LMFDB - Embedded newform 6039.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.76243 q^{2} +1.10617 q^{4} -2.60834 q^{5} +0.445474 q^{7} +1.57532 q^{8} +O(q^{10})$	\(q-1.76243 q^{2} +1.10617 q^{4} -2.60834 q^{5} +0.445474 q^{7} +1.57532 q^{8} +4.59703 q^{10} +1.00000 q^{11} +0.460729 q^{13} -0.785118 q^{14} -4.98873 q^{16} +7.17973 q^{17} -1.30151 q^{19} -2.88526 q^{20} -1.76243 q^{22} +7.00186 q^{23} +1.80346 q^{25} -0.812004 q^{26} +0.492768 q^{28} +4.54077 q^{29} -3.93161 q^{31} +5.64165 q^{32} -12.6538 q^{34} -1.16195 q^{35} +10.6095 q^{37} +2.29382 q^{38} -4.10898 q^{40} +4.26931 q^{41} -0.692654 q^{43} +1.10617 q^{44} -12.3403 q^{46} +2.26311 q^{47} -6.80155 q^{49} -3.17848 q^{50} +0.509643 q^{52} +4.25208 q^{53} -2.60834 q^{55} +0.701765 q^{56} -8.00279 q^{58} +0.781791 q^{59} -1.00000 q^{61} +6.92919 q^{62} +0.0344327 q^{64} -1.20174 q^{65} +2.09190 q^{67} +7.94198 q^{68} +2.04786 q^{70} +4.03237 q^{71} -3.26985 q^{73} -18.6985 q^{74} -1.43968 q^{76} +0.445474 q^{77} -1.20571 q^{79} +13.0123 q^{80} -7.52436 q^{82} -3.69137 q^{83} -18.7272 q^{85} +1.22075 q^{86} +1.57532 q^{88} -3.24777 q^{89} +0.205243 q^{91} +7.74522 q^{92} -3.98857 q^{94} +3.39478 q^{95} +6.26891 q^{97} +11.9873 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.76243	−1.24623	−0.623114	−	0.782131i	$-0.714133\pi$
$2$	−1.76243	−1.24623	−0.623114	+	0.782131i	$0.714133\pi$
$3$	0	0
$3$	0	0
$4$	1.10617	0.553083
$5$	−2.60834	−1.16649	−0.583244	−	0.812297i	$-0.698217\pi$
$5$	−2.60834	−1.16649	−0.583244	+	0.812297i	$0.698217\pi$
$6$	0	0
$7$	0.445474	0.168373	0.0841867	−	0.996450i	$-0.473171\pi$
$7$	0.445474	0.168373	0.0841867	+	0.996450i	$0.473171\pi$
$8$	1.57532	0.556960
$9$	0	0
$10$	4.59703	1.45371
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.460729	0.127783	0.0638917	−	0.997957i	$-0.479649\pi$
$13$	0.460729	0.127783	0.0638917	+	0.997957i	$0.479649\pi$
$14$	−0.785118	−0.209832
$15$	0	0
$16$	−4.98873	−1.24718
$17$	7.17973	1.74134	0.870671	−	0.491866i	$-0.163685\pi$
$17$	7.17973	1.74134	0.870671	+	0.491866i	$0.163685\pi$
$18$	0	0
$19$	−1.30151	−0.298586	−0.149293	−	0.988793i	$-0.547700\pi$
$19$	−1.30151	−0.298586	−0.149293	+	0.988793i	$0.547700\pi$
$20$	−2.88526	−0.645164
$21$	0	0
$22$	−1.76243	−0.375752
$23$	7.00186	1.45999	0.729995	−	0.683453i	$-0.239523\pi$
$23$	7.00186	1.45999	0.729995	+	0.683453i	$0.239523\pi$
$24$	0	0
$25$	1.80346	0.360692
$26$	−0.812004	−0.159247
$27$	0	0
$28$	0.492768	0.0931245
$29$	4.54077	0.843199	0.421600	−	0.906782i	$-0.361469\pi$
$29$	4.54077	0.843199	0.421600	+	0.906782i	$0.361469\pi$
$30$	0	0
$31$	−3.93161	−0.706137	−0.353069	−	0.935597i	$-0.614862\pi$
$31$	−3.93161	−0.706137	−0.353069	+	0.935597i	$0.614862\pi$
$32$	5.64165	0.997312
$33$	0	0
$34$	−12.6538	−2.17011
$35$	−1.16195	−0.196405
$36$	0	0
$37$	10.6095	1.74419	0.872095	−	0.489337i	$-0.162761\pi$
$37$	10.6095	1.74419	0.872095	+	0.489337i	$0.162761\pi$
$38$	2.29382	0.372106
$39$	0	0
$40$	−4.10898	−0.649687
$41$	4.26931	0.666754	0.333377	−	0.942794i	$-0.391812\pi$
$41$	4.26931	0.666754	0.333377	+	0.942794i	$0.391812\pi$
$42$	0	0
$43$	−0.692654	−0.105629	−0.0528143	−	0.998604i	$-0.516819\pi$
$43$	−0.692654	−0.105629	−0.0528143	+	0.998604i	$0.516819\pi$
$44$	1.10617	0.166761
$45$	0	0
$46$	−12.3403	−1.81948
$47$	2.26311	0.330108	0.165054	−	0.986285i	$-0.447220\pi$
$47$	2.26311	0.330108	0.165054	+	0.986285i	$0.447220\pi$
$48$	0	0
$49$	−6.80155	−0.971650
$50$	−3.17848	−0.449504
$51$	0	0
$52$	0.509643	0.0706748
$53$	4.25208	0.584068	0.292034	−	0.956408i	$-0.405668\pi$
$53$	4.25208	0.584068	0.292034	+	0.956408i	$0.405668\pi$
$54$	0	0
$55$	−2.60834	−0.351709
$56$	0.701765	0.0937773
$57$	0	0
$58$	−8.00279	−1.05082
$59$	0.781791	0.101780	0.0508902	−	0.998704i	$-0.483794\pi$
$59$	0.781791	0.101780	0.0508902	+	0.998704i	$0.483794\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	6.92919	0.880008
$63$	0	0
$64$	0.0344327	0.00430408
$65$	−1.20174	−0.149058
$66$	0	0
$67$	2.09190	0.255567	0.127783	−	0.991802i	$-0.459214\pi$
$67$	2.09190	0.255567	0.127783	+	0.991802i	$0.459214\pi$
$68$	7.94198	0.963106
$69$	0	0
$70$	2.04786	0.244766
$71$	4.03237	0.478554	0.239277	−	0.970951i	$-0.423090\pi$
$71$	4.03237	0.478554	0.239277	+	0.970951i	$0.423090\pi$
$72$	0	0
$73$	−3.26985	−0.382707	−0.191353	−	0.981521i	$-0.561288\pi$
$73$	−3.26985	−0.382707	−0.191353	+	0.981521i	$0.561288\pi$
$74$	−18.6985	−2.17366
$75$	0	0
$76$	−1.43968	−0.165143
$77$	0.445474	0.0507665
$78$	0	0
$79$	−1.20571	−0.135653	−0.0678267	−	0.997697i	$-0.521606\pi$
$79$	−1.20571	−0.135653	−0.0678267	+	0.997697i	$0.521606\pi$
$80$	13.0123	1.45482
$81$	0	0
$82$	−7.52436	−0.830927
$83$	−3.69137	−0.405180	−0.202590	−	0.979264i	$-0.564936\pi$
$83$	−3.69137	−0.405180	−0.202590	+	0.979264i	$0.564936\pi$
$84$	0	0
$85$	−18.7272	−2.03125
$86$	1.22075	0.131637
