LMFDB - Embedded newform 8001.2.a.v.1.18

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:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$-2.35941$ 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)

$f(q)$	$=$	$q+2.35941 q^{2} +3.56679 q^{4} +3.48367 q^{5} +1.00000 q^{7} +3.69670 q^{8} +O(q^{10})$	\(q+2.35941 q^{2} +3.56679 q^{4} +3.48367 q^{5} +1.00000 q^{7} +3.69670 q^{8} +8.21939 q^{10} +3.61589 q^{11} -1.92931 q^{13} +2.35941 q^{14} +1.58843 q^{16} +6.15261 q^{17} -4.23441 q^{19} +12.4255 q^{20} +8.53136 q^{22} +7.99661 q^{23} +7.13595 q^{25} -4.55202 q^{26} +3.56679 q^{28} -2.42362 q^{29} -1.78229 q^{31} -3.64565 q^{32} +14.5165 q^{34} +3.48367 q^{35} +5.76457 q^{37} -9.99070 q^{38} +12.8781 q^{40} -6.30711 q^{41} -2.65444 q^{43} +12.8971 q^{44} +18.8672 q^{46} -3.09597 q^{47} +1.00000 q^{49} +16.8366 q^{50} -6.88145 q^{52} -10.6256 q^{53} +12.5966 q^{55} +3.69670 q^{56} -5.71830 q^{58} -1.12226 q^{59} -7.38616 q^{61} -4.20514 q^{62} -11.7784 q^{64} -6.72107 q^{65} -10.9599 q^{67} +21.9451 q^{68} +8.21939 q^{70} +11.9564 q^{71} +12.4914 q^{73} +13.6009 q^{74} -15.1033 q^{76} +3.61589 q^{77} -8.46939 q^{79} +5.53356 q^{80} -14.8810 q^{82} -8.32052 q^{83} +21.4336 q^{85} -6.26291 q^{86} +13.3669 q^{88} -5.58500 q^{89} -1.92931 q^{91} +28.5223 q^{92} -7.30465 q^{94} -14.7513 q^{95} +3.96775 q^{97} +2.35941 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	2.35941	1.66835	0.834176	−	0.551499i	$-0.185944\pi$
$2$	2.35941	1.66835	0.834176	+	0.551499i	$0.185944\pi$
$3$	0	0
$3$	0	0
$4$	3.56679	1.78340
$5$	3.48367	1.55794	0.778972	−	0.627059i	$-0.215741\pi$
$5$	3.48367	1.55794	0.778972	+	0.627059i	$0.215741\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.69670	1.30698
$9$	0	0
$10$	8.21939	2.59920
$11$	3.61589	1.09023	0.545117	−	0.838360i	$-0.316485\pi$
$11$	3.61589	1.09023	0.545117	+	0.838360i	$0.316485\pi$
$12$	0	0
$13$	−1.92931	−0.535094	−0.267547	−	0.963545i	$-0.586213\pi$
$13$	−1.92931	−0.535094	−0.267547	+	0.963545i	$0.586213\pi$
$14$	2.35941	0.630578
$15$	0	0
$16$	1.58843	0.397107
$17$	6.15261	1.49223	0.746113	−	0.665819i	$-0.231918\pi$
$17$	6.15261	1.49223	0.746113	+	0.665819i	$0.231918\pi$
$18$	0	0
$19$	−4.23441	−0.971441	−0.485721	−	0.874114i	$-0.661443\pi$
$19$	−4.23441	−0.971441	−0.485721	+	0.874114i	$0.661443\pi$
$20$	12.4255	2.77843
$21$	0	0
$22$	8.53136	1.81889
$23$	7.99661	1.66741	0.833704	−	0.552211i	$-0.186216\pi$
$23$	7.99661	1.66741	0.833704	+	0.552211i	$0.186216\pi$
$24$	0	0
$25$	7.13595	1.42719
$26$	−4.55202	−0.892725
$27$	0	0
$28$	3.56679	0.674061
$29$	−2.42362	−0.450055	−0.225027	−	0.974352i	$-0.572247\pi$
$29$	−2.42362	−0.450055	−0.225027	+	0.974352i	$0.572247\pi$
$30$	0	0
$31$	−1.78229	−0.320109	−0.160054	−	0.987108i	$-0.551167\pi$
$31$	−1.78229	−0.320109	−0.160054	+	0.987108i	$0.551167\pi$
$32$	−3.64565	−0.644467
$33$	0	0
$34$	14.5165	2.48956
$35$	3.48367	0.588847
$36$	0	0
$37$	5.76457	0.947689	0.473844	−	0.880609i	$-0.342866\pi$
$37$	5.76457	0.947689	0.473844	+	0.880609i	$0.342866\pi$
$38$	−9.99070	−1.62071
$39$	0	0
$40$	12.8781	2.03620
$41$	−6.30711	−0.985004	−0.492502	−	0.870311i	$-0.663918\pi$
$41$	−6.30711	−0.985004	−0.492502	+	0.870311i	$0.663918\pi$
$42$	0	0
$43$	−2.65444	−0.404799	−0.202399	−	0.979303i	$-0.564874\pi$
$43$	−2.65444	−0.404799	−0.202399	+	0.979303i	$0.564874\pi$
$44$	12.8971	1.94432
$45$	0	0
$46$	18.8672	2.78182
$47$	−3.09597	−0.451594	−0.225797	−	0.974174i	$-0.572499\pi$
$47$	−3.09597	−0.451594	−0.225797	+	0.974174i	$0.572499\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	16.8366	2.38105
$51$	0	0
$52$	−6.88145	−0.954285
$53$	−10.6256	−1.45953	−0.729767	−	0.683696i	$-0.760371\pi$
$53$	−10.6256	−1.45953	−0.729767	+	0.683696i	$0.760371\pi$
$54$	0	0
$55$	12.5966	1.69852
$56$	3.69670	0.493992
$57$	0	0
$58$	−5.71830	−0.750850
$59$	−1.12226	−0.146106	−0.0730529	−	0.997328i	$-0.523274\pi$
$59$	−1.12226	−0.146106	−0.0730529	+	0.997328i	$0.523274\pi$
$60$	0	0
$61$	−7.38616	−0.945701	−0.472851	−	0.881143i	$-0.656775\pi$
$61$	−7.38616	−0.945701	−0.472851	+	0.881143i	$0.656775\pi$
$62$	−4.20514	−0.534054
$63$	0	0
$64$	−11.7784	−1.47230
$65$	−6.72107	−0.833646
$66$	0	0
$67$	−10.9599	−1.33897	−0.669485	−	0.742826i	$-0.733485\pi$
$67$	−10.9599	−1.33897	−0.669485	+	0.742826i	$0.733485\pi$
$68$	21.9451	2.66123
$69$	0	0
$70$	8.21939	0.982405
$71$	11.9564	1.41897	0.709483	−	0.704722i	$-0.248928\pi$
$71$	11.9564	1.41897	0.709483	+	0.704722i	$0.248928\pi$
$72$	0	0
$73$	12.4914	1.46201	0.731006	−	0.682371i	$-0.239051\pi$
$73$	12.4914	1.46201	0.731006	+	0.682371i	$0.239051\pi$
$74$	13.6009	1.58108
$75$	0	0
$76$	−15.1033	−1.73247
$77$	3.61589	0.412069
$78$	0	0
$79$	−8.46939	−0.952881	−0.476440	−	0.879207i	$-0.658073\pi$
$79$	−8.46939	−0.952881	−0.476440	+	0.879207i	$0.658073\pi$
