LMFDB - Embedded newform 6008.2.a.d.1.13

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:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-1.49420 q^{3} -0.747906 q^{5} +4.82128 q^{7} -0.767369 q^{9} +O(q^{10})$	$q-1.49420 q^{3} -0.747906 q^{5} +4.82128 q^{7} -0.767369 q^{9} -4.01141 q^{11} -0.852818 q^{13} +1.11752 q^{15} +0.486730 q^{17} -8.29511 q^{19} -7.20396 q^{21} -7.15640 q^{23} -4.44064 q^{25} +5.62920 q^{27} -10.0711 q^{29} +8.27802 q^{31} +5.99385 q^{33} -3.60587 q^{35} +2.91086 q^{37} +1.27428 q^{39} +5.00035 q^{41} +6.36752 q^{43} +0.573920 q^{45} -3.87355 q^{47} +16.2448 q^{49} -0.727271 q^{51} +8.86908 q^{53} +3.00016 q^{55} +12.3946 q^{57} -11.5251 q^{59} +6.20609 q^{61} -3.69970 q^{63} +0.637828 q^{65} +8.17537 q^{67} +10.6931 q^{69} +9.36399 q^{71} +7.81685 q^{73} +6.63520 q^{75} -19.3402 q^{77} +14.8320 q^{79} -6.10904 q^{81} -12.7924 q^{83} -0.364028 q^{85} +15.0482 q^{87} +2.87724 q^{89} -4.11168 q^{91} -12.3690 q^{93} +6.20397 q^{95} -12.6262 q^{97} +3.07823 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * 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
$3$	−1.49420	−0.862676	−0.431338	−	0.902190i	$-0.641958\pi$
$3$	−1.49420	−0.862676	−0.431338	+	0.902190i	$0.641958\pi$
$4$	0	0
$5$	−0.747906	−0.334474	−0.167237	−	0.985917i	$-0.553484\pi$
$5$	−0.747906	−0.334474	−0.167237	+	0.985917i	$0.553484\pi$
$6$	0	0
$7$	4.82128	1.82227	0.911137	−	0.412104i	$-0.135206\pi$
$7$	4.82128	1.82227	0.911137	+	0.412104i	$0.135206\pi$
$8$	0	0
$9$	−0.767369	−0.255790
$10$	0	0
$11$	−4.01141	−1.20949	−0.604743	−	0.796420i	$-0.706724\pi$
$11$	−4.01141	−1.20949	−0.604743	+	0.796420i	$0.706724\pi$
$12$	0	0
$13$	−0.852818	−0.236529	−0.118265	−	0.992982i	$-0.537733\pi$
$13$	−0.852818	−0.236529	−0.118265	+	0.992982i	$0.537733\pi$
$14$	0	0
$15$	1.11752	0.288543
$16$	0	0
$17$	0.486730	0.118049	0.0590247	−	0.998257i	$-0.481201\pi$
$17$	0.486730	0.118049	0.0590247	+	0.998257i	$0.481201\pi$
$18$	0	0
$19$	−8.29511	−1.90303	−0.951515	−	0.307603i	$-0.900473\pi$
$19$	−8.29511	−1.90303	−0.951515	+	0.307603i	$0.900473\pi$
$20$	0	0
$21$	−7.20396	−1.57203
$22$	0	0
$23$	−7.15640	−1.49221	−0.746106	−	0.665827i	$-0.768079\pi$
$23$	−7.15640	−1.49221	−0.746106	+	0.665827i	$0.768079\pi$
$24$	0	0
$25$	−4.44064	−0.888127
$26$	0	0
$27$	5.62920	1.08334
$28$	0	0
$29$	−10.0711	−1.87016	−0.935078	−	0.354441i	$-0.884671\pi$
$29$	−10.0711	−1.87016	−0.935078	+	0.354441i	$0.884671\pi$
$30$	0	0
$31$	8.27802	1.48678	0.743388	−	0.668860i	$-0.233217\pi$
$31$	8.27802	1.48678	0.743388	+	0.668860i	$0.233217\pi$
$32$	0	0
$33$	5.99385	1.04340
$34$	0	0
$35$	−3.60587	−0.609503
$36$	0	0
$37$	2.91086	0.478543	0.239272	−	0.970953i	$-0.423091\pi$
$37$	2.91086	0.478543	0.239272	+	0.970953i	$0.423091\pi$
$38$	0	0
$39$	1.27428	0.204048
$40$	0	0
$41$	5.00035	0.780923	0.390462	−	0.920619i	$-0.372315\pi$
$41$	5.00035	0.780923	0.390462	+	0.920619i	$0.372315\pi$
$42$	0	0
$43$	6.36752	0.971037	0.485519	−	0.874226i	$-0.338631\pi$
$43$	6.36752	0.971037	0.485519	+	0.874226i	$0.338631\pi$
$44$	0	0
$45$	0.573920	0.0855549
$46$	0	0
$47$	−3.87355	−0.565015	−0.282508	−	0.959265i	$-0.591166\pi$
$47$	−3.87355	−0.565015	−0.282508	+	0.959265i	$0.591166\pi$
$48$	0	0
$49$	16.2448	2.32068
$50$	0	0
$51$	−0.727271	−0.101838
$52$	0	0
$53$	8.86908	1.21826	0.609131	−	0.793069i	$-0.291518\pi$
$53$	8.86908	1.21826	0.609131	+	0.793069i	$0.291518\pi$
$54$	0	0
$55$	3.00016	0.404542
$56$	0	0
$57$	12.3946	1.64170
$58$	0	0
$59$	−11.5251	−1.50044	−0.750219	−	0.661190i	$-0.770052\pi$
$59$	−11.5251	−1.50044	−0.750219	+	0.661190i	$0.770052\pi$
$60$	0	0
$61$	6.20609	0.794609	0.397305	−	0.917687i	$-0.369946\pi$
$61$	6.20609	0.794609	0.397305	+	0.917687i	$0.369946\pi$
$62$	0	0
$63$	−3.69970	−0.466119
$64$	0	0
$65$	0.637828	0.0791128
$66$	0	0
$67$	8.17537	0.998781	0.499391	−	0.866377i	$-0.333557\pi$
$67$	8.17537	0.998781	0.499391	+	0.866377i	$0.333557\pi$
$68$	0	0
$69$	10.6931	1.28730
$70$	0	0
$71$	9.36399	1.11130	0.555651	−	0.831416i	$-0.312469\pi$
$71$	9.36399	1.11130	0.555651	+	0.831416i	$0.312469\pi$
$72$	0	0
$73$	7.81685	0.914893	0.457446	−	0.889237i	$-0.348764\pi$
$73$	7.81685	0.914893	0.457446	+	0.889237i	$0.348764\pi$
$74$	0	0
$75$	6.63520	0.766166
$76$	0	0
$77$	−19.3402	−2.20402
$78$	0	0
$79$	14.8320	1.66873	0.834367	−	0.551209i	$-0.185833\pi$
$79$	14.8320	1.66873	0.834367	+	0.551209i	$0.185833\pi$
$80$	0	0
$81$	−6.10904	−0.678782
$82$	0	0
