LMFDB - Embedded newform 8001.2.a.r.1.10

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.0296587$ 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)

$f(q)$	$=$	$q-0.0296587 q^{2} -1.99912 q^{4} -3.07938 q^{5} -1.00000 q^{7} +0.118609 q^{8} +O(q^{10})$	\(q-0.0296587 q^{2} -1.99912 q^{4} -3.07938 q^{5} -1.00000 q^{7} +0.118609 q^{8} +0.0913304 q^{10} +4.20887 q^{11} -6.43617 q^{13} +0.0296587 q^{14} +3.99472 q^{16} +0.978108 q^{17} +0.452098 q^{19} +6.15605 q^{20} -0.124830 q^{22} -1.87821 q^{23} +4.48258 q^{25} +0.190888 q^{26} +1.99912 q^{28} -6.69207 q^{29} -1.28170 q^{31} -0.355696 q^{32} -0.0290094 q^{34} +3.07938 q^{35} +6.10620 q^{37} -0.0134086 q^{38} -0.365241 q^{40} +3.48251 q^{41} +4.11590 q^{43} -8.41404 q^{44} +0.0557052 q^{46} +0.343427 q^{47} +1.00000 q^{49} -0.132948 q^{50} +12.8667 q^{52} -2.22047 q^{53} -12.9607 q^{55} -0.118609 q^{56} +0.198478 q^{58} -1.48973 q^{59} +7.08310 q^{61} +0.0380136 q^{62} -7.97890 q^{64} +19.8194 q^{65} +0.989393 q^{67} -1.95535 q^{68} -0.0913304 q^{70} +8.17613 q^{71} +9.17564 q^{73} -0.181102 q^{74} -0.903798 q^{76} -4.20887 q^{77} +9.02266 q^{79} -12.3013 q^{80} -0.103287 q^{82} +5.68319 q^{83} -3.01197 q^{85} -0.122072 q^{86} +0.499209 q^{88} -2.01684 q^{89} +6.43617 q^{91} +3.75476 q^{92} -0.0101856 q^{94} -1.39218 q^{95} +10.8384 q^{97} -0.0296587 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * 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 $

$2$	−0.0296587	−0.0209719	−0.0104859	−	0.999945i	$-0.503338\pi$
$2$	−0.0296587	−0.0209719	−0.0104859	+	0.999945i	$0.503338\pi$
$3$	0	0
$3$	0	0
$4$	−1.99912	−0.999560
$5$	−3.07938	−1.37714	−0.688570	−	0.725170i	$-0.741761\pi$
$5$	−3.07938	−1.37714	−0.688570	+	0.725170i	$0.741761\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.118609	0.0419345
$9$	0	0
$10$	0.0913304	0.0288812
$11$	4.20887	1.26902	0.634511	−	0.772913i	$-0.281201\pi$
$11$	4.20887	1.26902	0.634511	+	0.772913i	$0.281201\pi$
$12$	0	0
$13$	−6.43617	−1.78507	−0.892536	−	0.450977i	$-0.851076\pi$
$13$	−6.43617	−1.78507	−0.892536	+	0.450977i	$0.851076\pi$
$14$	0.0296587	0.00792662
$15$	0	0
$16$	3.99472	0.998681
$17$	0.978108	0.237226	0.118613	−	0.992941i	$-0.462155\pi$
$17$	0.978108	0.237226	0.118613	+	0.992941i	$0.462155\pi$
$18$	0	0
$19$	0.452098	0.103718	0.0518592	−	0.998654i	$-0.483485\pi$
$19$	0.452098	0.103718	0.0518592	+	0.998654i	$0.483485\pi$
$20$	6.15605	1.37654
$21$	0	0
$22$	−0.124830	−0.0266138
$23$	−1.87821	−0.391633	−0.195817	−	0.980641i	$-0.562736\pi$
$23$	−1.87821	−0.391633	−0.195817	+	0.980641i	$0.562736\pi$
$24$	0	0
$25$	4.48258	0.896517
$26$	0.190888	0.0374363
$27$	0	0
$28$	1.99912	0.377798
$29$	−6.69207	−1.24269	−0.621343	−	0.783539i	$-0.713413\pi$
$29$	−6.69207	−1.24269	−0.621343	+	0.783539i	$0.713413\pi$
$30$	0	0
$31$	−1.28170	−0.230200	−0.115100	−	0.993354i	$-0.536719\pi$
$31$	−1.28170	−0.230200	−0.115100	+	0.993354i	$0.536719\pi$
$32$	−0.355696	−0.0628787
$33$	0	0
$34$	−0.0290094	−0.00497507
$35$	3.07938	0.520510
$36$	0	0
$37$	6.10620	1.00385	0.501927	−	0.864910i	$-0.332625\pi$
$37$	6.10620	1.00385	0.501927	+	0.864910i	$0.332625\pi$
$38$	−0.0134086	−0.00217517
$39$	0	0
$40$	−0.365241	−0.0577497
$41$	3.48251	0.543877	0.271939	−	0.962315i	$-0.412335\pi$
$41$	3.48251	0.543877	0.271939	+	0.962315i	$0.412335\pi$
$42$	0	0
$43$	4.11590	0.627669	0.313835	−	0.949478i	$-0.398386\pi$
$43$	4.11590	0.627669	0.313835	+	0.949478i	$0.398386\pi$
$44$	−8.41404	−1.26846
$45$	0	0
$46$	0.0557052	0.00821328
$47$	0.343427	0.0500940	0.0250470	−	0.999686i	$-0.492026\pi$
$47$	0.343427	0.0500940	0.0250470	+	0.999686i	$0.492026\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.132948	−0.0188016
$51$	0	0
$52$	12.8667	1.78429
$53$	−2.22047	−0.305005	−0.152503	−	0.988303i	$-0.548733\pi$
$53$	−2.22047	−0.305005	−0.152503	+	0.988303i	$0.548733\pi$
$54$	0	0
$55$	−12.9607	−1.74762
$56$	−0.118609	−0.0158497
$57$	0	0
$58$	0.198478	0.0260614
$59$	−1.48973	−0.193946	−0.0969729	−	0.995287i	$-0.530916\pi$
$59$	−1.48973	−0.193946	−0.0969729	+	0.995287i	$0.530916\pi$
$60$	0	0
$61$	7.08310	0.906898	0.453449	−	0.891282i	$-0.350193\pi$
$61$	7.08310	0.906898	0.453449	+	0.891282i	$0.350193\pi$
$62$	0.0380136	0.00482773
$63$	0	0
$64$	−7.97890	−0.997362
$65$	19.8194	2.45829
$66$	0	0
$67$	0.989393	0.120874	0.0604368	−	0.998172i	$-0.480751\pi$
$67$	0.989393	0.120874	0.0604368	+	0.998172i	$0.480751\pi$
$68$	−1.95535	−0.237122
$69$	0	0
$70$	−0.0913304	−0.0109161
$71$	8.17613	0.970328	0.485164	−	0.874423i	$-0.338760\pi$
$71$	8.17613	0.970328	0.485164	+	0.874423i	$0.338760\pi$
$72$	0	0
$73$	9.17564	1.07393	0.536964	−	0.843605i	$-0.319571\pi$
$73$	9.17564	1.07393	0.536964	+	0.843605i	$0.319571\pi$
$74$	−0.181102	−0.0210527
$75$	0	0
$76$	−0.903798	−0.103673
$77$	−4.20887	−0.479646
$78$	0	0
$79$	9.02266	1.01513	0.507564	−	0.861614i	$-0.330546\pi$
$79$	9.02266	1.01513	0.507564	+	0.861614i	$0.330546\pi$
