LMFDB - Embedded newform 160.2.f.a.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,2,Mod(49,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("160.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	160.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.27760643234$
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} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	160.49
Dual form			160.2.f.a.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} +(1.41421 - 1.73205i) q^{5} +2.44949i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} +(1.41421 - 1.73205i) q^{5} +2.44949i q^{7} -1.00000 q^{9} -3.46410i q^{11} +(2.00000 - 2.44949i) q^{15} +4.89898i q^{17} +3.46410i q^{19} +3.46410i q^{21} -2.44949i q^{23} +(-1.00000 - 4.89898i) q^{25} -5.65685 q^{27} -4.00000 q^{31} -4.89898i q^{33} +(4.24264 + 3.46410i) q^{35} -8.48528 q^{37} -4.24264 q^{43} +(-1.41421 + 1.73205i) q^{45} +7.34847i q^{47} +1.00000 q^{49} +6.92820i q^{51} +5.65685 q^{53} +(-6.00000 - 4.89898i) q^{55} +4.89898i q^{57} -10.3923i q^{59} -3.46410i q^{61} -2.44949i q^{63} +4.24264 q^{67} -3.46410i q^{69} +12.0000 q^{71} +4.89898i q^{73} +(-1.41421 - 6.92820i) q^{75} +8.48528 q^{77} +4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +(8.48528 + 6.92820i) q^{85} +6.00000 q^{89} -5.65685 q^{93} +(6.00000 + 4.89898i) q^{95} -4.89898i q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	1.41421			0.816497			0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421			0.816497			0.408248	+	0.912871i	$0.366140\pi$
$4$		0			0
$5$	1.41421	−	1.73205i	0.632456	−	0.774597i
$5$	1.41421	−	1.73205i	0.632456	−	0.774597i
$6$		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
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		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
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$			3.46410i			0.755929i
$22$		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
$27$	−5.65685			−1.08866
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		−	4.89898i		−	0.852803i
$34$		0			0
$35$	4.24264	+	3.46410i	0.717137	+	0.585540i
$36$		0			0
$37$	−8.48528			−1.39497			−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528			−1.39497			−0.697486	+	0.716599i	$0.745698\pi$
$38$		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
$43$	−4.24264			−0.646997			−0.323498	−	0.946229i	$-0.604859\pi$
$43$	−4.24264			−0.646997			−0.323498	+	0.946229i	$0.604859\pi$
$44$		0			0
$45$	−1.41421	+	1.73205i	−0.210819	+	0.258199i
$46$		0			0
$47$			7.34847i			1.07188i	0.844255	+	0.535942i	$0.180044\pi$
$47$			7.34847i			1.07188i	−0.844255	+	0.535942i	$0.819956\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$			6.92820i			0.970143i
$52$		0			0
$53$	5.65685			0.777029			0.388514	−	0.921443i	$-0.372988\pi$
$53$	5.65685			0.777029			0.388514	+	0.921443i	$0.372988\pi$
$54$		0			0
$55$	−6.00000	−	4.89898i	−0.809040	−	0.660578i
$56$		0			0
$57$			4.89898i			0.648886i
$58$		0			0
$59$		−	10.3923i		−	1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$		−	10.3923i		−	1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$		0			0
$61$		−	3.46410i		−	0.443533i	−0.975100	−	0.221766i	$-0.928818\pi$
$61$		−	3.46410i		−	0.443533i	0.975100	−	0.221766i	$-0.0711822\pi$
$62$		0			0
$63$		−	2.44949i		−	0.308607i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.24264			0.518321			0.259161	−	0.965834i	$-0.416554\pi$
$67$	4.24264			0.518321			0.259161	+	0.965834i	$0.416554\pi$
$68$		0			0
$69$		−	3.46410i		−	0.417029i
$70$		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	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$	−1.41421	−	6.92820i	−0.163299	−	0.800000i
$76$		0			0
$77$	8.48528			0.966988
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	9.89949			1.08661			0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949			1.08661			0.543305	+	0.839535i	$0.317173\pi$
$84$		0			0
$85$	8.48528	+	6.92820i	0.920358	+	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−5.65685			−0.586588
$94$		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
$99$			3.46410i			0.348155i
$100$		0			0
$101$		−	13.8564i		−	1.37876i	−0.724398	−	0.689382i	$-0.757882\pi$
$101$		−	13.8564i		−	1.37876i	0.724398	−	0.689382i	$-0.242118\pi$
$102$		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
