LMFDB - Embedded newform 6008.2.a.b.1.26

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:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.26
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)

$f(q)$	$=$	$q-0.0368230 q^{3} -3.59158 q^{5} -0.547211 q^{7} -2.99864 q^{9} +O(q^{10})$	$q-0.0368230 q^{3} -3.59158 q^{5} -0.547211 q^{7} -2.99864 q^{9} +1.30756 q^{11} +0.987833 q^{13} +0.132253 q^{15} +1.11658 q^{17} +5.02241 q^{19} +0.0201500 q^{21} -2.20016 q^{23} +7.89946 q^{25} +0.220888 q^{27} +0.0479181 q^{29} +3.20026 q^{31} -0.0481482 q^{33} +1.96535 q^{35} -6.60748 q^{37} -0.0363750 q^{39} -0.410505 q^{41} -7.81229 q^{43} +10.7699 q^{45} +9.58834 q^{47} -6.70056 q^{49} -0.0411157 q^{51} -1.16958 q^{53} -4.69620 q^{55} -0.184940 q^{57} +7.75541 q^{59} +8.47931 q^{61} +1.64089 q^{63} -3.54788 q^{65} +5.28110 q^{67} +0.0810166 q^{69} +14.7035 q^{71} -10.4830 q^{73} -0.290882 q^{75} -0.715509 q^{77} +5.56063 q^{79} +8.98780 q^{81} -0.439874 q^{83} -4.01027 q^{85} -0.00176449 q^{87} +11.3734 q^{89} -0.540553 q^{91} -0.117843 q^{93} -18.0384 q^{95} -17.5419 q^{97} -3.92090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.0368230	−0.0212598	−0.0106299	−	0.999944i	$-0.503384\pi$
$3$	−0.0368230	−0.0212598	−0.0106299	+	0.999944i	$0.503384\pi$
$4$	0	0
$5$	−3.59158	−1.60620	−0.803102	−	0.595842i	$-0.796819\pi$
$5$	−3.59158	−1.60620	−0.803102	+	0.595842i	$0.796819\pi$
$6$	0	0
$7$	−0.547211	−0.206826	−0.103413	−	0.994638i	$-0.532976\pi$
$7$	−0.547211	−0.206826	−0.103413	+	0.994638i	$0.532976\pi$
$8$	0	0
$9$	−2.99864	−0.999548
$10$	0	0
$11$	1.30756	0.394243	0.197122	−	0.980379i	$-0.436841\pi$
$11$	1.30756	0.394243	0.197122	+	0.980379i	$0.436841\pi$
$12$	0	0
$13$	0.987833	0.273976	0.136988	−	0.990573i	$-0.456258\pi$
$13$	0.987833	0.273976	0.136988	+	0.990573i	$0.456258\pi$
$14$	0	0
$15$	0.132253	0.0341476
$16$	0	0
$17$	1.11658	0.270809	0.135405	−	0.990790i	$-0.456767\pi$
$17$	1.11658	0.270809	0.135405	+	0.990790i	$0.456767\pi$
$18$	0	0
$19$	5.02241	1.15222	0.576110	−	0.817373i	$-0.304570\pi$
$19$	5.02241	1.15222	0.576110	+	0.817373i	$0.304570\pi$
$20$	0	0
$21$	0.0201500	0.00439708
$22$	0	0
$23$	−2.20016	−0.458765	−0.229383	−	0.973336i	$-0.573671\pi$
$23$	−2.20016	−0.458765	−0.229383	+	0.973336i	$0.573671\pi$
$24$	0	0
$25$	7.89946	1.57989
$26$	0	0
$27$	0.220888	0.0425100
$28$	0	0
$29$	0.0479181	0.00889817	0.00444909	−	0.999990i	$-0.498584\pi$
$29$	0.0479181	0.00889817	0.00444909	+	0.999990i	$0.498584\pi$
$30$	0	0
$31$	3.20026	0.574784	0.287392	−	0.957813i	$-0.407212\pi$
$31$	3.20026	0.574784	0.287392	+	0.957813i	$0.407212\pi$
$32$	0	0
$33$	−0.0481482	−0.00838153
$34$	0	0
$35$	1.96535	0.332205
$36$	0	0
$37$	−6.60748	−1.08626	−0.543131	−	0.839648i	$-0.682761\pi$
$37$	−6.60748	−1.08626	−0.543131	+	0.839648i	$0.682761\pi$
$38$	0	0
$39$	−0.0363750	−0.00582467
$40$	0	0
$41$	−0.410505	−0.0641101	−0.0320551	−	0.999486i	$-0.510205\pi$
$41$	−0.410505	−0.0641101	−0.0320551	+	0.999486i	$0.510205\pi$
$42$	0	0
$43$	−7.81229	−1.19136	−0.595682	−	0.803221i	$-0.703118\pi$
$43$	−7.81229	−1.19136	−0.595682	+	0.803221i	$0.703118\pi$
$44$	0	0
$45$	10.7699	1.60548
$46$	0	0
$47$	9.58834	1.39860	0.699302	−	0.714827i	$-0.253494\pi$
$47$	9.58834	1.39860	0.699302	+	0.714827i	$0.253494\pi$
$48$	0	0
$49$	−6.70056	−0.957223
$50$	0	0
$51$	−0.0411157	−0.00575735
$52$	0	0
$53$	−1.16958	−0.160655	−0.0803274	−	0.996769i	$-0.525597\pi$
$53$	−1.16958	−0.160655	−0.0803274	+	0.996769i	$0.525597\pi$
$54$	0	0
$55$	−4.69620	−0.633235
$56$	0	0
$57$	−0.184940	−0.0244959
$58$	0	0
$59$	7.75541	1.00967	0.504834	−	0.863216i	$-0.331554\pi$
$59$	7.75541	1.00967	0.504834	+	0.863216i	$0.331554\pi$
$60$	0	0
$61$	8.47931	1.08566	0.542832	−	0.839841i	$-0.317352\pi$
$61$	8.47931	1.08566	0.542832	+	0.839841i	$0.317352\pi$
$62$	0	0
$63$	1.64089	0.206733
$64$	0	0
$65$	−3.54788	−0.440061
$66$	0	0
$67$	5.28110	0.645189	0.322594	−	0.946537i	$-0.395445\pi$
$67$	5.28110	0.645189	0.322594	+	0.946537i	$0.395445\pi$
$68$	0	0
$69$	0.0810166	0.00975325
$70$	0	0
$71$	14.7035	1.74498	0.872491	−	0.488631i	$-0.162504\pi$
$71$	14.7035	1.74498	0.872491	+	0.488631i	$0.162504\pi$
$72$	0	0
$73$	−10.4830	−1.22694	−0.613469	−	0.789719i	$-0.710227\pi$
$73$	−10.4830	−1.22694	−0.613469	+	0.789719i	$0.710227\pi$
$74$	0	0
$75$	−0.290882	−0.0335882
$76$	0	0
$77$	−0.715509	−0.0815399
$78$	0	0
$79$	5.56063	0.625620	0.312810	−	0.949816i	$-0.398730\pi$
$79$	5.56063	0.625620	0.312810	+	0.949816i	$0.398730\pi$
$80$	0	0
