LMFDB - Embedded newform 8048.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 $ x^12 - 4x^11 - 22x^10 + 67x^9 + 180x^8 - 383x^7 - 674x^6 + 875x^5 + 1077x^4 - 704x^3 - 467x^2 + 210*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.75875$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73829 q^{3} -1.48375 q^{5} -2.26909 q^{7} +4.49825 q^{9} +O(q^{10})$	$q-2.73829 q^{3} -1.48375 q^{5} -2.26909 q^{7} +4.49825 q^{9} -4.54237 q^{11} +1.28908 q^{13} +4.06294 q^{15} +3.35584 q^{17} +6.02589 q^{19} +6.21343 q^{21} -4.30148 q^{23} -2.79849 q^{25} -4.10265 q^{27} -5.74625 q^{29} +10.2624 q^{31} +12.4384 q^{33} +3.36676 q^{35} +4.70900 q^{37} -3.52989 q^{39} +6.50770 q^{41} -12.5570 q^{43} -6.67427 q^{45} -8.97508 q^{47} -1.85124 q^{49} -9.18926 q^{51} -5.39473 q^{53} +6.73974 q^{55} -16.5007 q^{57} -12.2282 q^{59} -5.71895 q^{61} -10.2069 q^{63} -1.91267 q^{65} +8.64755 q^{67} +11.7787 q^{69} -13.7302 q^{71} +3.68126 q^{73} +7.66309 q^{75} +10.3070 q^{77} +3.72712 q^{79} -2.26049 q^{81} +3.13641 q^{83} -4.97922 q^{85} +15.7349 q^{87} -12.6454 q^{89} -2.92504 q^{91} -28.1015 q^{93} -8.94091 q^{95} +14.6647 q^{97} -20.4327 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * 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
$3$	−2.73829	−1.58095	−0.790477	−	0.612491i	$-0.790167\pi$
$3$	−2.73829	−1.58095	−0.790477	+	0.612491i	$0.790167\pi$
$4$	0	0
$5$	−1.48375	−0.663552	−0.331776	−	0.943358i	$-0.607648\pi$
$5$	−1.48375	−0.663552	−0.331776	+	0.943358i	$0.607648\pi$
$6$	0	0
$7$	−2.26909	−0.857635	−0.428818	−	0.903391i	$-0.641070\pi$
$7$	−2.26909	−0.857635	−0.428818	+	0.903391i	$0.641070\pi$
$8$	0	0
$9$	4.49825	1.49942
$10$	0	0
$11$	−4.54237	−1.36958	−0.684789	−	0.728742i	$-0.740105\pi$
$11$	−4.54237	−1.36958	−0.684789	+	0.728742i	$0.740105\pi$
$12$	0	0
$13$	1.28908	0.357527	0.178764	−	0.983892i	$-0.442790\pi$
$13$	1.28908	0.357527	0.178764	+	0.983892i	$0.442790\pi$
$14$	0	0
$15$	4.06294	1.04905
$16$	0	0
$17$	3.35584	0.813910	0.406955	−	0.913448i	$-0.366591\pi$
$17$	3.35584	0.813910	0.406955	+	0.913448i	$0.366591\pi$
$18$	0	0
$19$	6.02589	1.38243	0.691217	−	0.722647i	$-0.257075\pi$
$19$	6.02589	1.38243	0.691217	+	0.722647i	$0.257075\pi$
$20$	0	0
$21$	6.21343	1.35588
$22$	0	0
$23$	−4.30148	−0.896921	−0.448460	−	0.893803i	$-0.648028\pi$
$23$	−4.30148	−0.896921	−0.448460	+	0.893803i	$0.648028\pi$
$24$	0	0
$25$	−2.79849	−0.559698
$26$	0	0
$27$	−4.10265	−0.789555
$28$	0	0
$29$	−5.74625	−1.06705	−0.533526	−	0.845784i	$-0.679133\pi$
$29$	−5.74625	−1.06705	−0.533526	+	0.845784i	$0.679133\pi$
$30$	0	0
$31$	10.2624	1.84318	0.921592	−	0.388161i	$-0.126889\pi$
$31$	10.2624	1.84318	0.921592	+	0.388161i	$0.126889\pi$
$32$	0	0
$33$	12.4384	2.16524
$34$	0	0
$35$	3.36676	0.569086
$36$	0	0
$37$	4.70900	0.774155	0.387077	−	0.922047i	$-0.373485\pi$
$37$	4.70900	0.774155	0.387077	+	0.922047i	$0.373485\pi$
$38$	0	0
$39$	−3.52989	−0.565234
$40$	0	0
$41$	6.50770	1.01633	0.508166	−	0.861259i	$-0.330324\pi$
$41$	6.50770	1.01633	0.508166	+	0.861259i	$0.330324\pi$
$42$	0	0
$43$	−12.5570	−1.91492	−0.957460	−	0.288567i	$-0.906821\pi$
$43$	−12.5570	−1.91492	−0.957460	+	0.288567i	$0.906821\pi$
$44$	0	0
$45$	−6.67427	−0.994942
$46$	0	0
$47$	−8.97508	−1.30915	−0.654575	−	0.755997i	$-0.727152\pi$
$47$	−8.97508	−1.30915	−0.654575	+	0.755997i	$0.727152\pi$
$48$	0	0
$49$	−1.85124	−0.264462
$50$	0	0
$51$	−9.18926	−1.28675
$52$	0	0
$53$	−5.39473	−0.741023	−0.370511	−	0.928828i	$-0.620818\pi$
$53$	−5.39473	−0.741023	−0.370511	+	0.928828i	$0.620818\pi$
$54$	0	0
$55$	6.73974	0.908786
$56$	0	0
$57$	−16.5007	−2.18557
$58$	0	0
$59$	−12.2282	−1.59197	−0.795986	−	0.605315i	$-0.793047\pi$
$59$	−12.2282	−1.59197	−0.795986	+	0.605315i	$0.793047\pi$
$60$	0	0
$61$	−5.71895	−0.732236	−0.366118	−	0.930568i	$-0.619313\pi$
$61$	−5.71895	−0.732236	−0.366118	+	0.930568i	$0.619313\pi$
$62$	0	0
$63$	−10.2069	−1.28595
$64$	0	0
$65$	−1.91267	−0.237238
$66$	0	0
$67$	8.64755	1.05647	0.528233	−	0.849099i	$-0.322855\pi$
$67$	8.64755	1.05647	0.528233	+	0.849099i	$0.322855\pi$
$68$	0	0
$69$	11.7787	1.41799
$70$	0	0
$71$	−13.7302	−1.62947	−0.814735	−	0.579833i	$-0.803118\pi$
$71$	−13.7302	−1.62947	−0.814735	+	0.579833i	$0.803118\pi$
$72$	0	0
$73$	3.68126	0.430859	0.215430	−	0.976519i	$-0.430885\pi$
$73$	3.68126	0.430859	0.215430	+	0.976519i	$0.430885\pi$
$74$	0	0
$75$	7.66309	0.884857
$76$	0	0
$77$	10.3070	1.17460
$78$	0	0
$79$	3.72712	0.419334	0.209667	−	0.977773i	$-0.432762\pi$