$107$	−1.41421			−0.136717			−0.0683586	−	0.997661i	$-0.521776\pi$
$107$	−1.41421			−0.136717			−0.0683586	+	0.997661i	$0.521776\pi$
$108$		0			0
$109$			3.46410i			0.331801i	0.986143	+	0.165900i	$0.0530530\pi$
$109$			3.46410i			0.331801i	−0.986143	+	0.165900i	$0.946947\pi$
$110$		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
$115$	−4.24264	−	3.46410i	−0.395628	−	0.323029i
$116$		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
$125$	−9.89949	−	5.19615i	−0.885438	−	0.464758i
$126$		0			0
$127$		−	17.1464i		−	1.52150i	−0.649045	−	0.760750i	$-0.724831\pi$
$127$		−	17.1464i		−	1.52150i	0.649045	−	0.760750i	$-0.275169\pi$
$128$		0			0
$129$	−6.00000			−0.528271
$130$		0			0
$131$			3.46410i			0.302660i	0.988483	+	0.151330i	$0.0483556\pi$
$131$			3.46410i			0.302660i	−0.988483	+	0.151330i	$0.951644\pi$
$132$		0			0
$133$	−8.48528			−0.735767
$134$		0			0
$135$	−8.00000	+	9.79796i	−0.688530	+	0.843274i
$136$		0			0
$137$		−	9.79796i		−	0.837096i	−0.908195	−	0.418548i	$-0.862539\pi$
$137$		−	9.79796i		−	0.837096i	0.908195	−	0.418548i	$-0.137461\pi$
$138$		0			0
$139$			10.3923i			0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$			10.3923i			0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$		0			0
$141$			10.3923i			0.875190i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.41421			0.116642
$148$		0			0
$149$			17.3205i			1.41895i	0.704730	+	0.709476i	$0.251068\pi$
$149$			17.3205i			1.41895i	−0.704730	+	0.709476i	$0.748932\pi$
$150$		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
$155$	−5.65685	+	6.92820i	−0.454369	+	0.556487i
$156$		0			0
$157$	8.48528			0.677199			0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528			0.677199			0.338600	+	0.940931i	$0.390047\pi$
$158$		0			0
$159$	8.00000			0.634441
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	−21.2132			−1.66155			−0.830773	−	0.556611i	$-0.812101\pi$
$163$	−21.2132			−1.66155			−0.830773	+	0.556611i	$0.812101\pi$
$164$		0			0
$165$	−8.48528	−	6.92820i	−0.660578	−	0.539360i
$166$		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
$171$		−	3.46410i		−	0.264906i
$172$		0			0
$173$	2.82843			0.215041			0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843			0.215041			0.107521	+	0.994203i	$0.465709\pi$
$174$		0			0
$175$	12.0000	−	2.44949i	0.907115	−	0.185164i
$176$		0			0
$177$		−	14.6969i		−	1.10469i
$178$		0			0
$179$		−	3.46410i		−	0.258919i	−0.991585	−	0.129460i	$-0.958676\pi$
$179$		−	3.46410i		−	0.258919i	0.991585	−	0.129460i	$-0.0413242\pi$
$180$		0			0
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$		0			0
$183$		−	4.89898i		−	0.362143i
$184$		0			0
$185$	−12.0000	+	14.6969i	−0.882258	+	1.08054i
$186$		0			0
$187$	16.9706			1.24101
$188$		0			0
$189$		−	13.8564i		−	1.00791i
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		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
$197$	−5.65685			−0.403034			−0.201517	−	0.979485i	$-0.564587\pi$
$197$	−5.65685			−0.403034			−0.201517	+	0.979485i	$0.564587\pi$
$198$		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
$211$		−	24.2487i		−	1.66935i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$		−	24.2487i		−	1.66935i	0.550743	−	0.834675i	$-0.314345\pi$
$212$		0			0
$213$	16.9706			1.16280
$214$		0			0
$215$	−6.00000	+	7.34847i	−0.409197	+	0.501161i
$216$		0			0
$217$		−	9.79796i		−	0.665129i
$218$		0			0
$219$			6.92820i			0.468165i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			22.0454i			1.47627i	0.674653	+	0.738135i	$0.264293\pi$
$223$			22.0454i			1.47627i	−0.674653	+	0.738135i	$0.735707\pi$
$224$		0			0
$225$	1.00000	+	4.89898i	0.0666667	+	0.326599i
$226$		0			0
$227$	1.41421			0.0938647			0.0469323	−	0.998898i	$-0.485055\pi$
$227$	1.41421			0.0938647			0.0469323	+	0.998898i	$0.485055\pi$
$228$		0			0
$229$		−	27.7128i		−	1.83131i	−0.401960	−	0.915657i	$-0.631671\pi$
$229$		−	27.7128i		−	1.83131i	0.401960	−	0.915657i	$-0.368329\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$		0			0
$233$		−	14.6969i		−	0.962828i	−0.876493	−	0.481414i	$-0.840123\pi$
$233$		−	14.6969i		−	0.962828i	0.876493	−	0.481414i	$-0.159877\pi$
$234$		0			0
$235$	12.7279	+	10.3923i	0.830278	+	0.677919i
$236$		0			0
$237$	5.65685			0.367452
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		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