$87$	0	0
$88$	1.57532	0.167930
$89$	−3.24777	−0.344263	−0.172131	−	0.985074i	$-0.555065\pi$
$89$	−3.24777	−0.344263	−0.172131	+	0.985074i	$0.555065\pi$
$90$	0	0
$91$	0.205243	0.0215153
$92$	7.74522	0.807495
$93$	0	0
$94$	−3.98857	−0.411390
$95$	3.39478	0.348297
$96$	0	0
$97$	6.26891	0.636511	0.318256	−	0.948005i	$-0.396903\pi$
$97$	6.26891	0.636511	0.318256	+	0.948005i	$0.396903\pi$
$98$	11.9873	1.21090
$99$	0	0
$100$	1.99493	0.199493
$101$	−11.2406	−1.11849	−0.559243	−	0.829004i	$-0.688908\pi$
$101$	−11.2406	−1.11849	−0.559243	+	0.829004i	$0.688908\pi$
$102$	0	0
$103$	7.78924	0.767497	0.383748	−	0.923438i	$-0.374633\pi$
$103$	7.78924	0.767497	0.383748	+	0.923438i	$0.374633\pi$
$104$	0.725797	0.0711702
$105$	0	0
$106$	−7.49400	−0.727882
$107$	−10.3243	−0.998087	−0.499043	−	0.866577i	$-0.666315\pi$
$107$	−10.3243	−0.998087	−0.499043	+	0.866577i	$0.666315\pi$
$108$	0	0
$109$	−1.01114	−0.0968495	−0.0484248	−	0.998827i	$-0.515420\pi$
$109$	−1.01114	−0.0968495	−0.0484248	+	0.998827i	$0.515420\pi$
$110$	4.59703	0.438310
$111$	0	0
$112$	−2.22235	−0.209992
$113$	7.45592	0.701394	0.350697	−	0.936489i	$-0.385945\pi$
$113$	7.45592	0.701394	0.350697	+	0.936489i	$0.385945\pi$
$114$	0	0
$115$	−18.2633	−1.70306
$116$	5.02284	0.466359
$117$	0	0
$118$	−1.37785	−0.126842
$119$	3.19839	0.293196
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.76243	0.159563
$123$	0	0
$124$	−4.34901	−0.390552
$125$	8.33768	0.745744
$126$	0	0
$127$	8.28012	0.734742	0.367371	−	0.930075i	$-0.380258\pi$
$127$	8.28012	0.734742	0.367371	+	0.930075i	$0.380258\pi$
$128$	−11.3440	−1.00268
$129$	0	0
$130$	2.11799	0.185760
$131$	−12.3029	−1.07491	−0.537454	−	0.843293i	$-0.680614\pi$
$131$	−12.3029	−1.07491	−0.537454	+	0.843293i	$0.680614\pi$
$132$	0	0
$133$	−0.579787	−0.0502740
$134$	−3.68684	−0.318494
$135$	0	0
$136$	11.3104	0.969858
$137$	9.42223	0.804995	0.402498	−	0.915421i	$-0.368142\pi$
$137$	9.42223	0.804995	0.402498	+	0.915421i	$0.368142\pi$
$138$	0	0
$139$	0.907470	0.0769706	0.0384853	−	0.999259i	$-0.487747\pi$
$139$	0.907470	0.0769706	0.0384853	+	0.999259i	$0.487747\pi$
$140$	−1.28531	−0.108629
$141$	0	0
$142$	−7.10678	−0.596387
$143$	0.460729	0.0385281
$144$	0	0
$145$	−11.8439	−0.983581
$146$	5.76288	0.476940
$147$	0	0
$148$	11.7359	0.964681
$149$	−3.25268	−0.266470	−0.133235	−	0.991084i	$-0.542536\pi$
$149$	−3.25268	−0.266470	−0.133235	+	0.991084i	$0.542536\pi$
$150$	0	0
$151$	−1.00609	−0.0818744	−0.0409372	−	0.999162i	$-0.513034\pi$
$151$	−1.00609	−0.0818744	−0.0409372	+	0.999162i	$0.513034\pi$
$152$	−2.05029	−0.166301
$153$	0	0
$154$	−0.785118	−0.0632666
$155$	10.2550	0.823700
$156$	0	0
$157$	−9.25586	−0.738698	−0.369349	−	0.929291i	$-0.620419\pi$
$157$	−9.25586	−0.738698	−0.369349	+	0.929291i	$0.620419\pi$
$158$	2.12499	0.169055
$159$	0	0
$160$	−14.7154	−1.16335
$161$	3.11915	0.245823
$162$	0	0
$163$	−6.70380	−0.525082	−0.262541	−	0.964921i	$-0.584560\pi$
$163$	−6.70380	−0.525082	−0.262541	+	0.964921i	$0.584560\pi$
$164$	4.72256	0.368770
$165$	0	0
$166$	6.50578	0.504947
$167$	−16.7005	−1.29232	−0.646162	−	0.763201i	$-0.723627\pi$
$167$	−16.7005	−1.29232	−0.646162	+	0.763201i	$0.723627\pi$
$168$	0	0
$169$	−12.7877	−0.983671
$170$	33.0054	2.53140
$171$	0	0
$172$	−0.766190	−0.0584214
$173$	−1.12829	−0.0857821	−0.0428911	−	0.999080i	$-0.513657\pi$
$173$	−1.12829	−0.0857821	−0.0428911	+	0.999080i	$0.513657\pi$
$174$	0	0
$175$	0.803395	0.0607310
$176$	−4.98873	−0.376040
$177$	0	0
$178$	5.72397	0.429030
$179$	−10.6083	−0.792903	−0.396452	−	0.918056i	$-0.629759\pi$
$179$	−10.6083	−0.792903	−0.396452	+	0.918056i	$0.629759\pi$
$180$	0	0
$181$	5.53432	0.411363	0.205682	−	0.978619i	$-0.434059\pi$
$181$	5.53432	0.411363	0.205682	+	0.978619i	$0.434059\pi$
$182$	−0.361727	−0.0268130
$183$	0	0
$184$	11.0302	0.813156
$185$	−27.6732	−2.03457
$186$	0	0
$187$	7.17973	0.525034
$188$	2.50337	0.182577
$189$	0	0
$190$	−5.98306	−0.434057
$191$	−19.4440	−1.40692	−0.703458	−	0.710736i	$-0.748362\pi$
$191$	−19.4440	−1.40692	−0.703458	+	0.710736i	$0.748362\pi$
$192$	0	0
$193$	24.1671	1.73958	0.869792	−	0.493419i	$-0.164253\pi$
$193$	24.1671	1.73958	0.869792	+	0.493419i	$0.164253\pi$
$194$	−11.0485	−0.793238
$195$	0	0
$196$	−7.52365	−0.537403
$197$	22.5089	1.60370	0.801848	−	0.597528i	$-0.203850\pi$
$197$	22.5089	1.60370	0.801848	+	0.597528i	$0.203850\pi$
$198$	0	0
$199$	−3.51237	−0.248985	−0.124492	−	0.992221i	$-0.539730\pi$
$199$	−3.51237	−0.248985	−0.124492	+	0.992221i	$0.539730\pi$
$200$	2.84103	0.200891
$201$	0	0
$202$	19.8109	1.39389
$203$	2.02280	0.141972
$204$	0	0
$205$	−11.1358	−0.777760
$206$	−13.7280	−0.956476
$207$	0	0
$208$	−2.29845	−0.159369
$209$	−1.30151	−0.0900271
$210$	0	0
$211$	−23.4578	−1.61490	−0.807450	−	0.589937i	$-0.799153\pi$
$211$	−23.4578	−1.61490	−0.807450	+	0.589937i	$0.799153\pi$