$80$	5.53356	0.618671
$81$	0	0
$82$	−14.8810	−1.64333
$83$	−8.32052	−0.913296	−0.456648	−	0.889647i	$-0.650950\pi$
$83$	−8.32052	−0.913296	−0.456648	+	0.889647i	$0.650950\pi$
$84$	0	0
$85$	21.4336	2.32480
$86$	−6.26291	−0.675347
$87$	0	0
$88$	13.3669	1.42491
$89$	−5.58500	−0.592009	−0.296005	−	0.955187i	$-0.595654\pi$
$89$	−5.58500	−0.592009	−0.296005	+	0.955187i	$0.595654\pi$
$90$	0	0
$91$	−1.92931	−0.202246
$92$	28.5223	2.97365
$93$	0	0
$94$	−7.30465	−0.753417
$95$	−14.7513	−1.51345
$96$	0	0
$97$	3.96775	0.402864	0.201432	−	0.979502i	$-0.435440\pi$
$97$	3.96775	0.402864	0.201432	+	0.979502i	$0.435440\pi$
$98$	2.35941	0.238336
$99$	0	0
$100$	25.4525	2.54525
$101$	−9.08165	−0.903658	−0.451829	−	0.892105i	$-0.649228\pi$
$101$	−9.08165	−0.903658	−0.451829	+	0.892105i	$0.649228\pi$
$102$	0	0
$103$	4.21785	0.415597	0.207799	−	0.978172i	$-0.433370\pi$
$103$	4.21785	0.415597	0.207799	+	0.978172i	$0.433370\pi$
$104$	−7.13208	−0.699358
$105$	0	0
$106$	−25.0700	−2.43501
$107$	17.9401	1.73433	0.867166	−	0.498019i	$-0.165939\pi$
$107$	17.9401	1.73433	0.867166	+	0.498019i	$0.165939\pi$
$108$	0	0
$109$	5.45243	0.522248	0.261124	−	0.965305i	$-0.415907\pi$
$109$	5.45243	0.522248	0.261124	+	0.965305i	$0.415907\pi$
$110$	29.7204	2.83373
$111$	0	0
$112$	1.58843	0.150092
$113$	−0.459752	−0.0432498	−0.0216249	−	0.999766i	$-0.506884\pi$
$113$	−0.459752	−0.0432498	−0.0216249	+	0.999766i	$0.506884\pi$
$114$	0	0
$115$	27.8575	2.59773
$116$	−8.64455	−0.802626
$117$	0	0
$118$	−2.64787	−0.243756
$119$	6.15261	0.564009
$120$	0	0
$121$	2.07469	0.188608
$122$	−17.4270	−1.57776
$123$	0	0
$124$	−6.35706	−0.570881
$125$	7.44093	0.665537
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−20.4988	−1.81185
$129$	0	0
$130$	−15.8577	−1.39082
$131$	−2.56902	−0.224457	−0.112228	−	0.993682i	$-0.535799\pi$
$131$	−2.56902	−0.224457	−0.112228	+	0.993682i	$0.535799\pi$
$132$	0	0
$133$	−4.23441	−0.367170
$134$	−25.8589	−2.23387
$135$	0	0
$136$	22.7443	1.95031
$137$	3.98364	0.340345	0.170172	−	0.985414i	$-0.445568\pi$
$137$	3.98364	0.340345	0.170172	+	0.985414i	$0.445568\pi$
$138$	0	0
$139$	10.9220	0.926390	0.463195	−	0.886256i	$-0.346703\pi$
$139$	10.9220	0.926390	0.463195	+	0.886256i	$0.346703\pi$
$140$	12.4255	1.05015
$141$	0	0
$142$	28.2101	2.36734
$143$	−6.97618	−0.583377
$144$	0	0
$145$	−8.44309	−0.701160
$146$	29.4724	2.43915
$147$	0	0
$148$	20.5610	1.69011
$149$	1.64936	0.135121	0.0675605	−	0.997715i	$-0.478478\pi$
$149$	1.64936	0.135121	0.0675605	+	0.997715i	$0.478478\pi$
$150$	0	0
$151$	9.57181	0.778943	0.389472	−	0.921038i	$-0.372658\pi$
$151$	9.57181	0.778943	0.389472	+	0.921038i	$0.372658\pi$
$152$	−15.6534	−1.26966
$153$	0	0
$154$	8.53136	0.687477
$155$	−6.20891	−0.498711
$156$	0	0
$157$	5.24867	0.418889	0.209445	−	0.977820i	$-0.432834\pi$
$157$	5.24867	0.418889	0.209445	+	0.977820i	$0.432834\pi$
$158$	−19.9827	−1.58974
$159$	0	0
$160$	−12.7003	−1.00404
$161$	7.99661	0.630221
$162$	0	0
$163$	−2.06308	−0.161593	−0.0807963	−	0.996731i	$-0.525746\pi$
$163$	−2.06308	−0.161593	−0.0807963	+	0.996731i	$0.525746\pi$
$164$	−22.4961	−1.75665
$165$	0	0
$166$	−19.6315	−1.52370
$167$	19.3404	1.49660	0.748302	−	0.663358i	$-0.230870\pi$
$167$	19.3404	1.49660	0.748302	+	0.663358i	$0.230870\pi$
$168$	0	0
$169$	−9.27777	−0.713674
$170$	50.5707	3.87859
$171$	0	0
$172$	−9.46785	−0.721917
$173$	−12.9591	−0.985265	−0.492632	−	0.870238i	$-0.663965\pi$
$173$	−12.9591	−0.985265	−0.492632	+	0.870238i	$0.663965\pi$
$174$	0	0
$175$	7.13595	0.539427
$176$	5.74359	0.432940
$177$	0	0
$178$	−13.1773	−0.987679
$179$	−26.4041	−1.97354	−0.986768	−	0.162141i	$-0.948160\pi$
$179$	−26.4041	−1.97354	−0.986768	+	0.162141i	$0.948160\pi$
$180$	0	0
$181$	3.13555	0.233063	0.116532	−	0.993187i	$-0.462822\pi$
$181$	3.13555	0.233063	0.116532	+	0.993187i	$0.462822\pi$
$182$	−4.55202	−0.337418
$183$	0	0
$184$	29.5611	2.17927
$185$	20.0818	1.47645
$186$	0	0
$187$	22.2472	1.62687
$188$	−11.0427	−0.805371
$189$	0	0
$190$	−34.8043	−2.52497
$191$	6.70700	0.485301	0.242651	−	0.970114i	$-0.421983\pi$
$191$	6.70700	0.485301	0.242651	+	0.970114i	$0.421983\pi$
$192$	0	0
$193$	10.0351	0.722345	0.361172	−	0.932499i	$-0.382377\pi$
$193$	10.0351	0.722345	0.361172	+	0.932499i	$0.382377\pi$
$194$	9.36154	0.672119
$195$	0	0
$196$	3.56679	0.254771
$197$	−7.25191	−0.516677	−0.258339	−	0.966054i	$-0.583175\pi$
$197$	−7.25191	−0.516677	−0.258339	+	0.966054i	$0.583175\pi$
$198$	0	0
$199$	−27.0499	−1.91751	−0.958757	−	0.284226i	$-0.908264\pi$
$199$	−27.0499	−1.91751	−0.958757	+	0.284226i	$0.908264\pi$
$200$	26.3795	1.86531
$201$	0	0
$202$	−21.4273	−1.50762
$203$	−2.42362	−0.170105
$204$	0	0
$205$	−21.9719	−1.53458
$206$	9.95162	0.693362
$207$	0	0
$208$	−3.06457	−0.212490
$209$	−15.3112	−1.05910
