LMFDB - Embedded newform 6008.2.a.e.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34723 q^{3} +0.0749039 q^{5} -4.25247 q^{7} -1.18497 q^{9} +O(q^{10})$	$q-1.34723 q^{3} +0.0749039 q^{5} -4.25247 q^{7} -1.18497 q^{9} +1.12257 q^{11} -0.485811 q^{13} -0.100913 q^{15} +6.54264 q^{17} -3.86478 q^{19} +5.72905 q^{21} -8.49829 q^{23} -4.99439 q^{25} +5.63812 q^{27} -1.99956 q^{29} -6.58629 q^{31} -1.51236 q^{33} -0.318526 q^{35} +5.71347 q^{37} +0.654499 q^{39} +4.59627 q^{41} -6.88895 q^{43} -0.0887591 q^{45} +2.43998 q^{47} +11.0835 q^{49} -8.81444 q^{51} -10.3855 q^{53} +0.0840846 q^{55} +5.20675 q^{57} -5.14682 q^{59} -5.48484 q^{61} +5.03906 q^{63} -0.0363891 q^{65} -8.24807 q^{67} +11.4492 q^{69} +14.5585 q^{71} -14.0072 q^{73} +6.72859 q^{75} -4.77368 q^{77} -1.73234 q^{79} -4.04092 q^{81} -11.8143 q^{83} +0.490069 q^{85} +2.69386 q^{87} -6.39261 q^{89} +2.06590 q^{91} +8.87324 q^{93} -0.289487 q^{95} -12.5205 q^{97} -1.33021 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−1.34723	−0.777823	−0.388912	−	0.921275i	$-0.627149\pi$
$3$	−1.34723	−0.777823	−0.388912	+	0.921275i	$0.627149\pi$
$4$	0	0
$5$	0.0749039	0.0334980	0.0167490	−	0.999860i	$-0.494668\pi$
$5$	0.0749039	0.0334980	0.0167490	+	0.999860i	$0.494668\pi$
$6$	0	0
$7$	−4.25247	−1.60728	−0.803641	−	0.595114i	$-0.797107\pi$
$7$	−4.25247	−1.60728	−0.803641	+	0.595114i	$0.797107\pi$
$8$	0	0
$9$	−1.18497	−0.394991
$10$	0	0
$11$	1.12257	0.338467	0.169233	−	0.985576i	$-0.445871\pi$
$11$	1.12257	0.338467	0.169233	+	0.985576i	$0.445871\pi$
$12$	0	0
$13$	−0.485811	−0.134740	−0.0673698	−	0.997728i	$-0.521461\pi$
$13$	−0.485811	−0.134740	−0.0673698	+	0.997728i	$0.521461\pi$
$14$	0	0
$15$	−0.100913	−0.0260555
$16$	0	0
$17$	6.54264	1.58682	0.793412	−	0.608685i	$-0.208303\pi$
$17$	6.54264	1.58682	0.793412	+	0.608685i	$0.208303\pi$
$18$	0	0
$19$	−3.86478	−0.886641	−0.443321	−	0.896363i	$-0.646200\pi$
$19$	−3.86478	−0.886641	−0.443321	+	0.896363i	$0.646200\pi$
$20$	0	0
$21$	5.72905	1.25018
$22$	0	0
$23$	−8.49829	−1.77202	−0.886008	−	0.463669i	$-0.846533\pi$
$23$	−8.49829	−1.77202	−0.886008	+	0.463669i	$0.846533\pi$
$24$	0	0
$25$	−4.99439	−0.998878
$26$	0	0
$27$	5.63812	1.08506
$28$	0	0
$29$	−1.99956	−0.371309	−0.185654	−	0.982615i	$-0.559440\pi$
$29$	−1.99956	−0.371309	−0.185654	+	0.982615i	$0.559440\pi$
$30$	0	0
$31$	−6.58629	−1.18293	−0.591466	−	0.806330i	$-0.701451\pi$
$31$	−6.58629	−1.18293	−0.591466	+	0.806330i	$0.701451\pi$
$32$	0	0
$33$	−1.51236	−0.263267
$34$	0	0
$35$	−0.318526	−0.0538408
$36$	0	0
$37$	5.71347	0.939289	0.469645	−	0.882856i	$-0.344382\pi$
$37$	5.71347	0.939289	0.469645	+	0.882856i	$0.344382\pi$
$38$	0	0
$39$	0.654499	0.104804
$40$	0	0
$41$	4.59627	0.717816	0.358908	−	0.933373i	$-0.383149\pi$
$41$	4.59627	0.717816	0.358908	+	0.933373i	$0.383149\pi$
$42$	0	0
$43$	−6.88895	−1.05055	−0.525277	−	0.850931i	$-0.676038\pi$
$43$	−6.88895	−1.05055	−0.525277	+	0.850931i	$0.676038\pi$
$44$	0	0
$45$	−0.0887591	−0.0132314
$46$	0	0
$47$	2.43998	0.355908	0.177954	−	0.984039i	$-0.443052\pi$
$47$	2.43998	0.355908	0.177954	+	0.984039i	$0.443052\pi$
$48$	0	0
$49$	11.0835	1.58336
$50$	0	0
$51$	−8.81444	−1.23427
$52$	0	0
$53$	−10.3855	−1.42656	−0.713280	−	0.700879i	$-0.752791\pi$
$53$	−10.3855	−1.42656	−0.713280	+	0.700879i	$0.752791\pi$
$54$	0	0
$55$	0.0840846	0.0113380
$56$	0	0
$57$	5.20675	0.689650
$58$	0	0
$59$	−5.14682	−0.670059	−0.335029	−	0.942208i	$-0.608746\pi$
$59$	−5.14682	−0.670059	−0.335029	+	0.942208i	$0.608746\pi$
$60$	0	0
$61$	−5.48484	−0.702262	−0.351131	−	0.936326i	$-0.614203\pi$
$61$	−5.48484	−0.702262	−0.351131	+	0.936326i	$0.614203\pi$
$62$	0	0
$63$	5.03906	0.634862
$64$	0	0
$65$	−0.0363891	−0.00451351
$66$	0	0
$67$	−8.24807	−1.00766	−0.503831	−	0.863802i	$-0.668077\pi$
$67$	−8.24807	−1.00766	−0.503831	+	0.863802i	$0.668077\pi$
$68$	0	0
$69$	11.4492	1.37832
$70$	0	0
$71$	14.5585	1.72777	0.863886	−	0.503688i	$-0.168024\pi$
$71$	14.5585	1.72777	0.863886	+	0.503688i	$0.168024\pi$
$72$	0	0
$73$	−14.0072	−1.63942	−0.819710	−	0.572779i	$-0.805865\pi$
$73$	−14.0072	−1.63942	−0.819710	+	0.572779i	$0.805865\pi$
$74$	0	0
$75$	6.72859	0.776950
$76$	0	0
$77$	−4.77368	−0.544012
$78$	0	0
$79$	−1.73234	−0.194903	−0.0974517	−	0.995240i	$-0.531069\pi$
$79$	−1.73234	−0.194903	−0.0974517	+	0.995240i	$0.531069\pi$
$80$	0	0
$81$	−4.04092	−0.448991
$82$	0	0
$83$	−11.8143	−1.29679	−0.648394	−	0.761305i	$-0.724559\pi$
