LMFDB - Embedded newform 6008.2.a.b.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.98092 q^{3} +3.05916 q^{5} +0.297244 q^{7} +5.88586 q^{9} +O(q^{10})$	$q-2.98092 q^{3} +3.05916 q^{5} +0.297244 q^{7} +5.88586 q^{9} -0.132067 q^{11} +3.60162 q^{13} -9.11910 q^{15} +6.86161 q^{17} -2.04931 q^{19} -0.886059 q^{21} -8.92554 q^{23} +4.35846 q^{25} -8.60250 q^{27} -4.43490 q^{29} -2.21683 q^{31} +0.393681 q^{33} +0.909317 q^{35} -5.90972 q^{37} -10.7361 q^{39} -9.20315 q^{41} +0.709880 q^{43} +18.0058 q^{45} +0.369774 q^{47} -6.91165 q^{49} -20.4539 q^{51} -11.1347 q^{53} -0.404015 q^{55} +6.10883 q^{57} +5.23739 q^{59} +0.397250 q^{61} +1.74954 q^{63} +11.0179 q^{65} -4.96553 q^{67} +26.6063 q^{69} -3.02363 q^{71} -9.74973 q^{73} -12.9922 q^{75} -0.0392562 q^{77} -4.16717 q^{79} +7.98576 q^{81} +4.18223 q^{83} +20.9908 q^{85} +13.2201 q^{87} +10.9084 q^{89} +1.07056 q^{91} +6.60819 q^{93} -6.26918 q^{95} -2.82202 q^{97} -0.777329 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.98092	−1.72103	−0.860516	−	0.509423i	$-0.829859\pi$
$3$	−2.98092	−1.72103	−0.860516	+	0.509423i	$0.829859\pi$
$4$	0	0
$5$	3.05916	1.36810	0.684049	−	0.729436i	$-0.260218\pi$
$5$	3.05916	1.36810	0.684049	+	0.729436i	$0.260218\pi$
$6$	0	0
$7$	0.297244	0.112348	0.0561738	−	0.998421i	$-0.482110\pi$
$7$	0.297244	0.112348	0.0561738	+	0.998421i	$0.482110\pi$
$8$	0	0
$9$	5.88586	1.96195
$10$	0	0
$11$	−0.132067	−0.0398198	−0.0199099	−	0.999802i	$-0.506338\pi$
$11$	−0.132067	−0.0398198	−0.0199099	+	0.999802i	$0.506338\pi$
$12$	0	0
$13$	3.60162	0.998909	0.499455	−	0.866340i	$-0.333534\pi$
$13$	3.60162	0.998909	0.499455	+	0.866340i	$0.333534\pi$
$14$	0	0
$15$	−9.11910	−2.35454
$16$	0	0
$17$	6.86161	1.66419	0.832093	−	0.554636i	$-0.187143\pi$
$17$	6.86161	1.66419	0.832093	+	0.554636i	$0.187143\pi$
$18$	0	0
$19$	−2.04931	−0.470145	−0.235072	−	0.971978i	$-0.575533\pi$
$19$	−2.04931	−0.470145	−0.235072	+	0.971978i	$0.575533\pi$
$20$	0	0
$21$	−0.886059	−0.193354
$22$	0	0
$23$	−8.92554	−1.86110	−0.930552	−	0.366161i	$-0.880672\pi$
$23$	−8.92554	−1.86110	−0.930552	+	0.366161i	$0.880672\pi$
$24$	0	0
$25$	4.35846	0.871691
$26$	0	0
$27$	−8.60250	−1.65555
$28$	0	0
$29$	−4.43490	−0.823541	−0.411770	−	0.911288i	$-0.635089\pi$
$29$	−4.43490	−0.823541	−0.411770	+	0.911288i	$0.635089\pi$
$30$	0	0
$31$	−2.21683	−0.398155	−0.199077	−	0.979984i	$-0.563795\pi$
$31$	−2.21683	−0.398155	−0.199077	+	0.979984i	$0.563795\pi$
$32$	0	0
$33$	0.393681	0.0685311
$34$	0	0
$35$	0.909317	0.153703
$36$	0	0
$37$	−5.90972	−0.971552	−0.485776	−	0.874083i	$-0.661463\pi$
$37$	−5.90972	−0.971552	−0.485776	+	0.874083i	$0.661463\pi$
$38$	0	0
$39$	−10.7361	−1.71916
$40$	0	0
$41$	−9.20315	−1.43729	−0.718645	−	0.695377i	$-0.755237\pi$
$41$	−9.20315	−1.43729	−0.718645	+	0.695377i	$0.755237\pi$
$42$	0	0
$43$	0.709880	0.108256	0.0541278	−	0.998534i	$-0.482762\pi$
$43$	0.709880	0.108256	0.0541278	+	0.998534i	$0.482762\pi$
$44$	0	0
$45$	18.0058	2.68414
$46$	0	0
$47$	0.369774	0.0539370	0.0269685	−	0.999636i	$-0.491415\pi$
$47$	0.369774	0.0539370	0.0269685	+	0.999636i	$0.491415\pi$
$48$	0	0
$49$	−6.91165	−0.987378
$50$	0	0
$51$	−20.4539	−2.86412
$52$	0	0
$53$	−11.1347	−1.52947	−0.764736	−	0.644344i	$-0.777131\pi$
$53$	−11.1347	−1.52947	−0.764736	+	0.644344i	$0.777131\pi$
$54$	0	0
$55$	−0.404015	−0.0544773
$56$	0	0
$57$	6.10883	0.809134
$58$	0	0
$59$	5.23739	0.681851	0.340925	−	0.940090i	$-0.389260\pi$
$59$	5.23739	0.681851	0.340925	+	0.940090i	$0.389260\pi$
$60$	0	0
$61$	0.397250	0.0508626	0.0254313	−	0.999677i	$-0.491904\pi$
$61$	0.397250	0.0508626	0.0254313	+	0.999677i	$0.491904\pi$
$62$	0	0
$63$	1.74954	0.220421
$64$	0	0
$65$	11.0179	1.36661
$66$	0	0
$67$	−4.96553	−0.606636	−0.303318	−	0.952889i	$-0.598094\pi$
$67$	−4.96553	−0.606636	−0.303318	+	0.952889i	$0.598094\pi$
$68$	0	0
$69$	26.6063	3.20302
$70$	0	0
$71$	−3.02363	−0.358839	−0.179419	−	0.983773i	$-0.557422\pi$
$71$	−3.02363	−0.358839	−0.179419	+	0.983773i	$0.557422\pi$
$72$	0	0
$73$	−9.74973	−1.14112	−0.570560	−	0.821256i	$-0.693274\pi$
$73$	−9.74973	−1.14112	−0.570560	+	0.821256i	$0.693274\pi$
$74$	0	0
$75$	−12.9922	−1.50021
$76$	0	0
$77$	−0.0392562	−0.00447366
$78$	0	0
$79$	−4.16717	−0.468844	−0.234422	−	0.972135i	$-0.575320\pi$
$79$	−4.16717	−0.468844	−0.234422	+	0.972135i	$0.575320\pi$
$80$	0	0
$81$	7.98576	0.887306
$82$	0	0
$83$	4.18223	0.459059	0.229530	−	0.973302i	$-0.426281\pi$
