LMFDB - Embedded newform 8048.2.a.v.1.3

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:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.84528 q^{3} -2.72112 q^{5} +3.71681 q^{7} +5.09562 q^{9} +O(q^{10})$	$q-2.84528 q^{3} -2.72112 q^{5} +3.71681 q^{7} +5.09562 q^{9} +3.83632 q^{11} +2.85172 q^{13} +7.74235 q^{15} +8.15606 q^{17} -3.41495 q^{19} -10.5754 q^{21} -0.915770 q^{23} +2.40450 q^{25} -5.96262 q^{27} -9.16823 q^{29} -5.69042 q^{31} -10.9154 q^{33} -10.1139 q^{35} -8.85477 q^{37} -8.11396 q^{39} -10.6339 q^{41} +11.1515 q^{43} -13.8658 q^{45} +3.44500 q^{47} +6.81471 q^{49} -23.2063 q^{51} -4.95726 q^{53} -10.4391 q^{55} +9.71648 q^{57} -2.81555 q^{59} -3.59989 q^{61} +18.9395 q^{63} -7.75989 q^{65} +9.01369 q^{67} +2.60562 q^{69} +7.15270 q^{71} -7.82872 q^{73} -6.84147 q^{75} +14.2589 q^{77} -13.4571 q^{79} +1.67848 q^{81} +13.5797 q^{83} -22.1936 q^{85} +26.0862 q^{87} +6.47898 q^{89} +10.5993 q^{91} +16.1908 q^{93} +9.29248 q^{95} +16.0682 q^{97} +19.5484 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−2.84528	−1.64272	−0.821362	−	0.570408i	$-0.806785\pi$
$3$	−2.84528	−1.64272	−0.821362	+	0.570408i	$0.806785\pi$
$4$	0	0
$5$	−2.72112	−1.21692	−0.608461	−	0.793584i	$-0.708213\pi$
$5$	−2.72112	−1.21692	−0.608461	+	0.793584i	$0.708213\pi$
$6$	0	0
$7$	3.71681	1.40482	0.702412	−	0.711771i	$-0.252106\pi$
$7$	3.71681	1.40482	0.702412	+	0.711771i	$0.252106\pi$
$8$	0	0
$9$	5.09562	1.69854
$10$	0	0
$11$	3.83632	1.15670	0.578348	−	0.815790i	$-0.303698\pi$
$11$	3.83632	1.15670	0.578348	+	0.815790i	$0.303698\pi$
$12$	0	0
$13$	2.85172	0.790926	0.395463	−	0.918482i	$-0.370584\pi$
$13$	2.85172	0.790926	0.395463	+	0.918482i	$0.370584\pi$
$14$	0	0
$15$	7.74235	1.99907
$16$	0	0
$17$	8.15606	1.97814	0.989068	−	0.147460i	$-0.0471098\pi$
$17$	8.15606	1.97814	0.989068	+	0.147460i	$0.0471098\pi$
$18$	0	0
$19$	−3.41495	−0.783443	−0.391721	−	0.920084i	$-0.628120\pi$
$19$	−3.41495	−0.783443	−0.391721	+	0.920084i	$0.628120\pi$
$20$	0	0
$21$	−10.5754	−2.30774
$22$	0	0
$23$	−0.915770	−0.190951	−0.0954756	−	0.995432i	$-0.530437\pi$
$23$	−0.915770	−0.190951	−0.0954756	+	0.995432i	$0.530437\pi$
$24$	0	0
$25$	2.40450	0.480900
$26$	0	0
$27$	−5.96262	−1.14751
$28$	0	0
$29$	−9.16823	−1.70250	−0.851249	−	0.524763i	$-0.824154\pi$
$29$	−9.16823	−1.70250	−0.851249	+	0.524763i	$0.824154\pi$
$30$	0	0
$31$	−5.69042	−1.02203	−0.511014	−	0.859572i	$-0.670730\pi$
$31$	−5.69042	−1.02203	−0.511014	+	0.859572i	$0.670730\pi$
$32$	0	0
$33$	−10.9154	−1.90013
$34$	0	0
$35$	−10.1139	−1.70956
$36$	0	0
$37$	−8.85477	−1.45571	−0.727857	−	0.685729i	$-0.759484\pi$
$37$	−8.85477	−1.45571	−0.727857	+	0.685729i	$0.759484\pi$
$38$	0	0
$39$	−8.11396	−1.29927
$40$	0	0
$41$	−10.6339	−1.66074	−0.830368	−	0.557215i	$-0.811870\pi$
$41$	−10.6339	−1.66074	−0.830368	+	0.557215i	$0.811870\pi$
$42$	0	0
$43$	11.1515	1.70058	0.850292	−	0.526312i	$-0.176425\pi$
$43$	11.1515	1.70058	0.850292	+	0.526312i	$0.176425\pi$
$44$	0	0
$45$	−13.8658	−2.06699
$46$	0	0
$47$	3.44500	0.502504	0.251252	−	0.967922i	$-0.419158\pi$
$47$	3.44500	0.502504	0.251252	+	0.967922i	$0.419158\pi$
$48$	0	0
$49$	6.81471	0.973530
$50$	0	0
$51$	−23.2063	−3.24953
$52$	0	0
$53$	−4.95726	−0.680932	−0.340466	−	0.940257i	$-0.610585\pi$
$53$	−4.95726	−0.680932	−0.340466	+	0.940257i	$0.610585\pi$
$54$	0	0
$55$	−10.4391	−1.40761
$56$	0	0
$57$	9.71648	1.28698
$58$	0	0
$59$	−2.81555	−0.366554	−0.183277	−	0.983061i	$-0.558671\pi$
$59$	−2.81555	−0.366554	−0.183277	+	0.983061i	$0.558671\pi$
$60$	0	0
$61$	−3.59989	−0.460919	−0.230460	−	0.973082i	$-0.574023\pi$
$61$	−3.59989	−0.460919	−0.230460	+	0.973082i	$0.574023\pi$
$62$	0	0
$63$	18.9395	2.38615
$64$	0	0
$65$	−7.75989	−0.962496
$66$	0	0
$67$	9.01369	1.10120	0.550598	−	0.834770i	$-0.314400\pi$
$67$	9.01369	1.10120	0.550598	+	0.834770i	$0.314400\pi$
$68$	0	0
$69$	2.60562	0.313680
$70$	0	0
$71$	7.15270	0.848869	0.424435	−	0.905459i	$-0.360473\pi$
$71$	7.15270	0.848869	0.424435	+	0.905459i	$0.360473\pi$
$72$	0	0
$73$	−7.82872	−0.916282	−0.458141	−	0.888880i	$-0.651485\pi$
$73$	−7.82872	−0.916282	−0.458141	+	0.888880i	$0.651485\pi$
$74$	0	0
$75$	−6.84147	−0.789985
$76$	0	0
$77$	14.2589	1.62495
$78$	0	0
$79$	−13.4571	−1.51405	−0.757024	−	0.653387i	$-0.773347\pi$
$79$	−13.4571	−1.51405	−0.757024	+	0.653387i	$0.773347\pi$
$80$	0	0
$81$	1.67848	0.186497
$82$	0	0
$83$	13.5797	1.49056	0.745280	−	0.666751i	$-0.232316\pi$
