LMFDB - Embedded newform 8001.2.a.o.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
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			1.10
Root			$-0.910949$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.910949 q^{2} -1.17017 q^{4} +2.06751 q^{5} +1.00000 q^{7} -2.88787 q^{8} +O(q^{10})$	\(q+0.910949 q^{2} -1.17017 q^{4} +2.06751 q^{5} +1.00000 q^{7} -2.88787 q^{8} +1.88339 q^{10} -4.76209 q^{11} +2.03773 q^{13} +0.910949 q^{14} -0.290355 q^{16} -0.854179 q^{17} +2.23714 q^{19} -2.41934 q^{20} -4.33802 q^{22} -5.99633 q^{23} -0.725414 q^{25} +1.85627 q^{26} -1.17017 q^{28} +5.46515 q^{29} +3.43070 q^{31} +5.51123 q^{32} -0.778113 q^{34} +2.06751 q^{35} +1.22426 q^{37} +2.03792 q^{38} -5.97068 q^{40} +7.05781 q^{41} -5.15064 q^{43} +5.57246 q^{44} -5.46235 q^{46} -7.66065 q^{47} +1.00000 q^{49} -0.660815 q^{50} -2.38450 q^{52} +1.39018 q^{53} -9.84565 q^{55} -2.88787 q^{56} +4.97848 q^{58} -8.77497 q^{59} +6.04889 q^{61} +3.12519 q^{62} +5.60116 q^{64} +4.21303 q^{65} -5.42409 q^{67} +0.999536 q^{68} +1.88339 q^{70} +1.57089 q^{71} -9.67049 q^{73} +1.11524 q^{74} -2.61784 q^{76} -4.76209 q^{77} -4.49693 q^{79} -0.600311 q^{80} +6.42930 q^{82} -8.88129 q^{83} -1.76602 q^{85} -4.69197 q^{86} +13.7523 q^{88} -10.1548 q^{89} +2.03773 q^{91} +7.01673 q^{92} -6.97846 q^{94} +4.62530 q^{95} +1.09551 q^{97} +0.910949 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * 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 $

$2$	0.910949	0.644138	0.322069	−	0.946716i	$-0.395622\pi$
$2$	0.910949	0.644138	0.322069	+	0.946716i	$0.395622\pi$
$3$	0	0
$3$	0	0
$4$	−1.17017	−0.585086
$5$	2.06751	0.924617	0.462309	−	0.886719i	$-0.347021\pi$
$5$	2.06751	0.924617	0.462309	+	0.886719i	$0.347021\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.88787	−1.02101
$9$	0	0
$10$	1.88339	0.595581
$11$	−4.76209	−1.43582	−0.717912	−	0.696134i	$-0.754902\pi$
$11$	−4.76209	−1.43582	−0.717912	+	0.696134i	$0.754902\pi$
$12$	0	0
$13$	2.03773	0.565166	0.282583	−	0.959243i	$-0.408809\pi$
$13$	2.03773	0.565166	0.282583	+	0.959243i	$0.408809\pi$
$14$	0.910949	0.243461
$15$	0	0
$16$	−0.290355	−0.0725888
$17$	−0.854179	−0.207169	−0.103584	−	0.994621i	$-0.533031\pi$
$17$	−0.854179	−0.207169	−0.103584	+	0.994621i	$0.533031\pi$
$18$	0	0
$19$	2.23714	0.513235	0.256617	−	0.966513i	$-0.417392\pi$
$19$	2.23714	0.513235	0.256617	+	0.966513i	$0.417392\pi$
$20$	−2.41934	−0.540980
$21$	0	0
$22$	−4.33802	−0.924869
$23$	−5.99633	−1.25032	−0.625160	−	0.780496i	$-0.714966\pi$
$23$	−5.99633	−1.25032	−0.625160	+	0.780496i	$0.714966\pi$
$24$	0	0
$25$	−0.725414	−0.145083
$26$	1.85627	0.364045
$27$	0	0
$28$	−1.17017	−0.221142
$29$	5.46515	1.01485	0.507427	−	0.861695i	$-0.330597\pi$
$29$	5.46515	1.01485	0.507427	+	0.861695i	$0.330597\pi$
$30$	0	0
$31$	3.43070	0.616172	0.308086	−	0.951358i	$-0.400312\pi$
$31$	3.43070	0.616172	0.308086	+	0.951358i	$0.400312\pi$
$32$	5.51123	0.974257
$33$	0	0
$34$	−0.778113	−0.133445
$35$	2.06751	0.349472
$36$	0	0
$37$	1.22426	0.201268	0.100634	−	0.994924i	$-0.467913\pi$
$37$	1.22426	0.201268	0.100634	+	0.994924i	$0.467913\pi$
$38$	2.03792	0.330594
$39$	0	0
$40$	−5.97068	−0.944048
$41$	7.05781	1.10224	0.551122	−	0.834425i	$-0.314200\pi$
$41$	7.05781	1.10224	0.551122	+	0.834425i	$0.314200\pi$
$42$	0	0
$43$	−5.15064	−0.785465	−0.392732	−	0.919653i	$-0.628470\pi$
$43$	−5.15064	−0.785465	−0.392732	+	0.919653i	$0.628470\pi$
$44$	5.57246	0.840080
$45$	0	0
$46$	−5.46235	−0.805380
$47$	−7.66065	−1.11742	−0.558710	−	0.829363i	$-0.688704\pi$
$47$	−7.66065	−1.11742	−0.558710	+	0.829363i	$0.688704\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.660815	−0.0934534
$51$	0	0
$52$	−2.38450	−0.330670
$53$	1.39018	0.190956	0.0954779	−	0.995432i	$-0.469562\pi$
$53$	1.39018	0.190956	0.0954779	+	0.995432i	$0.469562\pi$
$54$	0	0
$55$	−9.84565	−1.32759
$56$	−2.88787	−0.385907
$57$	0	0
$58$	4.97848	0.653706
$59$	−8.77497	−1.14240	−0.571202	−	0.820809i	$-0.693523\pi$
$59$	−8.77497	−1.14240	−0.571202	+	0.820809i	$0.693523\pi$
$60$	0	0
$61$	6.04889	0.774481	0.387240	−	0.921979i	$-0.373428\pi$
$61$	6.04889	0.774481	0.387240	+	0.921979i	$0.373428\pi$
$62$	3.12519	0.396900
$63$	0	0
$64$	5.60116	0.700145
$65$	4.21303	0.522562
$66$	0	0
$67$	−5.42409	−0.662658	−0.331329	−	0.943515i	$-0.607497\pi$
$67$	−5.42409	−0.662658	−0.331329	+	0.943515i	$0.607497\pi$
$68$	0.999536	0.121212
$69$	0	0
$70$	1.88339	0.225109
$71$	1.57089	0.186431	0.0932153	−	0.995646i	$-0.470286\pi$
$71$	1.57089	0.186431	0.0932153	+	0.995646i	$0.470286\pi$
$72$	0	0
$73$	−9.67049	−1.13185	−0.565923	−	0.824458i	$-0.691480\pi$
$73$	−9.67049	−1.13185	−0.565923	+	0.824458i	$0.691480\pi$
$74$	1.11524	0.129644
$75$	0	0
$76$	−2.61784	−0.300287
$77$	−4.76209	−0.542690
$78$	0	0
$79$	−4.49693	−0.505944	−0.252972	−	0.967474i	$-0.581408\pi$