$83$	−11.8143	−1.29679	−0.648394	+	0.761305i	$0.724559\pi$
$84$	0	0
$85$	0.490069	0.0531555
$86$	0	0
$87$	2.69386	0.288813
$88$	0	0
$89$	−6.39261	−0.677615	−0.338808	−	0.940856i	$-0.610024\pi$
$89$	−6.39261	−0.677615	−0.338808	+	0.940856i	$0.610024\pi$
$90$	0	0
$91$	2.06590	0.216565
$92$	0	0
$93$	8.87324	0.920112
$94$	0	0
$95$	−0.289487	−0.0297007
$96$	0	0
$97$	−12.5205	−1.27126	−0.635630	−	0.771994i	$-0.719260\pi$
$97$	−12.5205	−1.27126	−0.635630	+	0.771994i	$0.719260\pi$
$98$	0	0
$99$	−1.33021	−0.133691
$100$	0	0
$101$	14.1855	1.41151	0.705755	−	0.708456i	$-0.250608\pi$
$101$	14.1855	1.41151	0.705755	+	0.708456i	$0.250608\pi$
$102$	0	0
$103$	18.3729	1.81034	0.905169	−	0.425051i	$-0.139744\pi$
$103$	18.3729	1.81034	0.905169	+	0.425051i	$0.139744\pi$
$104$	0	0
$105$	0.429128	0.0418786
$106$	0	0
$107$	20.4669	1.97861	0.989304	−	0.145866i	$-0.0465968\pi$
$107$	20.4669	1.97861	0.989304	+	0.145866i	$0.0465968\pi$
$108$	0	0
$109$	−0.248709	−0.0238220	−0.0119110	−	0.999929i	$-0.503791\pi$
$109$	−0.248709	−0.0238220	−0.0119110	+	0.999929i	$0.503791\pi$
$110$	0	0
$111$	−7.69736	−0.730601
$112$	0	0
$113$	8.74307	0.822479	0.411240	−	0.911527i	$-0.365096\pi$
$113$	8.74307	0.822479	0.411240	+	0.911527i	$0.365096\pi$
$114$	0	0
$115$	−0.636555	−0.0593591
$116$	0	0
$117$	0.575673	0.0532210
$118$	0	0
$119$	−27.8224	−2.55047
$120$	0	0
$121$	−9.73984	−0.885440
$122$	0	0
$123$	−6.19223	−0.558334
$124$	0	0
$125$	−0.748618	−0.0669585
$126$	0	0
$127$	3.99042	0.354092	0.177046	−	0.984203i	$-0.443346\pi$
$127$	3.99042	0.354092	0.177046	+	0.984203i	$0.443346\pi$
$128$	0	0
$129$	9.28100	0.817146
$130$	0	0
$131$	−17.2850	−1.51020	−0.755101	−	0.655609i	$-0.772412\pi$
$131$	−17.2850	−1.51020	−0.755101	+	0.655609i	$0.772412\pi$
$132$	0	0
$133$	16.4349	1.42508
$134$	0	0
$135$	0.422317	0.0363473
$136$	0	0
$137$	18.7194	1.59930	0.799652	−	0.600464i	$-0.205017\pi$
$137$	18.7194	1.59930	0.799652	+	0.600464i	$0.205017\pi$
$138$	0	0
$139$	−15.8730	−1.34633	−0.673166	−	0.739491i	$-0.735066\pi$
$139$	−15.8730	−1.34633	−0.673166	+	0.739491i	$0.735066\pi$
$140$	0	0
$141$	−3.28721	−0.276833
$142$	0	0
$143$	−0.545355	−0.0456049
$144$	0	0
$145$	−0.149775	−0.0124381
$146$	0	0
$147$	−14.9320	−1.23157
$148$	0	0
$149$	9.08337	0.744138	0.372069	−	0.928205i	$-0.378648\pi$
$149$	9.08337	0.744138	0.372069	+	0.928205i	$0.378648\pi$
$150$	0	0
$151$	−0.180064	−0.0146534	−0.00732669	−	0.999973i	$-0.502332\pi$
$151$	−0.180064	−0.0146534	−0.00732669	+	0.999973i	$0.502332\pi$
$152$	0	0
$153$	−7.75285	−0.626781
$154$	0	0
$155$	−0.493338	−0.0396259
$156$	0	0
$157$	10.7585	0.858622	0.429311	−	0.903157i	$-0.358756\pi$
$157$	10.7585	0.858622	0.429311	+	0.903157i	$0.358756\pi$
$158$	0	0
$159$	13.9917	1.10961
$160$	0	0
$161$	36.1387	2.84813
$162$	0	0
$163$	−0.259605	−0.0203338	−0.0101669	−	0.999948i	$-0.503236\pi$
$163$	−0.259605	−0.0203338	−0.0101669	+	0.999948i	$0.503236\pi$
$164$	0	0
$165$	−0.113281	−0.00881894
$166$	0	0
$167$	19.8973	1.53970	0.769851	−	0.638224i	$-0.220331\pi$
$167$	19.8973	1.53970	0.769851	+	0.638224i	$0.220331\pi$
$168$	0	0
$169$	−12.7640	−0.981845
$170$	0	0
$171$	4.57966	0.350215
$172$	0	0
$173$	5.36173	0.407645	0.203822	−	0.979008i	$-0.434663\pi$
$173$	5.36173	0.407645	0.203822	+	0.979008i	$0.434663\pi$
$174$	0	0
$175$	21.2385	1.60548
$176$	0	0
$177$	6.93395	0.521187
$178$	0	0
$179$	9.66315	0.722258	0.361129	−	0.932516i	$-0.382391\pi$
$179$	9.66315	0.722258	0.361129	+	0.932516i	$0.382391\pi$
$180$	0	0
$181$	3.25833	0.242190	0.121095	−	0.992641i	$-0.461359\pi$
$181$	3.25833	0.242190	0.121095	+	0.992641i	$0.461359\pi$
$182$	0	0
$183$	7.38934	0.546236
$184$	0	0
$185$	0.427961	0.0314643
$186$	0	0
$187$	7.34456	0.537087
$188$	0	0
$189$	−23.9759	−1.74399
$190$	0	0
$191$	8.85083	0.640423	0.320212	−	0.947346i	$-0.396246\pi$
$191$	8.85083	0.640423	0.320212	+	0.947346i	$0.396246\pi$
$192$	0	0
$193$	−11.4779	−0.826195	−0.413098	−	0.910687i	$-0.635553\pi$
$193$	−11.4779	−0.826195	−0.413098	+	0.910687i	$0.635553\pi$
$194$	0	0
$195$	0.0490245	0.00351072
$196$	0	0
$197$	11.6058	0.826882	0.413441	−	0.910531i	$-0.364327\pi$
$197$	11.6058	0.826882	0.413441	+	0.910531i	$0.364327\pi$
$198$	0	0
$199$	5.37936	0.381333	0.190666	−	0.981655i	$-0.438935\pi$
$199$	5.37936	0.381333	0.190666	+	0.981655i	$0.438935\pi$
$200$	0	0
$201$	11.1120	0.783783
$202$	0	0
$203$	8.50306	0.596798
$204$	0	0
$205$	0.344278	0.0240454
$206$	0	0
$207$	10.0702	0.699931
$208$	0	0
$209$	−4.33848	−0.300099
