LMFDB - Embedded newform 8281.2.a.ck.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 3 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 3
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			1.4
Root			$-0.332375$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.332375 q^{2} +1.45984 q^{3} -1.88953 q^{4} -1.44562 q^{5} -0.485214 q^{6} +1.29278 q^{8} -0.868875 q^{9} +O(q^{10})$	\(q-0.332375 q^{2} +1.45984 q^{3} -1.88953 q^{4} -1.44562 q^{5} -0.485214 q^{6} +1.29278 q^{8} -0.868875 q^{9} +0.480489 q^{10} +5.95516 q^{11} -2.75840 q^{12} -2.11037 q^{15} +3.34936 q^{16} +4.32871 q^{17} +0.288793 q^{18} -1.95753 q^{19} +2.73154 q^{20} -1.97935 q^{22} -0.540163 q^{23} +1.88725 q^{24} -2.91018 q^{25} -5.64793 q^{27} +7.15857 q^{29} +0.701436 q^{30} +6.10800 q^{31} -3.69881 q^{32} +8.69356 q^{33} -1.43876 q^{34} +1.64176 q^{36} -8.02881 q^{37} +0.650636 q^{38} -1.86887 q^{40} +7.55362 q^{41} +4.24839 q^{43} -11.2524 q^{44} +1.25606 q^{45} +0.179537 q^{46} -6.26084 q^{47} +4.88953 q^{48} +0.967272 q^{50} +6.31922 q^{51} -2.77905 q^{53} +1.87723 q^{54} -8.60891 q^{55} -2.85768 q^{57} -2.37933 q^{58} +0.851152 q^{59} +3.98760 q^{60} +6.77905 q^{61} -2.03015 q^{62} -5.46933 q^{64} -2.88953 q^{66} +0.987106 q^{67} -8.17922 q^{68} -0.788550 q^{69} -3.76223 q^{71} -1.12327 q^{72} +9.13519 q^{73} +2.66858 q^{74} -4.24839 q^{75} +3.69881 q^{76} -0.131125 q^{79} -4.84191 q^{80} -5.63843 q^{81} -2.51064 q^{82} -2.66812 q^{83} -6.25768 q^{85} -1.41206 q^{86} +10.4503 q^{87} +7.69873 q^{88} -9.71739 q^{89} -0.417485 q^{90} +1.02065 q^{92} +8.91668 q^{93} +2.08095 q^{94} +2.82985 q^{95} -5.39966 q^{96} -6.58319 q^{97} -5.17429 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9	$ 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9} - 6 q^{10} + 18 q^{12} - 2 q^{16} + 8 q^{17} - 18 q^{22} - 12 q^{23} + 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} + 34 q^{38} + 4 q^{40} - 8 q^{43} + 18 q^{48} + 16 q^{51} + 20 q^{53} + 12 q^{55} + 12 q^{61} - 22 q^{62} - 44 q^{64} - 2 q^{66} + 2 q^{68} + 28 q^{69} - 42 q^{74} + 8 q^{75} - 20 q^{79} + 24 q^{81} - 16 q^{82} + 68 q^{87} + 4 q^{88} + 108 q^{90} + 6 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) $ 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9 - 6 * q^10 + 18 * q^12 - 2 * q^16 + 8 * q^17 - 18 * q^22 - 12 * q^23 + 16 * q^27 - 8 * q^29 + 38 * q^30 + 28 * q^36 + 34 * q^38 + 4 * q^40 - 8 * q^43 + 18 * q^48 + 16 * q^51 + 20 * q^53 + 12 * q^55 + 12 * q^61 - 22 * q^62 - 44 * q^64 - 2 * q^66 + 2 * q^68 + 28 * q^69 - 42 * q^74 + 8 * q^75 - 20 * q^79 + 24 * q^81 - 16 * q^82 + 68 * q^87 + 4 * q^88 + 108 * q^90 + 6 * q^92 + 26 * q^94 - 16 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.332375	−0.235025	−0.117512	−	0.993071i	$-0.537492\pi$
$2$	−0.332375	−0.235025	−0.117512	+	0.993071i	$0.537492\pi$
$3$	1.45984	0.842837	0.421419	−	0.906866i	$-0.361532\pi$
$3$	1.45984	0.842837	0.421419	+	0.906866i	$0.361532\pi$
$4$	−1.88953	−0.944763
$5$	−1.44562	−0.646502	−0.323251	−	0.946313i	$-0.604776\pi$
$5$	−1.44562	−0.646502	−0.323251	+	0.946313i	$0.604776\pi$
$6$	−0.485214	−0.198088
$7$	0	0
$7$	0	0
$8$	1.29278	0.457068
$9$	−0.868875	−0.289625
$10$	0.480489	0.151944
$11$	5.95516	1.79555	0.897774	−	0.440456i	$-0.145183\pi$
$11$	5.95516	1.79555	0.897774	+	0.440456i	$0.145183\pi$
$12$	−2.75840	−0.796282
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.11037	−0.544896
$16$	3.34936	0.837341
$17$	4.32871	1.04987	0.524933	−	0.851143i	$-0.324090\pi$
$17$	4.32871	1.04987	0.524933	+	0.851143i	$0.324090\pi$
$18$	0.288793	0.0680691
$19$	−1.95753	−0.449089	−0.224545	−	0.974464i	$-0.572089\pi$
$19$	−1.95753	−0.449089	−0.224545	+	0.974464i	$0.572089\pi$
$20$	2.73154	0.610791
$21$	0	0
$22$	−1.97935	−0.421998
$23$	−0.540163	−0.112632	−0.0563158	−	0.998413i	$-0.517935\pi$
$23$	−0.540163	−0.112632	−0.0563158	+	0.998413i	$0.517935\pi$
$24$	1.88725	0.385234
$25$	−2.91018	−0.582036
$26$	0	0
$27$	−5.64793	−1.08694
$28$	0	0
$29$	7.15857	1.32931	0.664656	−	0.747149i	$-0.268578\pi$
$29$	7.15857	1.32931	0.664656	+	0.747149i	$0.268578\pi$
$30$	0.701436	0.128064
$31$	6.10800	1.09703	0.548514	−	0.836141i	$-0.315194\pi$
$31$	6.10800	1.09703	0.548514	+	0.836141i	$0.315194\pi$
$32$	−3.69881	−0.653864
$33$	8.69356	1.51336
$34$	−1.43876	−0.246745
$35$	0	0
$36$	1.64176	0.273627
$37$	−8.02881	−1.31993	−0.659964	−	0.751297i	$-0.729429\pi$
$37$	−8.02881	−1.31993	−0.659964	+	0.751297i	$0.729429\pi$
$38$	0.650636	0.105547
$39$	0	0
$40$	−1.86887	−0.295495
$41$	7.55362	1.17968	0.589839	−	0.807521i	$-0.299191\pi$
$41$	7.55362	1.17968	0.589839	+	0.807521i	$0.299191\pi$
$42$	0	0
$43$	4.24839	0.647873	0.323936	−	0.946079i	$-0.394994\pi$
$43$	4.24839	0.647873	0.323936	+	0.946079i	$0.394994\pi$
$44$	−11.2524	−1.69637
$45$	1.25606	0.187243
$46$	0.179537	0.0264713
$47$	−6.26084	−0.913237	−0.456618	−	0.889663i	$-0.650940\pi$
$47$	−6.26084	−0.913237	−0.456618	+	0.889663i	$0.650940\pi$
$48$	4.88953	0.705742
$49$	0	0
$50$	0.967272	0.136793
$51$	6.31922	0.884867
$52$	0	0
$53$	−2.77905	−0.381732	−0.190866	−	0.981616i	$-0.561130\pi$
$53$	−2.77905	−0.381732	−0.190866	+	0.981616i	$0.561130\pi$
$54$	1.87723	0.255459
$55$	−8.60891	−1.16082
$56$	0	0
$57$	−2.85768	−0.378509
$58$	−2.37933	−0.312421
$59$	0.851152	0.110811	0.0554053	−	0.998464i	$-0.482355\pi$
$59$	0.851152	0.110811	0.0554053	+	0.998464i	$0.482355\pi$
$60$	3.98760	0.514798
$61$	6.77905	0.867969	0.433984	−	0.900920i	$-0.357107\pi$
$61$	6.77905	0.867969	0.433984	+	0.900920i	$0.357107\pi$
$62$	−2.03015	−0.257829
$63$	0	0
$64$	−5.46933	−0.683667
