LMFDB - Embedded newform 8016.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.76045$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.76045 q^{5} -3.28542 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.76045 q^{5} -3.28542 q^{7} +1.00000 q^{9} +3.56961 q^{11} -4.67902 q^{13} -2.76045 q^{15} +6.82998 q^{17} -3.92978 q^{19} -3.28542 q^{21} +6.73062 q^{23} +2.62008 q^{25} +1.00000 q^{27} +6.59666 q^{29} -7.05778 q^{31} +3.56961 q^{33} +9.06924 q^{35} -3.23882 q^{37} -4.67902 q^{39} -6.47113 q^{41} +6.00186 q^{43} -2.76045 q^{45} -2.08378 q^{47} +3.79399 q^{49} +6.82998 q^{51} +2.26395 q^{53} -9.85372 q^{55} -3.92978 q^{57} +2.05527 q^{59} +4.28411 q^{61} -3.28542 q^{63} +12.9162 q^{65} -7.96623 q^{67} +6.73062 q^{69} +3.87844 q^{71} +4.15348 q^{73} +2.62008 q^{75} -11.7277 q^{77} -1.39322 q^{79} +1.00000 q^{81} +4.70968 q^{83} -18.8538 q^{85} +6.59666 q^{87} +3.98357 q^{89} +15.3726 q^{91} -7.05778 q^{93} +10.8480 q^{95} -12.4233 q^{97} +3.56961 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9	$ 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9 + q^11 - 6 * q^13 - 10 * q^15 - 9 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 12 * q^25 + 10 * q^27 - 13 * q^29 - 23 * q^31 + q^33 - q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 10 * q^45 - 10 * q^47 + 7 * q^49 - 9 * q^51 - 26 * q^53 - 11 * q^55 - 2 * q^57 + 10 * q^59 - 10 * q^61 - q^63 - 22 * q^65 + 5 * q^67 - 7 * q^69 - 25 * q^71 - 8 * q^73 + 12 * q^75 - 46 * q^77 - 26 * q^79 + 10 * q^81 + 14 * q^83 + 9 * q^85 - 13 * q^87 - 31 * q^89 + 3 * q^91 - 23 * q^93 + 5 * q^95 - 32 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.76045	−1.23451	−0.617255	−	0.786763i	$-0.711755\pi$
$5$	−2.76045	−1.23451	−0.617255	+	0.786763i	$0.711755\pi$
$6$	0	0
$7$	−3.28542	−1.24177	−0.620886	−	0.783901i	$-0.713227\pi$
$7$	−3.28542	−1.24177	−0.620886	+	0.783901i	$0.713227\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.56961	1.07628	0.538138	−	0.842857i	$-0.319128\pi$
$11$	3.56961	1.07628	0.538138	+	0.842857i	$0.319128\pi$
$12$	0	0
$13$	−4.67902	−1.29773	−0.648864	−	0.760904i	$-0.724756\pi$
$13$	−4.67902	−1.29773	−0.648864	+	0.760904i	$0.724756\pi$
$14$	0	0
$15$	−2.76045	−0.712745
$16$	0	0
$17$	6.82998	1.65651	0.828257	−	0.560348i	$-0.189333\pi$
$17$	6.82998	1.65651	0.828257	+	0.560348i	$0.189333\pi$
$18$	0	0
$19$	−3.92978	−0.901554	−0.450777	−	0.892637i	$-0.648853\pi$
$19$	−3.92978	−0.901554	−0.450777	+	0.892637i	$0.648853\pi$
$20$	0	0
$21$	−3.28542	−0.716938
$22$	0	0
$23$	6.73062	1.40343	0.701715	−	0.712457i	$-0.252418\pi$
$23$	6.73062	1.40343	0.701715	+	0.712457i	$0.252418\pi$
$24$	0	0
$25$	2.62008	0.524016
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.59666	1.22497	0.612484	−	0.790483i	$-0.290170\pi$
$29$	6.59666	1.22497	0.612484	+	0.790483i	$0.290170\pi$
$30$	0	0
$31$	−7.05778	−1.26762	−0.633808	−	0.773491i	$-0.718509\pi$
$31$	−7.05778	−1.26762	−0.633808	+	0.773491i	$0.718509\pi$
$32$	0	0
$33$	3.56961	0.621389
$34$	0	0
$35$	9.06924	1.53298
$36$	0	0
$37$	−3.23882	−0.532458	−0.266229	−	0.963910i	$-0.585778\pi$
$37$	−3.23882	−0.532458	−0.266229	+	0.963910i	$0.585778\pi$
$38$	0	0
$39$	−4.67902	−0.749244
$40$	0	0
$41$	−6.47113	−1.01062	−0.505310	−	0.862938i	$-0.668622\pi$
$41$	−6.47113	−1.01062	−0.505310	+	0.862938i	$0.668622\pi$
$42$	0	0
$43$	6.00186	0.915275	0.457638	−	0.889139i	$-0.348696\pi$
$43$	6.00186	0.915275	0.457638	+	0.889139i	$0.348696\pi$
$44$	0	0
$45$	−2.76045	−0.411503
$46$	0	0
$47$	−2.08378	−0.303951	−0.151976	−	0.988384i	$-0.548563\pi$
$47$	−2.08378	−0.303951	−0.151976	+	0.988384i	$0.548563\pi$
$48$	0	0
$49$	3.79399	0.541999
$50$	0	0
$51$	6.82998	0.956389
$52$	0	0
$53$	2.26395	0.310978	0.155489	−	0.987838i	$-0.450305\pi$
$53$	2.26395	0.310978	0.155489	+	0.987838i	$0.450305\pi$
$54$	0	0
$55$	−9.85372	−1.32867
$56$	0	0
$57$	−3.92978	−0.520512
$58$	0	0
$59$	2.05527	0.267573	0.133786	−	0.991010i	$-0.457286\pi$
$59$	2.05527	0.267573	0.133786	+	0.991010i	$0.457286\pi$
$60$	0	0
$61$	4.28411	0.548524	0.274262	−	0.961655i	$-0.411566\pi$
$61$	4.28411	0.548524	0.274262	+	0.961655i	$0.411566\pi$
$62$	0	0
$63$	−3.28542	−0.413924
$64$	0	0
$65$	12.9162	1.60206
$66$	0	0
$67$	−7.96623	−0.973229	−0.486615	−	0.873617i	$-0.661768\pi$
$67$	−7.96623	−0.973229	−0.486615	+	0.873617i	$0.661768\pi$
$68$	0	0
$69$	6.73062	0.810271
$70$	0	0
$71$	3.87844	0.460286	0.230143	−	0.973157i	$-0.426081\pi$
$71$	3.87844	0.460286	0.230143	+	0.973157i	$0.426081\pi$
$72$	0	0
$73$	4.15348	0.486128	0.243064	−	0.970010i	$-0.421848\pi$