$80$	−12.3013	−1.37532
$81$	0	0
$82$	−0.103287	−0.0114061
$83$	5.68319	0.623811	0.311906	−	0.950113i	$-0.399033\pi$
$83$	5.68319	0.623811	0.311906	+	0.950113i	$0.399033\pi$
$84$	0	0
$85$	−3.01197	−0.326694
$86$	−0.122072	−0.0131634
$87$	0	0
$88$	0.499209	0.0532158
$89$	−2.01684	−0.213785	−0.106892	−	0.994271i	$-0.534090\pi$
$89$	−2.01684	−0.213785	−0.106892	+	0.994271i	$0.534090\pi$
$90$	0	0
$91$	6.43617	0.674694
$92$	3.75476	0.391461
$93$	0	0
$94$	−0.0101856	−0.00105056
$95$	−1.39218	−0.142835
$96$	0	0
$97$	10.8384	1.10047	0.550235	−	0.835010i	$-0.314538\pi$
$97$	10.8384	1.10047	0.550235	+	0.835010i	$0.314538\pi$
$98$	−0.0296587	−0.00299598
$99$	0	0
$100$	−8.96122	−0.896122
$101$	−9.76783	−0.971936	−0.485968	−	0.873977i	$-0.661533\pi$
$101$	−9.76783	−0.971936	−0.485968	+	0.873977i	$0.661533\pi$
$102$	0	0
$103$	6.28650	0.619428	0.309714	−	0.950830i	$-0.399767\pi$
$103$	6.28650	0.619428	0.309714	+	0.950830i	$0.399767\pi$
$104$	−0.763385	−0.0748561
$105$	0	0
$106$	0.0658563	0.00639653
$107$	14.2861	1.38109	0.690545	−	0.723289i	$-0.257371\pi$
$107$	14.2861	1.38109	0.690545	+	0.723289i	$0.257371\pi$
$108$	0	0
$109$	−6.40981	−0.613948	−0.306974	−	0.951718i	$-0.599317\pi$
$109$	−6.40981	−0.613948	−0.306974	+	0.951718i	$0.599317\pi$
$110$	0.384398	0.0366509
$111$	0	0
$112$	−3.99472	−0.377466
$113$	−8.85904	−0.833388	−0.416694	−	0.909047i	$-0.636811\pi$
$113$	−8.85904	−0.833388	−0.416694	+	0.909047i	$0.636811\pi$
$114$	0	0
$115$	5.78372	0.539334
$116$	13.3783	1.24214
$117$	0	0
$118$	0.0441833	0.00406740
$119$	−0.978108	−0.0896630
$120$	0	0
$121$	6.71461	0.610419
$122$	−0.210075	−0.0190193
$123$	0	0
$124$	2.56228	0.230099
$125$	1.59332	0.142511
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0.948035	0.0837952
$129$	0	0
$130$	−0.587817	−0.0515550
$131$	−4.30603	−0.376220	−0.188110	−	0.982148i	$-0.560236\pi$
$131$	−4.30603	−0.376220	−0.188110	+	0.982148i	$0.560236\pi$
$132$	0	0
$133$	−0.452098	−0.0392019
$134$	−0.0293441	−0.00253494
$135$	0	0
$136$	0.116012	0.00994795
$137$	−3.16745	−0.270614	−0.135307	−	0.990804i	$-0.543202\pi$
$137$	−3.16745	−0.270614	−0.135307	+	0.990804i	$0.543202\pi$
$138$	0	0
$139$	7.21212	0.611724	0.305862	−	0.952076i	$-0.401055\pi$
$139$	7.21212	0.611724	0.305862	+	0.952076i	$0.401055\pi$
$140$	−6.15605	−0.520281
$141$	0	0
$142$	−0.242493	−0.0203496
$143$	−27.0890	−2.26530
$144$	0	0
$145$	20.6074	1.71135
$146$	−0.272138	−0.0225223
$147$	0	0
$148$	−12.2070	−1.00341
$149$	−8.99048	−0.736529	−0.368264	−	0.929721i	$-0.620048\pi$
$149$	−8.99048	−0.736529	−0.368264	+	0.929721i	$0.620048\pi$
$150$	0	0
$151$	−2.86869	−0.233451	−0.116725	−	0.993164i	$-0.537240\pi$
$151$	−2.86869	−0.233451	−0.116725	+	0.993164i	$0.537240\pi$
$152$	0.0536227	0.00434938
$153$	0	0
$154$	0.124830	0.0100591
$155$	3.94685	0.317018
$156$	0	0
$157$	−0.748120	−0.0597065	−0.0298532	−	0.999554i	$-0.509504\pi$
$157$	−0.748120	−0.0597065	−0.0298532	+	0.999554i	$0.509504\pi$
$158$	−0.267600	−0.0212891
$159$	0	0
$160$	1.09532	0.0865928
$161$	1.87821	0.148023
$162$	0	0
$163$	13.1598	1.03075	0.515376	−	0.856964i	$-0.327652\pi$
$163$	13.1598	1.03075	0.515376	+	0.856964i	$0.327652\pi$
$164$	−6.96196	−0.543638
$165$	0	0
$166$	−0.168556	−0.0130825
$167$	−20.0284	−1.54984	−0.774921	−	0.632058i	$-0.782211\pi$
$167$	−20.0284	−1.54984	−0.774921	+	0.632058i	$0.782211\pi$
$168$	0	0
$169$	28.4242	2.18648
$170$	0.0893309	0.00685137
$171$	0	0
$172$	−8.22818	−0.627393
$173$	−14.6850	−1.11648	−0.558240	−	0.829680i	$-0.688523\pi$
$173$	−14.6850	−1.11648	−0.558240	+	0.829680i	$0.688523\pi$
$174$	0	0
$175$	−4.48258	−0.338851
$176$	16.8133	1.26735
$177$	0	0
$178$	0.0598168	0.00448346
$179$	−1.42580	−0.106569	−0.0532846	−	0.998579i	$-0.516969\pi$
$179$	−1.42580	−0.106569	−0.0532846	+	0.998579i	$0.516969\pi$
$180$	0	0
$181$	−16.0798	−1.19520	−0.597600	−	0.801794i	$-0.703879\pi$
$181$	−16.0798	−1.19520	−0.597600	+	0.801794i	$0.703879\pi$
$182$	−0.190888	−0.0141496
$183$	0	0
$184$	−0.222772	−0.0164229
$185$	−18.8033	−1.38245
$186$	0	0
$187$	4.11673	0.301045
$188$	−0.686553	−0.0500720
$189$	0	0
$190$	0.0412903	0.00299551
$191$	−8.95356	−0.647857	−0.323929	−	0.946082i	$-0.605004\pi$
$191$	−8.95356	−0.647857	−0.323929	+	0.946082i	$0.605004\pi$
$192$	0	0
$193$	4.34936	0.313074	0.156537	−	0.987672i	$-0.449967\pi$
$193$	4.34936	0.313074	0.156537	+	0.987672i	$0.449967\pi$
$194$	−0.321452	−0.0230789
$195$	0	0
$196$	−1.99912	−0.142794
$197$	−12.1307	−0.864278	−0.432139	−	0.901807i	$-0.642241\pi$
$197$	−12.1307	−0.864278	−0.432139	+	0.901807i	$0.642241\pi$
$198$	0	0
$199$	−15.3936	−1.09122	−0.545612	−	0.838038i	$-0.683703\pi$
$199$	−15.3936	−1.09122	−0.545612	+	0.838038i	$0.683703\pi$
$200$	0.531673	0.0375950
$201$	0	0
$202$	0.289701	0.0203833
$203$	6.69207	0.469691
$204$	0	0
$205$	−10.7240	−0.748995
$206$	−0.186449	−0.0129905
$207$	0	0
$208$	−25.7107	−1.78272