$65$	0	0
$66$	−2.88953	−0.355676
$67$	0.987106	0.120594	0.0602971	−	0.998180i	$-0.480795\pi$
$67$	0.987106	0.120594	0.0602971	+	0.998180i	$0.480795\pi$
$68$	−8.17922	−0.991876
$69$	−0.788550	−0.0949302
$70$	0	0
$71$	−3.76223	−0.446494	−0.223247	−	0.974762i	$-0.571666\pi$
$71$	−3.76223	−0.446494	−0.223247	+	0.974762i	$0.571666\pi$
$72$	−1.12327	−0.132378
$73$	9.13519	1.06919	0.534597	−	0.845107i	$-0.320463\pi$
$73$	9.13519	1.06919	0.534597	+	0.845107i	$0.320463\pi$
$74$	2.66858	0.310216
$75$	−4.24839	−0.490561
$76$	3.69881	0.424283
$77$	0	0
$78$	0	0
$79$	−0.131125	−0.0147527	−0.00737636	−	0.999973i	$-0.502348\pi$
$79$	−0.131125	−0.0147527	−0.00737636	+	0.999973i	$0.502348\pi$
$80$	−4.84191	−0.541342
$81$	−5.63843	−0.626492
$82$	−2.51064	−0.277253
$83$	−2.66812	−0.292865	−0.146432	−	0.989221i	$-0.546779\pi$
$83$	−2.66812	−0.292865	−0.146432	+	0.989221i	$0.546779\pi$
$84$	0	0
$85$	−6.25768	−0.678741
$86$	−1.41206	−0.152266
$87$	10.4503	1.12039
$88$	7.69873	0.820687
$89$	−9.71739	−1.03004	−0.515021	−	0.857178i	$-0.672216\pi$
$89$	−9.71739	−1.03004	−0.515021	+	0.857178i	$0.672216\pi$
$90$	−0.417485	−0.0440068
$91$	0	0
$92$	1.02065	0.106410
$93$	8.91668	0.924617
$94$	2.08095	0.214633
$95$	2.82985	0.290337
$96$	−5.39966	−0.551101
$97$	−6.58319	−0.668422	−0.334211	−	0.942498i	$-0.608470\pi$
$97$	−6.58319	−0.668422	−0.334211	+	0.942498i	$0.608470\pi$
$98$	0	0
$99$	−5.17429	−0.520036
$100$	5.49886	0.549886
$101$	−0.0708289	−0.00704774	−0.00352387	−	0.999994i	$-0.501122\pi$
$101$	−0.0708289	−0.00704774	−0.00352387	+	0.999994i	$0.501122\pi$
$102$	−2.10035	−0.207966
$103$	6.33821	0.624522	0.312261	−	0.949996i	$-0.398914\pi$
$103$	6.33821	0.624522	0.312261	+	0.949996i	$0.398914\pi$
$104$	0	0
$105$	0	0
$106$	0.923689	0.0897166
$107$	7.74953	0.749175	0.374588	−	0.927192i	$-0.377784\pi$
$107$	7.74953	0.749175	0.374588	+	0.927192i	$0.377784\pi$
$108$	10.6719	1.02691
$109$	0.0335623	0.00321468	0.00160734	−	0.999999i	$-0.499488\pi$
$109$	0.0335623	0.00321468	0.00160734	+	0.999999i	$0.499488\pi$
$110$	2.86139	0.272823
$111$	−11.7208	−1.11249
$112$	0	0
$113$	−9.19987	−0.865451	−0.432725	−	0.901526i	$-0.642448\pi$
$113$	−9.19987	−0.865451	−0.432725	+	0.901526i	$0.642448\pi$
$114$	0.949823	0.0889591
$115$	0.780871	0.0728166
$116$	−13.5263	−1.25589
$117$	0	0
$118$	−0.282902	−0.0260432
$119$	0	0
$120$	−2.72825	−0.249054
$121$	24.4639	2.22399
$122$	−2.25319	−0.203994
$123$	11.0271	0.994276
$124$	−11.5412	−1.03643
$125$	11.4351	1.02279
$126$	0	0
$127$	14.3952	1.27737	0.638683	−	0.769470i	$-0.279480\pi$
$127$	14.3952	1.27737	0.638683	+	0.769470i	$0.279480\pi$
$128$	9.21550	0.814542
$129$	6.20195	0.546052
$130$	0	0
$131$	−9.46828	−0.827248	−0.413624	−	0.910448i	$-0.635737\pi$
$131$	−9.46828	−0.827248	−0.413624	+	0.910448i	$0.635737\pi$
$132$	−16.4267	−1.42976
$133$	0	0
$134$	−0.328090	−0.0283426
$135$	8.16477	0.702711
$136$	5.59609	0.479860
$137$	−16.6063	−1.41877	−0.709384	−	0.704822i	$-0.751027\pi$
$137$	−16.6063	−1.41877	−0.709384	+	0.704822i	$0.751027\pi$
$138$	0.262094	0.0223110
$139$	18.4778	1.56726	0.783632	−	0.621225i	$-0.213365\pi$
$139$	18.4778	1.56726	0.783632	+	0.621225i	$0.213365\pi$
$140$	0	0
$141$	−9.13980	−0.769710
$142$	1.25047	0.104937
$143$	0	0
$144$	−2.91018	−0.242515
$145$	−10.3486	−0.859402
$146$	−3.03631	−0.251287
$147$	0	0
$148$	15.1707	1.24702
$149$	3.08080	0.252389	0.126195	−	0.992006i	$-0.459724\pi$
$149$	3.08080	0.252389	0.126195	+	0.992006i	$0.459724\pi$
$150$	1.41206	0.115294
$151$	−2.54885	−0.207422	−0.103711	−	0.994607i	$-0.533072\pi$
$151$	−2.54885	−0.207422	−0.103711	+	0.994607i	$0.533072\pi$
$152$	−2.53067	−0.205264
$153$	−3.76111	−0.304068
$154$	0	0
$155$	−8.82985	−0.709231
$156$	0	0
$157$	−9.40904	−0.750923	−0.375461	−	0.926838i	$-0.622516\pi$
$157$	−9.40904	−0.750923	−0.375461	+	0.926838i	$0.622516\pi$
$158$	0.0435828	0.00346726
$159$	−4.05697	−0.321738
$160$	5.34708	0.422724
$161$	0	0
$162$	1.87408	0.147241
$163$	−0.695157	−0.0544489	−0.0272244	−	0.999629i	$-0.508667\pi$
$163$	−0.695157	−0.0544489	−0.0272244	+	0.999629i	$0.508667\pi$
$164$	−14.2728	−1.11452
$165$	−12.5676	−0.978387
$166$	0.886819	0.0688305
$167$	−13.9840	−1.08211	−0.541056	−	0.840986i	$-0.681975\pi$
$167$	−13.9840	−1.08211	−0.541056	+	0.840986i	$0.681975\pi$
$168$	0	0
$169$	0	0
$170$	2.07990	0.159521
$171$	1.70085	0.130067
$172$	−8.02744	−0.612087
$173$	−5.43648	−0.413328	−0.206664	−	0.978412i	$-0.566261\pi$
$173$	−5.43648	−0.413328	−0.206664	+	0.978412i	$0.566261\pi$
$174$	−3.47344	−0.263321
$175$	0	0
$176$	19.9460	1.50349
$177$	1.24254	0.0933953
$178$	3.22982	0.242085
$179$	5.35824	0.400493	0.200247	−	0.979745i	$-0.435826\pi$
$179$	5.35824	0.400493	0.200247	+	0.979745i	$0.435826\pi$
$180$	−2.37337	−0.176900
$181$	7.54016	0.560456	0.280228	−	0.959933i	$-0.409590\pi$
$181$	7.54016	0.560456	0.280228	+	0.959933i	$0.409590\pi$
$182$	0	0
$183$	9.89632	0.731557
$184$	−0.698313	−0.0514803
$185$	11.6066	0.853336
$186$	−2.96369	−0.217308
$187$	25.7782	1.88509
$188$	11.8300	0.862793
$189$	0	0
$190$	−0.940574	−0.0682364
$191$	13.5463	0.980178	0.490089	−	0.871672i	$-0.336964\pi$
$191$	13.5463	0.980178	0.490089	+	0.871672i	$0.336964\pi$
$192$	−7.98434	−0.576220
$193$	−18.5562	−1.33571	−0.667853	−	0.744293i	$-0.732787\pi$
$193$	−18.5562	−1.33571	−0.667853	+	0.744293i	$0.732787\pi$
$194$	2.18809	0.157096
$195$	0	0
$196$	0	0
$197$	2.66812	0.190096	0.0950480	−	0.995473i	$-0.469700\pi$
$197$	2.66812	0.190096	0.0950480	+	0.995473i	$0.469700\pi$
$198$	1.71981	0.122221
$199$	20.1999	1.43193	0.715965	−	0.698136i	$-0.245987\pi$