$79$	3.72712	0.419334	0.209667	+	0.977773i	$0.432762\pi$
$80$	0	0
$81$	−2.26049	−0.251166
$82$	0	0
$83$	3.13641	0.344266	0.172133	−	0.985074i	$-0.444934\pi$
$83$	3.13641	0.344266	0.172133	+	0.985074i	$0.444934\pi$
$84$	0	0
$85$	−4.97922	−0.540072
$86$	0	0
$87$	15.7349	1.68696
$88$	0	0
$89$	−12.6454	−1.34041	−0.670206	−	0.742176i	$-0.733794\pi$
$89$	−12.6454	−1.34041	−0.670206	+	0.742176i	$0.733794\pi$
$90$	0	0
$91$	−2.92504	−0.306628
$92$	0	0
$93$	−28.1015	−2.91399
$94$	0	0
$95$	−8.94091	−0.917318
$96$	0	0
$97$	14.6647	1.48897	0.744485	−	0.667639i	$-0.232695\pi$
$97$	14.6647	1.48897	0.744485	+	0.667639i	$0.232695\pi$
$98$	0	0
$99$	−20.4327	−2.05357
$100$	0	0
$101$	−13.8041	−1.37356	−0.686779	−	0.726867i	$-0.740976\pi$
$101$	−13.8041	−1.37356	−0.686779	+	0.726867i	$0.740976\pi$
$102$	0	0
$103$	9.19189	0.905704	0.452852	−	0.891586i	$-0.350407\pi$
$103$	9.19189	0.905704	0.452852	+	0.891586i	$0.350407\pi$
$104$	0	0
$105$	−9.21917	−0.899699
$106$	0	0
$107$	−12.1898	−1.17843	−0.589217	−	0.807975i	$-0.700564\pi$
$107$	−12.1898	−1.17843	−0.589217	+	0.807975i	$0.700564\pi$
$108$	0	0
$109$	−14.4565	−1.38468	−0.692341	−	0.721570i	$-0.743421\pi$
$109$	−14.4565	−1.38468	−0.692341	+	0.721570i	$0.743421\pi$
$110$	0	0
$111$	−12.8946	−1.22390
$112$	0	0
$113$	−8.33155	−0.783766	−0.391883	−	0.920015i	$-0.628176\pi$
$113$	−8.33155	−0.783766	−0.391883	+	0.920015i	$0.628176\pi$
$114$	0	0
$115$	6.38232	0.595154
$116$	0	0
$117$	5.79862	0.536082
$118$	0	0
$119$	−7.61469	−0.698037
$120$	0	0
$121$	9.63315	0.875741
$122$	0	0
$123$	−17.8200	−1.60677
$124$	0	0
$125$	11.5710	1.03494
$126$	0	0
$127$	13.1126	1.16356	0.581779	−	0.813347i	$-0.302357\pi$
$127$	13.1126	1.16356	0.581779	+	0.813347i	$0.302357\pi$
$128$	0	0
$129$	34.3847	3.02740
$130$	0	0
$131$	9.60813	0.839467	0.419733	−	0.907647i	$-0.362124\pi$
$131$	9.60813	0.839467	0.419733	+	0.907647i	$0.362124\pi$
$132$	0	0
$133$	−13.6733	−1.18562
$134$	0	0
$135$	6.08730	0.523911
$136$	0	0
$137$	10.4046	0.888924	0.444462	−	0.895798i	$-0.353395\pi$
$137$	10.4046	0.888924	0.444462	+	0.895798i	$0.353395\pi$
$138$	0	0
$139$	−6.09136	−0.516663	−0.258331	−	0.966056i	$-0.583173\pi$
$139$	−6.09136	−0.516663	−0.258331	+	0.966056i	$0.583173\pi$
$140$	0	0
$141$	24.5764	2.06971
$142$	0	0
$143$	−5.85550	−0.489661
$144$	0	0
$145$	8.52599	0.708045
$146$	0	0
$147$	5.06922	0.418103
$148$	0	0
$149$	5.59684	0.458511	0.229256	−	0.973366i	$-0.426371\pi$
$149$	5.59684	0.458511	0.229256	+	0.973366i	$0.426371\pi$
$150$	0	0
$151$	−19.6914	−1.60246	−0.801230	−	0.598356i	$-0.795821\pi$
$151$	−19.6914	−1.60246	−0.801230	+	0.598356i	$0.795821\pi$
$152$	0	0
$153$	15.0954	1.22039
$154$	0	0
$155$	−15.2268	−1.22305
$156$	0	0
$157$	−13.2281	−1.05572	−0.527859	−	0.849332i	$-0.677005\pi$
$157$	−13.2281	−1.05572	−0.527859	+	0.849332i	$0.677005\pi$
$158$	0	0
$159$	14.7723	1.17152
$160$	0	0
$161$	9.76044	0.769231
$162$	0	0
$163$	10.0382	0.786249	0.393125	−	0.919485i	$-0.371394\pi$
$163$	10.0382	0.786249	0.393125	+	0.919485i	$0.371394\pi$
$164$	0	0
$165$	−18.4554	−1.43675
$166$	0	0
$167$	17.3143	1.33982	0.669911	−	0.742442i	$-0.266332\pi$
$167$	17.3143	1.33982	0.669911	+	0.742442i	$0.266332\pi$
$168$	0	0
$169$	−11.3383	−0.872174
$170$	0	0
$171$	27.1060	2.07285
$172$	0	0
$173$	−5.01182	−0.381042	−0.190521	−	0.981683i	$-0.561018\pi$
$173$	−5.01182	−0.381042	−0.190521	+	0.981683i	$0.561018\pi$
$174$	0	0
$175$	6.35002	0.480017
$176$	0	0
$177$	33.4843	2.51683
$178$	0	0
$179$	5.32616	0.398096	0.199048	−	0.979990i	$-0.436215\pi$
$179$	5.32616	0.398096	0.199048	+	0.979990i	$0.436215\pi$
$180$	0	0
$181$	10.6765	0.793579	0.396789	−	0.917910i	$-0.370124\pi$
$181$	10.6765	0.793579	0.396789	+	0.917910i	$0.370124\pi$
$182$	0	0
$183$	15.6602	1.15763
$184$	0	0
$185$	−6.98697	−0.513692
$186$	0	0
$187$	−15.2435	−1.11471
$188$	0	0
$189$	9.30928	0.677150
$190$	0	0
$191$	9.95164	0.720075	0.360038	−	0.932938i	$-0.382764\pi$
$191$	9.95164	0.720075	0.360038	+	0.932938i	$0.382764\pi$
$192$	0	0
$193$	−16.8905	−1.21580	−0.607902	−	0.794012i	$-0.707989\pi$
$193$	−16.8905	−1.21580	−0.607902	+	0.794012i	$0.707989\pi$
$194$	0	0
$195$	5.23747	0.375063
$196$	0	0
$197$	−16.4888	−1.17478	−0.587388	−	0.809305i	$-0.699844\pi$
$197$	−16.4888	−1.17478	−0.587388	+	0.809305i	$0.699844\pi$
$198$	0	0
$199$	−2.08774	−0.147996	−0.0739979	−	0.997258i	$-0.523576\pi$
$199$	−2.08774	−0.147996	−0.0739979	+	0.997258i	$0.523576\pi$
$200$	0	0