$73$	4.15348	0.486128	0.243064	+	0.970010i	$0.421848\pi$
$74$	0	0
$75$	2.62008	0.302541
$76$	0	0
$77$	−11.7277	−1.33649
$78$	0	0
$79$	−1.39322	−0.156750	−0.0783749	−	0.996924i	$-0.524973\pi$
$79$	−1.39322	−0.156750	−0.0783749	+	0.996924i	$0.524973\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.70968	0.516954	0.258477	−	0.966017i	$-0.416779\pi$
$83$	4.70968	0.516954	0.258477	+	0.966017i	$0.416779\pi$
$84$	0	0
$85$	−18.8538	−2.04498
$86$	0	0
$87$	6.59666	0.707236
$88$	0	0
$89$	3.98357	0.422258	0.211129	−	0.977458i	$-0.432286\pi$
$89$	3.98357	0.422258	0.211129	+	0.977458i	$0.432286\pi$
$90$	0	0
$91$	15.3726	1.61148
$92$	0	0
$93$	−7.05778	−0.731858
$94$	0	0
$95$	10.8480	1.11298
$96$	0	0
$97$	−12.4233	−1.26140	−0.630699	−	0.776028i	$-0.717232\pi$
$97$	−12.4233	−1.26140	−0.630699	+	0.776028i	$0.717232\pi$
$98$	0	0
$99$	3.56961	0.358759
$100$	0	0
$101$	−15.3884	−1.53121	−0.765603	−	0.643314i	$-0.777559\pi$
$101$	−15.3884	−1.53121	−0.765603	+	0.643314i	$0.777559\pi$
$102$	0	0
$103$	3.07497	0.302986	0.151493	−	0.988458i	$-0.451592\pi$
$103$	3.07497	0.302986	0.151493	+	0.988458i	$0.451592\pi$
$104$	0	0
$105$	9.06924	0.885067
$106$	0	0
$107$	14.4266	1.39467	0.697337	−	0.716743i	$-0.254368\pi$
$107$	14.4266	1.39467	0.697337	+	0.716743i	$0.254368\pi$
$108$	0	0
$109$	16.6703	1.59673	0.798363	−	0.602176i	$-0.205700\pi$
$109$	16.6703	1.59673	0.798363	+	0.602176i	$0.205700\pi$
$110$	0	0
$111$	−3.23882	−0.307415
$112$	0	0
$113$	−12.1684	−1.14470	−0.572352	−	0.820008i	$-0.693969\pi$
$113$	−12.1684	−1.14470	−0.572352	+	0.820008i	$0.693969\pi$
$114$	0	0
$115$	−18.5795	−1.73255
$116$	0	0
$117$	−4.67902	−0.432576
$118$	0	0
$119$	−22.4394	−2.05701
$120$	0	0
$121$	1.74209	0.158372
$122$	0	0
$123$	−6.47113	−0.583482
$124$	0	0
$125$	6.56965	0.587607
$126$	0	0
$127$	−2.60972	−0.231575	−0.115787	−	0.993274i	$-0.536939\pi$
$127$	−2.60972	−0.231575	−0.115787	+	0.993274i	$0.536939\pi$
$128$	0	0
$129$	6.00186	0.528434
$130$	0	0
$131$	8.14124	0.711303	0.355652	−	0.934619i	$-0.384259\pi$
$131$	8.14124	0.711303	0.355652	+	0.934619i	$0.384259\pi$
$132$	0	0
$133$	12.9110	1.11952
$134$	0	0
$135$	−2.76045	−0.237582
$136$	0	0
$137$	−10.8400	−0.926124	−0.463062	−	0.886326i	$-0.653249\pi$
$137$	−10.8400	−0.926124	−0.463062	+	0.886326i	$0.653249\pi$
$138$	0	0
$139$	−10.4840	−0.889244	−0.444622	−	0.895718i	$-0.646662\pi$
$139$	−10.4840	−0.889244	−0.444622	+	0.895718i	$0.646662\pi$
$140$	0	0
$141$	−2.08378	−0.175486
$142$	0	0
$143$	−16.7023	−1.39671
$144$	0	0
$145$	−18.2097	−1.51224
$146$	0	0
$147$	3.79399	0.312923
$148$	0	0
$149$	−19.5436	−1.60108	−0.800539	−	0.599281i	$-0.795453\pi$
$149$	−19.5436	−1.60108	−0.800539	+	0.599281i	$0.795453\pi$
$150$	0	0
$151$	−12.7540	−1.03791	−0.518955	−	0.854802i	$-0.673679\pi$
$151$	−12.7540	−1.03791	−0.518955	+	0.854802i	$0.673679\pi$
$152$	0	0
$153$	6.82998	0.552171
$154$	0	0
$155$	19.4827	1.56488
$156$	0	0
$157$	8.06348	0.643535	0.321768	−	0.946819i	$-0.395723\pi$
$157$	8.06348	0.643535	0.321768	+	0.946819i	$0.395723\pi$
$158$	0	0
$159$	2.26395	0.179543
$160$	0	0
$161$	−22.1129	−1.74274
$162$	0	0
$163$	6.62005	0.518522	0.259261	−	0.965807i	$-0.416521\pi$
$163$	6.62005	0.518522	0.259261	+	0.965807i	$0.416521\pi$
$164$	0	0
$165$	−9.85372	−0.767111
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	8.89327	0.684098
$170$	0	0
$171$	−3.92978	−0.300518
$172$	0	0
$173$	−7.89738	−0.600427	−0.300213	−	0.953872i	$-0.597058\pi$
$173$	−7.89738	−0.600427	−0.300213	+	0.953872i	$0.597058\pi$
$174$	0	0
$175$	−8.60807	−0.650709
$176$	0	0
$177$	2.05527	0.154483
$178$	0	0
$179$	13.2574	0.990901	0.495450	−	0.868636i	$-0.335003\pi$
$179$	13.2574	0.990901	0.495450	+	0.868636i	$0.335003\pi$
$180$	0	0
$181$	−23.3400	−1.73485	−0.867425	−	0.497567i	$-0.834227\pi$
$181$	−23.3400	−1.73485	−0.867425	+	0.497567i	$0.834227\pi$
$182$	0	0
$183$	4.28411	0.316691
$184$	0	0
$185$	8.94059	0.657325
$186$	0	0
$187$	24.3804	1.78287
$188$	0	0
$189$	−3.28542	−0.238979
$190$	0	0
$191$	−12.6160	−0.912864	−0.456432	−	0.889758i	$-0.650873\pi$
$191$	−12.6160	−0.912864	−0.456432	+	0.889758i	$0.650873\pi$
$192$	0	0
$193$	−26.9494	−1.93986	−0.969928	−	0.243390i	$-0.921740\pi$
$193$	−26.9494	−1.93986	−0.969928	+	0.243390i	$0.921740\pi$
$194$	0	0
$195$	12.9162	0.924949
$196$	0	0
$197$	−4.63420	−0.330173	−0.165086	−	0.986279i	$-0.552790\pi$
$197$	−4.63420	−0.330173	−0.165086	+	0.986279i	$0.552790\pi$
$198$	0	0
$199$	4.52367	0.320674	0.160337	−	0.987062i	$-0.448742\pi$
$199$	4.52367	0.320674	0.160337	+	0.987062i	$0.448742\pi$
$200$	0	0
$201$	−7.96623	−0.561894
