LMFDB - Embedded newform 8048.2.a.s.1.14

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:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+0.273229 q^{3} -1.56489 q^{5} +3.17063 q^{7} -2.92535 q^{9} +O(q^{10})$	$q+0.273229 q^{3} -1.56489 q^{5} +3.17063 q^{7} -2.92535 q^{9} +3.30009 q^{11} +0.195440 q^{13} -0.427573 q^{15} +3.22493 q^{17} -2.90990 q^{19} +0.866307 q^{21} +2.63682 q^{23} -2.55111 q^{25} -1.61897 q^{27} -3.03001 q^{29} -2.63018 q^{31} +0.901678 q^{33} -4.96170 q^{35} +0.803108 q^{37} +0.0533999 q^{39} -5.44544 q^{41} -10.2018 q^{43} +4.57785 q^{45} -13.3528 q^{47} +3.05291 q^{49} +0.881143 q^{51} -4.98403 q^{53} -5.16428 q^{55} -0.795067 q^{57} -8.67007 q^{59} +8.05465 q^{61} -9.27520 q^{63} -0.305843 q^{65} +10.0964 q^{67} +0.720454 q^{69} -7.01891 q^{71} +12.0037 q^{73} -0.697037 q^{75} +10.4634 q^{77} -7.62704 q^{79} +8.33369 q^{81} +16.1065 q^{83} -5.04667 q^{85} -0.827885 q^{87} +8.66032 q^{89} +0.619669 q^{91} -0.718639 q^{93} +4.55368 q^{95} -16.2797 q^{97} -9.65390 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * 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
$3$	0.273229	0.157749	0.0788743	−	0.996885i	$-0.474867\pi$
$3$	0.273229	0.157749	0.0788743	+	0.996885i	$0.474867\pi$
$4$	0	0
$5$	−1.56489	−0.699841	−0.349921	−	0.936779i	$-0.613791\pi$
$5$	−1.56489	−0.699841	−0.349921	+	0.936779i	$0.613791\pi$
$6$	0	0
$7$	3.17063	1.19839	0.599193	−	0.800605i	$-0.295488\pi$
$7$	3.17063	1.19839	0.599193	+	0.800605i	$0.295488\pi$
$8$	0	0
$9$	−2.92535	−0.975115
$10$	0	0
$11$	3.30009	0.995014	0.497507	−	0.867460i	$-0.334249\pi$
$11$	3.30009	0.995014	0.497507	+	0.867460i	$0.334249\pi$
$12$	0	0
$13$	0.195440	0.0542054	0.0271027	−	0.999633i	$-0.491372\pi$
$13$	0.195440	0.0542054	0.0271027	+	0.999633i	$0.491372\pi$
$14$	0	0
$15$	−0.427573	−0.110399
$16$	0	0
$17$	3.22493	0.782161	0.391080	−	0.920357i	$-0.372101\pi$
$17$	3.22493	0.782161	0.391080	+	0.920357i	$0.372101\pi$
$18$	0	0
$19$	−2.90990	−0.667576	−0.333788	−	0.942648i	$-0.608327\pi$
$19$	−2.90990	−0.667576	−0.333788	+	0.942648i	$0.608327\pi$
$20$	0	0
$21$	0.866307	0.189044
$22$	0	0
$23$	2.63682	0.549814	0.274907	−	0.961471i	$-0.411353\pi$
$23$	2.63682	0.549814	0.274907	+	0.961471i	$0.411353\pi$
$24$	0	0
$25$	−2.55111	−0.510222
$26$	0	0
$27$	−1.61897	−0.311572
$28$	0	0
$29$	−3.03001	−0.562658	−0.281329	−	0.959611i	$-0.590775\pi$
$29$	−3.03001	−0.562658	−0.281329	+	0.959611i	$0.590775\pi$
$30$	0	0
$31$	−2.63018	−0.472394	−0.236197	−	0.971705i	$-0.575901\pi$
$31$	−2.63018	−0.472394	−0.236197	+	0.971705i	$0.575901\pi$
$32$	0	0
$33$	0.901678	0.156962
$34$	0	0
$35$	−4.96170	−0.838680
$36$	0	0
$37$	0.803108	0.132030	0.0660151	−	0.997819i	$-0.478971\pi$
$37$	0.803108	0.132030	0.0660151	+	0.997819i	$0.478971\pi$
$38$	0	0
$39$	0.0533999	0.00855082
$40$	0	0
$41$	−5.44544	−0.850435	−0.425218	−	0.905091i	$-0.639802\pi$
$41$	−5.44544	−0.850435	−0.425218	+	0.905091i	$0.639802\pi$
$42$	0	0
$43$	−10.2018	−1.55576	−0.777879	−	0.628414i	$-0.783704\pi$
$43$	−10.2018	−1.55576	−0.777879	+	0.628414i	$0.783704\pi$
$44$	0	0
$45$	4.57785	0.682426
$46$	0	0
$47$	−13.3528	−1.94771	−0.973854	−	0.227174i	$-0.927051\pi$
$47$	−13.3528	−1.94771	−0.973854	+	0.227174i	$0.927051\pi$
$48$	0	0
$49$	3.05291	0.436130
$50$	0	0
$51$	0.881143	0.123385
$52$	0	0
$53$	−4.98403	−0.684609	−0.342305	−	0.939589i	$-0.611208\pi$
$53$	−4.98403	−0.684609	−0.342305	+	0.939589i	$0.611208\pi$
$54$	0	0
$55$	−5.16428	−0.696352
$56$	0	0
$57$	−0.795067	−0.105309
$58$	0	0
$59$	−8.67007	−1.12875	−0.564374	−	0.825520i	$-0.690882\pi$
$59$	−8.67007	−1.12875	−0.564374	+	0.825520i	$0.690882\pi$
$60$	0	0
$61$	8.05465	1.03129	0.515646	−	0.856802i	$-0.327552\pi$
$61$	8.05465	1.03129	0.515646	+	0.856802i	$0.327552\pi$
$62$	0	0
$63$	−9.27520	−1.16856
$64$	0	0
$65$	−0.305843	−0.0379352
$66$	0	0
$67$	10.0964	1.23347	0.616735	−	0.787171i	$-0.288455\pi$
$67$	10.0964	1.23347	0.616735	+	0.787171i	$0.288455\pi$
$68$	0	0
$69$	0.720454	0.0867324
$70$	0	0
$71$	−7.01891	−0.832992	−0.416496	−	0.909138i	$-0.636742\pi$
$71$	−7.01891	−0.832992	−0.416496	+	0.909138i	$0.636742\pi$
$72$	0	0
$73$	12.0037	1.40493	0.702463	−	0.711720i	$-0.252084\pi$
$73$	12.0037	1.40493	0.702463	+	0.711720i	$0.252084\pi$
$74$	0	0
$75$	−0.697037	−0.0804869
$76$	0	0
$77$	10.4634	1.19241
$78$	0	0
$79$	−7.62704	−0.858110	−0.429055	−	0.903278i	$-0.641153\pi$
$79$	−7.62704	−0.858110	−0.429055	+	0.903278i	$0.641153\pi$
$80$	0	0
$81$	8.33369	0.925965