$243$	9.89949			0.635053
$244$		0			0
$245$	1.41421	−	1.73205i	0.0903508	−	0.110657i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	14.0000			0.887214
$250$		0			0
$251$			10.3923i			0.655956i	0.944685	+	0.327978i	$0.106367\pi$
$251$			10.3923i			0.655956i	−0.944685	+	0.327978i	$0.893633\pi$
$252$		0			0
$253$	−8.48528			−0.533465
$254$		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
$259$		−	20.7846i		−	1.29149i
$260$		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
$267$	8.48528			0.519291
$268$		0			0
$269$			10.3923i			0.633630i	0.948487	+	0.316815i	$0.102613\pi$
$269$			10.3923i			0.633630i	−0.948487	+	0.316815i	$0.897387\pi$
$270$		0			0
$271$	−20.0000			−1.21491			−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000			−1.21491			−0.607457	+	0.794353i	$0.707810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.9706	+	3.46410i	−1.02336	+	0.208893i
$276$		0			0
$277$	25.4558			1.52949			0.764747	−	0.644331i	$-0.222864\pi$
$277$	25.4558			1.52949			0.764747	+	0.644331i	$0.222864\pi$
$278$		0			0
$279$	4.00000			0.239474
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	21.2132			1.26099			0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132			1.26099			0.630497	+	0.776192i	$0.282851\pi$
$284$		0			0
$285$	8.48528	+	6.92820i	0.502625	+	0.410391i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		−	6.92820i		−	0.406138i
$292$		0			0
$293$	−19.7990			−1.15667			−0.578335	−	0.815800i	$-0.696297\pi$
$293$	−19.7990			−1.15667			−0.578335	+	0.815800i	$0.696297\pi$
$294$		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
$301$		−	10.3923i		−	0.599002i
$302$		0			0
$303$		−	19.5959i		−	1.12576i
$304$		0			0
$305$	−6.00000	−	4.89898i	−0.343559	−	0.280515i
$306$		0			0
$307$	−29.6985			−1.69498			−0.847491	−	0.530810i	$-0.821888\pi$
$307$	−29.6985			−1.69498			−0.847491	+	0.530810i	$0.821888\pi$
$308$		0			0
$309$			10.3923i			0.591198i
$310$		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	$0.0893093\pi$
$313$			9.79796i			0.553813i	−0.960897	+	0.276907i	$0.910691\pi$
$314$		0			0
$315$	−4.24264	−	3.46410i	−0.239046	−	0.195180i
$316$		0			0
$317$	−28.2843			−1.58860			−0.794301	−	0.607524i	$-0.792163\pi$
$317$	−28.2843			−1.58860			−0.794301	+	0.607524i	$0.792163\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−2.00000			−0.111629
$322$		0			0
$323$	−16.9706			−0.944267
$324$		0			0
$325$		0			0
$326$		0			0
$327$			4.89898i			0.270914i
$328$		0			0
$329$	−18.0000			−0.992372
$330$		0			0
$331$			31.1769i			1.71364i	0.515617	+	0.856819i	$0.327563\pi$
$331$			31.1769i			1.71364i	−0.515617	+	0.856819i	$0.672437\pi$
$332$		0			0
$333$	8.48528			0.464991
$334$		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
$341$			13.8564i			0.750366i
$342$		0			0
$343$			19.5959i			1.05808i
$344$		0			0
$345$	−6.00000	−	4.89898i	−0.323029	−	0.263752i
$346$		0			0
$347$	−15.5563			−0.835109			−0.417554	−	0.908652i	$-0.637113\pi$
$347$	−15.5563			−0.835109			−0.417554	+	0.908652i	$0.637113\pi$
$348$		0			0
$349$			13.8564i			0.741716i	0.928689	+	0.370858i	$0.120936\pi$
$349$			13.8564i			0.741716i	−0.928689	+	0.370858i	$0.879064\pi$
$350$		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
$355$	16.9706	−	20.7846i	0.900704	−	1.10313i
$356$		0			0
$357$	−16.9706			−0.898177
$358$		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
$363$	−1.41421			−0.0742270
$364$		0			0
$365$	8.48528	+	6.92820i	0.444140	+	0.362639i
$366$		0			0
$367$			12.2474i			0.639312i	0.947534	+	0.319656i	$0.103567\pi$
$367$			12.2474i			0.639312i	−0.947534	+	0.319656i	$0.896433\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			13.8564i			0.719389i
$372$		0			0
$373$	−8.48528			−0.439351			−0.219676	−	0.975573i	$-0.570500\pi$
$373$	−8.48528			−0.439351			−0.219676	+	0.975573i	$0.570500\pi$
$374$		0			0
$375$	−14.0000	−	7.34847i	−0.722957	−	0.379473i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			24.2487i			1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$			24.2487i			1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$		0			0
$381$		−	24.2487i		−	1.24230i
$382$		0			0
$383$		−	26.9444i		−	1.37679i	−0.725334	−	0.688397i	$-0.758315\pi$
$383$		−	26.9444i		−	1.37679i	0.725334	−	0.688397i	$-0.241685\pi$
