LMFDB - Embedded newform 1280.2.c.i.769.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1280,2,Mod(769,1280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1280.769"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,4,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.1
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	1280.769
Dual form			1280.2.c.i.769.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} +(-1.73205 - 1.41421i) q^{5} +2.44949i q^{7} +1.00000 q^{9} +3.46410 q^{11} +(-2.00000 + 2.44949i) q^{15} +4.89898i q^{17} +3.46410 q^{19} +3.46410 q^{21} -2.44949i q^{23} +(1.00000 + 4.89898i) q^{25} -5.65685i q^{27} +4.00000 q^{31} -4.89898i q^{33} +(3.46410 - 4.24264i) q^{35} +8.48528i q^{37} -4.24264i q^{43} +(-1.73205 - 1.41421i) q^{45} -7.34847i q^{47} +1.00000 q^{49} +6.92820 q^{51} -5.65685i q^{53} +(-6.00000 - 4.89898i) q^{55} -4.89898i q^{57} +10.3923 q^{59} +3.46410 q^{61} +2.44949i q^{63} -4.24264i q^{67} -3.46410 q^{69} +12.0000 q^{71} -4.89898i q^{73} +(6.92820 - 1.41421i) q^{75} +8.48528i q^{77} -4.00000 q^{79} -5.00000 q^{81} -9.89949i q^{83} +(6.92820 - 8.48528i) q^{85} -6.00000 q^{89} -5.65685i q^{93} +(-6.00000 - 4.89898i) q^{95} -4.89898i q^{97} +3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} - 8 q^{15} + 4 q^{25} + 16 q^{31} + 4 q^{49} - 24 q^{55} + 48 q^{71} - 16 q^{79} - 20 q^{81} - 24 q^{89} - 24 q^{95}+O(q^{100}) $ 4 * q + 4 * q^9 - 8 * q^15 + 4 * q^25 + 16 * q^31 + 4 * q^49 - 24 * q^55 + 48 * q^71 - 16 * q^79 - 20 * q^81 - 24 * q^89 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	1.41421i		−	0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$		−	1.41421i		−	0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$		0			0
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$6$		0			0
$7$			2.44949i			0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$			2.44949i			0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	3.46410			1.04447			0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410			1.04447			0.522233	+	0.852803i	$0.325099\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−2.00000	+	2.44949i	−0.516398	+	0.632456i
$16$		0			0
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$	3.46410			0.794719			0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410			0.794719			0.397360	+	0.917663i	$0.369927\pi$
$20$		0			0
$21$	3.46410			0.755929
$22$		0			0
$23$		−	2.44949i		−	0.510754i	−0.966842	−	0.255377i	$-0.917800\pi$
$23$		−	2.44949i		−	0.510754i	0.966842	−	0.255377i	$-0.0821996\pi$
$24$		0			0
$25$	1.00000	+	4.89898i	0.200000	+	0.979796i
$26$		0			0
$27$		−	5.65685i		−	1.08866i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		−	4.89898i		−	0.852803i
$34$		0			0
$35$	3.46410	−	4.24264i	0.585540	−	0.717137i
$36$		0			0
$37$			8.48528i			1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$			8.48528i			1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		−	4.24264i		−	0.646997i	−0.946229	−	0.323498i	$-0.895141\pi$
$43$		−	4.24264i		−	0.646997i	0.946229	−	0.323498i	$-0.104859\pi$
$44$		0			0
$45$	−1.73205	−	1.41421i	−0.258199	−	0.210819i
$46$		0			0
$47$		−	7.34847i		−	1.07188i	−0.844255	−	0.535942i	$-0.819956\pi$
$47$		−	7.34847i		−	1.07188i	0.844255	−	0.535942i	$-0.180044\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	6.92820			0.970143
$52$		0			0
$53$		−	5.65685i		−	0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$		−	5.65685i		−	0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$		0			0
$55$	−6.00000	−	4.89898i	−0.809040	−	0.660578i
$56$		0			0
$57$		−	4.89898i		−	0.648886i
$58$		0			0
$59$	10.3923			1.35296			0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923			1.35296			0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$	3.46410			0.443533			0.221766	−	0.975100i	$-0.428818\pi$
$61$	3.46410			0.443533			0.221766	+	0.975100i	$0.428818\pi$
$62$		0			0
$63$			2.44949i			0.308607i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	4.24264i		−	0.518321i	−0.965834	−	0.259161i	$-0.916554\pi$
$67$		−	4.24264i		−	0.518321i	0.965834	−	0.259161i	$-0.0834459\pi$
$68$		0			0
$69$	−3.46410			−0.417029
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$		−	4.89898i		−	0.573382i	−0.958023	−	0.286691i	$-0.907445\pi$
$73$		−	4.89898i		−	0.573382i	0.958023	−	0.286691i	$-0.0925553\pi$
$74$		0			0
$75$	6.92820	−	1.41421i	0.800000	−	0.163299i
$76$		0			0
$77$			8.48528i			0.966988i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$		−	9.89949i		−	1.08661i	−0.839535	−	0.543305i	$-0.817173\pi$
$83$		−	9.89949i		−	1.08661i	0.839535	−	0.543305i	$-0.182827\pi$
$84$		0			0
$85$	6.92820	−	8.48528i	0.751469	−	0.920358i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	5.65685i		−	0.586588i
$94$		0			0
$95$	−6.00000	−	4.89898i	−0.615587	−	0.502625i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$	3.46410			0.348155
$100$		0			0
$101$	−13.8564			−1.37876			−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564			−1.37876			−0.689382	+	0.724398i	$0.742118\pi$