$82$	0	0
$83$	16.1065	1.76792	0.883961	−	0.467560i	$-0.154867\pi$
$83$	16.1065	1.76792	0.883961	+	0.467560i	$0.154867\pi$
$84$	0	0
$85$	−5.04667	−0.547388
$86$	0	0
$87$	−0.827885	−0.0887586
$88$	0	0
$89$	8.66032	0.917992	0.458996	−	0.888438i	$-0.348209\pi$
$89$	8.66032	0.917992	0.458996	+	0.888438i	$0.348209\pi$
$90$	0	0
$91$	0.619669	0.0649590
$92$	0	0
$93$	−0.718639	−0.0745194
$94$	0	0
$95$	4.55368	0.467198
$96$	0	0
$97$	−16.2797	−1.65295	−0.826475	−	0.562974i	$-0.809657\pi$
$97$	−16.2797	−1.65295	−0.826475	+	0.562974i	$0.809657\pi$
$98$	0	0
$99$	−9.65390	−0.970253
$100$	0	0
$101$	3.96778	0.394809	0.197405	−	0.980322i	$-0.436749\pi$
$101$	3.96778	0.394809	0.197405	+	0.980322i	$0.436749\pi$
$102$	0	0
$103$	6.41434	0.632024	0.316012	−	0.948755i	$-0.397656\pi$
$103$	6.41434	0.632024	0.316012	+	0.948755i	$0.397656\pi$
$104$	0	0
$105$	−1.35568	−0.132301
$106$	0	0
$107$	16.9674	1.64030	0.820150	−	0.572149i	$-0.193890\pi$
$107$	16.9674	1.64030	0.820150	+	0.572149i	$0.193890\pi$
$108$	0	0
$109$	−2.36545	−0.226569	−0.113285	−	0.993563i	$-0.536137\pi$
$109$	−2.36545	−0.226569	−0.113285	+	0.993563i	$0.536137\pi$
$110$	0	0
$111$	0.219432	0.0208276
$112$	0	0
$113$	2.35574	0.221610	0.110805	−	0.993842i	$-0.464657\pi$
$113$	2.35574	0.221610	0.110805	+	0.993842i	$0.464657\pi$
$114$	0	0
$115$	−4.12633	−0.384783
$116$	0	0
$117$	−0.571730	−0.0528565
$118$	0	0
$119$	10.2251	0.937331
$120$	0	0
$121$	−0.109424	−0.00994761
$122$	0	0
$123$	−1.48785	−0.134155
$124$	0	0
$125$	11.8167	1.05692
$126$	0	0
$127$	−14.1449	−1.25515	−0.627577	−	0.778555i	$-0.715953\pi$
$127$	−14.1449	−1.25515	−0.627577	+	0.778555i	$0.715953\pi$
$128$	0	0
$129$	−2.78742	−0.245419
$130$	0	0
$131$	−10.0712	−0.879929	−0.439964	−	0.898015i	$-0.645009\pi$
$131$	−10.0712	−0.879929	−0.439964	+	0.898015i	$0.645009\pi$
$132$	0	0
$133$	−9.22622	−0.800014
$134$	0	0
$135$	2.53352	0.218051
$136$	0	0
$137$	−11.9877	−1.02418	−0.512091	−	0.858931i	$-0.671129\pi$
$137$	−11.9877	−1.02418	−0.512091	+	0.858931i	$0.671129\pi$
$138$	0	0
$139$	−6.95764	−0.590140	−0.295070	−	0.955476i	$-0.595343\pi$
$139$	−6.95764	−0.590140	−0.295070	+	0.955476i	$0.595343\pi$
$140$	0	0
$141$	−3.64837	−0.307248
$142$	0	0
$143$	0.644970	0.0539351
$144$	0	0
$145$	4.74164	0.393771
$146$	0	0
$147$	0.834142	0.0687988
$148$	0	0
$149$	−6.30920	−0.516870	−0.258435	−	0.966029i	$-0.583207\pi$
$149$	−6.30920	−0.516870	−0.258435	+	0.966029i	$0.583207\pi$
$150$	0	0
$151$	−1.78363	−0.145150	−0.0725749	−	0.997363i	$-0.523122\pi$
$151$	−1.78363	−0.145150	−0.0725749	+	0.997363i	$0.523122\pi$
$152$	0	0
$153$	−9.43404	−0.762697
$154$	0	0
$155$	4.11594	0.330601
$156$	0	0
$157$	10.5775	0.844180	0.422090	−	0.906554i	$-0.361297\pi$
$157$	10.5775	0.844180	0.422090	+	0.906554i	$0.361297\pi$
$158$	0	0
$159$	−1.36178	−0.107996
$160$	0	0
$161$	8.36037	0.658890
$162$	0	0
$163$	−3.86264	−0.302545	−0.151273	−	0.988492i	$-0.548337\pi$
$163$	−3.86264	−0.302545	−0.151273	+	0.988492i	$0.548337\pi$
$164$	0	0
$165$	−1.41103	−0.109848
$166$	0	0
$167$	−24.2529	−1.87675	−0.938373	−	0.345625i	$-0.887667\pi$
$167$	−24.2529	−1.87675	−0.938373	+	0.345625i	$0.887667\pi$
$168$	0	0
$169$	−12.9618	−0.997062
$170$	0	0
$171$	8.51246	0.650964
$172$	0	0
$173$	13.8316	1.05160	0.525800	−	0.850608i	$-0.323766\pi$
$173$	13.8316	1.05160	0.525800	+	0.850608i	$0.323766\pi$
$174$	0	0
$175$	−8.08864	−0.611443
$176$	0	0
$177$	−2.36891	−0.178058
$178$	0	0
$179$	−13.1964	−0.986342	−0.493171	−	0.869932i	$-0.664162\pi$
$179$	−13.1964	−0.986342	−0.493171	+	0.869932i	$0.664162\pi$
$180$	0	0
$181$	−1.19633	−0.0889227	−0.0444614	−	0.999011i	$-0.514157\pi$
$181$	−1.19633	−0.0889227	−0.0444614	+	0.999011i	$0.514157\pi$
$182$	0	0
$183$	2.20076	0.162685
$184$	0	0
$185$	−1.25678	−0.0924001
$186$	0	0
$187$	10.6426	0.778261
$188$	0	0
$189$	−5.13317	−0.373383
$190$	0	0
$191$	17.3616	1.25624	0.628121	−	0.778116i	$-0.283824\pi$
$191$	17.3616	1.25624	0.628121	+	0.778116i	$0.283824\pi$
$192$	0	0
$193$	19.9733	1.43771	0.718855	−	0.695160i	$-0.244667\pi$
$193$	19.9733	1.43771	0.718855	+	0.695160i	$0.244667\pi$
$194$	0	0
$195$	−0.0835651	−0.00598422
$196$	0	0
$197$	−13.4724	−0.959869	−0.479934	−	0.877304i	$-0.659340\pi$
$197$	−13.4724	−0.959869	−0.479934	+	0.877304i	$0.659340\pi$
$198$	0	0
$199$	−7.20924	−0.511050	−0.255525	−	0.966802i	$-0.582248\pi$
$199$	−7.20924	−0.511050	−0.255525	+	0.966802i	$0.582248\pi$
$200$	0	0
$201$	2.75862	0.194578
$202$	0	0
$203$	−9.60704	−0.674282
$204$	0	0
