LMFDB - Embedded newform 8016.2.a.u.1.2

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.38569.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 5x^{3} + 4x - 1 $ x^5 - 5x^3 + 4x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.790734$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.61314 q^{5} -2.77861 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.61314 q^{5} -2.77861 q^{7} +1.00000 q^{9} -3.41479 q^{11} -6.91784 q^{13} -1.61314 q^{15} +3.58757 q^{17} +5.37474 q^{19} -2.77861 q^{21} +1.72034 q^{23} -2.39778 q^{25} +1.00000 q^{27} -7.25770 q^{29} +2.36382 q^{31} -3.41479 q^{33} +4.48229 q^{35} -5.21603 q^{37} -6.91784 q^{39} -1.85221 q^{41} -1.96645 q^{43} -1.61314 q^{45} -7.93197 q^{47} +0.720696 q^{49} +3.58757 q^{51} +0.173715 q^{53} +5.50854 q^{55} +5.37474 q^{57} +8.14440 q^{59} +6.08170 q^{61} -2.77861 q^{63} +11.1595 q^{65} +14.8238 q^{67} +1.72034 q^{69} -7.48522 q^{71} -3.99718 q^{73} -2.39778 q^{75} +9.48839 q^{77} +6.01895 q^{79} +1.00000 q^{81} +9.27564 q^{83} -5.78725 q^{85} -7.25770 q^{87} -8.88589 q^{89} +19.2220 q^{91} +2.36382 q^{93} -8.67021 q^{95} -15.7656 q^{97} -3.41479 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9} + 7 q^{11} - 8 q^{13} - q^{15} + 5 q^{17} + 20 q^{19} + 4 q^{21} - q^{23} - 6 q^{25} + 5 q^{27} - 5 q^{29} + 18 q^{31} + 7 q^{33} + 6 q^{35} - 17 q^{37} - 8 q^{39} + 6 q^{41} + 10 q^{43} - q^{45} + 3 q^{47} - 13 q^{49} + 5 q^{51} + 15 q^{53} + 9 q^{55} + 20 q^{57} + 17 q^{59} - 2 q^{61} + 4 q^{63} + 26 q^{65} + 24 q^{67} - q^{69} - q^{71} + 2 q^{73} - 6 q^{75} + 26 q^{77} + 6 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} - 5 q^{87} + 15 q^{91} + 18 q^{93} - q^{95} - 13 q^{97} + 7 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9 + 7 * q^11 - 8 * q^13 - q^15 + 5 * q^17 + 20 * q^19 + 4 * q^21 - q^23 - 6 * q^25 + 5 * q^27 - 5 * q^29 + 18 * q^31 + 7 * q^33 + 6 * q^35 - 17 * q^37 - 8 * q^39 + 6 * q^41 + 10 * q^43 - q^45 + 3 * q^47 - 13 * q^49 + 5 * q^51 + 15 * q^53 + 9 * q^55 + 20 * q^57 + 17 * q^59 - 2 * q^61 + 4 * q^63 + 26 * q^65 + 24 * q^67 - q^69 - q^71 + 2 * q^73 - 6 * q^75 + 26 * q^77 + 6 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 - 5 * q^87 + 15 * q^91 + 18 * q^93 - q^95 - 13 * q^97 + 7 * 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
$5$	−1.61314	−0.721418	−0.360709	−	0.932678i	$-0.617465\pi$
$5$	−1.61314	−0.721418	−0.360709	+	0.932678i	$0.617465\pi$
$6$	0	0
$7$	−2.77861	−1.05022	−0.525109	−	0.851035i	$-0.675975\pi$
$7$	−2.77861	−1.05022	−0.525109	+	0.851035i	$0.675975\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.41479	−1.02960	−0.514799	−	0.857311i	$-0.672134\pi$
$11$	−3.41479	−1.02960	−0.514799	+	0.857311i	$0.672134\pi$
$12$	0	0
$13$	−6.91784	−1.91866	−0.959332	−	0.282280i	$-0.908909\pi$
$13$	−6.91784	−1.91866	−0.959332	+	0.282280i	$0.908909\pi$
$14$	0	0
$15$	−1.61314	−0.416511
$16$	0	0
$17$	3.58757	0.870112	0.435056	−	0.900403i	$-0.356728\pi$
$17$	3.58757	0.870112	0.435056	+	0.900403i	$0.356728\pi$
$18$	0	0
$19$	5.37474	1.23305	0.616525	−	0.787335i	$-0.288540\pi$
$19$	5.37474	1.23305	0.616525	+	0.787335i	$0.288540\pi$
$20$	0	0
$21$	−2.77861	−0.606343
$22$	0	0
$23$	1.72034	0.358716	0.179358	−	0.983784i	$-0.442598\pi$
$23$	1.72034	0.358716	0.179358	+	0.983784i	$0.442598\pi$
$24$	0	0
$25$	−2.39778	−0.479556
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.25770	−1.34772	−0.673861	−	0.738858i	$-0.735365\pi$
$29$	−7.25770	−1.34772	−0.673861	+	0.738858i	$0.735365\pi$
$30$	0	0
$31$	2.36382	0.424555	0.212277	−	0.977209i	$-0.431912\pi$
$31$	2.36382	0.424555	0.212277	+	0.977209i	$0.431912\pi$
$32$	0	0
$33$	−3.41479	−0.594439
$34$	0	0
$35$	4.48229	0.757646
$36$	0	0
$37$	−5.21603	−0.857511	−0.428755	−	0.903421i	$-0.641048\pi$
$37$	−5.21603	−0.857511	−0.428755	+	0.903421i	$0.641048\pi$
$38$	0	0
$39$	−6.91784	−1.10774
$40$	0	0
$41$	−1.85221	−0.289267	−0.144634	−	0.989485i	$-0.546200\pi$
$41$	−1.85221	−0.289267	−0.144634	+	0.989485i	$0.546200\pi$
$42$	0	0
$43$	−1.96645	−0.299882	−0.149941	−	0.988695i	$-0.547908\pi$
$43$	−1.96645	−0.299882	−0.149941	+	0.988695i	$0.547908\pi$
$44$	0	0
$45$	−1.61314	−0.240473
$46$	0	0
$47$	−7.93197	−1.15700	−0.578498	−	0.815684i	$-0.696361\pi$
$47$	−7.93197	−1.15700	−0.578498	+	0.815684i	$0.696361\pi$
$48$	0	0
$49$	0.720696	0.102957
$50$	0	0
$51$	3.58757	0.502360
$52$	0	0
$53$	0.173715	0.0238616	0.0119308	−	0.999929i	$-0.496202\pi$
$53$	0.173715	0.0238616	0.0119308	+	0.999929i	$0.496202\pi$
$54$	0	0
$55$	5.50854	0.742772
$56$	0	0
$57$	5.37474	0.711902
$58$	0	0
$59$	8.14440	1.06031	0.530155	−	0.847900i	$-0.322134\pi$
$59$	8.14440	1.06031	0.530155	+	0.847900i	$0.322134\pi$
$60$	0	0
$61$	6.08170	0.778682	0.389341	−	0.921094i	$-0.372703\pi$
