LMFDB - Embedded newform 5733.2.a.bu.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.4507648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.66745$ 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)

$f(q)$	$=$	$q+2.44785 q^{2} +3.99195 q^{4} +0.910286 q^{5} +4.87599 q^{8} +O(q^{10})$	$q+2.44785 q^{2} +3.99195 q^{4} +0.910286 q^{5} +4.87599 q^{8} +2.22824 q^{10} -3.67837 q^{11} -1.00000 q^{13} +3.95177 q^{16} +7.18531 q^{17} +1.97802 q^{19} +3.63382 q^{20} -9.00407 q^{22} +0.596700 q^{23} -4.17138 q^{25} -2.44785 q^{26} +3.64900 q^{29} +7.08833 q^{31} -0.0786478 q^{32} +17.5885 q^{34} +0.710851 q^{37} +4.84189 q^{38} +4.43855 q^{40} +5.27529 q^{41} +11.0790 q^{43} -14.6839 q^{44} +1.46063 q^{46} +12.1135 q^{47} -10.2109 q^{50} -3.99195 q^{52} +11.4484 q^{53} -3.34837 q^{55} +8.93219 q^{58} +9.58986 q^{59} -6.98536 q^{61} +17.3511 q^{62} -8.09606 q^{64} -0.910286 q^{65} +1.22839 q^{67} +28.6834 q^{68} -11.3635 q^{71} -6.53419 q^{73} +1.74005 q^{74} +7.89616 q^{76} -11.5204 q^{79} +3.59725 q^{80} +12.9131 q^{82} +7.16403 q^{83} +6.54069 q^{85} +27.1198 q^{86} -17.9357 q^{88} +12.8490 q^{89} +2.38200 q^{92} +29.6520 q^{94} +1.80056 q^{95} -9.09062 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 6 q^{5}+O(q^{10}) $ 6 * q + 4 * q^4 + 6 * q^5	$ 6 q + 4 q^{4} + 6 q^{5} - 4 q^{10} - 4 q^{11} - 6 q^{13} + 16 q^{17} - 2 q^{19} + 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} - 6 q^{31} + 20 q^{32} + 8 q^{38} - 4 q^{40} - 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} + 30 q^{47} - 8 q^{50} - 4 q^{52} + 14 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} + 28 q^{68} - 8 q^{71} + 6 q^{73} + 12 q^{74} + 16 q^{76} - 22 q^{79} - 28 q^{80} + 40 q^{82} + 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} + 26 q^{89} - 20 q^{92} + 32 q^{94} + 6 q^{95} + 14 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 + 6 * q^5 - 4 * q^10 - 4 * q^11 - 6 * q^13 + 16 * q^17 - 2 * q^19 + 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 - 6 * q^31 + 20 * q^32 + 8 * q^38 - 4 * q^40 - 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 + 30 * q^47 - 8 * q^50 - 4 * q^52 + 14 * q^53 + 8 * q^55 - 8 * q^58 + 24 * q^59 + 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 + 28 * q^68 - 8 * q^71 + 6 * q^73 + 12 * q^74 + 16 * q^76 - 22 * q^79 - 28 * q^80 + 40 * q^82 + 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 + 26 * q^89 - 20 * q^92 + 32 * q^94 + 6 * q^95 + 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.44785	1.73089	0.865444	−	0.501005i	$-0.167036\pi$
$2$	2.44785	1.73089	0.865444	+	0.501005i	$0.167036\pi$
$3$	0	0
$3$	0	0
$4$	3.99195	1.99598
$5$	0.910286	0.407092	0.203546	−	0.979065i	$-0.434753\pi$
$5$	0.910286	0.407092	0.203546	+	0.979065i	$0.434753\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	4.87599	1.72392
$9$	0	0
$10$	2.22824	0.704632
$11$	−3.67837	−1.10907	−0.554534	−	0.832161i	$-0.687104\pi$
$11$	−3.67837	−1.10907	−0.554534	+	0.832161i	$0.687104\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	3.95177	0.987943
$17$	7.18531	1.74269	0.871347	−	0.490668i	$-0.163247\pi$
$17$	7.18531	1.74269	0.871347	+	0.490668i	$0.163247\pi$
$18$	0	0
$19$	1.97802	0.453789	0.226894	−	0.973919i	$-0.427143\pi$
$19$	1.97802	0.453789	0.226894	+	0.973919i	$0.427143\pi$
$20$	3.63382	0.812547
$21$	0	0
$22$	−9.00407	−1.91967
$23$	0.596700	0.124421	0.0622103	−	0.998063i	$-0.480185\pi$
$23$	0.596700	0.124421	0.0622103	+	0.998063i	$0.480185\pi$
$24$	0	0
$25$	−4.17138	−0.834276
$26$	−2.44785	−0.480062
$27$	0	0
$28$	0	0
$29$	3.64900	0.677602	0.338801	−	0.940858i	$-0.389979\pi$
$29$	3.64900	0.677602	0.338801	+	0.940858i	$0.389979\pi$
$30$	0	0
$31$	7.08833	1.27310	0.636551	−	0.771235i	$-0.280361\pi$
$31$	7.08833	1.27310	0.636551	+	0.771235i	$0.280361\pi$
$32$	−0.0786478	−0.0139031
$33$	0	0
$34$	17.5885	3.01641
$35$	0	0
$36$	0	0
$37$	0.710851	0.116863	0.0584316	−	0.998291i	$-0.481390\pi$
$37$	0.710851	0.116863	0.0584316	+	0.998291i	$0.481390\pi$
$38$	4.84189	0.785458
$39$	0	0
$40$	4.43855	0.701796
$41$	5.27529	0.823863	0.411931	−	0.911215i	$-0.364854\pi$
$41$	5.27529	0.823863	0.411931	+	0.911215i	$0.364854\pi$
$42$	0	0
$43$	11.0790	1.68954	0.844768	−	0.535132i	$-0.179738\pi$
$43$	11.0790	1.68954	0.844768	+	0.535132i	$0.179738\pi$
$44$	−14.6839	−2.21367
$45$	0	0
$46$	1.46063	0.215358
$47$	12.1135	1.76693	0.883467	−	0.468494i	$-0.155203\pi$
$47$	12.1135	1.76693	0.883467	+	0.468494i	$0.155203\pi$
$48$	0	0
$49$	0	0
$50$	−10.2109	−1.44404
$51$	0	0
$52$	−3.99195	−0.553584
$53$	11.4484	1.57256	0.786282	−	0.617868i	$-0.212003\pi$
$53$	11.4484	1.57256	0.786282	+	0.617868i	$0.212003\pi$
$54$	0	0
$55$	−3.34837	−0.451494
$56$	0	0
$57$	0	0
$58$	8.93219	1.17285
$59$	9.58986	1.24849	0.624247	−	0.781227i	$-0.285406\pi$
$59$	9.58986	1.24849	0.624247	+	0.781227i	$0.285406\pi$
$60$	0	0
$61$	−6.98536	−0.894384	−0.447192	−	0.894438i	$-0.647576\pi$
$61$	−6.98536	−0.894384	−0.447192	+	0.894438i	$0.647576\pi$
