LMFDB - Embedded newform 2001.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, 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("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.751024$ of defining polynomial
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000 q^{4} -1.58049 q^{5} -3.08254 q^{7} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{3} -2.00000 q^{4} -1.58049 q^{5} -3.08254 q^{7} +1.00000 q^{9} +6.03291 q^{11} +2.00000 q^{12} -2.36988 q^{13} +1.58049 q^{15} +4.00000 q^{16} +6.85002 q^{17} -5.13736 q^{19} +3.16098 q^{20} +3.08254 q^{21} +1.00000 q^{23} -2.50205 q^{25} -1.00000 q^{27} +6.16508 q^{28} +1.00000 q^{29} +0.0784427 q^{31} -6.03291 q^{33} +4.87193 q^{35} -2.00000 q^{36} +5.42641 q^{37} +2.36988 q^{39} +7.95446 q^{41} -2.05653 q^{43} -12.0658 q^{44} -1.58049 q^{45} +1.58049 q^{47} -4.00000 q^{48} +2.50205 q^{49} -6.85002 q^{51} +4.73975 q^{52} -5.00410 q^{53} -9.53496 q^{55} +5.13736 q^{57} -0.756479 q^{59} -3.16098 q^{60} -3.79110 q^{61} -3.08254 q^{63} -8.00000 q^{64} +3.74557 q^{65} +6.16508 q^{67} -13.7000 q^{68} -1.00000 q^{69} -14.1154 q^{71} -2.91746 q^{73} +2.50205 q^{75} +10.2747 q^{76} -18.5967 q^{77} -14.1415 q^{79} -6.32196 q^{80} +1.00000 q^{81} +5.48123 q^{83} -6.16508 q^{84} -10.8264 q^{85} -1.00000 q^{87} -13.5378 q^{89} +7.30524 q^{91} -2.00000 q^{92} -0.0784427 q^{93} +8.11954 q^{95} +2.53086 q^{97} +6.03291 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9} + 8 q^{12} + 2 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{21} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{28} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 8 q^{36} + 4 q^{37} + 6 q^{41} - 2 q^{45} + 2 q^{47} - 16 q^{48} + 4 q^{49} - 2 q^{51} - 8 q^{53} - 8 q^{55} - 4 q^{57} - 22 q^{59} - 4 q^{60} - 16 q^{61} - 2 q^{63} - 32 q^{64} - 10 q^{65} + 4 q^{67} - 4 q^{68} - 4 q^{69} - 22 q^{71} - 22 q^{73} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 20 q^{79} - 8 q^{80} + 4 q^{81} - 10 q^{83} - 4 q^{84} - 2 q^{85} - 4 q^{87} - 14 q^{89} - 26 q^{91} - 8 q^{92} - 2 q^{93} - 14 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9 + 8 * q^12 + 2 * q^15 + 16 * q^16 + 2 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^21 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 4 * q^28 + 4 * q^29 + 2 * q^31 + 4 * q^35 - 8 * q^36 + 4 * q^37 + 6 * q^41 - 2 * q^45 + 2 * q^47 - 16 * q^48 + 4 * q^49 - 2 * q^51 - 8 * q^53 - 8 * q^55 - 4 * q^57 - 22 * q^59 - 4 * q^60 - 16 * q^61 - 2 * q^63 - 32 * q^64 - 10 * q^65 + 4 * q^67 - 4 * q^68 - 4 * q^69 - 22 * q^71 - 22 * q^73 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 20 * q^79 - 8 * q^80 + 4 * q^81 - 10 * q^83 - 4 * q^84 - 2 * q^85 - 4 * q^87 - 14 * q^89 - 26 * q^91 - 8 * q^92 - 2 * q^93 - 14 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−1.58049	−0.706817	−0.353409	−	0.935469i	$-0.614977\pi$
$5$	−1.58049	−0.706817	−0.353409	+	0.935469i	$0.614977\pi$
$6$	0	0
$7$	−3.08254	−1.16509	−0.582545	−	0.812798i	$-0.697943\pi$
$7$	−3.08254	−1.16509	−0.582545	+	0.812798i	$0.697943\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.03291	1.81899	0.909495	−	0.415715i	$-0.136469\pi$
$11$	6.03291	1.81899	0.909495	+	0.415715i	$0.136469\pi$
$12$	2.00000	0.577350
$13$	−2.36988	−0.657286	−0.328643	−	0.944454i	$-0.606591\pi$
$13$	−2.36988	−0.657286	−0.328643	+	0.944454i	$0.606591\pi$
$14$	0	0
$15$	1.58049	0.408081
$16$	4.00000	1.00000
$17$	6.85002	1.66137	0.830687	−	0.556740i	$-0.187948\pi$
$17$	6.85002	1.66137	0.830687	+	0.556740i	$0.187948\pi$
$18$	0	0
$19$	−5.13736	−1.17859	−0.589295	−	0.807918i	$-0.700594\pi$
$19$	−5.13736	−1.17859	−0.589295	+	0.807918i	$0.700594\pi$
$20$	3.16098	0.706817
$21$	3.08254	0.672665
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.50205	−0.500410
$26$	0	0
$27$	−1.00000	−0.192450
$28$	6.16508	1.16509
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.0784427	0.0140887	0.00704436	−	0.999975i	$-0.497758\pi$
$31$	0.0784427	0.0140887	0.00704436	+	0.999975i	$0.497758\pi$
$32$	0	0
$33$	−6.03291	−1.05019
$34$	0	0
$35$	4.87193	0.823506
$36$	−2.00000	−0.333333
$37$	5.42641	0.892097	0.446048	−	0.895009i	$-0.352831\pi$
$37$	5.42641	0.892097	0.446048	+	0.895009i	$0.352831\pi$
$38$	0	0
$39$	2.36988	0.379484
$40$	0	0
$41$	7.95446	1.24228	0.621139	−	0.783700i	$-0.286670\pi$
$41$	7.95446	1.24228	0.621139	+	0.783700i	$0.286670\pi$
$42$	0	0
$43$	−2.05653	−0.313619	−0.156809	−	0.987629i	$-0.550121\pi$
$43$	−2.05653	−0.313619	−0.156809	+	0.987629i	$0.550121\pi$
$44$	−12.0658	−1.81899
$45$	−1.58049	−0.235606
$46$	0	0
$47$	1.58049	0.230538	0.115269	−	0.993334i	$-0.463227\pi$
$47$	1.58049	0.230538	0.115269	+	0.993334i	$0.463227\pi$
$48$	−4.00000	−0.577350
$49$	2.50205	0.357435
$50$	0	0
$51$	−6.85002	−0.959194
$52$	4.73975	0.657286
$53$	−5.00410	−0.687366	−0.343683	−	0.939086i	$-0.611674\pi$
$53$	−5.00410	−0.687366	−0.343683	+	0.939086i	$0.611674\pi$
$54$	0	0
$55$	−9.53496	−1.28569
$56$	0	0
$57$	5.13736	0.680459
$58$	0	0
$59$	−0.756479	−0.0984852	−0.0492426	−	0.998787i	$-0.515681\pi$
$59$	−0.756479	−0.0984852	−0.0492426	+	0.998787i	$0.515681\pi$
$60$	−3.16098	−0.408081
