LMFDB - Embedded newform 8048.2.a.v.1.25

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			1.25
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)

$f(q)$	$=$	$q+2.25613 q^{3} +2.34171 q^{5} -2.23682 q^{7} +2.09012 q^{9} +O(q^{10})$	$q+2.25613 q^{3} +2.34171 q^{5} -2.23682 q^{7} +2.09012 q^{9} -4.83651 q^{11} +0.834816 q^{13} +5.28320 q^{15} -5.54113 q^{17} +1.83270 q^{19} -5.04657 q^{21} +7.33684 q^{23} +0.483606 q^{25} -2.05280 q^{27} -3.35529 q^{29} -0.777446 q^{31} -10.9118 q^{33} -5.23799 q^{35} -7.78842 q^{37} +1.88345 q^{39} -9.17412 q^{41} +11.5622 q^{43} +4.89447 q^{45} +1.92508 q^{47} -1.99662 q^{49} -12.5015 q^{51} -2.04619 q^{53} -11.3257 q^{55} +4.13481 q^{57} -0.668827 q^{59} -8.64673 q^{61} -4.67524 q^{63} +1.95490 q^{65} -10.6650 q^{67} +16.5529 q^{69} +3.57854 q^{71} +5.69843 q^{73} +1.09108 q^{75} +10.8184 q^{77} +8.38518 q^{79} -10.9018 q^{81} -13.7104 q^{83} -12.9757 q^{85} -7.56997 q^{87} -8.41401 q^{89} -1.86734 q^{91} -1.75402 q^{93} +4.29165 q^{95} -3.23181 q^{97} -10.1089 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.25613	1.30258	0.651289	−	0.758830i	$-0.274229\pi$
$3$	2.25613	1.30258	0.651289	+	0.758830i	$0.274229\pi$
$4$	0	0
$5$	2.34171	1.04724	0.523622	−	0.851951i	$-0.324580\pi$
$5$	2.34171	1.04724	0.523622	+	0.851951i	$0.324580\pi$
$6$	0	0
$7$	−2.23682	−0.845440	−0.422720	−	0.906260i	$-0.638925\pi$
$7$	−2.23682	−0.845440	−0.422720	+	0.906260i	$0.638925\pi$
$8$	0	0
$9$	2.09012	0.696708
$10$	0	0
$11$	−4.83651	−1.45826	−0.729132	−	0.684374i	$-0.760076\pi$
$11$	−4.83651	−1.45826	−0.729132	+	0.684374i	$0.760076\pi$
$12$	0	0
$13$	0.834816	0.231536	0.115768	−	0.993276i	$-0.463067\pi$
$13$	0.834816	0.231536	0.115768	+	0.993276i	$0.463067\pi$
$14$	0	0
$15$	5.28320	1.36412
$16$	0	0
$17$	−5.54113	−1.34392	−0.671961	−	0.740587i	$-0.734548\pi$
$17$	−5.54113	−1.34392	−0.671961	+	0.740587i	$0.734548\pi$
$18$	0	0
$19$	1.83270	0.420450	0.210225	−	0.977653i	$-0.432580\pi$
$19$	1.83270	0.420450	0.210225	+	0.977653i	$0.432580\pi$
$20$	0	0
$21$	−5.04657	−1.10125
$22$	0	0
$23$	7.33684	1.52984	0.764919	−	0.644127i	$-0.222779\pi$
$23$	7.33684	1.52984	0.764919	+	0.644127i	$0.222779\pi$
$24$	0	0
$25$	0.483606	0.0967212
$26$	0	0
$27$	−2.05280	−0.395061
$28$	0	0
$29$	−3.35529	−0.623062	−0.311531	−	0.950236i	$-0.600842\pi$
$29$	−3.35529	−0.623062	−0.311531	+	0.950236i	$0.600842\pi$
$30$	0	0
$31$	−0.777446	−0.139633	−0.0698167	−	0.997560i	$-0.522241\pi$
$31$	−0.777446	−0.139633	−0.0698167	+	0.997560i	$0.522241\pi$
$32$	0	0
$33$	−10.9118	−1.89950
$34$	0	0
$35$	−5.23799	−0.885383
$36$	0	0
$37$	−7.78842	−1.28041	−0.640204	−	0.768205i	$-0.721150\pi$
$37$	−7.78842	−1.28041	−0.640204	+	0.768205i	$0.721150\pi$
$38$	0	0
$39$	1.88345	0.301594
$40$	0	0
$41$	−9.17412	−1.43276	−0.716379	−	0.697712i	$-0.754202\pi$
$41$	−9.17412	−1.43276	−0.716379	+	0.697712i	$0.754202\pi$
$42$	0	0
$43$	11.5622	1.76321	0.881606	−	0.471987i	$-0.156463\pi$
$43$	11.5622	1.76321	0.881606	+	0.471987i	$0.156463\pi$
$44$	0	0
$45$	4.89447	0.729624
$46$	0	0
$47$	1.92508	0.280801	0.140401	−	0.990095i	$-0.455161\pi$
$47$	1.92508	0.280801	0.140401	+	0.990095i	$0.455161\pi$
$48$	0	0
$49$	−1.99662	−0.285231
$50$	0	0
$51$	−12.5015	−1.75056
$52$	0	0
$53$	−2.04619	−0.281066	−0.140533	−	0.990076i	$-0.544882\pi$
$53$	−2.04619	−0.281066	−0.140533	+	0.990076i	$0.544882\pi$
$54$	0	0
$55$	−11.3257	−1.52716
$56$	0	0
$57$	4.13481	0.547669
$58$	0	0
$59$	−0.668827	−0.0870739	−0.0435369	−	0.999052i	$-0.513863\pi$
$59$	−0.668827	−0.0870739	−0.0435369	+	0.999052i	$0.513863\pi$
$60$	0	0
$61$	−8.64673	−1.10710	−0.553550	−	0.832816i	$-0.686727\pi$
$61$	−8.64673	−1.10710	−0.553550	+	0.832816i	$0.686727\pi$
$62$	0	0
$63$	−4.67524	−0.589025
$64$	0	0
$65$	1.95490	0.242475
$66$	0	0
$67$	−10.6650	−1.30294	−0.651470	−	0.758675i	$-0.725847\pi$
$67$	−10.6650	−1.30294	−0.651470	+	0.758675i	$0.725847\pi$
$68$	0	0
$69$	16.5529	1.99273
$70$	0	0
$71$	3.57854	0.424694	0.212347	−	0.977194i	$-0.431889\pi$
$71$	3.57854	0.424694	0.212347	+	0.977194i	$0.431889\pi$
$72$	0	0
$73$	5.69843	0.666951	0.333476	−	0.942759i	$-0.391779\pi$
$73$	5.69843	0.666951	0.333476	+	0.942759i	$0.391779\pi$
$74$	0	0
$75$	1.09108	0.125987
$76$	0	0
$77$	10.8184	1.23287
$78$	0	0
$79$	8.38518	0.943407	0.471703	−	0.881757i	$-0.343639\pi$
$79$	8.38518	0.943407	0.471703	+	0.881757i	$0.343639\pi$
$80$	0	0
$81$	−10.9018	−1.21131
$82$	0	0
$83$	−13.7104	−1.50491	−0.752454	−	0.658644i	$-0.771130\pi$