$62$	17.3511	2.20360
$63$	0	0
$64$	−8.09606	−1.01201
$65$	−0.910286	−0.112907
$66$	0	0
$67$	1.22839	0.150072	0.0750360	−	0.997181i	$-0.476093\pi$
$67$	1.22839	0.150072	0.0750360	+	0.997181i	$0.476093\pi$
$68$	28.6834	3.47837
$69$	0	0
$70$	0	0
$71$	−11.3635	−1.34859	−0.674297	−	0.738460i	$-0.735553\pi$
$71$	−11.3635	−1.34859	−0.674297	+	0.738460i	$0.735553\pi$
$72$	0	0
$73$	−6.53419	−0.764769	−0.382384	−	0.924003i	$-0.624897\pi$
$73$	−6.53419	−0.764769	−0.382384	+	0.924003i	$0.624897\pi$
$74$	1.74005	0.202277
$75$	0	0
$76$	7.89616	0.905752
$77$	0	0
$78$	0	0
$79$	−11.5204	−1.29615	−0.648074	−	0.761577i	$-0.724425\pi$
$79$	−11.5204	−1.29615	−0.648074	+	0.761577i	$0.724425\pi$
$80$	3.59725	0.402184
$81$	0	0
$82$	12.9131	1.42601
$83$	7.16403	0.786355	0.393177	−	0.919463i	$-0.371376\pi$
$83$	7.16403	0.786355	0.393177	+	0.919463i	$0.371376\pi$
$84$	0	0
$85$	6.54069	0.709437
$86$	27.1198	2.92440
$87$	0	0
$88$	−17.9357	−1.91195
$89$	12.8490	1.36199	0.680997	−	0.732286i	$-0.261547\pi$
$89$	12.8490	1.36199	0.680997	+	0.732286i	$0.261547\pi$
$90$	0	0
$91$	0	0
$92$	2.38200	0.248341
$93$	0	0
$94$	29.6520	3.05837
$95$	1.80056	0.184734
$96$	0	0
$97$	−9.09062	−0.923012	−0.461506	−	0.887137i	$-0.652691\pi$
$97$	−9.09062	−0.923012	−0.461506	+	0.887137i	$0.652691\pi$
$98$	0	0
$99$	0	0
$100$	−16.6519	−1.66519
$101$	−5.80645	−0.577763	−0.288882	−	0.957365i	$-0.593283\pi$
$101$	−5.80645	−0.577763	−0.288882	+	0.957365i	$0.593283\pi$
$102$	0	0
$103$	−12.8682	−1.26794	−0.633971	−	0.773357i	$-0.718576\pi$
$103$	−12.8682	−1.26794	−0.633971	+	0.773357i	$0.718576\pi$
$104$	−4.87599	−0.478130
$105$	0	0
$106$	28.0240	2.72193
$107$	−4.45747	−0.430920	−0.215460	−	0.976513i	$-0.569125\pi$
$107$	−4.45747	−0.430920	−0.215460	+	0.976513i	$0.569125\pi$
$108$	0	0
$109$	0.878896	0.0841830	0.0420915	−	0.999114i	$-0.486598\pi$
$109$	0.878896	0.0841830	0.0420915	+	0.999114i	$0.486598\pi$
$110$	−8.19628	−0.781485
$111$	0	0
$112$	0	0
$113$	5.36723	0.504906	0.252453	−	0.967609i	$-0.418763\pi$
$113$	5.36723	0.504906	0.252453	+	0.967609i	$0.418763\pi$
$114$	0	0
$115$	0.543168	0.0506507
$116$	14.5666	1.35248
$117$	0	0
$118$	23.4745	2.16100
$119$	0	0
$120$	0	0
$121$	2.53037	0.230034
$122$	−17.0991	−1.54808
$123$	0	0
$124$	28.2963	2.54108
$125$	−8.34858	−0.746720
$126$	0	0
$127$	6.61029	0.586568	0.293284	−	0.956025i	$-0.405252\pi$
$127$	6.61029	0.586568	0.293284	+	0.956025i	$0.405252\pi$
$128$	−19.6606	−1.73777
$129$	0	0
$130$	−2.22824	−0.195430
$131$	−19.9665	−1.74448	−0.872240	−	0.489079i	$-0.837333\pi$
$131$	−19.9665	−1.74448	−0.872240	+	0.489079i	$0.837333\pi$
$132$	0	0
$133$	0	0
$134$	3.00692	0.259758
$135$	0	0
$136$	35.0355	3.00427
$137$	−5.75451	−0.491641	−0.245821	−	0.969315i	$-0.579057\pi$
$137$	−5.75451	−0.491641	−0.245821	+	0.969315i	$0.579057\pi$
$138$	0	0
$139$	1.55138	0.131586	0.0657931	−	0.997833i	$-0.479042\pi$
$139$	1.55138	0.131586	0.0657931	+	0.997833i	$0.479042\pi$
$140$	0	0
$141$	0	0
$142$	−27.8160	−2.33427
$143$	3.67837	0.307600
$144$	0	0
$145$	3.32163	0.275847
$146$	−15.9947	−1.32373
$147$	0	0
$148$	2.83768	0.233256
$149$	−14.9337	−1.22341	−0.611707	−	0.791084i	$-0.709517\pi$
$149$	−14.9337	−1.22341	−0.611707	+	0.791084i	$0.709517\pi$
$150$	0	0
$151$	−7.90102	−0.642976	−0.321488	−	0.946914i	$-0.604183\pi$
$151$	−7.90102	−0.642976	−0.321488	+	0.946914i	$0.604183\pi$
$152$	9.64481	0.782297
$153$	0	0
$154$	0	0
$155$	6.45241	0.518270
$156$	0	0
$157$	−12.4948	−0.997194	−0.498597	−	0.866834i	$-0.666151\pi$
$157$	−12.4948	−0.997194	−0.498597	+	0.866834i	$0.666151\pi$
$158$	−28.2002	−2.24349
$159$	0	0
$160$	−0.0715920	−0.00565985
$161$	0	0
$162$	0	0
$163$	7.01427	0.549400	0.274700	−	0.961530i	$-0.411421\pi$
$163$	7.01427	0.549400	0.274700	+	0.961530i	$0.411421\pi$
$164$	21.0587	1.64441
$165$	0	0
$166$	17.5365	1.36109
$167$	4.82764	0.373574	0.186787	−	0.982400i	$-0.440193\pi$
$167$	4.82764	0.373574	0.186787	+	0.982400i	$0.440193\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	16.0106	1.22796
$171$	0	0
$172$	44.2270	3.37227
$173$	−22.2124	−1.68878	−0.844388	−	0.535732i	$-0.820036\pi$
$173$	−22.2124	−1.68878	−0.844388	+	0.535732i	$0.820036\pi$
$174$	0	0
$175$	0	0
$176$	−14.5361	−1.09570
$177$	0	0
$178$	31.4524	2.35746
$179$	−4.57505	−0.341956	−0.170978	−	0.985275i	$-0.554693\pi$
$179$	−4.57505	−0.341956	−0.170978	+	0.985275i	$0.554693\pi$
$180$	0	0
$181$	7.23332	0.537649	0.268824	−	0.963189i	$-0.413365\pi$
$181$	7.23332	0.537649	0.268824	+	0.963189i	$0.413365\pi$
$182$	0	0
$183$	0	0
$184$	2.90951	0.214492
$185$	0.647078	0.0475741
$186$	0	0
$187$	−26.4302	−1.93277
$188$	48.3565	3.52676
$189$	0	0
$190$	4.40751	0.319754
$191$	−1.97242	−0.142719	−0.0713596	−	0.997451i	$-0.522734\pi$
$191$	−1.97242	−0.142719	−0.0713596	+	0.997451i	$0.522734\pi$
$192$	0	0
$193$	−25.2073	−1.81446	−0.907230	−	0.420635i	$-0.861807\pi$
$193$	−25.2073	−1.81446	−0.907230	+	0.420635i	$0.861807\pi$
$194$	−22.2524	−1.59763
$195$	0	0
