LMFDB - Embedded newform 3864.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,2,Mod(1,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3864.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:	$30.8541953410$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	3864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.93543 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.93543 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.68133 q^{11} -6.61676 q^{13} +2.93543 q^{15} -4.18953 q^{17} +1.17313 q^{19} -1.00000 q^{21} -1.00000 q^{23} +3.61676 q^{25} +1.00000 q^{27} +1.68133 q^{29} -1.17313 q^{31} -3.68133 q^{33} -2.93543 q^{35} -2.31867 q^{37} -6.61676 q^{39} -7.17313 q^{41} -2.74590 q^{43} +2.93543 q^{45} +12.2171 q^{47} +1.00000 q^{49} -4.18953 q^{51} -13.1854 q^{53} -10.8063 q^{55} +1.17313 q^{57} -3.06457 q^{59} -0.745898 q^{61} -1.00000 q^{63} -19.4231 q^{65} +8.46004 q^{67} -1.00000 q^{69} -11.5040 q^{71} -10.3187 q^{73} +3.61676 q^{75} +3.68133 q^{77} -6.18953 q^{79} +1.00000 q^{81} +1.33508 q^{83} -12.2981 q^{85} +1.68133 q^{87} +6.14137 q^{89} +6.61676 q^{91} -1.17313 q^{93} +3.44364 q^{95} -6.69774 q^{97} -3.68133 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{17} - 2 q^{19} - 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 2 q^{29} + 2 q^{31} - 4 q^{33} - q^{35} - 14 q^{37} - 5 q^{39} - 16 q^{41} - 9 q^{43} + q^{45} + 10 q^{47} + 3 q^{49} - 4 q^{51} + q^{53} - 9 q^{55} - 2 q^{57} - 17 q^{59} - 3 q^{61} - 3 q^{63} - 20 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 38 q^{73} - 4 q^{75} + 4 q^{77} - 10 q^{79} + 3 q^{81} + 8 q^{83} - 15 q^{85} - 2 q^{87} - q^{89} + 5 q^{91} + 2 q^{93} + q^{95} - 10 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^17 - 2 * q^19 - 3 * q^21 - 3 * q^23 - 4 * q^25 + 3 * q^27 - 2 * q^29 + 2 * q^31 - 4 * q^33 - q^35 - 14 * q^37 - 5 * q^39 - 16 * q^41 - 9 * q^43 + q^45 + 10 * q^47 + 3 * q^49 - 4 * q^51 + q^53 - 9 * q^55 - 2 * q^57 - 17 * q^59 - 3 * q^61 - 3 * q^63 - 20 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 38 * q^73 - 4 * q^75 + 4 * q^77 - 10 * q^79 + 3 * q^81 + 8 * q^83 - 15 * q^85 - 2 * q^87 - q^89 + 5 * q^91 + 2 * q^93 + q^95 - 10 * q^97 - 4 * 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
$5$	2.93543	1.31277	0.656383	−	0.754428i	$-0.272086\pi$
$5$	2.93543	1.31277	0.656383	+	0.754428i	$0.272086\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.68133	−1.10996	−0.554981	−	0.831863i	$-0.687275\pi$
$11$	−3.68133	−1.10996	−0.554981	+	0.831863i	$0.687275\pi$
$12$	0	0
$13$	−6.61676	−1.83516	−0.917580	−	0.397551i	$-0.869860\pi$
$13$	−6.61676	−1.83516	−0.917580	+	0.397551i	$0.869860\pi$
$14$	0	0
$15$	2.93543	0.757925
$16$	0	0
$17$	−4.18953	−1.01611	−0.508056	−	0.861324i	$-0.669636\pi$
$17$	−4.18953	−1.01611	−0.508056	+	0.861324i	$0.669636\pi$
$18$	0	0
$19$	1.17313	0.269134	0.134567	−	0.990905i	$-0.457036\pi$
$19$	1.17313	0.269134	0.134567	+	0.990905i	$0.457036\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.61676	0.723353
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.68133	0.312215	0.156108	−	0.987740i	$-0.450105\pi$
$29$	1.68133	0.312215	0.156108	+	0.987740i	$0.450105\pi$
$30$	0	0
$31$	−1.17313	−0.210700	−0.105350	−	0.994435i	$-0.533596\pi$
$31$	−1.17313	−0.210700	−0.105350	+	0.994435i	$0.533596\pi$
$32$	0	0
$33$	−3.68133	−0.640837
$34$	0	0
$35$	−2.93543	−0.496179
$36$	0	0
$37$	−2.31867	−0.381187	−0.190593	−	0.981669i	$-0.561041\pi$
$37$	−2.31867	−0.381187	−0.190593	+	0.981669i	$0.561041\pi$
$38$	0	0
$39$	−6.61676	−1.05953
$40$	0	0
$41$	−7.17313	−1.12025	−0.560127	−	0.828407i	$-0.689248\pi$
$41$	−7.17313	−1.12025	−0.560127	+	0.828407i	$0.689248\pi$
$42$	0	0
$43$	−2.74590	−0.418746	−0.209373	−	0.977836i	$-0.567142\pi$
$43$	−2.74590	−0.418746	−0.209373	+	0.977836i	$0.567142\pi$
$44$	0	0
$45$	2.93543	0.437588
$46$	0	0
$47$	12.2171	1.78205	0.891025	−	0.453954i	$-0.149987\pi$
$47$	12.2171	1.78205	0.891025	+	0.453954i	$0.149987\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.18953	−0.586652
$52$	0	0
$53$	−13.1854	−1.81115	−0.905575	−	0.424187i	$-0.860560\pi$
$53$	−13.1854	−1.81115	−0.905575	+	0.424187i	$0.860560\pi$
$54$	0	0
$55$	−10.8063	−1.45712
$56$	0	0
$57$	1.17313	0.155385
$58$	0	0
$59$	−3.06457	−0.398973	−0.199486	−	0.979901i	$-0.563927\pi$
$59$	−3.06457	−0.398973	−0.199486	+	0.979901i	$0.563927\pi$
$60$	0	0
$61$	−0.745898	−0.0955025	−0.0477512	−	0.998859i	$-0.515205\pi$
$61$	−0.745898	−0.0955025	−0.0477512	+	0.998859i	$0.515205\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−19.4231	−2.40913
$66$	0	0
$67$	8.46004	1.03356	0.516779	−	0.856119i	$-0.327131\pi$