$210$	0	0
$211$	−15.0562	−1.03651	−0.518255	−	0.855226i	$-0.673418\pi$
$211$	−15.0562	−1.03651	−0.518255	+	0.855226i	$0.673418\pi$
$212$	−37.8992	−2.60293
$213$	0	0
$214$	42.3279	2.89348
$215$	−9.24720	−0.630654
$216$	0	0
$217$	−1.78229	−0.120990
$218$	12.8645	0.871293
$219$	0	0
$220$	44.9294	3.02914
$221$	−11.8703	−0.798481
$222$	0	0
$223$	10.0303	0.671676	0.335838	−	0.941920i	$-0.390981\pi$
$223$	10.0303	0.671676	0.335838	+	0.941920i	$0.390981\pi$
$224$	−3.64565	−0.243586
$225$	0	0
$226$	−1.08474	−0.0721559
$227$	−20.8364	−1.38296	−0.691479	−	0.722396i	$-0.743041\pi$
$227$	−20.8364	−1.38296	−0.691479	+	0.722396i	$0.743041\pi$
$228$	0	0
$229$	2.24221	0.148170	0.0740848	−	0.997252i	$-0.476396\pi$
$229$	2.24221	0.148170	0.0740848	+	0.997252i	$0.476396\pi$
$230$	65.7272	4.33393
$231$	0	0
$232$	−8.95940	−0.588213
$233$	20.5397	1.34560	0.672801	−	0.739823i	$-0.265091\pi$
$233$	20.5397	1.34560	0.672801	+	0.739823i	$0.265091\pi$
$234$	0	0
$235$	−10.7853	−0.703558
$236$	−4.00287	−0.260565
$237$	0	0
$238$	14.5165	0.940964
$239$	16.5949	1.07343	0.536716	−	0.843763i	$-0.319665\pi$
$239$	16.5949	1.07343	0.536716	+	0.843763i	$0.319665\pi$
$240$	0	0
$241$	5.78154	0.372422	0.186211	−	0.982510i	$-0.440379\pi$
$241$	5.78154	0.372422	0.186211	+	0.982510i	$0.440379\pi$
$242$	4.89504	0.314665
$243$	0	0
$244$	−26.3449	−1.68656
$245$	3.48367	0.222563
$246$	0	0
$247$	8.16949	0.519812
$248$	−6.58859	−0.418376
$249$	0	0
$250$	17.5562	1.11035
$251$	16.4271	1.03687	0.518436	−	0.855117i	$-0.326515\pi$
$251$	16.4271	1.03687	0.518436	+	0.855117i	$0.326515\pi$
$252$	0	0
$253$	28.9149	1.81786
$254$	2.35941	0.148042
$255$	0	0
$256$	−24.8081	−1.55051
$257$	−1.81936	−0.113489	−0.0567443	−	0.998389i	$-0.518072\pi$
$257$	−1.81936	−0.113489	−0.0567443	+	0.998389i	$0.518072\pi$
$258$	0	0
$259$	5.76457	0.358193
$260$	−23.9727	−1.48672
$261$	0	0
$262$	−6.06137	−0.374473
$263$	−10.7935	−0.665553	−0.332777	−	0.943006i	$-0.607986\pi$
$263$	−10.7935	−0.665553	−0.332777	+	0.943006i	$0.607986\pi$
$264$	0	0
$265$	−37.0159	−2.27387
$266$	−9.99070	−0.612569
$267$	0	0
$268$	−39.0918	−2.38791
$269$	−6.02788	−0.367526	−0.183763	−	0.982971i	$-0.558828\pi$
$269$	−6.02788	−0.367526	−0.183763	+	0.982971i	$0.558828\pi$
$270$	0	0
$271$	28.7213	1.74469	0.872347	−	0.488888i	$-0.162597\pi$
$271$	28.7213	1.74469	0.872347	+	0.488888i	$0.162597\pi$
$272$	9.77298	0.592574
$273$	0	0
$274$	9.39901	0.567815
$275$	25.8028	1.55597
$276$	0	0
$277$	15.9881	0.960632	0.480316	−	0.877095i	$-0.340522\pi$
$277$	15.9881	0.960632	0.480316	+	0.877095i	$0.340522\pi$
$278$	25.7694	1.54554
$279$	0	0
$280$	12.8781	0.769613
$281$	−16.7489	−0.999157	−0.499579	−	0.866269i	$-0.666512\pi$
$281$	−16.7489	−0.999157	−0.499579	+	0.866269i	$0.666512\pi$
$282$	0	0
$283$	−15.8469	−0.941998	−0.470999	−	0.882134i	$-0.656107\pi$
$283$	−15.8469	−0.941998	−0.470999	+	0.882134i	$0.656107\pi$
$284$	42.6461	2.53058
$285$	0	0
$286$	−16.4596	−0.973278
$287$	−6.30711	−0.372297
$288$	0	0
$289$	20.8546	1.22674
$290$	−19.9207	−1.16978
$291$	0	0
$292$	44.5544	2.60735
$293$	−25.3332	−1.47998	−0.739991	−	0.672616i	$-0.765170\pi$
$293$	−25.3332	−1.47998	−0.739991	+	0.672616i	$0.765170\pi$
$294$	0	0
$295$	−3.90958	−0.227625
$296$	21.3099	1.23861
$297$	0	0
$298$	3.89151	0.225429
$299$	−15.4279	−0.892220
$300$	0	0
$301$	−2.65444	−0.153000
$302$	22.5838	1.29955
$303$	0	0
$304$	−6.72607	−0.385766
$305$	−25.7309	−1.47335
$306$	0	0
$307$	26.3469	1.50370	0.751848	−	0.659337i	$-0.229163\pi$
$307$	26.3469	1.50370	0.751848	+	0.659337i	$0.229163\pi$
$308$	12.8971	0.734883
$309$	0	0
$310$	−14.6493	−0.832026
$311$	−28.9919	−1.64398	−0.821990	−	0.569502i	$-0.807136\pi$
$311$	−28.9919	−1.64398	−0.821990	+	0.569502i	$0.807136\pi$
$312$	0	0
$313$	17.9259	1.01323	0.506615	−	0.862172i	$-0.330897\pi$
$313$	17.9259	1.01323	0.506615	+	0.862172i	$0.330897\pi$
$314$	12.3837	0.698855
$315$	0	0
$316$	−30.2086	−1.69936
$317$	−5.92298	−0.332667	−0.166334	−	0.986070i	$-0.553193\pi$
$317$	−5.92298	−0.332667	−0.166334	+	0.986070i	$0.553193\pi$
$318$	0	0
$319$	−8.76355	−0.490665
$320$	−41.0322	−2.29377
$321$	0	0
$322$	18.8672	1.05143
$323$	−26.0527	−1.44961
$324$	0	0
$325$	−13.7674	−0.763680
$326$	−4.86764	−0.269593
$327$	0	0
$328$	−23.3155	−1.28738
$329$	−3.09597	−0.170686
$330$	0	0
$331$	20.8160	1.14415	0.572076	−	0.820201i	$-0.306138\pi$
$331$	20.8160	1.14415	0.572076	+	0.820201i	$0.306138\pi$
$332$	−29.6776	−1.62877
$333$	0	0
$334$	45.6318	2.49686
$335$	−38.1808	−2.08604
$336$	0	0
$337$	−17.2188	−0.937965	−0.468983	−	0.883207i	$-0.655379\pi$
$337$	−17.2188	−0.937965	−0.468983	+	0.883207i	$0.655379\pi$
$338$	−21.8900	−1.19066
$339$	0	0
$340$	76.4494	4.14605
$341$	−6.44457	−0.348993
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.81268	−0.529064