$83$	−12.7924	−1.40415	−0.702076	−	0.712102i	$-0.747743\pi$
$83$	−12.7924	−1.40415	−0.702076	+	0.712102i	$0.747743\pi$
$84$	0	0
$85$	−0.364028	−0.0394844
$86$	0	0
$87$	15.0482	1.61334
$88$	0	0
$89$	2.87724	0.304986	0.152493	−	0.988305i	$-0.451270\pi$
$89$	2.87724	0.304986	0.152493	+	0.988305i	$0.451270\pi$
$90$	0	0
$91$	−4.11168	−0.431021
$92$	0	0
$93$	−12.3690	−1.28261
$94$	0	0
$95$	6.20397	0.636513
$96$	0	0
$97$	−12.6262	−1.28199	−0.640997	−	0.767543i	$-0.721479\pi$
$97$	−12.6262	−1.28199	−0.640997	+	0.767543i	$0.721479\pi$
$98$	0	0
$99$	3.07823	0.309374
$100$	0	0
$101$	−9.78990	−0.974132	−0.487066	−	0.873365i	$-0.661933\pi$
$101$	−9.78990	−0.974132	−0.487066	+	0.873365i	$0.661933\pi$
$102$	0	0
$103$	1.39048	0.137008	0.0685038	−	0.997651i	$-0.478177\pi$
$103$	1.39048	0.137008	0.0685038	+	0.997651i	$0.478177\pi$
$104$	0	0
$105$	5.38788	0.525804
$106$	0	0
$107$	−1.05849	−0.102328	−0.0511642	−	0.998690i	$-0.516293\pi$
$107$	−1.05849	−0.102328	−0.0511642	+	0.998690i	$0.516293\pi$
$108$	0	0
$109$	−4.05522	−0.388420	−0.194210	−	0.980960i	$-0.562214\pi$
$109$	−4.05522	−0.388420	−0.194210	+	0.980960i	$0.562214\pi$
$110$	0	0
$111$	−4.34941	−0.412828
$112$	0	0
$113$	−0.837291	−0.0787657	−0.0393828	−	0.999224i	$-0.512539\pi$
$113$	−0.837291	−0.0787657	−0.0393828	+	0.999224i	$0.512539\pi$
$114$	0	0
$115$	5.35232	0.499106
$116$	0	0
$117$	0.654426	0.0605017
$118$	0	0
$119$	2.34666	0.215118
$120$	0	0
$121$	5.09143	0.462858
$122$	0	0
$123$	−7.47152	−0.673684
$124$	0	0
$125$	7.06071	0.631529
$126$	0	0
$127$	−0.470192	−0.0417228	−0.0208614	−	0.999782i	$-0.506641\pi$
$127$	−0.470192	−0.0417228	−0.0208614	+	0.999782i	$0.506641\pi$
$128$	0	0
$129$	−9.51434	−0.837691
$130$	0	0
$131$	19.6387	1.71584	0.857922	−	0.513780i	$-0.171755\pi$
$131$	19.6387	1.71584	0.857922	+	0.513780i	$0.171755\pi$
$132$	0	0
$133$	−39.9931	−3.46784
$134$	0	0
$135$	−4.21011	−0.362349
$136$	0	0
$137$	−16.6584	−1.42322	−0.711612	−	0.702573i	$-0.752035\pi$
$137$	−16.6584	−1.42322	−0.711612	+	0.702573i	$0.752035\pi$
$138$	0	0
$139$	−10.6038	−0.899400	−0.449700	−	0.893180i	$-0.648469\pi$
$139$	−10.6038	−0.899400	−0.449700	+	0.893180i	$0.648469\pi$
$140$	0	0
$141$	5.78785	0.487425
$142$	0	0
$143$	3.42100	0.286079
$144$	0	0
$145$	7.53224	0.625518
$146$	0	0
$147$	−24.2729	−2.00200
$148$	0	0
$149$	13.8731	1.13653	0.568266	−	0.822845i	$-0.307614\pi$
$149$	13.8731	1.13653	0.568266	+	0.822845i	$0.307614\pi$
$150$	0	0
$151$	19.8402	1.61457	0.807287	−	0.590159i	$-0.200935\pi$
$151$	19.8402	1.61457	0.807287	+	0.590159i	$0.200935\pi$
$152$	0	0
$153$	−0.373501	−0.0301958
$154$	0	0
$155$	−6.19119	−0.497288
$156$	0	0
$157$	12.7976	1.02136	0.510681	−	0.859770i	$-0.329393\pi$
$157$	12.7976	1.02136	0.510681	+	0.859770i	$0.329393\pi$
$158$	0	0
$159$	−13.2522	−1.05097
$160$	0	0
$161$	−34.5030	−2.71922
$162$	0	0
$163$	21.5402	1.68716	0.843580	−	0.537004i	$-0.180444\pi$
$163$	21.5402	1.68716	0.843580	+	0.537004i	$0.180444\pi$
$164$	0	0
$165$	−4.48284	−0.348988
$166$	0	0
$167$	5.93442	0.459219	0.229610	−	0.973283i	$-0.426255\pi$
$167$	5.93442	0.459219	0.229610	+	0.973283i	$0.426255\pi$
$168$	0	0
$169$	−12.2727	−0.944054
$170$	0	0
$171$	6.36541	0.486775
$172$	0	0
$173$	18.0580	1.37292	0.686462	−	0.727166i	$-0.259163\pi$
$173$	18.0580	1.37292	0.686462	+	0.727166i	$0.259163\pi$
$174$	0	0
$175$	−21.4096	−1.61841
$176$	0	0
$177$	17.2208	1.29439
$178$	0	0
$179$	5.44251	0.406792	0.203396	−	0.979097i	$-0.434802\pi$
$179$	5.44251	0.406792	0.203396	+	0.979097i	$0.434802\pi$
$180$	0	0
$181$	4.26956	0.317354	0.158677	−	0.987331i	$-0.449277\pi$
$181$	4.26956	0.317354	0.158677	+	0.987331i	$0.449277\pi$
$182$	0	0
$183$	−9.27314	−0.685490
$184$	0	0
$185$	−2.17705	−0.160060
$186$	0	0
$187$	−1.95247	−0.142779
$188$	0	0
$189$	27.1400	1.97414
$190$	0	0
$191$	6.64579	0.480872	0.240436	−	0.970665i	$-0.422710\pi$
$191$	6.64579	0.480872	0.240436	+	0.970665i	$0.422710\pi$
$192$	0	0
$193$	−20.6932	−1.48953	−0.744764	−	0.667328i	$-0.767438\pi$
$193$	−20.6932	−1.48953	−0.744764	+	0.667328i	$0.767438\pi$
$194$	0	0
$195$	−0.953042	−0.0682487
$196$	0	0
$197$	4.15490	0.296025	0.148012	−	0.988986i	$-0.452712\pi$
$197$	4.15490	0.296025	0.148012	+	0.988986i	$0.452712\pi$
$198$	0	0
$199$	4.57371	0.324222	0.162111	−	0.986773i	$-0.448170\pi$
$199$	4.57371	0.324222	0.162111	+	0.986773i	$0.448170\pi$
$200$	0	0
$201$	−12.2156	−0.861625
$202$	0	0
$203$	−48.5556	−3.40794
$204$	0	0
$205$	−3.73979	−0.261198