$205$	8.52153	0.595170
$206$	0	0
$207$	−7.71360	−0.536132
$208$	0	0
$209$	−9.60292	−0.664248
$210$	0	0
$211$	−20.6157	−1.41924	−0.709620	−	0.704584i	$-0.751133\pi$
$211$	−20.6157	−1.41924	−0.709620	+	0.704584i	$0.751133\pi$
$212$	0	0
$213$	−1.91777	−0.131403
$214$	0	0
$215$	15.9647	1.08878
$216$	0	0
$217$	−8.33932	−0.566110
$218$	0	0
$219$	3.27975	0.221625
$220$	0	0
$221$	0.630281	0.0423973
$222$	0	0
$223$	−11.7745	−0.788481	−0.394240	−	0.919007i	$-0.628992\pi$
$223$	−11.7745	−0.788481	−0.394240	+	0.919007i	$0.628992\pi$
$224$	0	0
$225$	7.46288	0.497526
$226$	0	0
$227$	−16.6101	−1.10245	−0.551226	−	0.834356i	$-0.685840\pi$
$227$	−16.6101	−1.10245	−0.551226	+	0.834356i	$0.685840\pi$
$228$	0	0
$229$	5.66445	0.374317	0.187159	−	0.982330i	$-0.440072\pi$
$229$	5.66445	0.374317	0.187159	+	0.982330i	$0.440072\pi$
$230$	0	0
$231$	2.85889	0.188101
$232$	0	0
$233$	−15.2449	−0.998727	−0.499364	−	0.866392i	$-0.666433\pi$
$233$	−15.2449	−0.998727	−0.499364	+	0.866392i	$0.666433\pi$
$234$	0	0
$235$	20.8957	1.36309
$236$	0	0
$237$	−2.08393	−0.135366
$238$	0	0
$239$	21.3369	1.38017	0.690084	−	0.723729i	$-0.257573\pi$
$239$	21.3369	1.38017	0.690084	+	0.723729i	$0.257573\pi$
$240$	0	0
$241$	−24.3725	−1.56997	−0.784986	−	0.619513i	$-0.787330\pi$
$241$	−24.3725	−1.56997	−0.784986	+	0.619513i	$0.787330\pi$
$242$	0	0
$243$	7.13392	0.457641
$244$	0	0
$245$	−4.77747	−0.305221
$246$	0	0
$247$	−0.568711	−0.0361862
$248$	0	0
$249$	4.40077	0.278887
$250$	0	0
$251$	−27.8094	−1.75531	−0.877657	−	0.479289i	$-0.840895\pi$
$251$	−27.8094	−1.75531	−0.877657	+	0.479289i	$0.840895\pi$
$252$	0	0
$253$	8.70172	0.547073
$254$	0	0
$255$	−1.37889	−0.0863497
$256$	0	0
$257$	−16.4633	−1.02696	−0.513478	−	0.858103i	$-0.671643\pi$
$257$	−16.4633	−1.02696	−0.513478	+	0.858103i	$0.671643\pi$
$258$	0	0
$259$	2.54636	0.158223
$260$	0	0
$261$	8.86382	0.548657
$262$	0	0
$263$	24.9586	1.53901	0.769506	−	0.638640i	$-0.220503\pi$
$263$	24.9586	1.53901	0.769506	+	0.638640i	$0.220503\pi$
$264$	0	0
$265$	7.79947	0.479118
$266$	0	0
$267$	2.36625	0.144812
$268$	0	0
$269$	−21.0368	−1.28263	−0.641317	−	0.767276i	$-0.721612\pi$
$269$	−21.0368	−1.28263	−0.641317	+	0.767276i	$0.721612\pi$
$270$	0	0
$271$	16.9702	1.03087	0.515434	−	0.856930i	$-0.327631\pi$
$271$	16.9702	1.03087	0.515434	+	0.856930i	$0.327631\pi$
$272$	0	0
$273$	0.169311	0.0102472
$274$	0	0
$275$	−8.41889	−0.507678
$276$	0	0
$277$	15.7817	0.948231	0.474116	−	0.880463i	$-0.342768\pi$
$277$	15.7817	0.948231	0.474116	+	0.880463i	$0.342768\pi$
$278$	0	0
$279$	7.69418	0.460638
$280$	0	0
$281$	−4.26075	−0.254175	−0.127087	−	0.991892i	$-0.540563\pi$
$281$	−4.26075	−0.254175	−0.127087	+	0.991892i	$0.540563\pi$
$282$	0	0
$283$	−14.3279	−0.851703	−0.425852	−	0.904793i	$-0.640025\pi$
$283$	−14.3279	−0.851703	−0.425852	+	0.904793i	$0.640025\pi$
$284$	0	0
$285$	1.24420	0.0736998
$286$	0	0
$287$	−17.2655	−1.01915
$288$	0	0
$289$	−6.59982	−0.388225
$290$	0	0
$291$	−4.44807	−0.260750
$292$	0	0
$293$	2.60963	0.152456	0.0762282	−	0.997090i	$-0.475712\pi$
$293$	2.60963	0.152456	0.0762282	+	0.997090i	$0.475712\pi$
$294$	0	0
$295$	13.5677	0.789944
$296$	0	0
$297$	−5.34276	−0.310018
$298$	0	0
$299$	0.515340	0.0298029
$300$	0	0
$301$	−32.3461	−1.86440
$302$	0	0
$303$	1.08411	0.0622806
$304$	0	0
$305$	−12.6047	−0.721741
$306$	0	0
$307$	−17.3932	−0.992682	−0.496341	−	0.868128i	$-0.665323\pi$
$307$	−17.3932	−0.992682	−0.496341	+	0.868128i	$0.665323\pi$
$308$	0	0
$309$	1.75258	0.0997009
$310$	0	0
$311$	−16.3257	−0.925743	−0.462872	−	0.886425i	$-0.653181\pi$
$311$	−16.3257	−0.925743	−0.462872	+	0.886425i	$0.653181\pi$
$312$	0	0
$313$	2.33693	0.132091	0.0660456	−	0.997817i	$-0.478962\pi$
$313$	2.33693	0.132091	0.0660456	+	0.997817i	$0.478962\pi$
$314$	0	0
$315$	14.5147	0.817810
$316$	0	0
$317$	14.7586	0.828924	0.414462	−	0.910067i	$-0.363970\pi$
$317$	14.7586	0.828924	0.414462	+	0.910067i	$0.363970\pi$
$318$	0	0
$319$	−9.99929	−0.559853
$320$	0	0
$321$	4.63598	0.258755
$322$	0	0
$323$	−9.38422	−0.522152
$324$	0	0
$325$	−0.498590	−0.0276568
$326$	0	0
$327$	−0.646309	−0.0357410
$328$	0	0
$329$	−42.3369	−2.33411
$330$	0	0
$331$	13.9749	0.768128	0.384064	−	0.923307i	$-0.374524\pi$
$331$	13.9749	0.768128	0.384064	+	0.923307i	$0.374524\pi$
$332$	0	0
$333$	−2.34937	−0.128745
$334$	0	0
$335$	−15.7998	−0.863233
$336$	0	0
$337$	29.1944	1.59032	0.795161	−	0.606399i	$-0.207386\pi$
$337$	29.1944	1.59032	0.795161	+	0.606399i	$0.207386\pi$
$338$	0	0
$339$	0.643656	0.0349586
