LMFDB - Embedded newform 6008.2.a.e.1.42

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			1.42
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)

$f(q)$	$=$	$q+2.26355 q^{3} -2.03106 q^{5} +2.84873 q^{7} +2.12365 q^{9} +O(q^{10})$	$q+2.26355 q^{3} -2.03106 q^{5} +2.84873 q^{7} +2.12365 q^{9} +5.94130 q^{11} +3.06346 q^{13} -4.59739 q^{15} +5.90773 q^{17} +2.58062 q^{19} +6.44823 q^{21} +1.10477 q^{23} -0.874809 q^{25} -1.98367 q^{27} +6.71326 q^{29} -6.73786 q^{31} +13.4484 q^{33} -5.78593 q^{35} +4.99862 q^{37} +6.93430 q^{39} -8.52835 q^{41} -5.24692 q^{43} -4.31325 q^{45} +10.6005 q^{47} +1.11525 q^{49} +13.3724 q^{51} -10.2822 q^{53} -12.0671 q^{55} +5.84136 q^{57} -2.18420 q^{59} -7.25388 q^{61} +6.04970 q^{63} -6.22207 q^{65} +7.02358 q^{67} +2.50070 q^{69} +11.8016 q^{71} -8.01357 q^{73} -1.98017 q^{75} +16.9252 q^{77} +11.6831 q^{79} -10.8611 q^{81} -4.69026 q^{83} -11.9989 q^{85} +15.1958 q^{87} -12.4148 q^{89} +8.72698 q^{91} -15.2515 q^{93} -5.24139 q^{95} -11.9835 q^{97} +12.6172 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.26355	1.30686	0.653430	−	0.756987i	$-0.273329\pi$
$3$	2.26355	1.30686	0.653430	+	0.756987i	$0.273329\pi$
$4$	0	0
$5$	−2.03106	−0.908316	−0.454158	−	0.890921i	$-0.650060\pi$
$5$	−2.03106	−0.908316	−0.454158	+	0.890921i	$0.650060\pi$
$6$	0	0
$7$	2.84873	1.07672	0.538359	−	0.842716i	$-0.319044\pi$
$7$	2.84873	1.07672	0.538359	+	0.842716i	$0.319044\pi$
$8$	0	0
$9$	2.12365	0.707883
$10$	0	0
$11$	5.94130	1.79137	0.895685	−	0.444689i	$-0.146686\pi$
$11$	5.94130	1.79137	0.895685	+	0.444689i	$0.146686\pi$
$12$	0	0
$13$	3.06346	0.849652	0.424826	−	0.905275i	$-0.360335\pi$
$13$	3.06346	0.849652	0.424826	+	0.905275i	$0.360335\pi$
$14$	0	0
$15$	−4.59739	−1.18704
$16$	0	0
$17$	5.90773	1.43283	0.716417	−	0.697672i	$-0.245781\pi$
$17$	5.90773	1.43283	0.716417	+	0.697672i	$0.245781\pi$
$18$	0	0
$19$	2.58062	0.592035	0.296017	−	0.955183i	$-0.404341\pi$
$19$	2.58062	0.592035	0.296017	+	0.955183i	$0.404341\pi$
$20$	0	0
$21$	6.44823	1.40712
$22$	0	0
$23$	1.10477	0.230361	0.115180	−	0.993345i	$-0.463255\pi$
$23$	1.10477	0.230361	0.115180	+	0.993345i	$0.463255\pi$
$24$	0	0
$25$	−0.874809	−0.174962
$26$	0	0
$27$	−1.98367	−0.381757
$28$	0	0
$29$	6.71326	1.24662	0.623310	−	0.781974i	$-0.285787\pi$
$29$	6.71326	1.24662	0.623310	+	0.781974i	$0.285787\pi$
$30$	0	0
$31$	−6.73786	−1.21015	−0.605077	−	0.796167i	$-0.706858\pi$
$31$	−6.73786	−1.21015	−0.605077	+	0.796167i	$0.706858\pi$
$32$	0	0
$33$	13.4484	2.34107
$34$	0	0
$35$	−5.78593	−0.978001
$36$	0	0
$37$	4.99862	0.821767	0.410884	−	0.911688i	$-0.365220\pi$
$37$	4.99862	0.821767	0.410884	+	0.911688i	$0.365220\pi$
$38$	0	0
$39$	6.93430	1.11038
$40$	0	0
$41$	−8.52835	−1.33190	−0.665952	−	0.745995i	$-0.731974\pi$
$41$	−8.52835	−1.33190	−0.665952	+	0.745995i	$0.731974\pi$
$42$	0	0
$43$	−5.24692	−0.800147	−0.400074	−	0.916483i	$-0.631015\pi$
$43$	−5.24692	−0.800147	−0.400074	+	0.916483i	$0.631015\pi$
$44$	0	0
$45$	−4.31325	−0.642981
$46$	0	0
$47$	10.6005	1.54624	0.773122	−	0.634258i	$-0.218694\pi$
$47$	10.6005	1.54624	0.773122	+	0.634258i	$0.218694\pi$
$48$	0	0
$49$	1.11525	0.159322
$50$	0	0
$51$	13.3724	1.87251
$52$	0	0
$53$	−10.2822	−1.41237	−0.706183	−	0.708030i	$-0.749584\pi$
$53$	−10.2822	−1.41237	−0.706183	+	0.708030i	$0.749584\pi$
$54$	0	0
$55$	−12.0671	−1.62713
$56$	0	0
$57$	5.84136	0.773707
$58$	0	0
$59$	−2.18420	−0.284359	−0.142180	−	0.989841i	$-0.545411\pi$
$59$	−2.18420	−0.284359	−0.142180	+	0.989841i	$0.545411\pi$
$60$	0	0
$61$	−7.25388	−0.928765	−0.464382	−	0.885635i	$-0.653724\pi$
$61$	−7.25388	−0.928765	−0.464382	+	0.885635i	$0.653724\pi$
$62$	0	0
$63$	6.04970	0.762190
$64$	0	0
$65$	−6.22207	−0.771753
$66$	0	0
$67$	7.02358	0.858066	0.429033	−	0.903289i	$-0.358854\pi$
$67$	7.02358	0.858066	0.429033	+	0.903289i	$0.358854\pi$
$68$	0	0
$69$	2.50070	0.301049
$70$	0	0
$71$	11.8016	1.40059	0.700296	−	0.713853i	$-0.253051\pi$
$71$	11.8016	1.40059	0.700296	+	0.713853i	$0.253051\pi$
$72$	0	0
$73$	−8.01357	−0.937918	−0.468959	−	0.883220i	$-0.655371\pi$
$73$	−8.01357	−0.937918	−0.468959	+	0.883220i	$0.655371\pi$
$74$	0	0
$75$	−1.98017	−0.228650
$76$	0	0
$77$	16.9252	1.92880
$78$	0	0
$79$	11.6831	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$79$	11.6831	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$80$	0	0
$81$	−10.8611	−1.20678
$82$	0	0
$83$	−4.69026	−0.514823	−0.257411	−	0.966302i	$-0.582870\pi$