$61$	6.08170	0.778682	0.389341	+	0.921094i	$0.372703\pi$
$62$	0	0
$63$	−2.77861	−0.350072
$64$	0	0
$65$	11.1595	1.38416
$66$	0	0
$67$	14.8238	1.81102	0.905510	−	0.424324i	$-0.139488\pi$
$67$	14.8238	1.81102	0.905510	+	0.424324i	$0.139488\pi$
$68$	0	0
$69$	1.72034	0.207105
$70$	0	0
$71$	−7.48522	−0.888333	−0.444166	−	0.895944i	$-0.646500\pi$
$71$	−7.48522	−0.888333	−0.444166	+	0.895944i	$0.646500\pi$
$72$	0	0
$73$	−3.99718	−0.467835	−0.233918	−	0.972256i	$-0.575155\pi$
$73$	−3.99718	−0.467835	−0.233918	+	0.972256i	$0.575155\pi$
$74$	0	0
$75$	−2.39778	−0.276872
$76$	0	0
$77$	9.48839	1.08130
$78$	0	0
$79$	6.01895	0.677184	0.338592	−	0.940933i	$-0.390049\pi$
$79$	6.01895	0.677184	0.338592	+	0.940933i	$0.390049\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.27564	1.01813	0.509067	−	0.860727i	$-0.329991\pi$
$83$	9.27564	1.01813	0.509067	+	0.860727i	$0.329991\pi$
$84$	0	0
$85$	−5.78725	−0.627715
$86$	0	0
$87$	−7.25770	−0.778107
$88$	0	0
$89$	−8.88589	−0.941903	−0.470951	−	0.882159i	$-0.656089\pi$
$89$	−8.88589	−0.941903	−0.470951	+	0.882159i	$0.656089\pi$
$90$	0	0
$91$	19.2220	2.01501
$92$	0	0
$93$	2.36382	0.245117
$94$	0	0
$95$	−8.67021	−0.889545
$96$	0	0
$97$	−15.7656	−1.60076	−0.800379	−	0.599494i	$-0.795369\pi$
$97$	−15.7656	−1.60076	−0.800379	+	0.599494i	$0.795369\pi$
$98$	0	0
$99$	−3.41479	−0.343200
$100$	0	0
$101$	0.521324	0.0518737	0.0259369	−	0.999664i	$-0.491743\pi$
$101$	0.521324	0.0518737	0.0259369	+	0.999664i	$0.491743\pi$
$102$	0	0
$103$	−12.2203	−1.20410	−0.602052	−	0.798457i	$-0.705650\pi$
$103$	−12.2203	−1.20410	−0.602052	+	0.798457i	$0.705650\pi$
$104$	0	0
$105$	4.48229	0.437427
$106$	0	0
$107$	−13.4469	−1.29996	−0.649980	−	0.759952i	$-0.725223\pi$
$107$	−13.4469	−1.29996	−0.649980	+	0.759952i	$0.725223\pi$
$108$	0	0
$109$	10.3423	0.990612	0.495306	−	0.868719i	$-0.335056\pi$
$109$	10.3423	0.990612	0.495306	+	0.868719i	$0.335056\pi$
$110$	0	0
$111$	−5.21603	−0.495084
$112$	0	0
$113$	−4.02353	−0.378502	−0.189251	−	0.981929i	$-0.560606\pi$
$113$	−4.02353	−0.378502	−0.189251	+	0.981929i	$0.560606\pi$
$114$	0	0
$115$	−2.77515	−0.258785
$116$	0	0
$117$	−6.91784	−0.639555
$118$	0	0
$119$	−9.96846	−0.913807
$120$	0	0
$121$	0.660811	0.0600738
$122$	0	0
$123$	−1.85221	−0.167008
$124$	0	0
$125$	11.9337	1.06738
$126$	0	0
$127$	−1.94300	−0.172414	−0.0862069	−	0.996277i	$-0.527475\pi$
$127$	−1.94300	−0.172414	−0.0862069	+	0.996277i	$0.527475\pi$
$128$	0	0
$129$	−1.96645	−0.173137
$130$	0	0
$131$	16.6952	1.45866	0.729332	−	0.684160i	$-0.239831\pi$
$131$	16.6952	1.45866	0.729332	+	0.684160i	$0.239831\pi$
$132$	0	0
$133$	−14.9343	−1.29497
$134$	0	0
$135$	−1.61314	−0.138837
$136$	0	0
$137$	6.18334	0.528278	0.264139	−	0.964485i	$-0.414912\pi$
$137$	6.18334	0.528278	0.264139	+	0.964485i	$0.414912\pi$
$138$	0	0
$139$	14.1406	1.19939	0.599697	−	0.800228i	$-0.295288\pi$
$139$	14.1406	1.19939	0.599697	+	0.800228i	$0.295288\pi$
$140$	0	0
$141$	−7.93197	−0.667992
$142$	0	0
$143$	23.6230	1.97545
$144$	0	0
$145$	11.7077	0.972271
$146$	0	0
$147$	0.720696	0.0594420
$148$	0	0
$149$	18.7763	1.53822	0.769108	−	0.639119i	$-0.220701\pi$
$149$	18.7763	1.53822	0.769108	+	0.639119i	$0.220701\pi$
$150$	0	0
$151$	13.9652	1.13647	0.568237	−	0.822865i	$-0.307626\pi$
$151$	13.9652	1.13647	0.568237	+	0.822865i	$0.307626\pi$
$152$	0	0
$153$	3.58757	0.290037
$154$	0	0
$155$	−3.81318	−0.306282
$156$	0	0
$157$	10.1676	0.811461	0.405731	−	0.913993i	$-0.367017\pi$
$157$	10.1676	0.811461	0.405731	+	0.913993i	$0.367017\pi$
$158$	0	0
$159$	0.173715	0.0137765
$160$	0	0
$161$	−4.78017	−0.376730
$162$	0	0
$163$	−7.01046	−0.549102	−0.274551	−	0.961573i	$-0.588529\pi$
$163$	−7.01046	−0.549102	−0.274551	+	0.961573i	$0.588529\pi$
$164$	0	0
$165$	5.50854	0.428839
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	34.8565	2.68127
$170$	0	0
$171$	5.37474	0.411017
$172$	0	0
$173$	−2.10797	−0.160266	−0.0801329	−	0.996784i	$-0.525534\pi$
$173$	−2.10797	−0.160266	−0.0801329	+	0.996784i	$0.525534\pi$
$174$	0	0
$175$	6.66250	0.503638
$176$	0	0
$177$	8.14440	0.612171
$178$	0	0
$179$	20.6203	1.54124	0.770618	−	0.637298i	$-0.219948\pi$
$179$	20.6203	1.54124	0.770618	+	0.637298i	$0.219948\pi$
$180$	0	0
$181$	−7.24625	−0.538610	−0.269305	−	0.963055i	$-0.586794\pi$
$181$	−7.24625	−0.538610	−0.269305	+	0.963055i	$0.586794\pi$
$182$	0	0
$183$	6.08170	0.449572
$184$	0	0
$185$	8.41420	0.618624
$186$	0	0
$187$	−12.2508	−0.895867
$188$	0	0
$189$	−2.77861	−0.202114
$190$	0	0
$191$	13.2153	0.956227	0.478113	−	0.878298i	$-0.341321\pi$
$191$	13.2153	0.956227	0.478113	+	0.878298i	$0.341321\pi$
$192$	0	0
$193$	16.9475	1.21991	0.609953	−	0.792438i	$-0.291188\pi$
$193$	16.9475	1.21991	0.609953	+	0.792438i	$0.291188\pi$
