LMFDB - Embedded newform 8048.2.a.y.1.19

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			1.19
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)

$f(q)$	$=$	$q+0.254882 q^{3} +3.76559 q^{5} +2.29648 q^{7} -2.93504 q^{9} +O(q^{10})$	$q+0.254882 q^{3} +3.76559 q^{5} +2.29648 q^{7} -2.93504 q^{9} +2.49668 q^{11} +6.47840 q^{13} +0.959781 q^{15} +5.17796 q^{17} -1.78611 q^{19} +0.585331 q^{21} +1.62486 q^{23} +9.17970 q^{25} -1.51273 q^{27} +3.13341 q^{29} +2.76790 q^{31} +0.636357 q^{33} +8.64762 q^{35} -0.433471 q^{37} +1.65123 q^{39} -3.69738 q^{41} -0.958002 q^{43} -11.0522 q^{45} -1.36228 q^{47} -1.72618 q^{49} +1.31977 q^{51} +9.03300 q^{53} +9.40148 q^{55} -0.455247 q^{57} +7.59760 q^{59} -7.79810 q^{61} -6.74025 q^{63} +24.3950 q^{65} -4.83170 q^{67} +0.414147 q^{69} -2.52115 q^{71} -4.72787 q^{73} +2.33974 q^{75} +5.73357 q^{77} -14.4664 q^{79} +8.41954 q^{81} -15.5950 q^{83} +19.4981 q^{85} +0.798648 q^{87} +2.46665 q^{89} +14.8775 q^{91} +0.705487 q^{93} -6.72578 q^{95} +3.53018 q^{97} -7.32784 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.254882	0.147156	0.0735780	−	0.997289i	$-0.476558\pi$
$3$	0.254882	0.147156	0.0735780	+	0.997289i	$0.476558\pi$
$4$	0	0
$5$	3.76559	1.68403	0.842013	−	0.539458i	$-0.181371\pi$
$5$	3.76559	1.68403	0.842013	+	0.539458i	$0.181371\pi$
$6$	0	0
$7$	2.29648	0.867988	0.433994	−	0.900916i	$-0.357104\pi$
$7$	2.29648	0.867988	0.433994	+	0.900916i	$0.357104\pi$
$8$	0	0
$9$	−2.93504	−0.978345
$10$	0	0
$11$	2.49668	0.752777	0.376388	−	0.926462i	$-0.377166\pi$
$11$	2.49668	0.752777	0.376388	+	0.926462i	$0.377166\pi$
$12$	0	0
$13$	6.47840	1.79679	0.898393	−	0.439193i	$-0.144736\pi$
$13$	6.47840	1.79679	0.898393	+	0.439193i	$0.144736\pi$
$14$	0	0
$15$	0.959781	0.247814
$16$	0	0
$17$	5.17796	1.25584	0.627920	−	0.778278i	$-0.283907\pi$
$17$	5.17796	1.25584	0.627920	+	0.778278i	$0.283907\pi$
$18$	0	0
$19$	−1.78611	−0.409762	−0.204881	−	0.978787i	$-0.565681\pi$
$19$	−1.78611	−0.409762	−0.204881	+	0.978787i	$0.565681\pi$
$20$	0	0
$21$	0.585331	0.127730
$22$	0	0
$23$	1.62486	0.338807	0.169403	−	0.985547i	$-0.445816\pi$
$23$	1.62486	0.338807	0.169403	+	0.985547i	$0.445816\pi$
$24$	0	0
$25$	9.17970	1.83594
$26$	0	0
$27$	−1.51273	−0.291125
$28$	0	0
$29$	3.13341	0.581859	0.290930	−	0.956744i	$-0.406035\pi$
$29$	3.13341	0.581859	0.290930	+	0.956744i	$0.406035\pi$
$30$	0	0
$31$	2.76790	0.497130	0.248565	−	0.968615i	$-0.420041\pi$
$31$	2.76790	0.497130	0.248565	+	0.968615i	$0.420041\pi$
$32$	0	0
$33$	0.636357	0.110776
$34$	0	0
$35$	8.64762	1.46171
$36$	0	0
$37$	−0.433471	−0.0712622	−0.0356311	−	0.999365i	$-0.511344\pi$
$37$	−0.433471	−0.0712622	−0.0356311	+	0.999365i	$0.511344\pi$
$38$	0	0
$39$	1.65123	0.264408
$40$	0	0
$41$	−3.69738	−0.577434	−0.288717	−	0.957415i	$-0.593229\pi$
$41$	−3.69738	−0.577434	−0.288717	+	0.957415i	$0.593229\pi$
$42$	0	0
$43$	−0.958002	−0.146094	−0.0730470	−	0.997328i	$-0.523272\pi$
$43$	−0.958002	−0.146094	−0.0730470	+	0.997328i	$0.523272\pi$
$44$	0	0
$45$	−11.0522	−1.64756
$46$	0	0
$47$	−1.36228	−0.198709	−0.0993546	−	0.995052i	$-0.531678\pi$
$47$	−1.36228	−0.198709	−0.0993546	+	0.995052i	$0.531678\pi$
$48$	0	0
$49$	−1.72618	−0.246596
$50$	0	0
$51$	1.31977	0.184804
$52$	0	0
$53$	9.03300	1.24078	0.620389	−	0.784294i	$-0.286975\pi$
$53$	9.03300	1.24078	0.620389	+	0.784294i	$0.286975\pi$
$54$	0	0
$55$	9.40148	1.26769
$56$	0	0
$57$	−0.455247	−0.0602990
$58$	0	0
$59$	7.59760	0.989123	0.494562	−	0.869143i	$-0.335329\pi$
$59$	7.59760	0.989123	0.494562	+	0.869143i	$0.335329\pi$
$60$	0	0
$61$	−7.79810	−0.998445	−0.499222	−	0.866474i	$-0.666381\pi$
$61$	−7.79810	−0.998445	−0.499222	+	0.866474i	$0.666381\pi$
$62$	0	0
$63$	−6.74025	−0.849192
$64$	0	0
$65$	24.3950	3.02583
$66$	0	0
$67$	−4.83170	−0.590286	−0.295143	−	0.955453i	$-0.595367\pi$
$67$	−4.83170	−0.590286	−0.295143	+	0.955453i	$0.595367\pi$
$68$	0	0
$69$	0.414147	0.0498574
$70$	0	0
$71$	−2.52115	−0.299205	−0.149603	−	0.988746i	$-0.547799\pi$
$71$	−2.52115	−0.299205	−0.149603	+	0.988746i	$0.547799\pi$
$72$	0	0
$73$	−4.72787	−0.553356	−0.276678	−	0.960963i	$-0.589233\pi$
$73$	−4.72787	−0.553356	−0.276678	+	0.960963i	$0.589233\pi$
$74$	0	0
$75$	2.33974	0.270170
$76$	0	0
$77$	5.73357	0.653401
$78$	0	0
$79$	−14.4664	−1.62760	−0.813798	−	0.581148i	$-0.802604\pi$
$79$	−14.4664	−1.62760	−0.813798	+	0.581148i	$0.802604\pi$
$80$	0	0
$81$	8.41954	0.935504
$82$	0	0
$83$	−15.5950	−1.71177	−0.855885	−	0.517165i	$-0.826987\pi$
