LMFDB - Embedded newform 4000.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, 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("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 13x^{2} + 41 $ x^4 - 13*x^2 + 41
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.31991$ of defining polynomial
Character	$\chi$	$=$	4000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} +0.618034 q^{7} -0.381966 q^{9} +O(q^{10})$	$q-1.61803 q^{3} +0.618034 q^{7} -0.381966 q^{9} -4.63981 q^{11} -4.63981 q^{13} -7.50738 q^{17} -7.50738 q^{19} -1.00000 q^{21} +4.61803 q^{23} +5.47214 q^{27} -10.0902 q^{29} +4.63981 q^{31} +7.50738 q^{33} +2.86756 q^{37} +7.50738 q^{39} -0.854102 q^{41} -3.32624 q^{43} +10.0902 q^{47} -6.61803 q^{49} +12.1472 q^{51} +9.27963 q^{53} +12.1472 q^{57} +10.3749 q^{59} +7.85410 q^{61} -0.236068 q^{63} +1.52786 q^{67} -7.47214 q^{69} -7.50738 q^{71} -10.3749 q^{73} -2.86756 q^{77} -2.86756 q^{79} -7.70820 q^{81} -7.56231 q^{83} +16.3262 q^{87} +9.85410 q^{89} -2.86756 q^{91} -7.50738 q^{93} +10.3749 q^{97} +1.77225 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 2 * q^7 - 6 * q^9	$ 4 q - 2 q^{3} - 2 q^{7} - 6 q^{9} - 4 q^{21} + 14 q^{23} + 4 q^{27} - 18 q^{29} + 10 q^{41} + 18 q^{43} + 18 q^{47} - 22 q^{49} + 18 q^{61} + 8 q^{63} + 24 q^{67} - 12 q^{69} - 4 q^{81} + 10 q^{83} + 34 q^{87} + 26 q^{89}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^7 - 6 * q^9 - 4 * q^21 + 14 * q^23 + 4 * q^27 - 18 * q^29 + 10 * q^41 + 18 * q^43 + 18 * q^47 - 22 * q^49 + 18 * q^61 + 8 * q^63 + 24 * q^67 - 12 * q^69 - 4 * q^81 + 10 * q^83 + 34 * q^87 + 26 * q^89

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−4.63981	−1.39896	−0.699478	−	0.714654i	$-0.746584\pi$
$11$	−4.63981	−1.39896	−0.699478	+	0.714654i	$0.746584\pi$
$12$	0	0
$13$	−4.63981	−1.28685	−0.643426	−	0.765508i	$-0.722488\pi$
$13$	−4.63981	−1.28685	−0.643426	+	0.765508i	$0.722488\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.50738	−1.82081	−0.910403	−	0.413723i	$-0.864228\pi$
$17$	−7.50738	−1.82081	−0.910403	+	0.413723i	$0.864228\pi$
$18$	0	0
$19$	−7.50738	−1.72231	−0.861155	−	0.508343i	$-0.830258\pi$
$19$	−7.50738	−1.72231	−0.861155	+	0.508343i	$0.830258\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.61803	0.962927	0.481463	−	0.876466i	$-0.340105\pi$
$23$	4.61803	0.962927	0.481463	+	0.876466i	$0.340105\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−10.0902	−1.87370	−0.936849	−	0.349735i	$-0.886272\pi$
$29$	−10.0902	−1.87370	−0.936849	+	0.349735i	$0.886272\pi$
$30$	0	0
$31$	4.63981	0.833335	0.416668	−	0.909059i	$-0.363198\pi$
$31$	4.63981	0.833335	0.416668	+	0.909059i	$0.363198\pi$
$32$	0	0
$33$	7.50738	1.30687
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.86756	0.471424	0.235712	−	0.971823i	$-0.424258\pi$
$37$	2.86756	0.471424	0.235712	+	0.971823i	$0.424258\pi$
$38$	0	0
$39$	7.50738	1.20214
$40$	0	0
$41$	−0.854102	−0.133388	−0.0666942	−	0.997773i	$-0.521245\pi$
$41$	−0.854102	−0.133388	−0.0666942	+	0.997773i	$0.521245\pi$
$42$	0	0
$43$	−3.32624	−0.507247	−0.253623	−	0.967303i	$-0.581622\pi$
$43$	−3.32624	−0.507247	−0.253623	+	0.967303i	$0.581622\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0902	1.47180	0.735901	−	0.677089i	$-0.236759\pi$
$47$	10.0902	1.47180	0.735901	+	0.677089i	$0.236759\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	12.1472	1.70095
$52$	0	0
$53$	9.27963	1.27465	0.637327	−	0.770593i	$-0.280040\pi$
$53$	9.27963	1.27465	0.637327	+	0.770593i	$0.280040\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.1472	1.60893
$58$	0	0
$59$	10.3749	1.35070	0.675351	−	0.737497i	$-0.263992\pi$
$59$	10.3749	1.35070	0.675351	+	0.737497i	$0.263992\pi$
$60$	0	0
$61$	7.85410	1.00561	0.502807	−	0.864398i	$-0.332301\pi$
$61$	7.85410	1.00561	0.502807	+	0.864398i	$0.332301\pi$
$62$	0	0
$63$	−0.236068	−0.0297418
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.52786	0.186658	0.0933292	−	0.995635i	$-0.470249\pi$
$67$	1.52786	0.186658	0.0933292	+	0.995635i	$0.470249\pi$
$68$	0	0
$69$	−7.47214	−0.899539
$70$	0	0
$71$	−7.50738	−0.890961	−0.445481	−	0.895292i	$-0.646967\pi$
$71$	−7.50738	−0.890961	−0.445481	+	0.895292i	$0.646967\pi$
$72$	0	0
$73$	−10.3749	−1.21429	−0.607147	−	0.794589i	$-0.707686\pi$
$73$	−10.3749	−1.21429	−0.607147	+	0.794589i	$0.707686\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.86756	−0.326789
$78$	0	0
$79$	−2.86756	−0.322626	−0.161313	−	0.986903i	$-0.551573\pi$
$79$	−2.86756	−0.322626	−0.161313	+	0.986903i	$0.551573\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−7.56231	−0.830071	−0.415035	−	0.909805i	$-0.636231\pi$