$345$	0	0
$346$	−30.5758	−1.64377
$347$	22.0536	1.18390	0.591949	−	0.805976i	$-0.298359\pi$
$347$	22.0536	1.18390	0.591949	+	0.805976i	$0.298359\pi$
$348$	0	0
$349$	−18.5505	−0.992984	−0.496492	−	0.868041i	$-0.665379\pi$
$349$	−18.5505	−0.992984	−0.496492	+	0.868041i	$0.665379\pi$
$350$	16.8366	0.899954
$351$	0	0
$352$	−13.1823	−0.702619
$353$	−13.0355	−0.693810	−0.346905	−	0.937900i	$-0.612767\pi$
$353$	−13.0355	−0.693810	−0.346905	+	0.937900i	$0.612767\pi$
$354$	0	0
$355$	41.6522	2.21067
$356$	−19.9205	−1.05579
$357$	0	0
$358$	−62.2980	−3.29255
$359$	−8.41399	−0.444073	−0.222037	−	0.975038i	$-0.571270\pi$
$359$	−8.41399	−0.444073	−0.222037	+	0.975038i	$0.571270\pi$
$360$	0	0
$361$	−1.06974	−0.0563021
$362$	7.39802	0.388831
$363$	0	0
$364$	−6.88145	−0.360686
$365$	43.5160	2.27773
$366$	0	0
$367$	−33.5002	−1.74870	−0.874349	−	0.485298i	$-0.838711\pi$
$367$	−33.5002	−1.74870	−0.874349	+	0.485298i	$0.838711\pi$
$368$	12.7021	0.662140
$369$	0	0
$370$	47.3812	2.46323
$371$	−10.6256	−0.551652
$372$	0	0
$373$	−10.5078	−0.544074	−0.272037	−	0.962287i	$-0.587697\pi$
$373$	−10.5078	−0.544074	−0.272037	+	0.962287i	$0.587697\pi$
$374$	52.4901	2.71420
$375$	0	0
$376$	−11.4449	−0.590225
$377$	4.67591	0.240822
$378$	0	0
$379$	−3.67184	−0.188610	−0.0943049	−	0.995543i	$-0.530063\pi$
$379$	−3.67184	−0.188610	−0.0943049	+	0.995543i	$0.530063\pi$
$380$	−52.6148	−2.69908
$381$	0	0
$382$	15.8245	0.809653
$383$	5.29752	0.270690	0.135345	−	0.990799i	$-0.456786\pi$
$383$	5.29752	0.270690	0.135345	+	0.990799i	$0.456786\pi$
$384$	0	0
$385$	12.5966	0.641981
$386$	23.6770	1.20513
$387$	0	0
$388$	14.1522	0.718467
$389$	9.52423	0.482898	0.241449	−	0.970414i	$-0.422377\pi$
$389$	9.52423	0.482898	0.241449	+	0.970414i	$0.422377\pi$
$390$	0	0
$391$	49.2000	2.48815
$392$	3.69670	0.186712
$393$	0	0
$394$	−17.1102	−0.861999
$395$	−29.5045	−1.48453
$396$	0	0
$397$	22.0243	1.10537	0.552683	−	0.833391i	$-0.313604\pi$
$397$	22.0243	1.10537	0.552683	+	0.833391i	$0.313604\pi$
$398$	−63.8216	−3.19909
$399$	0	0
$400$	11.3349	0.566747
$401$	31.7400	1.58502	0.792511	−	0.609857i	$-0.208773\pi$
$401$	31.7400	1.58502	0.792511	+	0.609857i	$0.208773\pi$
$402$	0	0
$403$	3.43859	0.171288
$404$	−32.3924	−1.61158
$405$	0	0
$406$	−5.71830	−0.283795
$407$	20.8441	1.03320
$408$	0	0
$409$	−22.5963	−1.11731	−0.558657	−	0.829399i	$-0.688683\pi$
$409$	−22.5963	−1.11731	−0.558657	+	0.829399i	$0.688683\pi$
$410$	−51.8405	−2.56022
$411$	0	0
$412$	15.0442	0.741175
$413$	−1.12226	−0.0552228
$414$	0	0
$415$	−28.9860	−1.42286
$416$	7.03359	0.344850
$417$	0	0
$418$	−36.1253	−1.76695
$419$	−35.0178	−1.71073	−0.855365	−	0.518025i	$-0.826667\pi$
$419$	−35.0178	−1.71073	−0.855365	+	0.518025i	$0.826667\pi$
$420$	0	0
$421$	−18.7984	−0.916177	−0.458089	−	0.888906i	$-0.651466\pi$
$421$	−18.7984	−0.916177	−0.458089	+	0.888906i	$0.651466\pi$
$422$	−35.5236	−1.72926
$423$	0	0
$424$	−39.2795	−1.90758
$425$	43.9047	2.12969
$426$	0	0
$427$	−7.38616	−0.357441
$428$	63.9885	3.09300
$429$	0	0
$430$	−21.8179	−1.05215
$431$	15.9195	0.766815	0.383408	−	0.923579i	$-0.374750\pi$
$431$	15.9195	0.766815	0.383408	+	0.923579i	$0.374750\pi$
$432$	0	0
$433$	−18.1858	−0.873952	−0.436976	−	0.899473i	$-0.643951\pi$
$433$	−18.1858	−0.873952	−0.436976	+	0.899473i	$0.643951\pi$
$434$	−4.20514	−0.201853
$435$	0	0
$436$	19.4477	0.931375
$437$	−33.8610	−1.61979
$438$	0	0
$439$	17.6641	0.843063	0.421531	−	0.906814i	$-0.361493\pi$
$439$	17.6641	0.843063	0.421531	+	0.906814i	$0.361493\pi$
$440$	46.5658	2.21994
$441$	0	0
$442$	−28.0068	−1.33215
$443$	12.2715	0.583036	0.291518	−	0.956565i	$-0.405840\pi$
$443$	12.2715	0.583036	0.291518	+	0.956565i	$0.405840\pi$
$444$	0	0
$445$	−19.4563	−0.922317
$446$	23.6654	1.12059
$447$	0	0
$448$	−11.7784	−0.556479
$449$	−5.56581	−0.262667	−0.131333	−	0.991338i	$-0.541926\pi$
$449$	−5.56581	−0.262667	−0.131333	+	0.991338i	$0.541926\pi$
$450$	0	0
$451$	−22.8058	−1.07388
$452$	−1.63984	−0.0771316
$453$	0	0
$454$	−49.1615	−2.30726
$455$	−6.72107	−0.315089
$456$	0	0
$457$	22.9214	1.07222	0.536108	−	0.844149i	$-0.319894\pi$
$457$	22.9214	1.07222	0.536108	+	0.844149i	$0.319894\pi$
$458$	5.29029	0.247199
$459$	0	0
$460$	99.3621	4.63278
$461$	−20.0267	−0.932736	−0.466368	−	0.884591i	$-0.654438\pi$
$461$	−20.0267	−0.932736	−0.466368	+	0.884591i	$0.654438\pi$
$462$	0	0
$463$	35.6784	1.65812	0.829058	−	0.559163i	$-0.188877\pi$
$463$	35.6784	1.65812	0.829058	+	0.559163i	$0.188877\pi$
$464$	−3.84975	−0.178720
$465$	0	0
$466$	48.4616	2.24494
$467$	−3.39355	−0.157035	−0.0785173	−	0.996913i	$-0.525019\pi$
$467$	−3.39355	−0.157035	−0.0785173	+	0.996913i	$0.525019\pi$
$468$	0	0
$469$	−10.9599	−0.506083
$470$	−25.4470	−1.17378
$471$	0	0
$472$	−4.14866	−0.190957
$473$	−9.59818	−0.441325
$474$	0	0
