LMFDB - Embedded newform 1620.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1620,2,Mod(1,1620)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1620.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +4.10083 q^{7} -3.81681 q^{11} +5.81681 q^{13} +3.81681 q^{17} -1.81681 q^{19} +2.10083 q^{23} +1.00000 q^{25} -7.20166 q^{29} -1.81681 q^{31} -4.10083 q^{35} -6.01847 q^{37} +11.0185 q^{41} +5.81681 q^{43} +11.9176 q^{47} +9.81681 q^{49} +4.20166 q^{53} +3.81681 q^{55} -4.20166 q^{59} -3.01847 q^{61} -5.81681 q^{65} +3.71598 q^{67} -2.01847 q^{71} +8.00000 q^{73} -15.6521 q^{77} +2.00000 q^{79} -3.89917 q^{83} -3.81681 q^{85} -3.00000 q^{89} +23.8538 q^{91} +1.81681 q^{95} +12.2017 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} + 6 q^{13} + 6 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{29} + 6 q^{31} - 3 q^{35} + 12 q^{37} + 3 q^{41} + 6 q^{43} + 15 q^{47} + 18 q^{49} - 6 q^{53} + 6 q^{59} + 21 q^{61} - 6 q^{65}+ \cdots + 18 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 6 * q^13 + 6 * q^19 - 3 * q^23 + 3 * q^25 - 3 * q^29 + 6 * q^31 - 3 * q^35 + 12 * q^37 + 3 * q^41 + 6 * q^43 + 15 * q^47 + 18 * q^49 - 6 * q^53 + 6 * q^59 + 21 * q^61 - 6 * q^65 + 9 * q^67 + 24 * q^71 + 24 * q^73 + 6 * q^77 + 6 * q^79 - 21 * q^83 - 9 * q^89 - 6 * q^95 + 18 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.10083	1.54997	0.774984	−	0.631981i	$-0.217758\pi$
$7$	4.10083	1.54997	0.774984	+	0.631981i	$0.217758\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.81681	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$11$	−3.81681	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$12$	0	0
$13$	5.81681	1.61329	0.806646	−	0.591034i	$-0.201280\pi$
$13$	5.81681	1.61329	0.806646	+	0.591034i	$0.201280\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.81681	0.925712	0.462856	−	0.886433i	$-0.346825\pi$
$17$	3.81681	0.925712	0.462856	+	0.886433i	$0.346825\pi$
$18$	0	0
$19$	−1.81681	−0.416805	−0.208402	−	0.978043i	$-0.566826\pi$
$19$	−1.81681	−0.416805	−0.208402	+	0.978043i	$0.566826\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.10083	0.438053	0.219027	−	0.975719i	$-0.429712\pi$
$23$	2.10083	0.438053	0.219027	+	0.975719i	$0.429712\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.20166	−1.33731	−0.668657	−	0.743571i	$-0.733131\pi$
$29$	−7.20166	−1.33731	−0.668657	+	0.743571i	$0.733131\pi$
$30$	0	0
$31$	−1.81681	−0.326309	−0.163154	−	0.986601i	$-0.552167\pi$
$31$	−1.81681	−0.326309	−0.163154	+	0.986601i	$0.552167\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.10083	−0.693167
$36$	0	0
$37$	−6.01847	−0.989431	−0.494715	−	0.869055i	$-0.664728\pi$
$37$	−6.01847	−0.989431	−0.494715	+	0.869055i	$0.664728\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.0185	1.72080	0.860398	−	0.509623i	$-0.170215\pi$
$41$	11.0185	1.72080	0.860398	+	0.509623i	$0.170215\pi$
$42$	0	0
$43$	5.81681	0.887055	0.443528	−	0.896261i	$-0.353727\pi$
$43$	5.81681	0.887055	0.443528	+	0.896261i	$0.353727\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.9176	1.73837	0.869183	−	0.494490i	$-0.164645\pi$
$47$	11.9176	1.73837	0.869183	+	0.494490i	$0.164645\pi$
$48$	0	0
$49$	9.81681	1.40240
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.20166	0.577143	0.288571	−	0.957458i	$-0.406820\pi$
$53$	4.20166	0.577143	0.288571	+	0.957458i	$0.406820\pi$
$54$	0	0
$55$	3.81681	0.514659
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.20166	−0.547010	−0.273505	−	0.961871i	$-0.588183\pi$
$59$	−4.20166	−0.547010	−0.273505	+	0.961871i	$0.588183\pi$
$60$	0	0
$61$	−3.01847	−0.386476	−0.193238	−	0.981152i	$-0.561899\pi$
$61$	−3.01847	−0.386476	−0.193238	+	0.981152i	$0.561899\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.81681	−0.721487
$66$	0	0
$67$	3.71598	0.453979	0.226990	−	0.973897i	$-0.427112\pi$
$67$	3.71598	0.453979	0.226990	+	0.973897i	$0.427112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.01847	−0.239548	−0.119774	−	0.992801i	$-0.538217\pi$
$71$	−2.01847	−0.239548	−0.119774	+	0.992801i	$0.538217\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.6521	−1.78372
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.89917	−0.427989	−0.213995	−	0.976835i	$-0.568648\pi$
$83$	−3.89917	−0.427989	−0.213995	+	0.976835i	$0.568648\pi$
$84$	0	0
$85$	−3.81681	−0.413991
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	23.8538	2.50055
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.81681	0.186401
$96$	0	0
$97$	12.2017	1.23889	0.619445	−	0.785040i	$-0.287357\pi$
