LMFDB - Embedded newform 4000.2.a.n.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4000,2,Mod(1,4000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,3,0,4,0,13,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.30040000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4$ x^6 - 11x^4 - 4x^3 + 29x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.81030$ of defining polynomial
Character	$\chi$	$=$	4000.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.81030 q^{3} +4.37760 q^{7} +0.277175 q^{9} +5.82433 q^{11} +5.34625 q^{13} +5.41151 q^{17} -1.59789 q^{19} +7.92476 q^{21} -0.340859 q^{23} -4.92912 q^{27} -4.96428 q^{29} -4.63796 q^{31} +10.5438 q^{33} +8.40977 q^{37} +9.67829 q^{39} -2.34244 q^{41} -7.10407 q^{43} +0.879427 q^{47} +12.1634 q^{49} +9.79645 q^{51} -1.30241 q^{53} -2.89265 q^{57} -11.0726 q^{59} -14.4423 q^{61} +1.21336 q^{63} -3.01420 q^{67} -0.617056 q^{69} +9.37319 q^{71} -13.7157 q^{73} +25.4966 q^{77} -5.85098 q^{79} -9.75470 q^{81} +8.54562 q^{83} -8.98682 q^{87} -18.0227 q^{89} +23.4037 q^{91} -8.39608 q^{93} -10.0919 q^{97} +1.61436 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 3 q^{7} + 4 q^{9} + 13 q^{11} + 7 q^{13} - 4 q^{17} + 9 q^{19} + 7 q^{23} - 12 q^{27} + 2 q^{29} + 12 q^{31} - 6 q^{33} + 21 q^{37} + 26 q^{39} - 5 q^{41} - 8 q^{43} + 19 q^{47} - 3 q^{49} + 18 q^{51}+ \cdots + 43 q^{99}+O(q^{100})$ 6 * q + 3 * q^7 + 4 * q^9 + 13 * q^11 + 7 * q^13 - 4 * q^17 + 9 * q^19 + 7 * q^23 - 12 * q^27 + 2 * q^29 + 12 * q^31 - 6 * q^33 + 21 * q^37 + 26 * q^39 - 5 * q^41 - 8 * q^43 + 19 * q^47 - 3 * q^49 + 18 * q^51 + 11 * q^53 - 28 * q^57 + 25 * q^59 - 6 * q^61 - 3 * q^63 - 12 * q^69 + 34 * q^71 - 20 * q^73 + 14 * q^77 + 16 * q^79 - 18 * q^81 - 4 * q^83 - 26 * q^87 - 3 * q^89 + 26 * q^91 + 16 * q^93 - 20 * q^97 + 43 * 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
$3$	1.81030	1.04518	0.522588	−	0.852586i	$-0.324967\pi$
$3$	1.81030	1.04518	0.522588	+	0.852586i	$0.324967\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.37760	1.65458	0.827289	−	0.561777i	$-0.189882\pi$
$7$	4.37760	1.65458	0.827289	+	0.561777i	$0.189882\pi$
$8$	0	0
$9$	0.277175	0.0923916
$10$	0	0
$11$	5.82433	1.75610	0.878051	−	0.478567i	$-0.158844\pi$
$11$	5.82433	1.75610	0.878051	+	0.478567i	$0.158844\pi$
$12$	0	0
$13$	5.34625	1.48278	0.741391	−	0.671073i	$-0.234166\pi$
$13$	5.34625	1.48278	0.741391	+	0.671073i	$0.234166\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.41151	1.31248	0.656242	−	0.754550i	$-0.272145\pi$
$17$	5.41151	1.31248	0.656242	+	0.754550i	$0.272145\pi$
$18$	0	0
$19$	−1.59789	−0.366580	−0.183290	−	0.983059i	$-0.558675\pi$
$19$	−1.59789	−0.366580	−0.183290	+	0.983059i	$0.558675\pi$
$20$	0	0
$21$	7.92476	1.72932
$22$	0	0
$23$	−0.340859	−0.0710740	−0.0355370	−	0.999368i	$-0.511314\pi$
$23$	−0.340859	−0.0710740	−0.0355370	+	0.999368i	$0.511314\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.92912	−0.948610
$28$	0	0
$29$	−4.96428	−0.921844	−0.460922	−	0.887441i	$-0.652481\pi$
$29$	−4.96428	−0.921844	−0.460922	+	0.887441i	$0.652481\pi$
$30$	0	0
$31$	−4.63796	−0.833002	−0.416501	−	0.909135i	$-0.636744\pi$
$31$	−4.63796	−0.833002	−0.416501	+	0.909135i	$0.636744\pi$
$32$	0	0
$33$	10.5438	1.83543
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.40977	1.38256	0.691278	−	0.722588i	$-0.257048\pi$
$37$	8.40977	1.38256	0.691278	+	0.722588i	$0.257048\pi$
$38$	0	0
$39$	9.67829	1.54977
$40$	0	0
$41$	−2.34244	−0.365828	−0.182914	−	0.983129i	$-0.558553\pi$
$41$	−2.34244	−0.365828	−0.182914	+	0.983129i	$0.558553\pi$
$42$	0	0
$43$	−7.10407	−1.08336	−0.541680	−	0.840585i	$-0.682212\pi$
$43$	−7.10407	−1.08336	−0.541680	+	0.840585i	$0.682212\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.879427	0.128278	0.0641388	−	0.997941i	$-0.479570\pi$
$47$	0.879427	0.128278	0.0641388	+	0.997941i	$0.479570\pi$
$48$	0	0
$49$	12.1634	1.73763
$50$	0	0
$51$	9.79645	1.37178
$52$	0	0
$53$	−1.30241	−0.178900	−0.0894502	−	0.995991i	$-0.528511\pi$
$53$	−1.30241	−0.178900	−0.0894502	+	0.995991i	$0.528511\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.89265	−0.383141
$58$	0	0
$59$	−11.0726	−1.44153	−0.720767	−	0.693178i	$-0.756210\pi$
$59$	−11.0726	−1.44153	−0.720767	+	0.693178i	$0.756210\pi$
$60$	0	0
$61$	−14.4423	−1.84915	−0.924574	−	0.381003i	$-0.875579\pi$
$61$	−14.4423	−1.84915	−0.924574	+	0.381003i	$0.875579\pi$
$62$	0	0
$63$	1.21336	0.152869
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.01420	−0.368243	−0.184122	−	0.982903i	$-0.558944\pi$