$102$		0			0
$103$			7.34847i			0.724066i	0.932165	+	0.362033i	$0.117917\pi$
$103$			7.34847i			0.724066i	−0.932165	+	0.362033i	$0.882083\pi$
$104$		0			0
$105$	−6.00000	−	4.89898i	−0.585540	−	0.478091i
$106$		0			0
$107$		−	1.41421i		−	0.136717i	−0.997661	−	0.0683586i	$-0.978224\pi$
$107$		−	1.41421i		−	0.136717i	0.997661	−	0.0683586i	$-0.0217762\pi$
$108$		0			0
$109$	−3.46410			−0.331801			−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410			−0.331801			−0.165900	+	0.986143i	$0.553053\pi$
$110$		0			0
$111$	12.0000			1.13899
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−3.46410	+	4.24264i	−0.323029	+	0.395628i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.0000			−1.10004
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.19615	−	9.89949i	0.464758	−	0.885438i
$126$		0			0
$127$			17.1464i			1.52150i	0.649045	+	0.760750i	$0.275169\pi$
$127$			17.1464i			1.52150i	−0.649045	+	0.760750i	$0.724831\pi$
$128$		0			0
$129$	−6.00000			−0.528271
$130$		0			0
$131$	3.46410			0.302660			0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410			0.302660			0.151330	+	0.988483i	$0.451644\pi$
$132$		0			0
$133$			8.48528i			0.735767i
$134$		0			0
$135$	−8.00000	+	9.79796i	−0.688530	+	0.843274i
$136$		0			0
$137$			9.79796i			0.837096i	0.908195	+	0.418548i	$0.137461\pi$
$137$			9.79796i			0.837096i	−0.908195	+	0.418548i	$0.862539\pi$
$138$		0			0
$139$	−10.3923			−0.881464			−0.440732	−	0.897639i	$-0.645281\pi$
$139$	−10.3923			−0.881464			−0.440732	+	0.897639i	$0.645281\pi$
$140$		0			0
$141$	−10.3923			−0.875190
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	1.41421i		−	0.116642i
$148$		0			0
$149$	17.3205			1.41895			0.709476	−	0.704730i	$-0.248932\pi$
$149$	17.3205			1.41895			0.709476	+	0.704730i	$0.248932\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$			4.89898i			0.396059i
$154$		0			0
$155$	−6.92820	−	5.65685i	−0.556487	−	0.454369i
$156$		0			0
$157$			8.48528i			0.677199i	0.940931	+	0.338600i	$0.109953\pi$
$157$			8.48528i			0.677199i	−0.940931	+	0.338600i	$0.890047\pi$
$158$		0			0
$159$	−8.00000			−0.634441
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$			21.2132i			1.66155i	0.556611	+	0.830773i	$0.312101\pi$
$163$			21.2132i			1.66155i	−0.556611	+	0.830773i	$0.687899\pi$
$164$		0			0
$165$	−6.92820	+	8.48528i	−0.539360	+	0.660578i
$166$		0			0
$167$		−	12.2474i		−	0.947736i	−0.880596	−	0.473868i	$-0.842857\pi$
$167$		−	12.2474i		−	0.947736i	0.880596	−	0.473868i	$-0.157143\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	3.46410			0.264906
$172$		0			0
$173$			2.82843i			0.215041i	0.994203	+	0.107521i	$0.0342912\pi$
$173$			2.82843i			0.215041i	−0.994203	+	0.107521i	$0.965709\pi$
$174$		0			0
$175$	−12.0000	+	2.44949i	−0.907115	+	0.185164i
$176$		0			0
$177$		−	14.6969i		−	1.10469i
$178$		0			0
$179$	−3.46410			−0.258919			−0.129460	−	0.991585i	$-0.541324\pi$
$179$	−3.46410			−0.258919			−0.129460	+	0.991585i	$0.541324\pi$
$180$		0			0
$181$	13.8564			1.02994			0.514969	−	0.857209i	$-0.327803\pi$
$181$	13.8564			1.02994			0.514969	+	0.857209i	$0.327803\pi$
$182$		0			0
$183$		−	4.89898i		−	0.362143i
$184$		0			0
$185$	12.0000	−	14.6969i	0.882258	−	1.08054i
$186$		0			0
$187$			16.9706i			1.24101i
$188$		0			0
$189$	13.8564			1.00791
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$			24.4949i			1.76318i	0.472015	+	0.881591i	$0.343527\pi$
$193$			24.4949i			1.76318i	−0.472015	+	0.881591i	$0.656473\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.65685i			0.403034i	0.979485	+	0.201517i	$0.0645872\pi$
$197$			5.65685i			0.403034i	−0.979485	+	0.201517i	$0.935413\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$	−6.00000			−0.423207
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	2.44949i		−	0.170251i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−24.2487			−1.66935			−0.834675	−	0.550743i	$-0.814345\pi$
$211$	−24.2487			−1.66935			−0.834675	+	0.550743i	$0.814345\pi$
$212$		0			0
$213$		−	16.9706i		−	1.16280i
$214$		0			0
$215$	−6.00000	+	7.34847i	−0.409197	+	0.501161i
$216$		0			0
$217$			9.79796i			0.665129i
$218$		0			0
$219$	−6.92820			−0.468165
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	22.0454i		−	1.47627i	−0.674653	−	0.738135i	$-0.735707\pi$
$223$		−	22.0454i		−	1.47627i	0.674653	−	0.738135i	$-0.264293\pi$
$224$		0			0
$225$	1.00000	+	4.89898i	0.0666667	+	0.326599i
$226$		0			0
$227$		−	1.41421i		−	0.0938647i	−0.998898	−	0.0469323i	$-0.985055\pi$
$227$		−	1.41421i		−	0.0938647i	0.998898	−	0.0469323i	$-0.0149445\pi$
$228$		0			0
$229$	−27.7128			−1.83131			−0.915657	−	0.401960i	$-0.868329\pi$
$229$	−27.7128			−1.83131			−0.915657	+	0.401960i	$0.868329\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$		0			0
$233$			14.6969i			0.962828i	0.876493	+	0.481414i	$0.159877\pi$
$233$			14.6969i			0.962828i	−0.876493	+	0.481414i	$0.840123\pi$
$234$		0			0