$194$	0	0
$195$	11.1595	0.799145
$196$	0	0
$197$	12.6151	0.898792	0.449396	−	0.893333i	$-0.351639\pi$
$197$	12.6151	0.898792	0.449396	+	0.893333i	$0.351639\pi$
$198$	0	0
$199$	2.81564	0.199595	0.0997976	−	0.995008i	$-0.468180\pi$
$199$	2.81564	0.199595	0.0997976	+	0.995008i	$0.468180\pi$
$200$	0	0
$201$	14.8238	1.04559
$202$	0	0
$203$	20.1664	1.41540
$204$	0	0
$205$	2.98788	0.208683
$206$	0	0
$207$	1.72034	0.119572
$208$	0	0
$209$	−18.3536	−1.26955
$210$	0	0
$211$	−11.1832	−0.769885	−0.384942	−	0.922941i	$-0.625779\pi$
$211$	−11.1832	−0.769885	−0.384942	+	0.922941i	$0.625779\pi$
$212$	0	0
$213$	−7.48522	−0.512879
$214$	0	0
$215$	3.17217	0.216340
$216$	0	0
$217$	−6.56815	−0.445875
$218$	0	0
$219$	−3.99718	−0.270105
$220$	0	0
$221$	−24.8182	−1.66945
$222$	0	0
$223$	5.71568	0.382750	0.191375	−	0.981517i	$-0.438705\pi$
$223$	5.71568	0.382750	0.191375	+	0.981517i	$0.438705\pi$
$224$	0	0
$225$	−2.39778	−0.159852
$226$	0	0
$227$	22.3863	1.48583	0.742915	−	0.669386i	$-0.233443\pi$
$227$	22.3863	1.48583	0.742915	+	0.669386i	$0.233443\pi$
$228$	0	0
$229$	23.2881	1.53892	0.769459	−	0.638696i	$-0.220526\pi$
$229$	23.2881	1.53892	0.769459	+	0.638696i	$0.220526\pi$
$230$	0	0
$231$	9.48839	0.624290
$232$	0	0
$233$	9.22181	0.604141	0.302071	−	0.953286i	$-0.402322\pi$
$233$	9.22181	0.604141	0.302071	+	0.953286i	$0.402322\pi$
$234$	0	0
$235$	12.7954	0.834678
$236$	0	0
$237$	6.01895	0.390973
$238$	0	0
$239$	0.000511214 0	3.30677e−5 0	1.65339e−5	−	1.00000i	$-0.499995\pi$
$239$	0.000511214 0	3.30677e−5 0	1.65339e−5		1.00000i	$0.499995\pi$
$240$	0	0
$241$	−19.3395	−1.24577	−0.622883	−	0.782315i	$-0.714039\pi$
$241$	−19.3395	−1.24577	−0.622883	+	0.782315i	$0.714039\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.16258	−0.0742748
$246$	0	0
$247$	−37.1816	−2.36581
$248$	0	0
$249$	9.27564	0.587820
$250$	0	0
$251$	8.84328	0.558183	0.279092	−	0.960264i	$-0.409967\pi$
$251$	8.84328	0.558183	0.279092	+	0.960264i	$0.409967\pi$
$252$	0	0
$253$	−5.87462	−0.369334
$254$	0	0
$255$	−5.78725	−0.362411
$256$	0	0
$257$	23.4052	1.45997	0.729987	−	0.683461i	$-0.239526\pi$
$257$	23.4052	1.45997	0.729987	+	0.683461i	$0.239526\pi$
$258$	0	0
$259$	14.4933	0.900573
$260$	0	0
$261$	−7.25770	−0.449240
$262$	0	0
$263$	−10.8833	−0.671091	−0.335546	−	0.942024i	$-0.608921\pi$
$263$	−10.8833	−0.671091	−0.335546	+	0.942024i	$0.608921\pi$
$264$	0	0
$265$	−0.280227	−0.0172142
$266$	0	0
$267$	−8.88589	−0.543808
$268$	0	0
$269$	−19.1446	−1.16727	−0.583633	−	0.812018i	$-0.698369\pi$
$269$	−19.1446	−1.16727	−0.583633	+	0.812018i	$0.698369\pi$
$270$	0	0
$271$	10.1077	0.613999	0.306999	−	0.951710i	$-0.400675\pi$
$271$	10.1077	0.613999	0.306999	+	0.951710i	$0.400675\pi$
$272$	0	0
$273$	19.2220	1.16337
$274$	0	0
$275$	8.18791	0.493750
$276$	0	0
$277$	23.1330	1.38993	0.694964	−	0.719044i	$-0.255420\pi$
$277$	23.1330	1.38993	0.694964	+	0.719044i	$0.255420\pi$
$278$	0	0
$279$	2.36382	0.141518
$280$	0	0
$281$	1.91354	0.114152	0.0570762	−	0.998370i	$-0.481822\pi$
$281$	1.91354	0.114152	0.0570762	+	0.998370i	$0.481822\pi$
$282$	0	0
$283$	−15.7243	−0.934710	−0.467355	−	0.884070i	$-0.654793\pi$
$283$	−15.7243	−0.934710	−0.467355	+	0.884070i	$0.654793\pi$
$284$	0	0
$285$	−8.67021	−0.513579
$286$	0	0
$287$	5.14659	0.303793
$288$	0	0
$289$	−4.12937	−0.242904
$290$	0	0
$291$	−15.7656	−0.924198
$292$	0	0
$293$	6.16688	0.360273	0.180137	−	0.983642i	$-0.442346\pi$
$293$	6.16688	0.360273	0.180137	+	0.983642i	$0.442346\pi$
$294$	0	0
$295$	−13.1381	−0.764928
$296$	0	0
$297$	−3.41479	−0.198146
$298$	0	0
$299$	−11.9011	−0.688256
$300$	0	0
$301$	5.46402	0.314941
$302$	0	0
$303$	0.521324	0.0299493
$304$	0	0
$305$	−9.81064	−0.561756
$306$	0	0
$307$	4.72463	0.269649	0.134825	−	0.990869i	$-0.456953\pi$
$307$	4.72463	0.269649	0.134825	+	0.990869i	$0.456953\pi$
$308$	0	0
$309$	−12.2203	−0.695190
$310$	0	0
$311$	−5.29036	−0.299989	−0.149994	−	0.988687i	$-0.547926\pi$
$311$	−5.29036	−0.299989	−0.149994	+	0.988687i	$0.547926\pi$
$312$	0	0
$313$	−11.9585	−0.675932	−0.337966	−	0.941158i	$-0.609739\pi$
$313$	−11.9585	−0.675932	−0.337966	+	0.941158i	$0.609739\pi$
$314$	0	0
$315$	4.48229	0.252549
$316$	0	0
$317$	28.2780	1.58825	0.794125	−	0.607754i	$-0.207929\pi$
$317$	28.2780	1.58825	0.794125	+	0.607754i	$0.207929\pi$
$318$	0	0
$319$	24.7835	1.38761
$320$	0	0
$321$	−13.4469	−0.750532
$322$	0	0
$323$	19.2822	1.07289
$324$	0	0
$325$	16.5874	0.920106
$326$	0	0
$327$	10.3423	0.571930
$328$	0	0
$329$	22.0399	1.21510
$330$	0	0
$331$	−19.4000	−1.06632	−0.533160	−	0.846015i	$-0.678995\pi$
$331$	−19.4000	−1.06632	−0.533160	+	0.846015i	$0.678995\pi$
$332$	0	0
$333$	−5.21603	−0.285837
$334$	0	0
$335$	−23.9129	−1.30650
$336$	0	0
