LMFDB - Embedded newform 8001.2.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 $ x^11 - 2x^10 - 15x^9 + 25x^8 + 88x^7 - 112x^6 - 247x^5 + 215x^4 + 313x^3 - 170x^2 - 121x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.50746$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.50746 q^{2} +0.272426 q^{4} -0.476857 q^{5} -1.00000 q^{7} +2.60424 q^{8} +O(q^{10})$	\(q-1.50746 q^{2} +0.272426 q^{4} -0.476857 q^{5} -1.00000 q^{7} +2.60424 q^{8} +0.718841 q^{10} +4.00870 q^{11} +1.63200 q^{13} +1.50746 q^{14} -4.47064 q^{16} +4.10247 q^{17} -1.35288 q^{19} -0.129908 q^{20} -6.04294 q^{22} +1.19852 q^{23} -4.77261 q^{25} -2.46017 q^{26} -0.272426 q^{28} -4.58834 q^{29} -3.68226 q^{31} +1.53080 q^{32} -6.18429 q^{34} +0.476857 q^{35} -3.21242 q^{37} +2.03940 q^{38} -1.24185 q^{40} +7.37574 q^{41} +3.11995 q^{43} +1.09207 q^{44} -1.80671 q^{46} +5.76016 q^{47} +1.00000 q^{49} +7.19450 q^{50} +0.444598 q^{52} -4.58069 q^{53} -1.91158 q^{55} -2.60424 q^{56} +6.91672 q^{58} -1.48560 q^{59} -3.84302 q^{61} +5.55084 q^{62} +6.63365 q^{64} -0.778229 q^{65} -16.2197 q^{67} +1.11762 q^{68} -0.718841 q^{70} -12.3537 q^{71} -13.3933 q^{73} +4.84258 q^{74} -0.368559 q^{76} -4.00870 q^{77} +6.36100 q^{79} +2.13185 q^{80} -11.1186 q^{82} -8.35059 q^{83} -1.95629 q^{85} -4.70319 q^{86} +10.4396 q^{88} +4.61788 q^{89} -1.63200 q^{91} +0.326507 q^{92} -8.68319 q^{94} +0.645129 q^{95} -9.03352 q^{97} -1.50746 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

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 $

$2$	−1.50746	−1.06593	−0.532966	−	0.846136i	$-0.678923\pi$
$2$	−1.50746	−1.06593	−0.532966	+	0.846136i	$0.678923\pi$
$3$	0	0
$3$	0	0
$4$	0.272426	0.136213
$5$	−0.476857	−0.213257	−0.106628	−	0.994299i	$-0.534006\pi$
$5$	−0.476857	−0.213257	−0.106628	+	0.994299i	$0.534006\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.60424	0.920739
$9$	0	0
$10$	0.718841	0.227317
$11$	4.00870	1.20867	0.604334	−	0.796731i	$-0.293439\pi$
$11$	4.00870	1.20867	0.604334	+	0.796731i	$0.293439\pi$
$12$	0	0
$13$	1.63200	0.452635	0.226317	−	0.974054i	$-0.427331\pi$
$13$	1.63200	0.452635	0.226317	+	0.974054i	$0.427331\pi$
$14$	1.50746	0.402885
$15$	0	0
$16$	−4.47064	−1.11766
$17$	4.10247	0.994995	0.497497	−	0.867466i	$-0.334252\pi$
$17$	4.10247	0.994995	0.497497	+	0.867466i	$0.334252\pi$
$18$	0	0
$19$	−1.35288	−0.310371	−0.155186	−	0.987885i	$-0.549598\pi$
$19$	−1.35288	−0.310371	−0.155186	+	0.987885i	$0.549598\pi$
$20$	−0.129908	−0.0290483
$21$	0	0
$22$	−6.04294	−1.28836
$23$	1.19852	0.249908	0.124954	−	0.992163i	$-0.460122\pi$
$23$	1.19852	0.249908	0.124954	+	0.992163i	$0.460122\pi$
$24$	0	0
$25$	−4.77261	−0.954522
$26$	−2.46017	−0.482478
$27$	0	0
$28$	−0.272426	−0.0514836
$29$	−4.58834	−0.852033	−0.426016	−	0.904716i	$-0.640083\pi$
$29$	−4.58834	−0.852033	−0.426016	+	0.904716i	$0.640083\pi$
$30$	0	0
$31$	−3.68226	−0.661353	−0.330676	−	0.943744i	$-0.607277\pi$
$31$	−3.68226	−0.661353	−0.330676	+	0.943744i	$0.607277\pi$
$32$	1.53080	0.270610
$33$	0	0
$34$	−6.18429	−1.06060
$35$	0.476857	0.0806035
$36$	0	0
$37$	−3.21242	−0.528119	−0.264059	−	0.964506i	$-0.585061\pi$
$37$	−3.21242	−0.528119	−0.264059	+	0.964506i	$0.585061\pi$
$38$	2.03940	0.330835
$39$	0	0
$40$	−1.24185	−0.196354
$41$	7.37574	1.15190	0.575949	−	0.817486i	$-0.304633\pi$
$41$	7.37574	1.15190	0.575949	+	0.817486i	$0.304633\pi$
$42$	0	0
$43$	3.11995	0.475788	0.237894	−	0.971291i	$-0.423543\pi$
$43$	3.11995	0.475788	0.237894	+	0.971291i	$0.423543\pi$
$44$	1.09207	0.164636
$45$	0	0
$46$	−1.80671	−0.266385
$47$	5.76016	0.840206	0.420103	−	0.907476i	$-0.361994\pi$
$47$	5.76016	0.840206	0.420103	+	0.907476i	$0.361994\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.19450	1.01746
$51$	0	0
$52$	0.444598	0.0616547
$53$	−4.58069	−0.629206	−0.314603	−	0.949223i	$-0.601871\pi$
$53$	−4.58069	−0.629206	−0.314603	+	0.949223i	$0.601871\pi$
$54$	0	0
$55$	−1.91158	−0.257757
$56$	−2.60424	−0.348007
$57$	0	0
$58$	6.91672	0.908209
$59$	−1.48560	−0.193408	−0.0967041	−	0.995313i	$-0.530830\pi$
$59$	−1.48560	−0.193408	−0.0967041	+	0.995313i	$0.530830\pi$
$60$	0	0
$61$	−3.84302	−0.492048	−0.246024	−	0.969264i	$-0.579124\pi$
$61$	−3.84302	−0.492048	−0.246024	+	0.969264i	$0.579124\pi$
$62$	5.55084	0.704958
$63$	0	0
$64$	6.63365	0.829207
$65$	−0.778229	−0.0965275
$66$	0	0
$67$	−16.2197	−1.98155	−0.990776	−	0.135513i	$-0.956732\pi$
$67$	−16.2197	−1.98155	−0.990776	+	0.135513i	$0.956732\pi$
$68$	1.11762	0.135531
$69$	0	0
$70$	−0.718841	−0.0859179
$71$	−12.3537	−1.46612	−0.733058	−	0.680166i	$-0.761908\pi$
$71$	−12.3537	−1.46612	−0.733058	+	0.680166i	$0.761908\pi$
$72$	0	0
$73$	−13.3933	−1.56756	−0.783782	−	0.621037i	$-0.786712\pi$
$73$	−13.3933	−1.56756	−0.783782	+	0.621037i	$0.786712\pi$
$74$	4.84258	0.562939
$75$	0	0
$76$	−0.368559	−0.0422766
$77$	−4.00870	−0.456834
