LMFDB - Embedded newform 5292.2.x.b.881.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(881,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

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

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

chi := DirichletCharacter("5292.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5292 = 2^{2} \cdot 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5292.x (of order $6$ , degree $2$ , 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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561$ x^16 - 2x^15 + 5x^14 - 17x^13 + 22x^12 - 31x^11 + 62x^10 - 52x^9 + 52x^8 - 156x^7 + 558x^6 - 837x^5 + 1782x^4 - 4131x^3 + 3645x^2 - 4374*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$3^{6}$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.3
Root			$-0.268067 - 1.71118i$ of defining polynomial
Character	$\chi$	$=$	5292.881
Dual form			5292.2.x.b.4409.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.842869 + 1.45989i) q^{5} +(3.38216 - 1.95269i) q^{11} +(5.24391 + 3.02757i) q^{13} +0.402488 q^{17} -0.168144i q^{19} +(7.69373 + 4.44198i) q^{23} +(1.07914 + 1.86913i) q^{25} +(6.15380 - 3.55290i) q^{29} +(-5.44527 - 3.14383i) q^{31} -6.26515 q^{37} +(1.64707 - 2.85281i) q^{41} +(1.80474 + 3.12590i) q^{43} +(-4.38482 - 7.59474i) q^{47} -5.71148i q^{53} +6.58345i q^{55} +(-2.25163 + 3.89994i) q^{59} +(-4.43678 + 2.56157i) q^{61} +(-8.83986 + 5.10369i) q^{65} +(2.95521 - 5.11857i) q^{67} -11.4308i q^{71} +6.99239i q^{73} +(-0.603968 - 1.04610i) q^{79} +(0.181350 + 0.314108i) q^{83} +(-0.339244 + 0.587588i) q^{85} +2.77052 q^{89} +(0.245471 + 0.141723i) q^{95} +(0.508914 - 0.293821i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 6 q^{11} + 3 q^{13} - 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} - 6 q^{31} - 2 q^{37} - 6 q^{41} - 2 q^{43} + 18 q^{47} + 15 q^{59} - 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} - 42 q^{89}+ \cdots + 3 q^{97}+O(q^{100})$ 16 * q - 6 * q^11 + 3 * q^13 - 18 * q^17 + 21 * q^23 - 8 * q^25 - 6 * q^29 - 6 * q^31 - 2 * q^37 - 6 * q^41 - 2 * q^43 + 18 * q^47 + 15 * q^59 - 3 * q^61 - 39 * q^65 - 7 * q^67 - q^79 + 6 * q^85 - 42 * q^89 + 6 * q^95 + 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.842869	+	1.45989i	−0.376942	+	0.652883i	−0.990616	−	0.136677i	$-0.956358\pi$
$5$	−0.842869	+	1.45989i	−0.376942	+	0.652883i	0.613673	+	0.789560i	$0.289691\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.38216	−	1.95269i	1.01976	−	0.588758i	0.105725	−	0.994395i	$-0.466284\pi$
$11$	3.38216	−	1.95269i	1.01976	−	0.588758i	0.914034	+	0.405637i	$0.132950\pi$
$12$		0			0
$13$	5.24391	+	3.02757i	1.45440	+	0.839698i	0.998727	−	0.0504496i	$-0.0160654\pi$
$13$	5.24391	+	3.02757i	1.45440	+	0.839698i	0.455673	+	0.890147i	$0.349399\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.402488			0.0976176			0.0488088	−	0.998808i	$-0.484458\pi$
$17$	0.402488			0.0976176			0.0488088	+	0.998808i	$0.484458\pi$
$18$		0			0
$19$		−	0.168144i		−	0.0385748i	−0.999814	−	0.0192874i	$-0.993860\pi$
$19$		−	0.168144i		−	0.0385748i	0.999814	−	0.0192874i	$-0.00613975\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.69373	+	4.44198i	1.60425	+	0.926216i	0.990623	+	0.136623i	$0.0436248\pi$