$81$	8.98780	0.998644
$82$	0	0
$83$	−0.439874	−0.0482824	−0.0241412	−	0.999709i	$-0.507685\pi$
$83$	−0.439874	−0.0482824	−0.0241412	+	0.999709i	$0.507685\pi$
$84$	0	0
$85$	−4.01027	−0.434975
$86$	0	0
$87$	−0.00176449	−0.000189173 0
$88$	0	0
$89$	11.3734	1.20558	0.602791	−	0.797899i	$-0.294055\pi$
$89$	11.3734	1.20558	0.602791	+	0.797899i	$0.294055\pi$
$90$	0	0
$91$	−0.540553	−0.0566654
$92$	0	0
$93$	−0.117843	−0.0122198
$94$	0	0
$95$	−18.0384	−1.85070
$96$	0	0
$97$	−17.5419	−1.78111	−0.890553	−	0.454880i	$-0.849682\pi$
$97$	−17.5419	−1.78111	−0.890553	+	0.454880i	$0.849682\pi$
$98$	0	0
$99$	−3.92090	−0.394065
$100$	0	0
$101$	2.46866	0.245641	0.122821	−	0.992429i	$-0.460806\pi$
$101$	2.46866	0.245641	0.122821	+	0.992429i	$0.460806\pi$
$102$	0	0
$103$	17.0932	1.68424	0.842119	−	0.539291i	$-0.181308\pi$
$103$	17.0932	1.68424	0.842119	+	0.539291i	$0.181308\pi$
$104$	0	0
$105$	−0.0723702	−0.00706261
$106$	0	0
$107$	−13.5655	−1.31143	−0.655714	−	0.755009i	$-0.727632\pi$
$107$	−13.5655	−1.31143	−0.655714	+	0.755009i	$0.727632\pi$
$108$	0	0
$109$	5.90938	0.566016	0.283008	−	0.959118i	$-0.408668\pi$
$109$	5.90938	0.566016	0.283008	+	0.959118i	$0.408668\pi$
$110$	0	0
$111$	0.243307	0.0230937
$112$	0	0
$113$	−8.81842	−0.829567	−0.414784	−	0.909920i	$-0.636143\pi$
$113$	−8.81842	−0.829567	−0.414784	+	0.909920i	$0.636143\pi$
$114$	0	0
$115$	7.90205	0.736870
$116$	0	0
$117$	−2.96216	−0.273852
$118$	0	0
$119$	−0.611002	−0.0560105
$120$	0	0
$121$	−9.29029	−0.844572
$122$	0	0
$123$	0.0151160	0.00136297
$124$	0	0
$125$	−10.4136	−0.931424
$126$	0	0
$127$	−16.1932	−1.43691	−0.718457	−	0.695571i	$-0.755151\pi$
$127$	−16.1932	−1.43691	−0.718457	+	0.695571i	$0.755151\pi$
$128$	0	0
$129$	0.287672	0.0253281
$130$	0	0
$131$	4.30546	0.376170	0.188085	−	0.982153i	$-0.439772\pi$
$131$	4.30546	0.376170	0.188085	+	0.982153i	$0.439772\pi$
$132$	0	0
$133$	−2.74831	−0.238309
$134$	0	0
$135$	−0.793338	−0.0682797
$136$	0	0
$137$	−18.2477	−1.55901	−0.779503	−	0.626399i	$-0.784528\pi$
$137$	−18.2477	−1.55901	−0.779503	+	0.626399i	$0.784528\pi$
$138$	0	0
$139$	−15.2483	−1.29334	−0.646672	−	0.762768i	$-0.723840\pi$
$139$	−15.2483	−1.29334	−0.646672	+	0.762768i	$0.723840\pi$
$140$	0	0
$141$	−0.353072	−0.0297340
$142$	0	0
$143$	1.29165	0.108013
$144$	0	0
$145$	−0.172102	−0.0142923
$146$	0	0
$147$	0.246735	0.0203504
$148$	0	0
$149$	10.5489	0.864199	0.432099	−	0.901826i	$-0.357773\pi$
$149$	10.5489	0.864199	0.432099	+	0.901826i	$0.357773\pi$
$150$	0	0
$151$	10.6271	0.864823	0.432411	−	0.901676i	$-0.357663\pi$
$151$	10.6271	0.864823	0.432411	+	0.901676i	$0.357663\pi$
$152$	0	0
$153$	−3.34821	−0.270687
$154$	0	0
$155$	−11.4940	−0.923220
$156$	0	0
$157$	−17.9067	−1.42911	−0.714556	−	0.699578i	$-0.753371\pi$
$157$	−17.9067	−1.42911	−0.714556	+	0.699578i	$0.753371\pi$
$158$	0	0
$159$	0.0430677	0.00341549
$160$	0	0
$161$	1.20395	0.0948846
$162$	0	0
$163$	−10.0276	−0.785421	−0.392710	−	0.919662i	$-0.628463\pi$
$163$	−10.0276	−0.785421	−0.392710	+	0.919662i	$0.628463\pi$
$164$	0	0
$165$	0.172928	0.0134625
$166$	0	0
$167$	−17.6913	−1.36899	−0.684497	−	0.729016i	$-0.739978\pi$
$167$	−17.6913	−1.36899	−0.684497	+	0.729016i	$0.739978\pi$
$168$	0	0
$169$	−12.0242	−0.924937
$170$	0	0
$171$	−15.0604	−1.15170
$172$	0	0
$173$	10.3232	0.784857	0.392428	−	0.919783i	$-0.371635\pi$
$173$	10.3232	0.784857	0.392428	+	0.919783i	$0.371635\pi$
$174$	0	0
$175$	−4.32267	−0.326763
$176$	0	0
$177$	−0.285578	−0.0214653
$178$	0	0
$179$	−0.442138	−0.0330469	−0.0165235	−	0.999863i	$-0.505260\pi$
$179$	−0.442138	−0.0330469	−0.0165235	+	0.999863i	$0.505260\pi$
$180$	0	0
$181$	0.333021	0.0247533	0.0123766	−	0.999923i	$-0.496060\pi$
$181$	0.333021	0.0247533	0.0123766	+	0.999923i	$0.496060\pi$
$182$	0	0
$183$	−0.312234	−0.0230810
$184$	0	0
$185$	23.7313	1.74476
$186$	0	0
$187$	1.45999	0.106765
$188$	0	0
$189$	−0.120872	−0.00879218
$190$	0	0
$191$	−14.5590	−1.05345	−0.526726	−	0.850035i	$-0.676581\pi$
$191$	−14.5590	−1.05345	−0.526726	+	0.850035i	$0.676581\pi$
$192$	0	0
$193$	−23.1033	−1.66301	−0.831504	−	0.555518i	$-0.812520\pi$
$193$	−23.1033	−1.66301	−0.831504	+	0.555518i	$0.812520\pi$
$194$	0	0
$195$	0.130644	0.00935560
$196$	0	0
$197$	17.1008	1.21838	0.609192	−	0.793023i	$-0.291494\pi$
$197$	17.1008	1.21838	0.609192	+	0.793023i	$0.291494\pi$
$198$	0	0
$199$	−9.75739	−0.691683	−0.345842	−	0.938293i	$-0.612407\pi$
$199$	−9.75739	−0.691683	−0.345842	+	0.938293i	$0.612407\pi$
$200$	0	0
$201$	−0.194466	−0.0137166
$202$	0	0