$83$	4.18223	0.459059	0.229530	+	0.973302i	$0.426281\pi$
$84$	0	0
$85$	20.9908	2.27677
$86$	0	0
$87$	13.2201	1.41734
$88$	0	0
$89$	10.9084	1.15629	0.578146	−	0.815933i	$-0.303776\pi$
$89$	10.9084	1.15629	0.578146	+	0.815933i	$0.303776\pi$
$90$	0	0
$91$	1.07056	0.112225
$92$	0	0
$93$	6.60819	0.685237
$94$	0	0
$95$	−6.26918	−0.643204
$96$	0	0
$97$	−2.82202	−0.286532	−0.143266	−	0.989684i	$-0.545761\pi$
$97$	−2.82202	−0.286532	−0.143266	+	0.989684i	$0.545761\pi$
$98$	0	0
$99$	−0.777329	−0.0781245
$100$	0	0
$101$	1.34157	0.133491	0.0667457	−	0.997770i	$-0.478738\pi$
$101$	1.34157	0.133491	0.0667457	+	0.997770i	$0.478738\pi$
$102$	0	0
$103$	3.83741	0.378111	0.189056	−	0.981966i	$-0.439457\pi$
$103$	3.83741	0.378111	0.189056	+	0.981966i	$0.439457\pi$
$104$	0	0
$105$	−2.71060	−0.264527
$106$	0	0
$107$	−14.4717	−1.39903	−0.699514	−	0.714619i	$-0.746600\pi$
$107$	−14.4717	−1.39903	−0.699514	+	0.714619i	$0.746600\pi$
$108$	0	0
$109$	−16.6270	−1.59258	−0.796290	−	0.604915i	$-0.793207\pi$
$109$	−16.6270	−1.59258	−0.796290	+	0.604915i	$0.793207\pi$
$110$	0	0
$111$	17.6164	1.67207
$112$	0	0
$113$	−9.49624	−0.893331	−0.446666	−	0.894701i	$-0.647389\pi$
$113$	−9.49624	−0.893331	−0.446666	+	0.894701i	$0.647389\pi$
$114$	0	0
$115$	−27.3046	−2.54617
$116$	0	0
$117$	21.1986	1.95981
$118$	0	0
$119$	2.03957	0.186967
$120$	0	0
$121$	−10.9826	−0.998414
$122$	0	0
$123$	27.4338	2.47362
$124$	0	0
$125$	−1.96259	−0.175539
$126$	0	0
$127$	6.61087	0.586620	0.293310	−	0.956017i	$-0.405243\pi$
$127$	6.61087	0.586620	0.293310	+	0.956017i	$0.405243\pi$
$128$	0	0
$129$	−2.11609	−0.186311
$130$	0	0
$131$	15.4245	1.34765	0.673824	−	0.738892i	$-0.264650\pi$
$131$	15.4245	1.34765	0.673824	+	0.738892i	$0.264650\pi$
$132$	0	0
$133$	−0.609146	−0.0528197
$134$	0	0
$135$	−26.3164	−2.26496
$136$	0	0
$137$	−18.1197	−1.54807	−0.774034	−	0.633144i	$-0.781764\pi$
$137$	−18.1197	−1.54807	−0.774034	+	0.633144i	$0.781764\pi$
$138$	0	0
$139$	−12.6817	−1.07565	−0.537823	−	0.843058i	$-0.680753\pi$
$139$	−12.6817	−1.07565	−0.537823	+	0.843058i	$0.680753\pi$
$140$	0	0
$141$	−1.10226	−0.0928274
$142$	0	0
$143$	−0.475656	−0.0397763
$144$	0	0
$145$	−13.5671	−1.12668
$146$	0	0
$147$	20.6030	1.69931
$148$	0	0
$149$	17.8738	1.46428	0.732141	−	0.681153i	$-0.238521\pi$
$149$	17.8738	1.46428	0.732141	+	0.681153i	$0.238521\pi$
$150$	0	0
$151$	12.0643	0.981775	0.490888	−	0.871223i	$-0.336673\pi$
$151$	12.0643	0.981775	0.490888	+	0.871223i	$0.336673\pi$
$152$	0	0
$153$	40.3865	3.26505
$154$	0	0
$155$	−6.78164	−0.544715
$156$	0	0
$157$	20.3399	1.62330	0.811651	−	0.584143i	$-0.198569\pi$
$157$	20.3399	1.62330	0.811651	+	0.584143i	$0.198569\pi$
$158$	0	0
$159$	33.1917	2.63227
$160$	0	0
$161$	−2.65306	−0.209091
$162$	0	0
$163$	16.8285	1.31811	0.659056	−	0.752094i	$-0.270956\pi$
$163$	16.8285	1.31811	0.659056	+	0.752094i	$0.270956\pi$
$164$	0	0
$165$	1.20433	0.0937573
$166$	0	0
$167$	15.7134	1.21594	0.607969	−	0.793960i	$-0.291984\pi$
$167$	15.7134	1.21594	0.607969	+	0.793960i	$0.291984\pi$
$168$	0	0
$169$	−0.0283398	−0.00217999
$170$	0	0
$171$	−12.0620	−0.922402
$172$	0	0
$173$	−7.03141	−0.534588	−0.267294	−	0.963615i	$-0.586129\pi$
$173$	−7.03141	−0.534588	−0.267294	+	0.963615i	$0.586129\pi$
$174$	0	0
$175$	1.29552	0.0979325
$176$	0	0
$177$	−15.6122	−1.17349
$178$	0	0
$179$	5.47536	0.409248	0.204624	−	0.978841i	$-0.434403\pi$
$179$	5.47536	0.409248	0.204624	+	0.978841i	$0.434403\pi$
$180$	0	0
$181$	−11.7222	−0.871305	−0.435653	−	0.900115i	$-0.643482\pi$
$181$	−11.7222	−0.871305	−0.435653	+	0.900115i	$0.643482\pi$
$182$	0	0
$183$	−1.18417	−0.0875362
$184$	0	0
$185$	−18.0788	−1.32918
$186$	0	0
$187$	−0.906194	−0.0662675
$188$	0	0
$189$	−2.55704	−0.185997
$190$	0	0
$191$	−24.1170	−1.74505	−0.872523	−	0.488573i	$-0.837517\pi$
$191$	−24.1170	−1.74505	−0.872523	+	0.488573i	$0.837517\pi$
$192$	0	0
$193$	8.15585	0.587071	0.293535	−	0.955948i	$-0.405168\pi$
$193$	8.15585	0.587071	0.293535	+	0.955948i	$0.405168\pi$
$194$	0	0
$195$	−32.8435	−2.35197
$196$	0	0
$197$	−9.76635	−0.695823	−0.347912	−	0.937527i	$-0.613109\pi$
$197$	−9.76635	−0.695823	−0.347912	+	0.937527i	$0.613109\pi$
$198$	0	0
$199$	−0.217197	−0.0153967	−0.00769834	−	0.999970i	$-0.502450\pi$
$199$	−0.217197	−0.0153967	−0.00769834	+	0.999970i	$0.502450\pi$
$200$	0	0
$201$	14.8018	1.04404
$202$	0	0
$203$	−1.31825	−0.0925229
$204$	0	0
$205$	−28.1539	−1.96635
$206$	0	0
$207$	−52.5345	−3.65140
