LMFDB - Embedded newform 8048.2.a.y.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q+0.224625 q^{3} -2.01216 q^{5} -1.98253 q^{7} -2.94954 q^{9} +O(q^{10})$	$q+0.224625 q^{3} -2.01216 q^{5} -1.98253 q^{7} -2.94954 q^{9} -5.16477 q^{11} +4.72296 q^{13} -0.451981 q^{15} +3.77632 q^{17} -1.20339 q^{19} -0.445326 q^{21} +1.63312 q^{23} -0.951215 q^{25} -1.33641 q^{27} -8.63975 q^{29} -6.56407 q^{31} -1.16013 q^{33} +3.98917 q^{35} +1.73314 q^{37} +1.06089 q^{39} +0.973128 q^{41} +6.75379 q^{43} +5.93495 q^{45} -11.9538 q^{47} -3.06956 q^{49} +0.848254 q^{51} -8.24121 q^{53} +10.3923 q^{55} -0.270310 q^{57} -11.2397 q^{59} +1.56229 q^{61} +5.84757 q^{63} -9.50335 q^{65} +8.49996 q^{67} +0.366839 q^{69} +0.0603593 q^{71} -3.47965 q^{73} -0.213666 q^{75} +10.2393 q^{77} +6.07716 q^{79} +8.54844 q^{81} -14.2254 q^{83} -7.59855 q^{85} -1.94070 q^{87} +0.498711 q^{89} -9.36343 q^{91} -1.47445 q^{93} +2.42141 q^{95} -8.81572 q^{97} +15.2337 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * 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
$3$	0.224625	0.129687	0.0648435	−	0.997895i	$-0.479345\pi$
$3$	0.224625	0.129687	0.0648435	+	0.997895i	$0.479345\pi$
$4$	0	0
$5$	−2.01216	−0.899865	−0.449933	−	0.893063i	$-0.648552\pi$
$5$	−2.01216	−0.899865	−0.449933	+	0.893063i	$0.648552\pi$
$6$	0	0
$7$	−1.98253	−0.749327	−0.374664	−	0.927161i	$-0.622242\pi$
$7$	−1.98253	−0.749327	−0.374664	+	0.927161i	$0.622242\pi$
$8$	0	0
$9$	−2.94954	−0.983181
$10$	0	0
$11$	−5.16477	−1.55724	−0.778618	−	0.627498i	$-0.784079\pi$
$11$	−5.16477	−1.55724	−0.778618	+	0.627498i	$0.784079\pi$
$12$	0	0
$13$	4.72296	1.30991	0.654957	−	0.755666i	$-0.272687\pi$
$13$	4.72296	1.30991	0.654957	+	0.755666i	$0.272687\pi$
$14$	0	0
$15$	−0.451981	−0.116701
$16$	0	0
$17$	3.77632	0.915891	0.457946	−	0.888980i	$-0.348585\pi$
$17$	3.77632	0.915891	0.457946	+	0.888980i	$0.348585\pi$
$18$	0	0
$19$	−1.20339	−0.276076	−0.138038	−	0.990427i	$-0.544080\pi$
$19$	−1.20339	−0.276076	−0.138038	+	0.990427i	$0.544080\pi$
$20$	0	0
$21$	−0.445326	−0.0971780
$22$	0	0
$23$	1.63312	0.340529	0.170265	−	0.985398i	$-0.445538\pi$
$23$	1.63312	0.340529	0.170265	+	0.985398i	$0.445538\pi$
$24$	0	0
$25$	−0.951215	−0.190243
$26$	0	0
$27$	−1.33641	−0.257193
$28$	0	0
$29$	−8.63975	−1.60436	−0.802181	−	0.597081i	$-0.796327\pi$
$29$	−8.63975	−1.60436	−0.802181	+	0.597081i	$0.796327\pi$
$30$	0	0
$31$	−6.56407	−1.17894	−0.589471	−	0.807789i	$-0.700664\pi$
$31$	−6.56407	−1.17894	−0.589471	+	0.807789i	$0.700664\pi$
$32$	0	0
$33$	−1.16013	−0.201953
$34$	0	0
$35$	3.98917	0.674293
$36$	0	0
$37$	1.73314	0.284926	0.142463	−	0.989800i	$-0.454498\pi$
$37$	1.73314	0.284926	0.142463	+	0.989800i	$0.454498\pi$
$38$	0	0
$39$	1.06089	0.169879
$40$	0	0
$41$	0.973128	0.151977	0.0759885	−	0.997109i	$-0.475789\pi$
$41$	0.973128	0.151977	0.0759885	+	0.997109i	$0.475789\pi$
$42$	0	0
$43$	6.75379	1.02994	0.514972	−	0.857207i	$-0.327802\pi$
$43$	6.75379	1.02994	0.514972	+	0.857207i	$0.327802\pi$
$44$	0	0
$45$	5.93495	0.884730
$46$	0	0
$47$	−11.9538	−1.74364	−0.871818	−	0.489830i	$-0.837059\pi$
$47$	−11.9538	−1.74364	−0.871818	+	0.489830i	$0.837059\pi$
$48$	0	0
$49$	−3.06956	−0.438509
$50$	0	0
$51$	0.848254	0.118779
$52$	0	0
$53$	−8.24121	−1.13202	−0.566009	−	0.824399i	$-0.691513\pi$
$53$	−8.24121	−1.13202	−0.566009	+	0.824399i	$0.691513\pi$
$54$	0	0
$55$	10.3923	1.40130
$56$	0	0
$57$	−0.270310	−0.0358035
$58$	0	0
$59$	−11.2397	−1.46328	−0.731642	−	0.681689i	$-0.761246\pi$
$59$	−11.2397	−1.46328	−0.731642	+	0.681689i	$0.761246\pi$
$60$	0	0
$61$	1.56229	0.200030	0.100015	−	0.994986i	$-0.468111\pi$
$61$	1.56229	0.200030	0.100015	+	0.994986i	$0.468111\pi$
$62$	0	0
$63$	5.84757	0.736724
$64$	0	0
$65$	−9.50335	−1.17875
$66$	0	0
$67$	8.49996	1.03843	0.519217	−	0.854642i	$-0.326223\pi$
$67$	8.49996	1.03843	0.519217	+	0.854642i	$0.326223\pi$
$68$	0	0
$69$	0.366839	0.0441622
$70$	0	0
$71$	0.0603593	0.00716333	0.00358166	−	0.999994i	$-0.498860\pi$
$71$	0.0603593	0.00716333	0.00358166	+	0.999994i	$0.498860\pi$
$72$	0	0
$73$	−3.47965	−0.407262	−0.203631	−	0.979048i	$-0.565274\pi$
$73$	−3.47965	−0.407262	−0.203631	+	0.979048i	$0.565274\pi$
$74$	0	0
$75$	−0.213666	−0.0246720
$76$	0	0
$77$	10.2393	1.16688
$78$	0	0
$79$	6.07716	0.683734	0.341867	−	0.939748i	$-0.388941\pi$
$79$	6.07716	0.683734	0.341867	+	0.939748i	$0.388941\pi$
$80$	0	0
$81$	8.54844	0.949827
$82$	0	0
$83$	−14.2254	−1.56144	−0.780719	−	0.624883i	$-0.785147\pi$