$337$	−35.0860	−1.91126	−0.955628	−	0.294577i	$-0.904821\pi$
$337$	−35.0860	−1.91126	−0.955628	+	0.294577i	$0.904821\pi$
$338$	0	0
$339$	−4.02353	−0.218528
$340$	0	0
$341$	−8.07196	−0.437121
$342$	0	0
$343$	17.4478	0.942091
$344$	0	0
$345$	−2.77515	−0.149409
$346$	0	0
$347$	33.0942	1.77659	0.888296	−	0.459271i	$-0.151889\pi$
$347$	33.0942	1.77659	0.888296	+	0.459271i	$0.151889\pi$
$348$	0	0
$349$	−26.1959	−1.40224	−0.701118	−	0.713045i	$-0.747316\pi$
$349$	−26.1959	−1.40224	−0.701118	+	0.713045i	$0.747316\pi$
$350$	0	0
$351$	−6.91784	−0.369247
$352$	0	0
$353$	−13.4533	−0.716046	−0.358023	−	0.933713i	$-0.616549\pi$
$353$	−13.4533	−0.716046	−0.358023	+	0.933713i	$0.616549\pi$
$354$	0	0
$355$	12.0747	0.640859
$356$	0	0
$357$	−9.96846	−0.527587
$358$	0	0
$359$	−24.1639	−1.27532	−0.637660	−	0.770318i	$-0.720098\pi$
$359$	−24.1639	−1.27532	−0.637660	+	0.770318i	$0.720098\pi$
$360$	0	0
$361$	9.88782	0.520412
$362$	0	0
$363$	0.660811	0.0346836
$364$	0	0
$365$	6.44802	0.337505
$366$	0	0
$367$	−18.5172	−0.966592	−0.483296	−	0.875457i	$-0.660561\pi$
$367$	−18.5172	−0.966592	−0.483296	+	0.875457i	$0.660561\pi$
$368$	0	0
$369$	−1.85221	−0.0964224
$370$	0	0
$371$	−0.482687	−0.0250599
$372$	0	0
$373$	27.1089	1.40364	0.701822	−	0.712353i	$-0.252370\pi$
$373$	27.1089	1.40364	0.701822	+	0.712353i	$0.252370\pi$
$374$	0	0
$375$	11.9337	0.616251
$376$	0	0
$377$	50.2076	2.58582
$378$	0	0
$379$	−19.7760	−1.01582	−0.507911	−	0.861409i	$-0.669582\pi$
$379$	−19.7760	−1.01582	−0.507911	+	0.861409i	$0.669582\pi$
$380$	0	0
$381$	−1.94300	−0.0995432
$382$	0	0
$383$	−26.9669	−1.37795	−0.688973	−	0.724787i	$-0.741938\pi$
$383$	−26.9669	−1.37795	−0.688973	+	0.724787i	$0.741938\pi$
$384$	0	0
$385$	−15.3061	−0.780072
$386$	0	0
$387$	−1.96645	−0.0999605
$388$	0	0
$389$	3.65831	0.185484	0.0927418	−	0.995690i	$-0.470437\pi$
$389$	3.65831	0.185484	0.0927418	+	0.995690i	$0.470437\pi$
$390$	0	0
$391$	6.17184	0.312124
$392$	0	0
$393$	16.6952	0.842160
$394$	0	0
$395$	−9.70941	−0.488533
$396$	0	0
$397$	−4.03329	−0.202425	−0.101213	−	0.994865i	$-0.532272\pi$
$397$	−4.03329	−0.202425	−0.101213	+	0.994865i	$0.532272\pi$
$398$	0	0
$399$	−14.9343	−0.747651
$400$	0	0
$401$	32.5483	1.62538	0.812692	−	0.582694i	$-0.198001\pi$
$401$	32.5483	1.62538	0.812692	+	0.582694i	$0.198001\pi$
$402$	0	0
$403$	−16.3525	−0.814578
$404$	0	0
$405$	−1.61314	−0.0801576
$406$	0	0
$407$	17.8117	0.882892
$408$	0	0
$409$	0.138153	0.00683122	0.00341561	−	0.999994i	$-0.498913\pi$
$409$	0.138153	0.00683122	0.00341561	+	0.999994i	$0.498913\pi$
$410$	0	0
$411$	6.18334	0.305002
$412$	0	0
$413$	−22.6301	−1.11356
$414$	0	0
$415$	−14.9629	−0.734500
$416$	0	0
$417$	14.1406	0.692470
$418$	0	0
$419$	23.2816	1.13738	0.568690	−	0.822552i	$-0.307450\pi$
$419$	23.2816	1.13738	0.568690	+	0.822552i	$0.307450\pi$
$420$	0	0
$421$	−34.5060	−1.68172	−0.840859	−	0.541255i	$-0.817950\pi$
$421$	−34.5060	−1.68172	−0.840859	+	0.541255i	$0.817950\pi$
$422$	0	0
$423$	−7.93197	−0.385665
$424$	0	0
$425$	−8.60219	−0.417267
$426$	0	0
$427$	−16.8987	−0.817785
$428$	0	0
$429$	23.6230	1.14053
$430$	0	0
$431$	31.8590	1.53459	0.767296	−	0.641293i	$-0.221602\pi$
$431$	31.8590	1.53459	0.767296	+	0.641293i	$0.221602\pi$
$432$	0	0
$433$	28.2106	1.35572	0.677858	−	0.735193i	$-0.262908\pi$
$433$	28.2106	1.35572	0.677858	+	0.735193i	$0.262908\pi$
$434$	0	0
$435$	11.7077	0.561341
$436$	0	0
$437$	9.24639	0.442315
$438$	0	0
$439$	31.9739	1.52603	0.763016	−	0.646380i	$-0.223718\pi$
$439$	31.9739	1.52603	0.763016	+	0.646380i	$0.223718\pi$
$440$	0	0
$441$	0.720696	0.0343189
$442$	0	0
$443$	−15.9146	−0.756125	−0.378063	−	0.925780i	$-0.623410\pi$
$443$	−15.9146	−0.756125	−0.378063	+	0.925780i	$0.623410\pi$
$444$	0	0
$445$	14.3342	0.679506
$446$	0	0
$447$	18.7763	0.888089
$448$	0	0
$449$	−0.915231	−0.0431924	−0.0215962	−	0.999767i	$-0.506875\pi$
$449$	−0.915231	−0.0431924	−0.0215962	+	0.999767i	$0.506875\pi$
$450$	0	0
$451$	6.32492	0.297829
$452$	0	0
$453$	13.9652	0.656144
$454$	0	0
$455$	−31.0078	−1.45367
$456$	0	0
$457$	−32.7336	−1.53121	−0.765606	−	0.643310i	$-0.777561\pi$
$457$	−32.7336	−1.53121	−0.765606	+	0.643310i	$0.777561\pi$
$458$	0	0
$459$	3.58757	0.167453
$460$	0	0
$461$	18.8469	0.877787	0.438894	−	0.898539i	$-0.355371\pi$
$461$	18.8469	0.877787	0.438894	+	0.898539i	$0.355371\pi$
$462$	0	0
$463$	14.9595	0.695229	0.347615	−	0.937638i	$-0.386992\pi$
$463$	14.9595	0.695229	0.347615	+	0.937638i	$0.386992\pi$
$464$	0	0
$465$	−3.81318	−0.176832
$466$	0	0
$467$	−43.0393	−1.99162	−0.995811	−	0.0914394i	$-0.970853\pi$
$467$	−43.0393	−1.99162	−0.995811	+	0.0914394i	$0.970853\pi$
$468$	0	0
$469$	−41.1897	−1.90197
$470$	0	0
$471$	10.1676	0.468497
$472$	0	0
$473$	6.71504	0.308758
$474$	0	0