$83$	13.5797	1.49056	0.745280	+	0.666751i	$0.232316\pi$
$84$	0	0
$85$	−22.1936	−2.40724
$86$	0	0
$87$	26.0862	2.79673
$88$	0	0
$89$	6.47898	0.686770	0.343385	−	0.939195i	$-0.388426\pi$
$89$	6.47898	0.686770	0.343385	+	0.939195i	$0.388426\pi$
$90$	0	0
$91$	10.5993	1.11111
$92$	0	0
$93$	16.1908	1.67891
$94$	0	0
$95$	9.29248	0.953389
$96$	0	0
$97$	16.0682	1.63148	0.815741	−	0.578417i	$-0.196329\pi$
$97$	16.0682	1.63148	0.815741	+	0.578417i	$0.196329\pi$
$98$	0	0
$99$	19.5484	1.96469
$100$	0	0
$101$	−19.2180	−1.91226	−0.956131	−	0.292940i	$-0.905366\pi$
$101$	−19.2180	−1.91226	−0.956131	+	0.292940i	$0.905366\pi$
$102$	0	0
$103$	0.857652	0.0845070	0.0422535	−	0.999107i	$-0.486546\pi$
$103$	0.857652	0.0845070	0.0422535	+	0.999107i	$0.486546\pi$
$104$	0	0
$105$	28.7769	2.80834
$106$	0	0
$107$	−17.7650	−1.71741	−0.858703	−	0.512474i	$-0.828729\pi$
$107$	−17.7650	−1.71741	−0.858703	+	0.512474i	$0.828729\pi$
$108$	0	0
$109$	−17.5952	−1.68532	−0.842659	−	0.538448i	$-0.819011\pi$
$109$	−17.5952	−1.68532	−0.842659	+	0.538448i	$0.819011\pi$
$110$	0	0
$111$	25.1943	2.39134
$112$	0	0
$113$	8.81848	0.829573	0.414786	−	0.909919i	$-0.363856\pi$
$113$	8.81848	0.829573	0.414786	+	0.909919i	$0.363856\pi$
$114$	0	0
$115$	2.49192	0.232373
$116$	0	0
$117$	14.5313	1.34342
$118$	0	0
$119$	30.3146	2.77893
$120$	0	0
$121$	3.71739	0.337944
$122$	0	0
$123$	30.2564	2.72813
$124$	0	0
$125$	7.06267	0.631705
$126$	0	0
$127$	−13.2789	−1.17831	−0.589155	−	0.808020i	$-0.700539\pi$
$127$	−13.2789	−1.17831	−0.589155	+	0.808020i	$0.700539\pi$
$128$	0	0
$129$	−31.7291	−2.79359
$130$	0	0
$131$	2.42642	0.211997	0.105999	−	0.994366i	$-0.466196\pi$
$131$	2.42642	0.211997	0.105999	+	0.994366i	$0.466196\pi$
$132$	0	0
$133$	−12.6927	−1.10060
$134$	0	0
$135$	16.2250	1.39643
$136$	0	0
$137$	−3.65954	−0.312656	−0.156328	−	0.987705i	$-0.549966\pi$
$137$	−3.65954	−0.312656	−0.156328	+	0.987705i	$0.549966\pi$
$138$	0	0
$139$	−3.40899	−0.289147	−0.144574	−	0.989494i	$-0.546181\pi$
$139$	−3.40899	−0.289147	−0.144574	+	0.989494i	$0.546181\pi$
$140$	0	0
$141$	−9.80198	−0.825475
$142$	0	0
$143$	10.9401	0.914861
$144$	0	0
$145$	24.9479	2.07181
$146$	0	0
$147$	−19.3898	−1.59924
$148$	0	0
$149$	10.7745	0.882678	0.441339	−	0.897340i	$-0.354504\pi$
$149$	10.7745	0.882678	0.441339	+	0.897340i	$0.354504\pi$
$150$	0	0
$151$	−10.4733	−0.852309	−0.426154	−	0.904650i	$-0.640132\pi$
$151$	−10.4733	−0.852309	−0.426154	+	0.904650i	$0.640132\pi$
$152$	0	0
$153$	41.5602	3.35994
$154$	0	0
$155$	15.4843	1.24373
$156$	0	0
$157$	2.27873	0.181862	0.0909310	−	0.995857i	$-0.471016\pi$
$157$	2.27873	0.181862	0.0909310	+	0.995857i	$0.471016\pi$
$158$	0	0
$159$	14.1048	1.11858
$160$	0	0
$161$	−3.40375	−0.268253
$162$	0	0
$163$	−15.0950	−1.18233	−0.591166	−	0.806550i	$-0.701332\pi$
$163$	−15.0950	−1.18233	−0.591166	+	0.806550i	$0.701332\pi$
$164$	0	0
$165$	29.7022	2.31231
$166$	0	0
$167$	10.3361	0.799830	0.399915	−	0.916552i	$-0.369040\pi$
$167$	10.3361	0.799830	0.399915	+	0.916552i	$0.369040\pi$
$168$	0	0
$169$	−4.86767	−0.374436
$170$	0	0
$171$	−17.4013	−1.33071
$172$	0	0
$173$	−15.6052	−1.18644	−0.593221	−	0.805040i	$-0.702144\pi$
$173$	−15.6052	−1.18644	−0.593221	+	0.805040i	$0.702144\pi$
$174$	0	0
$175$	8.93707	0.675579
$176$	0	0
$177$	8.01104	0.602147
$178$	0	0
$179$	5.86603	0.438448	0.219224	−	0.975675i	$-0.429648\pi$
$179$	5.86603	0.438448	0.219224	+	0.975675i	$0.429648\pi$
$180$	0	0
$181$	−17.7534	−1.31960	−0.659802	−	0.751440i	$-0.729360\pi$
$181$	−17.7534	−1.31960	−0.659802	+	0.751440i	$0.729360\pi$
$182$	0	0
$183$	10.2427	0.757162
$184$	0	0
$185$	24.0949	1.77149
$186$	0	0
$187$	31.2893	2.28810
$188$	0	0
$189$	−22.1620	−1.61205
$190$	0	0
$191$	17.9243	1.29695	0.648477	−	0.761234i	$-0.275406\pi$
$191$	17.9243	1.29695	0.648477	+	0.761234i	$0.275406\pi$
$192$	0	0
$193$	−14.7427	−1.06120	−0.530600	−	0.847623i	$-0.678033\pi$
$193$	−14.7427	−1.06120	−0.530600	+	0.847623i	$0.678033\pi$
$194$	0	0
$195$	22.0791	1.58111
$196$	0	0
$197$	−22.9129	−1.63248	−0.816239	−	0.577715i	$-0.803945\pi$
$197$	−22.9129	−1.63248	−0.816239	+	0.577715i	$0.803945\pi$
$198$	0	0
$199$	0.191759	0.0135934	0.00679672	−	0.999977i	$-0.497837\pi$
$199$	0.191759	0.0135934	0.00679672	+	0.999977i	$0.497837\pi$
$200$	0	0
$201$	−25.6465	−1.80896
$202$	0	0
$203$	−34.0766	−2.39171
$204$	0	0
$205$	28.9361	2.02099
$206$	0	0
$207$	−4.66641	−0.324338
$208$	0	0
$209$	−13.1008	−0.906204
$210$	0	0