$212$	4.70350	0.323038
$213$	0	0
$214$	18.1959	1.24384
$215$	1.80668	0.123214
$216$	0	0
$217$	−1.75143	−0.118895
$218$	1.78206	0.120697
$219$	0	0
$220$	−2.88526	−0.194524
$221$	3.30791	0.222514
$222$	0	0
$223$	−11.9825	−0.802406	−0.401203	−	0.915989i	$-0.631408\pi$
$223$	−11.9825	−0.802406	−0.401203	+	0.915989i	$0.631408\pi$
$224$	2.51321	0.167921
$225$	0	0
$226$	−13.1405	−0.874096
$227$	−3.89455	−0.258491	−0.129245	−	0.991613i	$-0.541255\pi$
$227$	−3.89455	−0.258491	−0.129245	+	0.991613i	$0.541255\pi$
$228$	0	0
$229$	−7.59707	−0.502029	−0.251014	−	0.967983i	$-0.580764\pi$
$229$	−7.59707	−0.502029	−0.251014	+	0.967983i	$0.580764\pi$
$230$	32.1878	2.12240
$231$	0	0
$232$	7.15317	0.469629
$233$	22.7328	1.48927	0.744636	−	0.667470i	$-0.232623\pi$
$233$	22.7328	1.48927	0.744636	+	0.667470i	$0.232623\pi$
$234$	0	0
$235$	−5.90296	−0.385067
$236$	0.864790	0.0562930
$237$	0	0
$238$	−5.63694	−0.365388
$239$	−20.1933	−1.30619	−0.653097	−	0.757274i	$-0.726531\pi$
$239$	−20.1933	−1.30619	−0.653097	+	0.757274i	$0.726531\pi$
$240$	0	0
$241$	7.53201	0.485179	0.242590	−	0.970129i	$-0.422003\pi$
$241$	7.53201	0.485179	0.242590	+	0.970129i	$0.422003\pi$
$242$	−1.76243	−0.113293
$243$	0	0
$244$	−1.10617	−0.0708150
$245$	17.7408	1.13342
$246$	0	0
$247$	−0.599642	−0.0381543
$248$	−6.19354	−0.393290
$249$	0	0
$250$	−14.6946	−0.929367
$251$	22.5266	1.42186	0.710932	−	0.703261i	$-0.248273\pi$
$251$	22.5266	1.42186	0.710932	+	0.703261i	$0.248273\pi$
$252$	0	0
$253$	7.00186	0.440203
$254$	−14.5931	−0.915655
$255$	0	0
$256$	19.9241	1.24526
$257$	14.5502	0.907613	0.453807	−	0.891100i	$-0.350066\pi$
$257$	14.5502	0.907613	0.453807	+	0.891100i	$0.350066\pi$
$258$	0	0
$259$	4.72625	0.293675
$260$	−1.32932	−0.0824412
$261$	0	0
$262$	21.6830	1.33958
$263$	17.1189	1.05560	0.527798	−	0.849370i	$-0.323018\pi$
$263$	17.1189	1.05560	0.527798	+	0.849370i	$0.323018\pi$
$264$	0	0
$265$	−11.0909	−0.681308
$266$	1.02184	0.0626528
$267$	0	0
$268$	2.31399	0.141349
$269$	0.796908	0.0485883	0.0242942	−	0.999705i	$-0.492266\pi$
$269$	0.796908	0.0485883	0.0242942	+	0.999705i	$0.492266\pi$
$270$	0	0
$271$	29.4136	1.78675	0.893373	−	0.449315i	$-0.148332\pi$
$271$	29.4136	1.78675	0.893373	+	0.449315i	$0.148332\pi$
$272$	−35.8177	−2.17177
$273$	0	0
$274$	−16.6060	−1.00321
$275$	1.80346	0.108753
$276$	0	0
$277$	−1.17118	−0.0703694	−0.0351847	−	0.999381i	$-0.511202\pi$
$277$	−1.17118	−0.0703694	−0.0351847	+	0.999381i	$0.511202\pi$
$278$	−1.59935	−0.0959229
$279$	0	0
$280$	−1.83045	−0.109390
$281$	−1.00109	−0.0597200	−0.0298600	−	0.999554i	$-0.509506\pi$
$281$	−1.00109	−0.0597200	−0.0298600	+	0.999554i	$0.509506\pi$
$282$	0	0
$283$	2.68772	0.159768	0.0798842	−	0.996804i	$-0.474545\pi$
$283$	2.68772	0.159768	0.0798842	+	0.996804i	$0.474545\pi$
$284$	4.46047	0.264680
$285$	0	0
$286$	−0.812004	−0.0480148
$287$	1.90187	0.112264
$288$	0	0
$289$	34.5486	2.03227
$290$	20.8740	1.22577
$291$	0	0
$292$	−3.61699	−0.211669
$293$	16.4929	0.963524	0.481762	−	0.876302i	$-0.339997\pi$
$293$	16.4929	0.963524	0.481762	+	0.876302i	$0.339997\pi$
$294$	0	0
$295$	−2.03918	−0.118726
$296$	16.7134	0.971444
$297$	0	0
$298$	5.73263	0.332082
$299$	3.22596	0.186562
$300$	0	0
$301$	−0.308559	−0.0177851
$302$	1.77316	0.102034
$303$	0	0
$304$	6.49286	0.372391
$305$	2.60834	0.149353
$306$	0	0
$307$	5.36231	0.306043	0.153022	−	0.988223i	$-0.451100\pi$
$307$	5.36231	0.306043	0.153022	+	0.988223i	$0.451100\pi$
$308$	0.492768	0.0280781
$309$	0	0
$310$	−18.0737	−1.02652
$311$	18.3626	1.04125	0.520623	−	0.853787i	$-0.325700\pi$
$311$	18.3626	1.04125	0.520623	+	0.853787i	$0.325700\pi$
$312$	0	0
$313$	−8.11837	−0.458877	−0.229439	−	0.973323i	$-0.573689\pi$
$313$	−8.11837	−0.458877	−0.229439	+	0.973323i	$0.573689\pi$
$314$	16.3128	0.920586
$315$	0	0
$316$	−1.33372	−0.0750275
$317$	1.09740	0.0616359	0.0308179	−	0.999525i	$-0.490189\pi$
$317$	1.09740	0.0616359	0.0308179	+	0.999525i	$0.490189\pi$
$318$	0	0
$319$	4.54077	0.254234
$320$	−0.0898123	−0.00502066
$321$	0	0
$322$	−5.49729	−0.306352
$323$	−9.34447	−0.519940
$324$	0	0
$325$	0.830907	0.0460904
$326$	11.8150	0.654371
$327$	0	0
$328$	6.72553	0.371355
$329$	1.00816	0.0555814
$330$	0	0
$331$	−18.3957	−1.01112	−0.505560	−	0.862791i	$-0.668714\pi$
$331$	−18.3957	−1.01112	−0.505560	+	0.862791i	$0.668714\pi$
$332$	−4.08326	−0.224098
$333$	0	0
$334$	29.4335	1.61053
$335$	−5.45640	−0.298115
$336$	0	0
$337$	16.5185	0.899819	0.449909	−	0.893074i	$-0.351456\pi$
$337$	16.5185	0.899819	0.449909	+	0.893074i	$0.351456\pi$
$338$	22.5375	1.22588
$339$	0	0
$340$	−20.7154	−1.12345
$341$	−3.93161	−0.212908
$342$	0	0
$343$	−6.14824	−0.331974
$344$	−1.09115	−0.0588310
$345$	0	0
$346$	1.98853	0.106904
$347$	23.2401	1.24759	0.623797	−	0.781586i	$-0.285589\pi$
$347$	23.2401	1.24759	0.623797	+	0.781586i	$0.285589\pi$