$83$	−13.7104	−1.50491	−0.752454	+	0.658644i	$0.771130\pi$
$84$	0	0
$85$	−12.9757	−1.40741
$86$	0	0
$87$	−7.56997	−0.811586
$88$	0	0
$89$	−8.41401	−0.891883	−0.445942	−	0.895062i	$-0.647131\pi$
$89$	−8.41401	−0.891883	−0.445942	+	0.895062i	$0.647131\pi$
$90$	0	0
$91$	−1.86734	−0.195750
$92$	0	0
$93$	−1.75402	−0.181883
$94$	0	0
$95$	4.29165	0.440314
$96$	0	0
$97$	−3.23181	−0.328140	−0.164070	−	0.986449i	$-0.552462\pi$
$97$	−3.23181	−0.328140	−0.164070	+	0.986449i	$0.552462\pi$
$98$	0	0
$99$	−10.1089	−1.01598
$100$	0	0
$101$	−9.70274	−0.965459	−0.482730	−	0.875769i	$-0.660355\pi$
$101$	−9.70274	−0.965459	−0.482730	+	0.875769i	$0.660355\pi$
$102$	0	0
$103$	−12.2046	−1.20256	−0.601279	−	0.799039i	$-0.705342\pi$
$103$	−12.2046	−1.20256	−0.601279	+	0.799039i	$0.705342\pi$
$104$	0	0
$105$	−11.8176	−1.15328
$106$	0	0
$107$	8.88079	0.858539	0.429269	−	0.903177i	$-0.358771\pi$
$107$	8.88079	0.858539	0.429269	+	0.903177i	$0.358771\pi$
$108$	0	0
$109$	6.62270	0.634340	0.317170	−	0.948369i	$-0.397267\pi$
$109$	6.62270	0.634340	0.317170	+	0.948369i	$0.397267\pi$
$110$	0	0
$111$	−17.5717	−1.66783
$112$	0	0
$113$	15.3908	1.44784	0.723922	−	0.689882i	$-0.242338\pi$
$113$	15.3908	1.44784	0.723922	+	0.689882i	$0.242338\pi$
$114$	0	0
$115$	17.1808	1.60211
$116$	0	0
$117$	1.74487	0.161313
$118$	0	0
$119$	12.3945	1.13620
$120$	0	0
$121$	12.3918	1.12653
$122$	0	0
$123$	−20.6980	−1.86628
$124$	0	0
$125$	−10.5761	−0.945954
$126$	0	0
$127$	−18.6489	−1.65482	−0.827410	−	0.561598i	$-0.810187\pi$
$127$	−18.6489	−1.65482	−0.827410	+	0.561598i	$0.810187\pi$
$128$	0	0
$129$	26.0857	2.29672
$130$	0	0
$131$	17.6208	1.53954	0.769769	−	0.638323i	$-0.220371\pi$
$131$	17.6208	1.53954	0.769769	+	0.638323i	$0.220371\pi$
$132$	0	0
$133$	−4.09943	−0.355466
$134$	0	0
$135$	−4.80706	−0.413726
$136$	0	0
$137$	20.4354	1.74591	0.872957	−	0.487797i	$-0.162199\pi$
$137$	20.4354	1.74591	0.872957	+	0.487797i	$0.162199\pi$
$138$	0	0
$139$	−3.68318	−0.312404	−0.156202	−	0.987725i	$-0.549925\pi$
$139$	−3.68318	−0.312404	−0.156202	+	0.987725i	$0.549925\pi$
$140$	0	0
$141$	4.34322	0.365765
$142$	0	0
$143$	−4.03760	−0.337641
$144$	0	0
$145$	−7.85711	−0.652498
$146$	0	0
$147$	−4.50463	−0.371535
$148$	0	0
$149$	−5.97233	−0.489272	−0.244636	−	0.969615i	$-0.578669\pi$
$149$	−5.97233	−0.489272	−0.244636	+	0.969615i	$0.578669\pi$
$150$	0	0
$151$	22.4868	1.82995	0.914974	−	0.403513i	$-0.132211\pi$
$151$	22.4868	1.82995	0.914974	+	0.403513i	$0.132211\pi$
$152$	0	0
$153$	−11.5816	−0.936321
$154$	0	0
$155$	−1.82055	−0.146230
$156$	0	0
$157$	7.15379	0.570934	0.285467	−	0.958388i	$-0.407851\pi$
$157$	7.15379	0.570934	0.285467	+	0.958388i	$0.407851\pi$
$158$	0	0
$159$	−4.61648	−0.366111
$160$	0	0
$161$	−16.4112	−1.29339
$162$	0	0
$163$	−19.2023	−1.50404	−0.752020	−	0.659141i	$-0.770920\pi$
$163$	−19.2023	−1.50404	−0.752020	+	0.659141i	$0.770920\pi$
$164$	0	0
$165$	−25.5523	−1.98924
$166$	0	0
$167$	13.7445	1.06358	0.531790	−	0.846876i	$-0.321519\pi$
$167$	13.7445	1.06358	0.531790	+	0.846876i	$0.321519\pi$
$168$	0	0
$169$	−12.3031	−0.946391
$170$	0	0
$171$	3.83057	0.292931
$172$	0	0
$173$	−12.6597	−0.962498	−0.481249	−	0.876584i	$-0.659817\pi$
$173$	−12.6597	−0.962498	−0.481249	+	0.876584i	$0.659817\pi$
$174$	0	0
$175$	−1.08174	−0.0817720
$176$	0	0
$177$	−1.50896	−0.113420
$178$	0	0
$179$	−26.2307	−1.96057	−0.980287	−	0.197581i	$-0.936691\pi$
$179$	−26.2307	−1.96057	−0.980287	+	0.197581i	$0.936691\pi$
$180$	0	0
$181$	−17.1620	−1.27564	−0.637822	−	0.770184i	$-0.720164\pi$
$181$	−17.1620	−1.27564	−0.637822	+	0.770184i	$0.720164\pi$
$182$	0	0
$183$	−19.5082	−1.44208
$184$	0	0
$185$	−18.2382	−1.34090
$186$	0	0
$187$	26.7997	1.95979
$188$	0	0
$189$	4.59175	0.334001
$190$	0	0
$191$	−17.3138	−1.25278	−0.626390	−	0.779510i	$-0.715468\pi$
$191$	−17.3138	−1.25278	−0.626390	+	0.779510i	$0.715468\pi$
$192$	0	0
$193$	−20.5094	−1.47630	−0.738151	−	0.674636i	$-0.764301\pi$
$193$	−20.5094	−1.47630	−0.738151	+	0.674636i	$0.764301\pi$
$194$	0	0
$195$	4.41050	0.315843
$196$	0	0
$197$	4.84527	0.345211	0.172606	−	0.984991i	$-0.444781\pi$
$197$	4.84527	0.345211	0.172606	+	0.984991i	$0.444781\pi$
$198$	0	0
$199$	−25.7280	−1.82381	−0.911905	−	0.410402i	$-0.865388\pi$
$199$	−25.7280	−1.82381	−0.911905	+	0.410402i	$0.865388\pi$
$200$	0	0
$201$	−24.0617	−1.69718
$202$	0	0
$203$	7.50519	0.526761
$204$	0	0
$205$	−21.4831	−1.50045
$206$	0	0
$207$	15.3349	1.06585
$208$	0	0