$67$	−3.01420	−0.368243	−0.184122	+	0.982903i	$0.558944\pi$
$68$	0	0
$69$	−0.617056	−0.0742848
$70$	0	0
$71$	9.37319	1.11239	0.556197	−	0.831051i	$-0.312260\pi$
$71$	9.37319	1.11239	0.556197	+	0.831051i	$0.312260\pi$
$72$	0	0
$73$	−13.7157	−1.60530	−0.802649	−	0.596451i	$-0.796577\pi$
$73$	−13.7157	−1.60530	−0.802649	+	0.596451i	$0.796577\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	25.4966	2.90561
$78$	0	0
$79$	−5.85098	−0.658287	−0.329143	−	0.944280i	$-0.606760\pi$
$79$	−5.85098	−0.658287	−0.329143	+	0.944280i	$0.606760\pi$
$80$	0	0
$81$	−9.75470	−1.08386
$82$	0	0
$83$	8.54562	0.938003	0.469002	−	0.883197i	$-0.344614\pi$
$83$	8.54562	0.938003	0.469002	+	0.883197i	$0.344614\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.98682	−0.963488
$88$	0	0
$89$	−18.0227	−1.91040	−0.955202	−	0.295956i	$-0.904362\pi$
$89$	−18.0227	−1.91040	−0.955202	+	0.295956i	$0.904362\pi$
$90$	0	0
$91$	23.4037	2.45338
$92$	0	0
$93$	−8.39608	−0.870633
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0919	−1.02468	−0.512340	−	0.858783i	$-0.671221\pi$
$97$	−10.0919	−1.02468	−0.512340	+	0.858783i	$0.671221\pi$
$98$	0	0
$99$	1.61436	0.162249
$100$	0	0
$101$	−6.64409	−0.661111	−0.330556	−	0.943787i	$-0.607236\pi$
$101$	−6.64409	−0.661111	−0.330556	+	0.943787i	$0.607236\pi$
$102$	0	0
$103$	−9.84923	−0.970474	−0.485237	−	0.874383i	$-0.661267\pi$
$103$	−9.84923	−0.970474	−0.485237	+	0.874383i	$0.661267\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.941408	−0.0910093	−0.0455047	−	0.998964i	$-0.514490\pi$
$107$	−0.941408	−0.0910093	−0.0455047	+	0.998964i	$0.514490\pi$
$108$	0	0
$109$	10.2276	0.979627	0.489813	−	0.871827i	$-0.337065\pi$
$109$	10.2276	0.979627	0.489813	+	0.871827i	$0.337065\pi$
$110$	0	0
$111$	15.2242	1.44501
$112$	0	0
$113$	−8.90474	−0.837687	−0.418844	−	0.908058i	$-0.637564\pi$
$113$	−8.90474	−0.837687	−0.418844	+	0.908058i	$0.637564\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.48185	0.136997
$118$	0	0
$119$	23.6894	2.17161
$120$	0	0
$121$	22.9228	2.08389
$122$	0	0
$123$	−4.24051	−0.382354
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.75678	0.155889	0.0779446	−	0.996958i	$-0.475164\pi$
$127$	1.75678	0.155889	0.0779446	+	0.996958i	$0.475164\pi$
$128$	0	0
$129$	−12.8605	−1.13230
$130$	0	0
$131$	6.14604	0.536982	0.268491	−	0.963282i	$-0.413475\pi$
$131$	6.14604	0.536982	0.268491	+	0.963282i	$0.413475\pi$
$132$	0	0
$133$	−6.99491	−0.606535
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.79739	0.153561	0.0767806	−	0.997048i	$-0.475536\pi$
$137$	1.79739	0.153561	0.0767806	+	0.997048i	$0.475536\pi$
$138$	0	0
$139$	−15.6284	−1.32559	−0.662793	−	0.748803i	$-0.730629\pi$
$139$	−15.6284	−1.32559	−0.662793	+	0.748803i	$0.730629\pi$
$140$	0	0
$141$	1.59202	0.134073
$142$	0	0
$143$	31.1383	2.60392
$144$	0	0
$145$	0	0
$146$	0	0
$147$	22.0193	1.81612
$148$	0	0
$149$	20.3836	1.66989	0.834944	−	0.550335i	$-0.185500\pi$
$149$	20.3836	1.66989	0.834944	+	0.550335i	$0.185500\pi$
$150$	0	0
$151$	14.2100	1.15639	0.578196	−	0.815898i	$-0.303757\pi$
$151$	14.2100	1.15639	0.578196	+	0.815898i	$0.303757\pi$
$152$	0	0
$153$	1.49994	0.121263
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.01209	−0.479817	−0.239908	−	0.970796i	$-0.577117\pi$
$157$	−6.01209	−0.479817	−0.239908	+	0.970796i	$0.577117\pi$
$158$	0	0
$159$	−2.35776	−0.186982
$160$	0	0
$161$	−1.49215	−0.117598
$162$	0	0
$163$	12.5491	0.982920	0.491460	−	0.870900i	$-0.336463\pi$
$163$	12.5491	0.982920	0.491460	+	0.870900i	$0.336463\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.32243	−0.334480	−0.167240	−	0.985916i	$-0.553485\pi$
$167$	−4.32243	−0.334480	−0.167240	+	0.985916i	$0.553485\pi$
$168$	0	0
$169$	15.5824	1.19864
$170$	0	0
$171$	−0.442894	−0.0338689
$172$	0	0
$173$	−8.22242	−0.625139	−0.312570	−	0.949895i	$-0.601190\pi$
$173$	−8.22242	−0.625139	−0.312570	+	0.949895i	$0.601190\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.0447	−1.50666
$178$	0	0
$179$	2.21704	0.165710	0.0828548	−	0.996562i	$-0.473596\pi$
$179$	2.21704	0.165710	0.0828548	+	0.996562i	$0.473596\pi$
$180$	0	0
$181$	15.6084	1.16016	0.580080	−	0.814559i	$-0.303021\pi$
$181$	15.6084	1.16016	0.580080	+	0.814559i	$0.303021\pi$
$182$	0	0
$183$	−26.1449	−1.93268
$184$	0	0
$185$	0	0
$186$	0	0
$187$	31.5184	2.30486
$188$	0	0
$189$	−21.5777	−1.56955
$190$	0	0
$191$	−0.823027	−0.0595521	−0.0297761	−	0.999557i	$-0.509479\pi$
$191$	−0.823027	−0.0595521	−0.0297761	+	0.999557i	$0.509479\pi$
$192$	0	0