$211$	−12.1248	−0.834705	−0.417353	−	0.908745i	$-0.637042\pi$
$211$	−12.1248	−0.834705	−0.417353	+	0.908745i	$0.637042\pi$
$212$	0	0
$213$	−20.3514	−1.39446
$214$	0	0
$215$	−30.3445	−2.06948
$216$	0	0
$217$	−21.1502	−1.43577
$218$	0	0
$219$	22.2749	1.50520
$220$	0	0
$221$	23.2588	1.56456
$222$	0	0
$223$	8.86947	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$223$	8.86947	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$224$	0	0
$225$	12.2524	0.816827
$226$	0	0
$227$	−10.3823	−0.689096	−0.344548	−	0.938769i	$-0.611968\pi$
$227$	−10.3823	−0.689096	−0.344548	+	0.938769i	$0.611968\pi$
$228$	0	0
$229$	−9.90139	−0.654303	−0.327151	−	0.944972i	$-0.606089\pi$
$229$	−9.90139	−0.654303	−0.327151	+	0.944972i	$0.606089\pi$
$230$	0	0
$231$	−40.5706	−2.66935
$232$	0	0
$233$	9.12344	0.597697	0.298848	−	0.954301i	$-0.403398\pi$
$233$	9.12344	0.597697	0.298848	+	0.954301i	$0.403398\pi$
$234$	0	0
$235$	−9.37425	−0.611509
$236$	0	0
$237$	38.2894	2.48716
$238$	0	0
$239$	13.2494	0.857034	0.428517	−	0.903534i	$-0.359036\pi$
$239$	13.2494	0.857034	0.428517	+	0.903534i	$0.359036\pi$
$240$	0	0
$241$	−11.6605	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$241$	−11.6605	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$242$	0	0
$243$	13.1121	0.841144
$244$	0	0
$245$	−18.5436	−1.18471
$246$	0	0
$247$	−9.73849	−0.619645
$248$	0	0
$249$	−38.6379	−2.44858
$250$	0	0
$251$	−6.89173	−0.435002	−0.217501	−	0.976060i	$-0.569791\pi$
$251$	−6.89173	−0.435002	−0.217501	+	0.976060i	$0.569791\pi$
$252$	0	0
$253$	−3.51319	−0.220872
$254$	0	0
$255$	63.1471	3.95443
$256$	0	0
$257$	−15.3937	−0.960230	−0.480115	−	0.877206i	$-0.659405\pi$
$257$	−15.3937	−0.960230	−0.480115	+	0.877206i	$0.659405\pi$
$258$	0	0
$259$	−32.9115	−2.04502
$260$	0	0
$261$	−46.7178	−2.89176
$262$	0	0
$263$	−17.4022	−1.07306	−0.536532	−	0.843880i	$-0.680266\pi$
$263$	−17.4022	−1.07306	−0.536532	+	0.843880i	$0.680266\pi$
$264$	0	0
$265$	13.4893	0.828641
$266$	0	0
$267$	−18.4345	−1.12817
$268$	0	0
$269$	26.1637	1.59523	0.797615	−	0.603166i	$-0.206095\pi$
$269$	26.1637	1.59523	0.797615	+	0.603166i	$0.206095\pi$
$270$	0	0
$271$	−25.9428	−1.57592	−0.787958	−	0.615729i	$-0.788861\pi$
$271$	−25.9428	−1.57592	−0.787958	+	0.615729i	$0.788861\pi$
$272$	0	0
$273$	−30.1581	−1.82525
$274$	0	0
$275$	9.22444	0.556254
$276$	0	0
$277$	−4.55601	−0.273744	−0.136872	−	0.990589i	$-0.543705\pi$
$277$	−4.55601	−0.273744	−0.136872	+	0.990589i	$0.543705\pi$
$278$	0	0
$279$	−28.9962	−1.73596
$280$	0	0
$281$	14.6290	0.872691	0.436346	−	0.899779i	$-0.356272\pi$
$281$	14.6290	0.872691	0.436346	+	0.899779i	$0.356272\pi$
$282$	0	0
$283$	0.505950	0.0300756	0.0150378	−	0.999887i	$-0.495213\pi$
$283$	0.505950	0.0300756	0.0150378	+	0.999887i	$0.495213\pi$
$284$	0	0
$285$	−26.4397	−1.56615
$286$	0	0
$287$	−39.5242	−2.33304
$288$	0	0
$289$	49.5214	2.91302
$290$	0	0
$291$	−45.7186	−2.68007
$292$	0	0
$293$	19.3805	1.13222	0.566109	−	0.824330i	$-0.308448\pi$
$293$	19.3805	1.13222	0.566109	+	0.824330i	$0.308448\pi$
$294$	0	0
$295$	7.66146	0.446068
$296$	0	0
$297$	−22.8746	−1.32732
$298$	0	0
$299$	−2.61152	−0.151028
$300$	0	0
$301$	41.4480	2.38902
$302$	0	0
$303$	54.6806	3.14132
$304$	0	0
$305$	9.79574	0.560903
$306$	0	0
$307$	30.6653	1.75016	0.875082	−	0.483975i	$-0.160807\pi$
$307$	30.6653	1.75016	0.875082	+	0.483975i	$0.160807\pi$
$308$	0	0
$309$	−2.44026	−0.138822
$310$	0	0
$311$	11.5530	0.655112	0.327556	−	0.944832i	$-0.393775\pi$
$311$	11.5530	0.655112	0.327556	+	0.944832i	$0.393775\pi$
$312$	0	0
$313$	10.3272	0.583729	0.291864	−	0.956460i	$-0.405724\pi$
$313$	10.3272	0.583729	0.291864	+	0.956460i	$0.405724\pi$
$314$	0	0
$315$	−51.5366	−2.90376
$316$	0	0
$317$	−10.6406	−0.597636	−0.298818	−	0.954310i	$-0.596592\pi$
$317$	−10.6406	−0.597636	−0.298818	+	0.954310i	$0.596592\pi$
$318$	0	0
$319$	−35.1723	−1.96927
$320$	0	0
$321$	50.5464	2.82122
$322$	0	0
$323$	−27.8525	−1.54976
$324$	0	0
$325$	6.85697	0.380356
$326$	0	0
$327$	50.0634	2.76851
$328$	0	0
$329$	12.8044	0.705930
$330$	0	0
$331$	−17.5605	−0.965215	−0.482608	−	0.875837i	$-0.660310\pi$
$331$	−17.5605	−0.965215	−0.482608	+	0.875837i	$0.660310\pi$
$332$	0	0
$333$	−45.1205	−2.47259
$334$	0	0
$335$	−24.5273	−1.34007
$336$	0	0
$337$	0.888584	0.0484043	0.0242021	−	0.999707i	$-0.492295\pi$
$337$	0.888584	0.0484043	0.0242021	+	0.999707i	$0.492295\pi$
$338$	0	0
$339$	−25.0910	−1.36276
$340$	0	0
$341$	−21.8303	−1.18218
