LMFDB - Embedded newform 5733.2.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5733,2,Mod(1,5733)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5733.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5733, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,6,0,0,0,0,-4,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +4.41421 q^{5} +2.82843 q^{8} -6.24264 q^{10} +4.24264 q^{11} +1.00000 q^{13} -4.00000 q^{16} -1.41421 q^{17} -1.24264 q^{19} -6.00000 q^{22} +0.171573 q^{23} +14.4853 q^{25} -1.41421 q^{26} -5.82843 q^{29} +5.24264 q^{31} +2.00000 q^{34} -6.24264 q^{37} +1.75736 q^{38} +12.4853 q^{40} +3.17157 q^{41} -5.00000 q^{43} -0.242641 q^{46} +4.41421 q^{47} -20.4853 q^{50} +5.82843 q^{53} +18.7279 q^{55} +8.24264 q^{58} +11.6569 q^{59} -6.00000 q^{61} -7.41421 q^{62} +8.00000 q^{64} +4.41421 q^{65} +2.48528 q^{67} -1.07107 q^{71} +0.757359 q^{73} +8.82843 q^{74} -1.48528 q^{79} -17.6569 q^{80} -4.48528 q^{82} +4.75736 q^{83} -6.24264 q^{85} +7.07107 q^{86} +12.0000 q^{88} +4.41421 q^{89} -6.24264 q^{94} -5.48528 q^{95} +13.7279 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} - 4 q^{10} + 2 q^{13} - 8 q^{16} + 6 q^{19} - 12 q^{22} + 6 q^{23} + 12 q^{25} - 6 q^{29} + 2 q^{31} + 4 q^{34} - 4 q^{37} + 12 q^{38} + 8 q^{40} + 12 q^{41} - 10 q^{43} + 8 q^{46} + 6 q^{47}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 - 4 * q^10 + 2 * q^13 - 8 * q^16 + 6 * q^19 - 12 * q^22 + 6 * q^23 + 12 * q^25 - 6 * q^29 + 2 * q^31 + 4 * q^34 - 4 * q^37 + 12 * q^38 + 8 * q^40 + 12 * q^41 - 10 * q^43 + 8 * q^46 + 6 * q^47 - 24 * q^50 + 6 * q^53 + 12 * q^55 + 8 * q^58 + 12 * q^59 - 12 * q^61 - 12 * q^62 + 16 * q^64 + 6 * q^65 - 12 * q^67 + 12 * q^71 + 10 * q^73 + 12 * q^74 + 14 * q^79 - 24 * q^80 + 8 * q^82 + 18 * q^83 - 4 * q^85 + 24 * q^88 + 6 * q^89 - 4 * q^94 + 6 * q^95 + 2 * q^97

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.41421	1.97410	0.987048	−	0.160424i	$-0.0512862\pi$
$5$	4.41421	1.97410	0.987048	+	0.160424i	$0.0512862\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	0	0
$10$	−6.24264	−1.97410
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	−1.24264	−0.285081	−0.142541	−	0.989789i	$-0.545527\pi$
$19$	−1.24264	−0.285081	−0.142541	+	0.989789i	$0.545527\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	0.171573	0.0357754	0.0178877	−	0.999840i	$-0.494306\pi$
$23$	0.171573	0.0357754	0.0178877	+	0.999840i	$0.494306\pi$
$24$	0	0
$25$	14.4853	2.89706
$26$	−1.41421	−0.277350
$27$	0	0
$28$	0	0
$29$	−5.82843	−1.08231	−0.541156	−	0.840922i	$-0.682013\pi$
$29$	−5.82843	−1.08231	−0.541156	+	0.840922i	$0.682013\pi$
$30$	0	0
$31$	5.24264	0.941606	0.470803	−	0.882238i	$-0.343964\pi$
$31$	5.24264	0.941606	0.470803	+	0.882238i	$0.343964\pi$
$32$	0	0
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	−6.24264	−1.02628	−0.513142	−	0.858304i	$-0.671519\pi$
$37$	−6.24264	−1.02628	−0.513142	+	0.858304i	$0.671519\pi$
$38$	1.75736	0.285081
$39$	0	0
$40$	12.4853	1.97410
$41$	3.17157	0.495316	0.247658	−	0.968847i	$-0.420339\pi$
$41$	3.17157	0.495316	0.247658	+	0.968847i	$0.420339\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	0	0
$46$	−0.242641	−0.0357754
$47$	4.41421	0.643879	0.321940	−	0.946760i	$-0.395665\pi$
$47$	4.41421	0.643879	0.321940	+	0.946760i	$0.395665\pi$
$48$	0	0
$49$	0	0
$50$	−20.4853	−2.89706
$51$	0	0
$52$	0	0
$53$	5.82843	0.800596	0.400298	−	0.916385i	$-0.368907\pi$
$53$	5.82843	0.800596	0.400298	+	0.916385i	$0.368907\pi$
$54$	0	0
$55$	18.7279	2.52527
$56$	0	0
$57$	0	0
$58$	8.24264	1.08231
$59$	11.6569	1.51759	0.758797	−	0.651328i	$-0.225788\pi$
$59$	11.6569	1.51759	0.758797	+	0.651328i	$0.225788\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−7.41421	−0.941606
$63$	0	0
$64$	8.00000	1.00000
$65$	4.41421	0.547516
$66$	0	0
$67$	2.48528	0.303625	0.151813	−	0.988409i	$-0.451489\pi$
$67$	2.48528	0.303625	0.151813	+	0.988409i	$0.451489\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.07107	−0.127112	−0.0635562	−	0.997978i	$-0.520244\pi$
$71$	−1.07107	−0.127112	−0.0635562	+	0.997978i	$0.520244\pi$
$72$	0	0
$73$	0.757359	0.0886422	0.0443211	−	0.999017i	$-0.485888\pi$
$73$	0.757359	0.0886422	0.0443211	+	0.999017i	$0.485888\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.48528	−0.167107	−0.0835536	−	0.996503i	$-0.526627\pi$
$79$	−1.48528	−0.167107	−0.0835536	+	0.996503i	$0.526627\pi$
$80$	−17.6569	−1.97410
$81$	0	0
$82$	−4.48528	−0.495316
$83$	4.75736	0.522188	0.261094	−	0.965313i	$-0.415917\pi$