$97$	12.2017	1.23889	0.619445	+	0.785040i	$0.287357\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.2017	1.61213	0.806063	−	0.591830i	$-0.201594\pi$
$101$	16.2017	1.61213	0.806063	+	0.591830i	$0.201594\pi$
$102$	0	0
$103$	−9.45043	−0.931179	−0.465589	−	0.885001i	$-0.654158\pi$
$103$	−9.45043	−0.931179	−0.465589	+	0.885001i	$0.654158\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.66887	0.451357	0.225678	−	0.974202i	$-0.427540\pi$
$107$	4.66887	0.451357	0.225678	+	0.974202i	$0.427540\pi$
$108$	0	0
$109$	8.81681	0.844497	0.422249	−	0.906480i	$-0.361241\pi$
$109$	8.81681	0.844497	0.422249	+	0.906480i	$0.361241\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.2017	−0.959692	−0.479846	−	0.877353i	$-0.659307\pi$
$113$	−10.2017	−0.959692	−0.479846	+	0.877353i	$0.659307\pi$
$114$	0	0
$115$	−2.10083	−0.195903
$116$	0	0
$117$	0	0
$118$	0	0
$119$	15.6521	1.43482
$120$	0	0
$121$	3.56804	0.324367
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.284020	0.0252027	0.0126014	−	0.999921i	$-0.495989\pi$
$127$	0.284020	0.0252027	0.0126014	+	0.999921i	$0.495989\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.2672	1.85813	0.929064	−	0.369920i	$-0.120615\pi$
$131$	21.2672	1.85813	0.929064	+	0.369920i	$0.120615\pi$
$132$	0	0
$133$	−7.45043	−0.646034
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.56804	−0.219402	−0.109701	−	0.993965i	$-0.534989\pi$
$137$	−2.56804	−0.219402	−0.109701	+	0.993965i	$0.534989\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−22.2017	−1.85660
$144$	0	0
$145$	7.20166	0.598065
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.4504	−1.18383	−0.591913	−	0.806002i	$-0.701627\pi$
$149$	−14.4504	−1.18383	−0.591913	+	0.806002i	$0.701627\pi$
$150$	0	0
$151$	−8.58651	−0.698760	−0.349380	−	0.936981i	$-0.613608\pi$
$151$	−8.58651	−0.698760	−0.349380	+	0.936981i	$0.613608\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.81681	0.145930
$156$	0	0
$157$	19.8353	1.58303	0.791514	−	0.611151i	$-0.209293\pi$
$157$	19.8353	1.58303	0.791514	+	0.611151i	$0.209293\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.61515	0.678969
$162$	0	0
$163$	−13.8168	−1.08222	−0.541108	−	0.840953i	$-0.681995\pi$
$163$	−13.8168	−1.08222	−0.541108	+	0.840953i	$0.681995\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.08236	0.470667	0.235334	−	0.971915i	$-0.424382\pi$
$167$	6.08236	0.470667	0.235334	+	0.971915i	$0.424382\pi$
$168$	0	0
$169$	20.8353	1.60271
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.36638	−0.331970	−0.165985	−	0.986128i	$-0.553080\pi$
$173$	−4.36638	−0.331970	−0.165985	+	0.986128i	$0.553080\pi$
$174$	0	0
$175$	4.10083	0.309994
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.38485	0.477226	0.238613	−	0.971115i	$-0.423307\pi$
$179$	6.38485	0.477226	0.238613	+	0.971115i	$0.423307\pi$
$180$	0	0
$181$	10.0656	0.748169	0.374084	−	0.927395i	$-0.377957\pi$
$181$	10.0656	0.748169	0.374084	+	0.927395i	$0.377957\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.01847	0.442487
$186$	0	0
$187$	−14.5680	−1.06532
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.6521	−1.13255	−0.566273	−	0.824218i	$-0.691615\pi$
$191$	−15.6521	−1.13255	−0.566273	+	0.824218i	$0.691615\pi$
$192$	0	0
$193$	13.4504	0.968183	0.484092	−	0.875017i	$-0.339150\pi$
$193$	13.4504	0.968183	0.484092	+	0.875017i	$0.339150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.79834	−0.128126	−0.0640632	−	0.997946i	$-0.520406\pi$
$197$	−1.79834	−0.128126	−0.0640632	+	0.997946i	$0.520406\pi$
$198$	0	0
$199$	−22.2201	−1.57514	−0.787572	−	0.616223i	$-0.788662\pi$
$199$	−22.2201	−1.57514	−0.787572	+	0.616223i	$0.788662\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−29.5328	−2.07280
$204$	0	0
$205$	−11.0185	−0.769563
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.93442	0.479664
$210$	0	0
$211$	−28.2201	−1.94275	−0.971377	−	0.237543i	$-0.923658\pi$
$211$	−28.2201	−1.94275	−0.971377	+	0.237543i	$0.923658\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.81681	−0.396703
$216$	0	0
$217$	−7.45043	−0.505768
$218$	0	0
$219$	0	0
$220$	0	0
$221$	22.2017	1.49345
$222$	0	0
$223$	1.91764	0.128415	0.0642074	−	0.997937i	$-0.479548\pi$
$223$	1.91764	0.128415	0.0642074	+	0.997937i	$0.479548\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.01847	−0.532205	−0.266102	−	0.963945i	$-0.585736\pi$
$227$	−8.01847	−0.532205	−0.266102	+	0.963945i	$0.585736\pi$
$228$	0	0
$229$	−18.8353	−1.24467	−0.622335	−	0.782751i	$-0.713816\pi$
$229$	−18.8353	−1.24467	−0.622335	+	0.782751i	$0.713816\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0185	−0.918381	−0.459190	−	0.888338i	$-0.651860\pi$