$23$	7.69373	+	4.44198i	1.60425	+	0.926216i	0.613630	+	0.789593i	$0.289709\pi$
$24$		0			0
$25$	1.07914	+	1.86913i	0.215829	+	0.373827i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.15380	−	3.55290i	1.14273	−	0.659757i	0.195627	−	0.980678i	$-0.437326\pi$
$29$	6.15380	−	3.55290i	1.14273	−	0.659757i	0.947106	+	0.320921i	$0.103993\pi$
$30$		0			0
$31$	−5.44527	−	3.14383i	−0.978000	−	0.564649i	−0.0763342	−	0.997082i	$-0.524322\pi$
$31$	−5.44527	−	3.14383i	−0.978000	−	0.564649i	−0.901666	+	0.432434i	$0.857655\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.26515			−1.02998			−0.514992	−	0.857195i	$-0.672205\pi$
$37$	−6.26515			−1.02998			−0.514992	+	0.857195i	$0.672205\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.64707	−	2.85281i	0.257229	−	0.445534i	−0.708269	−	0.705942i	$-0.750524\pi$
$41$	1.64707	−	2.85281i	0.257229	−	0.445534i	0.965499	+	0.260408i	$0.0838571\pi$
$42$		0			0
$43$	1.80474	+	3.12590i	0.275220	+	0.476695i	0.970191	−	0.242343i	$-0.0779161\pi$
$43$	1.80474	+	3.12590i	0.275220	+	0.476695i	−0.694971	+	0.719038i	$0.744583\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.38482	−	7.59474i	−0.639592	−	1.10781i	−0.985522	−	0.169546i	$-0.945770\pi$
$47$	−4.38482	−	7.59474i	−0.639592	−	1.10781i	0.345930	−	0.938260i	$-0.387563\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.71148i		−	0.784532i	−0.919852	−	0.392266i	$-0.871691\pi$
$53$		−	5.71148i		−	0.784532i	0.919852	−	0.392266i	$-0.128309\pi$
$54$		0			0
$55$			6.58345i			0.887712i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.25163	+	3.89994i	−0.293138	+	0.507729i	−0.974550	−	0.224171i	$-0.928033\pi$
$59$	−2.25163	+	3.89994i	−0.293138	+	0.507729i	0.681412	+	0.731900i	$0.261366\pi$
$60$		0			0
$61$	−4.43678	+	2.56157i	−0.568071	+	0.327976i	−0.756379	−	0.654134i	$-0.773033\pi$
$61$	−4.43678	+	2.56157i	−0.568071	+	0.327976i	0.188308	+	0.982110i	$0.439700\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.83986	+	5.10369i	−1.09645	+	0.633035i
$66$		0			0
$67$	2.95521	−	5.11857i	0.361036	−	0.625332i	−0.627096	−	0.778942i	$-0.715757\pi$
$67$	2.95521	−	5.11857i	0.361036	−	0.625332i	0.988132	+	0.153610i	$0.0490899\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.4308i		−	1.35658i	−0.734792	−	0.678292i	$-0.762720\pi$
$71$		−	11.4308i		−	1.35658i	0.734792	−	0.678292i	$-0.237280\pi$
$72$		0			0
$73$			6.99239i			0.818397i	0.912445	+	0.409199i	$0.134192\pi$
$73$			6.99239i			0.818397i	−0.912445	+	0.409199i	$0.865808\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.603968	−	1.04610i	−0.0679517	−	0.117696i	0.830048	−	0.557692i	$-0.188313\pi$
$79$	−0.603968	−	1.04610i	−0.0679517	−	0.117696i	−0.898000	+	0.439996i	$0.854980\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.181350	+	0.314108i	0.0199058	+	0.0344779i	0.875807	−	0.482662i	$-0.160330\pi$
$83$	0.181350	+	0.314108i	0.0199058	+	0.0344779i	−0.855901	+	0.517140i	$0.826997\pi$
$84$		0			0
$85$	−0.339244	+	0.587588i	−0.0367962	+	0.0637329i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.77052			0.293674			0.146837	−	0.989161i	$-0.453091\pi$
$89$	2.77052			0.293674			0.146837	+	0.989161i	$0.453091\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.245471	+	0.141723i	0.0251848	+	0.0145405i
$96$		0			0
$97$	0.508914	−	0.293821i	0.0516723	−	0.0298330i	−0.473941	−	0.880556i	$-0.657169\pi$