$83$	−7.56231	−0.830071	−0.415035	+	0.909805i	$0.636231\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.3262	1.75036
$88$	0	0
$89$	9.85410	1.04453	0.522266	−	0.852782i	$-0.325087\pi$
$89$	9.85410	1.04453	0.522266	+	0.852782i	$0.325087\pi$
$90$	0	0
$91$	−2.86756	−0.300602
$92$	0	0
$93$	−7.50738	−0.778479
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.3749	1.05342	0.526708	−	0.850047i	$-0.323426\pi$
$97$	10.3749	1.05342	0.526708	+	0.850047i	$0.323426\pi$
$98$	0	0
$99$	1.77225	0.178118
$100$	0	0
$101$	−2.90983	−0.289539	−0.144769	−	0.989465i	$-0.546244\pi$
$101$	−2.90983	−0.289539	−0.144769	+	0.989465i	$0.546244\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.8541	1.62935	0.814674	−	0.579920i	$-0.196916\pi$
$107$	16.8541	1.62935	0.814674	+	0.579920i	$0.196916\pi$
$108$	0	0
$109$	−9.32624	−0.893292	−0.446646	−	0.894711i	$-0.647382\pi$
$109$	−9.32624	−0.893292	−0.446646	+	0.894711i	$0.647382\pi$
$110$	0	0
$111$	−4.63981	−0.440392
$112$	0	0
$113$	−4.63981	−0.436477	−0.218238	−	0.975895i	$-0.570031\pi$
$113$	−4.63981	−0.436477	−0.218238	+	0.975895i	$0.570031\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.77225	0.163845
$118$	0	0
$119$	−4.63981	−0.425331
$120$	0	0
$121$	10.5279	0.957079
$122$	0	0
$123$	1.38197	0.124608
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.09017	0.185473	0.0927363	−	0.995691i	$-0.470439\pi$
$127$	2.09017	0.185473	0.0927363	+	0.995691i	$0.470439\pi$
$128$	0	0
$129$	5.38197	0.473856
$130$	0	0
$131$	−12.1472	−1.06130	−0.530652	−	0.847590i	$-0.678053\pi$
$131$	−12.1472	−1.06130	−0.530652	+	0.847590i	$0.678053\pi$
$132$	0	0
$133$	−4.63981	−0.402323
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.50738	−0.641398	−0.320699	−	0.947181i	$-0.603918\pi$
$137$	−7.50738	−0.641398	−0.320699	+	0.947181i	$0.603918\pi$
$138$	0	0
$139$	−12.1472	−1.03031	−0.515156	−	0.857097i	$-0.672266\pi$
$139$	−12.1472	−1.03031	−0.515156	+	0.857097i	$0.672266\pi$
$140$	0	0
$141$	−16.3262	−1.37492
$142$	0	0
$143$	21.5279	1.80025
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.7082	0.883198
$148$	0	0
$149$	−15.8541	−1.29882	−0.649409	−	0.760439i	$-0.724984\pi$
$149$	−15.8541	−1.29882	−0.649409	+	0.760439i	$0.724984\pi$
$150$	0	0
$151$	−6.41206	−0.521806	−0.260903	−	0.965365i	$-0.584020\pi$
$151$	−6.41206	−0.521806	−0.260903	+	0.965365i	$0.584020\pi$
$152$	0	0
$153$	2.86756	0.231829
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.7870	−1.33975	−0.669874	−	0.742475i	$-0.733652\pi$
$157$	−16.7870	−1.33975	−0.669874	+	0.742475i	$0.733652\pi$
$158$	0	0
$159$	−15.0148	−1.19075
$160$	0	0
$161$	2.85410	0.224935
$162$	0	0
$163$	−15.2705	−1.19608	−0.598039	−	0.801467i	$-0.704053\pi$
$163$	−15.2705	−1.19608	−0.598039	+	0.801467i	$0.704053\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.3820	0.958145	0.479073	−	0.877775i	$-0.340973\pi$
$167$	12.3820	0.958145	0.479073	+	0.877775i	$0.340973\pi$
$168$	0	0
$169$	8.52786	0.655990
$170$	0	0
$171$	2.86756	0.219288
$172$	0	0
$173$	22.5221	1.71233	0.856163	−	0.516706i	$-0.172842\pi$
$173$	22.5221	1.71233	0.856163	+	0.516706i	$0.172842\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.7870	−1.26179
$178$	0	0
$179$	10.3749	0.775459	0.387730	−	0.921773i	$-0.373259\pi$
$179$	10.3749	0.775459	0.387730	+	0.921773i	$0.373259\pi$
$180$	0	0
$181$	−9.90983	−0.736592	−0.368296	−	0.929709i	$-0.620059\pi$
$181$	−9.90983	−0.736592	−0.368296	+	0.929709i	$0.620059\pi$
$182$	0	0
$183$	−12.7082	−0.939417
$184$	0	0
$185$	0	0
$186$	0	0
$187$	34.8328	2.54723
$188$	0	0
$189$	3.38197	0.246002
$190$	0	0
$191$	22.5221	1.62964	0.814822	−	0.579711i	$-0.196835\pi$
$191$	22.5221	1.62964	0.814822	+	0.579711i	$0.196835\pi$
$192$	0	0
$193$	2.86756	0.206412	0.103206	−	0.994660i	$-0.467090\pi$
$193$	2.86756	0.206412	0.103206	+	0.994660i	$0.467090\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.50738	0.534878	0.267439	−	0.963575i	$-0.413823\pi$
$197$	7.50738	0.534878	0.267439	+	0.963575i	$0.413823\pi$
$198$	0	0
$199$	−1.77225	−0.125632	−0.0628158	−	0.998025i	$-0.520008\pi$
$199$	−1.77225	−0.125632	−0.0628158	+	0.998025i	$0.520008\pi$
$200$	0	0
$201$	−2.47214	−0.174371
$202$	0	0
$203$	−6.23607	−0.437686
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.76393	−0.122602
$208$	0	0
$209$	34.8328	2.40944
$210$	0	0
$211$	21.4268	1.47508	0.737541	−	0.675302i	$-0.235987\pi$
$211$	21.4268	1.47508	0.737541	+	0.675302i	$0.235987\pi$
$212$	0	0
$213$	12.1472	0.832312