$61$	−3.79110	−0.485401	−0.242701	−	0.970101i	$-0.578033\pi$
$61$	−3.79110	−0.485401	−0.242701	+	0.970101i	$0.578033\pi$
$62$	0	0
$63$	−3.08254	−0.388363
$64$	−8.00000	−1.00000
$65$	3.74557	0.464581
$66$	0	0
$67$	6.16508	0.753184	0.376592	−	0.926379i	$-0.377096\pi$
$67$	6.16508	0.753184	0.376592	+	0.926379i	$0.377096\pi$
$68$	−13.7000	−1.66137
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−14.1154	−1.67520	−0.837598	−	0.546288i	$-0.816041\pi$
$71$	−14.1154	−1.67520	−0.837598	+	0.546288i	$0.816041\pi$
$72$	0	0
$73$	−2.91746	−0.341463	−0.170731	−	0.985318i	$-0.554613\pi$
$73$	−2.91746	−0.341463	−0.170731	+	0.985318i	$0.554613\pi$
$74$	0	0
$75$	2.50205	0.288912
$76$	10.2747	1.17859
$77$	−18.5967	−2.11929
$78$	0	0
$79$	−14.1415	−1.59104	−0.795519	−	0.605929i	$-0.792802\pi$
$79$	−14.1415	−1.59104	−0.795519	+	0.605929i	$0.792802\pi$
$80$	−6.32196	−0.706817
$81$	1.00000	0.111111
$82$	0	0
$83$	5.48123	0.601643	0.300821	−	0.953680i	$-0.402739\pi$
$83$	5.48123	0.601643	0.300821	+	0.953680i	$0.402739\pi$
$84$	−6.16508	−0.672665
$85$	−10.8264	−1.17429
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−13.5378	−1.43500	−0.717500	−	0.696559i	$-0.754714\pi$
$89$	−13.5378	−1.43500	−0.717500	+	0.696559i	$0.754714\pi$
$90$	0	0
$91$	7.30524	0.765797
$92$	−2.00000	−0.208514
$93$	−0.0784427	−0.00813413
$94$	0	0
$95$	8.11954	0.833048
$96$	0	0
$97$	2.53086	0.256970	0.128485	−	0.991711i	$-0.458989\pi$
$97$	2.53086	0.256970	0.128485	+	0.991711i	$0.458989\pi$
$98$	0	0
$99$	6.03291	0.606330
$100$	5.00410	0.500410
$101$	−9.11545	−0.907021	−0.453510	−	0.891251i	$-0.649829\pi$
$101$	−9.11545	−0.907021	−0.453510	+	0.891251i	$0.649829\pi$
$102$	0	0
$103$	−5.74385	−0.565958	−0.282979	−	0.959126i	$-0.591323\pi$
$103$	−5.74385	−0.565958	−0.282979	+	0.959126i	$0.591323\pi$
$104$	0	0
$105$	−4.87193	−0.475451
$106$	0	0
$107$	−15.8032	−1.52775	−0.763876	−	0.645363i	$-0.776706\pi$
$107$	−15.8032	−1.52775	−0.763876	+	0.645363i	$0.776706\pi$
$108$	2.00000	0.192450
$109$	−5.81410	−0.556890	−0.278445	−	0.960452i	$-0.589819\pi$
$109$	−5.81410	−0.556890	−0.278445	+	0.960452i	$0.589819\pi$
$110$	0	0
$111$	−5.42641	−0.515052
$112$	−12.3302	−1.16509
$113$	5.53496	0.520685	0.260342	−	0.965516i	$-0.416165\pi$
$113$	5.53496	0.520685	0.260342	+	0.965516i	$0.416165\pi$
$114$	0	0
$115$	−1.58049	−0.147382
$116$	−2.00000	−0.185695
$117$	−2.36988	−0.219095
$118$	0	0
$119$	−21.1154	−1.93565
$120$	0	0
$121$	25.3960	2.30872
$122$	0	0
$123$	−7.95446	−0.717230
$124$	−0.156885	−0.0140887
$125$	11.8569	1.06052
$126$	0	0
$127$	−0.543487	−0.0482267	−0.0241133	−	0.999709i	$-0.507676\pi$
$127$	−0.543487	−0.0482267	−0.0241133	+	0.999709i	$0.507676\pi$
$128$	0	0
$129$	2.05653	0.181068
$130$	0	0
$131$	16.2227	1.41738	0.708692	−	0.705518i	$-0.249286\pi$
$131$	16.2227	1.41738	0.708692	+	0.705518i	$0.249286\pi$
$132$	12.0658	1.05019
$133$	15.8361	1.37316
$134$	0	0
$135$	1.58049	0.136027
$136$	0	0
$137$	−0.0548158	−0.00468323	−0.00234161	−	0.999997i	$-0.500745\pi$
$137$	−0.0548158	−0.00468323	−0.00234161	+	0.999997i	$0.500745\pi$
$138$	0	0
$139$	17.5679	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$139$	17.5679	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$140$	−9.74385	−0.823506
$141$	−1.58049	−0.133101
$142$	0	0
$143$	−14.2972	−1.19560
$144$	4.00000	0.333333
$145$	−1.58049	−0.131253
$146$	0	0
$147$	−2.50205	−0.206365
$148$	−10.8528	−0.892097
$149$	−14.3740	−1.17756	−0.588781	−	0.808293i	$-0.700392\pi$
$149$	−14.3740	−1.17756	−0.588781	+	0.808293i	$0.700392\pi$
$150$	0	0
$151$	−14.1939	−1.15508	−0.577541	−	0.816362i	$-0.695988\pi$
$151$	−14.1939	−1.15508	−0.577541	+	0.816362i	$0.695988\pi$
$152$	0	0
$153$	6.85002	0.553791
$154$	0	0
$155$	−0.123978	−0.00995815
$156$	−4.73975	−0.379484
$157$	−5.96546	−0.476096	−0.238048	−	0.971253i	$-0.576508\pi$
$157$	−5.96546	−0.476096	−0.238048	+	0.971253i	$0.576508\pi$
$158$	0	0
$159$	5.00410	0.396851
$160$	0	0
$161$	−3.08254	−0.242938
$162$	0	0
$163$	−22.0220	−1.72490	−0.862448	−	0.506146i	$-0.831070\pi$
$163$	−22.0220	−1.72490	−0.862448	+	0.506146i	$0.831070\pi$
$164$	−15.9089	−1.24228
$165$	9.53496	0.742295
$166$	0	0
$167$	19.1333	1.48058	0.740291	−	0.672286i	$-0.234687\pi$
$167$	19.1333	1.48058	0.740291	+	0.672286i	$0.234687\pi$
$168$	0	0
$169$	−7.38368	−0.567976
$170$	0	0
$171$	−5.13736	−0.392863
$172$	4.11307	0.313619
$173$	5.03119	0.382514	0.191257	−	0.981540i	$-0.438744\pi$
$173$	5.03119	0.382514	0.191257	+	0.981540i	$0.438744\pi$
$174$	0	0
$175$	7.71266	0.583022
$176$	24.1316	1.81899
$177$	0.756479	0.0568604
$178$	0	0
$179$	3.93946	0.294449	0.147224	−	0.989103i	$-0.452966\pi$
$179$	3.93946	0.294449	0.147224	+	0.989103i	$0.452966\pi$
$180$	3.16098	0.235606
$181$	−3.32606	−0.247224	−0.123612	−	0.992331i	$-0.539448\pi$
$181$	−3.32606	−0.247224	−0.123612	+	0.992331i	$0.539448\pi$
$182$	0	0
$183$	3.79110	0.280247
$184$	0	0
$185$	−8.57639	−0.630549
$186$	0	0
$187$	41.3255	3.02202
$188$	−3.16098	−0.230538
$189$	3.08254	0.224222
$190$	0	0
$191$	−10.5610	−0.764164	−0.382082	−	0.924128i	$-0.624793\pi$
$191$	−10.5610	−0.764164	−0.382082	+	0.924128i	$0.624793\pi$
$192$	8.00000	0.577350