$97$	0.508914	−	0.293821i	0.0516723	−	0.0298330i	0.525614	+	0.850723i	$0.323836\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.92329	+	11.9915i	0.688893	+	1.19320i	0.972196	+	0.234167i	$0.0752364\pi$
$101$	6.92329	+	11.9915i	0.688893	+	1.19320i	−0.283303	+	0.959030i	$0.591430\pi$
$102$		0			0
$103$	10.4610	+	6.03967i	1.03075	+	0.595106i	0.917201	−	0.398425i	$-0.130443\pi$
$103$	10.4610	+	6.03967i	1.03075	+	0.595106i	0.113554	+	0.993532i	$0.463777\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	18.3942i		−	1.77824i	−0.457678	−	0.889118i	$-0.651319\pi$
$107$		−	18.3942i		−	1.77824i	0.457678	−	0.889118i	$-0.348681\pi$
$108$		0			0
$109$	11.0207			1.05559			0.527796	−	0.849371i	$-0.323018\pi$
$109$	11.0207			1.05559			0.527796	+	0.849371i	$0.323018\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.36811	+	4.25398i	0.693133	+	0.400181i	0.804785	−	0.593567i	$-0.202281\pi$
$113$	7.36811	+	4.25398i	0.693133	+	0.400181i	−0.111652	+	0.993747i	$0.535614\pi$
$114$		0			0
$115$	−12.9696	+	7.48801i	−1.20942	+	0.698260i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.12600	−	3.68234i	0.193273	−	0.334758i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0670			−1.07930
$126$		0			0
$127$	−10.6312			−0.943365			−0.471682	−	0.881769i	$-0.656353\pi$
$127$	−10.6312			−0.943365			−0.471682	+	0.881769i	$0.656353\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.16740	−	5.48610i	0.276737	−	0.479322i	−0.693835	−	0.720134i	$-0.744080\pi$
$131$	3.16740	−	5.48610i	0.276737	−	0.479322i	0.970572	+	0.240812i	$0.0774136\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.4158	−	8.32296i	1.23162	−	0.711078i	0.264255	−	0.964453i	$-0.414874\pi$
$137$	14.4158	−	8.32296i	1.23162	−	0.711078i	0.967368	+	0.253375i	$0.0815406\pi$
$138$		0			0
$139$	−4.24007	−	2.44800i	−0.359638	−	0.207637i	0.309284	−	0.950970i	$-0.399911\pi$
$139$	−4.24007	−	2.44800i	−0.359638	−	0.207637i	−0.668922	+	0.743333i	$0.733244\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	23.6477			1.97752
$144$		0			0
$145$			11.9785i			0.994761i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.57864	−	2.64348i	−0.375097	−	0.216562i	0.300586	−	0.953755i	$-0.402818\pi$
$149$	−4.57864	−	2.64348i	−0.375097	−	0.216562i	−0.675683	+	0.737192i	$0.736151\pi$
$150$		0			0
$151$	7.29163	+	12.6295i	0.593385	+	1.02777i	0.993773	+	0.111427i	$0.0355421\pi$
$151$	7.29163	+	12.6295i	0.593385	+	1.02777i	−0.400388	+	0.916346i	$0.631125\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.17930	−	5.29967i	0.737299	−	0.425680i
$156$		0			0
$157$	15.4160	+	8.90044i	1.23033	+	0.710332i	0.967099	−	0.254400i	$-0.0818781\pi$
$157$	15.4160	+	8.90044i	1.23033	+	0.710332i	0.263232	+	0.964732i	$0.415211\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.0964456			−0.00755420			−0.00377710	−	0.999993i	$-0.501202\pi$
$163$	−0.0964456			−0.00755420			−0.00377710	+	0.999993i	$0.501202\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.47872	−	4.29327i	0.191809	−	0.332224i	−0.754041	−	0.656828i	$-0.771898\pi$
$167$	2.47872	−	4.29327i	0.191809	−	0.332224i	0.945850	+	0.324604i	$0.105231\pi$
$168$		0			0
$169$	11.8324	+	20.4943i	0.910185	+	1.57649i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.40033	+	12.8177i	0.562637	+	0.974515i	0.997265	+	0.0739055i	$0.0235463\pi$
