LMFDB - Embedded newform 5120.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5120,2,Mod(1,5120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5120, 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("5120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.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:	$40.8834058349$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 $ x^8 - 12x^6 - 8x^5 + 21x^4 + 12x^3 - 10x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 80)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.731397$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.169718 q^{3} +1.00000 q^{5} +2.66881 q^{7} -2.97120 q^{9} +4.94571 q^{11} -4.15881 q^{13} +0.169718 q^{15} -1.85116 q^{17} -4.87701 q^{19} +0.452946 q^{21} +0.707288 q^{23} +1.00000 q^{25} -1.01342 q^{27} -4.94847 q^{29} -6.84272 q^{31} +0.839377 q^{33} +2.66881 q^{35} +0.137894 q^{37} -0.705826 q^{39} -10.2052 q^{41} -6.27690 q^{43} -2.97120 q^{45} -1.89428 q^{47} +0.122561 q^{49} -0.314175 q^{51} -10.5203 q^{53} +4.94571 q^{55} -0.827717 q^{57} +1.35704 q^{59} +9.18991 q^{61} -7.92956 q^{63} -4.15881 q^{65} -4.94538 q^{67} +0.120040 q^{69} +7.86777 q^{71} +15.6564 q^{73} +0.169718 q^{75} +13.1992 q^{77} -6.70212 q^{79} +8.74159 q^{81} -5.47763 q^{83} -1.85116 q^{85} -0.839845 q^{87} -10.5055 q^{89} -11.0991 q^{91} -1.16133 q^{93} -4.87701 q^{95} +4.79937 q^{97} -14.6947 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59}+ \cdots - 40 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^15 - 16 * q^19 - 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 + 8 * q^45 - 20 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^55 - 16 * q^59 - 20 * q^63 - 36 * q^67 - 16 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 - 16 * q^95 - 40 * q^99

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
$2$	0	0
$3$	0.169718	0.0979869	0.0489934	−	0.998799i	$-0.484399\pi$
$3$	0.169718	0.0979869	0.0489934	+	0.998799i	$0.484399\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.66881	1.00872	0.504358	−	0.863495i	$-0.331729\pi$
$7$	2.66881	1.00872	0.504358	+	0.863495i	$0.331729\pi$
$8$	0	0
$9$	−2.97120	−0.990399
$10$	0	0
$11$	4.94571	1.49119	0.745594	−	0.666401i	$-0.232166\pi$
$11$	4.94571	1.49119	0.745594	+	0.666401i	$0.232166\pi$
$12$	0	0
$13$	−4.15881	−1.15345	−0.576723	−	0.816939i	$-0.695669\pi$
$13$	−4.15881	−1.15345	−0.576723	+	0.816939i	$0.695669\pi$
$14$	0	0
$15$	0.169718	0.0438211
$16$	0	0
$17$	−1.85116	−0.448971	−0.224486	−	0.974477i	$-0.572070\pi$
$17$	−1.85116	−0.448971	−0.224486	+	0.974477i	$0.572070\pi$
$18$	0	0
$19$	−4.87701	−1.11886	−0.559431	−	0.828877i	$-0.688980\pi$
$19$	−4.87701	−1.11886	−0.559431	+	0.828877i	$0.688980\pi$
$20$	0	0
$21$	0.452946	0.0988409
$22$	0	0
$23$	0.707288	0.147480	0.0737399	−	0.997278i	$-0.476507\pi$
$23$	0.707288	0.147480	0.0737399	+	0.997278i	$0.476507\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.01342	−0.195033
$28$	0	0
$29$	−4.94847	−0.918907	−0.459454	−	0.888202i	$-0.651955\pi$
$29$	−4.94847	−0.918907	−0.459454	+	0.888202i	$0.651955\pi$
$30$	0	0
$31$	−6.84272	−1.22899	−0.614494	−	0.788921i	$-0.710640\pi$
$31$	−6.84272	−1.22899	−0.614494	+	0.788921i	$0.710640\pi$
$32$	0	0
$33$	0.839377	0.146117
$34$	0	0
$35$	2.66881	0.451112
$36$	0	0
$37$	0.137894	0.0226697	0.0113348	−	0.999936i	$-0.496392\pi$
$37$	0.137894	0.0226697	0.0113348	+	0.999936i	$0.496392\pi$
$38$	0	0
$39$	−0.705826	−0.113023
$40$	0	0
$41$	−10.2052	−1.59379	−0.796896	−	0.604117i	$-0.793526\pi$
$41$	−10.2052	−1.59379	−0.796896	+	0.604117i	$0.793526\pi$
$42$	0	0
$43$	−6.27690	−0.957218	−0.478609	−	0.878028i	$-0.658859\pi$
$43$	−6.27690	−0.957218	−0.478609	+	0.878028i	$0.658859\pi$
$44$	0	0
$45$	−2.97120	−0.442920
$46$	0	0
$47$	−1.89428	−0.276310	−0.138155	−	0.990411i	$-0.544117\pi$
$47$	−1.89428	−0.276310	−0.138155	+	0.990411i	$0.544117\pi$
$48$	0	0
$49$	0.122561	0.0175087
$50$	0	0
$51$	−0.314175	−0.0439933
$52$	0	0
$53$	−10.5203	−1.44507	−0.722537	−	0.691332i	$-0.757024\pi$
$53$	−10.5203	−1.44507	−0.722537	+	0.691332i	$0.757024\pi$
$54$	0	0
$55$	4.94571	0.666879
$56$	0	0
$57$	−0.827717	−0.109634
$58$	0	0
$59$	1.35704	0.176672	0.0883359	−	0.996091i	$-0.471845\pi$
$59$	1.35704	0.176672	0.0883359	+	0.996091i	$0.471845\pi$
$60$	0	0
$61$	9.18991	1.17665	0.588324	−	0.808625i	$-0.299788\pi$
$61$	9.18991	1.17665	0.588324	+	0.808625i	$0.299788\pi$
$62$	0	0
$63$	−7.92956	−0.999031
$64$	0	0
$65$	−4.15881	−0.515837
$66$	0	0
$67$	−4.94538	−0.604174	−0.302087	−	0.953280i	$-0.597683\pi$
$67$	−4.94538	−0.604174	−0.302087	+	0.953280i	$0.597683\pi$
$68$	0	0
$69$	0.120040	0.0144511
$70$	0	0
$71$	7.86777	0.933733	0.466866	−	0.884328i	$-0.345383\pi$
$71$	7.86777	0.933733	0.466866	+	0.884328i	$0.345383\pi$
$72$	0	0
$73$	15.6564	1.83244	0.916220	−	0.400675i	$-0.131224\pi$