$83$	4.75736	0.522188	0.261094	+	0.965313i	$0.415917\pi$
$84$	0	0
$85$	−6.24264	−0.677109
$86$	7.07107	0.762493
$87$	0	0
$88$	12.0000	1.27920
$89$	4.41421	0.467906	0.233953	−	0.972248i	$-0.424834\pi$
$89$	4.41421	0.467906	0.233953	+	0.972248i	$0.424834\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−6.24264	−0.643879
$95$	−5.48528	−0.562778
$96$	0	0
$97$	13.7279	1.39386	0.696930	−	0.717139i	$-0.254549\pi$
$97$	13.7279	1.39386	0.696930	+	0.717139i	$0.254549\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.75736	0.174864	0.0874319	−	0.996170i	$-0.472134\pi$
$101$	1.75736	0.174864	0.0874319	+	0.996170i	$0.472134\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−8.24264	−0.800596
$107$	−8.14214	−0.787130	−0.393565	−	0.919297i	$-0.628758\pi$
$107$	−8.14214	−0.787130	−0.393565	+	0.919297i	$0.628758\pi$
$108$	0	0
$109$	−8.72792	−0.835983	−0.417992	−	0.908451i	$-0.637266\pi$
$109$	−8.72792	−0.835983	−0.417992	+	0.908451i	$0.637266\pi$
$110$	−26.4853	−2.52527
$111$	0	0
$112$	0	0
$113$	20.3137	1.91095	0.955476	−	0.295067i	$-0.0953421\pi$
$113$	20.3137	1.91095	0.955476	+	0.295067i	$0.0953421\pi$
$114$	0	0
$115$	0.757359	0.0706241
$116$	0	0
$117$	0	0
$118$	−16.4853	−1.51759
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	8.48528	0.768221
$123$	0	0
$124$	0	0
$125$	41.8701	3.74497
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	−6.24264	−0.547516
$131$	2.82843	0.247121	0.123560	−	0.992337i	$-0.460569\pi$
$131$	2.82843	0.247121	0.123560	+	0.992337i	$0.460569\pi$
$132$	0	0
$133$	0	0
$134$	−3.51472	−0.303625
$135$	0	0
$136$	−4.00000	−0.342997
$137$	7.41421	0.633439	0.316720	−	0.948519i	$-0.397419\pi$
$137$	7.41421	0.633439	0.316720	+	0.948519i	$0.397419\pi$
$138$	0	0
$139$	2.24264	0.190218	0.0951092	−	0.995467i	$-0.469680\pi$
$139$	2.24264	0.190218	0.0951092	+	0.995467i	$0.469680\pi$
$140$	0	0
$141$	0	0
$142$	1.51472	0.127112
$143$	4.24264	0.354787
$144$	0	0
$145$	−25.7279	−2.13659
$146$	−1.07107	−0.0886422
$147$	0	0
$148$	0	0
$149$	−7.75736	−0.635508	−0.317754	−	0.948173i	$-0.602929\pi$
$149$	−7.75736	−0.635508	−0.317754	+	0.948173i	$0.602929\pi$
$150$	0	0
$151$	−18.2426	−1.48457	−0.742283	−	0.670087i	$-0.766257\pi$
$151$	−18.2426	−1.48457	−0.742283	+	0.670087i	$0.766257\pi$
$152$	−3.51472	−0.285081
$153$	0	0
$154$	0	0
$155$	23.1421	1.85882
$156$	0	0
$157$	12.2426	0.977069	0.488535	−	0.872545i	$-0.337532\pi$
$157$	12.2426	0.977069	0.488535	+	0.872545i	$0.337532\pi$
$158$	2.10051	0.167107
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.48528	−0.664619	−0.332309	−	0.943170i	$-0.607828\pi$
$163$	−8.48528	−0.664619	−0.332309	+	0.943170i	$0.607828\pi$
$164$	0	0
$165$	0	0
$166$	−6.72792	−0.522188
$167$	−21.3848	−1.65480	−0.827402	−	0.561610i	$-0.810182\pi$
$167$	−21.3848	−1.65480	−0.827402	+	0.561610i	$0.810182\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	8.82843	0.677109
$171$	0	0
$172$	0	0
$173$	−0.727922	−0.0553429	−0.0276714	−	0.999617i	$-0.508809\pi$
$173$	−0.727922	−0.0553429	−0.0276714	+	0.999617i	$0.508809\pi$
$174$	0	0
$175$	0	0
$176$	−16.9706	−1.27920
$177$	0	0
$178$	−6.24264	−0.467906
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−6.72792	−0.500083	−0.250041	−	0.968235i	$-0.580444\pi$
$181$	−6.72792	−0.500083	−0.250041	+	0.968235i	$0.580444\pi$
$182$	0	0
$183$	0	0
$184$	0.485281	0.0357754
$185$	−27.5563	−2.02598
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	0	0
$190$	7.75736	0.562778
$191$	−20.8284	−1.50709	−0.753546	−	0.657395i	$-0.771658\pi$
$191$	−20.8284	−1.50709	−0.753546	+	0.657395i	$0.771658\pi$
$192$	0	0
$193$	2.48528	0.178894	0.0894472	−	0.995992i	$-0.471490\pi$
$193$	2.48528	0.178894	0.0894472	+	0.995992i	$0.471490\pi$
$194$	−19.4142	−1.39386
$195$	0	0
$196$	0	0
$197$	−23.6569	−1.68548	−0.842741	−	0.538320i	$-0.819059\pi$
$197$	−23.6569	−1.68548	−0.842741	+	0.538320i	$0.819059\pi$
$198$	0	0
$199$	12.2426	0.867858	0.433929	−	0.900947i	$-0.357127\pi$
$199$	12.2426	0.867858	0.433929	+	0.900947i	$0.357127\pi$
$200$	40.9706	2.89706
$201$	0	0
$202$	−2.48528	−0.174864
$203$	0	0
$204$	0	0
$205$	14.0000	0.977802
$206$	−11.3137	−0.788263
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−5.27208	−0.364677
$210$	0	0
$211$	17.9706	1.23714	0.618572	−	0.785728i	$-0.287711\pi$
$211$	17.9706	1.23714	0.618572	+	0.785728i	$0.287711\pi$
$212$	0	0
$213$	0	0
$214$	11.5147	0.787130
$215$	−22.0711	−1.50523
$216$	0	0
$217$	0	0
$218$	12.3431	0.835983
$219$	0	0