$199$	20.1999	1.43193	0.715965	+	0.698136i	$0.245987\pi$
$200$	−3.76223	−0.266030
$201$	1.44101	0.101641
$202$	0.0235418	0.00165639
$203$	0	0
$204$	−11.9403	−0.835990
$205$	−10.9197	−0.762663
$206$	−2.10666	−0.146778
$207$	0.469334	0.0326210
$208$	0	0
$209$	−11.6574	−0.806361
$210$	0	0
$211$	13.1268	0.903683	0.451842	−	0.892098i	$-0.350767\pi$
$211$	13.1268	0.903683	0.451842	+	0.892098i	$0.350767\pi$
$212$	5.25109	0.360647
$213$	−5.49224	−0.376322
$214$	−2.57575	−0.176075
$215$	−6.14156	−0.418851
$216$	−7.30155	−0.496807
$217$	0	0
$218$	−0.0111553	−0.000755530 0
$219$	13.3359	0.901157
$220$	16.2668	1.09670
$221$	0	0
$222$	3.89569	0.261462
$223$	2.22334	0.148886	0.0744428	−	0.997225i	$-0.476282\pi$
$223$	2.22334	0.148886	0.0744428	+	0.997225i	$0.476282\pi$
$224$	0	0
$225$	2.52858	0.168572
$226$	3.05781	0.203402
$227$	−27.1045	−1.79899	−0.899495	−	0.436931i	$-0.856065\pi$
$227$	−27.1045	−1.79899	−0.899495	+	0.436931i	$0.856065\pi$
$228$	5.39966	0.357601
$229$	18.9887	1.25481	0.627406	−	0.778693i	$-0.284117\pi$
$229$	18.9887	1.25481	0.627406	+	0.778693i	$0.284117\pi$
$230$	−0.259542	−0.0171137
$231$	0	0
$232$	9.25447	0.607586
$233$	21.7400	1.42424	0.712118	−	0.702059i	$-0.247736\pi$
$233$	21.7400	1.42424	0.712118	+	0.702059i	$0.247736\pi$
$234$	0	0
$235$	9.05080	0.590409
$236$	−1.60827	−0.104690
$237$	−0.191421	−0.0124342
$238$	0	0
$239$	−19.9695	−1.29172	−0.645861	−	0.763455i	$-0.723501\pi$
$239$	−19.9695	−1.29172	−0.645861	+	0.763455i	$0.723501\pi$
$240$	−7.06841	−0.456264
$241$	−3.23048	−0.208094	−0.104047	−	0.994572i	$-0.533179\pi$
$241$	−3.23048	−0.208094	−0.104047	+	0.994572i	$0.533179\pi$
$242$	−8.13120	−0.522694
$243$	8.71259	0.558913
$244$	−12.8092	−0.820025
$245$	0	0
$246$	−3.66512	−0.233680
$247$	0	0
$248$	7.89632	0.501417
$249$	−3.89503	−0.246837
$250$	−3.80075	−0.240381
$251$	12.4916	0.788466	0.394233	−	0.919011i	$-0.371010\pi$
$251$	12.4916	0.788466	0.394233	+	0.919011i	$0.371010\pi$
$252$	0	0
$253$	−3.21675	−0.202236
$254$	−4.78460	−0.300213
$255$	−9.13519	−0.572068
$256$	7.87566	0.492229
$257$	5.82757	0.363514	0.181757	−	0.983343i	$-0.441822\pi$
$257$	5.82757	0.363514	0.181757	+	0.983343i	$0.441822\pi$
$258$	−2.06138	−0.128336
$259$	0	0
$260$	0	0
$261$	−6.21990	−0.385002
$262$	3.14702	0.194424
$263$	17.5147	1.08000	0.540002	−	0.841664i	$-0.318424\pi$
$263$	17.5147	1.08000	0.540002	+	0.841664i	$0.318424\pi$
$264$	11.2389	0.691706
$265$	4.01746	0.246791
$266$	0	0
$267$	−14.1858	−0.868157
$268$	−1.86516	−0.113933
$269$	22.3287	1.36141	0.680703	−	0.732560i	$-0.261675\pi$
$269$	22.3287	1.36141	0.680703	+	0.732560i	$0.261675\pi$
$270$	−2.71377	−0.165155
$271$	26.3695	1.60183	0.800916	−	0.598777i	$-0.204346\pi$
$271$	26.3695	1.60183	0.800916	+	0.598777i	$0.204346\pi$
$272$	14.4984	0.879097
$273$	0	0
$274$	5.51951	0.333446
$275$	−17.3306	−1.04507
$276$	1.48999	0.0896866
$277$	−9.37618	−0.563360	−0.281680	−	0.959508i	$-0.590892\pi$
$277$	−9.37618	−0.563360	−0.281680	+	0.959508i	$0.590892\pi$
$278$	−6.14156	−0.368346
$279$	−5.30709	−0.317727
$280$	0	0
$281$	17.7754	1.06039	0.530195	−	0.847876i	$-0.322119\pi$
$281$	17.7754	1.06039	0.530195	+	0.847876i	$0.322119\pi$
$282$	3.03785	0.180901
$283$	−9.60662	−0.571055	−0.285527	−	0.958371i	$-0.592169\pi$
$283$	−9.60662	−0.571055	−0.285527	+	0.958371i	$0.592169\pi$
$284$	7.10883	0.421832
$285$	4.13113	0.244707
$286$	0	0
$287$	0	0
$288$	3.21380	0.189375
$289$	1.73775	0.102221
$290$	3.43961	0.201981
$291$	−9.61039	−0.563371
$292$	−17.2612	−1.01013
$293$	11.6338	0.679654	0.339827	−	0.940488i	$-0.389631\pi$
$293$	11.6338	0.679654	0.339827	+	0.940488i	$0.389631\pi$
$294$	0	0
$295$	−1.23044	−0.0716392
$296$	−10.3795	−0.603297
$297$	−33.6343	−1.95166
$298$	−1.02398	−0.0593177
$299$	0	0
$300$	8.02744	0.463464
$301$	0	0
$302$	0.847174	0.0487494
$303$	−0.103399	−0.00594010
$304$	−6.55649	−0.376041
$305$	−9.79995	−0.561143
$306$	1.25010	0.0714635
$307$	−13.8280	−0.789204	−0.394602	−	0.918852i	$-0.629118\pi$
$307$	−13.8280	−0.789204	−0.394602	+	0.918852i	$0.629118\pi$
$308$	0	0
$309$	9.25275	0.526371
$310$	2.93483	0.166687
$311$	30.7144	1.74165	0.870827	−	0.491590i	$-0.163584\pi$
$311$	30.7144	1.74165	0.870827	+	0.491590i	$0.163584\pi$
$312$	0	0
$313$	11.0867	0.626657	0.313328	−	0.949645i	$-0.398556\pi$
$313$	11.0867	0.626657	0.313328	+	0.949645i	$0.398556\pi$
$314$	3.12733	0.176486
$315$	0	0
$316$	0.247764	0.0139378
$317$	23.8834	1.34142	0.670712	−	0.741718i	$-0.265989\pi$
$317$	23.8834	1.34142	0.670712	+	0.741718i	$0.265989\pi$
$318$	1.34844	0.0756165
$319$	42.6304	2.38684
$320$	7.90659	0.441992
$321$	11.3130	0.631433
$322$	0	0
$323$	−8.47360	−0.471484
$324$	10.6540	0.591887
$325$	0	0
$326$	0.231053	0.0127968
$327$	0.0489954	0.00270945
$328$	9.76519	0.539192
$329$	0	0
$330$	4.17716	0.229945
$331$	18.3240	1.00718	0.503589	−	0.863943i	$-0.332013\pi$
$331$	18.3240	1.00718	0.503589	+	0.863943i	$0.332013\pi$
$332$	5.04149	0.276688
$333$	6.97603	0.382284
$334$	4.64793	0.254323
$335$	−1.42698	−0.0779643
$336$	0	0
$337$	−7.21762	−0.393169	−0.196584	−	0.980487i	$-0.562985\pi$
$337$	−7.21762	−0.393169	−0.196584	+	0.980487i	$0.562985\pi$
$338$	0	0
$339$	−13.4303	−0.729434
$340$	11.8241	0.641249
$341$	36.3741	1.96977
$342$	−0.565321	−0.0305691
$343$	0	0
$344$	5.49224	0.296122
$345$	1.13994	0.0613725
$346$	1.80695	0.0971423
$347$	−21.0782	−1.13154	−0.565770	−	0.824563i	$-0.691421\pi$
$347$	−21.0782	−1.13154	−0.565770	+	0.824563i	$0.691421\pi$
$348$	−19.7462	−1.05851
$349$	30.7629	1.64670	0.823350	−	0.567534i	$-0.192102\pi$
$349$	30.7629	1.64670	0.823350	+	0.567534i	$0.192102\pi$