$83$	−4.69026	−0.514823	−0.257411	+	0.966302i	$0.582870\pi$
$84$	0	0
$85$	−11.9989	−1.30147
$86$	0	0
$87$	15.1958	1.62916
$88$	0	0
$89$	−12.4148	−1.31597	−0.657984	−	0.753032i	$-0.728590\pi$
$89$	−12.4148	−1.31597	−0.657984	+	0.753032i	$0.728590\pi$
$90$	0	0
$91$	8.72698	0.914836
$92$	0	0
$93$	−15.2515	−1.58150
$94$	0	0
$95$	−5.24139	−0.537755
$96$	0	0
$97$	−11.9835	−1.21675	−0.608373	−	0.793652i	$-0.708177\pi$
$97$	−11.9835	−1.21675	−0.608373	+	0.793652i	$0.708177\pi$
$98$	0	0
$99$	12.6172	1.26808
$100$	0	0
$101$	6.56478	0.653220	0.326610	−	0.945159i	$-0.394094\pi$
$101$	6.56478	0.653220	0.326610	+	0.945159i	$0.394094\pi$
$102$	0	0
$103$	16.0425	1.58072	0.790359	−	0.612644i	$-0.209894\pi$
$103$	16.0425	1.58072	0.790359	+	0.612644i	$0.209894\pi$
$104$	0	0
$105$	−13.0967	−1.27811
$106$	0	0
$107$	−16.0761	−1.55413	−0.777067	−	0.629418i	$-0.783294\pi$
$107$	−16.0761	−1.55413	−0.777067	+	0.629418i	$0.783294\pi$
$108$	0	0
$109$	−7.58352	−0.726370	−0.363185	−	0.931717i	$-0.618311\pi$
$109$	−7.58352	−0.726370	−0.363185	+	0.931717i	$0.618311\pi$
$110$	0	0
$111$	11.3146	1.07393
$112$	0	0
$113$	−5.95235	−0.559950	−0.279975	−	0.960007i	$-0.590326\pi$
$113$	−5.95235	−0.559950	−0.279975	+	0.960007i	$0.590326\pi$
$114$	0	0
$115$	−2.24385	−0.209240
$116$	0	0
$117$	6.50572	0.601454
$118$	0	0
$119$	16.8295	1.54276
$120$	0	0
$121$	24.2991	2.20901
$122$	0	0
$123$	−19.3043	−1.74061
$124$	0	0
$125$	11.9321	1.06724
$126$	0	0
$127$	3.80739	0.337852	0.168926	−	0.985629i	$-0.445970\pi$
$127$	3.80739	0.337852	0.168926	+	0.985629i	$0.445970\pi$
$128$	0	0
$129$	−11.8766	−1.04568
$130$	0	0
$131$	−18.0150	−1.57398	−0.786988	−	0.616968i	$-0.788361\pi$
$131$	−18.0150	−1.57398	−0.786988	+	0.616968i	$0.788361\pi$
$132$	0	0
$133$	7.35149	0.637455
$134$	0	0
$135$	4.02894	0.346756
$136$	0	0
$137$	11.1942	0.956385	0.478192	−	0.878255i	$-0.341292\pi$
$137$	11.1942	0.956385	0.478192	+	0.878255i	$0.341292\pi$
$138$	0	0
$139$	18.1484	1.53933	0.769665	−	0.638449i	$-0.220423\pi$
$139$	18.1484	1.53933	0.769665	+	0.638449i	$0.220423\pi$
$140$	0	0
$141$	23.9948	2.02072
$142$	0	0
$143$	18.2010	1.52204
$144$	0	0
$145$	−13.6350	−1.13233
$146$	0	0
$147$	2.52443	0.208212
$148$	0	0
$149$	−6.10842	−0.500421	−0.250210	−	0.968191i	$-0.580500\pi$
$149$	−6.10842	−0.500421	−0.250210	+	0.968191i	$0.580500\pi$
$150$	0	0
$151$	−1.68762	−0.137337	−0.0686683	−	0.997640i	$-0.521875\pi$
$151$	−1.68762	−0.137337	−0.0686683	+	0.997640i	$0.521875\pi$
$152$	0	0
$153$	12.5459	1.01428
$154$	0	0
$155$	13.6850	1.09920
$156$	0	0
$157$	−1.12522	−0.0898023	−0.0449012	−	0.998991i	$-0.514297\pi$
$157$	−1.12522	−0.0898023	−0.0449012	+	0.998991i	$0.514297\pi$
$158$	0	0
$159$	−23.2742	−1.84576
$160$	0	0
$161$	3.14719	0.248033
$162$	0	0
$163$	−10.1769	−0.797114	−0.398557	−	0.917144i	$-0.630489\pi$
$163$	−10.1769	−0.797114	−0.398557	+	0.917144i	$0.630489\pi$
$164$	0	0
$165$	−27.3145	−2.12643
$166$	0	0
$167$	0.563580	0.0436111	0.0218056	−	0.999762i	$-0.493059\pi$
$167$	0.563580	0.0436111	0.0218056	+	0.999762i	$0.493059\pi$
$168$	0	0
$169$	−3.61518	−0.278091
$170$	0	0
$171$	5.48033	0.419091
$172$	0	0
$173$	15.1488	1.15174	0.575869	−	0.817542i	$-0.304664\pi$
$173$	15.1488	1.15174	0.575869	+	0.817542i	$0.304664\pi$
$174$	0	0
$175$	−2.49209	−0.188384
$176$	0	0
$177$	−4.94405	−0.371618
$178$	0	0
$179$	−22.6839	−1.69548	−0.847738	−	0.530415i	$-0.822036\pi$
$179$	−22.6839	−1.69548	−0.847738	+	0.530415i	$0.822036\pi$
$180$	0	0
$181$	−23.4230	−1.74102	−0.870508	−	0.492154i	$-0.836210\pi$
$181$	−23.4230	−1.74102	−0.870508	+	0.492154i	$0.836210\pi$
$182$	0	0
$183$	−16.4195	−1.21377
$184$	0	0
$185$	−10.1525	−0.746425
$186$	0	0
$187$	35.0996	2.56674
$188$	0	0
$189$	−5.65092	−0.411044
$190$	0	0
$191$	15.6670	1.13363	0.566813	−	0.823847i	$-0.308176\pi$
$191$	15.6670	1.13363	0.566813	+	0.823847i	$0.308176\pi$
$192$	0	0
$193$	10.9595	0.788882	0.394441	−	0.918921i	$-0.370938\pi$
$193$	10.9595	0.788882	0.394441	+	0.918921i	$0.370938\pi$
$194$	0	0
$195$	−14.0840	−1.00857
$196$	0	0
$197$	−7.32665	−0.522002	−0.261001	−	0.965339i	$-0.584053\pi$
$197$	−7.32665	−0.522002	−0.261001	+	0.965339i	$0.584053\pi$
$198$	0	0
$199$	1.96696	0.139434	0.0697172	−	0.997567i	$-0.477790\pi$
$199$	1.96696	0.139434	0.0697172	+	0.997567i	$0.477790\pi$
$200$	0	0
$201$	15.8982	1.12137
$202$	0	0
$203$	19.1243	1.34226
$204$	0	0
$205$	17.3216	1.20979
$206$	0	0