$206$	0	0
$207$	5.49160	0.381692
$208$	0	0
$209$	33.2751	2.30169
$210$	0	0
$211$	−10.1439	−0.698335	−0.349167	−	0.937060i	$-0.613536\pi$
$211$	−10.1439	−0.698335	−0.349167	+	0.937060i	$0.613536\pi$
$212$	0	0
$213$	−13.9917	−0.958693
$214$	0	0
$215$	−4.76230	−0.324786
$216$	0	0
$217$	39.9107	2.70931
$218$	0	0
$219$	−11.6799	−0.789256
$220$	0	0
$221$	−0.415092	−0.0279221
$222$	0	0
$223$	9.50048	0.636199	0.318099	−	0.948057i	$-0.396955\pi$
$223$	9.50048	0.636199	0.318099	+	0.948057i	$0.396955\pi$
$224$	0	0
$225$	3.40761	0.227174
$226$	0	0
$227$	12.8723	0.854366	0.427183	−	0.904165i	$-0.359506\pi$
$227$	12.8723	0.854366	0.427183	+	0.904165i	$0.359506\pi$
$228$	0	0
$229$	−2.52063	−0.166568	−0.0832839	−	0.996526i	$-0.526541\pi$
$229$	−2.52063	−0.166568	−0.0832839	+	0.996526i	$0.526541\pi$
$230$	0	0
$231$	28.8980	1.90135
$232$	0	0
$233$	22.2894	1.46023	0.730114	−	0.683325i	$-0.239467\pi$
$233$	22.2894	1.46023	0.730114	+	0.683325i	$0.239467\pi$
$234$	0	0
$235$	2.89705	0.188983
$236$	0	0
$237$	−22.1620	−1.43958
$238$	0	0
$239$	13.5934	0.879283	0.439642	−	0.898173i	$-0.355105\pi$
$239$	13.5934	0.879283	0.439642	+	0.898173i	$0.355105\pi$
$240$	0	0
$241$	−16.5145	−1.06379	−0.531895	−	0.846810i	$-0.678520\pi$
$241$	−16.5145	−1.06379	−0.531895	+	0.846810i	$0.678520\pi$
$242$	0	0
$243$	−7.75948	−0.497771
$244$	0	0
$245$	−12.1496	−0.776207
$246$	0	0
$247$	7.07422	0.450122
$248$	0	0
$249$	19.1144	1.21133
$250$	0	0
$251$	−18.1260	−1.14410	−0.572050	−	0.820218i	$-0.693852\pi$
$251$	−18.1260	−1.14410	−0.572050	+	0.820218i	$0.693852\pi$
$252$	0	0
$253$	28.7073	1.80481
$254$	0	0
$255$	0.543931	0.0340623
$256$	0	0
$257$	12.8349	0.800618	0.400309	−	0.916380i	$-0.368903\pi$
$257$	12.8349	0.800618	0.400309	+	0.916380i	$0.368903\pi$
$258$	0	0
$259$	14.0341	0.872036
$260$	0	0
$261$	7.72825	0.478367
$262$	0	0
$263$	19.6302	1.21045	0.605226	−	0.796054i	$-0.293083\pi$
$263$	19.6302	1.21045	0.605226	+	0.796054i	$0.293083\pi$
$264$	0	0
$265$	−6.63324	−0.407477
$266$	0	0
$267$	−4.29916	−0.263105
$268$	0	0
$269$	18.3249	1.11729	0.558645	−	0.829407i	$-0.311321\pi$
$269$	18.3249	1.11729	0.558645	+	0.829407i	$0.311321\pi$
$270$	0	0
$271$	27.8073	1.68917	0.844587	−	0.535418i	$-0.179846\pi$
$271$	27.8073	1.68917	0.844587	+	0.535418i	$0.179846\pi$
$272$	0	0
$273$	6.14366	0.371831
$274$	0	0
$275$	17.8132	1.07418
$276$	0	0
$277$	−27.2708	−1.63855	−0.819273	−	0.573404i	$-0.805623\pi$
$277$	−27.2708	−1.63855	−0.819273	+	0.573404i	$0.805623\pi$
$278$	0	0
$279$	−6.35230	−0.380302
$280$	0	0
$281$	15.2027	0.906915	0.453457	−	0.891278i	$-0.350190\pi$
$281$	15.2027	0.906915	0.453457	+	0.891278i	$0.350190\pi$
$282$	0	0
$283$	−7.17054	−0.426244	−0.213122	−	0.977026i	$-0.568363\pi$
$283$	−7.17054	−0.426244	−0.213122	+	0.977026i	$0.568363\pi$
$284$	0	0
$285$	−9.26996	−0.549105
$286$	0	0
$287$	24.1081	1.42306
$288$	0	0
$289$	−16.7631	−0.986064
$290$	0	0
$291$	18.8660	1.10595
$292$	0	0
$293$	−27.7694	−1.62231	−0.811153	−	0.584835i	$-0.801159\pi$
$293$	−27.7694	−1.62231	−0.811153	+	0.584835i	$0.801159\pi$
$294$	0	0
$295$	8.61968	0.501857
$296$	0	0
$297$	−22.5810	−1.31029
$298$	0	0
$299$	6.10311	0.352952
$300$	0	0
$301$	30.6996	1.76950
$302$	0	0
$303$	14.6281	0.840360
$304$	0	0
$305$	−4.64158	−0.265776
$306$	0	0
$307$	−10.3901	−0.592994	−0.296497	−	0.955034i	$-0.595818\pi$
$307$	−10.3901	−0.592994	−0.296497	+	0.955034i	$0.595818\pi$
$308$	0	0
$309$	−2.07765	−0.118193
$310$	0	0
$311$	9.96708	0.565181	0.282591	−	0.959241i	$-0.408806\pi$
$311$	9.96708	0.565181	0.282591	+	0.959241i	$0.408806\pi$
$312$	0	0
$313$	−30.9587	−1.74989	−0.874945	−	0.484222i	$-0.839103\pi$
$313$	−30.9587	−1.74989	−0.874945	+	0.484222i	$0.839103\pi$
$314$	0	0
$315$	2.76703	0.155904
$316$	0	0
$317$	−9.46427	−0.531566	−0.265783	−	0.964033i	$-0.585631\pi$
$317$	−9.46427	−0.531566	−0.265783	+	0.964033i	$0.585631\pi$
$318$	0	0
$319$	40.3993	2.26193
$320$	0	0
$321$	1.58160	0.0882763
$322$	0	0
$323$	−4.03748	−0.224651
$324$	0	0
$325$	3.78705	0.210068
$326$	0	0
$327$	6.05931	0.335081
$328$	0	0
$329$	−18.6755	−1.02961
$330$	0	0
$331$	2.35335	0.129352	0.0646758	−	0.997906i	$-0.479399\pi$
$331$	2.35335	0.129352	0.0646758	+	0.997906i	$0.479399\pi$
$332$	0	0
$333$	−2.23371	−0.122406
$334$	0	0
$335$	−6.11441	−0.334066
$336$	0	0
$337$	−28.8222	−1.57004	−0.785021	−	0.619469i	$-0.787348\pi$
$337$	−28.8222	−1.57004	−0.785021	+	0.619469i	$0.787348\pi$
$338$	0	0
$339$	1.25108	0.0679493
$340$	0	0