$202$	0	0
$203$	−21.6728	−1.52113
$204$	0	0
$205$	17.8632	1.24762
$206$	0	0
$207$	6.73062	0.467810
$208$	0	0
$209$	−14.0278	−0.970321
$210$	0	0
$211$	−16.5460	−1.13907	−0.569536	−	0.821966i	$-0.692877\pi$
$211$	−16.5460	−1.13907	−0.569536	+	0.821966i	$0.692877\pi$
$212$	0	0
$213$	3.87844	0.265746
$214$	0	0
$215$	−16.5678	−1.12992
$216$	0	0
$217$	23.1878	1.57409
$218$	0	0
$219$	4.15348	0.280666
$220$	0	0
$221$	−31.9577	−2.14971
$222$	0	0
$223$	−16.8807	−1.13042	−0.565208	−	0.824948i	$-0.691204\pi$
$223$	−16.8807	−1.13042	−0.565208	+	0.824948i	$0.691204\pi$
$224$	0	0
$225$	2.62008	0.174672
$226$	0	0
$227$	−3.39218	−0.225147	−0.112574	−	0.993643i	$-0.535909\pi$
$227$	−3.39218	−0.225147	−0.112574	+	0.993643i	$0.535909\pi$
$228$	0	0
$229$	−8.50012	−0.561703	−0.280852	−	0.959751i	$-0.590617\pi$
$229$	−8.50012	−0.561703	−0.280852	+	0.959751i	$0.590617\pi$
$230$	0	0
$231$	−11.7277	−0.771623
$232$	0	0
$233$	−9.20099	−0.602777	−0.301389	−	0.953501i	$-0.597450\pi$
$233$	−9.20099	−0.602777	−0.301389	+	0.953501i	$0.597450\pi$
$234$	0	0
$235$	5.75218	0.375231
$236$	0	0
$237$	−1.39322	−0.0904995
$238$	0	0
$239$	16.4125	1.06163	0.530817	−	0.847486i	$-0.321885\pi$
$239$	16.4125	1.06163	0.530817	+	0.847486i	$0.321885\pi$
$240$	0	0
$241$	−9.43359	−0.607671	−0.303835	−	0.952725i	$-0.598267\pi$
$241$	−9.43359	−0.607671	−0.303835	+	0.952725i	$0.598267\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.4731	−0.669104
$246$	0	0
$247$	18.3875	1.16997
$248$	0	0
$249$	4.70968	0.298464
$250$	0	0
$251$	19.5625	1.23477	0.617386	−	0.786660i	$-0.288192\pi$
$251$	19.5625	1.23477	0.617386	+	0.786660i	$0.288192\pi$
$252$	0	0
$253$	24.0257	1.51048
$254$	0	0
$255$	−18.8538	−1.18067
$256$	0	0
$257$	−13.7550	−0.858012	−0.429006	−	0.903302i	$-0.641136\pi$
$257$	−13.7550	−0.858012	−0.429006	+	0.903302i	$0.641136\pi$
$258$	0	0
$259$	10.6409	0.661192
$260$	0	0
$261$	6.59666	0.408323
$262$	0	0
$263$	−27.7159	−1.70904	−0.854518	−	0.519421i	$-0.826148\pi$
$263$	−27.7159	−1.70904	−0.854518	+	0.519421i	$0.826148\pi$
$264$	0	0
$265$	−6.24952	−0.383905
$266$	0	0
$267$	3.98357	0.243791
$268$	0	0
$269$	−23.5754	−1.43742	−0.718708	−	0.695312i	$-0.755266\pi$
$269$	−23.5754	−1.43742	−0.718708	+	0.695312i	$0.755266\pi$
$270$	0	0
$271$	11.1851	0.679446	0.339723	−	0.940526i	$-0.389667\pi$
$271$	11.1851	0.679446	0.339723	+	0.940526i	$0.389667\pi$
$272$	0	0
$273$	15.3726	0.930390
$274$	0	0
$275$	9.35265	0.563986
$276$	0	0
$277$	−29.0865	−1.74764	−0.873818	−	0.486253i	$-0.838363\pi$
$277$	−29.0865	−1.74764	−0.873818	+	0.486253i	$0.838363\pi$
$278$	0	0
$279$	−7.05778	−0.422538
$280$	0	0
$281$	−7.20697	−0.429932	−0.214966	−	0.976622i	$-0.568964\pi$
$281$	−7.20697	−0.429932	−0.214966	+	0.976622i	$0.568964\pi$
$282$	0	0
$283$	9.79768	0.582412	0.291206	−	0.956660i	$-0.405944\pi$
$283$	9.79768	0.582412	0.291206	+	0.956660i	$0.405944\pi$
$284$	0	0
$285$	10.8480	0.642578
$286$	0	0
$287$	21.2604	1.25496
$288$	0	0
$289$	29.6487	1.74404
$290$	0	0
$291$	−12.4233	−0.728268
$292$	0	0
$293$	−7.13122	−0.416610	−0.208305	−	0.978064i	$-0.566795\pi$
$293$	−7.13122	−0.416610	−0.208305	+	0.978064i	$0.566795\pi$
$294$	0	0
$295$	−5.67346	−0.330321
$296$	0	0
$297$	3.56961	0.207130
$298$	0	0
$299$	−31.4927	−1.82127
$300$	0	0
$301$	−19.7186	−1.13656
$302$	0	0
$303$	−15.3884	−0.884042
$304$	0	0
$305$	−11.8261	−0.677159
$306$	0	0
$307$	22.1521	1.26429	0.632143	−	0.774852i	$-0.282176\pi$
$307$	22.1521	1.26429	0.632143	+	0.774852i	$0.282176\pi$
$308$	0	0
$309$	3.07497	0.174929
$310$	0	0
$311$	2.88112	0.163373	0.0816867	−	0.996658i	$-0.473969\pi$
$311$	2.88112	0.163373	0.0816867	+	0.996658i	$0.473969\pi$
$312$	0	0
$313$	−2.35289	−0.132993	−0.0664965	−	0.997787i	$-0.521182\pi$
$313$	−2.35289	−0.132993	−0.0664965	+	0.997787i	$0.521182\pi$
$314$	0	0
$315$	9.06924	0.510994
$316$	0	0
$317$	−21.9928	−1.23524	−0.617620	−	0.786477i	$-0.711903\pi$
$317$	−21.9928	−1.23524	−0.617620	+	0.786477i	$0.711903\pi$
$318$	0	0
$319$	23.5475	1.31841
$320$	0	0
$321$	14.4266	0.805216
$322$	0	0
$323$	−26.8403	−1.49344
$324$	0	0
$325$	−12.2594	−0.680030
$326$	0	0
$327$	16.6703	0.921870
$328$	0	0
$329$	6.84611	0.377438
$330$	0	0
$331$	23.8674	1.31187	0.655935	−	0.754817i	$-0.272274\pi$
$331$	23.8674	1.31187	0.655935	+	0.754817i	$0.272274\pi$
$332$	0	0
$333$	−3.23882	−0.177486
$334$	0	0
$335$	21.9904	1.20146
$336$	0	0
$337$	16.2392	0.884604	0.442302	−	0.896866i	$-0.354162\pi$
$337$	16.2392	0.884604	0.442302	+	0.896866i	$0.354162\pi$
$338$	0	0
$339$	−12.1684	−0.660895
$340$	0	0
$341$	−25.1935	−1.36431
$342$	0	0