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.86756	0.194663
$218$	0	0
$219$	16.7870	1.13436
$220$	0	0
$221$	34.8328	2.34311
$222$	0	0
$223$	2.67376	0.179048	0.0895242	−	0.995985i	$-0.471465\pi$
$223$	2.67376	0.179048	0.0895242	+	0.995985i	$0.471465\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.85410	−0.255806	−0.127903	−	0.991787i	$-0.540825\pi$
$227$	−3.85410	−0.255806	−0.127903	+	0.991787i	$0.540825\pi$
$228$	0	0
$229$	−9.67376	−0.639260	−0.319630	−	0.947542i	$-0.603559\pi$
$229$	−9.67376	−0.639260	−0.319630	+	0.947542i	$0.603559\pi$
$230$	0	0
$231$	4.63981	0.305277
$232$	0	0
$233$	−7.50738	−0.491824	−0.245912	−	0.969292i	$-0.579088\pi$
$233$	−7.50738	−0.491824	−0.245912	+	0.969292i	$0.579088\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.63981	0.301388
$238$	0	0
$239$	−2.86756	−0.185487	−0.0927436	−	0.995690i	$-0.529564\pi$
$239$	−2.86756	−0.185487	−0.0927436	+	0.995690i	$0.529564\pi$
$240$	0	0
$241$	−5.56231	−0.358300	−0.179150	−	0.983822i	$-0.557335\pi$
$241$	−5.56231	−0.358300	−0.179150	+	0.983822i	$0.557335\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	34.8328	2.21636
$248$	0	0
$249$	12.2361	0.775429
$250$	0	0
$251$	−20.7499	−1.30972	−0.654860	−	0.755750i	$-0.727273\pi$
$251$	−20.7499	−1.30972	−0.654860	+	0.755750i	$0.727273\pi$
$252$	0	0
$253$	−21.4268	−1.34709
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.73512	−0.357747	−0.178874	−	0.983872i	$-0.557245\pi$
$257$	−5.73512	−0.357747	−0.178874	+	0.983872i	$0.557245\pi$
$258$	0	0
$259$	1.77225	0.110122
$260$	0	0
$261$	3.85410	0.238563
$262$	0	0
$263$	2.67376	0.164871	0.0824356	−	0.996596i	$-0.473730\pi$
$263$	2.67376	0.164871	0.0824356	+	0.996596i	$0.473730\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−15.9443	−0.975774
$268$	0	0
$269$	12.4721	0.760440	0.380220	−	0.924896i	$-0.375848\pi$
$269$	12.4721	0.760440	0.380220	+	0.924896i	$0.375848\pi$
$270$	0	0
$271$	9.27963	0.563697	0.281849	−	0.959459i	$-0.409052\pi$
$271$	9.27963	0.563697	0.281849	+	0.959459i	$0.409052\pi$
$272$	0	0
$273$	4.63981	0.280814
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.50738	0.451074	0.225537	−	0.974235i	$-0.427586\pi$
$277$	7.50738	0.451074	0.225537	+	0.974235i	$0.427586\pi$
$278$	0	0
$279$	−1.77225	−0.106102
$280$	0	0
$281$	2.43769	0.145421	0.0727103	−	0.997353i	$-0.476835\pi$
$281$	2.43769	0.145421	0.0727103	+	0.997353i	$0.476835\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.527864	−0.0311588
$288$	0	0
$289$	39.3607	2.31533
$290$	0	0
$291$	−16.7870	−0.984071
$292$	0	0
$293$	−19.6546	−1.14823	−0.574116	−	0.818774i	$-0.694654\pi$
$293$	−19.6546	−1.14823	−0.574116	+	0.818774i	$0.694654\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−25.3897	−1.47326
$298$	0	0
$299$	−21.4268	−1.23914
$300$	0	0
$301$	−2.05573	−0.118490
$302$	0	0
$303$	4.70820	0.270479
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.3820	0.763749	0.381875	−	0.924214i	$-0.375279\pi$
$307$	13.3820	0.763749	0.381875	+	0.924214i	$0.375279\pi$
$308$	0	0
$309$	−12.9443	−0.736374
$310$	0	0
$311$	−1.09531	−0.0621094	−0.0310547	−	0.999518i	$-0.509887\pi$
$311$	−1.09531	−0.0621094	−0.0310547	+	0.999518i	$0.509887\pi$
$312$	0	0
$313$	18.5593	1.04903	0.524515	−	0.851401i	$-0.324246\pi$
$313$	18.5593	1.04903	0.524515	+	0.851401i	$0.324246\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.6174	1.32649	0.663244	−	0.748404i	$-0.269179\pi$
$317$	23.6174	1.32649	0.663244	+	0.748404i	$0.269179\pi$
$318$	0	0
$319$	46.8165	2.62122
$320$	0	0
$321$	−27.2705	−1.52209
$322$	0	0
$323$	56.3607	3.13599
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.0902	0.834488
$328$	0	0
$329$	6.23607	0.343806
$330$	0	0
$331$	8.60269	0.472846	0.236423	−	0.971650i	$-0.424025\pi$
$331$	8.60269	0.472846	0.236423	+	0.971650i	$0.424025\pi$
$332$	0	0
$333$	−1.09531	−0.0600227
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.4268	1.16719	0.583596	−	0.812044i	$-0.301645\pi$
$337$	21.4268	1.16719	0.583596	+	0.812044i	$0.301645\pi$
$338$	0	0
$339$	7.50738	0.407745
$340$	0	0
$341$	−21.5279	−1.16580
$342$	0	0
$343$	−8.41641	−0.454443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.3262	1.19854	0.599268	−	0.800549i	$-0.295459\pi$
$347$	22.3262	1.19854	0.599268	+	0.800549i	$0.295459\pi$
$348$	0	0
$349$	−10.6180	−0.568370	−0.284185	−	0.958769i	$-0.591723\pi$
$349$	−10.6180	−0.568370	−0.284185	+	0.958769i	$0.591723\pi$
$350$	0	0
$351$	−25.3897	−1.35520
$352$	0	0
$353$	−10.3749	−0.552202	−0.276101	−	0.961129i	$-0.589042\pi$
