LMFDB - Embedded newform 8001.2.a.t.1.2

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:	$16$
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} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.24781$ 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)

$f(q)$	$=$	$q-2.24781 q^{2} +3.05263 q^{4} +0.411504 q^{5} -1.00000 q^{7} -2.36611 q^{8} +O(q^{10})$	\(q-2.24781 q^{2} +3.05263 q^{4} +0.411504 q^{5} -1.00000 q^{7} -2.36611 q^{8} -0.924982 q^{10} +2.04490 q^{11} +3.22161 q^{13} +2.24781 q^{14} -0.786714 q^{16} +4.69699 q^{17} -6.31686 q^{19} +1.25617 q^{20} -4.59653 q^{22} +2.31545 q^{23} -4.83066 q^{25} -7.24154 q^{26} -3.05263 q^{28} -4.12771 q^{29} -10.3067 q^{31} +6.50059 q^{32} -10.5579 q^{34} -0.411504 q^{35} -6.28463 q^{37} +14.1991 q^{38} -0.973663 q^{40} -3.24764 q^{41} +9.97662 q^{43} +6.24232 q^{44} -5.20469 q^{46} -5.22263 q^{47} +1.00000 q^{49} +10.8584 q^{50} +9.83437 q^{52} +3.24257 q^{53} +0.841485 q^{55} +2.36611 q^{56} +9.27829 q^{58} +0.616953 q^{59} +5.38762 q^{61} +23.1674 q^{62} -13.0386 q^{64} +1.32571 q^{65} +11.7719 q^{67} +14.3382 q^{68} +0.924982 q^{70} +16.1415 q^{71} +0.280081 q^{73} +14.1266 q^{74} -19.2830 q^{76} -2.04490 q^{77} -5.61511 q^{79} -0.323736 q^{80} +7.30005 q^{82} -0.433042 q^{83} +1.93283 q^{85} -22.4255 q^{86} -4.83844 q^{88} +4.62086 q^{89} -3.22161 q^{91} +7.06822 q^{92} +11.7395 q^{94} -2.59941 q^{95} +18.3025 q^{97} -2.24781 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * 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 $

$2$	−2.24781	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$2$	−2.24781	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$3$	0	0
$3$	0	0
$4$	3.05263	1.52631
$5$	0.411504	0.184030	0.0920152	−	0.995758i	$-0.470669\pi$
$5$	0.411504	0.184030	0.0920152	+	0.995758i	$0.470669\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.36611	−0.836545
$9$	0	0
$10$	−0.924982	−0.292505
$11$	2.04490	0.616560	0.308280	−	0.951296i	$-0.400247\pi$
$11$	2.04490	0.616560	0.308280	+	0.951296i	$0.400247\pi$
$12$	0	0
$13$	3.22161	0.893513	0.446756	−	0.894656i	$-0.352579\pi$
$13$	3.22161	0.893513	0.446756	+	0.894656i	$0.352579\pi$
$14$	2.24781	0.600751
$15$	0	0
$16$	−0.786714	−0.196679
$17$	4.69699	1.13919	0.569594	−	0.821926i	$-0.307101\pi$
$17$	4.69699	1.13919	0.569594	+	0.821926i	$0.307101\pi$
$18$	0	0
$19$	−6.31686	−1.44919	−0.724593	−	0.689177i	$-0.757972\pi$
$19$	−6.31686	−1.44919	−0.724593	+	0.689177i	$0.757972\pi$
$20$	1.25617	0.280888
$21$	0	0
$22$	−4.59653	−0.979984
$23$	2.31545	0.482805	0.241403	−	0.970425i	$-0.422393\pi$
$23$	2.31545	0.482805	0.241403	+	0.970425i	$0.422393\pi$
$24$	0	0
$25$	−4.83066	−0.966133
$26$	−7.24154	−1.42018
$27$	0	0
$28$	−3.05263	−0.576893
$29$	−4.12771	−0.766497	−0.383248	−	0.923645i	$-0.625195\pi$
$29$	−4.12771	−0.766497	−0.383248	+	0.923645i	$0.625195\pi$
$30$	0	0
$31$	−10.3067	−1.85113	−0.925566	−	0.378585i	$-0.876411\pi$
$31$	−10.3067	−1.85113	−0.925566	+	0.378585i	$0.876411\pi$
$32$	6.50059	1.14915
$33$	0	0
$34$	−10.5579	−1.81067
$35$	−0.411504	−0.0695570
$36$	0	0
$37$	−6.28463	−1.03319	−0.516593	−	0.856231i	$-0.672800\pi$
$37$	−6.28463	−1.03319	−0.516593	+	0.856231i	$0.672800\pi$
$38$	14.1991	2.30339
$39$	0	0
$40$	−0.973663	−0.153950
$41$	−3.24764	−0.507195	−0.253598	−	0.967310i	$-0.581614\pi$
$41$	−3.24764	−0.507195	−0.253598	+	0.967310i	$0.581614\pi$
$42$	0	0
$43$	9.97662	1.52142	0.760710	−	0.649091i	$-0.224851\pi$
$43$	9.97662	1.52142	0.760710	+	0.649091i	$0.224851\pi$
$44$	6.24232	0.941064
$45$	0	0
$46$	−5.20469	−0.767389
$47$	−5.22263	−0.761799	−0.380899	−	0.924616i	$-0.624386\pi$
$47$	−5.22263	−0.761799	−0.380899	+	0.924616i	$0.624386\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.8584	1.53561
$51$	0	0
$52$	9.83437	1.36378
$53$	3.24257	0.445401	0.222701	−	0.974887i	$-0.428513\pi$
$53$	3.24257	0.445401	0.222701	+	0.974887i	$0.428513\pi$
$54$	0	0
$55$	0.841485	0.113466
$56$	2.36611	0.316184
$57$	0	0
$58$	9.27829	1.21830
$59$	0.616953	0.0803204	0.0401602	−	0.999193i	$-0.487213\pi$
$59$	0.616953	0.0803204	0.0401602	+	0.999193i	$0.487213\pi$
$60$	0	0
$61$	5.38762	0.689814	0.344907	−	0.938637i	$-0.387910\pi$
$61$	5.38762	0.689814	0.344907	+	0.938637i	$0.387910\pi$
$62$	23.1674	2.94226
$63$	0	0
$64$	−13.0386	−1.62983
$65$	1.32571	0.164433
$66$	0	0
$67$	11.7719	1.43817	0.719083	−	0.694925i	$-0.244562\pi$
$67$	11.7719	1.43817	0.719083	+	0.694925i	$0.244562\pi$
$68$	14.3382	1.73876
$69$	0	0
$70$	0.924982	0.110556
$71$	16.1415	1.91564	0.957822	−	0.287364i	$-0.0927789\pi$
$71$	16.1415	1.91564	0.957822	+	0.287364i	$0.0927789\pi$
$72$	0	0
$73$	0.280081	0.0327810	0.0163905	−	0.999866i	$-0.494783\pi$
$73$	0.280081	0.0327810	0.0163905	+	0.999866i	$0.494783\pi$
$74$	14.1266	1.64219
$75$	0	0
$76$	−19.2830	−2.21191
$77$	−2.04490	−0.233038
$78$	0	0