$78$	0	0
$79$	6.36100	0.715669	0.357834	−	0.933785i	$-0.383515\pi$
$79$	6.36100	0.715669	0.357834	+	0.933785i	$0.383515\pi$
$80$	2.13185	0.238348
$81$	0	0
$82$	−11.1186	−1.22785
$83$	−8.35059	−0.916596	−0.458298	−	0.888799i	$-0.651541\pi$
$83$	−8.35059	−0.916596	−0.458298	+	0.888799i	$0.651541\pi$
$84$	0	0
$85$	−1.95629	−0.212189
$86$	−4.70319	−0.507158
$87$	0	0
$88$	10.4396	1.11287
$89$	4.61788	0.489495	0.244747	−	0.969587i	$-0.421295\pi$
$89$	4.61788	0.489495	0.244747	+	0.969587i	$0.421295\pi$
$90$	0	0
$91$	−1.63200	−0.171080
$92$	0.326507	0.0340407
$93$	0	0
$94$	−8.68319	−0.895603
$95$	0.645129	0.0661888
$96$	0	0
$97$	−9.03352	−0.917215	−0.458607	−	0.888639i	$-0.651652\pi$
$97$	−9.03352	−0.917215	−0.458607	+	0.888639i	$0.651652\pi$
$98$	−1.50746	−0.152276
$99$	0	0
$100$	−1.30018	−0.130018
$101$	−16.4724	−1.63906	−0.819530	−	0.573036i	$-0.805766\pi$
$101$	−16.4724	−1.63906	−0.819530	+	0.573036i	$0.805766\pi$
$102$	0	0
$103$	14.2483	1.40393	0.701964	−	0.712213i	$-0.252307\pi$
$103$	14.2483	1.40393	0.701964	+	0.712213i	$0.252307\pi$
$104$	4.25012	0.416759
$105$	0	0
$106$	6.90519	0.670692
$107$	9.61629	0.929641	0.464821	−	0.885405i	$-0.346119\pi$
$107$	9.61629	0.929641	0.464821	+	0.885405i	$0.346119\pi$
$108$	0	0
$109$	−6.83501	−0.654675	−0.327338	−	0.944907i	$-0.606151\pi$
$109$	−6.83501	−0.654675	−0.327338	+	0.944907i	$0.606151\pi$
$110$	2.88162	0.274752
$111$	0	0
$112$	4.47064	0.422435
$113$	−4.90814	−0.461719	−0.230859	−	0.972987i	$-0.574154\pi$
$113$	−4.90814	−0.461719	−0.230859	+	0.972987i	$0.574154\pi$
$114$	0	0
$115$	−0.571521	−0.0532946
$116$	−1.24998	−0.116058
$117$	0	0
$118$	2.23947	0.206160
$119$	−4.10247	−0.376073
$120$	0	0
$121$	5.06969	0.460881
$122$	5.79318	0.524490
$123$	0	0
$124$	−1.00314	−0.0900847
$125$	4.66013	0.416815
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−13.0615	−1.15449
$129$	0	0
$130$	1.17315	0.102892
$131$	14.5390	1.27028	0.635139	−	0.772398i	$-0.280943\pi$
$131$	14.5390	1.27028	0.635139	+	0.772398i	$0.280943\pi$
$132$	0	0
$133$	1.35288	0.117309
$134$	24.4505	2.11220
$135$	0	0
$136$	10.6838	0.916130
$137$	10.6630	0.911001	0.455501	−	0.890235i	$-0.349460\pi$
$137$	10.6630	0.911001	0.455501	+	0.890235i	$0.349460\pi$
$138$	0	0
$139$	21.4441	1.81887	0.909433	−	0.415852i	$-0.136516\pi$
$139$	21.4441	1.81887	0.909433	+	0.415852i	$0.136516\pi$
$140$	0.129908	0.0109792
$141$	0	0
$142$	18.6227	1.56278
$143$	6.54219	0.547086
$144$	0	0
$145$	2.18798	0.181702
$146$	20.1898	1.67092
$147$	0	0
$148$	−0.875146	−0.0719365
$149$	3.28032	0.268734	0.134367	−	0.990932i	$-0.457100\pi$
$149$	3.28032	0.268734	0.134367	+	0.990932i	$0.457100\pi$
$150$	0	0
$151$	−13.0042	−1.05827	−0.529133	−	0.848539i	$-0.677483\pi$
$151$	−13.0042	−1.05827	−0.529133	+	0.848539i	$0.677483\pi$
$152$	−3.52322	−0.285771
$153$	0	0
$154$	6.04294	0.486954
$155$	1.75591	0.141038
$156$	0	0
$157$	12.0308	0.960165	0.480083	−	0.877223i	$-0.340607\pi$
$157$	12.0308	0.960165	0.480083	+	0.877223i	$0.340607\pi$
$158$	−9.58894	−0.762855
$159$	0	0
$160$	−0.729974	−0.0577095
$161$	−1.19852	−0.0944564
$162$	0	0
$163$	−12.2322	−0.958098	−0.479049	−	0.877788i	$-0.659018\pi$
$163$	−12.2322	−0.958098	−0.479049	+	0.877788i	$0.659018\pi$
$164$	2.00934	0.156903
$165$	0	0
$166$	12.5881	0.977030
$167$	−18.1321	−1.40310	−0.701550	−	0.712620i	$-0.747509\pi$
$167$	−18.1321	−1.40310	−0.701550	+	0.712620i	$0.747509\pi$
$168$	0	0
$169$	−10.3366	−0.795122
$170$	2.94902	0.226180
$171$	0	0
$172$	0.849954	0.0648084
$173$	24.7895	1.88471	0.942357	−	0.334610i	$-0.108605\pi$
$173$	24.7895	1.88471	0.942357	+	0.334610i	$0.108605\pi$
$174$	0	0
$175$	4.77261	0.360775
$176$	−17.9214	−1.35088
$177$	0	0
$178$	−6.96126	−0.521768
$179$	−8.22815	−0.615001	−0.307501	−	0.951548i	$-0.599493\pi$
$179$	−8.22815	−0.615001	−0.307501	+	0.951548i	$0.599493\pi$
$180$	0	0
$181$	7.76719	0.577331	0.288665	−	0.957430i	$-0.406788\pi$
$181$	7.76719	0.577331	0.288665	+	0.957430i	$0.406788\pi$
$182$	2.46017	0.182360
$183$	0	0
$184$	3.12123	0.230100
$185$	1.53186	0.112625
$186$	0	0
$187$	16.4456	1.20262
$188$	1.56922	0.114447
$189$	0	0
$190$	−0.972504	−0.0705528
$191$	−3.84260	−0.278041	−0.139021	−	0.990289i	$-0.544395\pi$
$191$	−3.84260	−0.278041	−0.139021	+	0.990289i	$0.544395\pi$
$192$	0	0
$193$	1.48825	0.107126	0.0535632	−	0.998564i	$-0.482942\pi$
$193$	1.48825	0.107126	0.0535632	+	0.998564i	$0.482942\pi$
$194$	13.6176	0.977690
$195$	0	0
$196$	0.272426	0.0194590
$197$	−18.1534	−1.29337	−0.646687	−	0.762756i	$-0.723846\pi$
$197$	−18.1534	−1.29337	−0.646687	+	0.762756i	$0.723846\pi$
$198$	0	0
$199$	11.0920	0.786289	0.393144	−	0.919477i	$-0.371387\pi$
$199$	11.0920	0.786289	0.393144	+	0.919477i	$0.371387\pi$
$200$	−12.4290	−0.878865
$201$	0	0
$202$	24.8314	1.74713
$203$	4.58834	0.322038
$204$	0	0
$205$	−3.51717	−0.245650
$206$	−21.4787	−1.49649