$196$	0	0
$197$	3.04497	0.216945	0.108473	−	0.994099i	$-0.465404\pi$
$197$	3.04497	0.216945	0.108473	+	0.994099i	$0.465404\pi$
$198$	0	0
$199$	−4.05691	−0.287587	−0.143794	−	0.989608i	$-0.545930\pi$
$199$	−4.05691	−0.287587	−0.143794	+	0.989608i	$0.545930\pi$
$200$	−20.3396	−1.43823
$201$	0	0
$202$	−14.2133	−1.00004
$203$	0	0
$204$	0	0
$205$	4.80203	0.335388
$206$	−31.4994	−2.19467
$207$	0	0
$208$	−3.95177	−0.274006
$209$	−7.27588	−0.503283
$210$	0	0
$211$	16.4116	1.12982	0.564910	−	0.825152i	$-0.308911\pi$
$211$	16.4116	1.12982	0.564910	+	0.825152i	$0.308911\pi$
$212$	45.7016	3.13880
$213$	0	0
$214$	−10.9112	−0.745875
$215$	10.0851	0.687797
$216$	0	0
$217$	0	0
$218$	2.15140	0.145711
$219$	0	0
$220$	−13.3665	−0.901170
$221$	−7.18531	−0.483336
$222$	0	0
$223$	16.1205	1.07951	0.539755	−	0.841822i	$-0.318517\pi$
$223$	16.1205	1.07951	0.539755	+	0.841822i	$0.318517\pi$
$224$	0	0
$225$	0	0
$226$	13.1381	0.873936
$227$	−8.67804	−0.575982	−0.287991	−	0.957633i	$-0.592987\pi$
$227$	−8.67804	−0.575982	−0.287991	+	0.957633i	$0.592987\pi$
$228$	0	0
$229$	20.7592	1.37180	0.685902	−	0.727694i	$-0.259408\pi$
$229$	20.7592	1.37180	0.685902	+	0.727694i	$0.259408\pi$
$230$	1.32959	0.0876707
$231$	0	0
$232$	17.7925	1.16813
$233$	9.84468	0.644946	0.322473	−	0.946579i	$-0.395486\pi$
$233$	9.84468	0.644946	0.322473	+	0.946579i	$0.395486\pi$
$234$	0	0
$235$	11.0267	0.719305
$236$	38.2823	2.49196
$237$	0	0
$238$	0	0
$239$	−1.13539	−0.0734424	−0.0367212	−	0.999326i	$-0.511691\pi$
$239$	−1.13539	−0.0734424	−0.0367212	+	0.999326i	$0.511691\pi$
$240$	0	0
$241$	21.8208	1.40560	0.702802	−	0.711386i	$-0.251932\pi$
$241$	21.8208	1.40560	0.702802	+	0.711386i	$0.251932\pi$
$242$	6.19396	0.398163
$243$	0	0
$244$	−27.8852	−1.78517
$245$	0	0
$246$	0	0
$247$	−1.97802	−0.125858
$248$	34.5626	2.19473
$249$	0	0
$250$	−20.4360	−1.29249
$251$	9.44377	0.596086	0.298043	−	0.954552i	$-0.403666\pi$
$251$	9.44377	0.596086	0.298043	+	0.954552i	$0.403666\pi$
$252$	0	0
$253$	−2.19488	−0.137991
$254$	16.1810	1.01528
$255$	0	0
$256$	−31.9341	−1.99588
$257$	−3.62752	−0.226279	−0.113139	−	0.993579i	$-0.536091\pi$
$257$	−3.62752	−0.226279	−0.113139	+	0.993579i	$0.536091\pi$
$258$	0	0
$259$	0	0
$260$	−3.63382	−0.225360
$261$	0	0
$262$	−48.8749	−3.01950
$263$	−6.59584	−0.406717	−0.203359	−	0.979104i	$-0.565186\pi$
$263$	−6.59584	−0.406717	−0.203359	+	0.979104i	$0.565186\pi$
$264$	0	0
$265$	10.4214	0.640179
$266$	0	0
$267$	0	0
$268$	4.90369	0.299540
$269$	23.5313	1.43473	0.717365	−	0.696698i	$-0.245348\pi$
$269$	23.5313	1.43473	0.717365	+	0.696698i	$0.245348\pi$
$270$	0	0
$271$	−2.69767	−0.163872	−0.0819358	−	0.996638i	$-0.526110\pi$
$271$	−2.69767	−0.163872	−0.0819358	+	0.996638i	$0.526110\pi$
$272$	28.3947	1.72168
$273$	0	0
$274$	−14.0862	−0.850976
$275$	15.3439	0.925269
$276$	0	0
$277$	−24.7549	−1.48738	−0.743690	−	0.668525i	$-0.766926\pi$
$277$	−24.7549	−1.48738	−0.743690	+	0.668525i	$0.766926\pi$
$278$	3.79754	0.227761
$279$	0	0
$280$	0	0
$281$	4.05377	0.241828	0.120914	−	0.992663i	$-0.461418\pi$
$281$	4.05377	0.241828	0.120914	+	0.992663i	$0.461418\pi$
$282$	0	0
$283$	−12.9237	−0.768235	−0.384118	−	0.923284i	$-0.625494\pi$
$283$	−12.9237	−0.768235	−0.384118	+	0.923284i	$0.625494\pi$
$284$	−45.3624	−2.69176
$285$	0	0
$286$	9.00407	0.532422
$287$	0	0
$288$	0	0
$289$	34.6287	2.03698
$290$	8.13085	0.477460
$291$	0	0
$292$	−26.0842	−1.52646
$293$	−23.5553	−1.37611	−0.688057	−	0.725656i	$-0.741536\pi$
$293$	−23.5553	−1.37611	−0.688057	+	0.725656i	$0.741536\pi$
$294$	0	0
$295$	8.72952	0.508252
$296$	3.46610	0.201463
$297$	0	0
$298$	−36.5553	−2.11759
$299$	−0.596700	−0.0345081
$300$	0	0
$301$	0	0
$302$	−19.3405	−1.11292
$303$	0	0
$304$	7.81669	0.448318
$305$	−6.35868	−0.364097
$306$	0	0
$307$	−19.9551	−1.13890	−0.569450	−	0.822026i	$-0.692844\pi$
$307$	−19.9551	−1.13890	−0.569450	+	0.822026i	$0.692844\pi$
$308$	0	0
$309$	0	0
$310$	15.7945	0.897068
$311$	6.48069	0.367486	0.183743	−	0.982974i	$-0.441179\pi$
$311$	6.48069	0.367486	0.183743	+	0.982974i	$0.441179\pi$
$312$	0	0
$313$	−16.1154	−0.910897	−0.455449	−	0.890262i	$-0.650521\pi$
$313$	−16.1154	−0.910897	−0.455449	+	0.890262i	$0.650521\pi$
$314$	−30.5854	−1.72603
$315$	0	0
$316$	−45.9889	−2.58708
$317$	−14.5303	−0.816106	−0.408053	−	0.912958i	$-0.633792\pi$
$317$	−14.5303	−0.816106	−0.408053	+	0.912958i	$0.633792\pi$
$318$	0	0
$319$	−13.4224	−0.751508
$320$	−7.36974	−0.411981
$321$	0	0
$322$	0	0
$323$	14.2127	0.790815
$324$	0	0
$325$	4.17138	0.231386
$326$	17.1699	0.950951
$327$	0	0
$328$	25.7223	1.42028
$329$	0	0
$330$	0	0
$331$	21.0945	1.15946	0.579730	−	0.814809i	$-0.303158\pi$
$331$	21.0945	1.15946	0.579730	+	0.814809i	$0.303158\pi$
$332$	28.5985	1.56955
$333$	0	0
$334$	11.8173	0.646615
$335$	1.11819	0.0610932
$336$	0	0
$337$	−32.8693	−1.79050	−0.895251	−	0.445562i	$-0.853004\pi$
$337$	−32.8693	−1.79050	−0.895251	+	0.445562i	$0.853004\pi$
$338$	2.44785	0.133145
$339$	0	0
$340$	26.1101	1.41602