$348$	0	0
$349$	−6.86990	−0.367737	−0.183869	−	0.982951i	$-0.558862\pi$
$349$	−6.86990	−0.367737	−0.183869	+	0.982951i	$0.558862\pi$
$350$	−1.41593	−0.0756846
$351$	0	0
$352$	5.64165	0.300701
$353$	9.86711	0.525173	0.262587	−	0.964908i	$-0.415424\pi$
$353$	9.86711	0.525173	0.262587	+	0.964908i	$0.415424\pi$
$354$	0	0
$355$	−10.5178	−0.558227
$356$	−3.59257	−0.190406
$357$	0	0
$358$	18.6964	0.988138
$359$	2.52570	0.133301	0.0666507	−	0.997776i	$-0.478769\pi$
$359$	2.52570	0.133301	0.0666507	+	0.997776i	$0.478769\pi$
$360$	0	0
$361$	−17.3061	−0.910846
$362$	−9.75387	−0.512652
$363$	0	0
$364$	0.227033	0.0118998
$365$	8.52889	0.446422
$366$	0	0
$367$	−15.2502	−0.796055	−0.398027	−	0.917374i	$-0.630305\pi$
$367$	−15.2502	−0.796055	−0.398027	+	0.917374i	$0.630305\pi$
$368$	−34.9304	−1.82087
$369$	0	0
$370$	48.7721	2.53554
$371$	1.89419	0.0983415
$372$	0	0
$373$	−9.94971	−0.515177	−0.257588	−	0.966255i	$-0.582928\pi$
$373$	−9.94971	−0.515177	−0.257588	+	0.966255i	$0.582928\pi$
$374$	−12.6538	−0.654312
$375$	0	0
$376$	3.56512	0.183857
$377$	2.09206	0.107747
$378$	0	0
$379$	3.14074	0.161329	0.0806645	−	0.996741i	$-0.474296\pi$
$379$	3.14074	0.161329	0.0806645	+	0.996741i	$0.474296\pi$
$380$	3.75519	0.192637
$381$	0	0
$382$	34.2687	1.75334
$383$	14.9584	0.764341	0.382170	−	0.924092i	$-0.375177\pi$
$383$	14.9584	0.764341	0.382170	+	0.924092i	$0.375177\pi$
$384$	0	0
$385$	−1.16195	−0.0592185
$386$	−42.5928	−2.16792
$387$	0	0
$388$	6.93445	0.352044
$389$	25.0810	1.27166	0.635829	−	0.771830i	$-0.280658\pi$
$389$	25.0810	1.27166	0.635829	+	0.771830i	$0.280658\pi$
$390$	0	0
$391$	50.2715	2.54234
$392$	−10.7146	−0.541171
$393$	0	0
$394$	−39.6705	−1.99857
$395$	3.14491	0.158238
$396$	0	0
$397$	−28.6070	−1.43574	−0.717871	−	0.696176i	$-0.754883\pi$
$397$	−28.6070	−1.43574	−0.717871	+	0.696176i	$0.754883\pi$
$398$	6.19030	0.310292
$399$	0	0
$400$	−8.99698	−0.449849
$401$	−31.5367	−1.57487	−0.787433	−	0.616401i	$-0.788590\pi$
$401$	−31.5367	−1.57487	−0.787433	+	0.616401i	$0.788590\pi$
$402$	0	0
$403$	−1.81141	−0.0902326
$404$	−12.4340	−0.618615
$405$	0	0
$406$	−3.56504	−0.176930
$407$	10.6095	0.525893
$408$	0	0
$409$	−25.8163	−1.27653	−0.638267	−	0.769815i	$-0.720348\pi$
$409$	−25.8163	−1.27653	−0.638267	+	0.769815i	$0.720348\pi$
$410$	19.6261	0.969266
$411$	0	0
$412$	8.61619	0.424489
$413$	0.348268	0.0171371
$414$	0	0
$415$	9.62836	0.472637
$416$	2.59927	0.127440
$417$	0	0
$418$	2.29382	0.112194
$419$	26.6984	1.30430	0.652152	−	0.758088i	$-0.273866\pi$
$419$	26.6984	1.30430	0.652152	+	0.758088i	$0.273866\pi$
$420$	0	0
$421$	2.77548	0.135268	0.0676342	−	0.997710i	$-0.478455\pi$
$421$	2.77548	0.135268	0.0676342	+	0.997710i	$0.478455\pi$
$422$	41.3427	2.01253
$423$	0	0
$424$	6.69839	0.325303
$425$	12.9484	0.628088
$426$	0	0
$427$	−0.445474	−0.0215580
$428$	−11.4204	−0.552025
$429$	0	0
$430$	−3.18415	−0.153553
$431$	−1.46124	−0.0703855	−0.0351928	−	0.999381i	$-0.511205\pi$
$431$	−1.46124	−0.0703855	−0.0351928	+	0.999381i	$0.511205\pi$
$432$	0	0
$433$	11.1600	0.536318	0.268159	−	0.963375i	$-0.413585\pi$
$433$	11.1600	0.536318	0.268159	+	0.963375i	$0.413585\pi$
$434$	3.08677	0.148170
$435$	0	0
$436$	−1.11849	−0.0535658
$437$	−9.11297	−0.435932
$438$	0	0
$439$	10.6867	0.510048	0.255024	−	0.966935i	$-0.417917\pi$
$439$	10.6867	0.510048	0.255024	+	0.966935i	$0.417917\pi$
$440$	−4.10898	−0.195888
$441$	0	0
$442$	−5.82997	−0.277304
$443$	2.78268	0.132209	0.0661046	−	0.997813i	$-0.478943\pi$
$443$	2.78268	0.132209	0.0661046	+	0.997813i	$0.478943\pi$
$444$	0	0
$445$	8.47130	0.401578
$446$	21.1183	0.999981
$447$	0	0
$448$	0.0153389	0.000724693 0
$449$	22.2839	1.05164	0.525821	−	0.850595i	$-0.323758\pi$
$449$	22.2839	1.05164	0.525821	+	0.850595i	$0.323758\pi$
$450$	0	0
$451$	4.26931	0.201034
$452$	8.24748	0.387929
$453$	0	0
$454$	6.86389	0.322138
$455$	−0.535345	−0.0250973
$456$	0	0
$457$	7.73630	0.361889	0.180944	−	0.983493i	$-0.442085\pi$
$457$	7.73630	0.361889	0.180944	+	0.983493i	$0.442085\pi$
$458$	13.3893	0.625642
$459$	0	0
$460$	−20.2022	−0.941933
$461$	2.48702	0.115832	0.0579160	−	0.998321i	$-0.481554\pi$
$461$	2.48702	0.115832	0.0579160	+	0.998321i	$0.481554\pi$
$462$	0	0
$463$	6.47557	0.300945	0.150473	−	0.988614i	$-0.451920\pi$
$463$	6.47557	0.300945	0.150473	+	0.988614i	$0.451920\pi$
$464$	−22.6527	−1.05162
$465$	0	0
$466$	−40.0649	−1.85597
$467$	0.203369	0.00941078	0.00470539	−	0.999989i	$-0.498502\pi$
$467$	0.203369	0.00941078	0.00470539	+	0.999989i	$0.498502\pi$
$468$	0	0
$469$	0.931889	0.0430306
$470$	10.4036	0.479881
$471$	0	0
$472$	1.23157	0.0566877
$473$	−0.692654	−0.0318482
$474$	0	0
$475$	−2.34722	−0.107698
$476$	3.53795	0.162162
$477$	0	0
$478$	35.5893	1.62782
$479$	17.7842	0.812580	0.406290	−	0.913744i	$-0.366822\pi$
$479$	17.7842	0.812580	0.406290	+	0.913744i	$0.366822\pi$