$193$	−4.12278	−0.296764	−0.148382	−	0.988930i	$-0.547407\pi$
$193$	−4.12278	−0.296764	−0.148382	+	0.988930i	$0.547407\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.3961	1.31067	0.655333	−	0.755340i	$-0.272528\pi$
$197$	18.3961	1.31067	0.655333	+	0.755340i	$0.272528\pi$
$198$	0	0
$199$	1.01893	0.0722300	0.0361150	−	0.999348i	$-0.488502\pi$
$199$	1.01893	0.0722300	0.0361150	+	0.999348i	$0.488502\pi$
$200$	0	0
$201$	−5.45660	−0.384879
$202$	0	0
$203$	−21.7316	−1.52526
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.0944776	−0.00656665
$208$	0	0
$209$	−9.30662	−0.643752
$210$	0	0
$211$	−8.14609	−0.560800	−0.280400	−	0.959883i	$-0.590467\pi$
$211$	−8.14609	−0.560800	−0.280400	+	0.959883i	$0.590467\pi$
$212$	0	0
$213$	16.9683	1.16265
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.3031	−1.37827
$218$	0	0
$219$	−24.8294	−1.67782
$220$	0	0
$221$	28.9313	1.94613
$222$	0	0
$223$	−12.4073	−0.830856	−0.415428	−	0.909626i	$-0.636368\pi$
$223$	−12.4073	−0.830856	−0.415428	+	0.909626i	$0.636368\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.5589	−1.63003	−0.815016	−	0.579438i	$-0.803272\pi$
$227$	−24.5589	−1.63003	−0.815016	+	0.579438i	$0.803272\pi$
$228$	0	0
$229$	−5.35261	−0.353711	−0.176855	−	0.984237i	$-0.556592\pi$
$229$	−5.35261	−0.353711	−0.176855	+	0.984237i	$0.556592\pi$
$230$	0	0
$231$	46.1564	3.03687
$232$	0	0
$233$	−0.350300	−0.0229489	−0.0114745	−	0.999934i	$-0.503653\pi$
$233$	−0.350300	−0.0229489	−0.0114745	+	0.999934i	$0.503653\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.5920	−0.688025
$238$	0	0
$239$	8.90513	0.576025	0.288012	−	0.957627i	$-0.407006\pi$
$239$	8.90513	0.576025	0.288012	+	0.957627i	$0.407006\pi$
$240$	0	0
$241$	−8.09465	−0.521422	−0.260711	−	0.965417i	$-0.583957\pi$
$241$	−8.09465	−0.521422	−0.260711	+	0.965417i	$0.583957\pi$
$242$	0	0
$243$	−2.87154	−0.184209
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.54270	−0.543559
$248$	0	0
$249$	15.4701	0.980378
$250$	0	0
$251$	15.0480	0.949821	0.474910	−	0.880034i	$-0.342480\pi$
$251$	15.0480	0.949821	0.474910	+	0.880034i	$0.342480\pi$
$252$	0	0
$253$	−1.98528	−0.124813
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.1390	−0.819586	−0.409793	−	0.912179i	$-0.634399\pi$
$257$	−13.1390	−0.819586	−0.409793	+	0.912179i	$0.634399\pi$
$258$	0	0
$259$	36.8146	2.28755
$260$	0	0
$261$	−1.37597	−0.0851706
$262$	0	0
$263$	16.3912	1.01072	0.505362	−	0.862908i	$-0.331359\pi$
$263$	16.3912	1.01072	0.505362	+	0.862908i	$0.331359\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−32.6265	−1.99671
$268$	0	0
$269$	1.56270	0.0952793	0.0476397	−	0.998865i	$-0.484830\pi$
$269$	1.56270	0.0952793	0.0476397	+	0.998865i	$0.484830\pi$
$270$	0	0
$271$	30.1625	1.83224	0.916121	−	0.400902i	$-0.131303\pi$
$271$	30.1625	1.83224	0.916121	+	0.400902i	$0.131303\pi$
$272$	0	0
$273$	42.3677	2.56421
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.9905	−1.08095	−0.540473	−	0.841361i	$-0.681755\pi$
$277$	−17.9905	−1.08095	−0.540473	+	0.841361i	$0.681755\pi$
$278$	0	0
$279$	−1.28553	−0.0769624
$280$	0	0
$281$	−1.25760	−0.0750221	−0.0375111	−	0.999296i	$-0.511943\pi$
$281$	−1.25760	−0.0750221	−0.0375111	+	0.999296i	$0.511943\pi$
$282$	0	0
$283$	17.4528	1.03746	0.518729	−	0.854938i	$-0.326405\pi$
$283$	17.4528	1.03746	0.518729	+	0.854938i	$0.326405\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.2543	−0.605290
$288$	0	0
$289$	12.2845	0.722616
$290$	0	0
$291$	−18.2694	−1.07097
$292$	0	0
$293$	24.0216	1.40336	0.701678	−	0.712494i	$-0.252435\pi$
$293$	24.0216	1.40336	0.701678	+	0.712494i	$0.252435\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−28.7088	−1.66586
$298$	0	0
$299$	−1.82232	−0.105387
$300$	0	0
$301$	−31.0988	−1.79250
$302$	0	0
$303$	−12.0278	−0.690977
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.1584	−0.636845	−0.318423	−	0.947949i	$-0.603153\pi$
$307$	−11.1584	−0.636845	−0.318423	+	0.947949i	$0.603153\pi$
$308$	0	0
$309$	−17.8300	−1.01432
$310$	0	0
$311$	19.8088	1.12326	0.561628	−	0.827390i	$-0.310175\pi$
$311$	19.8088	1.12326	0.561628	+	0.827390i	$0.310175\pi$
$312$	0	0
$313$	3.29732	0.186376	0.0931879	−	0.995649i	$-0.470294\pi$
$313$	3.29732	0.186376	0.0931879	+	0.995649i	$0.470294\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.5911	0.988017	0.494009	−	0.869457i	$-0.335531\pi$
$317$	17.5911	0.988017	0.494009	+	0.869457i	$0.335531\pi$
$318$	0	0
$319$	−28.9136	−1.61885
$320$	0	0
$321$	−1.70423	−0.0951207
$322$	0	0
$323$	−8.64698	−0.481131