$209$	1.90282	0.131621
$210$	0	0
$211$	19.6993	1.35616	0.678078	−	0.734990i	$-0.262813\pi$
$211$	19.6993	1.35616	0.678078	+	0.734990i	$0.262813\pi$
$212$	4.43899	0.304871
$213$	0	0
$214$	−0.423707	−0.0289640
$215$	−12.6744	−0.864389
$216$	0	0
$217$	1.28170	0.0870076
$218$	0.190107	0.0128756
$219$	0	0
$220$	25.9100	1.74685
$221$	−6.29526	−0.423465
$222$	0	0
$223$	−10.1764	−0.681462	−0.340731	−	0.940161i	$-0.610675\pi$
$223$	−10.1764	−0.681462	−0.340731	+	0.940161i	$0.610675\pi$
$224$	0.355696	0.0237659
$225$	0	0
$226$	0.262747	0.0174777
$227$	3.99422	0.265105	0.132553	−	0.991176i	$-0.457683\pi$
$227$	3.99422	0.265105	0.132553	+	0.991176i	$0.457683\pi$
$228$	0	0
$229$	−10.8957	−0.720010	−0.360005	−	0.932950i	$-0.617225\pi$
$229$	−10.8957	−0.720010	−0.360005	+	0.932950i	$0.617225\pi$
$230$	−0.171537	−0.0113108
$231$	0	0
$232$	−0.793737	−0.0521114
$233$	−25.3409	−1.66014	−0.830070	−	0.557660i	$-0.811699\pi$
$233$	−25.3409	−1.66014	−0.830070	+	0.557660i	$0.811699\pi$
$234$	0	0
$235$	−1.05754	−0.0689865
$236$	2.97814	0.193860
$237$	0	0
$238$	0.0290094	0.00188040
$239$	−2.55633	−0.165355	−0.0826776	−	0.996576i	$-0.526347\pi$
$239$	−2.55633	−0.165355	−0.0826776	+	0.996576i	$0.526347\pi$
$240$	0	0
$241$	14.5627	0.938065	0.469032	−	0.883181i	$-0.344603\pi$
$241$	14.5627	0.938065	0.469032	+	0.883181i	$0.344603\pi$
$242$	−0.199147	−0.0128016
$243$	0	0
$244$	−14.1600	−0.906499
$245$	−3.07938	−0.196734
$246$	0	0
$247$	−2.90978	−0.185145
$248$	−0.152021	−0.00965334
$249$	0	0
$250$	−0.0472559	−0.00298873
$251$	6.85143	0.432458	0.216229	−	0.976343i	$-0.430624\pi$
$251$	6.85143	0.432458	0.216229	+	0.976343i	$0.430624\pi$
$252$	0	0
$253$	−7.90514	−0.496992
$254$	−0.0296587	−0.00186095
$255$	0	0
$256$	15.9297	0.995605
$257$	−6.82464	−0.425709	−0.212855	−	0.977084i	$-0.568276\pi$
$257$	−6.82464	−0.425709	−0.212855	+	0.977084i	$0.568276\pi$
$258$	0	0
$259$	−6.10620	−0.379421
$260$	−39.6214	−2.45721
$261$	0	0
$262$	0.127711	0.00789003
$263$	−9.84480	−0.607056	−0.303528	−	0.952822i	$-0.598165\pi$
$263$	−9.84480	−0.607056	−0.303528	+	0.952822i	$0.598165\pi$
$264$	0	0
$265$	6.83768	0.420035
$266$	0.0134086	0.000822136 0
$267$	0	0
$268$	−1.97792	−0.120820
$269$	−4.81113	−0.293340	−0.146670	−	0.989186i	$-0.546855\pi$
$269$	−4.81113	−0.293340	−0.146670	+	0.989186i	$0.546855\pi$
$270$	0	0
$271$	−25.2057	−1.53114	−0.765568	−	0.643356i	$-0.777542\pi$
$271$	−25.2057	−1.53114	−0.765568	+	0.643356i	$0.777542\pi$
$272$	3.90727	0.236913
$273$	0	0
$274$	0.0939424	0.00567527
$275$	18.8666	1.13770
$276$	0	0
$277$	20.4070	1.22614	0.613070	−	0.790028i	$-0.289934\pi$
$277$	20.4070	1.22614	0.613070	+	0.790028i	$0.289934\pi$
$278$	−0.213902	−0.0128290
$279$	0	0
$280$	0.365241	0.0218273
$281$	16.1617	0.964126	0.482063	−	0.876136i	$-0.339888\pi$
$281$	16.1617	0.964126	0.482063	+	0.876136i	$0.339888\pi$
$282$	0	0
$283$	3.55148	0.211114	0.105557	−	0.994413i	$-0.466338\pi$
$283$	3.55148	0.211114	0.105557	+	0.994413i	$0.466338\pi$
$284$	−16.3451	−0.969901
$285$	0	0
$286$	0.803424	0.0475075
$287$	−3.48251	−0.205566
$288$	0	0
$289$	−16.0433	−0.943724
$290$	−0.611189	−0.0358903
$291$	0	0
$292$	−18.3432	−1.07346
$293$	−21.0187	−1.22793	−0.613963	−	0.789335i	$-0.710426\pi$
$293$	−21.0187	−1.22793	−0.613963	+	0.789335i	$0.710426\pi$
$294$	0	0
$295$	4.58743	0.267091
$296$	0.724249	0.0420961
$297$	0	0
$298$	0.266646	0.0154464
$299$	12.0885	0.699094
$300$	0	0
$301$	−4.11590	−0.237237
$302$	0.0850816	0.00489590
$303$	0	0
$304$	1.80601	0.103582
$305$	−21.8116	−1.24893
$306$	0	0
$307$	16.9198	0.965666	0.482833	−	0.875712i	$-0.339608\pi$
$307$	16.9198	0.965666	0.482833	+	0.875712i	$0.339608\pi$
$308$	8.41404	0.479435
$309$	0	0
$310$	−0.117058	−0.00664846
$311$	−13.5782	−0.769948	−0.384974	−	0.922927i	$-0.625790\pi$
$311$	−13.5782	−0.769948	−0.384974	+	0.922927i	$0.625790\pi$
$312$	0	0
$313$	−31.4249	−1.77624	−0.888120	−	0.459612i	$-0.847989\pi$
$313$	−31.4249	−1.77624	−0.888120	+	0.459612i	$0.847989\pi$
$314$	0.0221883	0.00125216
$315$	0	0
$316$	−18.0374	−1.01468
$317$	17.5662	0.986614	0.493307	−	0.869855i	$-0.335788\pi$
$317$	17.5662	0.986614	0.493307	+	0.869855i	$0.335788\pi$
$318$	0	0
$319$	−28.1661	−1.57700
$320$	24.5701	1.37351
$321$	0	0
$322$	−0.0557052	−0.00310433
$323$	0.442200	0.0246047
$324$	0	0
$325$	−28.8507	−1.60035
$326$	−0.390301	−0.0216168
$327$	0	0
$328$	0.413056	0.0228072
$329$	−0.343427	−0.0189338
$330$	0	0
$331$	26.2992	1.44553	0.722766	−	0.691093i	$-0.242870\pi$
$331$	26.2992	1.44553	0.722766	+	0.691093i	$0.242870\pi$
$332$	−11.3614	−0.623537
$333$	0	0
$334$	0.594015	0.0325031
$335$	−3.04672	−0.166460
$336$	0	0
$337$	−1.31181	−0.0714590	−0.0357295	−	0.999361i	$-0.511375\pi$
$337$	−1.31181	−0.0714590	−0.0357295	+	0.999361i	$0.511375\pi$
$338$	−0.843026	−0.0458546
$339$	0	0
$340$	6.02128	0.326550
$341$	−5.39452	−0.292130
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.488182	0.0263210