$201$	−23.6795	−1.67022
$202$	0	0
$203$	13.0388	0.915141
$204$	0	0
$205$	−9.65578	−0.674389
$206$	0	0
$207$	−19.3491	−1.34486
$208$	0	0
$209$	−27.3718	−1.89335
$210$	0	0
$211$	−18.2647	−1.25739	−0.628697	−	0.777651i	$-0.716411\pi$
$211$	−18.2647	−1.25739	−0.628697	+	0.777651i	$0.716411\pi$
$212$	0	0
$213$	37.5972	2.57612
$214$	0	0
$215$	18.6314	1.27065
$216$	0	0
$217$	−23.2863	−1.58078
$218$	0	0
$219$	−10.0804	−0.681168
$220$	0	0
$221$	4.32595	0.290995
$222$	0	0
$223$	1.67601	0.112234	0.0561169	−	0.998424i	$-0.482128\pi$
$223$	1.67601	0.112234	0.0561169	+	0.998424i	$0.482128\pi$
$224$	0	0
$225$	−12.5883	−0.839221
$226$	0	0
$227$	6.46448	0.429063	0.214531	−	0.976717i	$-0.431178\pi$
$227$	6.46448	0.429063	0.214531	+	0.976717i	$0.431178\pi$
$228$	0	0
$229$	−16.3663	−1.08151	−0.540757	−	0.841179i	$-0.681862\pi$
$229$	−16.3663	−1.08151	−0.540757	+	0.841179i	$0.681862\pi$
$230$	0	0
$231$	−28.2237	−1.85698
$232$	0	0
$233$	−18.7072	−1.22555	−0.612774	−	0.790258i	$-0.709946\pi$
$233$	−18.7072	−1.22555	−0.612774	+	0.790258i	$0.709946\pi$
$234$	0	0
$235$	13.3168	0.868690
$236$	0	0
$237$	−10.2060	−0.662948
$238$	0	0
$239$	9.78654	0.633039	0.316519	−	0.948586i	$-0.397486\pi$
$239$	9.78654	0.633039	0.316519	+	0.948586i	$0.397486\pi$
$240$	0	0
$241$	5.40661	0.348270	0.174135	−	0.984722i	$-0.444287\pi$
$241$	5.40661	0.348270	0.174135	+	0.984722i	$0.444287\pi$
$242$	0	0
$243$	18.4978	1.18664
$244$	0	0
$245$	2.74677	0.175485
$246$	0	0
$247$	7.76787	0.494258
$248$	0	0
$249$	−8.58841	−0.544269
$250$	0	0
$251$	11.6955	0.738213	0.369106	−	0.929387i	$-0.379664\pi$
$251$	11.6955	0.738213	0.369106	+	0.929387i	$0.379664\pi$
$252$	0	0
$253$	19.5389	1.22840
$254$	0	0
$255$	13.6346	0.853829
$256$	0	0
$257$	9.81646	0.612334	0.306167	−	0.951978i	$-0.400953\pi$
$257$	9.81646	0.612334	0.306167	+	0.951978i	$0.400953\pi$
$258$	0	0
$259$	−10.6851	−0.663942
$260$	0	0
$261$	−25.8481	−1.59996
$262$	0	0
$263$	−18.5073	−1.14121	−0.570604	−	0.821225i	$-0.693291\pi$
$263$	−18.5073	−1.14121	−0.570604	+	0.821225i	$0.693291\pi$
$264$	0	0
$265$	8.00442	0.491707
$266$	0	0
$267$	34.6269	2.11913
$268$	0	0
$269$	−32.5020	−1.98168	−0.990840	−	0.135043i	$-0.956883\pi$
$269$	−32.5020	−1.98168	−0.990840	+	0.135043i	$0.956883\pi$
$270$	0	0
$271$	−14.5799	−0.885668	−0.442834	−	0.896604i	$-0.646027\pi$
$271$	−14.5799	−0.885668	−0.442834	+	0.896604i	$0.646027\pi$
$272$	0	0
$273$	8.00963	0.484765
$274$	0	0
$275$	12.7118	0.766550
$276$	0	0
$277$	7.33979	0.441005	0.220503	−	0.975386i	$-0.429230\pi$
$277$	7.33979	0.441005	0.220503	+	0.975386i	$0.429230\pi$
$278$	0	0
$279$	46.1629	2.76370
$280$	0	0
$281$	21.5248	1.28406	0.642030	−	0.766680i	$-0.278093\pi$
$281$	21.5248	1.28406	0.642030	+	0.766680i	$0.278093\pi$
$282$	0	0
$283$	−2.57829	−0.153264	−0.0766318	−	0.997059i	$-0.524417\pi$
$283$	−2.57829	−0.153264	−0.0766318	+	0.997059i	$0.524417\pi$
$284$	0	0
$285$	24.4828	1.45024
$286$	0	0
$287$	−14.7665	−0.871641
$288$	0	0
$289$	−5.73837	−0.337551
$290$	0	0
$291$	−40.1561	−2.35399
$292$	0	0
$293$	9.15409	0.534787	0.267394	−	0.963587i	$-0.413838\pi$
$293$	9.15409	0.534787	0.267394	+	0.963587i	$0.413838\pi$
$294$	0	0
$295$	18.1435	1.05636
$296$	0	0
$297$	18.6358	1.08136
$298$	0	0
$299$	−5.54497	−0.320674
$300$	0	0
$301$	28.4929	1.64230
$302$	0	0
$303$	37.7996	2.17153
$304$	0	0
$305$	8.48548	0.485877
$306$	0	0
$307$	2.44659	0.139634	0.0698171	−	0.997560i	$-0.477758\pi$
$307$	2.44659	0.139634	0.0698171	+	0.997560i	$0.477758\pi$
$308$	0	0
$309$	−25.1701	−1.43188
$310$	0	0
$311$	33.6140	1.90608	0.953038	−	0.302850i	$-0.0979382\pi$
$311$	33.6140	1.90608	0.953038	+	0.302850i	$0.0979382\pi$
$312$	0	0
$313$	14.8099	0.837103	0.418552	−	0.908193i	$-0.362538\pi$
$313$	14.8099	0.837103	0.418552	+	0.908193i	$0.362538\pi$
$314$	0	0
$315$	15.1445	0.853297
$316$	0	0
$317$	−3.52762	−0.198131	−0.0990653	−	0.995081i	$-0.531585\pi$
$317$	−3.52762	−0.198131	−0.0990653	+	0.995081i	$0.531585\pi$
$318$	0	0
$319$	26.1016	1.46141
$320$	0	0
$321$	33.3793	1.86305
$322$	0	0
$323$	20.2219	1.12518
$324$	0	0
$325$	−3.60749	−0.200107
$326$	0	0
$327$	39.5862	2.18912
$328$	0	0
$329$	20.3653	1.12277
$330$	0	0
$331$	27.4454	1.50854	0.754268	−	0.656566i	$-0.227992\pi$
$331$	27.4454	1.50854	0.754268	+	0.656566i	$0.227992\pi$
$332$	0	0
$333$	21.1823	1.16078
$334$	0	0
$335$	−12.8308	−0.701021
$336$	0	0
$337$	24.6271	1.34152	0.670761	−	0.741673i	$-0.265967\pi$
$337$	24.6271	1.34152	0.670761	+	0.741673i	$0.265967\pi$