$341$	−33.2066	−1.79824
$342$	0	0
$343$	44.5716	2.40664
$344$	0	0
$345$	−7.99743	−0.430567
$346$	0	0
$347$	12.9605	0.695754	0.347877	−	0.937540i	$-0.386903\pi$
$347$	12.9605	0.695754	0.347877	+	0.937540i	$0.386903\pi$
$348$	0	0
$349$	−7.32567	−0.392134	−0.196067	−	0.980591i	$-0.562817\pi$
$349$	−7.32567	−0.392134	−0.196067	+	0.980591i	$0.562817\pi$
$350$	0	0
$351$	−4.80068	−0.256241
$352$	0	0
$353$	12.3987	0.659918	0.329959	−	0.943995i	$-0.392965\pi$
$353$	12.3987	0.659918	0.329959	+	0.943995i	$0.392965\pi$
$354$	0	0
$355$	−7.00339	−0.371701
$356$	0	0
$357$	−3.50638	−0.185577
$358$	0	0
$359$	8.03529	0.424086	0.212043	−	0.977260i	$-0.431988\pi$
$359$	8.03529	0.424086	0.212043	+	0.977260i	$0.431988\pi$
$360$	0	0
$361$	49.8089	2.62152
$362$	0	0
$363$	−7.60762	−0.399296
$364$	0	0
$365$	−5.84627	−0.306008
$366$	0	0
$367$	27.0080	1.40981	0.704903	−	0.709304i	$-0.250991\pi$
$367$	27.0080	1.40981	0.704903	+	0.709304i	$0.250991\pi$
$368$	0	0
$369$	−3.83711	−0.199752
$370$	0	0
$371$	42.7604	2.22001
$372$	0	0
$373$	−20.8883	−1.08156	−0.540779	−	0.841165i	$-0.681870\pi$
$373$	−20.8883	−1.08156	−0.540779	+	0.841165i	$0.681870\pi$
$374$	0	0
$375$	−10.5501	−0.544805
$376$	0	0
$377$	8.58881	0.442346
$378$	0	0
$379$	14.3608	0.737663	0.368832	−	0.929496i	$-0.379758\pi$
$379$	14.3608	0.737663	0.368832	+	0.929496i	$0.379758\pi$
$380$	0	0
$381$	0.702561	0.0359933
$382$	0	0
$383$	−25.0288	−1.27891	−0.639455	−	0.768828i	$-0.720840\pi$
$383$	−25.0288	−1.27891	−0.639455	+	0.768828i	$0.720840\pi$
$384$	0	0
$385$	14.4646	0.737185
$386$	0	0
$387$	−4.88623	−0.248381
$388$	0	0
$389$	18.7857	0.952473	0.476236	−	0.879317i	$-0.342001\pi$
$389$	18.7857	0.952473	0.476236	+	0.879317i	$0.342001\pi$
$390$	0	0
$391$	−3.48323	−0.176155
$392$	0	0
$393$	−29.3442	−1.48022
$394$	0	0
$395$	−11.0930	−0.558148
$396$	0	0
$397$	16.6319	0.834733	0.417366	−	0.908738i	$-0.362953\pi$
$397$	16.6319	0.834733	0.417366	+	0.908738i	$0.362953\pi$
$398$	0	0
$399$	59.7576	2.99162
$400$	0	0
$401$	−37.0783	−1.85160	−0.925801	−	0.378011i	$-0.876608\pi$
$401$	−37.0783	−1.85160	−0.925801	+	0.378011i	$0.876608\pi$
$402$	0	0
$403$	−7.05965	−0.351666
$404$	0	0
$405$	4.56899	0.227035
$406$	0	0
$407$	−11.6767	−0.578791
$408$	0	0
$409$	16.0289	0.792578	0.396289	−	0.918126i	$-0.370298\pi$
$409$	16.0289	0.792578	0.396289	+	0.918126i	$0.370298\pi$
$410$	0	0
$411$	24.8910	1.22778
$412$	0	0
$413$	−55.5656	−2.73421
$414$	0	0
$415$	9.56754	0.469652
$416$	0	0
$417$	15.8441	0.775891
$418$	0	0
$419$	−23.8589	−1.16558	−0.582791	−	0.812622i	$-0.698039\pi$
$419$	−23.8589	−1.16558	−0.582791	+	0.812622i	$0.698039\pi$
$420$	0	0
$421$	17.7418	0.864684	0.432342	−	0.901710i	$-0.357687\pi$
$421$	17.7418	0.864684	0.432342	+	0.901710i	$0.357687\pi$
$422$	0	0
$423$	2.97244	0.144525
$424$	0	0
$425$	−2.16139	−0.104843
$426$	0	0
$427$	29.9213	1.44799
$428$	0	0
$429$	−5.11166	−0.246793
$430$	0	0
$431$	24.3216	1.17153	0.585766	−	0.810480i	$-0.300794\pi$
$431$	24.3216	1.17153	0.585766	+	0.810480i	$0.300794\pi$
$432$	0	0
$433$	4.85569	0.233350	0.116675	−	0.993170i	$-0.462776\pi$
$433$	4.85569	0.233350	0.116675	+	0.993170i	$0.462776\pi$
$434$	0	0
$435$	−11.2547	−0.539620
$436$	0	0
$437$	59.3632	2.83972
$438$	0	0
$439$	−10.3522	−0.494084	−0.247042	−	0.969005i	$-0.579459\pi$
$439$	−10.3522	−0.494084	−0.247042	+	0.969005i	$0.579459\pi$
$440$	0	0
$441$	−12.4657	−0.593606
$442$	0	0
$443$	−7.36817	−0.350072	−0.175036	−	0.984562i	$-0.556004\pi$
$443$	−7.36817	−0.350072	−0.175036	+	0.984562i	$0.556004\pi$
$444$	0	0
$445$	−2.15190	−0.102010
$446$	0	0
$447$	−20.7292	−0.980459
$448$	0	0
$449$	−39.2032	−1.85011	−0.925057	−	0.379827i	$-0.875983\pi$
$449$	−39.2032	−1.85011	−0.925057	+	0.379827i	$0.875983\pi$
$450$	0	0
$451$	−20.0585	−0.944516
$452$	0	0
$453$	−29.6452	−1.39286
$454$	0	0
$455$	3.07515	0.144165
$456$	0	0
$457$	−7.23578	−0.338476	−0.169238	−	0.985575i	$-0.554131\pi$
$457$	−7.23578	−0.338476	−0.169238	+	0.985575i	$0.554131\pi$
$458$	0	0
$459$	2.73990	0.127888
$460$	0	0
$461$	35.0108	1.63062	0.815308	−	0.579028i	$-0.196568\pi$
$461$	35.0108	1.63062	0.815308	+	0.579028i	$0.196568\pi$
$462$	0	0
$463$	19.9227	0.925886	0.462943	−	0.886388i	$-0.346793\pi$
$463$	19.9227	0.925886	0.462943	+	0.886388i	$0.346793\pi$
$464$	0	0
$465$	9.25086	0.428999
$466$	0	0
$467$	−12.3048	−0.569399	−0.284699	−	0.958617i	$-0.591894\pi$
$467$	−12.3048	−0.569399	−0.284699	+	0.958617i	$0.591894\pi$
$468$	0	0