$220$	0	0
$221$	−1.41421	−0.0951303
$222$	0	0
$223$	−9.24264	−0.618933	−0.309466	−	0.950910i	$-0.600150\pi$
$223$	−9.24264	−0.618933	−0.309466	+	0.950910i	$0.600150\pi$
$224$	0	0
$225$	0	0
$226$	−28.7279	−1.91095
$227$	21.1716	1.40521	0.702603	−	0.711582i	$-0.252021\pi$
$227$	21.1716	1.40521	0.702603	+	0.711582i	$0.252021\pi$
$228$	0	0
$229$	21.4558	1.41784	0.708921	−	0.705288i	$-0.249182\pi$
$229$	21.4558	1.41784	0.708921	+	0.705288i	$0.249182\pi$
$230$	−1.07107	−0.0706241
$231$	0	0
$232$	−16.4853	−1.08231
$233$	3.34315	0.219017	0.109508	−	0.993986i	$-0.465072\pi$
$233$	3.34315	0.219017	0.109508	+	0.993986i	$0.465072\pi$
$234$	0	0
$235$	19.4853	1.27108
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.4853	−1.32508	−0.662541	−	0.749025i	$-0.730522\pi$
$239$	−20.4853	−1.32508	−0.662541	+	0.749025i	$0.730522\pi$
$240$	0	0
$241$	−22.2132	−1.43088	−0.715439	−	0.698675i	$-0.753773\pi$
$241$	−22.2132	−1.43088	−0.715439	+	0.698675i	$0.753773\pi$
$242$	−9.89949	−0.636364
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.24264	−0.0790673
$248$	14.8284	0.941606
$249$	0	0
$250$	−59.2132	−3.74497
$251$	19.4142	1.22541	0.612707	−	0.790310i	$-0.290081\pi$
$251$	19.4142	1.22541	0.612707	+	0.790310i	$0.290081\pi$
$252$	0	0
$253$	0.727922	0.0457641
$254$	−2.82843	−0.177471
$255$	0	0
$256$	0	0
$257$	−16.5858	−1.03459	−0.517296	−	0.855806i	$-0.673062\pi$
$257$	−16.5858	−1.03459	−0.517296	+	0.855806i	$0.673062\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−4.00000	−0.247121
$263$	31.9706	1.97139	0.985695	−	0.168541i	$-0.0539055\pi$
$263$	31.9706	1.97139	0.985695	+	0.168541i	$0.0539055\pi$
$264$	0	0
$265$	25.7279	1.58045
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.8284	0.904105	0.452053	−	0.891991i	$-0.350692\pi$
$269$	14.8284	0.904105	0.452053	+	0.891991i	$0.350692\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	5.65685	0.342997
$273$	0	0
$274$	−10.4853	−0.633439
$275$	61.4558	3.70593
$276$	0	0
$277$	9.48528	0.569915	0.284958	−	0.958540i	$-0.408020\pi$
$277$	9.48528	0.569915	0.284958	+	0.958540i	$0.408020\pi$
$278$	−3.17157	−0.190218
$279$	0	0
$280$	0	0
$281$	15.5563	0.928014	0.464007	−	0.885832i	$-0.346411\pi$
$281$	15.5563	0.928014	0.464007	+	0.885832i	$0.346411\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	36.3848	2.13659
$291$	0	0
$292$	0	0
$293$	−21.3848	−1.24931	−0.624656	−	0.780900i	$-0.714761\pi$
$293$	−21.3848	−1.24931	−0.624656	+	0.780900i	$0.714761\pi$
$294$	0	0
$295$	51.4558	2.99588
$296$	−17.6569	−1.02628
$297$	0	0
$298$	10.9706	0.635508
$299$	0.171573	0.00992232
$300$	0	0
$301$	0	0
$302$	25.7990	1.48457
$303$	0	0
$304$	4.97056	0.285081
$305$	−26.4853	−1.51654
$306$	0	0
$307$	−4.75736	−0.271517	−0.135758	−	0.990742i	$-0.543347\pi$
$307$	−4.75736	−0.271517	−0.135758	+	0.990742i	$0.543347\pi$
$308$	0	0
$309$	0	0
$310$	−32.7279	−1.85882
$311$	−4.58579	−0.260036	−0.130018	−	0.991512i	$-0.541504\pi$
$311$	−4.58579	−0.260036	−0.130018	+	0.991512i	$0.541504\pi$
$312$	0	0
$313$	19.2132	1.08599	0.542997	−	0.839734i	$-0.317289\pi$
$313$	19.2132	1.08599	0.542997	+	0.839734i	$0.317289\pi$
$314$	−17.3137	−0.977069
$315$	0	0
$316$	0	0
$317$	−11.3137	−0.635441	−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137	−0.635441	−0.317721	+	0.948184i	$0.602917\pi$
$318$	0	0
$319$	−24.7279	−1.38450
$320$	35.3137	1.97410
$321$	0	0
$322$	0	0
$323$	1.75736	0.0977821
$324$	0	0
$325$	14.4853	0.803499
$326$	12.0000	0.664619
$327$	0	0
$328$	8.97056	0.495316
$329$	0	0
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	0	0
$334$	30.2426	1.65480
$335$	10.9706	0.599386
$336$	0	0
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	−1.41421	−0.0769231
$339$	0	0
$340$	0	0
$341$	22.2426	1.20451
$342$	0	0
$343$	0	0
$344$	−14.1421	−0.762493
$345$	0	0
$346$	1.02944	0.0553429
$347$	−5.65685	−0.303676	−0.151838	−	0.988405i	$-0.548519\pi$
$347$	−5.65685	−0.303676	−0.151838	+	0.988405i	$0.548519\pi$
$348$	0	0
$349$	0.272078	0.0145640	0.00728200	−	0.999973i	$-0.497682\pi$
$349$	0.272078	0.0145640	0.00728200	+	0.999973i	$0.497682\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.48528	0.451626	0.225813	−	0.974171i	$-0.427496\pi$
$353$	8.48528	0.451626	0.225813	+	0.974171i	$0.427496\pi$
$354$	0	0
$355$	−4.72792	−0.250932
$356$	0	0
$357$	0	0
$358$	−12.7279	−0.672692
$359$	8.10051	0.427528	0.213764	−	0.976885i	$-0.431428\pi$
$359$	8.10051	0.427528	0.213764	+	0.976885i	$0.431428\pi$
$360$	0	0
$361$	−17.4558	−0.918729