$173$	7.40033	+	12.8177i	0.562637	+	0.974515i	−0.434629	+	0.900610i	$0.643120\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			0.684450i			0.0511582i	0.999673	+	0.0255791i	$0.00814297\pi$
$179$			0.684450i			0.0511582i	−0.999673	+	0.0255791i	$0.991857\pi$
$180$		0			0
$181$		−	7.84745i		−	0.583297i	−0.956526	−	0.291648i	$-0.905796\pi$
$181$		−	7.84745i		−	0.583297i	0.956526	−	0.291648i	$-0.0942037\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.28070	−	9.14644i	0.388245	−	0.672459i
$186$		0			0
$187$	1.36128	−	0.785934i	0.0995464	−	0.0574732i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.9694	+	9.79729i	−1.22786	+	0.708907i	−0.966582	−	0.256356i	$-0.917478\pi$
$191$	−16.9694	+	9.79729i	−1.22786	+	0.708907i	−0.261281	+	0.965263i	$0.584145\pi$
$192$		0			0
$193$	−9.18116	+	15.9022i	−0.660875	+	1.14467i	0.319512	+	0.947582i	$0.396481\pi$
$193$	−9.18116	+	15.9022i	−0.660875	+	1.14467i	−0.980386	+	0.197086i	$0.936852\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.92313i		−	0.422006i	−0.977485	−	0.211003i	$-0.932327\pi$
$197$		−	5.92313i		−	0.422006i	0.977485	−	0.211003i	$-0.0676730\pi$
$198$		0			0
$199$			15.7348i			1.11541i	0.830039	+	0.557706i	$0.188318\pi$
$199$			15.7348i			1.11541i	−0.830039	+	0.557706i	$0.811682\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.77653	+	4.80909i	0.193921	+	0.335881i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.328332	−	0.568688i	−0.0227112	−	0.0393370i
$210$		0			0
$211$	5.06619	−	8.77489i	0.348771	−	0.604088i	−0.637261	−	0.770648i	$-0.719933\pi$
$211$	5.06619	−	8.77489i	0.348771	−	0.604088i	0.986031	+	0.166560i	$0.0532659\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.08462			−0.414968
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.11061	+	1.21856i	0.141975	+	0.0819693i
$222$		0			0
$223$	−13.3944	+	7.73325i	−0.896955	+	0.517857i	−0.876211	−	0.481928i	$-0.839937\pi$
$223$	−13.3944	+	7.73325i	−0.896955	+	0.517857i	−0.0207437	+	0.999785i	$0.506603\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.0360	+	24.3110i	0.931600	+	1.61358i	0.780588	+	0.625046i	$0.214920\pi$
$227$	14.0360	+	24.3110i	0.931600	+	1.61358i	0.151011	+	0.988532i	$0.451747\pi$
$228$		0			0
$229$	−14.7453	−	8.51319i	−0.974396	−	0.562568i	−0.0738222	−	0.997271i	$-0.523520\pi$
$229$	−14.7453	−	8.51319i	−0.974396	−	0.562568i	−0.900573	+	0.434704i	$0.856853\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			18.4769i			1.21047i	0.796049	+	0.605233i	$0.206920\pi$
$233$			18.4769i			1.21047i	−0.796049	+	0.605233i	$0.793080\pi$
$234$		0			0
$235$	14.7833			0.964358
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.06656	+	3.50253i	0.392413	+	0.226560i	0.683205	−	0.730226i	$-0.260585\pi$
$239$	6.06656	+	3.50253i	0.392413	+	0.226560i	−0.290792	+	0.956786i	$0.593919\pi$
$240$		0			0
$241$	5.38459	−	3.10879i	0.346852	−	0.200255i	−0.316446	−	0.948611i	$-0.602490\pi$
$241$	5.38459	−	3.10879i	0.346852	−	0.200255i	0.663298	+	0.748355i	$0.269156\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.509067	−	0.881730i	0.0323912	−	0.0561031i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.81844			0.619734			0.309867	−	0.950780i	$-0.399715\pi$