$67$	8.46004	1.03356	0.516779	+	0.856119i	$0.327131\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−11.5040	−1.36528	−0.682639	−	0.730756i	$-0.739168\pi$
$71$	−11.5040	−1.36528	−0.682639	+	0.730756i	$0.739168\pi$
$72$	0	0
$73$	−10.3187	−1.20771	−0.603854	−	0.797095i	$-0.706369\pi$
$73$	−10.3187	−1.20771	−0.603854	+	0.797095i	$0.706369\pi$
$74$	0	0
$75$	3.61676	0.417628
$76$	0	0
$77$	3.68133	0.419527
$78$	0	0
$79$	−6.18953	−0.696377	−0.348188	−	0.937425i	$-0.613203\pi$
$79$	−6.18953	−0.696377	−0.348188	+	0.937425i	$0.613203\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.33508	0.146544	0.0732718	−	0.997312i	$-0.476656\pi$
$83$	1.33508	0.146544	0.0732718	+	0.997312i	$0.476656\pi$
$84$	0	0
$85$	−12.2981	−1.33392
$86$	0	0
$87$	1.68133	0.180258
$88$	0	0
$89$	6.14137	0.650984	0.325492	−	0.945545i	$-0.394470\pi$
$89$	6.14137	0.650984	0.325492	+	0.945545i	$0.394470\pi$
$90$	0	0
$91$	6.61676	0.693625
$92$	0	0
$93$	−1.17313	−0.121648
$94$	0	0
$95$	3.44364	0.353310
$96$	0	0
$97$	−6.69774	−0.680052	−0.340026	−	0.940416i	$-0.610436\pi$
$97$	−6.69774	−0.680052	−0.340026	+	0.940416i	$0.610436\pi$
$98$	0	0
$99$	−3.68133	−0.369988
$100$	0	0
$101$	−17.6608	−1.75731	−0.878655	−	0.477456i	$-0.841559\pi$
$101$	−17.6608	−1.75731	−0.878655	+	0.477456i	$0.841559\pi$
$102$	0	0
$103$	5.01641	0.494281	0.247141	−	0.968980i	$-0.420509\pi$
$103$	5.01641	0.494281	0.247141	+	0.968980i	$0.420509\pi$
$104$	0	0
$105$	−2.93543	−0.286469
$106$	0	0
$107$	6.48763	0.627183	0.313591	−	0.949558i	$-0.398468\pi$
$107$	6.48763	0.627183	0.313591	+	0.949558i	$0.398468\pi$
$108$	0	0
$109$	15.1526	1.45135	0.725676	−	0.688037i	$-0.241527\pi$
$109$	15.1526	1.45135	0.725676	+	0.688037i	$0.241527\pi$
$110$	0	0
$111$	−2.31867	−0.220078
$112$	0	0
$113$	−4.90262	−0.461200	−0.230600	−	0.973049i	$-0.574069\pi$
$113$	−4.90262	−0.461200	−0.230600	+	0.973049i	$0.574069\pi$
$114$	0	0
$115$	−2.93543	−0.273730
$116$	0	0
$117$	−6.61676	−0.611720
$118$	0	0
$119$	4.18953	0.384054
$120$	0	0
$121$	2.55220	0.232018
$122$	0	0
$123$	−7.17313	−0.646779
$124$	0	0
$125$	−4.06040	−0.363173
$126$	0	0
$127$	−1.51237	−0.134201	−0.0671007	−	0.997746i	$-0.521375\pi$
$127$	−1.51237	−0.134201	−0.0671007	+	0.997746i	$0.521375\pi$
$128$	0	0
$129$	−2.74590	−0.241763
$130$	0	0
$131$	13.2939	1.16150	0.580748	−	0.814084i	$-0.302760\pi$
$131$	13.2939	1.16150	0.580748	+	0.814084i	$0.302760\pi$
$132$	0	0
$133$	−1.17313	−0.101723
$134$	0	0
$135$	2.93543	0.252642
$136$	0	0
$137$	−17.8021	−1.52094	−0.760469	−	0.649374i	$-0.775031\pi$
$137$	−17.8021	−1.52094	−0.760469	+	0.649374i	$0.775031\pi$
$138$	0	0
$139$	4.29809	0.364560	0.182280	−	0.983247i	$-0.441652\pi$
$139$	4.29809	0.364560	0.182280	+	0.983247i	$0.441652\pi$
$140$	0	0
$141$	12.2171	1.02887
$142$	0	0
$143$	24.3585	2.03696
$144$	0	0
$145$	4.93543	0.409865
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	6.85446	0.561539	0.280770	−	0.959775i	$-0.409410\pi$
$149$	6.85446	0.561539	0.280770	+	0.959775i	$0.409410\pi$
$150$	0	0
$151$	−11.2663	−0.916842	−0.458421	−	0.888735i	$-0.651585\pi$
$151$	−11.2663	−0.916842	−0.458421	+	0.888735i	$0.651585\pi$
$152$	0	0
$153$	−4.18953	−0.338704
$154$	0	0
$155$	−3.44364	−0.276599
$156$	0	0
$157$	13.2663	1.05877	0.529385	−	0.848382i	$-0.322423\pi$
$157$	13.2663	1.05877	0.529385	+	0.848382i	$0.322423\pi$
$158$	0	0
$159$	−13.1854	−1.04567
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	11.1854	0.876105	0.438053	−	0.898949i	$-0.355668\pi$
$163$	11.1854	0.876105	0.438053	+	0.898949i	$0.355668\pi$
$164$	0	0
$165$	−10.8063	−0.841269
$166$	0	0
$167$	20.9149	1.61844	0.809220	−	0.587506i	$-0.199890\pi$
$167$	20.9149	1.61844	0.809220	+	0.587506i	$0.199890\pi$
$168$	0	0
$169$	30.7816	2.36781
$170$	0	0
$171$	1.17313	0.0897113
$172$	0	0
$173$	2.15672	0.163972	0.0819862	−	0.996633i	$-0.473874\pi$
$173$	2.15672	0.163972	0.0819862	+	0.996633i	$0.473874\pi$
$174$	0	0
$175$	−3.61676	−0.273402
$176$	0	0
$177$	−3.06457	−0.230347
$178$	0	0
$179$	−9.47645	−0.708303	−0.354152	−	0.935188i	$-0.615230\pi$
$179$	−9.47645	−0.708303	−0.354152	+	0.935188i	$0.615230\pi$
$180$	0	0
$181$	−2.95601	−0.219718	−0.109859	−	0.993947i	$-0.535040\pi$
$181$	−2.95601	−0.219718	−0.109859	+	0.993947i	$0.535040\pi$
$182$	0	0
$183$	−0.745898	−0.0551384
$184$	0	0
$185$	−6.80630	−0.500409
$186$	0	0
$187$	15.4231	1.12785
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	11.2663	0.815204	0.407602	−	0.913160i	$-0.366365\pi$
$191$	11.2663	0.815204	0.407602	+	0.913160i	$0.366365\pi$
$192$	0	0
$193$	−4.76647	−0.343098	−0.171549	−	0.985176i	$-0.554877\pi$
$193$	−4.76647	−0.343098	−0.171549	+	0.985176i	$0.554877\pi$
$194$	0	0
$195$	−19.4231	−1.39091
$196$	0	0
$197$	27.9711	1.99286	0.996429	−	0.0844385i	$-0.0269097\pi$