$79$	−4.49693	−0.505944	−0.252972	+	0.967474i	$0.581408\pi$
$80$	−0.600311	−0.0671168
$81$	0	0
$82$	6.42930	0.709998
$83$	−8.88129	−0.974848	−0.487424	−	0.873165i	$-0.662063\pi$
$83$	−8.88129	−0.974848	−0.487424	+	0.873165i	$0.662063\pi$
$84$	0	0
$85$	−1.76602	−0.191552
$86$	−4.69197	−0.505948
$87$	0	0
$88$	13.7523	1.46600
$89$	−10.1548	−1.07641	−0.538204	−	0.842815i	$-0.680897\pi$
$89$	−10.1548	−1.07641	−0.538204	+	0.842815i	$0.680897\pi$
$90$	0	0
$91$	2.03773	0.213613
$92$	7.01673	0.731545
$93$	0	0
$94$	−6.97846	−0.719773
$95$	4.62530	0.474546
$96$	0	0
$97$	1.09551	0.111233	0.0556163	−	0.998452i	$-0.482288\pi$
$97$	1.09551	0.111233	0.0556163	+	0.998452i	$0.482288\pi$
$98$	0.910949	0.0920198
$99$	0	0
$100$	0.848859	0.0848859
$101$	−3.53703	−0.351948	−0.175974	−	0.984395i	$-0.556307\pi$
$101$	−3.53703	−0.351948	−0.175974	+	0.984395i	$0.556307\pi$
$102$	0	0
$103$	1.77000	0.174404	0.0872018	−	0.996191i	$-0.472208\pi$
$103$	1.77000	0.174404	0.0872018	+	0.996191i	$0.472208\pi$
$104$	−5.88470	−0.577042
$105$	0	0
$106$	1.26638	0.123002
$107$	−11.4134	−1.10338	−0.551689	−	0.834050i	$-0.686017\pi$
$107$	−11.4134	−1.10338	−0.551689	+	0.834050i	$0.686017\pi$
$108$	0	0
$109$	−15.2946	−1.46496	−0.732480	−	0.680788i	$-0.761637\pi$
$109$	−15.2946	−1.46496	−0.732480	+	0.680788i	$0.761637\pi$
$110$	−8.96889	−0.855150
$111$	0	0
$112$	−0.290355	−0.0274360
$113$	15.9081	1.49651	0.748254	−	0.663413i	$-0.230893\pi$
$113$	15.9081	1.49651	0.748254	+	0.663413i	$0.230893\pi$
$114$	0	0
$115$	−12.3975	−1.15607
$116$	−6.39517	−0.593776
$117$	0	0
$118$	−7.99355	−0.735866
$119$	−0.854179	−0.0783024
$120$	0	0
$121$	11.6775	1.06159
$122$	5.51023	0.498873
$123$	0	0
$124$	−4.01451	−0.360514
$125$	−11.8373	−1.05876
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−5.92009	−0.523267
$129$	0	0
$130$	3.83786	0.336602
$131$	−14.7361	−1.28750	−0.643749	−	0.765237i	$-0.722622\pi$
$131$	−14.7361	−1.28750	−0.643749	+	0.765237i	$0.722622\pi$
$132$	0	0
$133$	2.23714	0.193985
$134$	−4.94107	−0.426844
$135$	0	0
$136$	2.46675	0.211522
$137$	−7.61686	−0.650752	−0.325376	−	0.945585i	$-0.605491\pi$
$137$	−7.61686	−0.650752	−0.325376	+	0.945585i	$0.605491\pi$
$138$	0	0
$139$	5.10352	0.432875	0.216437	−	0.976296i	$-0.430556\pi$
$139$	5.10352	0.432875	0.216437	+	0.976296i	$0.430556\pi$
$140$	−2.41934	−0.204471
$141$	0	0
$142$	1.43100	0.120087
$143$	−9.70387	−0.811478
$144$	0	0
$145$	11.2992	0.938351
$146$	−8.80933	−0.729065
$147$	0	0
$148$	−1.43260	−0.117759
$149$	0.345030	0.0282659	0.0141330	−	0.999900i	$-0.495501\pi$
$149$	0.345030	0.0282659	0.0141330	+	0.999900i	$0.495501\pi$
$150$	0	0
$151$	6.54380	0.532527	0.266264	−	0.963900i	$-0.414211\pi$
$151$	6.54380	0.532527	0.266264	+	0.963900i	$0.414211\pi$
$152$	−6.46056	−0.524020
$153$	0	0
$154$	−4.33802	−0.349568
$155$	7.09300	0.569724
$156$	0	0
$157$	5.25736	0.419583	0.209792	−	0.977746i	$-0.432721\pi$
$157$	5.25736	0.419583	0.209792	+	0.977746i	$0.432721\pi$
$158$	−4.09647	−0.325898
$159$	0	0
$160$	11.3945	0.900815
$161$	−5.99633	−0.472577
$162$	0	0
$163$	−4.27474	−0.334824	−0.167412	−	0.985887i	$-0.553541\pi$
$163$	−4.27474	−0.334824	−0.167412	+	0.985887i	$0.553541\pi$
$164$	−8.25885	−0.644907
$165$	0	0
$166$	−8.09040	−0.627937
$167$	−15.9971	−1.23790	−0.618948	−	0.785432i	$-0.712441\pi$
$167$	−15.9971	−1.23790	−0.618948	+	0.785432i	$0.712441\pi$
$168$	0	0
$169$	−8.84764	−0.680588
$170$	−1.60876	−0.123386
$171$	0	0
$172$	6.02713	0.459564
$173$	−7.16537	−0.544773	−0.272387	−	0.962188i	$-0.587813\pi$
$173$	−7.16537	−0.544773	−0.272387	+	0.962188i	$0.587813\pi$
$174$	0	0
$175$	−0.725414	−0.0548361
$176$	1.38270	0.104225
$177$	0	0
$178$	−9.25051	−0.693355
$179$	−0.409220	−0.0305866	−0.0152933	−	0.999883i	$-0.504868\pi$
$179$	−0.409220	−0.0305866	−0.0152933	+	0.999883i	$0.504868\pi$
$180$	0	0
$181$	−10.5918	−0.787279	−0.393639	−	0.919265i	$-0.628784\pi$
$181$	−10.5918	−0.787279	−0.393639	+	0.919265i	$0.628784\pi$
$182$	1.85627	0.137596
$183$	0	0
$184$	17.3166	1.27660
$185$	2.53117	0.186096
$186$	0	0
$187$	4.06767	0.297458
$188$	8.96427	0.653787
$189$	0	0
$190$	4.21341	0.305673
$191$	−14.3830	−1.04072	−0.520358	−	0.853948i	$-0.674201\pi$
$191$	−14.3830	−1.04072	−0.520358	+	0.853948i	$0.674201\pi$
$192$	0	0
$193$	4.91728	0.353954	0.176977	−	0.984215i	$-0.443368\pi$
$193$	4.91728	0.353954	0.176977	+	0.984215i	$0.443368\pi$
$194$	0.997957	0.0716491
$195$	0	0
$196$	−1.17017	−0.0835837
$197$	18.4666	1.31569	0.657846	−	0.753152i	$-0.271468\pi$
$197$	18.4666	1.31569	0.657846	+	0.753152i	$0.271468\pi$
$198$	0	0
$199$	−15.1069	−1.07090	−0.535450	−	0.844567i	$-0.679858\pi$
$199$	−15.1069	−1.07090	−0.535450	+	0.844567i	$0.679858\pi$
$200$	2.09490	0.148132
$201$	0	0
$202$	−3.22206	−0.226703
$203$	5.46515	0.383578
$204$	0	0
$205$	14.5921	1.01915
$206$	1.61238	0.112340
$207$	0	0
$208$	−0.591666	−0.0410247