$193$	8.00819	0.576442	0.288221	−	0.957564i	$-0.406936\pi$
$193$	8.00819	0.576442	0.288221	+	0.957564i	$0.406936\pi$
$194$	0	0
$195$	−3.74557	−0.268226
$196$	−5.00410	−0.357435
$197$	18.8443	1.34260	0.671300	−	0.741186i	$-0.265736\pi$
$197$	18.8443	1.34260	0.671300	+	0.741186i	$0.265736\pi$
$198$	0	0
$199$	−0.807829	−0.0572655	−0.0286327	−	0.999590i	$-0.509115\pi$
$199$	−0.807829	−0.0572655	−0.0286327	+	0.999590i	$0.509115\pi$
$200$	0	0
$201$	−6.16508	−0.434851
$202$	0	0
$203$	−3.08254	−0.216352
$204$	13.7000	0.959194
$205$	−12.5720	−0.878064
$206$	0	0
$207$	1.00000	0.0695048
$208$	−9.47951	−0.657286
$209$	−30.9932	−2.14384
$210$	0	0
$211$	−16.2517	−1.11881	−0.559407	−	0.828893i	$-0.688971\pi$
$211$	−16.2517	−1.11881	−0.559407	+	0.828893i	$0.688971\pi$
$212$	10.0082	0.687366
$213$	14.1154	0.967174
$214$	0	0
$215$	3.25033	0.221671
$216$	0	0
$217$	−0.241803	−0.0164146
$218$	0	0
$219$	2.91746	0.197144
$220$	19.0699	1.28569
$221$	−16.2337	−1.09200
$222$	0	0
$223$	9.39869	0.629383	0.314691	−	0.949194i	$-0.398099\pi$
$223$	9.39869	0.629383	0.314691	+	0.949194i	$0.398099\pi$
$224$	0	0
$225$	−2.50205	−0.166803
$226$	0	0
$227$	19.9145	1.32177	0.660887	−	0.750485i	$-0.270180\pi$
$227$	19.9145	1.32177	0.660887	+	0.750485i	$0.270180\pi$
$228$	−10.2747	−0.680459
$229$	13.9682	0.923043	0.461522	−	0.887129i	$-0.347304\pi$
$229$	13.9682	0.923043	0.461522	+	0.887129i	$0.347304\pi$
$230$	0	0
$231$	18.5967	1.22357
$232$	0	0
$233$	−24.1813	−1.58417	−0.792084	−	0.610413i	$-0.791004\pi$
$233$	−24.1813	−1.58417	−0.792084	+	0.610413i	$0.791004\pi$
$234$	0	0
$235$	−2.49795	−0.162948
$236$	1.51296	0.0984852
$237$	14.1415	0.918586
$238$	0	0
$239$	−6.94066	−0.448954	−0.224477	−	0.974479i	$-0.572067\pi$
$239$	−6.94066	−0.448954	−0.224477	+	0.974479i	$0.572067\pi$
$240$	6.32196	0.408081
$241$	−8.10963	−0.522387	−0.261194	−	0.965286i	$-0.584116\pi$
$241$	−8.10963	−0.522387	−0.261194	+	0.965286i	$0.584116\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	7.58221	0.485401
$245$	−3.95446	−0.252641
$246$	0	0
$247$	12.1749	0.774670
$248$	0	0
$249$	−5.48123	−0.347359
$250$	0	0
$251$	−16.7178	−1.05522	−0.527611	−	0.849486i	$-0.676912\pi$
$251$	−16.7178	−1.05522	−0.527611	+	0.849486i	$0.676912\pi$
$252$	6.16508	0.388363
$253$	6.03291	0.379286
$254$	0	0
$255$	10.8264	0.677975
$256$	16.0000	1.00000
$257$	5.06410	0.315890	0.157945	−	0.987448i	$-0.449513\pi$
$257$	5.06410	0.315890	0.157945	+	0.987448i	$0.449513\pi$
$258$	0	0
$259$	−16.7271	−1.03937
$260$	−7.49114	−0.464581
$261$	1.00000	0.0618984
$262$	0	0
$263$	7.51585	0.463447	0.231724	−	0.972782i	$-0.425563\pi$
$263$	7.51585	0.463447	0.231724	+	0.972782i	$0.425563\pi$
$264$	0	0
$265$	7.90893	0.485842
$266$	0	0
$267$	13.5378	0.828498
$268$	−12.3302	−0.753184
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−30.4016	−1.84676	−0.923382	−	0.383882i	$-0.874587\pi$
$271$	−30.4016	−1.84676	−0.923382	+	0.383882i	$0.874587\pi$
$272$	27.4001	1.66137
$273$	−7.30524	−0.442133
$274$	0	0
$275$	−15.0946	−0.910240
$276$	2.00000	0.120386
$277$	−1.57196	−0.0944499	−0.0472250	−	0.998884i	$-0.515038\pi$
$277$	−1.57196	−0.0944499	−0.0472250	+	0.998884i	$0.515038\pi$
$278$	0	0
$279$	0.0784427	0.00469624
$280$	0	0
$281$	−15.0179	−0.895893	−0.447946	−	0.894060i	$-0.647844\pi$
$281$	−15.0179	−0.895893	−0.447946	+	0.894060i	$0.647844\pi$
$282$	0	0
$283$	0.0866358	0.00514997	0.00257498	−	0.999997i	$-0.499180\pi$
$283$	0.0866358	0.00514997	0.00257498	+	0.999997i	$0.499180\pi$
$284$	28.2309	1.67520
$285$	−8.11954	−0.480960
$286$	0	0
$287$	−24.5199	−1.44737
$288$	0	0
$289$	29.9227	1.76016
$290$	0	0
$291$	−2.53086	−0.148362
$292$	5.83492	0.341463
$293$	−23.0809	−1.34840	−0.674201	−	0.738548i	$-0.735512\pi$
$293$	−23.0809	−1.34840	−0.674201	+	0.738548i	$0.735512\pi$
$294$	0	0
$295$	1.19561	0.0696110
$296$	0	0
$297$	−6.03291	−0.350065
$298$	0	0
$299$	−2.36988	−0.137054
$300$	−5.00410	−0.288912
$301$	6.33935	0.365394
$302$	0	0
$303$	9.11545	0.523669
$304$	−20.5494	−1.17859
$305$	5.99181	0.343090
$306$	0	0
$307$	−20.8956	−1.19258	−0.596289	−	0.802770i	$-0.703359\pi$
$307$	−20.8956	−1.19258	−0.596289	+	0.802770i	$0.703359\pi$
$308$	37.1933	2.11929
$309$	5.74385	0.326756
$310$	0	0
$311$	−4.85282	−0.275178	−0.137589	−	0.990489i	$-0.543935\pi$
$311$	−4.85282	−0.275178	−0.137589	+	0.990489i	$0.543935\pi$
$312$	0	0
$313$	29.2004	1.65050	0.825251	−	0.564766i	$-0.191034\pi$
$313$	29.2004	1.65050	0.825251	+	0.564766i	$0.191034\pi$
$314$	0	0
$315$	4.87193	0.274502
$316$	28.2829	1.59104
$317$	22.2512	1.24975	0.624875	−	0.780725i	$-0.285150\pi$
$317$	22.2512	1.24975	0.624875	+	0.780725i	$0.285150\pi$
$318$	0	0
$319$	6.03291	0.337778
$320$	12.6439	0.706817
$321$	15.8032	0.882048
$322$	0	0
$323$	−35.1910	−1.95808
$324$	−2.00000	−0.111111
$325$	5.92955	0.328912
$326$	0	0
$327$	5.81410	0.321520
$328$	0	0
$329$	−4.87193	−0.268598
$330$	0	0
$331$	−17.4001	−0.956394	−0.478197	−	0.878253i	$-0.658710\pi$
$331$	−17.4001	−0.956394	−0.478197	+	0.878253i	$0.658710\pi$
$332$	−10.9625	−0.601643
$333$	5.42641	0.297366
$334$	0	0
$335$	−9.74385	−0.532363
$336$	12.3302	0.672665