$203$	−0.0262213	−0.00184038
$204$	0	0
$205$	1.47436	0.102974
$206$	0	0
$207$	6.59750	0.458558
$208$	0	0
$209$	6.56708	0.454255
$210$	0	0
$211$	−9.30499	−0.640582	−0.320291	−	0.947319i	$-0.603781\pi$
$211$	−9.30499	−0.640582	−0.320291	+	0.947319i	$0.603781\pi$
$212$	0	0
$213$	−0.541427	−0.0370979
$214$	0	0
$215$	28.0585	1.91357
$216$	0	0
$217$	−1.75122	−0.118880
$218$	0	0
$219$	0.386015	0.0260845
$220$	0	0
$221$	1.10299	0.0741952
$222$	0	0
$223$	−15.9472	−1.06790	−0.533951	−	0.845516i	$-0.679293\pi$
$223$	−15.9472	−1.06790	−0.533951	+	0.845516i	$0.679293\pi$
$224$	0	0
$225$	−23.6877	−1.57918
$226$	0	0
$227$	−22.8186	−1.51453	−0.757263	−	0.653110i	$-0.773464\pi$
$227$	−22.8186	−1.51453	−0.757263	+	0.653110i	$0.773464\pi$
$228$	0	0
$229$	−13.3410	−0.881597	−0.440798	−	0.897606i	$-0.645305\pi$
$229$	−13.3410	−0.881597	−0.440798	+	0.897606i	$0.645305\pi$
$230$	0	0
$231$	0.0263472	0.00173352
$232$	0	0
$233$	5.87184	0.384677	0.192339	−	0.981329i	$-0.438393\pi$
$233$	5.87184	0.384677	0.192339	+	0.981329i	$0.438393\pi$
$234$	0	0
$235$	−34.4373	−2.24644
$236$	0	0
$237$	−0.204759	−0.0133006
$238$	0	0
$239$	−11.1257	−0.719662	−0.359831	−	0.933018i	$-0.617166\pi$
$239$	−11.1257	−0.719662	−0.359831	+	0.933018i	$0.617166\pi$
$240$	0	0
$241$	−23.6564	−1.52384	−0.761921	−	0.647670i	$-0.775744\pi$
$241$	−23.6564	−1.52384	−0.761921	+	0.647670i	$0.775744\pi$
$242$	0	0
$243$	−0.993623	−0.0637409
$244$	0	0
$245$	24.0656	1.53750
$246$	0	0
$247$	4.96130	0.315680
$248$	0	0
$249$	0.0161975	0.00102647
$250$	0	0
$251$	0.674814	0.0425939	0.0212969	−	0.999773i	$-0.493220\pi$
$251$	0.674814	0.0425939	0.0212969	+	0.999773i	$0.493220\pi$
$252$	0	0
$253$	−2.87684	−0.180865
$254$	0	0
$255$	0.147670	0.00924749
$256$	0	0
$257$	28.1791	1.75776	0.878881	−	0.477042i	$-0.158291\pi$
$257$	28.1791	1.75776	0.878881	+	0.477042i	$0.158291\pi$
$258$	0	0
$259$	3.61568	0.224668
$260$	0	0
$261$	−0.143689	−0.00889415
$262$	0	0
$263$	25.1677	1.55190	0.775952	−	0.630792i	$-0.217270\pi$
$263$	25.1677	1.55190	0.775952	+	0.630792i	$0.217270\pi$
$264$	0	0
$265$	4.20066	0.258044
$266$	0	0
$267$	−0.418805	−0.0256304
$268$	0	0
$269$	−25.5178	−1.55585	−0.777923	−	0.628359i	$-0.783727\pi$
$269$	−25.5178	−1.55585	−0.777923	+	0.628359i	$0.783727\pi$
$270$	0	0
$271$	−12.0696	−0.733177	−0.366589	−	0.930383i	$-0.619474\pi$
$271$	−12.0696	−0.733177	−0.366589	+	0.930383i	$0.619474\pi$
$272$	0	0
$273$	0.0199048	0.00120469
$274$	0	0
$275$	10.3290	0.622862
$276$	0	0
$277$	−5.71005	−0.343084	−0.171542	−	0.985177i	$-0.554875\pi$
$277$	−5.71005	−0.343084	−0.171542	+	0.985177i	$0.554875\pi$
$278$	0	0
$279$	−9.59644	−0.574524
$280$	0	0
$281$	−0.212448	−0.0126736	−0.00633680	−	0.999980i	$-0.502017\pi$
$281$	−0.212448	−0.0126736	−0.00633680	+	0.999980i	$0.502017\pi$
$282$	0	0
$283$	−15.3052	−0.909802	−0.454901	−	0.890542i	$-0.650325\pi$
$283$	−15.3052	−0.909802	−0.454901	+	0.890542i	$0.650325\pi$
$284$	0	0
$285$	0.664228	0.0393455
$286$	0	0
$287$	0.224633	0.0132597
$288$	0	0
$289$	−15.7533	−0.926662
$290$	0	0
$291$	0.645944	0.0378659
$292$	0	0
$293$	23.9617	1.39986	0.699929	−	0.714212i	$-0.253215\pi$
$293$	23.9617	1.39986	0.699929	+	0.714212i	$0.253215\pi$
$294$	0	0
$295$	−27.8542	−1.62173
$296$	0	0
$297$	0.288824	0.0167593
$298$	0	0
$299$	−2.17339	−0.125690
$300$	0	0
$301$	4.27497	0.246405
$302$	0	0
$303$	−0.0909037	−0.00522228
$304$	0	0
$305$	−30.4541	−1.74380
$306$	0	0
$307$	30.9475	1.76627	0.883133	−	0.469123i	$-0.155430\pi$
$307$	30.9475	1.76627	0.883133	+	0.469123i	$0.155430\pi$
$308$	0	0
$309$	−0.629422	−0.0358066
$310$	0	0
$311$	−27.8105	−1.57699	−0.788495	−	0.615041i	$-0.789140\pi$
$311$	−27.8105	−1.57699	−0.788495	+	0.615041i	$0.789140\pi$
$312$	0	0
$313$	32.2987	1.82563	0.912814	−	0.408375i	$-0.133904\pi$
$313$	32.2987	1.82563	0.912814	+	0.408375i	$0.133904\pi$
$314$	0	0
$315$	−5.89339	−0.332055
$316$	0	0
$317$	−32.6102	−1.83157	−0.915787	−	0.401665i	$-0.868432\pi$
$317$	−32.6102	−1.83157	−0.915787	+	0.401665i	$0.868432\pi$
$318$	0	0
$319$	0.0626557	0.00350805
$320$	0	0
$321$	0.499524	0.0278807
$322$	0	0
$323$	5.60790	0.312032
$324$	0	0
$325$	7.80335	0.432852
$326$	0	0
$327$	−0.217601	−0.0120334
$328$	0	0
$329$	−5.24684	−0.289268
$330$	0	0
$331$	−16.4948	−0.906636	−0.453318	−	0.891349i	$-0.649760\pi$
$331$	−16.4948	−0.906636	−0.453318	+	0.891349i	$0.649760\pi$
$332$	0	0
$333$	19.8135	1.08577
$334$	0	0
$335$	−18.9675	−1.03630
$336$	0	0
$337$	−2.05633	−0.112015	−0.0560077	−	0.998430i	$-0.517837\pi$