$83$	−14.2254	−1.56144	−0.780719	+	0.624883i	$0.785147\pi$
$84$	0	0
$85$	−7.59855	−0.824179
$86$	0	0
$87$	−1.94070	−0.208065
$88$	0	0
$89$	0.498711	0.0528633	0.0264317	−	0.999651i	$-0.491586\pi$
$89$	0.498711	0.0528633	0.0264317	+	0.999651i	$0.491586\pi$
$90$	0	0
$91$	−9.36343	−0.981554
$92$	0	0
$93$	−1.47445	−0.152894
$94$	0	0
$95$	2.42141	0.248431
$96$	0	0
$97$	−8.81572	−0.895100	−0.447550	−	0.894259i	$-0.647703\pi$
$97$	−8.81572	−0.895100	−0.447550	+	0.894259i	$0.647703\pi$
$98$	0	0
$99$	15.2337	1.53105
$100$	0	0
$101$	7.76484	0.772630	0.386315	−	0.922367i	$-0.373748\pi$
$101$	7.76484	0.772630	0.386315	+	0.922367i	$0.373748\pi$
$102$	0	0
$103$	3.42751	0.337723	0.168861	−	0.985640i	$-0.445991\pi$
$103$	3.42751	0.337723	0.168861	+	0.985640i	$0.445991\pi$
$104$	0	0
$105$	0.896066	0.0874471
$106$	0	0
$107$	−4.05457	−0.391970	−0.195985	−	0.980607i	$-0.562790\pi$
$107$	−4.05457	−0.391970	−0.195985	+	0.980607i	$0.562790\pi$
$108$	0	0
$109$	3.19030	0.305576	0.152788	−	0.988259i	$-0.451175\pi$
$109$	3.19030	0.305576	0.152788	+	0.988259i	$0.451175\pi$
$110$	0	0
$111$	0.389305	0.0369512
$112$	0	0
$113$	7.25475	0.682469	0.341235	−	0.939978i	$-0.389155\pi$
$113$	7.25475	0.682469	0.341235	+	0.939978i	$0.389155\pi$
$114$	0	0
$115$	−3.28610	−0.306430
$116$	0	0
$117$	−13.9306	−1.28788
$118$	0	0
$119$	−7.48667	−0.686302
$120$	0	0
$121$	15.6748	1.42498
$122$	0	0
$123$	0.218588	0.0197095
$124$	0	0
$125$	11.9748	1.07106
$126$	0	0
$127$	−10.4834	−0.930249	−0.465125	−	0.885245i	$-0.653990\pi$
$127$	−10.4834	−0.930249	−0.465125	+	0.885245i	$0.653990\pi$
$128$	0	0
$129$	1.51707	0.133570
$130$	0	0
$131$	−18.5971	−1.62484	−0.812419	−	0.583074i	$-0.801850\pi$
$131$	−18.5971	−1.62484	−0.812419	+	0.583074i	$0.801850\pi$
$132$	0	0
$133$	2.38575	0.206871
$134$	0	0
$135$	2.68908	0.231439
$136$	0	0
$137$	7.85356	0.670975	0.335488	−	0.942045i	$-0.391099\pi$
$137$	7.85356	0.670975	0.335488	+	0.942045i	$0.391099\pi$
$138$	0	0
$139$	12.9436	1.09786	0.548930	−	0.835868i	$-0.315035\pi$
$139$	12.9436	1.09786	0.548930	+	0.835868i	$0.315035\pi$
$140$	0	0
$141$	−2.68511	−0.226127
$142$	0	0
$143$	−24.3930	−2.03984
$144$	0	0
$145$	17.3846	1.44371
$146$	0	0
$147$	−0.689499	−0.0568689
$148$	0	0
$149$	15.4675	1.26715	0.633574	−	0.773682i	$-0.281587\pi$
$149$	15.4675	1.26715	0.633574	+	0.773682i	$0.281587\pi$
$150$	0	0
$151$	−6.76326	−0.550387	−0.275193	−	0.961389i	$-0.588742\pi$
$151$	−6.76326	−0.550387	−0.275193	+	0.961389i	$0.588742\pi$
$152$	0	0
$153$	−11.1384	−0.900487
$154$	0	0
$155$	13.2080	1.06089
$156$	0	0
$157$	24.3594	1.94409	0.972044	−	0.234797i	$-0.0754425\pi$
$157$	24.3594	1.94409	0.972044	+	0.234797i	$0.0754425\pi$
$158$	0	0
$159$	−1.85118	−0.146808
$160$	0	0
$161$	−3.23771	−0.255168
$162$	0	0
$163$	−0.132237	−0.0103576	−0.00517879	−	0.999987i	$-0.501648\pi$
$163$	−0.132237	−0.0103576	−0.00517879	+	0.999987i	$0.501648\pi$
$164$	0	0
$165$	2.33437	0.181731
$166$	0	0
$167$	−9.44130	−0.730590	−0.365295	−	0.930892i	$-0.619032\pi$
$167$	−9.44130	−0.730590	−0.365295	+	0.930892i	$0.619032\pi$
$168$	0	0
$169$	9.30636	0.715873
$170$	0	0
$171$	3.54944	0.271433
$172$	0	0
$173$	21.5044	1.63495	0.817474	−	0.575965i	$-0.195374\pi$
$173$	21.5044	1.63495	0.817474	+	0.575965i	$0.195374\pi$
$174$	0	0
$175$	1.88581	0.142554
$176$	0	0
$177$	−2.52471	−0.189769
$178$	0	0
$179$	10.9909	0.821496	0.410748	−	0.911749i	$-0.365268\pi$
$179$	10.9909	0.821496	0.410748	+	0.911749i	$0.365268\pi$
$180$	0	0
$181$	0.150153	0.0111608	0.00558038	−	0.999984i	$-0.498224\pi$
$181$	0.150153	0.0111608	0.00558038	+	0.999984i	$0.498224\pi$
$182$	0	0
$183$	0.350928	0.0259413
$184$	0	0
$185$	−3.48735	−0.256395
$186$	0	0
$187$	−19.5038	−1.42626
$188$	0	0
$189$	2.64949	0.192722
$190$	0	0
$191$	14.1390	1.02307	0.511533	−	0.859264i	$-0.329078\pi$
$191$	14.1390	1.02307	0.511533	+	0.859264i	$0.329078\pi$
$192$	0	0
$193$	−7.84115	−0.564419	−0.282209	−	0.959353i	$-0.591067\pi$
$193$	−7.84115	−0.564419	−0.282209	+	0.959353i	$0.591067\pi$
$194$	0	0
$195$	−2.13469	−0.152868
$196$	0	0
$197$	−5.24987	−0.374038	−0.187019	−	0.982356i	$-0.559883\pi$
$197$	−5.24987	−0.374038	−0.187019	+	0.982356i	$0.559883\pi$
$198$	0	0
$199$	24.6850	1.74987	0.874936	−	0.484239i	$-0.160904\pi$
$199$	24.6850	1.74987	0.874936	+	0.484239i	$0.160904\pi$
$200$	0	0
$201$	1.90930	0.134672
$202$	0	0
$203$	17.1286	1.20219
$204$	0	0
$205$	−1.95809	−0.136759
$206$	0	0
$207$	−4.81696	−0.334802
$208$	0	0
$209$	6.21521	0.429915
