LMFDB - Embedded newform 9464.2.a.bf.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,-4,0,-6,0,-4,0,10,0,0,0,-6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.38595$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.38595 q^{3} -0.877521 q^{5} -1.00000 q^{7} +2.69277 q^{9} +3.18434 q^{11} -2.09372 q^{15} -0.971244 q^{17} +1.58529 q^{19} -2.38595 q^{21} -3.26443 q^{23} -4.22996 q^{25} -0.733037 q^{27} -9.73068 q^{29} -3.94150 q^{31} +7.59768 q^{33} +0.877521 q^{35} +5.16838 q^{37} -4.51950 q^{41} -7.62643 q^{43} -2.36296 q^{45} +4.12013 q^{47} +1.00000 q^{49} -2.31734 q^{51} -2.01686 q^{53} -2.79432 q^{55} +3.78243 q^{57} +1.04081 q^{59} +5.01011 q^{61} -2.69277 q^{63} -7.45668 q^{67} -7.78877 q^{69} -4.07774 q^{71} +12.6820 q^{73} -10.0925 q^{75} -3.18434 q^{77} -9.79759 q^{79} -9.82730 q^{81} +6.81690 q^{83} +0.852287 q^{85} -23.2169 q^{87} -4.63133 q^{89} -9.40424 q^{93} -1.39113 q^{95} +4.10631 q^{97} +8.57469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 4 q^{5} - 6 q^{7} - 4 q^{9} + 10 q^{11} - 6 q^{15} + 2 q^{17} + 14 q^{19} - 2 q^{21} - 6 q^{23} - 2 q^{25} + 2 q^{27} - 20 q^{29} - 4 q^{31} + 4 q^{33} + 4 q^{35} - 4 q^{37} + 10 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 - 4 * q^9 + 10 * q^11 - 6 * q^15 + 2 * q^17 + 14 * q^19 - 2 * q^21 - 6 * q^23 - 2 * q^25 + 2 * q^27 - 20 * q^29 - 4 * q^31 + 4 * q^33 + 4 * q^35 - 4 * q^37 + 10 * q^41 - 12 * q^43 + 6 * q^45 - 6 * q^47 + 6 * q^49 - 8 * q^51 + 10 * q^53 - 24 * q^55 - 8 * q^59 + 4 * q^61 + 4 * q^63 - 18 * q^67 + 12 * q^71 + 14 * q^73 - 10 * q^75 - 10 * q^77 - 10 * q^79 - 18 * q^81 + 28 * q^83 + 20 * q^85 - 32 * q^87 - 6 * q^89 + 12 * q^93 - 26 * q^95 + 8 * q^97 + 2 * q^99

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
$3$	2.38595	1.37753	0.688765	−	0.724985i	$-0.258153\pi$
$3$	2.38595	1.37753	0.688765	+	0.724985i	$0.258153\pi$
$4$	0	0
$5$	−0.877521	−0.392439	−0.196220	−	0.980560i	$-0.562867\pi$
$5$	−0.877521	−0.392439	−0.196220	+	0.980560i	$0.562867\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.69277	0.897590
$10$	0	0
$11$	3.18434	0.960114	0.480057	−	0.877237i	$-0.340616\pi$
$11$	3.18434	0.960114	0.480057	+	0.877237i	$0.340616\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.09372	−0.540597
$16$	0	0
$17$	−0.971244	−0.235561	−0.117781	−	0.993040i	$-0.537578\pi$
$17$	−0.971244	−0.235561	−0.117781	+	0.993040i	$0.537578\pi$
$18$	0	0
$19$	1.58529	0.363691	0.181845	−	0.983327i	$-0.441793\pi$
$19$	1.58529	0.363691	0.181845	+	0.983327i	$0.441793\pi$
$20$	0	0
$21$	−2.38595	−0.520658
$22$	0	0
$23$	−3.26443	−0.680680	−0.340340	−	0.940302i	$-0.610542\pi$
$23$	−3.26443	−0.680680	−0.340340	+	0.940302i	$0.610542\pi$
$24$	0	0
$25$	−4.22996	−0.845991
$26$	0	0
$27$	−0.733037	−0.141073
$28$	0	0
$29$	−9.73068	−1.80694	−0.903471	−	0.428649i	$-0.858990\pi$
$29$	−9.73068	−1.80694	−0.903471	+	0.428649i	$0.858990\pi$
$30$	0	0
$31$	−3.94150	−0.707915	−0.353957	−	0.935262i	$-0.615164\pi$
$31$	−3.94150	−0.707915	−0.353957	+	0.935262i	$0.615164\pi$
$32$	0	0
$33$	7.59768	1.32259
$34$	0	0
$35$	0.877521	0.148328
$36$	0	0
$37$	5.16838	0.849677	0.424838	−	0.905269i	$-0.360331\pi$
$37$	5.16838	0.849677	0.424838	+	0.905269i	$0.360331\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.51950	−0.705827	−0.352913	−	0.935656i	$-0.614809\pi$
$41$	−4.51950	−0.705827	−0.352913	+	0.935656i	$0.614809\pi$
$42$	0	0
$43$	−7.62643	−1.16302	−0.581510	−	0.813539i	$-0.697538\pi$
$43$	−7.62643	−1.16302	−0.581510	+	0.813539i	$0.697538\pi$
$44$	0	0
$45$	−2.36296	−0.352250
$46$	0	0
$47$	4.12013	0.600982	0.300491	−	0.953785i	$-0.402849\pi$
$47$	4.12013	0.600982	0.300491	+	0.953785i	$0.402849\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.31734	−0.324493
$52$	0	0
$53$	−2.01686	−0.277037	−0.138519	−	0.990360i	$-0.544234\pi$
$53$	−2.01686	−0.277037	−0.138519	+	0.990360i	$0.544234\pi$
$54$	0	0
$55$	−2.79432	−0.376786
$56$	0	0
$57$	3.78243	0.500995
$58$	0	0
$59$	1.04081	0.135502	0.0677508	−	0.997702i	$-0.478418\pi$
$59$	1.04081	0.135502	0.0677508	+	0.997702i	$0.478418\pi$
$60$	0	0
$61$	5.01011	0.641479	0.320740	−	0.947167i	$-0.396069\pi$
$61$	5.01011	0.641479	0.320740	+	0.947167i	$0.396069\pi$
$62$	0	0
$63$	−2.69277	−0.339257
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.45668	−0.910979	−0.455490	−	0.890241i	$-0.650536\pi$
$67$	−7.45668	−0.910979	−0.455490	+	0.890241i	$0.650536\pi$
$68$	0	0
$69$	−7.78877	−0.937658
$70$	0	0
$71$	−4.07774	−0.483938	−0.241969	−	0.970284i	$-0.577793\pi$
$71$	−4.07774	−0.483938	−0.241969	+	0.970284i	$0.577793\pi$
$72$	0	0
$73$	12.6820	1.48432	0.742159	−	0.670224i	$-0.233802\pi$