$341$	−26.0735	−1.41196
$342$	0	0
$343$	0	0
$344$	54.0213	2.91263
$345$	0	0
$346$	−54.3725	−2.92308
$347$	−15.9160	−0.854418	−0.427209	−	0.904153i	$-0.640503\pi$
$347$	−15.9160	−0.854418	−0.427209	+	0.904153i	$0.640503\pi$
$348$	0	0
$349$	29.6245	1.58576	0.792882	−	0.609375i	$-0.208580\pi$
$349$	29.6245	1.58576	0.792882	+	0.609375i	$0.208580\pi$
$350$	0	0
$351$	0	0
$352$	0.289295	0.0154195
$353$	−28.7078	−1.52796	−0.763980	−	0.645240i	$-0.776757\pi$
$353$	−28.7078	−1.52796	−0.763980	+	0.645240i	$0.776757\pi$
$354$	0	0
$355$	−10.3440	−0.549002
$356$	51.2927	2.71851
$357$	0	0
$358$	−11.1990	−0.591887
$359$	−27.2931	−1.44048	−0.720238	−	0.693727i	$-0.755968\pi$
$359$	−27.2931	−1.44048	−0.720238	+	0.693727i	$0.755968\pi$
$360$	0	0
$361$	−15.0874	−0.794076
$362$	17.7061	0.930610
$363$	0	0
$364$	0	0
$365$	−5.94798	−0.311332
$366$	0	0
$367$	9.98326	0.521122	0.260561	−	0.965457i	$-0.416093\pi$
$367$	9.98326	0.521122	0.260561	+	0.965457i	$0.416093\pi$
$368$	2.35802	0.122921
$369$	0	0
$370$	1.58395	0.0823455
$371$	0	0
$372$	0	0
$373$	4.68608	0.242636	0.121318	−	0.992614i	$-0.461288\pi$
$373$	4.68608	0.242636	0.121318	+	0.992614i	$0.461288\pi$
$374$	−64.6971	−3.34541
$375$	0	0
$376$	59.0653	3.04606
$377$	−3.64900	−0.187933
$378$	0	0
$379$	−2.45019	−0.125858	−0.0629288	−	0.998018i	$-0.520044\pi$
$379$	−2.45019	−0.125858	−0.0629288	+	0.998018i	$0.520044\pi$
$380$	7.18777	0.368725
$381$	0	0
$382$	−4.82818	−0.247031
$383$	−10.7671	−0.550175	−0.275087	−	0.961419i	$-0.588707\pi$
$383$	−10.7671	−0.550175	−0.275087	+	0.961419i	$0.588707\pi$
$384$	0	0
$385$	0	0
$386$	−61.7035	−3.14063
$387$	0	0
$388$	−36.2893	−1.84231
$389$	−14.0372	−0.711714	−0.355857	−	0.934540i	$-0.615811\pi$
$389$	−14.0372	−0.711714	−0.355857	+	0.934540i	$0.615811\pi$
$390$	0	0
$391$	4.28748	0.216827
$392$	0	0
$393$	0	0
$394$	7.45363	0.375508
$395$	−10.4869	−0.527652
$396$	0	0
$397$	−26.4605	−1.32801	−0.664007	−	0.747726i	$-0.731146\pi$
$397$	−26.4605	−1.32801	−0.664007	+	0.747726i	$0.731146\pi$
$398$	−9.93070	−0.497781
$399$	0	0
$400$	−16.4843	−0.824217
$401$	19.0156	0.949593	0.474796	−	0.880096i	$-0.342522\pi$
$401$	19.0156	0.949593	0.474796	+	0.880096i	$0.342522\pi$
$402$	0	0
$403$	−7.08833	−0.353095
$404$	−23.1791	−1.15320
$405$	0	0
$406$	0	0
$407$	−2.61477	−0.129609
$408$	0	0
$409$	−12.7759	−0.631728	−0.315864	−	0.948804i	$-0.602294\pi$
$409$	−12.7759	−0.631728	−0.315864	+	0.948804i	$0.602294\pi$
$410$	11.7546	0.580520
$411$	0	0
$412$	−51.3693	−2.53078
$413$	0	0
$414$	0	0
$415$	6.52132	0.320119
$416$	0.0786478	0.00385603
$417$	0	0
$418$	−17.8102	−0.871127
$419$	12.9811	0.634170	0.317085	−	0.948397i	$-0.397296\pi$
$419$	12.9811	0.634170	0.317085	+	0.948397i	$0.397296\pi$
$420$	0	0
$421$	−11.6737	−0.568943	−0.284472	−	0.958684i	$-0.591818\pi$
$421$	−11.6737	−0.568943	−0.284472	+	0.958684i	$0.591818\pi$
$422$	40.1731	1.95559
$423$	0	0
$424$	55.8225	2.71098
$425$	−29.9726	−1.45389
$426$	0	0
$427$	0	0
$428$	−17.7940	−0.860107
$429$	0	0
$430$	24.6868	1.19050
$431$	0.467684	0.0225275	0.0112638	−	0.999937i	$-0.496415\pi$
$431$	0.467684	0.0225275	0.0112638	+	0.999937i	$0.496415\pi$
$432$	0	0
$433$	9.27593	0.445773	0.222886	−	0.974844i	$-0.428452\pi$
$433$	9.27593	0.445773	0.222886	+	0.974844i	$0.428452\pi$
$434$	0	0
$435$	0	0
$436$	3.50851	0.168027
$437$	1.18029	0.0564607
$438$	0	0
$439$	35.7077	1.70424	0.852119	−	0.523348i	$-0.175317\pi$
$439$	35.7077	1.70424	0.852119	+	0.523348i	$0.175317\pi$
$440$	−16.3266	−0.778340
$441$	0	0
$442$	−17.5885	−0.836601
$443$	36.3458	1.72684	0.863421	−	0.504484i	$-0.168317\pi$
$443$	36.3458	1.72684	0.863421	+	0.504484i	$0.168317\pi$
$444$	0	0
$445$	11.6963	0.554457
$446$	39.4605	1.86851
$447$	0	0
$448$	0	0
$449$	40.4910	1.91089	0.955444	−	0.295171i	$-0.0953767\pi$
$449$	40.4910	1.91089	0.955444	+	0.295171i	$0.0953767\pi$
$450$	0	0
$451$	−19.4045	−0.913720
$452$	21.4257	1.00778
$453$	0	0
$454$	−21.2425	−0.996960
$455$	0	0
$456$	0	0
$457$	5.01988	0.234820	0.117410	−	0.993084i	$-0.462541\pi$
$457$	5.01988	0.234820	0.117410	+	0.993084i	$0.462541\pi$
$458$	50.8153	2.37444
$459$	0	0
$460$	2.16830	0.101098
$461$	−17.3627	−0.808661	−0.404331	−	0.914613i	$-0.632495\pi$
$461$	−17.3627	−0.808661	−0.404331	+	0.914613i	$0.632495\pi$
$462$	0	0
$463$	35.4306	1.64660	0.823301	−	0.567606i	$-0.192130\pi$
$463$	35.4306	1.64660	0.823301	+	0.567606i	$0.192130\pi$
$464$	14.4200	0.669433
$465$	0	0
$466$	24.0983	1.11633
$467$	35.4958	1.64255	0.821275	−	0.570533i	$-0.193263\pi$
$467$	35.4958	1.64255	0.821275	+	0.570533i	$0.193263\pi$
$468$	0	0
$469$	0	0
$470$	26.9918	1.24504
$471$	0	0
$472$	46.7601	2.15231
$473$	−40.7527	−1.87381
$474$	0	0
$475$	−8.25107	−0.378585
$476$	0	0
$477$	0	0
$478$	−2.77926	−0.127121
$479$	−24.0270	−1.09782	−0.548911	−	0.835881i	$-0.684958\pi$
$479$	−24.0270	−1.09782	−0.548911	+	0.835881i	$0.684958\pi$
$480$	0	0
$481$	−0.710851	−0.0324120
$482$	53.4140	2.43294
$483$	0	0
$484$	10.1011	0.459142
$485$	−8.27506	−0.375751