$337$	−2.05633	−0.112015	−0.0560077	+	0.998430i	$0.517837\pi$
$338$	0	0
$339$	0.324721	0.0176364
$340$	0	0
$341$	4.18453	0.226605
$342$	0	0
$343$	7.49709	0.404805
$344$	0	0
$345$	−0.290978	−0.0156657
$346$	0	0
$347$	−11.7232	−0.629334	−0.314667	−	0.949202i	$-0.601893\pi$
$347$	−11.7232	−0.629334	−0.314667	+	0.949202i	$0.601893\pi$
$348$	0	0
$349$	9.49831	0.508433	0.254217	−	0.967147i	$-0.418182\pi$
$349$	9.49831	0.508433	0.254217	+	0.967147i	$0.418182\pi$
$350$	0	0
$351$	0.218201	0.0116467
$352$	0	0
$353$	28.6099	1.52275	0.761377	−	0.648310i	$-0.224524\pi$
$353$	28.6099	1.52275	0.761377	+	0.648310i	$0.224524\pi$
$354$	0	0
$355$	−52.8087	−2.80280
$356$	0	0
$357$	0.0224990	0.00119077
$358$	0	0
$359$	9.76360	0.515303	0.257651	−	0.966238i	$-0.417051\pi$
$359$	9.76360	0.515303	0.257651	+	0.966238i	$0.417051\pi$
$360$	0	0
$361$	6.22456	0.327609
$362$	0	0
$363$	0.342097	0.0179554
$364$	0	0
$365$	37.6504	1.97071
$366$	0	0
$367$	−12.7863	−0.667437	−0.333719	−	0.942673i	$-0.608304\pi$
$367$	−12.7863	−0.667437	−0.333719	+	0.942673i	$0.608304\pi$
$368$	0	0
$369$	1.23096	0.0640812
$370$	0	0
$371$	0.640009	0.0332276
$372$	0	0
$373$	24.6501	1.27633	0.638166	−	0.769899i	$-0.279693\pi$
$373$	24.6501	1.27633	0.638166	+	0.769899i	$0.279693\pi$
$374$	0	0
$375$	0.383462	0.0198019
$376$	0	0
$377$	0.0473351	0.00243788
$378$	0	0
$379$	14.3795	0.738626	0.369313	−	0.929305i	$-0.379593\pi$
$379$	14.3795	0.738626	0.369313	+	0.929305i	$0.379593\pi$
$380$	0	0
$381$	0.596283	0.0305485
$382$	0	0
$383$	−25.9455	−1.32575	−0.662877	−	0.748728i	$-0.730665\pi$
$383$	−25.9455	−1.32575	−0.662877	+	0.748728i	$0.730665\pi$
$384$	0	0
$385$	2.56981	0.130970
$386$	0	0
$387$	23.4263	1.19082
$388$	0	0
$389$	−31.0273	−1.57314	−0.786572	−	0.617498i	$-0.788146\pi$
$389$	−31.0273	−1.57314	−0.786572	+	0.617498i	$0.788146\pi$
$390$	0	0
$391$	−2.45665	−0.124238
$392$	0	0
$393$	−0.158540	−0.00799730
$394$	0	0
$395$	−19.9715	−1.00487
$396$	0	0
$397$	9.06988	0.455204	0.227602	−	0.973754i	$-0.426911\pi$
$397$	9.06988	0.455204	0.227602	+	0.973754i	$0.426911\pi$
$398$	0	0
$399$	0.101201	0.00506640
$400$	0	0
$401$	−30.2787	−1.51204	−0.756022	−	0.654546i	$-0.772860\pi$
$401$	−30.2787	−1.51204	−0.756022	+	0.654546i	$0.772860\pi$
$402$	0	0
$403$	3.16133	0.157477
$404$	0	0
$405$	−32.2804	−1.60403
$406$	0	0
$407$	−8.63966	−0.428252
$408$	0	0
$409$	−3.15860	−0.156183	−0.0780913	−	0.996946i	$-0.524883\pi$
$409$	−3.15860	−0.156183	−0.0780913	+	0.996946i	$0.524883\pi$
$410$	0	0
$411$	0.671935	0.0331441
$412$	0	0
$413$	−4.24384	−0.208826
$414$	0	0
$415$	1.57984	0.0775514
$416$	0	0
$417$	0.561489	0.0274962
$418$	0	0
$419$	−14.9861	−0.732117	−0.366058	−	0.930592i	$-0.619293\pi$
$419$	−14.9861	−0.732117	−0.366058	+	0.930592i	$0.619293\pi$
$420$	0	0
$421$	22.3764	1.09056	0.545279	−	0.838255i	$-0.316424\pi$
$421$	22.3764	1.09056	0.545279	+	0.838255i	$0.316424\pi$
$422$	0	0
$423$	−28.7520	−1.39797
$424$	0	0
$425$	8.82035	0.427850
$426$	0	0
$427$	−4.63997	−0.224544
$428$	0	0
$429$	−0.0475624	−0.00229634
$430$	0	0
$431$	23.3150	1.12304	0.561522	−	0.827462i	$-0.310216\pi$
$431$	23.3150	1.12304	0.561522	+	0.827462i	$0.310216\pi$
$432$	0	0
$433$	31.8072	1.52856	0.764278	−	0.644887i	$-0.223096\pi$
$433$	31.8072	1.52856	0.764278	+	0.644887i	$0.223096\pi$
$434$	0	0
$435$	0.00633731	0.000303851 0
$436$	0	0
$437$	−11.0501	−0.528598
$438$	0	0
$439$	25.9005	1.23616	0.618081	−	0.786115i	$-0.287911\pi$
$439$	25.9005	1.23616	0.618081	+	0.786115i	$0.287911\pi$
$440$	0	0
$441$	20.0926	0.956790
$442$	0	0
$443$	−14.0153	−0.665887	−0.332943	−	0.942947i	$-0.608042\pi$
$443$	−14.0153	−0.665887	−0.332943	+	0.942947i	$0.608042\pi$
$444$	0	0
$445$	−40.8486	−1.93641
$446$	0	0
$447$	−0.388442	−0.0183727
$448$	0	0
$449$	35.2077	1.66155	0.830776	−	0.556607i	$-0.187897\pi$
$449$	35.2077	1.66155	0.830776	+	0.556607i	$0.187897\pi$
$450$	0	0
$451$	−0.536759	−0.0252750
$452$	0	0
$453$	−0.391323	−0.0183860
$454$	0	0
$455$	1.94144	0.0910161
$456$	0	0
$457$	−10.5389	−0.492991	−0.246495	−	0.969144i	$-0.579279\pi$
$457$	−10.5389	−0.492991	−0.246495	+	0.969144i	$0.579279\pi$
$458$	0	0
$459$	0.246639	0.0115121
$460$	0	0
$461$	12.3500	0.575199	0.287599	−	0.957751i	$-0.407143\pi$
$461$	12.3500	0.575199	0.287599	+	0.957751i	$0.407143\pi$
$462$	0	0
$463$	−25.1695	−1.16972	−0.584862	−	0.811133i	$-0.698851\pi$
$463$	−25.1695	−1.16972	−0.584862	+	0.811133i	$0.698851\pi$
$464$	0	0
$465$	0.423244	0.0196275
$466$	0	0
$467$	10.6410	0.492407	0.246203	−	0.969218i	$-0.420817\pi$