$353$	−10.3749	−0.552202	−0.276101	+	0.961129i	$0.589042\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.50738	0.397332
$358$	0	0
$359$	8.18431	0.431952	0.215976	−	0.976399i	$-0.430707\pi$
$359$	8.18431	0.431952	0.215976	+	0.976399i	$0.430707\pi$
$360$	0	0
$361$	37.3607	1.96635
$362$	0	0
$363$	−17.0344	−0.894076
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−32.9787	−1.72147	−0.860737	−	0.509049i	$-0.829997\pi$
$367$	−32.9787	−1.72147	−0.860737	+	0.509049i	$0.829997\pi$
$368$	0	0
$369$	0.326238	0.0169833
$370$	0	0
$371$	5.73512	0.297753
$372$	0	0
$373$	−27.1619	−1.40639	−0.703196	−	0.710996i	$-0.748244\pi$
$373$	−27.1619	−1.40639	−0.703196	+	0.710996i	$0.748244\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	46.8165	2.41117
$378$	0	0
$379$	−5.73512	−0.294594	−0.147297	−	0.989092i	$-0.547057\pi$
$379$	−5.73512	−0.294594	−0.147297	+	0.989092i	$0.547057\pi$
$380$	0	0
$381$	−3.38197	−0.173263
$382$	0	0
$383$	−34.0902	−1.74193	−0.870963	−	0.491348i	$-0.836504\pi$
$383$	−34.0902	−1.74193	−0.870963	+	0.491348i	$0.836504\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.27051	0.0645836
$388$	0	0
$389$	5.14590	0.260907	0.130454	−	0.991454i	$-0.458357\pi$
$389$	5.14590	0.260907	0.130454	+	0.991454i	$0.458357\pi$
$390$	0	0
$391$	−34.6693	−1.75330
$392$	0	0
$393$	19.6546	0.991442
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.1101	−0.808541	−0.404270	−	0.914640i	$-0.632475\pi$
$397$	−16.1101	−0.808541	−0.404270	+	0.914640i	$0.632475\pi$
$398$	0	0
$399$	7.50738	0.375839
$400$	0	0
$401$	11.0902	0.553817	0.276908	−	0.960896i	$-0.410690\pi$
$401$	11.0902	0.553817	0.276908	+	0.960896i	$0.410690\pi$
$402$	0	0
$403$	−21.5279	−1.07238
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.3050	−0.659502
$408$	0	0
$409$	30.5066	1.50845	0.754227	−	0.656614i	$-0.228012\pi$
$409$	30.5066	1.50845	0.754227	+	0.656614i	$0.228012\pi$
$410$	0	0
$411$	12.1472	0.599177
$412$	0	0
$413$	6.41206	0.315517
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.6546	0.962488
$418$	0	0
$419$	−33.5740	−1.64020	−0.820099	−	0.572222i	$-0.806082\pi$
$419$	−33.5740	−1.64020	−0.820099	+	0.572222i	$0.806082\pi$
$420$	0	0
$421$	−3.61803	−0.176332	−0.0881661	−	0.996106i	$-0.528101\pi$
$421$	−3.61803	−0.176332	−0.0881661	+	0.996106i	$0.528101\pi$
$422$	0	0
$423$	−3.85410	−0.187393
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.85410	0.234906
$428$	0	0
$429$	−34.8328	−1.68174
$430$	0	0
$431$	−6.41206	−0.308858	−0.154429	−	0.988004i	$-0.549354\pi$
$431$	−6.41206	−0.308858	−0.154429	+	0.988004i	$0.549354\pi$
$432$	0	0
$433$	−2.86756	−0.137806	−0.0689031	−	0.997623i	$-0.521950\pi$
$433$	−2.86756	−0.137806	−0.0689031	+	0.997623i	$0.521950\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−34.6693	−1.65846
$438$	0	0
$439$	−22.5221	−1.07492	−0.537461	−	0.843288i	$-0.680617\pi$
$439$	−22.5221	−1.07492	−0.537461	+	0.843288i	$0.680617\pi$
$440$	0	0
$441$	2.52786	0.120374
$442$	0	0
$443$	32.7984	1.55830	0.779149	−	0.626839i	$-0.215652\pi$
$443$	32.7984	1.55830	0.779149	+	0.626839i	$0.215652\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	25.6525	1.21332
$448$	0	0
$449$	−12.4721	−0.588596	−0.294298	−	0.955714i	$-0.595086\pi$
$449$	−12.4721	−0.588596	−0.294298	+	0.955714i	$0.595086\pi$
$450$	0	0
$451$	3.96287	0.186604
$452$	0	0
$453$	10.3749	0.487457
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.8389	−1.30225	−0.651124	−	0.758971i	$-0.725702\pi$
$457$	−27.8389	−1.30225	−0.651124	+	0.758971i	$0.725702\pi$
$458$	0	0
$459$	−41.0814	−1.91751
$460$	0	0
$461$	32.5066	1.51398	0.756991	−	0.653425i	$-0.226669\pi$
$461$	32.5066	1.51398	0.756991	+	0.653425i	$0.226669\pi$
$462$	0	0
$463$	−15.7426	−0.731623	−0.365811	−	0.930689i	$-0.619208\pi$
$463$	−15.7426	−0.731623	−0.365811	+	0.930689i	$0.619208\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.4508	−1.36282	−0.681411	−	0.731901i	$-0.738634\pi$
$467$	−29.4508	−1.36282	−0.681411	+	0.731901i	$0.738634\pi$
$468$	0	0
$469$	0.944272	0.0436024
$470$	0	0
$471$	27.1619	1.25156
$472$	0	0
$473$	15.4331	0.709616
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.54450	−0.162292
$478$	0	0
$479$	−12.1472	−0.555019	−0.277510	−	0.960723i	$-0.589509\pi$
$479$	−12.1472	−0.555019	−0.277510	+	0.960723i	$0.589509\pi$
$480$	0	0
$481$	−13.3050	−0.606654
$482$	0	0
$483$	−4.61803	−0.210128
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.27051	0.284144	0.142072	−	0.989856i	$-0.454624\pi$
$487$	6.27051	0.284144	0.142072	+	0.989856i	$0.454624\pi$
$488$	0	0
$489$	24.7082	1.11734
$490$	0	0
$491$	37.5369	1.69402	0.847008	−	0.531581i	$-0.178402\pi$