$340$	0	0
$341$	−8.67981	−0.470038
$342$	0	0
$343$	−12.5148	−0.675734
$344$	0	0
$345$	−1.12743	−0.0606989
$346$	0	0
$347$	−1.83407	−0.0984581	−0.0492291	−	0.998788i	$-0.515676\pi$
$347$	−1.83407	−0.0984581	−0.0492291	+	0.998788i	$0.515676\pi$
$348$	0	0
$349$	−21.4816	−1.14988	−0.574941	−	0.818195i	$-0.694975\pi$
$349$	−21.4816	−1.14988	−0.574941	+	0.818195i	$0.694975\pi$
$350$	0	0
$351$	−0.316413	−0.0168889
$352$	0	0
$353$	9.95416	0.529806	0.264903	−	0.964275i	$-0.414660\pi$
$353$	9.95416	0.529806	0.264903	+	0.964275i	$0.414660\pi$
$354$	0	0
$355$	10.9838	0.582962
$356$	0	0
$357$	2.79378	0.147863
$358$	0	0
$359$	2.80557	0.148072	0.0740361	−	0.997256i	$-0.476412\pi$
$359$	2.80557	0.148072	0.0740361	+	0.997256i	$0.476412\pi$
$360$	0	0
$361$	−10.5325	−0.554342
$362$	0	0
$363$	−0.0298977	−0.00156922
$364$	0	0
$365$	−18.7845	−0.983225
$366$	0	0
$367$	−34.2479	−1.78773	−0.893863	−	0.448340i	$-0.852015\pi$
$367$	−34.2479	−1.78773	−0.893863	+	0.448340i	$0.852015\pi$
$368$	0	0
$369$	15.9298	0.829272
$370$	0	0
$371$	−15.8025	−0.820426
$372$	0	0
$373$	0.368279	0.0190687	0.00953437	−	0.999955i	$-0.496965\pi$
$373$	0.368279	0.0190687	0.00953437	+	0.999955i	$0.496965\pi$
$374$	0	0
$375$	3.22865	0.166727
$376$	0	0
$377$	−0.592186	−0.0304991
$378$	0	0
$379$	15.3368	0.787797	0.393899	−	0.919154i	$-0.371126\pi$
$379$	15.3368	0.787797	0.393899	+	0.919154i	$0.371126\pi$
$380$	0	0
$381$	−3.86478	−0.197999
$382$	0	0
$383$	29.4428	1.50446	0.752229	−	0.658901i	$-0.228979\pi$
$383$	29.4428	1.50446	0.752229	+	0.658901i	$0.228979\pi$
$384$	0	0
$385$	−16.3740	−0.834498
$386$	0	0
$387$	29.8438	1.51704
$388$	0	0
$389$	−0.976992	−0.0495354	−0.0247677	−	0.999693i	$-0.507885\pi$
$389$	−0.976992	−0.0495354	−0.0247677	+	0.999693i	$0.507885\pi$
$390$	0	0
$391$	8.50355	0.430043
$392$	0	0
$393$	−2.75175	−0.138808
$394$	0	0
$395$	11.9355	0.600540
$396$	0	0
$397$	5.25454	0.263718	0.131859	−	0.991269i	$-0.457905\pi$
$397$	5.25454	0.263718	0.131859	+	0.991269i	$0.457905\pi$
$398$	0	0
$399$	−2.52087	−0.126201
$400$	0	0
$401$	−14.8203	−0.740091	−0.370046	−	0.929014i	$-0.620658\pi$
$401$	−14.8203	−0.740091	−0.370046	+	0.929014i	$0.620658\pi$
$402$	0	0
$403$	−0.514042	−0.0256063
$404$	0	0
$405$	−13.0413	−0.648029
$406$	0	0
$407$	2.65033	0.131372
$408$	0	0
$409$	−11.8474	−0.585815	−0.292908	−	0.956141i	$-0.594623\pi$
$409$	−11.8474	−0.585815	−0.292908	+	0.956141i	$0.594623\pi$
$410$	0	0
$411$	−3.27540	−0.161563
$412$	0	0
$413$	−27.4896	−1.35268
$414$	0	0
$415$	−25.2050	−1.23726
$416$	0	0
$417$	−1.90103	−0.0930937
$418$	0	0
$419$	13.2165	0.645670	0.322835	−	0.946455i	$-0.395364\pi$
$419$	13.2165	0.645670	0.322835	+	0.946455i	$0.395364\pi$
$420$	0	0
$421$	30.4521	1.48414	0.742071	−	0.670321i	$-0.233844\pi$
$421$	30.4521	1.48414	0.742071	+	0.670321i	$0.233844\pi$
$422$	0	0
$423$	39.0616	1.89924
$424$	0	0
$425$	−8.22716	−0.399076
$426$	0	0
$427$	25.5383	1.23589
$428$	0	0
$429$	0.176224	0.00850819
$430$	0	0
$431$	−33.6636	−1.62152	−0.810760	−	0.585378i	$-0.800946\pi$
$431$	−33.6636	−1.62152	−0.810760	+	0.585378i	$0.800946\pi$
$432$	0	0
$433$	25.2655	1.21418	0.607092	−	0.794632i	$-0.292336\pi$
$433$	25.2655	1.21418	0.607092	+	0.794632i	$0.292336\pi$
$434$	0	0
$435$	1.29555	0.0621169
$436$	0	0
$437$	−7.67287	−0.367043
$438$	0	0
$439$	−6.10084	−0.291177	−0.145588	−	0.989345i	$-0.546508\pi$
$439$	−6.10084	−0.291177	−0.145588	+	0.989345i	$0.546508\pi$
$440$	0	0
$441$	−8.93081	−0.425277
$442$	0	0
$443$	18.6862	0.887810	0.443905	−	0.896074i	$-0.353593\pi$
$443$	18.6862	0.887810	0.443905	+	0.896074i	$0.353593\pi$
$444$	0	0
$445$	−13.5525	−0.642449
$446$	0	0
$447$	−1.72385	−0.0815355
$448$	0	0
$449$	14.1908	0.669706	0.334853	−	0.942270i	$-0.391313\pi$
$449$	14.1908	0.669706	0.334853	+	0.942270i	$0.391313\pi$
$450$	0	0
$451$	−17.9704	−0.846195
$452$	0	0
$453$	−0.487339	−0.0228972
$454$	0	0
$455$	−0.969716	−0.0454610
$456$	0	0
$457$	−38.8430	−1.81700	−0.908499	−	0.417886i	$-0.862771\pi$
$457$	−38.8430	−1.81700	−0.908499	+	0.417886i	$0.862771\pi$
$458$	0	0
$459$	−5.22108	−0.243699
$460$	0	0
$461$	34.4424	1.60414	0.802071	−	0.597229i	$-0.203732\pi$
$461$	34.4424	1.60414	0.802071	+	0.597229i	$0.203732\pi$
$462$	0	0
$463$	−37.3489	−1.73575	−0.867874	−	0.496784i	$-0.834514\pi$
$463$	−37.3489	−1.73575	−0.867874	+	0.496784i	$0.834514\pi$
$464$	0	0
$465$	1.12459	0.0521518
$466$	0	0
$467$	−11.7793	−0.545080	−0.272540	−	0.962144i	$-0.587864\pi$
$467$	−11.7793	−0.545080	−0.272540	+	0.962144i	$0.587864\pi$