$233$	−14.0185	−0.918381	−0.459190	+	0.888338i	$0.651860\pi$
$234$	0	0
$235$	−11.9176	−0.777421
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.2017	0.659891	0.329945	−	0.944000i	$-0.392970\pi$
$239$	10.2017	0.659891	0.329945	+	0.944000i	$0.392970\pi$
$240$	0	0
$241$	17.2201	1.10925	0.554623	−	0.832102i	$-0.312862\pi$
$241$	17.2201	1.10925	0.554623	+	0.832102i	$0.312862\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.81681	−0.627173
$246$	0	0
$247$	−10.5680	−0.672428
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.384851	−0.0242916	−0.0121458	−	0.999926i	$-0.503866\pi$
$251$	−0.384851	−0.0242916	−0.0121458	+	0.999926i	$0.503866\pi$
$252$	0	0
$253$	−8.01847	−0.504117
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.26724	0.578075	0.289037	−	0.957318i	$-0.406665\pi$
$257$	9.26724	0.578075	0.289037	+	0.957318i	$0.406665\pi$
$258$	0	0
$259$	−24.6807	−1.53359
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.8538	−1.59421	−0.797105	−	0.603840i	$-0.793636\pi$
$263$	−25.8538	−1.59421	−0.797105	+	0.603840i	$0.793636\pi$
$264$	0	0
$265$	−4.20166	−0.258106
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.0841	1.71231	0.856157	−	0.516715i	$-0.172845\pi$
$269$	28.0841	1.71231	0.856157	+	0.516715i	$0.172845\pi$
$270$	0	0
$271$	−21.4504	−1.30302	−0.651510	−	0.758640i	$-0.725864\pi$
$271$	−21.4504	−1.30302	−0.651510	+	0.758640i	$0.725864\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.81681	−0.230162
$276$	0	0
$277$	−7.81681	−0.469667	−0.234833	−	0.972036i	$-0.575454\pi$
$277$	−7.81681	−0.469667	−0.234833	+	0.972036i	$0.575454\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.18319	0.309203	0.154602	−	0.987977i	$-0.450591\pi$
$281$	5.18319	0.309203	0.154602	+	0.987977i	$0.450591\pi$
$282$	0	0
$283$	−18.1008	−1.07598	−0.537991	−	0.842950i	$-0.680817\pi$
$283$	−18.1008	−1.07598	−0.537991	+	0.842950i	$0.680817\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	45.1849	2.66718
$288$	0	0
$289$	−2.43196	−0.143056
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.5865	−0.968994	−0.484497	−	0.874793i	$-0.660997\pi$
$293$	−16.5865	−0.968994	−0.484497	+	0.874793i	$0.660997\pi$
$294$	0	0
$295$	4.20166	0.244630
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.2201	0.706708
$300$	0	0
$301$	23.8538	1.37491
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.01847	0.172837
$306$	0	0
$307$	−22.3025	−1.27287	−0.636435	−	0.771330i	$-0.719592\pi$
$307$	−22.3025	−1.27287	−0.636435	+	0.771330i	$0.719592\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.6521	−1.56801	−0.784003	−	0.620757i	$-0.786825\pi$
$311$	−27.6521	−1.56801	−0.784003	+	0.620757i	$0.786825\pi$
$312$	0	0
$313$	11.6521	0.658615	0.329308	−	0.944223i	$-0.393185\pi$
$313$	11.6521	0.658615	0.329308	+	0.944223i	$0.393185\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.6151	0.652372	0.326186	−	0.945306i	$-0.394236\pi$
$317$	11.6151	0.652372	0.326186	+	0.945306i	$0.394236\pi$
$318$	0	0
$319$	27.4874	1.53900
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.93442	−0.385841
$324$	0	0
$325$	5.81681	0.322659
$326$	0	0
$327$	0	0
$328$	0	0
$329$	48.8722	2.69441
$330$	0	0
$331$	−19.4320	−1.06808	−0.534039	−	0.845460i	$-0.679326\pi$
$331$	−19.4320	−1.06808	−0.534039	+	0.845460i	$0.679326\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.71598	−0.203026
$336$	0	0
$337$	−9.83528	−0.535762	−0.267881	−	0.963452i	$-0.586323\pi$
$337$	−9.83528	−0.535762	−0.267881	+	0.963452i	$0.586323\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.93442	0.375520
$342$	0	0
$343$	11.5513	0.623709
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.5865	1.21251	0.606254	−	0.795271i	$-0.292672\pi$
$347$	22.5865	1.21251	0.606254	+	0.795271i	$0.292672\pi$
$348$	0	0
$349$	5.86392	0.313888	0.156944	−	0.987607i	$-0.449836\pi$
$349$	5.86392	0.313888	0.156944	+	0.987607i	$0.449836\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	2.01847	0.107129
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.2488	−1.01591	−0.507956	−	0.861383i	$-0.669599\pi$
$359$	−19.2488	−1.01591	−0.507956	+	0.861383i	$0.669599\pi$
$360$	0	0
$361$	−15.6992	−0.826274
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	18.5865	0.970208	0.485104	−	0.874456i	$-0.338782\pi$
$367$	18.5865	0.970208	0.485104	+	0.874456i	$0.338782\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	17.2303	0.894553
$372$	0	0
$373$	0.201661	0.0104416	0.00522080	−	0.999986i	$-0.498338\pi$
$373$	0.201661	0.0104416	0.00522080	+	0.999986i	$0.498338\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−41.8907	−2.15748