$207$	0	0
$208$	−7.29607	−0.505891
$209$	−5.42328	−0.375136
$210$	0	0
$211$	24.7375	1.70300	0.851502	−	0.524352i	$-0.175692\pi$
$211$	24.7375	1.70300	0.851502	+	0.524352i	$0.175692\pi$
$212$	−1.24790	−0.0857060
$213$	0	0
$214$	−14.4961	−0.990935
$215$	−1.48777	−0.101465
$216$	0	0
$217$	3.68226	0.249968
$218$	10.3035	0.697840
$219$	0	0
$220$	−0.520763	−0.0351098
$221$	6.69522	0.450369
$222$	0	0
$223$	−18.5073	−1.23934	−0.619671	−	0.784862i	$-0.712734\pi$
$223$	−18.5073	−1.23934	−0.619671	+	0.784862i	$0.712734\pi$
$224$	−1.53080	−0.102281
$225$	0	0
$226$	7.39880	0.492161
$227$	−16.7025	−1.10859	−0.554293	−	0.832322i	$-0.687011\pi$
$227$	−16.7025	−1.10859	−0.554293	+	0.832322i	$0.687011\pi$
$228$	0	0
$229$	−4.47952	−0.296015	−0.148008	−	0.988986i	$-0.547286\pi$
$229$	−4.47952	−0.296015	−0.148008	+	0.988986i	$0.547286\pi$
$230$	0.861544	0.0568085
$231$	0	0
$232$	−11.9491	−0.784500
$233$	5.75975	0.377334	0.188667	−	0.982041i	$-0.439583\pi$
$233$	5.75975	0.377334	0.188667	+	0.982041i	$0.439583\pi$
$234$	0	0
$235$	−2.74677	−0.179180
$236$	−0.404715	−0.0263447
$237$	0	0
$238$	6.18429	0.400868
$239$	26.2440	1.69758	0.848792	−	0.528728i	$-0.177331\pi$
$239$	26.2440	1.69758	0.848792	+	0.528728i	$0.177331\pi$
$240$	0	0
$241$	−10.3161	−0.664517	−0.332258	−	0.943188i	$-0.607811\pi$
$241$	−10.3161	−0.664517	−0.332258	+	0.943188i	$0.607811\pi$
$242$	−7.64233	−0.491268
$243$	0	0
$244$	−1.04694	−0.0670233
$245$	−0.476857	−0.0304653
$246$	0	0
$247$	−2.20789	−0.140485
$248$	−9.58949	−0.608933
$249$	0	0
$250$	−7.02495	−0.444297
$251$	−18.5530	−1.17105	−0.585527	−	0.810653i	$-0.699112\pi$
$251$	−18.5530	−1.17105	−0.585527	+	0.810653i	$0.699112\pi$
$252$	0	0
$253$	4.80450	0.302056
$254$	−1.50746	−0.0945862
$255$	0	0
$256$	6.42242	0.401401
$257$	13.2064	0.823794	0.411897	−	0.911230i	$-0.364866\pi$
$257$	13.2064	0.823794	0.411897	+	0.911230i	$0.364866\pi$
$258$	0	0
$259$	3.21242	0.199610
$260$	−0.212010	−0.0131483
$261$	0	0
$262$	−21.9169	−1.35403
$263$	−14.0581	−0.866861	−0.433431	−	0.901187i	$-0.642697\pi$
$263$	−14.0581	−0.866861	−0.433431	+	0.901187i	$0.642697\pi$
$264$	0	0
$265$	2.18433	0.134183
$266$	−2.03940	−0.125044
$267$	0	0
$268$	−4.41866	−0.269913
$269$	8.79649	0.536332	0.268166	−	0.963373i	$-0.413583\pi$
$269$	8.79649	0.536332	0.268166	+	0.963373i	$0.413583\pi$
$270$	0	0
$271$	−12.1696	−0.739251	−0.369626	−	0.929181i	$-0.620514\pi$
$271$	−12.1696	−0.739251	−0.369626	+	0.929181i	$0.620514\pi$
$272$	−18.3406	−1.11206
$273$	0	0
$274$	−16.0740	−0.971066
$275$	−19.1320	−1.15370
$276$	0	0
$277$	13.4529	0.808305	0.404153	−	0.914692i	$-0.367566\pi$
$277$	13.4529	0.808305	0.404153	+	0.914692i	$0.367566\pi$
$278$	−32.3261	−1.93879
$279$	0	0
$280$	1.24185	0.0742148
$281$	16.9850	1.01324	0.506621	−	0.862169i	$-0.330894\pi$
$281$	16.9850	1.01324	0.506621	+	0.862169i	$0.330894\pi$
$282$	0	0
$283$	−1.02410	−0.0608765	−0.0304383	−	0.999537i	$-0.509690\pi$
$283$	−1.02410	−0.0608765	−0.0304383	+	0.999537i	$0.509690\pi$
$284$	−3.36547	−0.199704
$285$	0	0
$286$	−9.86207	−0.583157
$287$	−7.37574	−0.435376
$288$	0	0
$289$	−0.169756	−0.00998567
$290$	−3.29828	−0.193682
$291$	0	0
$292$	−3.64867	−0.213522
$293$	−8.49297	−0.496165	−0.248082	−	0.968739i	$-0.579800\pi$
$293$	−8.49297	−0.496165	−0.248082	+	0.968739i	$0.579800\pi$
$294$	0	0
$295$	0.708417	0.0412456
$296$	−8.36592	−0.486259
$297$	0	0
$298$	−4.94494	−0.286453
$299$	1.95598	0.113117
$300$	0	0
$301$	−3.11995	−0.179831
$302$	19.6032	1.12804
$303$	0	0
$304$	6.04822	0.346889
$305$	1.83257	0.104933
$306$	0	0
$307$	−17.2485	−0.984425	−0.492212	−	0.870475i	$-0.663812\pi$
$307$	−17.2485	−0.984425	−0.492212	+	0.870475i	$0.663812\pi$
$308$	−1.09207	−0.0622267
$309$	0	0
$310$	−2.64696	−0.150337
$311$	−22.6484	−1.28428	−0.642138	−	0.766589i	$-0.721952\pi$
$311$	−22.6484	−1.28428	−0.642138	+	0.766589i	$0.721952\pi$
$312$	0	0
$313$	10.9057	0.616429	0.308214	−	0.951317i	$-0.400269\pi$
$313$	10.9057	0.616429	0.308214	+	0.951317i	$0.400269\pi$
$314$	−18.1360	−1.02347
$315$	0	0
$316$	1.73290	0.0974833
$317$	16.2244	0.911254	0.455627	−	0.890171i	$-0.349415\pi$
$317$	16.2244	0.911254	0.455627	+	0.890171i	$0.349415\pi$
$318$	0	0
$319$	−18.3933	−1.02983
$320$	−3.16330	−0.176834
$321$	0	0
$322$	1.80671	0.100684
$323$	−5.55014	−0.308818
$324$	0	0
$325$	−7.78889	−0.432050
$326$	18.4395	1.02127
$327$	0	0
$328$	19.2082	1.06060
$329$	−5.76016	−0.317568
$330$	0	0
$331$	−8.26321	−0.454187	−0.227094	−	0.973873i	$-0.572922\pi$
$331$	−8.26321	−0.454187	−0.227094	+	0.973873i	$0.572922\pi$
$332$	−2.27491	−0.124852
$333$	0	0
$334$	27.3333	1.49561
$335$	7.73447	0.422579
$336$	0	0
$337$	−4.30968	−0.234763	−0.117382	−	0.993087i	$-0.537450\pi$
$337$	−4.30968	−0.234763	−0.117382	+	0.993087i	$0.537450\pi$
$338$	15.5820	0.847546
$339$	0	0
$340$	−0.532944	−0.0289029
$341$	−14.7611	−0.799357