$342$	0	0
$343$	−0.688696	−0.0371861
$344$	0	0
$345$	−7.09021	−0.381724
$346$	0	0
$347$	9.13438	0.490359	0.245180	−	0.969478i	$-0.421153\pi$
$347$	9.13438	0.490359	0.245180	+	0.969478i	$0.421153\pi$
$348$	0	0
$349$	6.69512	0.358381	0.179191	−	0.983814i	$-0.442652\pi$
$349$	6.69512	0.358381	0.179191	+	0.983814i	$0.442652\pi$
$350$	0	0
$351$	−17.0038	−0.907594
$352$	0	0
$353$	−18.8389	−1.00269	−0.501346	−	0.865247i	$-0.667162\pi$
$353$	−18.8389	−1.00269	−0.501346	+	0.865247i	$0.667162\pi$
$354$	0	0
$355$	−19.4634	−1.03301
$356$	0	0
$357$	−86.2535	−4.56502
$358$	0	0
$359$	−1.59151	−0.0839966	−0.0419983	−	0.999118i	$-0.513372\pi$
$359$	−1.59151	−0.0839966	−0.0419983	+	0.999118i	$0.513372\pi$
$360$	0	0
$361$	−7.33814	−0.386218
$362$	0	0
$363$	−10.5770	−0.555149
$364$	0	0
$365$	21.3029	1.11504
$366$	0	0
$367$	−34.5867	−1.80541	−0.902706	−	0.430258i	$-0.858423\pi$
$367$	−34.5867	−1.80541	−0.902706	+	0.430258i	$0.858423\pi$
$368$	0	0
$369$	−54.1863	−2.82083
$370$	0	0
$371$	−18.4252	−0.956589
$372$	0	0
$373$	17.0429	0.882449	0.441224	−	0.897397i	$-0.354544\pi$
$373$	17.0429	0.882449	0.441224	+	0.897397i	$0.354544\pi$
$374$	0	0
$375$	−20.0953	−1.03772
$376$	0	0
$377$	−26.1453	−1.34655
$378$	0	0
$379$	14.7397	0.757129	0.378565	−	0.925575i	$-0.376418\pi$
$379$	14.7397	0.757129	0.378565	+	0.925575i	$0.376418\pi$
$380$	0	0
$381$	37.7821	1.93564
$382$	0	0
$383$	13.3274	0.680999	0.340499	−	0.940245i	$-0.389404\pi$
$383$	13.3274	0.680999	0.340499	+	0.940245i	$0.389404\pi$
$384$	0	0
$385$	−38.8002	−1.97744
$386$	0	0
$387$	56.8237	2.88851
$388$	0	0
$389$	14.8579	0.753326	0.376663	−	0.926350i	$-0.377071\pi$
$389$	14.8579	0.753326	0.376663	+	0.926350i	$0.377071\pi$
$390$	0	0
$391$	−7.46908	−0.377728
$392$	0	0
$393$	−6.90384	−0.348253
$394$	0	0
$395$	36.6185	1.84248
$396$	0	0
$397$	4.63286	0.232517	0.116258	−	0.993219i	$-0.462910\pi$
$397$	4.63286	0.232517	0.116258	+	0.993219i	$0.462910\pi$
$398$	0	0
$399$	36.1144	1.80798
$400$	0	0
$401$	−20.4551	−1.02148	−0.510740	−	0.859735i	$-0.670629\pi$
$401$	−20.4551	−1.02148	−0.510740	+	0.859735i	$0.670629\pi$
$402$	0	0
$403$	−16.2275	−0.808349
$404$	0	0
$405$	−4.56734	−0.226953
$406$	0	0
$407$	−33.9698	−1.68382
$408$	0	0
$409$	4.52690	0.223841	0.111920	−	0.993717i	$-0.464300\pi$
$409$	4.52690	0.223841	0.111920	+	0.993717i	$0.464300\pi$
$410$	0	0
$411$	10.4124	0.513607
$412$	0	0
$413$	−10.4649	−0.514944
$414$	0	0
$415$	−36.9519	−1.81390
$416$	0	0
$417$	9.69955	0.474989
$418$	0	0
$419$	−2.10462	−0.102817	−0.0514086	−	0.998678i	$-0.516371\pi$
$419$	−2.10462	−0.102817	−0.0514086	+	0.998678i	$0.516371\pi$
$420$	0	0
$421$	38.7225	1.88722	0.943610	−	0.331058i	$-0.107406\pi$
$421$	38.7225	1.88722	0.943610	+	0.331058i	$0.107406\pi$
$422$	0	0
$423$	17.5544	0.853523
$424$	0	0
$425$	19.6112	0.951285
$426$	0	0
$427$	−13.3801	−0.647510
$428$	0	0
$429$	−31.1278	−1.50286
$430$	0	0
$431$	−14.5749	−0.702047	−0.351024	−	0.936367i	$-0.614166\pi$
$431$	−14.5749	−0.702047	−0.351024	+	0.936367i	$0.614166\pi$
$432$	0	0
$433$	36.6936	1.76338	0.881691	−	0.471828i	$-0.156406\pi$
$433$	36.6936	1.76338	0.881691	+	0.471828i	$0.156406\pi$
$434$	0	0
$435$	−70.9836	−3.40340
$436$	0	0
$437$	3.12731	0.149599
$438$	0	0
$439$	17.8012	0.849605	0.424803	−	0.905286i	$-0.360343\pi$
$439$	17.8012	0.849605	0.424803	+	0.905286i	$0.360343\pi$
$440$	0	0
$441$	34.7252	1.65358
$442$	0	0
$443$	8.35105	0.396770	0.198385	−	0.980124i	$-0.436430\pi$
$443$	8.35105	0.396770	0.198385	+	0.980124i	$0.436430\pi$
$444$	0	0
$445$	−17.6301	−0.835746
$446$	0	0
$447$	−30.6563	−1.45000
$448$	0	0
$449$	−21.7497	−1.02643	−0.513216	−	0.858260i	$-0.671546\pi$
$449$	−21.7497	−1.02643	−0.513216	+	0.858260i	$0.671546\pi$
$450$	0	0
$451$	−40.7951	−1.92097
$452$	0	0
$453$	29.7996	1.40011
$454$	0	0
$455$	−28.8421	−1.35214
$456$	0	0
$457$	13.4414	0.628761	0.314381	−	0.949297i	$-0.398203\pi$
$457$	13.4414	0.628761	0.314381	+	0.949297i	$0.398203\pi$
$458$	0	0
$459$	−48.6315	−2.26993
$460$	0	0
$461$	12.1681	0.566726	0.283363	−	0.959013i	$-0.408550\pi$
$461$	12.1681	0.566726	0.283363	+	0.959013i	$0.408550\pi$
$462$	0	0
$463$	7.02984	0.326704	0.163352	−	0.986568i	$-0.447769\pi$
$463$	7.02984	0.326704	0.163352	+	0.986568i	$0.447769\pi$
$464$	0	0
$465$	−44.0572	−2.04310
$466$	0	0
$467$	7.89249	0.365221	0.182610	−	0.983185i	$-0.441545\pi$
$467$	7.89249	0.365221	0.182610	+	0.983185i	$0.441545\pi$
$468$	0	0
$469$	33.5022	1.54699
$470$	0	0