$475$	−12.8874	−0.591316
$476$	0	0
$477$	0.173715	0.00795386
$478$	0	0
$479$	0.537405	0.0245546	0.0122773	−	0.999925i	$-0.496092\pi$
$479$	0.537405	0.0245546	0.0122773	+	0.999925i	$0.496092\pi$
$480$	0	0
$481$	36.0837	1.64527
$482$	0	0
$483$	−4.78017	−0.217505
$484$	0	0
$485$	25.4322	1.15482
$486$	0	0
$487$	−19.9431	−0.903706	−0.451853	−	0.892092i	$-0.649237\pi$
$487$	−19.9431	−0.903706	−0.451853	+	0.892092i	$0.649237\pi$
$488$	0	0
$489$	−7.01046	−0.317024
$490$	0	0
$491$	−7.30076	−0.329479	−0.164739	−	0.986337i	$-0.552678\pi$
$491$	−7.30076	−0.329479	−0.164739	+	0.986337i	$0.552678\pi$
$492$	0	0
$493$	−26.0375	−1.17267
$494$	0	0
$495$	5.50854	0.247591
$496$	0	0
$497$	20.7985	0.932942
$498$	0	0
$499$	−35.5070	−1.58951	−0.794756	−	0.606929i	$-0.792401\pi$
$499$	−35.5070	−1.58951	−0.794756	+	0.606929i	$0.792401\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−39.2092	−1.74825	−0.874125	−	0.485701i	$-0.838564\pi$
$503$	−39.2092	−1.74825	−0.874125	+	0.485701i	$0.838564\pi$
$504$	0	0
$505$	−0.840969	−0.0374226
$506$	0	0
$507$	34.8565	1.54803
$508$	0	0
$509$	−0.416674	−0.0184688	−0.00923438	−	0.999957i	$-0.502939\pi$
$509$	−0.416674	−0.0184688	−0.00923438	+	0.999957i	$0.502939\pi$
$510$	0	0
$511$	11.1066	0.491329
$512$	0	0
$513$	5.37474	0.237301
$514$	0	0
$515$	19.7131	0.868663
$516$	0	0
$517$	27.0860	1.19124
$518$	0	0
$519$	−2.10797	−0.0925295
$520$	0	0
$521$	−35.9614	−1.57550	−0.787749	−	0.615996i	$-0.788754\pi$
$521$	−35.9614	−1.57550	−0.787749	+	0.615996i	$0.788754\pi$
$522$	0	0
$523$	−30.8956	−1.35097	−0.675485	−	0.737373i	$-0.736066\pi$
$523$	−30.8956	−1.35097	−0.675485	+	0.737373i	$0.736066\pi$
$524$	0	0
$525$	6.66250	0.290775
$526$	0	0
$527$	8.48036	0.369410
$528$	0	0
$529$	−20.0404	−0.871323
$530$	0	0
$531$	8.14440	0.353437
$532$	0	0
$533$	12.8133	0.555006
$534$	0	0
$535$	21.6917	0.937815
$536$	0	0
$537$	20.6203	0.889833
$538$	0	0
$539$	−2.46103	−0.106004
$540$	0	0
$541$	−14.1391	−0.607886	−0.303943	−	0.952690i	$-0.598303\pi$
$541$	−14.1391	−0.607886	−0.303943	+	0.952690i	$0.598303\pi$
$542$	0	0
$543$	−7.24625	−0.310966
$544$	0	0
$545$	−16.6836	−0.714646
$546$	0	0
$547$	−23.5859	−1.00846	−0.504230	−	0.863569i	$-0.668224\pi$
$547$	−23.5859	−1.00846	−0.504230	+	0.863569i	$0.668224\pi$
$548$	0	0
$549$	6.08170	0.259561
$550$	0	0
$551$	−39.0083	−1.66181
$552$	0	0
$553$	−16.7243	−0.711191
$554$	0	0
$555$	8.41420	0.357163
$556$	0	0
$557$	6.14011	0.260165	0.130082	−	0.991503i	$-0.458476\pi$
$557$	6.14011	0.260165	0.130082	+	0.991503i	$0.458476\pi$
$558$	0	0
$559$	13.6036	0.575372
$560$	0	0
$561$	−12.2508	−0.517229
$562$	0	0
$563$	14.2795	0.601807	0.300904	−	0.953655i	$-0.402712\pi$
$563$	14.2795	0.601807	0.300904	+	0.953655i	$0.402712\pi$
$564$	0	0
$565$	6.49052	0.273058
$566$	0	0
$567$	−2.77861	−0.116691
$568$	0	0
$569$	24.8067	1.03995	0.519976	−	0.854181i	$-0.325941\pi$
$569$	24.8067	1.03995	0.519976	+	0.854181i	$0.325941\pi$
$570$	0	0
$571$	2.69270	0.112686	0.0563429	−	0.998411i	$-0.482056\pi$
$571$	2.69270	0.112686	0.0563429	+	0.998411i	$0.482056\pi$
$572$	0	0
$573$	13.2153	0.552078
$574$	0	0
$575$	−4.12500	−0.172024
$576$	0	0
$577$	45.9555	1.91315	0.956576	−	0.291483i	$-0.0941488\pi$
$577$	45.9555	1.91315	0.956576	+	0.291483i	$0.0941488\pi$
$578$	0	0
$579$	16.9475	0.704313
$580$	0	0
$581$	−25.7734	−1.06926
$582$	0	0
$583$	−0.593201	−0.0245679
$584$	0	0
$585$	11.1595	0.461386
$586$	0	0
$587$	5.24633	0.216539	0.108270	−	0.994122i	$-0.465469\pi$
$587$	5.24633	0.216539	0.108270	+	0.994122i	$0.465469\pi$
$588$	0	0
$589$	12.7049	0.523497
$590$	0	0
$591$	12.6151	0.518918
$592$	0	0
$593$	−43.8456	−1.80052	−0.900261	−	0.435351i	$-0.856624\pi$
$593$	−43.8456	−1.80052	−0.900261	+	0.435351i	$0.856624\pi$
$594$	0	0
$595$	16.0805	0.659237
$596$	0	0
$597$	2.81564	0.115236
$598$	0	0
$599$	37.3958	1.52795	0.763975	−	0.645246i	$-0.223245\pi$
$599$	37.3958	1.52795	0.763975	+	0.645246i	$0.223245\pi$
$600$	0	0
$601$	−19.6305	−0.800745	−0.400373	−	0.916352i	$-0.631119\pi$
$601$	−19.6305	−0.800745	−0.400373	+	0.916352i	$0.631119\pi$
$602$	0	0
$603$	14.8238	0.603674
$604$	0	0
$605$	−1.06598	−0.0433383
$606$	0	0
$607$	−27.5629	−1.11874	−0.559372	−	0.828917i	$-0.688958\pi$
$607$	−27.5629	−1.11874	−0.559372	+	0.828917i	$0.688958\pi$
$608$	0	0
$609$	20.1664	0.817182
$610$	0	0
$611$	54.8721	2.21989
$612$	0	0
$613$	−33.2745	−1.34395	−0.671973	−	0.740576i	$-0.734553\pi$
$613$	−33.2745	−1.34395	−0.671973	+	0.740576i	$0.734553\pi$
$614$	0	0
$615$	2.98788	0.120483
$616$	0	0
$617$	29.9348	1.20513	0.602565	−	0.798070i	$-0.294145\pi$
$617$	29.9348	1.20513	0.602565	+	0.798070i	$0.294145\pi$
$618$	0	0
$619$	−12.5037	−0.502564	−0.251282	−	0.967914i	$-0.580852\pi$
