LMFDB - Embedded newform 6008.2.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-2.24563 q^{3} +3.86835 q^{5} -4.12735 q^{7} +2.04285 q^{9} +O(q^{10})$	$q-2.24563 q^{3} +3.86835 q^{5} -4.12735 q^{7} +2.04285 q^{9} -3.58555 q^{11} -2.00933 q^{13} -8.68688 q^{15} -1.02183 q^{17} +2.57219 q^{19} +9.26850 q^{21} -6.19989 q^{23} +9.96413 q^{25} +2.14940 q^{27} -4.21182 q^{29} -3.01713 q^{31} +8.05182 q^{33} -15.9660 q^{35} -3.89792 q^{37} +4.51221 q^{39} -5.80696 q^{41} +0.0570193 q^{43} +7.90247 q^{45} +1.46241 q^{47} +10.0350 q^{49} +2.29465 q^{51} +1.31159 q^{53} -13.8702 q^{55} -5.77618 q^{57} -7.55313 q^{59} -0.242841 q^{61} -8.43157 q^{63} -7.77278 q^{65} +10.5525 q^{67} +13.9227 q^{69} +5.81735 q^{71} +15.2698 q^{73} -22.3757 q^{75} +14.7988 q^{77} +11.6962 q^{79} -10.9553 q^{81} -3.56219 q^{83} -3.95279 q^{85} +9.45819 q^{87} -15.0875 q^{89} +8.29320 q^{91} +6.77535 q^{93} +9.95013 q^{95} +10.9573 q^{97} -7.32475 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.24563	−1.29651	−0.648257	−	0.761421i	$-0.724502\pi$
$3$	−2.24563	−1.29651	−0.648257	+	0.761421i	$0.724502\pi$
$4$	0	0
$5$	3.86835	1.72998	0.864989	−	0.501790i	$-0.167325\pi$
$5$	3.86835	1.72998	0.864989	+	0.501790i	$0.167325\pi$
$6$	0	0
$7$	−4.12735	−1.55999	−0.779996	−	0.625784i	$-0.784779\pi$
$7$	−4.12735	−1.55999	−0.779996	+	0.625784i	$0.784779\pi$
$8$	0	0
$9$	2.04285	0.680951
$10$	0	0
$11$	−3.58555	−1.08108	−0.540542	−	0.841317i	$-0.681781\pi$
$11$	−3.58555	−1.08108	−0.540542	+	0.841317i	$0.681781\pi$
$12$	0	0
$13$	−2.00933	−0.557287	−0.278644	−	0.960395i	$-0.589885\pi$
$13$	−2.00933	−0.557287	−0.278644	+	0.960395i	$0.589885\pi$
$14$	0	0
$15$	−8.68688	−2.24294
$16$	0	0
$17$	−1.02183	−0.247830	−0.123915	−	0.992293i	$-0.539545\pi$
$17$	−1.02183	−0.247830	−0.123915	+	0.992293i	$0.539545\pi$
$18$	0	0
$19$	2.57219	0.590101	0.295050	−	0.955482i	$-0.404664\pi$
$19$	2.57219	0.590101	0.295050	+	0.955482i	$0.404664\pi$
$20$	0	0
$21$	9.26850	2.02255
$22$	0	0
$23$	−6.19989	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$23$	−6.19989	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$24$	0	0
$25$	9.96413	1.99283
$26$	0	0
$27$	2.14940	0.413652
$28$	0	0
$29$	−4.21182	−0.782116	−0.391058	−	0.920366i	$-0.627891\pi$
$29$	−4.21182	−0.782116	−0.391058	+	0.920366i	$0.627891\pi$
$30$	0	0
$31$	−3.01713	−0.541892	−0.270946	−	0.962595i	$-0.587336\pi$
$31$	−3.01713	−0.541892	−0.270946	+	0.962595i	$0.587336\pi$
$32$	0	0
$33$	8.05182	1.40164
$34$	0	0
$35$	−15.9660	−2.69875
$36$	0	0
$37$	−3.89792	−0.640813	−0.320407	−	0.947280i	$-0.603820\pi$
$37$	−3.89792	−0.640813	−0.320407	+	0.947280i	$0.603820\pi$
$38$	0	0
$39$	4.51221	0.722531
$40$	0	0
$41$	−5.80696	−0.906894	−0.453447	−	0.891283i	$-0.649806\pi$
$41$	−5.80696	−0.906894	−0.453447	+	0.891283i	$0.649806\pi$
$42$	0	0
$43$	0.0570193	0.00869537	0.00434768	−	0.999991i	$-0.498616\pi$
$43$	0.0570193	0.00869537	0.00434768	+	0.999991i	$0.498616\pi$
$44$	0	0
$45$	7.90247	1.17803
$46$	0	0
$47$	1.46241	0.213314	0.106657	−	0.994296i	$-0.465985\pi$
$47$	1.46241	0.213314	0.106657	+	0.994296i	$0.465985\pi$
$48$	0	0
$49$	10.0350	1.43358
$50$	0	0
$51$	2.29465	0.321315
$52$	0	0
$53$	1.31159	0.180160	0.0900802	−	0.995935i	$-0.471288\pi$
$53$	1.31159	0.180160	0.0900802	+	0.995935i	$0.471288\pi$
$54$	0	0
$55$	−13.8702	−1.87025
$56$	0	0
$57$	−5.77618	−0.765074
$58$	0	0
$59$	−7.55313	−0.983333	−0.491667	−	0.870784i	$-0.663612\pi$
$59$	−7.55313	−0.983333	−0.491667	+	0.870784i	$0.663612\pi$
$60$	0	0
$61$	−0.242841	−0.0310927	−0.0155463	−	0.999879i	$-0.504949\pi$
$61$	−0.242841	−0.0310927	−0.0155463	+	0.999879i	$0.504949\pi$
$62$	0	0
$63$	−8.43157	−1.06228
$64$	0	0
$65$	−7.77278	−0.964095
$66$	0	0
$67$	10.5525	1.28920	0.644598	−	0.764522i	$-0.277025\pi$
$67$	10.5525	1.28920	0.644598	+	0.764522i	$0.277025\pi$
$68$	0	0
$69$	13.9227	1.67609
$70$	0	0
$71$	5.81735	0.690392	0.345196	−	0.938531i	$-0.387812\pi$
$71$	5.81735	0.690392	0.345196	+	0.938531i	$0.387812\pi$
$72$	0	0
$73$	15.2698	1.78719	0.893597	−	0.448869i	$-0.148173\pi$
$73$	15.2698	1.78719	0.893597	+	0.448869i	$0.148173\pi$
$74$	0	0
$75$	−22.3757	−2.58373
$76$	0	0
$77$	14.7988	1.68648
$78$	0	0
$79$	11.6962	1.31593	0.657965	−	0.753048i	$-0.271417\pi$
$79$	11.6962	1.31593	0.657965	+	0.753048i	$0.271417\pi$
$80$	0	0
$81$	−10.9553	−1.21726
$82$	0	0
$83$	−3.56219	−0.391001	−0.195500	−	0.980704i	$-0.562633\pi$