$79$	−5.61511	−0.631749	−0.315875	−	0.948801i	$-0.602298\pi$
$79$	−5.61511	−0.631749	−0.315875	+	0.948801i	$0.602298\pi$
$80$	−0.323736	−0.0361948
$81$	0	0
$82$	7.30005	0.806156
$83$	−0.433042	−0.0475325	−0.0237662	−	0.999718i	$-0.507566\pi$
$83$	−0.433042	−0.0475325	−0.0237662	+	0.999718i	$0.507566\pi$
$84$	0	0
$85$	1.93283	0.209645
$86$	−22.4255	−2.41820
$87$	0	0
$88$	−4.83844	−0.515780
$89$	4.62086	0.489810	0.244905	−	0.969547i	$-0.421243\pi$
$89$	4.62086	0.489810	0.244905	+	0.969547i	$0.421243\pi$
$90$	0	0
$91$	−3.22161	−0.337716
$92$	7.06822	0.736913
$93$	0	0
$94$	11.7395	1.21083
$95$	−2.59941	−0.266694
$96$	0	0
$97$	18.3025	1.85834	0.929169	−	0.369656i	$-0.120524\pi$
$97$	18.3025	1.85834	0.929169	+	0.369656i	$0.120524\pi$
$98$	−2.24781	−0.227063
$99$	0	0
$100$	−14.7462	−1.47462
$101$	2.42436	0.241233	0.120616	−	0.992699i	$-0.461513\pi$
$101$	2.42436	0.241233	0.120616	+	0.992699i	$0.461513\pi$
$102$	0	0
$103$	8.14077	0.802134	0.401067	−	0.916049i	$-0.368639\pi$
$103$	8.14077	0.802134	0.401067	+	0.916049i	$0.368639\pi$
$104$	−7.62266	−0.747463
$105$	0	0
$106$	−7.28866	−0.707938
$107$	−12.1165	−1.17135	−0.585675	−	0.810546i	$-0.699171\pi$
$107$	−12.1165	−1.17135	−0.585675	+	0.810546i	$0.699171\pi$
$108$	0	0
$109$	−16.1823	−1.54998	−0.774992	−	0.631971i	$-0.782246\pi$
$109$	−16.1823	−1.54998	−0.774992	+	0.631971i	$0.782246\pi$
$110$	−1.89149	−0.180347
$111$	0	0
$112$	0.786714	0.0743375
$113$	−1.15714	−0.108855	−0.0544275	−	0.998518i	$-0.517333\pi$
$113$	−1.15714	−0.108855	−0.0544275	+	0.998518i	$0.517333\pi$
$114$	0	0
$115$	0.952819	0.0888508
$116$	−12.6004	−1.16992
$117$	0	0
$118$	−1.38679	−0.127664
$119$	−4.69699	−0.430572
$120$	0	0
$121$	−6.81839	−0.619854
$122$	−12.1103	−1.09642
$123$	0	0
$124$	−31.4624	−2.82541
$125$	−4.04536	−0.361828
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	16.3071	1.44136
$129$	0	0
$130$	−2.97993	−0.261357
$131$	11.4264	0.998326	0.499163	−	0.866508i	$-0.333641\pi$
$131$	11.4264	0.998326	0.499163	+	0.866508i	$0.333641\pi$
$132$	0	0
$133$	6.31686	0.547741
$134$	−26.4609	−2.28588
$135$	0	0
$136$	−11.1136	−0.952981
$137$	11.1533	0.952891	0.476446	−	0.879204i	$-0.341925\pi$
$137$	11.1533	0.952891	0.476446	+	0.879204i	$0.341925\pi$
$138$	0	0
$139$	−2.77172	−0.235094	−0.117547	−	0.993067i	$-0.537503\pi$
$139$	−2.77172	−0.235094	−0.117547	+	0.993067i	$0.537503\pi$
$140$	−1.25617	−0.106166
$141$	0	0
$142$	−36.2829	−3.04480
$143$	6.58785	0.550904
$144$	0	0
$145$	−1.69857	−0.141059
$146$	−0.629567	−0.0521034
$147$	0	0
$148$	−19.1846	−1.57697
$149$	10.2652	0.840955	0.420478	−	0.907303i	$-0.361862\pi$
$149$	10.2652	0.840955	0.420478	+	0.907303i	$0.361862\pi$
$150$	0	0
$151$	21.5874	1.75676	0.878378	−	0.477966i	$-0.158626\pi$
$151$	21.5874	1.75676	0.878378	+	0.477966i	$0.158626\pi$
$152$	14.9463	1.21231
$153$	0	0
$154$	4.59653	0.370399
$155$	−4.24124	−0.340665
$156$	0	0
$157$	6.43652	0.513691	0.256845	−	0.966453i	$-0.417317\pi$
$157$	6.43652	0.513691	0.256845	+	0.966453i	$0.417317\pi$
$158$	12.6217	1.00413
$159$	0	0
$160$	2.67502	0.211479
$161$	−2.31545	−0.182483
$162$	0	0
$163$	−6.11197	−0.478726	−0.239363	−	0.970930i	$-0.576939\pi$
$163$	−6.11197	−0.478726	−0.239363	+	0.970930i	$0.576939\pi$
$164$	−9.91383	−0.774140
$165$	0	0
$166$	0.973393	0.0755500
$167$	16.0658	1.24321	0.621606	−	0.783330i	$-0.286480\pi$
$167$	16.0658	1.24321	0.621606	+	0.783330i	$0.286480\pi$
$168$	0	0
$169$	−2.62126	−0.201635
$170$	−4.34463	−0.333218
$171$	0	0
$172$	30.4549	2.32217
$173$	−10.7411	−0.816629	−0.408315	−	0.912841i	$-0.633883\pi$
$173$	−10.7411	−0.816629	−0.408315	+	0.912841i	$0.633883\pi$
$174$	0	0
$175$	4.83066	0.365164
$176$	−1.60875	−0.121264
$177$	0	0
$178$	−10.3868	−0.778522
$179$	−12.5968	−0.941528	−0.470764	−	0.882259i	$-0.656022\pi$
$179$	−12.5968	−0.941528	−0.470764	+	0.882259i	$0.656022\pi$
$180$	0	0
$181$	13.7179	1.01964	0.509821	−	0.860280i	$-0.329711\pi$
$181$	13.7179	1.01964	0.509821	+	0.860280i	$0.329711\pi$
$182$	7.24154	0.536779
$183$	0	0
$184$	−5.47860	−0.403888
$185$	−2.58615	−0.190138
$186$	0	0
$187$	9.60486	0.702377
$188$	−15.9428	−1.16274
$189$	0	0
$190$	5.84298	0.423894
$191$	6.08373	0.440203	0.220102	−	0.975477i	$-0.429361\pi$
$191$	6.08373	0.440203	0.220102	+	0.975477i	$0.429361\pi$
$192$	0	0
$193$	−25.0386	−1.80232	−0.901158	−	0.433490i	$-0.857282\pi$
$193$	−25.0386	−1.80232	−0.901158	+	0.433490i	$0.857282\pi$
$194$	−41.1405	−2.95371
$195$	0	0
$196$	3.05263	0.218045
$197$	−13.5198	−0.963246	−0.481623	−	0.876379i	$-0.659952\pi$
$197$	−13.5198	−0.963246	−0.481623	+	0.876379i	$0.659952\pi$
$198$	0	0
$199$	10.3890	0.736457	0.368228	−	0.929735i	$-0.379964\pi$
$199$	10.3890	0.736457	0.368228	+	0.929735i	$0.379964\pi$
$200$	11.4299	0.808213
$201$	0	0
$202$	−5.44949	−0.383425
$203$	4.12771	0.289709
$204$	0	0
$205$	−1.33642	−0.0933394