$207$	2.34614	0.163068
$208$	0	0
$209$	15.3322	1.06055
$210$	0	0
$211$	17.9708	1.23716	0.618579	−	0.785722i	$-0.287709\pi$
$211$	17.9708	1.23716	0.618579	+	0.785722i	$0.287709\pi$
$212$	0	0
$213$	26.7135	1.83038
$214$	0	0
$215$	10.6568	0.726787
$216$	0	0
$217$	−19.1943	−1.30300
$218$	0	0
$219$	−18.1391	−1.22573
$220$	0	0
$221$	18.0981	1.21741
$222$	0	0
$223$	0.871055	0.0583302	0.0291651	−	0.999575i	$-0.490715\pi$
$223$	0.871055	0.0583302	0.0291651	+	0.999575i	$0.490715\pi$
$224$	0	0
$225$	−1.85779	−0.123852
$226$	0	0
$227$	−12.0084	−0.797024	−0.398512	−	0.917163i	$-0.630473\pi$
$227$	−12.0084	−0.797024	−0.398512	+	0.917163i	$0.630473\pi$
$228$	0	0
$229$	−2.26262	−0.149518	−0.0747590	−	0.997202i	$-0.523819\pi$
$229$	−2.26262	−0.149518	−0.0747590	+	0.997202i	$0.523819\pi$
$230$	0	0
$231$	38.3109	2.52067
$232$	0	0
$233$	−5.68168	−0.372219	−0.186110	−	0.982529i	$-0.559588\pi$
$233$	−5.68168	−0.372219	−0.186110	+	0.982529i	$0.559588\pi$
$234$	0	0
$235$	−21.5302	−1.40448
$236$	0	0
$237$	26.4453	1.71780
$238$	0	0
$239$	−12.2544	−0.792670	−0.396335	−	0.918106i	$-0.629718\pi$
$239$	−12.2544	−0.792670	−0.396335	+	0.918106i	$0.629718\pi$
$240$	0	0
$241$	25.1614	1.62079	0.810393	−	0.585886i	$-0.199253\pi$
$241$	25.1614	1.62079	0.810393	+	0.585886i	$0.199253\pi$
$242$	0	0
$243$	−18.6335	−1.19534
$244$	0	0
$245$	−2.26515	−0.144715
$246$	0	0
$247$	7.90564	0.503024
$248$	0	0
$249$	−10.6166	−0.672801
$250$	0	0
$251$	14.6366	0.923854	0.461927	−	0.886918i	$-0.347158\pi$
$251$	14.6366	0.923854	0.461927	+	0.886918i	$0.347158\pi$
$252$	0	0
$253$	6.56378	0.412661
$254$	0	0
$255$	−27.1601	−1.70083
$256$	0	0
$257$	−2.45719	−0.153275	−0.0766377	−	0.997059i	$-0.524418\pi$
$257$	−2.45719	−0.153275	−0.0766377	+	0.997059i	$0.524418\pi$
$258$	0	0
$259$	14.2397	0.884812
$260$	0	0
$261$	14.2566	0.882461
$262$	0	0
$263$	−13.7817	−0.849814	−0.424907	−	0.905237i	$-0.639693\pi$
$263$	−13.7817	−0.849814	−0.424907	+	0.905237i	$0.639693\pi$
$264$	0	0
$265$	20.8837	1.28287
$266$	0	0
$267$	−28.1015	−1.71978
$268$	0	0
$269$	8.31451	0.506945	0.253472	−	0.967343i	$-0.418427\pi$
$269$	8.31451	0.506945	0.253472	+	0.967343i	$0.418427\pi$
$270$	0	0
$271$	11.2719	0.684716	0.342358	−	0.939570i	$-0.388774\pi$
$271$	11.2719	0.684716	0.342358	+	0.939570i	$0.388774\pi$
$272$	0	0
$273$	19.7539	1.19556
$274$	0	0
$275$	−5.19750	−0.313421
$276$	0	0
$277$	5.85187	0.351605	0.175803	−	0.984425i	$-0.443748\pi$
$277$	5.85187	0.351605	0.175803	+	0.984425i	$0.443748\pi$
$278$	0	0
$279$	−14.3088	−0.856647
$280$	0	0
$281$	−2.72235	−0.162402	−0.0812010	−	0.996698i	$-0.525876\pi$
$281$	−2.72235	−0.162402	−0.0812010	+	0.996698i	$0.525876\pi$
$282$	0	0
$283$	26.8627	1.59682	0.798410	−	0.602114i	$-0.205675\pi$
$283$	26.8627	1.59682	0.798410	+	0.602114i	$0.205675\pi$
$284$	0	0
$285$	−11.8641	−0.702770
$286$	0	0
$287$	−24.2949	−1.43409
$288$	0	0
$289$	17.9012	1.05301
$290$	0	0
$291$	−27.1253	−1.59012
$292$	0	0
$293$	3.57249	0.208707	0.104353	−	0.994540i	$-0.466723\pi$
$293$	3.57249	0.208707	0.104353	+	0.994540i	$0.466723\pi$
$294$	0	0
$295$	4.43624	0.258288
$296$	0	0
$297$	−11.7856	−0.683867
$298$	0	0
$299$	3.38442	0.195726
$300$	0	0
$301$	−14.9470	−0.861533
$302$	0	0
$303$	14.8597	0.853667
$304$	0	0
$305$	14.7330	0.843612
$306$	0	0
$307$	31.1639	1.77862	0.889310	−	0.457306i	$-0.151185\pi$
$307$	31.1639	1.77862	0.889310	+	0.457306i	$0.151185\pi$
$308$	0	0
$309$	36.3131	2.06578
$310$	0	0
$311$	−0.829003	−0.0470085	−0.0235042	−	0.999724i	$-0.507482\pi$
$311$	−0.829003	−0.0470085	−0.0235042	+	0.999724i	$0.507482\pi$
$312$	0	0
$313$	14.9858	0.847046	0.423523	−	0.905885i	$-0.360793\pi$
$313$	14.9858	0.847046	0.423523	+	0.905885i	$0.360793\pi$
$314$	0	0
$315$	−12.2873	−0.692310
$316$	0	0
$317$	−30.1502	−1.69340	−0.846701	−	0.532069i	$-0.821415\pi$
$317$	−30.1502	−1.69340	−0.846701	+	0.532069i	$0.821415\pi$
$318$	0	0
$319$	39.8855	2.23316
$320$	0	0
$321$	−36.3890	−2.03104
$322$	0	0
$323$	15.2456	0.848288
$324$	0	0
$325$	−2.67995	−0.148657
$326$	0	0
$327$	−17.1657	−0.949263
$328$	0	0
$329$	30.1980	1.66487
$330$	0	0
$331$	4.10175	0.225452	0.112726	−	0.993626i	$-0.464042\pi$
$331$	4.10175	0.225452	0.112726	+	0.993626i	$0.464042\pi$
$332$	0	0
$333$	10.6153	0.581715
$334$	0	0
$335$	−14.2653	−0.779395
$336$	0	0
$337$	15.0504	0.819848	0.409924	−	0.912120i	$-0.365555\pi$
$337$	15.0504	0.819848	0.409924	+	0.912120i	$0.365555\pi$
$338$	0	0
$339$	−13.4734	−0.731776