$210$	0	0
$211$	−6.54286	−0.450429	−0.225214	−	0.974309i	$-0.572308\pi$
$211$	−6.54286	−0.450429	−0.225214	+	0.974309i	$0.572308\pi$
$212$	0	0
$213$	0.0135582	0.000928991 0
$214$	0	0
$215$	−13.5897	−0.926810
$216$	0	0
$217$	13.0135	0.883413
$218$	0	0
$219$	−0.781615	−0.0528167
$220$	0	0
$221$	17.8354	1.19974
$222$	0	0
$223$	−14.7331	−0.986603	−0.493302	−	0.869858i	$-0.664210\pi$
$223$	−14.7331	−0.986603	−0.493302	+	0.869858i	$0.664210\pi$
$224$	0	0
$225$	2.80565	0.187043
$226$	0	0
$227$	21.2222	1.40857	0.704284	−	0.709918i	$-0.251268\pi$
$227$	21.2222	1.40857	0.704284	+	0.709918i	$0.251268\pi$
$228$	0	0
$229$	−11.9943	−0.792606	−0.396303	−	0.918120i	$-0.629707\pi$
$229$	−11.9943	−0.792606	−0.396303	+	0.918120i	$0.629707\pi$
$230$	0	0
$231$	2.30000	0.151329
$232$	0	0
$233$	21.6787	1.42022	0.710108	−	0.704092i	$-0.248646\pi$
$233$	21.6787	1.42022	0.710108	+	0.704092i	$0.248646\pi$
$234$	0	0
$235$	24.0529	1.56904
$236$	0	0
$237$	1.36508	0.0886714
$238$	0	0
$239$	4.50056	0.291117	0.145558	−	0.989350i	$-0.453502\pi$
$239$	4.50056	0.291117	0.145558	+	0.989350i	$0.453502\pi$
$240$	0	0
$241$	28.1175	1.81121	0.905605	−	0.424122i	$-0.139417\pi$
$241$	28.1175	1.81121	0.905605	+	0.424122i	$0.139417\pi$
$242$	0	0
$243$	5.92943	0.380373
$244$	0	0
$245$	6.17645	0.394599
$246$	0	0
$247$	−5.68355	−0.361635
$248$	0	0
$249$	−3.19537	−0.202498
$250$	0	0
$251$	15.1946	0.959073	0.479536	−	0.877522i	$-0.340805\pi$
$251$	15.1946	0.959073	0.479536	+	0.877522i	$0.340805\pi$
$252$	0	0
$253$	−8.43468	−0.530284
$254$	0	0
$255$	−1.70682	−0.106885
$256$	0	0
$257$	15.1876	0.947376	0.473688	−	0.880693i	$-0.342922\pi$
$257$	15.1876	0.947376	0.473688	+	0.880693i	$0.342922\pi$
$258$	0	0
$259$	−3.43600	−0.213503
$260$	0	0
$261$	25.4833	1.57738
$262$	0	0
$263$	−1.30783	−0.0806442	−0.0403221	−	0.999187i	$-0.512838\pi$
$263$	−1.30783	−0.0806442	−0.0403221	+	0.999187i	$0.512838\pi$
$264$	0	0
$265$	16.5826	1.01866
$266$	0	0
$267$	0.112023	0.00685569
$268$	0	0
$269$	9.71242	0.592177	0.296088	−	0.955161i	$-0.404318\pi$
$269$	9.71242	0.592177	0.296088	+	0.955161i	$0.404318\pi$
$270$	0	0
$271$	3.72847	0.226489	0.113244	−	0.993567i	$-0.463876\pi$
$271$	3.72847	0.226489	0.113244	+	0.993567i	$0.463876\pi$
$272$	0	0
$273$	−2.10326	−0.127295
$274$	0	0
$275$	4.91280	0.296253
$276$	0	0
$277$	7.54205	0.453158	0.226579	−	0.973993i	$-0.427246\pi$
$277$	7.54205	0.453158	0.226579	+	0.973993i	$0.427246\pi$
$278$	0	0
$279$	19.3610	1.15911
$280$	0	0
$281$	−32.1053	−1.91524	−0.957621	−	0.288033i	$-0.906999\pi$
$281$	−32.1053	−1.91524	−0.957621	+	0.288033i	$0.906999\pi$
$282$	0	0
$283$	5.00605	0.297579	0.148789	−	0.988869i	$-0.452462\pi$
$283$	5.00605	0.297579	0.148789	+	0.988869i	$0.452462\pi$
$284$	0	0
$285$	0.543907	0.0322183
$286$	0	0
$287$	−1.92926	−0.113881
$288$	0	0
$289$	−2.73943	−0.161143
$290$	0	0
$291$	−1.98023	−0.116083
$292$	0	0
$293$	−17.7890	−1.03924	−0.519621	−	0.854397i	$-0.673927\pi$
$293$	−17.7890	−1.03924	−0.519621	+	0.854397i	$0.673927\pi$
$294$	0	0
$295$	22.6161	1.31676
$296$	0	0
$297$	6.90227	0.400510
$298$	0	0
$299$	7.71316	0.446064
$300$	0	0
$301$	−13.3896	−0.771764
$302$	0	0
$303$	1.74417	0.100200
$304$	0	0
$305$	−3.14357	−0.180000
$306$	0	0
$307$	−18.6908	−1.06674	−0.533371	−	0.845881i	$-0.679075\pi$
$307$	−18.6908	−1.06674	−0.533371	+	0.845881i	$0.679075\pi$
$308$	0	0
$309$	0.769904	0.0437983
$310$	0	0
$311$	3.96417	0.224787	0.112394	−	0.993664i	$-0.464148\pi$
$311$	3.96417	0.224787	0.112394	+	0.993664i	$0.464148\pi$
$312$	0	0
$313$	23.3388	1.31919	0.659593	−	0.751623i	$-0.270729\pi$
$313$	23.3388	1.31919	0.659593	+	0.751623i	$0.270729\pi$
$314$	0	0
$315$	−11.7662	−0.662953
$316$	0	0
$317$	−1.46135	−0.0820778	−0.0410389	−	0.999158i	$-0.513067\pi$
$317$	−1.46135	−0.0820778	−0.0410389	+	0.999158i	$0.513067\pi$
$318$	0	0
$319$	44.6223	2.49837
$320$	0	0
$321$	−0.910756	−0.0508334
$322$	0	0
$323$	−4.54437	−0.252855
$324$	0	0
$325$	−4.49255	−0.249202
$326$	0	0
$327$	0.716621	0.0396292
$328$	0	0
$329$	23.6987	1.30655
$330$	0	0
$331$	12.9449	0.711515	0.355758	−	0.934578i	$-0.384223\pi$
$331$	12.9449	0.711515	0.355758	+	0.934578i	$0.384223\pi$
$332$	0	0
$333$	−5.11197	−0.280134
$334$	0	0
$335$	−17.1033	−0.934451
$336$	0	0
$337$	14.6080	0.795748	0.397874	−	0.917440i	$-0.369748\pi$
$337$	14.6080	0.795748	0.397874	+	0.917440i	$0.369748\pi$
$338$	0	0
$339$	1.62960	0.0885075
$340$	0	0
$341$	33.9019	1.83589
$342$	0	0
$343$	19.9632	1.07791
$344$	0	0