$197$	27.9711	1.99286	0.996429	+	0.0844385i	$0.0269097\pi$
$198$	0	0
$199$	13.3145	0.943840	0.471920	−	0.881641i	$-0.343561\pi$
$199$	13.3145	0.943840	0.471920	+	0.881641i	$0.343561\pi$
$200$	0	0
$201$	8.46004	0.596725
$202$	0	0
$203$	−1.68133	−0.118006
$204$	0	0
$205$	−21.0562	−1.47063
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−4.31867	−0.298729
$210$	0	0
$211$	−20.8185	−1.43321	−0.716604	−	0.697481i	$-0.754304\pi$
$211$	−20.8185	−1.43321	−0.716604	+	0.697481i	$0.754304\pi$
$212$	0	0
$213$	−11.5040	−0.788243
$214$	0	0
$215$	−8.06040	−0.549715
$216$	0	0
$217$	1.17313	0.0796371
$218$	0	0
$219$	−10.3187	−0.697271
$220$	0	0
$221$	27.7212	1.86473
$222$	0	0
$223$	−16.8943	−1.13132	−0.565662	−	0.824637i	$-0.691379\pi$
$223$	−16.8943	−1.13132	−0.565662	+	0.824637i	$0.691379\pi$
$224$	0	0
$225$	3.61676	0.241118
$226$	0	0
$227$	−29.2733	−1.94294	−0.971470	−	0.237162i	$-0.923783\pi$
$227$	−29.2733	−1.94294	−0.971470	+	0.237162i	$0.923783\pi$
$228$	0	0
$229$	−11.7899	−0.779098	−0.389549	−	0.921006i	$-0.627369\pi$
$229$	−11.7899	−0.779098	−0.389549	+	0.921006i	$0.627369\pi$
$230$	0	0
$231$	3.68133	0.242214
$232$	0	0
$233$	9.34731	0.612363	0.306181	−	0.951973i	$-0.400949\pi$
$233$	9.34731	0.612363	0.306181	+	0.951973i	$0.400949\pi$
$234$	0	0
$235$	35.8625	2.33941
$236$	0	0
$237$	−6.18953	−0.402053
$238$	0	0
$239$	−13.6279	−0.881518	−0.440759	−	0.897625i	$-0.645291\pi$
$239$	−13.6279	−0.881518	−0.440759	+	0.897625i	$0.645291\pi$
$240$	0	0
$241$	−19.0768	−1.22885	−0.614423	−	0.788977i	$-0.710611\pi$
$241$	−19.0768	−1.22885	−0.614423	+	0.788977i	$0.710611\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.93543	0.187538
$246$	0	0
$247$	−7.76231	−0.493904
$248$	0	0
$249$	1.33508	0.0846070
$250$	0	0
$251$	−20.3379	−1.28372	−0.641859	−	0.766823i	$-0.721837\pi$
$251$	−20.3379	−1.28372	−0.641859	+	0.766823i	$0.721837\pi$
$252$	0	0
$253$	3.68133	0.231443
$254$	0	0
$255$	−12.2981	−0.770136
$256$	0	0
$257$	19.5470	1.21931	0.609653	−	0.792668i	$-0.291309\pi$
$257$	19.5470	1.21931	0.609653	+	0.792668i	$0.291309\pi$
$258$	0	0
$259$	2.31867	0.144075
$260$	0	0
$261$	1.68133	0.104072
$262$	0	0
$263$	30.3103	1.86902	0.934508	−	0.355943i	$-0.115840\pi$
$263$	30.3103	1.86902	0.934508	+	0.355943i	$0.115840\pi$
$264$	0	0
$265$	−38.7047	−2.37761
$266$	0	0
$267$	6.14137	0.375846
$268$	0	0
$269$	−18.3585	−1.11934	−0.559669	−	0.828717i	$-0.689072\pi$
$269$	−18.3585	−1.11934	−0.559669	+	0.828717i	$0.689072\pi$
$270$	0	0
$271$	−17.1320	−1.04069	−0.520347	−	0.853955i	$-0.674197\pi$
$271$	−17.1320	−1.04069	−0.520347	+	0.853955i	$0.674197\pi$
$272$	0	0
$273$	6.61676	0.400465
$274$	0	0
$275$	−13.3145	−0.802895
$276$	0	0
$277$	−7.09215	−0.426126	−0.213063	−	0.977038i	$-0.568344\pi$
$277$	−7.09215	−0.426126	−0.213063	+	0.977038i	$0.568344\pi$
$278$	0	0
$279$	−1.17313	−0.0702333
$280$	0	0
$281$	22.2171	1.32536	0.662681	−	0.748902i	$-0.269418\pi$
$281$	22.2171	1.32536	0.662681	+	0.748902i	$0.269418\pi$
$282$	0	0
$283$	−4.83911	−0.287655	−0.143828	−	0.989603i	$-0.545941\pi$
$283$	−4.83911	−0.287655	−0.143828	+	0.989603i	$0.545941\pi$
$284$	0	0
$285$	3.44364	0.203983
$286$	0	0
$287$	7.17313	0.423416
$288$	0	0
$289$	0.552195	0.0324821
$290$	0	0
$291$	−6.69774	−0.392628
$292$	0	0
$293$	22.3791	1.30740	0.653700	−	0.756754i	$-0.273216\pi$
$293$	22.3791	1.30740	0.653700	+	0.756754i	$0.273216\pi$
$294$	0	0
$295$	−8.99583	−0.523758
$296$	0	0
$297$	−3.68133	−0.213612
$298$	0	0
$299$	6.61676	0.382657
$300$	0	0
$301$	2.74590	0.158271
$302$	0	0
$303$	−17.6608	−1.01458
$304$	0	0
$305$	−2.18953	−0.125372
$306$	0	0
$307$	−8.60142	−0.490909	−0.245454	−	0.969408i	$-0.578937\pi$
$307$	−8.60142	−0.490909	−0.245454	+	0.969408i	$0.578937\pi$
$308$	0	0
$309$	5.01641	0.285373
$310$	0	0
$311$	11.6660	0.661517	0.330759	−	0.943715i	$-0.392695\pi$
$311$	11.6660	0.661517	0.330759	+	0.943715i	$0.392695\pi$
$312$	0	0
$313$	21.8625	1.23574	0.617872	−	0.786279i	$-0.287995\pi$
$313$	21.8625	1.23574	0.617872	+	0.786279i	$0.287995\pi$
$314$	0	0
$315$	−2.93543	−0.165393
$316$	0	0
$317$	0.685500	0.0385015	0.0192507	−	0.999815i	$-0.493872\pi$
$317$	0.685500	0.0385015	0.0192507	+	0.999815i	$0.493872\pi$
$318$	0	0
$319$	−6.18953	−0.346547
$320$	0	0
$321$	6.48763	0.362104
$322$	0	0
$323$	−4.91486	−0.273470
$324$	0	0
$325$	−23.9313	−1.32747
$326$	0	0
$327$	15.1526	0.837938
$328$	0	0
$329$	−12.2171	−0.673552
$330$	0	0
$331$	24.8133	1.36386	0.681931	−	0.731416i	$-0.261140\pi$
$331$	24.8133	1.36386	0.681931	+	0.731416i	$0.261140\pi$
$332$	0	0
$333$	−2.31867	−0.127062
$334$	0	0
$335$	24.8339	1.35682
$336$	0	0
$337$	−4.96302	−0.270353	−0.135176	−	0.990822i	$-0.543160\pi$