$209$	−8.86388	−0.613127
$210$	0	0
$211$	25.8319	1.77834	0.889171	−	0.457574i	$-0.151282\pi$
$211$	25.8319	1.77834	0.889171	+	0.457574i	$0.151282\pi$
$212$	0	0
$213$	8.07365	0.553197
$214$	0	0
$215$	27.0752	1.84651
$216$	0	0
$217$	1.73901	0.118052
$218$	0	0
$219$	12.8564	0.868756
$220$	0	0
$221$	−4.62582	−0.311166
$222$	0	0
$223$	−12.2658	−0.821376	−0.410688	−	0.911776i	$-0.634712\pi$
$223$	−12.2658	−0.821376	−0.410688	+	0.911776i	$0.634712\pi$
$224$	0	0
$225$	1.01080	0.0673865
$226$	0	0
$227$	15.1894	1.00816	0.504078	−	0.863658i	$-0.331833\pi$
$227$	15.1894	1.00816	0.504078	+	0.863658i	$0.331833\pi$
$228$	0	0
$229$	2.63534	0.174148	0.0870741	−	0.996202i	$-0.472248\pi$
$229$	2.63534	0.174148	0.0870741	+	0.996202i	$0.472248\pi$
$230$	0	0
$231$	24.4078	1.60591
$232$	0	0
$233$	−9.32971	−0.611209	−0.305605	−	0.952158i	$-0.598859\pi$
$233$	−9.32971	−0.611209	−0.305605	+	0.952158i	$0.598859\pi$
$234$	0	0
$235$	4.50797	0.294067
$236$	0	0
$237$	18.9181	1.22886
$238$	0	0
$239$	24.0306	1.55441	0.777204	−	0.629249i	$-0.216637\pi$
$239$	24.0306	1.55441	0.777204	+	0.629249i	$0.216637\pi$
$240$	0	0
$241$	3.53600	0.227773	0.113887	−	0.993494i	$-0.463670\pi$
$241$	3.53600	0.227773	0.113887	+	0.993494i	$0.463670\pi$
$242$	0	0
$243$	−18.4374	−1.18276
$244$	0	0
$245$	−4.67550	−0.298706
$246$	0	0
$247$	1.52997	0.0973495
$248$	0	0
$249$	−30.9324	−1.96026
$250$	0	0
$251$	13.9214	0.878709	0.439354	−	0.898314i	$-0.355207\pi$
$251$	13.9214	0.878709	0.439354	+	0.898314i	$0.355207\pi$
$252$	0	0
$253$	−35.4847	−2.23091
$254$	0	0
$255$	−29.2749	−1.83327
$256$	0	0
$257$	21.9636	1.37005	0.685025	−	0.728519i	$-0.259791\pi$
$257$	21.9636	1.37005	0.685025	+	0.728519i	$0.259791\pi$
$258$	0	0
$259$	17.4213	1.08251
$260$	0	0
$261$	−7.01297	−0.434092
$262$	0	0
$263$	−14.8106	−0.913260	−0.456630	−	0.889657i	$-0.650944\pi$
$263$	−14.8106	−0.913260	−0.456630	+	0.889657i	$0.650944\pi$
$264$	0	0
$265$	−4.79160	−0.294345
$266$	0	0
$267$	−18.9831	−1.16175
$268$	0	0
$269$	−2.15656	−0.131488	−0.0657438	−	0.997837i	$-0.520942\pi$
$269$	−2.15656	−0.131488	−0.0657438	+	0.997837i	$0.520942\pi$
$270$	0	0
$271$	−10.6666	−0.647948	−0.323974	−	0.946066i	$-0.605019\pi$
$271$	−10.6666	−0.647948	−0.323974	+	0.946066i	$0.605019\pi$
$272$	0	0
$273$	−4.21296	−0.254980
$274$	0	0
$275$	−2.33897	−0.141045
$276$	0	0
$277$	9.51598	0.571760	0.285880	−	0.958265i	$-0.407714\pi$
$277$	9.51598	0.571760	0.285880	+	0.958265i	$0.407714\pi$
$278$	0	0
$279$	−1.62496	−0.0972838
$280$	0	0
$281$	−17.2666	−1.03004	−0.515020	−	0.857178i	$-0.672216\pi$
$281$	−17.2666	−1.03004	−0.515020	+	0.857178i	$0.672216\pi$
$282$	0	0
$283$	19.4806	1.15800	0.579001	−	0.815327i	$-0.303443\pi$
$283$	19.4806	1.15800	0.579001	+	0.815327i	$0.303443\pi$
$284$	0	0
$285$	9.68253	0.573544
$286$	0	0
$287$	20.5209	1.21131
$288$	0	0
$289$	13.7041	0.806124
$290$	0	0
$291$	−7.29138	−0.427428
$292$	0	0
$293$	17.4790	1.02113	0.510566	−	0.859838i	$-0.329436\pi$
$293$	17.4790	1.02113	0.510566	+	0.859838i	$0.329436\pi$
$294$	0	0
$295$	−1.56620	−0.0911876
$296$	0	0
$297$	9.92838	0.576103
$298$	0	0
$299$	6.12491	0.354213
$300$	0	0
$301$	−25.8625	−1.49069
$302$	0	0
$303$	−21.8907	−1.25759
$304$	0	0
$305$	−20.2481	−1.15941
$306$	0	0
$307$	10.1714	0.580513	0.290256	−	0.956949i	$-0.406259\pi$
$307$	10.1714	0.580513	0.290256	+	0.956949i	$0.406259\pi$
$308$	0	0
$309$	−27.5353	−1.56643
$310$	0	0
$311$	−20.7723	−1.17789	−0.588946	−	0.808172i	$-0.700457\pi$
$311$	−20.7723	−1.17789	−0.588946	+	0.808172i	$0.700457\pi$
$312$	0	0
$313$	−9.40277	−0.531476	−0.265738	−	0.964045i	$-0.585616\pi$
$313$	−9.40277	−0.531476	−0.265738	+	0.964045i	$0.585616\pi$
$314$	0	0
$315$	−10.9481	−0.616853
$316$	0	0
$317$	22.6489	1.27209	0.636046	−	0.771651i	$-0.280569\pi$
$317$	22.6489	1.27209	0.636046	+	0.771651i	$0.280569\pi$
$318$	0	0
$319$	16.2279	0.908588
$320$	0	0
$321$	20.0362	1.11831
$322$	0	0
$323$	−10.1552	−0.565052
$324$	0	0
$325$	0.403722	0.0223945
$326$	0	0
$327$	14.9417	0.826277
$328$	0	0
$329$	−4.30606	−0.237401
$330$	0	0
$331$	−17.8139	−0.979143	−0.489571	−	0.871963i	$-0.662847\pi$
$331$	−17.8139	−0.979143	−0.489571	+	0.871963i	$0.662847\pi$
$332$	0	0
$333$	−16.2788	−0.892071
$334$	0	0
$335$	−24.9744	−1.36450
$336$	0	0
$337$	4.74895	0.258692	0.129346	−	0.991600i	$-0.458712\pi$
$337$	4.74895	0.258692	0.129346	+	0.991600i	$0.458712\pi$
$338$	0	0
$339$	34.7236	1.88593
$340$	0	0
$341$	3.76013	0.203622
$342$	0	0