$362$	9.51472	0.500083
$363$	0	0
$364$	0	0
$365$	3.34315	0.174988
$366$	0	0
$367$	−1.75736	−0.0917334	−0.0458667	−	0.998948i	$-0.514605\pi$
$367$	−1.75736	−0.0917334	−0.0458667	+	0.998948i	$0.514605\pi$
$368$	−0.686292	−0.0357754
$369$	0	0
$370$	38.9706	2.02598
$371$	0	0
$372$	0	0
$373$	−8.48528	−0.439351	−0.219676	−	0.975573i	$-0.570500\pi$
$373$	−8.48528	−0.439351	−0.219676	+	0.975573i	$0.570500\pi$
$374$	8.48528	0.438763
$375$	0	0
$376$	12.4853	0.643879
$377$	−5.82843	−0.300179
$378$	0	0
$379$	32.2426	1.65619	0.828097	−	0.560585i	$-0.189424\pi$
$379$	32.2426	1.65619	0.828097	+	0.560585i	$0.189424\pi$
$380$	0	0
$381$	0	0
$382$	29.4558	1.50709
$383$	−3.51472	−0.179594	−0.0897969	−	0.995960i	$-0.528622\pi$
$383$	−3.51472	−0.179594	−0.0897969	+	0.995960i	$0.528622\pi$
$384$	0	0
$385$	0	0
$386$	−3.51472	−0.178894
$387$	0	0
$388$	0	0
$389$	−18.3431	−0.930034	−0.465017	−	0.885302i	$-0.653952\pi$
$389$	−18.3431	−0.930034	−0.465017	+	0.885302i	$0.653952\pi$
$390$	0	0
$391$	−0.242641	−0.0122709
$392$	0	0
$393$	0	0
$394$	33.4558	1.68548
$395$	−6.55635	−0.329886
$396$	0	0
$397$	−24.2132	−1.21523	−0.607613	−	0.794233i	$-0.707873\pi$
$397$	−24.2132	−1.21523	−0.607613	+	0.794233i	$0.707873\pi$
$398$	−17.3137	−0.867858
$399$	0	0
$400$	−57.9411	−2.89706
$401$	−17.6569	−0.881741	−0.440871	−	0.897571i	$-0.645330\pi$
$401$	−17.6569	−0.881741	−0.440871	+	0.897571i	$0.645330\pi$
$402$	0	0
$403$	5.24264	0.261155
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−26.4853	−1.31283
$408$	0	0
$409$	5.24264	0.259232	0.129616	−	0.991564i	$-0.458626\pi$
$409$	5.24264	0.259232	0.129616	+	0.991564i	$0.458626\pi$
$410$	−19.7990	−0.977802
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	21.0000	1.03085
$416$	0	0
$417$	0	0
$418$	7.45584	0.364677
$419$	32.8701	1.60581	0.802904	−	0.596109i	$-0.203287\pi$
$419$	32.8701	1.60581	0.802904	+	0.596109i	$0.203287\pi$
$420$	0	0
$421$	−18.7279	−0.912743	−0.456372	−	0.889789i	$-0.650851\pi$
$421$	−18.7279	−0.912743	−0.456372	+	0.889789i	$0.650851\pi$
$422$	−25.4142	−1.23714
$423$	0	0
$424$	16.4853	0.800596
$425$	−20.4853	−0.993682
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	31.2132	1.50523
$431$	23.6569	1.13951	0.569755	−	0.821814i	$-0.307038\pi$
$431$	23.6569	1.13951	0.569755	+	0.821814i	$0.307038\pi$
$432$	0	0
$433$	−8.97056	−0.431098	−0.215549	−	0.976493i	$-0.569154\pi$
$433$	−8.97056	−0.431098	−0.215549	+	0.976493i	$0.569154\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.213203	−0.0101989
$438$	0	0
$439$	17.5147	0.835932	0.417966	−	0.908463i	$-0.362743\pi$
$439$	17.5147	0.835932	0.417966	+	0.908463i	$0.362743\pi$
$440$	52.9706	2.52527
$441$	0	0
$442$	2.00000	0.0951303
$443$	26.3137	1.25020	0.625101	−	0.780544i	$-0.285058\pi$
$443$	26.3137	1.25020	0.625101	+	0.780544i	$0.285058\pi$
$444$	0	0
$445$	19.4853	0.923691
$446$	13.0711	0.618933
$447$	0	0
$448$	0	0
$449$	−26.8284	−1.26611	−0.633056	−	0.774106i	$-0.718200\pi$
$449$	−26.8284	−1.26611	−0.633056	+	0.774106i	$0.718200\pi$
$450$	0	0
$451$	13.4558	0.633611
$452$	0	0
$453$	0	0
$454$	−29.9411	−1.40521
$455$	0	0
$456$	0	0
$457$	35.2132	1.64720	0.823602	−	0.567168i	$-0.191961\pi$
$457$	35.2132	1.64720	0.823602	+	0.567168i	$0.191961\pi$
$458$	−30.3431	−1.41784
$459$	0	0
$460$	0	0
$461$	3.17157	0.147715	0.0738574	−	0.997269i	$-0.476469\pi$
$461$	3.17157	0.147715	0.0738574	+	0.997269i	$0.476469\pi$
$462$	0	0
$463$	4.24264	0.197172	0.0985861	−	0.995129i	$-0.468568\pi$
$463$	4.24264	0.197172	0.0985861	+	0.995129i	$0.468568\pi$
$464$	23.3137	1.08231
$465$	0	0
$466$	−4.72792	−0.219017
$467$	15.8995	0.735741	0.367870	−	0.929877i	$-0.380087\pi$
$467$	15.8995	0.735741	0.367870	+	0.929877i	$0.380087\pi$
$468$	0	0
$469$	0	0
$470$	−27.5563	−1.27108
$471$	0	0
$472$	32.9706	1.51759
$473$	−21.2132	−0.975384
$474$	0	0
$475$	−18.0000	−0.825897
$476$	0	0
$477$	0	0
$478$	28.9706	1.32508
$479$	36.2132	1.65462	0.827312	−	0.561743i	$-0.189869\pi$
$479$	36.2132	1.65462	0.827312	+	0.561743i	$0.189869\pi$
$480$	0	0
$481$	−6.24264	−0.284640
$482$	31.4142	1.43088
$483$	0	0
$484$	0	0
$485$	60.5980	2.75161
$486$	0	0
$487$	39.4558	1.78791	0.893957	−	0.448152i	$-0.147918\pi$
$487$	39.4558	1.78791	0.893957	+	0.448152i	$0.147918\pi$
$488$	−16.9706	−0.768221
$489$	0	0
$490$	0	0
$491$	28.6274	1.29194	0.645969	−	0.763364i	$-0.276454\pi$
$491$	28.6274	1.29194	0.645969	+	0.763364i	$0.276454\pi$
$492$	0	0
$493$	8.24264	0.371230
$494$	1.75736	0.0790673
$495$	0	0
$496$	−20.9706	−0.941606