$345$	−0.738138	−0.0397400
$346$	0	0
$347$	−17.3246	−0.930035	−0.465018	−	0.885301i	$-0.653952\pi$
$347$	−17.3246	−0.930035	−0.465018	+	0.885301i	$0.653952\pi$
$348$	0	0
$349$	22.8502	1.22314	0.611572	−	0.791189i	$-0.290538\pi$
$349$	22.8502	1.22314	0.611572	+	0.791189i	$0.290538\pi$
$350$	0	0
$351$	−6.31183	−0.336901
$352$	0	0
$353$	−17.7276	−0.943546	−0.471773	−	0.881720i	$-0.656386\pi$
$353$	−17.7276	−0.943546	−0.471773	+	0.881720i	$0.656386\pi$
$354$	0	0
$355$	−0.121452	−0.00644603
$356$	0	0
$357$	−1.68169	−0.0890045
$358$	0	0
$359$	21.0884	1.11300	0.556501	−	0.830847i	$-0.312143\pi$
$359$	21.0884	1.11300	0.556501	+	0.830847i	$0.312143\pi$
$360$	0	0
$361$	−17.5519	−0.923782
$362$	0	0
$363$	3.52095	0.184802
$364$	0	0
$365$	7.00161	0.366481
$366$	0	0
$367$	−21.3511	−1.11452	−0.557259	−	0.830339i	$-0.688147\pi$
$367$	−21.3511	−1.11452	−0.557259	+	0.830339i	$0.688147\pi$
$368$	0	0
$369$	−2.87028	−0.149421
$370$	0	0
$371$	16.3385	0.848252
$372$	0	0
$373$	15.7653	0.816297	0.408149	−	0.912916i	$-0.366175\pi$
$373$	15.7653	0.816297	0.408149	+	0.912916i	$0.366175\pi$
$374$	0	0
$375$	2.68983	0.138902
$376$	0	0
$377$	−40.8052	−2.10157
$378$	0	0
$379$	−16.1382	−0.828963	−0.414482	−	0.910058i	$-0.636037\pi$
$379$	−16.1382	−0.828963	−0.414482	+	0.910058i	$0.636037\pi$
$380$	0	0
$381$	−2.35482	−0.120641
$382$	0	0
$383$	6.23886	0.318791	0.159395	−	0.987215i	$-0.449046\pi$
$383$	6.23886	0.318791	0.159395	+	0.987215i	$0.449046\pi$
$384$	0	0
$385$	−20.6031	−1.05003
$386$	0	0
$387$	−19.9206	−1.01262
$388$	0	0
$389$	25.0716	1.27118	0.635591	−	0.772026i	$-0.280756\pi$
$389$	25.0716	1.27118	0.635591	+	0.772026i	$0.280756\pi$
$390$	0	0
$391$	6.16718	0.311888
$392$	0	0
$393$	−4.17737	−0.210721
$394$	0	0
$395$	−12.2282	−0.615268
$396$	0	0
$397$	−0.0124853	−0.000626621 0	−0.000313311	−	1.00000i	$-0.500100\pi$
$397$	−0.0124853	−0.000626621 0	−0.000313311		1.00000i	$0.500100\pi$
$398$	0	0
$399$	0.535899	0.0268285
$400$	0	0
$401$	−15.4353	−0.770803	−0.385402	−	0.922749i	$-0.625937\pi$
$401$	−15.4353	−0.770803	−0.385402	+	0.922749i	$0.625937\pi$
$402$	0	0
$403$	−31.0019	−1.54431
$404$	0	0
$405$	−17.2008	−0.854716
$406$	0	0
$407$	−8.95125	−0.443697
$408$	0	0
$409$	25.9121	1.28127	0.640635	−	0.767846i	$-0.278671\pi$
$409$	25.9121	1.28127	0.640635	+	0.767846i	$0.278671\pi$
$410$	0	0
$411$	1.76410	0.0870168
$412$	0	0
$413$	22.2831	1.09648
$414$	0	0
$415$	28.6237	1.40508
$416$	0	0
$417$	2.90745	0.142378
$418$	0	0
$419$	−34.8981	−1.70488	−0.852441	−	0.522823i	$-0.824879\pi$
$419$	−34.8981	−1.70488	−0.852441	+	0.522823i	$0.824879\pi$
$420$	0	0
$421$	−20.6385	−1.00586	−0.502929	−	0.864328i	$-0.667744\pi$
$421$	−20.6385	−1.00586	−0.502929	+	0.864328i	$0.667744\pi$
$422$	0	0
$423$	35.2581	1.71431
$424$	0	0
$425$	−3.59209	−0.174242
$426$	0	0
$427$	−3.09729	−0.149888
$428$	0	0
$429$	−5.47927	−0.264541
$430$	0	0
$431$	−29.0615	−1.39984	−0.699922	−	0.714219i	$-0.746782\pi$
$431$	−29.0615	−1.39984	−0.699922	+	0.714219i	$0.746782\pi$
$432$	0	0
$433$	−36.2641	−1.74274	−0.871372	−	0.490624i	$-0.836769\pi$
$433$	−36.2641	−1.74274	−0.871372	+	0.490624i	$0.836769\pi$
$434$	0	0
$435$	3.90500	0.187230
$436$	0	0
$437$	−1.96527	−0.0940118
$438$	0	0
$439$	−3.54940	−0.169404	−0.0847019	−	0.996406i	$-0.526994\pi$
$439$	−3.54940	−0.169404	−0.0847019	+	0.996406i	$0.526994\pi$
$440$	0	0
$441$	9.05381	0.431134
$442$	0	0
$443$	−12.2692	−0.582926	−0.291463	−	0.956582i	$-0.594142\pi$
$443$	−12.2692	−0.582926	−0.291463	+	0.956582i	$0.594142\pi$
$444$	0	0
$445$	−1.00349	−0.0475698
$446$	0	0
$447$	3.47439	0.164333
$448$	0	0
$449$	−11.3872	−0.537393	−0.268697	−	0.963225i	$-0.586593\pi$
$449$	−11.3872	−0.537393	−0.268697	+	0.963225i	$0.586593\pi$
$450$	0	0
$451$	−5.02598	−0.236664
$452$	0	0
$453$	−1.51920	−0.0713780
$454$	0	0
$455$	18.8407	0.883266
$456$	0	0
$457$	−36.2701	−1.69664	−0.848322	−	0.529481i	$-0.822387\pi$
$457$	−36.2701	−1.69664	−0.848322	+	0.529481i	$0.822387\pi$
$458$	0	0
$459$	−5.04672	−0.235561
$460$	0	0
$461$	11.6996	0.544906	0.272453	−	0.962169i	$-0.412165\pi$
$461$	11.6996	0.544906	0.272453	+	0.962169i	$0.412165\pi$
$462$	0	0
$463$	−27.7477	−1.28954	−0.644772	−	0.764375i	$-0.723048\pi$
$463$	−27.7477	−1.28954	−0.644772	+	0.764375i	$0.723048\pi$
$464$	0	0
$465$	2.96683	0.137584
$466$	0	0
$467$	−7.78383	−0.360193	−0.180096	−	0.983649i	$-0.557641\pi$
$467$	−7.78383	−0.360193	−0.180096	+	0.983649i	$0.557641\pi$
$468$	0	0
$469$	−16.8514	−0.778127
$470$	0	0