$73$	15.6564	1.83244	0.916220	+	0.400675i	$0.131224\pi$
$74$	0	0
$75$	0.169718	0.0195974
$76$	0	0
$77$	13.1992	1.50418
$78$	0	0
$79$	−6.70212	−0.754047	−0.377024	−	0.926204i	$-0.623052\pi$
$79$	−6.70212	−0.754047	−0.377024	+	0.926204i	$0.623052\pi$
$80$	0	0
$81$	8.74159	0.971288
$82$	0	0
$83$	−5.47763	−0.601248	−0.300624	−	0.953743i	$-0.597195\pi$
$83$	−5.47763	−0.601248	−0.300624	+	0.953743i	$0.597195\pi$
$84$	0	0
$85$	−1.85116	−0.200786
$86$	0	0
$87$	−0.839845	−0.0900408
$88$	0	0
$89$	−10.5055	−1.11358	−0.556790	−	0.830653i	$-0.687967\pi$
$89$	−10.5055	−1.11358	−0.556790	+	0.830653i	$0.687967\pi$
$90$	0	0
$91$	−11.0991	−1.16350
$92$	0	0
$93$	−1.16133	−0.120425
$94$	0	0
$95$	−4.87701	−0.500370
$96$	0	0
$97$	4.79937	0.487303	0.243651	−	0.969863i	$-0.421655\pi$
$97$	4.79937	0.487303	0.243651	+	0.969863i	$0.421655\pi$
$98$	0	0
$99$	−14.6947	−1.47687
$100$	0	0
$101$	−0.527472	−0.0524855	−0.0262427	−	0.999656i	$-0.508354\pi$
$101$	−0.527472	−0.0524855	−0.0262427	+	0.999656i	$0.508354\pi$
$102$	0	0
$103$	−10.3013	−1.01502	−0.507508	−	0.861647i	$-0.669433\pi$
$103$	−10.3013	−1.01502	−0.507508	+	0.861647i	$0.669433\pi$
$104$	0	0
$105$	0.452946	0.0442030
$106$	0	0
$107$	−20.5159	−1.98335	−0.991674	−	0.128770i	$-0.958897\pi$
$107$	−20.5159	−1.98335	−0.991674	+	0.128770i	$0.958897\pi$
$108$	0	0
$109$	−1.12612	−0.107862	−0.0539311	−	0.998545i	$-0.517175\pi$
$109$	−1.12612	−0.107862	−0.0539311	+	0.998545i	$0.517175\pi$
$110$	0	0
$111$	0.0234032	0.00222133
$112$	0	0
$113$	−0.842524	−0.0792580	−0.0396290	−	0.999214i	$-0.512618\pi$
$113$	−0.842524	−0.0792580	−0.0396290	+	0.999214i	$0.512618\pi$
$114$	0	0
$115$	0.707288	0.0659550
$116$	0	0
$117$	12.3566	1.14237
$118$	0	0
$119$	−4.94039	−0.452885
$120$	0	0
$121$	13.4600	1.22364
$122$	0	0
$123$	−1.73202	−0.156171
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	21.1693	1.87847	0.939234	−	0.343277i	$-0.111537\pi$
$127$	21.1693	1.87847	0.939234	+	0.343277i	$0.111537\pi$
$128$	0	0
$129$	−1.06530	−0.0937947
$130$	0	0
$131$	−6.60789	−0.577334	−0.288667	−	0.957430i	$-0.593212\pi$
$131$	−6.60789	−0.577334	−0.288667	+	0.957430i	$0.593212\pi$
$132$	0	0
$133$	−13.0158	−1.12861
$134$	0	0
$135$	−1.01342	−0.0872214
$136$	0	0
$137$	10.2840	0.878623	0.439312	−	0.898335i	$-0.355222\pi$
$137$	10.2840	0.878623	0.439312	+	0.898335i	$0.355222\pi$
$138$	0	0
$139$	−7.05110	−0.598066	−0.299033	−	0.954243i	$-0.596664\pi$
$139$	−7.05110	−0.598066	−0.299033	+	0.954243i	$0.596664\pi$
$140$	0	0
$141$	−0.321495	−0.0270747
$142$	0	0
$143$	−20.5683	−1.72001
$144$	0	0
$145$	−4.94847	−0.410948
$146$	0	0
$147$	0.0208008	0.00171563
$148$	0	0
$149$	12.4379	1.01895	0.509477	−	0.860484i	$-0.329839\pi$
$149$	12.4379	1.01895	0.509477	+	0.860484i	$0.329839\pi$
$150$	0	0
$151$	22.1838	1.80529	0.902645	−	0.430385i	$-0.141622\pi$
$151$	22.1838	1.80529	0.902645	+	0.430385i	$0.141622\pi$
$152$	0	0
$153$	5.50015	0.444661
$154$	0	0
$155$	−6.84272	−0.549620
$156$	0	0
$157$	−5.26352	−0.420075	−0.210037	−	0.977693i	$-0.567359\pi$
$157$	−5.26352	−0.420075	−0.210037	+	0.977693i	$0.567359\pi$
$158$	0	0
$159$	−1.78549	−0.141598
$160$	0	0
$161$	1.88762	0.148765
$162$	0	0
$163$	−2.99290	−0.234422	−0.117211	−	0.993107i	$-0.537395\pi$
$163$	−2.99290	−0.234422	−0.117211	+	0.993107i	$0.537395\pi$
$164$	0	0
$165$	0.839377	0.0653454
$166$	0	0
$167$	18.1604	1.40530	0.702648	−	0.711538i	$-0.252001\pi$
$167$	18.1604	1.40530	0.702648	+	0.711538i	$0.252001\pi$
$168$	0	0
$169$	4.29572	0.330440
$170$	0	0
$171$	14.4905	1.10812
$172$	0	0
$173$	12.0709	0.917734	0.458867	−	0.888505i	$-0.348255\pi$
$173$	12.0709	0.917734	0.458867	+	0.888505i	$0.348255\pi$
$174$	0	0
$175$	2.66881	0.201743
$176$	0	0
$177$	0.230315	0.0173115
$178$	0	0
$179$	−3.42945	−0.256329	−0.128165	−	0.991753i	$-0.540909\pi$
$179$	−3.42945	−0.256329	−0.128165	+	0.991753i	$0.540909\pi$
$180$	0	0
$181$	6.31578	0.469448	0.234724	−	0.972062i	$-0.424581\pi$
$181$	6.31578	0.469448	0.234724	+	0.972062i	$0.424581\pi$
$182$	0	0
$183$	1.55970	0.115296
$184$	0	0
$185$	0.137894	0.0101382
$186$	0	0
$187$	−9.15528	−0.669500
$188$	0	0
$189$	−2.70463	−0.196733
$190$	0	0
$191$	−7.75030	−0.560792	−0.280396	−	0.959884i	$-0.590466\pi$
$191$	−7.75030	−0.560792	−0.280396	+	0.959884i	$0.590466\pi$
$192$	0	0
$193$	−11.3388	−0.816181	−0.408091	−	0.912941i	$-0.633805\pi$
$193$	−11.3388	−0.816181	−0.408091	+	0.912941i	$0.633805\pi$
$194$	0	0
$195$	−0.705826	−0.0505453
$196$	0	0
$197$	−1.55565	−0.110836	−0.0554178	−	0.998463i	$-0.517649\pi$
$197$	−1.55565	−0.110836	−0.0554178	+	0.998463i	$0.517649\pi$
$198$	0	0
$199$	14.2722	1.01173	0.505864	−	0.862614i	$-0.331174\pi$
$199$	14.2722	1.01173	0.505864	+	0.862614i	$0.331174\pi$
$200$	0	0
$201$	−0.839321	−0.0592012
$202$	0	0
$203$	−13.2065	−0.926917
$204$	0	0
$205$	−10.2052	−0.712765
$206$	0	0
$207$	−2.10149	−0.146064