$497$	0	0
$498$	0	0
$499$	−38.7279	−1.73370	−0.866850	−	0.498569i	$-0.833859\pi$
$499$	−38.7279	−1.73370	−0.866850	+	0.498569i	$0.833859\pi$
$500$	0	0
$501$	0	0
$502$	−27.4558	−1.22541
$503$	16.6274	0.741380	0.370690	−	0.928757i	$-0.379121\pi$
$503$	16.6274	0.741380	0.370690	+	0.928757i	$0.379121\pi$
$504$	0	0
$505$	7.75736	0.345198
$506$	−1.02944	−0.0457641
$507$	0	0
$508$	0	0
$509$	−24.8995	−1.10365	−0.551825	−	0.833960i	$-0.686069\pi$
$509$	−24.8995	−1.10365	−0.551825	+	0.833960i	$0.686069\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	1.00000
$513$	0	0
$514$	23.4558	1.03459
$515$	35.3137	1.55611
$516$	0	0
$517$	18.7279	0.823653
$518$	0	0
$519$	0	0
$520$	12.4853	0.547516
$521$	−17.6569	−0.773561	−0.386780	−	0.922172i	$-0.626413\pi$
$521$	−17.6569	−0.773561	−0.386780	+	0.922172i	$0.626413\pi$
$522$	0	0
$523$	2.97056	0.129894	0.0649468	−	0.997889i	$-0.479312\pi$
$523$	2.97056	0.129894	0.0649468	+	0.997889i	$0.479312\pi$
$524$	0	0
$525$	0	0
$526$	−45.2132	−1.97139
$527$	−7.41421	−0.322968
$528$	0	0
$529$	−22.9706	−0.998720
$530$	−36.3848	−1.58045
$531$	0	0
$532$	0	0
$533$	3.17157	0.137376
$534$	0	0
$535$	−35.9411	−1.55387
$536$	7.02944	0.303625
$537$	0	0
$538$	−20.9706	−0.904105
$539$	0	0
$540$	0	0
$541$	35.2132	1.51393	0.756967	−	0.653453i	$-0.226680\pi$
$541$	35.2132	1.51393	0.756967	+	0.653453i	$0.226680\pi$
$542$	28.2843	1.21491
$543$	0	0
$544$	0	0
$545$	−38.5269	−1.65031
$546$	0	0
$547$	18.5147	0.791632	0.395816	−	0.918330i	$-0.370462\pi$
$547$	18.5147	0.791632	0.395816	+	0.918330i	$0.370462\pi$
$548$	0	0
$549$	0	0
$550$	−86.9117	−3.70593
$551$	7.24264	0.308547
$552$	0	0
$553$	0	0
$554$	−13.4142	−0.569915
$555$	0	0
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	−22.0000	−0.928014
$563$	−17.6569	−0.744148	−0.372074	−	0.928203i	$-0.621353\pi$
$563$	−17.6569	−0.744148	−0.372074	+	0.928203i	$0.621353\pi$
$564$	0	0
$565$	89.6690	3.77241
$566$	12.0000	0.504398
$567$	0	0
$568$	−3.02944	−0.127112
$569$	23.1421	0.970169	0.485084	−	0.874467i	$-0.338789\pi$
$569$	23.1421	0.970169	0.485084	+	0.874467i	$0.338789\pi$
$570$	0	0
$571$	−8.45584	−0.353866	−0.176933	−	0.984223i	$-0.556618\pi$
$571$	−8.45584	−0.353866	−0.176933	+	0.984223i	$0.556618\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.48528	0.103643
$576$	0	0
$577$	−26.9706	−1.12280	−0.561400	−	0.827545i	$-0.689737\pi$
$577$	−26.9706	−1.12280	−0.561400	+	0.827545i	$0.689737\pi$
$578$	21.2132	0.882353
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.7279	1.02413
$584$	2.14214	0.0886422
$585$	0	0
$586$	30.2426	1.24931
$587$	24.5563	1.01355	0.506775	−	0.862079i	$-0.330838\pi$
$587$	24.5563	1.01355	0.506775	+	0.862079i	$0.330838\pi$
$588$	0	0
$589$	−6.51472	−0.268434
$590$	−72.7696	−2.99588
$591$	0	0
$592$	24.9706	1.02628
$593$	30.5563	1.25480	0.627399	−	0.778698i	$-0.284119\pi$
$593$	30.5563	1.25480	0.627399	+	0.778698i	$0.284119\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−0.242641	−0.00992232
$599$	−10.7990	−0.441235	−0.220617	−	0.975360i	$-0.570807\pi$
$599$	−10.7990	−0.441235	−0.220617	+	0.975360i	$0.570807\pi$
$600$	0	0
$601$	5.02944	0.205155	0.102578	−	0.994725i	$-0.467291\pi$
$601$	5.02944	0.205155	0.102578	+	0.994725i	$0.467291\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	30.8995	1.25624
$606$	0	0
$607$	1.27208	0.0516321	0.0258160	−	0.999667i	$-0.491782\pi$
$607$	1.27208	0.0516321	0.0258160	+	0.999667i	$0.491782\pi$
$608$	0	0
$609$	0	0
$610$	37.4558	1.51654
$611$	4.41421	0.178580
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	6.72792	0.271517
$615$	0	0
$616$	0	0
$617$	−23.6569	−0.952389	−0.476195	−	0.879340i	$-0.657984\pi$
$617$	−23.6569	−0.952389	−0.476195	+	0.879340i	$0.657984\pi$
$618$	0	0
$619$	32.9706	1.32520	0.662599	−	0.748974i	$-0.269453\pi$
$619$	32.9706	1.32520	0.662599	+	0.748974i	$0.269453\pi$
$620$	0	0
$621$	0	0
$622$	6.48528	0.260036
$623$	0	0
$624$	0	0
$625$	112.397	4.49588
$626$	−27.1716	−1.08599
$627$	0	0
$628$	0	0
$629$	8.82843	0.352012
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	−4.20101	−0.167107
$633$	0	0
$634$	16.0000	0.635441
$635$	8.82843	0.350345
$636$	0	0
$637$	0	0
$638$	34.9706	1.38450
$639$	0	0
$640$	−49.9411	−1.97410
$641$	−26.6569	−1.05288	−0.526441	−	0.850212i	$-0.676474\pi$
$641$	−26.6569	−1.05288	−0.526441	+	0.850212i	$0.676474\pi$
$642$	0	0
$643$	−4.48528	−0.176882	−0.0884411	−	0.996081i	$-0.528189\pi$
$643$	−4.48528	−0.176882	−0.0884411	+	0.996081i	$0.528189\pi$
$644$	0	0