$73$	12.6820	1.48432	0.742159	+	0.670224i	$0.233802\pi$
$74$	0	0
$75$	−10.0925	−1.16538
$76$	0	0
$77$	−3.18434	−0.362889
$78$	0	0
$79$	−9.79759	−1.10232	−0.551158	−	0.834401i	$-0.685814\pi$
$79$	−9.79759	−1.10232	−0.551158	+	0.834401i	$0.685814\pi$
$80$	0	0
$81$	−9.82730	−1.09192
$82$	0	0
$83$	6.81690	0.748252	0.374126	−	0.927378i	$-0.377943\pi$
$83$	6.81690	0.748252	0.374126	+	0.927378i	$0.377943\pi$
$84$	0	0
$85$	0.852287	0.0924435
$86$	0	0
$87$	−23.2169	−2.48912
$88$	0	0
$89$	−4.63133	−0.490920	−0.245460	−	0.969407i	$-0.578939\pi$
$89$	−4.63133	−0.490920	−0.245460	+	0.969407i	$0.578939\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.40424	−0.975174
$94$	0	0
$95$	−1.39113	−0.142727
$96$	0	0
$97$	4.10631	0.416933	0.208466	−	0.978030i	$-0.433153\pi$
$97$	4.10631	0.416933	0.208466	+	0.978030i	$0.433153\pi$
$98$	0	0
$99$	8.57469	0.861789
$100$	0	0
$101$	−13.7349	−1.36667	−0.683335	−	0.730105i	$-0.739471\pi$
$101$	−13.7349	−1.36667	−0.683335	+	0.730105i	$0.739471\pi$
$102$	0	0
$103$	−2.35474	−0.232019	−0.116010	−	0.993248i	$-0.537010\pi$
$103$	−2.35474	−0.232019	−0.116010	+	0.993248i	$0.537010\pi$
$104$	0	0
$105$	2.09372	0.204326
$106$	0	0
$107$	4.24563	0.410441	0.205220	−	0.978716i	$-0.434209\pi$
$107$	4.24563	0.410441	0.205220	+	0.978716i	$0.434209\pi$
$108$	0	0
$109$	−18.8556	−1.80604	−0.903019	−	0.429600i	$-0.858655\pi$
$109$	−18.8556	−1.80604	−0.903019	+	0.429600i	$0.858655\pi$
$110$	0	0
$111$	12.3315	1.17046
$112$	0	0
$113$	18.7453	1.76341	0.881703	−	0.471805i	$-0.156397\pi$
$113$	18.7453	1.76341	0.881703	+	0.471805i	$0.156397\pi$
$114$	0	0
$115$	2.86460	0.267126
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.971244	0.0890338
$120$	0	0
$121$	−0.859992	−0.0781811
$122$	0	0
$123$	−10.7833	−0.972298
$124$	0	0
$125$	8.09948	0.724440
$126$	0	0
$127$	−20.7801	−1.84393	−0.921967	−	0.387268i	$-0.873419\pi$
$127$	−20.7801	−1.84393	−0.921967	+	0.387268i	$0.873419\pi$
$128$	0	0
$129$	−18.1963	−1.60210
$130$	0	0
$131$	1.78298	0.155780	0.0778898	−	0.996962i	$-0.475182\pi$
$131$	1.78298	0.155780	0.0778898	+	0.996962i	$0.475182\pi$
$132$	0	0
$133$	−1.58529	−0.137462
$134$	0	0
$135$	0.643256	0.0553626
$136$	0	0
$137$	−16.6374	−1.42143	−0.710715	−	0.703480i	$-0.751629\pi$
$137$	−16.6374	−1.42143	−0.710715	+	0.703480i	$0.751629\pi$
$138$	0	0
$139$	−1.55273	−0.131700	−0.0658502	−	0.997830i	$-0.520976\pi$
$139$	−1.55273	−0.131700	−0.0658502	+	0.997830i	$0.520976\pi$
$140$	0	0
$141$	9.83042	0.827871
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.53888	0.709115
$146$	0	0
$147$	2.38595	0.196790
$148$	0	0
$149$	−0.160852	−0.0131775	−0.00658875	−	0.999978i	$-0.502097\pi$
$149$	−0.160852	−0.0131775	−0.00658875	+	0.999978i	$0.502097\pi$
$150$	0	0
$151$	11.8322	0.962892	0.481446	−	0.876476i	$-0.340112\pi$
$151$	11.8322	0.962892	0.481446	+	0.876476i	$0.340112\pi$
$152$	0	0
$153$	−2.61534	−0.211437
$154$	0	0
$155$	3.45875	0.277813
$156$	0	0
$157$	−14.9635	−1.19421	−0.597107	−	0.802162i	$-0.703683\pi$
$157$	−14.9635	−1.19421	−0.597107	+	0.802162i	$0.703683\pi$
$158$	0	0
$159$	−4.81214	−0.381628
$160$	0	0
$161$	3.26443	0.257273
$162$	0	0
$163$	−12.2494	−0.959446	−0.479723	−	0.877420i	$-0.659263\pi$
$163$	−12.2494	−0.959446	−0.479723	+	0.877420i	$0.659263\pi$
$164$	0	0
$165$	−6.66712	−0.519035
$166$	0	0
$167$	−8.41151	−0.650902	−0.325451	−	0.945559i	$-0.605516\pi$
$167$	−8.41151	−0.650902	−0.325451	+	0.945559i	$0.605516\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.26882	0.326445
$172$	0	0
$173$	13.7418	1.04477	0.522384	−	0.852710i	$-0.325043\pi$
$173$	13.7418	1.04477	0.522384	+	0.852710i	$0.325043\pi$
$174$	0	0
$175$	4.22996	0.319755
$176$	0	0
$177$	2.48332	0.186658
$178$	0	0
$179$	21.8049	1.62977	0.814887	−	0.579619i	$-0.196799\pi$
$179$	21.8049	1.62977	0.814887	+	0.579619i	$0.196799\pi$
$180$	0	0
$181$	16.3948	1.21862	0.609309	−	0.792933i	$-0.291447\pi$
$181$	16.3948	1.21862	0.609309	+	0.792933i	$0.291447\pi$
$182$	0	0
$183$	11.9539	0.883657
$184$	0	0
$185$	−4.53536	−0.333447
$186$	0	0
$187$	−3.09277	−0.226166
$188$	0	0
$189$	0.733037	0.0533206
$190$	0	0
$191$	10.9596	0.793011	0.396506	−	0.918032i	$-0.370223\pi$
$191$	10.9596	0.793011	0.396506	+	0.918032i	$0.370223\pi$
$192$	0	0
$193$	−2.11026	−0.151900	−0.0759498	−	0.997112i	$-0.524199\pi$
$193$	−2.11026	−0.151900	−0.0759498	+	0.997112i	$0.524199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.82473	0.343748	0.171874	−	0.985119i	$-0.445018\pi$
$197$	4.82473	0.343748	0.171874	+	0.985119i	$0.445018\pi$
$198$	0	0
$199$	−7.39963	−0.524546	−0.262273	−	0.964994i	$-0.584472\pi$
$199$	−7.39963	−0.524546	−0.262273	+	0.964994i	$0.584472\pi$