$486$	0	0
$487$	20.8076	0.942880	0.471440	−	0.881898i	$-0.343734\pi$
$487$	20.8076	0.942880	0.471440	+	0.881898i	$0.343734\pi$
$488$	−34.0606	−1.54185
$489$	0	0
$490$	0	0
$491$	36.2195	1.63456	0.817281	−	0.576240i	$-0.195481\pi$
$491$	36.2195	1.63456	0.817281	+	0.576240i	$0.195481\pi$
$492$	0	0
$493$	26.2192	1.18085
$494$	−4.84189	−0.217847
$495$	0	0
$496$	28.0115	1.25775
$497$	0	0
$498$	0	0
$499$	−40.1828	−1.79883	−0.899416	−	0.437094i	$-0.856008\pi$
$499$	−40.1828	−1.79883	−0.899416	+	0.437094i	$0.856008\pi$
$500$	−33.3271	−1.49043
$501$	0	0
$502$	23.1169	1.03176
$503$	38.5636	1.71946	0.859732	−	0.510745i	$-0.170630\pi$
$503$	38.5636	1.71946	0.859732	+	0.510745i	$0.170630\pi$
$504$	0	0
$505$	−5.28553	−0.235203
$506$	−5.37273	−0.238847
$507$	0	0
$508$	26.3879	1.17078
$509$	4.33977	0.192357	0.0961785	−	0.995364i	$-0.469338\pi$
$509$	4.33977	0.192357	0.0961785	+	0.995364i	$0.469338\pi$
$510$	0	0
$511$	0	0
$512$	−38.8484	−1.71687
$513$	0	0
$514$	−8.87962	−0.391663
$515$	−11.7138	−0.516170
$516$	0	0
$517$	−44.5578	−1.95965
$518$	0	0
$519$	0	0
$520$	−4.43855	−0.194643
$521$	13.7844	0.603905	0.301952	−	0.953323i	$-0.402362\pi$
$521$	13.7844	0.603905	0.301952	+	0.953323i	$0.402362\pi$
$522$	0	0
$523$	29.4834	1.28922	0.644609	−	0.764512i	$-0.277020\pi$
$523$	29.4834	1.28922	0.644609	+	0.764512i	$0.277020\pi$
$524$	−79.7052	−3.48194
$525$	0	0
$526$	−16.1456	−0.703982
$527$	50.9318	2.21863
$528$	0	0
$529$	−22.6439	−0.984520
$530$	25.5099	1.10808
$531$	0	0
$532$	0	0
$533$	−5.27529	−0.228498
$534$	0	0
$535$	−4.05758	−0.175424
$536$	5.98963	0.258713
$537$	0	0
$538$	57.6011	2.48336
$539$	0	0
$540$	0	0
$541$	23.2519	0.999679	0.499840	−	0.866118i	$-0.333392\pi$
$541$	23.2519	0.999679	0.499840	+	0.866118i	$0.333392\pi$
$542$	−6.60348	−0.283644
$543$	0	0
$544$	−0.565109	−0.0242288
$545$	0.800047	0.0342703
$546$	0	0
$547$	1.18365	0.0506093	0.0253046	−	0.999680i	$-0.491944\pi$
$547$	1.18365	0.0506093	0.0253046	+	0.999680i	$0.491944\pi$
$548$	−22.9717	−0.981304
$549$	0	0
$550$	37.5594	1.60154
$551$	7.21780	0.307488
$552$	0	0
$553$	0	0
$554$	−60.5962	−2.57449
$555$	0	0
$556$	6.19303	0.262643
$557$	6.42824	0.272373	0.136187	−	0.990683i	$-0.456515\pi$
$557$	6.42824	0.272373	0.136187	+	0.990683i	$0.456515\pi$
$558$	0	0
$559$	−11.0790	−0.468593
$560$	0	0
$561$	0	0
$562$	9.92300	0.418577
$563$	11.4563	0.482824	0.241412	−	0.970423i	$-0.422390\pi$
$563$	11.4563	0.482824	0.241412	+	0.970423i	$0.422390\pi$
$564$	0	0
$565$	4.88571	0.205543
$566$	−31.6353	−1.32973
$567$	0	0
$568$	−55.4081	−2.32487
$569$	−38.3335	−1.60702	−0.803512	−	0.595288i	$-0.797038\pi$
$569$	−38.3335	−1.60702	−0.803512	+	0.595288i	$0.797038\pi$
$570$	0	0
$571$	6.25857	0.261913	0.130956	−	0.991388i	$-0.458195\pi$
$571$	6.25857	0.261913	0.130956	+	0.991388i	$0.458195\pi$
$572$	14.6839	0.613963
$573$	0	0
$574$	0	0
$575$	−2.48906	−0.103801
$576$	0	0
$577$	−0.0451932	−0.00188142	−0.000940708	−	1.00000i	$-0.500299\pi$
$577$	−0.0451932	−0.00188142	−0.000940708		1.00000i	$0.500299\pi$
$578$	84.7657	3.52579
$579$	0	0
$580$	13.2598	0.550583
$581$	0	0
$582$	0	0
$583$	−42.1115	−1.74408
$584$	−31.8606	−1.31840
$585$	0	0
$586$	−57.6597	−2.38190
$587$	2.71409	0.112023	0.0560113	−	0.998430i	$-0.482162\pi$
$587$	2.71409	0.112023	0.0560113	+	0.998430i	$0.482162\pi$
$588$	0	0
$589$	14.0209	0.577719
$590$	21.3685	0.879728
$591$	0	0
$592$	2.80912	0.115454
$593$	6.78615	0.278674	0.139337	−	0.990245i	$-0.455503\pi$
$593$	6.78615	0.278674	0.139337	+	0.990245i	$0.455503\pi$
$594$	0	0
$595$	0	0
$596$	−59.6145	−2.44190
$597$	0	0
$598$	−1.46063	−0.0597296
$599$	26.8337	1.09640	0.548198	−	0.836348i	$-0.315314\pi$
$599$	26.8337	1.09640	0.548198	+	0.836348i	$0.315314\pi$
$600$	0	0
$601$	−12.3356	−0.503178	−0.251589	−	0.967834i	$-0.580953\pi$
$601$	−12.3356	−0.503178	−0.251589	+	0.967834i	$0.580953\pi$
$602$	0	0
$603$	0	0
$604$	−31.5405	−1.28336
$605$	2.30336	0.0936450
$606$	0	0
$607$	17.4596	0.708665	0.354332	−	0.935120i	$-0.384708\pi$
$607$	17.4596	0.708665	0.354332	+	0.935120i	$0.384708\pi$
$608$	−0.155567	−0.00630907
$609$	0	0
$610$	−15.5651	−0.630211
$611$	−12.1135	−0.490059
$612$	0	0
$613$	1.05026	0.0424198	0.0212099	−	0.999775i	$-0.493248\pi$
$613$	1.05026	0.0424198	0.0212099	+	0.999775i	$0.493248\pi$
$614$	−48.8471	−1.97131
$615$	0	0
$616$	0	0
$617$	−10.5872	−0.426223	−0.213111	−	0.977028i	$-0.568360\pi$
$617$	−10.5872	−0.426223	−0.213111	+	0.977028i	$0.568360\pi$
$618$	0	0
$619$	−1.23805	−0.0497616	−0.0248808	−	0.999690i	$-0.507921\pi$
$619$	−1.23805	−0.0497616	−0.0248808	+	0.999690i	$0.507921\pi$
$620$	25.7577	1.03445
$621$	0	0
$622$	15.8637	0.636078
$623$	0	0
$624$	0	0
$625$	13.2573	0.530292
$626$	−39.4481	−1.57666
$627$	0	0
$628$	−49.8786	−1.99037
$629$	5.10769	0.203657
$630$	0	0
$631$	−21.2658	−0.846577	−0.423289	−	0.905995i	$-0.639124\pi$
$631$	−21.2658	−0.846577	−0.423289	+	0.905995i	$0.639124\pi$
$632$	−56.1735	−2.23446
$633$	0	0
$634$	−35.5681	−1.41259