$337$	−34.6141	−1.88555	−0.942776	−	0.333426i	$-0.891795\pi$
$337$	−34.6141	−1.88555	−0.942776	+	0.333426i	$0.891795\pi$
$338$	0	0
$339$	−5.53496	−0.300618
$340$	21.6528	1.17429
$341$	0.473237	0.0256272
$342$	0	0
$343$	13.8651	0.748646
$344$	0	0
$345$	1.58049	0.0850908
$346$	0	0
$347$	−6.08254	−0.326528	−0.163264	−	0.986582i	$-0.552202\pi$
$347$	−6.08254	−0.326528	−0.163264	+	0.986582i	$0.552202\pi$
$348$	2.00000	0.107211
$349$	−20.4911	−1.09687	−0.548433	−	0.836195i	$-0.684775\pi$
$349$	−20.4911	−1.09687	−0.548433	+	0.836195i	$0.684775\pi$
$350$	0	0
$351$	2.36988	0.126495
$352$	0	0
$353$	−16.4623	−0.876201	−0.438101	−	0.898926i	$-0.644349\pi$
$353$	−16.4623	−0.876201	−0.438101	+	0.898926i	$0.644349\pi$
$354$	0	0
$355$	22.3093	1.18406
$356$	27.0755	1.43500
$357$	21.1154	1.11755
$358$	0	0
$359$	−31.0398	−1.63822	−0.819109	−	0.573638i	$-0.805532\pi$
$359$	−31.0398	−1.63822	−0.819109	+	0.573638i	$0.805532\pi$
$360$	0	0
$361$	7.39242	0.389075
$362$	0	0
$363$	−25.3960	−1.33294
$364$	−14.6105	−0.765797
$365$	4.61102	0.241352
$366$	0	0
$367$	6.90483	0.360429	0.180215	−	0.983627i	$-0.442321\pi$
$367$	6.90483	0.360429	0.180215	+	0.983627i	$0.442321\pi$
$368$	4.00000	0.208514
$369$	7.95446	0.414093
$370$	0	0
$371$	15.4253	0.800843
$372$	0.156885	0.00813413
$373$	20.3093	1.05158	0.525789	−	0.850615i	$-0.323770\pi$
$373$	20.3093	1.05158	0.525789	+	0.850615i	$0.323770\pi$
$374$	0	0
$375$	−11.8569	−0.612289
$376$	0	0
$377$	−2.36988	−0.122055
$378$	0	0
$379$	−12.4414	−0.639073	−0.319536	−	0.947574i	$-0.603527\pi$
$379$	−12.4414	−0.639073	−0.319536	+	0.947574i	$0.603527\pi$
$380$	−16.2391	−0.833048
$381$	0.543487	0.0278437
$382$	0	0
$383$	−13.0535	−0.667004	−0.333502	−	0.942749i	$-0.608230\pi$
$383$	−13.0535	−0.667004	−0.333502	+	0.942749i	$0.608230\pi$
$384$	0	0
$385$	29.3919	1.49795
$386$	0	0
$387$	−2.05653	−0.104540
$388$	−5.06172	−0.256970
$389$	5.61231	0.284555	0.142278	−	0.989827i	$-0.454557\pi$
$389$	5.61231	0.284555	0.142278	+	0.989827i	$0.454557\pi$
$390$	0	0
$391$	6.85002	0.346420
$392$	0	0
$393$	−16.2227	−0.818327
$394$	0	0
$395$	22.3504	1.12457
$396$	−12.0658	−0.606330
$397$	3.27935	0.164586	0.0822929	−	0.996608i	$-0.473776\pi$
$397$	3.27935	0.164586	0.0822929	+	0.996608i	$0.473776\pi$
$398$	0	0
$399$	−15.8361	−0.792797
$400$	−10.0082	−0.500410
$401$	21.9644	1.09685	0.548424	−	0.836200i	$-0.315228\pi$
$401$	21.9644	1.09685	0.548424	+	0.836200i	$0.315228\pi$
$402$	0	0
$403$	−0.185900	−0.00926032
$404$	18.2309	0.907021
$405$	−1.58049	−0.0785352
$406$	0	0
$407$	32.7370	1.62271
$408$	0	0
$409$	32.7409	1.61893	0.809467	−	0.587165i	$-0.199756\pi$
$409$	32.7409	1.61893	0.809467	+	0.587165i	$0.199756\pi$
$410$	0	0
$411$	0.0548158	0.00270386
$412$	11.4877	0.565958
$413$	2.33188	0.114744
$414$	0	0
$415$	−8.66303	−0.425251
$416$	0	0
$417$	−17.5679	−0.860302
$418$	0	0
$419$	−10.1131	−0.494056	−0.247028	−	0.969008i	$-0.579454\pi$
$419$	−10.1131	−0.494056	−0.247028	+	0.969008i	$0.579454\pi$
$420$	9.74385	0.475451
$421$	−19.5169	−0.951197	−0.475599	−	0.879662i	$-0.657769\pi$
$421$	−19.5169	−0.951197	−0.475599	+	0.879662i	$0.657769\pi$
$422$	0	0
$423$	1.58049	0.0768461
$424$	0	0
$425$	−17.1391	−0.831367
$426$	0	0
$427$	11.6862	0.565536
$428$	31.6064	1.52775
$429$	14.2972	0.690278
$430$	0	0
$431$	−25.2432	−1.21592	−0.607961	−	0.793967i	$-0.708012\pi$
$431$	−25.2432	−1.21592	−0.607961	+	0.793967i	$0.708012\pi$
$432$	−4.00000	−0.192450
$433$	19.4330	0.933889	0.466945	−	0.884287i	$-0.345355\pi$
$433$	19.4330	0.933889	0.466945	+	0.884287i	$0.345355\pi$
$434$	0	0
$435$	1.58049	0.0757787
$436$	11.6282	0.556890
$437$	−5.13736	−0.245753
$438$	0	0
$439$	10.0808	0.481131	0.240566	−	0.970633i	$-0.422667\pi$
$439$	10.0808	0.481131	0.240566	+	0.970633i	$0.422667\pi$
$440$	0	0
$441$	2.50205	0.119145
$442$	0	0
$443$	24.1277	1.14634	0.573172	−	0.819435i	$-0.305713\pi$
$443$	24.1277	1.14634	0.573172	+	0.819435i	$0.305713\pi$
$444$	10.8528	0.515052
$445$	21.3963	1.01428
$446$	0	0
$447$	14.3740	0.679866
$448$	24.6603	1.16509
$449$	−13.9042	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$449$	−13.9042	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$450$	0	0
$451$	47.9885	2.25969
$452$	−11.0699	−0.520685
$453$	14.1939	0.666887
$454$	0	0
$455$	−11.5459	−0.541279
$456$	0	0
$457$	−2.45242	−0.114719	−0.0573596	−	0.998354i	$-0.518268\pi$
$457$	−2.45242	−0.114719	−0.0573596	+	0.998354i	$0.518268\pi$
$458$	0	0
$459$	−6.85002	−0.319731
$460$	3.16098	0.147382
$461$	−9.63983	−0.448972	−0.224486	−	0.974477i	$-0.572070\pi$
$461$	−9.63983	−0.448972	−0.224486	+	0.974477i	$0.572070\pi$
$462$	0	0
$463$	−36.7467	−1.70777	−0.853883	−	0.520465i	$-0.825759\pi$
$463$	−36.7467	−1.70777	−0.853883	+	0.520465i	$0.825759\pi$
$464$	4.00000	0.185695
$465$	0.123978	0.00574934
$466$	0	0
$467$	−15.9885	−0.739858	−0.369929	−	0.929060i	$-0.620618\pi$
$467$	−15.9885	−0.739858	−0.369929	+	0.929060i	$0.620618\pi$
$468$	4.73975	0.219095
$469$	−19.0041	−0.877528
$470$	0	0
$471$	5.96546	0.274874
$472$	0	0
$473$	−12.4069	−0.570469
$474$	0	0
$475$	12.8539	0.589778
$476$	42.2309	1.93565
$477$	−5.00410	−0.229122
$478$	0	0
$479$	38.7174	1.76904	0.884521	−	0.466500i	$-0.154485\pi$