$208$	0	0
$209$	0.270647	0.0187211
$210$	0	0
$211$	−13.6428	−0.939211	−0.469606	−	0.882876i	$-0.655604\pi$
$211$	−13.6428	−0.939211	−0.469606	+	0.882876i	$0.655604\pi$
$212$	0	0
$213$	9.01319	0.617574
$214$	0	0
$215$	2.17164	0.148104
$216$	0	0
$217$	−0.658940	−0.0447318
$218$	0	0
$219$	29.0631	1.96390
$220$	0	0
$221$	24.7129	1.66237
$222$	0	0
$223$	24.3483	1.63048	0.815241	−	0.579121i	$-0.196604\pi$
$223$	24.3483	1.63048	0.815241	+	0.579121i	$0.196604\pi$
$224$	0	0
$225$	25.6533	1.71022
$226$	0	0
$227$	12.9318	0.858312	0.429156	−	0.903230i	$-0.358811\pi$
$227$	12.9318	0.858312	0.429156	+	0.903230i	$0.358811\pi$
$228$	0	0
$229$	18.9500	1.25225	0.626125	−	0.779723i	$-0.284640\pi$
$229$	18.9500	1.25225	0.626125	+	0.779723i	$0.284640\pi$
$230$	0	0
$231$	0.117019	0.00769931
$232$	0	0
$233$	4.90095	0.321072	0.160536	−	0.987030i	$-0.448678\pi$
$233$	4.90095	0.321072	0.160536	+	0.987030i	$0.448678\pi$
$234$	0	0
$235$	1.13120	0.0737911
$236$	0	0
$237$	12.4220	0.806895
$238$	0	0
$239$	17.9952	1.16401	0.582006	−	0.813184i	$-0.302268\pi$
$239$	17.9952	1.16401	0.582006	+	0.813184i	$0.302268\pi$
$240$	0	0
$241$	6.92465	0.446056	0.223028	−	0.974812i	$-0.428406\pi$
$241$	6.92465	0.446056	0.223028	+	0.974812i	$0.428406\pi$
$242$	0	0
$243$	2.00263	0.128469
$244$	0	0
$245$	−21.1438	−1.35083
$246$	0	0
$247$	−7.38085	−0.469632
$248$	0	0
$249$	−12.4669	−0.790056
$250$	0	0
$251$	25.2014	1.59070	0.795349	−	0.606152i	$-0.207288\pi$
$251$	25.2014	1.59070	0.795349	+	0.606152i	$0.207288\pi$
$252$	0	0
$253$	1.17877	0.0741087
$254$	0	0
$255$	−62.5717	−3.91839
$256$	0	0
$257$	−10.7985	−0.673593	−0.336796	−	0.941578i	$-0.609343\pi$
$257$	−10.7985	−0.673593	−0.336796	+	0.941578i	$0.609343\pi$
$258$	0	0
$259$	−1.75663	−0.109152
$260$	0	0
$261$	−26.1032	−1.61575
$262$	0	0
$263$	−23.2806	−1.43554	−0.717771	−	0.696279i	$-0.754838\pi$
$263$	−23.2806	−1.43554	−0.717771	+	0.696279i	$0.754838\pi$
$264$	0	0
$265$	−34.0629	−2.09247
$266$	0	0
$267$	−32.5171	−1.99002
$268$	0	0
$269$	13.0946	0.798392	0.399196	−	0.916866i	$-0.369289\pi$
$269$	13.0946	0.798392	0.399196	+	0.916866i	$0.369289\pi$
$270$	0	0
$271$	−12.5126	−0.760086	−0.380043	−	0.924969i	$-0.624091\pi$
$271$	−12.5126	−0.760086	−0.380043	+	0.924969i	$0.624091\pi$
$272$	0	0
$273$	−3.19125	−0.193143
$274$	0	0
$275$	−0.575609	−0.0347105
$276$	0	0
$277$	20.2064	1.21409	0.607043	−	0.794669i	$-0.292355\pi$
$277$	20.2064	1.21409	0.607043	+	0.794669i	$0.292355\pi$
$278$	0	0
$279$	−13.0480	−0.781161
$280$	0	0
$281$	−7.31394	−0.436313	−0.218157	−	0.975914i	$-0.570004\pi$
$281$	−7.31394	−0.436313	−0.218157	+	0.975914i	$0.570004\pi$
$282$	0	0
$283$	−18.2538	−1.08507	−0.542536	−	0.840032i	$-0.682536\pi$
$283$	−18.2538	−1.08507	−0.542536	+	0.840032i	$0.682536\pi$
$284$	0	0
$285$	18.6879	1.10697
$286$	0	0
$287$	−2.73558	−0.161476
$288$	0	0
$289$	30.0817	1.76951
$290$	0	0
$291$	8.41219	0.493132
$292$	0	0
$293$	−14.3113	−0.836077	−0.418038	−	0.908429i	$-0.637282\pi$
$293$	−14.3113	−0.836077	−0.418038	+	0.908429i	$0.637282\pi$
$294$	0	0
$295$	16.0220	0.932838
$296$	0	0
$297$	1.13611	0.0659237
$298$	0	0
$299$	−32.1464	−1.85907
$300$	0	0
$301$	0.211008	0.0121623
$302$	0	0
$303$	−3.99912	−0.229743
$304$	0	0
$305$	1.21525	0.0695850
$306$	0	0
$307$	−6.15472	−0.351268	−0.175634	−	0.984456i	$-0.556198\pi$
$307$	−6.15472	−0.351268	−0.175634	+	0.984456i	$0.556198\pi$
$308$	0	0
$309$	−11.4390	−0.650742
$310$	0	0
$311$	−8.08231	−0.458306	−0.229153	−	0.973390i	$-0.573596\pi$
$311$	−8.08231	−0.458306	−0.229153	+	0.973390i	$0.573596\pi$
$312$	0	0
$313$	−24.6435	−1.39294	−0.696468	−	0.717588i	$-0.745246\pi$
$313$	−24.6435	−1.39294	−0.696468	+	0.717588i	$0.745246\pi$
$314$	0	0
$315$	5.35211	0.301557
$316$	0	0
$317$	−17.7784	−0.998535	−0.499268	−	0.866448i	$-0.666398\pi$
$317$	−17.7784	−0.998535	−0.499268	+	0.866448i	$0.666398\pi$
$318$	0	0
$319$	0.585705	0.0327932
$320$	0	0
$321$	43.1388	2.40777
$322$	0	0
$323$	−14.0616	−0.782408
$324$	0	0
$325$	15.6975	0.870740
$326$	0	0
$327$	49.5638	2.74088
$328$	0	0
$329$	0.109913	0.00605970
$330$	0	0
$331$	9.50558	0.522474	0.261237	−	0.965275i	$-0.415870\pi$
$331$	9.50558	0.522474	0.261237	+	0.965275i	$0.415870\pi$
$332$	0	0
$333$	−34.7838	−1.90614
$334$	0	0
$335$	−15.1903	−0.829937
$336$	0	0
$337$	−31.2528	−1.70245	−0.851224	−	0.524803i	$-0.824139\pi$
$337$	−31.2528	−1.70245	−0.851224	+	0.524803i	$0.824139\pi$
$338$	0	0
$339$	28.3075	1.53745