$491$	37.5369	1.69402	0.847008	+	0.531581i	$0.178402\pi$
$492$	0	0
$493$	75.7507	3.41164
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.63981	−0.208124
$498$	0	0
$499$	37.5369	1.68038	0.840191	−	0.542291i	$-0.182443\pi$
$499$	37.5369	1.68038	0.840191	+	0.542291i	$0.182443\pi$
$500$	0	0
$501$	−20.0344	−0.895073
$502$	0	0
$503$	31.1591	1.38931	0.694657	−	0.719341i	$-0.255556\pi$
$503$	31.1591	1.38931	0.694657	+	0.719341i	$0.255556\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−13.7984	−0.612807
$508$	0	0
$509$	−19.3050	−0.855677	−0.427838	−	0.903855i	$-0.640725\pi$
$509$	−19.3050	−0.855677	−0.427838	+	0.903855i	$0.640725\pi$
$510$	0	0
$511$	−6.41206	−0.283653
$512$	0	0
$513$	−41.0814	−1.81379
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−46.8165	−2.05899
$518$	0	0
$519$	−36.4416	−1.59961
$520$	0	0
$521$	−4.09017	−0.179194	−0.0895968	−	0.995978i	$-0.528558\pi$
$521$	−4.09017	−0.179194	−0.0895968	+	0.995978i	$0.528558\pi$
$522$	0	0
$523$	33.9787	1.48579	0.742893	−	0.669411i	$-0.233453\pi$
$523$	33.9787	1.48579	0.742893	+	0.669411i	$0.233453\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34.8328	−1.51734
$528$	0	0
$529$	−1.67376	−0.0727723
$530$	0	0
$531$	−3.96287	−0.171974
$532$	0	0
$533$	3.96287	0.171651
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.7870	−0.724413
$538$	0	0
$539$	30.7064	1.32262
$540$	0	0
$541$	−7.43769	−0.319771	−0.159886	−	0.987136i	$-0.551113\pi$
$541$	−7.43769	−0.319771	−0.159886	+	0.987136i	$0.551113\pi$
$542$	0	0
$543$	16.0344	0.688104
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−29.0902	−1.24381	−0.621903	−	0.783094i	$-0.713640\pi$
$547$	−29.0902	−1.24381	−0.621903	+	0.783094i	$0.713640\pi$
$548$	0	0
$549$	−3.00000	−0.128037
$550$	0	0
$551$	75.7507	3.22709
$552$	0	0
$553$	−1.77225	−0.0753638
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.3463	1.49767	0.748834	−	0.662757i	$-0.230614\pi$
$557$	35.3463	1.49767	0.748834	+	0.662757i	$0.230614\pi$
$558$	0	0
$559$	15.4331	0.652751
$560$	0	0
$561$	−56.3607	−2.37955
$562$	0	0
$563$	−8.94427	−0.376956	−0.188478	−	0.982077i	$-0.560355\pi$
$563$	−8.94427	−0.376956	−0.188478	+	0.982077i	$0.560355\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.76393	−0.200066
$568$	0	0
$569$	3.90983	0.163909	0.0819543	−	0.996636i	$-0.473884\pi$
$569$	3.90983	0.163909	0.0819543	+	0.996636i	$0.473884\pi$
$570$	0	0
$571$	−26.4850	−1.10836	−0.554181	−	0.832396i	$-0.686969\pi$
$571$	−26.4850	−1.10836	−0.554181	+	0.832396i	$0.686969\pi$
$572$	0	0
$573$	−36.4416	−1.52237
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.4702	−0.477513	−0.238756	−	0.971080i	$-0.576740\pi$
$577$	−11.4702	−0.477513	−0.238756	+	0.971080i	$0.576740\pi$
$578$	0	0
$579$	−4.63981	−0.192824
$580$	0	0
$581$	−4.67376	−0.193900
$582$	0	0
$583$	−43.0557	−1.78319
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.7771	1.47668	0.738339	−	0.674430i	$-0.235610\pi$
$587$	35.7771	1.47668	0.738339	+	0.674430i	$0.235610\pi$
$588$	0	0
$589$	−34.8328	−1.43526
$590$	0	0
$591$	−12.1472	−0.499669
$592$	0	0
$593$	−23.6174	−0.969852	−0.484926	−	0.874555i	$-0.661153\pi$
$593$	−23.6174	−0.969852	−0.484926	+	0.874555i	$0.661153\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.86756	0.117362
$598$	0	0
$599$	24.9713	1.02030	0.510150	−	0.860085i	$-0.329590\pi$
$599$	24.9713	1.02030	0.510150	+	0.860085i	$0.329590\pi$
$600$	0	0
$601$	26.5623	1.08350	0.541750	−	0.840540i	$-0.317762\pi$
$601$	26.5623	1.08350	0.541750	+	0.840540i	$0.317762\pi$
$602$	0	0
$603$	−0.583592	−0.0237657
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−40.3607	−1.63819	−0.819095	−	0.573658i	$-0.805524\pi$
$607$	−40.3607	−1.63819	−0.819095	+	0.573658i	$0.805524\pi$
$608$	0	0
$609$	10.0902	0.408874
$610$	0	0
$611$	−46.8165	−1.89399
$612$	0	0
$613$	−9.27963	−0.374801	−0.187400	−	0.982284i	$-0.560006\pi$
$613$	−9.27963	−0.374801	−0.187400	+	0.982284i	$0.560006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.7870	−0.675819	−0.337910	−	0.941179i	$-0.609720\pi$
$617$	−16.7870	−0.675819	−0.337910	+	0.941179i	$0.609720\pi$
$618$	0	0
$619$	26.0666	1.04771	0.523853	−	0.851809i	$-0.324494\pi$
$619$	26.0666	1.04771	0.523853	+	0.851809i	$0.324494\pi$
$620$	0	0
$621$	25.2705	1.01407
$622$	0	0
$623$	6.09017	0.243998
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−56.3607	−2.25083
$628$	0	0
$629$	−21.5279	−0.858372
$630$	0	0
$631$	−4.63981	−0.184708	−0.0923540	−	0.995726i	$-0.529439\pi$
$631$	−4.63981	−0.184708	−0.0923540	+	0.995726i	$0.529439\pi$
$632$	0	0