$340$	0	0
$341$	−40.0317	−2.16784
$342$	0	0
$343$	−16.7640	−0.905173
$344$	0	0
$345$	−5.07906	−0.273448
$346$	0	0
$347$	−16.0676	−0.862555	−0.431278	−	0.902219i	$-0.641937\pi$
$347$	−16.0676	−0.862555	−0.431278	+	0.902219i	$0.641937\pi$
$348$	0	0
$349$	10.9027	0.583607	0.291803	−	0.956478i	$-0.405745\pi$
$349$	10.9027	0.583607	0.291803	+	0.956478i	$0.405745\pi$
$350$	0	0
$351$	−6.07689	−0.324360
$352$	0	0
$353$	−9.01265	−0.479695	−0.239848	−	0.970811i	$-0.577097\pi$
$353$	−9.01265	−0.479695	−0.239848	+	0.970811i	$0.577097\pi$
$354$	0	0
$355$	−23.9697	−1.27218
$356$	0	0
$357$	38.0944	2.01617
$358$	0	0
$359$	34.8170	1.83757	0.918784	−	0.394760i	$-0.129172\pi$
$359$	34.8170	1.83757	0.918784	+	0.394760i	$0.129172\pi$
$360$	0	0
$361$	−12.3404	−0.649495
$362$	0	0
$363$	55.0021	2.88686
$364$	0	0
$365$	16.2760	0.851926
$366$	0	0
$367$	−35.2245	−1.83871	−0.919353	−	0.393434i	$-0.871287\pi$
$367$	−35.2245	−1.83871	−0.919353	+	0.393434i	$0.871287\pi$
$368$	0	0
$369$	−18.1112	−0.942832
$370$	0	0
$371$	−29.2911	−1.52072
$372$	0	0
$373$	−19.8146	−1.02596	−0.512981	−	0.858400i	$-0.671459\pi$
$373$	−19.8146	−1.02596	−0.512981	+	0.858400i	$0.671459\pi$
$374$	0	0
$375$	27.0088	1.39473
$376$	0	0
$377$	20.5658	1.05919
$378$	0	0
$379$	19.8202	1.01810	0.509048	−	0.860738i	$-0.329997\pi$
$379$	19.8202	1.01810	0.509048	+	0.860738i	$0.329997\pi$
$380$	0	0
$381$	8.61822	0.441525
$382$	0	0
$383$	14.2166	0.726434	0.363217	−	0.931705i	$-0.381678\pi$
$383$	14.2166	0.726434	0.363217	+	0.931705i	$0.381678\pi$
$384$	0	0
$385$	−34.3760	−1.75196
$386$	0	0
$387$	−11.1426	−0.566410
$388$	0	0
$389$	22.8295	1.15750	0.578751	−	0.815504i	$-0.303540\pi$
$389$	22.8295	1.15750	0.578751	+	0.815504i	$0.303540\pi$
$390$	0	0
$391$	6.52668	0.330068
$392$	0	0
$393$	−40.7778	−2.05697
$394$	0	0
$395$	−23.7291	−1.19394
$396$	0	0
$397$	17.5329	0.879951	0.439975	−	0.898010i	$-0.354987\pi$
$397$	17.5329	0.879951	0.439975	+	0.898010i	$0.354987\pi$
$398$	0	0
$399$	16.6404	0.833064
$400$	0	0
$401$	−30.2366	−1.50994	−0.754971	−	0.655758i	$-0.772349\pi$
$401$	−30.2366	−1.50994	−0.754971	+	0.655758i	$0.772349\pi$
$402$	0	0
$403$	−20.6412	−1.02821
$404$	0	0
$405$	22.0594	1.09614
$406$	0	0
$407$	29.6983	1.47209
$408$	0	0
$409$	20.6540	1.02127	0.510636	−	0.859797i	$-0.329410\pi$
$409$	20.6540	1.02127	0.510636	+	0.859797i	$0.329410\pi$
$410$	0	0
$411$	25.3386	1.24986
$412$	0	0
$413$	−6.22221	−0.306175
$414$	0	0
$415$	9.52618	0.467622
$416$	0	0
$417$	41.0798	2.01169
$418$	0	0
$419$	35.0027	1.70999	0.854996	−	0.518634i	$-0.173559\pi$
$419$	35.0027	1.70999	0.854996	+	0.518634i	$0.173559\pi$
$420$	0	0
$421$	−17.6641	−0.860894	−0.430447	−	0.902616i	$-0.641644\pi$
$421$	−17.6641	−0.860894	−0.430447	+	0.902616i	$0.641644\pi$
$422$	0	0
$423$	22.5118	1.09456
$424$	0	0
$425$	−5.16813	−0.250691
$426$	0	0
$427$	−20.6643	−1.00002
$428$	0	0
$429$	41.1988	1.98910
$430$	0	0
$431$	1.44029	0.0693761	0.0346881	−	0.999398i	$-0.488956\pi$
$431$	1.44029	0.0693761	0.0346881	+	0.999398i	$0.488956\pi$
$432$	0	0
$433$	−15.4980	−0.744788	−0.372394	−	0.928075i	$-0.621463\pi$
$433$	−15.4980	−0.744788	−0.372394	+	0.928075i	$0.621463\pi$
$434$	0	0
$435$	−30.8635	−1.47979
$436$	0	0
$437$	2.85099	0.136381
$438$	0	0
$439$	13.9160	0.664174	0.332087	−	0.943249i	$-0.392247\pi$
$439$	13.9160	0.664174	0.332087	+	0.943249i	$0.392247\pi$
$440$	0	0
$441$	2.36841	0.112781
$442$	0	0
$443$	−29.3742	−1.39561	−0.697806	−	0.716287i	$-0.745840\pi$
$443$	−29.3742	−1.39561	−0.697806	+	0.716287i	$0.745840\pi$
$444$	0	0
$445$	25.2152	1.19531
$446$	0	0
$447$	−13.8267	−0.653980
$448$	0	0
$449$	−11.9746	−0.565118	−0.282559	−	0.959250i	$-0.591183\pi$
$449$	−11.9746	−0.565118	−0.282559	+	0.959250i	$0.591183\pi$
$450$	0	0
$451$	−50.6695	−2.38593
$452$	0	0
$453$	−3.82001	−0.179480
$454$	0	0
$455$	−17.7250	−0.830960
$456$	0	0
$457$	−19.0554	−0.891376	−0.445688	−	0.895188i	$-0.647041\pi$
$457$	−19.0554	−0.891376	−0.445688	+	0.895188i	$0.647041\pi$
$458$	0	0
$459$	−11.7190	−0.546994
$460$	0	0
$461$	19.9012	0.926892	0.463446	−	0.886125i	$-0.346613\pi$
$461$	19.9012	0.926892	0.463446	+	0.886125i	$0.346613\pi$
$462$	0	0
$463$	−14.2887	−0.664053	−0.332027	−	0.943270i	$-0.607732\pi$
$463$	−14.2887	−0.664053	−0.332027	+	0.943270i	$0.607732\pi$
$464$	0	0
$465$	30.9766	1.43650
$466$	0	0
$467$	12.2255	0.565728	0.282864	−	0.959160i	$-0.408715\pi$
$467$	12.2255	0.565728	0.282864	+	0.959160i	$0.408715\pi$