$208$	0	0
$209$	−24.1203	−1.66843
$210$	0	0
$211$	−17.6374	−1.21421	−0.607106	−	0.794621i	$-0.707670\pi$
$211$	−17.6374	−1.21421	−0.607106	+	0.794621i	$0.707670\pi$
$212$	0	0
$213$	1.33530	0.0914935
$214$	0	0
$215$	−6.27690	−0.428081
$216$	0	0
$217$	−18.2619	−1.23970
$218$	0	0
$219$	2.65717	0.179555
$220$	0	0
$221$	7.69861	0.517865
$222$	0	0
$223$	−3.08673	−0.206703	−0.103351	−	0.994645i	$-0.532957\pi$
$223$	−3.08673	−0.206703	−0.103351	+	0.994645i	$0.532957\pi$
$224$	0	0
$225$	−2.97120	−0.198080
$226$	0	0
$227$	−11.7617	−0.780651	−0.390326	−	0.920677i	$-0.627638\pi$
$227$	−11.7617	−0.780651	−0.390326	+	0.920677i	$0.627638\pi$
$228$	0	0
$229$	14.1267	0.933520	0.466760	−	0.884384i	$-0.345421\pi$
$229$	14.1267	0.933520	0.466760	+	0.884384i	$0.345421\pi$
$230$	0	0
$231$	2.24014	0.147390
$232$	0	0
$233$	−13.9015	−0.910718	−0.455359	−	0.890308i	$-0.650489\pi$
$233$	−13.9015	−0.910718	−0.455359	+	0.890308i	$0.650489\pi$
$234$	0	0
$235$	−1.89428	−0.123570
$236$	0	0
$237$	−1.13747	−0.0738867
$238$	0	0
$239$	−10.7687	−0.696569	−0.348284	−	0.937389i	$-0.613236\pi$
$239$	−10.7687	−0.696569	−0.348284	+	0.937389i	$0.613236\pi$
$240$	0	0
$241$	12.4707	0.803305	0.401653	−	0.915792i	$-0.368436\pi$
$241$	12.4707	0.803305	0.401653	+	0.915792i	$0.368436\pi$
$242$	0	0
$243$	4.52387	0.290206
$244$	0	0
$245$	0.122561	0.00783014
$246$	0	0
$247$	20.2826	1.29055
$248$	0	0
$249$	−0.929654	−0.0589144
$250$	0	0
$251$	5.21976	0.329468	0.164734	−	0.986338i	$-0.447323\pi$
$251$	5.21976	0.329468	0.164734	+	0.986338i	$0.447323\pi$
$252$	0	0
$253$	3.49804	0.219920
$254$	0	0
$255$	−0.314175	−0.0196744
$256$	0	0
$257$	3.11011	0.194003	0.0970016	−	0.995284i	$-0.469075\pi$
$257$	3.11011	0.194003	0.0970016	+	0.995284i	$0.469075\pi$
$258$	0	0
$259$	0.368014	0.0228673
$260$	0	0
$261$	14.7029	0.910084
$262$	0	0
$263$	17.9512	1.10692	0.553458	−	0.832877i	$-0.313308\pi$
$263$	17.9512	1.10692	0.553458	+	0.832877i	$0.313308\pi$
$264$	0	0
$265$	−10.5203	−0.646257
$266$	0	0
$267$	−1.78297	−0.109116
$268$	0	0
$269$	−2.29720	−0.140063	−0.0700313	−	0.997545i	$-0.522310\pi$
$269$	−2.29720	−0.140063	−0.0700313	+	0.997545i	$0.522310\pi$
$270$	0	0
$271$	18.1808	1.10440	0.552201	−	0.833711i	$-0.313788\pi$
$271$	18.1808	1.10440	0.552201	+	0.833711i	$0.313788\pi$
$272$	0	0
$273$	−1.88372	−0.114008
$274$	0	0
$275$	4.94571	0.298237
$276$	0	0
$277$	−19.6116	−1.17835	−0.589174	−	0.808006i	$-0.700547\pi$
$277$	−19.6116	−1.17835	−0.589174	+	0.808006i	$0.700547\pi$
$278$	0	0
$279$	20.3310	1.21719
$280$	0	0
$281$	−10.7377	−0.640556	−0.320278	−	0.947324i	$-0.603776\pi$
$281$	−10.7377	−0.640556	−0.320278	+	0.947324i	$0.603776\pi$
$282$	0	0
$283$	−23.1477	−1.37599	−0.687994	−	0.725716i	$-0.741509\pi$
$283$	−23.1477	−1.37599	−0.687994	+	0.725716i	$0.741509\pi$
$284$	0	0
$285$	−0.827717	−0.0490297
$286$	0	0
$287$	−27.2359	−1.60768
$288$	0	0
$289$	−13.5732	−0.798425
$290$	0	0
$291$	0.814541	0.0477493
$292$	0	0
$293$	5.96872	0.348696	0.174348	−	0.984684i	$-0.444218\pi$
$293$	5.96872	0.348696	0.174348	+	0.984684i	$0.444218\pi$
$294$	0	0
$295$	1.35704	0.0790101
$296$	0	0
$297$	−5.01208	−0.290831
$298$	0	0
$299$	−2.94148	−0.170110
$300$	0	0
$301$	−16.7519	−0.965561
$302$	0	0
$303$	−0.0895217	−0.00514289
$304$	0	0
$305$	9.18991	0.526213
$306$	0	0
$307$	−17.8704	−1.01992	−0.509958	−	0.860199i	$-0.670339\pi$
$307$	−17.8704	−1.01992	−0.509958	+	0.860199i	$0.670339\pi$
$308$	0	0
$309$	−1.74832	−0.0994582
$310$	0	0
$311$	8.56815	0.485855	0.242928	−	0.970044i	$-0.421892\pi$
$311$	8.56815	0.485855	0.242928	+	0.970044i	$0.421892\pi$
$312$	0	0
$313$	19.1825	1.08426	0.542129	−	0.840295i	$-0.317618\pi$
$313$	19.1825	1.08426	0.542129	+	0.840295i	$0.317618\pi$
$314$	0	0
$315$	−7.92956	−0.446780
$316$	0	0
$317$	−13.3185	−0.748044	−0.374022	−	0.927420i	$-0.622022\pi$
$317$	−13.3185	−0.748044	−0.374022	+	0.927420i	$0.622022\pi$
$318$	0	0
$319$	−24.4737	−1.37026
$320$	0	0
$321$	−3.48193	−0.194342
$322$	0	0
$323$	9.02810	0.502337
$324$	0	0
$325$	−4.15881	−0.230689
$326$	0	0
$327$	−0.191122	−0.0105691
$328$	0	0
$329$	−5.05549	−0.278718
$330$	0	0
$331$	−18.1838	−0.999471	−0.499736	−	0.866178i	$-0.666569\pi$
$331$	−18.1838	−0.999471	−0.499736	+	0.866178i	$0.666569\pi$
$332$	0	0
$333$	−0.409711	−0.0224520
$334$	0	0
$335$	−4.94538	−0.270195
$336$	0	0
$337$	−3.31961	−0.180831	−0.0904153	−	0.995904i	$-0.528819\pi$
$337$	−3.31961	−0.180831	−0.0904153	+	0.995904i	$0.528819\pi$
$338$	0	0
$339$	−0.142992	−0.00776624
$340$	0	0
$341$	−33.8421	−1.83265
$342$	0	0
$343$	−18.3546	−0.991055
$344$	0	0
$345$	0.120040	0.00646272
$346$	0	0
$347$	25.2946	1.35789	0.678943	−	0.734191i	$-0.262438\pi$
$347$	25.2946	1.35789	0.678943	+	0.734191i	$0.262438\pi$
$348$	0	0
$349$	−5.21557	−0.279183	−0.139591	−	0.990209i	$-0.544579\pi$
$349$	−5.21557	−0.279183	−0.139591	+	0.990209i	$0.544579\pi$