$343$	10.5331	0.568733
$344$	0	0
$345$	−18.5795	−1.00029
$346$	0	0
$347$	−17.5565	−0.942481	−0.471240	−	0.882005i	$-0.656194\pi$
$347$	−17.5565	−0.942481	−0.471240	+	0.882005i	$0.656194\pi$
$348$	0	0
$349$	−28.2405	−1.51168	−0.755840	−	0.654756i	$-0.772771\pi$
$349$	−28.2405	−1.51168	−0.755840	+	0.654756i	$0.772771\pi$
$350$	0	0
$351$	−4.67902	−0.249748
$352$	0	0
$353$	−12.7708	−0.679719	−0.339860	−	0.940476i	$-0.610380\pi$
$353$	−12.7708	−0.679719	−0.339860	+	0.940476i	$0.610380\pi$
$354$	0	0
$355$	−10.7062	−0.568228
$356$	0	0
$357$	−22.4394	−1.18762
$358$	0	0
$359$	−11.4534	−0.604490	−0.302245	−	0.953230i	$-0.597736\pi$
$359$	−11.4534	−0.604490	−0.302245	+	0.953230i	$0.597736\pi$
$360$	0	0
$361$	−3.55682	−0.187201
$362$	0	0
$363$	1.74209	0.0914359
$364$	0	0
$365$	−11.4655	−0.600130
$366$	0	0
$367$	−23.9387	−1.24959	−0.624796	−	0.780788i	$-0.714818\pi$
$367$	−23.9387	−1.24959	−0.624796	+	0.780788i	$0.714818\pi$
$368$	0	0
$369$	−6.47113	−0.336873
$370$	0	0
$371$	−7.43803	−0.386163
$372$	0	0
$373$	−13.5603	−0.702123	−0.351062	−	0.936352i	$-0.614179\pi$
$373$	−13.5603	−0.702123	−0.351062	+	0.936352i	$0.614179\pi$
$374$	0	0
$375$	6.56965	0.339255
$376$	0	0
$377$	−30.8659	−1.58968
$378$	0	0
$379$	−23.0700	−1.18503	−0.592514	−	0.805560i	$-0.701864\pi$
$379$	−23.0700	−1.18503	−0.592514	+	0.805560i	$0.701864\pi$
$380$	0	0
$381$	−2.60972	−0.133700
$382$	0	0
$383$	−7.53908	−0.385229	−0.192614	−	0.981275i	$-0.561697\pi$
$383$	−7.53908	−0.385229	−0.192614	+	0.981275i	$0.561697\pi$
$384$	0	0
$385$	32.3736	1.64991
$386$	0	0
$387$	6.00186	0.305092
$388$	0	0
$389$	36.0867	1.82967	0.914833	−	0.403831i	$-0.132322\pi$
$389$	36.0867	1.82967	0.914833	+	0.403831i	$0.132322\pi$
$390$	0	0
$391$	45.9700	2.32480
$392$	0	0
$393$	8.14124	0.410671
$394$	0	0
$395$	3.84592	0.193509
$396$	0	0
$397$	−2.07582	−0.104182	−0.0520912	−	0.998642i	$-0.516589\pi$
$397$	−2.07582	−0.104182	−0.0520912	+	0.998642i	$0.516589\pi$
$398$	0	0
$399$	12.9110	0.646358
$400$	0	0
$401$	24.2629	1.21163	0.605817	−	0.795604i	$-0.292846\pi$
$401$	24.2629	1.21163	0.605817	+	0.795604i	$0.292846\pi$
$402$	0	0
$403$	33.0235	1.64502
$404$	0	0
$405$	−2.76045	−0.137168
$406$	0	0
$407$	−11.5613	−0.573072
$408$	0	0
$409$	−27.5537	−1.36244	−0.681222	−	0.732077i	$-0.738551\pi$
$409$	−27.5537	−1.36244	−0.681222	+	0.732077i	$0.738551\pi$
$410$	0	0
$411$	−10.8400	−0.534698
$412$	0	0
$413$	−6.75241	−0.332265
$414$	0	0
$415$	−13.0008	−0.638186
$416$	0	0
$417$	−10.4840	−0.513405
$418$	0	0
$419$	−19.5618	−0.955658	−0.477829	−	0.878453i	$-0.658576\pi$
$419$	−19.5618	−0.955658	−0.477829	+	0.878453i	$0.658576\pi$
$420$	0	0
$421$	−2.44705	−0.119262	−0.0596309	−	0.998220i	$-0.518992\pi$
$421$	−2.44705	−0.119262	−0.0596309	+	0.998220i	$0.518992\pi$
$422$	0	0
$423$	−2.08378	−0.101317
$424$	0	0
$425$	17.8951	0.868040
$426$	0	0
$427$	−14.0751	−0.681142
$428$	0	0
$429$	−16.7023	−0.806393
$430$	0	0
$431$	30.8346	1.48525	0.742626	−	0.669707i	$-0.233580\pi$
$431$	30.8346	1.48525	0.742626	+	0.669707i	$0.233580\pi$
$432$	0	0
$433$	−10.9613	−0.526765	−0.263383	−	0.964691i	$-0.584838\pi$
$433$	−10.9613	−0.526765	−0.263383	+	0.964691i	$0.584838\pi$
$434$	0	0
$435$	−18.2097	−0.873090
$436$	0	0
$437$	−26.4499	−1.26527
$438$	0	0
$439$	−27.1644	−1.29649	−0.648243	−	0.761433i	$-0.724496\pi$
$439$	−27.1644	−1.29649	−0.648243	+	0.761433i	$0.724496\pi$
$440$	0	0
$441$	3.79399	0.180666
$442$	0	0
$443$	−29.3873	−1.39623	−0.698116	−	0.715984i	$-0.745978\pi$
$443$	−29.3873	−1.39623	−0.698116	+	0.715984i	$0.745978\pi$
$444$	0	0
$445$	−10.9964	−0.521282
$446$	0	0
$447$	−19.5436	−0.924383
$448$	0	0
$449$	22.5354	1.06351	0.531755	−	0.846898i	$-0.321533\pi$
$449$	22.5354	1.06351	0.531755	+	0.846898i	$0.321533\pi$
$450$	0	0
$451$	−23.0994	−1.08771
$452$	0	0
$453$	−12.7540	−0.599237
$454$	0	0
$455$	−42.4352	−1.98939
$456$	0	0
$457$	−27.1416	−1.26963	−0.634815	−	0.772664i	$-0.718924\pi$
$457$	−27.1416	−1.26963	−0.634815	+	0.772664i	$0.718924\pi$
$458$	0	0
$459$	6.82998	0.318796
$460$	0	0
$461$	24.0917	1.12206	0.561031	−	0.827795i	$-0.310405\pi$
$461$	24.0917	1.12206	0.561031	+	0.827795i	$0.310405\pi$
$462$	0	0
$463$	20.1225	0.935173	0.467586	−	0.883947i	$-0.345124\pi$
$463$	20.1225	0.935173	0.467586	+	0.883947i	$0.345124\pi$
$464$	0	0
$465$	19.4827	0.903486
$466$	0	0
$467$	−22.4555	−1.03912	−0.519558	−	0.854435i	$-0.673903\pi$
$467$	−22.4555	−1.03912	−0.519558	+	0.854435i	$0.673903\pi$
$468$	0	0
$469$	26.1724	1.20853
$470$	0	0
$471$	8.06348	0.371545
$472$	0	0
$473$	21.4243	0.985089
$474$	0	0
$475$	−10.2963	−0.472429
$476$	0	0