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.12510	0.438076
$345$	0	0
$346$	−37.3691	−2.00898
$347$	22.1437	1.18873	0.594367	−	0.804194i	$-0.297403\pi$
$347$	22.1437	1.18873	0.594367	+	0.804194i	$0.297403\pi$
$348$	0	0
$349$	16.3191	0.873540	0.436770	−	0.899573i	$-0.356122\pi$
$349$	16.3191	0.873540	0.436770	+	0.899573i	$0.356122\pi$
$350$	−7.19450	−0.384562
$351$	0	0
$352$	6.13653	0.327078
$353$	−10.3526	−0.551011	−0.275505	−	0.961300i	$-0.588845\pi$
$353$	−10.3526	−0.551011	−0.275505	+	0.961300i	$0.588845\pi$
$354$	0	0
$355$	5.89095	0.312659
$356$	1.25803	0.0666755
$357$	0	0
$358$	12.4036	0.655550
$359$	−1.32049	−0.0696926	−0.0348463	−	0.999393i	$-0.511094\pi$
$359$	−1.32049	−0.0696926	−0.0348463	+	0.999393i	$0.511094\pi$
$360$	0	0
$361$	−17.1697	−0.903670
$362$	−11.7087	−0.615396
$363$	0	0
$364$	−0.444598	−0.0233033
$365$	6.38667	0.334294
$366$	0	0
$367$	2.22187	0.115981	0.0579904	−	0.998317i	$-0.481531\pi$
$367$	2.22187	0.115981	0.0579904	+	0.998317i	$0.481531\pi$
$368$	−5.35814	−0.279312
$369$	0	0
$370$	−2.30922	−0.120051
$371$	4.58069	0.237818
$372$	0	0
$373$	−21.2506	−1.10031	−0.550157	−	0.835061i	$-0.685432\pi$
$373$	−21.2506	−1.10031	−0.550157	+	0.835061i	$0.685432\pi$
$374$	−24.7910	−1.28191
$375$	0	0
$376$	15.0009	0.773610
$377$	−7.48815	−0.385660
$378$	0	0
$379$	38.2862	1.96663	0.983314	−	0.181914i	$-0.0582293\pi$
$379$	38.2862	1.96663	0.983314	+	0.181914i	$0.0582293\pi$
$380$	0.175750	0.00901577
$381$	0	0
$382$	5.79256	0.296373
$383$	−6.32795	−0.323343	−0.161672	−	0.986845i	$-0.551689\pi$
$383$	−6.32795	−0.323343	−0.161672	+	0.986845i	$0.551689\pi$
$384$	0	0
$385$	1.91158	0.0974230
$386$	−2.24347	−0.114190
$387$	0	0
$388$	−2.46096	−0.124936
$389$	−1.55850	−0.0790190	−0.0395095	−	0.999219i	$-0.512580\pi$
$389$	−1.55850	−0.0790190	−0.0395095	+	0.999219i	$0.512580\pi$
$390$	0	0
$391$	4.91688	0.248657
$392$	2.60424	0.131534
$393$	0	0
$394$	27.3654	1.37865
$395$	−3.03329	−0.152621
$396$	0	0
$397$	−2.78377	−0.139713	−0.0698567	−	0.997557i	$-0.522254\pi$
$397$	−2.78377	−0.139713	−0.0698567	+	0.997557i	$0.522254\pi$
$398$	−16.7207	−0.838131
$399$	0	0
$400$	21.3366	1.06683
$401$	−11.5664	−0.577599	−0.288800	−	0.957390i	$-0.593256\pi$
$401$	−11.5664	−0.577599	−0.288800	+	0.957390i	$0.593256\pi$
$402$	0	0
$403$	−6.00943	−0.299351
$404$	−4.48749	−0.223261
$405$	0	0
$406$	−6.91672	−0.343271
$407$	−12.8776	−0.638321
$408$	0	0
$409$	4.46557	0.220808	0.110404	−	0.993887i	$-0.464786\pi$
$409$	4.46557	0.220808	0.110404	+	0.993887i	$0.464786\pi$
$410$	5.30199	0.261846
$411$	0	0
$412$	3.88160	0.191233
$413$	1.48560	0.0731014
$414$	0	0
$415$	3.98203	0.195470
$416$	2.49827	0.122488
$417$	0	0
$418$	8.17536	0.399870
$419$	−8.52652	−0.416548	−0.208274	−	0.978071i	$-0.566785\pi$
$419$	−8.52652	−0.416548	−0.208274	+	0.978071i	$0.566785\pi$
$420$	0	0
$421$	−35.3870	−1.72466	−0.862329	−	0.506349i	$-0.830995\pi$
$421$	−35.3870	−1.72466	−0.862329	+	0.506349i	$0.830995\pi$
$422$	−37.2908	−1.81529
$423$	0	0
$424$	−11.9292	−0.579335
$425$	−19.5795	−0.949744
$426$	0	0
$427$	3.84302	0.185977
$428$	2.61972	0.126629
$429$	0	0
$430$	2.24275	0.108155
$431$	15.2058	0.732440	0.366220	−	0.930528i	$-0.380652\pi$
$431$	15.2058	0.732440	0.366220	+	0.930528i	$0.380652\pi$
$432$	0	0
$433$	−25.9711	−1.24809	−0.624047	−	0.781387i	$-0.714512\pi$
$433$	−25.9711	−1.24809	−0.624047	+	0.781387i	$0.714512\pi$
$434$	−5.55084	−0.266449
$435$	0	0
$436$	−1.86203	−0.0891752
$437$	−1.62145	−0.0775644
$438$	0	0
$439$	−35.4720	−1.69299	−0.846493	−	0.532400i	$-0.821290\pi$
$439$	−35.4720	−1.69299	−0.846493	+	0.532400i	$0.821290\pi$
$440$	−4.97821	−0.237327
$441$	0	0
$442$	−10.0928	−0.480063
$443$	−33.7956	−1.60568	−0.802839	−	0.596196i	$-0.796678\pi$
$443$	−33.7956	−1.60568	−0.802839	+	0.596196i	$0.796678\pi$
$444$	0	0
$445$	−2.20207	−0.104388
$446$	27.8990	1.32106
$447$	0	0
$448$	−6.63365	−0.313411
$449$	2.00057	0.0944127	0.0472063	−	0.998885i	$-0.484968\pi$
$449$	2.00057	0.0944127	0.0472063	+	0.998885i	$0.484968\pi$
$450$	0	0
$451$	29.5672	1.39226
$452$	−1.33710	−0.0628920
$453$	0	0
$454$	25.1783	1.18168
$455$	0.778229	0.0364840
$456$	0	0
$457$	−0.604245	−0.0282654	−0.0141327	−	0.999900i	$-0.504499\pi$
$457$	−0.604245	−0.0282654	−0.0141327	+	0.999900i	$0.504499\pi$
$458$	6.75269	0.315532
$459$	0	0
$460$	−0.155697	−0.00725941
$461$	−19.1495	−0.891882	−0.445941	−	0.895062i	$-0.647131\pi$
$461$	−19.1495	−0.891882	−0.445941	+	0.895062i	$0.647131\pi$
$462$	0	0
$463$	−27.3895	−1.27290	−0.636449	−	0.771319i	$-0.719597\pi$
$463$	−27.3895	−1.27290	−0.636449	+	0.771319i	$0.719597\pi$
$464$	20.5128	0.952282
$465$	0	0
$466$	−8.68258	−0.402213
$467$	−31.8395	−1.47336	−0.736679	−	0.676242i	$-0.763607\pi$
$467$	−31.8395	−1.47336	−0.736679	+	0.676242i	$0.763607\pi$
$468$	0	0
$469$	16.2197	0.748956
$470$	4.14064	0.190993
$471$	0	0