$235$	−10.3923	+	12.7279i	−0.677919	+	0.830278i
$236$		0			0
$237$			5.65685i			0.367452i
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	4.00000			0.257663			0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000			0.257663			0.128831	+	0.991667i	$0.458877\pi$
$242$		0			0
$243$		−	9.89949i		−	0.635053i
$244$		0			0
$245$	−1.73205	−	1.41421i	−0.110657	−	0.0903508i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−14.0000			−0.887214
$250$		0			0
$251$	−10.3923			−0.655956			−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923			−0.655956			−0.327978	+	0.944685i	$0.606367\pi$
$252$		0			0
$253$		−	8.48528i		−	0.533465i
$254$		0			0
$255$	−12.0000	−	9.79796i	−0.751469	−	0.613572i
$256$		0			0
$257$			9.79796i			0.611180i	0.952163	+	0.305590i	$0.0988537\pi$
$257$			9.79796i			0.611180i	−0.952163	+	0.305590i	$0.901146\pi$
$258$		0			0
$259$	−20.7846			−1.29149
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	7.34847i		−	0.453126i	−0.973997	−	0.226563i	$-0.927251\pi$
$263$		−	7.34847i		−	0.453126i	0.973997	−	0.226563i	$-0.0727489\pi$
$264$		0			0
$265$	−8.00000	+	9.79796i	−0.491436	+	0.601884i
$266$		0			0
$267$			8.48528i			0.519291i
$268$		0			0
$269$	−10.3923			−0.633630			−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923			−0.633630			−0.316815	+	0.948487i	$0.602613\pi$
$270$		0			0
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.46410	+	16.9706i	0.208893	+	1.02336i
$276$		0			0
$277$		−	25.4558i		−	1.52949i	−0.644331	−	0.764747i	$-0.722864\pi$
$277$		−	25.4558i		−	1.52949i	0.644331	−	0.764747i	$-0.277136\pi$
$278$		0			0
$279$	4.00000			0.239474
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$			21.2132i			1.26099i	0.776192	+	0.630497i	$0.217149\pi$
$283$			21.2132i			1.26099i	−0.776192	+	0.630497i	$0.782851\pi$
$284$		0			0
$285$	−6.92820	+	8.48528i	−0.410391	+	0.502625i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	−6.92820			−0.406138
$292$		0			0
$293$			19.7990i			1.15667i	0.815800	+	0.578335i	$0.196297\pi$
$293$			19.7990i			1.15667i	−0.815800	+	0.578335i	$0.803703\pi$
$294$		0			0
$295$	−18.0000	−	14.6969i	−1.04800	−	0.855689i
$296$		0			0
$297$		−	19.5959i		−	1.13707i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	10.3923			0.599002
$302$		0			0
$303$			19.5959i			1.12576i
$304$		0			0
$305$	−6.00000	−	4.89898i	−0.343559	−	0.280515i
$306$		0			0
$307$			29.6985i			1.69498i	0.530810	+	0.847491i	$0.321888\pi$
$307$			29.6985i			1.69498i	−0.530810	+	0.847491i	$0.678112\pi$
$308$		0			0
$309$	10.3923			0.591198
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$		−	9.79796i		−	0.553813i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$		−	9.79796i		−	0.553813i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$		0			0
$315$	3.46410	−	4.24264i	0.195180	−	0.239046i
$316$		0			0
$317$		−	28.2843i		−	1.58860i	−0.607524	−	0.794301i	$-0.707837\pi$
$317$		−	28.2843i		−	1.58860i	0.607524	−	0.794301i	$-0.292163\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−2.00000			−0.111629
$322$		0			0
$323$			16.9706i			0.944267i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			4.89898i			0.270914i
$328$		0			0
$329$	18.0000			0.992372
$330$		0			0
$331$	−31.1769			−1.71364			−0.856819	−	0.515617i	$-0.827563\pi$
$331$	−31.1769			−1.71364			−0.856819	+	0.515617i	$0.827563\pi$
$332$		0			0
$333$			8.48528i			0.464991i
$334$		0			0
$335$	−6.00000	+	7.34847i	−0.327815	+	0.401490i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.8564			0.750366
$342$		0			0
$343$			19.5959i			1.05808i
$344$		0			0
$345$	6.00000	+	4.89898i	0.323029	+	0.263752i
$346$		0			0
$347$		−	15.5563i		−	0.835109i	−0.908652	−	0.417554i	$-0.862887\pi$
$347$		−	15.5563i		−	0.835109i	0.908652	−	0.417554i	$-0.137113\pi$
$348$		0			0
$349$	−13.8564			−0.741716			−0.370858	−	0.928689i	$-0.620936\pi$
$349$	−13.8564			−0.741716			−0.370858	+	0.928689i	$0.620936\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	29.3939i		−	1.56448i	−0.622978	−	0.782239i	$-0.714078\pi$
$353$		−	29.3939i		−	1.56448i	0.622978	−	0.782239i	$-0.285922\pi$
$354$		0			0
$355$	−20.7846	−	16.9706i	−1.10313	−	0.900704i
$356$		0			0
$357$			16.9706i			0.898177i
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−7.00000			−0.368421
$362$		0			0
$363$		−	1.41421i		−	0.0742270i
$364$		0			0
$365$	−6.92820	+	8.48528i	−0.362639	+	0.444140i
$366$		0			0
$367$		−	12.2474i		−	0.639312i	−0.947534	−	0.319656i	$-0.896433\pi$
$367$		−	12.2474i		−	0.639312i	0.947534	−	0.319656i	$-0.103567\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.8564			0.719389
$372$		0			0
$373$			8.48528i			0.439351i	0.975573	+	0.219676i	$0.0704999\pi$
$373$			8.48528i			0.439351i	−0.975573	+	0.219676i	$0.929500\pi$
$374$		0			0
$375$	−14.0000	−	7.34847i	−0.722957	−	0.379473i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−24.2487			−1.24557			−0.622786	−	0.782392i	$-0.713999\pi$
$379$	−24.2487			−1.24557			−0.622786	+	0.782392i	$0.713999\pi$
$380$		0			0