$350$	0	0
$351$	0	0
$352$	−22.0270	−1.17404
$353$	−6.12173	−0.325827	−0.162913	−	0.986640i	$-0.552089\pi$
$353$	−6.12173	−0.325827	−0.162913	+	0.986640i	$0.552089\pi$
$354$	−0.412991	−0.0219502
$355$	5.43876	0.288659
$356$	18.3613	0.973145
$357$	0	0
$358$	−1.78095	−0.0941259
$359$	19.4287	1.02541	0.512703	−	0.858566i	$-0.328644\pi$
$359$	19.4287	1.02541	0.512703	+	0.858566i	$0.328644\pi$
$360$	1.62382	0.0855827
$361$	−15.1681	−0.798319
$362$	−2.50616	−0.131721
$363$	35.7133	1.87446
$364$	0	0
$365$	−13.2060	−0.691235
$366$	−3.28929	−0.171934
$367$	−5.40467	−0.282122	−0.141061	−	0.990001i	$-0.545051\pi$
$367$	−5.40467	−0.282122	−0.141061	+	0.990001i	$0.545051\pi$
$368$	−1.80920	−0.0943111
$369$	−6.56315	−0.341664
$370$	−3.85776	−0.200555
$371$	0	0
$372$	−16.8483	−0.873544
$373$	16.2507	0.841428	0.420714	−	0.907193i	$-0.361780\pi$
$373$	16.2507	0.841428	0.420714	+	0.907193i	$0.361780\pi$
$374$	−8.56803	−0.443042
$375$	16.6934	0.862045
$376$	−8.09390	−0.417411
$377$	0	0
$378$	0	0
$379$	−25.1730	−1.29305	−0.646525	−	0.762893i	$-0.723778\pi$
$379$	−25.1730	−1.29305	−0.646525	+	0.762893i	$0.723778\pi$
$380$	−5.34708	−0.274300
$381$	21.0146	1.07661
$382$	−4.50247	−0.230366
$383$	−3.81438	−0.194906	−0.0974529	−	0.995240i	$-0.531070\pi$
$383$	−3.81438	−0.194906	−0.0974529	+	0.995240i	$0.531070\pi$
$384$	13.4531	0.686527
$385$	0	0
$386$	6.16764	0.313924
$387$	−3.69132	−0.187640
$388$	12.4391	0.631500
$389$	−2.87096	−0.145563	−0.0727817	−	0.997348i	$-0.523188\pi$
$389$	−2.87096	−0.145563	−0.0727817	+	0.997348i	$0.523188\pi$
$390$	0	0
$391$	−2.33821	−0.118248
$392$	0	0
$393$	−13.8222	−0.697236
$394$	−0.886819	−0.0446773
$395$	0.189557	0.00953766
$396$	9.77696	0.491310
$397$	19.1184	0.959524	0.479762	−	0.877399i	$-0.340723\pi$
$397$	19.1184	0.959524	0.479762	+	0.877399i	$0.340723\pi$
$398$	−6.71394	−0.336539
$399$	0	0
$400$	−9.74725	−0.487362
$401$	−2.99824	−0.149725	−0.0748625	−	0.997194i	$-0.523852\pi$
$401$	−2.99824	−0.149725	−0.0748625	+	0.997194i	$0.523852\pi$
$402$	−0.478958	−0.0238882
$403$	0	0
$404$	0.133833	0.00665844
$405$	8.15104	0.405028
$406$	0	0
$407$	−47.8129	−2.37000
$408$	8.16937	0.404444
$409$	34.0805	1.68517	0.842587	−	0.538560i	$-0.181031\pi$
$409$	34.0805	1.68517	0.842587	+	0.538560i	$0.181031\pi$
$410$	3.62943	0.179245
$411$	−24.2424	−1.19579
$412$	−11.9762	−0.590026
$413$	0	0
$414$	−0.155995	−0.00766674
$415$	3.85710	0.189337
$416$	0	0
$417$	26.9746	1.32095
$418$	3.87464	0.189515
$419$	34.7759	1.69891	0.849457	−	0.527657i	$-0.176929\pi$
$419$	34.7759	1.69891	0.849457	+	0.527657i	$0.176929\pi$
$420$	0	0
$421$	24.1400	1.17651	0.588257	−	0.808674i	$-0.299814\pi$
$421$	24.1400	1.17651	0.588257	+	0.808674i	$0.299814\pi$
$422$	−4.36301	−0.212388
$423$	5.43988	0.264496
$424$	−3.59271	−0.174478
$425$	−12.5973	−0.611060
$426$	1.82549	0.0884451
$427$	0	0
$428$	−14.6429	−0.707793
$429$	0	0
$430$	2.04130	0.0984404
$431$	4.76477	0.229511	0.114755	−	0.993394i	$-0.463392\pi$
$431$	4.76477	0.229511	0.114755	+	0.993394i	$0.463392\pi$
$432$	−18.9170	−0.910143
$433$	22.0231	1.05836	0.529181	−	0.848509i	$-0.322499\pi$
$433$	22.0231	1.05836	0.529181	+	0.848509i	$0.322499\pi$
$434$	0	0
$435$	−15.1072	−0.724337
$436$	−0.0634168	−0.00303711
$437$	1.05739	0.0505817
$438$	−4.43252	−0.211794
$439$	3.43240	0.163819	0.0819097	−	0.996640i	$-0.473898\pi$
$439$	3.43240	0.163819	0.0819097	+	0.996640i	$0.473898\pi$
$440$	−11.1294	−0.530576
$441$	0	0
$442$	0	0
$443$	−8.70594	−0.413632	−0.206816	−	0.978380i	$-0.566310\pi$
$443$	−8.70594	−0.413632	−0.206816	+	0.978380i	$0.566310\pi$
$444$	22.1467	1.05104
$445$	14.0477	0.665923
$446$	−0.738982	−0.0349918
$447$	4.49747	0.212723
$448$	0	0
$449$	17.6120	0.831159	0.415580	−	0.909557i	$-0.363579\pi$
$449$	17.6120	0.831159	0.415580	+	0.909557i	$0.363579\pi$
$450$	−0.840438	−0.0396186
$451$	44.9830	2.11817
$452$	17.3834	0.817646
$453$	−3.72090	−0.174823
$454$	9.00887	0.422807
$455$	0	0
$456$	−3.69436	−0.173004
$457$	−9.07268	−0.424402	−0.212201	−	0.977226i	$-0.568063\pi$
$457$	−9.07268	−0.424402	−0.212201	+	0.977226i	$0.568063\pi$
$458$	−6.31139	−0.294912
$459$	−24.4483	−1.14115
$460$	−1.47548	−0.0687944
$461$	6.58319	0.306610	0.153305	−	0.988179i	$-0.451008\pi$
$461$	6.58319	0.306610	0.153305	+	0.988179i	$0.451008\pi$
$462$	0	0
$463$	3.47344	0.161424	0.0807121	−	0.996737i	$-0.474281\pi$
$463$	3.47344	0.161424	0.0807121	+	0.996737i	$0.474281\pi$
$464$	23.9766	1.11309
$465$	−12.8901	−0.597766
$466$	−7.22585	−0.334731
$467$	29.7854	1.37830	0.689152	−	0.724617i	$-0.257983\pi$
$467$	29.7854	1.37830	0.689152	+	0.724617i	$0.257983\pi$
$468$	0	0
$469$	0	0
$470$	−3.00826	−0.138761
$471$	−13.7357	−0.632906
$472$	1.10035	0.0506479
$473$	25.2998	1.16329
$474$	0.0636237	0.00292233
$475$	5.69677	0.261386
$476$	0	0
$477$	2.41465	0.110559
$478$	6.63738	0.303587
$479$	35.1855	1.60767	0.803833	−	0.594855i	$-0.202791\pi$
$479$	35.1855	1.60767	0.803833	+	0.594855i	$0.202791\pi$
$480$	7.80587	0.356288
$481$	0	0
$482$	1.07373	0.0489072
$483$	0	0
$484$	−46.2252	−2.10115
$485$	9.51680	0.432136
$486$	−2.89585	−0.131358
$487$	−1.80154	−0.0816355	−0.0408178	−	0.999167i	$-0.512996\pi$
$487$	−1.80154	−0.0816355	−0.0408178	+	0.999167i	$0.512996\pi$
$488$	8.76384	0.396721
$489$	−1.01482	−0.0458916
$490$	0	0
$491$	−8.19322	−0.369755	−0.184877	−	0.982762i	$-0.559189\pi$
$491$	−8.19322	−0.369755	−0.184877	+	0.982762i	$0.559189\pi$
$492$	−20.8359	−0.939356
$493$	30.9874	1.39560
$494$	0	0
$495$	7.48006	0.336204
$496$	20.4579	0.918587
$497$	0	0
$498$	1.29461	0.0580129