$209$	−10.6535	−0.736915
$210$	0	0
$211$	8.27452	0.569641	0.284821	−	0.958581i	$-0.408066\pi$
$211$	8.27452	0.569641	0.284821	+	0.958581i	$0.408066\pi$
$212$	−1.62675	−0.111726
$213$	0	0
$214$	−10.3971	−0.710728
$215$	−10.6490	−0.726254
$216$	0	0
$217$	3.43070	0.232891
$218$	−13.9326	−0.943637
$219$	0	0
$220$	11.5211	0.776753
$221$	−1.74059	−0.117085
$222$	0	0
$223$	9.20017	0.616089	0.308044	−	0.951372i	$-0.400325\pi$
$223$	9.20017	0.616089	0.308044	+	0.951372i	$0.400325\pi$
$224$	5.51123	0.368235
$225$	0	0
$226$	14.4915	0.963958
$227$	−4.61206	−0.306113	−0.153057	−	0.988217i	$-0.548912\pi$
$227$	−4.61206	−0.306113	−0.153057	+	0.988217i	$0.548912\pi$
$228$	0	0
$229$	8.47798	0.560240	0.280120	−	0.959965i	$-0.409626\pi$
$229$	8.47798	0.560240	0.280120	+	0.959965i	$0.409626\pi$
$230$	−11.2934	−0.744668
$231$	0	0
$232$	−15.7826	−1.03618
$233$	11.9704	0.784209	0.392104	−	0.919921i	$-0.371747\pi$
$233$	11.9704	0.784209	0.392104	+	0.919921i	$0.371747\pi$
$234$	0	0
$235$	−15.8384	−1.03319
$236$	10.2682	0.668404
$237$	0	0
$238$	−0.778113	−0.0504376
$239$	20.9024	1.35206	0.676031	−	0.736873i	$-0.263699\pi$
$239$	20.9024	1.35206	0.676031	+	0.736873i	$0.263699\pi$
$240$	0	0
$241$	5.81959	0.374873	0.187436	−	0.982277i	$-0.439982\pi$
$241$	5.81959	0.374873	0.187436	+	0.982277i	$0.439982\pi$
$242$	10.6376	0.683810
$243$	0	0
$244$	−7.07824	−0.453138
$245$	2.06751	0.132088
$246$	0	0
$247$	4.55869	0.290063
$248$	−9.90740	−0.629121
$249$	0	0
$250$	−10.7832	−0.681990
$251$	−0.00650335	−0.000410488 0	−0.000205244	−	1.00000i	$-0.500065\pi$
$251$	−0.00650335	−0.000410488 0	−0.000205244		1.00000i	$0.500065\pi$
$252$	0	0
$253$	28.5550	1.79524
$254$	−0.910949	−0.0571580
$255$	0	0
$256$	−16.5952	−1.03720
$257$	−21.6261	−1.34900	−0.674498	−	0.738276i	$-0.735640\pi$
$257$	−21.6261	−1.34900	−0.674498	+	0.738276i	$0.735640\pi$
$258$	0	0
$259$	1.22426	0.0760721
$260$	−4.92997	−0.305744
$261$	0	0
$262$	−13.4238	−0.829326
$263$	−6.14267	−0.378773	−0.189386	−	0.981903i	$-0.560650\pi$
$263$	−6.14267	−0.378773	−0.189386	+	0.981903i	$0.560650\pi$
$264$	0	0
$265$	2.87421	0.176561
$266$	2.03792	0.124953
$267$	0	0
$268$	6.34712	0.387712
$269$	20.1163	1.22651	0.613255	−	0.789885i	$-0.289860\pi$
$269$	20.1163	1.22651	0.613255	+	0.789885i	$0.289860\pi$
$270$	0	0
$271$	−13.3909	−0.813440	−0.406720	−	0.913553i	$-0.633327\pi$
$271$	−13.3909	−0.813440	−0.406720	+	0.913553i	$0.633327\pi$
$272$	0.248015	0.0150381
$273$	0	0
$274$	−6.93857	−0.419174
$275$	3.45449	0.208313
$276$	0	0
$277$	8.05697	0.484096	0.242048	−	0.970264i	$-0.422181\pi$
$277$	8.05697	0.484096	0.242048	+	0.970264i	$0.422181\pi$
$278$	4.64905	0.278831
$279$	0	0
$280$	−5.97068	−0.356816
$281$	−27.8734	−1.66279	−0.831395	−	0.555682i	$-0.812457\pi$
$281$	−27.8734	−1.66279	−0.831395	+	0.555682i	$0.812457\pi$
$282$	0	0
$283$	−10.1043	−0.600639	−0.300320	−	0.953839i	$-0.597093\pi$
$283$	−10.1043	−0.600639	−0.300320	+	0.953839i	$0.597093\pi$
$284$	−1.83821	−0.109078
$285$	0	0
$286$	−8.83973	−0.522704
$287$	7.05781	0.416609
$288$	0	0
$289$	−16.2704	−0.957081
$290$	10.2930	0.604428
$291$	0	0
$292$	11.3161	0.662227
$293$	−5.16465	−0.301722	−0.150861	−	0.988555i	$-0.548205\pi$
$293$	−5.16465	−0.301722	−0.150861	+	0.988555i	$0.548205\pi$
$294$	0	0
$295$	−18.1423	−1.05629
$296$	−3.53551	−0.205497
$297$	0	0
$298$	0.314305	0.0182072
$299$	−12.2189	−0.706638
$300$	0	0
$301$	−5.15064	−0.296878
$302$	5.96107	0.343021
$303$	0	0
$304$	−0.649565	−0.0372551
$305$	12.5061	0.716098
$306$	0	0
$307$	3.03374	0.173144	0.0865722	−	0.996246i	$-0.472409\pi$
$307$	3.03374	0.173144	0.0865722	+	0.996246i	$0.472409\pi$
$308$	5.57246	0.317520
$309$	0	0
$310$	6.46136	0.366981
$311$	−22.1371	−1.25528	−0.627640	−	0.778504i	$-0.715979\pi$
$311$	−22.1371	−1.25528	−0.627640	+	0.778504i	$0.715979\pi$
$312$	0	0
$313$	9.66850	0.546496	0.273248	−	0.961944i	$-0.411902\pi$
$313$	9.66850	0.546496	0.273248	+	0.961944i	$0.411902\pi$
$314$	4.78919	0.270270
$315$	0	0
$316$	5.26218	0.296021
$317$	9.91891	0.557101	0.278551	−	0.960422i	$-0.410146\pi$
$317$	9.91891	0.557101	0.278551	+	0.960422i	$0.410146\pi$
$318$	0	0
$319$	−26.0255	−1.45715
$320$	11.5804	0.647366
$321$	0	0
$322$	−5.46235	−0.304405
$323$	−1.91092	−0.106326
$324$	0	0
$325$	−1.47820	−0.0819958
$326$	−3.89408	−0.215673
$327$	0	0
$328$	−20.3820	−1.12541
$329$	−7.66065	−0.422345
$330$	0	0
$331$	22.7606	1.25104	0.625518	−	0.780210i	$-0.284888\pi$
$331$	22.7606	1.25104	0.625518	+	0.780210i	$0.284888\pi$
$332$	10.3926	0.570370
$333$	0	0
$334$	−14.5726	−0.797376
$335$	−11.2143	−0.612705
$336$	0	0
$337$	16.9411	0.922840	0.461420	−	0.887182i	$-0.347340\pi$
$337$	16.9411	0.922840	0.461420	+	0.887182i	$0.347340\pi$
$338$	−8.05975	−0.438393
$339$	0	0
$340$	2.06655	0.112074
$341$	−16.3373	−0.884715
$342$	0	0
$343$	1.00000	0.0539949