$340$	0	0
$341$	0.292771	0.0158544
$342$	0	0
$343$	−4.13515	−0.223277
$344$	0	0
$345$	81.3928	4.38204
$346$	0	0
$347$	−28.1380	−1.51053	−0.755264	−	0.655421i	$-0.772491\pi$
$347$	−28.1380	−1.51053	−0.755264	+	0.655421i	$0.772491\pi$
$348$	0	0
$349$	6.77830	0.362834	0.181417	−	0.983406i	$-0.441932\pi$
$349$	6.77830	0.362834	0.181417	+	0.983406i	$0.441932\pi$
$350$	0	0
$351$	−30.9829	−1.65375
$352$	0	0
$353$	−3.35060	−0.178334	−0.0891672	−	0.996017i	$-0.528421\pi$
$353$	−3.35060	−0.178334	−0.0891672	+	0.996017i	$0.528421\pi$
$354$	0	0
$355$	−9.24977	−0.490927
$356$	0	0
$357$	−6.07980	−0.321777
$358$	0	0
$359$	−17.0690	−0.900867	−0.450434	−	0.892810i	$-0.648731\pi$
$359$	−17.0690	−0.900867	−0.450434	+	0.892810i	$0.648731\pi$
$360$	0	0
$361$	−14.8003	−0.778964
$362$	0	0
$363$	32.7381	1.71830
$364$	0	0
$365$	−29.8260	−1.56116
$366$	0	0
$367$	−23.2072	−1.21140	−0.605702	−	0.795692i	$-0.707108\pi$
$367$	−23.2072	−1.21140	−0.605702	+	0.795692i	$0.707108\pi$
$368$	0	0
$369$	−54.1684	−2.81990
$370$	0	0
$371$	−3.30973	−0.171833
$372$	0	0
$373$	18.0032	0.932171	0.466085	−	0.884740i	$-0.345664\pi$
$373$	18.0032	0.932171	0.466085	+	0.884740i	$0.345664\pi$
$374$	0	0
$375$	5.85031	0.302109
$376$	0	0
$377$	−15.9728	−0.822643
$378$	0	0
$379$	−21.5805	−1.10852	−0.554259	−	0.832345i	$-0.686998\pi$
$379$	−21.5805	−1.10852	−0.554259	+	0.832345i	$0.686998\pi$
$380$	0	0
$381$	−19.7064	−1.00959
$382$	0	0
$383$	−28.4335	−1.45288	−0.726442	−	0.687227i	$-0.758828\pi$
$383$	−28.4335	−1.45288	−0.726442	+	0.687227i	$0.758828\pi$
$384$	0	0
$385$	−0.120091	−0.00612040
$386$	0	0
$387$	4.17825	0.212392
$388$	0	0
$389$	−2.10098	−0.106524	−0.0532619	−	0.998581i	$-0.516962\pi$
$389$	−2.10098	−0.106524	−0.0532619	+	0.998581i	$0.516962\pi$
$390$	0	0
$391$	−61.2436	−3.09722
$392$	0	0
$393$	−45.9793	−2.31935
$394$	0	0
$395$	−12.7480	−0.641424
$396$	0	0
$397$	23.9121	1.20011	0.600056	−	0.799958i	$-0.295145\pi$
$397$	23.9121	1.20011	0.600056	+	0.799958i	$0.295145\pi$
$398$	0	0
$399$	1.81581	0.0909044
$400$	0	0
$401$	11.2924	0.563918	0.281959	−	0.959426i	$-0.409016\pi$
$401$	11.2924	0.563918	0.281959	+	0.959426i	$0.409016\pi$
$402$	0	0
$403$	−7.98418	−0.397720
$404$	0	0
$405$	24.4297	1.21392
$406$	0	0
$407$	0.780480	0.0386870
$408$	0	0
$409$	9.18638	0.454237	0.227119	−	0.973867i	$-0.427069\pi$
$409$	9.18638	0.454237	0.227119	+	0.973867i	$0.427069\pi$
$410$	0	0
$411$	54.0132	2.66427
$412$	0	0
$413$	1.55678	0.0766043
$414$	0	0
$415$	12.7941	0.628038
$416$	0	0
$417$	37.8030	1.85122
$418$	0	0
$419$	−6.50817	−0.317945	−0.158973	−	0.987283i	$-0.550818\pi$
$419$	−6.50817	−0.317945	−0.158973	+	0.987283i	$0.550818\pi$
$420$	0	0
$421$	−22.9286	−1.11747	−0.558735	−	0.829346i	$-0.688713\pi$
$421$	−22.9286	−1.11747	−0.558735	+	0.829346i	$0.688713\pi$
$422$	0	0
$423$	2.17644	0.105822
$424$	0	0
$425$	29.9060	1.45066
$426$	0	0
$427$	0.118080	0.00571430
$428$	0	0
$429$	1.41789	0.0684564
$430$	0	0
$431$	1.03628	0.0499159	0.0249580	−	0.999689i	$-0.492055\pi$
$431$	1.03628	0.0499159	0.0249580	+	0.999689i	$0.492055\pi$
$432$	0	0
$433$	12.5581	0.603503	0.301751	−	0.953387i	$-0.402429\pi$
$433$	12.5581	0.603503	0.301751	+	0.953387i	$0.402429\pi$
$434$	0	0
$435$	40.4423	1.93906
$436$	0	0
$437$	18.2912	0.874988
$438$	0	0
$439$	−11.2191	−0.535457	−0.267729	−	0.963494i	$-0.586273\pi$
$439$	−11.2191	−0.535457	−0.267729	+	0.963494i	$0.586273\pi$
$440$	0	0
$441$	−40.6810	−1.93719
$442$	0	0
$443$	−11.5728	−0.549839	−0.274920	−	0.961467i	$-0.588651\pi$
$443$	−11.5728	−0.549839	−0.274920	+	0.961467i	$0.588651\pi$
$444$	0	0
$445$	33.3706	1.58192
$446$	0	0
$447$	−53.2804	−2.52008
$448$	0	0
$449$	−20.3019	−0.958107	−0.479053	−	0.877786i	$-0.659020\pi$
$449$	−20.3019	−0.958107	−0.479053	+	0.877786i	$0.659020\pi$
$450$	0	0
$451$	1.21543	0.0572326
$452$	0	0
$453$	−35.9625	−1.68967
$454$	0	0
$455$	3.27501	0.153535
$456$	0	0
$457$	18.1861	0.850708	0.425354	−	0.905027i	$-0.360150\pi$
$457$	18.1861	0.850708	0.425354	+	0.905027i	$0.360150\pi$
$458$	0	0
$459$	−59.0270	−2.75515
$460$	0	0
$461$	16.3340	0.760750	0.380375	−	0.924832i	$-0.375795\pi$
$461$	16.3340	0.760750	0.380375	+	0.924832i	$0.375795\pi$
$462$	0	0
$463$	−10.0699	−0.467988	−0.233994	−	0.972238i	$-0.575180\pi$
$463$	−10.0699	−0.467988	−0.233994	+	0.972238i	$0.575180\pi$
$464$	0	0
$465$	20.2155	0.937471
$466$	0	0
$467$	−5.21270	−0.241215	−0.120608	−	0.992700i	$-0.538484\pi$