$210$	0	0
$211$	−12.2353	−0.842310	−0.421155	−	0.906989i	$-0.638375\pi$
$211$	−12.2353	−0.842310	−0.421155	+	0.906989i	$0.638375\pi$
$212$	0	0
$213$	−19.6136	−1.34390
$214$	0	0
$215$	−0.516009	−0.0351915
$216$	0	0
$217$	28.0080	1.90131
$218$	0	0
$219$	18.8709	1.27518
$220$	0	0
$221$	−3.17849	−0.213808
$222$	0	0
$223$	−16.5361	−1.10734	−0.553670	−	0.832736i	$-0.686773\pi$
$223$	−16.5361	−1.10734	−0.553670	+	0.832736i	$0.686773\pi$
$224$	0	0
$225$	5.91822	0.394548
$226$	0	0
$227$	10.6412	0.706281	0.353141	−	0.935570i	$-0.385114\pi$
$227$	10.6412	0.706281	0.353141	+	0.935570i	$0.385114\pi$
$228$	0	0
$229$	17.7209	1.17103	0.585514	−	0.810662i	$-0.300893\pi$
$229$	17.7209	1.17103	0.585514	+	0.810662i	$0.300893\pi$
$230$	0	0
$231$	6.43124	0.423145
$232$	0	0
$233$	26.8468	1.75879	0.879397	−	0.476089i	$-0.157946\pi$
$233$	26.8468	1.75879	0.879397	+	0.476089i	$0.157946\pi$
$234$	0	0
$235$	0.182764	0.0119222
$236$	0	0
$237$	2.33386	0.151600
$238$	0	0
$239$	−15.0812	−0.975522	−0.487761	−	0.872977i	$-0.662186\pi$
$239$	−15.0812	−0.975522	−0.487761	+	0.872977i	$0.662186\pi$
$240$	0	0
$241$	−5.68364	−0.366116	−0.183058	−	0.983102i	$-0.558600\pi$
$241$	−5.68364	−0.366116	−0.183058	+	0.983102i	$0.558600\pi$
$242$	0	0
$243$	−11.4703	−0.735821
$244$	0	0
$245$	0.830196	0.0530393
$246$	0	0
$247$	1.87755	0.119466
$248$	0	0
$249$	15.9166	1.00867
$250$	0	0
$251$	−17.6892	−1.11653	−0.558266	−	0.829662i	$-0.688533\pi$
$251$	−17.6892	−1.11653	−0.558266	+	0.829662i	$0.688533\pi$
$252$	0	0
$253$	−9.53991	−0.599769
$254$	0	0
$255$	−0.660236	−0.0413456
$256$	0	0
$257$	2.60681	0.162608	0.0813041	−	0.996689i	$-0.474092\pi$
$257$	2.60681	0.162608	0.0813041	+	0.996689i	$0.474092\pi$
$258$	0	0
$259$	−24.2964	−1.50970
$260$	0	0
$261$	2.36942	0.146664
$262$	0	0
$263$	20.3704	1.25609	0.628045	−	0.778177i	$-0.283855\pi$
$263$	20.3704	1.25609	0.628045	+	0.778177i	$0.283855\pi$
$264$	0	0
$265$	−0.777915	−0.0477869
$266$	0	0
$267$	8.61231	0.527065
$268$	0	0
$269$	27.5197	1.67791	0.838954	−	0.544203i	$-0.183168\pi$
$269$	27.5197	1.67791	0.838954	+	0.544203i	$0.183168\pi$
$270$	0	0
$271$	15.9867	0.971124	0.485562	−	0.874202i	$-0.338615\pi$
$271$	15.9867	0.971124	0.485562	+	0.874202i	$0.338615\pi$
$272$	0	0
$273$	−2.78323	−0.168449
$274$	0	0
$275$	−5.60654	−0.338087
$276$	0	0
$277$	13.9604	0.838799	0.419399	−	0.907802i	$-0.362241\pi$
$277$	13.9604	0.838799	0.419399	+	0.907802i	$0.362241\pi$
$278$	0	0
$279$	7.80457	0.467247
$280$	0	0
$281$	5.69579	0.339782	0.169891	−	0.985463i	$-0.445658\pi$
$281$	5.69579	0.339782	0.169891	+	0.985463i	$0.445658\pi$
$282$	0	0
$283$	11.4291	0.679389	0.339695	−	0.940536i	$-0.389676\pi$
$283$	11.4291	0.679389	0.339695	+	0.940536i	$0.389676\pi$
$284$	0	0
$285$	0.390005	0.0231019
$286$	0	0
$287$	−19.5455	−1.15373
$288$	0	0
$289$	25.8062	1.51801
$290$	0	0
$291$	16.8679	0.988816
$292$	0	0
$293$	7.91999	0.462691	0.231345	−	0.972872i	$-0.425687\pi$
$293$	7.91999	0.462691	0.231345	+	0.972872i	$0.425687\pi$
$294$	0	0
$295$	−0.385517	−0.0224456
$296$	0	0
$297$	6.32917	0.367256
$298$	0	0
$299$	4.12856	0.238761
$300$	0	0
$301$	29.2950	1.68854
$302$	0	0
$303$	−19.1111	−1.09790
$304$	0	0
$305$	−0.410836	−0.0235244
$306$	0	0
$307$	14.1653	0.808454	0.404227	−	0.914659i	$-0.367541\pi$
$307$	14.1653	0.808454	0.404227	+	0.914659i	$0.367541\pi$
$308$	0	0
$309$	−24.7525	−1.40812
$310$	0	0
$311$	−16.0761	−0.911593	−0.455797	−	0.890084i	$-0.650646\pi$
$311$	−16.0761	−0.911593	−0.455797	+	0.890084i	$0.650646\pi$
$312$	0	0
$313$	−15.0069	−0.848242	−0.424121	−	0.905606i	$-0.639417\pi$
$313$	−15.0069	−0.848242	−0.424121	+	0.905606i	$0.639417\pi$
$314$	0	0
$315$	0.377445	0.0212666
$316$	0	0
$317$	−18.0571	−1.01419	−0.507094	−	0.861891i	$-0.669280\pi$
$317$	−18.0571	−1.01419	−0.507094	+	0.861891i	$0.669280\pi$
$318$	0	0
$319$	−2.24464	−0.125676
$320$	0	0
$321$	−27.5736	−1.53901
$322$	0	0
$323$	−25.2859	−1.40694
$324$	0	0
$325$	2.42633	0.134588
$326$	0	0
$327$	0.335068	0.0185293
$328$	0	0
$329$	−10.3759	−0.572044
$330$	0	0
$331$	−10.1118	−0.555793	−0.277896	−	0.960611i	$-0.589637\pi$
$331$	−10.1118	−0.555793	−0.277896	+	0.960611i	$0.589637\pi$
$332$	0	0
$333$	−6.77031	−0.371011
$334$	0	0
$335$	−0.617812	−0.0337547
$336$	0	0
$337$	−11.4575	−0.624131	−0.312066	−	0.950061i	$-0.601021\pi$
$337$	−11.4575	−0.624131	−0.312066	+	0.950061i	$0.601021\pi$
$338$	0	0
$339$	−11.7789	−0.639743
$340$	0	0
$341$	−7.39355	−0.400383
$342$	0	0
$343$	−17.3649	−0.937616