$251$	9.81844			0.619734			0.309867	+	0.950780i	$0.399715\pi$
$252$		0			0
$253$	34.6952			2.18127
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.667904	+	1.15684i	−0.0416627	+	0.0721619i	−0.886105	−	0.463485i	$-0.846599\pi$
$257$	−0.667904	+	1.15684i	−0.0416627	+	0.0721619i	0.844442	+	0.535647i	$0.179932\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	17.6238	−	10.1751i	1.08673	−	0.627424i	0.154026	−	0.988067i	$-0.450776\pi$
$263$	17.6238	−	10.1751i	1.08673	−	0.627424i	0.932704	+	0.360643i	$0.117443\pi$
$264$		0			0
$265$	8.33814	+	4.81402i	0.512208	+	0.295723i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−26.7228			−1.62932			−0.814659	−	0.579940i	$-0.803076\pi$
$269$	−26.7228			−1.62932			−0.814659	+	0.579940i	$0.803076\pi$
$270$		0			0
$271$			4.34764i			0.264100i	0.991243	+	0.132050i	$0.0421560\pi$
$271$			4.34764i			0.264100i	−0.991243	+	0.132050i	$0.957844\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.29968	+	4.21447i	0.440187	+	0.254142i
$276$		0			0
$277$	2.19901	+	3.80880i	0.132126	+	0.228849i	0.924496	−	0.381192i	$-0.124486\pi$
$277$	2.19901	+	3.80880i	0.132126	+	0.228849i	−0.792370	+	0.610041i	$0.791153\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.62273	+	2.66893i	−0.275769	+	0.159215i	−0.631506	−	0.775371i	$-0.717563\pi$
$281$	−4.62273	+	2.66893i	−0.275769	+	0.159215i	0.355738	+	0.934586i	$0.384230\pi$
$282$		0			0
$283$	15.5431	+	8.97381i	0.923941	+	0.533437i	0.884890	−	0.465800i	$-0.154233\pi$
$283$	15.5431	+	8.97381i	0.923941	+	0.533437i	0.0390505	+	0.999237i	$0.487567\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.8380			−0.990471
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.1126	+	22.7117i	−0.766048	+	1.32683i	0.173642	+	0.984809i	$0.444446\pi$
$293$	−13.1126	+	22.7117i	−0.766048	+	1.32683i	−0.939691	+	0.342026i	$0.888887\pi$
$294$		0			0
$295$	−3.79566	−	6.57428i	−0.220992	−	0.382769i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	26.8968	+	46.5867i	1.55548	+	2.69418i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	8.63628i		−	0.494512i
$306$		0			0
$307$		−	7.19520i		−	0.410652i	−0.978694	−	0.205326i	$-0.934175\pi$
$307$		−	7.19520i		−	0.410652i	0.978694	−	0.205326i	$-0.0658254\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.08721	+	1.88311i	−0.0616503	+	0.106781i	−0.895203	−	0.445658i	$-0.852970\pi$
$311$	−1.08721	+	1.88311i	−0.0616503	+	0.106781i	0.833553	+	0.552440i	$0.186303\pi$
$312$		0			0
$313$	10.2870	−	5.93922i	0.581457	−	0.335704i	−0.180255	−	0.983620i	$-0.557692\pi$
$313$	10.2870	−	5.93922i	0.581457	−	0.335704i	0.761712	+	0.647916i	$0.224359\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.09969	−	4.09901i	0.398758	−	0.230223i	−0.287190	−	0.957874i	$-0.592721\pi$
$317$	7.09969	−	4.09901i	0.398758	−	0.230223i	0.685948	+	0.727651i	$0.259388\pi$
$318$		0			0
$319$	13.8754	−	24.0329i	0.776875	−	1.34559i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	0.0676757i		−	0.00376558i
$324$		0			0
$325$			13.0688i			0.724924i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.58540	−	14.8704i	−0.471897	−	0.817349i	0.527586	−	0.849501i	$-0.323097\pi$
$331$	−8.58540	−	14.8704i	−0.471897	−	0.817349i	−0.999483	+	0.0321526i	$0.989764\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.98170	+	8.62856i	0.272179	+	0.471428i