$469$	39.4158	1.82005
$470$	0	0
$471$	−19.1222	−0.881105
$472$	0	0
$473$	−25.5427	−1.17446
$474$	0	0
$475$	36.8356	1.69013
$476$	0	0
$477$	−6.80586	−0.311619
$478$	0	0
$479$	17.6691	0.807323	0.403662	−	0.914908i	$-0.367737\pi$
$479$	17.6691	0.807323	0.403662	+	0.914908i	$0.367737\pi$
$480$	0	0
$481$	−2.48244	−0.113189
$482$	0	0
$483$	51.5544	2.34581
$484$	0	0
$485$	9.44320	0.428793
$486$	0	0
$487$	−24.2770	−1.10010	−0.550049	−	0.835132i	$-0.685391\pi$
$487$	−24.2770	−1.10010	−0.550049	+	0.835132i	$0.685391\pi$
$488$	0	0
$489$	−32.1854	−1.45547
$490$	0	0
$491$	6.70184	0.302450	0.151225	−	0.988499i	$-0.451678\pi$
$491$	6.70184	0.302450	0.151225	+	0.988499i	$0.451678\pi$
$492$	0	0
$493$	−4.90190	−0.220771
$494$	0	0
$495$	−2.30223	−0.103478
$496$	0	0
$497$	45.1465	2.02510
$498$	0	0
$499$	18.2516	0.817053	0.408527	−	0.912746i	$-0.366043\pi$
$499$	18.2516	0.817053	0.408527	+	0.912746i	$0.366043\pi$
$500$	0	0
$501$	−8.86720	−0.396157
$502$	0	0
$503$	−36.4875	−1.62690	−0.813448	−	0.581638i	$-0.802412\pi$
$503$	−36.4875	−1.62690	−0.813448	+	0.581638i	$0.802412\pi$
$504$	0	0
$505$	7.32193	0.325822
$506$	0	0
$507$	18.3379	0.814413
$508$	0	0
$509$	12.2553	0.543208	0.271604	−	0.962409i	$-0.412446\pi$
$509$	12.2553	0.543208	0.271604	+	0.962409i	$0.412446\pi$
$510$	0	0
$511$	37.6872	1.66719
$512$	0	0
$513$	−46.6948	−2.06163
$514$	0	0
$515$	−1.03995	−0.0458255
$516$	0	0
$517$	15.5384	0.683378
$518$	0	0
$519$	−26.9822	−1.18439
$520$	0	0
$521$	−16.3569	−0.716609	−0.358305	−	0.933605i	$-0.616645\pi$
$521$	−16.3569	−0.716609	−0.358305	+	0.933605i	$0.616645\pi$
$522$	0	0
$523$	−2.91161	−0.127316	−0.0636579	−	0.997972i	$-0.520277\pi$
$523$	−2.91161	−0.127316	−0.0636579	+	0.997972i	$0.520277\pi$
$524$	0	0
$525$	31.9902	1.39616
$526$	0	0
$527$	4.02916	0.175513
$528$	0	0
$529$	28.2141	1.22670
$530$	0	0
$531$	8.84398	0.383796
$532$	0	0
$533$	−4.26439	−0.184711
$534$	0	0
$535$	0.791654	0.0342262
$536$	0	0
$537$	−8.13219	−0.350930
$538$	0	0
$539$	−65.1645	−2.80683
$540$	0	0
$541$	7.77209	0.334148	0.167074	−	0.985944i	$-0.446568\pi$
$541$	7.77209	0.334148	0.167074	+	0.985944i	$0.446568\pi$
$542$	0	0
$543$	−6.37957	−0.273773
$544$	0	0
$545$	3.03293	0.129916
$546$	0	0
$547$	14.5678	0.622872	0.311436	−	0.950267i	$-0.399190\pi$
$547$	14.5678	0.622872	0.311436	+	0.950267i	$0.399190\pi$
$548$	0	0
$549$	−4.76236	−0.203253
$550$	0	0
$551$	83.5409	3.55896
$552$	0	0
$553$	71.5094	3.04089
$554$	0	0
$555$	3.25295	0.138080
$556$	0	0
$557$	27.1625	1.15091	0.575456	−	0.817833i	$-0.304825\pi$
$557$	27.1625	1.15091	0.575456	+	0.817833i	$0.304825\pi$
$558$	0	0
$559$	−5.43033	−0.229679
$560$	0	0
$561$	2.91739	0.123172
$562$	0	0
$563$	−0.00404256	−0.000170373 0	−8.51867e−5	−	1.00000i	$-0.500027\pi$
$563$	−0.00404256	−0.000170373 0	−8.51867e−5		1.00000i	$0.500027\pi$
$564$	0	0
$565$	0.626215	0.0263451
$566$	0	0
$567$	−29.4534	−1.23693
$568$	0	0
$569$	24.9028	1.04398	0.521990	−	0.852952i	$-0.325190\pi$
$569$	24.9028	1.04398	0.521990	+	0.852952i	$0.325190\pi$
$570$	0	0
$571$	15.5110	0.649114	0.324557	−	0.945866i	$-0.394785\pi$
$571$	15.5110	0.649114	0.324557	+	0.945866i	$0.394785\pi$
$572$	0	0
$573$	−9.93013	−0.414837
$574$	0	0
$575$	31.7790	1.32527
$576$	0	0
$577$	15.7162	0.654272	0.327136	−	0.944977i	$-0.393916\pi$
$577$	15.7162	0.654272	0.327136	+	0.944977i	$0.393916\pi$
$578$	0	0
$579$	30.9198	1.28498
$580$	0	0
$581$	−61.6759	−2.55875
$582$	0	0
$583$	−35.5776	−1.47347
$584$	0	0
$585$	−0.489449	−0.0202362
$586$	0	0
$587$	27.6243	1.14018	0.570088	−	0.821584i	$-0.306909\pi$
$587$	27.6243	1.14018	0.570088	+	0.821584i	$0.306909\pi$
$588$	0	0
$589$	−68.6671	−2.82938
$590$	0	0
$591$	−6.20825	−0.255373
$592$	0	0
$593$	−0.837913	−0.0344089	−0.0172045	−	0.999852i	$-0.505477\pi$
$593$	−0.837913	−0.0344089	−0.0172045	+	0.999852i	$0.505477\pi$
$594$	0	0
$595$	−1.75508	−0.0719514
$596$	0	0
$597$	−6.83404	−0.279698
$598$	0	0
$599$	33.7927	1.38073	0.690367	−	0.723460i	$-0.257449\pi$
$599$	33.7927	1.38073	0.690367	+	0.723460i	$0.257449\pi$
$600$	0	0
$601$	17.3297	0.706894	0.353447	−	0.935455i	$-0.385009\pi$
$601$	17.3297	0.706894	0.353447	+	0.935455i	$0.385009\pi$
$602$	0	0
$603$	−6.27353	−0.255478
$604$	0	0
$605$	−3.80792	−0.154814
$606$	0	0
$607$	42.7875	1.73669	0.868346	−	0.495959i	$-0.165183\pi$
$607$	42.7875	1.73669	0.868346	+	0.495959i	$0.165183\pi$
$608$	0	0
$609$	72.5518	2.93995
$610$	0	0
$611$	3.30343	0.133642
$612$	0	0