$378$	0	0
$379$	−2.36638	−0.121553	−0.0607764	−	0.998151i	$-0.519358\pi$
$379$	−2.36638	−0.121553	−0.0607764	+	0.998151i	$0.519358\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.8538	1.32107	0.660533	−	0.750797i	$-0.270331\pi$
$383$	25.8538	1.32107	0.660533	+	0.750797i	$0.270331\pi$
$384$	0	0
$385$	15.6521	0.797704
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.7882	−0.901896	−0.450948	−	0.892550i	$-0.648914\pi$
$389$	−17.7882	−0.901896	−0.450948	+	0.892550i	$0.648914\pi$
$390$	0	0
$391$	8.01847	0.405512
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.00000	−0.100631
$396$	0	0
$397$	13.6706	0.686106	0.343053	−	0.939316i	$-0.388539\pi$
$397$	13.6706	0.686106	0.343053	+	0.939316i	$0.388539\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.46890	−0.372979	−0.186490	−	0.982457i	$-0.559711\pi$
$401$	−7.46890	−0.372979	−0.186490	+	0.982457i	$0.559711\pi$
$402$	0	0
$403$	−10.5680	−0.526432
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.9714	1.13865
$408$	0	0
$409$	−29.3042	−1.44900	−0.724499	−	0.689276i	$-0.757929\pi$
$409$	−29.3042	−1.44900	−0.724499	+	0.689276i	$0.757929\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−17.2303	−0.847848
$414$	0	0
$415$	3.89917	0.191403
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.2201	−1.76947	−0.884735	−	0.466095i	$-0.845660\pi$
$419$	−36.2201	−1.76947	−0.884735	+	0.466095i	$0.845660\pi$
$420$	0	0
$421$	−26.0369	−1.26896	−0.634481	−	0.772938i	$-0.718786\pi$
$421$	−26.0369	−1.26896	−0.634481	+	0.772938i	$0.718786\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.81681	0.185142
$426$	0	0
$427$	−12.3782	−0.599025
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.0369	1.06148	0.530741	−	0.847534i	$-0.321914\pi$
$431$	22.0369	1.06148	0.530741	+	0.847534i	$0.321914\pi$
$432$	0	0
$433$	−0.183190	−0.00880354	−0.00440177	−	0.999990i	$-0.501401\pi$
$433$	−0.183190	−0.00880354	−0.00440177	+	0.999990i	$0.501401\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.81681	−0.182583
$438$	0	0
$439$	−22.6050	−1.07888	−0.539438	−	0.842025i	$-0.681363\pi$
$439$	−22.6050	−1.07888	−0.539438	+	0.842025i	$0.681363\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.9176	−1.42143	−0.710715	−	0.703480i	$-0.751628\pi$
$443$	−29.9176	−1.42143	−0.710715	+	0.703480i	$0.751628\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.1647	0.574089	0.287044	−	0.957917i	$-0.407327\pi$
$449$	12.1647	0.574089	0.287044	+	0.957917i	$0.407327\pi$
$450$	0	0
$451$	−42.0554	−1.98031
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−23.8538	−1.11828
$456$	0	0
$457$	−34.0554	−1.59305	−0.796523	−	0.604609i	$-0.793329\pi$
$457$	−34.0554	−1.59305	−0.796523	+	0.604609i	$0.793329\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	38.5058	1.79340	0.896698	−	0.442643i	$-0.145959\pi$
$461$	38.5058	1.79340	0.896698	+	0.442643i	$0.145959\pi$
$462$	0	0
$463$	−0.0184711	−0.000858425 0	−0.000429212	−	1.00000i	$-0.500137\pi$
$463$	−0.0184711	−0.000858425 0	−0.000429212		1.00000i	$0.500137\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.5865	−0.489885	−0.244943	−	0.969538i	$-0.578769\pi$
$467$	−10.5865	−0.489885	−0.244943	+	0.969538i	$0.578769\pi$
$468$	0	0
$469$	15.2386	0.703653
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22.2017	−1.02083
$474$	0	0
$475$	−1.81681	−0.0833610
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.76970	−0.309316	−0.154658	−	0.987968i	$-0.549427\pi$
$479$	−6.76970	−0.309316	−0.154658	+	0.987968i	$0.549427\pi$
$480$	0	0
$481$	−35.0083	−1.59624
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.2017	−0.554049
$486$	0	0
$487$	31.2857	1.41769	0.708845	−	0.705364i	$-0.249216\pi$
$487$	31.2857	1.41769	0.708845	+	0.705364i	$0.249216\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.1647	0.819762	0.409881	−	0.912139i	$-0.365570\pi$
$491$	18.1647	0.819762	0.409881	+	0.912139i	$0.365570\pi$
$492$	0	0
$493$	−27.4874	−1.23797
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.27741	−0.371292
$498$	0	0
$499$	−2.53110	−0.113308	−0.0566538	−	0.998394i	$-0.518043\pi$
$499$	−2.53110	−0.113308	−0.0566538	+	0.998394i	$0.518043\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.6873	0.833227	0.416614	−	0.909084i	$-0.363217\pi$
$503$	18.6873	0.833227	0.416614	+	0.909084i	$0.363217\pi$
$504$	0	0
$505$	−16.2017	−0.720964
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.86392	0.437211	0.218605	−	0.975813i	$-0.429849\pi$
$509$	9.86392	0.437211	0.218605	+	0.975813i	$0.429849\pi$
$510$	0	0