$206$	−18.2989	−1.27494
$207$	0	0
$208$	−2.53448	−0.175735
$209$	−12.9173	−0.893510
$210$	0	0
$211$	7.85673	0.540879	0.270440	−	0.962737i	$-0.412831\pi$
$211$	7.85673	0.540879	0.270440	+	0.962737i	$0.412831\pi$
$212$	9.89836	0.679822
$213$	0	0
$214$	27.2356	1.86179
$215$	4.10543	0.279988
$216$	0	0
$217$	10.3067	0.699662
$218$	36.3747	2.46360
$219$	0	0
$220$	2.56874	0.173184
$221$	15.1318	1.01788
$222$	0	0
$223$	4.24353	0.284167	0.142084	−	0.989855i	$-0.454620\pi$
$223$	4.24353	0.284167	0.142084	+	0.989855i	$0.454620\pi$
$224$	−6.50059	−0.434339
$225$	0	0
$226$	2.60104	0.173018
$227$	−0.763960	−0.0507058	−0.0253529	−	0.999679i	$-0.508071\pi$
$227$	−0.763960	−0.0507058	−0.0253529	+	0.999679i	$0.508071\pi$
$228$	0	0
$229$	4.87924	0.322429	0.161215	−	0.986919i	$-0.448459\pi$
$229$	4.87924	0.322429	0.161215	+	0.986919i	$0.448459\pi$
$230$	−2.14175	−0.141223
$231$	0	0
$232$	9.76660	0.641209
$233$	−8.04174	−0.526832	−0.263416	−	0.964682i	$-0.584849\pi$
$233$	−8.04174	−0.526832	−0.263416	+	0.964682i	$0.584849\pi$
$234$	0	0
$235$	−2.14914	−0.140194
$236$	1.88333	0.122594
$237$	0	0
$238$	10.5579	0.684368
$239$	1.12936	0.0730524	0.0365262	−	0.999333i	$-0.488371\pi$
$239$	1.12936	0.0730524	0.0365262	+	0.999333i	$0.488371\pi$
$240$	0	0
$241$	7.47452	0.481476	0.240738	−	0.970590i	$-0.422611\pi$
$241$	7.47452	0.481476	0.240738	+	0.970590i	$0.422611\pi$
$242$	15.3264	0.985219
$243$	0	0
$244$	16.4464	1.05287
$245$	0.411504	0.0262901
$246$	0	0
$247$	−20.3504	−1.29487
$248$	24.3867	1.54855
$249$	0	0
$250$	9.09319	0.575104
$251$	23.9109	1.50924	0.754621	−	0.656161i	$-0.227821\pi$
$251$	23.9109	1.50924	0.754621	+	0.656161i	$0.227821\pi$
$252$	0	0
$253$	4.73486	0.297678
$254$	2.24781	0.141040
$255$	0	0
$256$	−10.5780	−0.661124
$257$	4.44829	0.277477	0.138738	−	0.990329i	$-0.455695\pi$
$257$	4.44829	0.277477	0.138738	+	0.990329i	$0.455695\pi$
$258$	0	0
$259$	6.28463	0.390508
$260$	4.04689	0.250977
$261$	0	0
$262$	−25.6842	−1.58678
$263$	−7.03143	−0.433576	−0.216788	−	0.976219i	$-0.569558\pi$
$263$	−7.03143	−0.433576	−0.216788	+	0.976219i	$0.569558\pi$
$264$	0	0
$265$	1.33433	0.0819674
$266$	−14.1991	−0.870600
$267$	0	0
$268$	35.9352	2.19509
$269$	−20.5496	−1.25293	−0.626467	−	0.779448i	$-0.715500\pi$
$269$	−20.5496	−1.25293	−0.626467	+	0.779448i	$0.715500\pi$
$270$	0	0
$271$	−12.4480	−0.756162	−0.378081	−	0.925772i	$-0.623416\pi$
$271$	−12.4480	−0.756162	−0.378081	+	0.925772i	$0.623416\pi$
$272$	−3.69519	−0.224054
$273$	0	0
$274$	−25.0705	−1.51456
$275$	−9.87822	−0.595679
$276$	0	0
$277$	24.9530	1.49928	0.749641	−	0.661844i	$-0.230226\pi$
$277$	24.9530	1.49928	0.749641	+	0.661844i	$0.230226\pi$
$278$	6.23029	0.373668
$279$	0	0
$280$	0.973663	0.0581875
$281$	31.3699	1.87137	0.935686	−	0.352835i	$-0.114782\pi$
$281$	31.3699	1.87137	0.935686	+	0.352835i	$0.114782\pi$
$282$	0	0
$283$	−5.77614	−0.343356	−0.171678	−	0.985153i	$-0.554919\pi$
$283$	−5.77614	−0.343356	−0.171678	+	0.985153i	$0.554919\pi$
$284$	49.2740	2.92387
$285$	0	0
$286$	−14.8082	−0.875628
$287$	3.24764	0.191702
$288$	0	0
$289$	5.06171	0.297748
$290$	3.81806	0.224204
$291$	0	0
$292$	0.854983	0.0500341
$293$	9.75230	0.569735	0.284868	−	0.958567i	$-0.408050\pi$
$293$	9.75230	0.569735	0.284868	+	0.958567i	$0.408050\pi$
$294$	0	0
$295$	0.253879	0.0147814
$296$	14.8701	0.864306
$297$	0	0
$298$	−23.0741	−1.33665
$299$	7.45947	0.431393
$300$	0	0
$301$	−9.97662	−0.575043
$302$	−48.5242	−2.79226
$303$	0	0
$304$	4.96956	0.285024
$305$	2.21703	0.126947
$306$	0	0
$307$	23.1658	1.32214	0.661070	−	0.750324i	$-0.270103\pi$
$307$	23.1658	1.32214	0.661070	+	0.750324i	$0.270103\pi$
$308$	−6.24232	−0.355689
$309$	0	0
$310$	9.53349	0.541466
$311$	−2.45175	−0.139026	−0.0695129	−	0.997581i	$-0.522145\pi$
$311$	−2.45175	−0.139026	−0.0695129	+	0.997581i	$0.522145\pi$
$312$	0	0
$313$	−31.6925	−1.79137	−0.895684	−	0.444691i	$-0.853313\pi$
$313$	−31.6925	−1.79137	−0.895684	+	0.444691i	$0.853313\pi$
$314$	−14.4681	−0.816479
$315$	0	0
$316$	−17.1408	−0.964248
$317$	27.7770	1.56011	0.780057	−	0.625708i	$-0.215190\pi$
$317$	27.7770	1.56011	0.780057	+	0.625708i	$0.215190\pi$
$318$	0	0
$319$	−8.44075	−0.472591
$320$	−5.36546	−0.299938
$321$	0	0
$322$	5.20469	0.290046
$323$	−29.6702	−1.65089
$324$	0	0
$325$	−15.5625	−0.863252
$326$	13.7385	0.760906
$327$	0	0
$328$	7.68425	0.424292
$329$	5.22263	0.287933
$330$	0	0
$331$	11.0970	0.609947	0.304974	−	0.952361i	$-0.401352\pi$
$331$	11.0970	0.609947	0.304974	+	0.952361i	$0.401352\pi$
$332$	−1.32192	−0.0725495
$333$	0	0
$334$	−36.1129	−1.97601
$335$	4.84419	0.264666
$336$	0	0
$337$	−25.2778	−1.37697	−0.688485	−	0.725251i	$-0.741724\pi$
$337$	−25.2778	−1.37697	−0.688485	+	0.725251i	$0.741724\pi$
$338$	5.89208	0.320487
$339$	0	0
$340$	5.90022	0.319984
$341$	−21.0761	−1.14133
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−23.6057	−1.27274