$337$	−4.96302	−0.270353	−0.135176	+	0.990822i	$0.543160\pi$
$338$	0	0
$339$	−4.90262	−0.266274
$340$	0	0
$341$	4.31867	0.233869
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.93543	−0.158038
$346$	0	0
$347$	13.1731	0.707171	0.353585	−	0.935402i	$-0.384962\pi$
$347$	13.1731	0.707171	0.353585	+	0.935402i	$0.384962\pi$
$348$	0	0
$349$	−15.7899	−0.845213	−0.422607	−	0.906313i	$-0.638885\pi$
$349$	−15.7899	−0.845213	−0.422607	+	0.906313i	$0.638885\pi$
$350$	0	0
$351$	−6.61676	−0.353177
$352$	0	0
$353$	29.0028	1.54367	0.771833	−	0.635826i	$-0.219340\pi$
$353$	29.0028	1.54367	0.771833	+	0.635826i	$0.219340\pi$
$354$	0	0
$355$	−33.7693	−1.79229
$356$	0	0
$357$	4.18953	0.221734
$358$	0	0
$359$	−5.19370	−0.274113	−0.137057	−	0.990563i	$-0.543764\pi$
$359$	−5.19370	−0.274113	−0.137057	+	0.990563i	$0.543764\pi$
$360$	0	0
$361$	−17.6238	−0.927567
$362$	0	0
$363$	2.55220	0.133956
$364$	0	0
$365$	−30.2898	−1.58544
$366$	0	0
$367$	10.5839	0.552478	0.276239	−	0.961089i	$-0.410912\pi$
$367$	10.5839	0.552478	0.276239	+	0.961089i	$0.410912\pi$
$368$	0	0
$369$	−7.17313	−0.373418
$370$	0	0
$371$	13.1854	0.684550
$372$	0	0
$373$	−23.0112	−1.19147	−0.595737	−	0.803180i	$-0.703140\pi$
$373$	−23.0112	−1.19147	−0.595737	+	0.803180i	$0.703140\pi$
$374$	0	0
$375$	−4.06040	−0.209678
$376$	0	0
$377$	−11.1250	−0.572965
$378$	0	0
$379$	27.7417	1.42500	0.712498	−	0.701674i	$-0.247564\pi$
$379$	27.7417	1.42500	0.712498	+	0.701674i	$0.247564\pi$
$380$	0	0
$381$	−1.51237	−0.0774812
$382$	0	0
$383$	−10.5275	−0.537928	−0.268964	−	0.963150i	$-0.586681\pi$
$383$	−10.5275	−0.537928	−0.268964	+	0.963150i	$0.586681\pi$
$384$	0	0
$385$	10.8063	0.550740
$386$	0	0
$387$	−2.74590	−0.139582
$388$	0	0
$389$	−25.0028	−1.26769	−0.633847	−	0.773458i	$-0.718525\pi$
$389$	−25.0028	−1.26769	−0.633847	+	0.773458i	$0.718525\pi$
$390$	0	0
$391$	4.18953	0.211874
$392$	0	0
$393$	13.2939	0.670590
$394$	0	0
$395$	−18.1690	−0.914180
$396$	0	0
$397$	13.6454	0.684843	0.342422	−	0.939546i	$-0.388753\pi$
$397$	13.6454	0.684843	0.342422	+	0.939546i	$0.388753\pi$
$398$	0	0
$399$	−1.17313	−0.0587298
$400$	0	0
$401$	−5.46421	−0.272870	−0.136435	−	0.990649i	$-0.543564\pi$
$401$	−5.46421	−0.272870	−0.136435	+	0.990649i	$0.543564\pi$
$402$	0	0
$403$	7.76231	0.386668
$404$	0	0
$405$	2.93543	0.145863
$406$	0	0
$407$	8.53579	0.423103
$408$	0	0
$409$	−22.2223	−1.09882	−0.549412	−	0.835551i	$-0.685148\pi$
$409$	−22.2223	−1.09882	−0.549412	+	0.835551i	$0.685148\pi$
$410$	0	0
$411$	−17.8021	−0.878114
$412$	0	0
$413$	3.06457	0.150798
$414$	0	0
$415$	3.91903	0.192377
$416$	0	0
$417$	4.29809	0.210479
$418$	0	0
$419$	−5.00106	−0.244318	−0.122159	−	0.992511i	$-0.538982\pi$
$419$	−5.00106	−0.244318	−0.122159	+	0.992511i	$0.538982\pi$
$420$	0	0
$421$	−6.83388	−0.333063	−0.166532	−	0.986036i	$-0.553257\pi$
$421$	−6.83388	−0.333063	−0.166532	+	0.986036i	$0.553257\pi$
$422$	0	0
$423$	12.2171	0.594017
$424$	0	0
$425$	−15.1526	−0.735007
$426$	0	0
$427$	0.745898	0.0360965
$428$	0	0
$429$	24.3585	1.17604
$430$	0	0
$431$	−16.9547	−0.816678	−0.408339	−	0.912830i	$-0.633892\pi$
$431$	−16.9547	−0.816678	−0.408339	+	0.912830i	$0.633892\pi$
$432$	0	0
$433$	−15.5574	−0.747642	−0.373821	−	0.927501i	$-0.621953\pi$
$433$	−15.5574	−0.747642	−0.373821	+	0.927501i	$0.621953\pi$
$434$	0	0
$435$	4.93543	0.236636
$436$	0	0
$437$	−1.17313	−0.0561183
$438$	0	0
$439$	−33.4147	−1.59480	−0.797399	−	0.603453i	$-0.793791\pi$
$439$	−33.4147	−1.59480	−0.797399	+	0.603453i	$0.793791\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	25.4506	1.20920	0.604598	−	0.796531i	$-0.293334\pi$
$443$	25.4506	1.20920	0.604598	+	0.796531i	$0.293334\pi$
$444$	0	0
$445$	18.0276	0.854589
$446$	0	0
$447$	6.85446	0.324205
$448$	0	0
$449$	−39.0562	−1.84318	−0.921589	−	0.388168i	$-0.873108\pi$
$449$	−39.0562	−1.84318	−0.921589	+	0.388168i	$0.873108\pi$
$450$	0	0
$451$	26.4067	1.24344
$452$	0	0
$453$	−11.2663	−0.529339
$454$	0	0
$455$	19.4231	0.910567
$456$	0	0
$457$	−31.2182	−1.46032	−0.730162	−	0.683274i	$-0.760556\pi$
$457$	−31.2182	−1.46032	−0.730162	+	0.683274i	$0.760556\pi$
$458$	0	0
$459$	−4.18953	−0.195551
$460$	0	0
$461$	0.806297	0.0375530	0.0187765	−	0.999824i	$-0.494023\pi$
$461$	0.806297	0.0375530	0.0187765	+	0.999824i	$0.494023\pi$
$462$	0	0
$463$	4.15672	0.193179	0.0965896	−	0.995324i	$-0.469207\pi$
$463$	4.15672	0.193179	0.0965896	+	0.995324i	$0.469207\pi$
$464$	0	0
$465$	−3.44364	−0.159695
$466$	0	0
$467$	8.69774	0.402483	0.201242	−	0.979542i	$-0.435502\pi$
$467$	8.69774	0.402483	0.201242	+	0.979542i	$0.435502\pi$
$468$	0	0
$469$	−8.46004	−0.390648
$470$	0	0
$471$	13.2663	0.611281
$472$	0	0
$473$	10.1086	0.464792
$474$	0	0
$475$	4.24292	0.194679
$476$	0	0