$343$	20.1239	1.08659
$344$	0	0
$345$	38.7620	2.08688
$346$	0	0
$347$	−14.3445	−0.770050	−0.385025	−	0.922906i	$-0.625807\pi$
$347$	−14.3445	−0.770050	−0.385025	+	0.922906i	$0.625807\pi$
$348$	0	0
$349$	1.44132	0.0771523	0.0385762	−	0.999256i	$-0.487718\pi$
$349$	1.44132	0.0771523	0.0385762	+	0.999256i	$0.487718\pi$
$350$	0	0
$351$	−1.71371	−0.0914710
$352$	0	0
$353$	−4.80296	−0.255636	−0.127818	−	0.991798i	$-0.540797\pi$
$353$	−4.80296	−0.255636	−0.127818	+	0.991798i	$0.540797\pi$
$354$	0	0
$355$	8.37990	0.444759
$356$	0	0
$357$	27.9637	1.48000
$358$	0	0
$359$	−32.2320	−1.70114	−0.850570	−	0.525861i	$-0.823743\pi$
$359$	−32.2320	−1.70114	−0.850570	+	0.525861i	$0.823743\pi$
$360$	0	0
$361$	−15.6412	−0.823222
$362$	0	0
$363$	27.9576	1.46739
$364$	0	0
$365$	13.3441	0.698461
$366$	0	0
$367$	14.3212	0.747559	0.373780	−	0.927518i	$-0.378062\pi$
$367$	14.3212	0.747559	0.373780	+	0.927518i	$0.378062\pi$
$368$	0	0
$369$	−19.1751	−0.998214
$370$	0	0
$371$	4.57698	0.237625
$372$	0	0
$373$	6.84129	0.354228	0.177114	−	0.984190i	$-0.443324\pi$
$373$	6.84129	0.354228	0.177114	+	0.984190i	$0.443324\pi$
$374$	0	0
$375$	−23.8610	−1.23218
$376$	0	0
$377$	−2.80105	−0.144261
$378$	0	0
$379$	17.0591	0.876266	0.438133	−	0.898910i	$-0.355640\pi$
$379$	17.0591	0.876266	0.438133	+	0.898910i	$0.355640\pi$
$380$	0	0
$381$	−42.0743	−2.15553
$382$	0	0
$383$	17.0812	0.872810	0.436405	−	0.899750i	$-0.356252\pi$
$383$	17.0812	0.872810	0.436405	+	0.899750i	$0.356252\pi$
$384$	0	0
$385$	25.3336	1.29112
$386$	0	0
$387$	24.1663	1.22844
$388$	0	0
$389$	−12.0141	−0.609137	−0.304568	−	0.952491i	$-0.598512\pi$
$389$	−12.0141	−0.609137	−0.304568	+	0.952491i	$0.598512\pi$
$390$	0	0
$391$	−40.6544	−2.05598
$392$	0	0
$393$	39.7549	2.00537
$394$	0	0
$395$	19.6357	0.987978
$396$	0	0
$397$	−32.1001	−1.61106	−0.805530	−	0.592555i	$-0.798119\pi$
$397$	−32.1001	−1.61106	−0.805530	+	0.592555i	$0.798119\pi$
$398$	0	0
$399$	−9.24885	−0.463022
$400$	0	0
$401$	−5.36837	−0.268084	−0.134042	−	0.990976i	$-0.542796\pi$
$401$	−5.36837	−0.268084	−0.134042	+	0.990976i	$0.542796\pi$
$402$	0	0
$403$	−0.649024	−0.0323302
$404$	0	0
$405$	−25.5287	−1.26853
$406$	0	0
$407$	37.6688	1.86717
$408$	0	0
$409$	29.4937	1.45837	0.729184	−	0.684318i	$-0.239900\pi$
$409$	29.4937	1.45837	0.729184	+	0.684318i	$0.239900\pi$
$410$	0	0
$411$	46.1049	2.27419
$412$	0	0
$413$	1.49605	0.0736158
$414$	0	0
$415$	−32.1057	−1.57601
$416$	0	0
$417$	−8.30974	−0.406930
$418$	0	0
$419$	28.6861	1.40141	0.700705	−	0.713451i	$-0.252869\pi$
$419$	28.6861	1.40141	0.700705	+	0.713451i	$0.252869\pi$
$420$	0	0
$421$	27.8197	1.35585	0.677924	−	0.735132i	$-0.262880\pi$
$421$	27.8197	1.35585	0.677924	+	0.735132i	$0.262880\pi$
$422$	0	0
$423$	4.02365	0.195636
$424$	0	0
$425$	−2.67972	−0.129986
$426$	0	0
$427$	19.3412	0.935987
$428$	0	0
$429$	−9.10934	−0.439803
$430$	0	0
$431$	21.4518	1.03330	0.516649	−	0.856197i	$-0.327179\pi$
$431$	21.4518	1.03330	0.516649	+	0.856197i	$0.327179\pi$
$432$	0	0
$433$	31.3986	1.50892	0.754459	−	0.656347i	$-0.227899\pi$
$433$	31.3986	1.50892	0.754459	+	0.656347i	$0.227899\pi$
$434$	0	0
$435$	−17.7267	−0.849929
$436$	0	0
$437$	13.4462	0.643221
$438$	0	0
$439$	−22.7221	−1.08447	−0.542234	−	0.840228i	$-0.682421\pi$
$439$	−22.7221	−1.08447	−0.542234	+	0.840228i	$0.682421\pi$
$440$	0	0
$441$	−4.17318	−0.198723
$442$	0	0
$443$	12.8897	0.612410	0.306205	−	0.951966i	$-0.400941\pi$
$443$	12.8897	0.612410	0.306205	+	0.951966i	$0.400941\pi$
$444$	0	0
$445$	−19.7032	−0.934020
$446$	0	0
$447$	−13.4744	−0.637315
$448$	0	0
$449$	29.8866	1.41043	0.705217	−	0.708991i	$-0.250850\pi$
$449$	29.8866	1.41043	0.705217	+	0.708991i	$0.250850\pi$
$450$	0	0
$451$	44.3707	2.08934
$452$	0	0
$453$	50.7331	2.38365
$454$	0	0
$455$	−4.37276	−0.204998
$456$	0	0
$457$	9.07693	0.424601	0.212301	−	0.977204i	$-0.431904\pi$
$457$	9.07693	0.424601	0.212301	+	0.977204i	$0.431904\pi$
$458$	0	0
$459$	11.3748	0.530931
$460$	0	0
$461$	0.239249	0.0111429	0.00557147	−	0.999984i	$-0.498227\pi$
$461$	0.239249	0.0111429	0.00557147	+	0.999984i	$0.498227\pi$
$462$	0	0
$463$	11.2908	0.524726	0.262363	−	0.964969i	$-0.415498\pi$
$463$	11.2908	0.524726	0.262363	+	0.964969i	$0.415498\pi$
$464$	0	0
$465$	−4.10741	−0.190476
$466$	0	0
$467$	−13.9345	−0.644809	−0.322405	−	0.946602i	$-0.604491\pi$
$467$	−13.9345	−0.644809	−0.322405	+	0.946602i	$0.604491\pi$
$468$	0	0
$469$	23.8558	1.10156
$470$	0	0
$471$	16.1399	0.743686
$472$	0	0