$499$	−36.5164	−1.63470	−0.817350	−	0.576141i	$-0.804558\pi$
$499$	−36.5164	−1.63470	−0.817350	+	0.576141i	$0.804558\pi$
$500$	−21.6070	−0.966293
$501$	−20.4143	−0.912045
$502$	−4.15192	−0.185309
$503$	−3.02972	−0.135089	−0.0675443	−	0.997716i	$-0.521516\pi$
$503$	−3.02972	−0.135089	−0.0675443	+	0.997716i	$0.521516\pi$
$504$	0	0
$505$	0.102392	0.00455637
$506$	1.06917	0.0475304
$507$	0	0
$508$	−27.2001	−1.20681
$509$	29.3447	1.30068	0.650341	−	0.759642i	$-0.274626\pi$
$509$	29.3447	1.30068	0.650341	+	0.759642i	$0.274626\pi$
$510$	3.03631	0.134450
$511$	0	0
$512$	−21.0487	−0.930229
$513$	11.0560	0.488135
$514$	−1.93694	−0.0854348
$515$	−9.16265	−0.403755
$516$	−11.7188	−0.515890
$517$	−37.2843	−1.63976
$518$	0	0
$519$	−7.93637	−0.348368
$520$	0	0
$521$	−29.6838	−1.30047	−0.650236	−	0.759732i	$-0.725330\pi$
$521$	−29.6838	−1.30047	−0.650236	+	0.759732i	$0.725330\pi$
$522$	2.06734	0.0904851
$523$	20.5727	0.899583	0.449791	−	0.893134i	$-0.351498\pi$
$523$	20.5727	0.899583	0.449791	+	0.893134i	$0.351498\pi$
$524$	17.8906	0.781553
$525$	0	0
$526$	−5.82146	−0.253828
$527$	26.4398	1.15173
$528$	29.1179	1.26719
$529$	−22.7082	−0.987314
$530$	−1.33530	−0.0580019
$531$	−0.739544	−0.0320935
$532$	0	0
$533$	0	0
$534$	4.71501	0.204039
$535$	−11.2029	−0.484343
$536$	1.27611	0.0551197
$537$	7.82216	0.337551
$538$	−7.42151	−0.319964
$539$	0	0
$540$	−15.4275	−0.663896
$541$	−34.0668	−1.46465	−0.732324	−	0.680957i	$-0.761564\pi$
$541$	−34.0668	−1.46465	−0.732324	+	0.680957i	$0.761564\pi$
$542$	−8.76457	−0.376470
$543$	11.0074	0.472373
$544$	−16.0111	−0.686470
$545$	−0.0485183	−0.00207830
$546$	0	0
$547$	−0.850931	−0.0363832	−0.0181916	−	0.999835i	$-0.505791\pi$
$547$	−0.850931	−0.0363832	−0.0181916	+	0.999835i	$0.505791\pi$
$548$	31.3780	1.34040
$549$	−5.89015	−0.251385
$550$	5.76026	0.245618
$551$	−14.0131	−0.596980
$552$	−1.01942	−0.0433895
$553$	0	0
$554$	3.11641	0.132404
$555$	16.9438	0.719224
$556$	−34.9143	−1.48069
$557$	17.7281	0.751162	0.375581	−	0.926789i	$-0.377443\pi$
$557$	17.7281	0.751162	0.375581	+	0.926789i	$0.377443\pi$
$558$	1.76394	0.0746737
$559$	0	0
$560$	0	0
$561$	37.6319	1.58882
$562$	−5.90809	−0.249218
$563$	−24.1806	−1.01909	−0.509545	−	0.860444i	$-0.670186\pi$
$563$	−24.1806	−1.01909	−0.509545	+	0.860444i	$0.670186\pi$
$564$	17.2699	0.727194
$565$	13.2995	0.559515
$566$	3.19301	0.134212
$567$	0	0
$568$	−4.86375	−0.204078
$569$	−42.7749	−1.79322	−0.896608	−	0.442825i	$-0.853976\pi$
$569$	−42.7749	−1.79322	−0.896608	+	0.442825i	$0.853976\pi$
$570$	−1.37308	−0.0575122
$571$	7.36280	0.308124	0.154062	−	0.988061i	$-0.450765\pi$
$571$	7.36280	0.308124	0.154062	+	0.988061i	$0.450765\pi$
$572$	0	0
$573$	19.7754	0.826131
$574$	0	0
$575$	1.57197	0.0655557
$576$	4.75217	0.198007
$577$	8.19393	0.341118	0.170559	−	0.985347i	$-0.445443\pi$
$577$	8.19393	0.341118	0.170559	+	0.985347i	$0.445443\pi$
$578$	−0.577585	−0.0240244
$579$	−27.0891	−1.12578
$580$	19.5539	0.811932
$581$	0	0
$582$	3.19426	0.132406
$583$	−16.5497	−0.685419
$584$	11.8098	0.488694
$585$	0	0
$586$	−3.86679	−0.159736
$587$	−39.1141	−1.61441	−0.807205	−	0.590271i	$-0.799021\pi$
$587$	−39.1141	−1.61441	−0.807205	+	0.590271i	$0.799021\pi$
$588$	0	0
$589$	−11.9566	−0.492664
$590$	0.408969	0.0168370
$591$	3.89503	0.160220
$592$	−26.8914	−1.10523
$593$	1.21338	0.0498276	0.0249138	−	0.999690i	$-0.492069\pi$
$593$	1.21338	0.0498276	0.0249138	+	0.999690i	$0.492069\pi$
$594$	11.1792	0.458689
$595$	0	0
$596$	−5.82125	−0.238448
$597$	29.4885	1.20688
$598$	0	0
$599$	32.6638	1.33461	0.667303	−	0.744786i	$-0.267449\pi$
$599$	32.6638	1.33461	0.667303	+	0.744786i	$0.267449\pi$
$600$	−5.49224	−0.224220
$601$	2.50114	0.102024	0.0510118	−	0.998698i	$-0.483755\pi$
$601$	2.50114	0.102024	0.0510118	+	0.998698i	$0.483755\pi$
$602$	0	0
$603$	−0.857671	−0.0349271
$604$	4.81611	0.195965
$605$	−35.3656	−1.43781
$606$	0.0343672	0.00139607
$607$	12.6456	0.513271	0.256635	−	0.966508i	$-0.417386\pi$
$607$	12.6456	0.513271	0.256635	+	0.966508i	$0.417386\pi$
$608$	7.24055	0.293643
$609$	0	0
$610$	3.25726	0.131883
$611$	0	0
$612$	7.10672	0.287272
$613$	−20.0280	−0.808923	−0.404462	−	0.914555i	$-0.632541\pi$
$613$	−20.0280	−0.808923	−0.404462	+	0.914555i	$0.632541\pi$
$614$	4.59608	0.185483
$615$	−15.9409	−0.642801
$616$	0	0
$617$	45.2926	1.82341	0.911705	−	0.410846i	$-0.134767\pi$
$617$	45.2926	1.82341	0.911705	+	0.410846i	$0.134767\pi$
$618$	−3.07539	−0.123710
$619$	4.43315	0.178183	0.0890917	−	0.996023i	$-0.471604\pi$
$619$	4.43315	0.178183	0.0890917	+	0.996023i	$0.471604\pi$
$620$	16.6842	0.670055
$621$	3.05080	0.122424
$622$	−10.2087	−0.409332
$623$	0	0
$624$	0	0
$625$	−1.97997	−0.0791988
$626$	−3.68494	−0.147280
$627$	−17.0179	−0.679631
$628$	17.7786	0.709444
$629$	−34.7544	−1.38575
$630$	0	0
$631$	−19.7358	−0.785672	−0.392836	−	0.919609i	$-0.628506\pi$
$631$	−19.7358	−0.785672	−0.392836	+	0.919609i	$0.628506\pi$
$632$	−0.169516	−0.00674300
$633$	19.1629	0.761658
$634$	−7.93824	−0.315268
$635$	−20.8100	−0.825819
$636$	7.66574	0.303967
$637$	0	0
$638$	−14.1693	−0.560968
$639$	3.26891	0.129316
$640$	−13.3221	−0.526603
$641$	39.6425	1.56579	0.782893	−	0.622157i	$-0.213743\pi$
$641$	39.6425	1.56579	0.782893	+	0.622157i	$0.213743\pi$
$642$	−3.76018	−0.148402
$643$	20.8300	0.821453	0.410727	−	0.911759i	$-0.365275\pi$
$643$	20.8300	0.821453	0.410727	+	0.911759i	$0.365275\pi$
$644$	0	0
$645$	−8.96568	−0.353023
$646$	2.81642	0.110810
$647$	−15.7441	−0.618965	−0.309482	−	0.950905i	$-0.600156\pi$
$647$	−15.7441	−0.618965	−0.309482	+	0.950905i	$0.600156\pi$
$648$	−7.28927	−0.286350
$649$	5.06874	0.198966