$477$	2.26395	0.103659
$478$	0	0
$479$	−30.8200	−1.40820	−0.704100	−	0.710100i	$-0.748649\pi$
$479$	−30.8200	−1.40820	−0.704100	+	0.710100i	$0.748649\pi$
$480$	0	0
$481$	15.1545	0.690986
$482$	0	0
$483$	−22.1129	−1.00617
$484$	0	0
$485$	34.2940	1.55721
$486$	0	0
$487$	−12.9347	−0.586126	−0.293063	−	0.956093i	$-0.594675\pi$
$487$	−12.9347	−0.586126	−0.293063	+	0.956093i	$0.594675\pi$
$488$	0	0
$489$	6.62005	0.299369
$490$	0	0
$491$	25.0286	1.12952	0.564761	−	0.825254i	$-0.308968\pi$
$491$	25.0286	1.12952	0.564761	+	0.825254i	$0.308968\pi$
$492$	0	0
$493$	45.0551	2.02918
$494$	0	0
$495$	−9.85372	−0.442892
$496$	0	0
$497$	−12.7423	−0.571571
$498$	0	0
$499$	30.0740	1.34630	0.673149	−	0.739507i	$-0.264941\pi$
$499$	30.0740	1.34630	0.673149	+	0.739507i	$0.264941\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	21.8981	0.976390	0.488195	−	0.872735i	$-0.337655\pi$
$503$	21.8981	0.976390	0.488195	+	0.872735i	$0.337655\pi$
$504$	0	0
$505$	42.4790	1.89029
$506$	0	0
$507$	8.89327	0.394964
$508$	0	0
$509$	25.2118	1.11749	0.558746	−	0.829339i	$-0.311283\pi$
$509$	25.2118	1.11749	0.558746	+	0.829339i	$0.311283\pi$
$510$	0	0
$511$	−13.6459	−0.603660
$512$	0	0
$513$	−3.92978	−0.173504
$514$	0	0
$515$	−8.48831	−0.374040
$516$	0	0
$517$	−7.43829	−0.327135
$518$	0	0
$519$	−7.89738	−0.346656
$520$	0	0
$521$	−7.00487	−0.306889	−0.153445	−	0.988157i	$-0.549037\pi$
$521$	−7.00487	−0.306889	−0.153445	+	0.988157i	$0.549037\pi$
$522$	0	0
$523$	−6.56843	−0.287217	−0.143609	−	0.989635i	$-0.545871\pi$
$523$	−6.56843	−0.287217	−0.143609	+	0.989635i	$0.545871\pi$
$524$	0	0
$525$	−8.60807	−0.375687
$526$	0	0
$527$	−48.2046	−2.09982
$528$	0	0
$529$	22.3012	0.969618
$530$	0	0
$531$	2.05527	0.0891909
$532$	0	0
$533$	30.2786	1.31151
$534$	0	0
$535$	−39.8240	−1.72174
$536$	0	0
$537$	13.2574	0.572097
$538$	0	0
$539$	13.5431	0.583341
$540$	0	0
$541$	3.71126	0.159559	0.0797797	−	0.996813i	$-0.474578\pi$
$541$	3.71126	0.159559	0.0797797	+	0.996813i	$0.474578\pi$
$542$	0	0
$543$	−23.3400	−1.00162
$544$	0	0
$545$	−46.0176	−1.97118
$546$	0	0
$547$	−25.4721	−1.08911	−0.544555	−	0.838725i	$-0.683302\pi$
$547$	−25.4721	−1.08911	−0.544555	+	0.838725i	$0.683302\pi$
$548$	0	0
$549$	4.28411	0.182841
$550$	0	0
$551$	−25.9234	−1.10438
$552$	0	0
$553$	4.57732	0.194648
$554$	0	0
$555$	8.94059	0.379507
$556$	0	0
$557$	22.4869	0.952799	0.476400	−	0.879229i	$-0.341942\pi$
$557$	22.4869	0.952799	0.476400	+	0.879229i	$0.341942\pi$
$558$	0	0
$559$	−28.0829	−1.18778
$560$	0	0
$561$	24.3804	1.02934
$562$	0	0
$563$	22.8366	0.962449	0.481224	−	0.876597i	$-0.340192\pi$
$563$	22.8366	0.962449	0.481224	+	0.876597i	$0.340192\pi$
$564$	0	0
$565$	33.5902	1.41315
$566$	0	0
$567$	−3.28542	−0.137975
$568$	0	0
$569$	−15.5492	−0.651858	−0.325929	−	0.945394i	$-0.605677\pi$
$569$	−15.5492	−0.651858	−0.325929	+	0.945394i	$0.605677\pi$
$570$	0	0
$571$	−34.2826	−1.43468	−0.717340	−	0.696723i	$-0.754641\pi$
$571$	−34.2826	−1.43468	−0.717340	+	0.696723i	$0.754641\pi$
$572$	0	0
$573$	−12.6160	−0.527043
$574$	0	0
$575$	17.6348	0.735420
$576$	0	0
$577$	24.7169	1.02898	0.514489	−	0.857497i	$-0.327982\pi$
$577$	24.7169	1.02898	0.514489	+	0.857497i	$0.327982\pi$
$578$	0	0
$579$	−26.9494	−1.11998
$580$	0	0
$581$	−15.4733	−0.641940
$582$	0	0
$583$	8.08141	0.334698
$584$	0	0
$585$	12.9162	0.534020
$586$	0	0
$587$	2.34201	0.0966652	0.0483326	−	0.998831i	$-0.484609\pi$
$587$	2.34201	0.0966652	0.0483326	+	0.998831i	$0.484609\pi$
$588$	0	0
$589$	27.7355	1.14282
$590$	0	0
$591$	−4.63420	−0.190625
$592$	0	0
$593$	4.76332	0.195606	0.0978030	−	0.995206i	$-0.468818\pi$
$593$	4.76332	0.195606	0.0978030	+	0.995206i	$0.468818\pi$
$594$	0	0
$595$	61.9428	2.53941
$596$	0	0
$597$	4.52367	0.185141
$598$	0	0
$599$	39.8326	1.62752	0.813759	−	0.581203i	$-0.197418\pi$
$599$	39.8326	1.62752	0.813759	+	0.581203i	$0.197418\pi$
$600$	0	0
$601$	20.9145	0.853119	0.426560	−	0.904459i	$-0.359725\pi$
$601$	20.9145	0.853119	0.426560	+	0.904459i	$0.359725\pi$
$602$	0	0
$603$	−7.96623	−0.324410
$604$	0	0
$605$	−4.80895	−0.195511
$606$	0	0
$607$	13.4992	0.547917	0.273959	−	0.961741i	$-0.411667\pi$
$607$	13.4992	0.547917	0.273959	+	0.961741i	$0.411667\pi$
$608$	0	0
$609$	−21.6728	−0.878227
$610$	0	0
$611$	9.75007	0.394446
$612$	0	0
$613$	−23.5647	−0.951770	−0.475885	−	0.879507i	$-0.657872\pi$
$613$	−23.5647	−0.951770	−0.475885	+	0.879507i	$0.657872\pi$
$614$	0	0
$615$	17.8632	0.720314
$616$	0	0
$617$	−16.1288	−0.649321	−0.324660	−	0.945831i	$-0.605250\pi$
$617$	−16.1288	−0.649321	−0.324660	+	0.945831i	$0.605250\pi$