$473$	−55.9205	−2.57123
$474$	0	0
$475$	0.886305	0.0406665
$476$	0	0
$477$	−4.27680	−0.195821
$478$	0	0
$479$	−39.9989	−1.82760	−0.913798	−	0.406169i	$-0.866864\pi$
$479$	−39.9989	−1.82760	−0.913798	+	0.406169i	$0.866864\pi$
$480$	0	0
$481$	−6.50190	−0.296461
$482$	0	0
$483$	−37.0259	−1.68474
$484$	0	0
$485$	−7.56795	−0.343643
$486$	0	0
$487$	4.91070	0.222525	0.111263	−	0.993791i	$-0.464511\pi$
$487$	4.91070	0.222525	0.111263	+	0.993791i	$0.464511\pi$
$488$	0	0
$489$	−43.3229	−1.95913
$490$	0	0
$491$	−7.23885	−0.326685	−0.163342	−	0.986569i	$-0.552228\pi$
$491$	−7.23885	−0.326685	−0.163342	+	0.986569i	$0.552228\pi$
$492$	0	0
$493$	18.5921	0.837346
$494$	0	0
$495$	−23.6721	−1.06398
$496$	0	0
$497$	−8.00456	−0.359054
$498$	0	0
$499$	10.8028	0.483600	0.241800	−	0.970326i	$-0.422262\pi$
$499$	10.8028	0.483600	0.241800	+	0.970326i	$0.422262\pi$
$500$	0	0
$501$	31.0094	1.38540
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−22.7210	−1.01107
$506$	0	0
$507$	−27.7574	−1.23275
$508$	0	0
$509$	11.6925	0.518261	0.259131	−	0.965842i	$-0.416564\pi$
$509$	11.6925	0.518261	0.259131	+	0.965842i	$0.416564\pi$
$510$	0	0
$511$	−12.7464	−0.563867
$512$	0	0
$513$	−3.76216	−0.166104
$514$	0	0
$515$	−28.5797	−1.25937
$516$	0	0
$517$	−9.31065	−0.409482
$518$	0	0
$519$	−28.5619	−1.25373
$520$	0	0
$521$	2.55894	0.112109	0.0560546	−	0.998428i	$-0.482148\pi$
$521$	2.55894	0.112109	0.0560546	+	0.998428i	$0.482148\pi$
$522$	0	0
$523$	−6.24097	−0.272898	−0.136449	−	0.990647i	$-0.543569\pi$
$523$	−6.24097	−0.272898	−0.136449	+	0.990647i	$0.543569\pi$
$524$	0	0
$525$	−2.44055	−0.106514
$526$	0	0
$527$	4.30793	0.187656
$528$	0	0
$529$	30.8293	1.34040
$530$	0	0
$531$	−1.39793	−0.0606651
$532$	0	0
$533$	−7.65870	−0.331735
$534$	0	0
$535$	20.7962	0.899100
$536$	0	0
$537$	−59.1798	−2.55380
$538$	0	0
$539$	9.65666	0.415942
$540$	0	0
$541$	−28.9363	−1.24407	−0.622034	−	0.782990i	$-0.713693\pi$
$541$	−28.9363	−1.24407	−0.622034	+	0.782990i	$0.713693\pi$
$542$	0	0
$543$	−38.7198	−1.66162
$544$	0	0
$545$	15.5084	0.664309
$546$	0	0
$547$	8.36518	0.357669	0.178835	−	0.983879i	$-0.442767\pi$
$547$	8.36518	0.357669	0.178835	+	0.983879i	$0.442767\pi$
$548$	0	0
$549$	−18.0727	−0.771326
$550$	0	0
$551$	−6.14924	−0.261966
$552$	0	0
$553$	−18.7562	−0.797594
$554$	0	0
$555$	−41.1478	−1.74663
$556$	0	0
$557$	−9.53757	−0.404120	−0.202060	−	0.979373i	$-0.564764\pi$
$557$	−9.53757	−0.404120	−0.202060	+	0.979373i	$0.564764\pi$
$558$	0	0
$559$	9.65227	0.408247
$560$	0	0
$561$	60.4637	2.55278
$562$	0	0
$563$	2.53621	0.106888	0.0534442	−	0.998571i	$-0.482980\pi$
$563$	2.53621	0.106888	0.0534442	+	0.998571i	$0.482980\pi$
$564$	0	0
$565$	36.0408	1.51625
$566$	0	0
$567$	24.3853	1.02409
$568$	0	0
$569$	−31.1709	−1.30675	−0.653375	−	0.757034i	$-0.726648\pi$
$569$	−31.1709	−1.30675	−0.653375	+	0.757034i	$0.726648\pi$
$570$	0	0
$571$	40.4824	1.69413	0.847067	−	0.531486i	$-0.178366\pi$
$571$	40.4824	1.69413	0.847067	+	0.531486i	$0.178366\pi$
$572$	0	0
$573$	−39.0621	−1.63184
$574$	0	0
$575$	3.54814	0.147968
$576$	0	0
$577$	35.8137	1.49094	0.745472	−	0.666537i	$-0.232224\pi$
$577$	35.8137	1.49094	0.745472	+	0.666537i	$0.232224\pi$
$578$	0	0
$579$	−46.2720	−1.92300
$580$	0	0
$581$	30.6677	1.27231
$582$	0	0
$583$	9.89644	0.409869
$584$	0	0
$585$	4.08598	0.168934
$586$	0	0
$587$	31.1701	1.28653	0.643264	−	0.765644i	$-0.277580\pi$
$587$	31.1701	1.28653	0.643264	+	0.765644i	$0.277580\pi$
$588$	0	0
$589$	−1.42483	−0.0587089
$590$	0	0
$591$	10.9316	0.449665
$592$	0	0
$593$	22.6221	0.928977	0.464488	−	0.885579i	$-0.346238\pi$
$593$	22.6221	0.928977	0.464488	+	0.885579i	$0.346238\pi$
$594$	0	0
$595$	29.0244	1.18988
$596$	0	0
$597$	−58.0457	−2.37565
$598$	0	0
$599$	27.2164	1.11203	0.556016	−	0.831172i	$-0.312329\pi$
$599$	27.2164	1.11203	0.556016	+	0.831172i	$0.312329\pi$
$600$	0	0
$601$	5.31940	0.216983	0.108492	−	0.994097i	$-0.465398\pi$
$601$	5.31940	0.216983	0.108492	+	0.994097i	$0.465398\pi$
$602$	0	0
$603$	−22.2912	−0.907769
$604$	0	0
$605$	29.0181	1.17975
$606$	0	0
$607$	−36.9274	−1.49884	−0.749419	−	0.662096i	$-0.769667\pi$
$607$	−36.9274	−1.49884	−0.749419	+	0.662096i	$0.769667\pi$
$608$	0	0
$609$	16.9327	0.686147
$610$	0	0
$611$	1.60708	0.0650157
$612$	0	0
$613$	−45.7494	−1.84780	−0.923900	−	0.382634i	$-0.875017\pi$
$613$	−45.7494	−1.84780	−0.923900	+	0.382634i	$0.875017\pi$
$614$	0	0
$615$	−48.4687	−1.95445
$616$	0	0