$344$	0	0
$345$	0.857586	0.0461709
$346$	0	0
$347$	−18.9404	−1.01677	−0.508386	−	0.861129i	$-0.669758\pi$
$347$	−18.9404	−1.01677	−0.508386	+	0.861129i	$0.669758\pi$
$348$	0	0
$349$	−12.8108	−0.685745	−0.342873	−	0.939382i	$-0.611400\pi$
$349$	−12.8108	−0.685745	−0.342873	+	0.939382i	$0.611400\pi$
$350$	0	0
$351$	−2.73906	−0.146200
$352$	0	0
$353$	22.8842	1.21800	0.609002	−	0.793169i	$-0.291570\pi$
$353$	22.8842	1.21800	0.609002	+	0.793169i	$0.291570\pi$
$354$	0	0
$355$	1.09049	0.0578769
$356$	0	0
$357$	37.4831	1.98382
$358$	0	0
$359$	−0.0696918	−0.00367819	−0.00183910	−	0.999998i	$-0.500585\pi$
$359$	−0.0696918	−0.00367819	−0.00183910	+	0.999998i	$0.500585\pi$
$360$	0	0
$361$	−4.06347	−0.213867
$362$	0	0
$363$	13.1218	0.688716
$364$	0	0
$365$	−1.04919	−0.0549173
$366$	0	0
$367$	−17.6779	−0.922778	−0.461389	−	0.887198i	$-0.652649\pi$
$367$	−17.6779	−0.922778	−0.461389	+	0.887198i	$0.652649\pi$
$368$	0	0
$369$	−5.44645	−0.283531
$370$	0	0
$371$	44.1641	2.29288
$372$	0	0
$373$	8.47111	0.438617	0.219309	−	0.975656i	$-0.429620\pi$
$373$	8.47111	0.438617	0.219309	+	0.975656i	$0.429620\pi$
$374$	0	0
$375$	1.00856	0.0520819
$376$	0	0
$377$	0.971407	0.0500300
$378$	0	0
$379$	−13.7732	−0.707482	−0.353741	−	0.935343i	$-0.615091\pi$
$379$	−13.7732	−0.707482	−0.353741	+	0.935343i	$0.615091\pi$
$380$	0	0
$381$	−5.37601	−0.275421
$382$	0	0
$383$	8.75113	0.447162	0.223581	−	0.974685i	$-0.428225\pi$
$383$	8.75113	0.447162	0.223581	+	0.974685i	$0.428225\pi$
$384$	0	0
$385$	−0.357567	−0.0182233
$386$	0	0
$387$	8.16322	0.414960
$388$	0	0
$389$	2.08771	0.105851	0.0529255	−	0.998598i	$-0.483145\pi$
$389$	2.08771	0.105851	0.0529255	+	0.998598i	$0.483145\pi$
$390$	0	0
$391$	−55.6013	−2.81188
$392$	0	0
$393$	23.2869	1.17467
$394$	0	0
$395$	−0.129759	−0.00652888
$396$	0	0
$397$	27.3393	1.37212	0.686059	−	0.727546i	$-0.259339\pi$
$397$	27.3393	1.37212	0.686059	+	0.727546i	$0.259339\pi$
$398$	0	0
$399$	−22.1415	−1.10846
$400$	0	0
$401$	9.56123	0.477465	0.238732	−	0.971085i	$-0.423268\pi$
$401$	9.56123	0.477465	0.238732	+	0.971085i	$0.423268\pi$
$402$	0	0
$403$	3.19969	0.159388
$404$	0	0
$405$	−0.302681	−0.0150403
$406$	0	0
$407$	6.41376	0.317918
$408$	0	0
$409$	−15.1693	−0.750074	−0.375037	−	0.927010i	$-0.622370\pi$
$409$	−15.1693	−0.750074	−0.375037	+	0.927010i	$0.622370\pi$
$410$	0	0
$411$	−25.2193	−1.24398
$412$	0	0
$413$	21.8867	1.07697
$414$	0	0
$415$	−0.884937	−0.0434398
$416$	0	0
$417$	21.3846	1.04721
$418$	0	0
$419$	11.6001	0.566703	0.283352	−	0.959016i	$-0.408554\pi$
$419$	11.6001	0.566703	0.283352	+	0.959016i	$0.408554\pi$
$420$	0	0
$421$	17.5008	0.852936	0.426468	−	0.904503i	$-0.359758\pi$
$421$	17.5008	0.852936	0.426468	+	0.904503i	$0.359758\pi$
$422$	0	0
$423$	−2.89131	−0.140580
$424$	0	0
$425$	−32.6765	−1.58504
$426$	0	0
$427$	23.3241	1.12873
$428$	0	0
$429$	0.734719	0.0354726
$430$	0	0
$431$	11.1444	0.536808	0.268404	−	0.963306i	$-0.413504\pi$
$431$	11.1444	0.536808	0.268404	+	0.963306i	$0.413504\pi$
$432$	0	0
$433$	−37.4822	−1.80128	−0.900639	−	0.434568i	$-0.856901\pi$
$433$	−37.4822	−1.80128	−0.900639	+	0.434568i	$0.856901\pi$
$434$	0	0
$435$	0.201781	0.00967465
$436$	0	0
$437$	32.8440	1.57114
$438$	0	0
$439$	−5.57251	−0.265961	−0.132981	−	0.991119i	$-0.542455\pi$
$439$	−5.57251	−0.265961	−0.132981	+	0.991119i	$0.542455\pi$
$440$	0	0
$441$	−13.1336	−0.625411
$442$	0	0
$443$	16.3877	0.778603	0.389301	−	0.921110i	$-0.372716\pi$
$443$	16.3877	0.778603	0.389301	+	0.921110i	$0.372716\pi$
$444$	0	0
$445$	−0.478831	−0.0226988
$446$	0	0
$447$	−12.2374	−0.578808
$448$	0	0
$449$	1.50787	0.0711609	0.0355805	−	0.999367i	$-0.488672\pi$
$449$	1.50787	0.0711609	0.0355805	+	0.999367i	$0.488672\pi$
$450$	0	0
$451$	5.15962	0.242957
$452$	0	0
$453$	0.242587	0.0113977
$454$	0	0
$455$	0.154744	0.00725449
$456$	0	0
$457$	39.8861	1.86579	0.932896	−	0.360147i	$-0.117273\pi$
$457$	39.8861	1.86579	0.932896	+	0.360147i	$0.117273\pi$
$458$	0	0
$459$	36.8882	1.72179
$460$	0	0
$461$	21.8959	1.01980	0.509898	−	0.860235i	$-0.329683\pi$
$461$	21.8959	1.01980	0.509898	+	0.860235i	$0.329683\pi$
$462$	0	0
$463$	−19.7923	−0.919827	−0.459913	−	0.887964i	$-0.652120\pi$
$463$	−19.7923	−0.919827	−0.459913	+	0.887964i	$0.652120\pi$
$464$	0	0
$465$	0.664640	0.0308219
$466$	0	0
$467$	38.4128	1.77753	0.888767	−	0.458359i	$-0.151563\pi$
$467$	38.4128	1.77753	0.888767	+	0.458359i	$0.151563\pi$
$468$	0	0
$469$	35.0747	1.61960
$470$	0	0