$345$	0	0
$346$	0.435538	0.0234146
$347$	12.5440	0.673397	0.336699	−	0.941612i	$-0.390690\pi$
$347$	12.5440	0.673397	0.336699	+	0.941612i	$0.390690\pi$
$348$	0	0
$349$	−10.9299	−0.585067	−0.292533	−	0.956255i	$-0.594498\pi$
$349$	−10.9299	−0.585067	−0.292533	+	0.956255i	$0.594498\pi$
$350$	0.132948	0.00710634
$351$	0	0
$352$	−1.49708	−0.0797945
$353$	−24.2674	−1.29162	−0.645810	−	0.763498i	$-0.723480\pi$
$353$	−24.2674	−1.29162	−0.645810	+	0.763498i	$0.723480\pi$
$354$	0	0
$355$	−25.1774	−1.33628
$356$	4.03191	0.213691
$357$	0	0
$358$	0.0422873	0.00223496
$359$	−8.77538	−0.463147	−0.231573	−	0.972817i	$-0.574387\pi$
$359$	−8.77538	−0.463147	−0.231573	+	0.972817i	$0.574387\pi$
$360$	0	0
$361$	−18.7956	−0.989243
$362$	0.476905	0.0250656
$363$	0	0
$364$	−12.8667	−0.674397
$365$	−28.2553	−1.47895
$366$	0	0
$367$	−28.2522	−1.47475	−0.737376	−	0.675483i	$-0.763935\pi$
$367$	−28.2522	−1.47475	−0.737376	+	0.675483i	$0.763935\pi$
$368$	−7.50292	−0.391117
$369$	0	0
$370$	0.557682	0.0289925
$371$	2.22047	0.115281
$372$	0	0
$373$	16.7923	0.869475	0.434737	−	0.900557i	$-0.356841\pi$
$373$	16.7923	0.869475	0.434737	+	0.900557i	$0.356841\pi$
$374$	−0.122097	−0.00631348
$375$	0	0
$376$	0.0407335	0.00210067
$377$	43.0713	2.21828
$378$	0	0
$379$	−25.8015	−1.32533	−0.662666	−	0.748915i	$-0.730575\pi$
$379$	−25.8015	−1.32533	−0.662666	+	0.748915i	$0.730575\pi$
$380$	2.78314	0.142772
$381$	0	0
$382$	0.265551	0.0135868
$383$	14.3700	0.734272	0.367136	−	0.930167i	$-0.380338\pi$
$383$	14.3700	0.734272	0.367136	+	0.930167i	$0.380338\pi$
$384$	0	0
$385$	12.9607	0.660539
$386$	−0.128996	−0.00656575
$387$	0	0
$388$	−21.6672	−1.09999
$389$	22.5454	1.14309	0.571547	−	0.820569i	$-0.306343\pi$
$389$	22.5454	1.14309	0.571547	+	0.820569i	$0.306343\pi$
$390$	0	0
$391$	−1.83709	−0.0929056
$392$	0.118609	0.00599064
$393$	0	0
$394$	0.359781	0.0181255
$395$	−27.7842	−1.39797
$396$	0	0
$397$	0.445695	0.0223688	0.0111844	−	0.999937i	$-0.496440\pi$
$397$	0.445695	0.0223688	0.0111844	+	0.999937i	$0.496440\pi$
$398$	0.456554	0.0228850
$399$	0	0
$400$	17.9067	0.895334
$401$	−6.19494	−0.309361	−0.154680	−	0.987965i	$-0.549435\pi$
$401$	−6.19494	−0.309361	−0.154680	+	0.987965i	$0.549435\pi$
$402$	0	0
$403$	8.24924	0.410924
$404$	19.5271	0.971508
$405$	0	0
$406$	−0.198478	−0.00985030
$407$	25.7002	1.27391
$408$	0	0
$409$	−26.3504	−1.30294	−0.651472	−	0.758673i	$-0.725848\pi$
$409$	−26.3504	−1.30294	−0.651472	+	0.758673i	$0.725848\pi$
$410$	0.318059	0.0157078
$411$	0	0
$412$	−12.5675	−0.619155
$413$	1.48973	0.0733046
$414$	0	0
$415$	−17.5007	−0.859076
$416$	2.28932	0.112243
$417$	0	0
$418$	−0.0564352	−0.00276034
$419$	−6.10307	−0.298154	−0.149077	−	0.988826i	$-0.547630\pi$
$419$	−6.10307	−0.298154	−0.149077	+	0.988826i	$0.547630\pi$
$420$	0	0
$421$	10.1315	0.493777	0.246888	−	0.969044i	$-0.420592\pi$
$421$	10.1315	0.493777	0.246888	+	0.969044i	$0.420592\pi$
$422$	−0.584256	−0.0284411
$423$	0	0
$424$	−0.263367	−0.0127902
$425$	4.38445	0.212677
$426$	0	0
$427$	−7.08310	−0.342775
$428$	−28.5597	−1.38048
$429$	0	0
$430$	0.375907	0.0181278
$431$	39.8669	1.92032	0.960160	−	0.279451i	$-0.0901525\pi$
$431$	39.8669	1.92032	0.960160	+	0.279451i	$0.0901525\pi$
$432$	0	0
$433$	13.2786	0.638128	0.319064	−	0.947733i	$-0.396632\pi$
$433$	13.2786	0.638128	0.319064	+	0.947733i	$0.396632\pi$
$434$	−0.0380136	−0.00182471
$435$	0	0
$436$	12.8140	0.613678
$437$	−0.849134	−0.0406196
$438$	0	0
$439$	−12.8890	−0.615158	−0.307579	−	0.951523i	$-0.599519\pi$
$439$	−12.8890	−0.615158	−0.307579	+	0.951523i	$0.599519\pi$
$440$	−1.53725	−0.0732857
$441$	0	0
$442$	0.186709	0.00888085
$443$	25.3649	1.20512	0.602561	−	0.798073i	$-0.294147\pi$
$443$	25.3649	1.20512	0.602561	+	0.798073i	$0.294147\pi$
$444$	0	0
$445$	6.21062	0.294411
$446$	0.301819	0.0142915
$447$	0	0
$448$	7.97890	0.376967
$449$	−23.2619	−1.09780	−0.548899	−	0.835889i	$-0.684953\pi$
$449$	−23.2619	−1.09780	−0.548899	+	0.835889i	$0.684953\pi$
$450$	0	0
$451$	14.6575	0.690192
$452$	17.7103	0.833022
$453$	0	0
$454$	−0.118463	−0.00555975
$455$	−19.8194	−0.929148
$456$	0	0
$457$	−38.4218	−1.79730	−0.898649	−	0.438669i	$-0.855450\pi$
$457$	−38.4218	−1.79730	−0.898649	+	0.438669i	$0.855450\pi$
$458$	0.323153	0.0150999
$459$	0	0
$460$	−11.5623	−0.539097
$461$	−14.8206	−0.690262	−0.345131	−	0.938555i	$-0.612166\pi$
$461$	−14.8206	−0.690262	−0.345131	+	0.938555i	$0.612166\pi$
$462$	0	0
$463$	35.2074	1.63623	0.818114	−	0.575056i	$-0.195020\pi$
$463$	35.2074	1.63623	0.818114	+	0.575056i	$0.195020\pi$
$464$	−26.7330	−1.24105
$465$	0	0
$466$	0.751579	0.0348162
$467$	−23.2168	−1.07435	−0.537173	−	0.843472i	$-0.680508\pi$
$467$	−23.2168	−1.07435	−0.537173	+	0.843472i	$0.680508\pi$
$468$	0	0
$469$	−0.989393	−0.0456859
$470$	0.0313654	0.00144678
$471$	0	0
$472$	−0.176694	−0.00813302
$473$	17.3233	0.796527
$474$	0	0
$475$	2.02657	0.0929852