$617$	−20.8664	−0.840052	−0.420026	−	0.907512i	$-0.637979\pi$
$617$	−20.8664	−0.840052	−0.420026	+	0.907512i	$0.637979\pi$
$618$	0	0
$619$	−1.81322	−0.0728795	−0.0364398	−	0.999336i	$-0.511602\pi$
$619$	−1.81322	−0.0728795	−0.0364398	+	0.999336i	$0.511602\pi$
$620$	0	0
$621$	−15.0611	−0.604379
$622$	0	0
$623$	18.8207	0.754034
$624$	0	0
$625$	−27.1842	−1.08737
$626$	0	0
$627$	−19.9981	−0.798646
$628$	0	0
$629$	43.1566	1.72077
$630$	0	0
$631$	−16.7480	−0.666729	−0.333364	−	0.942798i	$-0.608184\pi$
$631$	−16.7480	−0.666729	−0.333364	+	0.942798i	$0.608184\pi$
$632$	0	0
$633$	58.2802	2.31643
$634$	0	0
$635$	−43.6703	−1.73300
$636$	0	0
$637$	−1.66681	−0.0660413
$638$	0	0
$639$	7.47959	0.295888
$640$	0	0
$641$	18.6738	0.737571	0.368786	−	0.929514i	$-0.379774\pi$
$641$	18.6738	0.737571	0.368786	+	0.929514i	$0.379774\pi$
$642$	0	0
$643$	−5.70773	−0.225091	−0.112545	−	0.993647i	$-0.535900\pi$
$643$	−5.70773	−0.225091	−0.112545	+	0.993647i	$0.535900\pi$
$644$	0	0
$645$	61.0852	2.40523
$646$	0	0
$647$	21.1702	0.832286	0.416143	−	0.909299i	$-0.363382\pi$
$647$	21.1702	0.832286	0.416143	+	0.909299i	$0.363382\pi$
$648$	0	0
$649$	3.23479	0.126977
$650$	0	0
$651$	3.92343	0.153772
$652$	0	0
$653$	−8.92038	−0.349082	−0.174541	−	0.984650i	$-0.555844\pi$
$653$	−8.92038	−0.349082	−0.174541	+	0.984650i	$0.555844\pi$
$654$	0	0
$655$	41.2628	1.61227
$656$	0	0
$657$	11.9104	0.464670
$658$	0	0
$659$	−18.1114	−0.705522	−0.352761	−	0.935714i	$-0.614757\pi$
$659$	−18.1114	−0.705522	−0.352761	+	0.935714i	$0.614757\pi$
$660$	0	0
$661$	30.7241	1.19503	0.597515	−	0.801858i	$-0.296155\pi$
$661$	30.7241	1.19503	0.597515	+	0.801858i	$0.296155\pi$
$662$	0	0
$663$	−10.4365	−0.405318
$664$	0	0
$665$	−9.59968	−0.372259
$666$	0	0
$667$	−24.6172	−0.953183
$668$	0	0
$669$	−27.6732	−1.06991
$670$	0	0
$671$	41.8200	1.61444
$672$	0	0
$673$	2.89455	0.111577	0.0557883	−	0.998443i	$-0.482233\pi$
$673$	2.89455	0.111577	0.0557883	+	0.998443i	$0.482233\pi$
$674$	0	0
$675$	−0.992745	−0.0382108
$676$	0	0
$677$	27.8588	1.07070	0.535351	−	0.844630i	$-0.320179\pi$
$677$	27.8588	1.07070	0.535351	+	0.844630i	$0.320179\pi$
$678$	0	0
$679$	7.22899	0.277423
$680$	0	0
$681$	34.2692	1.31320
$682$	0	0
$683$	28.1880	1.07859	0.539293	−	0.842118i	$-0.318692\pi$
$683$	28.1880	1.07859	0.539293	+	0.842118i	$0.318692\pi$
$684$	0	0
$685$	47.8538	1.82840
$686$	0	0
$687$	5.94567	0.226841
$688$	0	0
$689$	−1.70820	−0.0650771
$690$	0	0
$691$	−4.40269	−0.167486	−0.0837432	−	0.996487i	$-0.526688\pi$
$691$	−4.40269	−0.167486	−0.0837432	+	0.996487i	$0.526688\pi$
$692$	0	0
$693$	22.6119	0.858954
$694$	0	0
$695$	−8.62495	−0.327163
$696$	0	0
$697$	50.8350	1.92551
$698$	0	0
$699$	−21.0490	−0.796148
$700$	0	0
$701$	−25.3240	−0.956475	−0.478238	−	0.878230i	$-0.658724\pi$
$701$	−25.3240	−0.956475	−0.478238	+	0.878230i	$0.658724\pi$
$702$	0	0
$703$	−14.2738	−0.538348
$704$	0	0
$705$	10.1706	0.383046
$706$	0	0
$707$	21.7033	0.816238
$708$	0	0
$709$	16.7502	0.629066	0.314533	−	0.949247i	$-0.398152\pi$
$709$	16.7502	0.629066	0.314533	+	0.949247i	$0.398152\pi$
$710$	0	0
$711$	17.5261	0.657279
$712$	0	0
$713$	−5.70400	−0.213616
$714$	0	0
$715$	−9.45488	−0.353592
$716$	0	0
$717$	54.2161	2.02474
$718$	0	0
$719$	7.99858	0.298297	0.149148	−	0.988815i	$-0.452347\pi$
$719$	7.99858	0.298297	0.149148	+	0.988815i	$0.452347\pi$
$720$	0	0
$721$	27.2996	1.01669
$722$	0	0
$723$	7.97767	0.296693
$724$	0	0
$725$	−1.62264	−0.0602633
$726$	0	0
$727$	−34.6208	−1.28401	−0.642007	−	0.766699i	$-0.721898\pi$
$727$	−34.6208	−1.28401	−0.642007	+	0.766699i	$0.721898\pi$
$728$	0	0
$729$	−8.89189	−0.329329
$730$	0	0
$731$	−64.0674	−2.36962
$732$	0	0
$733$	−1.95052	−0.0720441	−0.0360221	−	0.999351i	$-0.511469\pi$
$733$	−1.95052	−0.0720441	−0.0360221	+	0.999351i	$0.511469\pi$
$734$	0	0
$735$	−10.5485	−0.389088
$736$	0	0
$737$	51.5815	1.90003
$738$	0	0
$739$	37.1600	1.36695	0.683476	−	0.729973i	$-0.260467\pi$
$739$	37.1600	1.36695	0.683476	+	0.729973i	$0.260467\pi$
$740$	0	0
$741$	3.45181	0.126805
$742$	0	0
$743$	27.7445	1.01785	0.508924	−	0.860811i	$-0.330043\pi$
$743$	27.7445	1.01785	0.508924	+	0.860811i	$0.330043\pi$
$744$	0	0
$745$	−13.9855	−0.512388
$746$	0	0
$747$	−28.6564	−1.04848
$748$	0	0
$749$	−19.8648	−0.725843
$750$	0	0
$751$	−47.0130	−1.71553	−0.857764	−	0.514043i	$-0.828147\pi$
$751$	−47.0130	−1.71553	−0.857764	+	0.514043i	$0.828147\pi$
$752$	0	0
$753$	31.4084	1.14459
$754$	0	0
$755$	52.6575	1.91640
$756$	0	0