$468$	0	0
$469$	20.0083	0.923896
$470$	0	0
$471$	−2.54699	−0.117359
$472$	0	0
$473$	−31.1735	−1.43336
$474$	0	0
$475$	−2.25755	−0.103583
$476$	0	0
$477$	−21.8357	−0.999789
$478$	0	0
$479$	−31.2654	−1.42855	−0.714277	−	0.699863i	$-0.753244\pi$
$479$	−31.2654	−1.42855	−0.714277	+	0.699863i	$0.753244\pi$
$480$	0	0
$481$	15.3131	0.698217
$482$	0	0
$483$	7.12382	0.324145
$484$	0	0
$485$	24.3393	1.10519
$486$	0	0
$487$	−20.0752	−0.909694	−0.454847	−	0.890570i	$-0.650306\pi$
$487$	−20.0752	−0.909694	−0.454847	+	0.890570i	$0.650306\pi$
$488$	0	0
$489$	−23.0358	−1.04172
$490$	0	0
$491$	−24.5106	−1.10615	−0.553075	−	0.833132i	$-0.686546\pi$
$491$	−24.5106	−1.10615	−0.553075	+	0.833132i	$0.686546\pi$
$492$	0	0
$493$	39.6601	1.78620
$494$	0	0
$495$	−25.6263	−1.15182
$496$	0	0
$497$	33.6195	1.50804
$498$	0	0
$499$	−37.9735	−1.69993	−0.849965	−	0.526840i	$-0.823377\pi$
$499$	−37.9735	−1.69993	−0.849965	+	0.526840i	$0.823377\pi$
$500$	0	0
$501$	1.27569	0.0569936
$502$	0	0
$503$	−0.416077	−0.0185520	−0.00927599	−	0.999957i	$-0.502953\pi$
$503$	−0.416077	−0.0185520	−0.00927599	+	0.999957i	$0.502953\pi$
$504$	0	0
$505$	−13.3334	−0.593330
$506$	0	0
$507$	−8.18314	−0.363426
$508$	0	0
$509$	−24.0031	−1.06392	−0.531959	−	0.846770i	$-0.678544\pi$
$509$	−24.0031	−1.06392	−0.531959	+	0.846770i	$0.678544\pi$
$510$	0	0
$511$	−22.8285	−1.00987
$512$	0	0
$513$	−5.11909	−0.226013
$514$	0	0
$515$	−32.5833	−1.43579
$516$	0	0
$517$	62.9809	2.76990
$518$	0	0
$519$	34.2899	1.50516
$520$	0	0
$521$	−14.4262	−0.632025	−0.316012	−	0.948755i	$-0.602344\pi$
$521$	−14.4262	−0.632025	−0.316012	+	0.948755i	$0.602344\pi$
$522$	0	0
$523$	41.7897	1.82734	0.913669	−	0.406460i	$-0.133237\pi$
$523$	41.7897	1.82734	0.913669	+	0.406460i	$0.133237\pi$
$524$	0	0
$525$	−5.64097	−0.246192
$526$	0	0
$527$	−39.8054	−1.73395
$528$	0	0
$529$	−21.7795	−0.946934
$530$	0	0
$531$	−4.63848	−0.201293
$532$	0	0
$533$	−26.1263	−1.13166
$534$	0	0
$535$	32.6515	1.41165
$536$	0	0
$537$	−51.3461	−2.21575
$538$	0	0
$539$	6.62607	0.285405
$540$	0	0
$541$	31.1607	1.33970	0.669852	−	0.742495i	$-0.266358\pi$
$541$	31.1607	1.33970	0.669852	+	0.742495i	$0.266358\pi$
$542$	0	0
$543$	−53.0190	−2.27526
$544$	0	0
$545$	15.4026	0.659773
$546$	0	0
$547$	23.8696	1.02059	0.510295	−	0.859999i	$-0.329536\pi$
$547$	23.8696	1.02059	0.510295	+	0.859999i	$0.329536\pi$
$548$	0	0
$549$	−15.4047	−0.657456
$550$	0	0
$551$	17.3244	0.738043
$552$	0	0
$553$	33.2820	1.41529
$554$	0	0
$555$	−22.9806	−0.975472
$556$	0	0
$557$	4.87995	0.206770	0.103385	−	0.994641i	$-0.467033\pi$
$557$	4.87995	0.206770	0.103385	+	0.994641i	$0.467033\pi$
$558$	0	0
$559$	−16.0737	−0.679847
$560$	0	0
$561$	79.4496	3.35437
$562$	0	0
$563$	34.0486	1.43498	0.717488	−	0.696571i	$-0.245292\pi$
$563$	34.0486	1.43498	0.717488	+	0.696571i	$0.245292\pi$
$564$	0	0
$565$	12.0896	0.508612
$566$	0	0
$567$	−30.9402	−1.29937
$568$	0	0
$569$	−37.1262	−1.55641	−0.778205	−	0.628010i	$-0.783870\pi$
$569$	−37.1262	−1.55641	−0.778205	+	0.628010i	$0.783870\pi$
$570$	0	0
$571$	2.54908	0.106676	0.0533379	−	0.998577i	$-0.483014\pi$
$571$	2.54908	0.106676	0.0533379	+	0.998577i	$0.483014\pi$
$572$	0	0
$573$	35.4630	1.48149
$574$	0	0
$575$	−0.966463	−0.0403043
$576$	0	0
$577$	7.33113	0.305199	0.152599	−	0.988288i	$-0.451236\pi$
$577$	7.33113	0.305199	0.152599	+	0.988288i	$0.451236\pi$
$578$	0	0
$579$	24.8073	1.03096
$580$	0	0
$581$	−13.3613	−0.554319
$582$	0	0
$583$	−61.0895	−2.53007
$584$	0	0
$585$	−13.2135	−0.546310
$586$	0	0
$587$	−35.0156	−1.44525	−0.722625	−	0.691240i	$-0.757065\pi$
$587$	−35.0156	−1.44525	−0.722625	+	0.691240i	$0.757065\pi$
$588$	0	0
$589$	−17.3879	−0.716454
$590$	0	0
$591$	−16.5842	−0.682183
$592$	0	0
$593$	−23.5914	−0.968781	−0.484391	−	0.874852i	$-0.660959\pi$
$593$	−23.5914	−0.968781	−0.484391	+	0.874852i	$0.660959\pi$
$594$	0	0
$595$	−34.1817	−1.40131
$596$	0	0
$597$	4.45232	0.182221
$598$	0	0
$599$	−26.1880	−1.07001	−0.535006	−	0.844848i	$-0.679691\pi$
$599$	−26.1880	−1.07001	−0.535006	+	0.844848i	$0.679691\pi$
$600$	0	0
$601$	1.98654	0.0810328	0.0405164	−	0.999179i	$-0.487100\pi$
$601$	1.98654	0.0810328	0.0405164	+	0.999179i	$0.487100\pi$
$602$	0	0
$603$	14.9156	0.607410
$604$	0	0
$605$	−49.3528	−2.00648
$606$	0	0
$607$	8.92706	0.362338	0.181169	−	0.983452i	$-0.442012\pi$
$607$	8.92706	0.362338	0.181169	+	0.983452i	$0.442012\pi$
$608$	0	0
$609$	43.2887	1.75415
$610$	0	0