$650$	0	0
$651$	0	0
$652$	1.31352	0.0514413
$653$	−27.0264	−1.05762	−0.528812	−	0.848739i	$-0.677362\pi$
$653$	−27.0264	−1.05762	−0.528812	+	0.848739i	$0.677362\pi$
$654$	−0.0162849	−0.000636789 0
$655$	13.6876	0.534817
$656$	25.2998	0.987792
$657$	−7.93734	−0.309665
$658$	0	0
$659$	−6.79491	−0.264692	−0.132346	−	0.991204i	$-0.542251\pi$
$659$	−6.79491	−0.264692	−0.132346	+	0.991204i	$0.542251\pi$
$660$	23.7468	0.924344
$661$	7.20526	0.280252	0.140126	−	0.990134i	$-0.455249\pi$
$661$	7.20526	0.280252	0.140126	+	0.990134i	$0.455249\pi$
$662$	−6.09044	−0.236712
$663$	0	0
$664$	−3.44930	−0.133859
$665$	0	0
$666$	−2.31866	−0.0898463
$667$	−3.86679	−0.149723
$668$	26.4231	1.02234
$669$	3.24571	0.125486
$670$	0.474293	0.0183236
$671$	40.3703	1.55848
$672$	0	0
$673$	8.32130	0.320763	0.160381	−	0.987055i	$-0.448728\pi$
$673$	8.32130	0.320763	0.160381	+	0.987055i	$0.448728\pi$
$674$	2.39896	0.0924044
$675$	16.4365	0.632640
$676$	0	0
$677$	−29.9956	−1.15283	−0.576413	−	0.817159i	$-0.695548\pi$
$677$	−29.9956	−1.15283	−0.576413	+	0.817159i	$0.695548\pi$
$678$	4.46391	0.171435
$679$	0	0
$680$	−8.08982	−0.310231
$681$	−39.5682	−1.51626
$682$	−12.0899	−0.462944
$683$	−36.0839	−1.38071	−0.690356	−	0.723469i	$-0.742546\pi$
$683$	−36.0839	−1.38071	−0.690356	+	0.723469i	$0.742546\pi$
$684$	−3.21380	−0.122883
$685$	24.0064	0.917236
$686$	0	0
$687$	27.7205	1.05760
$688$	14.2294	0.542491
$689$	0	0
$690$	−0.378889	−0.0144241
$691$	−25.7677	−0.980248	−0.490124	−	0.871653i	$-0.663049\pi$
$691$	−25.7677	−0.980248	−0.490124	+	0.871653i	$0.663049\pi$
$692$	10.2724	0.390497
$693$	0	0
$694$	7.00589	0.265940
$695$	−26.7119	−1.01324
$696$	13.5100	0.512096
$697$	32.6974	1.23850
$698$	−10.2248	−0.387016
$699$	31.7369	1.20040
$700$	0	0
$701$	41.7872	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$701$	41.7872	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$702$	0	0
$703$	15.7167	0.592765
$704$	−32.5708	−1.22756
$705$	13.2127	0.497619
$706$	2.03471	0.0765774
$707$	0	0
$708$	−2.34782	−0.0882364
$709$	0.343847	0.0129135	0.00645673	−	0.999979i	$-0.497945\pi$
$709$	0.343847	0.0129135	0.00645673	+	0.999979i	$0.497945\pi$
$710$	−1.80771	−0.0678421
$711$	0.113931	0.00427276
$712$	−12.5625	−0.470799
$713$	−3.29931	−0.123560
$714$	0	0
$715$	0	0
$716$	−10.1245	−0.378372
$717$	−29.1523	−1.08871
$718$	−6.45761	−0.240996
$719$	8.78010	0.327443	0.163721	−	0.986507i	$-0.447650\pi$
$719$	8.78010	0.327443	0.163721	+	0.986507i	$0.447650\pi$
$720$	4.20702	0.156786
$721$	0	0
$722$	5.04149	0.187625
$723$	−4.71598	−0.175389
$724$	−14.2473	−0.529498
$725$	−20.8327	−0.773707
$726$	−11.8702	−0.440546
$727$	−17.3658	−0.644064	−0.322032	−	0.946729i	$-0.604366\pi$
$727$	−17.3658	−0.644064	−0.322032	+	0.946729i	$0.604366\pi$
$728$	0	0
$729$	29.6343	1.09757
$730$	4.38936	0.162458
$731$	18.3900	0.680180
$732$	−18.6994	−0.691148
$733$	−9.05895	−0.334600	−0.167300	−	0.985906i	$-0.553505\pi$
$733$	−9.05895	−0.334600	−0.167300	+	0.985906i	$0.553505\pi$
$734$	1.79638	0.0663056
$735$	0	0
$736$	1.99796	0.0736458
$737$	5.87837	0.216533
$738$	2.18143	0.0802995
$739$	−7.07843	−0.260384	−0.130192	−	0.991489i	$-0.541559\pi$
$739$	−7.07843	−0.260384	−0.130192	+	0.991489i	$0.541559\pi$
$740$	−21.9310	−0.806201
$741$	0	0
$742$	0	0
$743$	−14.6779	−0.538479	−0.269240	−	0.963073i	$-0.586772\pi$
$743$	−14.6779	−0.538479	−0.269240	+	0.963073i	$0.586772\pi$
$744$	11.5273	0.422613
$745$	−4.45367	−0.163170
$746$	−5.40132	−0.197756
$747$	2.31827	0.0848209
$748$	−48.7085	−1.78096
$749$	0	0
$750$	−5.54848	−0.202602
$751$	−31.7113	−1.15716	−0.578580	−	0.815626i	$-0.696393\pi$
$751$	−31.7113	−1.15716	−0.578580	+	0.815626i	$0.696393\pi$
$752$	−20.9698	−0.764691
$753$	18.2358	0.664548
$754$	0	0
$755$	3.68467	0.134099
$756$	0	0
$757$	15.5317	0.564510	0.282255	−	0.959339i	$-0.408918\pi$
$757$	15.5317	0.564510	0.282255	+	0.959339i	$0.408918\pi$
$758$	8.36688	0.303899
$759$	−4.69594	−0.170452
$760$	3.65839	0.132704
$761$	0.250369	0.00907587	0.00453794	−	0.999990i	$-0.498556\pi$
$761$	0.250369	0.00907587	0.00453794	+	0.999990i	$0.498556\pi$
$762$	−6.98474	−0.253030
$763$	0	0
$764$	−25.5961	−0.926036
$765$	5.43714	0.196580
$766$	1.26781	0.0458077
$767$	0	0
$768$	11.4972	0.414869
$769$	−24.0146	−0.865988	−0.432994	−	0.901397i	$-0.642543\pi$
$769$	−24.0146	−0.865988	−0.432994	+	0.901397i	$0.642543\pi$
$770$	0	0
$771$	8.50731	0.306383
$772$	35.0625	1.26193
$773$	30.5062	1.09723	0.548616	−	0.836074i	$-0.315155\pi$
$773$	30.5062	1.09723	0.548616	+	0.836074i	$0.315155\pi$
$774$	1.22690	0.0441001
$775$	−17.7754	−0.638510
$776$	−8.51064	−0.305514
$777$	0	0
$778$	0.954237	0.0342110
$779$	−14.7865	−0.529780
$780$	0	0
$781$	−22.4047	−0.801702
$782$	0.777163	0.0277913
$783$	−40.4311	−1.44489
$784$	0	0
$785$	13.6019	0.485473
$786$	4.59414	0.163868
$787$	19.8492	0.707548	0.353774	−	0.935331i	$-0.384898\pi$
$787$	19.8492	0.707548	0.353774	+	0.935331i	$0.384898\pi$
$788$	−5.04149	−0.179596
$789$	25.5686	0.910268
$790$	−0.0630042	−0.00224159
$791$	0	0
$792$	−6.68923	−0.237691
$793$	0	0
$794$	−6.35448	−0.225512
$795$	5.86484	0.208004
$796$	−38.1682	−1.35284
$797$	52.2894	1.85219	0.926093	−	0.377296i	$-0.123146\pi$
$797$	52.2894	1.85219	0.926093	+	0.377296i	$0.123146\pi$
$798$	0	0
$799$	−27.1014	−0.958777
$800$	10.7642	0.380572
$801$	8.44319	0.298326
$802$	0.996542	0.0351891
$803$	54.4015	1.91979
$804$	−2.72283	−0.0960269
$805$	0	0
$806$	0	0
$807$	32.5963	1.14744
$808$	−0.0915664	−0.00322129
$809$	−2.36460	−0.0831349	−0.0415674	−	0.999136i	$-0.513235\pi$
$809$	−2.36460	−0.0831349	−0.0415674	+	0.999136i	$0.513235\pi$