$83$	−15.5950	−1.71177	−0.855885	+	0.517165i	$0.826987\pi$
$84$	0	0
$85$	19.4981	2.11487
$86$	0	0
$87$	0.798648	0.0856241
$88$	0	0
$89$	2.46665	0.261465	0.130732	−	0.991418i	$-0.458267\pi$
$89$	2.46665	0.261465	0.130732	+	0.991418i	$0.458267\pi$
$90$	0	0
$91$	14.8775	1.55959
$92$	0	0
$93$	0.705487	0.0731556
$94$	0	0
$95$	−6.72578	−0.690050
$96$	0	0
$97$	3.53018	0.358435	0.179218	−	0.983809i	$-0.442643\pi$
$97$	3.53018	0.358435	0.179218	+	0.983809i	$0.442643\pi$
$98$	0	0
$99$	−7.32784	−0.736475
$100$	0	0
$101$	−12.2339	−1.21732	−0.608661	−	0.793430i	$-0.708293\pi$
$101$	−12.2339	−1.21732	−0.608661	+	0.793430i	$0.708293\pi$
$102$	0	0
$103$	−17.9722	−1.77085	−0.885425	−	0.464783i	$-0.846132\pi$
$103$	−17.9722	−1.77085	−0.885425	+	0.464783i	$0.846132\pi$
$104$	0	0
$105$	2.20412	0.215100
$106$	0	0
$107$	1.94819	0.188338	0.0941691	−	0.995556i	$-0.469981\pi$
$107$	1.94819	0.188338	0.0941691	+	0.995556i	$0.469981\pi$
$108$	0	0
$109$	1.61401	0.154594	0.0772969	−	0.997008i	$-0.475371\pi$
$109$	1.61401	0.154594	0.0772969	+	0.997008i	$0.475371\pi$
$110$	0	0
$111$	−0.110484	−0.0104867
$112$	0	0
$113$	7.16562	0.674085	0.337043	−	0.941489i	$-0.390573\pi$
$113$	7.16562	0.674085	0.337043	+	0.941489i	$0.390573\pi$
$114$	0	0
$115$	6.11856	0.570559
$116$	0	0
$117$	−19.0143	−1.75788
$118$	0	0
$119$	11.8911	1.09005
$120$	0	0
$121$	−4.76660	−0.433327
$122$	0	0
$123$	−0.942394	−0.0849728
$124$	0	0
$125$	15.7391	1.40774
$126$	0	0
$127$	−7.54603	−0.669602	−0.334801	−	0.942289i	$-0.608669\pi$
$127$	−7.54603	−0.669602	−0.334801	+	0.942289i	$0.608669\pi$
$128$	0	0
$129$	−0.244177	−0.0214986
$130$	0	0
$131$	−15.2282	−1.33049	−0.665246	−	0.746624i	$-0.731674\pi$
$131$	−15.2282	−1.33049	−0.665246	+	0.746624i	$0.731674\pi$
$132$	0	0
$133$	−4.10177	−0.355669
$134$	0	0
$135$	−5.69633	−0.490262
$136$	0	0
$137$	19.6590	1.67958	0.839790	−	0.542912i	$-0.182678\pi$
$137$	19.6590	1.67958	0.839790	+	0.542912i	$0.182678\pi$
$138$	0	0
$139$	0.206999	0.0175574	0.00877872	−	0.999961i	$-0.497206\pi$
$139$	0.206999	0.0175574	0.00877872	+	0.999961i	$0.497206\pi$
$140$	0	0
$141$	−0.347221	−0.0292413
$142$	0	0
$143$	16.1745	1.35258
$144$	0	0
$145$	11.7991	0.979866
$146$	0	0
$147$	−0.439970	−0.0362882
$148$	0	0
$149$	−5.86694	−0.480639	−0.240319	−	0.970694i	$-0.577252\pi$
$149$	−5.86694	−0.480639	−0.240319	+	0.970694i	$0.577252\pi$
$150$	0	0
$151$	15.8060	1.28628	0.643138	−	0.765751i	$-0.277632\pi$
$151$	15.8060	1.28628	0.643138	+	0.765751i	$0.277632\pi$
$152$	0	0
$153$	−15.1975	−1.22864
$154$	0	0
$155$	10.4228	0.837179
$156$	0	0
$157$	16.6152	1.32604	0.663018	−	0.748603i	$-0.269275\pi$
$157$	16.6152	1.32604	0.663018	+	0.748603i	$0.269275\pi$
$158$	0	0
$159$	2.30235	0.182588
$160$	0	0
$161$	3.73146	0.294080
$162$	0	0
$163$	4.08622	0.320058	0.160029	−	0.987112i	$-0.448841\pi$
$163$	4.08622	0.320058	0.160029	+	0.987112i	$0.448841\pi$
$164$	0	0
$165$	2.39626	0.186549
$166$	0	0
$167$	−21.4703	−1.66142	−0.830709	−	0.556707i	$-0.812065\pi$
$167$	−21.4703	−1.66142	−0.830709	+	0.556707i	$0.812065\pi$
$168$	0	0
$169$	28.9697	2.22844
$170$	0	0
$171$	5.24231	0.400889
$172$	0	0
$173$	−15.1278	−1.15015	−0.575073	−	0.818102i	$-0.695026\pi$
$173$	−15.1278	−1.15015	−0.575073	+	0.818102i	$0.695026\pi$
$174$	0	0
$175$	21.0810	1.59357
$176$	0	0
$177$	1.93649	0.145555
$178$	0	0
$179$	−8.10224	−0.605590	−0.302795	−	0.953056i	$-0.597920\pi$
$179$	−8.10224	−0.605590	−0.302795	+	0.953056i	$0.597920\pi$
$180$	0	0
$181$	−20.7758	−1.54425	−0.772127	−	0.635468i	$-0.780807\pi$
$181$	−20.7758	−1.54425	−0.772127	+	0.635468i	$0.780807\pi$
$182$	0	0
$183$	−1.98759	−0.146927
$184$	0	0
$185$	−1.63228	−0.120007
$186$	0	0
$187$	12.9277	0.945367
$188$	0	0
$189$	−3.47396	−0.252693
$190$	0	0
$191$	8.04226	0.581917	0.290959	−	0.956736i	$-0.406026\pi$
$191$	8.04226	0.581917	0.290959	+	0.956736i	$0.406026\pi$
$192$	0	0
$193$	14.5805	1.04953	0.524764	−	0.851248i	$-0.324154\pi$
$193$	14.5805	1.04953	0.524764	+	0.851248i	$0.324154\pi$
$194$	0	0
$195$	6.21785	0.445269
$196$	0	0
$197$	−0.351844	−0.0250678	−0.0125339	−	0.999921i	$-0.503990\pi$
$197$	−0.351844	−0.0250678	−0.0125339	+	0.999921i	$0.503990\pi$
$198$	0	0
$199$	4.10629	0.291087	0.145544	−	0.989352i	$-0.453507\pi$
$199$	4.10629	0.291087	0.145544	+	0.989352i	$0.453507\pi$
$200$	0	0
$201$	−1.23151	−0.0868641
$202$	0	0
$203$	7.19581	0.505047
$204$	0	0
$205$	−13.9228	−0.972413
$206$	0	0
$207$	−4.76902	−0.331470
$208$	0	0