$471$	5.47171	0.252123
$472$	0	0
$473$	−34.8817	−1.60386
$474$	0	0
$475$	1.14468	0.0525215
$476$	0	0
$477$	24.3078	1.11298
$478$	0	0
$479$	12.5687	0.574278	0.287139	−	0.957889i	$-0.407296\pi$
$479$	12.5687	0.574278	0.287139	+	0.957889i	$0.407296\pi$
$480$	0	0
$481$	8.18554	0.373229
$482$	0	0
$483$	−0.727270	−0.0330920
$484$	0	0
$485$	17.7386	0.805470
$486$	0	0
$487$	−8.07349	−0.365845	−0.182922	−	0.983127i	$-0.558556\pi$
$487$	−8.07349	−0.365845	−0.182922	+	0.983127i	$0.558556\pi$
$488$	0	0
$489$	−0.0297036	−0.00134324
$490$	0	0
$491$	−10.1487	−0.458004	−0.229002	−	0.973426i	$-0.573546\pi$
$491$	−10.1487	−0.458004	−0.229002	+	0.973426i	$0.573546\pi$
$492$	0	0
$493$	−32.6264	−1.46942
$494$	0	0
$495$	−30.6526	−1.37773
$496$	0	0
$497$	−0.119664	−0.00536767
$498$	0	0
$499$	3.50745	0.157015	0.0785075	−	0.996914i	$-0.474985\pi$
$499$	3.50745	0.157015	0.0785075	+	0.996914i	$0.474985\pi$
$500$	0	0
$501$	−2.12075	−0.0947480
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−15.6241	−0.695263
$506$	0	0
$507$	2.09044	0.0928395
$508$	0	0
$509$	11.5909	0.513758	0.256879	−	0.966444i	$-0.417306\pi$
$509$	11.5909	0.513758	0.256879	+	0.966444i	$0.417306\pi$
$510$	0	0
$511$	6.89852	0.305173
$512$	0	0
$513$	1.60822	0.0710048
$514$	0	0
$515$	−6.89670	−0.303905
$516$	0	0
$517$	61.7384	2.71525
$518$	0	0
$519$	4.83041	0.212032
$520$	0	0
$521$	16.2425	0.711597	0.355799	−	0.934563i	$-0.384209\pi$
$521$	16.2425	0.711597	0.355799	+	0.934563i	$0.384209\pi$
$522$	0	0
$523$	7.53200	0.329351	0.164676	−	0.986348i	$-0.447342\pi$
$523$	7.53200	0.329351	0.164676	+	0.986348i	$0.447342\pi$
$524$	0	0
$525$	0.423600	0.0184874
$526$	0	0
$527$	−24.7880	−1.07978
$528$	0	0
$529$	−20.3329	−0.884040
$530$	0	0
$531$	33.1520	1.43867
$532$	0	0
$533$	4.59604	0.199077
$534$	0	0
$535$	8.15844	0.352720
$536$	0	0
$537$	2.46882	0.106537
$538$	0	0
$539$	15.8536	0.682862
$540$	0	0
$541$	−16.6480	−0.715753	−0.357876	−	0.933769i	$-0.616499\pi$
$541$	−16.6480	−0.715753	−0.357876	+	0.933769i	$0.616499\pi$
$542$	0	0
$543$	0.0337280	0.00144741
$544$	0	0
$545$	−6.41940	−0.274977
$546$	0	0
$547$	17.4585	0.746471	0.373235	−	0.927737i	$-0.378248\pi$
$547$	17.4585	0.746471	0.373235	+	0.927737i	$0.378248\pi$
$548$	0	0
$549$	−4.60803	−0.196666
$550$	0	0
$551$	10.3970	0.442925
$552$	0	0
$553$	−12.0482	−0.512340
$554$	0	0
$555$	−0.783345	−0.0332511
$556$	0	0
$557$	−17.5491	−0.743581	−0.371790	−	0.928317i	$-0.621256\pi$
$557$	−17.5491	−0.743581	−0.371790	+	0.928317i	$0.621256\pi$
$558$	0	0
$559$	31.8979	1.34914
$560$	0	0
$561$	−4.38103	−0.184967
$562$	0	0
$563$	30.4348	1.28267	0.641336	−	0.767260i	$-0.278380\pi$
$563$	30.4348	1.28267	0.641336	+	0.767260i	$0.278380\pi$
$564$	0	0
$565$	−14.5977	−0.614130
$566$	0	0
$567$	−16.9476	−0.711731
$568$	0	0
$569$	10.9429	0.458749	0.229374	−	0.973338i	$-0.426332\pi$
$569$	10.9429	0.458749	0.229374	+	0.973338i	$0.426332\pi$
$570$	0	0
$571$	45.1429	1.88917	0.944586	−	0.328264i	$-0.106464\pi$
$571$	45.1429	1.88917	0.944586	+	0.328264i	$0.106464\pi$
$572$	0	0
$573$	3.17598	0.132678
$574$	0	0
$575$	−1.55345	−0.0647832
$576$	0	0
$577$	12.4974	0.520273	0.260136	−	0.965572i	$-0.416232\pi$
$577$	12.4974	0.520273	0.260136	+	0.965572i	$0.416232\pi$
$578$	0	0
$579$	−1.76132	−0.0731978
$580$	0	0
$581$	28.2023	1.17003
$582$	0	0
$583$	42.5640	1.76282
$584$	0	0
$585$	28.0305	1.15892
$586$	0	0
$587$	−32.6269	−1.34666	−0.673328	−	0.739344i	$-0.735136\pi$
$587$	−32.6269	−1.34666	−0.673328	+	0.739344i	$0.735136\pi$
$588$	0	0
$589$	7.89912	0.325477
$590$	0	0
$591$	−1.17925	−0.0485079
$592$	0	0
$593$	−29.9167	−1.22853	−0.614267	−	0.789098i	$-0.710548\pi$
$593$	−29.9167	−1.22853	−0.614267	+	0.789098i	$0.710548\pi$
$594$	0	0
$595$	15.0644	0.617579
$596$	0	0
$597$	5.54485	0.226936
$598$	0	0
$599$	7.59802	0.310447	0.155223	−	0.987879i	$-0.450390\pi$
$599$	7.59802	0.310447	0.155223	+	0.987879i	$0.450390\pi$
$600$	0	0
$601$	5.11553	0.208667	0.104333	−	0.994542i	$-0.466729\pi$
$601$	5.11553	0.208667	0.104333	+	0.994542i	$0.466729\pi$
$602$	0	0
$603$	−25.0710	−1.02097
$604$	0	0
$605$	−31.5402	−1.28229
$606$	0	0
$607$	7.85941	0.319004	0.159502	−	0.987198i	$-0.449011\pi$
$607$	7.85941	0.319004	0.159502	+	0.987198i	$0.449011\pi$
$608$	0	0
$609$	3.84750	0.155909
$610$	0	0
$611$	−56.4571	−2.28401
$612$	0	0
$613$	31.1245	1.25711	0.628553	−	0.777767i	$-0.283648\pi$
$613$	31.1245	1.25711	0.628553	+	0.777767i	$0.283648\pi$
$614$	0	0
$615$	−0.439835	−0.0177359
$616$	0	0