$471$	−14.4942	−0.667856
$472$	0	0
$473$	−7.73331	−0.355578
$474$	0	0
$475$	19.3022	0.885647
$476$	0	0
$477$	12.3065	0.563478
$478$	0	0
$479$	−1.06727	−0.0487646	−0.0243823	−	0.999703i	$-0.507762\pi$
$479$	−1.06727	−0.0487646	−0.0243823	+	0.999703i	$0.507762\pi$
$480$	0	0
$481$	−2.77567	−0.126560
$482$	0	0
$483$	−48.6872	−2.21534
$484$	0	0
$485$	−0.937831	−0.0425847
$486$	0	0
$487$	12.8081	0.580393	0.290196	−	0.956967i	$-0.406279\pi$
$487$	12.8081	0.580393	0.290196	+	0.956967i	$0.406279\pi$
$488$	0	0
$489$	0.349747	0.0158161
$490$	0	0
$491$	5.67129	0.255942	0.127971	−	0.991778i	$-0.459154\pi$
$491$	5.67129	0.255942	0.127971	+	0.991778i	$0.459154\pi$
$492$	0	0
$493$	−13.0824	−0.589201
$494$	0	0
$495$	−0.0996380	−0.00447840
$496$	0	0
$497$	−61.9094	−2.77702
$498$	0	0
$499$	16.9907	0.760609	0.380304	−	0.924861i	$-0.375819\pi$
$499$	16.9907	0.760609	0.380304	+	0.924861i	$0.375819\pi$
$500$	0	0
$501$	−26.8063	−1.19762
$502$	0	0
$503$	−6.19038	−0.276015	−0.138008	−	0.990431i	$-0.544070\pi$
$503$	−6.19038	−0.276015	−0.138008	+	0.990431i	$0.544070\pi$
$504$	0	0
$505$	1.06255	0.0472828
$506$	0	0
$507$	17.1960	0.763702
$508$	0	0
$509$	−0.283868	−0.0125822	−0.00629111	−	0.999980i	$-0.502003\pi$
$509$	−0.283868	−0.0125822	−0.00629111	+	0.999980i	$0.502003\pi$
$510$	0	0
$511$	59.5652	2.63501
$512$	0	0
$513$	−21.7901	−0.962056
$514$	0	0
$515$	1.37620	0.0606428
$516$	0	0
$517$	2.73904	0.120463
$518$	0	0
$519$	−7.22348	−0.317076
$520$	0	0
$521$	−36.1674	−1.58452	−0.792261	−	0.610183i	$-0.791096\pi$
$521$	−36.1674	−1.58452	−0.792261	+	0.610183i	$0.791096\pi$
$522$	0	0
$523$	5.62713	0.246057	0.123029	−	0.992403i	$-0.460739\pi$
$523$	5.62713	0.246057	0.123029	+	0.992403i	$0.460739\pi$
$524$	0	0
$525$	−28.6131	−1.24878
$526$	0	0
$527$	−43.0917	−1.87710
$528$	0	0
$529$	49.2210	2.14004
$530$	0	0
$531$	6.09884	0.264667
$532$	0	0
$533$	−2.23292	−0.0967183
$534$	0	0
$535$	1.53305	0.0662795
$536$	0	0
$537$	−13.0185	−0.561789
$538$	0	0
$539$	12.4420	0.535913
$540$	0	0
$541$	−20.2787	−0.871850	−0.435925	−	0.899983i	$-0.643579\pi$
$541$	−20.2787	−0.871850	−0.435925	+	0.899983i	$0.643579\pi$
$542$	0	0
$543$	−4.38972	−0.188381
$544$	0	0
$545$	−0.0186293	−0.000797991 0
$546$	0	0
$547$	−33.7625	−1.44358	−0.721791	−	0.692111i	$-0.756681\pi$
$547$	−33.7625	−1.44358	−0.721791	+	0.692111i	$0.756681\pi$
$548$	0	0
$549$	6.49939	0.277387
$550$	0	0
$551$	7.72785	0.329218
$552$	0	0
$553$	7.36672	0.313265
$554$	0	0
$555$	−0.576562	−0.0244737
$556$	0	0
$557$	−33.0877	−1.40197	−0.700985	−	0.713176i	$-0.747256\pi$
$557$	−33.0877	−1.40197	−0.700985	+	0.713176i	$0.747256\pi$
$558$	0	0
$559$	3.34673	0.141551
$560$	0	0
$561$	−9.89480	−0.417759
$562$	0	0
$563$	−37.2463	−1.56974	−0.784872	−	0.619658i	$-0.787271\pi$
$563$	−37.2463	−1.56974	−0.784872	+	0.619658i	$0.787271\pi$
$564$	0	0
$565$	0.654890	0.0275514
$566$	0	0
$567$	17.1839	0.721655
$568$	0	0
$569$	32.9211	1.38012	0.690062	−	0.723750i	$-0.257583\pi$
$569$	32.9211	1.38012	0.690062	+	0.723750i	$0.257583\pi$
$570$	0	0
$571$	35.9815	1.50578	0.752890	−	0.658147i	$-0.228659\pi$
$571$	35.9815	1.50578	0.752890	+	0.658147i	$0.228659\pi$
$572$	0	0
$573$	−11.9241	−0.498136
$574$	0	0
$575$	42.4438	1.77003
$576$	0	0
$577$	−20.8605	−0.868433	−0.434217	−	0.900808i	$-0.642975\pi$
$577$	−20.8605	−0.868433	−0.434217	+	0.900808i	$0.642975\pi$
$578$	0	0
$579$	15.4633	0.642634
$580$	0	0
$581$	50.2399	2.08430
$582$	0	0
$583$	−11.6584	−0.482843
$584$	0	0
$585$	0.0431201	0.00178280
$586$	0	0
$587$	−16.3158	−0.673426	−0.336713	−	0.941607i	$-0.609315\pi$
$587$	−16.3158	−0.673426	−0.336713	+	0.941607i	$0.609315\pi$
$588$	0	0
$589$	25.4546	1.04884
$590$	0	0
$591$	−15.6357	−0.643168
$592$	0	0
$593$	−9.87672	−0.405588	−0.202794	−	0.979221i	$-0.565002\pi$
$593$	−9.87672	−0.405588	−0.202794	+	0.979221i	$0.565002\pi$
$594$	0	0
$595$	−2.08400	−0.0854358
$596$	0	0
$597$	−7.24723	−0.296610
$598$	0	0
$599$	35.7750	1.46173	0.730863	−	0.682524i	$-0.239118\pi$
$599$	35.7750	1.46173	0.730863	+	0.682524i	$0.239118\pi$
$600$	0	0
$601$	−15.7084	−0.640761	−0.320380	−	0.947289i	$-0.603811\pi$
$601$	−15.7084	−0.640761	−0.320380	+	0.947289i	$0.603811\pi$
$602$	0	0
$603$	9.77374	0.398017
$604$	0	0
$605$	−0.729552	−0.0296605
$606$	0	0
$607$	9.93044	0.403064	0.201532	−	0.979482i	$-0.435408\pi$
$607$	9.93044	0.403064	0.201532	+	0.979482i	$0.435408\pi$
$608$	0	0
$609$	−11.4556	−0.464203
$610$	0	0
$611$	−1.18537	−0.0479549
$612$	0	0