$472$	−3.86885	−0.178078
$473$	12.5069	0.575070
$474$	0	0
$475$	6.45675	0.296256
$476$	−1.11762	−0.0512259
$477$	0	0
$478$	−39.5617	−1.80951
$479$	−9.14232	−0.417723	−0.208862	−	0.977945i	$-0.566976\pi$
$479$	−9.14232	−0.417723	−0.208862	+	0.977945i	$0.566976\pi$
$480$	0	0
$481$	−5.24266	−0.239045
$482$	15.5510	0.708330
$483$	0	0
$484$	1.38111	0.0627779
$485$	4.30770	0.195602
$486$	0	0
$487$	21.8589	0.990522	0.495261	−	0.868744i	$-0.335072\pi$
$487$	21.8589	0.990522	0.495261	+	0.868744i	$0.335072\pi$
$488$	−10.0082	−0.453048
$489$	0	0
$490$	0.718841	0.0324739
$491$	15.5197	0.700395	0.350198	−	0.936676i	$-0.386114\pi$
$491$	15.5197	0.700395	0.350198	+	0.936676i	$0.386114\pi$
$492$	0	0
$493$	−18.8235	−0.847768
$494$	3.32830	0.149747
$495$	0	0
$496$	16.4620	0.739167
$497$	12.3537	0.554140
$498$	0	0
$499$	−7.52941	−0.337063	−0.168531	−	0.985696i	$-0.553902\pi$
$499$	−7.52941	−0.337063	−0.168531	+	0.985696i	$0.553902\pi$
$500$	1.26954	0.0567756
$501$	0	0
$502$	27.9678	1.24826
$503$	19.6788	0.877433	0.438717	−	0.898625i	$-0.355433\pi$
$503$	19.6788	0.877433	0.438717	+	0.898625i	$0.355433\pi$
$504$	0	0
$505$	7.85495	0.349541
$506$	−7.24257	−0.321972
$507$	0	0
$508$	0.272426	0.0120869
$509$	−16.4188	−0.727750	−0.363875	−	0.931448i	$-0.618546\pi$
$509$	−16.4188	−0.727750	−0.363875	+	0.931448i	$0.618546\pi$
$510$	0	0
$511$	13.3933	0.592483
$512$	16.4416	0.726622
$513$	0	0
$514$	−19.9081	−0.878109
$515$	−6.79440	−0.299397
$516$	0	0
$517$	23.0908	1.01553
$518$	−4.84258	−0.212771
$519$	0	0
$520$	−2.02670	−0.0888766
$521$	1.77228	0.0776452	0.0388226	−	0.999246i	$-0.487639\pi$
$521$	1.77228	0.0776452	0.0388226	+	0.999246i	$0.487639\pi$
$522$	0	0
$523$	14.4841	0.633347	0.316673	−	0.948535i	$-0.397434\pi$
$523$	14.4841	0.633347	0.316673	+	0.948535i	$0.397434\pi$
$524$	3.96080	0.173028
$525$	0	0
$526$	21.1920	0.924016
$527$	−15.1063	−0.658042
$528$	0	0
$529$	−21.5636	−0.937546
$530$	−3.29279	−0.143030
$531$	0	0
$532$	0.368559	0.0159790
$533$	12.0372	0.521389
$534$	0	0
$535$	−4.58559	−0.198252
$536$	−42.2400	−1.82449
$537$	0	0
$538$	−13.2603	−0.571693
$539$	4.00870	0.172667
$540$	0	0
$541$	−11.7221	−0.503974	−0.251987	−	0.967731i	$-0.581084\pi$
$541$	−11.7221	−0.503974	−0.251987	+	0.967731i	$0.581084\pi$
$542$	18.3452	0.787992
$543$	0	0
$544$	6.28007	0.269256
$545$	3.25932	0.139614
$546$	0	0
$547$	18.1429	0.775736	0.387868	−	0.921715i	$-0.373212\pi$
$547$	18.1429	0.775736	0.387868	+	0.921715i	$0.373212\pi$
$548$	2.90487	0.124090
$549$	0	0
$550$	28.8406	1.22977
$551$	6.20746	0.264447
$552$	0	0
$553$	−6.36100	−0.270497
$554$	−20.2796	−0.861599
$555$	0	0
$556$	5.84192	0.247753
$557$	−7.98907	−0.338508	−0.169254	−	0.985572i	$-0.554136\pi$
$557$	−7.98907	−0.338508	−0.169254	+	0.985572i	$0.554136\pi$
$558$	0	0
$559$	5.09175	0.215358
$560$	−2.13185	−0.0900872
$561$	0	0
$562$	−25.6042	−1.08005
$563$	−42.9567	−1.81041	−0.905205	−	0.424975i	$-0.860283\pi$
$563$	−42.9567	−1.81041	−0.905205	+	0.424975i	$0.860283\pi$
$564$	0	0
$565$	2.34048	0.0984647
$566$	1.54379	0.0648903
$567$	0	0
$568$	−32.1721	−1.34991
$569$	−17.8243	−0.747235	−0.373617	−	0.927583i	$-0.621883\pi$
$569$	−17.8243	−0.747235	−0.373617	+	0.927583i	$0.621883\pi$
$570$	0	0
$571$	10.8091	0.452347	0.226174	−	0.974087i	$-0.427378\pi$
$571$	10.8091	0.452347	0.226174	+	0.974087i	$0.427378\pi$
$572$	1.78226	0.0745201
$573$	0	0
$574$	11.1186	0.464082
$575$	−5.72005	−0.238543
$576$	0	0
$577$	−31.7439	−1.32152	−0.660758	−	0.750599i	$-0.729765\pi$
$577$	−31.7439	−1.32152	−0.660758	+	0.750599i	$0.729765\pi$
$578$	0.255900	0.0106440
$579$	0	0
$580$	0.596062	0.0247501
$581$	8.35059	0.346441
$582$	0	0
$583$	−18.3626	−0.760502
$584$	−34.8793	−1.44332
$585$	0	0
$586$	12.8028	0.528878
$587$	20.7124	0.854893	0.427446	−	0.904041i	$-0.359413\pi$
$587$	20.7124	0.854893	0.427446	+	0.904041i	$0.359413\pi$
$588$	0	0
$589$	4.98164	0.205265
$590$	−1.06791	−0.0439650
$591$	0	0
$592$	14.3616	0.590256
$593$	6.25211	0.256743	0.128372	−	0.991726i	$-0.459025\pi$
$593$	6.25211	0.256743	0.128372	+	0.991726i	$0.459025\pi$
$594$	0	0
$595$	1.95629	0.0802001
$596$	0.893644	0.0366051
$597$	0	0
$598$	−2.94855	−0.120575
$599$	−14.4524	−0.590507	−0.295254	−	0.955419i	$-0.595404\pi$
$599$	−14.4524	−0.590507	−0.295254	+	0.955419i	$0.595404\pi$
$600$	0	0
$601$	7.20061	0.293719	0.146859	−	0.989157i	$-0.453084\pi$
$601$	7.20061	0.293719	0.146859	+	0.989157i	$0.453084\pi$
$602$	4.70319	0.191688
$603$	0	0
$604$	−3.54267	−0.144149
$605$	−2.41751	−0.0982860
$606$	0	0
$607$	−23.5412	−0.955510	−0.477755	−	0.878493i	$-0.658549\pi$
$607$	−23.5412	−0.955510	−0.477755	+	0.878493i	$0.658549\pi$
$608$	−2.07099	−0.0839897
$609$	0	0
$610$	−2.76252	−0.111851
$611$	9.40057	0.380306
$612$	0	0
$613$	−23.0450	−0.930780	−0.465390	−	0.885106i	$-0.654086\pi$
$613$	−23.0450	−0.930780	−0.465390	+	0.885106i	$0.654086\pi$