$384$		0			0
$385$	12.0000	−	14.6969i	0.611577	−	0.749025i
$386$		0			0
$387$	4.24264			0.215666
$388$		0			0
$389$			3.46410i			0.175637i	0.996136	+	0.0878185i	$0.0279895\pi$
$389$			3.46410i			0.175637i	−0.996136	+	0.0878185i	$0.972010\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$			4.89898i			0.247121i
$394$		0			0
$395$	5.65685	−	6.92820i	0.284627	−	0.348596i
$396$		0			0
$397$	−16.9706			−0.851728			−0.425864	−	0.904787i	$-0.640030\pi$
$397$	−16.9706			−0.851728			−0.425864	+	0.904787i	$0.640030\pi$
$398$		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
$405$	−7.07107	+	8.66025i	−0.351364	+	0.430331i
$406$		0			0
$407$			29.3939i			1.45700i
$408$		0			0
$409$	32.0000			1.58230			0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000			1.58230			0.791149	+	0.611623i	$0.209483\pi$
$410$		0			0
$411$		−	13.8564i		−	0.683486i
$412$		0			0
$413$	25.4558			1.25260
$414$		0			0
$415$	14.0000	−	17.1464i	0.687233	−	0.841685i
$416$		0			0
$417$			14.6969i			0.719712i
$418$		0			0
$419$			10.3923i			0.507697i	0.967244	+	0.253849i	$0.0816965\pi$
$419$			10.3923i			0.507697i	−0.967244	+	0.253849i	$0.918303\pi$
$420$		0			0
$421$		−	24.2487i		−	1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$		−	24.2487i		−	1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$		0			0
$423$		−	7.34847i		−	0.357295i
$424$		0			0
$425$	24.0000	−	4.89898i	1.16417	−	0.237635i
$426$		0			0
$427$	8.48528			0.410632
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		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
$437$	8.48528			0.405906
$438$		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
$443$	−1.41421			−0.0671913			−0.0335957	−	0.999436i	$-0.510696\pi$
$443$	−1.41421			−0.0671913			−0.0335957	+	0.999436i	$0.510696\pi$
$444$		0			0
$445$	8.48528	−	10.3923i	0.402241	−	0.492642i
$446$		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
$453$	22.6274			1.06313
$454$		0			0
$455$		0			0
$456$		0			0
$457$			19.5959i			0.916658i	0.888783	+	0.458329i	$0.151552\pi$
$457$			19.5959i			0.916658i	−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		−	27.7128i		−	1.29352i
$460$		0			0
$461$			13.8564i			0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$			13.8564i			0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$		0			0
$463$			17.1464i			0.796862i	0.917198	+	0.398431i	$0.130445\pi$
$463$			17.1464i			0.796862i	−0.917198	+	0.398431i	$0.869555\pi$
$464$		0			0
$465$	−8.00000	+	9.79796i	−0.370991	+	0.454369i
$466$		0			0
$467$	7.07107			0.327210			0.163605	−	0.986526i	$-0.447688\pi$
$467$	7.07107			0.327210			0.163605	+	0.986526i	$0.447688\pi$
$468$		0			0
$469$			10.3923i			0.479872i
$470$		0			0
$471$	12.0000			0.552931
$472$		0			0
$473$			14.6969i			0.675766i
$474$		0			0
$475$	16.9706	−	3.46410i	0.778663	−	0.158944i
$476$		0			0
$477$	−5.65685			−0.259010
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	8.48528			0.386094
$484$		0			0
$485$	−8.48528	−	6.92820i	−0.385297	−	0.314594i
$486$		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
$491$		−	24.2487i		−	1.09433i	−0.837025	−	0.547165i	$-0.815707\pi$
$491$		−	24.2487i		−	1.09433i	0.837025	−	0.547165i	$-0.184293\pi$
$492$		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
$499$		−	17.3205i		−	0.775372i	−0.921791	−	0.387686i	$-0.873274\pi$
$499$		−	17.3205i		−	0.775372i	0.921791	−	0.387686i	$-0.126726\pi$
$500$		0			0
$501$		−	17.3205i		−	0.773823i
$502$		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
$507$	−18.3848			−0.816497
$508$		0			0
$509$		−	27.7128i		−	1.22835i	−0.789170	−	0.614174i	$-0.789489\pi$
$509$		−	27.7128i		−	1.22835i	0.789170	−	0.614174i	$-0.210511\pi$
$510$		0			0
$511$	−12.0000			−0.530849
$512$		0			0
$513$		−	19.5959i		−	0.865181i
$514$		0			0
$515$	12.7279	+	10.3923i	0.560859	+	0.457940i
$516$		0			0
$517$	25.4558			1.11955
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	−12.7279			−0.556553			−0.278277	−	0.960501i	$-0.589763\pi$
$523$	−12.7279			−0.556553			−0.278277	+	0.960501i	$0.589763\pi$
$524$		0			0
$525$	16.9706	−	3.46410i	0.740656	−	0.151186i
$526$		0			0
$527$		−	19.5959i		−	0.853612i
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$			10.3923i			0.450988i
$532$		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
$539$		−	3.46410i		−	0.149209i