$475$	−30.2166	−1.38643
$476$	21.9451	1.00585
$477$	0	0
$478$	39.1540	1.79086
$479$	4.08460	0.186630	0.0933151	−	0.995637i	$-0.470254\pi$
$479$	4.08460	0.186630	0.0933151	+	0.995637i	$0.470254\pi$
$480$	0	0
$481$	−11.1216	−0.507103
$482$	13.6410	0.621331
$483$	0	0
$484$	7.39999	0.336363
$485$	13.8223	0.627640
$486$	0	0
$487$	14.3210	0.648947	0.324473	−	0.945895i	$-0.394813\pi$
$487$	14.3210	0.648947	0.324473	+	0.945895i	$0.394813\pi$
$488$	−27.3044	−1.23601
$489$	0	0
$490$	8.21939	0.371314
$491$	−3.60157	−0.162537	−0.0812683	−	0.996692i	$-0.525897\pi$
$491$	−3.60157	−0.162537	−0.0812683	+	0.996692i	$0.525897\pi$
$492$	0	0
$493$	−14.9116	−0.671584
$494$	19.2751	0.867230
$495$	0	0
$496$	−2.83104	−0.127117
$497$	11.9564	0.536319
$498$	0	0
$499$	8.65491	0.387447	0.193723	−	0.981056i	$-0.437944\pi$
$499$	8.65491	0.387447	0.193723	+	0.981056i	$0.437944\pi$
$500$	26.5403	1.18692
$501$	0	0
$502$	38.7583	1.72987
$503$	2.22712	0.0993025	0.0496512	−	0.998767i	$-0.484189\pi$
$503$	2.22712	0.0993025	0.0496512	+	0.998767i	$0.484189\pi$
$504$	0	0
$505$	−31.6375	−1.40785
$506$	68.2220	3.03284
$507$	0	0
$508$	3.56679	0.158251
$509$	5.34661	0.236984	0.118492	−	0.992955i	$-0.462194\pi$
$509$	5.34661	0.236984	0.118492	+	0.992955i	$0.462194\pi$
$510$	0	0
$511$	12.4914	0.552588
$512$	−17.5348	−0.774934
$513$	0	0
$514$	−4.29261	−0.189339
$515$	14.6936	0.647477
$516$	0	0
$517$	−11.1947	−0.492343
$518$	13.6009	0.597591
$519$	0	0
$520$	−24.8458	−1.08956
$521$	26.5389	1.16269	0.581346	−	0.813656i	$-0.302526\pi$
$521$	26.5389	1.16269	0.581346	+	0.813656i	$0.302526\pi$
$522$	0	0
$523$	−38.0208	−1.66253	−0.831266	−	0.555874i	$-0.812384\pi$
$523$	−38.0208	−1.66253	−0.831266	+	0.555874i	$0.812384\pi$
$524$	−9.16318	−0.400295
$525$	0	0
$526$	−25.4662	−1.11038
$527$	−10.9657	−0.477675
$528$	0	0
$529$	40.9458	1.78025
$530$	−87.3356	−3.79362
$531$	0	0
$532$	−15.1033	−0.654810
$533$	12.1684	0.527070
$534$	0	0
$535$	62.4973	2.70199
$536$	−40.5156	−1.75001
$537$	0	0
$538$	−14.2222	−0.613163
$539$	3.61589	0.155748
$540$	0	0
$541$	−29.8452	−1.28314	−0.641572	−	0.767063i	$-0.721717\pi$
$541$	−29.8452	−1.28314	−0.641572	+	0.767063i	$0.721717\pi$
$542$	67.7651	2.91076
$543$	0	0
$544$	−22.4303	−0.961690
$545$	18.9945	0.813633
$546$	0	0
$547$	35.9013	1.53503	0.767515	−	0.641031i	$-0.221493\pi$
$547$	35.9013	1.53503	0.767515	+	0.641031i	$0.221493\pi$
$548$	14.2088	0.606970
$549$	0	0
$550$	60.8793	2.59590
$551$	10.2626	0.437202
$552$	0	0
$553$	−8.46939	−0.360155
$554$	37.7224	1.60267
$555$	0	0
$556$	38.9564	1.65212
$557$	−14.2853	−0.605286	−0.302643	−	0.953104i	$-0.597869\pi$
$557$	−14.2853	−0.605286	−0.302643	+	0.953104i	$0.597869\pi$
$558$	0	0
$559$	5.12124	0.216605
$560$	5.53356	0.233836
$561$	0	0
$562$	−39.5175	−1.66695
$563$	13.7483	0.579420	0.289710	−	0.957114i	$-0.406441\pi$
$563$	13.7483	0.579420	0.289710	+	0.957114i	$0.406441\pi$
$564$	0	0
$565$	−1.60162	−0.0673808
$566$	−37.3892	−1.57158
$567$	0	0
$568$	44.1993	1.85456
$569$	30.5444	1.28049	0.640244	−	0.768172i	$-0.278833\pi$
$569$	30.5444	1.28049	0.640244	+	0.768172i	$0.278833\pi$
$570$	0	0
$571$	−14.1134	−0.590628	−0.295314	−	0.955400i	$-0.595424\pi$
$571$	−14.1134	−0.590628	−0.295314	+	0.955400i	$0.595424\pi$
$572$	−24.8826	−1.04039
$573$	0	0
$574$	−14.8810	−0.621122
$575$	57.0634	2.37971
$576$	0	0
$577$	−15.2871	−0.636410	−0.318205	−	0.948022i	$-0.603080\pi$
$577$	−15.2871	−0.636410	−0.318205	+	0.948022i	$0.603080\pi$
$578$	49.2044	2.04663
$579$	0	0
$580$	−30.1148	−1.25045
$581$	−8.32052	−0.345194
$582$	0	0
$583$	−38.4209	−1.59123
$584$	46.1771	1.91082
$585$	0	0
$586$	−59.7713	−2.46913
$587$	33.0709	1.36498	0.682492	−	0.730893i	$-0.260896\pi$
$587$	33.0709	1.36498	0.682492	+	0.730893i	$0.260896\pi$
$588$	0	0
$589$	7.54695	0.310967
$590$	−9.22429	−0.379758
$591$	0	0
$592$	9.15660	0.376334
$593$	17.8911	0.734699	0.367349	−	0.930083i	$-0.380265\pi$
$593$	17.8911	0.734699	0.367349	+	0.930083i	$0.380265\pi$
$594$	0	0
$595$	21.4336	0.878694
$596$	5.88293	0.240974
$597$	0	0
$598$	−36.4007	−1.48854
$599$	2.10132	0.0858574	0.0429287	−	0.999078i	$-0.486331\pi$
$599$	2.10132	0.0858574	0.0429287	+	0.999078i	$0.486331\pi$
$600$	0	0
$601$	40.6507	1.65818	0.829089	−	0.559117i	$-0.188860\pi$
$601$	40.6507	1.65818	0.829089	+	0.559117i	$0.188860\pi$
$602$	−6.26291	−0.255257
$603$	0	0
$604$	34.1407	1.38916
$605$	7.22754	0.293841
$606$	0	0
$607$	38.0029	1.54249	0.771246	−	0.636537i	$-0.219634\pi$
$607$	38.0029	1.54249	0.771246	+	0.636537i	$0.219634\pi$
$608$	15.4372	0.626061
$609$	0	0
$610$	−60.7097	−2.45806
$611$	5.97308	0.241645
$612$	0	0
$613$	41.6896	1.68383	0.841913	−	0.539614i	$-0.181430\pi$
$613$	41.6896	1.68383	0.841913	+	0.539614i	$0.181430\pi$
$614$	62.1630	2.50869
$615$	0	0
$616$	13.3669	0.538567