$614$	26.0014	1.04933
$615$	0	0
$616$	−10.4396	−0.420625
$617$	−9.66256	−0.389000	−0.194500	−	0.980903i	$-0.562308\pi$
$617$	−9.66256	−0.389000	−0.194500	+	0.980903i	$0.562308\pi$
$618$	0	0
$619$	−27.1279	−1.09036	−0.545180	−	0.838319i	$-0.683539\pi$
$619$	−27.1279	−1.09036	−0.545180	+	0.838319i	$0.683539\pi$
$620$	0.478355	0.0192112
$621$	0	0
$622$	34.1415	1.36895
$623$	−4.61788	−0.185012
$624$	0	0
$625$	21.6408	0.865633
$626$	−16.4399	−0.657072
$627$	0	0
$628$	3.27751	0.130787
$629$	−13.1788	−0.525475
$630$	0	0
$631$	−26.6857	−1.06234	−0.531171	−	0.847264i	$-0.678248\pi$
$631$	−26.6857	−1.06234	−0.531171	+	0.847264i	$0.678248\pi$
$632$	16.5656	0.658944
$633$	0	0
$634$	−24.4576	−0.971335
$635$	−0.476857	−0.0189235
$636$	0	0
$637$	1.63200	0.0646621
$638$	27.7271	1.09772
$639$	0	0
$640$	6.22849	0.246203
$641$	17.7561	0.701322	0.350661	−	0.936502i	$-0.385957\pi$
$641$	17.7561	0.701322	0.350661	+	0.936502i	$0.385957\pi$
$642$	0	0
$643$	12.2052	0.481327	0.240663	−	0.970609i	$-0.422635\pi$
$643$	12.2052	0.481327	0.240663	+	0.970609i	$0.422635\pi$
$644$	−0.326507	−0.0128662
$645$	0	0
$646$	8.36659	0.329179
$647$	−15.9746	−0.628025	−0.314013	−	0.949419i	$-0.601673\pi$
$647$	−15.9746	−0.628025	−0.314013	+	0.949419i	$0.601673\pi$
$648$	0	0
$649$	−5.95531	−0.233766
$650$	11.7414	0.460536
$651$	0	0
$652$	−3.33236	−0.130505
$653$	34.1997	1.33834	0.669168	−	0.743111i	$-0.266651\pi$
$653$	34.1997	1.33834	0.669168	+	0.743111i	$0.266651\pi$
$654$	0	0
$655$	−6.93302	−0.270895
$656$	−32.9743	−1.28743
$657$	0	0
$658$	8.68319	0.338506
$659$	39.1898	1.52662	0.763309	−	0.646034i	$-0.223573\pi$
$659$	39.1898	1.52662	0.763309	+	0.646034i	$0.223573\pi$
$660$	0	0
$661$	−27.8774	−1.08431	−0.542153	−	0.840280i	$-0.682391\pi$
$661$	−27.8774	−1.08431	−0.542153	+	0.840280i	$0.682391\pi$
$662$	12.4564	0.484133
$663$	0	0
$664$	−21.7470	−0.843946
$665$	−0.645129	−0.0250170
$666$	0	0
$667$	−5.49920	−0.212930
$668$	−4.93964	−0.191120
$669$	0	0
$670$	−11.6594	−0.450441
$671$	−15.4055	−0.594723
$672$	0	0
$673$	34.4987	1.32983	0.664915	−	0.746919i	$-0.268468\pi$
$673$	34.4987	1.32983	0.664915	+	0.746919i	$0.268468\pi$
$674$	6.49666	0.250242
$675$	0	0
$676$	−2.81595	−0.108306
$677$	3.63530	0.139716	0.0698579	−	0.997557i	$-0.477745\pi$
$677$	3.63530	0.139716	0.0698579	+	0.997557i	$0.477745\pi$
$678$	0	0
$679$	9.03352	0.346675
$680$	−5.09465	−0.195371
$681$	0	0
$682$	22.2517	0.852060
$683$	46.3431	1.77327	0.886635	−	0.462470i	$-0.153037\pi$
$683$	46.3431	1.77327	0.886635	+	0.462470i	$0.153037\pi$
$684$	0	0
$685$	−5.08472	−0.194277
$686$	1.50746	0.0575550
$687$	0	0
$688$	−13.9481	−0.531768
$689$	−7.47568	−0.284801
$690$	0	0
$691$	−7.88660	−0.300020	−0.150010	−	0.988684i	$-0.547931\pi$
$691$	−7.88660	−0.300020	−0.150010	+	0.988684i	$0.547931\pi$
$692$	6.75331	0.256722
$693$	0	0
$694$	−33.3806	−1.26711
$695$	−10.2258	−0.387885
$696$	0	0
$697$	30.2588	1.14613
$698$	−24.6003	−0.931135
$699$	0	0
$700$	1.30018	0.0491422
$701$	18.7987	0.710018	0.355009	−	0.934863i	$-0.384478\pi$
$701$	18.7987	0.710018	0.355009	+	0.934863i	$0.384478\pi$
$702$	0	0
$703$	4.34601	0.163913
$704$	26.5923	1.00224
$705$	0	0
$706$	15.6060	0.587340
$707$	16.4724	0.619507
$708$	0	0
$709$	−11.3077	−0.424669	−0.212335	−	0.977197i	$-0.568107\pi$
$709$	−11.3077	−0.424669	−0.212335	+	0.977197i	$0.568107\pi$
$710$	−8.88036	−0.333274
$711$	0	0
$712$	12.0261	0.450697
$713$	−4.41325	−0.165277
$714$	0	0
$715$	−3.11969	−0.116670
$716$	−2.24156	−0.0837711
$717$	0	0
$718$	1.99058	0.0742876
$719$	−22.5236	−0.839988	−0.419994	−	0.907527i	$-0.637968\pi$
$719$	−22.5236	−0.839988	−0.419994	+	0.907527i	$0.637968\pi$
$720$	0	0
$721$	−14.2483	−0.530635
$722$	25.8826	0.963251
$723$	0	0
$724$	2.11598	0.0786399
$725$	21.8983	0.813283
$726$	0	0
$727$	−4.82852	−0.179080	−0.0895399	−	0.995983i	$-0.528540\pi$
$727$	−4.82852	−0.179080	−0.0895399	+	0.995983i	$0.528540\pi$
$728$	−4.25012	−0.157520
$729$	0	0
$730$	−9.62763	−0.356334
$731$	12.7995	0.473406
$732$	0	0
$733$	28.2647	1.04398	0.521991	−	0.852951i	$-0.325190\pi$
$733$	28.2647	1.04398	0.521991	+	0.852951i	$0.325190\pi$
$734$	−3.34938	−0.123628
$735$	0	0
$736$	1.83469	0.0676277
$737$	−65.0199	−2.39504
$738$	0	0
$739$	−47.6494	−1.75281	−0.876407	−	0.481572i	$-0.840066\pi$
$739$	−47.6494	−1.75281	−0.876407	+	0.481572i	$0.840066\pi$
$740$	0.417319	0.0153410
$741$	0	0
$742$	−6.90519	−0.253498
$743$	50.6859	1.85949	0.929743	−	0.368208i	$-0.120029\pi$
$743$	50.6859	1.85949	0.929743	+	0.368208i	$0.120029\pi$
$744$	0	0
$745$	−1.56424	−0.0573095
$746$	32.0343	1.17286
$747$	0	0
$748$	4.48020	0.163812
$749$	−9.61629	−0.351371
$750$	0	0
$751$	−26.9407	−0.983082	−0.491541	−	0.870855i	$-0.663566\pi$
$751$	−26.9407	−0.983082	−0.491541	+	0.870855i	$0.663566\pi$
$752$	−25.7516	−0.939063
$753$	0	0
$754$	11.2881	0.411087
$755$	6.20113	0.225682
$756$	0	0