$83$	−3.56219	−0.391001	−0.195500	+	0.980704i	$0.562633\pi$
$84$	0	0
$85$	−3.95279	−0.428741
$86$	0	0
$87$	9.45819	1.01402
$88$	0	0
$89$	−15.0875	−1.59927	−0.799634	−	0.600488i	$-0.794973\pi$
$89$	−15.0875	−1.59927	−0.799634	+	0.600488i	$0.794973\pi$
$90$	0	0
$91$	8.29320	0.869364
$92$	0	0
$93$	6.77535	0.702571
$94$	0	0
$95$	9.95013	1.02086
$96$	0	0
$97$	10.9573	1.11255	0.556275	−	0.830998i	$-0.312230\pi$
$97$	10.9573	1.11255	0.556275	+	0.830998i	$0.312230\pi$
$98$	0	0
$99$	−7.32475	−0.736165
$100$	0	0
$101$	2.94428	0.292966	0.146483	−	0.989213i	$-0.453205\pi$
$101$	2.94428	0.292966	0.146483	+	0.989213i	$0.453205\pi$
$102$	0	0
$103$	−0.250675	−0.0246997	−0.0123499	−	0.999924i	$-0.503931\pi$
$103$	−0.250675	−0.0246997	−0.0123499	+	0.999924i	$0.503931\pi$
$104$	0	0
$105$	35.8538	3.49897
$106$	0	0
$107$	−18.9718	−1.83407	−0.917036	−	0.398803i	$-0.869426\pi$
$107$	−18.9718	−1.83407	−0.917036	+	0.398803i	$0.869426\pi$
$108$	0	0
$109$	0.387808	0.0371452	0.0185726	−	0.999828i	$-0.494088\pi$
$109$	0.387808	0.0371452	0.0185726	+	0.999828i	$0.494088\pi$
$110$	0	0
$111$	8.75328	0.830824
$112$	0	0
$113$	4.86584	0.457740	0.228870	−	0.973457i	$-0.426497\pi$
$113$	4.86584	0.457740	0.228870	+	0.973457i	$0.426497\pi$
$114$	0	0
$115$	−23.9833	−2.23646
$116$	0	0
$117$	−4.10476	−0.379485
$118$	0	0
$119$	4.21745	0.386613
$120$	0	0
$121$	1.85617	0.168742
$122$	0	0
$123$	13.0403	1.17580
$124$	0	0
$125$	19.2030	1.71757
$126$	0	0
$127$	17.6795	1.56880	0.784402	−	0.620253i	$-0.212970\pi$
$127$	17.6795	1.56880	0.784402	+	0.620253i	$0.212970\pi$
$128$	0	0
$129$	−0.128044	−0.0112737
$130$	0	0
$131$	7.53080	0.657969	0.328985	−	0.944335i	$-0.393294\pi$
$131$	7.53080	0.657969	0.328985	+	0.944335i	$0.393294\pi$
$132$	0	0
$133$	−10.6163	−0.920553
$134$	0	0
$135$	8.31463	0.715609
$136$	0	0
$137$	1.03680	0.0885796	0.0442898	−	0.999019i	$-0.485898\pi$
$137$	1.03680	0.0885796	0.0442898	+	0.999019i	$0.485898\pi$
$138$	0	0
$139$	−0.219182	−0.0185907	−0.00929537	−	0.999957i	$-0.502959\pi$
$139$	−0.219182	−0.0185907	−0.00929537	+	0.999957i	$0.502959\pi$
$140$	0	0
$141$	−3.28402	−0.276564
$142$	0	0
$143$	7.20454	0.602474
$144$	0	0
$145$	−16.2928	−1.35304
$146$	0	0
$147$	−22.5350	−1.85865
$148$	0	0
$149$	−20.6938	−1.69530	−0.847651	−	0.530554i	$-0.821984\pi$
$149$	−20.6938	−1.69530	−0.847651	+	0.530554i	$0.821984\pi$
$150$	0	0
$151$	1.09109	0.0887919	0.0443959	−	0.999014i	$-0.485864\pi$
$151$	1.09109	0.0887919	0.0443959	+	0.999014i	$0.485864\pi$
$152$	0	0
$153$	−2.08745	−0.168760
$154$	0	0
$155$	−11.6713	−0.937461
$156$	0	0
$157$	19.3894	1.54744	0.773720	−	0.633528i	$-0.218394\pi$
$157$	19.3894	1.54744	0.773720	+	0.633528i	$0.218394\pi$
$158$	0	0
$159$	−2.94534	−0.233581
$160$	0	0
$161$	25.5891	2.01671
$162$	0	0
$163$	8.33118	0.652548	0.326274	−	0.945275i	$-0.394207\pi$
$163$	8.33118	0.652548	0.326274	+	0.945275i	$0.394207\pi$
$164$	0	0
$165$	31.1472	2.42481
$166$	0	0
$167$	8.82303	0.682746	0.341373	−	0.939928i	$-0.389108\pi$
$167$	8.82303	0.682746	0.341373	+	0.939928i	$0.389108\pi$
$168$	0	0
$169$	−8.96260	−0.689431
$170$	0	0
$171$	5.25460	0.401829
$172$	0	0
$173$	16.0022	1.21663	0.608314	−	0.793696i	$-0.291846\pi$
$173$	16.0022	1.21663	0.608314	+	0.793696i	$0.291846\pi$
$174$	0	0
$175$	−41.1255	−3.10879
$176$	0	0
$177$	16.9615	1.27491
$178$	0	0
$179$	13.2053	0.987009	0.493505	−	0.869743i	$-0.335716\pi$
$179$	13.2053	0.987009	0.493505	+	0.869743i	$0.335716\pi$
$180$	0	0
$181$	9.72832	0.723101	0.361550	−	0.932353i	$-0.382248\pi$
$181$	9.72832	0.723101	0.361550	+	0.932353i	$0.382248\pi$
$182$	0	0
$183$	0.545332	0.0403121
$184$	0	0
$185$	−15.0785	−1.10859
$186$	0	0
$187$	3.66382	0.267925
$188$	0	0
$189$	−8.87133	−0.645294
$190$	0	0
$191$	−5.04712	−0.365196	−0.182598	−	0.983188i	$-0.558451\pi$
$191$	−5.04712	−0.365196	−0.182598	+	0.983188i	$0.558451\pi$
$192$	0	0
$193$	−1.49467	−0.107589	−0.0537943	−	0.998552i	$-0.517132\pi$
$193$	−1.49467	−0.107589	−0.0537943	+	0.998552i	$0.517132\pi$
$194$	0	0
$195$	17.4548	1.24996
$196$	0	0
$197$	10.7990	0.769400	0.384700	−	0.923042i	$-0.374305\pi$
$197$	10.7990	0.769400	0.384700	+	0.923042i	$0.374305\pi$
$198$	0	0
$199$	−5.16057	−0.365823	−0.182911	−	0.983129i	$-0.558552\pi$
$199$	−5.16057	−0.365823	−0.182911	+	0.983129i	$0.558552\pi$
$200$	0	0
$201$	−23.6971	−1.67146
$202$	0	0
$203$	17.3837	1.22009
$204$	0	0
$205$	−22.4633	−1.56891
$206$	0	0
$207$	−12.6655	−0.880310
$208$	0	0
$209$	−9.22271	−0.637948