$618$	0	0
$619$	0.943288	0.0379139	0.0189570	−	0.999820i	$-0.493965\pi$
$619$	0.943288	0.0379139	0.0189570	+	0.999820i	$0.493965\pi$
$620$	0	0
$621$	6.73062	0.270090
$622$	0	0
$623$	−13.0877	−0.524348
$624$	0	0
$625$	−31.2356	−1.24942
$626$	0	0
$627$	−14.0278	−0.560215
$628$	0	0
$629$	−22.1211	−0.882025
$630$	0	0
$631$	36.8489	1.46693	0.733466	−	0.679726i	$-0.237901\pi$
$631$	36.8489	1.46693	0.733466	+	0.679726i	$0.237901\pi$
$632$	0	0
$633$	−16.5460	−0.657644
$634$	0	0
$635$	7.20399	0.285882
$636$	0	0
$637$	−17.7522	−0.703368
$638$	0	0
$639$	3.87844	0.153429
$640$	0	0
$641$	17.5876	0.694669	0.347334	−	0.937741i	$-0.387087\pi$
$641$	17.5876	0.694669	0.347334	+	0.937741i	$0.387087\pi$
$642$	0	0
$643$	14.9489	0.589526	0.294763	−	0.955570i	$-0.404759\pi$
$643$	14.9489	0.589526	0.294763	+	0.955570i	$0.404759\pi$
$644$	0	0
$645$	−16.5678	−0.652358
$646$	0	0
$647$	5.40249	0.212394	0.106197	−	0.994345i	$-0.466133\pi$
$647$	5.40249	0.212394	0.106197	+	0.994345i	$0.466133\pi$
$648$	0	0
$649$	7.33649	0.287982
$650$	0	0
$651$	23.1878	0.908801
$652$	0	0
$653$	49.0618	1.91994	0.959968	−	0.280108i	$-0.0903704\pi$
$653$	49.0618	1.91994	0.959968	+	0.280108i	$0.0903704\pi$
$654$	0	0
$655$	−22.4735	−0.878112
$656$	0	0
$657$	4.15348	0.162043
$658$	0	0
$659$	8.42635	0.328244	0.164122	−	0.986440i	$-0.447521\pi$
$659$	8.42635	0.328244	0.164122	+	0.986440i	$0.447521\pi$
$660$	0	0
$661$	−30.3323	−1.17979	−0.589895	−	0.807480i	$-0.700831\pi$
$661$	−30.3323	−1.17979	−0.589895	+	0.807480i	$0.700831\pi$
$662$	0	0
$663$	−31.9577	−1.24113
$664$	0	0
$665$	−35.6401	−1.38206
$666$	0	0
$667$	44.3996	1.71916
$668$	0	0
$669$	−16.8807	−0.652646
$670$	0	0
$671$	15.2926	0.590364
$672$	0	0
$673$	15.4705	0.596343	0.298172	−	0.954512i	$-0.403623\pi$
$673$	15.4705	0.596343	0.298172	+	0.954512i	$0.403623\pi$
$674$	0	0
$675$	2.62008	0.100847
$676$	0	0
$677$	46.3103	1.77985	0.889925	−	0.456106i	$-0.150756\pi$
$677$	46.3103	1.77985	0.889925	+	0.456106i	$0.150756\pi$
$678$	0	0
$679$	40.8159	1.56637
$680$	0	0
$681$	−3.39218	−0.129989
$682$	0	0
$683$	−26.0920	−0.998381	−0.499190	−	0.866492i	$-0.666369\pi$
$683$	−26.0920	−0.998381	−0.499190	+	0.866492i	$0.666369\pi$
$684$	0	0
$685$	29.9233	1.14331
$686$	0	0
$687$	−8.50012	−0.324300
$688$	0	0
$689$	−10.5931	−0.403564
$690$	0	0
$691$	−25.3282	−0.963531	−0.481765	−	0.876300i	$-0.660004\pi$
$691$	−25.3282	−0.963531	−0.481765	+	0.876300i	$0.660004\pi$
$692$	0	0
$693$	−11.7277	−0.445497
$694$	0	0
$695$	28.9406	1.09778
$696$	0	0
$697$	−44.1977	−1.67411
$698$	0	0
$699$	−9.20099	−0.348014
$700$	0	0
$701$	20.4763	0.773381	0.386690	−	0.922210i	$-0.373618\pi$
$701$	20.4763	0.773381	0.386690	+	0.922210i	$0.373618\pi$
$702$	0	0
$703$	12.7278	0.480040
$704$	0	0
$705$	5.75218	0.216640
$706$	0	0
$707$	50.5575	1.90141
$708$	0	0
$709$	46.1601	1.73358	0.866789	−	0.498675i	$-0.166180\pi$
$709$	46.1601	1.73358	0.866789	+	0.498675i	$0.166180\pi$
$710$	0	0
$711$	−1.39322	−0.0522499
$712$	0	0
$713$	−47.5032	−1.77901
$714$	0	0
$715$	46.1058	1.72426
$716$	0	0
$717$	16.4125	0.612935
$718$	0	0
$719$	7.65472	0.285473	0.142736	−	0.989761i	$-0.454410\pi$
$719$	7.65472	0.285473	0.142736	+	0.989761i	$0.454410\pi$
$720$	0	0
$721$	−10.1026	−0.376240
$722$	0	0
$723$	−9.43359	−0.350839
$724$	0	0
$725$	17.2838	0.641903
$726$	0	0
$727$	42.0382	1.55911	0.779556	−	0.626333i	$-0.215445\pi$
$727$	42.0382	1.55911	0.779556	+	0.626333i	$0.215445\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.9926	1.51617
$732$	0	0
$733$	26.5447	0.980451	0.490226	−	0.871596i	$-0.336914\pi$
$733$	26.5447	0.980451	0.490226	+	0.871596i	$0.336914\pi$
$734$	0	0
$735$	−10.4731	−0.386307
$736$	0	0
$737$	−28.4363	−1.04746
$738$	0	0
$739$	−16.0329	−0.589778	−0.294889	−	0.955531i	$-0.595283\pi$
$739$	−16.0329	−0.589778	−0.294889	+	0.955531i	$0.595283\pi$
$740$	0	0
$741$	18.3875	0.675483
$742$	0	0
$743$	−23.4034	−0.858589	−0.429295	−	0.903165i	$-0.641238\pi$
$743$	−23.4034	−0.858589	−0.429295	+	0.903165i	$0.641238\pi$
$744$	0	0
$745$	53.9492	1.97655
$746$	0	0
$747$	4.70968	0.172318
$748$	0	0
$749$	−47.3975	−1.73187
$750$	0	0
$751$	−2.71579	−0.0991006	−0.0495503	−	0.998772i	$-0.515779\pi$
$751$	−2.71579	−0.0991006	−0.0495503	+	0.998772i	$0.515779\pi$
$752$	0	0
$753$	19.5625	0.712896
$754$	0	0
$755$	35.2069	1.28131
$756$	0	0
$757$	50.7627	1.84500	0.922500	−	0.385996i	$-0.126142\pi$
$757$	50.7627	1.84500	0.922500	+	0.385996i	$0.126142\pi$
$758$	0	0
$759$	24.0257	0.872076
$760$	0	0
$761$	−30.9925	−1.12348	−0.561739	−	0.827314i	$-0.689868\pi$
$761$	−30.9925	−1.12348	−0.561739	+	0.827314i	$0.689868\pi$