$350$	0	0
$351$	4.21463	0.224960
$352$	0	0
$353$	33.0951	1.76148	0.880738	−	0.473604i	$-0.157047\pi$
$353$	33.0951	1.76148	0.880738	+	0.473604i	$0.157047\pi$
$354$	0	0
$355$	7.86777	0.417578
$356$	0	0
$357$	−0.838474	−0.0443767
$358$	0	0
$359$	−6.52522	−0.344388	−0.172194	−	0.985063i	$-0.555086\pi$
$359$	−6.52522	−0.344388	−0.172194	+	0.985063i	$0.555086\pi$
$360$	0	0
$361$	4.78519	0.251852
$362$	0	0
$363$	2.28441	0.119901
$364$	0	0
$365$	15.6564	0.819492
$366$	0	0
$367$	−11.0338	−0.575959	−0.287980	−	0.957636i	$-0.592984\pi$
$367$	−11.0338	−0.575959	−0.287980	+	0.957636i	$0.592984\pi$
$368$	0	0
$369$	30.3218	1.57849
$370$	0	0
$371$	−28.0767	−1.45767
$372$	0	0
$373$	9.67984	0.501203	0.250602	−	0.968090i	$-0.419372\pi$
$373$	9.67984	0.501203	0.250602	+	0.968090i	$0.419372\pi$
$374$	0	0
$375$	0.169718	0.00876421
$376$	0	0
$377$	20.5797	1.05991
$378$	0	0
$379$	−14.2937	−0.734219	−0.367109	−	0.930178i	$-0.619653\pi$
$379$	−14.2937	−0.734219	−0.367109	+	0.930178i	$0.619653\pi$
$380$	0	0
$381$	3.59281	0.184065
$382$	0	0
$383$	−29.5283	−1.50883	−0.754413	−	0.656400i	$-0.772078\pi$
$383$	−29.5283	−1.50883	−0.754413	+	0.656400i	$0.772078\pi$
$384$	0	0
$385$	13.1992	0.672692
$386$	0	0
$387$	18.6499	0.948027
$388$	0	0
$389$	−1.40141	−0.0710545	−0.0355272	−	0.999369i	$-0.511311\pi$
$389$	−1.40141	−0.0710545	−0.0355272	+	0.999369i	$0.511311\pi$
$390$	0	0
$391$	−1.30930	−0.0662142
$392$	0	0
$393$	−1.12148	−0.0565711
$394$	0	0
$395$	−6.70212	−0.337220
$396$	0	0
$397$	−24.0450	−1.20678	−0.603392	−	0.797444i	$-0.706185\pi$
$397$	−24.0450	−1.20678	−0.603392	+	0.797444i	$0.706185\pi$
$398$	0	0
$399$	−2.20902	−0.110589
$400$	0	0
$401$	−26.7791	−1.33728	−0.668642	−	0.743585i	$-0.733124\pi$
$401$	−26.7791	−1.33728	−0.668642	+	0.743585i	$0.733124\pi$
$402$	0	0
$403$	28.4576	1.41757
$404$	0	0
$405$	8.74159	0.434373
$406$	0	0
$407$	0.681985	0.0338048
$408$	0	0
$409$	13.1970	0.652550	0.326275	−	0.945275i	$-0.394206\pi$
$409$	13.1970	0.652550	0.326275	+	0.945275i	$0.394206\pi$
$410$	0	0
$411$	1.74539	0.0860935
$412$	0	0
$413$	3.62169	0.178212
$414$	0	0
$415$	−5.47763	−0.268886
$416$	0	0
$417$	−1.19670	−0.0586026
$418$	0	0
$419$	14.0356	0.685684	0.342842	−	0.939393i	$-0.388610\pi$
$419$	14.0356	0.685684	0.342842	+	0.939393i	$0.388610\pi$
$420$	0	0
$421$	−22.3346	−1.08852	−0.544261	−	0.838916i	$-0.683190\pi$
$421$	−22.3346	−1.08852	−0.544261	+	0.838916i	$0.683190\pi$
$422$	0	0
$423$	5.62829	0.273657
$424$	0	0
$425$	−1.85116	−0.0897943
$426$	0	0
$427$	24.5262	1.18690
$428$	0	0
$429$	−3.49081	−0.168538
$430$	0	0
$431$	−0.285215	−0.0137383	−0.00686917	−	0.999976i	$-0.502187\pi$
$431$	−0.285215	−0.0137383	−0.00686917	+	0.999976i	$0.502187\pi$
$432$	0	0
$433$	18.1101	0.870318	0.435159	−	0.900354i	$-0.356692\pi$
$433$	18.1101	0.870318	0.435159	+	0.900354i	$0.356692\pi$
$434$	0	0
$435$	−0.839845	−0.0402675
$436$	0	0
$437$	−3.44945	−0.165010
$438$	0	0
$439$	−11.5931	−0.553308	−0.276654	−	0.960970i	$-0.589226\pi$
$439$	−11.5931	−0.553308	−0.276654	+	0.960970i	$0.589226\pi$
$440$	0	0
$441$	−0.364153	−0.0173406
$442$	0	0
$443$	−32.0821	−1.52427	−0.762133	−	0.647420i	$-0.775848\pi$
$443$	−32.0821	−1.52427	−0.762133	+	0.647420i	$0.775848\pi$
$444$	0	0
$445$	−10.5055	−0.498008
$446$	0	0
$447$	2.11094	0.0998440
$448$	0	0
$449$	12.1999	0.575747	0.287873	−	0.957669i	$-0.407052\pi$
$449$	12.1999	0.575747	0.287873	+	0.957669i	$0.407052\pi$
$450$	0	0
$451$	−50.4722	−2.37664
$452$	0	0
$453$	3.76499	0.176895
$454$	0	0
$455$	−11.0991	−0.520333
$456$	0	0
$457$	−1.70660	−0.0798314	−0.0399157	−	0.999203i	$-0.512709\pi$
$457$	−1.70660	−0.0798314	−0.0399157	+	0.999203i	$0.512709\pi$
$458$	0	0
$459$	1.87600	0.0875642
$460$	0	0
$461$	6.71341	0.312675	0.156337	−	0.987704i	$-0.450031\pi$
$461$	6.71341	0.312675	0.156337	+	0.987704i	$0.450031\pi$
$462$	0	0
$463$	−11.1761	−0.519398	−0.259699	−	0.965690i	$-0.583623\pi$
$463$	−11.1761	−0.519398	−0.259699	+	0.965690i	$0.583623\pi$
$464$	0	0
$465$	−1.16133	−0.0538556
$466$	0	0
$467$	2.91995	0.135119	0.0675595	−	0.997715i	$-0.478479\pi$
$467$	2.91995	0.135119	0.0675595	+	0.997715i	$0.478479\pi$
$468$	0	0
$469$	−13.1983	−0.609441
$470$	0	0
$471$	−0.893315	−0.0411618
$472$	0	0
$473$	−31.0437	−1.42739
$474$	0	0
$475$	−4.87701	−0.223772
$476$	0	0
$477$	31.2579	1.43120
$478$	0	0
$479$	41.6214	1.90173	0.950864	−	0.309608i	$-0.100198\pi$
$479$	41.6214	1.90173	0.950864	+	0.309608i	$0.100198\pi$
$480$	0	0
$481$	−0.573477	−0.0261483
$482$	0	0
$483$	0.320364	0.0145770
$484$	0	0
$485$	4.79937	0.217928
$486$	0	0
$487$	8.25627	0.374127	0.187064	−	0.982348i	$-0.440103\pi$
$487$	8.25627	0.374127	0.187064	+	0.982348i	$0.440103\pi$
$488$	0	0
$489$	−0.507950	−0.0229703
$490$	0	0
$491$	−6.06007	−0.273487	−0.136744	−	0.990606i	$-0.543664\pi$
$491$	−6.06007	−0.273487	−0.136744	+	0.990606i	$0.543664\pi$