$633$	−34.6693	−1.37798
$634$	0	0
$635$	0	0
$636$	0	0
$637$	30.7064	1.21663
$638$	0	0
$639$	2.86756	0.113439
$640$	0	0
$641$	7.72949	0.305297	0.152648	−	0.988281i	$-0.451220\pi$
$641$	7.72949	0.305297	0.152648	+	0.988281i	$0.451220\pi$
$642$	0	0
$643$	6.90983	0.272497	0.136249	−	0.990675i	$-0.456495\pi$
$643$	6.90983	0.272497	0.136249	+	0.990675i	$0.456495\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.9443	0.823404	0.411702	−	0.911318i	$-0.364934\pi$
$647$	20.9443	0.823404	0.411702	+	0.911318i	$0.364934\pi$
$648$	0	0
$649$	−48.1378	−1.88957
$650$	0	0
$651$	−4.63981	−0.181849
$652$	0	0
$653$	−27.1619	−1.06293	−0.531464	−	0.847081i	$-0.678358\pi$
$653$	−27.1619	−1.06293	−0.531464	+	0.847081i	$0.678358\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.96287	0.154606
$658$	0	0
$659$	11.0519	0.430520	0.215260	−	0.976557i	$-0.430940\pi$
$659$	11.0519	0.430520	0.215260	+	0.976557i	$0.430940\pi$
$660$	0	0
$661$	7.43769	0.289293	0.144646	−	0.989483i	$-0.453796\pi$
$661$	7.43769	0.289293	0.144646	+	0.989483i	$0.453796\pi$
$662$	0	0
$663$	−56.3607	−2.18887
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−46.5967	−1.80423
$668$	0	0
$669$	−4.32624	−0.167262
$670$	0	0
$671$	−36.4416	−1.40681
$672$	0	0
$673$	23.1991	0.894258	0.447129	−	0.894469i	$-0.352446\pi$
$673$	23.1991	0.894258	0.447129	+	0.894469i	$0.352446\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.73512	−0.220419	−0.110209	−	0.993908i	$-0.535152\pi$
$677$	−5.73512	−0.220419	−0.110209	+	0.993908i	$0.535152\pi$
$678$	0	0
$679$	6.41206	0.246072
$680$	0	0
$681$	6.23607	0.238967
$682$	0	0
$683$	11.6869	0.447187	0.223594	−	0.974682i	$-0.428221\pi$
$683$	11.6869	0.447187	0.223594	+	0.974682i	$0.428221\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.6525	0.597179
$688$	0	0
$689$	−43.0557	−1.64029
$690$	0	0
$691$	−23.6174	−0.898450	−0.449225	−	0.893419i	$-0.648300\pi$
$691$	−23.6174	−0.898450	−0.449225	+	0.893419i	$0.648300\pi$
$692$	0	0
$693$	1.09531	0.0416074
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.41206	0.242874
$698$	0	0
$699$	12.1472	0.459449
$700$	0	0
$701$	9.05573	0.342030	0.171015	−	0.985268i	$-0.445295\pi$
$701$	9.05573	0.342030	0.171015	+	0.985268i	$0.445295\pi$
$702$	0	0
$703$	−21.5279	−0.811939
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.79837	−0.0676348
$708$	0	0
$709$	−38.6180	−1.45033	−0.725165	−	0.688575i	$-0.758237\pi$
$709$	−38.6180	−1.45033	−0.725165	+	0.688575i	$0.758237\pi$
$710$	0	0
$711$	1.09531	0.0410774
$712$	0	0
$713$	21.4268	0.802440
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.63981	0.173277
$718$	0	0
$719$	−47.9118	−1.78681	−0.893405	−	0.449253i	$-0.851690\pi$
$719$	−47.9118	−1.78681	−0.893405	+	0.449253i	$0.851690\pi$
$720$	0	0
$721$	4.94427	0.184134
$722$	0	0
$723$	9.00000	0.334714
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−46.0902	−1.70939	−0.854695	−	0.519131i	$-0.826256\pi$
$727$	−46.0902	−1.70939	−0.854695	+	0.519131i	$0.826256\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	24.9713	0.923597
$732$	0	0
$733$	28.9342	1.06871	0.534354	−	0.845261i	$-0.320555\pi$
$733$	28.9342	1.06871	0.534354	+	0.845261i	$0.320555\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.08900	−0.261127
$738$	0	0
$739$	−0.676940	−0.0249016	−0.0124508	−	0.999922i	$-0.503963\pi$
$739$	−0.676940	−0.0249016	−0.0124508	+	0.999922i	$0.503963\pi$
$740$	0	0
$741$	−56.3607	−2.07046
$742$	0	0
$743$	45.8885	1.68349	0.841744	−	0.539877i	$-0.181529\pi$
$743$	45.8885	1.68349	0.841744	+	0.539877i	$0.181529\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.88854	0.105686
$748$	0	0
$749$	10.4164	0.380607
$750$	0	0
$751$	0.676940	0.0247019	0.0123509	−	0.999924i	$-0.496068\pi$
$751$	0.676940	0.0247019	0.0123509	+	0.999924i	$0.496068\pi$
$752$	0	0
$753$	33.5740	1.22350
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.9489	1.59735	0.798676	−	0.601762i	$-0.205534\pi$
$757$	43.9489	1.59735	0.798676	+	0.601762i	$0.205534\pi$
$758$	0	0
$759$	34.6693	1.25842
$760$	0	0
$761$	−5.85410	−0.212211	−0.106106	−	0.994355i	$-0.533838\pi$
$761$	−5.85410	−0.212211	−0.106106	+	0.994355i	$0.533838\pi$
$762$	0	0
$763$	−5.76393	−0.208668
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−48.1378	−1.73815
$768$	0	0
$769$	−9.79837	−0.353338	−0.176669	−	0.984270i	$-0.556532\pi$
$769$	−9.79837	−0.353338	−0.176669	+	0.984270i	$0.556532\pi$
$770$	0	0
$771$	9.27963	0.334198
$772$	0	0
$773$	22.1038	0.795017	0.397508	−	0.917599i	$-0.369875\pi$
$773$	22.1038	0.795017	0.397508	+	0.917599i	$0.369875\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.86756	−0.102873