$209$	−4.45935	−0.308460
$210$	0	0
$211$	−19.3899	−1.33486	−0.667428	−	0.744675i	$-0.732605\pi$
$211$	−19.3899	−1.33486	−0.667428	+	0.744675i	$0.732605\pi$
$212$	0	0
$213$	−0.642594	−0.0440298
$214$	0	0
$215$	−3.60745	−0.246026
$216$	0	0
$217$	6.35643	0.431503
$218$	0	0
$219$	−1.20505	−0.0814296
$220$	0	0
$221$	33.5449	2.25647
$222$	0	0
$223$	−2.39184	−0.160170	−0.0800848	−	0.996788i	$-0.525519\pi$
$223$	−2.39184	−0.160170	−0.0800848	+	0.996788i	$0.525519\pi$
$224$	0	0
$225$	−26.9427	−1.79618
$226$	0	0
$227$	4.87067	0.323278	0.161639	−	0.986850i	$-0.448322\pi$
$227$	4.87067	0.323278	0.161639	+	0.986850i	$0.448322\pi$
$228$	0	0
$229$	−5.11633	−0.338097	−0.169048	−	0.985608i	$-0.554069\pi$
$229$	−5.11633	−0.338097	−0.169048	+	0.985608i	$0.554069\pi$
$230$	0	0
$231$	1.46138	0.0961519
$232$	0	0
$233$	−1.04425	−0.0684111	−0.0342056	−	0.999415i	$-0.510890\pi$
$233$	−1.04425	−0.0684111	−0.0342056	+	0.999415i	$0.510890\pi$
$234$	0	0
$235$	−5.12980	−0.334631
$236$	0	0
$237$	−3.68722	−0.239511
$238$	0	0
$239$	23.2524	1.50407	0.752035	−	0.659123i	$-0.229072\pi$
$239$	23.2524	1.50407	0.752035	+	0.659123i	$0.229072\pi$
$240$	0	0
$241$	13.7651	0.886686	0.443343	−	0.896352i	$-0.353792\pi$
$241$	13.7651	0.886686	0.443343	+	0.896352i	$0.353792\pi$
$242$	0	0
$243$	6.68418	0.428790
$244$	0	0
$245$	−6.50008	−0.415275
$246$	0	0
$247$	−11.5712	−0.736255
$248$	0	0
$249$	−3.97487	−0.251897
$250$	0	0
$251$	−15.6552	−0.988150	−0.494075	−	0.869419i	$-0.664493\pi$
$251$	−15.6552	−0.988150	−0.494075	+	0.869419i	$0.664493\pi$
$252$	0	0
$253$	4.05675	0.255046
$254$	0	0
$255$	4.96971	0.311215
$256$	0	0
$257$	−2.61866	−0.163348	−0.0816739	−	0.996659i	$-0.526027\pi$
$257$	−2.61866	−0.163348	−0.0816739	+	0.996659i	$0.526027\pi$
$258$	0	0
$259$	−0.995458	−0.0618548
$260$	0	0
$261$	−9.19666	−0.569259
$262$	0	0
$263$	16.9242	1.04359	0.521794	−	0.853072i	$-0.325263\pi$
$263$	16.9242	1.04359	0.521794	+	0.853072i	$0.325263\pi$
$264$	0	0
$265$	34.0146	2.08950
$266$	0	0
$267$	0.628705	0.0384761
$268$	0	0
$269$	−7.53746	−0.459567	−0.229784	−	0.973242i	$-0.573802\pi$
$269$	−7.53746	−0.459567	−0.229784	+	0.973242i	$0.573802\pi$
$270$	0	0
$271$	−20.8582	−1.26705	−0.633523	−	0.773724i	$-0.718392\pi$
$271$	−20.8582	−1.26705	−0.633523	+	0.773724i	$0.718392\pi$
$272$	0	0
$273$	3.79201	0.229503
$274$	0	0
$275$	22.9188	1.38205
$276$	0	0
$277$	−22.5550	−1.35520	−0.677600	−	0.735430i	$-0.736980\pi$
$277$	−22.5550	−1.35520	−0.677600	+	0.735430i	$0.736980\pi$
$278$	0	0
$279$	−8.12388	−0.486364
$280$	0	0
$281$	−4.34033	−0.258923	−0.129461	−	0.991584i	$-0.541325\pi$
$281$	−4.34033	−0.258923	−0.129461	+	0.991584i	$0.541325\pi$
$282$	0	0
$283$	−18.6361	−1.10780	−0.553902	−	0.832582i	$-0.686862\pi$
$283$	−18.6361	−1.10780	−0.553902	+	0.832582i	$0.686862\pi$
$284$	0	0
$285$	−1.71428	−0.101545
$286$	0	0
$287$	−8.49096	−0.501206
$288$	0	0
$289$	9.81127	0.577133
$290$	0	0
$291$	0.899777	0.0527459
$292$	0	0
$293$	−11.4643	−0.669752	−0.334876	−	0.942262i	$-0.608694\pi$
$293$	−11.4643	−0.669752	−0.334876	+	0.942262i	$0.608694\pi$
$294$	0	0
$295$	28.6095	1.66571
$296$	0	0
$297$	−3.77680	−0.219152
$298$	0	0
$299$	10.5265	0.608763
$300$	0	0
$301$	−2.20003	−0.126808
$302$	0	0
$303$	−3.11821	−0.179136
$304$	0	0
$305$	−29.3645	−1.68141
$306$	0	0
$307$	8.47828	0.483881	0.241940	−	0.970291i	$-0.422216\pi$
$307$	8.47828	0.483881	0.241940	+	0.970291i	$0.422216\pi$
$308$	0	0
$309$	−4.58077	−0.260591
$310$	0	0
$311$	−15.2474	−0.864602	−0.432301	−	0.901729i	$-0.642298\pi$
$311$	−15.2474	−0.864602	−0.432301	+	0.901729i	$0.642298\pi$
$312$	0	0
$313$	20.6356	1.16639	0.583196	−	0.812331i	$-0.301802\pi$
$313$	20.6356	1.16639	0.583196	+	0.812331i	$0.301802\pi$
$314$	0	0
$315$	−25.3811	−1.43006
$316$	0	0
$317$	−9.37620	−0.526620	−0.263310	−	0.964711i	$-0.584814\pi$
$317$	−9.37620	−0.526620	−0.263310	+	0.964711i	$0.584814\pi$
$318$	0	0
$319$	7.82311	0.438010
$320$	0	0
$321$	0.496557	0.0277151
$322$	0	0
$323$	−9.24842	−0.514596
$324$	0	0
$325$	59.4698	3.29879
$326$	0	0
$327$	0.411381	0.0227494
$328$	0	0
$329$	−3.12846	−0.172477
$330$	0	0
$331$	17.8337	0.980226	0.490113	−	0.871659i	$-0.336955\pi$
$331$	17.8337	0.980226	0.490113	+	0.871659i	$0.336955\pi$
$332$	0	0
$333$	1.27225	0.0697191
$334$	0	0
$335$	−18.1942	−0.994056
$336$	0	0
$337$	−0.315403	−0.0171811	−0.00859054	−	0.999963i	$-0.502734\pi$
$337$	−0.315403	−0.0171811	−0.00859054	+	0.999963i	$0.502734\pi$
$338$	0	0
$339$	1.82639	0.0991957
$340$	0	0