$613$	45.4527	1.83582	0.917908	−	0.396793i	$-0.129877\pi$
$613$	45.4527	1.83582	0.917908	+	0.396793i	$0.129877\pi$
$614$	0	0
$615$	−0.463822	−0.0187031
$616$	0	0
$617$	−27.1035	−1.09115	−0.545574	−	0.838063i	$-0.683688\pi$
$617$	−27.1035	−1.09115	−0.545574	+	0.838063i	$0.683688\pi$
$618$	0	0
$619$	28.9676	1.16431	0.582153	−	0.813079i	$-0.302211\pi$
$619$	28.9676	1.16431	0.582153	+	0.813079i	$0.302211\pi$
$620$	0	0
$621$	−47.9144	−1.92274
$622$	0	0
$623$	27.1844	1.08912
$624$	0	0
$625$	24.9159	0.996635
$626$	0	0
$627$	5.84492	0.233424
$628$	0	0
$629$	37.3812	1.49049
$630$	0	0
$631$	20.8230	0.828950	0.414475	−	0.910061i	$-0.363965\pi$
$631$	20.8230	0.828950	0.414475	+	0.910061i	$0.363965\pi$
$632$	0	0
$633$	16.4837	0.655168
$634$	0	0
$635$	0.298898	0.0118614
$636$	0	0
$637$	−5.38448	−0.213341
$638$	0	0
$639$	−17.2514	−0.682454
$640$	0	0
$641$	−33.1908	−1.31096	−0.655480	−	0.755213i	$-0.727533\pi$
$641$	−33.1908	−1.31096	−0.655480	+	0.755213i	$0.727533\pi$
$642$	0	0
$643$	−2.76533	−0.109054	−0.0545270	−	0.998512i	$-0.517365\pi$
$643$	−2.76533	−0.109054	−0.0545270	+	0.998512i	$0.517365\pi$
$644$	0	0
$645$	0.695183	0.0273728
$646$	0	0
$647$	5.18596	0.203881	0.101941	−	0.994790i	$-0.467495\pi$
$647$	5.18596	0.203881	0.101941	+	0.994790i	$0.467495\pi$
$648$	0	0
$649$	−5.77765	−0.226793
$650$	0	0
$651$	−37.7332	−1.47888
$652$	0	0
$653$	12.7366	0.498423	0.249212	−	0.968449i	$-0.419829\pi$
$653$	12.7366	0.498423	0.249212	+	0.968449i	$0.419829\pi$
$654$	0	0
$655$	−1.29472	−0.0505888
$656$	0	0
$657$	16.5982	0.647556
$658$	0	0
$659$	13.9110	0.541895	0.270947	−	0.962594i	$-0.412663\pi$
$659$	13.9110	0.541895	0.270947	+	0.962594i	$0.412663\pi$
$660$	0	0
$661$	14.1006	0.548452	0.274226	−	0.961665i	$-0.411578\pi$
$661$	14.1006	0.548452	0.274226	+	0.961665i	$0.411578\pi$
$662$	0	0
$663$	4.28215	0.166305
$664$	0	0
$665$	1.23103	0.0477375
$666$	0	0
$667$	16.9928	0.657965
$668$	0	0
$669$	22.2779	0.861314
$670$	0	0
$671$	−6.15710	−0.237692
$672$	0	0
$673$	−34.9497	−1.34721	−0.673607	−	0.739090i	$-0.735256\pi$
$673$	−34.9497	−1.34721	−0.673607	+	0.739090i	$0.735256\pi$
$674$	0	0
$675$	−28.1590	−1.08384
$676$	0	0
$677$	−20.0280	−0.769738	−0.384869	−	0.922971i	$-0.625753\pi$
$677$	−20.0280	−0.769738	−0.384869	+	0.922971i	$0.625753\pi$
$678$	0	0
$679$	53.2429	2.04327
$680$	0	0
$681$	−14.3361	−0.549362
$682$	0	0
$683$	20.5757	0.787308	0.393654	−	0.919259i	$-0.371211\pi$
$683$	20.5757	0.787308	0.393654	+	0.919259i	$0.371211\pi$
$684$	0	0
$685$	1.40215	0.0535735
$686$	0	0
$687$	−23.8741	−0.910853
$688$	0	0
$689$	5.04539	0.192214
$690$	0	0
$691$	−3.16945	−0.120571	−0.0602857	−	0.998181i	$-0.519201\pi$
$691$	−3.16945	−0.120571	−0.0602857	+	0.998181i	$0.519201\pi$
$692$	0	0
$693$	5.65668	0.214880
$694$	0	0
$695$	−1.18895	−0.0450995
$696$	0	0
$697$	30.0717	1.13905
$698$	0	0
$699$	−36.1688	−1.36803
$700$	0	0
$701$	−21.7344	−0.820897	−0.410448	−	0.911884i	$-0.634628\pi$
$701$	−21.7344	−0.820897	−0.410448	+	0.911884i	$0.634628\pi$
$702$	0	0
$703$	−22.0813	−0.832813
$704$	0	0
$705$	−0.246225	−0.00927337
$706$	0	0
$707$	−60.3234	−2.26869
$708$	0	0
$709$	−11.1719	−0.419569	−0.209784	−	0.977748i	$-0.567276\pi$
$709$	−11.1719	−0.419569	−0.209784	+	0.977748i	$0.567276\pi$
$710$	0	0
$711$	2.05278	0.0769851
$712$	0	0
$713$	55.9722	2.09618
$714$	0	0
$715$	−0.0408492	−0.00152767
$716$	0	0
$717$	20.3178	0.758783
$718$	0	0
$719$	−21.5011	−0.801855	−0.400927	−	0.916110i	$-0.631312\pi$
$719$	−21.5011	−0.801855	−0.400927	+	0.916110i	$0.631312\pi$
$720$	0	0
$721$	−78.1303	−2.90972
$722$	0	0
$723$	7.65717	0.284773
$724$	0	0
$725$	9.98657	0.370892
$726$	0	0
$727$	8.71945	0.323386	0.161693	−	0.986841i	$-0.448304\pi$
$727$	8.71945	0.323386	0.161693	+	0.986841i	$0.448304\pi$
$728$	0	0
$729$	27.5759	1.02133
$730$	0	0
$731$	−45.0719	−1.66705
$732$	0	0
$733$	4.45443	0.164528	0.0822640	−	0.996611i	$-0.473785\pi$
$733$	4.45443	0.164528	0.0822640	+	0.996611i	$0.473785\pi$
$734$	0	0
$735$	−1.11846	−0.0412552
$736$	0	0
$737$	−9.25901	−0.341060
$738$	0	0
$739$	−9.99944	−0.367835	−0.183918	−	0.982942i	$-0.558878\pi$
$739$	−9.99944	−0.367835	−0.183918	+	0.982942i	$0.558878\pi$
$740$	0	0
$741$	−2.52949	−0.0929233
$742$	0	0
$743$	30.6417	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$743$	30.6417	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$744$	0	0
$745$	0.680379	0.0249272
$746$	0	0
$747$	13.9996	0.512219
$748$	0	0
$749$	−87.0348	−3.18018