$480$	0	0
$481$	4.88810	0.222878
$482$	−13.2746	−0.604644
$483$	0	0
$484$	1.10617	0.0502803
$485$	−16.3515	−0.742482
$486$	0	0
$487$	24.6431	1.11669	0.558344	−	0.829610i	$-0.311437\pi$
$487$	24.6431	1.11669	0.558344	+	0.829610i	$0.311437\pi$
$488$	−1.57532	−0.0713115
$489$	0	0
$490$	−31.2669	−1.41250
$491$	−9.00021	−0.406174	−0.203087	−	0.979161i	$-0.565097\pi$
$491$	−9.00021	−0.406174	−0.203087	+	0.979161i	$0.565097\pi$
$492$	0	0
$493$	32.6015	1.46830
$494$	1.05683	0.0475490
$495$	0	0
$496$	19.6137	0.880682
$497$	1.79632	0.0805758
$498$	0	0
$499$	43.5832	1.95105	0.975527	−	0.219881i	$-0.0705670\pi$
$499$	43.5832	1.95105	0.975527	+	0.219881i	$0.0705670\pi$
$500$	9.22285	0.412458
$501$	0	0
$502$	−39.7015	−1.77197
$503$	−14.3711	−0.640775	−0.320387	−	0.947287i	$-0.603813\pi$
$503$	−14.3711	−0.640775	−0.320387	+	0.947287i	$0.603813\pi$
$504$	0	0
$505$	29.3195	1.30470
$506$	−12.3403	−0.548594
$507$	0	0
$508$	9.15918	0.406373
$509$	−8.58744	−0.380632	−0.190316	−	0.981723i	$-0.560951\pi$
$509$	−8.58744	−0.380632	−0.190316	+	0.981723i	$0.560951\pi$
$510$	0	0
$511$	−1.45663	−0.0644376
$512$	−12.4270	−0.549199
$513$	0	0
$514$	−25.6436	−1.13109
$515$	−20.3170	−0.895275
$516$	0	0
$517$	2.26311	0.0995313
$518$	−8.32970	−0.365986
$519$	0	0
$520$	−1.89313	−0.0830192
$521$	−30.6885	−1.34449	−0.672243	−	0.740331i	$-0.734669\pi$
$521$	−30.6885	−1.34449	−0.672243	+	0.740331i	$0.734669\pi$
$522$	0	0
$523$	−36.5838	−1.59970	−0.799849	−	0.600202i	$-0.795087\pi$
$523$	−36.5838	−1.59970	−0.799849	+	0.600202i	$0.795087\pi$
$524$	−13.6090	−0.594514
$525$	0	0
$526$	−30.1709	−1.31551
$527$	−28.2279	−1.22963
$528$	0	0
$529$	26.0261	1.13157
$530$	19.5469	0.849064
$531$	0	0
$532$	−0.641341	−0.0278057
$533$	1.96700	0.0852000
$534$	0	0
$535$	26.9293	1.16426
$536$	3.29542	0.142340
$537$	0	0
$538$	−1.40450	−0.0605521
$539$	−6.80155	−0.292964
$540$	0	0
$541$	−14.3466	−0.616809	−0.308405	−	0.951255i	$-0.599795\pi$
$541$	−14.3466	−0.616809	−0.308405	+	0.951255i	$0.599795\pi$
$542$	−51.8394	−2.22669
$543$	0	0
$544$	40.5056	1.73666
$545$	2.63740	0.112974
$546$	0	0
$547$	24.5634	1.05025	0.525127	−	0.851024i	$-0.324018\pi$
$547$	24.5634	1.05025	0.525127	+	0.851024i	$0.324018\pi$
$548$	10.4225	0.445229
$549$	0	0
$550$	−3.17848	−0.135531
$551$	−5.90984	−0.251767
$552$	0	0
$553$	−0.537114	−0.0228404
$554$	2.06413	0.0876963
$555$	0	0
$556$	1.00381	0.0425711
$557$	15.4512	0.654690	0.327345	−	0.944905i	$-0.393846\pi$
$557$	15.4512	0.654690	0.327345	+	0.944905i	$0.393846\pi$
$558$	0	0
$559$	−0.319126	−0.0134976
$560$	5.79666	0.244953
$561$	0	0
$562$	1.76435	0.0744248
$563$	6.81794	0.287342	0.143671	−	0.989626i	$-0.454109\pi$
$563$	6.81794	0.287342	0.143671	+	0.989626i	$0.454109\pi$
$564$	0	0
$565$	−19.4476	−0.818167
$566$	−4.73692	−0.199108
$567$	0	0
$568$	6.35228	0.266536
$569$	−38.7091	−1.62277	−0.811385	−	0.584512i	$-0.801286\pi$
$569$	−38.7091	−1.62277	−0.811385	+	0.584512i	$0.801286\pi$
$570$	0	0
$571$	26.5728	1.11204	0.556019	−	0.831170i	$-0.312328\pi$
$571$	26.5728	1.11204	0.556019	+	0.831170i	$0.312328\pi$
$572$	0.509643	0.0213092
$573$	0	0
$574$	−3.35191	−0.139906
$575$	12.6276	0.526607
$576$	0	0
$577$	−1.53413	−0.0638666	−0.0319333	−	0.999490i	$-0.510166\pi$
$577$	−1.53413	−0.0638666	−0.0319333	+	0.999490i	$0.510166\pi$
$578$	−60.8895	−2.53267
$579$	0	0
$580$	−13.1013	−0.544002
$581$	−1.64441	−0.0682216
$582$	0	0
$583$	4.25208	0.176103
$584$	−5.15106	−0.213152
$585$	0	0
$586$	−29.0676	−1.20077
$587$	27.6498	1.14123	0.570614	−	0.821218i	$-0.306705\pi$
$587$	27.6498	1.14123	0.570614	+	0.821218i	$0.306705\pi$
$588$	0	0
$589$	5.11701	0.210843
$590$	3.59391	0.147959
$591$	0	0
$592$	−52.9279	−2.17532
$593$	−28.4469	−1.16817	−0.584087	−	0.811691i	$-0.698547\pi$
$593$	−28.4469	−1.16817	−0.584087	+	0.811691i	$0.698547\pi$
$594$	0	0
$595$	−8.34250	−0.342009
$596$	−3.59800	−0.147380
$597$	0	0
$598$	−5.68554	−0.232499
$599$	6.49331	0.265309	0.132655	−	0.991162i	$-0.457650\pi$
$599$	6.49331	0.265309	0.132655	+	0.991162i	$0.457650\pi$
$600$	0	0
$601$	14.1177	0.575874	0.287937	−	0.957649i	$-0.407031\pi$
$601$	14.1177	0.575874	0.287937	+	0.957649i	$0.407031\pi$
$602$	0.543815	0.0221642
$603$	0	0
$604$	−1.11290	−0.0452833
$605$	−2.60834	−0.106044
$606$	0	0
$607$	−5.44041	−0.220820	−0.110410	−	0.993886i	$-0.535216\pi$
$607$	−5.44041	−0.220820	−0.110410	+	0.993886i	$0.535216\pi$
$608$	−7.34264	−0.297783
$609$	0	0
$610$	−4.59703	−0.186128
$611$	1.04268	0.0421823
$612$	0	0
$613$	−0.745272	−0.0301012	−0.0150506	−	0.999887i	$-0.504791\pi$
$613$	−0.745272	−0.0301012	−0.0150506	+	0.999887i	$0.504791\pi$
$614$	−9.45070	−0.381399
$615$	0	0
$616$	0.701765	0.0282749
$617$	14.4259	0.580764	0.290382	−	0.956911i	$-0.406218\pi$
$617$	14.4259	0.580764	0.290382	+	0.956911i	$0.406218\pi$
$618$	0	0