$345$	0	0
$346$	24.1439	1.29798
$347$	30.7483	1.65066	0.825328	−	0.564654i	$-0.190990\pi$
$347$	30.7483	1.65066	0.825328	+	0.564654i	$0.190990\pi$
$348$	0	0
$349$	−1.49682	−0.0801230	−0.0400615	−	0.999197i	$-0.512755\pi$
$349$	−1.49682	−0.0801230	−0.0400615	+	0.999197i	$0.512755\pi$
$350$	−10.8584	−0.580406
$351$	0	0
$352$	13.2930	0.708522
$353$	3.69916	0.196886	0.0984432	−	0.995143i	$-0.468614\pi$
$353$	3.69916	0.196886	0.0984432	+	0.995143i	$0.468614\pi$
$354$	0	0
$355$	6.64230	0.352537
$356$	14.1058	0.747604
$357$	0	0
$358$	28.3151	1.49650
$359$	29.9029	1.57821	0.789107	−	0.614256i	$-0.210544\pi$
$359$	29.9029	1.57821	0.789107	+	0.614256i	$0.210544\pi$
$360$	0	0
$361$	20.9027	1.10014
$362$	−30.8352	−1.62066
$363$	0	0
$364$	−9.83437	−0.515461
$365$	0.115255	0.00603270
$366$	0	0
$367$	25.1806	1.31442	0.657209	−	0.753708i	$-0.271737\pi$
$367$	25.1806	1.31442	0.657209	+	0.753708i	$0.271737\pi$
$368$	−1.82160	−0.0949574
$369$	0	0
$370$	5.81317	0.302212
$371$	−3.24257	−0.168346
$372$	0	0
$373$	22.9269	1.18711	0.593555	−	0.804794i	$-0.297724\pi$
$373$	22.9269	1.18711	0.593555	+	0.804794i	$0.297724\pi$
$374$	−21.5899	−1.11639
$375$	0	0
$376$	12.3573	0.637279
$377$	−13.2979	−0.684875
$378$	0	0
$379$	−14.3270	−0.735927	−0.367963	−	0.929840i	$-0.619945\pi$
$379$	−14.3270	−0.735927	−0.367963	+	0.929840i	$0.619945\pi$
$380$	−7.93505	−0.407059
$381$	0	0
$382$	−13.6750	−0.699676
$383$	3.07591	0.157172	0.0785859	−	0.996907i	$-0.474960\pi$
$383$	3.07591	0.157172	0.0785859	+	0.996907i	$0.474960\pi$
$384$	0	0
$385$	−0.841485	−0.0428860
$386$	56.2819	2.86467
$387$	0	0
$388$	55.8708	2.83641
$389$	−38.6677	−1.96053	−0.980266	−	0.197685i	$-0.936658\pi$
$389$	−38.6677	−1.96053	−0.980266	+	0.197685i	$0.936658\pi$
$390$	0	0
$391$	10.8757	0.550006
$392$	−2.36611	−0.119506
$393$	0	0
$394$	30.3899	1.53102
$395$	−2.31064	−0.116261
$396$	0	0
$397$	−17.9712	−0.901948	−0.450974	−	0.892537i	$-0.648923\pi$
$397$	−17.9712	−0.901948	−0.450974	+	0.892537i	$0.648923\pi$
$398$	−23.3524	−1.17055
$399$	0	0
$400$	3.80035	0.190018
$401$	−6.72517	−0.335839	−0.167919	−	0.985801i	$-0.553705\pi$
$401$	−6.72517	−0.335839	−0.167919	+	0.985801i	$0.553705\pi$
$402$	0	0
$403$	−33.2040	−1.65401
$404$	7.40067	0.368197
$405$	0	0
$406$	−9.27829	−0.460474
$407$	−12.8514	−0.637021
$408$	0	0
$409$	15.2011	0.751644	0.375822	−	0.926692i	$-0.377360\pi$
$409$	15.2011	0.751644	0.375822	+	0.926692i	$0.377360\pi$
$410$	3.00400	0.148357
$411$	0	0
$412$	24.8508	1.22431
$413$	−0.616953	−0.0303582
$414$	0	0
$415$	−0.178199	−0.00874742
$416$	20.9423	1.02678
$417$	0	0
$418$	29.0356	1.42018
$419$	1.10561	0.0540125	0.0270062	−	0.999635i	$-0.491403\pi$
$419$	1.10561	0.0540125	0.0270062	+	0.999635i	$0.491403\pi$
$420$	0	0
$421$	−11.2115	−0.546415	−0.273207	−	0.961955i	$-0.588085\pi$
$421$	−11.2115	−0.546415	−0.273207	+	0.961955i	$0.588085\pi$
$422$	−17.6604	−0.859694
$423$	0	0
$424$	−7.67226	−0.372598
$425$	−22.6896	−1.10061
$426$	0	0
$427$	−5.38762	−0.260725
$428$	−36.9873	−1.78785
$429$	0	0
$430$	−9.22820	−0.445023
$431$	−13.2110	−0.636349	−0.318175	−	0.948032i	$-0.603070\pi$
$431$	−13.2110	−0.636349	−0.318175	+	0.948032i	$0.603070\pi$
$432$	0	0
$433$	−39.4773	−1.89716	−0.948580	−	0.316538i	$-0.897479\pi$
$433$	−39.4773	−1.89716	−0.948580	+	0.316538i	$0.897479\pi$
$434$	−23.1674	−1.11207
$435$	0	0
$436$	−49.3986	−2.36576
$437$	−14.6264	−0.699675
$438$	0	0
$439$	−27.2830	−1.30215	−0.651074	−	0.759014i	$-0.725681\pi$
$439$	−27.2830	−1.30215	−0.651074	+	0.759014i	$0.725681\pi$
$440$	−1.99104	−0.0949192
$441$	0	0
$442$	−34.0134	−1.61785
$443$	−4.29862	−0.204233	−0.102117	−	0.994772i	$-0.532562\pi$
$443$	−4.29862	−0.204233	−0.102117	+	0.994772i	$0.532562\pi$
$444$	0	0
$445$	1.90150	0.0901399
$446$	−9.53862	−0.451667
$447$	0	0
$448$	13.0386	0.616018
$449$	22.9521	1.08318	0.541588	−	0.840644i	$-0.317823\pi$
$449$	22.9521	1.08318	0.541588	+	0.840644i	$0.317823\pi$
$450$	0	0
$451$	−6.64108	−0.312716
$452$	−3.53233	−0.166147
$453$	0	0
$454$	1.71723	0.0805938
$455$	−1.32571	−0.0621500
$456$	0	0
$457$	22.2524	1.04092	0.520462	−	0.853885i	$-0.325760\pi$
$457$	22.2524	1.04092	0.520462	+	0.853885i	$0.325760\pi$
$458$	−10.9676	−0.512481
$459$	0	0
$460$	2.90860	0.135614
$461$	16.2002	0.754521	0.377260	−	0.926107i	$-0.376866\pi$
$461$	16.2002	0.754521	0.377260	+	0.926107i	$0.376866\pi$
$462$	0	0
$463$	−10.0838	−0.468636	−0.234318	−	0.972160i	$-0.575286\pi$
$463$	−10.0838	−0.468636	−0.234318	+	0.972160i	$0.575286\pi$
$464$	3.24733	0.150753
$465$	0	0
$466$	18.0763	0.837367
$467$	16.2682	0.752801	0.376400	−	0.926457i	$-0.377162\pi$
$467$	16.2682	0.752801	0.376400	+	0.926457i	$0.377162\pi$
$468$	0	0
$469$	−11.7719	−0.543575
$470$	4.83084	0.222830
$471$	0	0
$472$	−1.45977	−0.0671916
$473$	20.4012	0.938047
$474$	0	0
$475$	30.5146	1.40011