$778$	0	0
$779$	6.41206	0.229736
$780$	0	0
$781$	34.8328	1.24642
$782$	0	0
$783$	−55.2148	−1.97322
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.4377	−0.479002	−0.239501	−	0.970896i	$-0.576984\pi$
$787$	−13.4377	−0.479002	−0.239501	+	0.970896i	$0.576984\pi$
$788$	0	0
$789$	−4.32624	−0.154018
$790$	0	0
$791$	−2.86756	−0.101959
$792$	0	0
$793$	−36.4416	−1.29408
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.44919	−0.0867548	−0.0433774	−	0.999059i	$-0.513812\pi$
$797$	−2.44919	−0.0867548	−0.0433774	+	0.999059i	$0.513812\pi$
$798$	0	0
$799$	−75.7507	−2.67987
$800$	0	0
$801$	−3.76393	−0.132992
$802$	0	0
$803$	48.1378	1.69874
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20.1803	−0.710382
$808$	0	0
$809$	−4.61803	−0.162361	−0.0811807	−	0.996699i	$-0.525869\pi$
$809$	−4.61803	−0.162361	−0.0811807	+	0.996699i	$0.525869\pi$
$810$	0	0
$811$	−2.44919	−0.0860027	−0.0430014	−	0.999075i	$-0.513692\pi$
$811$	−2.44919	−0.0860027	−0.0430014	+	0.999075i	$0.513692\pi$
$812$	0	0
$813$	−15.0148	−0.526590
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.9713	0.873636
$818$	0	0
$819$	1.09531	0.0382733
$820$	0	0
$821$	4.43769	0.154877	0.0774383	−	0.996997i	$-0.475326\pi$
$821$	4.43769	0.154877	0.0774383	+	0.996997i	$0.475326\pi$
$822$	0	0
$823$	26.2492	0.914990	0.457495	−	0.889212i	$-0.348747\pi$
$823$	26.2492	0.914990	0.457495	+	0.889212i	$0.348747\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.83282	−0.237600	−0.118800	−	0.992918i	$-0.537905\pi$
$827$	−6.83282	−0.237600	−0.118800	+	0.992918i	$0.537905\pi$
$828$	0	0
$829$	1.14590	0.0397987	0.0198993	−	0.999802i	$-0.493665\pi$
$829$	1.14590	0.0397987	0.0198993	+	0.999802i	$0.493665\pi$
$830$	0	0
$831$	−12.1472	−0.421381
$832$	0	0
$833$	49.6841	1.72145
$834$	0	0
$835$	0	0
$836$	0	0
$837$	25.3897	0.877596
$838$	0	0
$839$	29.3526	1.01336	0.506681	−	0.862133i	$-0.330872\pi$
$839$	29.3526	1.01336	0.506681	+	0.862133i	$0.330872\pi$
$840$	0	0
$841$	72.8115	2.51074
$842$	0	0
$843$	−3.94427	−0.135848
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.50658	0.223569
$848$	0	0
$849$	6.47214	0.222123
$850$	0	0
$851$	13.2425	0.453947
$852$	0	0
$853$	33.5740	1.14955	0.574776	−	0.818311i	$-0.305089\pi$
$853$	33.5740	1.14955	0.574776	+	0.818311i	$0.305089\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.83044	−0.233323	−0.116662	−	0.993172i	$-0.537219\pi$
$857$	−6.83044	−0.233323	−0.116662	+	0.993172i	$0.537219\pi$
$858$	0	0
$859$	15.0148	0.512297	0.256148	−	0.966637i	$-0.417546\pi$
$859$	15.0148	0.512297	0.256148	+	0.966637i	$0.417546\pi$
$860$	0	0
$861$	0.854102	0.0291077
$862$	0	0
$863$	22.5066	0.766133	0.383066	−	0.923721i	$-0.374868\pi$
$863$	22.5066	0.766133	0.383066	+	0.923721i	$0.374868\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−63.6869	−2.16292
$868$	0	0
$869$	13.3050	0.451340
$870$	0	0
$871$	−7.08900	−0.240202
$872$	0	0
$873$	−3.96287	−0.134123
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.41206	−0.216520	−0.108260	−	0.994123i	$-0.534528\pi$
$877$	−6.41206	−0.216520	−0.108260	+	0.994123i	$0.534528\pi$
$878$	0	0
$879$	31.8018	1.07265
$880$	0	0
$881$	−16.9787	−0.572027	−0.286014	−	0.958226i	$-0.592330\pi$
$881$	−16.9787	−0.572027	−0.286014	+	0.958226i	$0.592330\pi$
$882$	0	0
$883$	−32.1459	−1.08180	−0.540898	−	0.841088i	$-0.681915\pi$
$883$	−32.1459	−1.08180	−0.540898	+	0.841088i	$0.681915\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.5623	−0.858298	−0.429149	−	0.903234i	$-0.641187\pi$
$887$	−25.5623	−0.858298	−0.429149	+	0.903234i	$0.641187\pi$
$888$	0	0
$889$	1.29180	0.0433254
$890$	0	0
$891$	35.7646	1.19816
$892$	0	0
$893$	−75.7507	−2.53490
$894$	0	0
$895$	0	0
$896$	0	0
$897$	34.6693	1.15757
$898$	0	0
$899$	−46.8165	−1.56142
$900$	0	0
$901$	−69.6656	−2.32090
$902$	0	0
$903$	3.32624	0.110690
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.20163	0.106308	0.0531541	−	0.998586i	$-0.483073\pi$
$907$	3.20163	0.106308	0.0531541	+	0.998586i	$0.483073\pi$
$908$	0	0
$909$	1.11146	0.0368647
$910$	0	0
$911$	34.6693	1.14865	0.574323	−	0.818629i	$-0.305265\pi$
$911$	34.6693	1.14865	0.574323	+	0.818629i	$0.305265\pi$
$912$	0	0
$913$	35.0877	1.16123
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.50738	−0.247915
$918$	0	0
$919$	−18.9776	−0.626014	−0.313007	−	0.949751i	$-0.601336\pi$
$919$	−18.9776	−0.626014	−0.313007	+	0.949751i	$0.601336\pi$
$920$	0	0
$921$	−21.6525	−0.713473
$922$	0	0
$923$	34.8328	1.14654
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.05573	−0.100363