$540$		0			0
$541$			41.5692i			1.78720i	0.448864	+	0.893600i	$0.351829\pi$
$541$			41.5692i			1.78720i	−0.448864	+	0.893600i	$0.648171\pi$
$542$		0			0
$543$			19.5959i			0.840941i
$544$		0			0
$545$	6.00000	+	4.89898i	0.257012	+	0.209849i
$546$		0			0
$547$	−4.24264			−0.181402			−0.0907011	−	0.995878i	$-0.528911\pi$
$547$	−4.24264			−0.181402			−0.0907011	+	0.995878i	$0.528911\pi$
$548$		0			0
$549$			3.46410i			0.147844i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			9.79796i			0.416652i
$554$		0			0
$555$	−16.9706	+	20.7846i	−0.720360	+	0.882258i
$556$		0			0
$557$	−14.1421			−0.599222			−0.299611	−	0.954062i	$-0.596857\pi$
$557$	−14.1421			−0.599222			−0.299611	+	0.954062i	$0.596857\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$	41.0122			1.72846			0.864229	−	0.503099i	$-0.167807\pi$
$563$	41.0122			1.72846			0.864229	+	0.503099i	$0.167807\pi$
$564$		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	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$		−	3.46410i		−	0.144968i	−0.997370	−	0.0724841i	$-0.976907\pi$
$571$		−	3.46410i		−	0.144968i	0.997370	−	0.0724841i	$-0.0230926\pi$
$572$		0			0
$573$	−33.9411			−1.41791
$574$		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
$579$			34.6410i			1.43963i
$580$		0			0
$581$			24.2487i			1.00601i
$582$		0			0
$583$		−	19.5959i		−	0.811580i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.89949			−0.408596			−0.204298	−	0.978909i	$-0.565491\pi$
$587$	−9.89949			−0.408596			−0.204298	+	0.978909i	$0.565491\pi$
$588$		0			0
$589$		−	13.8564i		−	0.570943i
$590$		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
$595$	−16.9706	+	20.7846i	−0.695725	+	0.852086i
$596$		0			0
$597$	−5.65685			−0.231520
$598$		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	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$		0			0
$603$	−4.24264			−0.172774
$604$		0			0
$605$	−1.41421	+	1.73205i	−0.0574960	+	0.0704179i
$606$		0			0
$607$			7.34847i			0.298265i	0.988817	+	0.149133i	$0.0476481\pi$
$607$			7.34847i			0.298265i	−0.988817	+	0.149133i	$0.952352\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−33.9411			−1.37087			−0.685435	−	0.728134i	$-0.740388\pi$
$613$	−33.9411			−1.37087			−0.685435	+	0.728134i	$0.740388\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	34.2929i		−	1.38058i	−0.723534	−	0.690289i	$-0.757483\pi$
$617$		−	34.2929i		−	1.38058i	0.723534	−	0.690289i	$-0.242517\pi$
$618$		0			0
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$		0			0
$621$			13.8564i			0.556038i
$622$		0			0
$623$			14.6969i			0.588820i
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$	16.9706			0.677739
$628$		0			0
$629$		−	41.5692i		−	1.65747i
$630$		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
$635$	−29.6985	−	24.2487i	−1.17855	−	0.962281i
$636$		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
$643$	−29.6985			−1.17119			−0.585597	−	0.810602i	$-0.699140\pi$
$643$	−29.6985			−1.17119			−0.585597	+	0.810602i	$0.699140\pi$
$644$		0			0
$645$	−8.48528	+	10.3923i	−0.334108	+	0.409197i
$646$		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
$651$		−	13.8564i		−	0.543075i
$652$		0			0
$653$	11.3137			0.442740			0.221370	−	0.975190i	$-0.428947\pi$
$653$	11.3137			0.442740			0.221370	+	0.975190i	$0.428947\pi$
$654$		0			0
$655$	6.00000	+	4.89898i	0.234439	+	0.191419i
$656$		0			0
$657$		−	4.89898i		−	0.191127i
$658$		0			0
$659$			24.2487i			0.944596i	0.881439	+	0.472298i	$0.156575\pi$
$659$			24.2487i			0.944596i	−0.881439	+	0.472298i	$0.843425\pi$
$660$		0			0
$661$			10.3923i			0.404214i	0.979363	+	0.202107i	$0.0647788\pi$
$661$			10.3923i			0.404214i	−0.979363	+	0.202107i	$0.935221\pi$
$662$		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
$669$			31.1769i			1.20537i
$670$		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
$675$	5.65685	+	27.7128i	0.217732	+	1.06667i
$676$		0			0
$677$	22.6274			0.869642			0.434821	−	0.900517i	$-0.356812\pi$
$677$	22.6274			0.869642			0.434821	+	0.900517i	$0.356812\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$		0			0
$681$	2.00000			0.0766402
$682$		0			0
$683$	−15.5563			−0.595247			−0.297624	−	0.954683i	$-0.596194\pi$
$683$	−15.5563			−0.595247			−0.297624	+	0.954683i	$0.596194\pi$
$684$		0			0