$757$	−31.0594	−1.12887	−0.564436	−	0.825477i	$-0.690906\pi$
$757$	−31.0594	−1.12887	−0.564436	+	0.825477i	$0.690906\pi$
$758$	0	0
$759$	−80.0582	−2.90593
$760$	0	0
$761$	9.98902	0.362102	0.181051	−	0.983474i	$-0.442050\pi$
$761$	9.98902	0.362102	0.181051	+	0.983474i	$0.442050\pi$
$762$	0	0
$763$	−14.8138	−0.536296
$764$	0	0
$765$	−27.1209	−0.980557
$766$	0	0
$767$	−0.558348	−0.0201608
$768$	0	0
$769$	34.7684	1.25378	0.626890	−	0.779108i	$-0.284328\pi$
$769$	34.7684	1.25378	0.626890	+	0.779108i	$0.284328\pi$
$770$	0	0
$771$	49.5527	1.78460
$772$	0	0
$773$	−40.4865	−1.45620	−0.728100	−	0.685471i	$-0.759596\pi$
$773$	−40.4865	−1.45620	−0.728100	+	0.685471i	$0.759596\pi$
$774$	0	0
$775$	−0.375978	−0.0135055
$776$	0	0
$777$	39.3048	1.41005
$778$	0	0
$779$	−16.8134	−0.602403
$780$	0	0
$781$	−17.3076	−0.619316
$782$	0	0
$783$	6.88773	0.246147
$784$	0	0
$785$	16.7521	0.597908
$786$	0	0
$787$	19.3012	0.688013	0.344007	−	0.938967i	$-0.388216\pi$
$787$	19.3012	0.688013	0.344007	+	0.938967i	$0.388216\pi$
$788$	0	0
$789$	−33.4146	−1.18959
$790$	0	0
$791$	−34.4265	−1.22407
$792$	0	0
$793$	−7.21843	−0.256334
$794$	0	0
$795$	−10.8105	−0.383408
$796$	0	0
$797$	15.1738	0.537482	0.268741	−	0.963212i	$-0.413392\pi$
$797$	15.1738	0.537482	0.268741	+	0.963212i	$0.413392\pi$
$798$	0	0
$799$	−10.6671	−0.377375
$800$	0	0
$801$	−17.5863	−0.621383
$802$	0	0
$803$	−27.5605	−0.972590
$804$	0	0
$805$	−38.4303	−1.35449
$806$	0	0
$807$	−4.86547	−0.171273
$808$	0	0
$809$	−19.0124	−0.668440	−0.334220	−	0.942495i	$-0.608473\pi$
$809$	−19.0124	−0.668440	−0.334220	+	0.942495i	$0.608473\pi$
$810$	0	0
$811$	−13.9235	−0.488921	−0.244460	−	0.969659i	$-0.578611\pi$
$811$	−13.9235	−0.488921	−0.244460	+	0.969659i	$0.578611\pi$
$812$	0	0
$813$	−24.0652	−0.844003
$814$	0	0
$815$	−44.9662	−1.57510
$816$	0	0
$817$	21.1900	0.741343
$818$	0	0
$819$	−3.90297	−0.136381
$820$	0	0
$821$	−28.6868	−1.00117	−0.500587	−	0.865686i	$-0.666883\pi$
$821$	−28.6868	−1.00117	−0.500587	+	0.865686i	$0.666883\pi$
$822$	0	0
$823$	−12.4739	−0.434814	−0.217407	−	0.976081i	$-0.569760\pi$
$823$	−12.4739	−0.434814	−0.217407	+	0.976081i	$0.569760\pi$
$824$	0	0
$825$	−5.27701	−0.183722
$826$	0	0
$827$	47.9770	1.66832	0.834162	−	0.551519i	$-0.185952\pi$
$827$	47.9770	1.66832	0.834162	+	0.551519i	$0.185952\pi$
$828$	0	0
$829$	25.2740	0.877802	0.438901	−	0.898535i	$-0.355368\pi$
$829$	25.2740	0.877802	0.438901	+	0.898535i	$0.355368\pi$
$830$	0	0
$831$	21.4693	0.744761
$832$	0	0
$833$	11.0635	0.383328
$834$	0	0
$835$	32.1856	1.11383
$836$	0	0
$837$	1.59594	0.0551637
$838$	0	0
$839$	6.16546	0.212855	0.106428	−	0.994320i	$-0.466059\pi$
$839$	6.16546	0.212855	0.106428	+	0.994320i	$0.466059\pi$
$840$	0	0
$841$	−17.7420	−0.611794
$842$	0	0
$843$	−38.9557	−1.34171
$844$	0	0
$845$	−28.8103	−0.991103
$846$	0	0
$847$	−27.7184	−0.952415
$848$	0	0
$849$	43.9508	1.50839
$850$	0	0
$851$	−57.1424	−1.95882
$852$	0	0
$853$	49.8675	1.70743	0.853715	−	0.520741i	$-0.174344\pi$
$853$	49.8675	1.70743	0.853715	+	0.520741i	$0.174344\pi$
$854$	0	0
$855$	8.97009	0.306771
$856$	0	0
$857$	−21.3701	−0.729990	−0.364995	−	0.931009i	$-0.618929\pi$
$857$	−21.3701	−0.729990	−0.364995	+	0.931009i	$0.618929\pi$
$858$	0	0
$859$	−14.6587	−0.500149	−0.250075	−	0.968227i	$-0.580455\pi$
$859$	−14.6587	−0.500149	−0.250075	+	0.968227i	$0.580455\pi$
$860$	0	0
$861$	46.2978	1.57783
$862$	0	0
$863$	−47.8017	−1.62719	−0.813594	−	0.581434i	$-0.802492\pi$
$863$	−47.8017	−1.62719	−0.813594	+	0.581434i	$0.802492\pi$
$864$	0	0
$865$	−29.6453	−1.00797
$866$	0	0
$867$	30.9183	1.05004
$868$	0	0
$869$	−40.5550	−1.37574
$870$	0	0
$871$	−8.90333	−0.301678
$872$	0	0
$873$	−6.75488	−0.228618
$874$	0	0
$875$	23.6568	0.799747
$876$	0	0
$877$	48.2557	1.62948	0.814740	−	0.579826i	$-0.196880\pi$
$877$	48.2557	1.62948	0.814740	+	0.579826i	$0.196880\pi$
$878$	0	0
$879$	39.4349	1.33010
$880$	0	0
$881$	−42.1880	−1.42135	−0.710674	−	0.703521i	$-0.751610\pi$
$881$	−42.1880	−1.42135	−0.710674	+	0.703521i	$0.751610\pi$
$882$	0	0
$883$	−13.6532	−0.459468	−0.229734	−	0.973253i	$-0.573786\pi$
$883$	−13.6532	−0.459468	−0.229734	+	0.973253i	$0.573786\pi$
$884$	0	0
$885$	−3.53355	−0.118779
$886$	0	0
$887$	−18.4101	−0.618151	−0.309075	−	0.951038i	$-0.600020\pi$
$887$	−18.4101	−0.618151	−0.309075	+	0.951038i	$0.600020\pi$
$888$	0	0
$889$	41.7143	1.39905
$890$	0	0
$891$	52.7265	1.76640
$892$	0	0
$893$	3.52809	0.118063
$894$	0	0
$895$	−61.4246	−2.05320
$896$	0	0