$467$	10.6410	0.492407	0.246203	+	0.969218i	$0.420817\pi$
$468$	0	0
$469$	−2.88987	−0.133442
$470$	0	0
$471$	0.659380	0.0303826
$472$	0	0
$473$	−10.2150	−0.469687
$474$	0	0
$475$	39.6743	1.82038
$476$	0	0
$477$	3.50717	0.160582
$478$	0	0
$479$	27.1910	1.24239	0.621193	−	0.783657i	$-0.286648\pi$
$479$	27.1910	1.24239	0.621193	+	0.783657i	$0.286648\pi$
$480$	0	0
$481$	−6.52709	−0.297610
$482$	0	0
$483$	−0.0443331	−0.00201723
$484$	0	0
$485$	63.0030	2.86082
$486$	0	0
$487$	−13.2201	−0.599061	−0.299531	−	0.954087i	$-0.596830\pi$
$487$	−13.2201	−0.599061	−0.299531	+	0.954087i	$0.596830\pi$
$488$	0	0
$489$	0.369246	0.0166979
$490$	0	0
$491$	41.1468	1.85693	0.928464	−	0.371422i	$-0.121130\pi$
$491$	41.1468	1.85693	0.928464	+	0.371422i	$0.121130\pi$
$492$	0	0
$493$	0.0535042	0.00240971
$494$	0	0
$495$	14.0822	0.632949
$496$	0	0
$497$	−8.04590	−0.360908
$498$	0	0
$499$	33.7054	1.50886	0.754430	−	0.656380i	$-0.227913\pi$
$499$	33.7054	1.50886	0.754430	+	0.656380i	$0.227913\pi$
$500$	0	0
$501$	0.651447	0.0291045
$502$	0	0
$503$	−43.2694	−1.92929	−0.964645	−	0.263554i	$-0.915105\pi$
$503$	−43.2694	−1.92929	−0.964645	+	0.263554i	$0.915105\pi$
$504$	0	0
$505$	−8.86641	−0.394550
$506$	0	0
$507$	0.442767	0.0196640
$508$	0	0
$509$	13.3195	0.590375	0.295188	−	0.955439i	$-0.404618\pi$
$509$	13.3195	0.590375	0.295188	+	0.955439i	$0.404618\pi$
$510$	0	0
$511$	5.73639	0.253763
$512$	0	0
$513$	1.10939	0.0489808
$514$	0	0
$515$	−61.3915	−2.70523
$516$	0	0
$517$	12.5373	0.551390
$518$	0	0
$519$	−0.380131	−0.0166859
$520$	0	0
$521$	10.5245	0.461087	0.230544	−	0.973062i	$-0.425950\pi$
$521$	10.5245	0.461087	0.230544	+	0.973062i	$0.425950\pi$
$522$	0	0
$523$	−42.6580	−1.86531	−0.932653	−	0.360775i	$-0.882512\pi$
$523$	−42.6580	−1.86531	−0.932653	+	0.360775i	$0.882512\pi$
$524$	0	0
$525$	0.159174	0.00694691
$526$	0	0
$527$	3.57334	0.155657
$528$	0	0
$529$	−18.1593	−0.789535
$530$	0	0
$531$	−23.2557	−1.00921
$532$	0	0
$533$	−0.405511	−0.0175646
$534$	0	0
$535$	48.7217	2.10642
$536$	0	0
$537$	0.0162809	0.000702571 0
$538$	0	0
$539$	−8.76137	−0.377379
$540$	0	0
$541$	−24.8753	−1.06947	−0.534737	−	0.845018i	$-0.679589\pi$
$541$	−24.8753	−1.06947	−0.534737	+	0.845018i	$0.679589\pi$
$542$	0	0
$543$	−0.0122629	−0.000526250 0
$544$	0	0
$545$	−21.2240	−0.909137
$546$	0	0
$547$	38.4320	1.64323	0.821616	−	0.570041i	$-0.193073\pi$
$547$	38.4320	1.64323	0.821616	+	0.570041i	$0.193073\pi$
$548$	0	0
$549$	−25.4264	−1.08517
$550$	0	0
$551$	0.240664	0.0102526
$552$	0	0
$553$	−3.04284	−0.129395
$554$	0	0
$555$	−0.873859	−0.0370932
$556$	0	0
$557$	6.34246	0.268739	0.134369	−	0.990931i	$-0.457099\pi$
$557$	6.34246	0.268739	0.134369	+	0.990931i	$0.457099\pi$
$558$	0	0
$559$	−7.71724	−0.326405
$560$	0	0
$561$	−0.0537612	−0.00226980
$562$	0	0
$563$	16.2363	0.684280	0.342140	−	0.939649i	$-0.388848\pi$
$563$	16.2363	0.684280	0.342140	+	0.939649i	$0.388848\pi$
$564$	0	0
$565$	31.6721	1.33245
$566$	0	0
$567$	−4.91822	−0.206546
$568$	0	0
$569$	−6.88953	−0.288824	−0.144412	−	0.989518i	$-0.546129\pi$
$569$	−6.88953	−0.288824	−0.144412	+	0.989518i	$0.546129\pi$
$570$	0	0
$571$	14.1366	0.591597	0.295798	−	0.955250i	$-0.404414\pi$
$571$	14.1366	0.591597	0.295798	+	0.955250i	$0.404414\pi$
$572$	0	0
$573$	0.536106	0.0223962
$574$	0	0
$575$	−17.3801	−0.724799
$576$	0	0
$577$	6.91701	0.287959	0.143979	−	0.989581i	$-0.454010\pi$
$577$	6.91701	0.287959	0.143979	+	0.989581i	$0.454010\pi$
$578$	0	0
$579$	0.850732	0.0353552
$580$	0	0
$581$	0.240704	0.00998607
$582$	0	0
$583$	−1.52930	−0.0633371
$584$	0	0
$585$	10.6388	0.439862
$586$	0	0
$587$	29.4326	1.21481	0.607407	−	0.794391i	$-0.292210\pi$
$587$	29.4326	1.21481	0.607407	+	0.794391i	$0.292210\pi$
$588$	0	0
$589$	16.0730	0.662277
$590$	0	0
$591$	−0.629705	−0.0259026
$592$	0	0
$593$	−24.5283	−1.00725	−0.503627	−	0.863921i	$-0.668002\pi$
$593$	−24.5283	−1.00725	−0.503627	+	0.863921i	$0.668002\pi$
$594$	0	0
$595$	2.19446	0.0899643
$596$	0	0
$597$	0.359297	0.0147050
$598$	0	0
$599$	−32.5940	−1.33176	−0.665878	−	0.746061i	$-0.731943\pi$
$599$	−32.5940	−1.33176	−0.665878	+	0.746061i	$0.731943\pi$
$600$	0	0
$601$	18.3281	0.747618	0.373809	−	0.927506i	$-0.378052\pi$
$601$	18.3281	0.747618	0.373809	+	0.927506i	$0.378052\pi$
$602$	0	0
$603$	−15.8361	−0.644897
$604$	0	0
$605$	33.3668	1.35656
$606$	0	0
$607$	−36.6995	−1.48959	−0.744794	−	0.667294i	$-0.767452\pi$
$607$	−36.6995	−1.48959	−0.744794	+	0.667294i	$0.767452\pi$
$608$	0	0
$609$	0.000965548 0	3.91260e−5 0
$610$	0	0