$762$	0	0
$763$	−54.7690	−1.98277
$764$	0	0
$765$	−18.8538	−0.681661
$766$	0	0
$767$	−9.61664	−0.347237
$768$	0	0
$769$	−22.5039	−0.811510	−0.405755	−	0.913982i	$-0.632991\pi$
$769$	−22.5039	−0.811510	−0.405755	+	0.913982i	$0.632991\pi$
$770$	0	0
$771$	−13.7550	−0.495374
$772$	0	0
$773$	43.8384	1.57676	0.788378	−	0.615191i	$-0.210921\pi$
$773$	43.8384	1.57676	0.788378	+	0.615191i	$0.210921\pi$
$774$	0	0
$775$	−18.4920	−0.664251
$776$	0	0
$777$	10.6409	0.381739
$778$	0	0
$779$	25.4301	0.911128
$780$	0	0
$781$	13.8445	0.495395
$782$	0	0
$783$	6.59666	0.235745
$784$	0	0
$785$	−22.2588	−0.794451
$786$	0	0
$787$	9.46999	0.337569	0.168784	−	0.985653i	$-0.446016\pi$
$787$	9.46999	0.337569	0.168784	+	0.985653i	$0.446016\pi$
$788$	0	0
$789$	−27.7159	−0.986713
$790$	0	0
$791$	39.9782	1.42146
$792$	0	0
$793$	−20.0455	−0.711835
$794$	0	0
$795$	−6.24952	−0.221648
$796$	0	0
$797$	−22.7648	−0.806369	−0.403185	−	0.915119i	$-0.632097\pi$
$797$	−22.7648	−0.806369	−0.403185	+	0.915119i	$0.632097\pi$
$798$	0	0
$799$	−14.2322	−0.503499
$800$	0	0
$801$	3.98357	0.140753
$802$	0	0
$803$	14.8263	0.523208
$804$	0	0
$805$	61.0416	2.15143
$806$	0	0
$807$	−23.5754	−0.829893
$808$	0	0
$809$	−35.7421	−1.25663	−0.628313	−	0.777961i	$-0.716254\pi$
$809$	−35.7421	−1.25663	−0.628313	+	0.777961i	$0.716254\pi$
$810$	0	0
$811$	−33.6233	−1.18067	−0.590337	−	0.807157i	$-0.701005\pi$
$811$	−33.6233	−1.18067	−0.590337	+	0.807157i	$0.701005\pi$
$812$	0	0
$813$	11.1851	0.392278
$814$	0	0
$815$	−18.2743	−0.640121
$816$	0	0
$817$	−23.5860	−0.825170
$818$	0	0
$819$	15.3726	0.537161
$820$	0	0
$821$	2.76011	0.0963284	0.0481642	−	0.998839i	$-0.484663\pi$
$821$	2.76011	0.0963284	0.0481642	+	0.998839i	$0.484663\pi$
$822$	0	0
$823$	32.1807	1.12175	0.560874	−	0.827901i	$-0.310465\pi$
$823$	32.1807	1.12175	0.560874	+	0.827901i	$0.310465\pi$
$824$	0	0
$825$	9.35265	0.325618
$826$	0	0
$827$	17.7153	0.616020	0.308010	−	0.951383i	$-0.400337\pi$
$827$	17.7153	0.616020	0.308010	+	0.951383i	$0.400337\pi$
$828$	0	0
$829$	7.34169	0.254987	0.127494	−	0.991839i	$-0.459307\pi$
$829$	7.34169	0.254987	0.127494	+	0.991839i	$0.459307\pi$
$830$	0	0
$831$	−29.0865	−1.00900
$832$	0	0
$833$	25.9129	0.897830
$834$	0	0
$835$	−2.76045	−0.0955293
$836$	0	0
$837$	−7.05778	−0.243953
$838$	0	0
$839$	−22.6895	−0.783328	−0.391664	−	0.920108i	$-0.628100\pi$
$839$	−22.6895	−0.783328	−0.391664	+	0.920108i	$0.628100\pi$
$840$	0	0
$841$	14.5159	0.500549
$842$	0	0
$843$	−7.20697	−0.248221
$844$	0	0
$845$	−24.5494	−0.844526
$846$	0	0
$847$	−5.72349	−0.196662
$848$	0	0
$849$	9.79768	0.336256
$850$	0	0
$851$	−21.7992	−0.747268
$852$	0	0
$853$	−31.9393	−1.09358	−0.546790	−	0.837270i	$-0.684150\pi$
$853$	−31.9393	−1.09358	−0.546790	+	0.837270i	$0.684150\pi$
$854$	0	0
$855$	10.8480	0.370992
$856$	0	0
$857$	47.2805	1.61507	0.807536	−	0.589818i	$-0.200801\pi$
$857$	47.2805	1.61507	0.807536	+	0.589818i	$0.200801\pi$
$858$	0	0
$859$	−34.8933	−1.19054	−0.595272	−	0.803524i	$-0.702956\pi$
$859$	−34.8933	−1.19054	−0.595272	+	0.803524i	$0.702956\pi$
$860$	0	0
$861$	21.2604	0.724552
$862$	0	0
$863$	−19.8779	−0.676650	−0.338325	−	0.941029i	$-0.609860\pi$
$863$	−19.8779	−0.676650	−0.338325	+	0.941029i	$0.609860\pi$
$864$	0	0
$865$	21.8003	0.741233
$866$	0	0
$867$	29.6487	1.00692
$868$	0	0
$869$	−4.97325	−0.168706
$870$	0	0
$871$	37.2742	1.26299
$872$	0	0
$873$	−12.4233	−0.420466
$874$	0	0
$875$	−21.5841	−0.729675
$876$	0	0
$877$	53.0800	1.79238	0.896192	−	0.443667i	$-0.146323\pi$
$877$	53.0800	1.79238	0.896192	+	0.443667i	$0.146323\pi$
$878$	0	0
$879$	−7.13122	−0.240530
$880$	0	0
$881$	−41.2225	−1.38882	−0.694411	−	0.719578i	$-0.744335\pi$
$881$	−41.2225	−1.38882	−0.694411	+	0.719578i	$0.744335\pi$
$882$	0	0
$883$	−5.19468	−0.174815	−0.0874074	−	0.996173i	$-0.527858\pi$
$883$	−5.19468	−0.174815	−0.0874074	+	0.996173i	$0.527858\pi$
$884$	0	0
$885$	−5.67346	−0.190711
$886$	0	0
$887$	11.6436	0.390953	0.195477	−	0.980708i	$-0.437375\pi$
$887$	11.6436	0.390953	0.195477	+	0.980708i	$0.437375\pi$
$888$	0	0
$889$	8.57402	0.287563
$890$	0	0
$891$	3.56961	0.119586
$892$	0	0
$893$	8.18881	0.274028
$894$	0	0
$895$	−36.5962	−1.22328
$896$	0	0
$897$	−31.4927	−1.05151
$898$	0	0
$899$	−46.5578	−1.55279
$900$	0	0
$901$	15.4628	0.515139
$902$	0	0
$903$	−19.7186	−0.656195
$904$	0	0
$905$	64.4290	2.14169
$906$	0	0
$907$	31.0401	1.03067	0.515335	−	0.856989i	$-0.327667\pi$
$907$	31.0401	1.03067	0.515335	+	0.856989i	$0.327667\pi$
$908$	0	0
$909$	−15.3884	−0.510402
$910$	0	0
$911$	41.3323	1.36940	0.684700	−	0.728825i	$-0.259933\pi$