$757$	25.9571	0.943429	0.471714	−	0.881751i	$-0.343635\pi$
$757$	25.9571	0.943429	0.471714	+	0.881751i	$0.343635\pi$
$758$	−57.7148	−2.09629
$759$	0	0
$760$	1.68007	0.0609426
$761$	8.68948	0.314994	0.157497	−	0.987520i	$-0.449658\pi$
$761$	8.68948	0.314994	0.157497	+	0.987520i	$0.449658\pi$
$762$	0	0
$763$	6.83501	0.247444
$764$	−1.04682	−0.0378728
$765$	0	0
$766$	9.53911	0.344662
$767$	−2.42449	−0.0875433
$768$	0	0
$769$	32.4677	1.17082	0.585408	−	0.810739i	$-0.300934\pi$
$769$	32.4677	1.17082	0.585408	+	0.810739i	$0.300934\pi$
$770$	−2.88162	−0.103846
$771$	0	0
$772$	0.405437	0.0145920
$773$	27.0037	0.971257	0.485628	−	0.874165i	$-0.338591\pi$
$773$	27.0037	0.971257	0.485628	+	0.874165i	$0.338591\pi$
$774$	0	0
$775$	17.5740	0.631275
$776$	−23.5255	−0.844516
$777$	0	0
$778$	2.34937	0.0842290
$779$	−9.97848	−0.357516
$780$	0	0
$781$	−49.5224	−1.77205
$782$	−7.41198	−0.265052
$783$	0	0
$784$	−4.47064	−0.159666
$785$	−5.73699	−0.204762
$786$	0	0
$787$	−39.7783	−1.41794	−0.708971	−	0.705238i	$-0.750840\pi$
$787$	−39.7783	−1.41794	−0.708971	+	0.705238i	$0.750840\pi$
$788$	−4.94544	−0.176174
$789$	0	0
$790$	4.57255	0.162684
$791$	4.90814	0.174513
$792$	0	0
$793$	−6.27180	−0.222718
$794$	4.19641	0.148925
$795$	0	0
$796$	3.02174	0.107103
$797$	−50.6467	−1.79400	−0.896999	−	0.442033i	$-0.854258\pi$
$797$	−50.6467	−1.79400	−0.896999	+	0.442033i	$0.854258\pi$
$798$	0	0
$799$	23.6309	0.836000
$800$	−7.30592	−0.258303
$801$	0	0
$802$	17.4359	0.615682
$803$	−53.6896	−1.89466
$804$	0	0
$805$	0.571521	0.0201435
$806$	9.05896	0.319088
$807$	0	0
$808$	−42.8980	−1.50915
$809$	23.5469	0.827866	0.413933	−	0.910307i	$-0.364155\pi$
$809$	23.5469	0.827866	0.413933	+	0.910307i	$0.364155\pi$
$810$	0	0
$811$	17.7270	0.622478	0.311239	−	0.950332i	$-0.399256\pi$
$811$	17.7270	0.622478	0.311239	+	0.950332i	$0.399256\pi$
$812$	1.24998	0.0438657
$813$	0	0
$814$	19.4125	0.680407
$815$	5.83300	0.204321
$816$	0	0
$817$	−4.22091	−0.147671
$818$	−6.73165	−0.235367
$819$	0	0
$820$	−0.958169	−0.0334607
$821$	−51.7090	−1.80466	−0.902328	−	0.431050i	$-0.858143\pi$
$821$	−51.7090	−1.80466	−0.902328	+	0.431050i	$0.858143\pi$
$822$	0	0
$823$	−1.57490	−0.0548976	−0.0274488	−	0.999623i	$-0.508738\pi$
$823$	−1.57490	−0.0548976	−0.0274488	+	0.999623i	$0.508738\pi$
$824$	37.1061	1.29265
$825$	0	0
$826$	−2.23947	−0.0779212
$827$	23.4464	0.815310	0.407655	−	0.913136i	$-0.366347\pi$
$827$	23.4464	0.815310	0.407655	+	0.913136i	$0.366347\pi$
$828$	0	0
$829$	26.4105	0.917274	0.458637	−	0.888624i	$-0.348338\pi$
$829$	26.4105	0.917274	0.458637	+	0.888624i	$0.348338\pi$
$830$	−6.00274	−0.208358
$831$	0	0
$832$	10.8261	0.375328
$833$	4.10247	0.142142
$834$	0	0
$835$	8.64640	0.299221
$836$	−1.47744	−0.0510984
$837$	0	0
$838$	12.8534	0.444012
$839$	−31.9023	−1.10139	−0.550695	−	0.834707i	$-0.685637\pi$
$839$	−31.9023	−1.10139	−0.550695	+	0.834707i	$0.685637\pi$
$840$	0	0
$841$	−7.94718	−0.274041
$842$	53.3444	1.83837
$843$	0	0
$844$	6.73914	0.231971
$845$	4.92907	0.169565
$846$	0	0
$847$	−5.06969	−0.174197
$848$	20.4786	0.703238
$849$	0	0
$850$	29.5152	1.01236
$851$	−3.85014	−0.131981
$852$	0	0
$853$	−6.53389	−0.223716	−0.111858	−	0.993724i	$-0.535680\pi$
$853$	−6.53389	−0.223716	−0.111858	+	0.993724i	$0.535680\pi$
$854$	−5.79318	−0.198239
$855$	0	0
$856$	25.0431	0.855957
$857$	10.2262	0.349320	0.174660	−	0.984629i	$-0.444117\pi$
$857$	10.2262	0.349320	0.174660	+	0.984629i	$0.444117\pi$
$858$	0	0
$859$	−41.8291	−1.42719	−0.713595	−	0.700559i	$-0.752934\pi$
$859$	−41.8291	−1.42719	−0.713595	+	0.700559i	$0.752934\pi$
$860$	−0.405306	−0.0138208
$861$	0	0
$862$	−22.9221	−0.780731
$863$	−21.2131	−0.722104	−0.361052	−	0.932546i	$-0.617582\pi$
$863$	−21.2131	−0.722104	−0.361052	+	0.932546i	$0.617582\pi$
$864$	0	0
$865$	−11.8211	−0.401928
$866$	39.1504	1.33038
$867$	0	0
$868$	1.00314	0.0340488
$869$	25.4994	0.865007
$870$	0	0
$871$	−26.4705	−0.896919
$872$	−17.8000	−0.602785
$873$	0	0
$874$	2.44426	0.0826784
$875$	−4.66013	−0.157541
$876$	0	0
$877$	36.4081	1.22941	0.614706	−	0.788756i	$-0.289274\pi$
$877$	36.4081	1.22941	0.614706	+	0.788756i	$0.289274\pi$
$878$	53.4725	1.80461
$879$	0	0
$880$	8.54596	0.288084
$881$	−38.7081	−1.30411	−0.652055	−	0.758172i	$-0.726093\pi$
$881$	−38.7081	−1.30411	−0.652055	+	0.758172i	$0.726093\pi$
$882$	0	0
$883$	−46.9430	−1.57976	−0.789878	−	0.613264i	$-0.789856\pi$
$883$	−46.9430	−1.57976	−0.789878	+	0.613264i	$0.789856\pi$
$884$	1.82395	0.0613461
$885$	0	0
$886$	50.9454	1.71154
$887$	6.17773	0.207428	0.103714	−	0.994607i	$-0.466927\pi$
$887$	6.17773	0.207428	0.103714	+	0.994607i	$0.466927\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	3.31952	0.111271
$891$	0	0
$892$	−5.04187	−0.168814
$893$	−7.79279	−0.260776
$894$	0	0
$895$	3.92365	0.131153
$896$	13.0615	0.436356
$897$	0	0
$898$	−3.01577	−0.100638