$617$	−31.3419	−1.26178	−0.630888	−	0.775874i	$-0.717309\pi$
$617$	−31.3419	−1.26178	−0.630888	+	0.775874i	$0.717309\pi$
$618$	0	0
$619$	24.6489	0.990724	0.495362	−	0.868687i	$-0.335035\pi$
$619$	24.6489	0.990724	0.495362	+	0.868687i	$0.335035\pi$
$620$	−22.1459	−0.889400
$621$	0	0
$622$	−68.4037	−2.74274
$623$	−5.58500	−0.223758
$624$	0	0
$625$	−9.75799	−0.390320
$626$	42.2944	1.69042
$627$	0	0
$628$	18.7209	0.747046
$629$	35.4671	1.41417
$630$	0	0
$631$	−29.6796	−1.18153	−0.590763	−	0.806845i	$-0.701173\pi$
$631$	−29.6796	−1.18153	−0.590763	+	0.806845i	$0.701173\pi$
$632$	−31.3088	−1.24540
$633$	0	0
$634$	−13.9747	−0.555006
$635$	3.48367	0.138245
$636$	0	0
$637$	−1.92931	−0.0764420
$638$	−20.6768	−0.818601
$639$	0	0
$640$	−71.4110	−2.82277
$641$	8.73835	0.345144	0.172572	−	0.984997i	$-0.444792\pi$
$641$	8.73835	0.345144	0.172572	+	0.984997i	$0.444792\pi$
$642$	0	0
$643$	−10.7720	−0.424807	−0.212403	−	0.977182i	$-0.568129\pi$
$643$	−10.7720	−0.424807	−0.212403	+	0.977182i	$0.568129\pi$
$644$	28.5223	1.12393
$645$	0	0
$646$	−61.4688	−2.41846
$647$	−35.8867	−1.41085	−0.705426	−	0.708783i	$-0.749244\pi$
$647$	−35.8867	−1.41085	−0.705426	+	0.708783i	$0.749244\pi$
$648$	0	0
$649$	−4.05797	−0.159289
$650$	−32.4830	−1.27409
$651$	0	0
$652$	−7.35857	−0.288184
$653$	−12.3102	−0.481735	−0.240868	−	0.970558i	$-0.577432\pi$
$653$	−12.3102	−0.481735	−0.240868	+	0.970558i	$0.577432\pi$
$654$	0	0
$655$	−8.94963	−0.349691
$656$	−10.0184	−0.391152
$657$	0	0
$658$	−7.30465	−0.284765
$659$	−44.5047	−1.73366	−0.866829	−	0.498605i	$-0.833846\pi$
$659$	−44.5047	−1.73366	−0.866829	+	0.498605i	$0.833846\pi$
$660$	0	0
$661$	−28.3434	−1.10243	−0.551215	−	0.834363i	$-0.685836\pi$
$661$	−28.3434	−1.10243	−0.551215	+	0.834363i	$0.685836\pi$
$662$	49.1134	1.90885
$663$	0	0
$664$	−30.7585	−1.19366
$665$	−14.7513	−0.572031
$666$	0	0
$667$	−19.3807	−0.750426
$668$	68.9831	2.66904
$669$	0	0
$670$	−90.0840	−3.48025
$671$	−26.7076	−1.03103
$672$	0	0
$673$	−1.59797	−0.0615973	−0.0307987	−	0.999526i	$-0.509805\pi$
$673$	−1.59797	−0.0615973	−0.0307987	+	0.999526i	$0.509805\pi$
$674$	−40.6260	−1.56486
$675$	0	0
$676$	−33.0919	−1.27276
$677$	−40.4380	−1.55416	−0.777079	−	0.629403i	$-0.783299\pi$
$677$	−40.4380	−1.55416	−0.777079	+	0.629403i	$0.783299\pi$
$678$	0	0
$679$	3.96775	0.152268
$680$	79.2338	3.03848
$681$	0	0
$682$	−15.2054	−0.582243
$683$	22.0689	0.844444	0.422222	−	0.906492i	$-0.361250\pi$
$683$	22.0689	0.844444	0.422222	+	0.906492i	$0.361250\pi$
$684$	0	0
$685$	13.8777	0.530238
$686$	2.35941	0.0900825
$687$	0	0
$688$	−4.21639	−0.160749
$689$	20.5000	0.780987
$690$	0	0
$691$	−44.7054	−1.70067	−0.850337	−	0.526239i	$-0.823602\pi$
$691$	−44.7054	−1.70067	−0.850337	+	0.526239i	$0.823602\pi$
$692$	−46.2225	−1.75712
$693$	0	0
$694$	52.0333	1.97516
$695$	38.0485	1.44326
$696$	0	0
$697$	−38.8051	−1.46985
$698$	−43.7681	−1.65665
$699$	0	0
$700$	25.4525	0.962012
$701$	−27.9205	−1.05454	−0.527271	−	0.849697i	$-0.676785\pi$
$701$	−27.9205	−1.05454	−0.527271	+	0.849697i	$0.676785\pi$
$702$	0	0
$703$	−24.4096	−0.920624
$704$	−42.5896	−1.60515
$705$	0	0
$706$	−30.7560	−1.15752
$707$	−9.08165	−0.341551
$708$	0	0
$709$	−25.5996	−0.961412	−0.480706	−	0.876882i	$-0.659620\pi$
$709$	−25.5996	−0.961412	−0.480706	+	0.876882i	$0.659620\pi$
$710$	98.2745	3.68818
$711$	0	0
$712$	−20.6461	−0.773745
$713$	−14.2523	−0.533752
$714$	0	0
$715$	−24.3027	−0.908869
$716$	−94.1780	−3.51960
$717$	0	0
$718$	−19.8520	−0.740871
$719$	12.8401	0.478854	0.239427	−	0.970914i	$-0.423040\pi$
$719$	12.8401	0.478854	0.239427	+	0.970914i	$0.423040\pi$
$720$	0	0
$721$	4.21785	0.157081
$722$	−2.52395	−0.0939317
$723$	0	0
$724$	11.1838	0.415644
$725$	−17.2948	−0.642314
$726$	0	0
$727$	31.1884	1.15671	0.578357	−	0.815784i	$-0.303694\pi$
$727$	31.1884	1.15671	0.578357	+	0.815784i	$0.303694\pi$
$728$	−7.13208	−0.264332
$729$	0	0
$730$	102.672	3.80006
$731$	−16.3317	−0.604051
$732$	0	0
$733$	2.93743	0.108497	0.0542483	−	0.998527i	$-0.482724\pi$
$733$	2.93743	0.108497	0.0542483	+	0.998527i	$0.482724\pi$
$734$	−79.0406	−2.91744
$735$	0	0
$736$	−29.1529	−1.07459
$737$	−39.6300	−1.45979
$738$	0	0
$739$	29.1689	1.07300	0.536498	−	0.843902i	$-0.319747\pi$
$739$	29.1689	1.07300	0.536498	+	0.843902i	$0.319747\pi$
$740$	71.6278	2.63309
$741$	0	0
$742$	−25.0700	−0.920349
$743$	49.1554	1.80334	0.901668	−	0.432430i	$-0.142344\pi$
$743$	49.1554	1.80334	0.901668	+	0.432430i	$0.142344\pi$
$744$	0	0
$745$	5.74583	0.210511
$746$	−24.7922	−0.907707
$747$	0	0
$748$	79.3511	2.90136
$749$	17.9401	0.655516
$750$	0	0
$751$	−39.2079	−1.43072	−0.715358	−	0.698758i	$-0.753736\pi$
$751$	−39.2079	−1.43072	−0.715358	+	0.698758i	$0.753736\pi$
$752$	−4.91773	−0.179331
$753$	0	0
$754$	11.0324	0.401775
$755$	33.3450	1.21355
$756$	0	0