$911$	41.3323	1.36940	0.684700	+	0.728825i	$0.259933\pi$
$912$	0	0
$913$	16.8117	0.556386
$914$	0	0
$915$	−11.8261	−0.390958
$916$	0	0
$917$	−26.7474	−0.883277
$918$	0	0
$919$	−21.9163	−0.722952	−0.361476	−	0.932381i	$-0.617727\pi$
$919$	−21.9163	−0.722952	−0.361476	+	0.932381i	$0.617727\pi$
$920$	0	0
$921$	22.1521	0.729935
$922$	0	0
$923$	−18.1473	−0.597326
$924$	0	0
$925$	−8.48596	−0.279017
$926$	0	0
$927$	3.07497	0.100995
$928$	0	0
$929$	−34.4080	−1.12889	−0.564445	−	0.825471i	$-0.690910\pi$
$929$	−34.4080	−1.12889	−0.564445	+	0.825471i	$0.690910\pi$
$930$	0	0
$931$	−14.9096	−0.488641
$932$	0	0
$933$	2.88112	0.0943236
$934$	0	0
$935$	−67.3007	−2.20097
$936$	0	0
$937$	−5.68708	−0.185789	−0.0928943	−	0.995676i	$-0.529612\pi$
$937$	−5.68708	−0.185789	−0.0928943	+	0.995676i	$0.529612\pi$
$938$	0	0
$939$	−2.35289	−0.0767835
$940$	0	0
$941$	−15.9057	−0.518510	−0.259255	−	0.965809i	$-0.583477\pi$
$941$	−15.9057	−0.518510	−0.259255	+	0.965809i	$0.583477\pi$
$942$	0	0
$943$	−43.5547	−1.41834
$944$	0	0
$945$	9.06924	0.295022
$946$	0	0
$947$	58.3487	1.89608	0.948039	−	0.318155i	$-0.103063\pi$
$947$	58.3487	1.89608	0.948039	+	0.318155i	$0.103063\pi$
$948$	0	0
$949$	−19.4342	−0.630861
$950$	0	0
$951$	−21.9928	−0.713166
$952$	0	0
$953$	22.3113	0.722733	0.361367	−	0.932424i	$-0.382310\pi$
$953$	22.3113	0.722733	0.361367	+	0.932424i	$0.382310\pi$
$954$	0	0
$955$	34.8259	1.12694
$956$	0	0
$957$	23.5475	0.761182
$958$	0	0
$959$	35.6140	1.15004
$960$	0	0
$961$	18.8123	0.606849
$962$	0	0
$963$	14.4266	0.464891
$964$	0	0
$965$	74.3923	2.39477
$966$	0	0
$967$	−52.4306	−1.68606	−0.843028	−	0.537870i	$-0.819229\pi$
$967$	−52.4306	−1.68606	−0.843028	+	0.537870i	$0.819229\pi$
$968$	0	0
$969$	−26.8403	−0.862236
$970$	0	0
$971$	9.36218	0.300447	0.150223	−	0.988652i	$-0.452001\pi$
$971$	9.36218	0.300447	0.150223	+	0.988652i	$0.452001\pi$
$972$	0	0
$973$	34.4445	1.10424
$974$	0	0
$975$	−12.2594	−0.392616
$976$	0	0
$977$	39.5632	1.26574	0.632869	−	0.774259i	$-0.281877\pi$
$977$	39.5632	1.26574	0.632869	+	0.774259i	$0.281877\pi$
$978$	0	0
$979$	14.2198	0.454466
$980$	0	0
$981$	16.6703	0.532242
$982$	0	0
$983$	43.1031	1.37477	0.687387	−	0.726291i	$-0.258758\pi$
$983$	43.1031	1.37477	0.687387	+	0.726291i	$0.258758\pi$
$984$	0	0
$985$	12.7925	0.407602
$986$	0	0
$987$	6.84611	0.217914
$988$	0	0
$989$	40.3962	1.28453
$990$	0	0
$991$	−7.48044	−0.237624	−0.118812	−	0.992917i	$-0.537909\pi$
$991$	−7.48044	−0.237624	−0.118812	+	0.992917i	$0.537909\pi$
$992$	0	0
$993$	23.8674	0.757409
$994$	0	0
$995$	−12.4874	−0.395876
$996$	0	0
$997$	2.46249	0.0779879	0.0389940	−	0.999239i	$-0.487585\pi$
$997$	2.46249	0.0779879	0.0389940	+	0.999239i	$0.487585\pi$
$998$	0	0
$999$	−3.23882	−0.102472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bd.1.3		10
4.3	odd	2		4008.2.a.j.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.3	✓	10	4.3	odd	2
8016.2.a.bd.1.3		10	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.76045 q^{5} -3.28542 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.76045 q^{5} -3.28542 q^{7} +1.00000 q^{9} +3.56961 q^{11} -4.67902 q^{13} -2.76045 q^{15} +6.82998 q^{17} -3.92978 q^{19} -3.28542 q^{21} +6.73062 q^{23} +2.62008 q^{25} +1.00000 q^{27} +6.59666 q^{29} -7.05778 q^{31} +3.56961 q^{33} +9.06924 q^{35} -3.23882 q^{37} -4.67902 q^{39} -6.47113 q^{41} +6.00186 q^{43} -2.76045 q^{45} -2.08378 q^{47} +3.79399 q^{49} +6.82998 q^{51} +2.26395 q^{53} -9.85372 q^{55} -3.92978 q^{57} +2.05527 q^{59} +4.28411 q^{61} -3.28542 q^{63} +12.9162 q^{65} -7.96623 q^{67} +6.73062 q^{69} +3.87844 q^{71} +4.15348 q^{73} +2.62008 q^{75} -11.7277 q^{77} -1.39322 q^{79} +1.00000 q^{81} +4.70968 q^{83} -18.8538 q^{85} +6.59666 q^{87} +3.98357 q^{89} +15.3726 q^{91} -7.05778 q^{93} +10.8480 q^{95} -12.4233 q^{97} +3.56961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9	\( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9 + q^11 - 6 * q^13 - 10 * q^15 - 9 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 12 * q^25 + 10 * q^27 - 13 * q^29 - 23 * q^31 + q^33 - q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 10 * q^45 - 10 * q^47 + 7 * q^49 - 9 * q^51 - 26 * q^53 - 11 * q^55 - 2 * q^57 + 10 * q^59 - 10 * q^61 - q^63 - 22 * q^65 + 5 * q^67 - 7 * q^69 - 25 * q^71 - 8 * q^73 + 12 * q^75 - 46 * q^77 - 26 * q^79 + 10 * q^81 + 14 * q^83 + 9 * q^85 - 13 * q^87 - 31 * q^89 + 3 * q^91 - 23 * q^93 + 5 * q^95 - 32 * q^97 + q^99

Introduction

L-functions

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