$200$	0	0
$201$	−17.7913	−1.25490
$202$	0	0
$203$	9.73068	0.682960
$204$	0	0
$205$	3.96595	0.276994
$206$	0	0
$207$	−8.79035	−0.610972
$208$	0	0
$209$	5.04810	0.349185
$210$	0	0
$211$	2.14005	0.147327	0.0736634	−	0.997283i	$-0.476531\pi$
$211$	2.14005	0.147327	0.0736634	+	0.997283i	$0.476531\pi$
$212$	0	0
$213$	−9.72928	−0.666640
$214$	0	0
$215$	6.69236	0.456415
$216$	0	0
$217$	3.94150	0.267567
$218$	0	0
$219$	30.2587	2.04469
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−25.3180	−1.69542	−0.847710	−	0.530460i	$-0.822019\pi$
$223$	−25.3180	−1.69542	−0.847710	+	0.530460i	$0.822019\pi$
$224$	0	0
$225$	−11.3903	−0.759353
$226$	0	0
$227$	−8.70636	−0.577862	−0.288931	−	0.957350i	$-0.593300\pi$
$227$	−8.70636	−0.577862	−0.288931	+	0.957350i	$0.593300\pi$
$228$	0	0
$229$	13.7265	0.907076	0.453538	−	0.891237i	$-0.350162\pi$
$229$	13.7265	0.907076	0.453538	+	0.891237i	$0.350162\pi$
$230$	0	0
$231$	−7.59768	−0.499891
$232$	0	0
$233$	−16.8245	−1.10221	−0.551106	−	0.834435i	$-0.685794\pi$
$233$	−16.8245	−1.10221	−0.551106	+	0.834435i	$0.685794\pi$
$234$	0	0
$235$	−3.61550	−0.235849
$236$	0	0
$237$	−23.3766	−1.51847
$238$	0	0
$239$	4.27266	0.276376	0.138188	−	0.990406i	$-0.455872\pi$
$239$	4.27266	0.276376	0.138188	+	0.990406i	$0.455872\pi$
$240$	0	0
$241$	25.4646	1.64032	0.820159	−	0.572135i	$-0.193885\pi$
$241$	25.4646	1.64032	0.820159	+	0.572135i	$0.193885\pi$
$242$	0	0
$243$	−21.2484	−1.36308
$244$	0	0
$245$	−0.877521	−0.0560628
$246$	0	0
$247$	0	0
$248$	0	0
$249$	16.2648	1.03074
$250$	0	0
$251$	−23.7136	−1.49679	−0.748395	−	0.663253i	$-0.769175\pi$
$251$	−23.7136	−1.49679	−0.748395	+	0.663253i	$0.769175\pi$
$252$	0	0
$253$	−10.3950	−0.653531
$254$	0	0
$255$	2.03352	0.127344
$256$	0	0
$257$	−3.63543	−0.226772	−0.113386	−	0.993551i	$-0.536170\pi$
$257$	−3.63543	−0.226772	−0.113386	+	0.993551i	$0.536170\pi$
$258$	0	0
$259$	−5.16838	−0.321148
$260$	0	0
$261$	−26.2025	−1.62189
$262$	0	0
$263$	−15.9588	−0.984063	−0.492031	−	0.870577i	$-0.663745\pi$
$263$	−15.9588	−0.984063	−0.492031	+	0.870577i	$0.663745\pi$
$264$	0	0
$265$	1.76984	0.108720
$266$	0	0
$267$	−11.0501	−0.676257
$268$	0	0
$269$	15.7887	0.962654	0.481327	−	0.876541i	$-0.340155\pi$
$269$	15.7887	0.962654	0.481327	+	0.876541i	$0.340155\pi$
$270$	0	0
$271$	7.74180	0.470281	0.235140	−	0.971961i	$-0.424445\pi$
$271$	7.74180	0.470281	0.235140	+	0.971961i	$0.424445\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.4696	−0.812248
$276$	0	0
$277$	31.6295	1.90043	0.950215	−	0.311594i	$-0.100863\pi$
$277$	31.6295	1.90043	0.950215	+	0.311594i	$0.100863\pi$
$278$	0	0
$279$	−10.6136	−0.635417
$280$	0	0
$281$	7.36786	0.439530	0.219765	−	0.975553i	$-0.429471\pi$
$281$	7.36786	0.439530	0.219765	+	0.975553i	$0.429471\pi$
$282$	0	0
$283$	10.9435	0.650523	0.325261	−	0.945624i	$-0.394548\pi$
$283$	10.9435	0.650523	0.325261	+	0.945624i	$0.394548\pi$
$284$	0	0
$285$	−3.31916	−0.196610
$286$	0	0
$287$	4.51950	0.266777
$288$	0	0
$289$	−16.0567	−0.944511
$290$	0	0
$291$	9.79746	0.574337
$292$	0	0
$293$	−31.8523	−1.86083	−0.930415	−	0.366508i	$-0.880553\pi$
$293$	−31.8523	−1.86083	−0.930415	+	0.366508i	$0.880553\pi$
$294$	0	0
$295$	−0.913331	−0.0531762
$296$	0	0
$297$	−2.33424	−0.135446
$298$	0	0
$299$	0	0
$300$	0	0
$301$	7.62643	0.439580
$302$	0	0
$303$	−32.7707	−1.88263
$304$	0	0
$305$	−4.39648	−0.251742
$306$	0	0
$307$	32.6251	1.86201	0.931007	−	0.365002i	$-0.118932\pi$
$307$	32.6251	1.86201	0.931007	+	0.365002i	$0.118932\pi$
$308$	0	0
$309$	−5.61830	−0.319614
$310$	0	0
$311$	6.86932	0.389524	0.194762	−	0.980851i	$-0.437607\pi$
$311$	6.86932	0.389524	0.194762	+	0.980851i	$0.437607\pi$
$312$	0	0
$313$	−2.13137	−0.120472	−0.0602360	−	0.998184i	$-0.519185\pi$
$313$	−2.13137	−0.120472	−0.0602360	+	0.998184i	$0.519185\pi$
$314$	0	0
$315$	2.36296	0.133138
$316$	0	0
$317$	5.52242	0.310170	0.155085	−	0.987901i	$-0.450435\pi$
$317$	5.52242	0.310170	0.155085	+	0.987901i	$0.450435\pi$
$318$	0	0
$319$	−30.9858	−1.73487
$320$	0	0
$321$	10.1299	0.565394
$322$	0	0
$323$	−1.53971	−0.0856715
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−44.9886	−2.48787
$328$	0	0
$329$	−4.12013	−0.227150
$330$	0	0
$331$	9.47833	0.520976	0.260488	−	0.965477i	$-0.416117\pi$
$331$	9.47833	0.520976	0.260488	+	0.965477i	$0.416117\pi$
$332$	0	0
$333$	13.9173	0.762661
$334$	0	0
$335$	6.54340	0.357504
$336$	0	0
$337$	12.8597	0.700511	0.350255	−	0.936654i	$-0.386095\pi$
$337$	12.8597	0.700511	0.350255	+	0.936654i	$0.386095\pi$
$338$	0	0
$339$	44.7253	2.42915
$340$	0	0
$341$	−12.5511	−0.679679
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.83481	0.367974