$476$	1.95535	0.0896235
$477$	0	0
$478$	0.0758174	0.00346781
$479$	28.7854	1.31524	0.657620	−	0.753350i	$-0.271563\pi$
$479$	28.7854	1.31524	0.657620	+	0.753350i	$0.271563\pi$
$480$	0	0
$481$	−39.3006	−1.79195
$482$	−0.431910	−0.0196730
$483$	0	0
$484$	−13.4233	−0.610151
$485$	−33.3755	−1.51550
$486$	0	0
$487$	40.6311	1.84117	0.920584	−	0.390544i	$-0.127713\pi$
$487$	40.6311	1.84117	0.920584	+	0.390544i	$0.127713\pi$
$488$	0.840117	0.0380303
$489$	0	0
$490$	0.0913304	0.00412589
$491$	−6.94851	−0.313582	−0.156791	−	0.987632i	$-0.550115\pi$
$491$	−6.94851	−0.313582	−0.156791	+	0.987632i	$0.550115\pi$
$492$	0	0
$493$	−6.54556	−0.294797
$494$	0.0863002	0.00388283
$495$	0	0
$496$	−5.12004	−0.229897
$497$	−8.17613	−0.366750
$498$	0	0
$499$	5.09072	0.227892	0.113946	−	0.993487i	$-0.463651\pi$
$499$	5.09072	0.227892	0.113946	+	0.993487i	$0.463651\pi$
$500$	−3.18525	−0.142449
$501$	0	0
$502$	−0.203204	−0.00906946
$503$	34.4535	1.53621	0.768104	−	0.640325i	$-0.221201\pi$
$503$	34.4535	1.53621	0.768104	+	0.640325i	$0.221201\pi$
$504$	0	0
$505$	30.0789	1.33849
$506$	0.234456	0.0104228
$507$	0	0
$508$	−1.99912	−0.0886966
$509$	−33.6813	−1.49290	−0.746450	−	0.665442i	$-0.768243\pi$
$509$	−33.6813	−1.49290	−0.746450	+	0.665442i	$0.768243\pi$
$510$	0	0
$511$	−9.17564	−0.405907
$512$	−2.36852	−0.104675
$513$	0	0
$514$	0.202410	0.00892792
$515$	−19.3585	−0.853039
$516$	0	0
$517$	1.44544	0.0635705
$518$	0.181102	0.00795717
$519$	0	0
$520$	2.35075	0.103087
$521$	−36.6294	−1.60476	−0.802382	−	0.596811i	$-0.796434\pi$
$521$	−36.6294	−1.60476	−0.802382	+	0.596811i	$0.796434\pi$
$522$	0	0
$523$	−28.7191	−1.25580	−0.627900	−	0.778294i	$-0.716085\pi$
$523$	−28.7191	−1.25580	−0.627900	+	0.778294i	$0.716085\pi$
$524$	8.60828	0.376055
$525$	0	0
$526$	0.291984	0.0127311
$527$	−1.25364	−0.0546095
$528$	0	0
$529$	−19.4723	−0.846623
$530$	−0.202797	−0.00880892
$531$	0	0
$532$	0.903798	0.0391846
$533$	−22.4140	−0.970859
$534$	0	0
$535$	−43.9924	−1.90196
$536$	0.117351	0.00506877
$537$	0	0
$538$	0.142692	0.00615188
$539$	4.20887	0.181289
$540$	0	0
$541$	4.29971	0.184859	0.0924294	−	0.995719i	$-0.470537\pi$
$541$	4.29971	0.184859	0.0924294	+	0.995719i	$0.470537\pi$
$542$	0.747567	0.0321107
$543$	0	0
$544$	−0.347909	−0.0149165
$545$	19.7382	0.845493
$546$	0	0
$547$	−19.4300	−0.830767	−0.415383	−	0.909646i	$-0.636353\pi$
$547$	−19.4300	−0.830767	−0.415383	+	0.909646i	$0.636353\pi$
$548$	6.33212	0.270495
$549$	0	0
$550$	−0.559559	−0.0238597
$551$	−3.02547	−0.128889
$552$	0	0
$553$	−9.02266	−0.383682
$554$	−0.605246	−0.0257144
$555$	0	0
$556$	−14.4179	−0.611455
$557$	39.6587	1.68039	0.840197	−	0.542281i	$-0.182439\pi$
$557$	39.6587	1.68039	0.840197	+	0.542281i	$0.182439\pi$
$558$	0	0
$559$	−26.4906	−1.12043
$560$	12.3013	0.519824
$561$	0	0
$562$	−0.479335	−0.0202195
$563$	−1.06429	−0.0448545	−0.0224272	−	0.999748i	$-0.507139\pi$
$563$	−1.06429	−0.0448545	−0.0224272	+	0.999748i	$0.507139\pi$
$564$	0	0
$565$	27.2804	1.14769
$566$	−0.105332	−0.00442745
$567$	0	0
$568$	0.969760	0.0406902
$569$	−4.34569	−0.182181	−0.0910903	−	0.995843i	$-0.529035\pi$
$569$	−4.34569	−0.182181	−0.0910903	+	0.995843i	$0.529035\pi$
$570$	0	0
$571$	0.617402	0.0258374	0.0129187	−	0.999917i	$-0.495888\pi$
$571$	0.617402	0.0258374	0.0129187	+	0.999917i	$0.495888\pi$
$572$	54.1542	2.26430
$573$	0	0
$574$	0.103287	0.00431111
$575$	−8.41922	−0.351106
$576$	0	0
$577$	−45.6275	−1.89950	−0.949748	−	0.313015i	$-0.898661\pi$
$577$	−45.6275	−1.89950	−0.949748	+	0.313015i	$0.898661\pi$
$578$	0.475823	0.0197916
$579$	0	0
$580$	−41.1967	−1.71060
$581$	−5.68319	−0.235778
$582$	0	0
$583$	−9.34569	−0.387059
$584$	1.08831	0.0450346
$585$	0	0
$586$	0.623388	0.0257519
$587$	38.7194	1.59812	0.799061	−	0.601250i	$-0.205331\pi$
$587$	38.7194	1.59812	0.799061	+	0.601250i	$0.205331\pi$
$588$	0	0
$589$	−0.579455	−0.0238760
$590$	−0.136057	−0.00560139
$591$	0	0
$592$	24.3926	1.00253
$593$	−30.6824	−1.25998	−0.629988	−	0.776605i	$-0.716940\pi$
$593$	−30.6824	−1.25998	−0.629988	+	0.776605i	$0.716940\pi$
$594$	0	0
$595$	3.01197	0.123479
$596$	17.9731	0.736205
$597$	0	0
$598$	−0.358528	−0.0146613
$599$	−36.2093	−1.47947	−0.739736	−	0.672897i	$-0.765050\pi$
$599$	−36.2093	−1.47947	−0.739736	+	0.672897i	$0.765050\pi$
$600$	0	0
$601$	40.5462	1.65391	0.826956	−	0.562266i	$-0.190070\pi$
$601$	40.5462	1.65391	0.826956	+	0.562266i	$0.190070\pi$
$602$	0.122072	0.00497529
$603$	0	0
$604$	5.73486	0.233348
$605$	−20.6768	−0.840633
$606$	0	0
$607$	33.8247	1.37290	0.686451	−	0.727176i	$-0.259168\pi$
$607$	33.8247	1.37290	0.686451	+	0.727176i	$0.259168\pi$
$608$	−0.160809	−0.00652167
$609$	0	0
$610$	0.646902	0.0261923
$611$	−2.21036	−0.0894214
$612$	0	0
$613$	5.81836	0.235001	0.117501	−	0.993073i	$-0.462512\pi$
$613$	5.81836	0.235001	0.117501	+	0.993073i	$0.462512\pi$
$614$	−0.501820	−0.0202518
$615$	0	0
$616$	−0.499209	−0.0201137