$476$	−14.3382	−0.657189
$477$	0	0
$478$	−2.53859	−0.116112
$479$	23.7884	1.08692	0.543459	−	0.839436i	$-0.317114\pi$
$479$	23.7884	1.08692	0.543459	+	0.839436i	$0.317114\pi$
$480$	0	0
$481$	−20.2466	−0.923165
$482$	−16.8013	−0.765276
$483$	0	0
$484$	−20.8140	−0.946092
$485$	7.53156	0.341991
$486$	0	0
$487$	−0.625010	−0.0283219	−0.0141609	−	0.999900i	$-0.504508\pi$
$487$	−0.625010	−0.0283219	−0.0141609	+	0.999900i	$0.504508\pi$
$488$	−12.7477	−0.577060
$489$	0	0
$490$	−0.924982	−0.0417864
$491$	−30.3033	−1.36757	−0.683785	−	0.729683i	$-0.739667\pi$
$491$	−30.3033	−1.36757	−0.683785	+	0.729683i	$0.739667\pi$
$492$	0	0
$493$	−19.3878	−0.873183
$494$	45.7438	2.05811
$495$	0	0
$496$	8.10840	0.364078
$497$	−16.1415	−0.724045
$498$	0	0
$499$	7.98387	0.357407	0.178703	−	0.983903i	$-0.442810\pi$
$499$	7.98387	0.357407	0.178703	+	0.983903i	$0.442810\pi$
$500$	−12.3490	−0.552264
$501$	0	0
$502$	−53.7470	−2.39885
$503$	24.1008	1.07460	0.537301	−	0.843391i	$-0.319444\pi$
$503$	24.1008	1.07460	0.537301	+	0.843391i	$0.319444\pi$
$504$	0	0
$505$	0.997635	0.0443942
$506$	−10.6431	−0.473141
$507$	0	0
$508$	−3.05263	−0.135439
$509$	42.2969	1.87478	0.937389	−	0.348284i	$-0.113236\pi$
$509$	42.2969	1.87478	0.937389	+	0.348284i	$0.113236\pi$
$510$	0	0
$511$	−0.280081	−0.0123900
$512$	−8.83700	−0.390544
$513$	0	0
$514$	−9.99890	−0.441032
$515$	3.34996	0.147617
$516$	0	0
$517$	−10.6797	−0.469695
$518$	−14.1266	−0.620688
$519$	0	0
$520$	−3.13676	−0.137556
$521$	−15.0203	−0.658052	−0.329026	−	0.944321i	$-0.606720\pi$
$521$	−15.0203	−0.658052	−0.329026	+	0.944321i	$0.606720\pi$
$522$	0	0
$523$	−33.2467	−1.45378	−0.726888	−	0.686756i	$-0.759034\pi$
$523$	−33.2467	−1.45378	−0.726888	+	0.686756i	$0.759034\pi$
$524$	34.8804	1.52376
$525$	0	0
$526$	15.8053	0.689143
$527$	−48.4103	−2.10879
$528$	0	0
$529$	−17.6387	−0.766899
$530$	−2.99932	−0.130282
$531$	0	0
$532$	19.2830	0.836025
$533$	−10.4626	−0.453185
$534$	0	0
$535$	−4.98601	−0.215564
$536$	−27.8535	−1.20309
$537$	0	0
$538$	46.1916	1.99146
$539$	2.04490	0.0880800
$540$	0	0
$541$	14.9063	0.640873	0.320436	−	0.947270i	$-0.396170\pi$
$541$	14.9063	0.640873	0.320436	+	0.947270i	$0.396170\pi$
$542$	27.9807	1.20187
$543$	0	0
$544$	30.5332	1.30910
$545$	−6.65909	−0.285244
$546$	0	0
$547$	10.7221	0.458446	0.229223	−	0.973374i	$-0.426382\pi$
$547$	10.7221	0.458446	0.229223	+	0.973374i	$0.426382\pi$
$548$	34.0469	1.45441
$549$	0	0
$550$	22.2043	0.946795
$551$	26.0742	1.11080
$552$	0	0
$553$	5.61511	0.238779
$554$	−56.0896	−2.38302
$555$	0	0
$556$	−8.46103	−0.358828
$557$	6.82733	0.289283	0.144642	−	0.989484i	$-0.453797\pi$
$557$	6.82733	0.289283	0.144642	+	0.989484i	$0.453797\pi$
$558$	0	0
$559$	32.1407	1.35941
$560$	0.323736	0.0136804
$561$	0	0
$562$	−70.5134	−2.97443
$563$	−24.7198	−1.04182	−0.520908	−	0.853613i	$-0.674407\pi$
$563$	−24.7198	−1.04182	−0.520908	+	0.853613i	$0.674407\pi$
$564$	0	0
$565$	−0.476170	−0.0200326
$566$	12.9836	0.545743
$567$	0	0
$568$	−38.1925	−1.60252
$569$	−29.7916	−1.24893	−0.624464	−	0.781053i	$-0.714683\pi$
$569$	−29.7916	−1.24893	−0.624464	+	0.781053i	$0.714683\pi$
$570$	0	0
$571$	14.1130	0.590611	0.295305	−	0.955403i	$-0.404579\pi$
$571$	14.1130	0.590611	0.295305	+	0.955403i	$0.404579\pi$
$572$	20.1103	0.840853
$573$	0	0
$574$	−7.30005	−0.304698
$575$	−11.1852	−0.466454
$576$	0	0
$577$	6.50777	0.270922	0.135461	−	0.990783i	$-0.456748\pi$
$577$	6.50777	0.270922	0.135461	+	0.990783i	$0.456748\pi$
$578$	−11.3777	−0.473252
$579$	0	0
$580$	−5.18511	−0.215300
$581$	0.433042	0.0179656
$582$	0	0
$583$	6.63072	0.274617
$584$	−0.662701	−0.0274228
$585$	0	0
$586$	−21.9213	−0.905559
$587$	33.5990	1.38678	0.693389	−	0.720564i	$-0.256117\pi$
$587$	33.5990	1.38678	0.693389	+	0.720564i	$0.256117\pi$
$588$	0	0
$589$	65.1058	2.68264
$590$	−0.570670	−0.0234941
$591$	0	0
$592$	4.94420	0.203206
$593$	−27.5225	−1.13021	−0.565107	−	0.825017i	$-0.691165\pi$
$593$	−27.5225	−1.13021	−0.565107	+	0.825017i	$0.691165\pi$
$594$	0	0
$595$	−1.93283	−0.0792384
$596$	31.3358	1.28356
$597$	0	0
$598$	−16.7674	−0.685672
$599$	47.3256	1.93367	0.966836	−	0.255399i	$-0.0822067\pi$
$599$	47.3256	1.93367	0.966836	+	0.255399i	$0.0822067\pi$
$600$	0	0
$601$	−6.05581	−0.247022	−0.123511	−	0.992343i	$-0.539415\pi$
$601$	−6.05581	−0.247022	−0.123511	+	0.992343i	$0.539415\pi$
$602$	22.4255	0.913996
$603$	0	0
$604$	65.8983	2.68136
$605$	−2.80580	−0.114072
$606$	0	0
$607$	33.6790	1.36699	0.683493	−	0.729957i	$-0.260460\pi$
$607$	33.6790	1.36699	0.683493	+	0.729957i	$0.260460\pi$
$608$	−41.0633	−1.66534
$609$	0	0
$610$	−4.98345	−0.201774
$611$	−16.8253	−0.680677
$612$	0	0
$613$	18.7162	0.755939	0.377969	−	0.925818i	$-0.376622\pi$
$613$	18.7162	0.755939	0.377969	+	0.925818i	$0.376622\pi$
$614$	−52.0721	−2.10146
$615$	0	0
$616$	4.83844	0.194946