$471$	−6.48361	−0.298749
$472$	0	0
$473$	42.7807	1.96706
$474$	0	0
$475$	−8.21123	−0.376757
$476$	0	0
$477$	−25.2603	−1.15659
$478$	0	0
$479$	11.7262	0.535782	0.267891	−	0.963449i	$-0.413673\pi$
$479$	11.7262	0.535782	0.267891	+	0.963449i	$0.413673\pi$
$480$	0	0
$481$	−25.2514	−1.15136
$482$	0	0
$483$	9.68461	0.440665
$484$	0	0
$485$	−43.7236	−1.98539
$486$	0	0
$487$	−17.2492	−0.781637	−0.390818	−	0.920468i	$-0.627808\pi$
$487$	−17.2492	−0.781637	−0.390818	+	0.920468i	$0.627808\pi$
$488$	0	0
$489$	42.9495	1.94224
$490$	0	0
$491$	−12.0401	−0.543363	−0.271681	−	0.962387i	$-0.587580\pi$
$491$	−12.0401	−0.543363	−0.271681	+	0.962387i	$0.587580\pi$
$492$	0	0
$493$	−74.7766	−3.36777
$494$	0	0
$495$	−53.1937	−2.39088
$496$	0	0
$497$	26.5853	1.19251
$498$	0	0
$499$	43.6442	1.95378	0.976891	−	0.213739i	$-0.0685642\pi$
$499$	43.6442	1.95378	0.976891	+	0.213739i	$0.0685642\pi$
$500$	0	0
$501$	−29.4091	−1.31390
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	52.2945	2.32707
$506$	0	0
$507$	13.8499	0.615094
$508$	0	0
$509$	21.4932	0.952669	0.476334	−	0.879264i	$-0.341965\pi$
$509$	21.4932	0.952669	0.476334	+	0.879264i	$0.341965\pi$
$510$	0	0
$511$	−29.0979	−1.28721
$512$	0	0
$513$	20.3620	0.899006
$514$	0	0
$515$	−2.33377	−0.102838
$516$	0	0
$517$	13.2161	0.581244
$518$	0	0
$519$	44.4012	1.94900
$520$	0	0
$521$	−5.27284	−0.231007	−0.115504	−	0.993307i	$-0.536848\pi$
$521$	−5.27284	−0.231007	−0.115504	+	0.993307i	$0.536848\pi$
$522$	0	0
$523$	22.2996	0.975093	0.487546	−	0.873097i	$-0.337892\pi$
$523$	22.2996	0.975093	0.487546	+	0.873097i	$0.337892\pi$
$524$	0	0
$525$	−25.4285	−1.10979
$526$	0	0
$527$	−46.4114	−2.02171
$528$	0	0
$529$	−22.1614	−0.963538
$530$	0	0
$531$	−14.3470	−0.622607
$532$	0	0
$533$	−30.3250	−1.31352
$534$	0	0
$535$	48.3407	2.08995
$536$	0	0
$537$	−16.6905	−0.720248
$538$	0	0
$539$	26.1434	1.12608
$540$	0	0
$541$	7.94842	0.341729	0.170865	−	0.985294i	$-0.445344\pi$
$541$	7.94842	0.341729	0.170865	+	0.985294i	$0.445344\pi$
$542$	0	0
$543$	50.5135	2.16774
$544$	0	0
$545$	47.8788	2.05090
$546$	0	0
$547$	19.5507	0.835925	0.417963	−	0.908464i	$-0.362744\pi$
$547$	19.5507	0.835925	0.417963	+	0.908464i	$0.362744\pi$
$548$	0	0
$549$	−18.3437	−0.782889
$550$	0	0
$551$	31.3090	1.33381
$552$	0	0
$553$	−50.0177	−2.12697
$554$	0	0
$555$	−68.5567	−2.91007
$556$	0	0
$557$	5.55069	0.235190	0.117595	−	0.993062i	$-0.462481\pi$
$557$	5.55069	0.235190	0.117595	+	0.993062i	$0.462481\pi$
$558$	0	0
$559$	31.8009	1.34504
$560$	0	0
$561$	−89.0269	−3.75872
$562$	0	0
$563$	−8.19830	−0.345517	−0.172759	−	0.984964i	$-0.555268\pi$
$563$	−8.19830	−0.345517	−0.172759	+	0.984964i	$0.555268\pi$
$564$	0	0
$565$	−23.9961	−1.00953
$566$	0	0
$567$	6.23858	0.261996
$568$	0	0
$569$	3.01819	0.126529	0.0632645	−	0.997997i	$-0.479849\pi$
$569$	3.01819	0.126529	0.0632645	+	0.997997i	$0.479849\pi$
$570$	0	0
$571$	−34.0007	−1.42289	−0.711443	−	0.702744i	$-0.751958\pi$
$571$	−34.0007	−1.42289	−0.711443	+	0.702744i	$0.751958\pi$
$572$	0	0
$573$	−50.9995	−2.13054
$574$	0	0
$575$	−2.20197	−0.0918284
$576$	0	0
$577$	−27.0544	−1.12629	−0.563144	−	0.826358i	$-0.690409\pi$
$577$	−27.0544	−1.12629	−0.563144	+	0.826358i	$0.690409\pi$
$578$	0	0
$579$	41.9470	1.74326
$580$	0	0
$581$	50.4731	2.09398
$582$	0	0
$583$	−19.0177	−0.787631
$584$	0	0
$585$	−39.5414	−1.63484
$586$	0	0
$587$	17.7581	0.732953	0.366477	−	0.930427i	$-0.380564\pi$
$587$	17.7581	0.732953	0.366477	+	0.930427i	$0.380564\pi$
$588$	0	0
$589$	19.4325	0.800701
$590$	0	0
$591$	65.1936	2.68171
$592$	0	0
$593$	−43.5559	−1.78863	−0.894314	−	0.447441i	$-0.852336\pi$
$593$	−43.5559	−1.78863	−0.894314	+	0.447441i	$0.852336\pi$
$594$	0	0
$595$	−82.4896	−3.38174
$596$	0	0
$597$	−0.545608	−0.0223303
$598$	0	0
$599$	−32.7236	−1.33705	−0.668524	−	0.743690i	$-0.733074\pi$
$599$	−32.7236	−1.33705	−0.668524	+	0.743690i	$0.733074\pi$
$600$	0	0
$601$	−22.7206	−0.926792	−0.463396	−	0.886151i	$-0.653369\pi$
$601$	−22.7206	−0.926792	−0.463396	+	0.886151i	$0.653369\pi$
$602$	0	0
$603$	45.9303	1.87043
$604$	0	0
$605$	−10.1155	−0.411252
$606$	0	0
$607$	−10.2469	−0.415911	−0.207955	−	0.978138i	$-0.566681\pi$
$607$	−10.2469	−0.415911	−0.207955	+	0.978138i	$0.566681\pi$
$608$	0	0
$609$	96.9575	3.92891
$610$	0	0
$611$	9.82418	0.397444
$612$	0	0
$613$	22.3683	0.903449	0.451725	−	0.892157i	$-0.350809\pi$
$613$	22.3683	0.903449	0.451725	+	0.892157i	$0.350809\pi$
$614$	0	0