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.5150	1.02388
$328$	0	0
$329$	3.84978	0.212245
$330$	0	0
$331$	15.0330	0.826288	0.413144	−	0.910666i	$-0.364431\pi$
$331$	15.0330	0.826288	0.413144	+	0.910666i	$0.364431\pi$
$332$	0	0
$333$	2.33098	0.127737
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−31.7741	−1.73084	−0.865422	−	0.501043i	$-0.832950\pi$
$337$	−31.7741	−1.73084	−0.865422	+	0.501043i	$0.832950\pi$
$338$	0	0
$339$	−16.1202	−0.875530
$340$	0	0
$341$	−27.0130	−1.46284
$342$	0	0
$343$	22.6032	1.22046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.6632	1.43136	0.715679	−	0.698430i	$-0.246118\pi$
$347$	26.6632	1.43136	0.715679	+	0.698430i	$0.246118\pi$
$348$	0	0
$349$	4.44619	0.237999	0.119000	−	0.992894i	$-0.462031\pi$
$349$	4.44619	0.237999	0.119000	+	0.992894i	$0.462031\pi$
$350$	0	0
$351$	−26.3523	−1.40658
$352$	0	0
$353$	0.316976	0.0168710	0.00843548	−	0.999964i	$-0.497315\pi$
$353$	0.316976	0.0168710	0.00843548	+	0.999964i	$0.497315\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	42.8849	2.26971
$358$	0	0
$359$	22.0361	1.16302	0.581509	−	0.813540i	$-0.302462\pi$
$359$	22.0361	1.16302	0.581509	+	0.813540i	$0.302462\pi$
$360$	0	0
$361$	−16.4468	−0.865619
$362$	0	0
$363$	41.4971	2.17803
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.44627	−0.232093	−0.116047	−	0.993244i	$-0.537022\pi$
$367$	−4.44627	−0.232093	−0.116047	+	0.993244i	$0.537022\pi$
$368$	0	0
$369$	−0.649266	−0.0337994
$370$	0	0
$371$	−5.70145	−0.296005
$372$	0	0
$373$	23.3308	1.20802	0.604012	−	0.796975i	$-0.293568\pi$
$373$	23.3308	1.20802	0.604012	+	0.796975i	$0.293568\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26.5403	−1.36689
$378$	0	0
$379$	19.0905	0.980614	0.490307	−	0.871550i	$-0.336885\pi$
$379$	19.0905	0.980614	0.490307	+	0.871550i	$0.336885\pi$
$380$	0	0
$381$	3.18030	0.162932
$382$	0	0
$383$	0.371944	0.0190055	0.00950273	−	0.999955i	$-0.496975\pi$
$383$	0.371944	0.0190055	0.00950273	+	0.999955i	$0.496975\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.96907	−0.100093
$388$	0	0
$389$	−22.9338	−1.16279	−0.581395	−	0.813621i	$-0.697493\pi$
$389$	−22.9338	−1.16279	−0.581395	+	0.813621i	$0.697493\pi$
$390$	0	0
$391$	−1.84456	−0.0932836
$392$	0	0
$393$	11.1262	0.561240
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.2189	0.713628	0.356814	−	0.934176i	$-0.383863\pi$
$397$	14.2189	0.713628	0.356814	+	0.934176i	$0.383863\pi$
$398$	0	0
$399$	−12.6629	−0.633936
$400$	0	0
$401$	0.847710	0.0423326	0.0211663	−	0.999776i	$-0.493262\pi$
$401$	0.847710	0.0423326	0.0211663	+	0.999776i	$0.493262\pi$
$402$	0	0
$403$	−24.7957	−1.23516
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.9813	2.42791
$408$	0	0
$409$	−32.7324	−1.61851	−0.809256	−	0.587457i	$-0.800129\pi$
$409$	−32.7324	−1.61851	−0.809256	+	0.587457i	$0.800129\pi$
$410$	0	0
$411$	3.25381	0.160498
$412$	0	0
$413$	−48.4715	−2.38513
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−28.2921	−1.38547
$418$	0	0
$419$	24.3482	1.18949	0.594743	−	0.803916i	$-0.297254\pi$
$419$	24.3482	1.18949	0.594743	+	0.803916i	$0.297254\pi$
$420$	0	0
$421$	−5.79282	−0.282325	−0.141162	−	0.989986i	$-0.545084\pi$
$421$	−5.79282	−0.282325	−0.141162	+	0.989986i	$0.545084\pi$
$422$	0	0
$423$	0.243755	0.0118518
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−63.2226	−3.05956
$428$	0	0
$429$	56.3696	2.72155
$430$	0	0
$431$	−27.0917	−1.30496	−0.652480	−	0.757806i	$-0.726272\pi$
$431$	−27.0917	−1.30496	−0.652480	+	0.757806i	$0.726272\pi$
$432$	0	0
$433$	−35.6206	−1.71182	−0.855908	−	0.517129i	$-0.827001\pi$
$433$	−35.6206	−1.71182	−0.855908	+	0.517129i	$0.827001\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.544654	0.0260543
$438$	0	0
$439$	−35.2574	−1.68275	−0.841373	−	0.540455i	$-0.818252\pi$
$439$	−35.2574	−1.68275	−0.841373	+	0.540455i	$0.818252\pi$
$440$	0	0
$441$	3.37138	0.160542
$442$	0	0
$443$	−33.9602	−1.61350	−0.806748	−	0.590896i	$-0.798775\pi$
$443$	−33.9602	−1.61350	−0.806748	+	0.590896i	$0.798775\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	36.9003	1.74533
$448$	0	0
$449$	13.2446	0.625050	0.312525	−	0.949910i	$-0.398825\pi$
$449$	13.2446	0.625050	0.312525	+	0.949910i	$0.398825\pi$
$450$	0	0
$451$	−13.6432	−0.642431
$452$	0	0
$453$	25.7243	1.20863
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.86349	−0.133949	−0.0669743	−	0.997755i	$-0.521335\pi$
$457$	−2.86349	−0.133949	−0.0669743	+	0.997755i	$0.521335\pi$
$458$	0	0
$459$	−26.6740	−1.24504
$460$	0	0
$461$	22.8968	1.06641	0.533206	−	0.845986i	$-0.320987\pi$