$611$	9.47168	0.383183
$612$	0	0
$613$	31.1587	1.25849	0.629245	−	0.777207i	$-0.283364\pi$
$613$	31.1587	1.25849	0.629245	+	0.777207i	$0.283364\pi$
$614$	0	0
$615$	−0.0542905	−0.00218920
$616$	0	0
$617$	9.30996	0.374805	0.187402	−	0.982283i	$-0.439993\pi$
$617$	9.30996	0.374805	0.187402	+	0.982283i	$0.439993\pi$
$618$	0	0
$619$	−6.72625	−0.270351	−0.135175	−	0.990822i	$-0.543160\pi$
$619$	−6.72625	−0.270351	−0.135175	+	0.990822i	$0.543160\pi$
$620$	0	0
$621$	−0.485990	−0.0195021
$622$	0	0
$623$	−6.22367	−0.249346
$624$	0	0
$625$	−2.09586	−0.0838346
$626$	0	0
$627$	−0.241820	−0.00965736
$628$	0	0
$629$	−7.37775	−0.294170
$630$	0	0
$631$	10.1334	0.403405	0.201702	−	0.979447i	$-0.435353\pi$
$631$	10.1334	0.403405	0.201702	+	0.979447i	$0.435353\pi$
$632$	0	0
$633$	0.342638	0.0136186
$634$	0	0
$635$	58.1592	2.30798
$636$	0	0
$637$	−6.61904	−0.262256
$638$	0	0
$639$	−44.0905	−1.74419
$640$	0	0
$641$	−7.01212	−0.276962	−0.138481	−	0.990365i	$-0.544222\pi$
$641$	−7.01212	−0.276962	−0.138481	+	0.990365i	$0.544222\pi$
$642$	0	0
$643$	−35.7876	−1.41133	−0.705663	−	0.708547i	$-0.749351\pi$
$643$	−35.7876	−1.41133	−0.705663	+	0.708547i	$0.749351\pi$
$644$	0	0
$645$	−1.03320	−0.0406821
$646$	0	0
$647$	0.566397	0.0222674	0.0111337	−	0.999938i	$-0.496456\pi$
$647$	0.566397	0.0222674	0.0111337	+	0.999938i	$0.496456\pi$
$648$	0	0
$649$	10.1406	0.398055
$650$	0	0
$651$	0.0644851	0.00252737
$652$	0	0
$653$	−29.1741	−1.14167	−0.570836	−	0.821064i	$-0.693381\pi$
$653$	−29.1741	−1.14167	−0.570836	+	0.821064i	$0.693381\pi$
$654$	0	0
$655$	−15.4634	−0.604206
$656$	0	0
$657$	31.4347	1.22638
$658$	0	0
$659$	−35.5997	−1.38677	−0.693383	−	0.720569i	$-0.743881\pi$
$659$	−35.5997	−1.38677	−0.693383	+	0.720569i	$0.743881\pi$
$660$	0	0
$661$	−17.7417	−0.690071	−0.345035	−	0.938590i	$-0.612133\pi$
$661$	−17.7417	−0.690071	−0.345035	+	0.938590i	$0.612133\pi$
$662$	0	0
$663$	−0.0406155	−0.00157738
$664$	0	0
$665$	9.87079	0.382773
$666$	0	0
$667$	−0.105428	−0.00408217
$668$	0	0
$669$	0.587223	0.0227034
$670$	0	0
$671$	11.0872	0.428016
$672$	0	0
$673$	15.9294	0.614034	0.307017	−	0.951704i	$-0.400669\pi$
$673$	15.9294	0.614034	0.307017	+	0.951704i	$0.400669\pi$
$674$	0	0
$675$	1.74490	0.0671611
$676$	0	0
$677$	18.3273	0.704377	0.352188	−	0.935929i	$-0.385438\pi$
$677$	18.3273	0.704377	0.352188	+	0.935929i	$0.385438\pi$
$678$	0	0
$679$	9.59909	0.368379
$680$	0	0
$681$	0.840252	0.0321985
$682$	0	0
$683$	13.9610	0.534205	0.267102	−	0.963668i	$-0.413934\pi$
$683$	13.9610	0.534205	0.267102	+	0.963668i	$0.413934\pi$
$684$	0	0
$685$	65.5381	2.50408
$686$	0	0
$687$	0.491255	0.0187426
$688$	0	0
$689$	−1.15535	−0.0440155
$690$	0	0
$691$	12.3418	0.469504	0.234752	−	0.972055i	$-0.424572\pi$
$691$	12.3418	0.469504	0.234752	+	0.972055i	$0.424572\pi$
$692$	0	0
$693$	2.14556	0.0815030
$694$	0	0
$695$	54.7655	2.07737
$696$	0	0
$697$	−0.458360	−0.0173616
$698$	0	0
$699$	−0.216219	−0.00817816
$700$	0	0
$701$	41.7980	1.57869	0.789345	−	0.613950i	$-0.210420\pi$
$701$	41.7980	1.57869	0.789345	+	0.613950i	$0.210420\pi$
$702$	0	0
$703$	−33.1854	−1.25161
$704$	0	0
$705$	1.26809	0.0477589
$706$	0	0
$707$	−1.35088	−0.0508051
$708$	0	0
$709$	5.58664	0.209811	0.104905	−	0.994482i	$-0.466546\pi$
$709$	5.58664	0.209811	0.104905	+	0.994482i	$0.466546\pi$
$710$	0	0
$711$	−16.6744	−0.625338
$712$	0	0
$713$	−7.04109	−0.263691
$714$	0	0
$715$	−4.63906	−0.173491
$716$	0	0
$717$	0.409682	0.0152999
$718$	0	0
$719$	−32.2923	−1.20430	−0.602150	−	0.798383i	$-0.705689\pi$
$719$	−32.2923	−1.20430	−0.602150	+	0.798383i	$0.705689\pi$
$720$	0	0
$721$	−9.35356	−0.348345
$722$	0	0
$723$	0.871100	0.0323966
$724$	0	0
$725$	0.378527	0.0140581
$726$	0	0
$727$	−22.9665	−0.851779	−0.425890	−	0.904775i	$-0.640039\pi$
$727$	−22.9665	−0.851779	−0.425890	+	0.904775i	$0.640039\pi$
$728$	0	0
$729$	−26.9268	−0.997289
$730$	0	0
$731$	−8.72302	−0.322632
$732$	0	0
$733$	−45.3379	−1.67460	−0.837298	−	0.546747i	$-0.815866\pi$
$733$	−45.3379	−1.67460	−0.837298	+	0.546747i	$0.815866\pi$
$734$	0	0
$735$	−0.886169	−0.0326868
$736$	0	0
$737$	6.90534	0.254361
$738$	0	0
$739$	−15.5760	−0.572974	−0.286487	−	0.958084i	$-0.592487\pi$
$739$	−15.5760	−0.572974	−0.286487	+	0.958084i	$0.592487\pi$
$740$	0	0
$741$	−0.182690	−0.00671129
$742$	0	0
$743$	−4.24312	−0.155665	−0.0778326	−	0.996966i	$-0.524800\pi$
$743$	−4.24312	−0.155665	−0.0778326	+	0.996966i	$0.524800\pi$
$744$	0	0
$745$	−37.8872	−1.38808
$746$	0	0
$747$	1.31902	0.0482606
$748$	0	0
$749$	7.42320	0.271238