$492$	0	0
$493$	9.16038	0.412563
$494$	0	0
$495$	−14.6947	−0.660476
$496$	0	0
$497$	20.9976	0.941871
$498$	0	0
$499$	−21.3952	−0.957780	−0.478890	−	0.877875i	$-0.658961\pi$
$499$	−21.3952	−0.957780	−0.478890	+	0.877875i	$0.658961\pi$
$500$	0	0
$501$	3.08215	0.137700
$502$	0	0
$503$	−18.6439	−0.831291	−0.415646	−	0.909527i	$-0.636444\pi$
$503$	−18.6439	−0.831291	−0.415646	+	0.909527i	$0.636444\pi$
$504$	0	0
$505$	−0.527472	−0.0234722
$506$	0	0
$507$	0.729062	0.0323788
$508$	0	0
$509$	16.4393	0.728658	0.364329	−	0.931270i	$-0.381298\pi$
$509$	16.4393	0.728658	0.364329	+	0.931270i	$0.381298\pi$
$510$	0	0
$511$	41.7839	1.84841
$512$	0	0
$513$	4.94246	0.218215
$514$	0	0
$515$	−10.3013	−0.453929
$516$	0	0
$517$	−9.36858	−0.412030
$518$	0	0
$519$	2.04865	0.0899259
$520$	0	0
$521$	36.9052	1.61684	0.808422	−	0.588603i	$-0.200322\pi$
$521$	36.9052	1.61684	0.808422	+	0.588603i	$0.200322\pi$
$522$	0	0
$523$	−8.54409	−0.373607	−0.186803	−	0.982397i	$-0.559813\pi$
$523$	−8.54409	−0.373607	−0.186803	+	0.982397i	$0.559813\pi$
$524$	0	0
$525$	0.452946	0.0197682
$526$	0	0
$527$	12.6669	0.551780
$528$	0	0
$529$	−22.4997	−0.978250
$530$	0	0
$531$	−4.03204	−0.174976
$532$	0	0
$533$	42.4417	1.83835
$534$	0	0
$535$	−20.5159	−0.886981
$536$	0	0
$537$	−0.582041	−0.0251169
$538$	0	0
$539$	0.606151	0.0261088
$540$	0	0
$541$	−40.1951	−1.72812	−0.864061	−	0.503387i	$-0.832087\pi$
$541$	−40.1951	−1.72812	−0.864061	+	0.503387i	$0.832087\pi$
$542$	0	0
$543$	1.07190	0.0459997
$544$	0	0
$545$	−1.12612	−0.0482375
$546$	0	0
$547$	−33.0307	−1.41229	−0.706145	−	0.708067i	$-0.749567\pi$
$547$	−33.0307	−1.41229	−0.706145	+	0.708067i	$0.749567\pi$
$548$	0	0
$549$	−27.3050	−1.16535
$550$	0	0
$551$	24.1337	1.02813
$552$	0	0
$553$	−17.8867	−0.760620
$554$	0	0
$555$	0.0234032	0.000993410 0
$556$	0	0
$557$	6.92285	0.293331	0.146665	−	0.989186i	$-0.453146\pi$
$557$	6.92285	0.293331	0.146665	+	0.989186i	$0.453146\pi$
$558$	0	0
$559$	26.1044	1.10410
$560$	0	0
$561$	−1.55382	−0.0656022
$562$	0	0
$563$	1.81887	0.0766561	0.0383280	−	0.999265i	$-0.487797\pi$
$563$	1.81887	0.0766561	0.0383280	+	0.999265i	$0.487797\pi$
$564$	0	0
$565$	−0.842524	−0.0354453
$566$	0	0
$567$	23.3297	0.979754
$568$	0	0
$569$	11.4799	0.481261	0.240631	−	0.970617i	$-0.422646\pi$
$569$	11.4799	0.481261	0.240631	+	0.970617i	$0.422646\pi$
$570$	0	0
$571$	40.5976	1.69896	0.849479	−	0.527623i	$-0.176917\pi$
$571$	40.5976	1.69896	0.849479	+	0.527623i	$0.176917\pi$
$572$	0	0
$573$	−1.31537	−0.0549503
$574$	0	0
$575$	0.707288	0.0294960
$576$	0	0
$577$	−20.3419	−0.846842	−0.423421	−	0.905933i	$-0.639171\pi$
$577$	−20.3419	−0.846842	−0.423421	+	0.905933i	$0.639171\pi$
$578$	0	0
$579$	−1.92439	−0.0799751
$580$	0	0
$581$	−14.6188	−0.606489
$582$	0	0
$583$	−52.0303	−2.15488
$584$	0	0
$585$	12.3566	0.510884
$586$	0	0
$587$	36.5059	1.50676	0.753379	−	0.657586i	$-0.228422\pi$
$587$	36.5059	1.50676	0.753379	+	0.657586i	$0.228422\pi$
$588$	0	0
$589$	33.3720	1.37507
$590$	0	0
$591$	−0.264022	−0.0108604
$592$	0	0
$593$	−4.02945	−0.165470	−0.0827349	−	0.996572i	$-0.526365\pi$
$593$	−4.02945	−0.165470	−0.0827349	+	0.996572i	$0.526365\pi$
$594$	0	0
$595$	−4.94039	−0.202536
$596$	0	0
$597$	2.42225	0.0991360
$598$	0	0
$599$	31.6701	1.29400	0.647002	−	0.762489i	$-0.276023\pi$
$599$	31.6701	1.29400	0.647002	+	0.762489i	$0.276023\pi$
$600$	0	0
$601$	−19.4667	−0.794065	−0.397032	−	0.917805i	$-0.629960\pi$
$601$	−19.4667	−0.794065	−0.397032	+	0.917805i	$0.629960\pi$
$602$	0	0
$603$	14.6937	0.598373
$604$	0	0
$605$	13.4600	0.547228
$606$	0	0
$607$	−13.6128	−0.552528	−0.276264	−	0.961082i	$-0.589096\pi$
$607$	−13.6128	−0.552528	−0.276264	+	0.961082i	$0.589096\pi$
$608$	0	0
$609$	−2.24139	−0.0908257
$610$	0	0
$611$	7.87797	0.318709
$612$	0	0
$613$	15.7657	0.636771	0.318386	−	0.947961i	$-0.396859\pi$
$613$	15.7657	0.636771	0.318386	+	0.947961i	$0.396859\pi$
$614$	0	0
$615$	−1.73202	−0.0698416
$616$	0	0
$617$	1.96695	0.0791863	0.0395932	−	0.999216i	$-0.487394\pi$
$617$	1.96695	0.0791863	0.0395932	+	0.999216i	$0.487394\pi$
$618$	0	0
$619$	−11.0895	−0.445723	−0.222862	−	0.974850i	$-0.571540\pi$
$619$	−11.0895	−0.445723	−0.222862	+	0.974850i	$0.571540\pi$
$620$	0	0
$621$	−0.716781	−0.0287634
$622$	0	0
$623$	−28.0372	−1.12329
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.09365	−0.163484
$628$	0	0
$629$	−0.255264	−0.0101780
$630$	0	0
$631$	−0.220729	−0.00878708	−0.00439354	−	0.999990i	$-0.501399\pi$
$631$	−0.220729	−0.00878708	−0.00439354	+	0.999990i	$0.501399\pi$
$632$	0	0
$633$	−2.99339	−0.118977
$634$	0	0
$635$	21.1693	0.840077
$636$	0	0
$637$	−0.509709	−0.0201954
$638$	0	0
$639$	−23.3767	−0.924767
$640$	0	0
$641$	−19.2037	−0.758502	−0.379251	−	0.925294i	$-0.623818\pi$
$641$	−19.2037	−0.758502	−0.379251	+	0.925294i	$0.623818\pi$
$642$	0	0
$643$	10.1415	0.399940	0.199970	−	0.979802i	$-0.435915\pi$