$619$	15.6601	0.629434	0.314717	−	0.949186i	$-0.398090\pi$
$619$	15.6601	0.629434	0.314717	+	0.949186i	$0.398090\pi$
$620$	11.3437	0.455574
$621$	0	0
$622$	−32.3628	−1.29763
$623$	−1.44680	−0.0579648
$624$	0	0
$625$	−30.7648	−1.23059
$626$	14.3081	0.571866
$627$	0	0
$628$	−10.2385	−0.408561
$629$	76.1733	3.03723
$630$	0	0
$631$	13.1654	0.524106	0.262053	−	0.965053i	$-0.415600\pi$
$631$	13.1654	0.524106	0.262053	+	0.965053i	$0.415600\pi$
$632$	−1.89939	−0.0755535
$633$	0	0
$634$	−1.93408	−0.0768123
$635$	−21.5974	−0.857067
$636$	0	0
$637$	−3.13368	−0.124161
$638$	−8.00279	−0.316834
$639$	0	0
$640$	29.5890	1.16961
$641$	33.8595	1.33737	0.668686	−	0.743545i	$-0.266857\pi$
$641$	33.8595	1.33737	0.668686	+	0.743545i	$0.266857\pi$
$642$	0	0
$643$	3.58819	0.141505	0.0707523	−	0.997494i	$-0.477460\pi$
$643$	3.58819	0.141505	0.0707523	+	0.997494i	$0.477460\pi$
$644$	3.45030	0.135961
$645$	0	0
$646$	16.4690	0.647964
$647$	−5.86243	−0.230476	−0.115238	−	0.993338i	$-0.536763\pi$
$647$	−5.86243	−0.230476	−0.115238	+	0.993338i	$0.536763\pi$
$648$	0	0
$649$	0.781791	0.0306880
$650$	−1.46442	−0.0574392
$651$	0	0
$652$	−7.41551	−0.290414
$653$	−35.7177	−1.39774	−0.698872	−	0.715247i	$-0.746314\pi$
$653$	−35.7177	−1.39774	−0.698872	+	0.715247i	$0.746314\pi$
$654$	0	0
$655$	32.0902	1.25387
$656$	−21.2984	−0.831564
$657$	0	0
$658$	−1.77681	−0.0692671
$659$	33.0938	1.28915	0.644575	−	0.764541i	$-0.277034\pi$
$659$	33.0938	1.28915	0.644575	+	0.764541i	$0.277034\pi$
$660$	0	0
$661$	12.8098	0.498244	0.249122	−	0.968472i	$-0.419858\pi$
$661$	12.8098	0.498244	0.249122	+	0.968472i	$0.419858\pi$
$662$	32.4212	1.26009
$663$	0	0
$664$	−5.81509	−0.225669
$665$	1.51229	0.0586439
$666$	0	0
$667$	31.7938	1.23106
$668$	−18.4735	−0.714762
$669$	0	0
$670$	9.61654	0.371519
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	25.5029	0.983065	0.491532	−	0.870859i	$-0.336437\pi$
$673$	25.5029	0.983065	0.491532	+	0.870859i	$0.336437\pi$
$674$	−29.1127	−1.12138
$675$	0	0
$676$	−14.1453	−0.544052
$677$	−4.20164	−0.161482	−0.0807410	−	0.996735i	$-0.525729\pi$
$677$	−4.20164	−0.161482	−0.0807410	+	0.996735i	$0.525729\pi$
$678$	0	0
$679$	2.79264	0.107172
$680$	−29.5014	−1.13133
$681$	0	0
$682$	6.92919	0.265332
$683$	32.8180	1.25575	0.627873	−	0.778316i	$-0.283926\pi$
$683$	32.8180	1.25575	0.627873	+	0.778316i	$0.283926\pi$
$684$	0	0
$685$	−24.5764	−0.939016
$686$	10.8358	0.413715
$687$	0	0
$688$	3.45546	0.131738
$689$	1.95906	0.0746342
$690$	0	0
$691$	12.7344	0.484440	0.242220	−	0.970221i	$-0.422124\pi$
$691$	12.7344	0.484440	0.242220	+	0.970221i	$0.422124\pi$
$692$	−1.24807	−0.0474446
$693$	0	0
$694$	−40.9591	−1.55479
$695$	−2.36700	−0.0897852
$696$	0	0
$697$	30.6525	1.16105
$698$	12.1077	0.458284
$699$	0	0
$700$	0.888689	0.0335893
$701$	−4.55473	−0.172030	−0.0860149	−	0.996294i	$-0.527413\pi$
$701$	−4.55473	−0.172030	−0.0860149	+	0.996294i	$0.527413\pi$
$702$	0	0
$703$	−13.8083	−0.520790
$704$	0.0344327	0.00129773
$705$	0	0
$706$	−17.3901	−0.654485
$707$	−5.00742	−0.188323
$708$	0	0
$709$	3.45236	0.129656	0.0648280	−	0.997896i	$-0.479350\pi$
$709$	3.45236	0.129656	0.0648280	+	0.997896i	$0.479350\pi$
$710$	18.5369	0.695678
$711$	0	0
$712$	−5.11628	−0.191741
$713$	−27.5286	−1.03095
$714$	0	0
$715$	−1.20174	−0.0449426
$716$	−11.7346	−0.438541
$717$	0	0
$718$	−4.45138	−0.166124
$719$	41.8676	1.56140	0.780699	−	0.624907i	$-0.214863\pi$
$719$	41.8676	1.56140	0.780699	+	0.624907i	$0.214863\pi$
$720$	0	0
$721$	3.46991	0.129226
$722$	30.5008	1.13512
$723$	0	0
$724$	6.12188	0.227518
$725$	8.18910	0.304135
$726$	0	0
$727$	7.39909	0.274417	0.137208	−	0.990542i	$-0.456187\pi$
$727$	7.39909	0.274417	0.137208	+	0.990542i	$0.456187\pi$
$728$	0.323324	0.0119832
$729$	0	0
$730$	−15.0316	−0.556344
$731$	−4.97307	−0.183936
$732$	0	0
$733$	−17.6047	−0.650245	−0.325123	−	0.945672i	$-0.605406\pi$
$733$	−17.6047	−0.650245	−0.325123	+	0.945672i	$0.605406\pi$
$734$	26.8775	0.992065
$735$	0	0
$736$	39.5021	1.45607
$737$	2.09190	0.0770562
$738$	0	0
$739$	35.9313	1.32175	0.660877	−	0.750494i	$-0.270184\pi$
$739$	35.9313	1.32175	0.660877	+	0.750494i	$0.270184\pi$
$740$	−30.6111	−1.12529
$741$	0	0
$742$	−3.33838	−0.122556
$743$	16.3515	0.599877	0.299938	−	0.953959i	$-0.403034\pi$
$743$	16.3515	0.599877	0.299938	+	0.953959i	$0.403034\pi$
$744$	0	0
$745$	8.48411	0.310834
$746$	17.5357	0.642027
$747$	0	0
$748$	7.94198	0.290387
$749$	−4.59921	−0.168051
$750$	0	0
$751$	−24.8600	−0.907154	−0.453577	−	0.891217i	$-0.649852\pi$
$751$	−24.8600	−0.907154	−0.453577	+	0.891217i	$0.649852\pi$
$752$	−11.2900	−0.411705
$753$	0	0
$754$	−3.68712	−0.134277
$755$	2.62423	0.0955054
$756$	0	0
$757$	−3.75712	−0.136555	−0.0682773	−	0.997666i	$-0.521750\pi$
$757$	−3.75712	−0.136555	−0.0682773	+	0.997666i	$0.521750\pi$
$758$	−5.53534	−0.201053
$759$	0	0