$645$	0	0
$646$	−2.48528	−0.0977821
$647$	−50.5269	−1.98642	−0.993209	−	0.116344i	$-0.962882\pi$
$647$	−50.5269	−1.98642	−0.993209	+	0.116344i	$0.962882\pi$
$648$	0	0
$649$	49.4558	1.94131
$650$	−20.4853	−0.803499
$651$	0	0
$652$	0	0
$653$	5.31371	0.207941	0.103971	−	0.994580i	$-0.466845\pi$
$653$	5.31371	0.207941	0.103971	+	0.994580i	$0.466845\pi$
$654$	0	0
$655$	12.4853	0.487840
$656$	−12.6863	−0.495316
$657$	0	0
$658$	0	0
$659$	26.6569	1.03840	0.519202	−	0.854652i	$-0.326229\pi$
$659$	26.6569	1.03840	0.519202	+	0.854652i	$0.326229\pi$
$660$	0	0
$661$	19.2426	0.748452	0.374226	−	0.927338i	$-0.377908\pi$
$661$	19.2426	0.748452	0.374226	+	0.927338i	$0.377908\pi$
$662$	25.4558	0.989369
$663$	0	0
$664$	13.4558	0.522188
$665$	0	0
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	0	0
$670$	−15.5147	−0.599386
$671$	−25.4558	−0.982712
$672$	0	0
$673$	40.9411	1.57816	0.789082	−	0.614288i	$-0.210557\pi$
$673$	40.9411	1.57816	0.789082	+	0.614288i	$0.210557\pi$
$674$	46.6690	1.79762
$675$	0	0
$676$	0	0
$677$	29.3553	1.12822	0.564109	−	0.825701i	$-0.309220\pi$
$677$	29.3553	1.12822	0.564109	+	0.825701i	$0.309220\pi$
$678$	0	0
$679$	0	0
$680$	−17.6569	−0.677109
$681$	0	0
$682$	−31.4558	−1.20451
$683$	26.8284	1.02656	0.513281	−	0.858221i	$-0.328430\pi$
$683$	26.8284	1.02656	0.513281	+	0.858221i	$0.328430\pi$
$684$	0	0
$685$	32.7279	1.25047
$686$	0	0
$687$	0	0
$688$	20.0000	0.762493
$689$	5.82843	0.222045
$690$	0	0
$691$	30.6985	1.16783	0.583913	−	0.811816i	$-0.301521\pi$
$691$	30.6985	1.16783	0.583913	+	0.811816i	$0.301521\pi$
$692$	0	0
$693$	0	0
$694$	8.00000	0.303676
$695$	9.89949	0.375509
$696$	0	0
$697$	−4.48528	−0.169892
$698$	−0.384776	−0.0145640
$699$	0	0
$700$	0	0
$701$	−28.7990	−1.08772	−0.543861	−	0.839175i	$-0.683038\pi$
$701$	−28.7990	−1.08772	−0.543861	+	0.839175i	$0.683038\pi$
$702$	0	0
$703$	7.75736	0.292574
$704$	33.9411	1.27920
$705$	0	0
$706$	−12.0000	−0.451626
$707$	0	0
$708$	0	0
$709$	−24.7279	−0.928677	−0.464338	−	0.885658i	$-0.653708\pi$
$709$	−24.7279	−0.928677	−0.464338	+	0.885658i	$0.653708\pi$
$710$	6.68629	0.250932
$711$	0	0
$712$	12.4853	0.467906
$713$	0.899495	0.0336864
$714$	0	0
$715$	18.7279	0.700385
$716$	0	0
$717$	0	0
$718$	−11.4558	−0.427528
$719$	−10.2426	−0.381986	−0.190993	−	0.981591i	$-0.561171\pi$
$719$	−10.2426	−0.381986	−0.190993	+	0.981591i	$0.561171\pi$
$720$	0	0
$721$	0	0
$722$	24.6863	0.918729
$723$	0	0
$724$	0	0
$725$	−84.4264	−3.13552
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	0	0
$729$	0	0
$730$	−4.72792	−0.174988
$731$	7.07107	0.261533
$732$	0	0
$733$	−42.6985	−1.57710	−0.788552	−	0.614968i	$-0.789169\pi$
$733$	−42.6985	−1.57710	−0.788552	+	0.614968i	$0.789169\pi$
$734$	2.48528	0.0917334
$735$	0	0
$736$	0	0
$737$	10.5442	0.388399
$738$	0	0
$739$	41.6985	1.53390	0.766952	−	0.641705i	$-0.221773\pi$
$739$	41.6985	1.53390	0.766952	+	0.641705i	$0.221773\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.3431	0.672945	0.336472	−	0.941693i	$-0.390766\pi$
$743$	18.3431	0.672945	0.336472	+	0.941693i	$0.390766\pi$
$744$	0	0
$745$	−34.2426	−1.25455
$746$	12.0000	0.439351
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.48528	−0.0541987	−0.0270993	−	0.999633i	$-0.508627\pi$
$751$	−1.48528	−0.0541987	−0.0270993	+	0.999633i	$0.508627\pi$
$752$	−17.6569	−0.643879
$753$	0	0
$754$	8.24264	0.300179
$755$	−80.5269	−2.93067
$756$	0	0
$757$	4.51472	0.164090	0.0820451	−	0.996629i	$-0.473855\pi$
$757$	4.51472	0.164090	0.0820451	+	0.996629i	$0.473855\pi$
$758$	−45.5980	−1.65619
$759$	0	0
$760$	−15.5147	−0.562778
$761$	43.2426	1.56754	0.783772	−	0.621048i	$-0.213293\pi$
$761$	43.2426	1.56754	0.783772	+	0.621048i	$0.213293\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	4.97056	0.179594
$767$	11.6569	0.420905
$768$	0	0
$769$	−9.78680	−0.352921	−0.176460	−	0.984308i	$-0.556465\pi$
$769$	−9.78680	−0.352921	−0.176460	+	0.984308i	$0.556465\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.1716	−0.977294	−0.488647	−	0.872482i	$-0.662509\pi$
$773$	−27.1716	−0.977294	−0.488647	+	0.872482i	$0.662509\pi$
$774$	0	0
$775$	75.9411	2.72789
$776$	38.8284	1.39386
$777$	0	0
$778$	25.9411	0.930034
$779$	−3.94113	−0.141205
$780$	0	0
$781$	−4.54416	−0.162603
$782$	0.343146	0.0122709
$783$	0	0
$784$	0	0
$785$	54.0416	1.92883
$786$	0	0
$787$	33.2426	1.18497	0.592486	−	0.805581i	$-0.298147\pi$
$787$	33.2426	1.18497	0.592486	+	0.805581i	$0.298147\pi$
$788$	0	0
$789$	0	0
$790$	9.27208	0.329886
$791$	0	0