$338$	0	0
$339$	22.8142	1.23910
$340$	0	0
$341$	−46.6157	−2.52438
$342$	0	0
$343$	20.0842	1.08445
$344$	0	0
$345$	−17.4767	−0.940912
$346$	0	0
$347$	−32.6383	−1.75212	−0.876058	−	0.482205i	$-0.839836\pi$
$347$	−32.6383	−1.75212	−0.876058	+	0.482205i	$0.839836\pi$
$348$	0	0
$349$	14.9088	0.798048	0.399024	−	0.916940i	$-0.369349\pi$
$349$	14.9088	0.798048	0.399024	+	0.916940i	$0.369349\pi$
$350$	0	0
$351$	−5.28866	−0.282288
$352$	0	0
$353$	29.3756	1.56351	0.781753	−	0.623588i	$-0.214326\pi$
$353$	29.3756	1.56351	0.781753	+	0.623588i	$0.214326\pi$
$354$	0	0
$355$	20.3721	1.08124
$356$	0	0
$357$	20.8513	1.10357
$358$	0	0
$359$	−20.9128	−1.10374	−0.551868	−	0.833932i	$-0.686085\pi$
$359$	−20.9128	−1.10374	−0.551868	+	0.833932i	$0.686085\pi$
$360$	0	0
$361$	17.3114	0.911124
$362$	0	0
$363$	−26.3784	−1.38451
$364$	0	0
$365$	−5.46207	−0.285898
$366$	0	0
$367$	14.1787	0.740124	0.370062	−	0.929007i	$-0.379336\pi$
$367$	14.1787	0.740124	0.370062	+	0.929007i	$0.379336\pi$
$368$	0	0
$369$	29.2732	1.52390
$370$	0	0
$371$	12.2411	0.635527
$372$	0	0
$373$	6.24233	0.323215	0.161608	−	0.986855i	$-0.448332\pi$
$373$	6.24233	0.323215	0.161608	+	0.986855i	$0.448332\pi$
$374$	0	0
$375$	−31.6848	−1.63620
$376$	0	0
$377$	−7.40739	−0.381500
$378$	0	0
$379$	−0.864107	−0.0443862	−0.0221931	−	0.999754i	$-0.507065\pi$
$379$	−0.864107	−0.0443862	−0.0221931	+	0.999754i	$0.507065\pi$
$380$	0	0
$381$	−35.9062	−1.83953
$382$	0	0
$383$	−7.99182	−0.408363	−0.204181	−	0.978933i	$-0.565453\pi$
$383$	−7.99182	−0.408363	−0.204181	+	0.978933i	$0.565453\pi$
$384$	0	0
$385$	−15.2931	−0.779407
$386$	0	0
$387$	−56.4844	−2.87126
$388$	0	0
$389$	−22.0666	−1.11882	−0.559410	−	0.828891i	$-0.688972\pi$
$389$	−22.0666	−1.11882	−0.559410	+	0.828891i	$0.688972\pi$
$390$	0	0
$391$	−14.4351	−0.730013
$392$	0	0
$393$	−26.3099	−1.32716
$394$	0	0
$395$	−5.53011	−0.278250
$396$	0	0
$397$	−16.9314	−0.849765	−0.424882	−	0.905249i	$-0.639685\pi$
$397$	−16.9314	−0.849765	−0.424882	+	0.905249i	$0.639685\pi$
$398$	0	0
$399$	37.4415	1.87442
$400$	0	0
$401$	9.21200	0.460026	0.230013	−	0.973188i	$-0.426123\pi$
$401$	9.21200	0.460026	0.230013	+	0.973188i	$0.426123\pi$
$402$	0	0
$403$	13.2291	0.658988
$404$	0	0
$405$	3.35400	0.166662
$406$	0	0
$407$	−21.3900	−1.06026
$408$	0	0
$409$	−2.42103	−0.119712	−0.0598561	−	0.998207i	$-0.519064\pi$
$409$	−2.42103	−0.119712	−0.0598561	+	0.998207i	$0.519064\pi$
$410$	0	0
$411$	−28.4908	−1.40535
$412$	0	0
$413$	27.7468	1.36533
$414$	0	0
$415$	−4.65364	−0.228438
$416$	0	0
$417$	16.6799	0.816820
$418$	0	0
$419$	17.8392	0.871500	0.435750	−	0.900068i	$-0.356483\pi$
$419$	17.8392	0.871500	0.435750	+	0.900068i	$0.356483\pi$
$420$	0	0
$421$	8.45330	0.411989	0.205994	−	0.978553i	$-0.433957\pi$
$421$	8.45330	0.411989	0.205994	+	0.978553i	$0.433957\pi$
$422$	0	0
$423$	−40.3722	−1.96296
$424$	0	0
$425$	−9.39127	−0.455544
$426$	0	0
$427$	12.9768	0.627991
$428$	0	0
$429$	16.0341	0.774132
$430$	0	0
$431$	1.94384	0.0936314	0.0468157	−	0.998904i	$-0.485093\pi$
$431$	1.94384	0.0936314	0.0468157	+	0.998904i	$0.485093\pi$
$432$	0	0
$433$	−5.76438	−0.277018	−0.138509	−	0.990361i	$-0.544231\pi$
$433$	−5.76438	−0.277018	−0.138509	+	0.990361i	$0.544231\pi$
$434$	0	0
$435$	−23.3467	−1.11939
$436$	0	0
$437$	−25.9203	−1.23993
$438$	0	0
$439$	6.55943	0.313064	0.156532	−	0.987673i	$-0.449968\pi$
$439$	6.55943	0.313064	0.156532	+	0.987673i	$0.449968\pi$
$440$	0	0
$441$	−8.32732	−0.396539
$442$	0	0
$443$	−4.19833	−0.199469	−0.0997344	−	0.995014i	$-0.531799\pi$
$443$	−4.19833	−0.199469	−0.0997344	+	0.995014i	$0.531799\pi$
$444$	0	0
$445$	18.7626	0.889433
$446$	0	0
$447$	−15.3258	−0.724885
$448$	0	0
$449$	37.3731	1.76374	0.881872	−	0.471489i	$-0.156283\pi$
$449$	37.3731	1.76374	0.881872	+	0.471489i	$0.156283\pi$
$450$	0	0
$451$	−29.5604	−1.39194
$452$	0	0
$453$	53.9207	2.53342
$454$	0	0
$455$	4.34003	0.203464
$456$	0	0
$457$	−11.8271	−0.553246	−0.276623	−	0.960978i	$-0.589215\pi$
$457$	−11.8271	−0.553246	−0.276623	+	0.960978i	$0.589215\pi$
$458$	0	0
$459$	−13.7678	−0.642627
$460$	0	0
$461$	23.1984	1.08046	0.540230	−	0.841518i	$-0.318337\pi$
$461$	23.1984	1.08046	0.540230	+	0.841518i	$0.318337\pi$
$462$	0	0
$463$	20.0392	0.931299	0.465650	−	0.884969i	$-0.345821\pi$
$463$	20.0392	0.931299	0.465650	+	0.884969i	$0.345821\pi$
$464$	0	0
$465$	41.6955	1.93358
$466$	0	0
$467$	−7.81377	−0.361578	−0.180789	−	0.983522i	$-0.557865\pi$
$467$	−7.81377	−0.361578	−0.180789	+	0.983522i	$0.557865\pi$