$617$	−24.9170	−1.00312	−0.501559	−	0.865123i	$-0.667240\pi$
$617$	−24.9170	−1.00312	−0.501559	+	0.865123i	$0.667240\pi$
$618$	0	0
$619$	4.68746	0.188405	0.0942024	−	0.995553i	$-0.469970\pi$
$619$	4.68746	0.188405	0.0942024	+	0.995553i	$0.469970\pi$
$620$	0	0
$621$	−2.18252	−0.0875817
$622$	0	0
$623$	−0.988712	−0.0396119
$624$	0	0
$625$	−19.3391	−0.773565
$626$	0	0
$627$	1.39609	0.0557544
$628$	0	0
$629$	6.54488	0.260961
$630$	0	0
$631$	23.6768	0.942560	0.471280	−	0.881984i	$-0.343792\pi$
$631$	23.6768	0.942560	0.471280	+	0.881984i	$0.343792\pi$
$632$	0	0
$633$	−1.46969	−0.0584148
$634$	0	0
$635$	21.0942	0.837099
$636$	0	0
$637$	−14.4974	−0.574409
$638$	0	0
$639$	−0.178032	−0.00704285
$640$	0	0
$641$	−18.5880	−0.734180	−0.367090	−	0.930185i	$-0.619646\pi$
$641$	−18.5880	−0.734180	−0.367090	+	0.930185i	$0.619646\pi$
$642$	0	0
$643$	43.3788	1.71069	0.855346	−	0.518058i	$-0.173345\pi$
$643$	43.3788	1.71069	0.855346	+	0.518058i	$0.173345\pi$
$644$	0	0
$645$	−3.05258	−0.120195
$646$	0	0
$647$	9.53143	0.374719	0.187360	−	0.982291i	$-0.440007\pi$
$647$	9.53143	0.374719	0.187360	+	0.982291i	$0.440007\pi$
$648$	0	0
$649$	58.0504	2.27868
$650$	0	0
$651$	2.92315	0.114567
$652$	0	0
$653$	44.8203	1.75395	0.876977	−	0.480533i	$-0.159557\pi$
$653$	44.8203	1.75395	0.876977	+	0.480533i	$0.159557\pi$
$654$	0	0
$655$	37.4204	1.46214
$656$	0	0
$657$	10.2634	0.400413
$658$	0	0
$659$	−35.1924	−1.37090	−0.685450	−	0.728119i	$-0.740395\pi$
$659$	−35.1924	−1.37090	−0.685450	+	0.728119i	$0.740395\pi$
$660$	0	0
$661$	−5.52829	−0.215026	−0.107513	−	0.994204i	$-0.534289\pi$
$661$	−5.52829	−0.215026	−0.107513	+	0.994204i	$0.534289\pi$
$662$	0	0
$663$	4.00627	0.155591
$664$	0	0
$665$	−4.80052	−0.186156
$666$	0	0
$667$	−14.1098	−0.546332
$668$	0	0
$669$	−3.30942	−0.127950
$670$	0	0
$671$	−8.06885	−0.311494
$672$	0	0
$673$	−49.6266	−1.91297	−0.956483	−	0.291788i	$-0.905750\pi$
$673$	−49.6266	−1.91297	−0.956483	+	0.291788i	$0.905750\pi$
$674$	0	0
$675$	1.27122	0.0489291
$676$	0	0
$677$	−9.48396	−0.364498	−0.182249	−	0.983252i	$-0.558338\pi$
$677$	−9.48396	−0.364498	−0.182249	+	0.983252i	$0.558338\pi$
$678$	0	0
$679$	17.4775	0.670723
$680$	0	0
$681$	4.76703	0.182673
$682$	0	0
$683$	−16.4149	−0.628098	−0.314049	−	0.949407i	$-0.601686\pi$
$683$	−16.4149	−0.628098	−0.314049	+	0.949407i	$0.601686\pi$
$684$	0	0
$685$	−15.8026	−0.603787
$686$	0	0
$687$	−2.69422	−0.102791
$688$	0	0
$689$	−38.9229	−1.48285
$690$	0	0
$691$	−31.9720	−1.21627	−0.608135	−	0.793833i	$-0.708082\pi$
$691$	−31.9720	−1.21627	−0.608135	+	0.793833i	$0.708082\pi$
$692$	0	0
$693$	−30.2013	−1.14725
$694$	0	0
$695$	−26.0446	−0.987927
$696$	0	0
$697$	3.67484	0.139194
$698$	0	0
$699$	4.86956	0.184184
$700$	0	0
$701$	44.4116	1.67740	0.838701	−	0.544593i	$-0.183316\pi$
$701$	44.4116	1.67740	0.838701	+	0.544593i	$0.183316\pi$
$702$	0	0
$703$	−2.08563	−0.0786612
$704$	0	0
$705$	5.40287	0.203484
$706$	0	0
$707$	−15.3940	−0.578953
$708$	0	0
$709$	32.6935	1.22783	0.613914	−	0.789373i	$-0.289594\pi$
$709$	32.6935	1.22783	0.613914	+	0.789373i	$0.289594\pi$
$710$	0	0
$711$	−17.9248	−0.672234
$712$	0	0
$713$	−10.7199	−0.401464
$714$	0	0
$715$	49.0826	1.83558
$716$	0	0
$717$	1.01094	0.0377541
$718$	0	0
$719$	−38.9669	−1.45322	−0.726610	−	0.687050i	$-0.758905\pi$
$719$	−38.9669	−1.45322	−0.726610	+	0.687050i	$0.758905\pi$
$720$	0	0
$721$	−6.79516	−0.253065
$722$	0	0
$723$	6.31589	0.234891
$724$	0	0
$725$	8.21826	0.305218
$726$	0	0
$727$	19.5384	0.724639	0.362319	−	0.932054i	$-0.381985\pi$
$727$	19.5384	0.724639	0.362319	+	0.932054i	$0.381985\pi$
$728$	0	0
$729$	−24.3134	−0.900497
$730$	0	0
$731$	25.5045	0.943316
$732$	0	0
$733$	43.6080	1.61070	0.805349	−	0.592801i	$-0.201978\pi$
$733$	43.6080	1.61070	0.805349	+	0.592801i	$0.201978\pi$
$734$	0	0
$735$	1.38738	0.0511744
$736$	0	0
$737$	−43.9003	−1.61709
$738$	0	0
$739$	−14.1061	−0.518902	−0.259451	−	0.965756i	$-0.583542\pi$
$739$	−14.1061	−0.518902	−0.259451	+	0.965756i	$0.583542\pi$
$740$	0	0
$741$	−1.27666	−0.0468994
$742$	0	0
$743$	26.9756	0.989638	0.494819	−	0.868996i	$-0.335234\pi$
$743$	26.9756	0.989638	0.494819	+	0.868996i	$0.335234\pi$
$744$	0	0
$745$	−31.1231	−1.14026
$746$	0	0
$747$	41.9584	1.53518
$748$	0	0
$749$	8.03832	0.293714
$750$	0	0
$751$	30.5184	1.11363	0.556816	−	0.830636i	$-0.312023\pi$
$751$	30.5184	1.11363	0.556816	+	0.830636i	$0.312023\pi$
$752$	0	0
$753$	3.41307	0.124379
$754$	0	0
$755$	13.6088	0.495274
$756$	0	0