$792$	0	0
$793$	−6.00000	−0.213066
$794$	34.2426	1.21523
$795$	0	0
$796$	0	0
$797$	−35.6569	−1.26303	−0.631515	−	0.775363i	$-0.717567\pi$
$797$	−35.6569	−1.26303	−0.631515	+	0.775363i	$0.717567\pi$
$798$	0	0
$799$	−6.24264	−0.220849
$800$	0	0
$801$	0	0
$802$	24.9706	0.881741
$803$	3.21320	0.113391
$804$	0	0
$805$	0	0
$806$	−7.41421	−0.261155
$807$	0	0
$808$	4.97056	0.174864
$809$	−0.514719	−0.0180965	−0.00904827	−	0.999959i	$-0.502880\pi$
$809$	−0.514719	−0.0180965	−0.00904827	+	0.999959i	$0.502880\pi$
$810$	0	0
$811$	−45.9411	−1.61321	−0.806606	−	0.591090i	$-0.798698\pi$
$811$	−45.9411	−1.61321	−0.806606	+	0.591090i	$0.798698\pi$
$812$	0	0
$813$	0	0
$814$	37.4558	1.31283
$815$	−37.4558	−1.31202
$816$	0	0
$817$	6.21320	0.217372
$818$	−7.41421	−0.259232
$819$	0	0
$820$	0	0
$821$	−8.14214	−0.284162	−0.142081	−	0.989855i	$-0.545379\pi$
$821$	−8.14214	−0.284162	−0.142081	+	0.989855i	$0.545379\pi$
$822$	0	0
$823$	−6.48528	−0.226063	−0.113031	−	0.993591i	$-0.536056\pi$
$823$	−6.48528	−0.226063	−0.113031	+	0.993591i	$0.536056\pi$
$824$	22.6274	0.788263
$825$	0	0
$826$	0	0
$827$	−1.45584	−0.0506247	−0.0253123	−	0.999680i	$-0.508058\pi$
$827$	−1.45584	−0.0506247	−0.0253123	+	0.999680i	$0.508058\pi$
$828$	0	0
$829$	−25.2721	−0.877736	−0.438868	−	0.898552i	$-0.644620\pi$
$829$	−25.2721	−0.877736	−0.438868	+	0.898552i	$0.644620\pi$
$830$	−29.6985	−1.03085
$831$	0	0
$832$	8.00000	0.277350
$833$	0	0
$834$	0	0
$835$	−94.3970	−3.26674
$836$	0	0
$837$	0	0
$838$	−46.4853	−1.60581
$839$	−38.8284	−1.34051	−0.670253	−	0.742133i	$-0.733814\pi$
$839$	−38.8284	−1.34051	−0.670253	+	0.742133i	$0.733814\pi$
$840$	0	0
$841$	4.97056	0.171399
$842$	26.4853	0.912743
$843$	0	0
$844$	0	0
$845$	4.41421	0.151854
$846$	0	0
$847$	0	0
$848$	−23.3137	−0.800596
$849$	0	0
$850$	28.9706	0.993682
$851$	−1.07107	−0.0367157
$852$	0	0
$853$	−14.2721	−0.488667	−0.244333	−	0.969691i	$-0.578569\pi$
$853$	−14.2721	−0.488667	−0.244333	+	0.969691i	$0.578569\pi$
$854$	0	0
$855$	0	0
$856$	−23.0294	−0.787130
$857$	18.7696	0.641156	0.320578	−	0.947222i	$-0.396123\pi$
$857$	18.7696	0.641156	0.320578	+	0.947222i	$0.396123\pi$
$858$	0	0
$859$	2.97056	0.101354	0.0506771	−	0.998715i	$-0.483862\pi$
$859$	2.97056	0.101354	0.0506771	+	0.998715i	$0.483862\pi$
$860$	0	0
$861$	0	0
$862$	−33.4558	−1.13951
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	−3.21320	−0.109252
$866$	12.6863	0.431098
$867$	0	0
$868$	0	0
$869$	−6.30152	−0.213764
$870$	0	0
$871$	2.48528	0.0842105
$872$	−24.6863	−0.835983
$873$	0	0
$874$	0.301515	0.0101989
$875$	0	0
$876$	0	0
$877$	−1.75736	−0.0593418	−0.0296709	−	0.999560i	$-0.509446\pi$
$877$	−1.75736	−0.0593418	−0.0296709	+	0.999560i	$0.509446\pi$
$878$	−24.7696	−0.835932
$879$	0	0
$880$	−74.9117	−2.52527
$881$	−49.1127	−1.65465	−0.827324	−	0.561724i	$-0.810138\pi$
$881$	−49.1127	−1.65465	−0.827324	+	0.561724i	$0.810138\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	−37.2132	−1.25020
$887$	−43.1127	−1.44758	−0.723791	−	0.690019i	$-0.757602\pi$
$887$	−43.1127	−1.44758	−0.723791	+	0.690019i	$0.757602\pi$
$888$	0	0
$889$	0	0
$890$	−27.5563	−0.923691
$891$	0	0
$892$	0	0
$893$	−5.48528	−0.183558
$894$	0	0
$895$	39.7279	1.32796
$896$	0	0
$897$	0	0
$898$	37.9411	1.26611
$899$	−30.5563	−1.01911
$900$	0	0
$901$	−8.24264	−0.274602
$902$	−19.0294	−0.633611
$903$	0	0
$904$	57.4558	1.91095
$905$	−29.6985	−0.987211
$906$	0	0
$907$	−31.9706	−1.06157	−0.530783	−	0.847508i	$-0.678102\pi$
$907$	−31.9706	−1.06157	−0.530783	+	0.847508i	$0.678102\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.0294	0.332290	0.166145	−	0.986101i	$-0.446868\pi$
$911$	10.0294	0.332290	0.166145	+	0.986101i	$0.446868\pi$
$912$	0	0
$913$	20.1838	0.667985
$914$	−49.7990	−1.64720
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	2.14214	0.0706241
$921$	0	0
$922$	−4.48528	−0.147715
$923$	−1.07107	−0.0352546
$924$	0	0
$925$	−90.4264	−2.97320
$926$	−6.00000	−0.197172
$927$	0	0
$928$	0	0
$929$	17.1005	0.561049	0.280525	−	0.959847i	$-0.409492\pi$
$929$	17.1005	0.561049	0.280525	+	0.959847i	$0.409492\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−22.4853	−0.735741
$935$	−26.4853	−0.866161
$936$	0	0
$937$	−2.78680	−0.0910407	−0.0455203	−	0.998963i	$-0.514495\pi$
$937$	−2.78680	−0.0910407	−0.0455203	+	0.998963i	$0.514495\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25.9289	0.845259	0.422630	−	0.906303i	$-0.361107\pi$
$941$	25.9289	0.845259	0.422630	+	0.906303i	$0.361107\pi$
$942$	0	0