$336$		0			0
$337$	3.95399	−	6.84850i	0.215387	−	0.373062i	−0.738005	−	0.674795i	$-0.764232\pi$
$337$	3.95399	−	6.84850i	0.215387	−	0.373062i	0.953392	+	0.301733i	$0.0975653\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.5557			−1.32977
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.443850	−	0.256257i	−0.0238271	−	0.0137566i	0.488039	−	0.872822i	$-0.337712\pi$
$347$	−0.443850	−	0.256257i	−0.0238271	−	0.0137566i	−0.511866	+	0.859065i	$0.671046\pi$
$348$		0			0
$349$	−5.74612	+	3.31752i	−0.307583	+	0.177583i	−0.645844	−	0.763469i	$-0.723494\pi$
$349$	−5.74612	+	3.31752i	−0.307583	+	0.177583i	0.338262	+	0.941052i	$0.390161\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.03437	+	15.6480i	0.480851	+	0.832858i	0.999759	−	0.0219721i	$-0.00699449\pi$
$353$	9.03437	+	15.6480i	0.480851	+	0.832858i	−0.518908	+	0.854830i	$0.673661\pi$
$354$		0			0
$355$	16.6877	+	9.63465i	0.885691	+	0.511354i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.76296i			0.0930453i	0.998917	+	0.0465227i	$0.0148140\pi$
$359$			1.76296i			0.0930453i	−0.998917	+	0.0465227i	$0.985186\pi$
$360$		0			0
$361$	18.9717			0.998512
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.2081	−	5.89367i	−0.534318	−	0.308489i
$366$		0			0
$367$	−28.9614	+	16.7209i	−1.51177	+	0.872822i	−0.511867	+	0.859065i	$0.671046\pi$
$367$	−28.9614	+	16.7209i	−1.51177	+	0.872822i	−0.999905	+	0.0137576i	$0.995621\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−12.7844	+	22.1433i	−0.661952	+	1.14653i	0.318150	+	0.948040i	$0.396938\pi$
$373$	−12.7844	+	22.1433i	−0.661952	+	1.14653i	−0.980102	+	0.198494i	$0.936395\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	43.0267			2.21599
$378$		0			0
$379$	25.7920			1.32485			0.662423	−	0.749130i	$-0.269528\pi$
$379$	25.7920			1.32485			0.662423	+	0.749130i	$0.269528\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.4158	−	28.4330i	0.838808	−	1.45286i	−0.0520838	−	0.998643i	$-0.516586\pi$
$383$	16.4158	−	28.4330i	0.838808	−	1.45286i	0.890892	−	0.454215i	$-0.150080\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.4542	+	10.0772i	−0.884965	+	0.510935i	−0.872292	−	0.488985i	$-0.837367\pi$
$389$	−17.4542	+	10.0772i	−0.884965	+	0.510935i	−0.0126730	+	0.999920i	$0.504034\pi$
$390$		0			0
$391$	3.09663	+	1.78784i	0.156603	+	0.0904150i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.03626			0.102456
$396$		0			0
$397$		−	34.8864i		−	1.75090i	−0.483311	−	0.875449i	$-0.660566\pi$
$397$		−	34.8864i		−	1.75090i	0.483311	−	0.875449i	$-0.339434\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	8.36793	+	4.83122i	0.417874	+	0.241260i	0.694167	−	0.719814i	$-0.255773\pi$
$401$	8.36793	+	4.83122i	0.417874	+	0.241260i	−0.276293	+	0.961073i	$0.589106\pi$
$402$		0			0
$403$	−19.0364	−	32.9719i	−0.948268	−	1.64245i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.1897	+	12.2339i	−1.05034	+	0.606412i
$408$		0			0
$409$	−32.1202	−	18.5446i	−1.58824	−	0.916973i	−0.993597	−	0.112986i	$-0.963958\pi$
$409$	−32.1202	−	18.5446i	−1.58824	−	0.916973i	−0.594647	−	0.803987i	$-0.702708\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.611419			−0.0300134
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.84193	−	3.19031i	0.0899841	−	0.155857i	−0.817520	−	0.575900i	$-0.804652\pi$