$617$	−10.0870	−0.406086	−0.203043	−	0.979170i	$-0.565083\pi$
$617$	−10.0870	−0.406086	−0.203043	+	0.979170i	$0.565083\pi$
$618$	0	0
$619$	9.04320	0.363477	0.181739	−	0.983347i	$-0.441828\pi$
$619$	9.04320	0.363477	0.181739	+	0.983347i	$0.441828\pi$
$620$	−12.9469	−0.519961
$621$	0	0
$622$	5.51105	0.220973
$623$	−4.62086	−0.185131
$624$	0	0
$625$	22.4886	0.899545
$626$	71.2386	2.84727
$627$	0	0
$628$	19.6483	0.784053
$629$	−29.5188	−1.17699
$630$	0	0
$631$	39.3731	1.56742	0.783710	−	0.621127i	$-0.213325\pi$
$631$	39.3731	1.56742	0.783710	+	0.621127i	$0.213325\pi$
$632$	13.2859	0.528486
$633$	0	0
$634$	−62.4374	−2.47971
$635$	−0.411504	−0.0163301
$636$	0	0
$637$	3.22161	0.127645
$638$	18.9732	0.751155
$639$	0	0
$640$	6.71046	0.265254
$641$	19.0128	0.750959	0.375480	−	0.926831i	$-0.377478\pi$
$641$	19.0128	0.750959	0.375480	+	0.926831i	$0.377478\pi$
$642$	0	0
$643$	7.38745	0.291333	0.145666	−	0.989334i	$-0.453467\pi$
$643$	7.38745	0.291333	0.145666	+	0.989334i	$0.453467\pi$
$644$	−7.06822	−0.278527
$645$	0	0
$646$	66.6929	2.62400
$647$	−7.69455	−0.302504	−0.151252	−	0.988495i	$-0.548331\pi$
$647$	−7.69455	−0.302504	−0.151252	+	0.988495i	$0.548331\pi$
$648$	0	0
$649$	1.26160	0.0495223
$650$	34.9815	1.37209
$651$	0	0
$652$	−18.6576	−0.730687
$653$	−16.9199	−0.662128	−0.331064	−	0.943608i	$-0.607408\pi$
$653$	−16.9199	−0.662128	−0.331064	+	0.943608i	$0.607408\pi$
$654$	0	0
$655$	4.70200	0.183722
$656$	2.55496	0.0997544
$657$	0	0
$658$	−11.7395	−0.457652
$659$	47.8532	1.86410	0.932049	−	0.362333i	$-0.118020\pi$
$659$	47.8532	1.86410	0.932049	+	0.362333i	$0.118020\pi$
$660$	0	0
$661$	−24.8740	−0.967486	−0.483743	−	0.875210i	$-0.660723\pi$
$661$	−24.8740	−0.967486	−0.483743	+	0.875210i	$0.660723\pi$
$662$	−24.9439	−0.969473
$663$	0	0
$664$	1.02462	0.0397631
$665$	2.59941	0.100801
$666$	0	0
$667$	−9.55752	−0.370069
$668$	49.0431	1.89753
$669$	0	0
$670$	−10.8888	−0.420671
$671$	11.0171	0.425312
$672$	0	0
$673$	−19.5653	−0.754188	−0.377094	−	0.926175i	$-0.623077\pi$
$673$	−19.5653	−0.754188	−0.377094	+	0.926175i	$0.623077\pi$
$674$	56.8196	2.18861
$675$	0	0
$676$	−8.00173	−0.307759
$677$	−10.9550	−0.421035	−0.210518	−	0.977590i	$-0.567515\pi$
$677$	−10.9550	−0.421035	−0.210518	+	0.977590i	$0.567515\pi$
$678$	0	0
$679$	−18.3025	−0.702386
$680$	−4.57328	−0.175377
$681$	0	0
$682$	47.3749	1.81408
$683$	36.8029	1.40822	0.704112	−	0.710089i	$-0.251345\pi$
$683$	36.8029	1.40822	0.704112	+	0.710089i	$0.251345\pi$
$684$	0	0
$685$	4.58964	0.175361
$686$	2.24781	0.0858216
$687$	0	0
$688$	−7.84875	−0.299231
$689$	10.4463	0.397972
$690$	0	0
$691$	−13.7842	−0.524374	−0.262187	−	0.965017i	$-0.584444\pi$
$691$	−13.7842	−0.524374	−0.262187	+	0.965017i	$0.584444\pi$
$692$	−32.7885	−1.24643
$693$	0	0
$694$	−69.1162	−2.62362
$695$	−1.14057	−0.0432645
$696$	0	0
$697$	−15.2541	−0.577791
$698$	3.36456	0.127351
$699$	0	0
$700$	14.7462	0.557355
$701$	−5.57306	−0.210492	−0.105246	−	0.994446i	$-0.533563\pi$
$701$	−5.57306	−0.210492	−0.105246	+	0.994446i	$0.533563\pi$
$702$	0	0
$703$	39.6991	1.49728
$704$	−26.6627	−1.00489
$705$	0	0
$706$	−8.31499	−0.312939
$707$	−2.42436	−0.0911774
$708$	0	0
$709$	52.4026	1.96802	0.984010	−	0.178112i	$-0.0569989\pi$
$709$	52.4026	1.96802	0.984010	+	0.178112i	$0.0569989\pi$
$710$	−14.9306	−0.560335
$711$	0	0
$712$	−10.9334	−0.409748
$713$	−23.8646	−0.893737
$714$	0	0
$715$	2.71093	0.101383
$716$	−38.4533	−1.43707
$717$	0	0
$718$	−67.2158	−2.50847
$719$	4.36820	0.162906	0.0814532	−	0.996677i	$-0.474044\pi$
$719$	4.36820	0.162906	0.0814532	+	0.996677i	$0.474044\pi$
$720$	0	0
$721$	−8.14077	−0.303178
$722$	−46.9851	−1.74861
$723$	0	0
$724$	41.8756	1.55630
$725$	19.9396	0.740538
$726$	0	0
$727$	14.3236	0.531233	0.265616	−	0.964079i	$-0.414425\pi$
$727$	14.3236	0.531233	0.265616	+	0.964079i	$0.414425\pi$
$728$	7.62266	0.282514
$729$	0	0
$730$	−0.259070	−0.00958860
$731$	46.8601	1.73318
$732$	0	0
$733$	−8.52787	−0.314984	−0.157492	−	0.987520i	$-0.550341\pi$
$733$	−8.52787	−0.314984	−0.157492	+	0.987520i	$0.550341\pi$
$734$	−56.6012	−2.08919
$735$	0	0
$736$	15.0518	0.554817
$737$	24.0723	0.886715
$738$	0	0
$739$	0.722415	0.0265745	0.0132872	−	0.999912i	$-0.495770\pi$
$739$	0.722415	0.0265745	0.0132872	+	0.999912i	$0.495770\pi$
$740$	−7.89456	−0.290210
$741$	0	0
$742$	7.28866	0.267575
$743$	−12.0501	−0.442074	−0.221037	−	0.975265i	$-0.570944\pi$
$743$	−12.0501	−0.442074	−0.221037	+	0.975265i	$0.570944\pi$
$744$	0	0
$745$	4.22416	0.154761
$746$	−51.5352	−1.88684
$747$	0	0
$748$	29.3201	1.07205
$749$	12.1165	0.442729
$750$	0	0
$751$	9.16914	0.334587	0.167293	−	0.985907i	$-0.446497\pi$
$751$	9.16914	0.334587	0.167293	+	0.985907i	$0.446497\pi$
$752$	4.10872	0.149829
$753$	0	0
$754$	29.8910	1.08857
$755$	8.88330	0.323297
$756$	0	0
$757$	−38.2682	−1.39088	−0.695441	−	0.718583i	$-0.744791\pi$