$210$	0	0
$211$	−21.2776	−1.46481	−0.732407	−	0.680868i	$-0.761603\pi$
$211$	−21.2776	−1.46481	−0.732407	+	0.680868i	$0.761603\pi$
$212$	0	0
$213$	−13.0636	−0.895104
$214$	0	0
$215$	0.220571	0.0150428
$216$	0	0
$217$	12.4527	0.845347
$218$	0	0
$219$	−34.2903	−2.31712
$220$	0	0
$221$	2.05319	0.138113
$222$	0	0
$223$	7.27161	0.486943	0.243472	−	0.969908i	$-0.421714\pi$
$223$	7.27161	0.486943	0.243472	+	0.969908i	$0.421714\pi$
$224$	0	0
$225$	20.3552	1.35702
$226$	0	0
$227$	−0.742723	−0.0492963	−0.0246481	−	0.999696i	$-0.507847\pi$
$227$	−0.742723	−0.0492963	−0.0246481	+	0.999696i	$0.507847\pi$
$228$	0	0
$229$	−9.08421	−0.600301	−0.300151	−	0.953892i	$-0.597037\pi$
$229$	−9.08421	−0.600301	−0.300151	+	0.953892i	$0.597037\pi$
$230$	0	0
$231$	−33.2327	−2.18655
$232$	0	0
$233$	18.0901	1.18512	0.592560	−	0.805526i	$-0.298117\pi$
$233$	18.0901	1.18512	0.592560	+	0.805526i	$0.298117\pi$
$234$	0	0
$235$	5.65709	0.369028
$236$	0	0
$237$	−26.2654	−1.70612
$238$	0	0
$239$	−2.08622	−0.134946	−0.0674730	−	0.997721i	$-0.521494\pi$
$239$	−2.08622	−0.134946	−0.0674730	+	0.997721i	$0.521494\pi$
$240$	0	0
$241$	−3.63280	−0.234009	−0.117004	−	0.993131i	$-0.537329\pi$
$241$	−3.63280	−0.234009	−0.117004	+	0.993131i	$0.537329\pi$
$242$	0	0
$243$	18.1534	1.16454
$244$	0	0
$245$	38.8190	2.48006
$246$	0	0
$247$	−5.16837	−0.328856
$248$	0	0
$249$	7.99935	0.506938
$250$	0	0
$251$	5.95049	0.375592	0.187796	−	0.982208i	$-0.439866\pi$
$251$	5.95049	0.375592	0.187796	+	0.982208i	$0.439866\pi$
$252$	0	0
$253$	22.2300	1.39759
$254$	0	0
$255$	8.87651	0.555869
$256$	0	0
$257$	−17.7496	−1.10719	−0.553594	−	0.832787i	$-0.686744\pi$
$257$	−17.7496	−1.10719	−0.553594	+	0.832787i	$0.686744\pi$
$258$	0	0
$259$	16.0881	0.999664
$260$	0	0
$261$	−8.60413	−0.532582
$262$	0	0
$263$	−20.0832	−1.23838	−0.619190	−	0.785241i	$-0.712539\pi$
$263$	−20.0832	−1.23838	−0.619190	+	0.785241i	$0.712539\pi$
$264$	0	0
$265$	5.07368	0.311674
$266$	0	0
$267$	33.8808	2.07347
$268$	0	0
$269$	−31.1407	−1.89868	−0.949340	−	0.314251i	$-0.898247\pi$
$269$	−31.1407	−1.89868	−0.949340	+	0.314251i	$0.898247\pi$
$270$	0	0
$271$	−6.52857	−0.396582	−0.198291	−	0.980143i	$-0.563539\pi$
$271$	−6.52857	−0.396582	−0.198291	+	0.980143i	$0.563539\pi$
$272$	0	0
$273$	−18.6235	−1.12714
$274$	0	0
$275$	−35.7269	−2.15441
$276$	0	0
$277$	−0.196105	−0.0117828	−0.00589142	−	0.999983i	$-0.501875\pi$
$277$	−0.196105	−0.0117828	−0.00589142	+	0.999983i	$0.501875\pi$
$278$	0	0
$279$	−6.16354	−0.369001
$280$	0	0
$281$	14.6199	0.872152	0.436076	−	0.899910i	$-0.356368\pi$
$281$	14.6199	0.872152	0.436076	+	0.899910i	$0.356368\pi$
$282$	0	0
$283$	−2.43022	−0.144462	−0.0722308	−	0.997388i	$-0.523012\pi$
$283$	−2.43022	−0.144462	−0.0722308	+	0.997388i	$0.523012\pi$
$284$	0	0
$285$	−22.3443	−1.32356
$286$	0	0
$287$	23.9674	1.41475
$288$	0	0
$289$	−15.9559	−0.938580
$290$	0	0
$291$	−24.6061	−1.44244
$292$	0	0
$293$	8.69893	0.508197	0.254098	−	0.967178i	$-0.418221\pi$
$293$	8.69893	0.508197	0.254098	+	0.967178i	$0.418221\pi$
$294$	0	0
$295$	−29.2181	−1.70115
$296$	0	0
$297$	−7.70678	−0.447193
$298$	0	0
$299$	12.4576	0.720442
$300$	0	0
$301$	−0.235339	−0.0135647
$302$	0	0
$303$	−6.61175	−0.379835
$304$	0	0
$305$	−0.939396	−0.0537896
$306$	0	0
$307$	−30.3653	−1.73304	−0.866519	−	0.499144i	$-0.833648\pi$
$307$	−30.3653	−1.73304	−0.866519	+	0.499144i	$0.833648\pi$
$308$	0	0
$309$	0.562923	0.0320236
$310$	0	0
$311$	9.29032	0.526806	0.263403	−	0.964686i	$-0.415155\pi$
$311$	9.29032	0.526806	0.263403	+	0.964686i	$0.415155\pi$
$312$	0	0
$313$	21.6115	1.22156	0.610778	−	0.791802i	$-0.290857\pi$
$313$	21.6115	1.22156	0.610778	+	0.791802i	$0.290857\pi$
$314$	0	0
$315$	−32.6163	−1.83772
$316$	0	0
$317$	2.00868	0.112819	0.0564095	−	0.998408i	$-0.482035\pi$
$317$	2.00868	0.112819	0.0564095	+	0.998408i	$0.482035\pi$
$318$	0	0
$319$	15.1017	0.845533
$320$	0	0
$321$	42.6036	2.37790
$322$	0	0
$323$	−2.62834	−0.146245
$324$	0	0
$325$	−20.0212	−1.11058
$326$	0	0
$327$	−0.870872	−0.0481593
$328$	0	0
$329$	−6.03586	−0.332768
$330$	0	0
$331$	29.0256	1.59539	0.797696	−	0.603060i	$-0.206052\pi$
$331$	29.0256	1.59539	0.797696	+	0.603060i	$0.206052\pi$
$332$	0	0
$333$	−7.96287	−0.436362
$334$	0	0
$335$	40.8208	2.23028
$336$	0	0
$337$	9.06693	0.493907	0.246953	−	0.969027i	$-0.420571\pi$
$337$	9.06693	0.493907	0.246953	+	0.969027i	$0.420571\pi$
$338$	0	0
$339$	−10.9269	−0.593466
$340$	0	0
$341$	10.8181	0.585830