$344$	14.8743	0.801971
$345$	0	0
$346$	−6.52729	−0.350909
$347$	−35.6135	−1.91183	−0.955917	−	0.293636i	$-0.905135\pi$
$347$	−35.6135	−1.91183	−0.955917	+	0.293636i	$0.905135\pi$
$348$	0	0
$349$	−0.202587	−0.0108443	−0.00542213	−	0.999985i	$-0.501726\pi$
$349$	−0.202587	−0.0108443	−0.00542213	+	0.999985i	$0.501726\pi$
$350$	−0.660815	−0.0353221
$351$	0	0
$352$	−26.2450	−1.39886
$353$	−14.0659	−0.748653	−0.374327	−	0.927297i	$-0.622126\pi$
$353$	−14.0659	−0.748653	−0.374327	+	0.927297i	$0.622126\pi$
$354$	0	0
$355$	3.24783	0.172377
$356$	11.8829	0.629791
$357$	0	0
$358$	−0.372779	−0.0197020
$359$	9.61467	0.507443	0.253722	−	0.967277i	$-0.418345\pi$
$359$	9.61467	0.507443	0.253722	+	0.967277i	$0.418345\pi$
$360$	0	0
$361$	−13.9952	−0.736590
$362$	−9.64855	−0.507116
$363$	0	0
$364$	−2.38450	−0.124982
$365$	−19.9938	−1.04652
$366$	0	0
$367$	−23.8796	−1.24651	−0.623254	−	0.782020i	$-0.714190\pi$
$367$	−23.8796	−1.24651	−0.623254	+	0.782020i	$0.714190\pi$
$368$	1.74106	0.0907592
$369$	0	0
$370$	2.30577	0.119871
$371$	1.39018	0.0721745
$372$	0	0
$373$	19.0393	0.985816	0.492908	−	0.870081i	$-0.335934\pi$
$373$	19.0393	0.985816	0.492908	+	0.870081i	$0.335934\pi$
$374$	3.70544	0.191604
$375$	0	0
$376$	22.1229	1.14090
$377$	11.1365	0.573560
$378$	0	0
$379$	−8.43150	−0.433097	−0.216548	−	0.976272i	$-0.569480\pi$
$379$	−8.43150	−0.433097	−0.216548	+	0.976272i	$0.569480\pi$
$380$	−5.41240	−0.277650
$381$	0	0
$382$	−13.1022	−0.670365
$383$	22.7597	1.16296	0.581482	−	0.813559i	$-0.302473\pi$
$383$	22.7597	1.16296	0.581482	+	0.813559i	$0.302473\pi$
$384$	0	0
$385$	−9.84565	−0.501781
$386$	4.47939	0.227995
$387$	0	0
$388$	−1.28194	−0.0650806
$389$	−15.4243	−0.782041	−0.391021	−	0.920382i	$-0.627878\pi$
$389$	−15.4243	−0.782041	−0.391021	+	0.920382i	$0.627878\pi$
$390$	0	0
$391$	5.12194	0.259027
$392$	−2.88787	−0.145859
$393$	0	0
$394$	16.8222	0.847488
$395$	−9.29743	−0.467804
$396$	0	0
$397$	−7.28576	−0.365662	−0.182831	−	0.983144i	$-0.558526\pi$
$397$	−7.28576	−0.365662	−0.182831	+	0.983144i	$0.558526\pi$
$398$	−13.7616	−0.689807
$399$	0	0
$400$	0.210628	0.0105314
$401$	−25.8458	−1.29068	−0.645340	−	0.763896i	$-0.723284\pi$
$401$	−25.8458	−1.29068	−0.645340	+	0.763896i	$0.723284\pi$
$402$	0	0
$403$	6.99086	0.348239
$404$	4.13894	0.205920
$405$	0	0
$406$	4.97848	0.247078
$407$	−5.83005	−0.288985
$408$	0	0
$409$	−18.8712	−0.933120	−0.466560	−	0.884490i	$-0.654507\pi$
$409$	−18.8712	−0.933120	−0.466560	+	0.884490i	$0.654507\pi$
$410$	13.2926	0.656476
$411$	0	0
$412$	−2.07121	−0.102041
$413$	−8.77497	−0.431788
$414$	0	0
$415$	−18.3621	−0.901361
$416$	11.2304	0.550617
$417$	0	0
$418$	−9.70476	−0.474675
$419$	−36.8364	−1.79958	−0.899788	−	0.436328i	$-0.856279\pi$
$419$	−36.8364	−1.79958	−0.899788	+	0.436328i	$0.856279\pi$
$420$	0	0
$421$	1.66960	0.0813711	0.0406855	−	0.999172i	$-0.487046\pi$
$421$	1.66960	0.0813711	0.0406855	+	0.999172i	$0.487046\pi$
$422$	7.53767	0.366928
$423$	0	0
$424$	−4.01465	−0.194969
$425$	0.619633	0.0300566
$426$	0	0
$427$	6.04889	0.292726
$428$	13.3557	0.645571
$429$	0	0
$430$	−9.70068	−0.467808
$431$	−22.4575	−1.08174	−0.540869	−	0.841107i	$-0.681905\pi$
$431$	−22.4575	−1.08174	−0.540869	+	0.841107i	$0.681905\pi$
$432$	0	0
$433$	34.2436	1.64564	0.822822	−	0.568299i	$-0.192398\pi$
$433$	34.2436	1.64564	0.822822	+	0.568299i	$0.192398\pi$
$434$	3.12519	0.150014
$435$	0	0
$436$	17.8974	0.857128
$437$	−13.4146	−0.641708
$438$	0	0
$439$	21.2700	1.01516	0.507581	−	0.861604i	$-0.330540\pi$
$439$	21.2700	1.01516	0.507581	+	0.861604i	$0.330540\pi$
$440$	28.4329	1.35549
$441$	0	0
$442$	−1.58559	−0.0754187
$443$	30.7912	1.46294	0.731468	−	0.681876i	$-0.238835\pi$
$443$	30.7912	1.46294	0.731468	+	0.681876i	$0.238835\pi$
$444$	0	0
$445$	−20.9951	−0.995265
$446$	8.38089	0.396846
$447$	0	0
$448$	5.60116	0.264630
$449$	−26.8456	−1.26692	−0.633462	−	0.773774i	$-0.718367\pi$
$449$	−26.8456	−1.26692	−0.633462	+	0.773774i	$0.718367\pi$
$450$	0	0
$451$	−33.6099	−1.58263
$452$	−18.6152	−0.875585
$453$	0	0
$454$	−4.20135	−0.197179
$455$	4.21303	0.197510
$456$	0	0
$457$	2.07317	0.0969789	0.0484895	−	0.998824i	$-0.484559\pi$
$457$	2.07317	0.0969789	0.0484895	+	0.998824i	$0.484559\pi$
$458$	7.72300	0.360872
$459$	0	0
$460$	14.5071	0.676399
$461$	−37.8084	−1.76091	−0.880455	−	0.474129i	$-0.842763\pi$
$461$	−37.8084	−1.76091	−0.880455	+	0.474129i	$0.842763\pi$
$462$	0	0
$463$	30.4977	1.41735	0.708674	−	0.705536i	$-0.249294\pi$
$463$	30.4977	1.41735	0.708674	+	0.705536i	$0.249294\pi$
$464$	−1.58683	−0.0736669
$465$	0	0
$466$	10.9045	0.505139
$467$	7.92034	0.366509	0.183255	−	0.983065i	$-0.441337\pi$
$467$	7.92034	0.366509	0.183255	+	0.983065i	$0.441337\pi$
$468$	0	0
$469$	−5.42409	−0.250461
$470$	−14.4280	−0.665515
$471$	0	0
$472$	25.3409	1.16641
$473$	24.5278	1.12779
$474$	0	0
$475$	−1.62285	−0.0744616