$511$	32.8066	1.45128
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.45043	0.416436
$516$	0	0
$517$	−45.4874	−2.00053
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.60498	0.420802	0.210401	−	0.977615i	$-0.432523\pi$
$521$	9.60498	0.420802	0.210401	+	0.977615i	$0.432523\pi$
$522$	0	0
$523$	4.32096	0.188942	0.0944712	−	0.995528i	$-0.469884\pi$
$523$	4.32096	0.188942	0.0944712	+	0.995528i	$0.469884\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.93442	−0.302068
$528$	0	0
$529$	−18.5865	−0.808109
$530$	0	0
$531$	0	0
$532$	0	0
$533$	64.0924	2.77615
$534$	0	0
$535$	−4.66887	−0.201853
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−37.4689	−1.61390
$540$	0	0
$541$	4.23030	0.181875	0.0909374	−	0.995857i	$-0.471014\pi$
$541$	4.23030	0.181875	0.0909374	+	0.995857i	$0.471014\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.81681	−0.377671
$546$	0	0
$547$	−10.3025	−0.440503	−0.220251	−	0.975443i	$-0.570688\pi$
$547$	−10.3025	−0.440503	−0.220251	+	0.975443i	$0.570688\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.0841	0.557399
$552$	0	0
$553$	8.20166	0.348770
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	33.8353	1.43108
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.51432	−0.400981	−0.200490	−	0.979696i	$-0.564254\pi$
$563$	−9.51432	−0.400981	−0.200490	+	0.979696i	$0.564254\pi$
$564$	0	0
$565$	10.2017	0.429187
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.2672	−0.891569	−0.445785	−	0.895140i	$-0.647075\pi$
$569$	−21.2672	−0.891569	−0.445785	+	0.895140i	$0.647075\pi$
$570$	0	0
$571$	16.4033	0.686458	0.343229	−	0.939252i	$-0.388479\pi$
$571$	16.4033	0.686458	0.343229	+	0.939252i	$0.388479\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.10083	0.0876107
$576$	0	0
$577$	34.3328	1.42929	0.714647	−	0.699485i	$-0.246587\pi$
$577$	34.3328	1.42929	0.714647	+	0.699485i	$0.246587\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−15.9898	−0.663370
$582$	0	0
$583$	−16.0369	−0.664182
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.6975	0.730454	0.365227	−	0.930918i	$-0.380991\pi$
$587$	17.6975	0.730454	0.365227	+	0.930918i	$0.380991\pi$
$588$	0	0
$589$	3.30080	0.136007
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−45.3227	−1.86118	−0.930589	−	0.366065i	$-0.880705\pi$
$593$	−45.3227	−1.86118	−0.930589	+	0.366065i	$0.880705\pi$
$594$	0	0
$595$	−15.6521	−0.641673
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.0369	−0.410098	−0.205049	−	0.978752i	$-0.565735\pi$
$599$	−10.0369	−0.410098	−0.205049	+	0.978752i	$0.565735\pi$
$600$	0	0
$601$	33.3042	1.35851	0.679253	−	0.733904i	$-0.262304\pi$
$601$	33.3042	1.35851	0.679253	+	0.733904i	$0.262304\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.56804	−0.145061
$606$	0	0
$607$	−3.75292	−0.152326	−0.0761632	−	0.997095i	$-0.524267\pi$
$607$	−3.75292	−0.152326	−0.0761632	+	0.997095i	$0.524267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	69.3227	2.80449
$612$	0	0
$613$	−3.45043	−0.139362	−0.0696808	−	0.997569i	$-0.522198\pi$
$613$	−3.45043	−0.139362	−0.0696808	+	0.997569i	$0.522198\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.4689	0.542238	0.271119	−	0.962546i	$-0.412606\pi$
$617$	13.4689	0.542238	0.271119	+	0.962546i	$0.412606\pi$
$618$	0	0
$619$	2.54957	0.102476	0.0512379	−	0.998686i	$-0.483683\pi$
$619$	2.54957	0.102476	0.0512379	+	0.998686i	$0.483683\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.3025	−0.492889
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−22.9714	−0.915928
$630$	0	0
$631$	30.2017	1.20231	0.601155	−	0.799133i	$-0.294708\pi$
$631$	30.2017	1.20231	0.601155	+	0.799133i	$0.294708\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.284020	−0.0112710
$636$	0	0
$637$	57.1025	2.26248
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.4135	0.648294	0.324147	−	0.946007i	$-0.394923\pi$
$641$	16.4135	0.648294	0.324147	+	0.946007i	$0.394923\pi$
$642$	0	0
$643$	−2.28402	−0.0900730	−0.0450365	−	0.998985i	$-0.514340\pi$
$643$	−2.28402	−0.0900730	−0.0450365	+	0.998985i	$0.514340\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.08236	0.239122	0.119561	−	0.992827i	$-0.461851\pi$
$647$	6.08236	0.239122	0.119561	+	0.992827i	$0.461851\pi$
$648$	0	0
$649$	16.0369	0.629505
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.5209	0.450849	0.225424	−	0.974261i	$-0.427623\pi$
$653$	11.5209	0.450849	0.225424	+	0.974261i	$0.427623\pi$
$654$	0	0
$655$	−21.2672	−0.830980
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.0656	−1.13223	−0.566117	−	0.824325i	$-0.691555\pi$