$757$	50.6563	1.84113	0.920567	−	0.390584i	$-0.127727\pi$
$757$	50.6563	1.84113	0.920567	+	0.390584i	$0.127727\pi$
$758$	−8.66336	−0.314667
$759$	0	0
$760$	−54.5311	−1.97805
$761$	−29.2786	−1.06135	−0.530674	−	0.847576i	$-0.678061\pi$
$761$	−29.2786	−1.06135	−0.530674	+	0.847576i	$0.678061\pi$
$762$	0	0
$763$	5.45243	0.197391
$764$	23.9225	0.865485
$765$	0	0
$766$	12.4990	0.451607
$767$	2.16519	0.0781803
$768$	0	0
$769$	−31.3773	−1.13149	−0.565747	−	0.824579i	$-0.691412\pi$
$769$	−31.3773	−1.13149	−0.565747	+	0.824579i	$0.691412\pi$
$770$	29.7204	1.07105
$771$	0	0
$772$	35.7933	1.28823
$773$	−9.28097	−0.333813	−0.166907	−	0.985973i	$-0.553378\pi$
$773$	−9.28097	−0.333813	−0.166907	+	0.985973i	$0.553378\pi$
$774$	0	0
$775$	−12.7183	−0.456856
$776$	14.6676	0.526536
$777$	0	0
$778$	22.4715	0.805643
$779$	26.7069	0.956874
$780$	0	0
$781$	43.2332	1.54700
$782$	116.083	4.15111
$783$	0	0
$784$	1.58843	0.0567296
$785$	18.2846	0.652606
$786$	0	0
$787$	−45.7571	−1.63107	−0.815533	−	0.578711i	$-0.803556\pi$
$787$	−45.7571	−1.63107	−0.815533	+	0.578711i	$0.803556\pi$
$788$	−25.8661	−0.921440
$789$	0	0
$790$	−69.6132	−2.47673
$791$	−0.459752	−0.0163469
$792$	0	0
$793$	14.2502	0.506039
$794$	51.9642	1.84414
$795$	0	0
$796$	−96.4813	−3.41969
$797$	−11.5364	−0.408639	−0.204320	−	0.978904i	$-0.565498\pi$
$797$	−11.5364	−0.408639	−0.204320	+	0.978904i	$0.565498\pi$
$798$	0	0
$799$	−19.0483	−0.673880
$800$	−26.0152	−0.919776
$801$	0	0
$802$	74.8876	2.64437
$803$	45.1677	1.59393
$804$	0	0
$805$	27.8575	0.981849
$806$	8.11302	0.285769
$807$	0	0
$808$	−33.5722	−1.18106
$809$	51.3289	1.80463	0.902314	−	0.431080i	$-0.141867\pi$
$809$	51.3289	1.80463	0.902314	+	0.431080i	$0.141867\pi$
$810$	0	0
$811$	−40.7749	−1.43180	−0.715901	−	0.698202i	$-0.753984\pi$
$811$	−40.7749	−1.43180	−0.715901	+	0.698202i	$0.753984\pi$
$812$	−8.64455	−0.303364
$813$	0	0
$814$	49.1796	1.72374
$815$	−7.18708	−0.251752
$816$	0	0
$817$	11.2400	0.393238
$818$	−53.3138	−1.86407
$819$	0	0
$820$	−78.3691	−2.73677
$821$	13.2030	0.460789	0.230395	−	0.973097i	$-0.425998\pi$
$821$	13.2030	0.460789	0.230395	+	0.973097i	$0.425998\pi$
$822$	0	0
$823$	45.2538	1.57745	0.788724	−	0.614747i	$-0.210742\pi$
$823$	45.2538	1.57745	0.788724	+	0.614747i	$0.210742\pi$
$824$	15.5921	0.543178
$825$	0	0
$826$	−2.64787	−0.0921310
$827$	−19.2982	−0.671064	−0.335532	−	0.942029i	$-0.608916\pi$
$827$	−19.2982	−0.671064	−0.335532	+	0.942029i	$0.608916\pi$
$828$	0	0
$829$	−50.2459	−1.74511	−0.872556	−	0.488514i	$-0.837539\pi$
$829$	−50.2459	−1.74511	−0.872556	+	0.488514i	$0.837539\pi$
$830$	−68.3896	−2.37384
$831$	0	0
$832$	22.7242	0.787821
$833$	6.15261	0.213175
$834$	0	0
$835$	67.3755	2.33162
$836$	−54.6119	−1.88879
$837$	0	0
$838$	−82.6212	−2.85410
$839$	−25.8205	−0.891423	−0.445711	−	0.895177i	$-0.647049\pi$
$839$	−25.8205	−0.891423	−0.445711	+	0.895177i	$0.647049\pi$
$840$	0	0
$841$	−23.1261	−0.797451
$842$	−44.3530	−1.52851
$843$	0	0
$844$	−53.7023	−1.84851
$845$	−32.3207	−1.11186
$846$	0	0
$847$	2.07469	0.0712872
$848$	−16.8780	−0.579591
$849$	0	0
$850$	103.589	3.55307
$851$	46.0970	1.58018
$852$	0	0
$853$	−17.2283	−0.589887	−0.294943	−	0.955515i	$-0.595301\pi$
$853$	−17.2283	−0.589887	−0.294943	+	0.955515i	$0.595301\pi$
$854$	−17.4270	−0.596338
$855$	0	0
$856$	66.3191	2.26674
$857$	−1.30072	−0.0444319	−0.0222159	−	0.999753i	$-0.507072\pi$
$857$	−1.30072	−0.0444319	−0.0222159	+	0.999753i	$0.507072\pi$
$858$	0	0
$859$	−43.2218	−1.47471	−0.737355	−	0.675505i	$-0.763926\pi$
$859$	−43.2218	−1.47471	−0.737355	+	0.675505i	$0.763926\pi$
$860$	−32.9828	−1.12471
$861$	0	0
$862$	37.5606	1.27932
$863$	−41.9644	−1.42849	−0.714243	−	0.699898i	$-0.753229\pi$
$863$	−41.9644	−1.42849	−0.714243	+	0.699898i	$0.753229\pi$
$864$	0	0
$865$	−45.1453	−1.53499
$866$	−42.9076	−1.45806
$867$	0	0
$868$	−6.35706	−0.215773
$869$	−30.6244	−1.03886
$870$	0	0
$871$	21.1451	0.716474
$872$	20.1560	0.682568
$873$	0	0
$874$	−79.8917	−2.70238
$875$	7.44093	0.251549
$876$	0	0
$877$	−10.1477	−0.342665	−0.171333	−	0.985213i	$-0.554807\pi$
$877$	−10.1477	−0.342665	−0.171333	+	0.985213i	$0.554807\pi$
$878$	41.6768	1.40652
$879$	0	0
$880$	20.0088	0.674495
$881$	−5.94565	−0.200314	−0.100157	−	0.994972i	$-0.531934\pi$
$881$	−5.94565	−0.200314	−0.100157	+	0.994972i	$0.531934\pi$
$882$	0	0
$883$	45.8661	1.54352	0.771758	−	0.635916i	$-0.219378\pi$
$883$	45.8661	1.54352	0.771758	+	0.635916i	$0.219378\pi$
$884$	−42.3388	−1.42401
$885$	0	0
$886$	28.9534	0.972710
$887$	46.5631	1.56344	0.781718	−	0.623632i	$-0.214344\pi$
$887$	46.5631	1.56344	0.781718	+	0.623632i	$0.214344\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−45.9053	−1.53875
$891$	0	0
$892$	35.7758	1.19786
$893$	13.1096	0.438697
$894$	0	0
$895$	−91.9831	−3.07466
$896$	−20.4988	−0.684817