$479$	38.7174	1.76904	0.884521	+	0.466500i	$0.154485\pi$
$480$	0	0
$481$	−12.8599	−0.586362
$482$	0	0
$483$	3.08254	0.140260
$484$	−50.7919	−2.30872
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−34.2084	−1.55013	−0.775064	−	0.631882i	$-0.782283\pi$
$487$	−34.2084	−1.55013	−0.775064	+	0.631882i	$0.782283\pi$
$488$	0	0
$489$	22.0220	0.995869
$490$	0	0
$491$	15.2723	0.689231	0.344615	−	0.938744i	$-0.388009\pi$
$491$	15.2723	0.689231	0.344615	+	0.938744i	$0.388009\pi$
$492$	15.9089	0.717230
$493$	6.85002	0.308509
$494$	0	0
$495$	−9.53496	−0.428564
$496$	0.313771	0.0140887
$497$	43.5114	1.95175
$498$	0	0
$499$	−32.6891	−1.46337	−0.731683	−	0.681645i	$-0.761265\pi$
$499$	−32.6891	−1.46337	−0.731683	+	0.681645i	$0.761265\pi$
$500$	−23.7138	−1.06052
$501$	−19.1333	−0.854815
$502$	0	0
$503$	19.2651	0.858988	0.429494	−	0.903070i	$-0.358692\pi$
$503$	19.2651	0.858988	0.429494	+	0.903070i	$0.358692\pi$
$504$	0	0
$505$	14.4069	0.641098
$506$	0	0
$507$	7.38368	0.327921
$508$	1.08697	0.0482267
$509$	−3.58802	−0.159036	−0.0795182	−	0.996833i	$-0.525338\pi$
$509$	−3.58802	−0.159036	−0.0795182	+	0.996833i	$0.525338\pi$
$510$	0	0
$511$	8.99319	0.397835
$512$	0	0
$513$	5.13736	0.226820
$514$	0	0
$515$	9.07810	0.400029
$516$	−4.11307	−0.181068
$517$	9.53496	0.419347
$518$	0	0
$519$	−5.03119	−0.220845
$520$	0	0
$521$	−20.9113	−0.916141	−0.458071	−	0.888916i	$-0.651459\pi$
$521$	−20.9113	−0.916141	−0.458071	+	0.888916i	$0.651459\pi$
$522$	0	0
$523$	15.6607	0.684793	0.342396	−	0.939556i	$-0.388761\pi$
$523$	15.6607	0.684793	0.342396	+	0.939556i	$0.388761\pi$
$524$	−32.4454	−1.41738
$525$	−7.71266	−0.336608
$526$	0	0
$527$	0.537334	0.0234066
$528$	−24.1316	−1.05019
$529$	1.00000	0.0434783
$530$	0	0
$531$	−0.756479	−0.0328284
$532$	−31.6722	−1.37316
$533$	−18.8511	−0.816532
$534$	0	0
$535$	24.9768	1.07984
$536$	0	0
$537$	−3.93946	−0.170000
$538$	0	0
$539$	15.0946	0.650172
$540$	−3.16098	−0.136027
$541$	−43.8213	−1.88402	−0.942012	−	0.335578i	$-0.891068\pi$
$541$	−43.8213	−1.88402	−0.942012	+	0.335578i	$0.891068\pi$
$542$	0	0
$543$	3.32606	0.142735
$544$	0	0
$545$	9.18913	0.393619
$546$	0	0
$547$	−0.650741	−0.0278237	−0.0139118	−	0.999903i	$-0.504428\pi$
$547$	−0.650741	−0.0278237	−0.0139118	+	0.999903i	$0.504428\pi$
$548$	0.109632	0.00468323
$549$	−3.79110	−0.161800
$550$	0	0
$551$	−5.13736	−0.218859
$552$	0	0
$553$	43.5916	1.85370
$554$	0	0
$555$	8.57639	0.364048
$556$	−35.1357	−1.49009
$557$	−20.1750	−0.854842	−0.427421	−	0.904053i	$-0.640578\pi$
$557$	−20.1750	−0.854842	−0.427421	+	0.904053i	$0.640578\pi$
$558$	0	0
$559$	4.87373	0.206137
$560$	19.4877	0.823506
$561$	−41.3255	−1.74476
$562$	0	0
$563$	−39.0644	−1.64637	−0.823184	−	0.567775i	$-0.807804\pi$
$563$	−39.0644	−1.64637	−0.823184	+	0.567775i	$0.807804\pi$
$564$	3.16098	0.133101
$565$	−8.74795	−0.368029
$566$	0	0
$567$	−3.08254	−0.129454
$568$	0	0
$569$	−17.9474	−0.752396	−0.376198	−	0.926539i	$-0.622769\pi$
$569$	−17.9474	−0.752396	−0.376198	+	0.926539i	$0.622769\pi$
$570$	0	0
$571$	4.81820	0.201635	0.100818	−	0.994905i	$-0.467854\pi$
$571$	4.81820	0.201635	0.100818	+	0.994905i	$0.467854\pi$
$572$	28.5945	1.19560
$573$	10.5610	0.441190
$574$	0	0
$575$	−2.50205	−0.104343
$576$	−8.00000	−0.333333
$577$	29.4011	1.22398	0.611991	−	0.790865i	$-0.290369\pi$
$577$	29.4011	1.22398	0.611991	+	0.790865i	$0.290369\pi$
$578$	0	0
$579$	−8.00819	−0.332809
$580$	3.16098	0.131253
$581$	−16.8961	−0.700968
$582$	0	0
$583$	−30.1893	−1.25031
$584$	0	0
$585$	3.74557	0.154860
$586$	0	0
$587$	−37.8697	−1.56305	−0.781524	−	0.623875i	$-0.785557\pi$
$587$	−37.8697	−1.56305	−0.781524	+	0.623875i	$0.785557\pi$
$588$	5.00410	0.206365
$589$	−0.402988	−0.0166048
$590$	0	0
$591$	−18.8443	−0.775151
$592$	21.7056	0.892097
$593$	−34.4867	−1.41620	−0.708100	−	0.706113i	$-0.750447\pi$
$593$	−34.4867	−1.41620	−0.708100	+	0.706113i	$0.750447\pi$
$594$	0	0
$595$	33.3728	1.36815
$596$	28.7479	1.17756
$597$	0.807829	0.0330622
$598$	0	0
$599$	−35.1675	−1.43690	−0.718452	−	0.695577i	$-0.755149\pi$
$599$	−35.1675	−1.43690	−0.718452	+	0.695577i	$0.755149\pi$
$600$	0	0
$601$	22.0714	0.900312	0.450156	−	0.892950i	$-0.351368\pi$
$601$	22.0714	0.900312	0.450156	+	0.892950i	$0.351368\pi$
$602$	0	0
$603$	6.16508	0.251061
$604$	28.3878	1.15508
$605$	−40.1381	−1.63185
$606$	0	0
$607$	−34.0532	−1.38218	−0.691088	−	0.722771i	$-0.742868\pi$
$607$	−34.0532	−1.38218	−0.691088	+	0.722771i	$0.742868\pi$
$608$	0	0
$609$	3.08254	0.124911
$610$	0	0
$611$	−3.74557	−0.151530
$612$	−13.7000	−0.553791
$613$	28.2447	1.14079	0.570396	−	0.821370i	$-0.306790\pi$
$613$	28.2447	1.14079	0.570396	+	0.821370i	$0.306790\pi$
$614$	0	0
$615$	12.5720	0.506950
$616$	0	0
$617$	−23.4226	−0.942959	−0.471479	−	0.881877i	$-0.656280\pi$
$617$	−23.4226	−0.942959	−0.471479	+	0.881877i	$0.656280\pi$
$618$	0	0
$619$	−5.66876	−0.227847	−0.113923	−	0.993490i	$-0.536342\pi$
$619$	−5.66876	−0.227847	−0.113923	+	0.993490i	$0.536342\pi$
$620$	0.247956	0.00995815
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	41.7307	1.67190
$624$	9.47951	0.379484
$625$	−6.22951	−0.249181
$626$	0	0