$619$	−12.5037	−0.502564	−0.251282	+	0.967914i	$0.580852\pi$
$620$	0	0
$621$	1.72034	0.0690350
$622$	0	0
$623$	24.6905	0.989202
$624$	0	0
$625$	−7.26177	−0.290471
$626$	0	0
$627$	−18.3536	−0.732973
$628$	0	0
$629$	−18.7129	−0.746131
$630$	0	0
$631$	14.2670	0.567960	0.283980	−	0.958830i	$-0.408345\pi$
$631$	14.2670	0.567960	0.283980	+	0.958830i	$0.408345\pi$
$632$	0	0
$633$	−11.1832	−0.444493
$634$	0	0
$635$	3.13434	0.124382
$636$	0	0
$637$	−4.98566	−0.197539
$638$	0	0
$639$	−7.48522	−0.296111
$640$	0	0
$641$	22.7348	0.897973	0.448986	−	0.893539i	$-0.351785\pi$
$641$	22.7348	0.897973	0.448986	+	0.893539i	$0.351785\pi$
$642$	0	0
$643$	34.2460	1.35053	0.675266	−	0.737575i	$-0.264029\pi$
$643$	34.2460	1.35053	0.675266	+	0.737575i	$0.264029\pi$
$644$	0	0
$645$	3.17217	0.124904
$646$	0	0
$647$	25.4985	1.00245	0.501224	−	0.865317i	$-0.332883\pi$
$647$	25.4985	1.00245	0.501224	+	0.865317i	$0.332883\pi$
$648$	0	0
$649$	−27.8114	−1.09169
$650$	0	0
$651$	−6.56815	−0.257426
$652$	0	0
$653$	−21.4770	−0.840460	−0.420230	−	0.907418i	$-0.638051\pi$
$653$	−21.4770	−0.840460	−0.420230	+	0.907418i	$0.638051\pi$
$654$	0	0
$655$	−26.9317	−1.05231
$656$	0	0
$657$	−3.99718	−0.155945
$658$	0	0
$659$	−14.9174	−0.581098	−0.290549	−	0.956860i	$-0.593838\pi$
$659$	−14.9174	−0.581098	−0.290549	+	0.956860i	$0.593838\pi$
$660$	0	0
$661$	17.4388	0.678291	0.339146	−	0.940734i	$-0.389862\pi$
$661$	17.4388	0.678291	0.339146	+	0.940734i	$0.389862\pi$
$662$	0	0
$663$	−24.8182	−0.963859
$664$	0	0
$665$	24.0912	0.934215
$666$	0	0
$667$	−12.4857	−0.483450
$668$	0	0
$669$	5.71568	0.220981
$670$	0	0
$671$	−20.7678	−0.801730
$672$	0	0
$673$	6.96187	0.268360	0.134180	−	0.990957i	$-0.457160\pi$
$673$	6.96187	0.268360	0.134180	+	0.990957i	$0.457160\pi$
$674$	0	0
$675$	−2.39778	−0.0922905
$676$	0	0
$677$	−14.3354	−0.550955	−0.275478	−	0.961308i	$-0.588836\pi$
$677$	−14.3354	−0.550955	−0.275478	+	0.961308i	$0.588836\pi$
$678$	0	0
$679$	43.8066	1.68114
$680$	0	0
$681$	22.3863	0.857845
$682$	0	0
$683$	25.1139	0.960956	0.480478	−	0.877007i	$-0.340463\pi$
$683$	25.1139	0.960956	0.480478	+	0.877007i	$0.340463\pi$
$684$	0	0
$685$	−9.97459	−0.381110
$686$	0	0
$687$	23.2881	0.888495
$688$	0	0
$689$	−1.20173	−0.0457824
$690$	0	0
$691$	48.3951	1.84104	0.920518	−	0.390700i	$-0.127767\pi$
$691$	48.3951	1.84104	0.920518	+	0.390700i	$0.127767\pi$
$692$	0	0
$693$	9.48839	0.360434
$694$	0	0
$695$	−22.8108	−0.865264
$696$	0	0
$697$	−6.64494	−0.251695
$698$	0	0
$699$	9.22181	0.348801
$700$	0	0
$701$	36.9203	1.39446	0.697231	−	0.716846i	$-0.254415\pi$
$701$	36.9203	1.39446	0.697231	+	0.716846i	$0.254415\pi$
$702$	0	0
$703$	−28.0348	−1.05735
$704$	0	0
$705$	12.7954	0.481902
$706$	0	0
$707$	−1.44856	−0.0544787
$708$	0	0
$709$	−31.2996	−1.17548	−0.587740	−	0.809050i	$-0.699982\pi$
$709$	−31.2996	−1.17548	−0.587740	+	0.809050i	$0.699982\pi$
$710$	0	0
$711$	6.01895	0.225728
$712$	0	0
$713$	4.06658	0.152295
$714$	0	0
$715$	−38.1072	−1.42513
$716$	0	0
$717$	0.000511214 0	1.90916e−5 0
$718$	0	0
$719$	25.0431	0.933951	0.466976	−	0.884270i	$-0.345344\pi$
$719$	25.0431	0.933951	0.466976	+	0.884270i	$0.345344\pi$
$720$	0	0
$721$	33.9556	1.26457
$722$	0	0
$723$	−19.3395	−0.719244
$724$	0	0
$725$	17.4024	0.646307
$726$	0	0
$727$	13.4493	0.498807	0.249403	−	0.968400i	$-0.419765\pi$
$727$	13.4493	0.498807	0.249403	+	0.968400i	$0.419765\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.05479	−0.260931
$732$	0	0
$733$	−6.83801	−0.252568	−0.126284	−	0.991994i	$-0.540305\pi$
$733$	−6.83801	−0.252568	−0.126284	+	0.991994i	$0.540305\pi$
$734$	0	0
$735$	−1.16258	−0.0428826
$736$	0	0
$737$	−50.6204	−1.86462
$738$	0	0
$739$	16.7041	0.614469	0.307234	−	0.951634i	$-0.400596\pi$
$739$	16.7041	0.614469	0.307234	+	0.951634i	$0.400596\pi$
$740$	0	0
$741$	−37.1816	−1.36590
$742$	0	0
$743$	39.7227	1.45728	0.728642	−	0.684894i	$-0.240152\pi$
$743$	39.7227	1.45728	0.728642	+	0.684894i	$0.240152\pi$
$744$	0	0
$745$	−30.2888	−1.10970
$746$	0	0
$747$	9.27564	0.339378
$748$	0	0
$749$	37.3637	1.36524
$750$	0	0
$751$	3.40539	0.124264	0.0621322	−	0.998068i	$-0.480210\pi$
$751$	3.40539	0.124264	0.0621322	+	0.998068i	$0.480210\pi$
$752$	0	0
$753$	8.84328	0.322267
$754$	0	0
$755$	−22.5279	−0.819874
$756$	0	0
$757$	−12.9591	−0.471007	−0.235503	−	0.971874i	$-0.575674\pi$
$757$	−12.9591	−0.471007	−0.235503	+	0.971874i	$0.575674\pi$
$758$	0	0
$759$	−5.87462	−0.213235
$760$	0	0
$761$	27.1687	0.984864	0.492432	−	0.870351i	$-0.336108\pi$
$761$	27.1687	0.984864	0.492432	+	0.870351i	$0.336108\pi$
$762$	0	0
$763$	−28.7372	−1.04036
$764$	0	0
$765$	−5.78725	−0.209238
$766$	0	0
$767$	−56.3417	−2.03438
$768$	0	0
$769$	−40.0895	−1.44566	−0.722832	−	0.691024i	$-0.757160\pi$