$477$	−13.1854	−0.603716
$478$	0	0
$479$	21.8656	0.999066	0.499533	−	0.866295i	$-0.333505\pi$
$479$	21.8656	0.999066	0.499533	+	0.866295i	$0.333505\pi$
$480$	0	0
$481$	15.3421	0.699539
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−19.6608	−0.892749
$486$	0	0
$487$	33.2939	1.50869	0.754346	−	0.656477i	$-0.227954\pi$
$487$	33.2939	1.50869	0.754346	+	0.656477i	$0.227954\pi$
$488$	0	0
$489$	11.1854	0.505820
$490$	0	0
$491$	29.4405	1.32863	0.664316	−	0.747452i	$-0.268723\pi$
$491$	29.4405	1.32863	0.664316	+	0.747452i	$0.268723\pi$
$492$	0	0
$493$	−7.04399	−0.317245
$494$	0	0
$495$	−10.8063	−0.485707
$496$	0	0
$497$	11.5040	0.516026
$498$	0	0
$499$	−39.0786	−1.74940	−0.874699	−	0.484667i	$-0.838941\pi$
$499$	−39.0786	−1.74940	−0.874699	+	0.484667i	$0.838941\pi$
$500$	0	0
$501$	20.9149	0.934407
$502$	0	0
$503$	37.4353	1.66916	0.834579	−	0.550889i	$-0.185711\pi$
$503$	37.4353	1.66916	0.834579	+	0.550889i	$0.185711\pi$
$504$	0	0
$505$	−51.8420	−2.30694
$506$	0	0
$507$	30.7816	1.36706
$508$	0	0
$509$	4.12914	0.183021	0.0915104	−	0.995804i	$-0.470831\pi$
$509$	4.12914	0.183021	0.0915104	+	0.995804i	$0.470831\pi$
$510$	0	0
$511$	10.3187	0.456471
$512$	0	0
$513$	1.17313	0.0517948
$514$	0	0
$515$	14.7253	0.648875
$516$	0	0
$517$	−44.9753	−1.97801
$518$	0	0
$519$	2.15672	0.0946695
$520$	0	0
$521$	31.0164	1.35885	0.679427	−	0.733743i	$-0.262229\pi$
$521$	31.0164	1.35885	0.679427	+	0.733743i	$0.262229\pi$
$522$	0	0
$523$	−20.3791	−0.891114	−0.445557	−	0.895253i	$-0.646994\pi$
$523$	−20.3791	−0.891114	−0.445557	+	0.895253i	$0.646994\pi$
$524$	0	0
$525$	−3.61676	−0.157848
$526$	0	0
$527$	4.91486	0.214095
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.06457	−0.132991
$532$	0	0
$533$	47.4629	2.05585
$534$	0	0
$535$	19.0440	0.823344
$536$	0	0
$537$	−9.47645	−0.408939
$538$	0	0
$539$	−3.68133	−0.158566
$540$	0	0
$541$	−8.12914	−0.349499	−0.174749	−	0.984613i	$-0.555912\pi$
$541$	−8.12914	−0.349499	−0.174749	+	0.984613i	$0.555912\pi$
$542$	0	0
$543$	−2.95601	−0.126854
$544$	0	0
$545$	44.4793	1.90528
$546$	0	0
$547$	30.2898	1.29510	0.647548	−	0.762024i	$-0.275794\pi$
$547$	30.2898	1.29510	0.647548	+	0.762024i	$0.275794\pi$
$548$	0	0
$549$	−0.745898	−0.0318342
$550$	0	0
$551$	1.97241	0.0840277
$552$	0	0
$553$	6.18953	0.263206
$554$	0	0
$555$	−6.80630	−0.288911
$556$	0	0
$557$	−2.97526	−0.126066	−0.0630328	−	0.998011i	$-0.520077\pi$
$557$	−2.97526	−0.126066	−0.0630328	+	0.998011i	$0.520077\pi$
$558$	0	0
$559$	18.1690	0.768465
$560$	0	0
$561$	15.4231	0.651162
$562$	0	0
$563$	−18.1882	−0.766541	−0.383271	−	0.923636i	$-0.625202\pi$
$563$	−18.1882	−0.766541	−0.383271	+	0.923636i	$0.625202\pi$
$564$	0	0
$565$	−14.3913	−0.605447
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	13.8625	0.581147	0.290574	−	0.956853i	$-0.406154\pi$
$569$	13.8625	0.581147	0.290574	+	0.956853i	$0.406154\pi$
$570$	0	0
$571$	−37.5522	−1.57151	−0.785755	−	0.618538i	$-0.787725\pi$
$571$	−37.5522	−1.57151	−0.785755	+	0.618538i	$0.787725\pi$
$572$	0	0
$573$	11.2663	0.470658
$574$	0	0
$575$	−3.61676	−0.150829
$576$	0	0
$577$	9.80736	0.408286	0.204143	−	0.978941i	$-0.434559\pi$
$577$	9.80736	0.408286	0.204143	+	0.978941i	$0.434559\pi$
$578$	0	0
$579$	−4.76647	−0.198088
$580$	0	0
$581$	−1.33508	−0.0553883
$582$	0	0
$583$	48.5397	2.01031
$584$	0	0
$585$	−19.4231	−0.803045
$586$	0	0
$587$	−1.21295	−0.0500638	−0.0250319	−	0.999687i	$-0.507969\pi$
$587$	−1.21295	−0.0500638	−0.0250319	+	0.999687i	$0.507969\pi$
$588$	0	0
$589$	−1.37623	−0.0567065
$590$	0	0
$591$	27.9711	1.15058
$592$	0	0
$593$	−32.9424	−1.35278	−0.676392	−	0.736542i	$-0.736457\pi$
$593$	−32.9424	−1.35278	−0.676392	+	0.736542i	$0.736457\pi$
$594$	0	0
$595$	12.2981	0.504173
$596$	0	0
$597$	13.3145	0.544926
$598$	0	0
$599$	7.92425	0.323776	0.161888	−	0.986809i	$-0.448242\pi$
$599$	7.92425	0.323776	0.161888	+	0.986809i	$0.448242\pi$
$600$	0	0
$601$	−0.0893124	−0.00364313	−0.00182156	−	0.999998i	$-0.500580\pi$
$601$	−0.0893124	−0.00364313	−0.00182156	+	0.999998i	$0.500580\pi$
$602$	0	0
$603$	8.46004	0.344520
$604$	0	0
$605$	7.49180	0.304585
$606$	0	0
$607$	−39.5592	−1.60566	−0.802829	−	0.596209i	$-0.796673\pi$
$607$	−39.5592	−1.60566	−0.802829	+	0.596209i	$0.796673\pi$
$608$	0	0
$609$	−1.68133	−0.0681310
$610$	0	0
$611$	−80.8378	−3.27035
$612$	0	0
$613$	−8.68940	−0.350962	−0.175481	−	0.984483i	$-0.556148\pi$
$613$	−8.68940	−0.350962	−0.175481	+	0.984483i	$0.556148\pi$
$614$	0	0
$615$	−21.0562	−0.849069
$616$	0	0
$617$	−20.3668	−0.819938	−0.409969	−	0.912100i	$-0.634460\pi$
$617$	−20.3668	−0.819938	−0.409969	+	0.912100i	$0.634460\pi$
$618$	0	0
$619$	5.56754	0.223778	0.111889	−	0.993721i	$-0.464310\pi$