$419$	1.84193	−	3.19031i	0.0899841	−	0.155857i	0.907504	+	0.420043i	$0.137985\pi$
$420$		0			0
$421$	−8.55139	−	14.8114i	−0.416769	−	0.721866i	0.578843	−	0.815439i	$-0.303504\pi$
$421$	−8.55139	−	14.8114i	−0.416769	−	0.721866i	−0.995612	+	0.0935732i	$0.970171\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.434342	+	0.752303i	0.0210687	+	0.0364921i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	31.5512i		−	1.51977i	−0.650058	−	0.759885i	$-0.725255\pi$
$431$		−	31.5512i		−	1.51977i	0.650058	−	0.759885i	$-0.274745\pi$
$432$		0			0
$433$		−	10.0692i		−	0.483893i	−0.970290	−	0.241947i	$-0.922214\pi$
$433$		−	10.0692i		−	0.483893i	0.970290	−	0.241947i	$-0.0777859\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.746890	−	1.29365i	0.0357286	−	0.0618837i
$438$		0			0
$439$	−24.1966	+	13.9699i	−1.15484	+	0.666748i	−0.950062	−	0.312060i	$-0.898981\pi$
$439$	−24.1966	+	13.9699i	−1.15484	+	0.666748i	−0.204779	+	0.978808i	$0.565648\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−30.1930	+	17.4319i	−1.43451	+	0.828215i	−0.997460	−	0.0712223i	$-0.977310\pi$
$443$	−30.1930	+	17.4319i	−1.43451	+	0.828215i	−0.437050	+	0.899437i	$0.643977\pi$
$444$		0			0
$445$	−2.33518	+	4.04466i	−0.110698	+	0.191735i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	23.2411i		−	1.09682i	−0.836211	−	0.548408i	$-0.815234\pi$
$449$		−	23.2411i		−	1.09682i	0.836211	−	0.548408i	$-0.184766\pi$
$450$		0			0
$451$		−	12.8649i		−	0.605783i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.10938	+	5.38560i	0.145451	+	0.251928i	0.929541	−	0.368719i	$-0.120203\pi$
$457$	3.10938	+	5.38560i	0.145451	+	0.251928i	−0.784090	+	0.620647i	$0.786870\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.17165	−	3.76140i	−0.101144	−	0.175186i	0.811012	−	0.585029i	$-0.198917\pi$
$461$	−2.17165	−	3.76140i	−0.101144	−	0.175186i	−0.912156	+	0.409843i	$0.865584\pi$
$462$		0			0
$463$	3.57451	−	6.19124i	0.166122	−	0.287731i	−0.770931	−	0.636918i	$-0.780209\pi$
$463$	3.57451	−	6.19124i	0.166122	−	0.287731i	0.937053	+	0.349187i	$0.113542\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.88890			−0.0874080			−0.0437040	−	0.999045i	$-0.513916\pi$
$467$	−1.88890			−0.0874080			−0.0437040	+	0.999045i	$0.513916\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.2078	+	7.04818i	0.561316	+	0.324076i
$474$		0			0
$475$	0.314283	−	0.181451i	0.0144203	−	0.00832555i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.22491	−	9.04981i	−0.238732	−	0.413497i	0.721618	−	0.692291i	$-0.243399\pi$
$479$	−5.22491	−	9.04981i	−0.238732	−	0.413497i	−0.960351	+	0.278794i	$0.910065\pi$
$480$		0			0
$481$	−32.8539	−	18.9682i	−1.49801	−	0.864875i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			0.990611i			0.0449813i
$486$		0			0
$487$	23.6596			1.07212			0.536060	−	0.844180i	$-0.319912\pi$
$487$	23.6596			1.07212			0.536060	+	0.844180i	$0.319912\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.6767	+	6.74152i	0.526960	+	0.304241i	0.739778	−	0.672851i	$-0.234931\pi$
$491$	11.6767	+	6.74152i	0.526960	+	0.304241i	−0.212817	+	0.977092i	$0.568264\pi$
$492$		0			0
$493$	2.47683	−	1.43000i	0.111551	−	0.0644039i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.04035	−	10.4622i	0.270403	−	0.468352i	−0.698562	−	0.715550i	$-0.746176\pi$