$381$	24.2487			1.24230
$382$		0			0
$383$			26.9444i			1.37679i	0.725334	+	0.688397i	$0.241685\pi$
$383$			26.9444i			1.37679i	−0.725334	+	0.688397i	$0.758315\pi$
$384$		0			0
$385$	12.0000	−	14.6969i	0.611577	−	0.749025i
$386$		0			0
$387$		−	4.24264i		−	0.215666i
$388$		0			0
$389$	3.46410			0.175637			0.0878185	−	0.996136i	$-0.472010\pi$
$389$	3.46410			0.175637			0.0878185	+	0.996136i	$0.472010\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$		−	4.89898i		−	0.247121i
$394$		0			0
$395$	6.92820	+	5.65685i	0.348596	+	0.284627i
$396$		0			0
$397$		−	16.9706i		−	0.851728i	−0.904787	−	0.425864i	$-0.859970\pi$
$397$		−	16.9706i		−	0.851728i	0.904787	−	0.425864i	$-0.140030\pi$
$398$		0			0
$399$	12.0000			0.600751
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	8.66025	+	7.07107i	0.430331	+	0.351364i
$406$		0			0
$407$			29.3939i			1.45700i
$408$		0			0
$409$	−32.0000			−1.58230			−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000			−1.58230			−0.791149	+	0.611623i	$0.790517\pi$
$410$		0			0
$411$	13.8564			0.683486
$412$		0			0
$413$			25.4558i			1.25260i
$414$		0			0
$415$	−14.0000	+	17.1464i	−0.687233	+	0.841685i
$416$		0			0
$417$			14.6969i			0.719712i
$418$		0			0
$419$	10.3923			0.507697			0.253849	−	0.967244i	$-0.418303\pi$
$419$	10.3923			0.507697			0.253849	+	0.967244i	$0.418303\pi$
$420$		0			0
$421$	−24.2487			−1.18181			−0.590905	−	0.806741i	$-0.701229\pi$
$421$	−24.2487			−1.18181			−0.590905	+	0.806741i	$0.701229\pi$
$422$		0			0
$423$		−	7.34847i		−	0.357295i
$424$		0			0
$425$	−24.0000	+	4.89898i	−1.16417	+	0.237635i
$426$		0			0
$427$			8.48528i			0.410632i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		−	4.89898i		−	0.235430i	−0.993047	−	0.117715i	$-0.962443\pi$
$433$		−	4.89898i		−	0.235430i	0.993047	−	0.117715i	$-0.0375569\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	8.48528i		−	0.405906i
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	1.41421i		−	0.0671913i	−0.999436	−	0.0335957i	$-0.989304\pi$
$443$		−	1.41421i		−	0.0671913i	0.999436	−	0.0335957i	$-0.0106958\pi$
$444$		0			0
$445$	10.3923	+	8.48528i	0.492642	+	0.402241i
$446$		0			0
$447$		−	24.4949i		−	1.15857i
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	22.6274i		−	1.06313i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	19.5959i		−	0.916658i	−0.888783	−	0.458329i	$-0.848448\pi$
$457$		−	19.5959i		−	0.916658i	0.888783	−	0.458329i	$-0.151552\pi$
$458$		0			0
$459$	27.7128			1.29352
$460$		0			0
$461$	−13.8564			−0.645357			−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564			−0.645357			−0.322679	+	0.946509i	$0.604583\pi$
$462$		0			0
$463$		−	17.1464i		−	0.796862i	−0.917198	−	0.398431i	$-0.869555\pi$
$463$		−	17.1464i		−	0.796862i	0.917198	−	0.398431i	$-0.130445\pi$
$464$		0			0
$465$	−8.00000	+	9.79796i	−0.370991	+	0.454369i
$466$		0			0
$467$		−	7.07107i		−	0.327210i	−0.986526	−	0.163605i	$-0.947688\pi$
$467$		−	7.07107i		−	0.327210i	0.986526	−	0.163605i	$-0.0523123\pi$
$468$		0			0
$469$	10.3923			0.479872
$470$		0			0
$471$	12.0000			0.552931
$472$		0			0
$473$		−	14.6969i		−	0.675766i
$474$		0			0
$475$	3.46410	+	16.9706i	0.158944	+	0.778663i
$476$		0			0
$477$		−	5.65685i		−	0.259010i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		−	8.48528i		−	0.386094i
$484$		0			0
$485$	−6.92820	+	8.48528i	−0.314594	+	0.385297i
$486$		0			0
$487$		−	7.34847i		−	0.332991i	−0.986042	−	0.166495i	$-0.946755\pi$
$487$		−	7.34847i		−	0.332991i	0.986042	−	0.166495i	$-0.0532451\pi$
$488$		0			0
$489$	30.0000			1.35665
$490$		0			0
$491$	24.2487			1.09433			0.547165	−	0.837025i	$-0.315707\pi$
$491$	24.2487			1.09433			0.547165	+	0.837025i	$0.315707\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−6.00000	−	4.89898i	−0.269680	−	0.220193i
$496$		0			0
$497$			29.3939i			1.31850i
$498$		0			0
$499$	−17.3205			−0.775372			−0.387686	−	0.921791i	$-0.626726\pi$
$499$	−17.3205			−0.775372			−0.387686	+	0.921791i	$0.626726\pi$
$500$		0			0
$501$	−17.3205			−0.773823
$502$		0			0
$503$		−	12.2474i		−	0.546087i	−0.962002	−	0.273043i	$-0.911970\pi$
$503$		−	12.2474i		−	0.546087i	0.962002	−	0.273043i	$-0.0880303\pi$
$504$		0			0
$505$	24.0000	+	19.5959i	1.06799	+	0.872007i
$506$		0			0
$507$		−	18.3848i		−	0.816497i
$508$		0			0
$509$	27.7128			1.22835			0.614174	−	0.789170i	$-0.289489\pi$
$509$	27.7128			1.22835			0.614174	+	0.789170i	$0.289489\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		0			0
$513$		−	19.5959i		−	0.865181i
$514$		0			0
$515$	10.3923	−	12.7279i	0.457940	−	0.560859i
$516$		0			0
$517$		−	25.4558i		−	1.11955i
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$		−	12.7279i		−	0.556553i	−0.960501	−	0.278277i	$-0.910237\pi$
$523$		−	12.7279i		−	0.556553i	0.960501	−	0.278277i	$-0.0897632\pi$
$524$		0			0
$525$	3.46410	+	16.9706i	0.151186	+	0.740656i
$526$		0			0
$527$			19.5959i			0.853612i
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$	10.3923			0.450988