$769$	−40.0895	−1.44566	−0.722832	+	0.691024i	$0.757160\pi$
$770$	0	0
$771$	23.4052	0.842917
$772$	0	0
$773$	53.0227	1.90709	0.953546	−	0.301246i	$-0.0974025\pi$
$773$	53.0227	1.90709	0.953546	+	0.301246i	$0.0974025\pi$
$774$	0	0
$775$	−5.66792	−0.203598
$776$	0	0
$777$	14.4933	0.519946
$778$	0	0
$779$	−9.95516	−0.356681
$780$	0	0
$781$	25.5605	0.914626
$782$	0	0
$783$	−7.25770	−0.259369
$784$	0	0
$785$	−16.4017	−0.585403
$786$	0	0
$787$	15.5222	0.553305	0.276653	−	0.960970i	$-0.410775\pi$
$787$	15.5222	0.553305	0.276653	+	0.960970i	$0.410775\pi$
$788$	0	0
$789$	−10.8833	−0.387455
$790$	0	0
$791$	11.1798	0.397509
$792$	0	0
$793$	−42.0722	−1.49403
$794$	0	0
$795$	−0.280227	−0.00993862
$796$	0	0
$797$	−22.7831	−0.807019	−0.403509	−	0.914976i	$-0.632210\pi$
$797$	−22.7831	−0.807019	−0.403509	+	0.914976i	$0.632210\pi$
$798$	0	0
$799$	−28.4565	−1.00672
$800$	0	0
$801$	−8.88589	−0.313968
$802$	0	0
$803$	13.6496	0.481682
$804$	0	0
$805$	7.71108	0.271780
$806$	0	0
$807$	−19.1446	−0.673921
$808$	0	0
$809$	20.6785	0.727017	0.363509	−	0.931591i	$-0.381579\pi$
$809$	20.6785	0.727017	0.363509	+	0.931591i	$0.381579\pi$
$810$	0	0
$811$	18.7227	0.657444	0.328722	−	0.944427i	$-0.393382\pi$
$811$	18.7227	0.657444	0.328722	+	0.944427i	$0.393382\pi$
$812$	0	0
$813$	10.1077	0.354492
$814$	0	0
$815$	11.3089	0.396132
$816$	0	0
$817$	−10.5692	−0.369769
$818$	0	0
$819$	19.2220	0.671671
$820$	0	0
$821$	−49.6746	−1.73366	−0.866829	−	0.498606i	$-0.833845\pi$
$821$	−49.6746	−1.73366	−0.866829	+	0.498606i	$0.833845\pi$
$822$	0	0
$823$	−8.42160	−0.293558	−0.146779	−	0.989169i	$-0.546891\pi$
$823$	−8.42160	−0.293558	−0.146779	+	0.989169i	$0.546891\pi$
$824$	0	0
$825$	8.18791	0.285067
$826$	0	0
$827$	−15.7215	−0.546691	−0.273346	−	0.961916i	$-0.588130\pi$
$827$	−15.7215	−0.546691	−0.273346	+	0.961916i	$0.588130\pi$
$828$	0	0
$829$	19.0607	0.662006	0.331003	−	0.943630i	$-0.392613\pi$
$829$	19.0607	0.662006	0.331003	+	0.943630i	$0.392613\pi$
$830$	0	0
$831$	23.1330	0.802476
$832$	0	0
$833$	2.58554	0.0895838
$834$	0	0
$835$	−1.61314	−0.0558250
$836$	0	0
$837$	2.36382	0.0817056
$838$	0	0
$839$	−31.9834	−1.10419	−0.552095	−	0.833781i	$-0.686172\pi$
$839$	−31.9834	−1.10419	−0.552095	+	0.833781i	$0.686172\pi$
$840$	0	0
$841$	23.6742	0.816353
$842$	0	0
$843$	1.91354	0.0659060
$844$	0	0
$845$	−56.2285	−1.93432
$846$	0	0
$847$	−1.83614	−0.0630905
$848$	0	0
$849$	−15.7243	−0.539655
$850$	0	0
$851$	−8.97337	−0.307603
$852$	0	0
$853$	−36.2696	−1.24185	−0.620924	−	0.783871i	$-0.713242\pi$
$853$	−36.2696	−1.24185	−0.620924	+	0.783871i	$0.713242\pi$
$854$	0	0
$855$	−8.67021	−0.296515
$856$	0	0
$857$	−44.1656	−1.50867	−0.754333	−	0.656492i	$-0.772040\pi$
$857$	−44.1656	−1.50867	−0.754333	+	0.656492i	$0.772040\pi$
$858$	0	0
$859$	21.9021	0.747290	0.373645	−	0.927572i	$-0.378108\pi$
$859$	21.9021	0.747290	0.373645	+	0.927572i	$0.378108\pi$
$860$	0	0
$861$	5.14659	0.175395
$862$	0	0
$863$	−27.7755	−0.945489	−0.472745	−	0.881200i	$-0.656737\pi$
$863$	−27.7755	−0.945489	−0.472745	+	0.881200i	$0.656737\pi$
$864$	0	0
$865$	3.40045	0.115619
$866$	0	0
$867$	−4.12937	−0.140241
$868$	0	0
$869$	−20.5535	−0.697228
$870$	0	0
$871$	−102.549	−3.47474
$872$	0	0
$873$	−15.7656	−0.533586
$874$	0	0
$875$	−33.1590	−1.12098
$876$	0	0
$877$	9.87255	0.333372	0.166686	−	0.986010i	$-0.446693\pi$
$877$	9.87255	0.333372	0.166686	+	0.986010i	$0.446693\pi$
$878$	0	0
$879$	6.16688	0.208004
$880$	0	0
$881$	−54.6502	−1.84121	−0.920606	−	0.390492i	$-0.872305\pi$
$881$	−54.6502	−1.84121	−0.920606	+	0.390492i	$0.872305\pi$
$882$	0	0
$883$	42.6946	1.43679	0.718394	−	0.695636i	$-0.244877\pi$
$883$	42.6946	1.43679	0.718394	+	0.695636i	$0.244877\pi$
$884$	0	0
$885$	−13.1381	−0.441631
$886$	0	0
$887$	35.7614	1.20075	0.600375	−	0.799718i	$-0.295018\pi$
$887$	35.7614	1.20075	0.600375	+	0.799718i	$0.295018\pi$
$888$	0	0
$889$	5.39886	0.181072
$890$	0	0
$891$	−3.41479	−0.114400
$892$	0	0
$893$	−42.6323	−1.42663
$894$	0	0
$895$	−33.2635	−1.11188
$896$	0	0
$897$	−11.9011	−0.397365
$898$	0	0
$899$	−17.1559	−0.572182
$900$	0	0
$901$	0.623214	0.0207623
$902$	0	0
$903$	5.46402	0.181831
$904$	0	0
$905$	11.6892	0.388563
$906$	0	0
$907$	32.7057	1.08598	0.542988	−	0.839741i	$-0.317293\pi$
$907$	32.7057	1.08598	0.542988	+	0.839741i	$0.317293\pi$
$908$	0	0
$909$	0.521324	0.0172912
$910$	0	0
$911$	2.49656	0.0827146	0.0413573	−	0.999144i	$-0.486832\pi$
$911$	2.49656	0.0827146	0.0413573	+	0.999144i	$0.486832\pi$
$912$	0	0
$913$	−31.6744	−1.04827
$914$	0	0
$915$	−9.81064	−0.324330
$916$	0	0
$917$	−46.3895	−1.53191
$918$	0	0
$919$	−17.2722	−0.569757	−0.284879	−	0.958564i	$-0.591953\pi$
$919$	−17.2722	−0.569757	−0.284879	+	0.958564i	$0.591953\pi$
$920$	0	0
$921$	4.72463	0.155682
$922$	0	0