$468$	0	0
$469$	−19.6221	−0.906062
$470$	0	0
$471$	36.2224	1.66904
$472$	0	0
$473$	57.0384	2.62263
$474$	0	0
$475$	−16.8634	−0.773746
$476$	0	0
$477$	−24.2668	−1.11110
$478$	0	0
$479$	5.44168	0.248637	0.124318	−	0.992242i	$-0.460326\pi$
$479$	5.44168	0.248637	0.124318	+	0.992242i	$0.460326\pi$
$480$	0	0
$481$	6.07029	0.276781
$482$	0	0
$483$	−26.7270	−1.21612
$484$	0	0
$485$	−21.7587	−0.988010
$486$	0	0
$487$	−0.0629078	−0.00285062	−0.00142531	−	0.999999i	$-0.500454\pi$
$487$	−0.0629078	−0.00285062	−0.00142531	+	0.999999i	$0.500454\pi$
$488$	0	0
$489$	−27.4874	−1.24302
$490$	0	0
$491$	−16.2044	−0.731296	−0.365648	−	0.930753i	$-0.619153\pi$
$491$	−16.2044	−0.731296	−0.365648	+	0.930753i	$0.619153\pi$
$492$	0	0
$493$	−19.2835	−0.868484
$494$	0	0
$495$	30.3170	1.36265
$496$	0	0
$497$	31.1550	1.39749
$498$	0	0
$499$	9.19623	0.411680	0.205840	−	0.978586i	$-0.434007\pi$
$499$	9.19623	0.411680	0.205840	+	0.978586i	$0.434007\pi$
$500$	0	0
$501$	−47.4117	−2.11820
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	20.4818	0.911427
$506$	0	0
$507$	31.0475	1.37887
$508$	0	0
$509$	−23.7710	−1.05363	−0.526815	−	0.849980i	$-0.676614\pi$
$509$	−23.7710	−1.05363	−0.526815	+	0.849980i	$0.676614\pi$
$510$	0	0
$511$	−8.35311	−0.369520
$512$	0	0
$513$	−24.7221	−1.09151
$514$	0	0
$515$	−13.6385	−0.600982
$516$	0	0
$517$	40.7682	1.79298
$518$	0	0
$519$	13.7238	0.602410
$520$	0	0
$521$	13.9106	0.609435	0.304717	−	0.952443i	$-0.401438\pi$
$521$	13.9106	0.609435	0.304717	+	0.952443i	$0.401438\pi$
$522$	0	0
$523$	28.8699	1.26239	0.631196	−	0.775623i	$-0.282564\pi$
$523$	28.8699	1.26239	0.631196	+	0.775623i	$0.282564\pi$
$524$	0	0
$525$	−17.3882	−0.758885
$526$	0	0
$527$	34.4390	1.50018
$528$	0	0
$529$	−4.49725	−0.195533
$530$	0	0
$531$	−55.0054	−2.38703
$532$	0	0
$533$	8.38896	0.363366
$534$	0	0
$535$	18.0866	0.781953
$536$	0	0
$537$	−14.5846	−0.629372
$538$	0	0
$539$	8.40900	0.362201
$540$	0	0
$541$	11.3229	0.486807	0.243404	−	0.969925i	$-0.421736\pi$
$541$	11.3229	0.486807	0.243404	+	0.969925i	$0.421736\pi$
$542$	0	0
$543$	−29.2354	−1.25461
$544$	0	0
$545$	21.4498	0.918810
$546$	0	0
$547$	−21.2578	−0.908917	−0.454459	−	0.890768i	$-0.650167\pi$
$547$	−21.2578	−0.908917	−0.454459	+	0.890768i	$0.650167\pi$
$548$	0	0
$549$	−25.7253	−1.09793
$550$	0	0
$551$	−34.6263	−1.47513
$552$	0	0
$553$	−8.45717	−0.359636
$554$	0	0
$555$	19.1324	0.812124
$556$	0	0
$557$	−4.20090	−0.177998	−0.0889990	−	0.996032i	$-0.528367\pi$
$557$	−4.20090	−0.177998	−0.0889990	+	0.996032i	$0.528367\pi$
$558$	0	0
$559$	−16.1870	−0.684636
$560$	0	0
$561$	41.7411	1.76231
$562$	0	0
$563$	14.1852	0.597836	0.298918	−	0.954279i	$-0.403374\pi$
$563$	14.1852	0.597836	0.298918	+	0.954279i	$0.403374\pi$
$564$	0	0
$565$	12.3619	0.520070
$566$	0	0
$567$	5.12926	0.215409
$568$	0	0
$569$	−41.7571	−1.75055	−0.875274	−	0.483627i	$-0.839319\pi$
$569$	−41.7571	−1.75055	−0.875274	+	0.483627i	$0.839319\pi$
$570$	0	0
$571$	−15.4943	−0.648417	−0.324209	−	0.945986i	$-0.605098\pi$
$571$	−15.4943	−0.648417	−0.324209	+	0.945986i	$0.605098\pi$
$572$	0	0
$573$	−27.2505	−1.13841
$574$	0	0
$575$	12.0377	0.502005
$576$	0	0
$577$	−34.1385	−1.42120	−0.710601	−	0.703595i	$-0.751577\pi$
$577$	−34.1385	−1.42120	−0.710601	+	0.703595i	$0.751577\pi$
$578$	0	0
$579$	46.2511	1.92213
$580$	0	0
$581$	−7.11680	−0.295254
$582$	0	0
$583$	24.5049	1.01489
$584$	0	0
$585$	−8.60369	−0.355719
$586$	0	0
$587$	10.6074	0.437812	0.218906	−	0.975746i	$-0.429751\pi$
$587$	10.6074	0.437812	0.218906	+	0.975746i	$0.429751\pi$
$588$	0	0
$589$	61.8402	2.54808
$590$	0	0
$591$	45.1511	1.85727
$592$	0	0
$593$	6.82095	0.280103	0.140051	−	0.990144i	$-0.455273\pi$
$593$	6.82095	0.280103	0.140051	+	0.990144i	$0.455273\pi$
$594$	0	0
$595$	11.2983	0.463184
$596$	0	0
$597$	5.71684	0.233975
$598$	0	0
$599$	−28.9243	−1.18182	−0.590908	−	0.806739i	$-0.701231\pi$
$599$	−28.9243	−1.18182	−0.590908	+	0.806739i	$0.701231\pi$
$600$	0	0
$601$	8.18366	0.333818	0.166909	−	0.985972i	$-0.446621\pi$
$601$	8.18366	0.333818	0.166909	+	0.985972i	$0.446621\pi$
$602$	0	0
$603$	38.8988	1.58408
$604$	0	0
$605$	−14.2932	−0.581100
$606$	0	0
$607$	−27.0315	−1.09717	−0.548587	−	0.836093i	$-0.684834\pi$
$607$	−27.0315	−1.09717	−0.548587	+	0.836093i	$0.684834\pi$
$608$	0	0
$609$	−35.7039	−1.44680
$610$	0	0
$611$	−11.5696	−0.468057
$612$	0	0
$613$	28.7511	1.16124	0.580622	−	0.814173i	$-0.302809\pi$
$613$	28.7511	1.16124	0.580622	+	0.814173i	$0.302809\pi$