$341$	6.91056	0.374228
$342$	0	0
$343$	−20.0395	−1.08203
$344$	0	0
$345$	1.55951	0.0839612
$346$	0	0
$347$	6.17570	0.331529	0.165765	−	0.986165i	$-0.446991\pi$
$347$	6.17570	0.331529	0.165765	+	0.986165i	$0.446991\pi$
$348$	0	0
$349$	10.5980	0.567300	0.283650	−	0.958928i	$-0.408455\pi$
$349$	10.5980	0.567300	0.283650	+	0.958928i	$0.408455\pi$
$350$	0	0
$351$	−9.80008	−0.523090
$352$	0	0
$353$	−22.2080	−1.18201	−0.591007	−	0.806666i	$-0.701269\pi$
$353$	−22.2080	−1.18201	−0.591007	+	0.806666i	$0.701269\pi$
$354$	0	0
$355$	−9.49362	−0.503869
$356$	0	0
$357$	3.03082	0.160408
$358$	0	0
$359$	21.4903	1.13422	0.567108	−	0.823643i	$-0.308062\pi$
$359$	21.4903	1.13422	0.567108	+	0.823643i	$0.308062\pi$
$360$	0	0
$361$	−15.8098	−0.832095
$362$	0	0
$363$	−1.21492	−0.0637667
$364$	0	0
$365$	−17.8032	−0.931865
$366$	0	0
$367$	3.59560	0.187689	0.0938445	−	0.995587i	$-0.470084\pi$
$367$	3.59560	0.187689	0.0938445	+	0.995587i	$0.470084\pi$
$368$	0	0
$369$	10.8519	0.564929
$370$	0	0
$371$	20.7441	1.07698
$372$	0	0
$373$	8.38095	0.433949	0.216975	−	0.976177i	$-0.430381\pi$
$373$	8.38095	0.433949	0.216975	+	0.976177i	$0.430381\pi$
$374$	0	0
$375$	4.01160	0.207158
$376$	0	0
$377$	20.2995	1.04548
$378$	0	0
$379$	10.3157	0.529884	0.264942	−	0.964264i	$-0.414647\pi$
$379$	10.3157	0.529884	0.264942	+	0.964264i	$0.414647\pi$
$380$	0	0
$381$	−1.92334	−0.0985359
$382$	0	0
$383$	−9.93463	−0.507636	−0.253818	−	0.967252i	$-0.581686\pi$
$383$	−9.93463	−0.507636	−0.253818	+	0.967252i	$0.581686\pi$
$384$	0	0
$385$	21.5903	1.10034
$386$	0	0
$387$	2.81177	0.142930
$388$	0	0
$389$	13.4919	0.684065	0.342033	−	0.939688i	$-0.388885\pi$
$389$	13.4919	0.684065	0.342033	+	0.939688i	$0.388885\pi$
$390$	0	0
$391$	8.41346	0.425487
$392$	0	0
$393$	−3.88139	−0.195790
$394$	0	0
$395$	−54.4746	−2.74091
$396$	0	0
$397$	−9.92914	−0.498329	−0.249165	−	0.968461i	$-0.580156\pi$
$397$	−9.92914	−0.498329	−0.249165	+	0.968461i	$0.580156\pi$
$398$	0	0
$399$	−1.04547	−0.0523388
$400$	0	0
$401$	20.4777	1.02261	0.511304	−	0.859400i	$-0.329163\pi$
$401$	20.4777	1.02261	0.511304	+	0.859400i	$0.329163\pi$
$402$	0	0
$403$	17.9316	0.893235
$404$	0	0
$405$	31.7046	1.57541
$406$	0	0
$407$	−1.08224	−0.0536446
$408$	0	0
$409$	5.70617	0.282152	0.141076	−	0.989999i	$-0.454944\pi$
$409$	5.70617	0.282152	0.141076	+	0.989999i	$0.454944\pi$
$410$	0	0
$411$	5.01071	0.247160
$412$	0	0
$413$	17.4477	0.858547
$414$	0	0
$415$	−58.7244	−2.88267
$416$	0	0
$417$	0.0527603	0.00258368
$418$	0	0
$419$	7.94903	0.388335	0.194168	−	0.980968i	$-0.437799\pi$
$419$	7.94903	0.388335	0.194168	+	0.980968i	$0.437799\pi$
$420$	0	0
$421$	6.37442	0.310670	0.155335	−	0.987862i	$-0.450354\pi$
$421$	6.37442	0.310670	0.155335	+	0.987862i	$0.450354\pi$
$422$	0	0
$423$	3.99835	0.194406
$424$	0	0
$425$	47.5321	2.30565
$426$	0	0
$427$	−17.9082	−0.866638
$428$	0	0
$429$	4.12258	0.199040
$430$	0	0
$431$	16.9371	0.815829	0.407915	−	0.913020i	$-0.366256\pi$
$431$	16.9371	0.815829	0.407915	+	0.913020i	$0.366256\pi$
$432$	0	0
$433$	−26.4676	−1.27195	−0.635977	−	0.771708i	$-0.719403\pi$
$433$	−26.4676	−1.27195	−0.635977	+	0.771708i	$0.719403\pi$
$434$	0	0
$435$	3.00739	0.144193
$436$	0	0
$437$	−2.90218	−0.138830
$438$	0	0
$439$	−26.2998	−1.25522	−0.627611	−	0.778527i	$-0.715967\pi$
$439$	−26.2998	−1.25522	−0.627611	+	0.778527i	$0.715967\pi$
$440$	0	0
$441$	5.06639	0.241256
$442$	0	0
$443$	−10.3165	−0.490151	−0.245075	−	0.969504i	$-0.578813\pi$
$443$	−10.3165	−0.490151	−0.245075	+	0.969504i	$0.578813\pi$
$444$	0	0
$445$	9.28842	0.440313
$446$	0	0
$447$	−1.49538	−0.0707288
$448$	0	0
$449$	32.7813	1.54704	0.773522	−	0.633769i	$-0.218493\pi$
$449$	32.7813	1.54704	0.773522	+	0.633769i	$0.218493\pi$
$450$	0	0
$451$	−9.23117	−0.434679
$452$	0	0
$453$	4.02866	0.189283
$454$	0	0
$455$	56.0227	2.62639
$456$	0	0
$457$	8.00367	0.374396	0.187198	−	0.982322i	$-0.440059\pi$
$457$	8.00367	0.374396	0.187198	+	0.982322i	$0.440059\pi$
$458$	0	0
$459$	−7.83286	−0.365607
$460$	0	0
$461$	−22.1178	−1.03013	−0.515065	−	0.857151i	$-0.672232\pi$
$461$	−22.1178	−1.03013	−0.515065	+	0.857151i	$0.672232\pi$
$462$	0	0
$463$	13.1882	0.612907	0.306454	−	0.951886i	$-0.400858\pi$
$463$	13.1882	0.612907	0.306454	+	0.951886i	$0.400858\pi$
$464$	0	0
$465$	2.65658	0.123196
$466$	0	0
$467$	−24.2615	−1.12269	−0.561344	−	0.827583i	$-0.689715\pi$
$467$	−24.2615	−1.12269	−0.561344	+	0.827583i	$0.689715\pi$
$468$	0	0
$469$	−11.0959	−0.512361
$470$	0	0