$760$	5.34786	0.193987
$761$	14.4823	0.524982	0.262491	−	0.964934i	$-0.415456\pi$
$761$	14.4823	0.524982	0.262491	+	0.964934i	$0.415456\pi$
$762$	0	0
$763$	−0.450436	−0.0163069
$764$	−21.5083	−0.778142
$765$	0	0
$766$	−26.3632	−0.952542
$767$	0.360194	0.0130058
$768$	0	0
$769$	−14.3649	−0.518011	−0.259006	−	0.965876i	$-0.583395\pi$
$769$	−14.3649	−0.518011	−0.259006	+	0.965876i	$0.583395\pi$
$770$	2.04786	0.0737997
$771$	0	0
$772$	26.7328	0.962134
$773$	14.5855	0.524605	0.262303	−	0.964986i	$-0.415518\pi$
$773$	14.5855	0.524605	0.262303	+	0.964986i	$0.415518\pi$
$774$	0	0
$775$	−7.09050	−0.254698
$776$	9.87555	0.354512
$777$	0	0
$778$	−44.2036	−1.58478
$779$	−5.55653	−0.199083
$780$	0	0
$781$	4.03237	0.144290
$782$	−88.6001	−3.16833
$783$	0	0
$784$	33.9311	1.21183
$785$	24.1425	0.861682
$786$	0	0
$787$	−22.6545	−0.807546	−0.403773	−	0.914859i	$-0.632301\pi$
$787$	−22.6545	−0.807546	−0.403773	+	0.914859i	$0.632301\pi$
$788$	24.8986	0.886977
$789$	0	0
$790$	−5.54270	−0.197200
$791$	3.32142	0.118096
$792$	0	0
$793$	−0.460729	−0.0163610
$794$	50.4178	1.78926
$795$	0	0
$796$	−3.88526	−0.137709
$797$	−44.4677	−1.57513	−0.787563	−	0.616234i	$-0.788658\pi$
$797$	−44.4677	−1.57513	−0.787563	+	0.616234i	$0.788658\pi$
$798$	0	0
$799$	16.2485	0.574831
$800$	10.1745	0.359723
$801$	0	0
$802$	55.5812	1.96264
$803$	−3.26985	−0.115390
$804$	0	0
$805$	−8.13582	−0.286750
$806$	3.19248	0.112450
$807$	0	0
$808$	−17.7076	−0.622952
$809$	40.3698	1.41933	0.709663	−	0.704541i	$-0.248847\pi$
$809$	40.3698	1.41933	0.709663	+	0.704541i	$0.248847\pi$
$810$	0	0
$811$	−1.99934	−0.0702064	−0.0351032	−	0.999384i	$-0.511176\pi$
$811$	−1.99934	−0.0702064	−0.0351032	+	0.999384i	$0.511176\pi$
$812$	2.23755	0.0785225
$813$	0	0
$814$	−18.6985	−0.655382
$815$	17.4858	0.612501
$816$	0	0
$817$	0.901493	0.0315392
$818$	45.4995	1.59085
$819$	0	0
$820$	−12.3181	−0.430166
$821$	16.9238	0.590644	0.295322	−	0.955398i	$-0.404573\pi$
$821$	16.9238	0.590644	0.295322	+	0.955398i	$0.404573\pi$
$822$	0	0
$823$	7.45641	0.259914	0.129957	−	0.991520i	$-0.458516\pi$
$823$	7.45641	0.259914	0.129957	+	0.991520i	$0.458516\pi$
$824$	12.2706	0.427465
$825$	0	0
$826$	−0.613798	−0.0213568
$827$	11.6244	0.404219	0.202109	−	0.979363i	$-0.435220\pi$
$827$	11.6244	0.404219	0.202109	+	0.979363i	$0.435220\pi$
$828$	0	0
$829$	−7.90717	−0.274627	−0.137314	−	0.990528i	$-0.543847\pi$
$829$	−7.90717	−0.274627	−0.137314	+	0.990528i	$0.543847\pi$
$830$	−16.9693	−0.589014
$831$	0	0
$832$	0.0158641	0.000549990 0
$833$	−48.8333	−1.69197
$834$	0	0
$835$	43.5606	1.50748
$836$	−1.43968	−0.0497924
$837$	0	0
$838$	−47.0542	−1.62546
$839$	8.94069	0.308667	0.154333	−	0.988019i	$-0.450677\pi$
$839$	8.94069	0.308667	0.154333	+	0.988019i	$0.450677\pi$
$840$	0	0
$841$	−8.38143	−0.289015
$842$	−4.89159	−0.168575
$843$	0	0
$844$	−25.9482	−0.893173
$845$	33.3548	1.14744
$846$	0	0
$847$	0.445474	0.0153067
$848$	−21.2125	−0.728439
$849$	0	0
$850$	−22.8206	−0.782741
$851$	74.2862	2.54650
$852$	0	0
$853$	−24.4661	−0.837705	−0.418852	−	0.908054i	$-0.637568\pi$
$853$	−24.4661	−0.837705	−0.418852	+	0.908054i	$0.637568\pi$
$854$	0.785118	0.0268662
$855$	0	0
$856$	−16.2641	−0.555895
$857$	−11.8837	−0.405938	−0.202969	−	0.979185i	$-0.565059\pi$
$857$	−11.8837	−0.405938	−0.202969	+	0.979185i	$0.565059\pi$
$858$	0	0
$859$	31.3070	1.06818	0.534091	−	0.845427i	$-0.320654\pi$
$859$	31.3070	1.06818	0.534091	+	0.845427i	$0.320654\pi$
$860$	1.99849	0.0681478
$861$	0	0
$862$	2.57534	0.0877164
$863$	8.82728	0.300484	0.150242	−	0.988649i	$-0.451995\pi$
$863$	8.82728	0.300484	0.150242	+	0.988649i	$0.451995\pi$
$864$	0	0
$865$	2.94296	0.100064
$866$	−19.6688	−0.668374
$867$	0	0
$868$	−1.93737	−0.0657587
$869$	−1.20571	−0.0409010
$870$	0	0
$871$	0.963801	0.0326571
$872$	−1.59287	−0.0539413
$873$	0	0
$874$	16.0610	0.543271
$875$	3.71422	0.125564
$876$	0	0
$877$	44.4877	1.50224	0.751121	−	0.660165i	$-0.229513\pi$
$877$	44.4877	1.50224	0.751121	+	0.660165i	$0.229513\pi$
$878$	−18.8346	−0.635636
$879$	0	0
$880$	13.0123	0.438645
$881$	19.5868	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$881$	19.5868	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$882$	0	0
$883$	−16.7337	−0.563133	−0.281566	−	0.959542i	$-0.590854\pi$
$883$	−16.7337	−0.563133	−0.281566	+	0.959542i	$0.590854\pi$
$884$	3.65910	0.123069
$885$	0	0
$886$	−4.90429	−0.164763
$887$	−31.3926	−1.05406	−0.527029	−	0.849847i	$-0.676694\pi$
$887$	−31.3926	−1.05406	−0.527029	+	0.849847i	$0.676694\pi$
$888$	0	0
$889$	3.68858	0.123711
$890$	−14.9301	−0.500458
$891$	0	0
$892$	−13.2546	−0.443797
$893$	−2.94545	−0.0985656
$894$	0	0
$895$	27.6702	0.924911
$896$	−5.05345	−0.168824
$897$	0	0
$898$	−39.2738	−1.31058
$899$	−17.8525	−0.595414
$900$	0	0
$901$	30.5288	1.01706
$902$	−7.52436	−0.250534
$903$	0	0
$904$	11.7455	0.390649