$467$	−5.21270	−0.241215	−0.120608	+	0.992700i	$0.538484\pi$
$468$	0	0
$469$	−1.47597	−0.0681541
$470$	0	0
$471$	−60.6316	−2.79376
$472$	0	0
$473$	−0.0937519	−0.00431071
$474$	0	0
$475$	−8.93184	−0.409821
$476$	0	0
$477$	−65.5374	−3.00075
$478$	0	0
$479$	5.14836	0.235235	0.117617	−	0.993059i	$-0.462474\pi$
$479$	5.14836	0.235235	0.117617	+	0.993059i	$0.462474\pi$
$480$	0	0
$481$	−21.2846	−0.970493
$482$	0	0
$483$	7.90856	0.359852
$484$	0	0
$485$	−8.63300	−0.392004
$486$	0	0
$487$	6.00118	0.271940	0.135970	−	0.990713i	$-0.456585\pi$
$487$	6.00118	0.271940	0.135970	+	0.990713i	$0.456585\pi$
$488$	0	0
$489$	−50.1645	−2.26851
$490$	0	0
$491$	14.9781	0.675951	0.337976	−	0.941155i	$-0.390258\pi$
$491$	14.9781	0.675951	0.337976	+	0.941155i	$0.390258\pi$
$492$	0	0
$493$	−30.4306	−1.37053
$494$	0	0
$495$	−2.37797	−0.106882
$496$	0	0
$497$	−0.898756	−0.0403147
$498$	0	0
$499$	20.4259	0.914391	0.457195	−	0.889366i	$-0.348854\pi$
$499$	20.4259	0.914391	0.457195	+	0.889366i	$0.348854\pi$
$500$	0	0
$501$	−46.8403	−2.09267
$502$	0	0
$503$	13.5387	0.603662	0.301831	−	0.953361i	$-0.402402\pi$
$503$	13.5387	0.603662	0.301831	+	0.953361i	$0.402402\pi$
$504$	0	0
$505$	4.10408	0.182629
$506$	0	0
$507$	0.0844786	0.00375183
$508$	0	0
$509$	10.3012	0.456594	0.228297	−	0.973591i	$-0.426684\pi$
$509$	10.3012	0.456594	0.228297	+	0.973591i	$0.426684\pi$
$510$	0	0
$511$	−2.89805	−0.128202
$512$	0	0
$513$	17.6292	0.778349
$514$	0	0
$515$	11.7392	0.517293
$516$	0	0
$517$	−0.0488350	−0.00214776
$518$	0	0
$519$	20.9600	0.920043
$520$	0	0
$521$	15.8230	0.693220	0.346610	−	0.938009i	$-0.387333\pi$
$521$	15.8230	0.693220	0.346610	+	0.938009i	$0.387333\pi$
$522$	0	0
$523$	−29.7200	−1.29956	−0.649782	−	0.760121i	$-0.725140\pi$
$523$	−29.7200	−1.29956	−0.649782	+	0.760121i	$0.725140\pi$
$524$	0	0
$525$	−3.86185	−0.168545
$526$	0	0
$527$	−15.2110	−0.662603
$528$	0	0
$529$	56.6652	2.46371
$530$	0	0
$531$	30.8266	1.33776
$532$	0	0
$533$	−33.1462	−1.43572
$534$	0	0
$535$	−44.2711	−1.91401
$536$	0	0
$537$	−16.3216	−0.704329
$538$	0	0
$539$	0.912802	0.0393172
$540$	0	0
$541$	−7.91261	−0.340190	−0.170095	−	0.985428i	$-0.554407\pi$
$541$	−7.91261	−0.340190	−0.170095	+	0.985428i	$0.554407\pi$
$542$	0	0
$543$	34.9429	1.49954
$544$	0	0
$545$	−50.8647	−2.17881
$546$	0	0
$547$	29.2010	1.24855	0.624273	−	0.781206i	$-0.285395\pi$
$547$	29.2010	1.24855	0.624273	+	0.781206i	$0.285395\pi$
$548$	0	0
$549$	2.33816	0.0997900
$550$	0	0
$551$	9.08851	0.387183
$552$	0	0
$553$	−1.23867	−0.0526735
$554$	0	0
$555$	53.8913	2.28756
$556$	0	0
$557$	−22.1567	−0.938810	−0.469405	−	0.882983i	$-0.655532\pi$
$557$	−22.1567	−0.938810	−0.469405	+	0.882983i	$0.655532\pi$
$558$	0	0
$559$	2.55672	0.108138
$560$	0	0
$561$	2.70129	0.114049
$562$	0	0
$563$	0.409756	0.0172692	0.00863458	−	0.999963i	$-0.497251\pi$
$563$	0.409756	0.0172692	0.00863458	+	0.999963i	$0.497251\pi$
$564$	0	0
$565$	−29.0505	−1.22216
$566$	0	0
$567$	2.37372	0.0996868
$568$	0	0
$569$	−4.87091	−0.204199	−0.102100	−	0.994774i	$-0.532556\pi$
$569$	−4.87091	−0.204199	−0.102100	+	0.994774i	$0.532556\pi$
$570$	0	0
$571$	−43.2933	−1.81177	−0.905884	−	0.423525i	$-0.860793\pi$
$571$	−43.2933	−1.81177	−0.905884	+	0.423525i	$0.860793\pi$
$572$	0	0
$573$	71.8908	3.00328
$574$	0	0
$575$	−38.9016	−1.62231
$576$	0	0
$577$	13.2488	0.551553	0.275776	−	0.961222i	$-0.411065\pi$
$577$	13.2488	0.551553	0.275776	+	0.961222i	$0.411065\pi$
$578$	0	0
$579$	−24.3119	−1.01037
$580$	0	0
$581$	1.24314	0.0515742
$582$	0	0
$583$	1.47053	0.0609032
$584$	0	0
$585$	64.8500	2.68122
$586$	0	0
$587$	5.15537	0.212785	0.106392	−	0.994324i	$-0.466070\pi$
$587$	5.15537	0.212785	0.106392	+	0.994324i	$0.466070\pi$
$588$	0	0
$589$	4.54298	0.187190
$590$	0	0
$591$	29.1127	1.19753
$592$	0	0
$593$	43.3018	1.77819	0.889095	−	0.457722i	$-0.151335\pi$
$593$	43.3018	1.77819	0.889095	+	0.457722i	$0.151335\pi$
$594$	0	0
$595$	6.23938	0.255790
$596$	0	0
$597$	0.647446	0.0264982
$598$	0	0
$599$	12.2368	0.499983	0.249992	−	0.968248i	$-0.419572\pi$
$599$	12.2368	0.499983	0.249992	+	0.968248i	$0.419572\pi$
$600$	0	0
$601$	−25.2617	−1.03045	−0.515224	−	0.857056i	$-0.672291\pi$
$601$	−25.2617	−1.03045	−0.515224	+	0.857056i	$0.672291\pi$
$602$	0	0
$603$	−29.2264	−1.19019
$604$	0	0
$605$	−33.5974	−1.36593
$606$	0	0
$607$	−16.9595	−0.688365	−0.344182	−	0.938903i	$-0.611844\pi$
$607$	−16.9595	−0.688365	−0.344182	+	0.938903i	$0.611844\pi$
$608$	0	0