$471$	4.23491	0.195134
$472$	0	0
$473$	−2.39182	−0.109976
$474$	0	0
$475$	−16.3960	−0.752299
$476$	0	0
$477$	−26.5122	−1.21391
$478$	0	0
$479$	21.4445	0.979824	0.489912	−	0.871772i	$-0.337029\pi$
$479$	21.4445	0.979824	0.489912	+	0.871772i	$0.337029\pi$
$480$	0	0
$481$	−2.80820	−0.128043
$482$	0	0
$483$	0.951081	0.0432757
$484$	0	0
$485$	13.2932	0.603614
$486$	0	0
$487$	−36.0633	−1.63418	−0.817092	−	0.576508i	$-0.804415\pi$
$487$	−36.0633	−1.63418	−0.817092	+	0.576508i	$0.804415\pi$
$488$	0	0
$489$	1.04150	0.0470984
$490$	0	0
$491$	25.1007	1.13278	0.566389	−	0.824138i	$-0.308340\pi$
$491$	25.1007	1.13278	0.566389	+	0.824138i	$0.308340\pi$
$492$	0	0
$493$	16.2247	0.730722
$494$	0	0
$495$	−27.5937	−1.24024
$496$	0	0
$497$	−5.78977	−0.259707
$498$	0	0
$499$	27.7534	1.24241	0.621207	−	0.783647i	$-0.286643\pi$
$499$	27.7534	1.24241	0.621207	+	0.783647i	$0.286643\pi$
$500$	0	0
$501$	−5.47237	−0.244488
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−46.0680	−2.05000
$506$	0	0
$507$	7.38384	0.327928
$508$	0	0
$509$	25.7904	1.14314	0.571569	−	0.820554i	$-0.306335\pi$
$509$	25.7904	1.14314	0.571569	+	0.820554i	$0.306335\pi$
$510$	0	0
$511$	−10.8575	−0.480306
$512$	0	0
$513$	2.70191	0.119292
$514$	0	0
$515$	−67.6759	−2.98215
$516$	0	0
$517$	−3.40118	−0.149584
$518$	0	0
$519$	−3.85580	−0.169251
$520$	0	0
$521$	−16.9650	−0.743251	−0.371625	−	0.928383i	$-0.621199\pi$
$521$	−16.9650	−0.743251	−0.371625	+	0.928383i	$0.621199\pi$
$522$	0	0
$523$	5.31401	0.232366	0.116183	−	0.993228i	$-0.462934\pi$
$523$	5.31401	0.232366	0.116183	+	0.993228i	$0.462934\pi$
$524$	0	0
$525$	5.37316	0.234504
$526$	0	0
$527$	14.3321	0.624315
$528$	0	0
$529$	−20.3598	−0.885210
$530$	0	0
$531$	−22.2992	−0.967704
$532$	0	0
$533$	−23.9531	−1.03752
$534$	0	0
$535$	7.33608	0.317166
$536$	0	0
$537$	−2.06511	−0.0891161
$538$	0	0
$539$	−4.30970	−0.185632
$540$	0	0
$541$	−7.28080	−0.313026	−0.156513	−	0.987676i	$-0.550025\pi$
$541$	−7.28080	−0.313026	−0.156513	+	0.987676i	$0.550025\pi$
$542$	0	0
$543$	−5.29538	−0.227246
$544$	0	0
$545$	6.07770	0.260340
$546$	0	0
$547$	−19.5475	−0.835791	−0.417896	−	0.908495i	$-0.637232\pi$
$547$	−19.5475	−0.835791	−0.417896	+	0.908495i	$0.637232\pi$
$548$	0	0
$549$	22.8877	0.976823
$550$	0	0
$551$	−5.59662	−0.238424
$552$	0	0
$553$	−33.2218	−1.41273
$554$	0	0
$555$	−0.416037	−0.0176598
$556$	0	0
$557$	32.3433	1.37043	0.685215	−	0.728341i	$-0.259708\pi$
$557$	32.3433	1.37043	0.685215	+	0.728341i	$0.259708\pi$
$558$	0	0
$559$	−6.20632	−0.262500
$560$	0	0
$561$	3.29503	0.139116
$562$	0	0
$563$	37.8155	1.59373	0.796867	−	0.604155i	$-0.206489\pi$
$563$	37.8155	1.59373	0.796867	+	0.604155i	$0.206489\pi$
$564$	0	0
$565$	26.9828	1.13518
$566$	0	0
$567$	19.3353	0.812007
$568$	0	0
$569$	−9.04535	−0.379201	−0.189600	−	0.981861i	$-0.560719\pi$
$569$	−9.04535	−0.379201	−0.189600	+	0.981861i	$0.560719\pi$
$570$	0	0
$571$	46.5230	1.94693	0.973464	−	0.228842i	$-0.0734940\pi$
$571$	46.5230	1.94693	0.973464	+	0.228842i	$0.0734940\pi$
$572$	0	0
$573$	2.04982	0.0856326
$574$	0	0
$575$	14.9157	0.622029
$576$	0	0
$577$	−26.5655	−1.10593	−0.552967	−	0.833203i	$-0.686505\pi$
$577$	−26.5655	−1.10593	−0.552967	+	0.833203i	$0.686505\pi$
$578$	0	0
$579$	3.71630	0.154444
$580$	0	0
$581$	−35.8136	−1.48580
$582$	0	0
$583$	22.5525	0.934029
$584$	0	0
$585$	−71.6003	−2.96031
$586$	0	0
$587$	−46.8735	−1.93467	−0.967337	−	0.253492i	$-0.918421\pi$
$587$	−46.8735	−1.93467	−0.967337	+	0.253492i	$0.918421\pi$
$588$	0	0
$589$	−4.94378	−0.203705
$590$	0	0
$591$	−0.0896786	−0.00368888
$592$	0	0
$593$	−43.6191	−1.79122	−0.895612	−	0.444836i	$-0.853262\pi$
$593$	−43.6191	−1.79122	−0.895612	+	0.444836i	$0.853262\pi$
$594$	0	0
$595$	44.7770	1.83568
$596$	0	0
$597$	1.04662	0.0428353
$598$	0	0
$599$	−27.7730	−1.13477	−0.567386	−	0.823452i	$-0.692045\pi$
$599$	−27.7730	−1.13477	−0.567386	+	0.823452i	$0.692045\pi$
$600$	0	0
$601$	−10.8823	−0.443900	−0.221950	−	0.975058i	$-0.571242\pi$
$601$	−10.8823	−0.443900	−0.221950	+	0.975058i	$0.571242\pi$
$602$	0	0
$603$	14.1812	0.577503
$604$	0	0
$605$	−17.9491	−0.729734
$606$	0	0
$607$	19.6998	0.799588	0.399794	−	0.916605i	$-0.369082\pi$
$607$	19.6998	0.799588	0.399794	+	0.916605i	$0.369082\pi$
$608$	0	0
$609$	1.83408	0.0743207
$610$	0	0
$611$	−8.82541	−0.357038
$612$	0	0
$613$	−11.7591	−0.474945	−0.237473	−	0.971394i	$-0.576319\pi$
$613$	−11.7591	−0.474945	−0.237473	+	0.971394i	$0.576319\pi$
$614$	0	0
$615$	−3.54867	−0.143096