$757$	−18.5901	−0.675669	−0.337835	−	0.941205i	$-0.609694\pi$
$757$	−18.5901	−0.675669	−0.337835	+	0.941205i	$0.609694\pi$
$758$	0	0
$759$	−1.89464	−0.0687710
$760$	0	0
$761$	−22.5248	−0.816524	−0.408262	−	0.912865i	$-0.633865\pi$
$761$	−22.5248	−0.816524	−0.408262	+	0.912865i	$0.633865\pi$
$762$	0	0
$763$	−6.32488	−0.228976
$764$	0	0
$765$	22.4123	0.810317
$766$	0	0
$767$	−53.0846	−1.91677
$768$	0	0
$769$	−14.0042	−0.505006	−0.252503	−	0.967596i	$-0.581254\pi$
$769$	−14.0042	−0.505006	−0.252503	+	0.967596i	$0.581254\pi$
$770$	0	0
$771$	3.41151	0.122862
$772$	0	0
$773$	8.89095	0.319785	0.159893	−	0.987134i	$-0.448885\pi$
$773$	8.89095	0.319785	0.159893	+	0.987134i	$0.448885\pi$
$774$	0	0
$775$	6.24384	0.224285
$776$	0	0
$777$	−0.771811	−0.0276886
$778$	0	0
$779$	−1.17105	−0.0419572
$780$	0	0
$781$	−0.311741	−0.0111550
$782$	0	0
$783$	11.5463	0.412631
$784$	0	0
$785$	−49.0149	−1.74942
$786$	0	0
$787$	42.3895	1.51102	0.755511	−	0.655136i	$-0.227389\pi$
$787$	42.3895	1.51102	0.755511	+	0.655136i	$0.227389\pi$
$788$	0	0
$789$	−0.293771	−0.0104585
$790$	0	0
$791$	−14.3828	−0.511393
$792$	0	0
$793$	7.37862	0.262022
$794$	0	0
$795$	3.72487	0.132107
$796$	0	0
$797$	−38.9081	−1.37820	−0.689098	−	0.724669i	$-0.741993\pi$
$797$	−38.9081	−1.37820	−0.689098	+	0.724669i	$0.741993\pi$
$798$	0	0
$799$	−45.1412	−1.59698
$800$	0	0
$801$	−1.47097	−0.0519742
$802$	0	0
$803$	17.9716	0.634204
$804$	0	0
$805$	6.51480	0.229616
$806$	0	0
$807$	2.18165	0.0767977
$808$	0	0
$809$	0.525574	0.0184782	0.00923910	−	0.999957i	$-0.497059\pi$
$809$	0.525574	0.0184782	0.00923910	+	0.999957i	$0.497059\pi$
$810$	0	0
$811$	32.8550	1.15370	0.576848	−	0.816852i	$-0.304283\pi$
$811$	32.8550	1.15370	0.576848	+	0.816852i	$0.304283\pi$
$812$	0	0
$813$	0.837507	0.0293727
$814$	0	0
$815$	0.266081	0.00932042
$816$	0	0
$817$	−8.12742	−0.284342
$818$	0	0
$819$	27.6178	0.965045
$820$	0	0
$821$	31.9309	1.11440	0.557198	−	0.830380i	$-0.311876\pi$
$821$	31.9309	1.11440	0.557198	+	0.830380i	$0.311876\pi$
$822$	0	0
$823$	−15.7444	−0.548815	−0.274407	−	0.961614i	$-0.588482\pi$
$823$	−15.7444	−0.548815	−0.274407	+	0.961614i	$0.588482\pi$
$824$	0	0
$825$	1.10354	0.0384202
$826$	0	0
$827$	6.54590	0.227623	0.113812	−	0.993502i	$-0.463694\pi$
$827$	6.54590	0.227623	0.113812	+	0.993502i	$0.463694\pi$
$828$	0	0
$829$	−24.0859	−0.836536	−0.418268	−	0.908324i	$-0.637363\pi$
$829$	−24.0859	−0.836536	−0.418268	+	0.908324i	$0.637363\pi$
$830$	0	0
$831$	1.69413	0.0587688
$832$	0	0
$833$	−11.5916	−0.401627
$834$	0	0
$835$	18.9974	0.657432
$836$	0	0
$837$	8.77232	0.303216
$838$	0	0
$839$	25.1163	0.867109	0.433555	−	0.901127i	$-0.357259\pi$
$839$	25.1163	0.867109	0.433555	+	0.901127i	$0.357259\pi$
$840$	0	0
$841$	45.6453	1.57398
$842$	0	0
$843$	−7.21164	−0.248382
$844$	0	0
$845$	−18.7259	−0.644190
$846$	0	0
$847$	−31.0758	−1.06778
$848$	0	0
$849$	1.12448	0.0385921
$850$	0	0
$851$	2.83042	0.0970256
$852$	0	0
$853$	42.6052	1.45878	0.729388	−	0.684100i	$-0.239805\pi$
$853$	42.6052	1.45878	0.729388	+	0.684100i	$0.239805\pi$
$854$	0	0
$855$	−7.14204	−0.244253
$856$	0	0
$857$	9.57277	0.327000	0.163500	−	0.986543i	$-0.447722\pi$
$857$	9.57277	0.327000	0.163500	+	0.986543i	$0.447722\pi$
$858$	0	0
$859$	−22.2607	−0.759525	−0.379762	−	0.925084i	$-0.623994\pi$
$859$	−22.2607	−0.759525	−0.379762	+	0.925084i	$0.623994\pi$
$860$	0	0
$861$	−0.433359	−0.0147688
$862$	0	0
$863$	−24.9858	−0.850526	−0.425263	−	0.905070i	$-0.639818\pi$
$863$	−24.9858	−0.850526	−0.425263	+	0.905070i	$0.639818\pi$
$864$	0	0
$865$	−43.2703	−1.47123
$866$	0	0
$867$	−0.615343	−0.0208981
$868$	0	0
$869$	−31.3871	−1.06473
$870$	0	0
$871$	40.1450	1.36026
$872$	0	0
$873$	26.0023	0.880046
$874$	0	0
$875$	−23.7404	−0.802573
$876$	0	0
$877$	−2.99183	−0.101027	−0.0505135	−	0.998723i	$-0.516086\pi$
$877$	−2.99183	−0.101027	−0.0505135	+	0.998723i	$0.516086\pi$
$878$	0	0
$879$	−3.99584	−0.134776
$880$	0	0
$881$	−7.79189	−0.262515	−0.131258	−	0.991348i	$-0.541902\pi$
$881$	−7.79189	−0.262515	−0.131258	+	0.991348i	$0.541902\pi$
$882$	0	0
$883$	−13.2503	−0.445909	−0.222955	−	0.974829i	$-0.571570\pi$
$883$	−13.2503	−0.445909	−0.222955	+	0.974829i	$0.571570\pi$
$884$	0	0
$885$	5.08012	0.170766
$886$	0	0
$887$	−22.6732	−0.761291	−0.380646	−	0.924721i	$-0.624298\pi$
$887$	−22.6732	−0.761291	−0.380646	+	0.924721i	$0.624298\pi$
$888$	0	0
$889$	20.7836	0.697061
$890$	0	0
$891$	−44.1507	−1.47910
$892$	0	0
$893$	14.3850	0.481376
$894$	0	0
$895$	−22.1154	−0.739235