$635$	6.01725	0.238787
$636$	0	0
$637$	0	0
$638$	−32.8559	−1.30078
$639$	0	0
$640$	−17.8968	−0.707433
$641$	−11.0506	−0.436474	−0.218237	−	0.975896i	$-0.570031\pi$
$641$	−11.0506	−0.436474	−0.218237	+	0.975896i	$0.570031\pi$
$642$	0	0
$643$	−6.50321	−0.256461	−0.128231	−	0.991744i	$-0.540930\pi$
$643$	−6.50321	−0.256461	−0.128231	+	0.991744i	$0.540930\pi$
$644$	0	0
$645$	0	0
$646$	34.7905	1.36881
$647$	−0.893941	−0.0351444	−0.0175722	−	0.999846i	$-0.505594\pi$
$647$	−0.893941	−0.0351444	−0.0175722	+	0.999846i	$0.505594\pi$
$648$	0	0
$649$	−35.2750	−1.38467
$650$	10.2109	0.400504
$651$	0	0
$652$	28.0006	1.09659
$653$	38.5477	1.50849	0.754244	−	0.656594i	$-0.228003\pi$
$653$	38.5477	1.50849	0.754244	+	0.656594i	$0.228003\pi$
$654$	0	0
$655$	−18.1752	−0.710164
$656$	20.8468	0.813930
$657$	0	0
$658$	0	0
$659$	10.8013	0.420759	0.210380	−	0.977620i	$-0.432530\pi$
$659$	10.8013	0.420759	0.210380	+	0.977620i	$0.432530\pi$
$660$	0	0
$661$	−40.6339	−1.58047	−0.790237	−	0.612801i	$-0.790043\pi$
$661$	−40.6339	−1.58047	−0.790237	+	0.612801i	$0.790043\pi$
$662$	51.6361	2.00690
$663$	0	0
$664$	34.9318	1.35562
$665$	0	0
$666$	0	0
$667$	2.17736	0.0843077
$668$	19.2717	0.745645
$669$	0	0
$670$	2.73716	0.105746
$671$	25.6947	0.991934
$672$	0	0
$673$	32.3136	1.24560	0.622799	−	0.782382i	$-0.285995\pi$
$673$	32.3136	1.24560	0.622799	+	0.782382i	$0.285995\pi$
$674$	−80.4589	−3.09916
$675$	0	0
$676$	3.99195	0.153537
$677$	−25.2158	−0.969123	−0.484562	−	0.874757i	$-0.661021\pi$
$677$	−25.2158	−0.969123	−0.484562	+	0.874757i	$0.661021\pi$
$678$	0	0
$679$	0	0
$680$	31.8923	1.22302
$681$	0	0
$682$	−63.8238	−2.44394
$683$	6.28731	0.240577	0.120289	−	0.992739i	$-0.461618\pi$
$683$	6.28731	0.240577	0.120289	+	0.992739i	$0.461618\pi$
$684$	0	0
$685$	−5.23826	−0.200143
$686$	0	0
$687$	0	0
$688$	43.7818	1.66917
$689$	−11.4484	−0.436151
$690$	0	0
$691$	−17.1531	−0.652533	−0.326267	−	0.945278i	$-0.605791\pi$
$691$	−17.1531	−0.652533	−0.326267	+	0.945278i	$0.605791\pi$
$692$	−88.6707	−3.37076
$693$	0	0
$694$	−38.9600	−1.47890
$695$	1.41220	0.0535678
$696$	0	0
$697$	37.9046	1.43574
$698$	72.5162	2.74478
$699$	0	0
$700$	0	0
$701$	23.6620	0.893702	0.446851	−	0.894609i	$-0.352545\pi$
$701$	23.6620	0.893702	0.446851	+	0.894609i	$0.352545\pi$
$702$	0	0
$703$	1.40608	0.0530312
$704$	29.7803	1.12239
$705$	0	0
$706$	−70.2722	−2.64473
$707$	0	0
$708$	0	0
$709$	−21.5030	−0.807563	−0.403782	−	0.914855i	$-0.632304\pi$
$709$	−21.5030	−0.807563	−0.403782	+	0.914855i	$0.632304\pi$
$710$	−25.3205	−0.950262
$711$	0	0
$712$	62.6517	2.34797
$713$	4.22961	0.158400
$714$	0	0
$715$	3.34837	0.125222
$716$	−18.2634	−0.682535
$717$	0	0
$718$	−66.8094	−2.49330
$719$	−12.8693	−0.479943	−0.239971	−	0.970780i	$-0.577138\pi$
$719$	−12.8693	−0.479943	−0.239971	+	0.970780i	$0.577138\pi$
$720$	0	0
$721$	0	0
$722$	−36.9317	−1.37446
$723$	0	0
$724$	28.8751	1.07313
$725$	−15.2214	−0.565307
$726$	0	0
$727$	16.4329	0.609463	0.304732	−	0.952438i	$-0.401433\pi$
$727$	16.4329	0.609463	0.304732	+	0.952438i	$0.401433\pi$
$728$	0	0
$729$	0	0
$730$	−14.5597	−0.538880
$731$	79.6063	2.94434
$732$	0	0
$733$	−21.7222	−0.802328	−0.401164	−	0.916006i	$-0.631394\pi$
$733$	−21.7222	−0.802328	−0.401164	+	0.916006i	$0.631394\pi$
$734$	24.4375	0.902004
$735$	0	0
$736$	−0.0469292	−0.00172983
$737$	−4.51848	−0.166440
$738$	0	0
$739$	1.58001	0.0581218	0.0290609	−	0.999578i	$-0.490748\pi$
$739$	1.58001	0.0581218	0.0290609	+	0.999578i	$0.490748\pi$
$740$	2.58310	0.0949568
$741$	0	0
$742$	0	0
$743$	−45.9718	−1.68654	−0.843271	−	0.537489i	$-0.819373\pi$
$743$	−45.9718	−1.68654	−0.843271	+	0.537489i	$0.819373\pi$
$744$	0	0
$745$	−13.5939	−0.498043
$746$	11.4708	0.419976
$747$	0	0
$748$	−105.508	−3.85776
$749$	0	0
$750$	0	0
$751$	−6.42286	−0.234373	−0.117187	−	0.993110i	$-0.537388\pi$
$751$	−6.42286	−0.234373	−0.117187	+	0.993110i	$0.537388\pi$
$752$	47.8698	1.74563
$753$	0	0
$754$	−8.93219	−0.325291
$755$	−7.19219	−0.261750
$756$	0	0
$757$	−6.10016	−0.221714	−0.110857	−	0.993836i	$-0.535360\pi$
$757$	−6.10016	−0.221714	−0.110857	+	0.993836i	$0.535360\pi$
$758$	−5.99768	−0.217846
$759$	0	0
$760$	8.77954	0.318467
$761$	18.3948	0.666812	0.333406	−	0.942783i	$-0.391802\pi$
$761$	18.3948	0.666812	0.333406	+	0.942783i	$0.391802\pi$
$762$	0	0
$763$	0	0
$764$	−7.87380	−0.284864
$765$	0	0
$766$	−26.3563	−0.952291
$767$	−9.58986	−0.346270
$768$	0	0
$769$	24.1850	0.872133	0.436066	−	0.899914i	$-0.356371\pi$
$769$	24.1850	0.872133	0.436066	+	0.899914i	$0.356371\pi$
$770$	0	0
$771$	0	0
$772$	−100.626	−3.62162
$773$	−10.7601	−0.387014	−0.193507	−	0.981099i	$-0.561986\pi$
$773$	−10.7601	−0.387014	−0.193507	+	0.981099i	$0.561986\pi$
$774$	0	0
$775$	−29.5681	−1.06212
$776$	−44.3258	−1.59120
$777$	0	0
$778$	−34.3609	−1.23190
$779$	10.4346	0.373860
$780$	0	0
$781$	41.7989	1.49568
$782$	10.4951	0.375303
$783$	0	0
$784$	0	0
$785$	−11.3738	−0.405950
$786$	0	0
$787$	14.7348	0.525238	0.262619	−	0.964900i	$-0.415414\pi$