$943$	0.544156	0.0177202
$944$	−46.6274	−1.51759
$945$	0	0
$946$	30.0000	0.975384
$947$	−20.8284	−0.676833	−0.338416	−	0.940996i	$-0.609891\pi$
$947$	−20.8284	−0.676833	−0.338416	+	0.940996i	$0.609891\pi$
$948$	0	0
$949$	0.757359	0.0245849
$950$	25.4558	0.825897
$951$	0	0
$952$	0	0
$953$	17.1421	0.555288	0.277644	−	0.960684i	$-0.410446\pi$
$953$	17.1421	0.555288	0.277644	+	0.960684i	$0.410446\pi$
$954$	0	0
$955$	−91.9411	−2.97514
$956$	0	0
$957$	0	0
$958$	−51.2132	−1.65462
$959$	0	0
$960$	0	0
$961$	−3.51472	−0.113378
$962$	8.82843	0.284640
$963$	0	0
$964$	0	0
$965$	10.9706	0.353155
$966$	0	0
$967$	−41.6985	−1.34093	−0.670466	−	0.741940i	$-0.733906\pi$
$967$	−41.6985	−1.34093	−0.670466	+	0.741940i	$0.733906\pi$
$968$	19.7990	0.636364
$969$	0	0
$970$	−85.6985	−2.75161
$971$	−7.45584	−0.239269	−0.119635	−	0.992818i	$-0.538172\pi$
$971$	−7.45584	−0.239269	−0.119635	+	0.992818i	$0.538172\pi$
$972$	0	0
$973$	0	0
$974$	−55.7990	−1.78791
$975$	0	0
$976$	24.0000	0.768221
$977$	24.0416	0.769160	0.384580	−	0.923092i	$-0.374346\pi$
$977$	24.0416	0.769160	0.384580	+	0.923092i	$0.374346\pi$
$978$	0	0
$979$	18.7279	0.598547
$980$	0	0
$981$	0	0
$982$	−40.4853	−1.29194
$983$	33.0416	1.05386	0.526932	−	0.849907i	$-0.323342\pi$
$983$	33.0416	1.05386	0.526932	+	0.849907i	$0.323342\pi$
$984$	0	0
$985$	−104.426	−3.32730
$986$	−11.6569	−0.371230
$987$	0	0
$988$	0	0
$989$	−0.857864	−0.0272785
$990$	0	0
$991$	34.9706	1.11088	0.555438	−	0.831558i	$-0.312551\pi$
$991$	34.9706	1.11088	0.555438	+	0.831558i	$0.312551\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	54.0416	1.71323
$996$	0	0
$997$	28.4853	0.902138	0.451069	−	0.892489i	$-0.351043\pi$
$997$	28.4853	0.902138	0.451069	+	0.892489i	$0.351043\pi$
$998$	54.7696	1.73370
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.s.1.1		2
3.2	odd	2		637.2.a.g.1.2		2
7.6	odd	2		819.2.a.h.1.1		2
21.2	odd	6		637.2.e.g.508.1		4
21.5	even	6		637.2.e.f.508.1		4
21.11	odd	6		637.2.e.g.79.1		4
21.17	even	6		637.2.e.f.79.1		4
21.20	even	2		91.2.a.c.1.2	✓	2
39.38	odd	2		8281.2.a.v.1.1		2
84.83	odd	2		1456.2.a.q.1.2		2
105.104	even	2		2275.2.a.j.1.1		2
168.83	odd	2		5824.2.a.bk.1.1		2
168.125	even	2		5824.2.a.bl.1.2		2
273.83	odd	4		1183.2.c.d.337.1		4
273.125	odd	4		1183.2.c.d.337.3		4
273.272	even	2		1183.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.c.1.2	✓	2	21.20	even	2
637.2.a.g.1.2		2	3.2	odd	2
637.2.e.f.79.1		4	21.17	even	6
637.2.e.f.508.1		4	21.5	even	6
637.2.e.g.79.1		4	21.11	odd	6
637.2.e.g.508.1		4	21.2	odd	6
819.2.a.h.1.1		2	7.6	odd	2
1183.2.a.d.1.1		2	273.272	even	2
1183.2.c.d.337.1		4	273.83	odd	4
1183.2.c.d.337.3		4	273.125	odd	4
1456.2.a.q.1.2		2	84.83	odd	2
2275.2.a.j.1.1		2	105.104	even	2
5733.2.a.s.1.1		2	1.1	even	1	trivial
5824.2.a.bk.1.1		2	168.83	odd	2
5824.2.a.bl.1.2		2	168.125	even	2
8281.2.a.v.1.1		2	39.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +4.41421 q^{5} +2.82843 q^{8} -6.24264 q^{10} +4.24264 q^{11} +1.00000 q^{13} -4.00000 q^{16} -1.41421 q^{17} -1.24264 q^{19} -6.00000 q^{22} +0.171573 q^{23} +14.4853 q^{25} -1.41421 q^{26} -5.82843 q^{29} +5.24264 q^{31} +2.00000 q^{34} -6.24264 q^{37} +1.75736 q^{38} +12.4853 q^{40} +3.17157 q^{41} -5.00000 q^{43} -0.242641 q^{46} +4.41421 q^{47} -20.4853 q^{50} +5.82843 q^{53} +18.7279 q^{55} +8.24264 q^{58} +11.6569 q^{59} -6.00000 q^{61} -7.41421 q^{62} +8.00000 q^{64} +4.41421 q^{65} +2.48528 q^{67} -1.07107 q^{71} +0.757359 q^{73} +8.82843 q^{74} -1.48528 q^{79} -17.6569 q^{80} -4.48528 q^{82} +4.75736 q^{83} -6.24264 q^{85} +7.07107 q^{86} +12.0000 q^{88} +4.41421 q^{89} -6.24264 q^{94} -5.48528 q^{95} +13.7279 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} - 4 q^{10} + 2 q^{13} - 8 q^{16} + 6 q^{19} - 12 q^{22} + 6 q^{23} + 12 q^{25} - 6 q^{29} + 2 q^{31} + 4 q^{34} - 4 q^{37} + 12 q^{38} + 8 q^{40} + 12 q^{41} - 10 q^{43} + 8 q^{46} + 6 q^{47}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 - 4 * q^10 + 2 * q^13 - 8 * q^16 + 6 * q^19 - 12 * q^22 + 6 * q^23 + 12 * q^25 - 6 * q^29 + 2 * q^31 + 4 * q^34 - 4 * q^37 + 12 * q^38 + 8 * q^40 + 12 * q^41 - 10 * q^43 + 8 * q^46 + 6 * q^47 - 24 * q^50 + 6 * q^53 + 12 * q^55 + 8 * q^58 + 12 * q^59 - 12 * q^61 - 12 * q^62 + 16 * q^64 + 6 * q^65 - 12 * q^67 + 12 * q^71 + 10 * q^73 + 12 * q^74 + 14 * q^79 - 24 * q^80 + 8 * q^82 + 18 * q^83 - 4 * q^85 + 24 * q^88 + 6 * q^89 - 4 * q^94 + 6 * q^95 + 2 * q^97

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