$685$	−16.9706	−	13.8564i	−0.648412	−	0.529426i
$686$		0			0
$687$		−	39.1918i		−	1.49526i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			3.46410i			0.131781i	0.997827	+	0.0658903i	$0.0209887\pi$
$691$			3.46410i			0.131781i	−0.997827	+	0.0658903i	$0.979011\pi$
$692$		0			0
$693$	−8.48528			−0.322329
$694$		0			0
$695$	18.0000	+	14.6969i	0.682779	+	0.557487i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	20.7846i		−	0.786146i
$700$		0			0
$701$		−	24.2487i		−	0.915861i	−0.888988	−	0.457931i	$-0.848591\pi$
$701$		−	24.2487i		−	0.915861i	0.888988	−	0.457931i	$-0.151409\pi$
$702$		0			0
$703$		−	29.3939i		−	1.10861i
$704$		0			0
$705$	18.0000	+	14.6969i	0.677919	+	0.553519i
$706$		0			0
$707$	33.9411			1.27649
$708$		0			0
$709$			41.5692i			1.56116i	0.625053	+	0.780582i	$0.285077\pi$
$709$			41.5692i			1.56116i	−0.625053	+	0.780582i	$0.714923\pi$
$710$		0			0
$711$	−4.00000			−0.150012
$712$		0			0
$713$			9.79796i			0.366936i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16.9706			0.633777
$718$		0			0
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$		0			0
$721$	−18.0000			−0.670355
$722$		0			0
$723$	5.65685			0.210381
$724$		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
$731$		−	20.7846i		−	0.768747i
$732$		0			0
$733$	42.4264			1.56706			0.783528	−	0.621357i	$-0.213418\pi$
$733$	42.4264			1.56706			0.783528	+	0.621357i	$0.213418\pi$
$734$		0			0
$735$	2.00000	−	2.44949i	0.0737711	−	0.0903508i
$736$		0			0
$737$		−	14.6969i		−	0.541369i
$738$		0			0
$739$		−	3.46410i		−	0.127429i	−0.997968	−	0.0637145i	$-0.979705\pi$
$739$		−	3.46410i		−	0.127429i	0.997968	−	0.0637145i	$-0.0202947\pi$
$740$		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
$747$	−9.89949			−0.362204
$748$		0			0
$749$		−	3.46410i		−	0.126576i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$			14.6969i			0.535586i
$754$		0			0
$755$	22.6274	−	27.7128i	0.823496	−	1.00857i
$756$		0			0
$757$	−25.4558			−0.925208			−0.462604	−	0.886565i	$-0.653085\pi$
$757$	−25.4558			−0.925208			−0.462604	+	0.886565i	$0.653085\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	−8.48528			−0.307188
$764$		0			0
$765$	−8.48528	−	6.92820i	−0.306786	−	0.250490i
$766$		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
$771$			13.8564i			0.499026i
$772$		0			0
$773$	28.2843			1.01731			0.508657	−	0.860969i	$-0.330142\pi$
$773$	28.2843			1.01731			0.508657	+	0.860969i	$0.330142\pi$
$774$		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
$781$		−	41.5692i		−	1.48746i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.0000	−	14.6969i	0.428298	−	0.524556i
$786$		0			0
$787$	21.2132			0.756169			0.378085	−	0.925771i	$-0.376583\pi$
$787$	21.2132			0.756169			0.378085	+	0.925771i	$0.376583\pi$
$788$		0			0
$789$		−	10.3923i		−	0.369976i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	11.3137	−	13.8564i	0.401256	−	0.491436i
$796$		0			0
$797$	39.5980			1.40263			0.701316	−	0.712850i	$-0.252596\pi$
$797$	39.5980			1.40263			0.701316	+	0.712850i	$0.252596\pi$
$798$		0			0
$799$	−36.0000			−1.27359
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$	16.9706			0.598878
$804$		0			0
$805$	8.48528	−	10.3923i	0.299067	−	0.366281i
$806$		0			0
$807$			14.6969i			0.517357i
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$			10.3923i			0.364923i	0.983213	+	0.182462i	$0.0584065\pi$
$811$			10.3923i			0.364923i	−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$	−28.2843			−0.991973
$814$		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
$821$		−	31.1769i		−	1.08808i	−0.839059	−	0.544041i	$-0.816894\pi$
$821$		−	31.1769i		−	1.08808i	0.839059	−	0.544041i	$-0.183106\pi$
$822$		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
$827$	35.3553			1.22943			0.614713	−	0.788751i	$-0.289272\pi$
$827$	35.3553			1.22943			0.614713	+	0.788751i	$0.289272\pi$
$828$		0			0
$829$			10.3923i			0.360940i	0.983581	+	0.180470i	$0.0577618\pi$
$829$			10.3923i			0.360940i	−0.983581	+	0.180470i	$0.942238\pi$
$830$		0			0
$831$	36.0000			1.24883
$832$		0			0
$833$			4.89898i			0.169740i
$834$		0			0
$835$	−21.2132	−	17.3205i	−0.734113	−	0.599401i
$836$		0			0
$837$	22.6274			0.782118
$838$		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
$843$	−16.9706			−0.584497
$844$		0			0
$845$	−18.3848	+	22.5167i	−0.632456	+	0.774597i