$609$	3.92959	0.159235
$610$	0	0
$611$	1.33178	0.0538782
$612$	0	0
$613$	28.3245	1.14402	0.572008	−	0.820248i	$-0.306165\pi$
$613$	28.3245	1.14402	0.572008	+	0.820248i	$0.306165\pi$
$614$	0	0
$615$	83.9244	3.38416
$616$	0	0
$617$	47.8076	1.92466	0.962331	−	0.271881i	$-0.0876456\pi$
$617$	47.8076	1.92466	0.962331	+	0.271881i	$0.0876456\pi$
$618$	0	0
$619$	34.9193	1.40353	0.701763	−	0.712411i	$-0.252397\pi$
$619$	34.9193	1.40353	0.701763	+	0.712411i	$0.252397\pi$
$620$	0	0
$621$	76.7820	3.08115
$622$	0	0
$623$	3.24247	0.129907
$624$	0	0
$625$	−27.7961	−1.11185
$626$	0	0
$627$	−0.806776	−0.0322195
$628$	0	0
$629$	−40.5502	−1.61684
$630$	0	0
$631$	12.9559	0.515766	0.257883	−	0.966176i	$-0.416975\pi$
$631$	12.9559	0.515766	0.257883	+	0.966176i	$0.416975\pi$
$632$	0	0
$633$	40.6681	1.61641
$634$	0	0
$635$	20.2237	0.802553
$636$	0	0
$637$	−24.8931	−0.986301
$638$	0	0
$639$	−17.7967	−0.704025
$640$	0	0
$641$	−6.61478	−0.261268	−0.130634	−	0.991431i	$-0.541701\pi$
$641$	−6.61478	−0.261268	−0.130634	+	0.991431i	$0.541701\pi$
$642$	0	0
$643$	−2.03171	−0.0801227	−0.0400613	−	0.999197i	$-0.512755\pi$
$643$	−2.03171	−0.0801227	−0.0400613	+	0.999197i	$0.512755\pi$
$644$	0	0
$645$	−6.47346	−0.254892
$646$	0	0
$647$	−22.8630	−0.898839	−0.449420	−	0.893321i	$-0.648369\pi$
$647$	−22.8630	−0.898839	−0.449420	+	0.893321i	$0.648369\pi$
$648$	0	0
$649$	−0.691688	−0.0271511
$650$	0	0
$651$	1.96424	0.0769848
$652$	0	0
$653$	−6.75304	−0.264267	−0.132133	−	0.991232i	$-0.542183\pi$
$653$	−6.75304	−0.264267	−0.132133	+	0.991232i	$0.542183\pi$
$654$	0	0
$655$	47.1861	1.84371
$656$	0	0
$657$	−57.3855	−2.23882
$658$	0	0
$659$	−4.72239	−0.183958	−0.0919790	−	0.995761i	$-0.529319\pi$
$659$	−4.72239	−0.183958	−0.0919790	+	0.995761i	$0.529319\pi$
$660$	0	0
$661$	47.4057	1.84387	0.921934	−	0.387346i	$-0.126608\pi$
$661$	47.4057	1.84387	0.921934	+	0.387346i	$0.126608\pi$
$662$	0	0
$663$	−73.6671	−2.86099
$664$	0	0
$665$	−1.86348	−0.0722625
$666$	0	0
$667$	39.5839	1.53269
$668$	0	0
$669$	−72.5802	−2.80611
$670$	0	0
$671$	−0.0524637	−0.00202534
$672$	0	0
$673$	11.4380	0.440902	0.220451	−	0.975398i	$-0.429247\pi$
$673$	11.4380	0.440902	0.220451	+	0.975398i	$0.429247\pi$
$674$	0	0
$675$	−37.4936	−1.44313
$676$	0	0
$677$	9.02734	0.346949	0.173475	−	0.984838i	$-0.444501\pi$
$677$	9.02734	0.346949	0.173475	+	0.984838i	$0.444501\pi$
$678$	0	0
$679$	−0.838828	−0.0321912
$680$	0	0
$681$	−38.5485	−1.47718
$682$	0	0
$683$	13.4228	0.513610	0.256805	−	0.966463i	$-0.417330\pi$
$683$	13.4228	0.513610	0.256805	+	0.966463i	$0.417330\pi$
$684$	0	0
$685$	−55.4309	−2.11791
$686$	0	0
$687$	−56.4883	−2.15516
$688$	0	0
$689$	−40.1030	−1.52780
$690$	0	0
$691$	36.9348	1.40506	0.702532	−	0.711652i	$-0.252053\pi$
$691$	36.9348	1.40506	0.702532	+	0.711652i	$0.252053\pi$
$692$	0	0
$693$	−0.231056	−0.00877711
$694$	0	0
$695$	−38.7953	−1.47159
$696$	0	0
$697$	−63.1484	−2.39192
$698$	0	0
$699$	−14.6093	−0.552576
$700$	0	0
$701$	45.7399	1.72757	0.863785	−	0.503860i	$-0.168087\pi$
$701$	45.7399	1.72757	0.863785	+	0.503860i	$0.168087\pi$
$702$	0	0
$703$	12.1109	0.456770
$704$	0	0
$705$	−3.37200	−0.126997
$706$	0	0
$707$	0.398774	0.0149975
$708$	0	0
$709$	−10.9620	−0.411687	−0.205843	−	0.978585i	$-0.565994\pi$
$709$	−10.9620	−0.411687	−0.205843	+	0.978585i	$0.565994\pi$
$710$	0	0
$711$	−24.5274	−0.919849
$712$	0	0
$713$	19.7864	0.741007
$714$	0	0
$715$	−1.45511	−0.0544179
$716$	0	0
$717$	−53.6422	−2.00330
$718$	0	0
$719$	46.8983	1.74901	0.874506	−	0.485015i	$-0.161186\pi$
$719$	46.8983	1.74901	0.874506	+	0.485015i	$0.161186\pi$
$720$	0	0
$721$	1.14065	0.0424799
$722$	0	0
$723$	−20.6418	−0.767677
$724$	0	0
$725$	−19.3293	−0.717873
$726$	0	0
$727$	−47.1700	−1.74944	−0.874719	−	0.484631i	$-0.838954\pi$
$727$	−47.1700	−1.74944	−0.874719	+	0.484631i	$0.838954\pi$
$728$	0	0
$729$	−29.9270	−1.10841
$730$	0	0
$731$	4.87092	0.180157
$732$	0	0
$733$	50.1170	1.85111	0.925557	−	0.378609i	$-0.123597\pi$
$733$	50.1170	1.85111	0.925557	+	0.378609i	$0.123597\pi$
$734$	0	0
$735$	63.0280	2.32482
$736$	0	0
$737$	0.655783	0.0241561
$738$	0	0
$739$	35.1130	1.29165	0.645827	−	0.763484i	$-0.276513\pi$
$739$	35.1130	1.29165	0.645827	+	0.763484i	$0.276513\pi$
$740$	0	0
$741$	22.0017	0.808252
$742$	0	0
$743$	−46.4629	−1.70456	−0.852280	−	0.523087i	$-0.824780\pi$
$743$	−46.4629	−1.70456	−0.852280	+	0.523087i	$0.824780\pi$
$744$	0	0
$745$	54.6789	2.00328
$746$	0	0