$611$	32.4743	1.31377
$612$	0	0
$613$	−28.3094	−1.14341	−0.571703	−	0.820460i	$-0.693717\pi$
$613$	−28.3094	−1.14341	−0.571703	+	0.820460i	$0.693717\pi$
$614$	0	0
$615$	39.2082	1.58103
$616$	0	0
$617$	−19.8166	−0.797787	−0.398894	−	0.916997i	$-0.630606\pi$
$617$	−19.8166	−0.797787	−0.398894	+	0.916997i	$0.630606\pi$
$618$	0	0
$619$	5.77849	0.232257	0.116129	−	0.993234i	$-0.462952\pi$
$619$	5.77849	0.232257	0.116129	+	0.993234i	$0.462952\pi$
$620$	0	0
$621$	−2.19149	−0.0879416
$622$	0	0
$623$	−35.3664	−1.41693
$624$	0	0
$625$	−19.8607	−0.794427
$626$	0	0
$627$	34.7053	1.38600
$628$	0	0
$629$	29.5305	1.17746
$630$	0	0
$631$	−48.0493	−1.91281	−0.956406	−	0.292040i	$-0.905666\pi$
$631$	−48.0493	−1.91281	−0.956406	+	0.292040i	$0.905666\pi$
$632$	0	0
$633$	40.6777	1.61679
$634$	0	0
$635$	−7.73303	−0.306876
$636$	0	0
$637$	3.41654	0.135368
$638$	0	0
$639$	25.0624	0.991455
$640$	0	0
$641$	28.2296	1.11500	0.557501	−	0.830176i	$-0.311760\pi$
$641$	28.2296	1.11500	0.557501	+	0.830176i	$0.311760\pi$
$642$	0	0
$643$	30.6110	1.20718	0.603589	−	0.797295i	$-0.293737\pi$
$643$	30.6110	1.20718	0.603589	+	0.797295i	$0.293737\pi$
$644$	0	0
$645$	24.1221	0.949808
$646$	0	0
$647$	−34.7903	−1.36775	−0.683873	−	0.729601i	$-0.739706\pi$
$647$	−34.7903	−1.36775	−0.683873	+	0.729601i	$0.739706\pi$
$648$	0	0
$649$	−12.9770	−0.509393
$650$	0	0
$651$	−43.4473	−1.70283
$652$	0	0
$653$	−16.4855	−0.645128	−0.322564	−	0.946548i	$-0.604545\pi$
$653$	−16.4855	−0.645128	−0.322564	+	0.946548i	$0.604545\pi$
$654$	0	0
$655$	36.5894	1.42967
$656$	0	0
$657$	−17.0180	−0.663936
$658$	0	0
$659$	−3.56024	−0.138687	−0.0693437	−	0.997593i	$-0.522091\pi$
$659$	−3.56024	−0.138687	−0.0693437	+	0.997593i	$0.522091\pi$
$660$	0	0
$661$	−35.1172	−1.36590	−0.682950	−	0.730466i	$-0.739303\pi$
$661$	−35.1172	−1.36590	−0.682950	+	0.730466i	$0.739303\pi$
$662$	0	0
$663$	40.9659	1.59099
$664$	0	0
$665$	−14.9313	−0.579010
$666$	0	0
$667$	7.41661	0.287172
$668$	0	0
$669$	1.97167	0.0762293
$670$	0	0
$671$	−43.0975	−1.66376
$672$	0	0
$673$	11.6622	0.449546	0.224773	−	0.974411i	$-0.427836\pi$
$673$	11.6622	0.449546	0.224773	+	0.974411i	$0.427836\pi$
$674$	0	0
$675$	1.73533	0.0667928
$676$	0	0
$677$	−19.2277	−0.738980	−0.369490	−	0.929235i	$-0.620468\pi$
$677$	−19.2277	−0.738980	−0.369490	+	0.929235i	$0.620468\pi$
$678$	0	0
$679$	−34.1379	−1.31009
$680$	0	0
$681$	−27.1815	−1.04160
$682$	0	0
$683$	3.35587	0.128409	0.0642043	−	0.997937i	$-0.479549\pi$
$683$	3.35587	0.128409	0.0642043	+	0.997937i	$0.479549\pi$
$684$	0	0
$685$	−22.7361	−0.868700
$686$	0	0
$687$	−5.12154	−0.195399
$688$	0	0
$689$	−31.4991	−1.20002
$690$	0	0
$691$	−26.9912	−1.02679	−0.513397	−	0.858152i	$-0.671613\pi$
$691$	−26.9912	−1.02679	−0.513397	+	0.858152i	$0.671613\pi$
$692$	0	0
$693$	35.9431	1.36536
$694$	0	0
$695$	−36.8605	−1.39820
$696$	0	0
$697$	−50.3832	−1.90840
$698$	0	0
$699$	−12.8608	−0.486438
$700$	0	0
$701$	9.34446	0.352935	0.176468	−	0.984306i	$-0.443533\pi$
$701$	9.34446	0.352935	0.176468	+	0.984306i	$0.443533\pi$
$702$	0	0
$703$	12.8995	0.486515
$704$	0	0
$705$	−48.7347	−1.83546
$706$	0	0
$707$	18.7013	0.703334
$708$	0	0
$709$	16.9833	0.637820	0.318910	−	0.947785i	$-0.396683\pi$
$709$	16.9833	0.637820	0.318910	+	0.947785i	$0.396683\pi$
$710$	0	0
$711$	24.8108	0.930478
$712$	0	0
$713$	−7.44378	−0.278772
$714$	0	0
$715$	−36.9672	−1.38250
$716$	0	0
$717$	−27.7384	−1.03591
$718$	0	0
$719$	−19.0904	−0.711951	−0.355976	−	0.934495i	$-0.615851\pi$
$719$	−19.0904	−0.711951	−0.355976	+	0.934495i	$0.615851\pi$
$720$	0	0
$721$	45.7008	1.70199
$722$	0	0
$723$	56.9540	2.11814
$724$	0	0
$725$	−5.87282	−0.218111
$726$	0	0
$727$	−13.6110	−0.504803	−0.252402	−	0.967623i	$-0.581220\pi$
$727$	−13.6110	−0.504803	−0.252402	+	0.967623i	$0.581220\pi$
$728$	0	0
$729$	−9.59471	−0.355360
$730$	0	0
$731$	−30.9973	−1.14648
$732$	0	0
$733$	14.2251	0.525416	0.262708	−	0.964875i	$-0.415384\pi$
$733$	14.2251	0.525416	0.262708	+	0.964875i	$0.415384\pi$
$734$	0	0
$735$	−5.12726	−0.189122
$736$	0	0
$737$	41.7292	1.53711
$738$	0	0
$739$	41.4200	1.52366	0.761830	−	0.647777i	$-0.224301\pi$
$739$	41.4200	1.52366	0.761830	+	0.647777i	$0.224301\pi$
$740$	0	0
$741$	17.8948	0.657382
$742$	0	0
$743$	2.14855	0.0788227	0.0394114	−	0.999223i	$-0.487452\pi$
$743$	2.14855	0.0788227	0.0394114	+	0.999223i	$0.487452\pi$
$744$	0	0
$745$	12.4065	0.454540
$746$	0	0
$747$	−9.96045	−0.364434
$748$	0	0