$619$	5.56754	0.223778	0.111889	+	0.993721i	$0.464310\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−6.14137	−0.246049
$624$	0	0
$625$	−30.0028	−1.20011
$626$	0	0
$627$	−4.31867	−0.172471
$628$	0	0
$629$	9.71414	0.387328
$630$	0	0
$631$	10.7306	0.427176	0.213588	−	0.976924i	$-0.431485\pi$
$631$	10.7306	0.427176	0.213588	+	0.976924i	$0.431485\pi$
$632$	0	0
$633$	−20.8185	−0.827462
$634$	0	0
$635$	−4.43947	−0.176175
$636$	0	0
$637$	−6.61676	−0.262166
$638$	0	0
$639$	−11.5040	−0.455093
$640$	0	0
$641$	19.9466	0.787844	0.393922	−	0.919144i	$-0.371118\pi$
$641$	19.9466	0.787844	0.393922	+	0.919144i	$0.371118\pi$
$642$	0	0
$643$	−14.6084	−0.576100	−0.288050	−	0.957615i	$-0.593007\pi$
$643$	−14.6084	−0.576100	−0.288050	+	0.957615i	$0.593007\pi$
$644$	0	0
$645$	−8.06040	−0.317378
$646$	0	0
$647$	28.0398	1.10236	0.551180	−	0.834387i	$-0.314178\pi$
$647$	28.0398	1.10236	0.551180	+	0.834387i	$0.314178\pi$
$648$	0	0
$649$	11.2817	0.442845
$650$	0	0
$651$	1.17313	0.0459785
$652$	0	0
$653$	22.7323	0.889585	0.444792	−	0.895634i	$-0.353277\pi$
$653$	22.7323	0.889585	0.444792	+	0.895634i	$0.353277\pi$
$654$	0	0
$655$	39.0234	1.52477
$656$	0	0
$657$	−10.3187	−0.402570
$658$	0	0
$659$	−13.7141	−0.534227	−0.267114	−	0.963665i	$-0.586070\pi$
$659$	−13.7141	−0.534227	−0.267114	+	0.963665i	$0.586070\pi$
$660$	0	0
$661$	−11.7141	−0.455627	−0.227814	−	0.973705i	$-0.573158\pi$
$661$	−11.7141	−0.455627	−0.227814	+	0.973705i	$0.573158\pi$
$662$	0	0
$663$	27.7212	1.07660
$664$	0	0
$665$	−3.44364	−0.133538
$666$	0	0
$667$	−1.68133	−0.0651014
$668$	0	0
$669$	−16.8943	−0.653171
$670$	0	0
$671$	2.74590	0.106004
$672$	0	0
$673$	−14.0192	−0.540402	−0.270201	−	0.962804i	$-0.587090\pi$
$673$	−14.0192	−0.540402	−0.270201	+	0.962804i	$0.587090\pi$
$674$	0	0
$675$	3.61676	0.139209
$676$	0	0
$677$	6.30332	0.242256	0.121128	−	0.992637i	$-0.461349\pi$
$677$	6.30332	0.242256	0.121128	+	0.992637i	$0.461349\pi$
$678$	0	0
$679$	6.69774	0.257036
$680$	0	0
$681$	−29.2733	−1.12176
$682$	0	0
$683$	−27.8984	−1.06750	−0.533752	−	0.845641i	$-0.679219\pi$
$683$	−27.8984	−1.06750	−0.533752	+	0.845641i	$0.679219\pi$
$684$	0	0
$685$	−52.2569	−1.99664
$686$	0	0
$687$	−11.7899	−0.449812
$688$	0	0
$689$	87.2444	3.32375
$690$	0	0
$691$	32.1414	1.22272	0.611358	−	0.791354i	$-0.290624\pi$
$691$	32.1414	1.22272	0.611358	+	0.791354i	$0.290624\pi$
$692$	0	0
$693$	3.68133	0.139842
$694$	0	0
$695$	12.6168	0.478581
$696$	0	0
$697$	30.0521	1.13830
$698$	0	0
$699$	9.34731	0.353548
$700$	0	0
$701$	−5.02342	−0.189732	−0.0948659	−	0.995490i	$-0.530242\pi$
$701$	−5.02342	−0.189732	−0.0948659	+	0.995490i	$0.530242\pi$
$702$	0	0
$703$	−2.72009	−0.102590
$704$	0	0
$705$	35.8625	1.35066
$706$	0	0
$707$	17.6608	0.664201
$708$	0	0
$709$	6.26217	0.235181	0.117590	−	0.993062i	$-0.462483\pi$
$709$	6.26217	0.235181	0.117590	+	0.993062i	$0.462483\pi$
$710$	0	0
$711$	−6.18953	−0.232126
$712$	0	0
$713$	1.17313	0.0439340
$714$	0	0
$715$	71.5027	2.67405
$716$	0	0
$717$	−13.6279	−0.508945
$718$	0	0
$719$	34.2140	1.27597	0.637984	−	0.770050i	$-0.279769\pi$
$719$	34.2140	1.27597	0.637984	+	0.770050i	$0.279769\pi$
$720$	0	0
$721$	−5.01641	−0.186821
$722$	0	0
$723$	−19.0768	−0.709474
$724$	0	0
$725$	6.08097	0.225842
$726$	0	0
$727$	16.5358	0.613278	0.306639	−	0.951826i	$-0.400795\pi$
$727$	16.5358	0.613278	0.306639	+	0.951826i	$0.400795\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.5040	0.425492
$732$	0	0
$733$	21.4590	0.792606	0.396303	−	0.918120i	$-0.370293\pi$
$733$	21.4590	0.792606	0.396303	+	0.918120i	$0.370293\pi$
$734$	0	0
$735$	2.93543	0.108275
$736$	0	0
$737$	−31.1442	−1.14721
$738$	0	0
$739$	2.23069	0.0820571	0.0410285	−	0.999158i	$-0.486937\pi$
$739$	2.23069	0.0820571	0.0410285	+	0.999158i	$0.486937\pi$
$740$	0	0
$741$	−7.76231	−0.285155
$742$	0	0
$743$	−33.5009	−1.22903	−0.614515	−	0.788905i	$-0.710648\pi$
$743$	−33.5009	−1.22903	−0.614515	+	0.788905i	$0.710648\pi$
$744$	0	0
$745$	20.1208	0.737169
$746$	0	0
$747$	1.33508	0.0488479
$748$	0	0
$749$	−6.48763	−0.237053
$750$	0	0
$751$	−22.8719	−0.834608	−0.417304	−	0.908767i	$-0.637025\pi$
$751$	−22.8719	−0.834608	−0.417304	+	0.908767i	$0.637025\pi$
$752$	0	0
$753$	−20.3379	−0.741155
$754$	0	0
$755$	−33.0716	−1.20360
$756$	0	0
$757$	16.4618	0.598315	0.299158	−	0.954204i	$-0.403294\pi$
$757$	16.4618	0.598315	0.299158	+	0.954204i	$0.403294\pi$
$758$	0	0
$759$	3.68133	0.133624
$760$	0	0
$761$	−30.5550	−1.10762	−0.553810	−	0.832643i	$-0.686826\pi$
$761$	−30.5550	−1.10762	−0.553810	+	0.832643i	$0.686826\pi$
$762$	0	0
$763$	−15.1526	−0.548559
$764$	0	0
$765$	−12.2981	−0.444639
$766$	0	0
$767$	20.2775	0.732179
$768$	0	0
$769$	−0.508203	−0.0183263	−0.00916314	−	0.999958i	$-0.502917\pi$