$342$	0	0
$343$	−12.5267	−0.676376
$344$	0	0
$345$	53.8577	2.89960
$346$	0	0
$347$	13.4505	0.722062	0.361031	−	0.932554i	$-0.382425\pi$
$347$	13.4505	0.722062	0.361031	+	0.932554i	$0.382425\pi$
$348$	0	0
$349$	21.2675	1.13842	0.569212	−	0.822191i	$-0.307248\pi$
$349$	21.2675	1.13842	0.569212	+	0.822191i	$0.307248\pi$
$350$	0	0
$351$	−4.31885	−0.230523
$352$	0	0
$353$	−19.4861	−1.03714	−0.518571	−	0.855034i	$-0.673536\pi$
$353$	−19.4861	−1.03714	−0.518571	+	0.855034i	$0.673536\pi$
$354$	0	0
$355$	22.5035	1.19436
$356$	0	0
$357$	−9.47083	−0.501249
$358$	0	0
$359$	23.1068	1.21953	0.609765	−	0.792582i	$-0.291264\pi$
$359$	23.1068	1.21953	0.609765	+	0.792582i	$0.291264\pi$
$360$	0	0
$361$	−12.3838	−0.651781
$362$	0	0
$363$	−4.16826	−0.218777
$364$	0	0
$365$	59.0689	3.09181
$366$	0	0
$367$	30.6813	1.60155	0.800775	−	0.598965i	$-0.204421\pi$
$367$	30.6813	1.60155	0.800775	+	0.598965i	$0.204421\pi$
$368$	0	0
$369$	−11.8628	−0.617550
$370$	0	0
$371$	−5.41338	−0.281049
$372$	0	0
$373$	30.9162	1.60078	0.800390	−	0.599480i	$-0.204626\pi$
$373$	30.9162	1.60078	0.800390	+	0.599480i	$0.204626\pi$
$374$	0	0
$375$	−43.1228	−2.22685
$376$	0	0
$377$	8.46293	0.435863
$378$	0	0
$379$	16.0759	0.825762	0.412881	−	0.910785i	$-0.364523\pi$
$379$	16.0759	0.825762	0.412881	+	0.910785i	$0.364523\pi$
$380$	0	0
$381$	−39.7016	−2.03398
$382$	0	0
$383$	28.5450	1.45858	0.729290	−	0.684205i	$-0.239851\pi$
$383$	28.5450	1.45858	0.729290	+	0.684205i	$0.239851\pi$
$384$	0	0
$385$	57.2470	2.91758
$386$	0	0
$387$	0.116482	0.00592112
$388$	0	0
$389$	24.4058	1.23743	0.618713	−	0.785617i	$-0.287654\pi$
$389$	24.4058	1.23743	0.618713	+	0.785617i	$0.287654\pi$
$390$	0	0
$391$	6.33523	0.320386
$392$	0	0
$393$	−16.9114	−0.853067
$394$	0	0
$395$	45.2452	2.27653
$396$	0	0
$397$	−26.5858	−1.33430	−0.667150	−	0.744923i	$-0.732486\pi$
$397$	−26.5858	−1.33430	−0.667150	+	0.744923i	$0.732486\pi$
$398$	0	0
$399$	23.8403	1.19351
$400$	0	0
$401$	31.9310	1.59456	0.797278	−	0.603612i	$-0.206272\pi$
$401$	31.9310	1.59456	0.797278	+	0.603612i	$0.206272\pi$
$402$	0	0
$403$	6.06239	0.301989
$404$	0	0
$405$	−42.3790	−2.10583
$406$	0	0
$407$	13.9762	0.692773
$408$	0	0
$409$	2.06458	0.102087	0.0510435	−	0.998696i	$-0.483745\pi$
$409$	2.06458	0.102087	0.0510435	+	0.998696i	$0.483745\pi$
$410$	0	0
$411$	−2.32826	−0.114845
$412$	0	0
$413$	31.1744	1.53399
$414$	0	0
$415$	−13.7798	−0.676423
$416$	0	0
$417$	0.492201	0.0241032
$418$	0	0
$419$	7.04738	0.344287	0.172144	−	0.985072i	$-0.444931\pi$
$419$	7.04738	0.344287	0.172144	+	0.985072i	$0.444931\pi$
$420$	0	0
$421$	−9.18236	−0.447521	−0.223760	−	0.974644i	$-0.571833\pi$
$421$	−9.18236	−0.447521	−0.223760	+	0.974644i	$0.571833\pi$
$422$	0	0
$423$	2.98748	0.145256
$424$	0	0
$425$	−10.1816	−0.493882
$426$	0	0
$427$	1.00229	0.0485043
$428$	0	0
$429$	−16.1787	−0.781117
$430$	0	0
$431$	26.3927	1.27129	0.635646	−	0.771980i	$-0.280734\pi$
$431$	26.3927	1.27129	0.635646	+	0.771980i	$0.280734\pi$
$432$	0	0
$433$	29.5283	1.41904	0.709520	−	0.704685i	$-0.248912\pi$
$433$	29.5283	1.41904	0.709520	+	0.704685i	$0.248912\pi$
$434$	0	0
$435$	36.5876	1.75424
$436$	0	0
$437$	−15.9473	−0.762862
$438$	0	0
$439$	9.83243	0.469276	0.234638	−	0.972083i	$-0.424610\pi$
$439$	9.83243	0.469276	0.234638	+	0.972083i	$0.424610\pi$
$440$	0	0
$441$	20.5001	0.976195
$442$	0	0
$443$	24.5381	1.16584	0.582921	−	0.812529i	$-0.301910\pi$
$443$	24.5381	1.16584	0.582921	+	0.812529i	$0.301910\pi$
$444$	0	0
$445$	−58.3636	−2.76670
$446$	0	0
$447$	46.4706	2.19798
$448$	0	0
$449$	−18.2902	−0.863170	−0.431585	−	0.902072i	$-0.642046\pi$
$449$	−18.2902	−0.863170	−0.431585	+	0.902072i	$0.642046\pi$
$450$	0	0
$451$	20.8211	0.980429
$452$	0	0
$453$	−2.45019	−0.115120
$454$	0	0
$455$	32.0810	1.50398
$456$	0	0
$457$	5.87844	0.274982	0.137491	−	0.990503i	$-0.456096\pi$
$457$	5.87844	0.274982	0.137491	+	0.990503i	$0.456096\pi$
$458$	0	0
$459$	−2.19632	−0.102515
$460$	0	0
$461$	−25.6372	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$461$	−25.6372	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$462$	0	0
$463$	−0.832038	−0.0386681	−0.0193340	−	0.999813i	$-0.506155\pi$
$463$	−0.832038	−0.0386681	−0.0193340	+	0.999813i	$0.506155\pi$
$464$	0	0
$465$	26.2094	1.21543
$466$	0	0
$467$	−26.3242	−1.21814	−0.609070	−	0.793117i	$-0.708457\pi$
$467$	−26.3242	−1.21814	−0.609070	+	0.793117i	$0.708457\pi$
$468$	0	0
$469$	−43.5540	−2.01114