$468$	0	0
$469$	32.0119	1.47817
$470$	0	0
$471$	2.89009	0.133168
$472$	0	0
$473$	−33.6668	−1.54800
$474$	0	0
$475$	7.42348	0.340612
$476$	0	0
$477$	14.5800	0.667573
$478$	0	0
$479$	−5.56492	−0.254268	−0.127134	−	0.991886i	$-0.540578\pi$
$479$	−5.56492	−0.254268	−0.127134	+	0.991886i	$0.540578\pi$
$480$	0	0
$481$	0.156960	0.00715674
$482$	0	0
$483$	2.28429	0.103939
$484$	0	0
$485$	25.4759	1.15680
$486$	0	0
$487$	3.32172	0.150522	0.0752608	−	0.997164i	$-0.476021\pi$
$487$	3.32172	0.150522	0.0752608	+	0.997164i	$0.476021\pi$
$488$	0	0
$489$	−1.05538	−0.0477261
$490$	0	0
$491$	33.3344	1.50436	0.752180	−	0.658958i	$-0.229003\pi$
$491$	33.3344	1.50436	0.752180	+	0.658958i	$0.229003\pi$
$492$	0	0
$493$	−9.77156	−0.440089
$494$	0	0
$495$	15.1073	0.679023
$496$	0	0
$497$	−22.2544	−0.998246
$498$	0	0
$499$	−23.6353	−1.05806	−0.529032	−	0.848602i	$-0.677445\pi$
$499$	−23.6353	−1.05806	−0.529032	+	0.848602i	$0.677445\pi$
$500$	0	0
$501$	−6.62658	−0.296054
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−6.20915	−0.276304
$506$	0	0
$507$	−3.54154	−0.157285
$508$	0	0
$509$	−32.4987	−1.44048	−0.720239	−	0.693726i	$-0.755968\pi$
$509$	−32.4987	−1.44048	−0.720239	+	0.693726i	$0.755968\pi$
$510$	0	0
$511$	38.0593	1.68364
$512$	0	0
$513$	4.71105	0.207998
$514$	0	0
$515$	−10.0378	−0.442316
$516$	0	0
$517$	−44.0655	−1.93800
$518$	0	0
$519$	3.77920	0.165888
$520$	0	0
$521$	17.9935	0.788309	0.394155	−	0.919044i	$-0.371037\pi$
$521$	17.9935	0.788309	0.394155	+	0.919044i	$0.371037\pi$
$522$	0	0
$523$	−15.7777	−0.689910	−0.344955	−	0.938619i	$-0.612106\pi$
$523$	−15.7777	−0.689910	−0.344955	+	0.938619i	$0.612106\pi$
$524$	0	0
$525$	−2.21005	−0.0964543
$526$	0	0
$527$	−8.48214	−0.369488
$528$	0	0
$529$	−16.0472	−0.697704
$530$	0	0
$531$	25.3630	1.10066
$532$	0	0
$533$	−1.06426	−0.0460982
$534$	0	0
$535$	−26.5521	−1.14795
$536$	0	0
$537$	−3.60562	−0.155594
$538$	0	0
$539$	10.0749	0.433955
$540$	0	0
$541$	−25.3870	−1.09147	−0.545737	−	0.837956i	$-0.683750\pi$
$541$	−25.3870	−1.09147	−0.545737	+	0.837956i	$0.683750\pi$
$542$	0	0
$543$	−0.326872	−0.0140274
$544$	0	0
$545$	3.70168	0.158562
$546$	0	0
$547$	−26.6779	−1.14066	−0.570332	−	0.821414i	$-0.693185\pi$
$547$	−26.6779	−1.14066	−0.570332	+	0.821414i	$0.693185\pi$
$548$	0	0
$549$	−23.5626	−1.00563
$550$	0	0
$551$	8.81701	0.375617
$552$	0	0
$553$	−24.1826	−1.02835
$554$	0	0
$555$	−0.343387	−0.0145760
$556$	0	0
$557$	39.4235	1.67043	0.835213	−	0.549926i	$-0.185344\pi$
$557$	39.4235	1.67043	0.835213	+	0.549926i	$0.185344\pi$
$558$	0	0
$559$	−1.99384	−0.0843304
$560$	0	0
$561$	2.90785	0.122770
$562$	0	0
$563$	31.1735	1.31381	0.656904	−	0.753974i	$-0.271866\pi$
$563$	31.1735	1.31381	0.656904	+	0.753974i	$0.271866\pi$
$564$	0	0
$565$	−3.68648	−0.155092
$566$	0	0
$567$	26.4231	1.10966
$568$	0	0
$569$	−45.9959	−1.92825	−0.964124	−	0.265452i	$-0.914479\pi$
$569$	−45.9959	−1.92825	−0.964124	+	0.265452i	$0.914479\pi$
$570$	0	0
$571$	−32.3266	−1.35283	−0.676414	−	0.736522i	$-0.736467\pi$
$571$	−32.3266	−1.35283	−0.676414	+	0.736522i	$0.736467\pi$
$572$	0	0
$573$	4.74369	0.198170
$574$	0	0
$575$	−6.72681	−0.280527
$576$	0	0
$577$	−12.9890	−0.540740	−0.270370	−	0.962756i	$-0.587146\pi$
$577$	−12.9890	−0.540740	−0.270370	+	0.962756i	$0.587146\pi$
$578$	0	0
$579$	5.45728	0.226797
$580$	0	0
$581$	51.0679	2.11865
$582$	0	0
$583$	−16.4477	−0.681196
$584$	0	0
$585$	0.894697	0.0369912
$586$	0	0
$587$	11.7191	0.483699	0.241849	−	0.970314i	$-0.422246\pi$
$587$	11.7191	0.483699	0.241849	+	0.970314i	$0.422246\pi$
$588$	0	0
$589$	7.65355	0.315359
$590$	0	0
$591$	−3.68104	−0.151418
$592$	0	0
$593$	28.2820	1.16140	0.580702	−	0.814116i	$-0.302778\pi$
$593$	28.2820	1.16140	0.580702	+	0.814116i	$0.302778\pi$
$594$	0	0
$595$	−16.0011	−0.655983
$596$	0	0
$597$	−1.96977	−0.0806174
$598$	0	0
$599$	−10.4437	−0.426717	−0.213359	−	0.976974i	$-0.568440\pi$
$599$	−10.4437	−0.426717	−0.213359	+	0.976974i	$0.568440\pi$
$600$	0	0
$601$	27.6419	1.12754	0.563769	−	0.825933i	$-0.309351\pi$
$601$	27.6419	1.12754	0.563769	+	0.825933i	$0.309351\pi$
$602$	0	0
$603$	−29.5354	−1.20278
$604$	0	0
$605$	0.171236	0.00696174
$606$	0	0
$607$	−25.6453	−1.04091	−0.520455	−	0.853889i	$-0.674238\pi$
$607$	−25.6453	−1.04091	−0.520455	+	0.853889i	$0.674238\pi$
$608$	0	0
$609$	−2.62492	−0.106367
$610$	0	0
$611$	−2.60968	−0.105576
$612$	0	0
$613$	40.5649	1.63840	0.819200	−	0.573507i	$-0.194418\pi$
$613$	40.5649	1.63840	0.819200	+	0.573507i	$0.194418\pi$