$614$	0	0
$615$	26.4404	1.06618
$616$	0	0
$617$	44.0195	1.77216	0.886079	−	0.463534i	$-0.153419\pi$
$617$	44.0195	1.77216	0.886079	+	0.463534i	$0.153419\pi$
$618$	0	0
$619$	15.2639	0.613508	0.306754	−	0.951789i	$-0.400757\pi$
$619$	15.2639	0.613508	0.306754	+	0.951789i	$0.400757\pi$
$620$	0	0
$621$	17.6475	0.708169
$622$	0	0
$623$	28.6936	1.14958
$624$	0	0
$625$	−3.17600	−0.127040
$626$	0	0
$627$	74.9521	2.99330
$628$	0	0
$629$	15.8026	0.630092
$630$	0	0
$631$	−36.0998	−1.43711	−0.718555	−	0.695470i	$-0.755196\pi$
$631$	−36.0998	−1.43711	−0.718555	+	0.695470i	$0.755196\pi$
$632$	0	0
$633$	50.0141	1.98788
$634$	0	0
$635$	−19.4558	−0.772081
$636$	0	0
$637$	−2.38640	−0.0945524
$638$	0	0
$639$	−61.7617	−2.44326
$640$	0	0
$641$	6.46253	0.255255	0.127627	−	0.991822i	$-0.459264\pi$
$641$	6.46253	0.255255	0.127627	+	0.991822i	$0.459264\pi$
$642$	0	0
$643$	−27.1699	−1.07148	−0.535738	−	0.844384i	$-0.679967\pi$
$643$	−27.1699	−1.07148	−0.535738	+	0.844384i	$0.679967\pi$
$644$	0	0
$645$	−51.0182	−2.00884
$646$	0	0
$647$	29.5369	1.16121	0.580607	−	0.814184i	$-0.302815\pi$
$647$	29.5369	1.16121	0.580607	+	0.814184i	$0.302815\pi$
$648$	0	0
$649$	55.5449	2.18033
$650$	0	0
$651$	63.7648	2.49914
$652$	0	0
$653$	44.1850	1.72909	0.864547	−	0.502552i	$-0.167605\pi$
$653$	44.1850	1.72909	0.864547	+	0.502552i	$0.167605\pi$
$654$	0	0
$655$	−14.2561	−0.557030
$656$	0	0
$657$	16.5592	0.646037
$658$	0	0
$659$	−35.6833	−1.39002	−0.695012	−	0.718999i	$-0.744601\pi$
$659$	−35.6833	−1.39002	−0.695012	+	0.718999i	$0.744601\pi$
$660$	0	0
$661$	−16.2086	−0.630441	−0.315221	−	0.949018i	$-0.602078\pi$
$661$	−16.2086	−0.630441	−0.315221	+	0.949018i	$0.602078\pi$
$662$	0	0
$663$	−11.8457	−0.460050
$664$	0	0
$665$	20.2877	0.786724
$666$	0	0
$667$	24.7174	0.957061
$668$	0	0
$669$	−4.58940	−0.177436
$670$	0	0
$671$	25.9776	1.00285
$672$	0	0
$673$	−24.6223	−0.949119	−0.474560	−	0.880223i	$-0.657393\pi$
$673$	−24.6223	−0.949119	−0.474560	+	0.880223i	$0.657393\pi$
$674$	0	0
$675$	11.4812	0.441913
$676$	0	0
$677$	2.20008	0.0845560	0.0422780	−	0.999106i	$-0.486538\pi$
$677$	2.20008	0.0845560	0.0422780	+	0.999106i	$0.486538\pi$
$678$	0	0
$679$	−33.2754	−1.27699
$680$	0	0
$681$	−17.7016	−0.678328
$682$	0	0
$683$	17.4127	0.666280	0.333140	−	0.942877i	$-0.391892\pi$
$683$	17.4127	0.666280	0.333140	+	0.942877i	$0.391892\pi$
$684$	0	0
$685$	−15.4378	−0.589848
$686$	0	0
$687$	44.8157	1.70982
$688$	0	0
$689$	−6.95425	−0.264936
$690$	0	0
$691$	20.1005	0.764661	0.382330	−	0.924026i	$-0.375122\pi$
$691$	20.1005	0.764661	0.382330	+	0.924026i	$0.375122\pi$
$692$	0	0
$693$	46.3637	1.76121
$694$	0	0
$695$	9.03805	0.342833
$696$	0	0
$697$	21.8388	0.827202
$698$	0	0
$699$	51.2257	1.93753
$700$	0	0
$701$	9.92749	0.374956	0.187478	−	0.982269i	$-0.439969\pi$
$701$	9.92749	0.374956	0.187478	+	0.982269i	$0.439969\pi$
$702$	0	0
$703$	28.3759	1.07022
$704$	0	0
$705$	−36.4652	−1.37336
$706$	0	0
$707$	31.3227	1.17801
$708$	0	0
$709$	35.7960	1.34435	0.672174	−	0.740393i	$-0.265361\pi$
$709$	35.7960	1.34435	0.672174	+	0.740393i	$0.265361\pi$
$710$	0	0
$711$	16.7655	0.628756
$712$	0	0
$713$	−44.1436	−1.65319
$714$	0	0
$715$	8.68808	0.324916
$716$	0	0
$717$	−26.7984	−1.00081
$718$	0	0
$719$	−25.1524	−0.938027	−0.469014	−	0.883191i	$-0.655390\pi$
$719$	−25.1524	−0.938027	−0.469014	+	0.883191i	$0.655390\pi$
$720$	0	0
$721$	−20.8572	−0.776763
$722$	0	0
$723$	−14.8049	−0.550600
$724$	0	0
$725$	16.0808	0.597227
$726$	0	0
$727$	26.6890	0.989839	0.494919	−	0.868939i	$-0.335198\pi$
$727$	26.6890	0.989839	0.494919	+	0.868939i	$0.335198\pi$
$728$	0	0
$729$	−43.8710	−1.62485
$730$	0	0
$731$	−42.1391	−1.55857
$732$	0	0
$733$	50.0102	1.84717	0.923584	−	0.383397i	$-0.125246\pi$
$733$	50.0102	1.84717	0.923584	+	0.383397i	$0.125246\pi$
$734$	0	0
$735$	−7.52145	−0.277433
$736$	0	0
$737$	−39.2804	−1.44691
$738$	0	0
$739$	−28.7422	−1.05730	−0.528650	−	0.848840i	$-0.677302\pi$
$739$	−28.7422	−1.05730	−0.528650	+	0.848840i	$0.677302\pi$
$740$	0	0
$741$	−21.2707	−0.781399
$742$	0	0
$743$	42.3309	1.55297	0.776485	−	0.630136i	$-0.217001\pi$
$743$	42.3309	1.55297	0.776485	+	0.630136i	$0.217001\pi$
$744$	0	0
$745$	−8.30431	−0.304246
$746$	0	0
$747$	14.1084	0.516198
$748$	0	0
$749$	27.6598	1.01067
$750$	0	0
$751$	40.1796	1.46617	0.733086	−	0.680135i	$-0.238079\pi$
$751$	40.1796	1.46617	0.733086	+	0.680135i	$0.238079\pi$
$752$	0	0
$753$	−32.0257	−1.16708
$754$	0	0