$617$	−43.1946	−1.73895	−0.869474	−	0.493979i	$-0.835542\pi$
$617$	−43.1946	−1.73895	−0.869474	+	0.493979i	$0.835542\pi$
$618$	0	0
$619$	−0.582136	−0.0233980	−0.0116990	−	0.999932i	$-0.503724\pi$
$619$	−0.582136	−0.0233980	−0.0116990	+	0.999932i	$0.503724\pi$
$620$	−7.89022	−0.316879
$621$	0	0
$622$	0.402711	0.0161472
$623$	2.01684	0.0808030
$624$	0	0
$625$	−27.3194	−1.09277
$626$	0.932021	0.0372511
$627$	0	0
$628$	1.49558	0.0596802
$629$	5.97253	0.238140
$630$	0	0
$631$	−30.1214	−1.19911	−0.599557	−	0.800332i	$-0.704657\pi$
$631$	−30.1214	−1.19911	−0.599557	+	0.800332i	$0.704657\pi$
$632$	1.07017	0.0425689
$633$	0	0
$634$	−0.520989	−0.0206911
$635$	−3.07938	−0.122201
$636$	0	0
$637$	−6.43617	−0.255010
$638$	0.835369	0.0330726
$639$	0	0
$640$	−2.91936	−0.115398
$641$	−31.1460	−1.23019	−0.615096	−	0.788452i	$-0.710883\pi$
$641$	−31.1460	−1.23019	−0.615096	+	0.788452i	$0.710883\pi$
$642$	0	0
$643$	2.91383	0.114910	0.0574551	−	0.998348i	$-0.481701\pi$
$643$	2.91383	0.114910	0.0574551	+	0.998348i	$0.481701\pi$
$644$	−3.75476	−0.147958
$645$	0	0
$646$	−0.0131151	−0.000516006 0
$647$	18.7263	0.736206	0.368103	−	0.929785i	$-0.380007\pi$
$647$	18.7263	0.736206	0.368103	+	0.929785i	$0.380007\pi$
$648$	0	0
$649$	−6.27007	−0.246122
$650$	0.855672	0.0335622
$651$	0	0
$652$	−26.3079	−1.03030
$653$	−13.6677	−0.534857	−0.267429	−	0.963578i	$-0.586174\pi$
$653$	−13.6677	−0.534857	−0.267429	+	0.963578i	$0.586174\pi$
$654$	0	0
$655$	13.2599	0.518108
$656$	13.9117	0.543160
$657$	0	0
$658$	0.0101856	0.000397076 0
$659$	6.46273	0.251752	0.125876	−	0.992046i	$-0.459826\pi$
$659$	6.46273	0.251752	0.125876	+	0.992046i	$0.459826\pi$
$660$	0	0
$661$	−27.1580	−1.05632	−0.528162	−	0.849144i	$-0.677118\pi$
$661$	−27.1580	−1.05632	−0.528162	+	0.849144i	$0.677118\pi$
$662$	−0.779998	−0.0303155
$663$	0	0
$664$	0.674076	0.0261592
$665$	1.39218	0.0539865
$666$	0	0
$667$	12.5691	0.486677
$668$	40.0391	1.54916
$669$	0	0
$670$	0.0903617	0.00349098
$671$	29.8119	1.15087
$672$	0	0
$673$	−1.23799	−0.0477211	−0.0238606	−	0.999715i	$-0.507596\pi$
$673$	−1.23799	−0.0477211	−0.0238606	+	0.999715i	$0.507596\pi$
$674$	0.0389066	0.00149863
$675$	0	0
$676$	−56.8235	−2.18552
$677$	3.72439	0.143140	0.0715700	−	0.997436i	$-0.477199\pi$
$677$	3.72439	0.143140	0.0715700	+	0.997436i	$0.477199\pi$
$678$	0	0
$679$	−10.8384	−0.415939
$680$	−0.357245	−0.0136997
$681$	0	0
$682$	0.159994	0.00612650
$683$	−1.26391	−0.0483621	−0.0241811	−	0.999708i	$-0.507698\pi$
$683$	−1.26391	−0.0483621	−0.0241811	+	0.999708i	$0.507698\pi$
$684$	0	0
$685$	9.75379	0.372673
$686$	0.0296587	0.00113237
$687$	0	0
$688$	16.4419	0.626841
$689$	14.2913	0.544456
$690$	0	0
$691$	−9.08197	−0.345495	−0.172747	−	0.984966i	$-0.555264\pi$
$691$	−9.08197	−0.345495	−0.172747	+	0.984966i	$0.555264\pi$
$692$	29.3571	1.11599
$693$	0	0
$694$	−0.372038	−0.0141224
$695$	−22.2089	−0.842430
$696$	0	0
$697$	3.40627	0.129022
$698$	0.324168	0.0122699
$699$	0	0
$700$	8.96122	0.338702
$701$	−11.3303	−0.427941	−0.213971	−	0.976840i	$-0.568640\pi$
$701$	−11.3303	−0.427941	−0.213971	+	0.976840i	$0.568640\pi$
$702$	0	0
$703$	2.76060	0.104118
$704$	−33.5822	−1.26568
$705$	0	0
$706$	0.719738	0.0270877
$707$	9.76783	0.367357
$708$	0	0
$709$	19.1018	0.717383	0.358691	−	0.933456i	$-0.383223\pi$
$709$	19.1018	0.717383	0.358691	+	0.933456i	$0.383223\pi$
$710$	0.746729	0.0280242
$711$	0	0
$712$	−0.239215	−0.00896495
$713$	2.40730	0.0901542
$714$	0	0
$715$	83.4174	3.11963
$716$	2.85035	0.106522
$717$	0	0
$718$	0.260266	0.00971305
$719$	−16.5348	−0.616645	−0.308322	−	0.951282i	$-0.599768\pi$
$719$	−16.5348	−0.616645	−0.308322	+	0.951282i	$0.599768\pi$
$720$	0	0
$721$	−6.28650	−0.234122
$722$	0.557453	0.0207463
$723$	0	0
$724$	32.1454	1.19468
$725$	−29.9978	−1.11409
$726$	0	0
$727$	−13.6415	−0.505934	−0.252967	−	0.967475i	$-0.581406\pi$
$727$	−13.6415	−0.505934	−0.252967	+	0.967475i	$0.581406\pi$
$728$	0.763385	0.0282929
$729$	0	0
$730$	0.838015	0.0310163
$731$	4.02579	0.148899
$732$	0	0
$733$	42.7851	1.58030	0.790152	−	0.612911i	$-0.210002\pi$
$733$	42.7851	1.58030	0.790152	+	0.612911i	$0.210002\pi$
$734$	0.837922	0.0309283
$735$	0	0
$736$	0.668070	0.0246254
$737$	4.16423	0.153391
$738$	0	0
$739$	24.3103	0.894270	0.447135	−	0.894467i	$-0.352444\pi$
$739$	24.3103	0.894270	0.447135	+	0.894467i	$0.352444\pi$
$740$	37.5901	1.38184
$741$	0	0
$742$	−0.0658563	−0.00241766
$743$	31.4154	1.15252	0.576260	−	0.817266i	$-0.304511\pi$
$743$	31.4154	1.15252	0.576260	+	0.817266i	$0.304511\pi$
$744$	0	0
$745$	27.6851	1.01430
$746$	−0.498039	−0.0182345
$747$	0	0
$748$	−8.22984	−0.300913
$749$	−14.2861	−0.522003
$750$	0	0
$751$	5.27111	0.192345	0.0961727	−	0.995365i	$-0.469340\pi$
$751$	5.27111	0.192345	0.0961727	+	0.995365i	$0.469340\pi$
$752$	1.37190	0.0500279
$753$	0	0
$754$	−1.27744	−0.0465215
$755$	8.83379	0.321495
$756$	0	0