$614$	0	0
$615$	2.32833	0.0938872
$616$	0	0
$617$	9.25795	0.372711	0.186356	−	0.982482i	$-0.440332\pi$
$617$	9.25795	0.372711	0.186356	+	0.982482i	$0.440332\pi$
$618$	0	0
$619$	12.5901	0.506039	0.253020	−	0.967461i	$-0.418576\pi$
$619$	12.5901	0.506039	0.253020	+	0.967461i	$0.418576\pi$
$620$	0	0
$621$	−4.26894	−0.171307
$622$	0	0
$623$	27.4587	1.10011
$624$	0	0
$625$	−5.73627	−0.229451
$626$	0	0
$627$	−2.62379	−0.104784
$628$	0	0
$629$	2.58997	0.103269
$630$	0	0
$631$	−7.58491	−0.301951	−0.150975	−	0.988538i	$-0.548241\pi$
$631$	−7.58491	−0.301951	−0.150975	+	0.988538i	$0.548241\pi$
$632$	0	0
$633$	−5.63279	−0.223883
$634$	0	0
$635$	22.1352	0.878408
$636$	0	0
$637$	0.596661	0.0236406
$638$	0	0
$639$	20.5328	0.812263
$640$	0	0
$641$	−33.3758	−1.31826	−0.659132	−	0.752027i	$-0.729076\pi$
$641$	−33.3758	−1.31826	−0.659132	+	0.752027i	$0.729076\pi$
$642$	0	0
$643$	3.11695	0.122921	0.0614603	−	0.998110i	$-0.480424\pi$
$643$	3.11695	0.122921	0.0614603	+	0.998110i	$0.480424\pi$
$644$	0	0
$645$	4.36201	0.171754
$646$	0	0
$647$	13.1598	0.517365	0.258683	−	0.965962i	$-0.416712\pi$
$647$	13.1598	0.517365	0.258683	+	0.965962i	$0.416712\pi$
$648$	0	0
$649$	−28.6120	−1.12312
$650$	0	0
$651$	−2.27854	−0.0893031
$652$	0	0
$653$	7.98659	0.312539	0.156270	−	0.987714i	$-0.450053\pi$
$653$	7.98659	0.312539	0.156270	+	0.987714i	$0.450053\pi$
$654$	0	0
$655$	15.7604	0.615810
$656$	0	0
$657$	−35.1150	−1.36997
$658$	0	0
$659$	16.1986	0.631007	0.315503	−	0.948924i	$-0.397827\pi$
$659$	16.1986	0.631007	0.315503	+	0.948924i	$0.397827\pi$
$660$	0	0
$661$	6.68776	0.260124	0.130062	−	0.991506i	$-0.458482\pi$
$661$	6.68776	0.260124	0.130062	+	0.991506i	$0.458482\pi$
$662$	0	0
$663$	0.172211	0.00668812
$664$	0	0
$665$	14.4380	0.559883
$666$	0	0
$667$	−7.98957	−0.309358
$668$	0	0
$669$	−3.21714	−0.124382
$670$	0	0
$671$	26.5811	1.02615
$672$	0	0
$673$	13.2712	0.511567	0.255783	−	0.966734i	$-0.417667\pi$
$673$	13.2712	0.511567	0.255783	+	0.966734i	$0.417667\pi$
$674$	0	0
$675$	4.13018	0.158971
$676$	0	0
$677$	1.34526	0.0517027	0.0258514	−	0.999666i	$-0.491770\pi$
$677$	1.34526	0.0517027	0.0258514	+	0.999666i	$0.491770\pi$
$678$	0	0
$679$	−51.6168	−1.98087
$680$	0	0
$681$	−4.53836	−0.173910
$682$	0	0
$683$	7.47329	0.285957	0.142979	−	0.989726i	$-0.454332\pi$
$683$	7.47329	0.285957	0.142979	+	0.989726i	$0.454332\pi$
$684$	0	0
$685$	18.7595	0.716765
$686$	0	0
$687$	1.54769	0.0590480
$688$	0	0
$689$	−0.974080	−0.0371095
$690$	0	0
$691$	−7.43204	−0.282728	−0.141364	−	0.989958i	$-0.545149\pi$
$691$	−7.43204	−0.282728	−0.141364	+	0.989958i	$0.545149\pi$
$692$	0	0
$693$	−30.6090	−1.16274
$694$	0	0
$695$	10.8880	0.413004
$696$	0	0
$697$	−17.5612	−0.665177
$698$	0	0
$699$	−4.16535	−0.157548
$700$	0	0
$701$	−3.58585	−0.135436	−0.0677179	−	0.997705i	$-0.521572\pi$
$701$	−3.58585	−0.135436	−0.0677179	+	0.997705i	$0.521572\pi$
$702$	0	0
$703$	−2.33696	−0.0881402
$704$	0	0
$705$	5.70931	0.215025
$706$	0	0
$707$	12.5804	0.473134
$708$	0	0
$709$	−24.8440	−0.933037	−0.466519	−	0.884511i	$-0.654492\pi$
$709$	−24.8440	−0.933037	−0.466519	+	0.884511i	$0.654492\pi$
$710$	0	0
$711$	22.3117	0.836756
$712$	0	0
$713$	−6.93529	−0.259729
$714$	0	0
$715$	−1.00931	−0.0377460
$716$	0	0
$717$	5.82985	0.217720
$718$	0	0
$719$	29.4265	1.09742	0.548712	−	0.836011i	$-0.315118\pi$
$719$	29.4265	1.09742	0.548712	+	0.836011i	$0.315118\pi$
$720$	0	0
$721$	20.3375	0.757409
$722$	0	0
$723$	−6.65927	−0.247661
$724$	0	0
$725$	7.72989	0.287081
$726$	0	0
$727$	−33.4148	−1.23929	−0.619643	−	0.784884i	$-0.712723\pi$
$727$	−33.4148	−1.23929	−0.619643	+	0.784884i	$0.712723\pi$
$728$	0	0
$729$	−23.0519	−0.853773
$730$	0	0
$731$	−32.9001	−1.21685
$732$	0	0
$733$	−9.30466	−0.343676	−0.171838	−	0.985125i	$-0.554970\pi$
$733$	−9.30466	−0.343676	−0.171838	+	0.985125i	$0.554970\pi$
$734$	0	0
$735$	−1.30534	−0.0481483
$736$	0	0
$737$	33.3190	1.22732
$738$	0	0
$739$	−38.7472	−1.42534	−0.712670	−	0.701500i	$-0.752514\pi$
$739$	−38.7472	−1.42534	−0.712670	+	0.701500i	$0.752514\pi$
$740$	0	0
$741$	−0.155388	−0.00570833
$742$	0	0
$743$	−18.4634	−0.677356	−0.338678	−	0.940902i	$-0.609980\pi$
$743$	−18.4634	−0.677356	−0.338678	+	0.940902i	$0.609980\pi$
$744$	0	0
$745$	9.87322	0.361727
$746$	0	0
$747$	−47.1172	−1.72393
$748$	0	0
$749$	53.7973	1.96571
$750$	0	0
$751$	−32.9926	−1.20392	−0.601958	−	0.798528i	$-0.705613\pi$
$751$	−32.9926	−1.20392	−0.601958	+	0.798528i	$0.705613\pi$
$752$	0	0