$461$	22.8968	1.06641	0.533206	+	0.845986i	$0.320987\pi$
$462$	0	0
$463$	−15.0334	−0.698659	−0.349330	−	0.937000i	$-0.613591\pi$
$463$	−15.0334	−0.698659	−0.349330	+	0.937000i	$0.613591\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.2490	0.751913	0.375957	−	0.926637i	$-0.377314\pi$
$467$	16.2490	0.751913	0.375957	+	0.926637i	$0.377314\pi$
$468$	0	0
$469$	−13.1950	−0.609287
$470$	0	0
$471$	−10.8837	−0.501493
$472$	0	0
$473$	−41.3764	−1.90249
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.360997	−0.0165289
$478$	0	0
$479$	16.8024	0.767722	0.383861	−	0.923391i	$-0.374594\pi$
$479$	16.8024	0.767722	0.383861	+	0.923391i	$0.374594\pi$
$480$	0	0
$481$	44.9607	2.05003
$482$	0	0
$483$	−2.70123	−0.122910
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.0426	−0.772276	−0.386138	−	0.922441i	$-0.626191\pi$
$487$	−17.0426	−0.772276	−0.386138	+	0.922441i	$0.626191\pi$
$488$	0	0
$489$	22.7176	1.02732
$490$	0	0
$491$	25.5972	1.15518	0.577592	−	0.816326i	$-0.303993\pi$
$491$	25.5972	1.15518	0.577592	+	0.816326i	$0.303993\pi$
$492$	0	0
$493$	−26.8643	−1.20991
$494$	0	0
$495$	0	0
$496$	0	0
$497$	41.0321	1.84054
$498$	0	0
$499$	2.10330	0.0941565	0.0470783	−	0.998891i	$-0.485009\pi$
$499$	2.10330	0.0941565	0.0470783	+	0.998891i	$0.485009\pi$
$500$	0	0
$501$	−7.82489	−0.349590
$502$	0	0
$503$	−1.80566	−0.0805105	−0.0402552	−	0.999189i	$-0.512817\pi$
$503$	−1.80566	−0.0805105	−0.0402552	+	0.999189i	$0.512817\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.2087	1.25279
$508$	0	0
$509$	−16.5594	−0.733980	−0.366990	−	0.930225i	$-0.619612\pi$
$509$	−16.5594	−0.733980	−0.366990	+	0.930225i	$0.619612\pi$
$510$	0	0
$511$	−60.0417	−2.65609
$512$	0	0
$513$	7.87618	0.347742
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.12207	0.225269
$518$	0	0
$519$	−14.8850	−0.653380
$520$	0	0
$521$	−22.9830	−1.00690	−0.503452	−	0.864023i	$-0.667937\pi$
$521$	−22.9830	−1.00690	−0.503452	+	0.864023i	$0.667937\pi$
$522$	0	0
$523$	9.47128	0.414150	0.207075	−	0.978325i	$-0.433606\pi$
$523$	9.47128	0.414150	0.207075	+	0.978325i	$0.433606\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.0984	−1.09330
$528$	0	0
$529$	−22.8838	−0.994948
$530$	0	0
$531$	−3.06905	−0.133186
$532$	0	0
$533$	−12.5233	−0.542443
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.01351	0.173196
$538$	0	0
$539$	70.8436	3.05145
$540$	0	0
$541$	13.2203	0.568386	0.284193	−	0.958767i	$-0.408274\pi$
$541$	13.2203	0.568386	0.284193	+	0.958767i	$0.408274\pi$
$542$	0	0
$543$	28.2558	1.21257
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.1211	1.41616	0.708079	−	0.706133i	$-0.249562\pi$
$547$	33.1211	1.41616	0.708079	+	0.706133i	$0.249562\pi$
$548$	0	0
$549$	−4.00304	−0.170846
$550$	0	0
$551$	7.93236	0.337930
$552$	0	0
$553$	−25.6133	−1.08919
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.75852	0.371110	0.185555	−	0.982634i	$-0.440592\pi$
$557$	8.75852	0.371110	0.185555	+	0.982634i	$0.440592\pi$
$558$	0	0
$559$	−37.9801	−1.60639
$560$	0	0
$561$	57.0577	2.40898
$562$	0	0
$563$	1.05986	0.0446678	0.0223339	−	0.999751i	$-0.492890\pi$
$563$	1.05986	0.0446678	0.0223339	+	0.999751i	$0.492890\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−42.7022	−1.79332
$568$	0	0
$569$	38.9002	1.63078	0.815390	−	0.578912i	$-0.196523\pi$
$569$	38.9002	1.63078	0.815390	+	0.578912i	$0.196523\pi$
$570$	0	0
$571$	33.4842	1.40127	0.700635	−	0.713520i	$-0.252900\pi$
$571$	33.4842	1.40127	0.700635	+	0.713520i	$0.252900\pi$
$572$	0	0
$573$	−1.48992	−0.0622424
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.5217	0.729436	0.364718	−	0.931118i	$-0.381165\pi$
$577$	17.5217	0.729436	0.364718	+	0.931118i	$0.381165\pi$
$578$	0	0
$579$	−7.46346	−0.310171
$580$	0	0
$581$	37.4093	1.55200
$582$	0	0
$583$	−7.58569	−0.314167
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.6889	0.895195	0.447598	−	0.894235i	$-0.352280\pi$
$587$	21.6889	0.895195	0.447598	+	0.894235i	$0.352280\pi$
$588$	0	0
$589$	7.41093	0.305362
$590$	0	0
$591$	33.3024	1.36988
$592$	0	0
$593$	34.8683	1.43187	0.715935	−	0.698167i	$-0.246001\pi$
$593$	34.8683	1.43187	0.715935	+	0.698167i	$0.246001\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.84456	0.0754930
$598$	0	0
$599$	14.6435	0.598317	0.299158	−	0.954203i	$-0.403294\pi$
$599$	14.6435	0.598317	0.299158	+	0.954203i	$0.403294\pi$
$600$	0	0
$601$	−9.69821	−0.395598	−0.197799	−	0.980243i	$-0.563379\pi$
$601$	−9.69821	−0.395598	−0.197799	+	0.980243i	$0.563379\pi$
$602$	0	0
$603$	−0.835460	−0.0340226