$757$	−38.2682	−1.39088	−0.695441	+	0.718583i	$0.744791\pi$
$758$	32.2042	1.16971
$759$	0	0
$760$	6.15049	0.223102
$761$	−44.8942	−1.62741	−0.813707	−	0.581275i	$-0.802554\pi$
$761$	−44.8942	−1.62741	−0.813707	+	0.581275i	$0.802554\pi$
$762$	0	0
$763$	16.1823	0.585839
$764$	18.5714	0.671889
$765$	0	0
$766$	−6.91405	−0.249815
$767$	1.98758	0.0717673
$768$	0	0
$769$	−11.8095	−0.425860	−0.212930	−	0.977067i	$-0.568301\pi$
$769$	−11.8095	−0.425860	−0.212930	+	0.977067i	$0.568301\pi$
$770$	1.89149	0.0681647
$771$	0	0
$772$	−76.4335	−2.75090
$773$	39.7407	1.42937	0.714686	−	0.699445i	$-0.246570\pi$
$773$	39.7407	1.42937	0.714686	+	0.699445i	$0.246570\pi$
$774$	0	0
$775$	49.7881	1.78844
$776$	−43.3056	−1.55458
$777$	0	0
$778$	86.9175	3.11614
$779$	20.5148	0.735021
$780$	0	0
$781$	33.0077	1.18111
$782$	−24.4464	−0.874200
$783$	0	0
$784$	−0.786714	−0.0280969
$785$	2.64866	0.0945347
$786$	0	0
$787$	39.5717	1.41058	0.705290	−	0.708919i	$-0.250817\pi$
$787$	39.5717	1.41058	0.705290	+	0.708919i	$0.250817\pi$
$788$	−41.2709	−1.47022
$789$	0	0
$790$	5.19388	0.184790
$791$	1.15714	0.0411433
$792$	0	0
$793$	17.3568	0.616358
$794$	40.3957	1.43359
$795$	0	0
$796$	31.7138	1.12406
$797$	9.84539	0.348742	0.174371	−	0.984680i	$-0.444211\pi$
$797$	9.84539	0.348742	0.174371	+	0.984680i	$0.444211\pi$
$798$	0	0
$799$	−24.5306	−0.867832
$800$	−31.4022	−1.11023
$801$	0	0
$802$	15.1169	0.533795
$803$	0.572737	0.0202114
$804$	0	0
$805$	−0.952819	−0.0335825
$806$	74.6362	2.62895
$807$	0	0
$808$	−5.73629	−0.201802
$809$	43.5123	1.52981	0.764905	−	0.644143i	$-0.222786\pi$
$809$	43.5123	1.52981	0.764905	+	0.644143i	$0.222786\pi$
$810$	0	0
$811$	−12.2355	−0.429647	−0.214824	−	0.976653i	$-0.568918\pi$
$811$	−12.2355	−0.429647	−0.214824	+	0.976653i	$0.568918\pi$
$812$	12.6004	0.442186
$813$	0	0
$814$	28.8875	1.01251
$815$	−2.51510	−0.0881002
$816$	0	0
$817$	−63.0209	−2.20482
$818$	−34.1690	−1.19469
$819$	0	0
$820$	−4.07958	−0.142465
$821$	20.0667	0.700334	0.350167	−	0.936687i	$-0.386125\pi$
$821$	20.0667	0.700334	0.350167	+	0.936687i	$0.386125\pi$
$822$	0	0
$823$	−1.42672	−0.0497323	−0.0248662	−	0.999691i	$-0.507916\pi$
$823$	−1.42672	−0.0497323	−0.0248662	+	0.999691i	$0.507916\pi$
$824$	−19.2619	−0.671021
$825$	0	0
$826$	1.38679	0.0482526
$827$	−31.7271	−1.10326	−0.551629	−	0.834089i	$-0.685994\pi$
$827$	−31.7271	−1.10326	−0.551629	+	0.834089i	$0.685994\pi$
$828$	0	0
$829$	−21.7979	−0.757073	−0.378537	−	0.925586i	$-0.623573\pi$
$829$	−21.7979	−0.757073	−0.378537	+	0.925586i	$0.623573\pi$
$830$	0.400556	0.0139035
$831$	0	0
$832$	−42.0053	−1.45627
$833$	4.69699	0.162741
$834$	0	0
$835$	6.61117	0.228789
$836$	−39.4318	−1.36378
$837$	0	0
$838$	−2.48519	−0.0858495
$839$	29.9953	1.03555	0.517776	−	0.855516i	$-0.326760\pi$
$839$	29.9953	1.03555	0.517776	+	0.855516i	$0.326760\pi$
$840$	0	0
$841$	−11.9620	−0.412483
$842$	25.2013	0.868492
$843$	0	0
$844$	23.9837	0.825552
$845$	−1.07866	−0.0371070
$846$	0	0
$847$	6.81839	0.234283
$848$	−2.55098	−0.0876008
$849$	0	0
$850$	51.0018	1.74935
$851$	−14.5518	−0.498828
$852$	0	0
$853$	44.5822	1.52647	0.763234	−	0.646123i	$-0.223611\pi$
$853$	44.5822	1.52647	0.763234	+	0.646123i	$0.223611\pi$
$854$	12.1103	0.414407
$855$	0	0
$856$	28.6690	0.979887
$857$	−16.9515	−0.579053	−0.289527	−	0.957170i	$-0.593498\pi$
$857$	−16.9515	−0.579053	−0.289527	+	0.957170i	$0.593498\pi$
$858$	0	0
$859$	−30.5547	−1.04251	−0.521256	−	0.853400i	$-0.674536\pi$
$859$	−30.5547	−1.04251	−0.521256	+	0.853400i	$0.674536\pi$
$860$	12.5323	0.427349
$861$	0	0
$862$	29.6957	1.01144
$863$	−21.0245	−0.715682	−0.357841	−	0.933782i	$-0.616487\pi$
$863$	−21.0245	−0.715682	−0.357841	+	0.933782i	$0.616487\pi$
$864$	0	0
$865$	−4.42000	−0.150285
$866$	88.7374	3.01542
$867$	0	0
$868$	31.4624	1.06790
$869$	−11.4823	−0.389511
$870$	0	0
$871$	37.9244	1.28502
$872$	38.2891	1.29663
$873$	0	0
$874$	32.8773	1.11209
$875$	4.04536	0.136758
$876$	0	0
$877$	−47.4470	−1.60217	−0.801086	−	0.598550i	$-0.795744\pi$
$877$	−47.4470	−1.60217	−0.801086	+	0.598550i	$0.795744\pi$
$878$	61.3269	2.06968
$879$	0	0
$880$	−0.662008	−0.0223163
$881$	54.6629	1.84164	0.920820	−	0.389988i	$-0.127521\pi$
$881$	54.6629	1.84164	0.920820	+	0.389988i	$0.127521\pi$
$882$	0	0
$883$	−16.6543	−0.560462	−0.280231	−	0.959933i	$-0.590411\pi$
$883$	−16.6543	−0.560462	−0.280231	+	0.959933i	$0.590411\pi$
$884$	46.1919	1.55360
$885$	0	0
$886$	9.66245	0.324616
$887$	46.7709	1.57041	0.785207	−	0.619234i	$-0.212557\pi$
$887$	46.7709	1.57041	0.785207	+	0.619234i	$0.212557\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−4.27421	−0.143272
$891$	0	0
$892$	12.9539	0.433729
$893$	32.9906	1.10399
$894$	0	0
$895$	−5.18363	−0.173270
$896$	−16.3071	−0.544783
$897$	0	0
$898$	−51.5918	−1.72164
$899$	42.5430	1.41889
$900$	0	0
$901$	15.2303	0.507395
$902$	14.9279	0.497043
$903$	0	0