$615$	−82.3314	−3.31992
$616$	0	0
$617$	6.86067	0.276200	0.138100	−	0.990418i	$-0.455900\pi$
$617$	6.86067	0.276200	0.138100	+	0.990418i	$0.455900\pi$
$618$	0	0
$619$	−15.0206	−0.603729	−0.301865	−	0.953351i	$-0.597609\pi$
$619$	−15.0206	−0.603729	−0.301865	+	0.953351i	$0.597609\pi$
$620$	0	0
$621$	5.46039	0.219118
$622$	0	0
$623$	24.0811	0.964791
$624$	0	0
$625$	−31.2409	−1.24964
$626$	0	0
$627$	37.2756	1.48864
$628$	0	0
$629$	−72.2200	−2.87960
$630$	0	0
$631$	−35.7080	−1.42151	−0.710756	−	0.703438i	$-0.751647\pi$
$631$	−35.7080	−1.42151	−0.710756	+	0.703438i	$0.751647\pi$
$632$	0	0
$633$	34.4984	1.37119
$634$	0	0
$635$	36.1334	1.43391
$636$	0	0
$637$	19.4337	0.769990
$638$	0	0
$639$	36.4474	1.44184
$640$	0	0
$641$	−13.2969	−0.525197	−0.262599	−	0.964905i	$-0.584580\pi$
$641$	−13.2969	−0.525197	−0.262599	+	0.964905i	$0.584580\pi$
$642$	0	0
$643$	−41.1855	−1.62420	−0.812099	−	0.583519i	$-0.801675\pi$
$643$	−41.1855	−1.62420	−0.812099	+	0.583519i	$0.801675\pi$
$644$	0	0
$645$	86.3386	3.39958
$646$	0	0
$647$	10.0464	0.394963	0.197482	−	0.980307i	$-0.436724\pi$
$647$	10.0464	0.394963	0.197482	+	0.980307i	$0.436724\pi$
$648$	0	0
$649$	−10.8014	−0.423991
$650$	0	0
$651$	60.1783	2.35857
$652$	0	0
$653$	−41.7793	−1.63495	−0.817476	−	0.575963i	$-0.804627\pi$
$653$	−41.7793	−1.63495	−0.817476	+	0.575963i	$0.804627\pi$
$654$	0	0
$655$	−6.60258	−0.257984
$656$	0	0
$657$	−39.8922	−1.55634
$658$	0	0
$659$	−22.6672	−0.882991	−0.441495	−	0.897264i	$-0.645552\pi$
$659$	−22.6672	−0.882991	−0.441495	+	0.897264i	$0.645552\pi$
$660$	0	0
$661$	30.7091	1.19445	0.597224	−	0.802075i	$-0.296271\pi$
$661$	30.7091	1.19445	0.597224	+	0.802075i	$0.296271\pi$
$662$	0	0
$663$	−66.1779	−2.57014
$664$	0	0
$665$	34.5384	1.33934
$666$	0	0
$667$	8.39599	0.325094
$668$	0	0
$669$	−25.2361	−0.975685
$670$	0	0
$671$	−13.8104	−0.533143
$672$	0	0
$673$	−9.82187	−0.378605	−0.189303	−	0.981919i	$-0.560623\pi$
$673$	−9.82187	−0.378605	−0.189303	+	0.981919i	$0.560623\pi$
$674$	0	0
$675$	−14.3371	−0.551836
$676$	0	0
$677$	−4.41084	−0.169522	−0.0847611	−	0.996401i	$-0.527013\pi$
$677$	−4.41084	−0.169522	−0.0847611	+	0.996401i	$0.527013\pi$
$678$	0	0
$679$	59.7227	2.29195
$680$	0	0
$681$	29.5405	1.13199
$682$	0	0
$683$	−8.09181	−0.309625	−0.154812	−	0.987944i	$-0.549477\pi$
$683$	−8.09181	−0.309625	−0.154812	+	0.987944i	$0.549477\pi$
$684$	0	0
$685$	9.95806	0.380478
$686$	0	0
$687$	28.1722	1.07484
$688$	0	0
$689$	−14.1367	−0.538567
$690$	0	0
$691$	−20.7906	−0.790914	−0.395457	−	0.918485i	$-0.629414\pi$
$691$	−20.7906	−0.790914	−0.395457	+	0.918485i	$0.629414\pi$
$692$	0	0
$693$	72.6580	2.76005
$694$	0	0
$695$	9.27629	0.351870
$696$	0	0
$697$	−86.7308	−3.28516
$698$	0	0
$699$	−25.9588	−0.981850
$700$	0	0
$701$	21.9468	0.828920	0.414460	−	0.910068i	$-0.363971\pi$
$701$	21.9468	0.828920	0.414460	+	0.910068i	$0.363971\pi$
$702$	0	0
$703$	30.2386	1.14047
$704$	0	0
$705$	26.6724	1.00454
$706$	0	0
$707$	−71.4297	−2.68639
$708$	0	0
$709$	−43.2940	−1.62594	−0.812970	−	0.582305i	$-0.802151\pi$
$709$	−43.2940	−1.62594	−0.812970	+	0.582305i	$0.802151\pi$
$710$	0	0
$711$	−68.5725	−2.57167
$712$	0	0
$713$	5.21111	0.195158
$714$	0	0
$715$	−29.7694	−1.11331
$716$	0	0
$717$	−37.6983	−1.40787
$718$	0	0
$719$	38.3358	1.42968	0.714842	−	0.699286i	$-0.246499\pi$
$719$	38.3358	1.42968	0.714842	+	0.699286i	$0.246499\pi$
$720$	0	0
$721$	3.18773	0.118717
$722$	0	0
$723$	33.1773	1.23388
$724$	0	0
$725$	−22.0450	−0.818730
$726$	0	0
$727$	47.6902	1.76873	0.884366	−	0.466794i	$-0.154591\pi$
$727$	47.6902	1.76873	0.884366	+	0.466794i	$0.154591\pi$
$728$	0	0
$729$	−42.3431	−1.56826
$730$	0	0
$731$	90.9521	3.36399
$732$	0	0
$733$	−1.46620	−0.0541553	−0.0270777	−	0.999633i	$-0.508620\pi$
$733$	−1.46620	−0.0541553	−0.0270777	+	0.999633i	$0.508620\pi$
$734$	0	0
$735$	52.7619	1.94615
$736$	0	0
$737$	34.5794	1.27375
$738$	0	0
$739$	−19.1867	−0.705793	−0.352896	−	0.935662i	$-0.614803\pi$
$739$	−19.1867	−0.705793	−0.352896	+	0.935662i	$0.614803\pi$
$740$	0	0
$741$	27.7087	1.01791
$742$	0	0
$743$	25.3389	0.929593	0.464796	−	0.885418i	$-0.346128\pi$
$743$	25.3389	0.929593	0.464796	+	0.885418i	$0.346128\pi$
$744$	0	0
$745$	−29.3186	−1.07415
$746$	0	0
$747$	69.1967	2.53178
$748$	0	0
$749$	−66.0291	−2.41265
$750$	0	0
$751$	−1.45873	−0.0532296	−0.0266148	−	0.999646i	$-0.508473\pi$
$751$	−1.45873	−0.0532296	−0.0266148	+	0.999646i	$0.508473\pi$
$752$	0	0
$753$	19.6089	0.714588