$476$	0.999536	0.0458137
$477$	0	0
$478$	19.0410	0.870915
$479$	34.0782	1.55707	0.778537	−	0.627599i	$-0.215962\pi$
$479$	34.0782	1.55707	0.778537	+	0.627599i	$0.215962\pi$
$480$	0	0
$481$	2.49472	0.113750
$482$	5.30135	0.241470
$483$	0	0
$484$	−13.6647	−0.621121
$485$	2.26498	0.102848
$486$	0	0
$487$	−10.7347	−0.486436	−0.243218	−	0.969972i	$-0.578203\pi$
$487$	−10.7347	−0.486436	−0.243218	+	0.969972i	$0.578203\pi$
$488$	−17.4684	−0.790756
$489$	0	0
$490$	1.88339	0.0850831
$491$	35.6179	1.60741	0.803707	−	0.595025i	$-0.202858\pi$
$491$	35.6179	1.60741	0.803707	+	0.595025i	$0.202858\pi$
$492$	0	0
$493$	−4.66822	−0.210246
$494$	4.15274	0.186841
$495$	0	0
$496$	−0.996122	−0.0447272
$497$	1.57089	0.0704641
$498$	0	0
$499$	−29.5279	−1.32185	−0.660925	−	0.750452i	$-0.729836\pi$
$499$	−29.5279	−1.32185	−0.660925	+	0.750452i	$0.729836\pi$
$500$	13.8517	0.619467
$501$	0	0
$502$	−0.00592422	−0.000264411 0
$503$	21.5076	0.958976	0.479488	−	0.877549i	$-0.340822\pi$
$503$	21.5076	0.958976	0.479488	+	0.877549i	$0.340822\pi$
$504$	0	0
$505$	−7.31284	−0.325417
$506$	26.0122	1.15638
$507$	0	0
$508$	1.17017	0.0519180
$509$	20.3805	0.903348	0.451674	−	0.892183i	$-0.350827\pi$
$509$	20.3805	0.903348	0.451674	+	0.892183i	$0.350827\pi$
$510$	0	0
$511$	−9.67049	−0.427797
$512$	−3.27723	−0.144834
$513$	0	0
$514$	−19.7002	−0.868941
$515$	3.65949	0.161257
$516$	0	0
$517$	36.4807	1.60442
$518$	1.11524	0.0490009
$519$	0	0
$520$	−12.1667	−0.533543
$521$	35.3393	1.54824	0.774122	−	0.633037i	$-0.218192\pi$
$521$	35.3393	1.54824	0.774122	+	0.633037i	$0.218192\pi$
$522$	0	0
$523$	41.3525	1.80822	0.904110	−	0.427300i	$-0.140535\pi$
$523$	41.3525	1.80822	0.904110	+	0.427300i	$0.140535\pi$
$524$	17.2437	0.753297
$525$	0	0
$526$	−5.59566	−0.243982
$527$	−2.93043	−0.127652
$528$	0	0
$529$	12.9559	0.563302
$530$	2.61826	0.113730
$531$	0	0
$532$	−2.61784	−0.113498
$533$	14.3819	0.622951
$534$	0	0
$535$	−23.5974	−1.02020
$536$	15.6640	0.676584
$537$	0	0
$538$	18.3249	0.790042
$539$	−4.76209	−0.205118
$540$	0	0
$541$	25.4880	1.09582	0.547908	−	0.836539i	$-0.315424\pi$
$541$	25.4880	1.09582	0.547908	+	0.836539i	$0.315424\pi$
$542$	−12.1984	−0.523968
$543$	0	0
$544$	−4.70758	−0.201836
$545$	−31.6218	−1.35453
$546$	0	0
$547$	30.4992	1.30405	0.652025	−	0.758198i	$-0.273920\pi$
$547$	30.4992	1.30405	0.652025	+	0.758198i	$0.273920\pi$
$548$	8.91303	0.380746
$549$	0	0
$550$	3.14686	0.134183
$551$	12.2263	0.520858
$552$	0	0
$553$	−4.49693	−0.191229
$554$	7.33949	0.311825
$555$	0	0
$556$	−5.97199	−0.253269
$557$	−36.9759	−1.56672	−0.783361	−	0.621568i	$-0.786496\pi$
$557$	−36.9759	−1.56672	−0.783361	+	0.621568i	$0.786496\pi$
$558$	0	0
$559$	−10.4956	−0.443918
$560$	−0.600311	−0.0253678
$561$	0	0
$562$	−25.3913	−1.07107
$563$	−36.8641	−1.55363	−0.776817	−	0.629726i	$-0.783167\pi$
$563$	−36.8641	−1.55363	−0.776817	+	0.629726i	$0.783167\pi$
$564$	0	0
$565$	32.8901	1.38370
$566$	−9.20452	−0.386895
$567$	0	0
$568$	−4.53652	−0.190348
$569$	−12.5246	−0.525057	−0.262529	−	0.964924i	$-0.584556\pi$
$569$	−12.5246	−0.525057	−0.262529	+	0.964924i	$0.584556\pi$
$570$	0	0
$571$	13.9998	0.585875	0.292937	−	0.956132i	$-0.405367\pi$
$571$	13.9998	0.585875	0.292937	+	0.956132i	$0.405367\pi$
$572$	11.3552	0.474784
$573$	0	0
$574$	6.42930	0.268354
$575$	4.34982	0.181400
$576$	0	0
$577$	−31.7993	−1.32382	−0.661911	−	0.749582i	$-0.730254\pi$
$577$	−31.7993	−1.32382	−0.661911	+	0.749582i	$0.730254\pi$
$578$	−14.8215	−0.616493
$579$	0	0
$580$	−13.2221	−0.549016
$581$	−8.88129	−0.368458
$582$	0	0
$583$	−6.62016	−0.274179
$584$	27.9271	1.15563
$585$	0	0
$586$	−4.70474	−0.194351
$587$	−0.697457	−0.0287871	−0.0143936	−	0.999896i	$-0.504582\pi$
$587$	−0.697457	−0.0287871	−0.0143936	+	0.999896i	$0.504582\pi$
$588$	0	0
$589$	7.67496	0.316241
$590$	−16.5267	−0.680395
$591$	0	0
$592$	−0.355471	−0.0146098
$593$	−8.07973	−0.331795	−0.165897	−	0.986143i	$-0.553052\pi$
$593$	−8.07973	−0.331795	−0.165897	+	0.986143i	$0.553052\pi$
$594$	0	0
$595$	−1.76602	−0.0723998
$596$	−0.403744	−0.0165380
$597$	0	0
$598$	−11.1308	−0.455173
$599$	−15.7153	−0.642108	−0.321054	−	0.947061i	$-0.604037\pi$
$599$	−15.7153	−0.642108	−0.321054	+	0.947061i	$0.604037\pi$
$600$	0	0
$601$	−40.1903	−1.63939	−0.819697	−	0.572797i	$-0.805858\pi$
$601$	−40.1903	−1.63939	−0.819697	+	0.572797i	$0.805858\pi$
$602$	−4.69197	−0.191230
$603$	0	0
$604$	−7.65737	−0.311574
$605$	24.1433	0.981564
$606$	0	0
$607$	−4.34490	−0.176354	−0.0881770	−	0.996105i	$-0.528104\pi$
$607$	−4.34490	−0.176354	−0.0881770	+	0.996105i	$0.528104\pi$
$608$	12.3294	0.500023
$609$	0	0
$610$	11.3924	0.461266
$611$	−15.6104	−0.631528
$612$	0	0
$613$	16.5086	0.666775	0.333388	−	0.942790i	$-0.391808\pi$
$613$	16.5086	0.666775	0.333388	+	0.942790i	$0.391808\pi$
$614$	2.76358	0.111529
$615$	0	0
$616$	13.7523	0.554095