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.94086	−0.119366	−0.0596829	−	0.998217i	$-0.519009\pi$
$607$	−2.94086	−0.119366	−0.0596829	+	0.998217i	$0.519009\pi$
$608$	0	0
$609$	−39.3407	−1.59417
$610$	0	0
$611$	4.70163	0.190208
$612$	0	0
$613$	−17.0019	−0.686702	−0.343351	−	0.939207i	$-0.611562\pi$
$613$	−17.0019	−0.686702	−0.343351	+	0.939207i	$0.611562\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.7996	0.716583	0.358292	−	0.933610i	$-0.383359\pi$
$617$	17.7996	0.716583	0.358292	+	0.933610i	$0.383359\pi$
$618$	0	0
$619$	27.5237	1.10627	0.553135	−	0.833092i	$-0.313431\pi$
$619$	27.5237	1.10627	0.553135	+	0.833092i	$0.313431\pi$
$620$	0	0
$621$	1.68014	0.0674215
$622$	0	0
$623$	−78.8962	−3.16091
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−16.8477	−0.672834
$628$	0	0
$629$	45.5096	1.81458
$630$	0	0
$631$	−47.3515	−1.88503	−0.942517	−	0.334158i	$-0.891548\pi$
$631$	−47.3515	−1.88503	−0.942517	+	0.334158i	$0.891548\pi$
$632$	0	0
$633$	−14.7468	−0.586134
$634$	0	0
$635$	0	0
$636$	0	0
$637$	65.0285	2.57652
$638$	0	0
$639$	2.59801	0.102776
$640$	0	0
$641$	−15.3183	−0.605037	−0.302518	−	0.953144i	$-0.597827\pi$
$641$	−15.3183	−0.605037	−0.302518	+	0.953144i	$0.597827\pi$
$642$	0	0
$643$	38.4573	1.51661	0.758304	−	0.651901i	$-0.226028\pi$
$643$	38.4573	1.51661	0.758304	+	0.651901i	$0.226028\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.46190	−0.136101	−0.0680507	−	0.997682i	$-0.521678\pi$
$647$	−3.46190	−0.136101	−0.0680507	+	0.997682i	$0.521678\pi$
$648$	0	0
$649$	−64.4906	−2.53148
$650$	0	0
$651$	−36.7547	−1.44053
$652$	0	0
$653$	−37.9039	−1.48329	−0.741647	−	0.670790i	$-0.765955\pi$
$653$	−37.9039	−1.48329	−0.741647	+	0.670790i	$0.765955\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.80164	−0.148316
$658$	0	0
$659$	−22.2876	−0.868203	−0.434102	−	0.900864i	$-0.642934\pi$
$659$	−22.2876	−0.868203	−0.434102	+	0.900864i	$0.642934\pi$
$660$	0	0
$661$	−10.1877	−0.396257	−0.198129	−	0.980176i	$-0.563486\pi$
$661$	−10.1877	−0.396257	−0.198129	+	0.980176i	$0.563486\pi$
$662$	0	0
$663$	52.3742	2.03405
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.69212	0.0655192
$668$	0	0
$669$	−22.4609	−0.868390
$670$	0	0
$671$	−84.1168	−3.24729
$672$	0	0
$673$	22.6737	0.874008	0.437004	−	0.899460i	$-0.356040\pi$
$673$	22.6737	0.874008	0.437004	+	0.899460i	$0.356040\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.30011	−0.318999	−0.159500	−	0.987198i	$-0.550988\pi$
$677$	−8.30011	−0.318999	−0.159500	+	0.987198i	$0.550988\pi$
$678$	0	0
$679$	−44.1784	−1.69541
$680$	0	0
$681$	−44.4589	−1.70367
$682$	0	0
$683$	8.62826	0.330151	0.165076	−	0.986281i	$-0.447213\pi$
$683$	8.62826	0.330151	0.165076	+	0.986281i	$0.447213\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9.68982	−0.369690
$688$	0	0
$689$	−6.96303	−0.265270
$690$	0	0
$691$	−42.6346	−1.62190	−0.810949	−	0.585117i	$-0.801049\pi$
$691$	−42.6346	−1.62190	−0.810949	+	0.585117i	$0.801049\pi$
$692$	0	0
$693$	7.06701	0.268454
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.6762	−0.480143
$698$	0	0
$699$	−0.634147	−0.0239857
$700$	0	0
$701$	−38.4312	−1.45153	−0.725763	−	0.687945i	$-0.758513\pi$
$701$	−38.4312	−1.45153	−0.725763	+	0.687945i	$0.758513\pi$
$702$	0	0
$703$	−13.4379	−0.506818
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−29.0852	−1.09386
$708$	0	0
$709$	19.9047	0.747537	0.373769	−	0.927522i	$-0.378065\pi$
$709$	19.9047	0.747537	0.373769	+	0.927522i	$0.378065\pi$
$710$	0	0
$711$	−1.62175	−0.0608202
$712$	0	0
$713$	1.58089	0.0592048
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.1209	0.602047
$718$	0	0
$719$	14.9260	0.556644	0.278322	−	0.960488i	$-0.410222\pi$
$719$	14.9260	0.556644	0.278322	+	0.960488i	$0.410222\pi$
$720$	0	0
$721$	−43.1160	−1.60572
$722$	0	0
$723$	−14.6537	−0.544978
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−25.6583	−0.951615	−0.475808	−	0.879549i	$-0.657844\pi$
$727$	−25.6583	−0.951615	−0.475808	+	0.879549i	$0.657844\pi$
$728$	0	0
$729$	24.0658	0.891325
$730$	0	0
$731$	−38.4437	−1.42189
$732$	0	0
$733$	45.1247	1.66672	0.833360	−	0.552731i	$-0.186414\pi$
$733$	45.1247	1.66672	0.833360	+	0.552731i	$0.186414\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.5557	−0.646672
$738$	0	0
$739$	−4.88625	−0.179743	−0.0898717	−	0.995953i	$-0.528646\pi$
$739$	−4.88625	−0.179743	−0.0898717	+	0.995953i	$0.528646\pi$
$740$	0	0
$741$	−15.4648	−0.568114
$742$	0	0
$743$	31.3357	1.14960	0.574798	−	0.818295i	$-0.305081\pi$