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−0.0248487	−0.000905537 0
$754$	0	0
$755$	−38.1682	−1.38908
$756$	0	0
$757$	43.9764	1.59835	0.799174	−	0.601099i	$-0.205270\pi$
$757$	43.9764	1.59835	0.799174	+	0.601099i	$0.205270\pi$
$758$	0	0
$759$	0.105934	0.00384515
$760$	0	0
$761$	8.95905	0.324765	0.162383	−	0.986728i	$-0.448082\pi$
$761$	8.95905	0.324765	0.162383	+	0.986728i	$0.448082\pi$
$762$	0	0
$763$	−3.23367	−0.117067
$764$	0	0
$765$	12.0254	0.434779
$766$	0	0
$767$	7.66106	0.276625
$768$	0	0
$769$	−40.5452	−1.46210	−0.731049	−	0.682325i	$-0.760969\pi$
$769$	−40.5452	−1.46210	−0.731049	+	0.682325i	$0.760969\pi$
$770$	0	0
$771$	−1.03764	−0.0373696
$772$	0	0
$773$	19.1455	0.688615	0.344307	−	0.938857i	$-0.388114\pi$
$773$	19.1455	0.688615	0.344307	+	0.938857i	$0.388114\pi$
$774$	0	0
$775$	25.2803	0.908096
$776$	0	0
$777$	−0.133140	−0.00477639
$778$	0	0
$779$	−2.06172	−0.0738689
$780$	0	0
$781$	19.2256	0.687947
$782$	0	0
$783$	0.0105846	0.000378261 0
$784$	0	0
$785$	64.3135	2.29545
$786$	0	0
$787$	27.6116	0.984247	0.492124	−	0.870525i	$-0.336221\pi$
$787$	27.6116	0.984247	0.492124	+	0.870525i	$0.336221\pi$
$788$	0	0
$789$	−0.926750	−0.0329932
$790$	0	0
$791$	4.82554	0.171576
$792$	0	0
$793$	8.37614	0.297446
$794$	0	0
$795$	−0.154681	−0.00548597
$796$	0	0
$797$	41.2018	1.45944	0.729721	−	0.683745i	$-0.239650\pi$
$797$	41.2018	1.45944	0.729721	+	0.683745i	$0.239650\pi$
$798$	0	0
$799$	10.7061	0.378755
$800$	0	0
$801$	−34.1049	−1.20504
$802$	0	0
$803$	−13.7071	−0.483712
$804$	0	0
$805$	−4.32409	−0.152404
$806$	0	0
$807$	0.939642	0.0330770
$808$	0	0
$809$	−50.1610	−1.76357	−0.881783	−	0.471655i	$-0.843657\pi$
$809$	−50.1610	−1.76357	−0.881783	+	0.471655i	$0.843657\pi$
$810$	0	0
$811$	19.6046	0.688411	0.344206	−	0.938894i	$-0.388148\pi$
$811$	19.6046	0.688411	0.344206	+	0.938894i	$0.388148\pi$
$812$	0	0
$813$	0.444440	0.0155872
$814$	0	0
$815$	36.0149	1.26155
$816$	0	0
$817$	−39.2365	−1.37271
$818$	0	0
$819$	1.62093	0.0566397
$820$	0	0
$821$	12.5097	0.436593	0.218297	−	0.975882i	$-0.429950\pi$
$821$	12.5097	0.436593	0.218297	+	0.975882i	$0.429950\pi$
$822$	0	0
$823$	22.4157	0.781363	0.390682	−	0.920526i	$-0.372239\pi$
$823$	22.4157	0.781363	0.390682	+	0.920526i	$0.372239\pi$
$824$	0	0
$825$	−0.380345	−0.0132419
$826$	0	0
$827$	−6.43398	−0.223731	−0.111866	−	0.993723i	$-0.535683\pi$
$827$	−6.43398	−0.223731	−0.111866	+	0.993723i	$0.535683\pi$
$828$	0	0
$829$	44.2022	1.53520	0.767602	−	0.640927i	$-0.221450\pi$
$829$	44.2022	1.53520	0.767602	+	0.640927i	$0.221450\pi$
$830$	0	0
$831$	0.210262	0.00729389
$832$	0	0
$833$	−7.48169	−0.259225
$834$	0	0
$835$	63.5397	2.19888
$836$	0	0
$837$	0.706900	0.0244340
$838$	0	0
$839$	31.3849	1.08353	0.541763	−	0.840531i	$-0.317757\pi$
$839$	31.3849	1.08353	0.541763	+	0.840531i	$0.317757\pi$
$840$	0	0
$841$	−28.9977	−0.999921
$842$	0	0
$843$	0.00782299	0.000269438 0
$844$	0	0
$845$	43.1858	1.48564
$846$	0	0
$847$	5.08375	0.174680
$848$	0	0
$849$	0.563585	0.0193422
$850$	0	0
$851$	14.5375	0.498339
$852$	0	0
$853$	−19.7815	−0.677305	−0.338653	−	0.940911i	$-0.609971\pi$
$853$	−19.7815	−0.677305	−0.338653	+	0.940911i	$0.609971\pi$
$854$	0	0
$855$	54.0907	1.84986
$856$	0	0
$857$	−28.0193	−0.957122	−0.478561	−	0.878054i	$-0.658842\pi$
$857$	−28.0193	−0.957122	−0.478561	+	0.878054i	$0.658842\pi$
$858$	0	0
$859$	16.4600	0.561606	0.280803	−	0.959765i	$-0.409399\pi$
$859$	16.4600	0.561606	0.280803	+	0.959765i	$0.409399\pi$
$860$	0	0
$861$	−0.00827166	−0.000281897 0
$862$	0	0
$863$	−4.60626	−0.156799	−0.0783995	−	0.996922i	$-0.524981\pi$
$863$	−4.60626	−0.156799	−0.0783995	+	0.996922i	$0.524981\pi$
$864$	0	0
$865$	−37.0765	−1.26064
$866$	0	0
$867$	0.580083	0.0197006
$868$	0	0
$869$	7.27085	0.246647
$870$	0	0
$871$	5.21684	0.176766
$872$	0	0
$873$	52.6018	1.78030
$874$	0	0
$875$	5.69845	0.192643
$876$	0	0
$877$	46.1424	1.55812	0.779059	−	0.626951i	$-0.215697\pi$
$877$	46.1424	1.55812	0.779059	+	0.626951i	$0.215697\pi$
$878$	0	0
$879$	−0.882343	−0.0297607
$880$	0	0
$881$	5.07723	0.171056	0.0855280	−	0.996336i	$-0.472742\pi$
$881$	5.07723	0.171056	0.0855280	+	0.996336i	$0.472742\pi$
$882$	0	0
$883$	−54.4257	−1.83157	−0.915786	−	0.401668i	$-0.868431\pi$
$883$	−54.4257	−1.83157	−0.915786	+	0.401668i	$0.868431\pi$
$884$	0	0
$885$	1.02568	0.0344777
$886$	0	0
$887$	−20.9554	−0.703613	−0.351806	−	0.936073i	$-0.614432\pi$
$887$	−20.9554	−0.703613	−0.351806	+	0.936073i	$0.614432\pi$
$888$	0	0
$889$	8.86109	0.297191
$890$	0	0