$346$	0	0
$347$	0.0885314	0.00475261	0.00237631	−	0.999997i	$-0.499244\pi$
$347$	0.0885314	0.00475261	0.00237631	+	0.999997i	$0.499244\pi$
$348$	0	0
$349$	−5.87025	−0.314227	−0.157114	−	0.987581i	$-0.550219\pi$
$349$	−5.87025	−0.314227	−0.157114	+	0.987581i	$0.550219\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.9660	0.743333	0.371667	−	0.928366i	$-0.378786\pi$
$353$	13.9660	0.743333	0.371667	+	0.928366i	$0.378786\pi$
$354$	0	0
$355$	3.57830	0.189916
$356$	0	0
$357$	2.31734	0.122647
$358$	0	0
$359$	−21.4343	−1.13126	−0.565629	−	0.824659i	$-0.691367\pi$
$359$	−21.4343	−1.13126	−0.565629	+	0.824659i	$0.691367\pi$
$360$	0	0
$361$	−16.4869	−0.867729
$362$	0	0
$363$	−2.05190	−0.107697
$364$	0	0
$365$	−11.1287	−0.582504
$366$	0	0
$367$	−3.70494	−0.193397	−0.0966983	−	0.995314i	$-0.530828\pi$
$367$	−3.70494	−0.193397	−0.0966983	+	0.995314i	$0.530828\pi$
$368$	0	0
$369$	−12.1700	−0.633543
$370$	0	0
$371$	2.01686	0.104710
$372$	0	0
$373$	−1.77919	−0.0921230	−0.0460615	−	0.998939i	$-0.514667\pi$
$373$	−1.77919	−0.0921230	−0.0460615	+	0.998939i	$0.514667\pi$
$374$	0	0
$375$	19.3250	0.997937
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0.317064	0.0162865	0.00814323	−	0.999967i	$-0.497408\pi$
$379$	0.317064	0.0162865	0.00814323	+	0.999967i	$0.497408\pi$
$380$	0	0
$381$	−49.5803	−2.54008
$382$	0	0
$383$	−26.2675	−1.34220	−0.671102	−	0.741365i	$-0.734179\pi$
$383$	−26.2675	−1.34220	−0.671102	+	0.741365i	$0.734179\pi$
$384$	0	0
$385$	2.79432	0.142412
$386$	0	0
$387$	−20.5362	−1.04392
$388$	0	0
$389$	−26.0221	−1.31937	−0.659686	−	0.751542i	$-0.729311\pi$
$389$	−26.0221	−1.31937	−0.659686	+	0.751542i	$0.729311\pi$
$390$	0	0
$391$	3.17056	0.160342
$392$	0	0
$393$	4.25410	0.214591
$394$	0	0
$395$	8.59759	0.432592
$396$	0	0
$397$	−32.3447	−1.62334	−0.811668	−	0.584119i	$-0.801440\pi$
$397$	−32.3447	−1.62334	−0.811668	+	0.584119i	$0.801440\pi$
$398$	0	0
$399$	−3.78243	−0.189358
$400$	0	0
$401$	29.2032	1.45834	0.729168	−	0.684334i	$-0.239907\pi$
$401$	29.2032	1.45834	0.729168	+	0.684334i	$0.239907\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	8.62366	0.428513
$406$	0	0
$407$	16.4579	0.815787
$408$	0	0
$409$	−13.4511	−0.665112	−0.332556	−	0.943084i	$-0.607911\pi$
$409$	−13.4511	−0.665112	−0.332556	+	0.943084i	$0.607911\pi$
$410$	0	0
$411$	−39.6961	−1.95806
$412$	0	0
$413$	−1.04081	−0.0512148
$414$	0	0
$415$	−5.98197	−0.293643
$416$	0	0
$417$	−3.70473	−0.181421
$418$	0	0
$419$	8.26689	0.403864	0.201932	−	0.979400i	$-0.435278\pi$
$419$	8.26689	0.403864	0.201932	+	0.979400i	$0.435278\pi$
$420$	0	0
$421$	−11.3220	−0.551802	−0.275901	−	0.961186i	$-0.588976\pi$
$421$	−11.3220	−0.551802	−0.275901	+	0.961186i	$0.588976\pi$
$422$	0	0
$423$	11.0945	0.539435
$424$	0	0
$425$	4.10832	0.199283
$426$	0	0
$427$	−5.01011	−0.242456
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.6290	−0.800989	−0.400495	−	0.916299i	$-0.631162\pi$
$431$	−16.6290	−0.800989	−0.400495	+	0.916299i	$0.631162\pi$
$432$	0	0
$433$	−23.2877	−1.11914	−0.559569	−	0.828784i	$-0.689033\pi$
$433$	−23.2877	−1.11914	−0.559569	+	0.828784i	$0.689033\pi$
$434$	0	0
$435$	20.3734	0.976828
$436$	0	0
$437$	−5.17507	−0.247557
$438$	0	0
$439$	−21.6370	−1.03268	−0.516338	−	0.856385i	$-0.672705\pi$
$439$	−21.6370	−1.03268	−0.516338	+	0.856385i	$0.672705\pi$
$440$	0	0
$441$	2.69277	0.128227
$442$	0	0
$443$	32.7018	1.55371	0.776855	−	0.629680i	$-0.216814\pi$
$443$	32.7018	1.55371	0.776855	+	0.629680i	$0.216814\pi$
$444$	0	0
$445$	4.06409	0.192656
$446$	0	0
$447$	−0.383785	−0.0181524
$448$	0	0
$449$	4.89620	0.231066	0.115533	−	0.993304i	$-0.463142\pi$
$449$	4.89620	0.231066	0.115533	+	0.993304i	$0.463142\pi$
$450$	0	0
$451$	−14.3916	−0.677674
$452$	0	0
$453$	28.2311	1.32641
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.7043	0.781396	0.390698	−	0.920519i	$-0.372234\pi$
$457$	16.7043	0.781396	0.390698	+	0.920519i	$0.372234\pi$
$458$	0	0
$459$	0.711958	0.0332314
$460$	0	0
$461$	−20.7451	−0.966195	−0.483098	−	0.875566i	$-0.660488\pi$
$461$	−20.7451	−0.966195	−0.483098	+	0.875566i	$0.660488\pi$
$462$	0	0
$463$	−27.0197	−1.25571	−0.627855	−	0.778330i	$-0.716067\pi$
$463$	−27.0197	−1.25571	−0.627855	+	0.778330i	$0.716067\pi$
$464$	0	0
$465$	8.25241	0.382697
$466$	0	0
$467$	−33.8384	−1.56586	−0.782928	−	0.622112i	$-0.786275\pi$
$467$	−33.8384	−1.56586	−0.782928	+	0.622112i	$0.786275\pi$
$468$	0	0
$469$	7.45668	0.344318
$470$	0	0
$471$	−35.7021	−1.64507
$472$	0	0
$473$	−24.2851	−1.11663
$474$	0	0
$475$	−6.70572	−0.307679
$476$	0	0
$477$	−5.43095	−0.248666
$478$	0	0
$479$	12.0454	0.550367	0.275183	−	0.961392i	$-0.411261\pi$