$749$	−45.7964	−1.67336
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	33.1306	1.20735
$754$	0	0
$755$	3.42765	0.124745
$756$	0	0
$757$	11.1801	0.406347	0.203173	−	0.979143i	$-0.434875\pi$
$757$	11.1801	0.406347	0.203173	+	0.979143i	$0.434875\pi$
$758$	0	0
$759$	14.8574	0.539290
$760$	0	0
$761$	−26.5107	−0.961011	−0.480506	−	0.876992i	$-0.659547\pi$
$761$	−26.5107	−0.961011	−0.480506	+	0.876992i	$0.659547\pi$
$762$	0	0
$763$	−21.6034	−0.782095
$764$	0	0
$765$	−25.4815	−0.921286
$766$	0	0
$767$	−6.69123	−0.241606
$768$	0	0
$769$	−25.7752	−0.929477	−0.464739	−	0.885448i	$-0.653852\pi$
$769$	−25.7752	−0.929477	−0.464739	+	0.885448i	$0.653852\pi$
$770$	0	0
$771$	−5.56197	−0.200309
$772$	0	0
$773$	−10.7241	−0.385719	−0.192860	−	0.981226i	$-0.561776\pi$
$773$	−10.7241	−0.385719	−0.192860	+	0.981226i	$0.561776\pi$
$774$	0	0
$775$	5.89434	0.211731
$776$	0	0
$777$	32.2322	1.15633
$778$	0	0
$779$	−22.0084	−0.788534
$780$	0	0
$781$	70.1169	2.50898
$782$	0	0
$783$	−13.3169	−0.475906
$784$	0	0
$785$	2.28539	0.0815689
$786$	0	0
$787$	9.80761	0.349603	0.174802	−	0.984604i	$-0.444072\pi$
$787$	9.80761	0.349603	0.174802	+	0.984604i	$0.444072\pi$
$788$	0	0
$789$	−31.1955	−1.11059
$790$	0	0
$791$	−16.9566	−0.602909
$792$	0	0
$793$	−22.2220	−0.789127
$794$	0	0
$795$	47.2712	1.67654
$796$	0	0
$797$	14.4353	0.511326	0.255663	−	0.966766i	$-0.417706\pi$
$797$	14.4353	0.511326	0.255663	+	0.966766i	$0.417706\pi$
$798$	0	0
$799$	62.6249	2.21551
$800$	0	0
$801$	−26.3647	−0.931550
$802$	0	0
$803$	−47.6111	−1.68016
$804$	0	0
$805$	−6.39212	−0.225293
$806$	0	0
$807$	18.8203	0.662506
$808$	0	0
$809$	−17.7583	−0.624348	−0.312174	−	0.950025i	$-0.601057\pi$
$809$	−17.7583	−0.624348	−0.312174	+	0.950025i	$0.601057\pi$
$810$	0	0
$811$	−42.2433	−1.48336	−0.741682	−	0.670752i	$-0.765972\pi$
$811$	−42.2433	−1.48336	−0.741682	+	0.670752i	$0.765972\pi$
$812$	0	0
$813$	25.5144	0.894828
$814$	0	0
$815$	20.6698	0.724032
$816$	0	0
$817$	−13.5403	−0.473715
$818$	0	0
$819$	18.5330	0.647596
$820$	0	0
$821$	−13.0015	−0.453756	−0.226878	−	0.973923i	$-0.572852\pi$
$821$	−13.0015	−0.453756	−0.226878	+	0.973923i	$0.572852\pi$
$822$	0	0
$823$	−29.7993	−1.03874	−0.519369	−	0.854550i	$-0.673833\pi$
$823$	−29.7993	−1.03874	−0.519369	+	0.854550i	$0.673833\pi$
$824$	0	0
$825$	−11.7648	−0.409598
$826$	0	0
$827$	32.0935	1.11600	0.557999	−	0.829841i	$-0.311569\pi$
$827$	32.0935	1.11600	0.557999	+	0.829841i	$0.311569\pi$
$828$	0	0
$829$	24.2528	0.842334	0.421167	−	0.906983i	$-0.361621\pi$
$829$	24.2528	0.842334	0.421167	+	0.906983i	$0.361621\pi$
$830$	0	0
$831$	13.2460	0.459498
$832$	0	0
$833$	6.58862	0.228282
$834$	0	0
$835$	−1.14466	−0.0396127
$836$	0	0
$837$	13.3657	0.461985
$838$	0	0
$839$	−7.51319	−0.259384	−0.129692	−	0.991554i	$-0.541399\pi$
$839$	−7.51319	−0.259384	−0.129692	+	0.991554i	$0.541399\pi$
$840$	0	0
$841$	16.0679	0.554064
$842$	0	0
$843$	−6.16217	−0.212237
$844$	0	0
$845$	7.34264	0.252595
$846$	0	0
$847$	69.2215	2.37848
$848$	0	0
$849$	60.8049	2.08682
$850$	0	0
$851$	5.52232	0.189303
$852$	0	0
$853$	−7.77565	−0.266233	−0.133117	−	0.991100i	$-0.542498\pi$
$853$	−7.77565	−0.266233	−0.133117	+	0.991100i	$0.542498\pi$
$854$	0	0
$855$	−11.1309	−0.380667
$856$	0	0
$857$	−31.5852	−1.07893	−0.539466	−	0.842008i	$-0.681374\pi$
$857$	−31.5852	−1.07893	−0.539466	+	0.842008i	$0.681374\pi$
$858$	0	0
$859$	47.5576	1.62264	0.811321	−	0.584600i	$-0.198749\pi$
$859$	47.5576	1.62264	0.811321	+	0.584600i	$0.198749\pi$
$860$	0	0
$861$	−54.9928	−1.87415
$862$	0	0
$863$	29.4008	1.00082	0.500408	−	0.865790i	$-0.333183\pi$
$863$	29.4008	1.00082	0.500408	+	0.865790i	$0.333183\pi$
$864$	0	0
$865$	−30.7680	−1.04614
$866$	0	0
$867$	40.5203	1.37614
$868$	0	0
$869$	69.4129	2.35467
$870$	0	0
$871$	21.5165	0.729058
$872$	0	0
$873$	−25.4488	−0.861313
$874$	0	0
$875$	33.9912	1.14911
$876$	0	0
$877$	38.9880	1.31653	0.658265	−	0.752786i	$-0.271291\pi$
$877$	38.9880	1.31653	0.658265	+	0.752786i	$0.271291\pi$
$878$	0	0
$879$	8.08649	0.272751
$880$	0	0
$881$	−24.9847	−0.841755	−0.420878	−	0.907117i	$-0.638278\pi$
$881$	−24.9847	−0.841755	−0.420878	+	0.907117i	$0.638278\pi$
$882$	0	0
$883$	28.7179	0.966433	0.483217	−	0.875501i	$-0.339468\pi$
$883$	28.7179	0.966433	0.483217	+	0.875501i	$0.339468\pi$
$884$	0	0
$885$	10.0416	0.337546
$886$	0	0
$887$	18.0783	0.607011	0.303506	−	0.952830i	$-0.401843\pi$
$887$	18.0783	0.607011	0.303506	+	0.952830i	$0.401843\pi$