$659$	−29.0656	−1.13223	−0.566117	+	0.824325i	$0.691555\pi$
$660$	0	0
$661$	40.9378	1.59230	0.796148	−	0.605102i	$-0.206868\pi$
$661$	40.9378	1.59230	0.796148	+	0.605102i	$0.206868\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.45043	0.288915
$666$	0	0
$667$	−15.1295	−0.585815
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.5209	0.444761
$672$	0	0
$673$	28.0185	1.08003	0.540016	−	0.841655i	$-0.318418\pi$
$673$	28.0185	1.08003	0.540016	+	0.841655i	$0.318418\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−51.6521	−1.98515	−0.992576	−	0.121630i	$-0.961188\pi$
$677$	−51.6521	−1.98515	−0.992576	+	0.121630i	$0.961188\pi$
$678$	0	0
$679$	50.0369	1.92024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13.8538	0.530099	0.265050	−	0.964235i	$-0.414612\pi$
$683$	13.8538	0.530099	0.265050	+	0.964235i	$0.414612\pi$
$684$	0	0
$685$	2.56804	0.0981198
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.4403	0.931100
$690$	0	0
$691$	12.9009	0.490772	0.245386	−	0.969425i	$-0.421085\pi$
$691$	12.9009	0.490772	0.245386	+	0.969425i	$0.421085\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	42.0554	1.59296
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.75123	0.0661430	0.0330715	−	0.999453i	$-0.489471\pi$
$701$	1.75123	0.0661430	0.0330715	+	0.999453i	$0.489471\pi$
$702$	0	0
$703$	10.9344	0.412399
$704$	0	0
$705$	0	0
$706$	0	0
$707$	66.4403	2.49874
$708$	0	0
$709$	14.3377	0.538465	0.269233	−	0.963075i	$-0.413230\pi$
$709$	14.3377	0.538465	0.269233	+	0.963075i	$0.413230\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.81681	−0.142941
$714$	0	0
$715$	22.2017	0.830295
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12.6050	−0.470087	−0.235043	−	0.971985i	$-0.575523\pi$
$719$	−12.6050	−0.470087	−0.235043	+	0.971985i	$0.575523\pi$
$720$	0	0
$721$	−38.7546	−1.44330
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.20166	−0.267463
$726$	0	0
$727$	−43.7899	−1.62408	−0.812038	−	0.583604i	$-0.801642\pi$
$727$	−43.7899	−1.62408	−0.812038	+	0.583604i	$0.801642\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22.2017	0.821158
$732$	0	0
$733$	7.13608	0.263577	0.131789	−	0.991278i	$-0.457928\pi$
$733$	7.13608	0.263577	0.131789	+	0.991278i	$0.457928\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.1832	−0.522445
$738$	0	0
$739$	−44.6419	−1.64218	−0.821090	−	0.570799i	$-0.806634\pi$
$739$	−44.6419	−1.64218	−0.821090	+	0.570799i	$0.806634\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.83359	−0.177327	−0.0886636	−	0.996062i	$-0.528260\pi$
$743$	−4.83359	−0.177327	−0.0886636	+	0.996062i	$0.528260\pi$
$744$	0	0
$745$	14.4504	0.529423
$746$	0	0
$747$	0	0
$748$	0	0
$749$	19.1462	0.699589
$750$	0	0
$751$	28.8824	1.05393	0.526967	−	0.849886i	$-0.323329\pi$
$751$	28.8824	1.05393	0.526967	+	0.849886i	$0.323329\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.58651	0.312495
$756$	0	0
$757$	30.5865	1.11169	0.555843	−	0.831287i	$-0.312396\pi$
$757$	30.5865	1.11169	0.555843	+	0.831287i	$0.312396\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.2386	−1.27740	−0.638699	−	0.769457i	$-0.720527\pi$
$761$	−35.2386	−1.27740	−0.638699	+	0.769457i	$0.720527\pi$
$762$	0	0
$763$	36.1562	1.30894
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.4403	−0.882487
$768$	0	0
$769$	−45.2386	−1.63135	−0.815673	−	0.578513i	$-0.803633\pi$
$769$	−45.2386	−1.63135	−0.815673	+	0.578513i	$0.803633\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.2672	0.980734	0.490367	−	0.871516i	$-0.336863\pi$
$773$	27.2672	0.980734	0.490367	+	0.871516i	$0.336863\pi$
$774$	0	0
$775$	−1.81681	−0.0652618
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20.0185	−0.717236
$780$	0	0
$781$	7.70412	0.275675
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.8353	−0.707952
$786$	0	0
$787$	13.4504	0.479456	0.239728	−	0.970840i	$-0.422942\pi$
$787$	13.4504	0.479456	0.239728	+	0.970840i	$0.422942\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−41.8353	−1.48749
$792$	0	0
$793$	−17.5579	−0.623498
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.21183	−0.113769	−0.0568844	−	0.998381i	$-0.518117\pi$
$797$	−3.21183	−0.113769	−0.0568844	+	0.998381i	$0.518117\pi$
$798$	0	0
$799$	45.4874	1.60923
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−30.5345	−1.07754
$804$	0	0
$805$	−8.61515	−0.303644
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43.4689	1.52829	0.764143	−	0.645047i	$-0.223162\pi$
$809$	43.4689	1.52829	0.764143	+	0.645047i	$0.223162\pi$
$810$	0	0
$811$	−18.4033	−0.646228	−0.323114	−	0.946360i	$-0.604730\pi$