$897$	0	0
$898$	−13.1320	−0.438220
$899$	4.31959	0.144066
$900$	0	0
$901$	−65.3749	−2.17795
$902$	−53.8082	−1.79162
$903$	0	0
$904$	−1.69957	−0.0565267
$905$	10.9232	0.363099
$906$	0	0
$907$	56.8250	1.88684	0.943421	−	0.331598i	$-0.107588\pi$
$907$	56.8250	1.88684	0.943421	+	0.331598i	$0.107588\pi$
$908$	−74.3191	−2.46636
$909$	0	0
$910$	−15.8577	−0.525679
$911$	21.2254	0.703230	0.351615	−	0.936145i	$-0.385633\pi$
$911$	21.2254	0.703230	0.351615	+	0.936145i	$0.385633\pi$
$912$	0	0
$913$	−30.0861	−0.995706
$914$	54.0808	1.78883
$915$	0	0
$916$	7.99751	0.264245
$917$	−2.56902	−0.0848366
$918$	0	0
$919$	27.1303	0.894945	0.447472	−	0.894298i	$-0.352324\pi$
$919$	27.1303	0.894945	0.447472	+	0.894298i	$0.352324\pi$
$920$	102.981	3.39518
$921$	0	0
$922$	−47.2511	−1.55613
$923$	−23.0676	−0.759281
$924$	0	0
$925$	41.1356	1.35253
$926$	84.1798	2.76632
$927$	0	0
$928$	8.83568	0.290045
$929$	−28.0177	−0.919231	−0.459615	−	0.888118i	$-0.652013\pi$
$929$	−28.0177	−0.919231	−0.459615	+	0.888118i	$0.652013\pi$
$930$	0	0
$931$	−4.23441	−0.138777
$932$	73.2610	2.39974
$933$	0	0
$934$	−8.00676	−0.261989
$935$	77.5018	2.53458
$936$	0	0
$937$	31.6573	1.03420	0.517100	−	0.855925i	$-0.327012\pi$
$937$	31.6573	1.03420	0.517100	+	0.855925i	$0.327012\pi$
$938$	−25.8589	−0.844324
$939$	0	0
$940$	−38.4691	−1.25472
$941$	−24.7886	−0.808086	−0.404043	−	0.914740i	$-0.632395\pi$
$941$	−24.7886	−0.808086	−0.404043	+	0.914740i	$0.632395\pi$
$942$	0	0
$943$	−50.4355	−1.64240
$944$	−1.78263	−0.0580197
$945$	0	0
$946$	−22.6460	−0.736285
$947$	−55.5768	−1.80600	−0.903001	−	0.429638i	$-0.858641\pi$
$947$	−55.5768	−1.80600	−0.903001	+	0.429638i	$0.858641\pi$
$948$	0	0
$949$	−24.0998	−0.782314
$950$	−71.2931	−2.31305
$951$	0	0
$952$	22.7443	0.737149
$953$	34.7345	1.12516	0.562580	−	0.826743i	$-0.309809\pi$
$953$	34.7345	1.12516	0.562580	+	0.826743i	$0.309809\pi$
$954$	0	0
$955$	23.3650	0.756072
$956$	59.1904	1.91436
$957$	0	0
$958$	9.63723	0.311365
$959$	3.98364	0.128638
$960$	0	0
$961$	−27.8234	−0.897530
$962$	−26.2404	−0.846025
$963$	0	0
$964$	20.6216	0.664176
$965$	34.9591	1.12537
$966$	0	0
$967$	44.2329	1.42244	0.711218	−	0.702972i	$-0.248144\pi$
$967$	44.2329	1.42244	0.711218	+	0.702972i	$0.248144\pi$
$968$	7.66951	0.246507
$969$	0	0
$970$	32.6125	1.04712
$971$	39.4528	1.26610	0.633050	−	0.774111i	$-0.281803\pi$
$971$	39.4528	1.26610	0.633050	+	0.774111i	$0.281803\pi$
$972$	0	0
$973$	10.9220	0.350142
$974$	33.7891	1.08267
$975$	0	0
$976$	−11.7324	−0.375545
$977$	11.1644	0.357180	0.178590	−	0.983924i	$-0.442846\pi$
$977$	11.1644	0.357180	0.178590	+	0.983924i	$0.442846\pi$
$978$	0	0
$979$	−20.1948	−0.645428
$980$	12.4255	0.396919
$981$	0	0
$982$	−8.49757	−0.271168
$983$	−15.1449	−0.483047	−0.241524	−	0.970395i	$-0.577647\pi$
$983$	−15.1449	−0.483047	−0.241524	+	0.970395i	$0.577647\pi$
$984$	0	0
$985$	−25.2633	−0.804954
$986$	−35.1825	−1.12044
$987$	0	0
$988$	29.1389	0.927032
$989$	−21.2265	−0.674965
$990$	0	0
$991$	−50.2569	−1.59646	−0.798232	−	0.602351i	$-0.794231\pi$
$991$	−50.2569	−1.59646	−0.798232	+	0.602351i	$0.794231\pi$
$992$	6.49761	0.206299
$993$	0	0
$994$	28.2101	0.894769
$995$	−94.2328	−2.98738
$996$	0	0
$997$	35.0681	1.11062	0.555308	−	0.831645i	$-0.312600\pi$
$997$	35.0681	1.11062	0.555308	+	0.831645i	$0.312600\pi$
$998$	20.4204	0.646398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.18		19
3.2	odd	2		2667.2.a.q.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.2	✓	19	3.2	odd	2
8001.2.a.v.1.18		19	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)

\(f(q)\)	\(=\)	\(q+2.35941 q^{2} +3.56679 q^{4} +3.48367 q^{5} +1.00000 q^{7} +3.69670 q^{8} +O(q^{10})\)	\(q+2.35941 q^{2} +3.56679 q^{4} +3.48367 q^{5} +1.00000 q^{7} +3.69670 q^{8} +8.21939 q^{10} +3.61589 q^{11} -1.92931 q^{13} +2.35941 q^{14} +1.58843 q^{16} +6.15261 q^{17} -4.23441 q^{19} +12.4255 q^{20} +8.53136 q^{22} +7.99661 q^{23} +7.13595 q^{25} -4.55202 q^{26} +3.56679 q^{28} -2.42362 q^{29} -1.78229 q^{31} -3.64565 q^{32} +14.5165 q^{34} +3.48367 q^{35} +5.76457 q^{37} -9.99070 q^{38} +12.8781 q^{40} -6.30711 q^{41} -2.65444 q^{43} +12.8971 q^{44} +18.8672 q^{46} -3.09597 q^{47} +1.00000 q^{49} +16.8366 q^{50} -6.88145 q^{52} -10.6256 q^{53} +12.5966 q^{55} +3.69670 q^{56} -5.71830 q^{58} -1.12226 q^{59} -7.38616 q^{61} -4.20514 q^{62} -11.7784 q^{64} -6.72107 q^{65} -10.9599 q^{67} +21.9451 q^{68} +8.21939 q^{70} +11.9564 q^{71} +12.4914 q^{73} +13.6009 q^{74} -15.1033 q^{76} +3.61589 q^{77} -8.46939 q^{79} +5.53356 q^{80} -14.8810 q^{82} -8.32052 q^{83} +21.4336 q^{85} -6.26291 q^{86} +13.3669 q^{88} -5.58500 q^{89} -1.92931 q^{91} +28.5223 q^{92} -7.30465 q^{94} -14.7513 q^{95} +3.96775 q^{97} +2.35941 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more