$470$	0	0
$471$	−43.5413	−2.00628
$472$	0	0
$473$	−0.204446	−0.00940042
$474$	0	0
$475$	25.6296	1.17597
$476$	0	0
$477$	2.67938	0.122680
$478$	0	0
$479$	17.0901	0.780865	0.390433	−	0.920631i	$-0.372325\pi$
$479$	17.0901	0.780865	0.390433	+	0.920631i	$0.372325\pi$
$480$	0	0
$481$	7.83219	0.357117
$482$	0	0
$483$	−57.4637	−2.61469
$484$	0	0
$485$	42.3868	1.92469
$486$	0	0
$487$	7.95573	0.360509	0.180254	−	0.983620i	$-0.442308\pi$
$487$	7.95573	0.360509	0.180254	+	0.983620i	$0.442308\pi$
$488$	0	0
$489$	−18.7087	−0.846039
$490$	0	0
$491$	35.0580	1.58215	0.791074	−	0.611721i	$-0.209522\pi$
$491$	35.0580	1.58215	0.791074	+	0.611721i	$0.209522\pi$
$492$	0	0
$493$	4.30377	0.193832
$494$	0	0
$495$	−28.3347	−1.27355
$496$	0	0
$497$	−24.0102	−1.07701
$498$	0	0
$499$	25.2841	1.13187	0.565935	−	0.824450i	$-0.308515\pi$
$499$	25.2841	1.13187	0.565935	+	0.824450i	$0.308515\pi$
$500$	0	0
$501$	−19.8133	−0.885191
$502$	0	0
$503$	19.1981	0.855999	0.428000	−	0.903779i	$-0.359218\pi$
$503$	19.1981	0.855999	0.428000	+	0.903779i	$0.359218\pi$
$504$	0	0
$505$	11.3895	0.506825
$506$	0	0
$507$	20.1267	0.893857
$508$	0	0
$509$	17.3575	0.769359	0.384679	−	0.923050i	$-0.374312\pi$
$509$	17.3575	0.769359	0.384679	+	0.923050i	$0.374312\pi$
$510$	0	0
$511$	−63.0238	−2.78801
$512$	0	0
$513$	5.52866	0.244096
$514$	0	0
$515$	−0.969699	−0.0427300
$516$	0	0
$517$	−5.24353	−0.230610
$518$	0	0
$519$	−35.9351	−1.57738
$520$	0	0
$521$	−40.5189	−1.77517	−0.887583	−	0.460647i	$-0.847617\pi$
$521$	−40.5189	−1.77517	−0.887583	+	0.460647i	$0.847617\pi$
$522$	0	0
$523$	−13.4974	−0.590199	−0.295100	−	0.955466i	$-0.595353\pi$
$523$	−13.4974	−0.590199	−0.295100	+	0.955466i	$0.595353\pi$
$524$	0	0
$525$	92.3526	4.03060
$526$	0	0
$527$	3.08299	0.134297
$528$	0	0
$529$	15.4386	0.671245
$530$	0	0
$531$	−15.4299	−0.669601
$532$	0	0
$533$	11.6681	0.505401
$534$	0	0
$535$	−73.3895	−3.17291
$536$	0	0
$537$	−29.6542	−1.27967
$538$	0	0
$539$	−35.9811	−1.54982
$540$	0	0
$541$	1.94426	0.0835905	0.0417952	−	0.999126i	$-0.486692\pi$
$541$	1.94426	0.0835905	0.0417952	+	0.999126i	$0.486692\pi$
$542$	0	0
$543$	−21.8462	−0.937511
$544$	0	0
$545$	1.50018	0.0642604
$546$	0	0
$547$	−4.07056	−0.174045	−0.0870223	−	0.996206i	$-0.527735\pi$
$547$	−4.07056	−0.174045	−0.0870223	+	0.996206i	$0.527735\pi$
$548$	0	0
$549$	−0.496089	−0.0211726
$550$	0	0
$551$	−10.8336	−0.461527
$552$	0	0
$553$	−48.2745	−2.05284
$554$	0	0
$555$	33.8607	1.43731
$556$	0	0
$557$	43.8994	1.86008	0.930039	−	0.367460i	$-0.119772\pi$
$557$	43.8994	1.86008	0.930039	+	0.367460i	$0.119772\pi$
$558$	0	0
$559$	−0.114571	−0.00484582
$560$	0	0
$561$	−8.22758	−0.347369
$562$	0	0
$563$	−22.4677	−0.946901	−0.473451	−	0.880820i	$-0.656992\pi$
$563$	−22.4677	−0.946901	−0.473451	+	0.880820i	$0.656992\pi$
$564$	0	0
$565$	18.8228	0.791880
$566$	0	0
$567$	45.2164	1.89891
$568$	0	0
$569$	20.7639	0.870467	0.435234	−	0.900318i	$-0.356666\pi$
$569$	20.7639	0.870467	0.435234	+	0.900318i	$0.356666\pi$
$570$	0	0
$571$	−46.0109	−1.92549	−0.962747	−	0.270403i	$-0.912843\pi$
$571$	−46.0109	−1.92549	−0.962747	+	0.270403i	$0.912843\pi$
$572$	0	0
$573$	11.3340	0.473483
$574$	0	0
$575$	−61.7765	−2.57626
$576$	0	0
$577$	8.66239	0.360620	0.180310	−	0.983610i	$-0.442290\pi$
$577$	8.66239	0.360620	0.180310	+	0.983610i	$0.442290\pi$
$578$	0	0
$579$	3.35647	0.139490
$580$	0	0
$581$	14.7024	0.609958
$582$	0	0
$583$	−4.70276	−0.194769
$584$	0	0
$585$	−15.8786	−0.656501
$586$	0	0
$587$	42.4086	1.75039	0.875195	−	0.483770i	$-0.160733\pi$
$587$	42.4086	1.75039	0.875195	+	0.483770i	$0.160733\pi$
$588$	0	0
$589$	−7.76062	−0.319771
$590$	0	0
$591$	−24.2507	−0.997539
$592$	0	0
$593$	−41.2466	−1.69379	−0.846897	−	0.531756i	$-0.821532\pi$
$593$	−41.2466	−1.69379	−0.846897	+	0.531756i	$0.821532\pi$
$594$	0	0
$595$	16.3146	0.668832
$596$	0	0
$597$	11.5887	0.474295
$598$	0	0
$599$	−16.7882	−0.685949	−0.342974	−	0.939345i	$-0.611434\pi$
$599$	−16.7882	−0.685949	−0.342974	+	0.939345i	$0.611434\pi$
$600$	0	0
$601$	17.7112	0.722455	0.361227	−	0.932478i	$-0.382358\pi$
$601$	17.7112	0.722455	0.361227	+	0.932478i	$0.382358\pi$
$602$	0	0
$603$	21.5572	0.877879
$604$	0	0
$605$	7.18030	0.291921
$606$	0	0
$607$	−3.59364	−0.145861	−0.0729306	−	0.997337i	$-0.523235\pi$
$607$	−3.59364	−0.145861	−0.0729306	+	0.997337i	$0.523235\pi$
$608$	0	0
$609$	−39.0373	−1.58187
$610$	0	0
$611$	−2.93845	−0.118877
$612$	0	0