$616$	0	0
$617$	19.7062	0.793341	0.396670	−	0.917961i	$-0.370166\pi$
$617$	19.7062	0.793341	0.396670	+	0.917961i	$0.370166\pi$
$618$	0	0
$619$	29.6273	1.19082	0.595412	−	0.803421i	$-0.296989\pi$
$619$	29.6273	1.19082	0.595412	+	0.803421i	$0.296989\pi$
$620$	0	0
$621$	−2.45798	−0.0986352
$622$	0	0
$623$	5.66463	0.226948
$624$	0	0
$625$	13.3684	0.534736
$626$	0	0
$627$	−1.13661	−0.0453917
$628$	0	0
$629$	−2.24450	−0.0894939
$630$	0	0
$631$	41.5798	1.65526	0.827632	−	0.561271i	$-0.189688\pi$
$631$	41.5798	1.65526	0.827632	+	0.561271i	$0.189688\pi$
$632$	0	0
$633$	−4.94213	−0.196432
$634$	0	0
$635$	−28.4153	−1.12763
$636$	0	0
$637$	−11.1829	−0.443081
$638$	0	0
$639$	7.39966	0.292726
$640$	0	0
$641$	1.84120	0.0727229	0.0363614	−	0.999339i	$-0.488423\pi$
$641$	1.84120	0.0727229	0.0363614	+	0.999339i	$0.488423\pi$
$642$	0	0
$643$	−13.0244	−0.513632	−0.256816	−	0.966460i	$-0.582673\pi$
$643$	−13.0244	−0.513632	−0.256816	+	0.966460i	$0.582673\pi$
$644$	0	0
$645$	−0.919472	−0.0362042
$646$	0	0
$647$	39.1412	1.53880	0.769400	−	0.638768i	$-0.220556\pi$
$647$	39.1412	1.53880	0.769400	+	0.638768i	$0.220556\pi$
$648$	0	0
$649$	18.9688	0.744589
$650$	0	0
$651$	1.62014	0.0634982
$652$	0	0
$653$	−0.681012	−0.0266500	−0.0133250	−	0.999911i	$-0.504242\pi$
$653$	−0.681012	−0.0266500	−0.0133250	+	0.999911i	$0.504242\pi$
$654$	0	0
$655$	−57.3432	−2.24058
$656$	0	0
$657$	13.8765	0.541373
$658$	0	0
$659$	51.0887	1.99013	0.995067	−	0.0992042i	$-0.0316297\pi$
$659$	51.0887	1.99013	0.995067	+	0.0992042i	$0.0316297\pi$
$660$	0	0
$661$	35.9346	1.39769	0.698846	−	0.715272i	$-0.253697\pi$
$661$	35.9346	1.39769	0.698846	+	0.715272i	$0.253697\pi$
$662$	0	0
$663$	8.54998	0.332054
$664$	0	0
$665$	−15.4456	−0.598955
$666$	0	0
$667$	5.09135	0.197138
$668$	0	0
$669$	−0.609637	−0.0235699
$670$	0	0
$671$	−19.4693	−0.751606
$672$	0	0
$673$	−24.3616	−0.939071	−0.469536	−	0.882914i	$-0.655579\pi$
$673$	−24.3616	−0.939071	−0.469536	+	0.882914i	$0.655579\pi$
$674$	0	0
$675$	−13.8864	−0.534489
$676$	0	0
$677$	6.63011	0.254816	0.127408	−	0.991850i	$-0.459334\pi$
$677$	6.63011	0.254816	0.127408	+	0.991850i	$0.459334\pi$
$678$	0	0
$679$	8.10698	0.311117
$680$	0	0
$681$	1.24145	0.0475723
$682$	0	0
$683$	40.7217	1.55817	0.779086	−	0.626917i	$-0.215683\pi$
$683$	40.7217	1.55817	0.779086	+	0.626917i	$0.215683\pi$
$684$	0	0
$685$	74.0277	2.82845
$686$	0	0
$687$	−1.30406	−0.0497529
$688$	0	0
$689$	58.5194	2.22941
$690$	0	0
$691$	40.2262	1.53028	0.765139	−	0.643865i	$-0.222670\pi$
$691$	40.2262	1.53028	0.765139	+	0.643865i	$0.222670\pi$
$692$	0	0
$693$	−16.8282	−0.639252
$694$	0	0
$695$	0.779475	0.0295672
$696$	0	0
$697$	−19.1449	−0.725164
$698$	0	0
$699$	−0.266160	−0.0100671
$700$	0	0
$701$	−10.9940	−0.415238	−0.207619	−	0.978210i	$-0.566571\pi$
$701$	−10.9940	−0.415238	−0.207619	+	0.978210i	$0.566571\pi$
$702$	0	0
$703$	0.774229	0.0292006
$704$	0	0
$705$	−1.30749	−0.0492430
$706$	0	0
$707$	−28.0950	−1.05662
$708$	0	0
$709$	1.57925	0.0593100	0.0296550	−	0.999560i	$-0.490559\pi$
$709$	1.57925	0.0593100	0.0296550	+	0.999560i	$0.490559\pi$
$710$	0	0
$711$	42.4594	1.59235
$712$	0	0
$713$	4.49745	0.168431
$714$	0	0
$715$	60.9065	2.27778
$716$	0	0
$717$	5.92660	0.221333
$718$	0	0
$719$	30.6811	1.14421	0.572107	−	0.820179i	$-0.306126\pi$
$719$	30.6811	1.14421	0.572107	+	0.820179i	$0.306126\pi$
$720$	0	0
$721$	−41.2727	−1.53708
$722$	0	0
$723$	3.50847	0.130481
$724$	0	0
$725$	28.7638	1.06826
$726$	0	0
$727$	10.2511	0.380192	0.190096	−	0.981766i	$-0.439120\pi$
$727$	10.2511	0.380192	0.190096	+	0.981766i	$0.439120\pi$
$728$	0	0
$729$	−23.5549	−0.872405
$730$	0	0
$731$	−4.96050	−0.183471
$732$	0	0
$733$	8.81655	0.325647	0.162823	−	0.986655i	$-0.447940\pi$
$733$	8.81655	0.325647	0.162823	+	0.986655i	$0.447940\pi$
$734$	0	0
$735$	−1.65675	−0.0611102
$736$	0	0
$737$	−12.0632	−0.444353
$738$	0	0
$739$	37.6509	1.38501	0.692505	−	0.721413i	$-0.256507\pi$
$739$	37.6509	1.38501	0.692505	+	0.721413i	$0.256507\pi$
$740$	0	0
$741$	−2.94928	−0.108344
$742$	0	0
$743$	−12.6691	−0.464785	−0.232392	−	0.972622i	$-0.574655\pi$
$743$	−12.6691	−0.464785	−0.232392	+	0.972622i	$0.574655\pi$
$744$	0	0
$745$	−22.0925	−0.809407
$746$	0	0
$747$	45.7718	1.67470
$748$	0	0
$749$	4.47397	0.163475
$750$	0	0
$751$	−27.5008	−1.00352	−0.501760	−	0.865007i	$-0.667314\pi$
$751$	−27.5008	−1.00352	−0.501760	+	0.865007i	$0.667314\pi$
$752$	0	0
$753$	−3.99023	−0.145412
$754$	0	0
$755$	59.5190	2.16612
$756$	0	0