$897$	13.8186	0.461390
$898$	0	0
$899$	2.60856	0.0870002
$900$	0	0
$901$	11.3382	0.377731
$902$	0	0
$903$	−58.3492	−1.94174
$904$	0	0
$905$	−40.1885	−1.33591
$906$	0	0
$907$	−56.5302	−1.87705	−0.938527	−	0.345206i	$-0.887809\pi$
$907$	−56.5302	−1.87705	−0.938527	+	0.345206i	$0.887809\pi$
$908$	0	0
$909$	−20.2799	−0.672643
$910$	0	0
$911$	15.5516	0.515247	0.257623	−	0.966245i	$-0.417061\pi$
$911$	15.5516	0.515247	0.257623	+	0.966245i	$0.417061\pi$
$912$	0	0
$913$	66.3104	2.19455
$914$	0	0
$915$	−45.6824	−1.51022
$916$	0	0
$917$	−39.4147	−1.30159
$918$	0	0
$919$	1.04563	0.0344921	0.0172461	−	0.999851i	$-0.494510\pi$
$919$	1.04563	0.0344921	0.0172461	+	0.999851i	$0.494510\pi$
$920$	0	0
$921$	22.9480	0.756163
$922$	0	0
$923$	2.98742	0.0983322
$924$	0	0
$925$	−3.76653	−0.123843
$926$	0	0
$927$	−25.5092	−0.837833
$928$	0	0
$929$	0.796090	0.0261189	0.0130594	−	0.999915i	$-0.495843\pi$
$929$	0.796090	0.0261189	0.0130594	+	0.999915i	$0.495843\pi$
$930$	0	0
$931$	−3.65920	−0.119925
$932$	0	0
$933$	−46.8651	−1.53430
$934$	0	0
$935$	62.7572	2.05238
$936$	0	0
$937$	−20.3836	−0.665904	−0.332952	−	0.942944i	$-0.608045\pi$
$937$	−20.3836	−0.665904	−0.332952	+	0.942944i	$0.608045\pi$
$938$	0	0
$939$	−21.2139	−0.692288
$940$	0	0
$941$	−12.8415	−0.418621	−0.209310	−	0.977849i	$-0.567122\pi$
$941$	−12.8415	−0.418621	−0.209310	+	0.977849i	$0.567122\pi$
$942$	0	0
$943$	−67.3091	−2.19189
$944$	0	0
$945$	10.7525	0.349780
$946$	0	0
$947$	−42.0108	−1.36517	−0.682585	−	0.730807i	$-0.739144\pi$
$947$	−42.0108	−1.36517	−0.682585	+	0.730807i	$0.739144\pi$
$948$	0	0
$949$	4.75714	0.154423
$950$	0	0
$951$	51.0990	1.65700
$952$	0	0
$953$	−15.5148	−0.502575	−0.251287	−	0.967913i	$-0.580854\pi$
$953$	−15.5148	−0.502575	−0.251287	+	0.967913i	$0.580854\pi$
$954$	0	0
$955$	−40.5438	−1.31197
$956$	0	0
$957$	36.6122	1.18351
$958$	0	0
$959$	−45.7104	−1.47607
$960$	0	0
$961$	−30.3956	−0.980503
$962$	0	0
$963$	18.5620	0.598151
$964$	0	0
$965$	−48.0272	−1.54605
$966$	0	0
$967$	29.9778	0.964022	0.482011	−	0.876165i	$-0.339906\pi$
$967$	29.9778	0.964022	0.482011	+	0.876165i	$0.339906\pi$
$968$	0	0
$969$	−22.9115	−0.736024
$970$	0	0
$971$	−3.14021	−0.100774	−0.0503870	−	0.998730i	$-0.516045\pi$
$971$	−3.14021	−0.100774	−0.0503870	+	0.998730i	$0.516045\pi$
$972$	0	0
$973$	8.23864	0.264119
$974$	0	0
$975$	0.910850	0.0291705
$976$	0	0
$977$	14.3894	0.460357	0.230178	−	0.973148i	$-0.426069\pi$
$977$	14.3894	0.460357	0.230178	+	0.973148i	$0.426069\pi$
$978$	0	0
$979$	40.6945	1.30060
$980$	0	0
$981$	13.8423	0.441950
$982$	0	0
$983$	59.6676	1.90310	0.951550	−	0.307495i	$-0.0994906\pi$
$983$	59.6676	1.90310	0.951550	+	0.307495i	$0.0994906\pi$
$984$	0	0
$985$	11.3462	0.361521
$986$	0	0
$987$	−9.71503	−0.309233
$988$	0	0
$989$	84.8297	2.69743
$990$	0	0
$991$	42.6315	1.35424	0.677118	−	0.735875i	$-0.263229\pi$
$991$	42.6315	1.35424	0.677118	+	0.735875i	$0.263229\pi$
$992$	0	0
$993$	−40.1906	−1.27541
$994$	0	0
$995$	−60.2475	−1.90997
$996$	0	0
$997$	38.5552	1.22105	0.610527	−	0.791995i	$-0.290958\pi$
$997$	38.5552	1.22105	0.610527	+	0.791995i	$0.290958\pi$
$998$	0	0
$999$	15.9880	0.505839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.25		28
4.3	odd	2		4024.2.a.d.1.4	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.4	✓	28	4.3	odd	2
8048.2.a.v.1.25		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)

\(f(q)\)	\(=\)	\(q+2.25613 q^{3} +2.34171 q^{5} -2.23682 q^{7} +2.09012 q^{9} +O(q^{10})\)	\(q+2.25613 q^{3} +2.34171 q^{5} -2.23682 q^{7} +2.09012 q^{9} -4.83651 q^{11} +0.834816 q^{13} +5.28320 q^{15} -5.54113 q^{17} +1.83270 q^{19} -5.04657 q^{21} +7.33684 q^{23} +0.483606 q^{25} -2.05280 q^{27} -3.35529 q^{29} -0.777446 q^{31} -10.9118 q^{33} -5.23799 q^{35} -7.78842 q^{37} +1.88345 q^{39} -9.17412 q^{41} +11.5622 q^{43} +4.89447 q^{45} +1.92508 q^{47} -1.99662 q^{49} -12.5015 q^{51} -2.04619 q^{53} -11.3257 q^{55} +4.13481 q^{57} -0.668827 q^{59} -8.64673 q^{61} -4.67524 q^{63} +1.95490 q^{65} -10.6650 q^{67} +16.5529 q^{69} +3.57854 q^{71} +5.69843 q^{73} +1.09108 q^{75} +10.8184 q^{77} +8.38518 q^{79} -10.9018 q^{81} -13.7104 q^{83} -12.9757 q^{85} -7.56997 q^{87} -8.41401 q^{89} -1.86734 q^{91} -1.75402 q^{93} +4.29165 q^{95} -3.23181 q^{97} -10.1089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 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

Introduction

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