$479$	12.0454	0.550367	0.275183	+	0.961392i	$0.411261\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	7.78877	0.354401
$484$	0	0
$485$	−3.60337	−0.163621
$486$	0	0
$487$	14.2491	0.645688	0.322844	−	0.946452i	$-0.395361\pi$
$487$	14.2491	0.645688	0.322844	+	0.946452i	$0.395361\pi$
$488$	0	0
$489$	−29.2264	−1.32167
$490$	0	0
$491$	25.6033	1.15546	0.577731	−	0.816227i	$-0.303938\pi$
$491$	25.6033	1.15546	0.577731	+	0.816227i	$0.303938\pi$
$492$	0	0
$493$	9.45087	0.425646
$494$	0	0
$495$	−7.52447	−0.338200
$496$	0	0
$497$	4.07774	0.182911
$498$	0	0
$499$	12.8345	0.574550	0.287275	−	0.957848i	$-0.407251\pi$
$499$	12.8345	0.574550	0.287275	+	0.957848i	$0.407251\pi$
$500$	0	0
$501$	−20.0695	−0.896638
$502$	0	0
$503$	−26.1240	−1.16481	−0.582406	−	0.812898i	$-0.697889\pi$
$503$	−26.1240	−1.16481	−0.582406	+	0.812898i	$0.697889\pi$
$504$	0	0
$505$	12.0526	0.536335
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.0296155	−0.00131268	−0.000656342	−	1.00000i	$-0.500209\pi$
$509$	−0.0296155	−0.00131268	−0.000656342		1.00000i	$0.500209\pi$
$510$	0	0
$511$	−12.6820	−0.561019
$512$	0	0
$513$	−1.16208	−0.0513070
$514$	0	0
$515$	2.06633	0.0910535
$516$	0	0
$517$	13.1199	0.577011
$518$	0	0
$519$	32.7873	1.43920
$520$	0	0
$521$	25.4799	1.11630	0.558148	−	0.829742i	$-0.311512\pi$
$521$	25.4799	1.11630	0.558148	+	0.829742i	$0.311512\pi$
$522$	0	0
$523$	−39.0099	−1.70578	−0.852892	−	0.522087i	$-0.825154\pi$
$523$	−39.0099	−1.70578	−0.852892	+	0.522087i	$0.825154\pi$
$524$	0	0
$525$	10.0925	0.440472
$526$	0	0
$527$	3.82816	0.166757
$528$	0	0
$529$	−12.3435	−0.536675
$530$	0	0
$531$	2.80266	0.121625
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−3.72563	−0.161073
$536$	0	0
$537$	52.0255	2.24506
$538$	0	0
$539$	3.18434	0.137159
$540$	0	0
$541$	−3.87645	−0.166662	−0.0833308	−	0.996522i	$-0.526556\pi$
$541$	−3.87645	−0.166662	−0.0833308	+	0.996522i	$0.526556\pi$
$542$	0	0
$543$	39.1173	1.67868
$544$	0	0
$545$	16.5462	0.708761
$546$	0	0
$547$	38.8022	1.65906	0.829532	−	0.558460i	$-0.188607\pi$
$547$	38.8022	1.65906	0.829532	+	0.558460i	$0.188607\pi$
$548$	0	0
$549$	13.4911	0.575785
$550$	0	0
$551$	−15.4260	−0.657168
$552$	0	0
$553$	9.79759	0.416636
$554$	0	0
$555$	−10.8212	−0.459333
$556$	0	0
$557$	3.84787	0.163039	0.0815197	−	0.996672i	$-0.474023\pi$
$557$	3.84787	0.163039	0.0815197	+	0.996672i	$0.474023\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.37920	−0.311550
$562$	0	0
$563$	−13.4656	−0.567506	−0.283753	−	0.958897i	$-0.591580\pi$
$563$	−13.4656	−0.567506	−0.283753	+	0.958897i	$0.591580\pi$
$564$	0	0
$565$	−16.4494	−0.692030
$566$	0	0
$567$	9.82730	0.412708
$568$	0	0
$569$	7.51845	0.315190	0.157595	−	0.987504i	$-0.449626\pi$
$569$	7.51845	0.315190	0.157595	+	0.987504i	$0.449626\pi$
$570$	0	0
$571$	4.48050	0.187503	0.0937515	−	0.995596i	$-0.470114\pi$
$571$	4.48050	0.187503	0.0937515	+	0.995596i	$0.470114\pi$
$572$	0	0
$573$	26.1492	1.09240
$574$	0	0
$575$	13.8084	0.575850
$576$	0	0
$577$	8.68801	0.361686	0.180843	−	0.983512i	$-0.442117\pi$
$577$	8.68801	0.361686	0.180843	+	0.983512i	$0.442117\pi$
$578$	0	0
$579$	−5.03497	−0.209246
$580$	0	0
$581$	−6.81690	−0.282813
$582$	0	0
$583$	−6.42237	−0.265988
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.0991	1.28360	0.641799	−	0.766873i	$-0.278188\pi$
$587$	31.0991	1.28360	0.641799	+	0.766873i	$0.278188\pi$
$588$	0	0
$589$	−6.24843	−0.257462
$590$	0	0
$591$	11.5116	0.473523
$592$	0	0
$593$	−45.7116	−1.87715	−0.938575	−	0.345076i	$-0.887853\pi$
$593$	−45.7116	−1.87715	−0.938575	+	0.345076i	$0.887853\pi$
$594$	0	0
$595$	−0.852287	−0.0349404
$596$	0	0
$597$	−17.6552	−0.722578
$598$	0	0
$599$	−10.2396	−0.418380	−0.209190	−	0.977875i	$-0.567083\pi$
$599$	−10.2396	−0.418380	−0.209190	+	0.977875i	$0.567083\pi$
$600$	0	0
$601$	−15.8699	−0.647346	−0.323673	−	0.946169i	$-0.604918\pi$
$601$	−15.8699	−0.647346	−0.323673	+	0.946169i	$0.604918\pi$
$602$	0	0
$603$	−20.0791	−0.817686
$604$	0	0
$605$	0.754661	0.0306813
$606$	0	0
$607$	−35.3481	−1.43474	−0.717368	−	0.696695i	$-0.754653\pi$
$607$	−35.3481	−1.43474	−0.717368	+	0.696695i	$0.754653\pi$
$608$	0	0
$609$	23.2169	0.940798
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.56595	−0.0632479	−0.0316240	−	0.999500i	$-0.510068\pi$
$613$	−1.56595	−0.0632479	−0.0316240	+	0.999500i	$0.510068\pi$
$614$	0	0
$615$	9.46258	0.381568
$616$	0	0
$617$	−9.54239	−0.384162	−0.192081	−	0.981379i	$-0.561524\pi$
$617$	−9.54239	−0.384162	−0.192081	+	0.981379i	$0.561524\pi$
$618$	0	0
$619$	25.6394	1.03053	0.515267	−	0.857030i	$-0.327693\pi$
$619$	25.6394	1.03053	0.515267	+	0.857030i	$0.327693\pi$
$620$	0	0