$753$	−7.59832	−0.276898
$754$	0	0
$755$	2.79119	0.101582
$756$	0	0
$757$	−0.277375	−0.0100814	−0.00504068	−	0.999987i	$-0.501605\pi$
$757$	−0.277375	−0.0100814	−0.00504068	+	0.999987i	$0.501605\pi$
$758$	0	0
$759$	2.37756	0.0863000
$760$	0	0
$761$	5.29684	0.192010	0.0960051	−	0.995381i	$-0.469393\pi$
$761$	5.29684	0.192010	0.0960051	+	0.995381i	$0.469393\pi$
$762$	0	0
$763$	−7.49998	−0.271517
$764$	0	0
$765$	14.7633	0.533767
$766$	0	0
$767$	−1.69448	−0.0611842
$768$	0	0
$769$	−30.4959	−1.09971	−0.549855	−	0.835260i	$-0.685317\pi$
$769$	−30.4959	−1.09971	−0.549855	+	0.835260i	$0.685317\pi$
$770$	0	0
$771$	−4.49826	−0.162001
$772$	0	0
$773$	29.3324	1.05501	0.527506	−	0.849551i	$-0.323127\pi$
$773$	29.3324	1.05501	0.527506	+	0.849551i	$0.323127\pi$
$774$	0	0
$775$	6.70987	0.241026
$776$	0	0
$777$	0.695738	0.0249595
$778$	0	0
$779$	15.8457	0.567730
$780$	0	0
$781$	−23.1630	−0.828838
$782$	0	0
$783$	4.90550	0.175308
$784$	0	0
$785$	−16.5527	−0.590792
$786$	0	0
$787$	41.6151	1.48342	0.741710	−	0.670721i	$-0.234015\pi$
$787$	41.6151	1.48342	0.741710	+	0.670721i	$0.234015\pi$
$788$	0	0
$789$	6.81939	0.242777
$790$	0	0
$791$	7.46919	0.265574
$792$	0	0
$793$	1.57420	0.0559016
$794$	0	0
$795$	2.13104	0.0755802
$796$	0	0
$797$	2.07135	0.0733709	0.0366854	−	0.999327i	$-0.488320\pi$
$797$	2.07135	0.0733709	0.0366854	+	0.999327i	$0.488320\pi$
$798$	0	0
$799$	−43.0619	−1.52342
$800$	0	0
$801$	−25.3344	−0.895148
$802$	0	0
$803$	39.6132	1.39792
$804$	0	0
$805$	−13.0831	−0.461118
$806$	0	0
$807$	−5.74784	−0.202334
$808$	0	0
$809$	16.5359	0.581370	0.290685	−	0.956819i	$-0.406117\pi$
$809$	16.5359	0.581370	0.290685	+	0.956819i	$0.406117\pi$
$810$	0	0
$811$	−40.1217	−1.40886	−0.704432	−	0.709772i	$-0.748798\pi$
$811$	−40.1217	−1.40886	−0.704432	+	0.709772i	$0.748798\pi$
$812$	0	0
$813$	4.63675	0.162618
$814$	0	0
$815$	6.04461	0.211734
$816$	0	0
$817$	29.6862	1.03859
$818$	0	0
$819$	−1.81275	−0.0633425
$820$	0	0
$821$	44.7515	1.56184	0.780920	−	0.624631i	$-0.214751\pi$
$821$	44.7515	1.56184	0.780920	+	0.624631i	$0.214751\pi$
$822$	0	0
$823$	10.6491	0.371204	0.185602	−	0.982625i	$-0.440577\pi$
$823$	10.6491	0.371204	0.185602	+	0.982625i	$0.440577\pi$
$824$	0	0
$825$	−2.30028	−0.0800855
$826$	0	0
$827$	−37.7042	−1.31110	−0.655552	−	0.755150i	$-0.727564\pi$
$827$	−37.7042	−1.31110	−0.655552	+	0.755150i	$0.727564\pi$
$828$	0	0
$829$	10.5401	0.366074	0.183037	−	0.983106i	$-0.441407\pi$
$829$	10.5401	0.366074	0.183037	+	0.983106i	$0.441407\pi$
$830$	0	0
$831$	4.31202	0.149582
$832$	0	0
$833$	9.84542	0.341123
$834$	0	0
$835$	37.9532	1.31342
$836$	0	0
$837$	4.25819	0.147184
$838$	0	0
$839$	16.4775	0.568868	0.284434	−	0.958696i	$-0.408194\pi$
$839$	16.4775	0.568868	0.284434	+	0.958696i	$0.408194\pi$
$840$	0	0
$841$	−19.8191	−0.683416
$842$	0	0
$843$	−1.16416	−0.0400957
$844$	0	0
$845$	20.2838	0.697785
$846$	0	0
$847$	−0.346942	−0.0119211
$848$	0	0
$849$	−3.91478	−0.134355
$850$	0	0
$851$	2.11765	0.0725920
$852$	0	0
$853$	38.0707	1.30352	0.651758	−	0.758427i	$-0.274032\pi$
$853$	38.0707	1.30352	0.651758	+	0.758427i	$0.274032\pi$
$854$	0	0
$855$	−13.3211	−0.455571
$856$	0	0
$857$	−8.09527	−0.276529	−0.138265	−	0.990395i	$-0.544152\pi$
$857$	−8.09527	−0.276529	−0.138265	+	0.990395i	$0.544152\pi$
$858$	0	0
$859$	34.3162	1.17085	0.585426	−	0.810726i	$-0.300927\pi$
$859$	34.3162	1.17085	0.585426	+	0.810726i	$0.300927\pi$
$860$	0	0
$861$	−4.71743	−0.160769
$862$	0	0
$863$	3.34482	0.113859	0.0569294	−	0.998378i	$-0.481869\pi$
$863$	3.34482	0.113859	0.0569294	+	0.998378i	$0.481869\pi$
$864$	0	0
$865$	−21.6450	−0.735953
$866$	0	0
$867$	−1.80326	−0.0612419
$868$	0	0
$869$	−25.1699	−0.853831
$870$	0	0
$871$	1.97324	0.0668607
$872$	0	0
$873$	47.6237	1.61182
$874$	0	0
$875$	37.4663	1.26659
$876$	0	0
$877$	39.3175	1.32766	0.663828	−	0.747885i	$-0.268931\pi$
$877$	39.3175	1.32766	0.663828	+	0.747885i	$0.268931\pi$
$878$	0	0
$879$	0.713026	0.0240498
$880$	0	0
$881$	12.1789	0.410318	0.205159	−	0.978729i	$-0.434229\pi$
$881$	12.1789	0.410318	0.205159	+	0.978729i	$0.434229\pi$
$882$	0	0
$883$	−48.7442	−1.64037	−0.820186	−	0.572096i	$-0.806130\pi$
$883$	−48.7442	−1.64037	−0.820186	+	0.572096i	$0.806130\pi$
$884$	0	0
$885$	3.70709	0.124613
$886$	0	0
$887$	20.3913	0.684672	0.342336	−	0.939578i	$-0.388782\pi$
$887$	20.3913	0.684672	0.342336	+	0.939578i	$0.388782\pi$
$888$	0	0
$889$	−44.8481	−1.50416
$890$	0	0
$891$	27.5019	0.921348
$892$	0	0
$893$	38.8553	1.30024
$894$	0	0