$904$	2.73793	0.0910621
$905$	5.64497	0.187645
$906$	0	0
$907$	47.9136	1.59095	0.795473	−	0.605989i	$-0.207223\pi$
$907$	47.9136	1.59095	0.795473	+	0.605989i	$0.207223\pi$
$908$	−2.33209	−0.0773931
$909$	0	0
$910$	2.97993	0.0987836
$911$	−5.06738	−0.167890	−0.0839449	−	0.996470i	$-0.526752\pi$
$911$	−5.06738	−0.167890	−0.0839449	+	0.996470i	$0.526752\pi$
$912$	0	0
$913$	−0.885526	−0.0293066
$914$	−50.0191	−1.65448
$915$	0	0
$916$	14.8945	0.492128
$917$	−11.4264	−0.377332
$918$	0	0
$919$	−5.65814	−0.186645	−0.0933224	−	0.995636i	$-0.529749\pi$
$919$	−5.65814	−0.186645	−0.0933224	+	0.995636i	$0.529749\pi$
$920$	−2.25447	−0.0743277
$921$	0	0
$922$	−36.4150	−1.19926
$923$	52.0015	1.71165
$924$	0	0
$925$	30.3589	0.998195
$926$	22.6665	0.744867
$927$	0	0
$928$	−26.8326	−0.880822
$929$	12.2715	0.402615	0.201308	−	0.979528i	$-0.435481\pi$
$929$	12.2715	0.402615	0.201308	+	0.979528i	$0.435481\pi$
$930$	0	0
$931$	−6.31686	−0.207027
$932$	−24.5484	−0.804111
$933$	0	0
$934$	−36.5677	−1.19653
$935$	3.95244	0.129259
$936$	0	0
$937$	31.1120	1.01638	0.508192	−	0.861244i	$-0.330314\pi$
$937$	31.1120	1.01638	0.508192	+	0.861244i	$0.330314\pi$
$938$	26.4609	0.863980
$939$	0	0
$940$	−6.56051	−0.213980
$941$	23.4344	0.763941	0.381970	−	0.924175i	$-0.375246\pi$
$941$	23.4344	0.763941	0.381970	+	0.924175i	$0.375246\pi$
$942$	0	0
$943$	−7.51975	−0.244877
$944$	−0.485365	−0.0157973
$945$	0	0
$946$	−45.8579	−1.49097
$947$	−9.84376	−0.319879	−0.159940	−	0.987127i	$-0.551130\pi$
$947$	−9.84376	−0.319879	−0.159940	+	0.987127i	$0.551130\pi$
$948$	0	0
$949$	0.902310	0.0292902
$950$	−68.5909	−2.22538
$951$	0	0
$952$	11.1136	0.360193
$953$	−29.8256	−0.966145	−0.483072	−	0.875580i	$-0.660479\pi$
$953$	−29.8256	−0.966145	−0.483072	+	0.875580i	$0.660479\pi$
$954$	0	0
$955$	2.50348	0.0810108
$956$	3.44753	0.111501
$957$	0	0
$958$	−53.4716	−1.72759
$959$	−11.1533	−0.360159
$960$	0	0
$961$	75.2275	2.42669
$962$	45.5104	1.46731
$963$	0	0
$964$	22.8169	0.734884
$965$	−10.3035	−0.331681
$966$	0	0
$967$	29.0012	0.932617	0.466309	−	0.884622i	$-0.345584\pi$
$967$	29.0012	0.932617	0.466309	+	0.884622i	$0.345584\pi$
$968$	16.1330	0.518535
$969$	0	0
$970$	−16.9295	−0.543573
$971$	−51.3928	−1.64928	−0.824638	−	0.565662i	$-0.808621\pi$
$971$	−51.3928	−1.64928	−0.824638	+	0.565662i	$0.808621\pi$
$972$	0	0
$973$	2.77172	0.0888572
$974$	1.40490	0.0450159
$975$	0	0
$976$	−4.23852	−0.135672
$977$	−0.390921	−0.0125067	−0.00625334	−	0.999980i	$-0.501991\pi$
$977$	−0.390921	−0.0125067	−0.00625334	+	0.999980i	$0.501991\pi$
$978$	0	0
$979$	9.44918	0.301997
$980$	1.25617	0.0401269
$981$	0	0
$982$	68.1160	2.17367
$983$	0.781530	0.0249269	0.0124635	−	0.999922i	$-0.496033\pi$
$983$	0.781530	0.0249269	0.0124635	+	0.999922i	$0.496033\pi$
$984$	0	0
$985$	−5.56346	−0.177267
$986$	43.5800	1.38787
$987$	0	0
$988$	−62.1223	−1.97637
$989$	23.1004	0.734550
$990$	0	0
$991$	−25.4455	−0.808303	−0.404152	−	0.914692i	$-0.632433\pi$
$991$	−25.4455	−0.808303	−0.404152	+	0.914692i	$0.632433\pi$
$992$	−66.9994	−2.12723
$993$	0	0
$994$	36.2829	1.15082
$995$	4.27512	0.135530
$996$	0	0
$997$	11.7621	0.372510	0.186255	−	0.982501i	$-0.440365\pi$
$997$	11.7621	0.372510	0.186255	+	0.982501i	$0.440365\pi$
$998$	−17.9462	−0.568076
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.2		16
3.2	odd	2		889.2.a.c.1.15	✓	16
21.20	even	2		6223.2.a.k.1.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.15	✓	16	3.2	odd	2
6223.2.a.k.1.15		16	21.20	even	2
8001.2.a.t.1.2		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)

\(f(q)\)	\(=\)	\(q-2.24781 q^{2} +3.05263 q^{4} +0.411504 q^{5} -1.00000 q^{7} -2.36611 q^{8} +O(q^{10})\)	\(q-2.24781 q^{2} +3.05263 q^{4} +0.411504 q^{5} -1.00000 q^{7} -2.36611 q^{8} -0.924982 q^{10} +2.04490 q^{11} +3.22161 q^{13} +2.24781 q^{14} -0.786714 q^{16} +4.69699 q^{17} -6.31686 q^{19} +1.25617 q^{20} -4.59653 q^{22} +2.31545 q^{23} -4.83066 q^{25} -7.24154 q^{26} -3.05263 q^{28} -4.12771 q^{29} -10.3067 q^{31} +6.50059 q^{32} -10.5579 q^{34} -0.411504 q^{35} -6.28463 q^{37} +14.1991 q^{38} -0.973663 q^{40} -3.24764 q^{41} +9.97662 q^{43} +6.24232 q^{44} -5.20469 q^{46} -5.22263 q^{47} +1.00000 q^{49} +10.8584 q^{50} +9.83437 q^{52} +3.24257 q^{53} +0.841485 q^{55} +2.36611 q^{56} +9.27829 q^{58} +0.616953 q^{59} +5.38762 q^{61} +23.1674 q^{62} -13.0386 q^{64} +1.32571 q^{65} +11.7719 q^{67} +14.3382 q^{68} +0.924982 q^{70} +16.1415 q^{71} +0.280081 q^{73} +14.1266 q^{74} -19.2830 q^{76} -2.04490 q^{77} -5.61511 q^{79} -0.323736 q^{80} +7.30005 q^{82} -0.433042 q^{83} +1.93283 q^{85} -22.4255 q^{86} -4.83844 q^{88} +4.62086 q^{89} -3.22161 q^{91} +7.06822 q^{92} +11.7395 q^{94} -2.59941 q^{95} +18.3025 q^{97} -2.24781 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Database

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