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−2.00000	+	2.44949i	−0.0864675	+	0.105901i
$536$		0			0
$537$			4.89898i			0.211407i
$538$		0			0
$539$	3.46410			0.149209
$540$		0			0
$541$	−41.5692			−1.78720			−0.893600	−	0.448864i	$-0.851829\pi$
$541$	−41.5692			−1.78720			−0.893600	+	0.448864i	$0.851829\pi$
$542$		0			0
$543$		−	19.5959i		−	0.840941i
$544$		0			0
$545$	6.00000	+	4.89898i	0.257012	+	0.209849i
$546$		0			0
$547$			4.24264i			0.181402i	0.995878	+	0.0907011i	$0.0289108\pi$
$547$			4.24264i			0.181402i	−0.995878	+	0.0907011i	$0.971089\pi$
$548$		0			0
$549$	3.46410			0.147844
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	9.79796i		−	0.416652i
$554$		0			0
$555$	−20.7846	−	16.9706i	−0.882258	−	0.720360i
$556$		0			0
$557$		−	14.1421i		−	0.599222i	−0.954062	−	0.299611i	$-0.903143\pi$
$557$		−	14.1421i		−	0.599222i	0.954062	−	0.299611i	$-0.0968568\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$		−	41.0122i		−	1.72846i	−0.503099	−	0.864229i	$-0.667807\pi$
$563$		−	41.0122i		−	1.72846i	0.503099	−	0.864229i	$-0.332193\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	12.2474i		−	0.514344i
$568$		0			0
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$		0			0
$571$	3.46410			0.144968			0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410			0.144968			0.0724841	+	0.997370i	$0.476907\pi$
$572$		0			0
$573$		−	33.9411i		−	1.41791i
$574$		0			0
$575$	12.0000	−	2.44949i	0.500435	−	0.102151i
$576$		0			0
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$		0			0
$579$	34.6410			1.43963
$580$		0			0
$581$	24.2487			1.00601
$582$		0			0
$583$		−	19.5959i		−	0.811580i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	9.89949i		−	0.408596i	−0.978909	−	0.204298i	$-0.934509\pi$
$587$		−	9.89949i		−	0.408596i	0.978909	−	0.204298i	$-0.0654911\pi$
$588$		0			0
$589$	13.8564			0.570943
$590$		0			0
$591$	8.00000			0.329076
$592$		0			0
$593$			9.79796i			0.402354i	0.979555	+	0.201177i	$0.0644766\pi$
$593$			9.79796i			0.402354i	−0.979555	+	0.201177i	$0.935523\pi$
$594$		0			0
$595$	20.7846	+	16.9706i	0.852086	+	0.695725i
$596$		0			0
$597$			5.65685i			0.231520i
$598$		0			0
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	28.0000			1.14214			0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000			1.14214			0.571072	+	0.820900i	$0.306528\pi$
$602$		0			0
$603$		−	4.24264i		−	0.172774i
$604$		0			0
$605$	−1.73205	−	1.41421i	−0.0704179	−	0.0574960i
$606$		0			0
$607$		−	7.34847i		−	0.298265i	−0.988817	−	0.149133i	$-0.952352\pi$
$607$		−	7.34847i		−	0.298265i	0.988817	−	0.149133i	$-0.0476481\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			33.9411i			1.37087i	0.728134	+	0.685435i	$0.240388\pi$
$613$			33.9411i			1.37087i	−0.728134	+	0.685435i	$0.759612\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			34.2929i			1.38058i	0.723534	+	0.690289i	$0.242517\pi$
$617$			34.2929i			1.38058i	−0.723534	+	0.690289i	$0.757483\pi$
$618$		0			0
$619$	10.3923			0.417702			0.208851	−	0.977947i	$-0.433028\pi$
$619$	10.3923			0.417702			0.208851	+	0.977947i	$0.433028\pi$
$620$		0			0
$621$	−13.8564			−0.556038
$622$		0			0
$623$		−	14.6969i		−	0.588820i
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$		−	16.9706i		−	0.677739i
$628$		0			0
$629$	−41.5692			−1.65747
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$			34.2929i			1.36302i
$634$		0			0
$635$	24.2487	−	29.6985i	0.962281	−	1.17855i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	12.0000			0.474713
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$			29.6985i			1.17119i	0.810602	+	0.585597i	$0.199140\pi$
$643$			29.6985i			1.17119i	−0.810602	+	0.585597i	$0.800860\pi$
$644$		0			0
$645$	10.3923	+	8.48528i	0.409197	+	0.334108i
$646$		0			0
$647$			12.2474i			0.481497i	0.970588	+	0.240748i	$0.0773929\pi$
$647$			12.2474i			0.481497i	−0.970588	+	0.240748i	$0.922607\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$		0			0
$651$	13.8564			0.543075
$652$		0			0
$653$			11.3137i			0.442740i	0.975190	+	0.221370i	$0.0710528\pi$
$653$			11.3137i			0.442740i	−0.975190	+	0.221370i	$0.928947\pi$
$654$		0			0
$655$	−6.00000	−	4.89898i	−0.234439	−	0.191419i
$656$		0			0
$657$		−	4.89898i		−	0.191127i
$658$		0			0
$659$	24.2487			0.944596			0.472298	−	0.881439i	$-0.343425\pi$
$659$	24.2487			0.944596			0.472298	+	0.881439i	$0.343425\pi$
$660$		0			0
$661$	10.3923			0.404214			0.202107	−	0.979363i	$-0.435221\pi$
$661$	10.3923			0.404214			0.202107	+	0.979363i	$0.435221\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.0000	−	14.6969i	0.465340	−	0.569923i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−31.1769			−1.20537
$670$		0			0
$671$	12.0000			0.463255
$672$		0			0
$673$			34.2929i			1.32189i	0.750433	+	0.660946i	$0.229845\pi$
$673$			34.2929i			1.32189i	−0.750433	+	0.660946i	$0.770155\pi$
$674$		0			0
$675$	27.7128	−	5.65685i	1.06667	−	0.217732i
$676$		0			0
$677$		−	22.6274i		−	0.869642i	−0.900517	−	0.434821i	$-0.856812\pi$