$923$	51.7816	1.70441
$924$	0	0
$925$	12.5069	0.411224
$926$	0	0
$927$	−12.2203	−0.401368
$928$	0	0
$929$	41.6554	1.36667	0.683335	−	0.730105i	$-0.260529\pi$
$929$	41.6554	1.36667	0.683335	+	0.730105i	$0.260529\pi$
$930$	0	0
$931$	3.87355	0.126951
$932$	0	0
$933$	−5.29036	−0.173199
$934$	0	0
$935$	19.7623	0.646295
$936$	0	0
$937$	−24.1561	−0.789145	−0.394573	−	0.918865i	$-0.629107\pi$
$937$	−24.1561	−0.789145	−0.394573	+	0.918865i	$0.629107\pi$
$938$	0	0
$939$	−11.9585	−0.390250
$940$	0	0
$941$	15.3192	0.499391	0.249696	−	0.968324i	$-0.419669\pi$
$941$	15.3192	0.499391	0.249696	+	0.968324i	$0.419669\pi$
$942$	0	0
$943$	−3.18644	−0.103765
$944$	0	0
$945$	4.48229	0.145809
$946$	0	0
$947$	−3.12793	−0.101644	−0.0508220	−	0.998708i	$-0.516184\pi$
$947$	−3.12793	−0.101644	−0.0508220	+	0.998708i	$0.516184\pi$
$948$	0	0
$949$	27.6519	0.897618
$950$	0	0
$951$	28.2780	0.916977
$952$	0	0
$953$	29.3610	0.951095	0.475548	−	0.879690i	$-0.342250\pi$
$953$	29.3610	0.951095	0.475548	+	0.879690i	$0.342250\pi$
$954$	0	0
$955$	−21.3182	−0.689840
$956$	0	0
$957$	24.7835	0.801138
$958$	0	0
$959$	−17.1811	−0.554807
$960$	0	0
$961$	−25.4124	−0.819753
$962$	0	0
$963$	−13.4469	−0.433320
$964$	0	0
$965$	−27.3386	−0.880062
$966$	0	0
$967$	30.1255	0.968770	0.484385	−	0.874855i	$-0.339043\pi$
$967$	30.1255	0.968770	0.484385	+	0.874855i	$0.339043\pi$
$968$	0	0
$969$	19.2822	0.619434
$970$	0	0
$971$	12.7688	0.409770	0.204885	−	0.978786i	$-0.434318\pi$
$971$	12.7688	0.409770	0.204885	+	0.978786i	$0.434318\pi$
$972$	0	0
$973$	−39.2914	−1.25962
$974$	0	0
$975$	16.5874	0.531223
$976$	0	0
$977$	53.0684	1.69781	0.848905	−	0.528545i	$-0.177262\pi$
$977$	53.0684	1.69781	0.848905	+	0.528545i	$0.177262\pi$
$978$	0	0
$979$	30.3435	0.969782
$980$	0	0
$981$	10.3423	0.330204
$982$	0	0
$983$	−34.9436	−1.11453	−0.557264	−	0.830335i	$-0.688149\pi$
$983$	−34.9436	−1.11453	−0.557264	+	0.830335i	$0.688149\pi$
$984$	0	0
$985$	−20.3500	−0.648405
$986$	0	0
$987$	22.0399	0.701537
$988$	0	0
$989$	−3.38298	−0.107572
$990$	0	0
$991$	61.2570	1.94589	0.972946	−	0.231034i	$-0.0742108\pi$
$991$	61.2570	1.94589	0.972946	+	0.231034i	$0.0742108\pi$
$992$	0	0
$993$	−19.4000	−0.615640
$994$	0	0
$995$	−4.54202	−0.143992
$996$	0	0
$997$	4.23271	0.134051	0.0670256	−	0.997751i	$-0.478649\pi$
$997$	4.23271	0.134051	0.0670256	+	0.997751i	$0.478649\pi$
$998$	0	0
$999$	−5.21603	−0.165028

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.u.1.2		5
4.3	odd	2		501.2.a.c.1.2	✓	5
12.11	even	2		1503.2.a.c.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.c.1.2	✓	5	4.3	odd	2
1503.2.a.c.1.4		5	12.11	even	2
8016.2.a.u.1.2		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.61314 q^{5} -2.77861 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.61314 q^{5} -2.77861 q^{7} +1.00000 q^{9} -3.41479 q^{11} -6.91784 q^{13} -1.61314 q^{15} +3.58757 q^{17} +5.37474 q^{19} -2.77861 q^{21} +1.72034 q^{23} -2.39778 q^{25} +1.00000 q^{27} -7.25770 q^{29} +2.36382 q^{31} -3.41479 q^{33} +4.48229 q^{35} -5.21603 q^{37} -6.91784 q^{39} -1.85221 q^{41} -1.96645 q^{43} -1.61314 q^{45} -7.93197 q^{47} +0.720696 q^{49} +3.58757 q^{51} +0.173715 q^{53} +5.50854 q^{55} +5.37474 q^{57} +8.14440 q^{59} +6.08170 q^{61} -2.77861 q^{63} +11.1595 q^{65} +14.8238 q^{67} +1.72034 q^{69} -7.48522 q^{71} -3.99718 q^{73} -2.39778 q^{75} +9.48839 q^{77} +6.01895 q^{79} +1.00000 q^{81} +9.27564 q^{83} -5.78725 q^{85} -7.25770 q^{87} -8.88589 q^{89} +19.2220 q^{91} +2.36382 q^{93} -8.67021 q^{95} -15.7656 q^{97} -3.41479 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9} + 7 q^{11} - 8 q^{13} - q^{15} + 5 q^{17} + 20 q^{19} + 4 q^{21} - q^{23} - 6 q^{25} + 5 q^{27} - 5 q^{29} + 18 q^{31} + 7 q^{33} + 6 q^{35} - 17 q^{37} - 8 q^{39} + 6 q^{41} + 10 q^{43} - q^{45} + 3 q^{47} - 13 q^{49} + 5 q^{51} + 15 q^{53} + 9 q^{55} + 20 q^{57} + 17 q^{59} - 2 q^{61} + 4 q^{63} + 26 q^{65} + 24 q^{67} - q^{69} - q^{71} + 2 q^{73} - 6 q^{75} + 26 q^{77} + 6 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} - 5 q^{87} + 15 q^{91} + 18 q^{93} - q^{95} - 13 q^{97} + 7 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9 + 7 * q^11 - 8 * q^13 - q^15 + 5 * q^17 + 20 * q^19 + 4 * q^21 - q^23 - 6 * q^25 + 5 * q^27 - 5 * q^29 + 18 * q^31 + 7 * q^33 + 6 * q^35 - 17 * q^37 - 8 * q^39 + 6 * q^41 + 10 * q^43 - q^45 + 3 * q^47 - 13 * q^49 + 5 * q^51 + 15 * q^53 + 9 * q^55 + 20 * q^57 + 17 * q^59 - 2 * q^61 + 4 * q^63 + 26 * q^65 + 24 * q^67 - q^69 - q^71 + 2 * q^73 - 6 * q^75 + 26 * q^77 + 6 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 - 5 * q^87 + 15 * q^91 + 18 * q^93 - q^95 - 13 * q^97 + 7 * q^99

Introduction

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