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	23.8314	0.868465
$754$	0	0
$755$	−0.0134875	−0.000490859 0
$756$	0	0
$757$	−46.2342	−1.68041	−0.840206	−	0.542268i	$-0.817566\pi$
$757$	−46.2342	−1.68041	−0.840206	+	0.542268i	$0.817566\pi$
$758$	0	0
$759$	12.8524	0.466514
$760$	0	0
$761$	31.2829	1.13401	0.567003	−	0.823716i	$-0.308103\pi$
$761$	31.2829	1.13401	0.567003	+	0.823716i	$0.308103\pi$
$762$	0	0
$763$	1.05763	0.0382887
$764$	0	0
$765$	−0.580719	−0.0209959
$766$	0	0
$767$	2.50038	0.0902835
$768$	0	0
$769$	20.8600	0.752229	0.376115	−	0.926573i	$-0.377260\pi$
$769$	20.8600	0.752229	0.376115	+	0.926573i	$0.377260\pi$
$770$	0	0
$771$	−3.51197	−0.126480
$772$	0	0
$773$	45.6621	1.64235	0.821176	−	0.570676i	$-0.193319\pi$
$773$	45.6621	1.64235	0.821176	+	0.570676i	$0.193319\pi$
$774$	0	0
$775$	32.8945	1.18160
$776$	0	0
$777$	32.7328	1.17428
$778$	0	0
$779$	−17.7636	−0.636446
$780$	0	0
$781$	16.3429	0.584793
$782$	0	0
$783$	−11.2737	−0.402891
$784$	0	0
$785$	0.805854	0.0287621
$786$	0	0
$787$	6.91001	0.246315	0.123158	−	0.992387i	$-0.460698\pi$
$787$	6.91001	0.246315	0.123158	+	0.992387i	$0.460698\pi$
$788$	0	0
$789$	−27.4435	−0.977016
$790$	0	0
$791$	−37.1796	−1.32196
$792$	0	0
$793$	2.66459	0.0946225
$794$	0	0
$795$	1.04803	0.0371698
$796$	0	0
$797$	−31.7434	−1.12441	−0.562205	−	0.826998i	$-0.690047\pi$
$797$	−31.7434	−1.12441	−0.562205	+	0.826998i	$0.690047\pi$
$798$	0	0
$799$	15.9639	0.564763
$800$	0	0
$801$	7.57507	0.267652
$802$	0	0
$803$	−15.7240	−0.554889
$804$	0	0
$805$	2.70693	0.0954068
$806$	0	0
$807$	−37.0754	−1.30512
$808$	0	0
$809$	3.73470	0.131305	0.0656526	−	0.997843i	$-0.479087\pi$
$809$	3.73470	0.131305	0.0656526	+	0.997843i	$0.479087\pi$
$810$	0	0
$811$	53.5714	1.88114	0.940572	−	0.339593i	$-0.110289\pi$
$811$	53.5714	1.88114	0.940572	+	0.339593i	$0.110289\pi$
$812$	0	0
$813$	−21.5378	−0.755363
$814$	0	0
$815$	−0.0194454	−0.000681142 0
$816$	0	0
$817$	26.6243	0.931466
$818$	0	0
$819$	−2.44803	−0.0855411
$820$	0	0
$821$	−4.05328	−0.141461	−0.0707303	−	0.997495i	$-0.522533\pi$
$821$	−4.05328	−0.141461	−0.0707303	+	0.997495i	$0.522533\pi$
$822$	0	0
$823$	16.0611	0.559856	0.279928	−	0.960021i	$-0.409689\pi$
$823$	16.0611	0.559856	0.279928	+	0.960021i	$0.409689\pi$
$824$	0	0
$825$	7.55329	0.262972
$826$	0	0
$827$	−4.25680	−0.148024	−0.0740118	−	0.997257i	$-0.523580\pi$
$827$	−4.25680	−0.148024	−0.0740118	+	0.997257i	$0.523580\pi$
$828$	0	0
$829$	22.5583	0.783482	0.391741	−	0.920076i	$-0.371873\pi$
$829$	22.5583	0.783482	0.391741	+	0.920076i	$0.371873\pi$
$830$	0	0
$831$	−18.8079	−0.652437
$832$	0	0
$833$	72.5153	2.51251
$834$	0	0
$835$	1.49039	0.0515770
$836$	0	0
$837$	−37.1343	−1.28355
$838$	0	0
$839$	−30.5380	−1.05429	−0.527145	−	0.849776i	$-0.676737\pi$
$839$	−30.5380	−1.05429	−0.527145	+	0.849776i	$0.676737\pi$
$840$	0	0
$841$	−25.0018	−0.862130
$842$	0	0
$843$	−7.67353	−0.264291
$844$	0	0
$845$	−0.956072	−0.0328899
$846$	0	0
$847$	41.4184	1.42315
$848$	0	0
$849$	−15.3976	−0.528445
$850$	0	0
$851$	−48.5548	−1.66444
$852$	0	0
$853$	3.74804	0.128330	0.0641652	−	0.997939i	$-0.479562\pi$
$853$	3.74804	0.128330	0.0641652	+	0.997939i	$0.479562\pi$
$854$	0	0
$855$	0.343034	0.0117315
$856$	0	0
$857$	1.28672	0.0439535	0.0219767	−	0.999758i	$-0.493004\pi$
$857$	1.28672	0.0439535	0.0219767	+	0.999758i	$0.493004\pi$
$858$	0	0
$859$	−8.31880	−0.283834	−0.141917	−	0.989879i	$-0.545327\pi$
$859$	−8.31880	−0.283834	−0.141917	+	0.989879i	$0.545327\pi$
$860$	0	0
$861$	26.3322	0.897401
$862$	0	0
$863$	5.84487	0.198962	0.0994808	−	0.995039i	$-0.468282\pi$
$863$	5.84487	0.198962	0.0994808	+	0.995039i	$0.468282\pi$
$864$	0	0
$865$	0.401614	0.0136553
$866$	0	0
$867$	−34.7668	−1.18074
$868$	0	0
$869$	−1.94467	−0.0659683
$870$	0	0
$871$	4.00700	0.135772
$872$	0	0
$873$	14.8364	0.502136
$874$	0	0
$875$	3.18348	0.107621
$876$	0	0
$877$	49.1131	1.65843	0.829217	−	0.558927i	$-0.188787\pi$
$877$	49.1131	1.65843	0.829217	+	0.558927i	$0.188787\pi$
$878$	0	0
$879$	−10.6700	−0.359892
$880$	0	0
$881$	18.4500	0.621597	0.310799	−	0.950476i	$-0.399404\pi$
$881$	18.4500	0.621597	0.310799	+	0.950476i	$0.399404\pi$
$882$	0	0
$883$	39.0972	1.31573	0.657864	−	0.753137i	$-0.271460\pi$
$883$	39.0972	1.31573	0.657864	+	0.753137i	$0.271460\pi$
$884$	0	0
$885$	0.519379	0.0174587
$886$	0	0
$887$	−48.2198	−1.61906	−0.809532	−	0.587076i	$-0.800279\pi$
$887$	−48.2198	−1.61906	−0.809532	+	0.587076i	$0.800279\pi$
$888$	0	0
$889$	−16.9691	−0.569126