$743$	31.3357	1.14960	0.574798	+	0.818295i	$0.305081\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.36863	0.0866637
$748$	0	0
$749$	−4.12111	−0.150582
$750$	0	0
$751$	2.75676	0.100596	0.0502978	−	0.998734i	$-0.483983\pi$
$751$	2.75676	0.100596	0.0502978	+	0.998734i	$0.483983\pi$
$752$	0	0
$753$	27.2413	0.992729
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.7888	1.01000	0.505000	−	0.863119i	$-0.331492\pi$
$757$	27.7888	1.01000	0.505000	+	0.863119i	$0.331492\pi$
$758$	0	0
$759$	−3.59394	−0.130452
$760$	0	0
$761$	38.2357	1.38604	0.693022	−	0.720916i	$-0.256279\pi$
$761$	38.2357	1.38604	0.693022	+	0.720916i	$0.256279\pi$
$762$	0	0
$763$	44.7724	1.62087
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−59.1970	−2.13748
$768$	0	0
$769$	41.3417	1.49082	0.745409	−	0.666607i	$-0.232254\pi$
$769$	41.3417	1.49082	0.745409	+	0.666607i	$0.232254\pi$
$770$	0	0
$771$	−23.7854	−0.856611
$772$	0	0
$773$	4.31886	0.155339	0.0776693	−	0.996979i	$-0.475252\pi$
$773$	4.31886	0.155339	0.0776693	+	0.996979i	$0.475252\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	66.6453	2.39089
$778$	0	0
$779$	3.74296	0.134105
$780$	0	0
$781$	54.5926	1.95348
$782$	0	0
$783$	24.4695	0.874470
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.48163	0.302337	0.151169	−	0.988508i	$-0.451696\pi$
$787$	8.48163	0.302337	0.151169	+	0.988508i	$0.451696\pi$
$788$	0	0
$789$	29.6729	1.05638
$790$	0	0
$791$	−38.9814	−1.38602
$792$	0	0
$793$	−77.2121	−2.74188
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.6851	−0.484750	−0.242375	−	0.970183i	$-0.577926\pi$
$797$	−13.6851	−0.484750	−0.242375	+	0.970183i	$0.577926\pi$
$798$	0	0
$799$	4.75903	0.168362
$800$	0	0
$801$	−4.99544	−0.176505
$802$	0	0
$803$	−79.8846	−2.81907
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.82895	0.0995836
$808$	0	0
$809$	−5.08132	−0.178650	−0.0893249	−	0.996003i	$-0.528471\pi$
$809$	−5.08132	−0.178650	−0.0893249	+	0.996003i	$0.528471\pi$
$810$	0	0
$811$	24.9990	0.877834	0.438917	−	0.898528i	$-0.355362\pi$
$811$	24.9990	0.877834	0.438917	+	0.898528i	$0.355362\pi$
$812$	0	0
$813$	54.6031	1.91501
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.3515	0.397138
$818$	0	0
$819$	6.48693	0.226672
$820$	0	0
$821$	−25.8738	−0.903002	−0.451501	−	0.892271i	$-0.649111\pi$
$821$	−25.8738	−0.903002	−0.451501	+	0.892271i	$0.649111\pi$
$822$	0	0
$823$	11.0152	0.383965	0.191982	−	0.981398i	$-0.438508\pi$
$823$	11.0152	0.383965	0.191982	+	0.981398i	$0.438508\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.5502	0.471185	0.235592	−	0.971852i	$-0.424297\pi$
$827$	13.5502	0.471185	0.235592	+	0.971852i	$0.424297\pi$
$828$	0	0
$829$	−38.5707	−1.33962	−0.669808	−	0.742534i	$-0.733624\pi$
$829$	−38.5707	−1.33962	−0.669808	+	0.742534i	$0.733624\pi$
$830$	0	0
$831$	−32.5682	−1.12978
$832$	0	0
$833$	65.8223	2.28061
$834$	0	0
$835$	0	0
$836$	0	0
$837$	22.8611	0.790194
$838$	0	0
$839$	−4.38927	−0.151535	−0.0757673	−	0.997126i	$-0.524141\pi$
$839$	−4.38927	−0.151535	−0.0757673	+	0.997126i	$0.524141\pi$
$840$	0	0
$841$	−4.35591	−0.150204
$842$	0	0
$843$	−2.27663	−0.0784113
$844$	0	0
$845$	0	0
$846$	0	0
$847$	100.347	3.44796
$848$	0	0
$849$	31.5947	1.08433
$850$	0	0
$851$	−2.86655	−0.0982639
$852$	0	0
$853$	35.3589	1.21067	0.605333	−	0.795972i	$-0.293040\pi$
$853$	35.3589	1.21067	0.605333	+	0.795972i	$0.293040\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−56.4133	−1.92704	−0.963520	−	0.267636i	$-0.913758\pi$
$857$	−56.4133	−1.92704	−0.963520	+	0.267636i	$0.913758\pi$
$858$	0	0
$859$	28.2629	0.964319	0.482160	−	0.876083i	$-0.339853\pi$
$859$	28.2629	0.964319	0.482160	+	0.876083i	$0.339853\pi$
$860$	0	0
$861$	−18.5633	−0.632635
$862$	0	0
$863$	11.3915	0.387772	0.193886	−	0.981024i	$-0.437891\pi$
$863$	11.3915	0.387772	0.193886	+	0.981024i	$0.437891\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	22.2385	0.755261
$868$	0	0
$869$	−34.0781	−1.15602
$870$	0	0
$871$	−16.1147	−0.546024
$872$	0	0
$873$	−2.79723	−0.0946718
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.1016	0.577479	0.288740	−	0.957408i	$-0.406764\pi$
$877$	17.1016	0.577479	0.288740	+	0.957408i	$0.406764\pi$
$878$	0	0
$879$	43.4862	1.46675
$880$	0	0
$881$	−53.3569	−1.79764	−0.898821	−	0.438317i	$-0.855575\pi$
$881$	−53.3569	−1.79764	−0.898821	+	0.438317i	$0.855575\pi$
$882$	0	0
$883$	−35.9395	−1.20946	−0.604730	−	0.796430i	$-0.706719\pi$
$883$	−35.9395	−1.20946	−0.604730	+	0.796430i	$0.706719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.4603	1.62714	0.813569	−	0.581468i	$-0.197521\pi$