$895$	20.6509	0.690283
$896$	0	0
$897$	0.140806	0.00470136
$898$	0	0
$899$	7.96945	0.265796
$900$	0	0
$901$	−16.0732	−0.535474
$902$	0	0
$903$	−8.83788	−0.294106
$904$	0	0
$905$	1.87213	0.0622318
$906$	0	0
$907$	−1.00195	−0.0332692	−0.0166346	−	0.999862i	$-0.505295\pi$
$907$	−1.00195	−0.0332692	−0.0166346	+	0.999862i	$0.505295\pi$
$908$	0	0
$909$	−11.6071	−0.384984
$910$	0	0
$911$	16.3169	0.540602	0.270301	−	0.962776i	$-0.412877\pi$
$911$	16.3169	0.540602	0.270301	+	0.962776i	$0.412877\pi$
$912$	0	0
$913$	53.1530	1.75911
$914$	0	0
$915$	−3.44396	−0.113854
$916$	0	0
$917$	−31.9322	−1.05449
$918$	0	0
$919$	−15.8595	−0.523155	−0.261578	−	0.965182i	$-0.584243\pi$
$919$	−15.8595	−0.523155	−0.261578	+	0.965182i	$0.584243\pi$
$920$	0	0
$921$	−4.75232	−0.156594
$922$	0	0
$923$	−1.37178	−0.0451526
$924$	0	0
$925$	−2.04882	−0.0673647
$926$	0	0
$927$	−18.7642	−0.616296
$928$	0	0
$929$	−27.7530	−0.910547	−0.455274	−	0.890352i	$-0.650459\pi$
$929$	−27.7530	−0.910547	−0.455274	+	0.890352i	$0.650459\pi$
$930$	0	0
$931$	−8.88365	−0.291150
$932$	0	0
$933$	−4.46064	−0.146035
$934$	0	0
$935$	−16.6545	−0.544659
$936$	0	0
$937$	48.6031	1.58780	0.793898	−	0.608051i	$-0.208049\pi$
$937$	48.6031	1.58780	0.793898	+	0.608051i	$0.208049\pi$
$938$	0	0
$939$	0.638517	0.0208372
$940$	0	0
$941$	26.9381	0.878158	0.439079	−	0.898449i	$-0.355305\pi$
$941$	26.9381	0.878158	0.439079	+	0.898449i	$0.355305\pi$
$942$	0	0
$943$	−14.3586	−0.467581
$944$	0	0
$945$	8.03286	0.261309
$946$	0	0
$947$	16.2330	0.527501	0.263750	−	0.964591i	$-0.415040\pi$
$947$	16.2330	0.527501	0.263750	+	0.964591i	$0.415040\pi$
$948$	0	0
$949$	2.34601	0.0761546
$950$	0	0
$951$	4.03246	0.130762
$952$	0	0
$953$	10.0132	0.324361	0.162180	−	0.986761i	$-0.448147\pi$
$953$	10.0132	0.324361	0.162180	+	0.986761i	$0.448147\pi$
$954$	0	0
$955$	−27.1690	−0.879169
$956$	0	0
$957$	−2.73209	−0.0883160
$958$	0	0
$959$	−38.0087	−1.22737
$960$	0	0
$961$	−24.0822	−0.776844
$962$	0	0
$963$	−49.6355	−1.59948
$964$	0	0
$965$	−31.2561	−1.00617
$966$	0	0
$967$	51.0143	1.64051	0.820255	−	0.571999i	$-0.193832\pi$
$967$	51.0143	1.64051	0.820255	+	0.571999i	$0.193832\pi$
$968$	0	0
$969$	−2.56404	−0.0823687
$970$	0	0
$971$	−34.3467	−1.10224	−0.551119	−	0.834427i	$-0.685799\pi$
$971$	−34.3467	−1.10224	−0.551119	+	0.834427i	$0.685799\pi$
$972$	0	0
$973$	−22.0601	−0.707215
$974$	0	0
$975$	−0.136229	−0.00436282
$976$	0	0
$977$	−52.5800	−1.68218	−0.841092	−	0.540892i	$-0.818087\pi$
$977$	−52.5800	−1.68218	−0.841092	+	0.540892i	$0.818087\pi$
$978$	0	0
$979$	28.5798	0.913415
$980$	0	0
$981$	6.91977	0.220931
$982$	0	0
$983$	46.0339	1.46825	0.734127	−	0.679012i	$-0.237592\pi$
$983$	46.0339	1.46825	0.734127	+	0.679012i	$0.237592\pi$
$984$	0	0
$985$	21.0829	0.671756
$986$	0	0
$987$	−11.5676	−0.368202
$988$	0	0
$989$	−26.9002	−0.855378
$990$	0	0
$991$	−31.8298	−1.01111	−0.505553	−	0.862796i	$-0.668711\pi$
$991$	−31.8298	−1.01111	−0.505553	+	0.862796i	$0.668711\pi$
$992$	0	0
$993$	3.81833	0.121171
$994$	0	0
$995$	11.2817	0.357654
$996$	0	0
$997$	10.0587	0.318563	0.159282	−	0.987233i	$-0.449082\pi$
$997$	10.0587	0.318563	0.159282	+	0.987233i	$0.449082\pi$
$998$	0	0
$999$	−1.30021	−0.0411368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.s.1.14		21
4.3	odd	2		2012.2.a.b.1.8	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.b.1.8	✓	21	4.3	odd	2
8048.2.a.s.1.14		21	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)

\(f(q)\)	\(=\)	\(q+0.273229 q^{3} -1.56489 q^{5} +3.17063 q^{7} -2.92535 q^{9} +O(q^{10})\)	\(q+0.273229 q^{3} -1.56489 q^{5} +3.17063 q^{7} -2.92535 q^{9} +3.30009 q^{11} +0.195440 q^{13} -0.427573 q^{15} +3.22493 q^{17} -2.90990 q^{19} +0.866307 q^{21} +2.63682 q^{23} -2.55111 q^{25} -1.61897 q^{27} -3.03001 q^{29} -2.63018 q^{31} +0.901678 q^{33} -4.96170 q^{35} +0.803108 q^{37} +0.0533999 q^{39} -5.44544 q^{41} -10.2018 q^{43} +4.57785 q^{45} -13.3528 q^{47} +3.05291 q^{49} +0.881143 q^{51} -4.98403 q^{53} -5.16428 q^{55} -0.795067 q^{57} -8.67007 q^{59} +8.05465 q^{61} -9.27520 q^{63} -0.305843 q^{65} +10.0964 q^{67} +0.720454 q^{69} -7.01891 q^{71} +12.0037 q^{73} -0.697037 q^{75} +10.4634 q^{77} -7.62704 q^{79} +8.33369 q^{81} +16.1065 q^{83} -5.04667 q^{85} -0.827885 q^{87} +8.66032 q^{89} +0.619669 q^{91} -0.718639 q^{93} +4.55368 q^{95} -16.2797 q^{97} -9.65390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * q^99

Introduction

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