$787$	14.7348	0.525238	0.262619	+	0.964900i	$0.415414\pi$
$788$	12.1554	0.433018
$789$	0	0
$790$	−25.6703	−0.913307
$791$	0	0
$792$	0	0
$793$	6.98536	0.248058
$794$	−64.7713	−2.29865
$795$	0	0
$796$	−16.1950	−0.574017
$797$	32.5732	1.15380	0.576901	−	0.816814i	$-0.304262\pi$
$797$	32.5732	1.15380	0.576901	+	0.816814i	$0.304262\pi$
$798$	0	0
$799$	87.0392	3.07922
$800$	0.328070	0.0115990
$801$	0	0
$802$	46.5472	1.64364
$803$	24.0351	0.848181
$804$	0	0
$805$	0	0
$806$	−17.3511	−0.611168
$807$	0	0
$808$	−28.3122	−0.996019
$809$	23.6926	0.832987	0.416493	−	0.909139i	$-0.363259\pi$
$809$	23.6926	0.832987	0.416493	+	0.909139i	$0.363259\pi$
$810$	0	0
$811$	−45.8568	−1.61025	−0.805125	−	0.593105i	$-0.797902\pi$
$811$	−45.8568	−1.61025	−0.805125	+	0.593105i	$0.797902\pi$
$812$	0	0
$813$	0	0
$814$	−6.40056	−0.224339
$815$	6.38500	0.223657
$816$	0	0
$817$	21.9145	0.766693
$818$	−31.2735	−1.09345
$819$	0	0
$820$	19.1695	0.669427
$821$	−13.2923	−0.463905	−0.231953	−	0.972727i	$-0.574511\pi$
$821$	−13.2923	−0.463905	−0.231953	+	0.972727i	$0.574511\pi$
$822$	0	0
$823$	−4.13033	−0.143974	−0.0719871	−	0.997406i	$-0.522934\pi$
$823$	−4.13033	−0.143974	−0.0719871	+	0.997406i	$0.522934\pi$
$824$	−62.7453	−2.18584
$825$	0	0
$826$	0	0
$827$	4.67317	0.162502	0.0812510	−	0.996694i	$-0.474108\pi$
$827$	4.67317	0.162502	0.0812510	+	0.996694i	$0.474108\pi$
$828$	0	0
$829$	−1.74971	−0.0607700	−0.0303850	−	0.999538i	$-0.509673\pi$
$829$	−1.74971	−0.0607700	−0.0303850	+	0.999538i	$0.509673\pi$
$830$	15.9632	0.554091
$831$	0	0
$832$	8.09606	0.280681
$833$	0	0
$834$	0	0
$835$	4.39454	0.152079
$836$	−29.0450	−1.00454
$837$	0	0
$838$	31.7758	1.09768
$839$	−15.4495	−0.533374	−0.266687	−	0.963783i	$-0.585929\pi$
$839$	−15.4495	−0.533374	−0.266687	+	0.963783i	$0.585929\pi$
$840$	0	0
$841$	−15.6848	−0.540855
$842$	−28.5755	−0.984777
$843$	0	0
$844$	65.5143	2.25510
$845$	0.910286	0.0313148
$846$	0	0
$847$	0	0
$848$	45.2416	1.55360
$849$	0	0
$850$	−73.3684	−2.51652
$851$	0.424165	0.0145402
$852$	0	0
$853$	0.602575	0.0206318	0.0103159	−	0.999947i	$-0.496716\pi$
$853$	0.602575	0.0206318	0.0103159	+	0.999947i	$0.496716\pi$
$854$	0	0
$855$	0	0
$856$	−21.7346	−0.742874
$857$	−43.1639	−1.47445	−0.737226	−	0.675647i	$-0.763865\pi$
$857$	−43.1639	−1.47445	−0.737226	+	0.675647i	$0.763865\pi$
$858$	0	0
$859$	−37.9408	−1.29452	−0.647261	−	0.762269i	$-0.724086\pi$
$859$	−37.9408	−1.29452	−0.647261	+	0.762269i	$0.724086\pi$
$860$	40.2592	1.37283
$861$	0	0
$862$	1.14482	0.0389927
$863$	−33.9067	−1.15420	−0.577098	−	0.816675i	$-0.695815\pi$
$863$	−33.9067	−1.15420	−0.577098	+	0.816675i	$0.695815\pi$
$864$	0	0
$865$	−20.2196	−0.687488
$866$	22.7061	0.771583
$867$	0	0
$868$	0	0
$869$	42.3763	1.43752
$870$	0	0
$871$	−1.22839	−0.0416225
$872$	4.28549	0.145125
$873$	0	0
$874$	2.88916	0.0977272
$875$	0	0
$876$	0	0
$877$	6.51036	0.219839	0.109920	−	0.993940i	$-0.464941\pi$
$877$	6.51036	0.219839	0.109920	+	0.993940i	$0.464941\pi$
$878$	87.4071	2.94985
$879$	0	0
$880$	−13.2320	−0.446050
$881$	−24.5160	−0.825966	−0.412983	−	0.910739i	$-0.635513\pi$
$881$	−24.5160	−0.825966	−0.412983	+	0.910739i	$0.635513\pi$
$882$	0	0
$883$	−28.7175	−0.966419	−0.483210	−	0.875505i	$-0.660529\pi$
$883$	−28.7175	−0.966419	−0.483210	+	0.875505i	$0.660529\pi$
$884$	−28.6834	−0.964727
$885$	0	0
$886$	88.9690	2.98897
$887$	6.94442	0.233171	0.116585	−	0.993181i	$-0.462805\pi$
$887$	6.94442	0.233171	0.116585	+	0.993181i	$0.462805\pi$
$888$	0	0
$889$	0	0
$890$	28.6307	0.959704
$891$	0	0
$892$	64.3523	2.15467
$893$	23.9607	0.801815
$894$	0	0
$895$	−4.16461	−0.139208
$896$	0	0
$897$	0	0
$898$	99.1158	3.30754
$899$	25.8653	0.862656
$900$	0	0
$901$	82.2605	2.74050
$902$	−47.4991	−1.58155
$903$	0	0
$904$	26.1705	0.870419
$905$	6.58439	0.218873
$906$	0	0
$907$	0.229549	0.00762204	0.00381102	−	0.999993i	$-0.498787\pi$
$907$	0.229549	0.00762204	0.00381102	+	0.999993i	$0.498787\pi$
$908$	−34.6423	−1.14965
$909$	0	0
$910$	0	0
$911$	26.6727	0.883706	0.441853	−	0.897087i	$-0.354321\pi$
$911$	26.6727	0.883706	0.441853	+	0.897087i	$0.354321\pi$
$912$	0	0
$913$	−26.3519	−0.872122
$914$	12.2879	0.406447
$915$	0	0
$916$	82.8696	2.73809
$917$	0	0
$918$	0	0
$919$	−43.2331	−1.42613	−0.713064	−	0.701099i	$-0.752693\pi$
$919$	−43.2331	−1.42613	−0.713064	+	0.701099i	$0.752693\pi$
$920$	2.64848	0.0873179
$921$	0	0
$922$	−42.5012	−1.39970
$923$	11.3635	0.374033
$924$	0	0
$925$	−2.96523	−0.0974961
$926$	86.7287	2.85008
$927$	0	0
$928$	−0.286986	−0.00942077
$929$	−23.8491	−0.782465	−0.391233	−	0.920292i	$-0.627951\pi$
$929$	−23.8491	−0.782465	−0.391233	+	0.920292i	$0.627951\pi$
$930$	0	0
$931$	0	0
$932$	39.2995	1.28730
$933$	0	0
$934$	86.8883	2.84307
$935$	−24.0590	−0.786815
$936$	0	0
$937$	27.2033	0.888694	0.444347	−	0.895855i	$-0.353436\pi$
$937$	27.2033	0.888694	0.444347	+	0.895855i	$0.353436\pi$
$938$	0	0
$939$	0	0
$940$	44.0182	1.43572
$941$	13.4515	0.438505	0.219253	−	0.975668i	$-0.429638\pi$