$613$	−3.60604	−0.145647	−0.0728234	−	0.997345i	$-0.523201\pi$
$613$	−3.60604	−0.145647	−0.0728234	+	0.997345i	$0.523201\pi$
$614$	0	0
$615$	50.4443	2.03411
$616$	0	0
$617$	−39.4534	−1.58833	−0.794167	−	0.607699i	$-0.792093\pi$
$617$	−39.4534	−1.58833	−0.794167	+	0.607699i	$0.792093\pi$
$618$	0	0
$619$	37.4639	1.50580	0.752900	−	0.658135i	$-0.228654\pi$
$619$	37.4639	1.50580	0.752900	+	0.658135i	$0.228654\pi$
$620$	0	0
$621$	−13.3260	−0.534756
$622$	0	0
$623$	62.2712	2.49484
$624$	0	0
$625$	24.4632	0.978528
$626$	0	0
$627$	20.7108	0.827110
$628$	0	0
$629$	3.98301	0.158813
$630$	0	0
$631$	9.38680	0.373683	0.186841	−	0.982390i	$-0.440175\pi$
$631$	9.38680	0.373683	0.186841	+	0.982390i	$0.440175\pi$
$632$	0	0
$633$	47.7817	1.89915
$634$	0	0
$635$	68.3906	2.71400
$636$	0	0
$637$	−20.1637	−0.798914
$638$	0	0
$639$	11.8840	0.470123
$640$	0	0
$641$	34.8033	1.37465	0.687323	−	0.726352i	$-0.258786\pi$
$641$	34.8033	1.37465	0.687323	+	0.726352i	$0.258786\pi$
$642$	0	0
$643$	46.7128	1.84217	0.921086	−	0.389360i	$-0.127304\pi$
$643$	46.7128	1.84217	0.921086	+	0.389360i	$0.127304\pi$
$644$	0	0
$645$	−0.495320	−0.0195032
$646$	0	0
$647$	−8.33210	−0.327568	−0.163784	−	0.986496i	$-0.552370\pi$
$647$	−8.33210	−0.327568	−0.163784	+	0.986496i	$0.552370\pi$
$648$	0	0
$649$	27.0821	1.06307
$650$	0	0
$651$	−27.9642	−1.09600
$652$	0	0
$653$	25.5258	0.998903	0.499452	−	0.866342i	$-0.333535\pi$
$653$	25.5258	0.998903	0.499452	+	0.866342i	$0.333535\pi$
$654$	0	0
$655$	29.1318	1.13827
$656$	0	0
$657$	31.1939	1.21699
$658$	0	0
$659$	3.62485	0.141204	0.0706020	−	0.997505i	$-0.477508\pi$
$659$	3.62485	0.141204	0.0706020	+	0.997505i	$0.477508\pi$
$660$	0	0
$661$	−46.9428	−1.82586	−0.912931	−	0.408113i	$-0.866187\pi$
$661$	−46.9428	−1.82586	−0.912931	+	0.408113i	$0.866187\pi$
$662$	0	0
$663$	−4.61070	−0.179065
$664$	0	0
$665$	−41.0677	−1.59254
$666$	0	0
$667$	26.1128	1.01109
$668$	0	0
$669$	−16.3293	−0.631329
$670$	0	0
$671$	0.870720	0.0336138
$672$	0	0
$673$	−25.3263	−0.976256	−0.488128	−	0.872772i	$-0.662320\pi$
$673$	−25.3263	−0.976256	−0.488128	+	0.872772i	$0.662320\pi$
$674$	0	0
$675$	21.4169	0.824337
$676$	0	0
$677$	7.06174	0.271405	0.135702	−	0.990750i	$-0.456671\pi$
$677$	7.06174	0.271405	0.135702	+	0.990750i	$0.456671\pi$
$678$	0	0
$679$	−45.2248	−1.73557
$680$	0	0
$681$	1.66788	0.0639133
$682$	0	0
$683$	14.4159	0.551610	0.275805	−	0.961214i	$-0.411056\pi$
$683$	14.4159	0.551610	0.275805	+	0.961214i	$0.411056\pi$
$684$	0	0
$685$	4.01070	0.153241
$686$	0	0
$687$	20.3998	0.778300
$688$	0	0
$689$	−2.63541	−0.100401
$690$	0	0
$691$	−10.9622	−0.417021	−0.208511	−	0.978020i	$-0.566862\pi$
$691$	−10.9622	−0.417021	−0.208511	+	0.978020i	$0.566862\pi$
$692$	0	0
$693$	30.2318	1.14841
$694$	0	0
$695$	−0.847871	−0.0321616
$696$	0	0
$697$	5.93372	0.224756
$698$	0	0
$699$	−40.6236	−1.53653
$700$	0	0
$701$	−22.9555	−0.867018	−0.433509	−	0.901149i	$-0.642725\pi$
$701$	−22.9555	−0.867018	−0.433509	+	0.901149i	$0.642725\pi$
$702$	0	0
$703$	−10.0262	−0.378144
$704$	0	0
$705$	−12.7037	−0.478450
$706$	0	0
$707$	−12.1521	−0.457025
$708$	0	0
$709$	52.0170	1.95354	0.976770	−	0.214288i	$-0.0687432\pi$
$709$	52.0170	1.95354	0.976770	+	0.214288i	$0.0687432\pi$
$710$	0	0
$711$	23.8937	0.896084
$712$	0	0
$713$	18.7058	0.700539
$714$	0	0
$715$	27.8697	1.04227
$716$	0	0
$717$	4.68487	0.174960
$718$	0	0
$719$	−47.6509	−1.77708	−0.888539	−	0.458801i	$-0.848279\pi$
$719$	−47.6509	−1.77708	−0.888539	+	0.458801i	$0.848279\pi$
$720$	0	0
$721$	1.03462	0.0385314
$722$	0	0
$723$	8.15791	0.303396
$724$	0	0
$725$	−41.9671	−1.55862
$726$	0	0
$727$	−45.2003	−1.67639	−0.838193	−	0.545374i	$-0.816387\pi$
$727$	−45.2003	−1.67639	−0.838193	+	0.545374i	$0.816387\pi$
$728$	0	0
$729$	−7.89981	−0.292586
$730$	0	0
$731$	−0.0582640	−0.00215497
$732$	0	0
$733$	−11.4146	−0.421609	−0.210804	−	0.977528i	$-0.567608\pi$
$733$	−11.4146	−0.421609	−0.210804	+	0.977528i	$0.567608\pi$
$734$	0	0
$735$	−87.1731	−3.21543
$736$	0	0
$737$	−37.8366	−1.39373
$738$	0	0
$739$	−43.0239	−1.58266	−0.791331	−	0.611388i	$-0.790611\pi$
$739$	−43.0239	−1.58266	−0.791331	+	0.611388i	$0.790611\pi$
$740$	0	0
$741$	11.6062	0.426366
$742$	0	0
$743$	−40.9621	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$743$	−40.9621	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$744$	0	0
$745$	−80.0508	−2.93284
$746$	0	0
$747$	−7.27702	−0.266252
$748$	0	0
$749$	78.3033	2.86114