$755$	29.2170	1.06332
$756$	0	0
$757$	−36.2258	−1.31665	−0.658325	−	0.752734i	$-0.728735\pi$
$757$	−36.2258	−1.31665	−0.658325	+	0.752734i	$0.728735\pi$
$758$	0	0
$759$	−53.5033	−1.94205
$760$	0	0
$761$	24.2959	0.880726	0.440363	−	0.897820i	$-0.354850\pi$
$761$	24.2959	0.880726	0.440363	+	0.897820i	$0.354850\pi$
$762$	0	0
$763$	32.8031	1.18755
$764$	0	0
$765$	−22.3978	−0.809793
$766$	0	0
$767$	−15.7631	−0.569173
$768$	0	0
$769$	−28.8393	−1.03997	−0.519985	−	0.854175i	$-0.674062\pi$
$769$	−28.8393	−1.03997	−0.519985	+	0.854175i	$0.674062\pi$
$770$	0	0
$771$	−26.8804	−0.968072
$772$	0	0
$773$	10.7214	0.385621	0.192810	−	0.981236i	$-0.438240\pi$
$773$	10.7214	0.385621	0.192810	+	0.981236i	$0.438240\pi$
$774$	0	0
$775$	−28.7193	−1.03163
$776$	0	0
$777$	29.2590	1.04966
$778$	0	0
$779$	39.2147	1.40501
$780$	0	0
$781$	62.3675	2.23169
$782$	0	0
$783$	23.5748	0.842496
$784$	0	0
$785$	19.6272	0.700524
$786$	0	0
$787$	39.1775	1.39653	0.698264	−	0.715841i	$-0.253956\pi$
$787$	39.1775	1.39653	0.698264	+	0.715841i	$0.253956\pi$
$788$	0	0
$789$	50.6784	1.80420
$790$	0	0
$791$	18.9050	0.672185
$792$	0	0
$793$	−7.37220	−0.261794
$794$	0	0
$795$	−21.9184	−0.777367
$796$	0	0
$797$	−12.9907	−0.460153	−0.230077	−	0.973172i	$-0.573898\pi$
$797$	−12.9907	−0.460153	−0.230077	+	0.973172i	$0.573898\pi$
$798$	0	0
$799$	−30.1189	−1.06553
$800$	0	0
$801$	−56.8822	−2.00984
$802$	0	0
$803$	−16.7217	−0.590095
$804$	0	0
$805$	−14.4820	−0.510425
$806$	0	0
$807$	88.9999	3.13294
$808$	0	0
$809$	45.8834	1.61317	0.806587	−	0.591115i	$-0.201312\pi$
$809$	45.8834	1.61317	0.806587	+	0.591115i	$0.201312\pi$
$810$	0	0
$811$	−8.78811	−0.308592	−0.154296	−	0.988025i	$-0.549311\pi$
$811$	−8.78811	−0.308592	−0.154296	+	0.988025i	$0.549311\pi$
$812$	0	0
$813$	39.9241	1.40020
$814$	0	0
$815$	−14.8941	−0.521718
$816$	0	0
$817$	−75.6669	−2.64725
$818$	0	0
$819$	−13.1576	−0.459763
$820$	0	0
$821$	29.7771	1.03923	0.519613	−	0.854402i	$-0.326076\pi$
$821$	29.7771	1.03923	0.519613	+	0.854402i	$0.326076\pi$
$822$	0	0
$823$	−6.40161	−0.223146	−0.111573	−	0.993756i	$-0.535589\pi$
$823$	−6.40161	−0.223146	−0.111573	+	0.993756i	$0.535589\pi$
$824$	0	0
$825$	−34.8086	−1.21188
$826$	0	0
$827$	−15.2358	−0.529799	−0.264900	−	0.964276i	$-0.585339\pi$
$827$	−15.2358	−0.529799	−0.264900	+	0.964276i	$0.585339\pi$
$828$	0	0
$829$	−0.979554	−0.0340213	−0.0170107	−	0.999855i	$-0.505415\pi$
$829$	−0.979554	−0.0340213	−0.0170107	+	0.999855i	$0.505415\pi$
$830$	0	0
$831$	−20.0985	−0.697209
$832$	0	0
$833$	−6.21244	−0.215248
$834$	0	0
$835$	−25.6901	−0.889042
$836$	0	0
$837$	−42.1031	−1.45530
$838$	0	0
$839$	16.0218	0.553133	0.276566	−	0.960995i	$-0.410803\pi$
$839$	16.0218	0.553133	0.276566	+	0.960995i	$0.410803\pi$
$840$	0	0
$841$	4.01938	0.138599
$842$	0	0
$843$	−58.9411	−2.03004
$844$	0	0
$845$	16.8231	0.578733
$846$	0	0
$847$	−21.8585	−0.751066
$848$	0	0
$849$	7.06012	0.242303
$850$	0	0
$851$	−20.2557	−0.694355
$852$	0	0
$853$	12.0533	0.412697	0.206349	−	0.978479i	$-0.433842\pi$
$853$	12.0533	0.412697	0.206349	+	0.978479i	$0.433842\pi$
$854$	0	0
$855$	−40.2184	−1.37544
$856$	0	0
$857$	16.2478	0.555014	0.277507	−	0.960724i	$-0.410492\pi$
$857$	16.2478	0.555014	0.277507	+	0.960724i	$0.410492\pi$
$858$	0	0
$859$	−48.6855	−1.66113	−0.830564	−	0.556923i	$-0.811982\pi$
$859$	−48.6855	−1.66113	−0.830564	+	0.556923i	$0.811982\pi$
$860$	0	0
$861$	40.4351	1.37803
$862$	0	0
$863$	28.0309	0.954183	0.477091	−	0.878854i	$-0.341691\pi$
$863$	28.0309	0.954183	0.477091	+	0.878854i	$0.341691\pi$
$864$	0	0
$865$	7.43628	0.252841
$866$	0	0
$867$	15.7133	0.533653
$868$	0	0
$869$	−16.9300	−0.574310
$870$	0	0
$871$	11.1474	0.377715
$872$	0	0
$873$	65.9653	2.23259
$874$	0	0
$875$	−26.2556	−0.887602
$876$	0	0
$877$	−4.43157	−0.149643	−0.0748217	−	0.997197i	$-0.523839\pi$
$877$	−4.43157	−0.149643	−0.0748217	+	0.997197i	$0.523839\pi$
$878$	0	0
$879$	−25.0666	−0.845474
$880$	0	0
$881$	12.1050	0.407829	0.203914	−	0.978989i	$-0.434634\pi$
$881$	12.1050	0.407829	0.203914	+	0.978989i	$0.434634\pi$
$882$	0	0
$883$	29.6896	0.999136	0.499568	−	0.866275i	$-0.333492\pi$
$883$	29.6896	0.999136	0.499568	+	0.866275i	$0.333492\pi$
$884$	0	0
$885$	−49.6823	−1.67005
$886$	0	0
$887$	41.2081	1.38363	0.691816	−	0.722074i	$-0.256811\pi$
$887$	41.2081	1.38363	0.691816	+	0.722074i	$0.256811\pi$
$888$	0	0
$889$	−29.7537	−0.997907
$890$	0	0
$891$	10.2680	0.343991
$892$	0	0
$893$	−54.0828	−1.80981
$894$	0	0
$895$	−7.90269	−0.264158