$621$	2.39295	0.0960256
$622$	0	0
$623$	4.63133	0.185550
$624$	0	0
$625$	14.0423	0.561693
$626$	0	0
$627$	12.0445	0.481012
$628$	0	0
$629$	−5.01976	−0.200151
$630$	0	0
$631$	30.5409	1.21581	0.607907	−	0.794009i	$-0.292009\pi$
$631$	30.5409	1.21581	0.607907	+	0.794009i	$0.292009\pi$
$632$	0	0
$633$	5.10605	0.202947
$634$	0	0
$635$	18.2350	0.723632
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.9804	−0.434378
$640$	0	0
$641$	−10.0657	−0.397572	−0.198786	−	0.980043i	$-0.563700\pi$
$641$	−10.0657	−0.397572	−0.198786	+	0.980043i	$0.563700\pi$
$642$	0	0
$643$	34.1859	1.34816	0.674080	−	0.738658i	$-0.264540\pi$
$643$	34.1859	1.34816	0.674080	+	0.738658i	$0.264540\pi$
$644$	0	0
$645$	15.9676	0.628725
$646$	0	0
$647$	−43.5258	−1.71117	−0.855587	−	0.517658i	$-0.826804\pi$
$647$	−43.5258	−1.71117	−0.855587	+	0.517658i	$0.826804\pi$
$648$	0	0
$649$	3.31428	0.130097
$650$	0	0
$651$	9.40424	0.368581
$652$	0	0
$653$	49.0602	1.91987	0.959937	−	0.280217i	$-0.0904064\pi$
$653$	49.0602	1.91987	0.959937	+	0.280217i	$0.0904064\pi$
$654$	0	0
$655$	−1.56460	−0.0611340
$656$	0	0
$657$	34.1497	1.33231
$658$	0	0
$659$	−37.6187	−1.46541	−0.732707	−	0.680544i	$-0.761744\pi$
$659$	−37.6187	−1.46541	−0.732707	+	0.680544i	$0.761744\pi$
$660$	0	0
$661$	−9.37504	−0.364647	−0.182323	−	0.983239i	$-0.558362\pi$
$661$	−9.37504	−0.364647	−0.182323	+	0.983239i	$0.558362\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.39113	0.0539456
$666$	0	0
$667$	31.7651	1.22995
$668$	0	0
$669$	−60.4076	−2.33549
$670$	0	0
$671$	15.9539	0.615893
$672$	0	0
$673$	−23.4831	−0.905206	−0.452603	−	0.891712i	$-0.649505\pi$
$673$	−23.4831	−0.905206	−0.452603	+	0.891712i	$0.649505\pi$
$674$	0	0
$675$	3.10072	0.119347
$676$	0	0
$677$	2.71645	0.104401	0.0522007	−	0.998637i	$-0.483376\pi$
$677$	2.71645	0.104401	0.0522007	+	0.998637i	$0.483376\pi$
$678$	0	0
$679$	−4.10631	−0.157586
$680$	0	0
$681$	−20.7730	−0.796022
$682$	0	0
$683$	−45.0155	−1.72247	−0.861235	−	0.508206i	$-0.830309\pi$
$683$	−45.0155	−1.72247	−0.861235	+	0.508206i	$0.830309\pi$
$684$	0	0
$685$	14.5997	0.557825
$686$	0	0
$687$	32.7509	1.24952
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−17.7336	−0.674619	−0.337310	−	0.941394i	$-0.609517\pi$
$691$	−17.7336	−0.674619	−0.337310	+	0.941394i	$0.609517\pi$
$692$	0	0
$693$	−8.57469	−0.325725
$694$	0	0
$695$	1.36255	0.0516844
$696$	0	0
$697$	4.38954	0.166266
$698$	0	0
$699$	−40.1426	−1.51833
$700$	0	0
$701$	7.97397	0.301173	0.150586	−	0.988597i	$-0.451884\pi$
$701$	7.97397	0.301173	0.150586	+	0.988597i	$0.451884\pi$
$702$	0	0
$703$	8.19339	0.309020
$704$	0	0
$705$	−8.62640	−0.324889
$706$	0	0
$707$	13.7349	0.516553
$708$	0	0
$709$	−32.3379	−1.21447	−0.607237	−	0.794521i	$-0.707722\pi$
$709$	−32.3379	−1.21447	−0.607237	+	0.794521i	$0.707722\pi$
$710$	0	0
$711$	−26.3827	−0.989427
$712$	0	0
$713$	12.8667	0.481863
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.1944	0.380716
$718$	0	0
$719$	28.7628	1.07267	0.536335	−	0.844005i	$-0.319808\pi$
$719$	28.7628	1.07267	0.536335	+	0.844005i	$0.319808\pi$
$720$	0	0
$721$	2.35474	0.0876951
$722$	0	0
$723$	60.7573	2.25959
$724$	0	0
$725$	41.1604	1.52866
$726$	0	0
$727$	27.5051	1.02011	0.510054	−	0.860143i	$-0.329626\pi$
$727$	27.5051	1.02011	0.510054	+	0.860143i	$0.329626\pi$
$728$	0	0
$729$	−21.2157	−0.785766
$730$	0	0
$731$	7.40713	0.273963
$732$	0	0
$733$	−6.53199	−0.241265	−0.120632	−	0.992697i	$-0.538492\pi$
$733$	−6.53199	−0.241265	−0.120632	+	0.992697i	$0.538492\pi$
$734$	0	0
$735$	−2.09372	−0.0772281
$736$	0	0
$737$	−23.7446	−0.874644
$738$	0	0
$739$	26.5907	0.978154	0.489077	−	0.872241i	$-0.337334\pi$
$739$	26.5907	0.978154	0.489077	+	0.872241i	$0.337334\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	37.3030	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$743$	37.3030	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$744$	0	0
$745$	0.141151	0.00517137
$746$	0	0
$747$	18.3563	0.671623
$748$	0	0
$749$	−4.24563	−0.155132
$750$	0	0
$751$	3.66994	0.133918	0.0669591	−	0.997756i	$-0.478670\pi$
$751$	3.66994	0.133918	0.0669591	+	0.997756i	$0.478670\pi$
$752$	0	0
$753$	−56.5796	−2.06187
$754$	0	0
$755$	−10.3830	−0.377876
$756$	0	0
$757$	−44.7042	−1.62480	−0.812401	−	0.583099i	$-0.801840\pi$
$757$	−44.7042	−1.62480	−0.812401	+	0.583099i	$0.801840\pi$
$758$	0	0
$759$	−24.8021	−0.900258
$760$	0	0
$761$	−39.4739	−1.43093	−0.715463	−	0.698650i	$-0.753784\pi$
$761$	−39.4739	−1.43093	−0.715463	+	0.698650i	$0.753784\pi$
$762$	0	0
$763$	18.8556	0.682618
$764$	0	0
$765$	2.29501	0.0829764
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.95318	0.250738	0.125369	−	0.992110i	$-0.459989\pi$