$643$	10.1415	0.399940	0.199970	+	0.979802i	$0.435915\pi$
$644$	0	0
$645$	−1.06530	−0.0419463
$646$	0	0
$647$	−26.4735	−1.04078	−0.520391	−	0.853928i	$-0.674214\pi$
$647$	−26.4735	−1.04078	−0.520391	+	0.853928i	$0.674214\pi$
$648$	0	0
$649$	6.71153	0.263451
$650$	0	0
$651$	−3.09938	−0.121474
$652$	0	0
$653$	−14.9547	−0.585222	−0.292611	−	0.956232i	$-0.594524\pi$
$653$	−14.9547	−0.585222	−0.292611	+	0.956232i	$0.594524\pi$
$654$	0	0
$655$	−6.60789	−0.258192
$656$	0	0
$657$	−46.5182	−1.81485
$658$	0	0
$659$	34.1237	1.32927	0.664636	−	0.747167i	$-0.268587\pi$
$659$	34.1237	1.32927	0.664636	+	0.747167i	$0.268587\pi$
$660$	0	0
$661$	33.1342	1.28877	0.644386	−	0.764701i	$-0.277113\pi$
$661$	33.1342	1.28877	0.644386	+	0.764701i	$0.277113\pi$
$662$	0	0
$663$	1.30659	0.0507439
$664$	0	0
$665$	−13.0158	−0.504732
$666$	0	0
$667$	−3.49999	−0.135520
$668$	0	0
$669$	−0.523874	−0.0202541
$670$	0	0
$671$	45.4506	1.75460
$672$	0	0
$673$	−41.8069	−1.61154	−0.805769	−	0.592230i	$-0.798248\pi$
$673$	−41.8069	−1.61154	−0.805769	+	0.592230i	$0.798248\pi$
$674$	0	0
$675$	−1.01342	−0.0390066
$676$	0	0
$677$	32.1521	1.23571	0.617853	−	0.786294i	$-0.288003\pi$
$677$	32.1521	1.23571	0.617853	+	0.786294i	$0.288003\pi$
$678$	0	0
$679$	12.8086	0.491550
$680$	0	0
$681$	−1.99617	−0.0764936
$682$	0	0
$683$	−48.7058	−1.86367	−0.931837	−	0.362877i	$-0.881794\pi$
$683$	−48.7058	−1.86367	−0.931837	+	0.362877i	$0.881794\pi$
$684$	0	0
$685$	10.2840	0.392932
$686$	0	0
$687$	2.39756	0.0914727
$688$	0	0
$689$	43.7519	1.66682
$690$	0	0
$691$	22.7675	0.866117	0.433059	−	0.901366i	$-0.357434\pi$
$691$	22.7675	0.866117	0.433059	+	0.901366i	$0.357434\pi$
$692$	0	0
$693$	−39.2173	−1.48974
$694$	0	0
$695$	−7.05110	−0.267463
$696$	0	0
$697$	18.8915	0.715567
$698$	0	0
$699$	−2.35934	−0.0892384
$700$	0	0
$701$	−42.4981	−1.60513	−0.802566	−	0.596563i	$-0.796533\pi$
$701$	−42.4981	−1.60513	−0.802566	+	0.596563i	$0.796533\pi$
$702$	0	0
$703$	−0.672512	−0.0253643
$704$	0	0
$705$	−0.321495	−0.0121082
$706$	0	0
$707$	−1.40773	−0.0529430
$708$	0	0
$709$	18.2700	0.686143	0.343071	−	0.939309i	$-0.388533\pi$
$709$	18.2700	0.686143	0.343071	+	0.939309i	$0.388533\pi$
$710$	0	0
$711$	19.9133	0.746807
$712$	0	0
$713$	−4.83977	−0.181251
$714$	0	0
$715$	−20.5683	−0.769210
$716$	0	0
$717$	−1.82764	−0.0682546
$718$	0	0
$719$	17.0356	0.635319	0.317659	−	0.948205i	$-0.397103\pi$
$719$	17.0356	0.635319	0.317659	+	0.948205i	$0.397103\pi$
$720$	0	0
$721$	−27.4922	−1.02386
$722$	0	0
$723$	2.11650	0.0787134
$724$	0	0
$725$	−4.94847	−0.183781
$726$	0	0
$727$	−31.7051	−1.17588	−0.587939	−	0.808905i	$-0.700061\pi$
$727$	−31.7051	−1.17588	−0.587939	+	0.808905i	$0.700061\pi$
$728$	0	0
$729$	−25.4570	−0.942852
$730$	0	0
$731$	11.6195	0.429763
$732$	0	0
$733$	5.48230	0.202493	0.101247	−	0.994861i	$-0.467717\pi$
$733$	5.48230	0.202493	0.101247	+	0.994861i	$0.467717\pi$
$734$	0	0
$735$	0.0208008	0.000767251 0
$736$	0	0
$737$	−24.4584	−0.900937
$738$	0	0
$739$	−15.9840	−0.587983	−0.293991	−	0.955808i	$-0.594984\pi$
$739$	−15.9840	−0.587983	−0.293991	+	0.955808i	$0.594984\pi$
$740$	0	0
$741$	3.44232	0.126457
$742$	0	0
$743$	−30.7210	−1.12704	−0.563521	−	0.826102i	$-0.690554\pi$
$743$	−30.7210	−1.12704	−0.563521	+	0.826102i	$0.690554\pi$
$744$	0	0
$745$	12.4379	0.455690
$746$	0	0
$747$	16.2751	0.595475
$748$	0	0
$749$	−54.7531	−2.00064
$750$	0	0
$751$	16.4695	0.600981	0.300491	−	0.953785i	$-0.402850\pi$
$751$	16.4695	0.600981	0.300491	+	0.953785i	$0.402850\pi$
$752$	0	0
$753$	0.885888	0.0322836
$754$	0	0
$755$	22.1838	0.807350
$756$	0	0
$757$	31.0756	1.12946	0.564730	−	0.825276i	$-0.308980\pi$
$757$	31.0756	1.12946	0.564730	+	0.825276i	$0.308980\pi$
$758$	0	0
$759$	0.593681	0.0215493
$760$	0	0
$761$	5.91749	0.214509	0.107254	−	0.994232i	$-0.465794\pi$
$761$	5.91749	0.214509	0.107254	+	0.994232i	$0.465794\pi$
$762$	0	0
$763$	−3.00539	−0.108802
$764$	0	0
$765$	5.50015	0.198858
$766$	0	0
$767$	−5.64368	−0.203782
$768$	0	0
$769$	10.7206	0.386596	0.193298	−	0.981140i	$-0.438082\pi$
$769$	10.7206	0.386596	0.193298	+	0.981140i	$0.438082\pi$
$770$	0	0
$771$	0.527842	0.0190098
$772$	0	0
$773$	23.8053	0.856219	0.428109	−	0.903727i	$-0.359180\pi$
$773$	23.8053	0.856219	0.428109	+	0.903727i	$0.359180\pi$
$774$	0	0
$775$	−6.84272	−0.245798
$776$	0	0
$777$	0.0624587	0.00224069
$778$	0	0
$779$	49.7710	1.78323
$780$	0	0
$781$	38.9117	1.39237
$782$	0	0
$783$	5.01488	0.179217
$784$	0	0
$785$	−5.26352	−0.187863
$786$	0	0
$787$	−36.0086	−1.28357	−0.641784	−	0.766886i	$-0.721805\pi$
$787$	−36.0086	−1.28357	−0.641784	+	0.766886i	$0.721805\pi$
$788$	0	0
$789$	3.04664	0.108463
$790$	0	0
$791$	−2.24854	−0.0799489
$792$	0	0
$793$	−38.2191	−1.35720
$794$	0	0
$795$	−1.78549	−0.0633247
$796$	0	0
$797$	10.1048	0.357931	0.178965	−	0.983855i	$-0.442725\pi$