$890$	0	0
$891$	−4.53621	−0.151969
$892$	0	0
$893$	−9.42999	−0.315563
$894$	0	0
$895$	0.723808	0.0241942
$896$	0	0
$897$	−5.56212	−0.185714
$898$	0	0
$899$	13.1697	0.439233
$900$	0	0
$901$	−67.9487	−2.26370
$902$	0	0
$903$	−39.4671	−1.31338
$904$	0	0
$905$	0.244062	0.00811288
$906$	0	0
$907$	30.4977	1.01266	0.506329	−	0.862340i	$-0.331002\pi$
$907$	30.4977	1.01266	0.506329	+	0.862340i	$0.331002\pi$
$908$	0	0
$909$	−16.8094	−0.557533
$910$	0	0
$911$	−15.5291	−0.514502	−0.257251	−	0.966345i	$-0.582817\pi$
$911$	−15.5291	−0.514502	−0.257251	+	0.966345i	$0.582817\pi$
$912$	0	0
$913$	−13.2623	−0.438920
$914$	0	0
$915$	0.553490	0.0182978
$916$	0	0
$917$	73.5041	2.42732
$918$	0	0
$919$	−54.7722	−1.80677	−0.903384	−	0.428832i	$-0.858925\pi$
$919$	−54.7722	−1.80677	−0.903384	+	0.428832i	$0.858925\pi$
$920$	0	0
$921$	−19.0838	−0.628834
$922$	0	0
$923$	−7.07266	−0.232799
$924$	0	0
$925$	−28.5353	−0.938235
$926$	0	0
$927$	−21.7714	−0.715067
$928$	0	0
$929$	13.9121	0.456440	0.228220	−	0.973610i	$-0.426709\pi$
$929$	13.9121	0.456440	0.228220	+	0.973610i	$0.426709\pi$
$930$	0	0
$931$	−42.8352	−1.40387
$932$	0	0
$933$	21.6582	0.709058
$934$	0	0
$935$	0.550136	0.0179914
$936$	0	0
$937$	−11.9858	−0.391559	−0.195779	−	0.980648i	$-0.562724\pi$
$937$	−11.9858	−0.391559	−0.195779	+	0.980648i	$0.562724\pi$
$938$	0	0
$939$	20.2178	0.659782
$940$	0	0
$941$	−58.8007	−1.91685	−0.958424	−	0.285349i	$-0.907891\pi$
$941$	−58.8007	−1.91685	−0.958424	+	0.285349i	$0.907891\pi$
$942$	0	0
$943$	−39.0604	−1.27198
$944$	0	0
$945$	−1.79589	−0.0584203
$946$	0	0
$947$	3.68932	0.119887	0.0599435	−	0.998202i	$-0.480908\pi$
$947$	3.68932	0.119887	0.0599435	+	0.998202i	$0.480908\pi$
$948$	0	0
$949$	6.80485	0.220895
$950$	0	0
$951$	24.3271	0.788859
$952$	0	0
$953$	25.3743	0.821954	0.410977	−	0.911646i	$-0.365188\pi$
$953$	25.3743	0.821954	0.410977	+	0.911646i	$0.365188\pi$
$954$	0	0
$955$	0.662961	0.0214529
$956$	0	0
$957$	3.02404	0.0977534
$958$	0	0
$959$	−79.6035	−2.57053
$960$	0	0
$961$	12.3792	0.399328
$962$	0	0
$963$	−24.2527	−0.781533
$964$	0	0
$965$	−0.859737	−0.0276759
$966$	0	0
$967$	30.0166	0.965270	0.482635	−	0.875822i	$-0.339680\pi$
$967$	30.0166	0.965270	0.482635	+	0.875822i	$0.339680\pi$
$968$	0	0
$969$	34.0659	1.09435
$970$	0	0
$971$	2.26870	0.0728061	0.0364030	−	0.999337i	$-0.488410\pi$
$971$	2.26870	0.0728061	0.0364030	+	0.999337i	$0.488410\pi$
$972$	0	0
$973$	67.4995	2.16393
$974$	0	0
$975$	−3.26882	−0.104686
$976$	0	0
$977$	−0.536664	−0.0171694	−0.00858470	−	0.999963i	$-0.502733\pi$
$977$	−0.536664	−0.0171694	−0.00858470	+	0.999963i	$0.502733\pi$
$978$	0	0
$979$	−7.17613	−0.229350
$980$	0	0
$981$	0.294714	0.00940948
$982$	0	0
$983$	10.8763	0.346899	0.173450	−	0.984843i	$-0.444509\pi$
$983$	10.8763	0.346899	0.173450	+	0.984843i	$0.444509\pi$
$984$	0	0
$985$	0.869322	0.0276989
$986$	0	0
$987$	13.9788	0.444949
$988$	0	0
$989$	58.5443	1.86160
$990$	0	0
$991$	−52.5065	−1.66792	−0.833961	−	0.551823i	$-0.813933\pi$
$991$	−52.5065	−1.66792	−0.833961	+	0.551823i	$0.813933\pi$
$992$	0	0
$993$	13.6229	0.432308
$994$	0	0
$995$	0.402935	0.0127739
$996$	0	0
$997$	12.8794	0.407895	0.203947	−	0.978982i	$-0.434623\pi$
$997$	12.8794	0.407895	0.203947	+	0.978982i	$0.434623\pi$
$998$	0	0
$999$	32.2132	1.01918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.15	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.15	✓	50	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34723 q^{3} +0.0749039 q^{5} -4.25247 q^{7} -1.18497 q^{9} +O(q^{10})\)	\(q-1.34723 q^{3} +0.0749039 q^{5} -4.25247 q^{7} -1.18497 q^{9} +1.12257 q^{11} -0.485811 q^{13} -0.100913 q^{15} +6.54264 q^{17} -3.86478 q^{19} +5.72905 q^{21} -8.49829 q^{23} -4.99439 q^{25} +5.63812 q^{27} -1.99956 q^{29} -6.58629 q^{31} -1.51236 q^{33} -0.318526 q^{35} +5.71347 q^{37} +0.654499 q^{39} +4.59627 q^{41} -6.88895 q^{43} -0.0887591 q^{45} +2.43998 q^{47} +11.0835 q^{49} -8.81444 q^{51} -10.3855 q^{53} +0.0840846 q^{55} +5.20675 q^{57} -5.14682 q^{59} -5.48484 q^{61} +5.03906 q^{63} -0.0363891 q^{65} -8.24807 q^{67} +11.4492 q^{69} +14.5585 q^{71} -14.0072 q^{73} +6.72859 q^{75} -4.77368 q^{77} -1.73234 q^{79} -4.04092 q^{81} -11.8143 q^{83} +0.490069 q^{85} +2.69386 q^{87} -6.39261 q^{89} +2.06590 q^{91} +8.87324 q^{93} -0.289487 q^{95} -12.5205 q^{97} -1.33021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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