$811$	−18.4033	−0.646228	−0.323114	+	0.946360i	$0.604730\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.8168	0.483982
$816$	0	0
$817$	−10.5680	−0.369729
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.3849	−0.536935	−0.268467	−	0.963289i	$-0.586517\pi$
$821$	−15.3849	−0.536935	−0.268467	+	0.963289i	$0.586517\pi$
$822$	0	0
$823$	−19.9546	−0.695573	−0.347787	−	0.937574i	$-0.613067\pi$
$823$	−19.9546	−0.695573	−0.347787	+	0.937574i	$0.613067\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.0571	−1.28860	−0.644301	−	0.764772i	$-0.722852\pi$
$827$	−37.0571	−1.28860	−0.644301	+	0.764772i	$0.722852\pi$
$828$	0	0
$829$	14.9815	0.520330	0.260165	−	0.965564i	$-0.416223\pi$
$829$	14.9815	0.520330	0.260165	+	0.965564i	$0.416223\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	37.4689	1.29822
$834$	0	0
$835$	−6.08236	−0.210489
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.54957	−0.226116	−0.113058	−	0.993588i	$-0.536065\pi$
$839$	−6.54957	−0.226116	−0.113058	+	0.993588i	$0.536065\pi$
$840$	0	0
$841$	22.8639	0.788411
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−20.8353	−0.716755
$846$	0	0
$847$	14.6319	0.502759
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.6438	−0.433423
$852$	0	0
$853$	−2.84545	−0.0974263	−0.0487131	−	0.998813i	$-0.515512\pi$
$853$	−2.84545	−0.0974263	−0.0487131	+	0.998813i	$0.515512\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.2201	0.622388	0.311194	−	0.950346i	$-0.399271\pi$
$857$	18.2201	0.622388	0.311194	+	0.950346i	$0.399271\pi$
$858$	0	0
$859$	−46.0554	−1.57139	−0.785695	−	0.618614i	$-0.787695\pi$
$859$	−46.0554	−1.57139	−0.785695	+	0.618614i	$0.787695\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.4521	1.24084	0.620422	−	0.784268i	$-0.286961\pi$
$863$	36.4521	1.24084	0.620422	+	0.784268i	$0.286961\pi$
$864$	0	0
$865$	4.36638	0.148461
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.63362	−0.258953
$870$	0	0
$871$	21.6151	0.732401
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.10083	−0.138633
$876$	0	0
$877$	52.7177	1.78015	0.890075	−	0.455815i	$-0.150652\pi$
$877$	52.7177	1.78015	0.890075	+	0.455815i	$0.150652\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	17.8824	0.602473	0.301237	−	0.953549i	$-0.402601\pi$
$881$	17.8824	0.602473	0.301237	+	0.953549i	$0.402601\pi$
$882$	0	0
$883$	29.3496	0.987693	0.493846	−	0.869549i	$-0.335591\pi$
$883$	29.3496	0.987693	0.493846	+	0.869549i	$0.335591\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.6807	0.560084	0.280042	−	0.959988i	$-0.409652\pi$
$887$	16.6807	0.560084	0.280042	+	0.959988i	$0.409652\pi$
$888$	0	0
$889$	1.16472	0.0390634
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.6521	−0.724560
$894$	0	0
$895$	−6.38485	−0.213422
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.0841	0.436378
$900$	0	0
$901$	16.0369	0.534268
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.0656	−0.334591
$906$	0	0
$907$	2.24708	0.0746130	0.0373065	−	0.999304i	$-0.488122\pi$
$907$	2.24708	0.0746130	0.0373065	+	0.999304i	$0.488122\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10.2959	−0.341118	−0.170559	−	0.985347i	$-0.554557\pi$
$911$	−10.2959	−0.341118	−0.170559	+	0.985347i	$0.554557\pi$
$912$	0	0
$913$	14.8824	0.492535
$914$	0	0
$915$	0	0
$916$	0	0
$917$	87.2133	2.88004
$918$	0	0
$919$	−14.0369	−0.463036	−0.231518	−	0.972831i	$-0.574369\pi$
$919$	−14.0369	−0.463036	−0.231518	+	0.972831i	$0.574369\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.7411	−0.386462
$924$	0	0
$925$	−6.01847	−0.197886
$926$	0	0
$927$	0	0
$928$	0	0
$929$	54.4403	1.78613	0.893064	−	0.449931i	$-0.148551\pi$
$929$	54.4403	1.78613	0.893064	+	0.449931i	$0.148551\pi$
$930$	0	0
$931$	−17.8353	−0.584528
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.5680	0.476426
$936$	0	0
$937$	−33.0656	−1.08021	−0.540103	−	0.841599i	$-0.681615\pi$
$937$	−33.0656	−1.08021	−0.540103	+	0.841599i	$0.681615\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.2017	−0.430362	−0.215181	−	0.976574i	$-0.569034\pi$
$941$	−13.2017	−0.430362	−0.215181	+	0.976574i	$0.569034\pi$
$942$	0	0
$943$	23.1479	0.753801
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.687342	−0.0223356	−0.0111678	−	0.999938i	$-0.503555\pi$
$947$	−0.687342	−0.0223356	−0.0111678	+	0.999938i	$0.503555\pi$
$948$	0	0
$949$	46.5345	1.51057
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−32.7882	−1.06211	−0.531057	−	0.847336i	$-0.678205\pi$
$953$	−32.7882	−1.06211	−0.531057	+	0.847336i	$0.678205\pi$
$954$	0	0
$955$	15.6521	0.506490