$627$	30.9932	1.23775
$628$	11.9309	0.476096
$629$	37.1710	1.48211
$630$	0	0
$631$	−17.9480	−0.714498	−0.357249	−	0.934009i	$-0.616285\pi$
$631$	−17.9480	−0.714498	−0.357249	+	0.934009i	$0.616285\pi$
$632$	0	0
$633$	16.2517	0.645948
$634$	0	0
$635$	0.858976	0.0340874
$636$	−10.0082	−0.396851
$637$	−5.92955	−0.234937
$638$	0	0
$639$	−14.1154	−0.558398
$640$	0	0
$641$	37.0502	1.46339	0.731697	−	0.681630i	$-0.238729\pi$
$641$	37.0502	1.46339	0.731697	+	0.681630i	$0.238729\pi$
$642$	0	0
$643$	15.5952	0.615013	0.307507	−	0.951546i	$-0.400505\pi$
$643$	15.5952	0.615013	0.307507	+	0.951546i	$0.400505\pi$
$644$	6.16508	0.242938
$645$	−3.25033	−0.127982
$646$	0	0
$647$	0.957902	0.0376590	0.0188295	−	0.999823i	$-0.494006\pi$
$647$	0.957902	0.0376590	0.0188295	+	0.999823i	$0.494006\pi$
$648$	0	0
$649$	−4.56377	−0.179144
$650$	0	0
$651$	0.241803	0.00947699
$652$	44.0440	1.72490
$653$	−14.8709	−0.581944	−0.290972	−	0.956732i	$-0.593979\pi$
$653$	−14.8709	−0.581944	−0.290972	+	0.956732i	$0.593979\pi$
$654$	0	0
$655$	−25.6398	−1.00183
$656$	31.8179	1.24228
$657$	−2.91746	−0.113821
$658$	0	0
$659$	−5.99937	−0.233702	−0.116851	−	0.993149i	$-0.537280\pi$
$659$	−5.99937	−0.233702	−0.116851	+	0.993149i	$0.537280\pi$
$660$	−19.0699	−0.742295
$661$	22.1339	0.860909	0.430454	−	0.902612i	$-0.358353\pi$
$661$	22.1339	0.860909	0.430454	+	0.902612i	$0.358353\pi$
$662$	0	0
$663$	16.2337	0.630465
$664$	0	0
$665$	−25.0288	−0.970576
$666$	0	0
$667$	1.00000	0.0387202
$668$	−38.2667	−1.48058
$669$	−9.39869	−0.363374
$670$	0	0
$671$	−22.8714	−0.882940
$672$	0	0
$673$	−41.2309	−1.58933	−0.794667	−	0.607046i	$-0.792355\pi$
$673$	−41.2309	−1.58933	−0.794667	+	0.607046i	$0.792355\pi$
$674$	0	0
$675$	2.50205	0.0963039
$676$	14.7674	0.567976
$677$	−11.5761	−0.444904	−0.222452	−	0.974944i	$-0.571406\pi$
$677$	−11.5761	−0.444904	−0.222452	+	0.974944i	$0.571406\pi$
$678$	0	0
$679$	−7.80147	−0.299393
$680$	0	0
$681$	−19.9145	−0.763127
$682$	0	0
$683$	43.3966	1.66053	0.830263	−	0.557372i	$-0.188190\pi$
$683$	43.3966	1.66053	0.830263	+	0.557372i	$0.188190\pi$
$684$	10.2747	0.392863
$685$	0.0866358	0.00331019
$686$	0	0
$687$	−13.9682	−0.532919
$688$	−8.22614	−0.313619
$689$	11.8591	0.451796
$690$	0	0
$691$	−7.28754	−0.277231	−0.138616	−	0.990346i	$-0.544265\pi$
$691$	−7.28754	−0.277231	−0.138616	+	0.990346i	$0.544265\pi$
$692$	−10.0624	−0.382514
$693$	−18.5967	−0.706429
$694$	0	0
$695$	−27.7658	−1.05322
$696$	0	0
$697$	54.4882	2.06389
$698$	0	0
$699$	24.1813	0.914619
$700$	−15.4253	−0.583022
$701$	50.4266	1.90459	0.952294	−	0.305184i	$-0.0987178\pi$
$701$	50.4266	1.90459	0.952294	+	0.305184i	$0.0987178\pi$
$702$	0	0
$703$	−27.8774	−1.05142
$704$	−48.2633	−1.81899
$705$	2.49795	0.0940783
$706$	0	0
$707$	28.0987	1.05676
$708$	−1.51296	−0.0568604
$709$	50.9442	1.91325	0.956625	−	0.291322i	$-0.0940951\pi$
$709$	50.9442	1.91325	0.956625	+	0.291322i	$0.0940951\pi$
$710$	0	0
$711$	−14.1415	−0.530346
$712$	0	0
$713$	0.0784427	0.00293770
$714$	0	0
$715$	22.5967	0.845068
$716$	−7.87892	−0.294449
$717$	6.94066	0.259204
$718$	0	0
$719$	13.7468	0.512668	0.256334	−	0.966588i	$-0.417485\pi$
$719$	13.7468	0.512668	0.256334	+	0.966588i	$0.417485\pi$
$720$	−6.32196	−0.235606
$721$	17.7056	0.659393
$722$	0	0
$723$	8.10963	0.301600
$724$	6.65212	0.247224
$725$	−2.50205	−0.0929237
$726$	0	0
$727$	−44.6736	−1.65685	−0.828426	−	0.560099i	$-0.810763\pi$
$727$	−44.6736	−1.65685	−0.828426	+	0.560099i	$0.810763\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.0873	−0.521037
$732$	−7.58221	−0.280247
$733$	−8.27471	−0.305633	−0.152817	−	0.988255i	$-0.548834\pi$
$733$	−8.27471	−0.305633	−0.152817	+	0.988255i	$0.548834\pi$
$734$	0	0
$735$	3.95446	0.145863
$736$	0	0
$737$	37.1933	1.37003
$738$	0	0
$739$	−43.6284	−1.60490	−0.802448	−	0.596722i	$-0.796469\pi$
$739$	−43.6284	−1.60490	−0.802448	+	0.596722i	$0.796469\pi$
$740$	17.1528	0.630549
$741$	−12.1749	−0.447256
$742$	0	0
$743$	16.3815	0.600980	0.300490	−	0.953785i	$-0.402850\pi$
$743$	16.3815	0.600980	0.300490	+	0.953785i	$0.402850\pi$
$744$	0	0
$745$	22.7179	0.832321
$746$	0	0
$747$	5.48123	0.200548
$748$	−82.6510	−3.02202
$749$	48.7140	1.77997
$750$	0	0
$751$	−16.7772	−0.612208	−0.306104	−	0.951998i	$-0.599026\pi$
$751$	−16.7772	−0.612208	−0.306104	+	0.951998i	$0.599026\pi$
$752$	6.32196	0.230538
$753$	16.7178	0.609232
$754$	0	0
$755$	22.4333	0.816432
$756$	−6.16508	−0.224222
$757$	30.6060	1.11239	0.556196	−	0.831051i	$-0.312260\pi$
$757$	30.6060	1.11239	0.556196	+	0.831051i	$0.312260\pi$
$758$	0	0
$759$	−6.03291	−0.218981
$760$	0	0
$761$	49.8457	1.80690	0.903452	−	0.428689i	$-0.141024\pi$
$761$	49.8457	1.80690	0.903452	+	0.428689i	$0.141024\pi$
$762$	0	0
$763$	17.9222	0.648827
$764$	21.1219	0.764164
$765$	−10.8264	−0.391429
$766$	0	0
$767$	1.79276	0.0647329
$768$	−16.0000	−0.577350
$769$	37.8744	1.36579	0.682893	−	0.730519i	$-0.260722\pi$
$769$	37.8744	1.36579	0.682893	+	0.730519i	$0.260722\pi$
$770$	0	0
$771$	−5.06410	−0.182379
$772$	−16.0164	−0.576442
$773$	23.3083	0.838340	0.419170	−	0.907908i	$-0.362321\pi$
$773$	23.3083	0.838340	0.419170	+	0.907908i	$0.362321\pi$
$774$	0	0