$810$	−2.70920	−0.0951917
$811$	23.6646	0.830978	0.415489	−	0.909598i	$-0.363610\pi$
$811$	23.6646	0.830978	0.415489	+	0.909598i	$0.363610\pi$
$812$	0	0
$813$	38.4952	1.35008
$814$	15.8918	0.557008
$815$	1.00493	0.0352013
$816$	21.1654	0.740936
$817$	−8.31636	−0.290953
$818$	−11.3275	−0.396058
$819$	0	0
$820$	20.6330	0.720536
$821$	−3.57753	−0.124857	−0.0624284	−	0.998049i	$-0.519885\pi$
$821$	−3.57753	−0.124857	−0.0624284	+	0.998049i	$0.519885\pi$
$822$	8.05759	0.281041
$823$	−29.9422	−1.04372	−0.521859	−	0.853032i	$-0.674761\pi$
$823$	−29.9422	−1.04372	−0.521859	+	0.853032i	$0.674761\pi$
$824$	8.19393	0.285449
$825$	−25.2998	−0.880827
$826$	0	0
$827$	−9.32620	−0.324304	−0.162152	−	0.986766i	$-0.551843\pi$
$827$	−9.32620	−0.324304	−0.162152	+	0.986766i	$0.551843\pi$
$828$	−0.886819	−0.0308191
$829$	38.2268	1.32767	0.663836	−	0.747878i	$-0.268927\pi$
$829$	38.2268	1.32767	0.663836	+	0.747878i	$0.268927\pi$
$830$	−1.28200	−0.0444990
$831$	−13.6877	−0.474821
$832$	0	0
$833$	0	0
$834$	−8.96568	−0.310456
$835$	20.2155	0.699587
$836$	22.0270	0.761820
$837$	−34.4975	−1.19241
$838$	−11.5587	−0.399287
$839$	−23.4981	−0.811244	−0.405622	−	0.914041i	$-0.632945\pi$
$839$	−23.4981	−0.811244	−0.405622	+	0.914041i	$0.632945\pi$
$840$	0	0
$841$	22.2451	0.767071
$842$	−8.02356	−0.276510
$843$	25.9491	0.893736
$844$	−24.8034	−0.853767
$845$	0	0
$846$	−1.80808	−0.0621632
$847$	0	0
$848$	−9.30806	−0.319640
$849$	−14.0241	−0.481306
$850$	4.18704	0.143614
$851$	4.33686	0.148666
$852$	10.3777	0.355535
$853$	−40.9295	−1.40140	−0.700700	−	0.713456i	$-0.747129\pi$
$853$	−40.9295	−1.40140	−0.700700	+	0.713456i	$0.747129\pi$
$854$	0	0
$855$	−2.45879	−0.0840888
$856$	10.0185	0.342424
$857$	−11.6620	−0.398366	−0.199183	−	0.979962i	$-0.563829\pi$
$857$	−11.6620	−0.398366	−0.199183	+	0.979962i	$0.563829\pi$
$858$	0	0
$859$	28.2776	0.964820	0.482410	−	0.875945i	$-0.339762\pi$
$859$	28.2776	0.964820	0.482410	+	0.875945i	$0.339762\pi$
$860$	11.6046	0.395715
$861$	0	0
$862$	−1.58369	−0.0539407
$863$	−11.4442	−0.389567	−0.194783	−	0.980846i	$-0.562400\pi$
$863$	−11.4442	−0.389567	−0.194783	+	0.980846i	$0.562400\pi$
$864$	20.8906	0.710713
$865$	7.85909	0.267217
$866$	−7.31993	−0.248741
$867$	2.53683	0.0861553
$868$	0	0
$869$	−0.780871	−0.0264892
$870$	5.02127	0.170237
$871$	0	0
$872$	0.0433887	0.00146933
$873$	5.71997	0.193592
$874$	−0.351449	−0.0118879
$875$	0	0
$876$	−25.1985	−0.851380
$877$	−56.3486	−1.90276	−0.951378	−	0.308026i	$-0.900332\pi$
$877$	−56.3486	−1.90276	−0.951378	+	0.308026i	$0.900332\pi$
$878$	−1.14084	−0.0385016
$879$	16.9835	0.572838
$880$	−28.8344	−0.972006
$881$	−1.16418	−0.0392221	−0.0196111	−	0.999808i	$-0.506243\pi$
$881$	−1.16418	−0.0392221	−0.0196111	+	0.999808i	$0.506243\pi$
$882$	0	0
$883$	−12.1881	−0.410162	−0.205081	−	0.978745i	$-0.565746\pi$
$883$	−12.1881	−0.410162	−0.205081	+	0.978745i	$0.565746\pi$
$884$	0	0
$885$	−1.79625	−0.0603802
$886$	2.89364	0.0972138
$887$	30.6641	1.02960	0.514799	−	0.857311i	$-0.327866\pi$
$887$	30.6641	1.02960	0.514799	+	0.857311i	$0.327866\pi$
$888$	−15.1524	−0.508481
$889$	0	0
$890$	−4.66910	−0.156509
$891$	−33.5778	−1.12490
$892$	−4.20105	−0.140662
$893$	12.2558	0.410125
$894$	−1.49485	−0.0499952
$895$	−7.74598	−0.258920
$896$	0	0
$897$	0	0
$898$	−5.85378	−0.195343
$899$	43.7245	1.45829
$900$	−4.77782	−0.159261
$901$	−12.0297	−0.400768
$902$	−14.9512	−0.497822
$903$	0	0
$904$	−11.8934	−0.395570
$905$	−10.9002	−0.362336
$906$	1.23674	0.0410878
$907$	−11.6479	−0.386763	−0.193382	−	0.981124i	$-0.561946\pi$
$907$	−11.6479	−0.386763	−0.193382	+	0.981124i	$0.561946\pi$
$908$	51.2147	1.69962
$909$	0.0615414	0.00204120
$910$	0	0
$911$	−26.5833	−0.880743	−0.440371	−	0.897816i	$-0.645153\pi$
$911$	−26.5833	−0.880743	−0.440371	+	0.897816i	$0.645153\pi$
$912$	−9.57141	−0.316941
$913$	−15.8891	−0.525852
$914$	3.01553	0.0997450
$915$	−14.3063	−0.472953
$916$	−35.8797	−1.18550
$917$	0	0
$918$	8.12600	0.268198
$919$	−45.7079	−1.50777	−0.753883	−	0.657009i	$-0.771821\pi$
$919$	−45.7079	−1.50777	−0.753883	+	0.657009i	$0.771821\pi$
$920$	1.00950	0.0332821
$921$	−20.1866	−0.665171
$922$	−2.18809	−0.0720609
$923$	0	0
$924$	0	0
$925$	23.3653	0.768246
$926$	−1.15448	−0.0379387
$927$	−5.50711	−0.180877
$928$	−26.4782	−0.869189
$929$	−11.0651	−0.363035	−0.181518	−	0.983388i	$-0.558101\pi$
$929$	−11.0651	−0.363035	−0.181518	+	0.983388i	$0.558101\pi$
$930$	4.28437	0.140490
$931$	0	0
$932$	−41.0784	−1.34557
$933$	44.8380	1.46793
$934$	−9.89994	−0.323936
$935$	−37.2655	−1.21871
$936$	0	0
$937$	−57.6584	−1.88362	−0.941808	−	0.336150i	$-0.890875\pi$
$937$	−57.6584	−1.88362	−0.941808	+	0.336150i	$0.890875\pi$
$938$	0	0
$939$	16.1848	0.528170
$940$	−17.1017	−0.557797
$941$	−16.9745	−0.553351	−0.276676	−	0.960963i	$-0.589233\pi$
$941$	−16.9745	−0.553351	−0.276676	+	0.960963i	$0.589233\pi$
$942$	4.56540	0.148749
$943$	−4.08018	−0.132869
$944$	2.85082	0.0927862
$945$	0	0
$946$	−8.40904	−0.273401
$947$	−16.6962	−0.542552	−0.271276	−	0.962502i	$-0.587446\pi$
$947$	−16.6962	−0.542552	−0.271276	+	0.962502i	$0.587446\pi$
$948$	0.361696	0.0117473
$949$	0	0
$950$	−1.89347	−0.0614322
$951$	34.8658	1.13060
$952$	0	0
$953$	−18.2473	−0.591089	−0.295545	−	0.955329i	$-0.595501\pi$
$953$	−18.2473	−0.591089	−0.295545	+	0.955329i	$0.595501\pi$
$954$	−0.802570	−0.0259842
$955$	−19.5829	−0.633687
$956$	37.7330	1.22037
$957$	62.2334	2.01172
$958$	−11.6948	−0.377841
$959$	0	0
$960$	11.5423	0.372527
$961$	6.30763	0.203472
$962$	0	0
$963$	−6.73337	−0.216980
$964$	6.10408	0.196599
$965$	26.8253	0.863537