$617$	35.3853	1.42456	0.712280	−	0.701896i	$-0.247663\pi$
$617$	35.3853	1.42456	0.712280	+	0.701896i	$0.247663\pi$
$618$	0	0
$619$	5.41450	0.217627	0.108814	−	0.994062i	$-0.465295\pi$
$619$	5.41450	0.217627	0.108814	+	0.994062i	$0.465295\pi$
$620$	−8.30003	−0.333337
$621$	0	0
$622$	−20.1658	−0.808574
$623$	−10.1548	−0.406844
$624$	0	0
$625$	−20.8467	−0.833868
$626$	8.80751	0.352019
$627$	0	0
$628$	−6.15202	−0.245492
$629$	−1.04574	−0.0416964
$630$	0	0
$631$	−4.17145	−0.166063	−0.0830315	−	0.996547i	$-0.526460\pi$
$631$	−4.17145	−0.166063	−0.0830315	+	0.996547i	$0.526460\pi$
$632$	12.9865	0.516576
$633$	0	0
$634$	9.03562	0.358850
$635$	−2.06751	−0.0820465
$636$	0	0
$637$	2.03773	0.0807380
$638$	−23.7079	−0.938606
$639$	0	0
$640$	−12.2398	−0.483822
$641$	−4.95692	−0.195787	−0.0978933	−	0.995197i	$-0.531210\pi$
$641$	−4.95692	−0.195787	−0.0978933	+	0.995197i	$0.531210\pi$
$642$	0	0
$643$	−14.0285	−0.553230	−0.276615	−	0.960981i	$-0.589213\pi$
$643$	−14.0285	−0.553230	−0.276615	+	0.960981i	$0.589213\pi$
$644$	7.01673	0.276498
$645$	0	0
$646$	−1.74075	−0.0684888
$647$	15.8380	0.622658	0.311329	−	0.950302i	$-0.399226\pi$
$647$	15.8380	0.622658	0.311329	+	0.950302i	$0.399226\pi$
$648$	0	0
$649$	41.7872	1.64029
$650$	−1.34657	−0.0528167
$651$	0	0
$652$	5.00219	0.195901
$653$	−29.1768	−1.14178	−0.570888	−	0.821028i	$-0.693401\pi$
$653$	−29.1768	−1.14178	−0.570888	+	0.821028i	$0.693401\pi$
$654$	0	0
$655$	−30.4670	−1.19044
$656$	−2.04927	−0.0800105
$657$	0	0
$658$	−6.97846	−0.272049
$659$	3.36477	0.131073	0.0655364	−	0.997850i	$-0.479124\pi$
$659$	3.36477	0.131073	0.0655364	+	0.997850i	$0.479124\pi$
$660$	0	0
$661$	−3.56055	−0.138489	−0.0692447	−	0.997600i	$-0.522059\pi$
$661$	−3.56055	−0.138489	−0.0692447	+	0.997600i	$0.522059\pi$
$662$	20.7337	0.805840
$663$	0	0
$664$	25.6480	0.995334
$665$	4.62530	0.179362
$666$	0	0
$667$	−32.7708	−1.26889
$668$	18.7194	0.724275
$669$	0	0
$670$	−10.2157	−0.394667
$671$	−28.8053	−1.11202
$672$	0	0
$673$	−6.79286	−0.261845	−0.130923	−	0.991393i	$-0.541794\pi$
$673$	−6.79286	−0.261845	−0.130923	+	0.991393i	$0.541794\pi$
$674$	15.4325	0.594437
$675$	0	0
$676$	10.3533	0.398202
$677$	−3.52953	−0.135651	−0.0678254	−	0.997697i	$-0.521606\pi$
$677$	−3.52953	−0.135651	−0.0678254	+	0.997697i	$0.521606\pi$
$678$	0	0
$679$	1.09551	0.0420419
$680$	5.10003	0.195577
$681$	0	0
$682$	−14.8825	−0.569879
$683$	−36.8948	−1.41174	−0.705871	−	0.708341i	$-0.749444\pi$
$683$	−36.8948	−1.41174	−0.705871	+	0.708341i	$0.749444\pi$
$684$	0	0
$685$	−15.7479	−0.601696
$686$	0.910949	0.0347802
$687$	0	0
$688$	1.49551	0.0570159
$689$	2.83282	0.107922
$690$	0	0
$691$	−0.939809	−0.0357520	−0.0178760	−	0.999840i	$-0.505690\pi$
$691$	−0.939809	−0.0357520	−0.0178760	+	0.999840i	$0.505690\pi$
$692$	8.38472	0.318739
$693$	0	0
$694$	−32.4421	−1.23149
$695$	10.5516	0.400244
$696$	0	0
$697$	−6.02863	−0.228351
$698$	−0.184547	−0.00698520
$699$	0	0
$700$	0.848859	0.0320839
$701$	15.3128	0.578356	0.289178	−	0.957275i	$-0.406618\pi$
$701$	15.3128	0.578356	0.289178	+	0.957275i	$0.406618\pi$
$702$	0	0
$703$	2.73885	0.103298
$704$	−26.6732	−1.00529
$705$	0	0
$706$	−12.8133	−0.482236
$707$	−3.53703	−0.133024
$708$	0	0
$709$	−20.5964	−0.773515	−0.386757	−	0.922182i	$-0.626405\pi$
$709$	−20.5964	−0.773515	−0.386757	+	0.922182i	$0.626405\pi$
$710$	2.95861	0.111035
$711$	0	0
$712$	29.3257	1.09903
$713$	−20.5716	−0.770413
$714$	0	0
$715$	−20.0628	−0.750307
$716$	0.478858	0.0178958
$717$	0	0
$718$	8.75848	0.326864
$719$	−25.9093	−0.966255	−0.483128	−	0.875550i	$-0.660499\pi$
$719$	−25.9093	−0.966255	−0.483128	+	0.875550i	$0.660499\pi$
$720$	0	0
$721$	1.77000	0.0659184
$722$	−12.7489	−0.474466
$723$	0	0
$724$	12.3942	0.460626
$725$	−3.96450	−0.147238
$726$	0	0
$727$	33.5208	1.24322	0.621609	−	0.783328i	$-0.286479\pi$
$727$	33.5208	1.24322	0.621609	+	0.783328i	$0.286479\pi$
$728$	−5.88470	−0.218102
$729$	0	0
$730$	−18.2133	−0.674106
$731$	4.39956	0.162724
$732$	0	0
$733$	8.91851	0.329413	0.164706	−	0.986343i	$-0.447332\pi$
$733$	8.91851	0.329413	0.164706	+	0.986343i	$0.447332\pi$
$734$	−21.7531	−0.802923
$735$	0	0
$736$	−33.0472	−1.21813
$737$	25.8300	0.951460
$738$	0	0
$739$	−17.5441	−0.645368	−0.322684	−	0.946507i	$-0.604585\pi$
$739$	−17.5441	−0.645368	−0.322684	+	0.946507i	$0.604585\pi$
$740$	−2.96191	−0.108882
$741$	0	0
$742$	1.26638	0.0464904
$743$	31.3395	1.14973	0.574867	−	0.818247i	$-0.305054\pi$
$743$	31.3395	1.14973	0.574867	+	0.818247i	$0.305054\pi$
$744$	0	0
$745$	0.713352	0.0261352
$746$	17.3438	0.635002
$747$	0	0
$748$	−4.75988	−0.174038
$749$	−11.4134	−0.417038
$750$	0	0
$751$	44.1522	1.61114	0.805568	−	0.592503i	$-0.201860\pi$
$751$	44.1522	1.61114	0.805568	+	0.592503i	$0.201860\pi$
$752$	2.22431	0.0811122
$753$	0	0
$754$	10.1448	0.369452
$755$	13.5294	0.492384
$756$	0	0