$896$	0	0
$897$	15.1837	0.506971
$898$	0	0
$899$	−58.9704	−1.96677
$900$	0	0
$901$	−18.1038	−0.603125
$902$	0	0
$903$	−78.0218	−2.59640
$904$	0	0
$905$	−15.8413	−0.526581
$906$	0	0
$907$	56.7444	1.88417	0.942083	−	0.335380i	$-0.108865\pi$
$907$	56.7444	1.88417	0.942083	+	0.335380i	$0.108865\pi$
$908$	0	0
$909$	−62.0942	−2.05953
$910$	0	0
$911$	−51.5438	−1.70772	−0.853861	−	0.520501i	$-0.825745\pi$
$911$	−51.5438	−1.70772	−0.853861	+	0.520501i	$0.825745\pi$
$912$	0	0
$913$	−14.2467	−0.471499
$914$	0	0
$915$	−23.2357	−0.768150
$916$	0	0
$917$	−21.8017	−0.719956
$918$	0	0
$919$	−9.67727	−0.319224	−0.159612	−	0.987180i	$-0.551024\pi$
$919$	−9.67727	−0.319224	−0.159612	+	0.987180i	$0.551024\pi$
$920$	0	0
$921$	−6.69947	−0.220755
$922$	0	0
$923$	−17.6993	−0.582580
$924$	0	0
$925$	−13.1781	−0.433293
$926$	0	0
$927$	41.3474	1.35803
$928$	0	0
$929$	−28.3825	−0.931201	−0.465601	−	0.884995i	$-0.654162\pi$
$929$	−28.3825	−0.931201	−0.465601	+	0.884995i	$0.654162\pi$
$930$	0	0
$931$	−11.1553	−0.365602
$932$	0	0
$933$	−92.0450	−3.01342
$934$	0	0
$935$	22.6175	0.739670
$936$	0	0
$937$	36.0495	1.17769	0.588844	−	0.808247i	$-0.299583\pi$
$937$	36.0495	1.17769	0.588844	+	0.808247i	$0.299583\pi$
$938$	0	0
$939$	−40.5538	−1.32342
$940$	0	0
$941$	−5.76404	−0.187902	−0.0939512	−	0.995577i	$-0.529950\pi$
$941$	−5.76404	−0.187902	−0.0939512	+	0.995577i	$0.529950\pi$
$942$	0	0
$943$	−27.9927	−0.911569
$944$	0	0
$945$	−13.8126	−0.449325
$946$	0	0
$947$	1.33942	0.0435252	0.0217626	−	0.999763i	$-0.493072\pi$
$947$	1.33942	0.0435252	0.0217626	+	0.999763i	$0.493072\pi$
$948$	0	0
$949$	4.74545	0.154044
$950$	0	0
$951$	9.65965	0.313236
$952$	0	0
$953$	−21.1859	−0.686280	−0.343140	−	0.939284i	$-0.611491\pi$
$953$	−21.1859	−0.686280	−0.343140	+	0.939284i	$0.611491\pi$
$954$	0	0
$955$	−14.7657	−0.477808
$956$	0	0
$957$	−71.4739	−2.31042
$958$	0	0
$959$	−23.6089	−0.762372
$960$	0	0
$961$	74.3171	2.39732
$962$	0	0
$963$	−54.8328	−1.76696
$964$	0	0
$965$	25.0612	0.806749
$966$	0	0
$967$	−45.9570	−1.47788	−0.738938	−	0.673773i	$-0.764672\pi$
$967$	−45.9570	−1.47788	−0.738938	+	0.673773i	$0.764672\pi$
$968$	0	0
$969$	−55.3735	−1.77885
$970$	0	0
$971$	11.7697	0.377707	0.188853	−	0.982005i	$-0.439523\pi$
$971$	11.7697	0.377707	0.188853	+	0.982005i	$0.439523\pi$
$972$	0	0
$973$	13.8218	0.443108
$974$	0	0
$975$	9.87836	0.316361
$976$	0	0
$977$	33.5288	1.07268	0.536341	−	0.844001i	$-0.319806\pi$
$977$	33.5288	1.07268	0.536341	+	0.844001i	$0.319806\pi$
$978$	0	0
$979$	57.4402	1.83580
$980$	0	0
$981$	−65.0290	−2.07622
$982$	0	0
$983$	−29.8579	−0.952320	−0.476160	−	0.879359i	$-0.657972\pi$
$983$	−29.8579	−0.952320	−0.476160	+	0.879359i	$0.657972\pi$
$984$	0	0
$985$	24.4652	0.779526
$986$	0	0
$987$	−55.7660	−1.77505
$988$	0	0
$989$	54.0136	1.71753
$990$	0	0
$991$	37.9384	1.20515	0.602577	−	0.798061i	$-0.294141\pi$
$991$	37.9384	1.20515	0.602577	+	0.798061i	$0.294141\pi$
$992$	0	0
$993$	−75.1536	−2.38493
$994$	0	0
$995$	3.09768	0.0982030
$996$	0	0
$997$	34.1960	1.08300	0.541499	−	0.840701i	$-0.317857\pi$
$997$	34.1960	1.08300	0.541499	+	0.840701i	$0.317857\pi$
$998$	0	0
$999$	−19.3194	−0.611238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.r.1.2		12
4.3	odd	2		1006.2.a.i.1.11	✓	12
12.11	even	2		9054.2.a.bj.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.i.1.11	✓	12	4.3	odd	2
8048.2.a.r.1.2		12	1.1	even	1	trivial
9054.2.a.bj.1.9		12	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73829 q^{3} -1.48375 q^{5} -2.26909 q^{7} +4.49825 q^{9} +O(q^{10})\)	\(q-2.73829 q^{3} -1.48375 q^{5} -2.26909 q^{7} +4.49825 q^{9} -4.54237 q^{11} +1.28908 q^{13} +4.06294 q^{15} +3.35584 q^{17} +6.02589 q^{19} +6.21343 q^{21} -4.30148 q^{23} -2.79849 q^{25} -4.10265 q^{27} -5.74625 q^{29} +10.2624 q^{31} +12.4384 q^{33} +3.36676 q^{35} +4.70900 q^{37} -3.52989 q^{39} +6.50770 q^{41} -12.5570 q^{43} -6.67427 q^{45} -8.97508 q^{47} -1.85124 q^{49} -9.18926 q^{51} -5.39473 q^{53} +6.73974 q^{55} -16.5007 q^{57} -12.2282 q^{59} -5.71895 q^{61} -10.2069 q^{63} -1.91267 q^{65} +8.64755 q^{67} +11.7787 q^{69} -13.7302 q^{71} +3.68126 q^{73} +7.66309 q^{75} +10.3070 q^{77} +3.72712 q^{79} -2.26049 q^{81} +3.13641 q^{83} -4.97922 q^{85} +15.7349 q^{87} -12.6454 q^{89} -2.92504 q^{91} -28.1015 q^{93} -8.94091 q^{95} +14.6647 q^{97} -20.4327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * q^99

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