$613$	−24.0079	−0.969669	−0.484834	−	0.874606i	$-0.661120\pi$
$613$	−24.0079	−0.969669	−0.484834	+	0.874606i	$0.661120\pi$
$614$	0	0
$615$	5.58800	0.225330
$616$	0	0
$617$	32.4862	1.30785	0.653923	−	0.756561i	$-0.273122\pi$
$617$	32.4862	1.30785	0.653923	+	0.756561i	$0.273122\pi$
$618$	0	0
$619$	16.2058	0.651365	0.325683	−	0.945479i	$-0.394406\pi$
$619$	16.2058	0.651365	0.325683	+	0.945479i	$0.394406\pi$
$620$	0	0
$621$	−40.2848	−1.61657
$622$	0	0
$623$	13.8720	0.555769
$624$	0	0
$625$	16.9224	0.676897
$626$	0	0
$627$	−49.7197	−1.98561
$628$	0	0
$629$	1.41680	0.0564917
$630$	0	0
$631$	−4.26920	−0.169954	−0.0849770	−	0.996383i	$-0.527082\pi$
$631$	−4.26920	−0.169954	−0.0849770	+	0.996383i	$0.527082\pi$
$632$	0	0
$633$	15.1570	0.602437
$634$	0	0
$635$	0.351660	0.0139552
$636$	0	0
$637$	−13.8538	−0.548909
$638$	0	0
$639$	−7.18563	−0.284259
$640$	0	0
$641$	−21.6829	−0.856425	−0.428212	−	0.903678i	$-0.640856\pi$
$641$	−21.6829	−0.856425	−0.428212	+	0.903678i	$0.640856\pi$
$642$	0	0
$643$	32.2355	1.27124	0.635622	−	0.772000i	$-0.280744\pi$
$643$	32.2355	1.27124	0.635622	+	0.772000i	$0.280744\pi$
$644$	0	0
$645$	7.11583	0.280186
$646$	0	0
$647$	2.19239	0.0861917	0.0430959	−	0.999071i	$-0.486278\pi$
$647$	2.19239	0.0861917	0.0430959	+	0.999071i	$0.486278\pi$
$648$	0	0
$649$	46.2318	1.81476
$650$	0	0
$651$	−59.6345	−2.33726
$652$	0	0
$653$	18.6376	0.729346	0.364673	−	0.931136i	$-0.381181\pi$
$653$	18.6376	0.729346	0.364673	+	0.931136i	$0.381181\pi$
$654$	0	0
$655$	−14.6879	−0.573905
$656$	0	0
$657$	−5.99840	−0.234020
$658$	0	0
$659$	−25.8042	−1.00519	−0.502594	−	0.864522i	$-0.667621\pi$
$659$	−25.8042	−1.00519	−0.502594	+	0.864522i	$0.667621\pi$
$660$	0	0
$661$	−22.7008	−0.882958	−0.441479	−	0.897272i	$-0.645546\pi$
$661$	−22.7008	−0.882958	−0.441479	+	0.897272i	$0.645546\pi$
$662$	0	0
$663$	0.620230	0.0240877
$664$	0	0
$665$	29.9111	1.15990
$666$	0	0
$667$	72.0728	2.79067
$668$	0	0
$669$	−14.1956	−0.548834
$670$	0	0
$671$	−24.8952	−0.961069
$672$	0	0
$673$	−26.1452	−1.00782	−0.503912	−	0.863755i	$-0.668106\pi$
$673$	−26.1452	−1.00782	−0.503912	+	0.863755i	$0.668106\pi$
$674$	0	0
$675$	−24.9972	−0.962144
$676$	0	0
$677$	−23.3490	−0.897375	−0.448687	−	0.893689i	$-0.648108\pi$
$677$	−23.3490	−0.897375	−0.448687	+	0.893689i	$0.648108\pi$
$678$	0	0
$679$	−60.8744	−2.33614
$680$	0	0
$681$	−19.2338	−0.737041
$682$	0	0
$683$	−19.0707	−0.729722	−0.364861	−	0.931062i	$-0.618883\pi$
$683$	−19.0707	−0.729722	−0.364861	+	0.931062i	$0.618883\pi$
$684$	0	0
$685$	12.4589	0.476031
$686$	0	0
$687$	3.76632	0.143694
$688$	0	0
$689$	−7.56371	−0.288154
$690$	0	0
$691$	27.6605	1.05226	0.526128	−	0.850405i	$-0.323643\pi$
$691$	27.6605	1.05226	0.526128	+	0.850405i	$0.323643\pi$
$692$	0	0
$693$	14.8410	0.563764
$694$	0	0
$695$	7.93063	0.300826
$696$	0	0
$697$	2.43382	0.0921875
$698$	0	0
$699$	−33.3048	−1.25970
$700$	0	0
$701$	47.8635	1.80778	0.903890	−	0.427764i	$-0.140699\pi$
$701$	47.8635	1.80778	0.903890	+	0.427764i	$0.140699\pi$
$702$	0	0
$703$	−24.1459	−0.910682
$704$	0	0
$705$	−4.32877	−0.163031
$706$	0	0
$707$	−47.1999	−1.77513
$708$	0	0
$709$	−7.47365	−0.280679	−0.140339	−	0.990103i	$-0.544819\pi$
$709$	−7.47365	−0.280679	−0.140339	+	0.990103i	$0.544819\pi$
$710$	0	0
$711$	−11.3816	−0.426845
$712$	0	0
$713$	−59.2409	−2.21859
$714$	0	0
$715$	−2.55859	−0.0956859
$716$	0	0
$717$	−20.3112	−0.758537
$718$	0	0
$719$	−27.8599	−1.03900	−0.519499	−	0.854471i	$-0.673881\pi$
$719$	−27.8599	−1.03900	−0.519499	+	0.854471i	$0.673881\pi$
$720$	0	0
$721$	6.70387	0.249665
$722$	0	0
$723$	24.6759	0.917706
$724$	0	0
$725$	44.7221	1.66094
$726$	0	0
$727$	−5.30595	−0.196787	−0.0983934	−	0.995148i	$-0.531370\pi$
$727$	−5.30595	−0.196787	−0.0983934	+	0.995148i	$0.531370\pi$
$728$	0	0
$729$	29.9213	1.10820
$730$	0	0
$731$	3.09926	0.114630
$732$	0	0
$733$	33.0424	1.22045	0.610225	−	0.792228i	$-0.291079\pi$
$733$	33.0424	1.22045	0.610225	+	0.792228i	$0.291079\pi$
$734$	0	0
$735$	18.1539	0.669615
$736$	0	0
$737$	−32.7948	−1.20801
$738$	0	0
$739$	25.0063	0.919871	0.459936	−	0.887952i	$-0.347872\pi$
$739$	25.0063	0.919871	0.459936	+	0.887952i	$0.347872\pi$
$740$	0	0
$741$	−10.5703	−0.388309
$742$	0	0
$743$	22.6235	0.829974	0.414987	−	0.909827i	$-0.363786\pi$
$743$	22.6235	0.829974	0.414987	+	0.909827i	$0.363786\pi$
$744$	0	0
$745$	−10.3758	−0.380140
$746$	0	0
$747$	9.81651	0.359167
$748$	0	0
$749$	−5.10330	−0.186470
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	27.0838	0.986989