$757$	−17.5802	−0.638964	−0.319482	−	0.947592i	$-0.603509\pi$
$757$	−17.5802	−0.638964	−0.319482	+	0.947592i	$0.603509\pi$
$758$	−7.68066	−0.278974
$759$	0	0
$760$	−13.3572	−0.484518
$761$	15.5785	0.564720	0.282360	−	0.959308i	$-0.408883\pi$
$761$	15.5785	0.564720	0.282360	+	0.959308i	$0.408883\pi$
$762$	0	0
$763$	−15.2946	−0.553703
$764$	16.8305	0.608908
$765$	0	0
$766$	20.7329	0.749110
$767$	−17.8811	−0.645648
$768$	0	0
$769$	−26.4832	−0.955009	−0.477505	−	0.878629i	$-0.658459\pi$
$769$	−26.4832	−0.955009	−0.477505	+	0.878629i	$0.658459\pi$
$770$	−8.96889	−0.323216
$771$	0	0
$772$	−5.75406	−0.207093
$773$	4.85564	0.174645	0.0873226	−	0.996180i	$-0.472169\pi$
$773$	4.85564	0.174645	0.0873226	+	0.996180i	$0.472169\pi$
$774$	0	0
$775$	−2.48868	−0.0893960
$776$	−3.16369	−0.113570
$777$	0	0
$778$	−14.0507	−0.503743
$779$	15.7893	0.565710
$780$	0	0
$781$	−7.48072	−0.267681
$782$	4.66582	0.166850
$783$	0	0
$784$	−0.290355	−0.0103698
$785$	10.8696	0.387954
$786$	0	0
$787$	−6.95194	−0.247810	−0.123905	−	0.992294i	$-0.539542\pi$
$787$	−6.95194	−0.247810	−0.123905	+	0.992294i	$0.539542\pi$
$788$	−21.6091	−0.769793
$789$	0	0
$790$	−8.46949	−0.301331
$791$	15.9081	0.565627
$792$	0	0
$793$	12.3260	0.437710
$794$	−6.63695	−0.235537
$795$	0	0
$796$	17.6777	0.626568
$797$	−26.4161	−0.935705	−0.467852	−	0.883807i	$-0.654972\pi$
$797$	−26.4161	−0.935705	−0.467852	+	0.883807i	$0.654972\pi$
$798$	0	0
$799$	6.54356	0.231495
$800$	−3.99792	−0.141348
$801$	0	0
$802$	−23.5442	−0.831376
$803$	46.0517	1.62513
$804$	0	0
$805$	−12.3975	−0.436953
$806$	6.36832	0.224314
$807$	0	0
$808$	10.2145	0.359344
$809$	20.0526	0.705010	0.352505	−	0.935810i	$-0.385330\pi$
$809$	20.0526	0.705010	0.352505	+	0.935810i	$0.385330\pi$
$810$	0	0
$811$	−6.59746	−0.231668	−0.115834	−	0.993269i	$-0.536954\pi$
$811$	−6.59746	−0.231668	−0.115834	+	0.993269i	$0.536954\pi$
$812$	−6.39517	−0.224426
$813$	0	0
$814$	−5.31088	−0.186146
$815$	−8.83807	−0.309584
$816$	0	0
$817$	−11.5227	−0.403128
$818$	−17.1907	−0.601058
$819$	0	0
$820$	−17.0752	−0.596293
$821$	−32.1085	−1.12060	−0.560298	−	0.828291i	$-0.689313\pi$
$821$	−32.1085	−1.12060	−0.560298	+	0.828291i	$0.689313\pi$
$822$	0	0
$823$	53.0767	1.85014	0.925069	−	0.379799i	$-0.124007\pi$
$823$	53.0767	1.85014	0.925069	+	0.379799i	$0.124007\pi$
$824$	−5.11153	−0.178069
$825$	0	0
$826$	−7.99355	−0.278131
$827$	5.72183	0.198968	0.0994838	−	0.995039i	$-0.468281\pi$
$827$	5.72183	0.198968	0.0994838	+	0.995039i	$0.468281\pi$
$828$	0	0
$829$	36.9131	1.28204	0.641022	−	0.767522i	$-0.278511\pi$
$829$	36.9131	1.28204	0.641022	+	0.767522i	$0.278511\pi$
$830$	−16.7270	−0.580601
$831$	0	0
$832$	11.4137	0.395698
$833$	−0.854179	−0.0295955
$834$	0	0
$835$	−33.0742	−1.14458
$836$	12.4664	0.431158
$837$	0	0
$838$	−33.5561	−1.15918
$839$	35.8828	1.23881	0.619406	−	0.785071i	$-0.287374\pi$
$839$	35.8828	1.23881	0.619406	+	0.785071i	$0.287374\pi$
$840$	0	0
$841$	0.867890	0.0299272
$842$	1.52092	0.0524142
$843$	0	0
$844$	−9.68261	−0.333289
$845$	−18.2926	−0.629283
$846$	0	0
$847$	11.6775	0.401243
$848$	−0.403646	−0.0138612
$849$	0	0
$850$	0.564454	0.0193606
$851$	−7.34109	−0.251649
$852$	0	0
$853$	38.1872	1.30750	0.653752	−	0.756709i	$-0.273194\pi$
$853$	38.1872	1.30750	0.653752	+	0.756709i	$0.273194\pi$
$854$	5.51023	0.188556
$855$	0	0
$856$	32.9605	1.12657
$857$	5.39552	0.184308	0.0921538	−	0.995745i	$-0.470625\pi$
$857$	5.39552	0.184308	0.0921538	+	0.995745i	$0.470625\pi$
$858$	0	0
$859$	−22.8095	−0.778250	−0.389125	−	0.921185i	$-0.627223\pi$
$859$	−22.8095	−0.778250	−0.389125	+	0.921185i	$0.627223\pi$
$860$	12.4611	0.424921
$861$	0	0
$862$	−20.4576	−0.696789
$863$	−15.7818	−0.537220	−0.268610	−	0.963249i	$-0.586564\pi$
$863$	−15.7818	−0.537220	−0.268610	+	0.963249i	$0.586564\pi$
$864$	0	0
$865$	−14.8145	−0.503707
$866$	31.1942	1.06002
$867$	0	0
$868$	−4.01451	−0.136261
$869$	21.4148	0.726446
$870$	0	0
$871$	−11.0529	−0.374512
$872$	44.1689	1.49575
$873$	0	0
$874$	−12.2200	−0.413349
$875$	−11.8373	−0.400175
$876$	0	0
$877$	−27.3621	−0.923952	−0.461976	−	0.886892i	$-0.652859\pi$
$877$	−27.3621	−0.923952	−0.461976	+	0.886892i	$0.652859\pi$
$878$	19.3759	0.653905
$879$	0	0
$880$	2.85873	0.0963679
$881$	39.0186	1.31457	0.657284	−	0.753643i	$-0.271705\pi$
$881$	39.0186	1.31457	0.657284	+	0.753643i	$0.271705\pi$
$882$	0	0
$883$	−47.1651	−1.58723	−0.793616	−	0.608419i	$-0.791804\pi$
$883$	−47.1651	−1.58723	−0.793616	+	0.608419i	$0.791804\pi$
$884$	2.03679	0.0685046
$885$	0	0
$886$	28.0493	0.942333
$887$	−1.26669	−0.0425313	−0.0212656	−	0.999774i	$-0.506770\pi$
$887$	−1.26669	−0.0425313	−0.0212656	+	0.999774i	$0.506770\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−19.1255	−0.641088
$891$	0	0
$892$	−10.7658	−0.360465
$893$	−17.1379	−0.573499
$894$	0	0
$895$	−0.846066	−0.0282809
$896$	−5.92009	−0.197776
$897$	0	0