$677$		−	22.6274i		−	0.869642i	0.900517	−	0.434821i	$-0.143188\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$		0			0
$681$	−2.00000			−0.0766402
$682$		0			0
$683$		−	15.5563i		−	0.595247i	−0.954683	−	0.297624i	$-0.903806\pi$
$683$		−	15.5563i		−	0.595247i	0.954683	−	0.297624i	$-0.0961940\pi$
$684$		0			0
$685$	13.8564	−	16.9706i	0.529426	−	0.648412i
$686$		0			0
$687$			39.1918i			1.49526i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	3.46410			0.131781			0.0658903	−	0.997827i	$-0.479011\pi$
$691$	3.46410			0.131781			0.0658903	+	0.997827i	$0.479011\pi$
$692$		0			0
$693$			8.48528i			0.322329i
$694$		0			0
$695$	18.0000	+	14.6969i	0.682779	+	0.557487i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	20.7846			0.786146
$700$		0			0
$701$	24.2487			0.915861			0.457931	−	0.888988i	$-0.348591\pi$
$701$	24.2487			0.915861			0.457931	+	0.888988i	$0.348591\pi$
$702$		0			0
$703$			29.3939i			1.10861i
$704$		0			0
$705$	18.0000	+	14.6969i	0.677919	+	0.553519i
$706$		0			0
$707$		−	33.9411i		−	1.27649i
$708$		0			0
$709$	41.5692			1.56116			0.780582	−	0.625053i	$-0.214923\pi$
$709$	41.5692			1.56116			0.780582	+	0.625053i	$0.214923\pi$
$710$		0			0
$711$	−4.00000			−0.150012
$712$		0			0
$713$		−	9.79796i		−	0.366936i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			16.9706i			0.633777i
$718$		0			0
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$		0			0
$721$	−18.0000			−0.670355
$722$		0			0
$723$		−	5.65685i		−	0.210381i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	46.5403i		−	1.72608i	−0.505132	−	0.863042i	$-0.668556\pi$
$727$		−	46.5403i		−	1.72608i	0.505132	−	0.863042i	$-0.331444\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	20.7846			0.768747
$732$		0			0
$733$			42.4264i			1.56706i	0.621357	+	0.783528i	$0.286582\pi$
$733$			42.4264i			1.56706i	−0.621357	+	0.783528i	$0.713418\pi$
$734$		0			0
$735$	−2.00000	+	2.44949i	−0.0737711	+	0.0903508i
$736$		0			0
$737$		−	14.6969i		−	0.541369i
$738$		0			0
$739$	−3.46410			−0.127429			−0.0637145	−	0.997968i	$-0.520295\pi$
$739$	−3.46410			−0.127429			−0.0637145	+	0.997968i	$0.520295\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			51.4393i			1.88712i	0.331195	+	0.943562i	$0.392548\pi$
$743$			51.4393i			1.88712i	−0.331195	+	0.943562i	$0.607452\pi$
$744$		0			0
$745$	−30.0000	−	24.4949i	−1.09911	−	0.897424i
$746$		0			0
$747$		−	9.89949i		−	0.362204i
$748$		0			0
$749$	3.46410			0.126576
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$			14.6969i			0.535586i
$754$		0			0
$755$	−27.7128	−	22.6274i	−1.00857	−	0.823496i
$756$		0			0
$757$			25.4558i			0.925208i	0.886565	+	0.462604i	$0.153085\pi$
$757$			25.4558i			0.925208i	−0.886565	+	0.462604i	$0.846915\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$		−	8.48528i		−	0.307188i
$764$		0			0
$765$	6.92820	−	8.48528i	0.250490	−	0.306786i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$	13.8564			0.499026
$772$		0			0
$773$		−	28.2843i		−	1.01731i	−0.860969	−	0.508657i	$-0.830142\pi$
$773$		−	28.2843i		−	1.01731i	0.860969	−	0.508657i	$-0.169858\pi$
$774$		0			0
$775$	4.00000	+	19.5959i	0.143684	+	0.703906i
$776$		0			0
$777$			29.3939i			1.05450i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	41.5692			1.48746
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.0000	−	14.6969i	0.428298	−	0.524556i
$786$		0			0
$787$		−	21.2132i		−	0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$		−	21.2132i		−	0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$		0			0
$789$	−10.3923			−0.369976
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	13.8564	+	11.3137i	0.491436	+	0.401256i
$796$		0			0
$797$			39.5980i			1.40263i	0.712850	+	0.701316i	$0.247404\pi$
$797$			39.5980i			1.40263i	−0.712850	+	0.701316i	$0.752596\pi$
$798$		0			0
$799$	36.0000			1.27359
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$		−	16.9706i		−	0.598878i
$804$		0			0
$805$	−10.3923	−	8.48528i	−0.366281	−	0.299067i
$806$		0			0
$807$			14.6969i			0.517357i
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$	−10.3923			−0.364923			−0.182462	−	0.983213i	$-0.558407\pi$
$811$	−10.3923			−0.364923			−0.182462	+	0.983213i	$0.558407\pi$
$812$		0			0
$813$		−	28.2843i		−	0.991973i
$814$		0			0
$815$	30.0000	−	36.7423i	1.05085	−	1.28703i
$816$		0			0
$817$		−	14.6969i		−	0.514181i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−31.1769			−1.08808			−0.544041	−	0.839059i	$-0.683106\pi$
$821$	−31.1769			−1.08808			−0.544041	+	0.839059i	$0.683106\pi$
$822$		0			0
$823$			26.9444i			0.939222i	0.882873	+	0.469611i	$0.155606\pi$
$823$			26.9444i			0.939222i	−0.882873	+	0.469611i	$0.844394\pi$
$824$		0			0
$825$	24.0000	−	4.89898i	0.835573	−	0.170561i
$826$		0			0
$827$			35.3553i			1.22943i	0.788751	+	0.614713i	$0.210728\pi$
$827$			35.3553i			1.22943i	−0.788751	+	0.614713i	$0.789272\pi$
$828$		0			0
$829$	−10.3923			−0.360940			−0.180470	−	0.983581i	$-0.557762\pi$