$747$	24.6160	0.900653
$748$	0	0
$749$	−4.30161	−0.157177
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−75.1232	−2.73764
$754$	0	0
$755$	36.9065	1.34316
$756$	0	0
$757$	24.7374	0.899097	0.449548	−	0.893256i	$-0.351585\pi$
$757$	24.7374	0.899097	0.449548	+	0.893256i	$0.351585\pi$
$758$	0	0
$759$	−3.51382	−0.127543
$760$	0	0
$761$	−2.95100	−0.106974	−0.0534869	−	0.998569i	$-0.517034\pi$
$761$	−2.95100	−0.106974	−0.0534869	+	0.998569i	$0.517034\pi$
$762$	0	0
$763$	−4.94229	−0.178923
$764$	0	0
$765$	123.549	4.46691
$766$	0	0
$767$	18.8631	0.681107
$768$	0	0
$769$	7.49197	0.270167	0.135084	−	0.990834i	$-0.456870\pi$
$769$	7.49197	0.270167	0.135084	+	0.990834i	$0.456870\pi$
$770$	0	0
$771$	32.1894	1.15927
$772$	0	0
$773$	35.3565	1.27169	0.635843	−	0.771819i	$-0.280653\pi$
$773$	35.3565	1.27169	0.635843	+	0.771819i	$0.280653\pi$
$774$	0	0
$775$	−9.66196	−0.347068
$776$	0	0
$777$	5.23636	0.187853
$778$	0	0
$779$	18.8601	0.675735
$780$	0	0
$781$	0.399323	0.0142889
$782$	0	0
$783$	38.1513	1.36342
$784$	0	0
$785$	62.2230	2.22084
$786$	0	0
$787$	−30.3186	−1.08074	−0.540372	−	0.841427i	$-0.681716\pi$
$787$	−30.3186	−1.08074	−0.540372	+	0.841427i	$0.681716\pi$
$788$	0	0
$789$	69.3975	2.47062
$790$	0	0
$791$	−2.82270	−0.100364
$792$	0	0
$793$	1.43074	0.0508071
$794$	0	0
$795$	101.539	3.60120
$796$	0	0
$797$	−49.6831	−1.75987	−0.879933	−	0.475099i	$-0.842412\pi$
$797$	−49.6831	−1.75987	−0.879933	+	0.475099i	$0.842412\pi$
$798$	0	0
$799$	2.53724	0.0897612
$800$	0	0
$801$	64.2055	2.26859
$802$	0	0
$803$	1.28762	0.0454391
$804$	0	0
$805$	−8.11614	−0.286056
$806$	0	0
$807$	−39.0339	−1.37406
$808$	0	0
$809$	29.6256	1.04158	0.520791	−	0.853685i	$-0.325637\pi$
$809$	29.6256	1.04158	0.520791	+	0.853685i	$0.325637\pi$
$810$	0	0
$811$	−52.0726	−1.82852	−0.914258	−	0.405134i	$-0.867225\pi$
$811$	−52.0726	−1.82852	−0.914258	+	0.405134i	$0.867225\pi$
$812$	0	0
$813$	37.2990	1.30813
$814$	0	0
$815$	51.4812	1.80331
$816$	0	0
$817$	−1.45477	−0.0508958
$818$	0	0
$819$	6.30116	0.220180
$820$	0	0
$821$	−24.0722	−0.840124	−0.420062	−	0.907495i	$-0.637992\pi$
$821$	−24.0722	−0.840124	−0.420062	+	0.907495i	$0.637992\pi$
$822$	0	0
$823$	−54.7874	−1.90977	−0.954884	−	0.296980i	$-0.904020\pi$
$823$	−54.7874	−1.90977	−0.954884	+	0.296980i	$0.904020\pi$
$824$	0	0
$825$	1.71584	0.0597380
$826$	0	0
$827$	16.2267	0.564257	0.282128	−	0.959377i	$-0.408960\pi$
$827$	16.2267	0.564257	0.282128	+	0.959377i	$0.408960\pi$
$828$	0	0
$829$	−24.2588	−0.842544	−0.421272	−	0.906934i	$-0.638416\pi$
$829$	−24.2588	−0.842544	−0.421272	+	0.906934i	$0.638416\pi$
$830$	0	0
$831$	−60.2337	−2.08948
$832$	0	0
$833$	−47.4250	−1.64318
$834$	0	0
$835$	48.0698	1.66352
$836$	0	0
$837$	19.0703	0.659166
$838$	0	0
$839$	41.6871	1.43920	0.719599	−	0.694390i	$-0.244326\pi$
$839$	41.6871	1.43920	0.719599	+	0.694390i	$0.244326\pi$
$840$	0	0
$841$	−9.33163	−0.321780
$842$	0	0
$843$	21.8022	0.750909
$844$	0	0
$845$	−0.0866960	−0.00298243
$846$	0	0
$847$	−3.26450	−0.112170
$848$	0	0
$849$	54.4129	1.86745
$850$	0	0
$851$	52.7474	1.80816
$852$	0	0
$853$	2.71850	0.0930798	0.0465399	−	0.998916i	$-0.485181\pi$
$853$	2.71850	0.0930798	0.0465399	+	0.998916i	$0.485181\pi$
$854$	0	0
$855$	−36.8995	−1.26194
$856$	0	0
$857$	−1.75359	−0.0599015	−0.0299507	−	0.999551i	$-0.509535\pi$
$857$	−1.75359	−0.0599015	−0.0299507	+	0.999551i	$0.509535\pi$
$858$	0	0
$859$	13.7566	0.469369	0.234685	−	0.972072i	$-0.424594\pi$
$859$	13.7566	0.469369	0.234685	+	0.972072i	$0.424594\pi$
$860$	0	0
$861$	8.15454	0.277906
$862$	0	0
$863$	9.16644	0.312029	0.156015	−	0.987755i	$-0.450135\pi$
$863$	9.16644	0.312029	0.156015	+	0.987755i	$0.450135\pi$
$864$	0	0
$865$	−21.5102	−0.731369
$866$	0	0
$867$	−89.6711	−3.04539
$868$	0	0
$869$	0.550347	0.0186692
$870$	0	0
$871$	−17.8839	−0.605974
$872$	0	0
$873$	−16.6100	−0.562163
$874$	0	0
$875$	−0.583368	−0.0197214
$876$	0	0
$877$	4.39218	0.148313	0.0741567	−	0.997247i	$-0.476374\pi$
$877$	4.39218	0.148313	0.0741567	+	0.997247i	$0.476374\pi$
$878$	0	0
$879$	42.6609	1.43892
$880$	0	0
$881$	−12.9220	−0.435352	−0.217676	−	0.976021i	$-0.569848\pi$
$881$	−12.9220	−0.435352	−0.217676	+	0.976021i	$0.569848\pi$
$882$	0	0
$883$	−40.8403	−1.37438	−0.687192	−	0.726476i	$-0.741157\pi$
$883$	−40.8403	−1.37438	−0.687192	+	0.726476i	$0.741157\pi$
$884$	0	0
$885$	−47.7603	−1.60544
$886$	0	0
$887$	−29.7519	−0.998972	−0.499486	−	0.866322i	$-0.666478\pi$