$797$	10.1048	0.357931	0.178965	+	0.983855i	$0.442725\pi$
$798$	0	0
$799$	3.50662	0.124055
$800$	0	0
$801$	31.2139	1.10289
$802$	0	0
$803$	77.4319	2.73251
$804$	0	0
$805$	1.88762	0.0665299
$806$	0	0
$807$	−0.389876	−0.0137243
$808$	0	0
$809$	12.4413	0.437412	0.218706	−	0.975791i	$-0.429817\pi$
$809$	12.4413	0.437412	0.218706	+	0.975791i	$0.429817\pi$
$810$	0	0
$811$	43.3448	1.52204	0.761020	−	0.648728i	$-0.224699\pi$
$811$	43.3448	1.52204	0.761020	+	0.648728i	$0.224699\pi$
$812$	0	0
$813$	3.08561	0.108217
$814$	0	0
$815$	−2.99290	−0.104837
$816$	0	0
$817$	30.6125	1.07099
$818$	0	0
$819$	32.9776	1.15233
$820$	0	0
$821$	12.5145	0.436758	0.218379	−	0.975864i	$-0.429923\pi$
$821$	12.5145	0.436758	0.218379	+	0.975864i	$0.429923\pi$
$822$	0	0
$823$	−11.7501	−0.409583	−0.204792	−	0.978806i	$-0.565652\pi$
$823$	−11.7501	−0.409583	−0.204792	+	0.978806i	$0.565652\pi$
$824$	0	0
$825$	0.839377	0.0292233
$826$	0	0
$827$	−4.81408	−0.167402	−0.0837010	−	0.996491i	$-0.526674\pi$
$827$	−4.81408	−0.167402	−0.0837010	+	0.996491i	$0.526674\pi$
$828$	0	0
$829$	35.1740	1.22165	0.610823	−	0.791768i	$-0.290839\pi$
$829$	35.1740	1.22165	0.610823	+	0.791768i	$0.290839\pi$
$830$	0	0
$831$	−3.32845	−0.115463
$832$	0	0
$833$	−0.226880	−0.00786092
$834$	0	0
$835$	18.1604	0.628467
$836$	0	0
$837$	6.93455	0.239693
$838$	0	0
$839$	3.26196	0.112615	0.0563076	−	0.998413i	$-0.482067\pi$
$839$	3.26196	0.112615	0.0563076	+	0.998413i	$0.482067\pi$
$840$	0	0
$841$	−4.51268	−0.155610
$842$	0	0
$843$	−1.82238	−0.0627661
$844$	0	0
$845$	4.29572	0.147777
$846$	0	0
$847$	35.9223	1.23430
$848$	0	0
$849$	−3.92859	−0.134829
$850$	0	0
$851$	0.0975311	0.00334332
$852$	0	0
$853$	5.68904	0.194789	0.0973945	−	0.995246i	$-0.468949\pi$
$853$	5.68904	0.194789	0.0973945	+	0.995246i	$0.468949\pi$
$854$	0	0
$855$	14.4905	0.495566
$856$	0	0
$857$	44.6563	1.52543	0.762714	−	0.646736i	$-0.223866\pi$
$857$	44.6563	1.52543	0.762714	+	0.646736i	$0.223866\pi$
$858$	0	0
$859$	7.39442	0.252294	0.126147	−	0.992012i	$-0.459739\pi$
$859$	7.39442	0.252294	0.126147	+	0.992012i	$0.459739\pi$
$860$	0	0
$861$	−4.62243	−0.157532
$862$	0	0
$863$	−36.9653	−1.25831	−0.629157	−	0.777278i	$-0.716600\pi$
$863$	−36.9653	−1.25831	−0.629157	+	0.777278i	$0.716600\pi$
$864$	0	0
$865$	12.0709	0.410423
$866$	0	0
$867$	−2.30362	−0.0782351
$868$	0	0
$869$	−33.1467	−1.12443
$870$	0	0
$871$	20.5669	0.696883
$872$	0	0
$873$	−14.2599	−0.482624
$874$	0	0
$875$	2.66881	0.0902223
$876$	0	0
$877$	−13.2963	−0.448985	−0.224493	−	0.974476i	$-0.572072\pi$
$877$	−13.2963	−0.448985	−0.224493	+	0.974476i	$0.572072\pi$
$878$	0	0
$879$	1.01300	0.0341677
$880$	0	0
$881$	10.3069	0.347248	0.173624	−	0.984812i	$-0.444452\pi$
$881$	10.3069	0.347248	0.173624	+	0.984812i	$0.444452\pi$
$882$	0	0
$883$	48.5606	1.63419	0.817097	−	0.576500i	$-0.195582\pi$
$883$	48.5606	1.63419	0.817097	+	0.576500i	$0.195582\pi$
$884$	0	0
$885$	0.230315	0.00774195
$886$	0	0
$887$	−6.79523	−0.228161	−0.114081	−	0.993471i	$-0.536392\pi$
$887$	−6.79523	−0.228161	−0.114081	+	0.993471i	$0.536392\pi$
$888$	0	0
$889$	56.4968	1.89484
$890$	0	0
$891$	43.2334	1.44837
$892$	0	0
$893$	9.23844	0.309153
$894$	0	0
$895$	−3.42945	−0.114634
$896$	0	0
$897$	−0.499223	−0.0166686
$898$	0	0
$899$	33.8610	1.12933
$900$	0	0
$901$	19.4747	0.648797
$902$	0	0
$903$	−2.84310	−0.0946123
$904$	0	0
$905$	6.31578	0.209944
$906$	0	0
$907$	5.40783	0.179564	0.0897820	−	0.995961i	$-0.471383\pi$
$907$	5.40783	0.179564	0.0897820	+	0.995961i	$0.471383\pi$
$908$	0	0
$909$	1.56722	0.0519815
$910$	0	0
$911$	−18.9169	−0.626743	−0.313372	−	0.949631i	$-0.601459\pi$
$911$	−18.9169	−0.626743	−0.313372	+	0.949631i	$0.601459\pi$
$912$	0	0
$913$	−27.0908	−0.896574
$914$	0	0
$915$	1.55970	0.0515619
$916$	0	0
$917$	−17.6352	−0.582366
$918$	0	0
$919$	−48.9075	−1.61331	−0.806655	−	0.591022i	$-0.798724\pi$
$919$	−48.9075	−1.61331	−0.806655	+	0.591022i	$0.798724\pi$
$920$	0	0
$921$	−3.03293	−0.0999384
$922$	0	0
$923$	−32.7206	−1.07701
$924$	0	0
$925$	0.137894	0.00453394
$926$	0	0
$927$	30.6071	1.00527
$928$	0	0
$929$	35.4660	1.16360	0.581801	−	0.813331i	$-0.302348\pi$
$929$	35.4660	1.16360	0.581801	+	0.813331i	$0.302348\pi$
$930$	0	0
$931$	−0.597731	−0.0195898
$932$	0	0
$933$	1.45417	0.0476074
$934$	0	0
$935$	−9.15528	−0.299410
$936$	0	0
$937$	−56.4991	−1.84575	−0.922873	−	0.385105i	$-0.874165\pi$
$937$	−56.4991	−1.84575	−0.922873	+	0.385105i	$0.874165\pi$
$938$	0	0
$939$	3.25562	0.106243
$940$	0	0
$941$	4.04514	0.131868	0.0659338	−	0.997824i	$-0.478997\pi$
$941$	4.04514	0.131868	0.0659338	+	0.997824i	$0.478997\pi$
$942$	0	0
$943$	−7.21805	−0.235052
$944$	0	0
$945$	−2.70463	−0.0879816
$946$	0	0
$947$	8.29692	0.269614	0.134807	−	0.990872i	$-0.456959\pi$
$947$	8.29692	0.269614	0.134807	+	0.990872i	$0.456959\pi$
$948$	0	0
$949$	−65.1119	−2.11362
$950$	0	0