$888$	0	0
$889$	10.8462	0.363771
$890$	0	0
$891$	−64.5289	−2.16180
$892$	0	0
$893$	27.3559	0.915430
$894$	0	0
$895$	46.0723	1.54003
$896$	0	0
$897$	7.66081	0.255787
$898$	0	0
$899$	−45.2330	−1.50860
$900$	0	0
$901$	−60.7443	−2.02369
$902$	0	0
$903$	−33.8333	−1.12590
$904$	0	0
$905$	47.5734	1.58139
$906$	0	0
$907$	9.19940	0.305461	0.152730	−	0.988268i	$-0.451193\pi$
$907$	9.19940	0.305461	0.152730	+	0.988268i	$0.451193\pi$
$908$	0	0
$909$	13.9413	0.462403
$910$	0	0
$911$	5.85991	0.194148	0.0970738	−	0.995277i	$-0.469052\pi$
$911$	5.85991	0.194148	0.0970738	+	0.995277i	$0.469052\pi$
$912$	0	0
$913$	−27.8662	−0.922238
$914$	0	0
$915$	33.3490	1.10248
$916$	0	0
$917$	−51.3198	−1.69473
$918$	0	0
$919$	31.2584	1.03112	0.515560	−	0.856854i	$-0.327584\pi$
$919$	31.2584	1.03112	0.515560	+	0.856854i	$0.327584\pi$
$920$	0	0
$921$	70.5410	2.32441
$922$	0	0
$923$	36.1538	1.19002
$924$	0	0
$925$	−4.37283	−0.143778
$926$	0	0
$927$	34.0687	1.11896
$928$	0	0
$929$	31.3616	1.02894	0.514470	−	0.857508i	$-0.327989\pi$
$929$	31.3616	1.02894	0.514470	+	0.857508i	$0.327989\pi$
$930$	0	0
$931$	2.87805	0.0943243
$932$	0	0
$933$	−1.87649	−0.0614335
$934$	0	0
$935$	−71.2893	−2.33141
$936$	0	0
$937$	5.35220	0.174849	0.0874244	−	0.996171i	$-0.472136\pi$
$937$	5.35220	0.174849	0.0874244	+	0.996171i	$0.472136\pi$
$938$	0	0
$939$	33.9210	1.10697
$940$	0	0
$941$	−13.4723	−0.439186	−0.219593	−	0.975592i	$-0.570473\pi$
$941$	−13.4723	−0.439186	−0.219593	+	0.975592i	$0.570473\pi$
$942$	0	0
$943$	−9.42186	−0.306818
$944$	0	0
$945$	11.4773	0.373358
$946$	0	0
$947$	−33.5555	−1.09041	−0.545204	−	0.838303i	$-0.683548\pi$
$947$	−33.5555	−1.09041	−0.545204	+	0.838303i	$0.683548\pi$
$948$	0	0
$949$	−24.5493	−0.796904
$950$	0	0
$951$	−68.2463	−2.21304
$952$	0	0
$953$	36.4805	1.18172	0.590860	−	0.806774i	$-0.298789\pi$
$953$	36.4805	1.18172	0.590860	+	0.806774i	$0.298789\pi$
$954$	0	0
$955$	−31.8206	−1.02969
$956$	0	0
$957$	90.2828	2.91843
$958$	0	0
$959$	31.8892	1.02976
$960$	0	0
$961$	14.3987	0.464475
$962$	0	0
$963$	−34.1400	−1.10014
$964$	0	0
$965$	−22.2594	−0.716554
$966$	0	0
$967$	46.5393	1.49660	0.748302	−	0.663359i	$-0.230870\pi$
$967$	46.5393	1.49660	0.748302	+	0.663359i	$0.230870\pi$
$968$	0	0
$969$	34.5091	1.10859
$970$	0	0
$971$	−39.2402	−1.25928	−0.629639	−	0.776888i	$-0.716797\pi$
$971$	−39.2402	−1.25928	−0.629639	+	0.776888i	$0.716797\pi$
$972$	0	0
$973$	51.6999	1.65742
$974$	0	0
$975$	−6.06618	−0.194273
$976$	0	0
$977$	42.5085	1.35997	0.679983	−	0.733228i	$-0.261987\pi$
$977$	42.5085	1.35997	0.679983	+	0.733228i	$0.261987\pi$
$978$	0	0
$979$	−73.7601	−2.35738
$980$	0	0
$981$	−16.1047	−0.514184
$982$	0	0
$983$	9.68313	0.308844	0.154422	−	0.988005i	$-0.450648\pi$
$983$	9.68313	0.308844	0.154422	+	0.988005i	$0.450648\pi$
$984$	0	0
$985$	14.8808	0.474143
$986$	0	0
$987$	68.3546	2.17575
$988$	0	0
$989$	−5.79664	−0.184322
$990$	0	0
$991$	26.7828	0.850783	0.425392	−	0.905009i	$-0.360136\pi$
$991$	26.7828	0.850783	0.425392	+	0.905009i	$0.360136\pi$
$992$	0	0
$993$	9.28450	0.294635
$994$	0	0
$995$	−3.99501	−0.126650
$996$	0	0
$997$	23.2881	0.737541	0.368770	−	0.929520i	$-0.379779\pi$
$997$	23.2881	0.737541	0.368770	+	0.929520i	$0.379779\pi$
$998$	0	0
$999$	−9.91558	−0.313715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.42	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.42	✓	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)

\(f(q)\)	\(=\)	\(q+2.26355 q^{3} -2.03106 q^{5} +2.84873 q^{7} +2.12365 q^{9} +O(q^{10})\)	\(q+2.26355 q^{3} -2.03106 q^{5} +2.84873 q^{7} +2.12365 q^{9} +5.94130 q^{11} +3.06346 q^{13} -4.59739 q^{15} +5.90773 q^{17} +2.58062 q^{19} +6.44823 q^{21} +1.10477 q^{23} -0.874809 q^{25} -1.98367 q^{27} +6.71326 q^{29} -6.73786 q^{31} +13.4484 q^{33} -5.78593 q^{35} +4.99862 q^{37} +6.93430 q^{39} -8.52835 q^{41} -5.24692 q^{43} -4.31325 q^{45} +10.6005 q^{47} +1.11525 q^{49} +13.3724 q^{51} -10.2822 q^{53} -12.0671 q^{55} +5.84136 q^{57} -2.18420 q^{59} -7.25388 q^{61} +6.04970 q^{63} -6.22207 q^{65} +7.02358 q^{67} +2.50070 q^{69} +11.8016 q^{71} -8.01357 q^{73} -1.98017 q^{75} +16.9252 q^{77} +11.6831 q^{79} -10.8611 q^{81} -4.69026 q^{83} -11.9989 q^{85} +15.1958 q^{87} -12.4148 q^{89} +8.72698 q^{91} -15.2515 q^{93} -5.24139 q^{95} -11.9835 q^{97} +12.6172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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