$829$	−10.3923			−0.360940			−0.180470	+	0.983581i	$0.557762\pi$
$830$		0			0
$831$	−36.0000			−1.24883
$832$		0			0
$833$			4.89898i			0.169740i
$834$		0			0
$835$	−17.3205	+	21.2132i	−0.599401	+	0.734113i
$836$		0			0
$837$		−	22.6274i		−	0.782118i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		−	16.9706i		−	0.584497i
$844$		0			0
$845$	−22.5167	−	18.3848i	−0.774597	−	0.632456i
$846$		0			0
$847$			2.44949i			0.0841655i
$848$		0			0
$849$	30.0000			1.02960
$850$		0			0
$851$	20.7846			0.712487
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$	−6.00000	−	4.89898i	−0.205196	−	0.167542i
$856$		0			0
$857$		−	39.1918i		−	1.33877i	−0.742917	−	0.669384i	$-0.766558\pi$
$857$		−	39.1918i		−	1.33877i	0.742917	−	0.669384i	$-0.233442\pi$
$858$		0			0
$859$	−3.46410			−0.118194			−0.0590968	−	0.998252i	$-0.518822\pi$
$859$	−3.46410			−0.118194			−0.0590968	+	0.998252i	$0.518822\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			2.44949i			0.0833816i	0.999131	+	0.0416908i	$0.0132744\pi$
$863$			2.44949i			0.0833816i	−0.999131	+	0.0416908i	$0.986726\pi$
$864$		0			0
$865$	4.00000	−	4.89898i	0.136004	−	0.166570i
$866$		0			0
$867$			9.89949i			0.336204i
$868$		0			0
$869$	−13.8564			−0.470046
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	4.89898i		−	0.165805i
$874$		0			0
$875$	24.2487	+	12.7279i	0.819756	+	0.430282i
$876$		0			0
$877$		−	8.48528i		−	0.286528i	−0.989685	−	0.143264i	$-0.954240\pi$
$877$		−	8.48528i		−	0.286528i	0.989685	−	0.143264i	$-0.0457597\pi$
$878$		0			0
$879$	28.0000			0.944417
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$			4.24264i			0.142776i	0.997449	+	0.0713881i	$0.0227429\pi$
$883$			4.24264i			0.142776i	−0.997449	+	0.0713881i	$0.977257\pi$
$884$		0			0
$885$	−20.7846	+	25.4558i	−0.698667	+	0.855689i
$886$		0			0
$887$		−	26.9444i		−	0.904704i	−0.891839	−	0.452352i	$-0.850585\pi$
$887$		−	26.9444i		−	0.904704i	0.891839	−	0.452352i	$-0.149415\pi$
$888$		0			0
$889$	−42.0000			−1.40863
$890$		0			0
$891$	−17.3205			−0.580259
$892$		0			0
$893$		−	25.4558i		−	0.851847i
$894$		0			0
$895$	6.00000	+	4.89898i	0.200558	+	0.163755i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	27.7128			0.923248
$902$		0			0
$903$		−	14.6969i		−	0.489083i
$904$		0			0
$905$	−24.0000	−	19.5959i	−0.797787	−	0.651390i
$906$		0			0
$907$			4.24264i			0.140875i	0.997516	+	0.0704373i	$0.0224395\pi$
$907$			4.24264i			0.140875i	−0.997516	+	0.0704373i	$0.977561\pi$
$908$		0			0
$909$	−13.8564			−0.459588
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		−	34.2929i		−	1.13493i
$914$		0			0
$915$	−6.92820	+	8.48528i	−0.229039	+	0.280515i
$916$		0			0
$917$			8.48528i			0.280209i
$918$		0			0
$919$	4.00000			0.131948			0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$		0			0
$921$	42.0000			1.38395
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−41.5692	+	8.48528i	−1.36679	+	0.278994i
$926$		0			0
$927$			7.34847i			0.241355i
$928$		0			0
$929$	−36.0000			−1.18112			−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000			−1.18112			−0.590561	+	0.806993i	$0.701093\pi$
$930$		0			0
$931$	3.46410			0.113531
$932$		0			0
$933$		−	33.9411i		−	1.11118i
$934$		0			0
$935$	24.0000	−	29.3939i	0.784884	−	0.961283i
$936$		0			0
$937$			4.89898i			0.160043i	0.996793	+	0.0800213i	$0.0254988\pi$
$937$			4.89898i			0.160043i	−0.996793	+	0.0800213i	$0.974501\pi$
$938$		0			0
$939$	−13.8564			−0.452187
$940$		0			0
$941$	13.8564			0.451706			0.225853	−	0.974161i	$-0.427483\pi$
$941$	13.8564			0.451706			0.225853	+	0.974161i	$0.427483\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−24.0000	−	19.5959i	−0.780720	−	0.637455i
$946$		0			0
$947$			41.0122i			1.33272i	0.745631	+	0.666359i	$0.232148\pi$
$947$			41.0122i			1.33272i	−0.745631	+	0.666359i	$0.767852\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−40.0000			−1.29709
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−41.5692	−	33.9411i	−1.34515	−	1.09831i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.0000			−0.775000
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		−	1.41421i		−	0.0455724i
$964$		0			0
$965$	34.6410	−	42.4264i	1.11513	−	1.36575i
$966$		0			0
$967$			17.1464i			0.551392i	0.961245	+	0.275696i	$0.0889083\pi$
$967$			17.1464i			0.551392i	−0.961245	+	0.275696i	$0.911092\pi$
$968$		0			0
$969$	24.0000			0.770991
$970$		0			0
$971$	−3.46410			−0.111168			−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410			−0.111168			−0.0555842	+	0.998454i	$0.517702\pi$
$972$		0			0
$973$		−	25.4558i		−	0.816077i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	44.0908i		−	1.41059i	−0.708914	−	0.705295i	$-0.750815\pi$
$977$		−	44.0908i		−	1.41059i	0.708914	−	0.705295i	$-0.249185\pi$
$978$		0			0
$979$	−20.7846			−0.664279
$980$		0			0
$981$	−3.46410			−0.110600
$982$		0			0
$983$		−	56.3383i		−	1.79691i	−0.439064	−	0.898456i	$-0.644690\pi$
$983$		−	56.3383i		−	1.79691i	0.439064	−	0.898456i	$-0.355310\pi$
$984$		0			0