$966$	0	0
$967$	−29.5845	−0.951374	−0.475687	−	0.879615i	$-0.657801\pi$
$967$	−29.5845	−0.951374	−0.475687	+	0.879615i	$0.657801\pi$
$968$	31.6265	1.01652
$969$	−12.3701	−0.397384
$970$	−3.16315	−0.101563
$971$	15.1301	0.485548	0.242774	−	0.970083i	$-0.421943\pi$
$971$	15.1301	0.485548	0.242774	+	0.970083i	$0.421943\pi$
$972$	−16.4627	−0.528040
$973$	0	0
$974$	0.598787	0.0191864
$975$	0	0
$976$	22.7055	0.726786
$977$	32.4636	1.03860	0.519301	−	0.854591i	$-0.326192\pi$
$977$	32.4636	1.03860	0.519301	+	0.854591i	$0.326192\pi$
$978$	0.337300	0.0107857
$979$	−57.8686	−1.84949
$980$	0	0
$981$	−0.0291614	−0.000931052 0
$982$	2.72322	0.0869016
$983$	−44.2945	−1.41278	−0.706388	−	0.707825i	$-0.749677\pi$
$983$	−44.2945	−1.41278	−0.706388	+	0.707825i	$0.749677\pi$
$984$	14.2556	0.454452
$985$	−3.85710	−0.122897
$986$	−10.2994	−0.328001
$987$	0	0
$988$	0	0
$989$	−2.29482	−0.0729710
$990$	−2.48619	−0.0790163
$991$	−8.52117	−0.270684	−0.135342	−	0.990799i	$-0.543213\pi$
$991$	−8.52117	−0.270684	−0.135342	+	0.990799i	$0.543213\pi$
$992$	−22.5923	−0.717307
$993$	26.7500	0.848887
$994$	0	0
$995$	−29.2014	−0.925746
$996$	7.35976	0.233203
$997$	−6.77905	−0.214695	−0.107347	−	0.994222i	$-0.534236\pi$
$997$	−6.77905	−0.214695	−0.107347	+	0.994222i	$0.534236\pi$
$998$	12.1372	0.384195
$999$	45.3462	1.43469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ck.1.4		8
7.2	even	3		1183.2.e.i.508.5		16
7.4	even	3		1183.2.e.i.170.5		16
7.6	odd	2		8281.2.a.cj.1.4		8
13.5	odd	4		637.2.c.f.246.5		8
13.8	odd	4		637.2.c.f.246.4		8
13.12	even	2	inner	8281.2.a.ck.1.5		8
91.5	even	12		637.2.r.f.116.5		16
91.18	odd	12		91.2.r.a.51.4	yes	16
91.25	even	6		1183.2.e.i.170.4		16
91.31	even	12		637.2.r.f.324.4		16
91.34	even	4		637.2.c.e.246.4		8
91.44	odd	12		91.2.r.a.25.5	yes	16
91.47	even	12		637.2.r.f.116.4		16
91.51	even	6		1183.2.e.i.508.4		16
91.60	odd	12		91.2.r.a.51.5	yes	16
91.73	even	12		637.2.r.f.324.5		16
91.83	even	4		637.2.c.e.246.5		8
91.86	odd	12		91.2.r.a.25.4	✓	16
91.90	odd	2		8281.2.a.cj.1.5		8
273.44	even	12		819.2.dl.e.298.4		16
273.86	even	12		819.2.dl.e.298.5		16
273.200	even	12		819.2.dl.e.415.5		16
273.242	even	12		819.2.dl.e.415.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.r.a.25.4	✓	16	91.86	odd	12
91.2.r.a.25.5	yes	16	91.44	odd	12
91.2.r.a.51.4	yes	16	91.18	odd	12
91.2.r.a.51.5	yes	16	91.60	odd	12
637.2.c.e.246.4		8	91.34	even	4
637.2.c.e.246.5		8	91.83	even	4
637.2.c.f.246.4		8	13.8	odd	4
637.2.c.f.246.5		8	13.5	odd	4
637.2.r.f.116.4		16	91.47	even	12
637.2.r.f.116.5		16	91.5	even	12
637.2.r.f.324.4		16	91.31	even	12
637.2.r.f.324.5		16	91.73	even	12
819.2.dl.e.298.4		16	273.44	even	12
819.2.dl.e.298.5		16	273.86	even	12
819.2.dl.e.415.4		16	273.242	even	12
819.2.dl.e.415.5		16	273.200	even	12
1183.2.e.i.170.4		16	91.25	even	6
1183.2.e.i.170.5		16	7.4	even	3
1183.2.e.i.508.4		16	91.51	even	6
1183.2.e.i.508.5		16	7.2	even	3
8281.2.a.cj.1.4		8	7.6	odd	2
8281.2.a.cj.1.5		8	91.90	odd	2
8281.2.a.ck.1.4		8	1.1	even	1	trivial
8281.2.a.ck.1.5		8	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-0.332375 q^{2} +1.45984 q^{3} -1.88953 q^{4} -1.44562 q^{5} -0.485214 q^{6} +1.29278 q^{8} -0.868875 q^{9} +O(q^{10})\)	\(q-0.332375 q^{2} +1.45984 q^{3} -1.88953 q^{4} -1.44562 q^{5} -0.485214 q^{6} +1.29278 q^{8} -0.868875 q^{9} +0.480489 q^{10} +5.95516 q^{11} -2.75840 q^{12} -2.11037 q^{15} +3.34936 q^{16} +4.32871 q^{17} +0.288793 q^{18} -1.95753 q^{19} +2.73154 q^{20} -1.97935 q^{22} -0.540163 q^{23} +1.88725 q^{24} -2.91018 q^{25} -5.64793 q^{27} +7.15857 q^{29} +0.701436 q^{30} +6.10800 q^{31} -3.69881 q^{32} +8.69356 q^{33} -1.43876 q^{34} +1.64176 q^{36} -8.02881 q^{37} +0.650636 q^{38} -1.86887 q^{40} +7.55362 q^{41} +4.24839 q^{43} -11.2524 q^{44} +1.25606 q^{45} +0.179537 q^{46} -6.26084 q^{47} +4.88953 q^{48} +0.967272 q^{50} +6.31922 q^{51} -2.77905 q^{53} +1.87723 q^{54} -8.60891 q^{55} -2.85768 q^{57} -2.37933 q^{58} +0.851152 q^{59} +3.98760 q^{60} +6.77905 q^{61} -2.03015 q^{62} -5.46933 q^{64} -2.88953 q^{66} +0.987106 q^{67} -8.17922 q^{68} -0.788550 q^{69} -3.76223 q^{71} -1.12327 q^{72} +9.13519 q^{73} +2.66858 q^{74} -4.24839 q^{75} +3.69881 q^{76} -0.131125 q^{79} -4.84191 q^{80} -5.63843 q^{81} -2.51064 q^{82} -2.66812 q^{83} -6.25768 q^{85} -1.41206 q^{86} +10.4503 q^{87} +7.69873 q^{88} -9.71739 q^{89} -0.417485 q^{90} +1.02065 q^{92} +8.91668 q^{93} +2.08095 q^{94} +2.82985 q^{95} -5.39966 q^{96} -6.58319 q^{97} -5.17429 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9	\( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9} - 6 q^{10} + 18 q^{12} - 2 q^{16} + 8 q^{17} - 18 q^{22} - 12 q^{23} + 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} + 34 q^{38} + 4 q^{40} - 8 q^{43} + 18 q^{48} + 16 q^{51} + 20 q^{53} + 12 q^{55} + 12 q^{61} - 22 q^{62} - 44 q^{64} - 2 q^{66} + 2 q^{68} + 28 q^{69} - 42 q^{74} + 8 q^{75} - 20 q^{79} + 24 q^{81} - 16 q^{82} + 68 q^{87} + 4 q^{88} + 108 q^{90} + 6 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) \) 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9 - 6 * q^10 + 18 * q^12 - 2 * q^16 + 8 * q^17 - 18 * q^22 - 12 * q^23 + 16 * q^27 - 8 * q^29 + 38 * q^30 + 28 * q^36 + 34 * q^38 + 4 * q^40 - 8 * q^43 + 18 * q^48 + 16 * q^51 + 20 * q^53 + 12 * q^55 + 12 * q^61 - 22 * q^62 - 44 * q^64 - 2 * q^66 + 2 * q^68 + 28 * q^69 - 42 * q^74 + 8 * q^75 - 20 * q^79 + 24 * q^81 - 16 * q^82 + 68 * q^87 + 4 * q^88 + 108 * q^90 + 6 * q^92 + 26 * q^94 - 16 * q^95

Introduction

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