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−13.3626	−0.486960
$754$	0	0
$755$	4.22073	0.153608
$756$	0	0
$757$	29.5531	1.07413	0.537063	−	0.843542i	$-0.319534\pi$
$757$	29.5531	1.07413	0.537063	+	0.843542i	$0.319534\pi$
$758$	0	0
$759$	−49.9204	−1.81199
$760$	0	0
$761$	−12.1186	−0.439299	−0.219649	−	0.975579i	$-0.570491\pi$
$761$	−12.1186	−0.439299	−0.219649	+	0.975579i	$0.570491\pi$
$762$	0	0
$763$	−1.60062	−0.0579463
$764$	0	0
$765$	−8.07497	−0.291951
$766$	0	0
$767$	15.1767	0.547999
$768$	0	0
$769$	−31.9512	−1.15219	−0.576096	−	0.817382i	$-0.695424\pi$
$769$	−31.9512	−1.15219	−0.576096	+	0.817382i	$0.695424\pi$
$770$	0	0
$771$	39.8589	1.43548
$772$	0	0
$773$	−29.9199	−1.07614	−0.538072	−	0.842899i	$-0.680847\pi$
$773$	−29.9199	−1.07614	−0.538072	+	0.842899i	$0.680847\pi$
$774$	0	0
$775$	−30.0630	−1.07990
$776$	0	0
$777$	−36.1278	−1.29608
$778$	0	0
$779$	−14.9366	−0.535159
$780$	0	0
$781$	−20.8584	−0.746372
$782$	0	0
$783$	−9.05289	−0.323524
$784$	0	0
$785$	75.0048	2.67704
$786$	0	0
$787$	10.0271	0.357429	0.178714	−	0.983901i	$-0.442806\pi$
$787$	10.0271	0.357429	0.178714	+	0.983901i	$0.442806\pi$
$788$	0	0
$789$	45.0993	1.60558
$790$	0	0
$791$	−20.0830	−0.714070
$792$	0	0
$793$	0.487948	0.0173275
$794$	0	0
$795$	−11.3936	−0.404090
$796$	0	0
$797$	5.87150	0.207979	0.103990	−	0.994578i	$-0.466839\pi$
$797$	5.87150	0.207979	0.103990	+	0.994578i	$0.466839\pi$
$798$	0	0
$799$	−1.49433	−0.0528655
$800$	0	0
$801$	−30.8214	−1.08902
$802$	0	0
$803$	−54.7506	−1.93211
$804$	0	0
$805$	98.9877	3.48886
$806$	0	0
$807$	69.9304	2.46167
$808$	0	0
$809$	44.5216	1.56530	0.782649	−	0.622464i	$-0.213868\pi$
$809$	44.5216	1.56530	0.782649	+	0.622464i	$0.213868\pi$
$810$	0	0
$811$	10.4644	0.367453	0.183727	−	0.982977i	$-0.441184\pi$
$811$	10.4644	0.367453	0.183727	+	0.982977i	$0.441184\pi$
$812$	0	0
$813$	14.6608	0.514175
$814$	0	0
$815$	32.2279	1.12889
$816$	0	0
$817$	0.146665	0.00513114
$818$	0	0
$819$	16.9418	0.591994
$820$	0	0
$821$	−33.1807	−1.15801	−0.579007	−	0.815323i	$-0.696560\pi$
$821$	−33.1807	−1.15801	−0.579007	+	0.815323i	$0.696560\pi$
$822$	0	0
$823$	−4.80329	−0.167432	−0.0837161	−	0.996490i	$-0.526679\pi$
$823$	−4.80329	−0.167432	−0.0837161	+	0.996490i	$0.526679\pi$
$824$	0	0
$825$	80.2293	2.79323
$826$	0	0
$827$	35.0800	1.21985	0.609926	−	0.792458i	$-0.291199\pi$
$827$	35.0800	1.21985	0.609926	+	0.792458i	$0.291199\pi$
$828$	0	0
$829$	−3.49480	−0.121379	−0.0606897	−	0.998157i	$-0.519330\pi$
$829$	−3.49480	−0.121379	−0.0606897	+	0.998157i	$0.519330\pi$
$830$	0	0
$831$	0.440380	0.0152766
$832$	0	0
$833$	−10.2541	−0.355283
$834$	0	0
$835$	34.1306	1.18114
$836$	0	0
$837$	−6.48501	−0.224155
$838$	0	0
$839$	−15.4940	−0.534913	−0.267457	−	0.963570i	$-0.586183\pi$
$839$	−15.4940	−0.534913	−0.267457	+	0.963570i	$0.586183\pi$
$840$	0	0
$841$	−11.2605	−0.388295
$842$	0	0
$843$	−32.8309	−1.13076
$844$	0	0
$845$	−34.6705	−1.19270
$846$	0	0
$847$	−7.66105	−0.263237
$848$	0	0
$849$	5.45737	0.187296
$850$	0	0
$851$	24.1666	0.828422
$852$	0	0
$853$	28.1189	0.962773	0.481386	−	0.876508i	$-0.340133\pi$
$853$	28.1189	0.962773	0.481386	+	0.876508i	$0.340133\pi$
$854$	0	0
$855$	20.3266	0.695156
$856$	0	0
$857$	−1.44535	−0.0493723	−0.0246861	−	0.999695i	$-0.507859\pi$
$857$	−1.44535	−0.0493723	−0.0246861	+	0.999695i	$0.507859\pi$
$858$	0	0
$859$	−3.29259	−0.112342	−0.0561708	−	0.998421i	$-0.517889\pi$
$859$	−3.29259	−0.112342	−0.0561708	+	0.998421i	$0.517889\pi$
$860$	0	0
$861$	−53.8218	−1.83424
$862$	0	0
$863$	−30.7547	−1.04690	−0.523451	−	0.852056i	$-0.675356\pi$
$863$	−30.7547	−1.04690	−0.523451	+	0.852056i	$0.675356\pi$
$864$	0	0
$865$	61.9023	2.10474
$866$	0	0
$867$	35.8310	1.21688
$868$	0	0
$869$	−41.9375	−1.42263
$870$	0	0
$871$	−21.2035	−0.718452
$872$	0	0
$873$	22.3842	0.757591
$874$	0	0
$875$	−79.2575	−2.67939
$876$	0	0
$877$	0.0561193	0.00189502	0.000947508	−	1.00000i	$-0.499698\pi$
$877$	0.0561193	0.00189502	0.000947508		1.00000i	$0.499698\pi$
$878$	0	0
$879$	−19.5346	−0.658885
$880$	0	0
$881$	14.5293	0.489505	0.244752	−	0.969586i	$-0.421293\pi$
$881$	14.5293	0.489505	0.244752	+	0.969586i	$0.421293\pi$
$882$	0	0
$883$	22.2369	0.748332	0.374166	−	0.927362i	$-0.377929\pi$
$883$	22.2369	0.748332	0.374166	+	0.927362i	$0.377929\pi$
$884$	0	0
$885$	65.6131	2.20556
$886$	0	0
$887$	−5.67185	−0.190442	−0.0952211	−	0.995456i	$-0.530356\pi$
$887$	−5.67185	−0.190442	−0.0952211	+	0.995456i	$0.530356\pi$
$888$	0	0