$896$	0	0
$897$	1.73257	0.0578487
$898$	0	0
$899$	56.7120	1.89145
$900$	0	0
$901$	−31.1214	−1.03681
$902$	0	0
$903$	−3.00764	−0.100088
$904$	0	0
$905$	−0.302131	−0.0100432
$906$	0	0
$907$	−10.9764	−0.364467	−0.182233	−	0.983255i	$-0.558333\pi$
$907$	−10.9764	−0.364467	−0.182233	+	0.983255i	$0.558333\pi$
$908$	0	0
$909$	−22.9027	−0.759635
$910$	0	0
$911$	−42.8706	−1.42036	−0.710182	−	0.704018i	$-0.751388\pi$
$911$	−42.8706	−1.42036	−0.710182	+	0.704018i	$0.751388\pi$
$912$	0	0
$913$	73.4707	2.43153
$914$	0	0
$915$	−0.706123	−0.0233437
$916$	0	0
$917$	36.8694	1.21754
$918$	0	0
$919$	12.4054	0.409215	0.204608	−	0.978844i	$-0.434408\pi$
$919$	12.4054	0.409215	0.204608	+	0.978844i	$0.434408\pi$
$920$	0	0
$921$	−4.19842	−0.138343
$922$	0	0
$923$	0.285074	0.00938334
$924$	0	0
$925$	−1.64859	−0.0542052
$926$	0	0
$927$	−10.1096	−0.332043
$928$	0	0
$929$	0.805797	0.0264373	0.0132187	−	0.999913i	$-0.495792\pi$
$929$	0.805797	0.0264373	0.0132187	+	0.999913i	$0.495792\pi$
$930$	0	0
$931$	3.69387	0.121062
$932$	0	0
$933$	0.890450	0.0291520
$934$	0	0
$935$	39.2448	1.28344
$936$	0	0
$937$	−37.2987	−1.21850	−0.609248	−	0.792980i	$-0.708529\pi$
$937$	−37.2987	−1.21850	−0.609248	+	0.792980i	$0.708529\pi$
$938$	0	0
$939$	5.24247	0.171081
$940$	0	0
$941$	−8.20339	−0.267423	−0.133711	−	0.991020i	$-0.542690\pi$
$941$	−8.20339	−0.267423	−0.133711	+	0.991020i	$0.542690\pi$
$942$	0	0
$943$	1.58923	0.0517526
$944$	0	0
$945$	−5.33119	−0.173424
$946$	0	0
$947$	22.1004	0.718167	0.359084	−	0.933305i	$-0.383089\pi$
$947$	22.1004	0.718167	0.359084	+	0.933305i	$0.383089\pi$
$948$	0	0
$949$	−16.4343	−0.533479
$950$	0	0
$951$	−0.328256	−0.0106444
$952$	0	0
$953$	50.2570	1.62798	0.813992	−	0.580877i	$-0.197290\pi$
$953$	50.2570	1.62798	0.813992	+	0.580877i	$0.197290\pi$
$954$	0	0
$955$	−28.4500	−0.920621
$956$	0	0
$957$	10.0233	0.324006
$958$	0	0
$959$	−15.5699	−0.502780
$960$	0	0
$961$	12.0871	0.389905
$962$	0	0
$963$	11.9591	0.385377
$964$	0	0
$965$	15.7777	0.507901
$966$	0	0
$967$	27.1579	0.873338	0.436669	−	0.899622i	$-0.356158\pi$
$967$	27.1579	0.873338	0.436669	+	0.899622i	$0.356158\pi$
$968$	0	0
$969$	−1.02078	−0.0327921
$970$	0	0
$971$	−56.0456	−1.79859	−0.899295	−	0.437343i	$-0.855919\pi$
$971$	−56.0456	−1.79859	−0.899295	+	0.437343i	$0.855919\pi$
$972$	0	0
$973$	−25.6611	−0.822657
$974$	0	0
$975$	−1.00914	−0.0323183
$976$	0	0
$977$	−32.9503	−1.05417	−0.527086	−	0.849812i	$-0.676715\pi$
$977$	−32.9503	−1.05417	−0.527086	+	0.849812i	$0.676715\pi$
$978$	0	0
$979$	−2.57573	−0.0823206
$980$	0	0
$981$	−9.40994	−0.300436
$982$	0	0
$983$	40.2054	1.28235	0.641176	−	0.767394i	$-0.278447\pi$
$983$	40.2054	1.28235	0.641176	+	0.767394i	$0.278447\pi$
$984$	0	0
$985$	10.5636	0.336584
$986$	0	0
$987$	5.32332	0.169443
$988$	0	0
$989$	11.0297	0.350726
$990$	0	0
$991$	−3.84136	−0.122025	−0.0610125	−	0.998137i	$-0.519433\pi$
$991$	−3.84136	−0.122025	−0.0610125	+	0.998137i	$0.519433\pi$
$992$	0	0
$993$	2.90774	0.0922743
$994$	0	0
$995$	−49.6701	−1.57465
$996$	0	0
$997$	−26.9615	−0.853879	−0.426939	−	0.904280i	$-0.640408\pi$
$997$	−26.9615	−0.853879	−0.426939	+	0.904280i	$0.640408\pi$
$998$	0	0
$999$	−2.31619	−0.0732810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.y.1.18		33
4.3	odd	2		4024.2.a.f.1.16	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.16	✓	33	4.3	odd	2
8048.2.a.y.1.18		33	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)

\(f(q)\)	\(=\)	\(q+0.224625 q^{3} -2.01216 q^{5} -1.98253 q^{7} -2.94954 q^{9} +O(q^{10})\)	\(q+0.224625 q^{3} -2.01216 q^{5} -1.98253 q^{7} -2.94954 q^{9} -5.16477 q^{11} +4.72296 q^{13} -0.451981 q^{15} +3.77632 q^{17} -1.20339 q^{19} -0.445326 q^{21} +1.63312 q^{23} -0.951215 q^{25} -1.33641 q^{27} -8.63975 q^{29} -6.56407 q^{31} -1.16013 q^{33} +3.98917 q^{35} +1.73314 q^{37} +1.06089 q^{39} +0.973128 q^{41} +6.75379 q^{43} +5.93495 q^{45} -11.9538 q^{47} -3.06956 q^{49} +0.848254 q^{51} -8.24121 q^{53} +10.3923 q^{55} -0.270310 q^{57} -11.2397 q^{59} +1.56229 q^{61} +5.84757 q^{63} -9.50335 q^{65} +8.49996 q^{67} +0.366839 q^{69} +0.0603593 q^{71} -3.47965 q^{73} -0.213666 q^{75} +10.2393 q^{77} +6.07716 q^{79} +8.54844 q^{81} -14.2254 q^{83} -7.59855 q^{85} -1.94070 q^{87} +0.498711 q^{89} -9.36343 q^{91} -1.47445 q^{93} +2.42141 q^{95} -8.81572 q^{97} +15.2337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * q^99

Introduction

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