$891$	11.7521	0.393709
$892$	0	0
$893$	48.1565	1.61150
$894$	0	0
$895$	1.58797	0.0530801
$896$	0	0
$897$	0.0800309	0.00267215
$898$	0	0
$899$	0.153351	0.00511453
$900$	0	0
$901$	−1.30593	−0.0435068
$902$	0	0
$903$	−0.157417	−0.00523852
$904$	0	0
$905$	−1.19607	−0.0397588
$906$	0	0
$907$	−46.8048	−1.55413	−0.777063	−	0.629422i	$-0.783292\pi$
$907$	−46.8048	−1.55413	−0.777063	+	0.629422i	$0.783292\pi$
$908$	0	0
$909$	−7.40265	−0.245530
$910$	0	0
$911$	−35.9833	−1.19218	−0.596090	−	0.802917i	$-0.703280\pi$
$911$	−35.9833	−1.19218	−0.596090	+	0.802917i	$0.703280\pi$
$912$	0	0
$913$	−0.575160	−0.0190350
$914$	0	0
$915$	1.12141	0.0370728
$916$	0	0
$917$	−2.35600	−0.0778018
$918$	0	0
$919$	−51.3043	−1.69237	−0.846186	−	0.532888i	$-0.821107\pi$
$919$	−51.3043	−1.69237	−0.846186	+	0.532888i	$0.821107\pi$
$920$	0	0
$921$	−1.13958	−0.0375504
$922$	0	0
$923$	14.5246	0.478082
$924$	0	0
$925$	−52.1955	−1.71618
$926$	0	0
$927$	−51.2563	−1.68348
$928$	0	0
$929$	38.2441	1.25475	0.627374	−	0.778718i	$-0.284130\pi$
$929$	38.2441	1.25475	0.627374	+	0.778718i	$0.284130\pi$
$930$	0	0
$931$	−33.6529	−1.10293
$932$	0	0
$933$	1.02407	0.0335265
$934$	0	0
$935$	−5.24366	−0.171486
$936$	0	0
$937$	−55.3953	−1.80969	−0.904843	−	0.425746i	$-0.860012\pi$
$937$	−55.3953	−1.80969	−0.904843	+	0.425746i	$0.860012\pi$
$938$	0	0
$939$	−1.18934	−0.0388125
$940$	0	0
$941$	−1.61479	−0.0526406	−0.0263203	−	0.999654i	$-0.508379\pi$
$941$	−1.61479	−0.0526406	−0.0263203	+	0.999654i	$0.508379\pi$
$942$	0	0
$943$	0.903177	0.0294115
$944$	0	0
$945$	0.434123	0.0141220
$946$	0	0
$947$	−27.3784	−0.889677	−0.444839	−	0.895611i	$-0.646739\pi$
$947$	−27.3784	−0.889677	−0.444839	+	0.895611i	$0.646739\pi$
$948$	0	0
$949$	−10.3554	−0.336151
$950$	0	0
$951$	1.20081	0.0389389
$952$	0	0
$953$	−43.3160	−1.40314	−0.701572	−	0.712599i	$-0.747518\pi$
$953$	−43.3160	−1.40314	−0.701572	+	0.712599i	$0.747518\pi$
$954$	0	0
$955$	52.2898	1.69206
$956$	0	0
$957$	−0.00230717	−7.45803e−5 0
$958$	0	0
$959$	9.98533	0.322443
$960$	0	0
$961$	−20.7583	−0.669624
$962$	0	0
$963$	40.6782	1.31084
$964$	0	0
$965$	82.9772	2.67113
$966$	0	0
$967$	−3.91608	−0.125933	−0.0629664	−	0.998016i	$-0.520056\pi$
$967$	−3.91608	−0.125933	−0.0629664	+	0.998016i	$0.520056\pi$
$968$	0	0
$969$	−0.206500	−0.00663373
$970$	0	0
$971$	27.4920	0.882260	0.441130	−	0.897443i	$-0.354578\pi$
$971$	27.4920	0.882260	0.441130	+	0.897443i	$0.354578\pi$
$972$	0	0
$973$	8.34403	0.267497
$974$	0	0
$975$	−0.287343	−0.00920234
$976$	0	0
$977$	−20.7942	−0.665265	−0.332632	−	0.943057i	$-0.607937\pi$
$977$	−20.7942	−0.665265	−0.332632	+	0.943057i	$0.607937\pi$
$978$	0	0
$979$	14.8714	0.475293
$980$	0	0
$981$	−17.7201	−0.565760
$982$	0	0
$983$	−11.2392	−0.358474	−0.179237	−	0.983806i	$-0.557363\pi$
$983$	−11.2392	−0.358474	−0.179237	+	0.983806i	$0.557363\pi$
$984$	0	0
$985$	−61.4190	−1.95697
$986$	0	0
$987$	0.193205	0.00614977
$988$	0	0
$989$	17.1883	0.546556
$990$	0	0
$991$	51.4475	1.63428	0.817142	−	0.576436i	$-0.195557\pi$
$991$	51.4475	1.63428	0.817142	+	0.576436i	$0.195557\pi$
$992$	0	0
$993$	0.607388	0.0192749
$994$	0	0
$995$	35.0445	1.11098
$996$	0	0
$997$	−49.3824	−1.56396	−0.781979	−	0.623305i	$-0.785789\pi$
$997$	−49.3824	−1.56396	−0.781979	+	0.623305i	$0.785789\pi$
$998$	0	0
$999$	−1.45952	−0.0461770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.26	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.26	✓	44	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)

\(f(q)\)	\(=\)	\(q-0.0368230 q^{3} -3.59158 q^{5} -0.547211 q^{7} -2.99864 q^{9} +O(q^{10})\)	\(q-0.0368230 q^{3} -3.59158 q^{5} -0.547211 q^{7} -2.99864 q^{9} +1.30756 q^{11} +0.987833 q^{13} +0.132253 q^{15} +1.11658 q^{17} +5.02241 q^{19} +0.0201500 q^{21} -2.20016 q^{23} +7.89946 q^{25} +0.220888 q^{27} +0.0479181 q^{29} +3.20026 q^{31} -0.0481482 q^{33} +1.96535 q^{35} -6.60748 q^{37} -0.0363750 q^{39} -0.410505 q^{41} -7.81229 q^{43} +10.7699 q^{45} +9.58834 q^{47} -6.70056 q^{49} -0.0411157 q^{51} -1.16958 q^{53} -4.69620 q^{55} -0.184940 q^{57} +7.75541 q^{59} +8.47931 q^{61} +1.64089 q^{63} -3.54788 q^{65} +5.28110 q^{67} +0.0810166 q^{69} +14.7035 q^{71} -10.4830 q^{73} -0.290882 q^{75} -0.715509 q^{77} +5.56063 q^{79} +8.98780 q^{81} -0.439874 q^{83} -4.01027 q^{85} -0.00176449 q^{87} +11.3734 q^{89} -0.540553 q^{91} -0.117843 q^{93} -18.0384 q^{95} -17.5419 q^{97} -3.92090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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