$754$	0	0
$755$	−14.8386	−0.540033
$756$	0	0
$757$	−26.9028	−0.977798	−0.488899	−	0.872340i	$-0.662601\pi$
$757$	−26.9028	−0.977798	−0.488899	+	0.872340i	$0.662601\pi$
$758$	0	0
$759$	−42.8944	−1.55697
$760$	0	0
$761$	28.0727	1.01763	0.508816	−	0.860875i	$-0.330083\pi$
$761$	28.0727	1.01763	0.508816	+	0.860875i	$0.330083\pi$
$762$	0	0
$763$	−19.5514	−0.707808
$764$	0	0
$765$	0.279344	0.0100997
$766$	0	0
$767$	9.82879	0.354897
$768$	0	0
$769$	−35.1068	−1.26598	−0.632992	−	0.774159i	$-0.718173\pi$
$769$	−35.1068	−1.26598	−0.632992	+	0.774159i	$0.718173\pi$
$770$	0	0
$771$	−19.1779	−0.690674
$772$	0	0
$773$	−5.48620	−0.197325	−0.0986624	−	0.995121i	$-0.531456\pi$
$773$	−5.48620	−0.197325	−0.0986624	+	0.995121i	$0.531456\pi$
$774$	0	0
$775$	−36.7597	−1.32045
$776$	0	0
$777$	−20.9697	−0.752285
$778$	0	0
$779$	−41.4785	−1.48612
$780$	0	0
$781$	−37.5628	−1.34410
$782$	0	0
$783$	−56.6922	−2.02602
$784$	0	0
$785$	−9.57143	−0.341619
$786$	0	0
$787$	−25.4160	−0.905981	−0.452991	−	0.891515i	$-0.649643\pi$
$787$	−25.4160	−0.905981	−0.452991	+	0.891515i	$0.649643\pi$
$788$	0	0
$789$	−29.3315	−1.04423
$790$	0	0
$791$	−4.03682	−0.143533
$792$	0	0
$793$	−5.29267	−0.187948
$794$	0	0
$795$	9.91139	0.351521
$796$	0	0
$797$	−40.8826	−1.44814	−0.724069	−	0.689728i	$-0.757730\pi$
$797$	−40.8826	−1.44814	−0.724069	+	0.689728i	$0.757730\pi$
$798$	0	0
$799$	−1.88537	−0.0666996
$800$	0	0
$801$	−2.20790	−0.0780123
$802$	0	0
$803$	−31.3566	−1.10655
$804$	0	0
$805$	25.8050	0.909508
$806$	0	0
$807$	−27.3811	−0.963860
$808$	0	0
$809$	−51.6625	−1.81636	−0.908179	−	0.418583i	$-0.862527\pi$
$809$	−51.6625	−1.81636	−0.908179	+	0.418583i	$0.862527\pi$
$810$	0	0
$811$	−55.0321	−1.93244	−0.966219	−	0.257721i	$-0.917028\pi$
$811$	−55.0321	−1.93244	−0.966219	+	0.257721i	$0.917028\pi$
$812$	0	0
$813$	−41.5497	−1.45721
$814$	0	0
$815$	−16.1101	−0.564311
$816$	0	0
$817$	−52.8193	−1.84791
$818$	0	0
$819$	3.15517	0.110251
$820$	0	0
$821$	−8.56648	−0.298972	−0.149486	−	0.988764i	$-0.547762\pi$
$821$	−8.56648	−0.298972	−0.149486	+	0.988764i	$0.547762\pi$
$822$	0	0
$823$	−4.90193	−0.170871	−0.0854353	−	0.996344i	$-0.527228\pi$
$823$	−4.90193	−0.170871	−0.0854353	+	0.996344i	$0.527228\pi$
$824$	0	0
$825$	−26.6165	−0.926668
$826$	0	0
$827$	19.1627	0.666354	0.333177	−	0.942864i	$-0.391879\pi$
$827$	19.1627	0.666354	0.333177	+	0.942864i	$0.391879\pi$
$828$	0	0
$829$	18.7658	0.651764	0.325882	−	0.945411i	$-0.394339\pi$
$829$	18.7658	0.651764	0.325882	+	0.945411i	$0.394339\pi$
$830$	0	0
$831$	40.7481	1.41353
$832$	0	0
$833$	7.90681	0.273955
$834$	0	0
$835$	−4.43839	−0.153597
$836$	0	0
$837$	46.5986	1.61068
$838$	0	0
$839$	−30.2856	−1.04558	−0.522788	−	0.852463i	$-0.675108\pi$
$839$	−30.2856	−1.04558	−0.522788	+	0.852463i	$0.675108\pi$
$840$	0	0
$841$	72.4271	2.49749
$842$	0	0
$843$	−22.7158	−0.782374
$844$	0	0
$845$	9.17883	0.315761
$846$	0	0
$847$	24.5472	0.843453
$848$	0	0
$849$	10.7142	0.367711
$850$	0	0
$851$	−20.8313	−0.714088
$852$	0	0
$853$	−4.94940	−0.169464	−0.0847322	−	0.996404i	$-0.527003\pi$
$853$	−4.94940	−0.169464	−0.0847322	+	0.996404i	$0.527003\pi$
$854$	0	0
$855$	−4.76073	−0.162814
$856$	0	0
$857$	−26.2353	−0.896180	−0.448090	−	0.893989i	$-0.647896\pi$
$857$	−26.2353	−0.896180	−0.448090	+	0.893989i	$0.647896\pi$
$858$	0	0
$859$	42.7727	1.45939	0.729694	−	0.683774i	$-0.239663\pi$
$859$	42.7727	1.45939	0.729694	+	0.683774i	$0.239663\pi$
$860$	0	0
$861$	−36.0223	−1.22764
$862$	0	0
$863$	−14.8540	−0.505637	−0.252818	−	0.967514i	$-0.581358\pi$
$863$	−14.8540	−0.505637	−0.252818	+	0.967514i	$0.581358\pi$
$864$	0	0
$865$	−13.5057	−0.459207
$866$	0	0
$867$	25.0474	0.850654
$868$	0	0
$869$	−59.4974	−2.01831
$870$	0	0
$871$	−6.97211	−0.236241
$872$	0	0
$873$	9.68893	0.327921
$874$	0	0
$875$	34.0417	1.15082
$876$	0	0
$877$	−17.3327	−0.585283	−0.292641	−	0.956222i	$-0.594534\pi$
$877$	−17.3327	−0.585283	−0.292641	+	0.956222i	$0.594534\pi$
$878$	0	0
$879$	41.4930	1.39952
$880$	0	0
$881$	−18.7894	−0.633030	−0.316515	−	0.948588i	$-0.602513\pi$
$881$	−18.7894	−0.633030	−0.316515	+	0.948588i	$0.602513\pi$
$882$	0	0
$883$	29.9198	1.00688	0.503440	−	0.864030i	$-0.332067\pi$
$883$	29.9198	1.00688	0.503440	+	0.864030i	$0.332067\pi$
$884$	0	0
$885$	−12.8795	−0.432940
$886$	0	0
$887$	30.1215	1.01138	0.505690	−	0.862715i	$-0.331238\pi$
$887$	30.1215	1.01138	0.505690	+	0.862715i	$0.331238\pi$
$888$	0	0
$889$	−2.26693	−0.0760304