$941$	13.4515	0.438505	0.219253	+	0.975668i	$0.429638\pi$
$942$	0	0
$943$	3.14777	0.102505
$944$	37.8970	1.23344
$945$	0	0
$946$	−99.7564	−3.24336
$947$	−15.0341	−0.488543	−0.244271	−	0.969707i	$-0.578549\pi$
$947$	−15.0341	−0.488543	−0.244271	+	0.969707i	$0.578549\pi$
$948$	0	0
$949$	6.53419	0.212109
$950$	−20.1974	−0.655289
$951$	0	0
$952$	0	0
$953$	−28.3775	−0.919237	−0.459618	−	0.888116i	$-0.652014\pi$
$953$	−28.3775	−0.919237	−0.459618	+	0.888116i	$0.652014\pi$
$954$	0	0
$955$	−1.79546	−0.0580999
$956$	−4.53243	−0.146589
$957$	0	0
$958$	−58.8144	−1.90021
$959$	0	0
$960$	0	0
$961$	19.2444	0.620787
$962$	−1.74005	−0.0561016
$963$	0	0
$964$	87.1077	2.80555
$965$	−22.9458	−0.738653
$966$	0	0
$967$	−52.0994	−1.67540	−0.837701	−	0.546129i	$-0.816101\pi$
$967$	−52.0994	−1.67540	−0.837701	+	0.546129i	$0.816101\pi$
$968$	12.3381	0.396561
$969$	0	0
$970$	−20.2561	−0.650384
$971$	−20.8418	−0.668845	−0.334423	−	0.942423i	$-0.608541\pi$
$971$	−20.8418	−0.668845	−0.334423	+	0.942423i	$0.608541\pi$
$972$	0	0
$973$	0	0
$974$	50.9337	1.63202
$975$	0	0
$976$	−27.6046	−0.883601
$977$	−34.6302	−1.10792	−0.553960	−	0.832544i	$-0.686884\pi$
$977$	−34.6302	−1.10792	−0.553960	+	0.832544i	$0.686884\pi$
$978$	0	0
$979$	−47.2634	−1.51054
$980$	0	0
$981$	0	0
$982$	88.6597	2.82924
$983$	−58.8628	−1.87743	−0.938717	−	0.344690i	$-0.887984\pi$
$983$	−58.8628	−1.87743	−0.938717	+	0.344690i	$0.887984\pi$
$984$	0	0
$985$	2.77180	0.0883168
$986$	64.1806	2.04393
$987$	0	0
$988$	−7.89616	−0.251210
$989$	6.61086	0.210213
$990$	0	0
$991$	−29.3747	−0.933118	−0.466559	−	0.884490i	$-0.654507\pi$
$991$	−29.3747	−0.933118	−0.466559	+	0.884490i	$0.654507\pi$
$992$	−0.557482	−0.0177001
$993$	0	0
$994$	0	0
$995$	−3.69295	−0.117075
$996$	0	0
$997$	7.50606	0.237719	0.118860	−	0.992911i	$-0.462076\pi$
$997$	7.50606	0.237719	0.118860	+	0.992911i	$0.462076\pi$
$998$	−98.3614	−3.11358
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bu.1.6		6
3.2	odd	2		637.2.a.m.1.1	✓	6
7.6	odd	2		5733.2.a.br.1.6		6
21.2	odd	6		637.2.e.o.508.6		12
21.5	even	6		637.2.e.n.508.6		12
21.11	odd	6		637.2.e.o.79.6		12
21.17	even	6		637.2.e.n.79.6		12
21.20	even	2		637.2.a.n.1.1	yes	6
39.38	odd	2		8281.2.a.cc.1.6		6
273.272	even	2		8281.2.a.cd.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.1	✓	6	3.2	odd	2
637.2.a.n.1.1	yes	6	21.20	even	2
637.2.e.n.79.6		12	21.17	even	6
637.2.e.n.508.6		12	21.5	even	6
637.2.e.o.79.6		12	21.11	odd	6
637.2.e.o.508.6		12	21.2	odd	6
5733.2.a.br.1.6		6	7.6	odd	2
5733.2.a.bu.1.6		6	1.1	even	1	trivial
8281.2.a.cc.1.6		6	39.38	odd	2
8281.2.a.cd.1.6		6	273.272	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.44785 q^{2} +3.99195 q^{4} +0.910286 q^{5} +4.87599 q^{8} +O(q^{10})\)	\(q+2.44785 q^{2} +3.99195 q^{4} +0.910286 q^{5} +4.87599 q^{8} +2.22824 q^{10} -3.67837 q^{11} -1.00000 q^{13} +3.95177 q^{16} +7.18531 q^{17} +1.97802 q^{19} +3.63382 q^{20} -9.00407 q^{22} +0.596700 q^{23} -4.17138 q^{25} -2.44785 q^{26} +3.64900 q^{29} +7.08833 q^{31} -0.0786478 q^{32} +17.5885 q^{34} +0.710851 q^{37} +4.84189 q^{38} +4.43855 q^{40} +5.27529 q^{41} +11.0790 q^{43} -14.6839 q^{44} +1.46063 q^{46} +12.1135 q^{47} -10.2109 q^{50} -3.99195 q^{52} +11.4484 q^{53} -3.34837 q^{55} +8.93219 q^{58} +9.58986 q^{59} -6.98536 q^{61} +17.3511 q^{62} -8.09606 q^{64} -0.910286 q^{65} +1.22839 q^{67} +28.6834 q^{68} -11.3635 q^{71} -6.53419 q^{73} +1.74005 q^{74} +7.89616 q^{76} -11.5204 q^{79} +3.59725 q^{80} +12.9131 q^{82} +7.16403 q^{83} +6.54069 q^{85} +27.1198 q^{86} -17.9357 q^{88} +12.8490 q^{89} +2.38200 q^{92} +29.6520 q^{94} +1.80056 q^{95} -9.09062 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 6 q^{5}+O(q^{10}) \) 6 * q + 4 * q^4 + 6 * q^5	\( 6 q + 4 q^{4} + 6 q^{5} - 4 q^{10} - 4 q^{11} - 6 q^{13} + 16 q^{17} - 2 q^{19} + 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} - 6 q^{31} + 20 q^{32} + 8 q^{38} - 4 q^{40} - 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} + 30 q^{47} - 8 q^{50} - 4 q^{52} + 14 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} + 28 q^{68} - 8 q^{71} + 6 q^{73} + 12 q^{74} + 16 q^{76} - 22 q^{79} - 28 q^{80} + 40 q^{82} + 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} + 26 q^{89} - 20 q^{92} + 32 q^{94} + 6 q^{95} + 14 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 + 6 * q^5 - 4 * q^10 - 4 * q^11 - 6 * q^13 + 16 * q^17 - 2 * q^19 + 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 - 6 * q^31 + 20 * q^32 + 8 * q^38 - 4 * q^40 - 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 + 30 * q^47 - 8 * q^50 - 4 * q^52 + 14 * q^53 + 8 * q^55 - 8 * q^58 + 24 * q^59 + 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 + 28 * q^68 - 8 * q^71 + 6 * q^73 + 12 * q^74 + 16 * q^76 - 22 * q^79 - 28 * q^80 + 40 * q^82 + 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 + 26 * q^89 - 20 * q^92 + 32 * q^94 + 6 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more