$928$	0	0
$929$	28.0344	0.919780	0.459890	−	0.887976i	$-0.347889\pi$
$929$	28.0344	0.919780	0.459890	+	0.887976i	$0.347889\pi$
$930$	0	0
$931$	49.6841	1.62833
$932$	0	0
$933$	1.77225	0.0580209
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.5593	0.606304	0.303152	−	0.952942i	$-0.401961\pi$
$937$	18.5593	0.606304	0.303152	+	0.952942i	$0.401961\pi$
$938$	0	0
$939$	−30.0295	−0.979976
$940$	0	0
$941$	35.8885	1.16993	0.584967	−	0.811057i	$-0.301108\pi$
$941$	35.8885	1.16993	0.584967	+	0.811057i	$0.301108\pi$
$942$	0	0
$943$	−3.94427	−0.128443
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.72949	0.121192	0.0605961	−	0.998162i	$-0.480700\pi$
$947$	3.72949	0.121192	0.0605961	+	0.998162i	$0.480700\pi$
$948$	0	0
$949$	48.1378	1.56262
$950$	0	0
$951$	−38.2138	−1.23917
$952$	0	0
$953$	−17.2054	−0.557337	−0.278668	−	0.960387i	$-0.589893\pi$
$953$	−17.2054	−0.557337	−0.278668	+	0.960387i	$0.589893\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−75.7507	−2.44867
$958$	0	0
$959$	−4.63981	−0.149827
$960$	0	0
$961$	−9.47214	−0.305553
$962$	0	0
$963$	−6.43769	−0.207452
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−31.3951	−1.00960	−0.504800	−	0.863237i	$-0.668434\pi$
$967$	−31.3951	−1.00960	−0.504800	+	0.863237i	$0.668434\pi$
$968$	0	0
$969$	−91.1935	−2.92956
$970$	0	0
$971$	12.1472	0.389822	0.194911	−	0.980821i	$-0.437558\pi$
$971$	12.1472	0.389822	0.194911	+	0.980821i	$0.437558\pi$
$972$	0	0
$973$	−7.50738	−0.240675
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.0814	−1.31431	−0.657155	−	0.753756i	$-0.728240\pi$
$977$	−41.0814	−1.31431	−0.657155	+	0.753756i	$0.728240\pi$
$978$	0	0
$979$	−45.7212	−1.46126
$980$	0	0
$981$	3.56231	0.113736
$982$	0	0
$983$	19.4164	0.619287	0.309644	−	0.950853i	$-0.399790\pi$
$983$	19.4164	0.619287	0.309644	+	0.950853i	$0.399790\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−10.0902	−0.321174
$988$	0	0
$989$	−15.3607	−0.488441
$990$	0	0
$991$	52.1333	1.65607	0.828034	−	0.560678i	$-0.189460\pi$
$991$	52.1333	1.65607	0.828034	+	0.560678i	$0.189460\pi$
$992$	0	0
$993$	−13.9194	−0.441720
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.2572	0.894916	0.447458	−	0.894305i	$-0.352329\pi$
$997$	28.2572	0.894916	0.447458	+	0.894305i	$0.352329\pi$
$998$	0	0
$999$	15.6917	0.496463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.a.c.1.1	✓	4
4.3	odd	2		4000.2.a.h.1.4	yes	4
5.2	odd	4		4000.2.c.e.1249.7		8
5.3	odd	4		4000.2.c.e.1249.1		8
5.4	even	2		4000.2.a.h.1.3	yes	4
8.3	odd	2		8000.2.a.bc.1.1		4
8.5	even	2		8000.2.a.bp.1.4		4
20.3	even	4		4000.2.c.e.1249.8		8
20.7	even	4		4000.2.c.e.1249.2		8
20.19	odd	2	inner	4000.2.a.c.1.2	yes	4
40.19	odd	2		8000.2.a.bp.1.3		4
40.29	even	2		8000.2.a.bc.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.c.1.1	✓	4	1.1	even	1	trivial
4000.2.a.c.1.2	yes	4	20.19	odd	2	inner
4000.2.a.h.1.3	yes	4	5.4	even	2
4000.2.a.h.1.4	yes	4	4.3	odd	2
4000.2.c.e.1249.1		8	5.3	odd	4
4000.2.c.e.1249.2		8	20.7	even	4
4000.2.c.e.1249.7		8	5.2	odd	4
4000.2.c.e.1249.8		8	20.3	even	4
8000.2.a.bc.1.1		4	8.3	odd	2
8000.2.a.bc.1.2		4	40.29	even	2
8000.2.a.bp.1.3		4	40.19	odd	2
8000.2.a.bp.1.4		4	8.5	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} +0.618034 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q-1.61803 q^{3} +0.618034 q^{7} -0.381966 q^{9} -4.63981 q^{11} -4.63981 q^{13} -7.50738 q^{17} -7.50738 q^{19} -1.00000 q^{21} +4.61803 q^{23} +5.47214 q^{27} -10.0902 q^{29} +4.63981 q^{31} +7.50738 q^{33} +2.86756 q^{37} +7.50738 q^{39} -0.854102 q^{41} -3.32624 q^{43} +10.0902 q^{47} -6.61803 q^{49} +12.1472 q^{51} +9.27963 q^{53} +12.1472 q^{57} +10.3749 q^{59} +7.85410 q^{61} -0.236068 q^{63} +1.52786 q^{67} -7.47214 q^{69} -7.50738 q^{71} -10.3749 q^{73} -2.86756 q^{77} -2.86756 q^{79} -7.70820 q^{81} -7.56231 q^{83} +16.3262 q^{87} +9.85410 q^{89} -2.86756 q^{91} -7.50738 q^{93} +10.3749 q^{97} +1.77225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 2 * q^7 - 6 * q^9	\( 4 q - 2 q^{3} - 2 q^{7} - 6 q^{9} - 4 q^{21} + 14 q^{23} + 4 q^{27} - 18 q^{29} + 10 q^{41} + 18 q^{43} + 18 q^{47} - 22 q^{49} + 18 q^{61} + 8 q^{63} + 24 q^{67} - 12 q^{69} - 4 q^{81} + 10 q^{83} + 34 q^{87} + 26 q^{89}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^7 - 6 * q^9 - 4 * q^21 + 14 * q^23 + 4 * q^27 - 18 * q^29 + 10 * q^41 + 18 * q^43 + 18 * q^47 - 22 * q^49 + 18 * q^61 + 8 * q^63 + 24 * q^67 - 12 * q^69 - 4 * q^81 + 10 * q^83 + 34 * q^87 + 26 * q^89

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