$887$	−29.7519	−0.998972	−0.499486	+	0.866322i	$0.666478\pi$
$888$	0	0
$889$	1.96504	0.0659054
$890$	0	0
$891$	−1.05466	−0.0353323
$892$	0	0
$893$	−0.757782	−0.0253582
$894$	0	0
$895$	16.7500	0.559891
$896$	0	0
$897$	95.8257	3.19953
$898$	0	0
$899$	9.83144	0.327897
$900$	0	0
$901$	−76.4022	−2.54533
$902$	0	0
$903$	−0.628996	−0.0209317
$904$	0	0
$905$	−35.8601	−1.19203
$906$	0	0
$907$	26.8703	0.892214	0.446107	−	0.894980i	$-0.352810\pi$
$907$	26.8703	0.892214	0.446107	+	0.894980i	$0.352810\pi$
$908$	0	0
$909$	7.89631	0.261904
$910$	0	0
$911$	−0.290314	−0.00961854	−0.00480927	−	0.999988i	$-0.501531\pi$
$911$	−0.290314	−0.00961854	−0.00480927	+	0.999988i	$0.501531\pi$
$912$	0	0
$913$	−0.552335	−0.0182796
$914$	0	0
$915$	−3.62256	−0.119758
$916$	0	0
$917$	4.58485	0.151405
$918$	0	0
$919$	20.2476	0.667908	0.333954	−	0.942589i	$-0.391617\pi$
$919$	20.2476	0.667908	0.333954	+	0.942589i	$0.391617\pi$
$920$	0	0
$921$	18.3467	0.604544
$922$	0	0
$923$	−10.8900	−0.358448
$924$	0	0
$925$	−25.7573	−0.846893
$926$	0	0
$927$	22.5864	0.741836
$928$	0	0
$929$	5.86011	0.192264	0.0961320	−	0.995369i	$-0.469353\pi$
$929$	5.86011	0.192264	0.0961320	+	0.995369i	$0.469353\pi$
$930$	0	0
$931$	14.1641	0.464211
$932$	0	0
$933$	24.0927	0.788759
$934$	0	0
$935$	−2.77219	−0.0906604
$936$	0	0
$937$	−33.8764	−1.10670	−0.553348	−	0.832950i	$-0.686650\pi$
$937$	−33.8764	−1.10670	−0.553348	+	0.832950i	$0.686650\pi$
$938$	0	0
$939$	73.4603	2.39729
$940$	0	0
$941$	53.5147	1.74453	0.872265	−	0.489033i	$-0.162650\pi$
$941$	53.5147	1.74453	0.872265	+	0.489033i	$0.162650\pi$
$942$	0	0
$943$	82.1431	2.67495
$944$	0	0
$945$	−7.82240	−0.254463
$946$	0	0
$947$	−47.2206	−1.53446	−0.767231	−	0.641371i	$-0.778366\pi$
$947$	−47.2206	−1.53446	−0.767231	+	0.641371i	$0.778366\pi$
$948$	0	0
$949$	−35.1148	−1.13987
$950$	0	0
$951$	52.9960	1.71851
$952$	0	0
$953$	−39.3831	−1.27574	−0.637872	−	0.770142i	$-0.720185\pi$
$953$	−39.3831	−1.27574	−0.637872	+	0.770142i	$0.720185\pi$
$954$	0	0
$955$	−73.7778	−2.38739
$956$	0	0
$957$	−1.74594	−0.0564382
$958$	0	0
$959$	−5.38596	−0.173922
$960$	0	0
$961$	−26.0857	−0.841473
$962$	0	0
$963$	−85.1781	−2.74483
$964$	0	0
$965$	24.9500	0.803170
$966$	0	0
$967$	−5.23083	−0.168212	−0.0841061	−	0.996457i	$-0.526803\pi$
$967$	−5.23083	−0.168212	−0.0841061	+	0.996457i	$0.526803\pi$
$968$	0	0
$969$	41.9164	1.34655
$970$	0	0
$971$	49.7138	1.59539	0.797697	−	0.603059i	$-0.206052\pi$
$971$	49.7138	1.59539	0.797697	+	0.603059i	$0.206052\pi$
$972$	0	0
$973$	−3.76955	−0.120846
$974$	0	0
$975$	−46.7929	−1.49857
$976$	0	0
$977$	16.7627	0.536286	0.268143	−	0.963379i	$-0.413590\pi$
$977$	16.7627	0.536286	0.268143	+	0.963379i	$0.413590\pi$
$978$	0	0
$979$	−1.44065	−0.0460433
$980$	0	0
$981$	−97.8643	−3.12457
$982$	0	0
$983$	−2.31366	−0.0737942	−0.0368971	−	0.999319i	$-0.511747\pi$
$983$	−2.31366	−0.0737942	−0.0368971	+	0.999319i	$0.511747\pi$
$984$	0	0
$985$	−29.8768	−0.951954
$986$	0	0
$987$	−0.327641	−0.0104289
$988$	0	0
$989$	−6.33606	−0.201475
$990$	0	0
$991$	−27.7561	−0.881702	−0.440851	−	0.897580i	$-0.645323\pi$
$991$	−27.7561	−0.881702	−0.440851	+	0.897580i	$0.645323\pi$
$992$	0	0
$993$	−28.3353	−0.899194
$994$	0	0
$995$	−0.664440	−0.0210642
$996$	0	0
$997$	38.8153	1.22929	0.614646	−	0.788803i	$-0.289299\pi$
$997$	38.8153	1.22929	0.614646	+	0.788803i	$0.289299\pi$
$998$	0	0
$999$	50.8384	1.60846

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.6	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-2.98092 q^{3} +3.05916 q^{5} +0.297244 q^{7} +5.88586 q^{9} +O(q^{10})\)	\(q-2.98092 q^{3} +3.05916 q^{5} +0.297244 q^{7} +5.88586 q^{9} -0.132067 q^{11} +3.60162 q^{13} -9.11910 q^{15} +6.86161 q^{17} -2.04931 q^{19} -0.886059 q^{21} -8.92554 q^{23} +4.35846 q^{25} -8.60250 q^{27} -4.43490 q^{29} -2.21683 q^{31} +0.393681 q^{33} +0.909317 q^{35} -5.90972 q^{37} -10.7361 q^{39} -9.20315 q^{41} +0.709880 q^{43} +18.0058 q^{45} +0.369774 q^{47} -6.91165 q^{49} -20.4539 q^{51} -11.1347 q^{53} -0.404015 q^{55} +6.10883 q^{57} +5.23739 q^{59} +0.397250 q^{61} +1.74954 q^{63} +11.0179 q^{65} -4.96553 q^{67} +26.6063 q^{69} -3.02363 q^{71} -9.74973 q^{73} -12.9922 q^{75} -0.0392562 q^{77} -4.16717 q^{79} +7.98576 q^{81} +4.18223 q^{83} +20.9908 q^{85} +13.2201 q^{87} +10.9084 q^{89} +1.07056 q^{91} +6.60819 q^{93} -6.26918 q^{95} -2.82202 q^{97} -0.777329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more