$775$	−0.196267	−0.00705013
$776$	0	0
$777$	16.7271	0.600082
$778$	0	0
$779$	−40.8649	−1.46414
$780$	7.49114	0.268226
$781$	−85.1572	−3.04716
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	10.0082	0.357435
$785$	9.42836	0.336513
$786$	0	0
$787$	38.9212	1.38739	0.693696	−	0.720268i	$-0.255981\pi$
$787$	38.9212	1.38739	0.693696	+	0.720268i	$0.255981\pi$
$788$	−37.6886	−1.34260
$789$	−7.51585	−0.267571
$790$	0	0
$791$	−17.0617	−0.606645
$792$	0	0
$793$	8.98445	0.319047
$794$	0	0
$795$	−7.90893	−0.280501
$796$	1.61566	0.0572655
$797$	28.8892	1.02331	0.511653	−	0.859192i	$-0.329033\pi$
$797$	28.8892	1.02331	0.511653	+	0.859192i	$0.329033\pi$
$798$	0	0
$799$	10.8264	0.383010
$800$	0	0
$801$	−13.5378	−0.478333
$802$	0	0
$803$	−17.6008	−0.621118
$804$	12.3302	0.434851
$805$	4.87193	0.171713
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	31.3008	1.10048	0.550239	−	0.835007i	$-0.314537\pi$
$809$	31.3008	1.10048	0.550239	+	0.835007i	$0.314537\pi$
$810$	0	0
$811$	−39.8370	−1.39886	−0.699432	−	0.714699i	$-0.746564\pi$
$811$	−39.8370	−1.39886	−0.699432	+	0.714699i	$0.746564\pi$
$812$	6.16508	0.216352
$813$	30.4016	1.06623
$814$	0	0
$815$	34.8056	1.21919
$816$	−27.4001	−0.959194
$817$	10.5651	0.369628
$818$	0	0
$819$	7.30524	0.255266
$820$	25.1439	0.878064
$821$	34.1067	1.19033	0.595166	−	0.803603i	$-0.297086\pi$
$821$	34.1067	1.19033	0.595166	+	0.803603i	$0.297086\pi$
$822$	0	0
$823$	−4.34152	−0.151336	−0.0756680	−	0.997133i	$-0.524109\pi$
$823$	−4.34152	−0.151336	−0.0756680	+	0.997133i	$0.524109\pi$
$824$	0	0
$825$	15.0946	0.525527
$826$	0	0
$827$	13.3533	0.464339	0.232169	−	0.972675i	$-0.425418\pi$
$827$	13.3533	0.464339	0.232169	+	0.972675i	$0.425418\pi$
$828$	−2.00000	−0.0695048
$829$	13.5973	0.472255	0.236127	−	0.971722i	$-0.424122\pi$
$829$	13.5973	0.472255	0.236127	+	0.971722i	$0.424122\pi$
$830$	0	0
$831$	1.57196	0.0545307
$832$	18.9590	0.657286
$833$	17.1391	0.593834
$834$	0	0
$835$	−30.2401	−1.04650
$836$	61.9864	2.14384
$837$	−0.0784427	−0.00271138
$838$	0	0
$839$	−34.0788	−1.17653	−0.588265	−	0.808668i	$-0.700189\pi$
$839$	−34.0788	−1.17653	−0.588265	+	0.808668i	$0.700189\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	15.0179	0.517244
$844$	32.5034	1.11881
$845$	11.6698	0.401455
$846$	0	0
$847$	−78.2841	−2.68987
$848$	−20.0164	−0.687366
$849$	−0.0866358	−0.00297333
$850$	0	0
$851$	5.42641	0.186015
$852$	−28.2309	−0.967174
$853$	−34.8414	−1.19295	−0.596473	−	0.802633i	$-0.703432\pi$
$853$	−34.8414	−1.19295	−0.596473	+	0.802633i	$0.703432\pi$
$854$	0	0
$855$	8.11954	0.277683
$856$	0	0
$857$	−12.3185	−0.420793	−0.210396	−	0.977616i	$-0.567475\pi$
$857$	−12.3185	−0.420793	−0.210396	+	0.977616i	$0.567475\pi$
$858$	0	0
$859$	32.9580	1.12451	0.562257	−	0.826963i	$-0.309933\pi$
$859$	32.9580	1.12451	0.562257	+	0.826963i	$0.309933\pi$
$860$	−6.50067	−0.221671
$861$	24.5199	0.835638
$862$	0	0
$863$	0.00343710	0.000117000 0	5.85001e−5	−	1.00000i	$-0.499981\pi$
$863$	0.00343710	0.000117000 0	5.85001e−5		1.00000i	$0.499981\pi$
$864$	0	0
$865$	−7.95175	−0.270368
$866$	0	0
$867$	−29.9227	−1.01623
$868$	0.483605	0.0164146
$869$	−85.3141	−2.89408
$870$	0	0
$871$	−14.6105	−0.495057
$872$	0	0
$873$	2.53086	0.0856566
$874$	0	0
$875$	−36.5494	−1.23560
$876$	−5.83492	−0.197144
$877$	56.4139	1.90496	0.952481	−	0.304599i	$-0.0985226\pi$
$877$	56.4139	1.90496	0.952481	+	0.304599i	$0.0985226\pi$
$878$	0	0
$879$	23.0809	0.778500
$880$	−38.1398	−1.28569
$881$	−33.4554	−1.12714	−0.563571	−	0.826068i	$-0.690573\pi$
$881$	−33.4554	−1.12714	−0.563571	+	0.826068i	$0.690573\pi$
$882$	0	0
$883$	−16.1830	−0.544601	−0.272300	−	0.962212i	$-0.587784\pi$
$883$	−16.1830	−0.544601	−0.272300	+	0.962212i	$0.587784\pi$
$884$	32.4674	1.09200
$885$	−1.19561	−0.0401899
$886$	0	0
$887$	−7.86901	−0.264215	−0.132108	−	0.991235i	$-0.542174\pi$
$887$	−7.86901	−0.264215	−0.132108	+	0.991235i	$0.542174\pi$
$888$	0	0
$889$	1.67532	0.0561884
$890$	0	0
$891$	6.03291	0.202110
$892$	−18.7974	−0.629383
$893$	−8.11954	−0.271710
$894$	0	0
$895$	−6.22628	−0.208122
$896$	0	0
$897$	2.36988	0.0791279
$898$	0	0
$899$	0.0784427	0.00261621
$900$	5.00410	0.166803
$901$	−34.2781	−1.14197
$902$	0	0
$903$	−6.33935	−0.210960
$904$	0	0
$905$	5.25681	0.174742
$906$	0	0
$907$	−56.0273	−1.86036	−0.930178	−	0.367109i	$-0.880348\pi$
$907$	−56.0273	−1.86036	−0.930178	+	0.367109i	$0.880348\pi$
$908$	−39.8291	−1.32177
$909$	−9.11545	−0.302340
$910$	0	0
$911$	−6.10153	−0.202153	−0.101076	−	0.994879i	$-0.532229\pi$
$911$	−6.10153	−0.202153	−0.101076	+	0.994879i	$0.532229\pi$
$912$	20.5494	0.680459
$913$	33.0677	1.09438
$914$	0	0
$915$	−5.99181	−0.198083
$916$	−27.9364	−0.923043
$917$	−50.0071	−1.65138
$918$	0	0
$919$	−22.9048	−0.755561	−0.377780	−	0.925895i	$-0.623313\pi$
$919$	−22.9048	−0.755561	−0.377780	+	0.925895i	$0.623313\pi$
$920$	0	0
$921$	20.8956	0.688535
$922$	0	0
$923$	33.4519	1.10108
$924$	−37.1933	−1.22357
$925$	−13.5771	−0.446414
$926$	0	0
$927$	−5.74385	−0.188653
$928$	0	0
$929$	38.8265	1.27386	0.636928	−	0.770923i	$-0.280205\pi$
$929$	38.8265	1.27386	0.636928	+	0.770923i	$0.280205\pi$
$930$	0	0
$931$	−12.8539	−0.421270