$905$	−14.4354	−0.479850
$906$	0	0
$907$	−19.7931	−0.657220	−0.328610	−	0.944466i	$-0.606580\pi$
$907$	−19.7931	−0.657220	−0.328610	+	0.944466i	$0.606580\pi$
$908$	−4.30802	−0.142967
$909$	0	0
$910$	0.943508	0.0312770
$911$	−39.7344	−1.31646	−0.658230	−	0.752817i	$-0.728695\pi$
$911$	−39.7344	−1.31646	−0.658230	+	0.752817i	$0.728695\pi$
$912$	0	0
$913$	−3.69137	−0.122166
$914$	−13.6347	−0.450996
$915$	0	0
$916$	−8.40362	−0.277663
$917$	−5.48062	−0.180986
$918$	0	0
$919$	44.8558	1.47966	0.739828	−	0.672796i	$-0.234907\pi$
$919$	44.8558	1.47966	0.739828	+	0.672796i	$0.234907\pi$
$920$	−28.7705	−0.948536
$921$	0	0
$922$	−4.38320	−0.144353
$923$	1.85783	0.0611512
$924$	0	0
$925$	19.1338	0.629115
$926$	−11.4127	−0.375046
$927$	0	0
$928$	25.6174	0.840933
$929$	52.6319	1.72680	0.863399	−	0.504522i	$-0.168331\pi$
$929$	52.6319	1.72680	0.863399	+	0.504522i	$0.168331\pi$
$930$	0	0
$931$	8.85226	0.290121
$932$	25.1462	0.823691
$933$	0	0
$934$	−0.358423	−0.0117280
$935$	−18.7272	−0.612446
$936$	0	0
$937$	21.3064	0.696051	0.348026	−	0.937485i	$-0.386852\pi$
$937$	21.3064	0.696051	0.348026	+	0.937485i	$0.386852\pi$
$938$	−1.64239	−0.0536259
$939$	0	0
$940$	−6.52966	−0.212974
$941$	57.5983	1.87765	0.938825	−	0.344395i	$-0.111916\pi$
$941$	57.5983	1.87765	0.938825	+	0.344395i	$0.111916\pi$
$942$	0	0
$943$	29.8931	0.973454
$944$	−3.90014	−0.126939
$945$	0	0
$946$	1.22075	0.0396902
$947$	21.9241	0.712438	0.356219	−	0.934402i	$-0.384066\pi$
$947$	21.9241	0.712438	0.356219	+	0.934402i	$0.384066\pi$
$948$	0	0
$949$	−1.50651	−0.0489035
$950$	4.13681	0.134216
$951$	0	0
$952$	5.03849	0.163298
$953$	28.8641	0.934999	0.467499	−	0.883993i	$-0.345155\pi$
$953$	28.8641	0.934999	0.467499	+	0.883993i	$0.345155\pi$
$954$	0	0
$955$	50.7166	1.64115
$956$	−22.3371	−0.722434
$957$	0	0
$958$	−31.3434	−1.01266
$959$	4.19736	0.135540
$960$	0	0
$961$	−15.5425	−0.501370
$962$	−8.61495	−0.277757
$963$	0	0
$964$	8.33165	0.268344
$965$	−63.0360	−2.02920
$966$	0	0
$967$	30.9502	0.995292	0.497646	−	0.867380i	$-0.334198\pi$
$967$	30.9502	0.995292	0.497646	+	0.867380i	$0.334198\pi$
$968$	1.57532	0.0506328
$969$	0	0
$970$	28.8184	0.925302
$971$	−46.9808	−1.50769	−0.753843	−	0.657055i	$-0.771802\pi$
$971$	−46.9808	−1.50769	−0.753843	+	0.657055i	$0.771802\pi$
$972$	0	0
$973$	0.404255	0.0129598
$974$	−43.4319	−1.39165
$975$	0	0
$976$	4.98873	0.159685
$977$	−41.2468	−1.31960	−0.659802	−	0.751440i	$-0.729360\pi$
$977$	−41.2468	−1.31960	−0.659802	+	0.751440i	$0.729360\pi$
$978$	0	0
$979$	−3.24777	−0.103799
$980$	19.6243	0.626874
$981$	0	0
$982$	15.8623	0.506185
$983$	−24.5208	−0.782091	−0.391045	−	0.920371i	$-0.627886\pi$
$983$	−24.5208	−0.782091	−0.391045	+	0.920371i	$0.627886\pi$
$984$	0	0
$985$	−58.7111	−1.87069
$986$	−57.4579	−1.82983
$987$	0	0
$988$	−0.663304	−0.0211025
$989$	−4.84987	−0.154217
$990$	0	0
$991$	−40.5553	−1.28828	−0.644140	−	0.764907i	$-0.722785\pi$
$991$	−40.5553	−1.28828	−0.644140	+	0.764907i	$0.722785\pi$
$992$	−22.1807	−0.704239
$993$	0	0
$994$	−3.16589	−0.100416
$995$	9.16146	0.290438
$996$	0	0
$997$	20.5113	0.649599	0.324800	−	0.945783i	$-0.394703\pi$
$997$	20.5113	0.649599	0.324800	+	0.945783i	$0.394703\pi$
$998$	−76.8125	−2.43146
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.6	yes	25
3.2	odd	2		6039.2.a.n.1.20	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.20	✓	25	3.2	odd	2
6039.2.a.o.1.6	yes	25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-1.76243 q^{2} +1.10617 q^{4} -2.60834 q^{5} +0.445474 q^{7} +1.57532 q^{8} +O(q^{10})\)	\(q-1.76243 q^{2} +1.10617 q^{4} -2.60834 q^{5} +0.445474 q^{7} +1.57532 q^{8} +4.59703 q^{10} +1.00000 q^{11} +0.460729 q^{13} -0.785118 q^{14} -4.98873 q^{16} +7.17973 q^{17} -1.30151 q^{19} -2.88526 q^{20} -1.76243 q^{22} +7.00186 q^{23} +1.80346 q^{25} -0.812004 q^{26} +0.492768 q^{28} +4.54077 q^{29} -3.93161 q^{31} +5.64165 q^{32} -12.6538 q^{34} -1.16195 q^{35} +10.6095 q^{37} +2.29382 q^{38} -4.10898 q^{40} +4.26931 q^{41} -0.692654 q^{43} +1.10617 q^{44} -12.3403 q^{46} +2.26311 q^{47} -6.80155 q^{49} -3.17848 q^{50} +0.509643 q^{52} +4.25208 q^{53} -2.60834 q^{55} +0.701765 q^{56} -8.00279 q^{58} +0.781791 q^{59} -1.00000 q^{61} +6.92919 q^{62} +0.0344327 q^{64} -1.20174 q^{65} +2.09190 q^{67} +7.94198 q^{68} +2.04786 q^{70} +4.03237 q^{71} -3.26985 q^{73} -18.6985 q^{74} -1.43968 q^{76} +0.445474 q^{77} -1.20571 q^{79} +13.0123 q^{80} -7.52436 q^{82} -3.69137 q^{83} -18.7272 q^{85} +1.22075 q^{86} +1.57532 q^{88} -3.24777 q^{89} +0.205243 q^{91} +7.74522 q^{92} -3.98857 q^{94} +3.39478 q^{95} +6.26891 q^{97} +11.9873 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Introduction

L-functions

Modular forms

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Database

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