$898$	−24.4550	−0.816074
$899$	18.7493	0.625324
$900$	0	0
$901$	−1.18746	−0.0395601
$902$	−30.6169	−1.01943
$903$	0	0
$904$	−45.9404	−1.52796
$905$	−21.8985	−0.727931
$906$	0	0
$907$	−14.3655	−0.476999	−0.238500	−	0.971143i	$-0.576656\pi$
$907$	−14.3655	−0.476999	−0.238500	+	0.971143i	$0.576656\pi$
$908$	5.39690	0.179102
$909$	0	0
$910$	3.83786	0.127224
$911$	15.5385	0.514812	0.257406	−	0.966303i	$-0.417132\pi$
$911$	15.5385	0.514812	0.257406	+	0.966303i	$0.417132\pi$
$912$	0	0
$913$	42.2935	1.39971
$914$	1.88855	0.0624678
$915$	0	0
$916$	−9.92069	−0.327789
$917$	−14.7361	−0.486628
$918$	0	0
$919$	0.672694	0.0221901	0.0110951	−	0.999938i	$-0.496468\pi$
$919$	0.672694	0.0221901	0.0110951	+	0.999938i	$0.496468\pi$
$920$	35.8022	1.18036
$921$	0	0
$922$	−34.4415	−1.13427
$923$	3.20106	0.105364
$924$	0	0
$925$	−0.888098	−0.0292005
$926$	27.7819	0.912968
$927$	0	0
$928$	30.1197	0.988728
$929$	−49.6218	−1.62804	−0.814020	−	0.580837i	$-0.802725\pi$
$929$	−49.6218	−1.62804	−0.814020	+	0.580837i	$0.802725\pi$
$930$	0	0
$931$	2.23714	0.0733193
$932$	−14.0075	−0.458829
$933$	0	0
$934$	7.21502	0.236083
$935$	8.40995	0.275035
$936$	0	0
$937$	−26.7103	−0.872586	−0.436293	−	0.899805i	$-0.643709\pi$
$937$	−26.7103	−0.872586	−0.436293	+	0.899805i	$0.643709\pi$
$938$	−4.94107	−0.161332
$939$	0	0
$940$	18.5337	0.604503
$941$	12.6965	0.413895	0.206948	−	0.978352i	$-0.433647\pi$
$941$	12.6965	0.413895	0.206948	+	0.978352i	$0.433647\pi$
$942$	0	0
$943$	−42.3209	−1.37816
$944$	2.54786	0.0829257
$945$	0	0
$946$	22.3436	0.726452
$947$	38.5444	1.25253	0.626263	−	0.779612i	$-0.284584\pi$
$947$	38.5444	1.25253	0.626263	+	0.779612i	$0.284584\pi$
$948$	0	0
$949$	−19.7059	−0.639680
$950$	−1.47834	−0.0479636
$951$	0	0
$952$	2.46675	0.0799479
$953$	20.6289	0.668235	0.334118	−	0.942531i	$-0.391562\pi$
$953$	20.6289	0.668235	0.334118	+	0.942531i	$0.391562\pi$
$954$	0	0
$955$	−29.7369	−0.962263
$956$	−24.4594	−0.791072
$957$	0	0
$958$	31.0435	1.00297
$959$	−7.61686	−0.245961
$960$	0	0
$961$	−19.2303	−0.620332
$962$	2.27257	0.0732705
$963$	0	0
$964$	−6.80992	−0.219333
$965$	10.1665	0.327272
$966$	0	0
$967$	−12.4906	−0.401669	−0.200835	−	0.979625i	$-0.564365\pi$
$967$	−12.4906	−0.401669	−0.200835	+	0.979625i	$0.564365\pi$
$968$	−33.7230	−1.08390
$969$	0	0
$970$	2.06328	0.0662480
$971$	−38.8901	−1.24804	−0.624022	−	0.781407i	$-0.714502\pi$
$971$	−38.8901	−1.24804	−0.624022	+	0.781407i	$0.714502\pi$
$972$	0	0
$973$	5.10352	0.163611
$974$	−9.77877	−0.313332
$975$	0	0
$976$	−1.75633	−0.0562186
$977$	17.5486	0.561428	0.280714	−	0.959791i	$-0.409429\pi$
$977$	17.5486	0.561428	0.280714	+	0.959791i	$0.409429\pi$
$978$	0	0
$979$	48.3581	1.54553
$980$	−2.41934	−0.0772829
$981$	0	0
$982$	32.4461	1.03540
$983$	−35.7325	−1.13969	−0.569845	−	0.821752i	$-0.692997\pi$
$983$	−35.7325	−1.13969	−0.569845	+	0.821752i	$0.692997\pi$
$984$	0	0
$985$	38.1799	1.21651
$986$	−4.25251	−0.135427
$987$	0	0
$988$	−5.33446	−0.169712
$989$	30.8849	0.982083
$990$	0	0
$991$	38.7774	1.23180	0.615902	−	0.787822i	$-0.288792\pi$
$991$	38.7774	1.23180	0.615902	+	0.787822i	$0.288792\pi$
$992$	18.9074	0.600310
$993$	0	0
$994$	1.43100	0.0453886
$995$	−31.2336	−0.990172
$996$	0	0
$997$	58.7906	1.86192	0.930958	−	0.365127i	$-0.118974\pi$
$997$	58.7906	1.86192	0.930958	+	0.365127i	$0.118974\pi$
$998$	−26.8984	−0.851455
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.10		13
3.2	odd	2		2667.2.a.l.1.4	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.4	✓	13	3.2	odd	2
8001.2.a.o.1.10		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.910949 q^{2} -1.17017 q^{4} +2.06751 q^{5} +1.00000 q^{7} -2.88787 q^{8} +O(q^{10})\)	\(q+0.910949 q^{2} -1.17017 q^{4} +2.06751 q^{5} +1.00000 q^{7} -2.88787 q^{8} +1.88339 q^{10} -4.76209 q^{11} +2.03773 q^{13} +0.910949 q^{14} -0.290355 q^{16} -0.854179 q^{17} +2.23714 q^{19} -2.41934 q^{20} -4.33802 q^{22} -5.99633 q^{23} -0.725414 q^{25} +1.85627 q^{26} -1.17017 q^{28} +5.46515 q^{29} +3.43070 q^{31} +5.51123 q^{32} -0.778113 q^{34} +2.06751 q^{35} +1.22426 q^{37} +2.03792 q^{38} -5.97068 q^{40} +7.05781 q^{41} -5.15064 q^{43} +5.57246 q^{44} -5.46235 q^{46} -7.66065 q^{47} +1.00000 q^{49} -0.660815 q^{50} -2.38450 q^{52} +1.39018 q^{53} -9.84565 q^{55} -2.88787 q^{56} +4.97848 q^{58} -8.77497 q^{59} +6.04889 q^{61} +3.12519 q^{62} +5.60116 q^{64} +4.21303 q^{65} -5.42409 q^{67} +0.999536 q^{68} +1.88339 q^{70} +1.57089 q^{71} -9.67049 q^{73} +1.11524 q^{74} -2.61784 q^{76} -4.76209 q^{77} -4.49693 q^{79} -0.600311 q^{80} +6.42930 q^{82} -8.88129 q^{83} -1.76602 q^{85} -4.69197 q^{86} +13.7523 q^{88} -10.1548 q^{89} +2.03773 q^{91} +7.01673 q^{92} -6.97846 q^{94} +4.62530 q^{95} +1.09551 q^{97} +0.910949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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