$890$	0	0
$891$	24.5059	0.820978
$892$	0	0
$893$	32.1315	1.07524
$894$	0	0
$895$	−4.07048	−0.136061
$896$	0	0
$897$	−9.11926	−0.304483
$898$	0	0
$899$	−83.3688	−2.78051
$900$	0	0
$901$	4.31685	0.143815
$902$	0	0
$903$	−45.8713	−1.52650
$904$	0	0
$905$	−3.19323	−0.106146
$906$	0	0
$907$	2.74774	0.0912373	0.0456187	−	0.998959i	$-0.485474\pi$
$907$	2.74774	0.0912373	0.0456187	+	0.998959i	$0.485474\pi$
$908$	0	0
$909$	7.51246	0.249173
$910$	0	0
$911$	5.14281	0.170389	0.0851944	−	0.996364i	$-0.472849\pi$
$911$	5.14281	0.170389	0.0851944	+	0.996364i	$0.472849\pi$
$912$	0	0
$913$	51.3157	1.69830
$914$	0	0
$915$	6.93544	0.229279
$916$	0	0
$917$	94.6839	3.12674
$918$	0	0
$919$	8.71960	0.287633	0.143817	−	0.989604i	$-0.454062\pi$
$919$	8.71960	0.287633	0.143817	+	0.989604i	$0.454062\pi$
$920$	0	0
$921$	15.5249	0.511562
$922$	0	0
$923$	−7.98578	−0.262855
$924$	0	0
$925$	−12.9261	−0.425007
$926$	0	0
$927$	−1.06701	−0.0350451
$928$	0	0
$929$	9.11020	0.298896	0.149448	−	0.988770i	$-0.452250\pi$
$929$	9.11020	0.298896	0.149448	+	0.988770i	$0.452250\pi$
$930$	0	0
$931$	−134.752	−4.41632
$932$	0	0
$933$	−14.8928	−0.487569
$934$	0	0
$935$	1.46027	0.0477559
$936$	0	0
$937$	12.9601	0.423389	0.211695	−	0.977336i	$-0.432102\pi$
$937$	12.9601	0.423389	0.211695	+	0.977336i	$0.432102\pi$
$938$	0	0
$939$	46.2585	1.50959
$940$	0	0
$941$	4.49902	0.146664	0.0733319	−	0.997308i	$-0.476637\pi$
$941$	4.49902	0.146664	0.0733319	+	0.997308i	$0.476637\pi$
$942$	0	0
$943$	−35.7845	−1.16530
$944$	0	0
$945$	−20.2981	−0.660299
$946$	0	0
$947$	20.8503	0.677544	0.338772	−	0.940869i	$-0.389988\pi$
$947$	20.8503	0.677544	0.338772	+	0.940869i	$0.389988\pi$
$948$	0	0
$949$	−6.66635	−0.216399
$950$	0	0
$951$	14.1415	0.458570
$952$	0	0
$953$	−22.2768	−0.721617	−0.360808	−	0.932640i	$-0.617499\pi$
$953$	−22.2768	−0.721617	−0.360808	+	0.932640i	$0.617499\pi$
$954$	0	0
$955$	−4.97042	−0.160839
$956$	0	0
$957$	−60.3647	−1.95131
$958$	0	0
$959$	−80.3149	−2.59350
$960$	0	0
$961$	37.5257	1.21051
$962$	0	0
$963$	0.812255	0.0261745
$964$	0	0
$965$	15.4766	0.498208
$966$	0	0
$967$	−25.8026	−0.829756	−0.414878	−	0.909877i	$-0.636176\pi$
$967$	−25.8026	−0.829756	−0.414878	+	0.909877i	$0.636176\pi$
$968$	0	0
$969$	6.03280	0.193801
$970$	0	0
$971$	11.4672	0.368000	0.184000	−	0.982926i	$-0.441095\pi$
$971$	11.4672	0.368000	0.184000	+	0.982926i	$0.441095\pi$
$972$	0	0
$973$	−51.1238	−1.63895
$974$	0	0
$975$	−5.65861	−0.181221
$976$	0	0
$977$	−29.6851	−0.949709	−0.474854	−	0.880064i	$-0.657499\pi$
$977$	−29.6851	−0.949709	−0.474854	+	0.880064i	$0.657499\pi$
$978$	0	0
$979$	−11.5418	−0.368877
$980$	0	0
$981$	3.11185	0.0993538
$982$	0	0
$983$	−22.5394	−0.718895	−0.359447	−	0.933165i	$-0.617035\pi$
$983$	−22.5394	−0.718895	−0.359447	+	0.933165i	$0.617035\pi$
$984$	0	0
$985$	−3.10748	−0.0990125
$986$	0	0
$987$	27.9049	0.888222
$988$	0	0
$989$	−45.5685	−1.44899
$990$	0	0
$991$	29.9451	0.951236	0.475618	−	0.879652i	$-0.342224\pi$
$991$	29.9451	0.951236	0.475618	+	0.879652i	$0.342224\pi$
$992$	0	0
$993$	−3.51637	−0.111589
$994$	0	0
$995$	−3.42071	−0.108444
$996$	0	0
$997$	−12.9260	−0.409372	−0.204686	−	0.978828i	$-0.565617\pi$
$997$	−12.9260	−0.409372	−0.204686	+	0.978828i	$0.565617\pi$
$998$	0	0
$999$	16.3858	0.518425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.13	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.13	✓	49	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)

\(f(q)\)	\(=\)	\(q-1.49420 q^{3} -0.747906 q^{5} +4.82128 q^{7} -0.767369 q^{9} +O(q^{10})\)	\(q-1.49420 q^{3} -0.747906 q^{5} +4.82128 q^{7} -0.767369 q^{9} -4.01141 q^{11} -0.852818 q^{13} +1.11752 q^{15} +0.486730 q^{17} -8.29511 q^{19} -7.20396 q^{21} -7.15640 q^{23} -4.44064 q^{25} +5.62920 q^{27} -10.0711 q^{29} +8.27802 q^{31} +5.99385 q^{33} -3.60587 q^{35} +2.91086 q^{37} +1.27428 q^{39} +5.00035 q^{41} +6.36752 q^{43} +0.573920 q^{45} -3.87355 q^{47} +16.2448 q^{49} -0.727271 q^{51} +8.86908 q^{53} +3.00016 q^{55} +12.3946 q^{57} -11.5251 q^{59} +6.20609 q^{61} -3.69970 q^{63} +0.637828 q^{65} +8.17537 q^{67} +10.6931 q^{69} +9.36399 q^{71} +7.81685 q^{73} +6.63520 q^{75} -19.3402 q^{77} +14.8320 q^{79} -6.10904 q^{81} -12.7924 q^{83} -0.364028 q^{85} +15.0482 q^{87} +2.87724 q^{89} -4.11168 q^{91} -12.3690 q^{93} +6.20397 q^{95} -12.6262 q^{97} +3.07823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

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