$757$	44.1949	1.60629	0.803145	−	0.595784i	$-0.203158\pi$
$757$	44.1949	1.60629	0.803145	+	0.595784i	$0.203158\pi$
$758$	0.765238	0.0277947
$759$	0	0
$760$	−0.165125	−0.00598970
$761$	−43.9305	−1.59248	−0.796240	−	0.604981i	$-0.793181\pi$
$761$	−43.9305	−1.59248	−0.796240	+	0.604981i	$0.793181\pi$
$762$	0	0
$763$	6.40981	0.232051
$764$	17.8993	0.647572
$765$	0	0
$766$	−0.426195	−0.0153990
$767$	9.58812	0.346207
$768$	0	0
$769$	7.00950	0.252769	0.126385	−	0.991981i	$-0.459663\pi$
$769$	7.00950	0.252769	0.126385	+	0.991981i	$0.459663\pi$
$770$	−0.384398	−0.0138527
$771$	0	0
$772$	−8.69490	−0.312936
$773$	−25.2979	−0.909902	−0.454951	−	0.890516i	$-0.650343\pi$
$773$	−25.2979	−0.909902	−0.454951	+	0.890516i	$0.650343\pi$
$774$	0	0
$775$	−5.74533	−0.206378
$776$	1.28553	0.0461477
$777$	0	0
$778$	−0.668666	−0.0239728
$779$	1.57444	0.0564100
$780$	0	0
$781$	34.4123	1.23137
$782$	0.0544856	0.00194840
$783$	0	0
$784$	3.99472	0.142669
$785$	2.30375	0.0822242
$786$	0	0
$787$	32.8781	1.17198	0.585989	−	0.810319i	$-0.300706\pi$
$787$	32.8781	1.17198	0.585989	+	0.810319i	$0.300706\pi$
$788$	24.2508	0.863898
$789$	0	0
$790$	0.824043	0.0293181
$791$	8.85904	0.314991
$792$	0	0
$793$	−45.5880	−1.61888
$794$	−0.0132187	−0.000469115 0
$795$	0	0
$796$	30.7737	1.09074
$797$	9.58152	0.339395	0.169697	−	0.985496i	$-0.445721\pi$
$797$	9.58152	0.339395	0.169697	+	0.985496i	$0.445721\pi$
$798$	0	0
$799$	0.335909	0.0118836
$800$	−1.59443	−0.0563718
$801$	0	0
$802$	0.183734	0.00648787
$803$	38.6191	1.36284
$804$	0	0
$805$	−5.78372	−0.203849
$806$	−0.244662	−0.00861784
$807$	0	0
$808$	−1.15855	−0.0407576
$809$	7.62861	0.268208	0.134104	−	0.990967i	$-0.457184\pi$
$809$	7.62861	0.268208	0.134104	+	0.990967i	$0.457184\pi$
$810$	0	0
$811$	53.0396	1.86247	0.931236	−	0.364416i	$-0.118731\pi$
$811$	53.0396	1.86247	0.931236	+	0.364416i	$0.118731\pi$
$812$	−13.3783	−0.469485
$813$	0	0
$814$	−0.762235	−0.0267163
$815$	−40.5239	−1.41949
$816$	0	0
$817$	1.86079	0.0651008
$818$	0.781518	0.0273251
$819$	0	0
$820$	21.4385	0.748666
$821$	−29.2651	−1.02136	−0.510679	−	0.859772i	$-0.670606\pi$
$821$	−29.2651	−1.02136	−0.510679	+	0.859772i	$0.670606\pi$
$822$	0	0
$823$	5.28174	0.184110	0.0920550	−	0.995754i	$-0.470656\pi$
$823$	5.28174	0.184110	0.0920550	+	0.995754i	$0.470656\pi$
$824$	0.745634	0.0259754
$825$	0	0
$826$	−0.0441833	−0.00153733
$827$	−23.0353	−0.801014	−0.400507	−	0.916294i	$-0.631166\pi$
$827$	−23.0353	−0.801014	−0.400507	+	0.916294i	$0.631166\pi$
$828$	0	0
$829$	−22.6811	−0.787748	−0.393874	−	0.919164i	$-0.628865\pi$
$829$	−22.6811	−0.787748	−0.393874	+	0.919164i	$0.628865\pi$
$830$	0.519048	0.0180164
$831$	0	0
$832$	51.3535	1.78036
$833$	0.978108	0.0338894
$834$	0	0
$835$	61.6750	2.13435
$836$	−3.80397	−0.131563
$837$	0	0
$838$	0.181009	0.00625285
$839$	39.4521	1.36204	0.681018	−	0.732267i	$-0.261538\pi$
$839$	39.4521	1.36204	0.681018	+	0.732267i	$0.261538\pi$
$840$	0	0
$841$	15.7838	0.544269
$842$	−0.300486	−0.0103554
$843$	0	0
$844$	−39.3813	−1.35556
$845$	−87.5290	−3.01109
$846$	0	0
$847$	−6.71461	−0.230717
$848$	−8.87017	−0.304603
$849$	0	0
$850$	−0.130037	−0.00446023
$851$	−11.4687	−0.393143
$852$	0	0
$853$	53.5165	1.83237	0.916186	−	0.400754i	$-0.131252\pi$
$853$	53.5165	1.83237	0.916186	+	0.400754i	$0.131252\pi$
$854$	0.210075	0.00718864
$855$	0	0
$856$	1.69446	0.0579153
$857$	−15.6792	−0.535593	−0.267796	−	0.963476i	$-0.586295\pi$
$857$	−15.6792	−0.535593	−0.267796	+	0.963476i	$0.586295\pi$
$858$	0	0
$859$	−31.0758	−1.06029	−0.530146	−	0.847907i	$-0.677863\pi$
$859$	−31.0758	−1.06029	−0.530146	+	0.847907i	$0.677863\pi$
$860$	25.3377	0.864009
$861$	0	0
$862$	−1.18240	−0.0402727
$863$	27.3878	0.932291	0.466145	−	0.884708i	$-0.345642\pi$
$863$	27.3878	0.932291	0.466145	+	0.884708i	$0.345642\pi$
$864$	0	0
$865$	45.2207	1.53755
$866$	−0.393825	−0.0133827
$867$	0	0
$868$	−2.56228	−0.0869693
$869$	37.9752	1.28822
$870$	0	0
$871$	−6.36790	−0.215768
$872$	−0.760259	−0.0257456
$873$	0	0
$874$	0.0251842	0.000851868 0
$875$	−1.59332	−0.0538642
$876$	0	0
$877$	−17.1942	−0.580608	−0.290304	−	0.956935i	$-0.593756\pi$
$877$	−17.1942	−0.580608	−0.290304	+	0.956935i	$0.593756\pi$
$878$	0.382270	0.0129010
$879$	0	0
$880$	−51.7745	−1.74532
$881$	56.6431	1.90835	0.954177	−	0.299241i	$-0.0967336\pi$
$881$	56.6431	1.90835	0.954177	+	0.299241i	$0.0967336\pi$
$882$	0	0
$883$	9.71797	0.327036	0.163518	−	0.986540i	$-0.447716\pi$
$883$	9.71797	0.327036	0.163518	+	0.986540i	$0.447716\pi$
$884$	12.5850	0.423279
$885$	0	0
$886$	−0.752289	−0.0252737
$887$	37.9696	1.27489	0.637447	−	0.770494i	$-0.279990\pi$
$887$	37.9696	1.27489	0.637447	+	0.770494i	$0.279990\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−0.184199	−0.00617436
$891$	0	0
$892$	20.3438	0.681162
$893$	0.155263	0.00519567
$894$	0	0
$895$	4.39058	0.146761
$896$	−0.948035	−0.0316716
$897$	0	0
$898$	0.689918	0.0230229
$899$	8.57724	0.286067