$754$	0	0
$755$	28.4992	1.03719
$756$	0	0
$757$	−20.9421	−0.761154	−0.380577	−	0.924749i	$-0.624275\pi$
$757$	−20.9421	−0.761154	−0.380577	+	0.924749i	$0.624275\pi$
$758$	0	0
$759$	9.99601	0.362832
$760$	0	0
$761$	−18.4858	−0.670110	−0.335055	−	0.942199i	$-0.608755\pi$
$761$	−18.4858	−0.670110	−0.335055	+	0.942199i	$0.608755\pi$
$762$	0	0
$763$	−65.3982	−2.36757
$764$	0	0
$765$	−113.090	−4.08879
$766$	0	0
$767$	−8.02919	−0.289917
$768$	0	0
$769$	−35.0896	−1.26536	−0.632682	−	0.774412i	$-0.718046\pi$
$769$	−35.0896	−1.26536	−0.632682	+	0.774412i	$0.718046\pi$
$770$	0	0
$771$	43.7993	1.57739
$772$	0	0
$773$	1.58201	0.0569009	0.0284504	−	0.999595i	$-0.490943\pi$
$773$	1.58201	0.0569009	0.0284504	+	0.999595i	$0.490943\pi$
$774$	0	0
$775$	−13.6826	−0.491493
$776$	0	0
$777$	93.6425	3.35941
$778$	0	0
$779$	36.3142	1.30109
$780$	0	0
$781$	27.4401	0.981883
$782$	0	0
$783$	54.6667	1.95363
$784$	0	0
$785$	−6.20069	−0.221312
$786$	0	0
$787$	−43.4639	−1.54932	−0.774660	−	0.632378i	$-0.782079\pi$
$787$	−43.4639	−1.54932	−0.774660	+	0.632378i	$0.782079\pi$
$788$	0	0
$789$	49.5141	1.76275
$790$	0	0
$791$	32.7766	1.16540
$792$	0	0
$793$	−10.2659	−0.364553
$794$	0	0
$795$	−38.3808	−1.36123
$796$	0	0
$797$	−5.25945	−0.186299	−0.0931496	−	0.995652i	$-0.529693\pi$
$797$	−5.25945	−0.186299	−0.0931496	+	0.995652i	$0.529693\pi$
$798$	0	0
$799$	28.0976	0.994022
$800$	0	0
$801$	33.0144	1.16651
$802$	0	0
$803$	−30.0335	−1.05986
$804$	0	0
$805$	9.26201	0.326443
$806$	0	0
$807$	−74.4431	−2.62052
$808$	0	0
$809$	9.20469	0.323620	0.161810	−	0.986822i	$-0.448267\pi$
$809$	9.20469	0.323620	0.161810	+	0.986822i	$0.448267\pi$
$810$	0	0
$811$	−4.18969	−0.147120	−0.0735600	−	0.997291i	$-0.523436\pi$
$811$	−4.18969	−0.147120	−0.0735600	+	0.997291i	$0.523436\pi$
$812$	0	0
$813$	73.8146	2.58879
$814$	0	0
$815$	41.0753	1.43881
$816$	0	0
$817$	−38.0817	−1.33231
$818$	0	0
$819$	54.0102	1.88727
$820$	0	0
$821$	−16.1785	−0.564635	−0.282317	−	0.959321i	$-0.591103\pi$
$821$	−16.1785	−0.564635	−0.282317	+	0.959321i	$0.591103\pi$
$822$	0	0
$823$	−29.2609	−1.01997	−0.509985	−	0.860183i	$-0.670349\pi$
$823$	−29.2609	−1.01997	−0.509985	+	0.860183i	$0.670349\pi$
$824$	0	0
$825$	−26.2461	−0.913772
$826$	0	0
$827$	−22.9175	−0.796919	−0.398460	−	0.917186i	$-0.630455\pi$
$827$	−22.9175	−0.796919	−0.398460	+	0.917186i	$0.630455\pi$
$828$	0	0
$829$	−8.02706	−0.278791	−0.139396	−	0.990237i	$-0.544516\pi$
$829$	−8.02706	−0.278791	−0.139396	+	0.990237i	$0.544516\pi$
$830$	0	0
$831$	12.9631	0.449686
$832$	0	0
$833$	55.5812	1.92577
$834$	0	0
$835$	−28.1257	−0.973331
$836$	0	0
$837$	33.9298	1.17279
$838$	0	0
$839$	32.2797	1.11442	0.557209	−	0.830372i	$-0.311872\pi$
$839$	32.2797	1.11442	0.557209	+	0.830372i	$0.311872\pi$
$840$	0	0
$841$	55.0564	1.89850
$842$	0	0
$843$	−41.6235	−1.43359
$844$	0	0
$845$	13.2455	0.455659
$846$	0	0
$847$	13.8168	0.474752
$848$	0	0
$849$	−1.43957	−0.0494059
$850$	0	0
$851$	8.10893	0.277971
$852$	0	0
$853$	−51.4452	−1.76145	−0.880726	−	0.473627i	$-0.842945\pi$
$853$	−51.4452	−1.76145	−0.880726	+	0.473627i	$0.842945\pi$
$854$	0	0
$855$	47.3509	1.61937
$856$	0	0
$857$	−41.5405	−1.41900	−0.709498	−	0.704707i	$-0.751078\pi$
$857$	−41.5405	−1.41900	−0.709498	+	0.704707i	$0.751078\pi$
$858$	0	0
$859$	17.3381	0.591567	0.295784	−	0.955255i	$-0.404419\pi$
$859$	17.3381	0.591567	0.295784	+	0.955255i	$0.404419\pi$
$860$	0	0
$861$	112.458	3.83254
$862$	0	0
$863$	−55.0368	−1.87347	−0.936737	−	0.350034i	$-0.886170\pi$
$863$	−55.0368	−1.87347	−0.936737	+	0.350034i	$0.886170\pi$
$864$	0	0
$865$	42.4636	1.44381
$866$	0	0
$867$	−140.902	−4.78529
$868$	0	0
$869$	−51.6260	−1.75129
$870$	0	0
$871$	25.7045	0.870965
$872$	0	0
$873$	81.8776	2.77114
$874$	0	0
$875$	26.2506	0.887434
$876$	0	0
$877$	−19.9695	−0.674322	−0.337161	−	0.941447i	$-0.609467\pi$
$877$	−19.9695	−0.674322	−0.337161	+	0.941447i	$0.609467\pi$
$878$	0	0
$879$	−55.1429	−1.85992
$880$	0	0
$881$	36.1829	1.21903	0.609517	−	0.792773i	$-0.291364\pi$
$881$	36.1829	1.21903	0.609517	+	0.792773i	$0.291364\pi$
$882$	0	0
$883$	−3.33445	−0.112213	−0.0561066	−	0.998425i	$-0.517869\pi$
$883$	−3.33445	−0.112213	−0.0561066	+	0.998425i	$0.517869\pi$
$884$	0	0
$885$	−21.7990	−0.732766
$886$	0	0
$887$	27.0791	0.909228	0.454614	−	0.890689i	$-0.349777\pi$
$887$	27.0791	0.909228	0.454614	+	0.890689i	$0.349777\pi$
$888$	0	0
$889$	−49.3551	−1.65532
$890$	0	0
$891$	6.43918	0.215721
$892$	0	0