$899$	16.8954	0.563494
$900$	0	0
$901$	−18.7921	−0.626057
$902$	−44.5712	−1.48406
$903$	0	0
$904$	−12.7820	−0.425122
$905$	−3.70384	−0.123120
$906$	0	0
$907$	21.8087	0.724144	0.362072	−	0.932150i	$-0.382069\pi$
$907$	21.8087	0.724144	0.362072	+	0.932150i	$0.382069\pi$
$908$	−4.55020	−0.151004
$909$	0	0
$910$	−1.17315	−0.0388894
$911$	−44.8491	−1.48592	−0.742958	−	0.669338i	$-0.766578\pi$
$911$	−44.8491	−1.48592	−0.742958	+	0.669338i	$0.766578\pi$
$912$	0	0
$913$	−33.4750	−1.10786
$914$	0.910873	0.0301290
$915$	0	0
$916$	−1.22034	−0.0403211
$917$	−14.5390	−0.480120
$918$	0	0
$919$	33.9710	1.12060	0.560300	−	0.828289i	$-0.310686\pi$
$919$	33.9710	1.12060	0.560300	+	0.828289i	$0.310686\pi$
$920$	−1.48838	−0.0490705
$921$	0	0
$922$	28.8671	0.950686
$923$	−20.1612	−0.663616
$924$	0	0
$925$	15.3316	0.504101
$926$	41.2885	1.35682
$927$	0	0
$928$	−7.02384	−0.230569
$929$	8.71751	0.286012	0.143006	−	0.989722i	$-0.454323\pi$
$929$	8.71751	0.286012	0.143006	+	0.989722i	$0.454323\pi$
$930$	0	0
$931$	−1.35288	−0.0443388
$932$	1.56910	0.0513977
$933$	0	0
$934$	47.9967	1.57050
$935$	−7.84218	−0.256467
$936$	0	0
$937$	−36.0777	−1.17861	−0.589304	−	0.807911i	$-0.700598\pi$
$937$	−36.0777	−1.17861	−0.589304	+	0.807911i	$0.700598\pi$
$938$	−24.4505	−0.798337
$939$	0	0
$940$	−0.748291	−0.0244066
$941$	−15.0349	−0.490124	−0.245062	−	0.969507i	$-0.578808\pi$
$941$	−15.0349	−0.490124	−0.245062	+	0.969507i	$0.578808\pi$
$942$	0	0
$943$	8.83996	0.287869
$944$	6.64156	0.216164
$945$	0	0
$946$	−18.8537	−0.612986
$947$	−37.6361	−1.22301	−0.611504	−	0.791241i	$-0.709435\pi$
$947$	−37.6361	−1.22301	−0.611504	+	0.791241i	$0.709435\pi$
$948$	0	0
$949$	−21.8578	−0.709534
$950$	−9.73328	−0.315789
$951$	0	0
$952$	−10.6838	−0.346265
$953$	−20.1261	−0.651948	−0.325974	−	0.945379i	$-0.605692\pi$
$953$	−20.1261	−0.651948	−0.325974	+	0.945379i	$0.605692\pi$
$954$	0	0
$955$	1.83237	0.0592942
$956$	7.14954	0.231233
$957$	0	0
$958$	13.7817	0.445265
$959$	−10.6630	−0.344326
$960$	0	0
$961$	−17.4410	−0.562613
$962$	7.90309	0.254806
$963$	0	0
$964$	−2.81036	−0.0905157
$965$	−0.709681	−0.0228454
$966$	0	0
$967$	1.02734	0.0330369	0.0165185	−	0.999864i	$-0.494742\pi$
$967$	1.02734	0.0330369	0.0165185	+	0.999864i	$0.494742\pi$
$968$	13.2027	0.424351
$969$	0	0
$970$	−6.49366	−0.208499
$971$	54.3876	1.74538	0.872691	−	0.488273i	$-0.162373\pi$
$971$	54.3876	1.74538	0.872691	+	0.488273i	$0.162373\pi$
$972$	0	0
$973$	−21.4441	−0.687466
$974$	−32.9514	−1.05583
$975$	0	0
$976$	17.1807	0.549942
$977$	36.8220	1.17804	0.589021	−	0.808118i	$-0.299514\pi$
$977$	36.8220	1.17804	0.589021	+	0.808118i	$0.299514\pi$
$978$	0	0
$979$	18.5117	0.591637
$980$	−0.129908	−0.00414976
$981$	0	0
$982$	−23.3953	−0.746574
$983$	4.03371	0.128655	0.0643276	−	0.997929i	$-0.479510\pi$
$983$	4.03371	0.128655	0.0643276	+	0.997929i	$0.479510\pi$
$984$	0	0
$985$	8.65655	0.275821
$986$	28.3756	0.903664
$987$	0	0
$988$	−0.601487	−0.0191358
$989$	3.73931	0.118903
$990$	0	0
$991$	−54.2295	−1.72266	−0.861329	−	0.508048i	$-0.830367\pi$
$991$	−54.2295	−1.72266	−0.861329	+	0.508048i	$0.830367\pi$
$992$	−5.63681	−0.178969
$993$	0	0
$994$	−18.6227	−0.590676
$995$	−5.28928	−0.167681
$996$	0	0
$997$	−34.0091	−1.07708	−0.538539	−	0.842600i	$-0.681024\pi$
$997$	−34.0091	−1.07708	−0.538539	+	0.842600i	$0.681024\pi$
$998$	11.3503	0.359286
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.m.1.4		11
3.2	odd	2		2667.2.a.k.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.k.1.8	✓	11	3.2	odd	2
8001.2.a.m.1.4		11	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.50746 q^{2} +0.272426 q^{4} -0.476857 q^{5} -1.00000 q^{7} +2.60424 q^{8} +O(q^{10})\)	\(q-1.50746 q^{2} +0.272426 q^{4} -0.476857 q^{5} -1.00000 q^{7} +2.60424 q^{8} +0.718841 q^{10} +4.00870 q^{11} +1.63200 q^{13} +1.50746 q^{14} -4.47064 q^{16} +4.10247 q^{17} -1.35288 q^{19} -0.129908 q^{20} -6.04294 q^{22} +1.19852 q^{23} -4.77261 q^{25} -2.46017 q^{26} -0.272426 q^{28} -4.58834 q^{29} -3.68226 q^{31} +1.53080 q^{32} -6.18429 q^{34} +0.476857 q^{35} -3.21242 q^{37} +2.03940 q^{38} -1.24185 q^{40} +7.37574 q^{41} +3.11995 q^{43} +1.09207 q^{44} -1.80671 q^{46} +5.76016 q^{47} +1.00000 q^{49} +7.19450 q^{50} +0.444598 q^{52} -4.58069 q^{53} -1.91158 q^{55} -2.60424 q^{56} +6.91672 q^{58} -1.48560 q^{59} -3.84302 q^{61} +5.55084 q^{62} +6.63365 q^{64} -0.778229 q^{65} -16.2197 q^{67} +1.11762 q^{68} -0.718841 q^{70} -12.3537 q^{71} -13.3933 q^{73} +4.84258 q^{74} -0.368559 q^{76} -4.00870 q^{77} +6.36100 q^{79} +2.13185 q^{80} -11.1186 q^{82} -8.35059 q^{83} -1.95629 q^{85} -4.70319 q^{86} +10.4396 q^{88} +4.61788 q^{89} -1.63200 q^{91} +0.326507 q^{92} -8.68319 q^{94} +0.645129 q^{95} -9.03352 q^{97} -1.50746 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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