$900$	0	0
$901$	−2.17186	−0.0723552
$902$	−0.434721	−0.0144746
$903$	0	0
$904$	−1.05076	−0.0349477
$905$	49.5158	1.64596
$906$	0	0
$907$	−40.9254	−1.35891	−0.679453	−	0.733719i	$-0.737783\pi$
$907$	−40.9254	−1.35891	−0.679453	+	0.733719i	$0.737783\pi$
$908$	−7.98492	−0.264989
$909$	0	0
$910$	0.587817	0.0194860
$911$	−24.4739	−0.810858	−0.405429	−	0.914127i	$-0.632878\pi$
$911$	−24.4739	−0.810858	−0.405429	+	0.914127i	$0.632878\pi$
$912$	0	0
$913$	23.9198	0.791631
$914$	1.13954	0.0376927
$915$	0	0
$916$	21.7819	0.719693
$917$	4.30603	0.142198
$918$	0	0
$919$	−18.2904	−0.603346	−0.301673	−	0.953411i	$-0.597545\pi$
$919$	−18.2904	−0.603346	−0.301673	+	0.953411i	$0.597545\pi$
$920$	0.685999	0.0226167
$921$	0	0
$922$	0.439558	0.0144761
$923$	−52.6229	−1.73210
$924$	0	0
$925$	27.3716	0.899972
$926$	−1.04421	−0.0343147
$927$	0	0
$928$	2.38034	0.0781385
$929$	−3.21214	−0.105387	−0.0526935	−	0.998611i	$-0.516781\pi$
$929$	−3.21214	−0.105387	−0.0526935	+	0.998611i	$0.516781\pi$
$930$	0	0
$931$	0.452098	0.0148169
$932$	50.6596	1.65941
$933$	0	0
$934$	0.688580	0.0225310
$935$	−12.6770	−0.414582
$936$	0	0
$937$	0.847525	0.0276874	0.0138437	−	0.999904i	$-0.495593\pi$
$937$	0.847525	0.0276874	0.0138437	+	0.999904i	$0.495593\pi$
$938$	0.0293441	0.000958119 0
$939$	0	0
$940$	2.11416	0.0689562
$941$	−20.7245	−0.675601	−0.337800	−	0.941218i	$-0.609683\pi$
$941$	−20.7245	−0.675601	−0.337800	+	0.941218i	$0.609683\pi$
$942$	0	0
$943$	−6.54088	−0.213000
$944$	−5.95104	−0.193690
$945$	0	0
$946$	−0.513787	−0.0167046
$947$	−53.5475	−1.74006	−0.870031	−	0.492998i	$-0.835901\pi$
$947$	−53.5475	−1.74006	−0.870031	+	0.492998i	$0.835901\pi$
$948$	0	0
$949$	−59.0560	−1.91704
$950$	−0.0601053	−0.00195007
$951$	0	0
$952$	−0.116012	−0.00375997
$953$	6.51677	0.211099	0.105549	−	0.994414i	$-0.466340\pi$
$953$	6.51677	0.211099	0.105549	+	0.994414i	$0.466340\pi$
$954$	0	0
$955$	27.5714	0.892191
$956$	5.11041	0.165283
$957$	0	0
$958$	−0.853738	−0.0275830
$959$	3.16745	0.102282
$960$	0	0
$961$	−29.3572	−0.947008
$962$	1.16560	0.0375805
$963$	0	0
$964$	−29.1125	−0.937652
$965$	−13.3933	−0.431147
$966$	0	0
$967$	55.2012	1.77515	0.887575	−	0.460662i	$-0.152388\pi$
$967$	55.2012	1.77515	0.887575	+	0.460662i	$0.152388\pi$
$968$	0.796411	0.0255976
$969$	0	0
$970$	0.989873	0.0317829
$971$	37.1275	1.19148	0.595740	−	0.803178i	$-0.296859\pi$
$971$	37.1275	1.19148	0.595740	+	0.803178i	$0.296859\pi$
$972$	0	0
$973$	−7.21212	−0.231210
$974$	−1.20506	−0.0386127
$975$	0	0
$976$	28.2950	0.905702
$977$	61.2366	1.95913	0.979566	−	0.201124i	$-0.0644595\pi$
$977$	61.2366	1.95913	0.979566	+	0.201124i	$0.0644595\pi$
$978$	0	0
$979$	−8.48862	−0.271298
$980$	6.15605	0.196648
$981$	0	0
$982$	0.206084	0.00657640
$983$	48.0106	1.53130	0.765651	−	0.643257i	$-0.222417\pi$
$983$	48.0106	1.53130	0.765651	+	0.643257i	$0.222417\pi$
$984$	0	0
$985$	37.3551	1.19023
$986$	0.194133	0.00618245
$987$	0	0
$988$	5.81699	0.185063
$989$	−7.73052	−0.245816
$990$	0	0
$991$	44.1047	1.40103	0.700517	−	0.713636i	$-0.252953\pi$
$991$	44.1047	1.40103	0.700517	+	0.713636i	$0.252953\pi$
$992$	0.455896	0.0144747
$993$	0	0
$994$	0.242493	0.00769142
$995$	47.4027	1.50277
$996$	0	0
$997$	−46.7633	−1.48101	−0.740504	−	0.672052i	$-0.765413\pi$
$997$	−46.7633	−1.48101	−0.740504	+	0.672052i	$0.765413\pi$
$998$	−0.150984	−0.00477932
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.10		16
3.2	odd	2		2667.2.a.o.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.7	✓	16	3.2	odd	2
8001.2.a.r.1.10		16	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)

\(f(q)\)	\(=\)	\(q-0.0296587 q^{2} -1.99912 q^{4} -3.07938 q^{5} -1.00000 q^{7} +0.118609 q^{8} +O(q^{10})\)	\(q-0.0296587 q^{2} -1.99912 q^{4} -3.07938 q^{5} -1.00000 q^{7} +0.118609 q^{8} +0.0913304 q^{10} +4.20887 q^{11} -6.43617 q^{13} +0.0296587 q^{14} +3.99472 q^{16} +0.978108 q^{17} +0.452098 q^{19} +6.15605 q^{20} -0.124830 q^{22} -1.87821 q^{23} +4.48258 q^{25} +0.190888 q^{26} +1.99912 q^{28} -6.69207 q^{29} -1.28170 q^{31} -0.355696 q^{32} -0.0290094 q^{34} +3.07938 q^{35} +6.10620 q^{37} -0.0134086 q^{38} -0.365241 q^{40} +3.48251 q^{41} +4.11590 q^{43} -8.41404 q^{44} +0.0557052 q^{46} +0.343427 q^{47} +1.00000 q^{49} -0.132948 q^{50} +12.8667 q^{52} -2.22047 q^{53} -12.9607 q^{55} -0.118609 q^{56} +0.198478 q^{58} -1.48973 q^{59} +7.08310 q^{61} +0.0380136 q^{62} -7.97890 q^{64} +19.8194 q^{65} +0.989393 q^{67} -1.95535 q^{68} -0.0913304 q^{70} +8.17613 q^{71} +9.17564 q^{73} -0.181102 q^{74} -0.903798 q^{76} -4.20887 q^{77} +9.02266 q^{79} -12.3013 q^{80} -0.103287 q^{82} +5.68319 q^{83} -3.01197 q^{85} -0.122072 q^{86} +0.499209 q^{88} -2.01684 q^{89} +6.43617 q^{91} +3.75476 q^{92} -0.0101856 q^{94} -1.39218 q^{95} +10.8384 q^{97} -0.0296587 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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