$757$	33.0830	1.20242	0.601210	−	0.799091i	$-0.294685\pi$
$757$	33.0830	1.20242	0.601210	+	0.799091i	$0.294685\pi$
$758$	0	0
$759$	1.03399	0.0375315
$760$	0	0
$761$	−27.6329	−1.00169	−0.500845	−	0.865537i	$-0.666977\pi$
$761$	−27.6329	−1.00169	−0.500845	+	0.865537i	$0.666977\pi$
$762$	0	0
$763$	3.70654	0.134186
$764$	0	0
$765$	−57.2276	−2.06907
$766$	0	0
$767$	49.2203	1.77724
$768$	0	0
$769$	−5.70938	−0.205885	−0.102943	−	0.994687i	$-0.532826\pi$
$769$	−5.70938	−0.205885	−0.102943	+	0.994687i	$0.532826\pi$
$770$	0	0
$771$	−0.667449	−0.0240376
$772$	0	0
$773$	26.3190	0.946630	0.473315	−	0.880893i	$-0.343057\pi$
$773$	26.3190	0.946630	0.473315	+	0.880893i	$0.343057\pi$
$774$	0	0
$775$	25.4085	0.912700
$776$	0	0
$777$	−0.253724	−0.00910230
$778$	0	0
$779$	6.60394	0.236611
$780$	0	0
$781$	−6.29449	−0.225235
$782$	0	0
$783$	−4.74001	−0.169394
$784$	0	0
$785$	62.5661	2.23308
$786$	0	0
$787$	−25.0879	−0.894285	−0.447143	−	0.894463i	$-0.647558\pi$
$787$	−25.0879	−0.894285	−0.447143	+	0.894463i	$0.647558\pi$
$788$	0	0
$789$	4.31366	0.153570
$790$	0	0
$791$	16.4557	0.585098
$792$	0	0
$793$	−50.5192	−1.79399
$794$	0	0
$795$	8.66970	0.307483
$796$	0	0
$797$	−23.0432	−0.816232	−0.408116	−	0.912930i	$-0.633814\pi$
$797$	−23.0432	−0.816232	−0.408116	+	0.912930i	$0.633814\pi$
$798$	0	0
$799$	−7.05384	−0.249547
$800$	0	0
$801$	−7.23972	−0.255803
$802$	0	0
$803$	−11.8040	−0.416553
$804$	0	0
$805$	14.0512	0.495239
$806$	0	0
$807$	−1.92116	−0.0676281
$808$	0	0
$809$	−38.8340	−1.36533	−0.682666	−	0.730731i	$-0.739179\pi$
$809$	−38.8340	−1.36533	−0.682666	+	0.730731i	$0.739179\pi$
$810$	0	0
$811$	−10.7626	−0.377927	−0.188963	−	0.981984i	$-0.560513\pi$
$811$	−10.7626	−0.377927	−0.188963	+	0.981984i	$0.560513\pi$
$812$	0	0
$813$	−5.31638	−0.186454
$814$	0	0
$815$	15.3871	0.538985
$816$	0	0
$817$	1.71110	0.0598638
$818$	0	0
$819$	−43.6661	−1.52582
$820$	0	0
$821$	45.7704	1.59740	0.798698	−	0.601732i	$-0.205522\pi$
$821$	45.7704	1.59740	0.798698	+	0.601732i	$0.205522\pi$
$822$	0	0
$823$	18.4088	0.641690	0.320845	−	0.947132i	$-0.396033\pi$
$823$	18.4088	0.641690	0.320845	+	0.947132i	$0.396033\pi$
$824$	0	0
$825$	5.84157	0.203377
$826$	0	0
$827$	17.1709	0.597092	0.298546	−	0.954395i	$-0.403498\pi$
$827$	17.1709	0.597092	0.298546	+	0.954395i	$0.403498\pi$
$828$	0	0
$829$	7.60351	0.264081	0.132040	−	0.991244i	$-0.457847\pi$
$829$	7.60351	0.264081	0.132040	+	0.991244i	$0.457847\pi$
$830$	0	0
$831$	−5.74886	−0.199426
$832$	0	0
$833$	−8.93807	−0.309686
$834$	0	0
$835$	−80.8483	−2.79787
$836$	0	0
$837$	−4.18709	−0.144727
$838$	0	0
$839$	−26.3593	−0.910022	−0.455011	−	0.890486i	$-0.650365\pi$
$839$	−26.3593	−0.910022	−0.455011	+	0.890486i	$0.650365\pi$
$840$	0	0
$841$	−19.1818	−0.661440
$842$	0	0
$843$	−1.10627	−0.0381020
$844$	0	0
$845$	109.088	3.75274
$846$	0	0
$847$	−10.9464	−0.376123
$848$	0	0
$849$	−4.75001	−0.163020
$850$	0	0
$851$	−0.704330	−0.0241441
$852$	0	0
$853$	31.0149	1.06193	0.530964	−	0.847394i	$-0.321830\pi$
$853$	31.0149	1.06193	0.530964	+	0.847394i	$0.321830\pi$
$854$	0	0
$855$	19.7404	0.675107
$856$	0	0
$857$	−7.68851	−0.262634	−0.131317	−	0.991340i	$-0.541921\pi$
$857$	−7.68851	−0.262634	−0.131317	+	0.991340i	$0.541921\pi$
$858$	0	0
$859$	34.9513	1.19252	0.596262	−	0.802790i	$-0.296652\pi$
$859$	34.9513	1.19252	0.596262	+	0.802790i	$0.296652\pi$
$860$	0	0
$861$	−2.16419	−0.0737554
$862$	0	0
$863$	23.1875	0.789310	0.394655	−	0.918829i	$-0.370864\pi$
$863$	23.1875	0.789310	0.394655	+	0.918829i	$0.370864\pi$
$864$	0	0
$865$	−56.9652	−1.93688
$866$	0	0
$867$	2.50071	0.0849286
$868$	0	0
$869$	−36.1179	−1.22522
$870$	0	0
$871$	−31.3017	−1.06062
$872$	0	0
$873$	−10.3612	−0.350673
$874$	0	0
$875$	36.1445	1.22191
$876$	0	0
$877$	27.6000	0.931987	0.465993	−	0.884788i	$-0.345697\pi$
$877$	27.6000	0.931987	0.465993	+	0.884788i	$0.345697\pi$
$878$	0	0
$879$	−2.92204	−0.0985580
$880$	0	0
$881$	−5.22776	−0.176128	−0.0880638	−	0.996115i	$-0.528068\pi$
$881$	−5.22776	−0.176128	−0.0880638	+	0.996115i	$0.528068\pi$
$882$	0	0
$883$	−21.3174	−0.717389	−0.358695	−	0.933455i	$-0.616778\pi$
$883$	−21.3174	−0.717389	−0.358695	+	0.933455i	$0.616778\pi$
$884$	0	0
$885$	7.29203	0.245119
$886$	0	0
$887$	−7.36692	−0.247357	−0.123678	−	0.992322i	$-0.539469\pi$
$887$	−7.36692	−0.247357	−0.123678	+	0.992322i	$0.539469\pi$
$888$	0	0
$889$	−17.3293	−0.581207
$890$	0	0
$891$	21.0209	0.704226
$892$	0	0
$893$	2.43319	0.0814236
$894$	0	0
$895$	−30.5097	−1.01983