$951$	−2.26040	−0.0732985
$952$	0	0
$953$	25.0238	0.810599	0.405299	−	0.914184i	$-0.367167\pi$
$953$	25.0238	0.810599	0.405299	+	0.914184i	$0.367167\pi$
$954$	0	0
$955$	−7.75030	−0.250794
$956$	0	0
$957$	−4.15363	−0.134268
$958$	0	0
$959$	27.4461	0.886281
$960$	0	0
$961$	15.8228	0.510412
$962$	0	0
$963$	60.9568	1.96431
$964$	0	0
$965$	−11.3388	−0.365007
$966$	0	0
$967$	−16.6523	−0.535502	−0.267751	−	0.963488i	$-0.586280\pi$
$967$	−16.6523	−0.535502	−0.267751	+	0.963488i	$0.586280\pi$
$968$	0	0
$969$	1.53223	0.0492224
$970$	0	0
$971$	43.3530	1.39126	0.695631	−	0.718399i	$-0.255125\pi$
$971$	43.3530	1.39126	0.695631	+	0.718399i	$0.255125\pi$
$972$	0	0
$973$	−18.8181	−0.603279
$974$	0	0
$975$	−0.705826	−0.0226045
$976$	0	0
$977$	13.4307	0.429687	0.214844	−	0.976648i	$-0.431076\pi$
$977$	13.4307	0.429687	0.214844	+	0.976648i	$0.431076\pi$
$978$	0	0
$979$	−51.9571	−1.66056
$980$	0	0
$981$	3.34591	0.106827
$982$	0	0
$983$	7.94549	0.253422	0.126711	−	0.991940i	$-0.459558\pi$
$983$	7.94549	0.253422	0.126711	+	0.991940i	$0.459558\pi$
$984$	0	0
$985$	−1.55565	−0.0495672
$986$	0	0
$987$	−0.858009	−0.0273107
$988$	0	0
$989$	−4.43958	−0.141170
$990$	0	0
$991$	25.0787	0.796652	0.398326	−	0.917244i	$-0.369591\pi$
$991$	25.0787	0.796652	0.398326	+	0.917244i	$0.369591\pi$
$992$	0	0
$993$	−3.08612	−0.0979350
$994$	0	0
$995$	14.2722	0.452458
$996$	0	0
$997$	−51.1689	−1.62054	−0.810268	−	0.586060i	$-0.800678\pi$
$997$	−51.1689	−1.62054	−0.810268	+	0.586060i	$0.800678\pi$
$998$	0	0
$999$	−0.139745	−0.00442134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.t.1.5		8
4.3	odd	2		5120.2.a.v.1.4		8
8.3	odd	2		5120.2.a.s.1.5		8
8.5	even	2		5120.2.a.u.1.4		8
32.3	odd	8		80.2.l.a.61.7	yes	16
32.5	even	8		640.2.l.a.481.4		16
32.11	odd	8		80.2.l.a.21.7	✓	16
32.13	even	8		640.2.l.a.161.4		16
32.19	odd	8		640.2.l.b.161.5		16
32.21	even	8		320.2.l.a.241.5		16
32.27	odd	8		640.2.l.b.481.5		16
32.29	even	8		320.2.l.a.81.5		16
96.11	even	8		720.2.t.c.181.2		16
96.29	odd	8		2880.2.t.c.721.6		16
96.35	even	8		720.2.t.c.541.2		16
96.53	odd	8		2880.2.t.c.2161.7		16
160.3	even	8		400.2.q.g.349.3		16
160.29	even	8		1600.2.l.i.401.4		16
160.43	even	8		400.2.q.h.149.6		16
160.53	odd	8		1600.2.q.g.49.5		16
160.67	even	8		400.2.q.h.349.6		16
160.93	odd	8		1600.2.q.h.849.4		16
160.99	odd	8		400.2.l.h.301.2		16
160.107	even	8		400.2.q.g.149.3		16
160.117	odd	8		1600.2.q.h.49.4		16
160.139	odd	8		400.2.l.h.101.2		16
160.149	even	8		1600.2.l.i.1201.4		16
160.157	odd	8		1600.2.q.g.849.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.7	✓	16	32.11	odd	8
80.2.l.a.61.7	yes	16	32.3	odd	8
320.2.l.a.81.5		16	32.29	even	8
320.2.l.a.241.5		16	32.21	even	8
400.2.l.h.101.2		16	160.139	odd	8
400.2.l.h.301.2		16	160.99	odd	8
400.2.q.g.149.3		16	160.107	even	8
400.2.q.g.349.3		16	160.3	even	8
400.2.q.h.149.6		16	160.43	even	8
400.2.q.h.349.6		16	160.67	even	8
640.2.l.a.161.4		16	32.13	even	8
640.2.l.a.481.4		16	32.5	even	8
640.2.l.b.161.5		16	32.19	odd	8
640.2.l.b.481.5		16	32.27	odd	8
720.2.t.c.181.2		16	96.11	even	8
720.2.t.c.541.2		16	96.35	even	8
1600.2.l.i.401.4		16	160.29	even	8
1600.2.l.i.1201.4		16	160.149	even	8
1600.2.q.g.49.5		16	160.53	odd	8
1600.2.q.g.849.5		16	160.157	odd	8
1600.2.q.h.49.4		16	160.117	odd	8
1600.2.q.h.849.4		16	160.93	odd	8
2880.2.t.c.721.6		16	96.29	odd	8
2880.2.t.c.2161.7		16	96.53	odd	8
5120.2.a.s.1.5		8	8.3	odd	2
5120.2.a.t.1.5		8	1.1	even	1	trivial
5120.2.a.u.1.4		8	8.5	even	2
5120.2.a.v.1.4		8	4.3	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:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q+0.169718 q^{3} +1.00000 q^{5} +2.66881 q^{7} -2.97120 q^{9} +4.94571 q^{11} -4.15881 q^{13} +0.169718 q^{15} -1.85116 q^{17} -4.87701 q^{19} +0.452946 q^{21} +0.707288 q^{23} +1.00000 q^{25} -1.01342 q^{27} -4.94847 q^{29} -6.84272 q^{31} +0.839377 q^{33} +2.66881 q^{35} +0.137894 q^{37} -0.705826 q^{39} -10.2052 q^{41} -6.27690 q^{43} -2.97120 q^{45} -1.89428 q^{47} +0.122561 q^{49} -0.314175 q^{51} -10.5203 q^{53} +4.94571 q^{55} -0.827717 q^{57} +1.35704 q^{59} +9.18991 q^{61} -7.92956 q^{63} -4.15881 q^{65} -4.94538 q^{67} +0.120040 q^{69} +7.86777 q^{71} +15.6564 q^{73} +0.169718 q^{75} +13.1992 q^{77} -6.70212 q^{79} +8.74159 q^{81} -5.47763 q^{83} -1.85116 q^{85} -0.839845 q^{87} -10.5055 q^{89} -11.0991 q^{91} -1.16133 q^{93} -4.87701 q^{95} +4.79937 q^{97} -14.6947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59}+ \cdots - 40 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^15 - 16 * q^19 - 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 + 8 * q^45 - 20 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^55 - 16 * q^59 - 20 * q^63 - 36 * q^67 - 16 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 - 16 * q^95 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more