$769$	6.95318	0.250738	0.125369	+	0.992110i	$0.459989\pi$
$770$	0	0
$771$	−8.67397	−0.312385
$772$	0	0
$773$	49.4334	1.77800	0.888998	−	0.457911i	$-0.151402\pi$
$773$	49.4334	1.77800	0.888998	+	0.457911i	$0.151402\pi$
$774$	0	0
$775$	16.6724	0.598890
$776$	0	0
$777$	−12.3315	−0.442391
$778$	0	0
$779$	−7.16472	−0.256703
$780$	0	0
$781$	−12.9849	−0.464636
$782$	0	0
$783$	7.13295	0.254911
$784$	0	0
$785$	13.1307	0.468657
$786$	0	0
$787$	−28.3548	−1.01074	−0.505370	−	0.862903i	$-0.668644\pi$
$787$	−28.3548	−1.01074	−0.505370	+	0.862903i	$0.668644\pi$
$788$	0	0
$789$	−38.0770	−1.35558
$790$	0	0
$791$	−18.7453	−0.666505
$792$	0	0
$793$	0	0
$794$	0	0
$795$	4.22275	0.149766
$796$	0	0
$797$	−17.7040	−0.627109	−0.313555	−	0.949570i	$-0.601520\pi$
$797$	−17.7040	−0.627109	−0.313555	+	0.949570i	$0.601520\pi$
$798$	0	0
$799$	−4.00165	−0.141568
$800$	0	0
$801$	−12.4711	−0.440645
$802$	0	0
$803$	40.3838	1.42511
$804$	0	0
$805$	−2.86460	−0.100964
$806$	0	0
$807$	37.6711	1.32609
$808$	0	0
$809$	9.35040	0.328743	0.164371	−	0.986399i	$-0.447440\pi$
$809$	9.35040	0.328743	0.164371	+	0.986399i	$0.447440\pi$
$810$	0	0
$811$	35.5948	1.24990	0.624951	−	0.780664i	$-0.285119\pi$
$811$	35.5948	1.24990	0.624951	+	0.780664i	$0.285119\pi$
$812$	0	0
$813$	18.4716	0.647826
$814$	0	0
$815$	10.7491	0.376524
$816$	0	0
$817$	−12.0901	−0.422980
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.5928	1.34690	0.673449	−	0.739234i	$-0.264812\pi$
$821$	38.5928	1.34690	0.673449	+	0.739234i	$0.264812\pi$
$822$	0	0
$823$	50.7203	1.76800	0.883999	−	0.467490i	$-0.154841\pi$
$823$	50.7203	1.76800	0.883999	+	0.467490i	$0.154841\pi$
$824$	0	0
$825$	−32.1379	−1.11890
$826$	0	0
$827$	−21.9754	−0.764158	−0.382079	−	0.924130i	$-0.624792\pi$
$827$	−21.9754	−0.764158	−0.382079	+	0.924130i	$0.624792\pi$
$828$	0	0
$829$	8.85481	0.307540	0.153770	−	0.988107i	$-0.450858\pi$
$829$	8.85481	0.307540	0.153770	+	0.988107i	$0.450858\pi$
$830$	0	0
$831$	75.4664	2.61790
$832$	0	0
$833$	−0.971244	−0.0336516
$834$	0	0
$835$	7.38128	0.255440
$836$	0	0
$837$	2.88927	0.0998677
$838$	0	0
$839$	29.9148	1.03277	0.516387	−	0.856355i	$-0.327277\pi$
$839$	29.9148	1.03277	0.516387	+	0.856355i	$0.327277\pi$
$840$	0	0
$841$	65.6862	2.26504
$842$	0	0
$843$	17.5794	0.605466
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.859992	0.0295497
$848$	0	0
$849$	26.1106	0.896115
$850$	0	0
$851$	−16.8718	−0.578358
$852$	0	0
$853$	−39.7867	−1.36227	−0.681135	−	0.732158i	$-0.738513\pi$
$853$	−39.7867	−1.36227	−0.681135	+	0.732158i	$0.738513\pi$
$854$	0	0
$855$	−3.74598	−0.128110
$856$	0	0
$857$	−17.0715	−0.583150	−0.291575	−	0.956548i	$-0.594179\pi$
$857$	−17.0715	−0.583150	−0.291575	+	0.956548i	$0.594179\pi$
$858$	0	0
$859$	−45.4327	−1.55014	−0.775071	−	0.631874i	$-0.782286\pi$
$859$	−45.4327	−1.55014	−0.775071	+	0.631874i	$0.782286\pi$
$860$	0	0
$861$	10.7833	0.367494
$862$	0	0
$863$	22.8572	0.778068	0.389034	−	0.921223i	$-0.372809\pi$
$863$	22.8572	0.778068	0.389034	+	0.921223i	$0.372809\pi$
$864$	0	0
$865$	−12.0587	−0.410008
$866$	0	0
$867$	−38.3105	−1.30109
$868$	0	0
$869$	−31.1988	−1.05835
$870$	0	0
$871$	0	0
$872$	0	0
$873$	11.0573	0.374234
$874$	0	0
$875$	−8.09948	−0.273812
$876$	0	0
$877$	−29.8620	−1.00837	−0.504185	−	0.863596i	$-0.668207\pi$
$877$	−29.8620	−1.00837	−0.504185	+	0.863596i	$0.668207\pi$
$878$	0	0
$879$	−75.9980	−2.56335
$880$	0	0
$881$	−24.9735	−0.841379	−0.420689	−	0.907205i	$-0.638212\pi$
$881$	−24.9735	−0.841379	−0.420689	+	0.907205i	$0.638212\pi$
$882$	0	0
$883$	−12.3025	−0.414011	−0.207005	−	0.978340i	$-0.566372\pi$
$883$	−12.3025	−0.414011	−0.207005	+	0.978340i	$0.566372\pi$
$884$	0	0
$885$	−2.17916	−0.0732518
$886$	0	0
$887$	−38.3862	−1.28888	−0.644441	−	0.764654i	$-0.722910\pi$
$887$	−38.3862	−1.28888	−0.644441	+	0.764654i	$0.722910\pi$
$888$	0	0
$889$	20.7801	0.696942
$890$	0	0
$891$	−31.2934	−1.04837
$892$	0	0
$893$	6.53160	0.218572
$894$	0	0
$895$	−19.1343	−0.639588
$896$	0	0
$897$	0	0
$898$	0	0
$899$	38.3535	1.27916
$900$	0	0
$901$	1.95887	0.0652593
$902$	0	0
$903$	18.1963	0.605535
$904$	0	0
$905$	−14.3868	−0.478233
$906$	0	0
$907$	15.1964	0.504589	0.252294	−	0.967651i	$-0.418815\pi$
$907$	15.1964	0.504589	0.252294	+	0.967651i	$0.418815\pi$
$908$	0	0
$909$	−36.9848	−1.22671
$910$	0	0
$911$	46.1592	1.52932	0.764661	−	0.644433i	$-0.222907\pi$
$911$	46.1592	1.52932	0.764661	+	0.644433i	$0.222907\pi$
$912$	0	0
$913$	21.7073	0.718407
$914$	0	0
$915$	−10.4898	−0.346782
$916$	0	0
$917$	−1.78298	−0.0588791
$918$	0	0
$919$	−17.4589	−0.575916	−0.287958	−	0.957643i	$-0.592976\pi$
$919$	−17.4589	−0.575916	−0.287958	+	0.957643i	$0.592976\pi$