$846$		0			0
$847$		−	2.44949i		−	0.0841655i
$848$		0			0
$849$	30.0000			1.02960
$850$		0			0
$851$			20.7846i			0.712487i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$	−6.00000	−	4.89898i	−0.205196	−	0.167542i
$856$		0			0
$857$			39.1918i			1.33877i	0.742917	+	0.669384i	$0.233442\pi$
$857$			39.1918i			1.33877i	−0.742917	+	0.669384i	$0.766558\pi$
$858$		0			0
$859$			3.46410i			0.118194i	0.998252	+	0.0590968i	$0.0188221\pi$
$859$			3.46410i			0.118194i	−0.998252	+	0.0590968i	$0.981178\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.44949i		−	0.0833816i	−0.999131	−	0.0416908i	$-0.986726\pi$
$863$		−	2.44949i		−	0.0833816i	0.999131	−	0.0416908i	$-0.0132744\pi$
$864$		0			0
$865$	4.00000	−	4.89898i	0.136004	−	0.166570i
$866$		0			0
$867$	−9.89949			−0.336204
$868$		0			0
$869$		−	13.8564i		−	0.470046i
$870$		0			0
$871$		0			0
$872$		0			0
$873$			4.89898i			0.165805i
$874$		0			0
$875$	12.7279	−	24.2487i	0.430282	−	0.819756i
$876$		0			0
$877$	−8.48528			−0.286528			−0.143264	−	0.989685i	$-0.545760\pi$
$877$	−8.48528			−0.286528			−0.143264	+	0.989685i	$0.545760\pi$
$878$		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
$883$	−4.24264			−0.142776			−0.0713881	−	0.997449i	$-0.522743\pi$
$883$	−4.24264			−0.142776			−0.0713881	+	0.997449i	$0.522743\pi$
$884$		0			0
$885$	−25.4558	−	20.7846i	−0.855689	−	0.698667i
$886$		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
$891$			17.3205i			0.580259i
$892$		0			0
$893$	−25.4558			−0.851847
$894$		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
$901$			27.7128i			0.923248i
$902$		0			0
$903$		−	14.6969i		−	0.489083i
$904$		0			0
$905$	24.0000	+	19.5959i	0.797787	+	0.651390i
$906$		0			0
$907$	4.24264			0.140875			0.0704373	−	0.997516i	$-0.477561\pi$
$907$	4.24264			0.140875			0.0704373	+	0.997516i	$0.477561\pi$
$908$		0			0
$909$			13.8564i			0.459588i
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$		−	34.2929i		−	1.13493i
$914$		0			0
$915$	−8.48528	−	6.92820i	−0.280515	−	0.229039i
$916$		0			0
$917$	−8.48528			−0.280209
$918$		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
$925$	8.48528	+	41.5692i	0.278994	+	1.36679i
$926$		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
$931$			3.46410i			0.113531i
$932$		0			0
$933$	33.9411			1.11118
$934$		0			0
$935$	24.0000	−	29.3939i	0.784884	−	0.961283i
$936$		0			0
$937$		−	4.89898i		−	0.160043i	−0.996793	−	0.0800213i	$-0.974501\pi$
$937$		−	4.89898i		−	0.160043i	0.996793	−	0.0800213i	$-0.0254988\pi$
$938$		0			0
$939$			13.8564i			0.452187i
$940$		0			0
$941$		−	13.8564i		−	0.451706i	−0.974161	−	0.225853i	$-0.927483\pi$
$941$		−	13.8564i		−	0.451706i	0.974161	−	0.225853i	$-0.0725169\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−24.0000	−	19.5959i	−0.780720	−	0.637455i
$946$		0			0
$947$	−41.0122			−1.33272			−0.666359	−	0.745631i	$-0.732148\pi$
$947$	−41.0122			−1.33272			−0.666359	+	0.745631i	$0.732148\pi$
$948$		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
$955$	−33.9411	+	41.5692i	−1.09831	+	1.34515i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000			0.775000
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	1.41421			0.0455724
$964$		0			0
$965$	42.4264	+	34.6410i	1.36575	+	1.11513i
$966$		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
$971$			3.46410i			0.111168i	0.998454	+	0.0555842i	$0.0177021\pi$
$971$			3.46410i			0.111168i	−0.998454	+	0.0555842i	$0.982298\pi$
$972$		0			0
$973$	−25.4558			−0.816077
$974$		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
$979$		−	20.7846i		−	0.664279i
$980$		0			0
$981$		−	3.46410i		−	0.110600i
$982$		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
$987$	−25.4558			−0.810268
$988$		0			0
$989$			10.3923i			0.330456i
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$			44.0908i			1.39918i
$994$		0			0
$995$	−5.65685	+	6.92820i	−0.179334	+	0.219639i
$996$		0			0
$997$	−50.9117			−1.61239			−0.806195	−	0.591650i	$-0.798477\pi$
$997$	−50.9117			−1.61239			−0.806195	+	0.591650i	$0.798477\pi$
$998$		0			0
$999$	48.0000			1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.2.f.a.49.3		4
3.2	odd	2		1440.2.d.c.1009.2		4
4.3	odd	2		40.2.f.a.29.4	yes	4
5.2	odd	4		800.2.d.f.401.1		4
5.3	odd	4		800.2.d.f.401.4		4
5.4	even	2	inner	160.2.f.a.49.1		4