$985$	8.00000	−	9.79796i	0.254901	−	0.312189i
$986$		0			0
$987$		−	25.4558i		−	0.810268i
$988$		0			0
$989$	−10.3923			−0.330456
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$			44.0908i			1.39918i
$994$		0			0
$995$	6.92820	+	5.65685i	0.219639	+	0.179334i
$996$		0			0
$997$			50.9117i			1.61239i	0.591650	+	0.806195i	$0.298477\pi$
$997$			50.9117i			1.61239i	−0.591650	+	0.806195i	$0.701523\pi$
$998$		0			0
$999$	48.0000			1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.2.c.i.769.1		4
4.3	odd	2		1280.2.c.k.769.3		4
5.2	odd	4		6400.2.a.co.1.1		4
5.3	odd	4		6400.2.a.co.1.4		4
5.4	even	2	inner	1280.2.c.i.769.3		4
8.3	odd	2		1280.2.c.k.769.2		4
8.5	even	2	inner	1280.2.c.i.769.4		4
16.3	odd	4		160.2.f.a.49.2		4
16.5	even	4		40.2.f.a.29.4	yes	4
16.11	odd	4		160.2.f.a.49.3		4
16.13	even	4		40.2.f.a.29.2	yes	4
20.3	even	4		6400.2.a.cm.1.1		4
20.7	even	4		6400.2.a.cm.1.4		4
20.19	odd	2		1280.2.c.k.769.1		4
40.3	even	4		6400.2.a.cm.1.3		4
40.13	odd	4		6400.2.a.co.1.2		4
40.19	odd	2		1280.2.c.k.769.4		4
40.27	even	4		6400.2.a.cm.1.2		4
40.29	even	2	inner	1280.2.c.i.769.2		4
40.37	odd	4		6400.2.a.co.1.3		4
48.5	odd	4		360.2.d.b.109.1		4
48.11	even	4		1440.2.d.c.1009.2		4
48.29	odd	4		360.2.d.b.109.3		4
48.35	even	4		1440.2.d.c.1009.3		4
80.3	even	4		800.2.d.f.401.2		4
80.13	odd	4		200.2.d.e.101.4		4
80.19	odd	4		160.2.f.a.49.4		4
80.27	even	4		800.2.d.f.401.1		4
80.29	even	4		40.2.f.a.29.3	yes	4
80.37	odd	4		200.2.d.e.101.2		4
80.43	even	4		800.2.d.f.401.4		4
80.53	odd	4		200.2.d.e.101.3		4
80.59	odd	4		160.2.f.a.49.1		4
80.67	even	4		800.2.d.f.401.3		4
80.69	even	4		40.2.f.a.29.1	✓	4
80.77	odd	4		200.2.d.e.101.1		4
240.29	odd	4		360.2.d.b.109.2		4
240.53	even	4		1800.2.k.m.901.2		4
240.59	even	4		1440.2.d.c.1009.4		4
240.77	even	4		1800.2.k.m.901.4		4
240.83	odd	4		7200.2.k.l.3601.3		4
240.107	odd	4		7200.2.k.l.3601.2		4
240.149	odd	4		360.2.d.b.109.4		4
240.173	even	4		1800.2.k.m.901.1		4
240.179	even	4		1440.2.d.c.1009.1		4
240.197	even	4		1800.2.k.m.901.3		4
240.203	odd	4		7200.2.k.l.3601.4		4
240.227	odd	4		7200.2.k.l.3601.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.f.a.29.1	✓	4	80.69	even	4
40.2.f.a.29.2	yes	4	16.13	even	4
40.2.f.a.29.3	yes	4	80.29	even	4
40.2.f.a.29.4	yes	4	16.5	even	4
160.2.f.a.49.1		4	80.59	odd	4
160.2.f.a.49.2		4	16.3	odd	4
160.2.f.a.49.3		4	16.11	odd	4
160.2.f.a.49.4		4	80.19	odd	4
200.2.d.e.101.1		4	80.77	odd	4
200.2.d.e.101.2		4	80.37	odd	4
200.2.d.e.101.3		4	80.53	odd	4
200.2.d.e.101.4		4	80.13	odd	4
360.2.d.b.109.1		4	48.5	odd	4
360.2.d.b.109.2		4	240.29	odd	4
360.2.d.b.109.3		4	48.29	odd	4
360.2.d.b.109.4		4	240.149	odd	4
800.2.d.f.401.1		4	80.27	even	4
800.2.d.f.401.2		4	80.3	even	4
800.2.d.f.401.3		4	80.67	even	4
800.2.d.f.401.4		4	80.43	even	4
1280.2.c.i.769.1		4	1.1	even	1	trivial
1280.2.c.i.769.2		4	40.29	even	2	inner
1280.2.c.i.769.3		4	5.4	even	2	inner
1280.2.c.i.769.4		4	8.5	even	2	inner
1280.2.c.k.769.1		4	20.19	odd	2
1280.2.c.k.769.2		4	8.3	odd	2
1280.2.c.k.769.3		4	4.3	odd	2
1280.2.c.k.769.4		4	40.19	odd	2
1440.2.d.c.1009.1		4	240.179	even	4
1440.2.d.c.1009.2		4	48.11	even	4
1440.2.d.c.1009.3		4	48.35	even	4
1440.2.d.c.1009.4		4	240.59	even	4
1800.2.k.m.901.1		4	240.173	even	4
1800.2.k.m.901.2		4	240.53	even	4
1800.2.k.m.901.3		4	240.197	even	4
1800.2.k.m.901.4		4	240.77	even	4
6400.2.a.cm.1.1		4	20.3	even	4
6400.2.a.cm.1.2		4	40.27	even	4
6400.2.a.cm.1.3		4	40.3	even	4
6400.2.a.cm.1.4		4	20.7	even	4
6400.2.a.co.1.1		4	5.2	odd	4
6400.2.a.co.1.2		4	40.13	odd	4
6400.2.a.co.1.3		4	40.37	odd	4
6400.2.a.co.1.4		4	5.3	odd	4
7200.2.k.l.3601.1		4	240.227	odd	4
7200.2.k.l.3601.2		4	240.107	odd	4
7200.2.k.l.3601.3		4	240.83	odd	4
7200.2.k.l.3601.4		4	240.203	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} +(-1.73205 - 1.41421i) q^{5} +2.44949i q^{7} +1.00000 q^{9} +3.46410 q^{11} +(-2.00000 + 2.44949i) q^{15} +4.89898i q^{17} +3.46410 q^{19} +3.46410 q^{21} -2.44949i q^{23} +(1.00000 + 4.89898i) q^{25} -5.65685i q^{27} +4.00000 q^{31} -4.89898i q^{33} +(3.46410 - 4.24264i) q^{35} +8.48528i q^{37} -4.24264i q^{43} +(-1.73205 - 1.41421i) q^{45} -7.34847i q^{47} +1.00000 q^{49} +6.92820 q^{51} -5.65685i q^{53} +(-6.00000 - 4.89898i) q^{55} -4.89898i q^{57} +10.3923 q^{59} +3.46410 q^{61} +2.44949i q^{63} -4.24264i q^{67} -3.46410 q^{69} +12.0000 q^{71} -4.89898i q^{73} +(6.92820 - 1.41421i) q^{75} +8.48528i q^{77} -4.00000 q^{79} -5.00000 q^{81} -9.89949i q^{83} +(6.92820 - 8.48528i) q^{85} -6.00000 q^{89} -5.65685i q^{93} +(-6.00000 - 4.89898i) q^{95} -4.89898i q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} - 8 q^{15} + 4 q^{25} + 16 q^{31} + 4 q^{49} - 24 q^{55} + 48 q^{71} - 16 q^{79} - 20 q^{81} - 24 q^{89} - 24 q^{95}+O(q^{100}) \) 4 * q + 4 * q^9 - 8 * q^15 + 4 * q^25 + 16 * q^31 + 4 * q^49 - 24 * q^55 + 48 * q^71 - 16 * q^79 - 20 * q^81 - 24 * q^89 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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