$769$	−0.508203	−0.0183263	−0.00916314	+	0.999958i	$0.502917\pi$
$770$	0	0
$771$	19.5470	0.703967
$772$	0	0
$773$	28.5275	1.02606	0.513031	−	0.858370i	$-0.328523\pi$
$773$	28.5275	1.02606	0.513031	+	0.858370i	$0.328523\pi$
$774$	0	0
$775$	−4.24292	−0.152410
$776$	0	0
$777$	2.31867	0.0831818
$778$	0	0
$779$	−8.41499	−0.301498
$780$	0	0
$781$	42.3502	1.51541
$782$	0	0
$783$	1.68133	0.0600859
$784$	0	0
$785$	38.9424	1.38992
$786$	0	0
$787$	−33.3285	−1.18803	−0.594017	−	0.804453i	$-0.702459\pi$
$787$	−33.3285	−1.18803	−0.594017	+	0.804453i	$0.702459\pi$
$788$	0	0
$789$	30.3103	1.07908
$790$	0	0
$791$	4.90262	0.174317
$792$	0	0
$793$	4.93543	0.175262
$794$	0	0
$795$	−38.7047	−1.37272
$796$	0	0
$797$	55.0961	1.95160	0.975801	−	0.218660i	$-0.0701685\pi$
$797$	55.0961	1.95160	0.975801	+	0.218660i	$0.0701685\pi$
$798$	0	0
$799$	−51.1840	−1.81076
$800$	0	0
$801$	6.14137	0.216995
$802$	0	0
$803$	37.9864	1.34051
$804$	0	0
$805$	2.93543	0.103460
$806$	0	0
$807$	−18.3585	−0.646250
$808$	0	0
$809$	13.6660	0.480470	0.240235	−	0.970715i	$-0.422775\pi$
$809$	13.6660	0.480470	0.240235	+	0.970715i	$0.422775\pi$
$810$	0	0
$811$	23.4887	0.824799	0.412400	−	0.911003i	$-0.364691\pi$
$811$	23.4887	0.824799	0.412400	+	0.911003i	$0.364691\pi$
$812$	0	0
$813$	−17.1320	−0.600845
$814$	0	0
$815$	32.8339	1.15012
$816$	0	0
$817$	−3.22129	−0.112699
$818$	0	0
$819$	6.61676	0.231208
$820$	0	0
$821$	−1.21117	−0.0422701	−0.0211350	−	0.999777i	$-0.506728\pi$
$821$	−1.21117	−0.0422701	−0.0211350	+	0.999777i	$0.506728\pi$
$822$	0	0
$823$	−12.5975	−0.439122	−0.219561	−	0.975599i	$-0.570462\pi$
$823$	−12.5975	−0.439122	−0.219561	+	0.975599i	$0.570462\pi$
$824$	0	0
$825$	−13.3145	−0.463551
$826$	0	0
$827$	−14.0726	−0.489354	−0.244677	−	0.969605i	$-0.578682\pi$
$827$	−14.0726	−0.489354	−0.244677	+	0.969605i	$0.578682\pi$
$828$	0	0
$829$	−9.06847	−0.314961	−0.157480	−	0.987522i	$-0.550337\pi$
$829$	−9.06847	−0.314961	−0.157480	+	0.987522i	$0.550337\pi$
$830$	0	0
$831$	−7.09215	−0.246024
$832$	0	0
$833$	−4.18953	−0.145159
$834$	0	0
$835$	61.3941	2.12463
$836$	0	0
$837$	−1.17313	−0.0405492
$838$	0	0
$839$	51.3009	1.77110	0.885552	−	0.464539i	$-0.153780\pi$
$839$	51.3009	1.77110	0.885552	+	0.464539i	$0.153780\pi$
$840$	0	0
$841$	−26.1731	−0.902522
$842$	0	0
$843$	22.2171	0.765198
$844$	0	0
$845$	90.3572	3.10838
$846$	0	0
$847$	−2.55220	−0.0876945
$848$	0	0
$849$	−4.83911	−0.166078
$850$	0	0
$851$	2.31867	0.0794830
$852$	0	0
$853$	−22.9753	−0.786658	−0.393329	−	0.919398i	$-0.628677\pi$
$853$	−22.9753	−0.786658	−0.393329	+	0.919398i	$0.628677\pi$
$854$	0	0
$855$	3.44364	0.117770
$856$	0	0
$857$	−22.4754	−0.767745	−0.383872	−	0.923386i	$-0.625410\pi$
$857$	−22.4754	−0.767745	−0.383872	+	0.923386i	$0.625410\pi$
$858$	0	0
$859$	14.3463	0.489488	0.244744	−	0.969588i	$-0.421296\pi$
$859$	14.3463	0.489488	0.244744	+	0.969588i	$0.421296\pi$
$860$	0	0
$861$	7.17313	0.244460
$862$	0	0
$863$	−38.8873	−1.32374	−0.661869	−	0.749619i	$-0.730237\pi$
$863$	−38.8873	−1.32374	−0.661869	+	0.749619i	$0.730237\pi$
$864$	0	0
$865$	6.33091	0.215257
$866$	0	0
$867$	0.552195	0.0187535
$868$	0	0
$869$	22.7857	0.772953
$870$	0	0
$871$	−55.9781	−1.89675
$872$	0	0
$873$	−6.69774	−0.226684
$874$	0	0
$875$	4.06040	0.137267
$876$	0	0
$877$	21.7969	0.736029	0.368015	−	0.929820i	$-0.380038\pi$
$877$	21.7969	0.736029	0.368015	+	0.929820i	$0.380038\pi$
$878$	0	0
$879$	22.3791	0.754827
$880$	0	0
$881$	−41.5079	−1.39844	−0.699219	−	0.714908i	$-0.746469\pi$
$881$	−41.5079	−1.39844	−0.699219	+	0.714908i	$0.746469\pi$
$882$	0	0
$883$	−37.5920	−1.26507	−0.632536	−	0.774531i	$-0.717986\pi$
$883$	−37.5920	−1.26507	−0.632536	+	0.774531i	$0.717986\pi$
$884$	0	0
$885$	−8.99583	−0.302392
$886$	0	0
$887$	−3.81748	−0.128178	−0.0640891	−	0.997944i	$-0.520414\pi$
$887$	−3.81748	−0.128178	−0.0640891	+	0.997944i	$0.520414\pi$
$888$	0	0
$889$	1.51237	0.0507233
$890$	0	0
$891$	−3.68133	−0.123329
$892$	0	0
$893$	14.3322	0.479610
$894$	0	0
$895$	−27.8175	−0.929836
$896$	0	0
$897$	6.61676	0.220927
$898$	0	0
$899$	−1.97241	−0.0657837
$900$	0	0
$901$	55.2405	1.84033
$902$	0	0
$903$	2.74590	0.0913778
$904$	0	0
$905$	−8.67716	−0.288439
$906$	0	0
$907$	−8.58084	−0.284922	−0.142461	−	0.989800i	$-0.545502\pi$
$907$	−8.58084	−0.284922	−0.142461	+	0.989800i	$0.545502\pi$
$908$	0	0
$909$	−17.6608	−0.585770
$910$	0	0
$911$	−18.5082	−0.613204	−0.306602	−	0.951838i	$-0.599192\pi$
$911$	−18.5082	−0.613204	−0.306602	+	0.951838i	$0.599192\pi$
$912$	0	0
$913$	−4.91486	−0.162658
$914$	0	0
$915$	−2.18953	−0.0723838
$916$	0	0
$917$	−13.2939	−0.439004
$918$	0	0
$919$	11.1648	0.368292	0.184146	−	0.982899i	$-0.441048\pi$