$920$	0	0
$921$	77.8420	2.56498
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−21.8620	−0.718819
$926$	0	0
$927$	−6.34077	−0.208258
$928$	0	0
$929$	55.4173	1.81818	0.909091	−	0.416599i	$-0.136778\pi$
$929$	55.4173	1.81818	0.909091	+	0.416599i	$0.136778\pi$
$930$	0	0
$931$	1.58529	0.0519558
$932$	0	0
$933$	16.3899	0.536581
$934$	0	0
$935$	2.71397	0.0887563
$936$	0	0
$937$	28.3045	0.924668	0.462334	−	0.886706i	$-0.347012\pi$
$937$	28.3045	0.924668	0.462334	+	0.886706i	$0.347012\pi$
$938$	0	0
$939$	−5.08534	−0.165954
$940$	0	0
$941$	20.5105	0.668622	0.334311	−	0.942463i	$-0.391496\pi$
$941$	20.5105	0.668622	0.334311	+	0.942463i	$0.391496\pi$
$942$	0	0
$943$	14.7536	0.480442
$944$	0	0
$945$	−0.643256	−0.0209251
$946$	0	0
$947$	6.16669	0.200390	0.100195	−	0.994968i	$-0.468053\pi$
$947$	6.16669	0.200390	0.100195	+	0.994968i	$0.468053\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	13.1762	0.427269
$952$	0	0
$953$	−41.1044	−1.33150	−0.665751	−	0.746174i	$-0.731889\pi$
$953$	−41.1044	−1.33150	−0.665751	+	0.746174i	$0.731889\pi$
$954$	0	0
$955$	−9.61731	−0.311209
$956$	0	0
$957$	−73.9306	−2.38984
$958$	0	0
$959$	16.6374	0.537250
$960$	0	0
$961$	−15.4646	−0.498857
$962$	0	0
$963$	11.4325	0.368407
$964$	0	0
$965$	1.85179	0.0596114
$966$	0	0
$967$	45.3751	1.45917	0.729583	−	0.683892i	$-0.239714\pi$
$967$	45.3751	1.45917	0.729583	+	0.683892i	$0.239714\pi$
$968$	0	0
$969$	−3.67366	−0.118015
$970$	0	0
$971$	41.2205	1.32283	0.661415	−	0.750020i	$-0.269956\pi$
$971$	41.2205	1.32283	0.661415	+	0.750020i	$0.269956\pi$
$972$	0	0
$973$	1.55273	0.0497781
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−57.2225	−1.83071	−0.915355	−	0.402648i	$-0.868090\pi$
$977$	−57.2225	−1.83071	−0.915355	+	0.402648i	$0.868090\pi$
$978$	0	0
$979$	−14.7477	−0.471339
$980$	0	0
$981$	−50.7738	−1.62108
$982$	0	0
$983$	45.1712	1.44074	0.720368	−	0.693592i	$-0.243973\pi$
$983$	45.1712	1.44074	0.720368	+	0.693592i	$0.243973\pi$
$984$	0	0
$985$	−4.23380	−0.134900
$986$	0	0
$987$	−9.83042	−0.312906
$988$	0	0
$989$	24.8959	0.791645
$990$	0	0
$991$	11.5814	0.367895	0.183947	−	0.982936i	$-0.441112\pi$
$991$	11.5814	0.367895	0.183947	+	0.982936i	$0.441112\pi$
$992$	0	0
$993$	22.6148	0.717661
$994$	0	0
$995$	6.49333	0.205852
$996$	0	0
$997$	40.0062	1.26701	0.633505	−	0.773738i	$-0.281615\pi$
$997$	40.0062	1.26701	0.633505	+	0.773738i	$0.281615\pi$
$998$	0	0
$999$	−3.78862	−0.119867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.bf.1.6		6
13.2	odd	12		728.2.bm.b.225.1	✓	12
13.7	odd	12		728.2.bm.b.673.1	yes	12
13.12	even	2		9464.2.a.bg.1.6		6
52.7	even	12		1456.2.cc.e.673.6		12
52.15	even	12		1456.2.cc.e.225.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.bm.b.225.1	✓	12	13.2	odd	12
728.2.bm.b.673.1	yes	12	13.7	odd	12
1456.2.cc.e.225.6		12	52.15	even	12
1456.2.cc.e.673.6		12	52.7	even	12
9464.2.a.bf.1.6		6	1.1	even	1	trivial
9464.2.a.bg.1.6		6	13.12	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.38595 q^{3} -0.877521 q^{5} -1.00000 q^{7} +2.69277 q^{9} +3.18434 q^{11} -2.09372 q^{15} -0.971244 q^{17} +1.58529 q^{19} -2.38595 q^{21} -3.26443 q^{23} -4.22996 q^{25} -0.733037 q^{27} -9.73068 q^{29} -3.94150 q^{31} +7.59768 q^{33} +0.877521 q^{35} +5.16838 q^{37} -4.51950 q^{41} -7.62643 q^{43} -2.36296 q^{45} +4.12013 q^{47} +1.00000 q^{49} -2.31734 q^{51} -2.01686 q^{53} -2.79432 q^{55} +3.78243 q^{57} +1.04081 q^{59} +5.01011 q^{61} -2.69277 q^{63} -7.45668 q^{67} -7.78877 q^{69} -4.07774 q^{71} +12.6820 q^{73} -10.0925 q^{75} -3.18434 q^{77} -9.79759 q^{79} -9.82730 q^{81} +6.81690 q^{83} +0.852287 q^{85} -23.2169 q^{87} -4.63133 q^{89} -9.40424 q^{93} -1.39113 q^{95} +4.10631 q^{97} +8.57469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 4 q^{5} - 6 q^{7} - 4 q^{9} + 10 q^{11} - 6 q^{15} + 2 q^{17} + 14 q^{19} - 2 q^{21} - 6 q^{23} - 2 q^{25} + 2 q^{27} - 20 q^{29} - 4 q^{31} + 4 q^{33} + 4 q^{35} - 4 q^{37} + 10 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 - 4 * q^9 + 10 * q^11 - 6 * q^15 + 2 * q^17 + 14 * q^19 - 2 * q^21 - 6 * q^23 - 2 * q^25 + 2 * q^27 - 20 * q^29 - 4 * q^31 + 4 * q^33 + 4 * q^35 - 4 * q^37 + 10 * q^41 - 12 * q^43 + 6 * q^45 - 6 * q^47 + 6 * q^49 - 8 * q^51 + 10 * q^53 - 24 * q^55 - 8 * q^59 + 4 * q^61 + 4 * q^63 - 18 * q^67 + 12 * q^71 + 14 * q^73 - 10 * q^75 - 10 * q^77 - 10 * q^79 - 18 * q^81 + 28 * q^83 + 20 * q^85 - 32 * q^87 - 6 * q^89 + 12 * q^93 - 26 * q^95 + 8 * q^97 + 2 * q^99

Introduction

L-functions

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Database

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