$932$	48.3625	1.58417
$933$	4.85282	0.158874
$934$	0	0
$935$	−65.3146	−2.13602
$936$	0	0
$937$	−19.2384	−0.628492	−0.314246	−	0.949342i	$-0.601752\pi$
$937$	−19.2384	−0.628492	−0.314246	+	0.949342i	$0.601752\pi$
$938$	0	0
$939$	−29.2004	−0.952918
$940$	4.99590	0.162948
$941$	−35.9529	−1.17203	−0.586016	−	0.810300i	$-0.699304\pi$
$941$	−35.9529	−1.17203	−0.586016	+	0.810300i	$0.699304\pi$
$942$	0	0
$943$	7.95446	0.259033
$944$	−3.02592	−0.0984852
$945$	−4.87193	−0.158484
$946$	0	0
$947$	−55.9193	−1.81713	−0.908567	−	0.417740i	$-0.862822\pi$
$947$	−55.9193	−1.81713	−0.908567	+	0.417740i	$0.862822\pi$
$948$	−28.2829	−0.918586
$949$	6.91402	0.224439
$950$	0	0
$951$	−22.2512	−0.721544
$952$	0	0
$953$	11.8943	0.385293	0.192646	−	0.981268i	$-0.438293\pi$
$953$	11.8943	0.385293	0.192646	+	0.981268i	$0.438293\pi$
$954$	0	0
$955$	16.6915	0.540124
$956$	13.8813	0.448954
$957$	−6.03291	−0.195016
$958$	0	0
$959$	0.168972	0.00545638
$960$	−12.6439	−0.408081
$961$	−30.9938	−0.999802
$962$	0	0
$963$	−15.8032	−0.509251
$964$	16.2193	0.522387
$965$	−12.6569	−0.407439
$966$	0	0
$967$	−6.93946	−0.223158	−0.111579	−	0.993756i	$-0.535591\pi$
$967$	−6.93946	−0.223158	−0.111579	+	0.993756i	$0.535591\pi$
$968$	0	0
$969$	35.1910	1.13050
$970$	0	0
$971$	46.0353	1.47734	0.738671	−	0.674066i	$-0.235454\pi$
$971$	46.0353	1.47734	0.738671	+	0.674066i	$0.235454\pi$
$972$	2.00000	0.0641500
$973$	−54.1536	−1.73609
$974$	0	0
$975$	−5.92955	−0.189897
$976$	−15.1644	−0.485401
$977$	−32.8489	−1.05093	−0.525465	−	0.850815i	$-0.676109\pi$
$977$	−32.8489	−1.05093	−0.525465	+	0.850815i	$0.676109\pi$
$978$	0	0
$979$	−81.6721	−2.61025
$980$	7.90893	0.252641
$981$	−5.81410	−0.185630
$982$	0	0
$983$	31.6616	1.00985	0.504924	−	0.863164i	$-0.331520\pi$
$983$	31.6616	1.00985	0.504924	+	0.863164i	$0.331520\pi$
$984$	0	0
$985$	−29.7832	−0.948973
$986$	0	0
$987$	4.87193	0.155075
$988$	−24.3498	−0.774670
$989$	−2.05653	−0.0653940
$990$	0	0
$991$	40.8796	1.29858	0.649291	−	0.760540i	$-0.275066\pi$
$991$	40.8796	1.29858	0.649291	+	0.760540i	$0.275066\pi$
$992$	0	0
$993$	17.4001	0.552174
$994$	0	0
$995$	1.27677	0.0404762
$996$	10.9625	0.347359
$997$	18.9650	0.600629	0.300314	−	0.953840i	$-0.402908\pi$
$997$	18.9650	0.600629	0.300314	+	0.953840i	$0.402908\pi$
$998$	0	0
$999$	−5.42641	−0.171684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.g.1.2	✓	4
3.2	odd	2		6003.2.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.g.1.2	✓	4	1.1	even	1	trivial
6003.2.a.g.1.3		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000 q^{4} -1.58049 q^{5} -3.08254 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000 q^{4} -1.58049 q^{5} -3.08254 q^{7} +1.00000 q^{9} +6.03291 q^{11} +2.00000 q^{12} -2.36988 q^{13} +1.58049 q^{15} +4.00000 q^{16} +6.85002 q^{17} -5.13736 q^{19} +3.16098 q^{20} +3.08254 q^{21} +1.00000 q^{23} -2.50205 q^{25} -1.00000 q^{27} +6.16508 q^{28} +1.00000 q^{29} +0.0784427 q^{31} -6.03291 q^{33} +4.87193 q^{35} -2.00000 q^{36} +5.42641 q^{37} +2.36988 q^{39} +7.95446 q^{41} -2.05653 q^{43} -12.0658 q^{44} -1.58049 q^{45} +1.58049 q^{47} -4.00000 q^{48} +2.50205 q^{49} -6.85002 q^{51} +4.73975 q^{52} -5.00410 q^{53} -9.53496 q^{55} +5.13736 q^{57} -0.756479 q^{59} -3.16098 q^{60} -3.79110 q^{61} -3.08254 q^{63} -8.00000 q^{64} +3.74557 q^{65} +6.16508 q^{67} -13.7000 q^{68} -1.00000 q^{69} -14.1154 q^{71} -2.91746 q^{73} +2.50205 q^{75} +10.2747 q^{76} -18.5967 q^{77} -14.1415 q^{79} -6.32196 q^{80} +1.00000 q^{81} +5.48123 q^{83} -6.16508 q^{84} -10.8264 q^{85} -1.00000 q^{87} -13.5378 q^{89} +7.30524 q^{91} -2.00000 q^{92} -0.0784427 q^{93} +8.11954 q^{95} +2.53086 q^{97} +6.03291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9} + 8 q^{12} + 2 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{21} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{28} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 8 q^{36} + 4 q^{37} + 6 q^{41} - 2 q^{45} + 2 q^{47} - 16 q^{48} + 4 q^{49} - 2 q^{51} - 8 q^{53} - 8 q^{55} - 4 q^{57} - 22 q^{59} - 4 q^{60} - 16 q^{61} - 2 q^{63} - 32 q^{64} - 10 q^{65} + 4 q^{67} - 4 q^{68} - 4 q^{69} - 22 q^{71} - 22 q^{73} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 20 q^{79} - 8 q^{80} + 4 q^{81} - 10 q^{83} - 4 q^{84} - 2 q^{85} - 4 q^{87} - 14 q^{89} - 26 q^{91} - 8 q^{92} - 2 q^{93} - 14 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9 + 8 * q^12 + 2 * q^15 + 16 * q^16 + 2 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^21 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 4 * q^28 + 4 * q^29 + 2 * q^31 + 4 * q^35 - 8 * q^36 + 4 * q^37 + 6 * q^41 - 2 * q^45 + 2 * q^47 - 16 * q^48 + 4 * q^49 - 2 * q^51 - 8 * q^53 - 8 * q^55 - 4 * q^57 - 22 * q^59 - 4 * q^60 - 16 * q^61 - 2 * q^63 - 32 * q^64 - 10 * q^65 + 4 * q^67 - 4 * q^68 - 4 * q^69 - 22 * q^71 - 22 * q^73 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 20 * q^79 - 8 * q^80 + 4 * q^81 - 10 * q^83 - 4 * q^84 - 2 * q^85 - 4 * q^87 - 14 * q^89 - 26 * q^91 - 8 * q^92 - 2 * q^93 - 14 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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