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.5311	−0.340067
$960$	0	0
$961$	−27.6992	−0.893523
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.4504	−0.432985
$966$	0	0
$967$	−5.77139	−0.185595	−0.0927977	−	0.995685i	$-0.529581\pi$
$967$	−5.77139	−0.185595	−0.0927977	+	0.995685i	$0.529581\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.1210	−1.31964	−0.659818	−	0.751426i	$-0.729367\pi$
$971$	−41.1210	−1.31964	−0.659818	+	0.751426i	$0.729367\pi$
$972$	0	0
$973$	−16.4033	−0.525866
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.1546	−0.420851	−0.210426	−	0.977610i	$-0.567485\pi$
$977$	−13.1546	−0.420851	−0.210426	+	0.977610i	$0.567485\pi$
$978$	0	0
$979$	11.4504	0.365957
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18.1378	0.578505	0.289253	−	0.957253i	$-0.406593\pi$
$983$	18.1378	0.578505	0.289253	+	0.957253i	$0.406593\pi$
$984$	0	0
$985$	1.79834	0.0572998
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.2201	0.388578
$990$	0	0
$991$	11.4320	0.363148	0.181574	−	0.983377i	$-0.441881\pi$
$991$	11.4320	0.363148	0.181574	+	0.983377i	$0.441881\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22.2201	0.704426
$996$	0	0
$997$	−8.53110	−0.270183	−0.135091	−	0.990833i	$-0.543133\pi$
$997$	−8.53110	−0.270183	−0.135091	+	0.990833i	$0.543133\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.a.i.1.3		3
3.2	odd	2		1620.2.a.j.1.3		3
4.3	odd	2		6480.2.a.bt.1.1		3
5.2	odd	4		8100.2.d.p.649.6		6
5.3	odd	4		8100.2.d.p.649.1		6
5.4	even	2		8100.2.a.v.1.1		3
9.2	odd	6		540.2.i.b.361.1		6
9.4	even	3		180.2.i.b.61.3	✓	6
9.5	odd	6		540.2.i.b.181.1		6
9.7	even	3		180.2.i.b.121.3	yes	6
12.11	even	2		6480.2.a.bw.1.1		3
15.2	even	4		8100.2.d.o.649.6		6
15.8	even	4		8100.2.d.o.649.1		6
15.14	odd	2		8100.2.a.u.1.1		3
36.7	odd	6		720.2.q.k.481.1		6
36.11	even	6		2160.2.q.i.1441.3		6
36.23	even	6		2160.2.q.i.721.3		6
36.31	odd	6		720.2.q.k.241.1		6
45.2	even	12		2700.2.s.c.1549.6		12
45.4	even	6		900.2.i.c.601.1		6
45.7	odd	12		900.2.s.c.49.2		12
45.13	odd	12		900.2.s.c.349.2		12
45.14	odd	6		2700.2.i.c.1801.3		6
45.22	odd	12		900.2.s.c.349.5		12
45.23	even	12		2700.2.s.c.2449.6		12
45.29	odd	6		2700.2.i.c.901.3		6
45.32	even	12		2700.2.s.c.2449.1		12
45.34	even	6		900.2.i.c.301.1		6
45.38	even	12		2700.2.s.c.1549.1		12
45.43	odd	12		900.2.s.c.49.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.b.61.3	✓	6	9.4	even	3
180.2.i.b.121.3	yes	6	9.7	even	3
540.2.i.b.181.1		6	9.5	odd	6
540.2.i.b.361.1		6	9.2	odd	6
720.2.q.k.241.1		6	36.31	odd	6
720.2.q.k.481.1		6	36.7	odd	6
900.2.i.c.301.1		6	45.34	even	6
900.2.i.c.601.1		6	45.4	even	6
900.2.s.c.49.2		12	45.7	odd	12
900.2.s.c.49.5		12	45.43	odd	12
900.2.s.c.349.2		12	45.13	odd	12
900.2.s.c.349.5		12	45.22	odd	12
1620.2.a.i.1.3		3	1.1	even	1	trivial
1620.2.a.j.1.3		3	3.2	odd	2
2160.2.q.i.721.3		6	36.23	even	6
2160.2.q.i.1441.3		6	36.11	even	6
2700.2.i.c.901.3		6	45.29	odd	6
2700.2.i.c.1801.3		6	45.14	odd	6
2700.2.s.c.1549.1		12	45.38	even	12
2700.2.s.c.1549.6		12	45.2	even	12
2700.2.s.c.2449.1		12	45.32	even	12
2700.2.s.c.2449.6		12	45.23	even	12
6480.2.a.bt.1.1		3	4.3	odd	2
6480.2.a.bw.1.1		3	12.11	even	2
8100.2.a.u.1.1		3	15.14	odd	2
8100.2.a.v.1.1		3	5.4	even	2
8100.2.d.o.649.1		6	15.8	even	4
8100.2.d.o.649.6		6	15.2	even	4
8100.2.d.p.649.1		6	5.3	odd	4
8100.2.d.p.649.6		6	5.2	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.10083 q^{7} -3.81681 q^{11} +5.81681 q^{13} +3.81681 q^{17} -1.81681 q^{19} +2.10083 q^{23} +1.00000 q^{25} -7.20166 q^{29} -1.81681 q^{31} -4.10083 q^{35} -6.01847 q^{37} +11.0185 q^{41} +5.81681 q^{43} +11.9176 q^{47} +9.81681 q^{49} +4.20166 q^{53} +3.81681 q^{55} -4.20166 q^{59} -3.01847 q^{61} -5.81681 q^{65} +3.71598 q^{67} -2.01847 q^{71} +8.00000 q^{73} -15.6521 q^{77} +2.00000 q^{79} -3.89917 q^{83} -3.81681 q^{85} -3.00000 q^{89} +23.8538 q^{91} +1.81681 q^{95} +12.2017 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} + 6 q^{13} + 6 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{29} + 6 q^{31} - 3 q^{35} + 12 q^{37} + 3 q^{41} + 6 q^{43} + 15 q^{47} + 18 q^{49} - 6 q^{53} + 6 q^{59} + 21 q^{61} - 6 q^{65}+ \cdots + 18 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 6 * q^13 + 6 * q^19 - 3 * q^23 + 3 * q^25 - 3 * q^29 + 6 * q^31 - 3 * q^35 + 12 * q^37 + 3 * q^41 + 6 * q^43 + 15 * q^47 + 18 * q^49 - 6 * q^53 + 6 * q^59 + 21 * q^61 - 6 * q^65 + 9 * q^67 + 24 * q^71 + 24 * q^73 + 6 * q^77 + 6 * q^79 - 21 * q^83 - 9 * q^89 - 6 * q^95 + 18 * q^97

Introduction

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