$889$	−72.9696	−2.44732
$890$	0	0
$891$	39.2808	1.31596
$892$	0	0
$893$	3.76158	0.125877
$894$	0	0
$895$	51.0827	1.70751
$896$	0	0
$897$	−27.9752	−0.934064
$898$	0	0
$899$	12.7076	0.423822
$900$	0	0
$901$	−1.34022	−0.0446492
$902$	0	0
$903$	0.528484	0.0175868
$904$	0	0
$905$	37.6326	1.25095
$906$	0	0
$907$	10.0051	0.332215	0.166108	−	0.986108i	$-0.446880\pi$
$907$	10.0051	0.332215	0.166108	+	0.986108i	$0.446880\pi$
$908$	0	0
$909$	6.01472	0.199496
$910$	0	0
$911$	29.7482	0.985600	0.492800	−	0.870142i	$-0.335973\pi$
$911$	29.7482	0.985600	0.492800	+	0.870142i	$0.335973\pi$
$912$	0	0
$913$	12.7724	0.422705
$914$	0	0
$915$	2.10953	0.0697391
$916$	0	0
$917$	−31.0823	−1.02643
$918$	0	0
$919$	−50.5021	−1.66591	−0.832956	−	0.553340i	$-0.813353\pi$
$919$	−50.5021	−1.66591	−0.832956	+	0.553340i	$0.813353\pi$
$920$	0	0
$921$	68.1892	2.24691
$922$	0	0
$923$	−11.6890	−0.384747
$924$	0	0
$925$	−38.8393	−1.27703
$926$	0	0
$927$	−0.512092	−0.0168193
$928$	0	0
$929$	8.62854	0.283093	0.141547	−	0.989932i	$-0.454792\pi$
$929$	8.62854	0.283093	0.141547	+	0.989932i	$0.454792\pi$
$930$	0	0
$931$	25.8120	0.845954
$932$	0	0
$933$	−20.8626	−0.683011
$934$	0	0
$935$	14.1729	0.463505
$936$	0	0
$937$	−0.825041	−0.0269529	−0.0134765	−	0.999909i	$-0.504290\pi$
$937$	−0.825041	−0.0269529	−0.0134765	+	0.999909i	$0.504290\pi$
$938$	0	0
$939$	−48.5315	−1.58377
$940$	0	0
$941$	−33.8598	−1.10380	−0.551898	−	0.833911i	$-0.686096\pi$
$941$	−33.8598	−1.10380	−0.551898	+	0.833911i	$0.686096\pi$
$942$	0	0
$943$	36.0025	1.17240
$944$	0	0
$945$	−34.3174	−1.11635
$946$	0	0
$947$	−57.1115	−1.85588	−0.927938	−	0.372735i	$-0.878420\pi$
$947$	−57.1115	−1.85588	−0.927938	+	0.372735i	$0.878420\pi$
$948$	0	0
$949$	−30.6820	−0.995981
$950$	0	0
$951$	−4.51076	−0.146271
$952$	0	0
$953$	10.0768	0.326418	0.163209	−	0.986592i	$-0.447816\pi$
$953$	10.0768	0.326418	0.163209	+	0.986592i	$0.447816\pi$
$954$	0	0
$955$	−19.5240	−0.631782
$956$	0	0
$957$	−33.9128	−1.09625
$958$	0	0
$959$	−4.27923	−0.138184
$960$	0	0
$961$	−21.8970	−0.706353
$962$	0	0
$963$	−38.7566	−1.24891
$964$	0	0
$965$	−5.78191	−0.186126
$966$	0	0
$967$	−38.6186	−1.24189	−0.620946	−	0.783853i	$-0.713251\pi$
$967$	−38.6186	−1.24189	−0.620946	+	0.783853i	$0.713251\pi$
$968$	0	0
$969$	5.90228	0.189608
$970$	0	0
$971$	12.5031	0.401244	0.200622	−	0.979669i	$-0.435704\pi$
$971$	12.5031	0.401244	0.200622	+	0.979669i	$0.435704\pi$
$972$	0	0
$973$	0.904640	0.0290014
$974$	0	0
$975$	44.9602	1.43988
$976$	0	0
$977$	12.1363	0.388276	0.194138	−	0.980974i	$-0.437809\pi$
$977$	12.1363	0.388276	0.194138	+	0.980974i	$0.437809\pi$
$978$	0	0
$979$	54.0968	1.72894
$980$	0	0
$981$	0.792233	0.0252941
$982$	0	0
$983$	−35.8508	−1.14346	−0.571731	−	0.820441i	$-0.693728\pi$
$983$	−35.8508	−1.14346	−0.571731	+	0.820441i	$0.693728\pi$
$984$	0	0
$985$	41.7745	1.33105
$986$	0	0
$987$	13.5543	0.431438
$988$	0	0
$989$	−0.353514	−0.0112411
$990$	0	0
$991$	40.9478	1.30075	0.650374	−	0.759614i	$-0.274612\pi$
$991$	40.9478	1.30075	0.650374	+	0.759614i	$0.274612\pi$
$992$	0	0
$993$	−65.1807	−2.06845
$994$	0	0
$995$	−19.9629	−0.632866
$996$	0	0
$997$	24.6576	0.780915	0.390457	−	0.920621i	$-0.372317\pi$
$997$	24.6576	0.780915	0.390457	+	0.920621i	$0.372317\pi$
$998$	0	0
$999$	−8.37818	−0.265074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.11	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.11	✓	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24563 q^{3} +3.86835 q^{5} -4.12735 q^{7} +2.04285 q^{9} +O(q^{10})\)	\(q-2.24563 q^{3} +3.86835 q^{5} -4.12735 q^{7} +2.04285 q^{9} -3.58555 q^{11} -2.00933 q^{13} -8.68688 q^{15} -1.02183 q^{17} +2.57219 q^{19} +9.26850 q^{21} -6.19989 q^{23} +9.96413 q^{25} +2.14940 q^{27} -4.21182 q^{29} -3.01713 q^{31} +8.05182 q^{33} -15.9660 q^{35} -3.89792 q^{37} +4.51221 q^{39} -5.80696 q^{41} +0.0570193 q^{43} +7.90247 q^{45} +1.46241 q^{47} +10.0350 q^{49} +2.29465 q^{51} +1.31159 q^{53} -13.8702 q^{55} -5.77618 q^{57} -7.55313 q^{59} -0.242841 q^{61} -8.43157 q^{63} -7.77278 q^{65} +10.5525 q^{67} +13.9227 q^{69} +5.81735 q^{71} +15.2698 q^{73} -22.3757 q^{75} +14.7988 q^{77} +11.6962 q^{79} -10.9553 q^{81} -3.56219 q^{83} -3.95279 q^{85} +9.45819 q^{87} -15.0875 q^{89} +8.29320 q^{91} +6.77535 q^{93} +9.95013 q^{95} +10.9573 q^{97} -7.32475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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