8.3	odd	2		40.2.f.a.29.2	yes	4
8.5	even	2	inner	160.2.f.a.49.2		4
12.11	even	2		360.2.d.b.109.1		4
15.2	even	4		7200.2.k.l.3601.2		4
15.8	even	4		7200.2.k.l.3601.4		4
15.14	odd	2		1440.2.d.c.1009.4		4
16.3	odd	4		1280.2.c.i.769.1		4
16.5	even	4		1280.2.c.k.769.2		4
16.11	odd	4		1280.2.c.i.769.4		4
16.13	even	4		1280.2.c.k.769.3		4
20.3	even	4		200.2.d.e.101.3		4
20.7	even	4		200.2.d.e.101.2		4
20.19	odd	2		40.2.f.a.29.1	✓	4
24.5	odd	2		1440.2.d.c.1009.3		4
24.11	even	2		360.2.d.b.109.3		4
40.3	even	4		200.2.d.e.101.4		4
40.13	odd	4		800.2.d.f.401.2		4
40.19	odd	2		40.2.f.a.29.3	yes	4
40.27	even	4		200.2.d.e.101.1		4
40.29	even	2	inner	160.2.f.a.49.4		4
40.37	odd	4		800.2.d.f.401.3		4
60.23	odd	4		1800.2.k.m.901.2		4
60.47	odd	4		1800.2.k.m.901.3		4
60.59	even	2		360.2.d.b.109.4		4
80.3	even	4		6400.2.a.co.1.4		4
80.13	odd	4		6400.2.a.cm.1.1		4
80.19	odd	4		1280.2.c.i.769.3		4
80.27	even	4		6400.2.a.co.1.3		4
80.29	even	4		1280.2.c.k.769.1		4
80.37	odd	4		6400.2.a.cm.1.2		4
80.43	even	4		6400.2.a.co.1.2		4
80.53	odd	4		6400.2.a.cm.1.3		4
80.59	odd	4		1280.2.c.i.769.2		4
80.67	even	4		6400.2.a.co.1.1		4
80.69	even	4		1280.2.c.k.769.4		4
80.77	odd	4		6400.2.a.cm.1.4		4
120.29	odd	2		1440.2.d.c.1009.1		4
120.53	even	4		7200.2.k.l.3601.3		4
120.59	even	2		360.2.d.b.109.2		4
120.77	even	4		7200.2.k.l.3601.1		4
120.83	odd	4		1800.2.k.m.901.1		4
120.107	odd	4		1800.2.k.m.901.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.f.a.29.1	✓	4	20.19	odd	2
40.2.f.a.29.2	yes	4	8.3	odd	2
40.2.f.a.29.3	yes	4	40.19	odd	2
40.2.f.a.29.4	yes	4	4.3	odd	2
160.2.f.a.49.1		4	5.4	even	2	inner
160.2.f.a.49.2		4	8.5	even	2	inner
160.2.f.a.49.3		4	1.1	even	1	trivial
160.2.f.a.49.4		4	40.29	even	2	inner
200.2.d.e.101.1		4	40.27	even	4
200.2.d.e.101.2		4	20.7	even	4
200.2.d.e.101.3		4	20.3	even	4
200.2.d.e.101.4		4	40.3	even	4
360.2.d.b.109.1		4	12.11	even	2
360.2.d.b.109.2		4	120.59	even	2
360.2.d.b.109.3		4	24.11	even	2
360.2.d.b.109.4		4	60.59	even	2
800.2.d.f.401.1		4	5.2	odd	4
800.2.d.f.401.2		4	40.13	odd	4
800.2.d.f.401.3		4	40.37	odd	4
800.2.d.f.401.4		4	5.3	odd	4
1280.2.c.i.769.1		4	16.3	odd	4
1280.2.c.i.769.2		4	80.59	odd	4
1280.2.c.i.769.3		4	80.19	odd	4
1280.2.c.i.769.4		4	16.11	odd	4
1280.2.c.k.769.1		4	80.29	even	4
1280.2.c.k.769.2		4	16.5	even	4
1280.2.c.k.769.3		4	16.13	even	4
1280.2.c.k.769.4		4	80.69	even	4
1440.2.d.c.1009.1		4	120.29	odd	2
1440.2.d.c.1009.2		4	3.2	odd	2
1440.2.d.c.1009.3		4	24.5	odd	2
1440.2.d.c.1009.4		4	15.14	odd	2
1800.2.k.m.901.1		4	120.83	odd	4
1800.2.k.m.901.2		4	60.23	odd	4
1800.2.k.m.901.3		4	60.47	odd	4
1800.2.k.m.901.4		4	120.107	odd	4
6400.2.a.cm.1.1		4	80.13	odd	4
6400.2.a.cm.1.2		4	80.37	odd	4
6400.2.a.cm.1.3		4	80.53	odd	4
6400.2.a.cm.1.4		4	80.77	odd	4
6400.2.a.co.1.1		4	80.67	even	4
6400.2.a.co.1.2		4	80.43	even	4
6400.2.a.co.1.3		4	80.27	even	4
6400.2.a.co.1.4		4	80.3	even	4
7200.2.k.l.3601.1		4	120.77	even	4
7200.2.k.l.3601.2		4	15.2	even	4
7200.2.k.l.3601.3		4	120.53	even	4
7200.2.k.l.3601.4		4	15.8	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	160.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +(1.41421 - 1.73205i) q^{5} +2.44949i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} +(1.41421 - 1.73205i) q^{5} +2.44949i q^{7} -1.00000 q^{9} -3.46410i q^{11} +(2.00000 - 2.44949i) q^{15} +4.89898i q^{17} +3.46410i q^{19} +3.46410i q^{21} -2.44949i q^{23} +(-1.00000 - 4.89898i) q^{25} -5.65685 q^{27} -4.00000 q^{31} -4.89898i q^{33} +(4.24264 + 3.46410i) q^{35} -8.48528 q^{37} -4.24264 q^{43} +(-1.41421 + 1.73205i) q^{45} +7.34847i q^{47} +1.00000 q^{49} +6.92820i q^{51} +5.65685 q^{53} +(-6.00000 - 4.89898i) q^{55} +4.89898i q^{57} -10.3923i q^{59} -3.46410i q^{61} -2.44949i q^{63} +4.24264 q^{67} -3.46410i q^{69} +12.0000 q^{71} +4.89898i q^{73} +(-1.41421 - 6.92820i) q^{75} +8.48528 q^{77} +4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +(8.48528 + 6.92820i) q^{85} +6.00000 q^{89} -5.65685 q^{93} +(6.00000 + 4.89898i) q^{95} -4.89898i q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 8 q^{15} - 4 q^{25} - 16 q^{31} + 4 q^{49} - 24 q^{55} + 48 q^{71} + 16 q^{79} - 20 q^{81} + 24 q^{89} + 24 q^{95}+O(q^{100}) \) 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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