$887$	48.4603	1.62714	0.813569	+	0.581468i	$0.197521\pi$
$888$	0	0
$889$	7.69049	0.257931
$890$	0	0
$891$	−56.8146	−1.90336
$892$	0	0
$893$	−1.40522	−0.0470241
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.29894	−0.110148
$898$	0	0
$899$	23.0241	0.767898
$900$	0	0
$901$	−7.04803	−0.234804
$902$	0	0
$903$	−56.2980	−1.87348
$904$	0	0
$905$	0	0
$906$	0	0
$907$	21.2305	0.704948	0.352474	−	0.935822i	$-0.385341\pi$
$907$	21.2305	0.704948	0.352474	+	0.935822i	$0.385341\pi$
$908$	0	0
$909$	−1.84157	−0.0610811
$910$	0	0
$911$	23.2599	0.770634	0.385317	−	0.922784i	$-0.374092\pi$
$911$	23.2599	0.770634	0.385317	+	0.922784i	$0.374092\pi$
$912$	0	0
$913$	49.7725	1.64723
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.9049	0.888478
$918$	0	0
$919$	−43.5439	−1.43638	−0.718191	−	0.695846i	$-0.755029\pi$
$919$	−43.5439	−1.43638	−0.718191	+	0.695846i	$0.755029\pi$
$920$	0	0
$921$	−20.2001	−0.665615
$922$	0	0
$923$	50.1114	1.64944
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.72996	−0.0896637
$928$	0	0
$929$	1.93781	0.0635776	0.0317888	−	0.999495i	$-0.489880\pi$
$929$	1.93781	0.0635776	0.0317888	+	0.999495i	$0.489880\pi$
$930$	0	0
$931$	−19.4357	−0.636980
$932$	0	0
$933$	35.8599	1.17400
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.2146	0.399032	0.199516	−	0.979895i	$-0.436063\pi$
$937$	12.2146	0.399032	0.199516	+	0.979895i	$0.436063\pi$
$938$	0	0
$939$	5.96913	0.194795
$940$	0	0
$941$	19.4761	0.634902	0.317451	−	0.948275i	$-0.397173\pi$
$941$	19.4761	0.634902	0.317451	+	0.948275i	$0.397173\pi$
$942$	0	0
$943$	0.798442	0.0260009
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.77050	−0.0575335	−0.0287667	−	0.999586i	$-0.509158\pi$
$947$	−1.77050	−0.0575335	−0.0287667	+	0.999586i	$0.509158\pi$
$948$	0	0
$949$	−73.3274	−2.38031
$950$	0	0
$951$	31.8452	1.03265
$952$	0	0
$953$	26.4519	0.856860	0.428430	−	0.903575i	$-0.359067\pi$
$953$	26.4519	0.856860	0.428430	+	0.903575i	$0.359067\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−52.3422	−1.69198
$958$	0	0
$959$	7.86824	0.254079
$960$	0	0
$961$	−9.48935	−0.306108
$962$	0	0
$963$	−0.260935	−0.00840850
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.5985	−0.951822	−0.475911	−	0.879493i	$-0.657882\pi$
$967$	−29.5985	−0.951822	−0.475911	+	0.879493i	$0.657882\pi$
$968$	0	0
$969$	−15.6536	−0.502866
$970$	0	0
$971$	3.26775	0.104867	0.0524335	−	0.998624i	$-0.483302\pi$
$971$	3.26775	0.104867	0.0524335	+	0.998624i	$0.483302\pi$
$972$	0	0
$973$	−68.4150	−2.19328
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.6591	−0.373009	−0.186504	−	0.982454i	$-0.559716\pi$
$977$	−11.6591	−0.373009	−0.186504	+	0.982454i	$0.559716\pi$
$978$	0	0
$979$	−104.970	−3.35486
$980$	0	0
$981$	2.83483	0.0905093
$982$	0	0
$983$	−37.8109	−1.20598	−0.602990	−	0.797749i	$-0.706024\pi$
$983$	−37.8109	−1.20598	−0.602990	+	0.797749i	$0.706024\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.96925	0.221834
$988$	0	0
$989$	2.42149	0.0769988
$990$	0	0
$991$	−10.0303	−0.318623	−0.159311	−	0.987228i	$-0.550927\pi$
$991$	−10.0303	−0.318623	−0.159311	+	0.987228i	$0.550927\pi$
$992$	0	0
$993$	27.2142	0.863616
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−45.2025	−1.43158	−0.715788	−	0.698317i	$-0.753933\pi$
$997$	−45.2025	−1.43158	−0.715788	+	0.698317i	$0.753933\pi$
$998$	0	0
$999$	−41.4528	−1.31151

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.a.n.1.5	yes	6
4.3	odd	2		4000.2.a.k.1.2	✓	6
5.2	odd	4		4000.2.c.g.1249.4		12
5.3	odd	4		4000.2.c.g.1249.9		12
5.4	even	2		4000.2.a.l.1.2	yes	6
8.3	odd	2		8000.2.a.bv.1.5		6
8.5	even	2		8000.2.a.bw.1.2		6
20.3	even	4		4000.2.c.f.1249.4		12
20.7	even	4		4000.2.c.f.1249.9		12
20.19	odd	2		4000.2.a.m.1.5	yes	6
40.19	odd	2		8000.2.a.bx.1.2		6
40.29	even	2		8000.2.a.bu.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.k.1.2	✓	6	4.3	odd	2
4000.2.a.l.1.2	yes	6	5.4	even	2
4000.2.a.m.1.5	yes	6	20.19	odd	2
4000.2.a.n.1.5	yes	6	1.1	even	1	trivial
4000.2.c.f.1249.4		12	20.3	even	4
4000.2.c.f.1249.9		12	20.7	even	4
4000.2.c.g.1249.4		12	5.2	odd	4
4000.2.c.g.1249.9		12	5.3	odd	4
8000.2.a.bu.1.5		6	40.29	even	2
8000.2.a.bv.1.5		6	8.3	odd	2
8000.2.a.bw.1.2		6	8.5	even	2
8000.2.a.bx.1.2		6	40.19	odd	2

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

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4000.2.a.n.1.5

Level	$4000$
Weight	$2$
Character	4000.1
Self dual	yes
Analytic conductor	$31.940$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$