$893$	−11.7645	−0.393683
$894$	0	0
$895$	−15.9622	−0.533557
$896$	0	0
$897$	7.43052	0.248098
$898$	0	0
$899$	52.1710	1.74000
$900$	0	0
$901$	−40.4317	−1.34698
$902$	0	0
$903$	−117.931	−3.92450
$904$	0	0
$905$	48.3093	1.60585
$906$	0	0
$907$	25.6784	0.852636	0.426318	−	0.904573i	$-0.359810\pi$
$907$	25.6784	0.852636	0.426318	+	0.904573i	$0.359810\pi$
$908$	0	0
$909$	−97.9276	−3.24805
$910$	0	0
$911$	−26.3249	−0.872183	−0.436092	−	0.899902i	$-0.643638\pi$
$911$	−26.3249	−0.872183	−0.436092	+	0.899902i	$0.643638\pi$
$912$	0	0
$913$	52.0960	1.72412
$914$	0	0
$915$	−27.8716	−0.921408
$916$	0	0
$917$	9.01854	0.297819
$918$	0	0
$919$	−43.2092	−1.42534	−0.712670	−	0.701499i	$-0.752514\pi$
$919$	−43.2092	−1.42534	−0.712670	+	0.701499i	$0.752514\pi$
$920$	0	0
$921$	−87.2515	−2.87503
$922$	0	0
$923$	20.3975	0.671393
$924$	0	0
$925$	−21.2913	−0.700053
$926$	0	0
$927$	4.37027	0.143538
$928$	0	0
$929$	48.7540	1.59957	0.799784	−	0.600288i	$-0.204948\pi$
$929$	48.7540	1.59957	0.799784	+	0.600288i	$0.204948\pi$
$930$	0	0
$931$	−23.2719	−0.762705
$932$	0	0
$933$	−32.8716	−1.07617
$934$	0	0
$935$	−85.1420	−2.78444
$936$	0	0
$937$	13.6113	0.444663	0.222332	−	0.974971i	$-0.428633\pi$
$937$	13.6113	0.444663	0.222332	+	0.974971i	$0.428633\pi$
$938$	0	0
$939$	−29.3838	−0.958904
$940$	0	0
$941$	−16.7996	−0.547652	−0.273826	−	0.961779i	$-0.588289\pi$
$941$	−16.7996	−0.547652	−0.273826	+	0.961779i	$0.588289\pi$
$942$	0	0
$943$	9.73821	0.317120
$944$	0	0
$945$	60.3054	1.96173
$946$	0	0
$947$	−39.8921	−1.29632	−0.648159	−	0.761505i	$-0.724461\pi$
$947$	−39.8921	−1.29632	−0.648159	+	0.761505i	$0.724461\pi$
$948$	0	0
$949$	−22.3253	−0.724711
$950$	0	0
$951$	30.2755	0.981750
$952$	0	0
$953$	−8.71093	−0.282175	−0.141087	−	0.989997i	$-0.545060\pi$
$953$	−8.71093	−0.282175	−0.141087	+	0.989997i	$0.545060\pi$
$954$	0	0
$955$	−48.7741	−1.57829
$956$	0	0
$957$	100.075	3.23497
$958$	0	0
$959$	−13.6018	−0.439226
$960$	0	0
$961$	1.38083	0.0445430
$962$	0	0
$963$	−90.5236	−2.91708
$964$	0	0
$965$	40.1165	1.29140
$966$	0	0
$967$	−29.5255	−0.949476	−0.474738	−	0.880127i	$-0.657457\pi$
$967$	−29.5255	−0.949476	−0.474738	+	0.880127i	$0.657457\pi$
$968$	0	0
$969$	79.2482	2.54582
$970$	0	0
$971$	−8.31884	−0.266964	−0.133482	−	0.991051i	$-0.542616\pi$
$971$	−8.31884	−0.266964	−0.133482	+	0.991051i	$0.542616\pi$
$972$	0	0
$973$	−12.6706	−0.406201
$974$	0	0
$975$	−19.5100	−0.624820
$976$	0	0
$977$	−25.5361	−0.816972	−0.408486	−	0.912765i	$-0.633943\pi$
$977$	−25.5361	−0.816972	−0.408486	+	0.912765i	$0.633943\pi$
$978$	0	0
$979$	24.8555	0.794384
$980$	0	0
$981$	−89.6586	−2.86258
$982$	0	0
$983$	44.6712	1.42479	0.712394	−	0.701779i	$-0.247611\pi$
$983$	44.6712	1.42479	0.712394	+	0.701779i	$0.247611\pi$
$984$	0	0
$985$	62.3488	1.98660
$986$	0	0
$987$	−36.4321	−1.15965
$988$	0	0
$989$	−10.2122	−0.324729
$990$	0	0
$991$	−1.18270	−0.0375696	−0.0187848	−	0.999824i	$-0.505980\pi$
$991$	−1.18270	−0.0375696	−0.0187848	+	0.999824i	$0.505980\pi$
$992$	0	0
$993$	49.9647	1.58558
$994$	0	0
$995$	−0.521800	−0.0165422
$996$	0	0
$997$	6.46140	0.204635	0.102317	−	0.994752i	$-0.467374\pi$
$997$	6.46140	0.204635	0.102317	+	0.994752i	$0.467374\pi$
$998$	0	0
$999$	52.7976	1.67044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.3		28
4.3	odd	2		4024.2.a.d.1.26	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.26	✓	28	4.3	odd	2
8048.2.a.v.1.3		28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.84528 q^{3} -2.72112 q^{5} +3.71681 q^{7} +5.09562 q^{9} +O(q^{10})\)	\(q-2.84528 q^{3} -2.72112 q^{5} +3.71681 q^{7} +5.09562 q^{9} +3.83632 q^{11} +2.85172 q^{13} +7.74235 q^{15} +8.15606 q^{17} -3.41495 q^{19} -10.5754 q^{21} -0.915770 q^{23} +2.40450 q^{25} -5.96262 q^{27} -9.16823 q^{29} -5.69042 q^{31} -10.9154 q^{33} -10.1139 q^{35} -8.85477 q^{37} -8.11396 q^{39} -10.6339 q^{41} +11.1515 q^{43} -13.8658 q^{45} +3.44500 q^{47} +6.81471 q^{49} -23.2063 q^{51} -4.95726 q^{53} -10.4391 q^{55} +9.71648 q^{57} -2.81555 q^{59} -3.59989 q^{61} +18.9395 q^{63} -7.75989 q^{65} +9.01369 q^{67} +2.60562 q^{69} +7.15270 q^{71} -7.82872 q^{73} -6.84147 q^{75} +14.2589 q^{77} -13.4571 q^{79} +1.67848 q^{81} +13.5797 q^{83} -22.1936 q^{85} +26.0862 q^{87} +6.47898 q^{89} +10.5993 q^{91} +16.1908 q^{93} +9.29248 q^{95} +16.0682 q^{97} +19.5484 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

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