7007	0.472010
$971$	−15.2003	−0.487800	−0.243900	−	0.969800i	$-0.578427\pi$
$971$	−15.2003	−0.487800	−0.243900	+	0.969800i	$0.578427\pi$
$972$	0	0
$973$	−3.52064	−0.112866
$974$	−6.64678	−0.212976
$975$	0	0
$976$	−8.85928	−0.283579
$977$	48.2850	1.54478	0.772388	−	0.635151i	$-0.219062\pi$
$977$	48.2850	1.54478	0.772388	+	0.635151i	$0.219062\pi$
$978$	0	0
$979$	54.0700	1.72809
$980$	−29.7823	−0.951361
$981$	0	0
$982$	24.5662	0.783938
$983$	−46.5893	−1.48597	−0.742983	−	0.669310i	$-0.766590\pi$
$983$	−46.5893	−1.48597	−0.742983	+	0.669310i	$0.766590\pi$
$984$	0	0
$985$	−36.0537	−1.14877
$986$	−5.95537	−0.189658
$987$	0	0
$988$	−5.11486	−0.162725
$989$	44.9670	1.42987
$990$	0	0
$991$	3.50448	0.111323	0.0556617	−	0.998450i	$-0.482273\pi$
$991$	3.50448	0.111323	0.0556617	+	0.998450i	$0.482273\pi$
$992$	42.6417	1.35387
$993$	0	0
$994$	−0.569950	−0.0180777
$995$	−30.8224	−0.977136
$996$	0	0
$997$	−10.9063	−0.345407	−0.172703	−	0.984974i	$-0.555250\pi$
$997$	−10.9063	−0.345407	−0.172703	+	0.984974i	$0.555250\pi$
$998$	−20.9060	−0.661769
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.f.1.8	yes	25
3.2	odd	2		4023.2.a.e.1.18	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.18	✓	25	3.2	odd	2
4023.2.a.f.1.8	yes	25	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-0.680129 q^{2} -1.53742 q^{4} -2.78190 q^{5} -0.191226 q^{7} +2.40591 q^{8} +O(q^{10})\)	\(q-0.680129 q^{2} -1.53742 q^{4} -2.78190 q^{5} -0.191226 q^{7} +2.40591 q^{8} +1.89205 q^{10} +3.81315 q^{11} +0.780186 q^{13} +0.130058 q^{14} +1.43852 q^{16} +5.90537 q^{17} +4.26424 q^{19} +4.27696 q^{20} -2.59344 q^{22} -5.94776 q^{23} +2.73895 q^{25} -0.530628 q^{26} +0.293995 q^{28} +1.48276 q^{29} -7.36446 q^{31} -5.79019 q^{32} -4.01642 q^{34} +0.531970 q^{35} +6.39900 q^{37} -2.90023 q^{38} -6.69298 q^{40} +3.39169 q^{41} -7.56033 q^{43} -5.86244 q^{44} +4.04524 q^{46} -4.15789 q^{47} -6.96343 q^{49} -1.86284 q^{50} -1.19948 q^{52} +3.46016 q^{53} -10.6078 q^{55} -0.460071 q^{56} -1.00847 q^{58} +11.4225 q^{59} -6.15860 q^{61} +5.00879 q^{62} +1.06104 q^{64} -2.17040 q^{65} -1.97910 q^{67} -9.07906 q^{68} -0.361808 q^{70} -4.38227 q^{71} +1.74168 q^{73} -4.35215 q^{74} -6.55594 q^{76} -0.729173 q^{77} -10.4604 q^{79} -4.00182 q^{80} -2.30679 q^{82} +4.97497 q^{83} -16.4281 q^{85} +5.14200 q^{86} +9.17409 q^{88} +14.1799 q^{89} -0.149192 q^{91} +9.14422 q^{92} +2.82790 q^{94} -11.8627 q^{95} +7.76970 q^{97} +4.73603 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

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Database

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