$896$	0	0
$897$	2.68301	0.0895831
$898$	0	0
$899$	8.67296	0.289259
$900$	0	0
$901$	46.7725	1.55822
$902$	0	0
$903$	−0.560748	−0.0186605
$904$	0	0
$905$	−78.2333	−2.60056
$906$	0	0
$907$	47.1488	1.56555	0.782775	−	0.622305i	$-0.213804\pi$
$907$	47.1488	1.56555	0.782775	+	0.622305i	$0.213804\pi$
$908$	0	0
$909$	35.9070	1.19096
$910$	0	0
$911$	16.1113	0.533790	0.266895	−	0.963726i	$-0.414002\pi$
$911$	16.1113	0.533790	0.266895	+	0.963726i	$0.414002\pi$
$912$	0	0
$913$	−38.9356	−1.28858
$914$	0	0
$915$	−7.48447	−0.247429
$916$	0	0
$917$	−34.9712	−1.15485
$918$	0	0
$919$	23.3023	0.768673	0.384336	−	0.923193i	$-0.374430\pi$
$919$	23.3023	0.768673	0.384336	+	0.923193i	$0.374430\pi$
$920$	0	0
$921$	2.16096	0.0712060
$922$	0	0
$923$	−16.3330	−0.537607
$924$	0	0
$925$	−3.97914	−0.130833
$926$	0	0
$927$	52.7489	1.73250
$928$	0	0
$929$	13.0596	0.428470	0.214235	−	0.976782i	$-0.431274\pi$
$929$	13.0596	0.428470	0.214235	+	0.976782i	$0.431274\pi$
$930$	0	0
$931$	3.08314	0.101046
$932$	0	0
$933$	−3.88629	−0.127231
$934$	0	0
$935$	48.6805	1.59202
$936$	0	0
$937$	53.6979	1.75423	0.877117	−	0.480277i	$-0.159464\pi$
$937$	53.6979	1.75423	0.877117	+	0.480277i	$0.159464\pi$
$938$	0	0
$939$	5.25964	0.171642
$940$	0	0
$941$	33.4900	1.09174	0.545871	−	0.837869i	$-0.316199\pi$
$941$	33.4900	1.09174	0.545871	+	0.837869i	$0.316199\pi$
$942$	0	0
$943$	−6.00772	−0.195638
$944$	0	0
$945$	−13.0815	−0.425542
$946$	0	0
$947$	−21.3996	−0.695394	−0.347697	−	0.937607i	$-0.613036\pi$
$947$	−21.3996	−0.695394	−0.347697	+	0.937607i	$0.613036\pi$
$948$	0	0
$949$	−30.6291	−0.994261
$950$	0	0
$951$	−2.38982	−0.0774952
$952$	0	0
$953$	−33.2342	−1.07656	−0.538281	−	0.842765i	$-0.680926\pi$
$953$	−33.2342	−1.07656	−0.538281	+	0.842765i	$0.680926\pi$
$954$	0	0
$955$	30.2839	0.979963
$956$	0	0
$957$	1.99397	0.0644558
$958$	0	0
$959$	45.1465	1.45786
$960$	0	0
$961$	−23.3387	−0.752862
$962$	0	0
$963$	−5.71799	−0.184260
$964$	0	0
$965$	54.9042	1.76743
$966$	0	0
$967$	−0.816852	−0.0262682	−0.0131341	−	0.999914i	$-0.504181\pi$
$967$	−0.816852	−0.0262682	−0.0131341	+	0.999914i	$0.504181\pi$
$968$	0	0
$969$	−2.35725	−0.0757259
$970$	0	0
$971$	26.2870	0.843591	0.421796	−	0.906691i	$-0.361400\pi$
$971$	26.2870	0.843591	0.421796	+	0.906691i	$0.361400\pi$
$972$	0	0
$973$	0.475369	0.0152396
$974$	0	0
$975$	15.1578	0.485437
$976$	0	0
$977$	39.6146	1.26738	0.633691	−	0.773586i	$-0.281539\pi$
$977$	39.6146	1.26738	0.633691	+	0.773586i	$0.281539\pi$
$978$	0	0
$979$	6.15844	0.196825
$980$	0	0
$981$	−4.73717	−0.151246
$982$	0	0
$983$	−27.3907	−0.873626	−0.436813	−	0.899552i	$-0.643893\pi$
$983$	−27.3907	−0.873626	−0.436813	+	0.899552i	$0.643893\pi$
$984$	0	0
$985$	−1.32490	−0.0422149
$986$	0	0
$987$	−0.797386	−0.0253811
$988$	0	0
$989$	−1.55662	−0.0494976
$990$	0	0
$991$	42.6080	1.35349	0.676743	−	0.736219i	$-0.263391\pi$
$991$	42.6080	1.35349	0.676743	+	0.736219i	$0.263391\pi$
$992$	0	0
$993$	4.54547	0.144246
$994$	0	0
$995$	15.4626	0.490199
$996$	0	0
$997$	−24.4457	−0.774204	−0.387102	−	0.922037i	$-0.626524\pi$
$997$	−24.4457	−0.774204	−0.387102	+	0.922037i	$0.626524\pi$
$998$	0	0
$999$	0.655726	0.0207462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.y.1.19		33
4.3	odd	2		4024.2.a.f.1.15	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.15	✓	33	4.3	odd	2
8048.2.a.y.1.19		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)

\(f(q)\)	\(=\)	\(q+0.254882 q^{3} +3.76559 q^{5} +2.29648 q^{7} -2.93504 q^{9} +O(q^{10})\)	\(q+0.254882 q^{3} +3.76559 q^{5} +2.29648 q^{7} -2.93504 q^{9} +2.49668 q^{11} +6.47840 q^{13} +0.959781 q^{15} +5.17796 q^{17} -1.78611 q^{19} +0.585331 q^{21} +1.62486 q^{23} +9.17970 q^{25} -1.51273 q^{27} +3.13341 q^{29} +2.76790 q^{31} +0.636357 q^{33} +8.64762 q^{35} -0.433471 q^{37} +1.65123 q^{39} -3.69738 q^{41} -0.958002 q^{43} -11.0522 q^{45} -1.36228 q^{47} -1.72618 q^{49} +1.31977 q^{51} +9.03300 q^{53} +9.40148 q^{55} -0.455247 q^{57} +7.59760 q^{59} -7.79810 q^{61} -6.74025 q^{63} +24.3950 q^{65} -4.83170 q^{67} +0.414147 q^{69} -2.52115 q^{71} -4.72787 q^{73} +2.33974 q^{75} +5.73357 q^{77} -14.4664 q^{79} +8.41954 q^{81} -15.5950 q^{83} +19.4981 q^{85} +0.798648 q^{87} +2.46665 q^{89} +14.8775 q^{91} +0.705487 q^{93} -6.72578 q^{95} +3.53018 q^{97} -7.32784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 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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