$919$	11.1648	0.368292	0.184146	+	0.982899i	$0.441048\pi$
$920$	0	0
$921$	−8.60142	−0.283426
$922$	0	0
$923$	76.1195	2.50550
$924$	0	0
$925$	−8.38608	−0.275733
$926$	0	0
$927$	5.01641	0.164760
$928$	0	0
$929$	−24.9271	−0.817831	−0.408916	−	0.912572i	$-0.634093\pi$
$929$	−24.9271	−0.817831	−0.408916	+	0.912572i	$0.634093\pi$
$930$	0	0
$931$	1.17313	0.0384477
$932$	0	0
$933$	11.6660	0.381927
$934$	0	0
$935$	45.2733	1.48060
$936$	0	0
$937$	−14.5358	−0.474864	−0.237432	−	0.971404i	$-0.576306\pi$
$937$	−14.5358	−0.474864	−0.237432	+	0.971404i	$0.576306\pi$
$938$	0	0
$939$	21.8625	0.713457
$940$	0	0
$941$	34.1208	1.11231	0.556153	−	0.831080i	$-0.312277\pi$
$941$	34.1208	1.11231	0.556153	+	0.831080i	$0.312277\pi$
$942$	0	0
$943$	7.17313	0.233589
$944$	0	0
$945$	−2.93543	−0.0954896
$946$	0	0
$947$	−17.7641	−0.577255	−0.288628	−	0.957441i	$-0.593199\pi$
$947$	−17.7641	−0.577255	−0.288628	+	0.957441i	$0.593199\pi$
$948$	0	0
$949$	68.2762	2.21634
$950$	0	0
$951$	0.685500	0.0222288
$952$	0	0
$953$	53.7763	1.74199	0.870993	−	0.491295i	$-0.163476\pi$
$953$	53.7763	1.74199	0.870993	+	0.491295i	$0.163476\pi$
$954$	0	0
$955$	33.0716	1.07017
$956$	0	0
$957$	−6.18953	−0.200079
$958$	0	0
$959$	17.8021	0.574861
$960$	0	0
$961$	−29.6238	−0.955606
$962$	0	0
$963$	6.48763	0.209061
$964$	0	0
$965$	−13.9917	−0.450408
$966$	0	0
$967$	50.7805	1.63299	0.816495	−	0.577352i	$-0.195914\pi$
$967$	50.7805	1.63299	0.816495	+	0.577352i	$0.195914\pi$
$968$	0	0
$969$	−4.91486	−0.157888
$970$	0	0
$971$	13.6883	0.439280	0.219640	−	0.975581i	$-0.429512\pi$
$971$	13.6883	0.439280	0.219640	+	0.975581i	$0.429512\pi$
$972$	0	0
$973$	−4.29809	−0.137791
$974$	0	0
$975$	−23.9313	−0.766414
$976$	0	0
$977$	−47.1473	−1.50838	−0.754188	−	0.656658i	$-0.771969\pi$
$977$	−47.1473	−1.50838	−0.754188	+	0.656658i	$0.771969\pi$
$978$	0	0
$979$	−22.6084	−0.722568
$980$	0	0
$981$	15.1526	0.483784
$982$	0	0
$983$	22.9888	0.733230	0.366615	−	0.930373i	$-0.380517\pi$
$983$	22.9888	0.733230	0.366615	+	0.930373i	$0.380517\pi$
$984$	0	0
$985$	82.1072	2.61615
$986$	0	0
$987$	−12.2171	−0.388875
$988$	0	0
$989$	2.74590	0.0873145
$990$	0	0
$991$	48.9135	1.55379	0.776895	−	0.629631i	$-0.216794\pi$
$991$	48.9135	1.55379	0.776895	+	0.629631i	$0.216794\pi$
$992$	0	0
$993$	24.8133	0.787426
$994$	0	0
$995$	39.0838	1.23904
$996$	0	0
$997$	−45.9536	−1.45537	−0.727683	−	0.685914i	$-0.759403\pi$
$997$	−45.9536	−1.45537	−0.727683	+	0.685914i	$0.759403\pi$
$998$	0	0
$999$	−2.31867	−0.0733595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.2.a.q.1.3	✓	3
4.3	odd	2		7728.2.a.bs.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.q.1.3	✓	3	1.1	even	1	trivial
7728.2.a.bs.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.93543 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.93543 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.68133 q^{11} -6.61676 q^{13} +2.93543 q^{15} -4.18953 q^{17} +1.17313 q^{19} -1.00000 q^{21} -1.00000 q^{23} +3.61676 q^{25} +1.00000 q^{27} +1.68133 q^{29} -1.17313 q^{31} -3.68133 q^{33} -2.93543 q^{35} -2.31867 q^{37} -6.61676 q^{39} -7.17313 q^{41} -2.74590 q^{43} +2.93543 q^{45} +12.2171 q^{47} +1.00000 q^{49} -4.18953 q^{51} -13.1854 q^{53} -10.8063 q^{55} +1.17313 q^{57} -3.06457 q^{59} -0.745898 q^{61} -1.00000 q^{63} -19.4231 q^{65} +8.46004 q^{67} -1.00000 q^{69} -11.5040 q^{71} -10.3187 q^{73} +3.61676 q^{75} +3.68133 q^{77} -6.18953 q^{79} +1.00000 q^{81} +1.33508 q^{83} -12.2981 q^{85} +1.68133 q^{87} +6.14137 q^{89} +6.61676 q^{91} -1.17313 q^{93} +3.44364 q^{95} -6.69774 q^{97} -3.68133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{17} - 2 q^{19} - 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 2 q^{29} + 2 q^{31} - 4 q^{33} - q^{35} - 14 q^{37} - 5 q^{39} - 16 q^{41} - 9 q^{43} + q^{45} + 10 q^{47} + 3 q^{49} - 4 q^{51} + q^{53} - 9 q^{55} - 2 q^{57} - 17 q^{59} - 3 q^{61} - 3 q^{63} - 20 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 38 q^{73} - 4 q^{75} + 4 q^{77} - 10 q^{79} + 3 q^{81} + 8 q^{83} - 15 q^{85} - 2 q^{87} - q^{89} + 5 q^{91} + 2 q^{93} + q^{95} - 10 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^17 - 2 * q^19 - 3 * q^21 - 3 * q^23 - 4 * q^25 + 3 * q^27 - 2 * q^29 + 2 * q^31 - 4 * q^33 - q^35 - 14 * q^37 - 5 * q^39 - 16 * q^41 - 9 * q^43 + q^45 + 10 * q^47 + 3 * q^49 - 4 * q^51 + q^53 - 9 * q^55 - 2 * q^57 - 17 * q^59 - 3 * q^61 - 3 * q^63 - 20 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 38 * q^73 - 4 * q^75 + 4 * q^77 - 10 * q^79 + 3 * q^81 + 8 * q^83 - 15 * q^85 - 2 * q^87 - q^89 + 5 * q^91 + 2 * q^93 + q^95 - 10 * q^97 - 4 * q^99

Introduction

L-functions

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