$499$	6.04035	−	10.4622i	0.270403	−	0.468352i	0.968965	+	0.247197i	$0.0795096\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20.5283			0.915310			0.457655	−	0.889130i	$-0.348690\pi$
$503$	20.5283			0.915310			0.457655	+	0.889130i	$0.348690\pi$
$504$		0			0
$505$	−23.3417			−1.03869
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.09043	−	7.08483i	0.181305	−	0.314029i	−0.761020	−	0.648728i	$-0.775301\pi$
$509$	4.09043	−	7.08483i	0.181305	−	0.314029i	0.942325	+	0.334699i	$0.108635\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−17.6345	+	10.1813i	−0.777070	+	0.448642i
$516$		0			0
$517$	−29.6603	−	17.1244i	−1.30446	−	0.753131i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.7491			−1.21571			−0.607856	−	0.794048i	$-0.707970\pi$
$521$	−27.7491			−1.21571			−0.607856	+	0.794048i	$0.707970\pi$
$522$		0			0
$523$			22.9604i			1.00399i	0.864872	+	0.501993i	$0.167400\pi$
$523$			22.9604i			1.00399i	−0.864872	+	0.501993i	$0.832600\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.19166	−	1.26535i	−0.0954700	−	0.0551196i
$528$		0			0
$529$	27.9623	+	48.4322i	1.21575	+	2.10575i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17.2742	−	9.97325i	0.748228	−	0.431990i
$534$		0			0
$535$	26.8536	+	15.5039i	1.16098	+	0.670292i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5.20810			0.223914			0.111957	−	0.993713i	$-0.464288\pi$
$541$	5.20810			0.223914			0.111957	+	0.993713i	$0.464288\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.28902	+	16.0890i	−0.397898	+	0.689179i
$546$		0			0
$547$	10.6224	+	18.3985i	0.454181	+	0.786664i	0.998641	−	0.0521229i	$-0.0165988\pi$
$547$	10.6224	+	18.3985i	0.454181	+	0.786664i	−0.544460	+	0.838787i	$0.683265\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.597397	−	1.03472i	−0.0254500	−	0.0440807i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	12.8109i		−	0.542813i	−0.962465	−	0.271407i	$-0.912511\pi$
$557$		−	12.8109i		−	0.542813i	0.962465	−	0.271407i	$-0.0874888\pi$
$558$		0			0
$559$			21.8559i			0.924406i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.7396	−	32.4580i	0.789781	−	1.36794i	−0.136319	−	0.990665i	$-0.543527\pi$
$563$	18.7396	−	32.4580i	0.789781	−	1.36794i	0.926101	−	0.377277i	$-0.123139\pi$
$564$		0			0
$565$	−12.4207	+	7.17109i	−0.522542	+	0.301690i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.94906	−	3.43469i	0.249397	−	0.143990i	−0.370091	−	0.928996i	$-0.620673\pi$
$569$	5.94906	−	3.43469i	0.249397	−	0.143990i	0.619488	+	0.785006i	$0.287340\pi$
$570$		0			0
$571$	−0.0847909	+	0.146862i	−0.00354839	+	0.00614599i	−0.867794	−	0.496924i	$-0.834463\pi$
$571$	−0.0847909	+	0.146862i	−0.00354839	+	0.00614599i	0.864246	+	0.503070i	$0.167796\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			19.1741i			0.799617i
$576$		0			0
$577$			6.24916i			0.260156i	0.991504	+	0.130078i	$0.0415228\pi$
$577$			6.24916i			0.260156i	−0.991504	+	0.130078i	$0.958477\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−11.1527	−	19.3171i	−0.461900	−	0.800033i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.7881	−	18.6855i	−0.445273	−	0.771235i	0.552799	−	0.833315i	$-0.313560\pi$
$587$	−10.7881	−	18.6855i	−0.445273	−	0.771235i	−0.998071	+	0.0620801i	$0.980227\pi$
$588$		0			0
$589$	−0.528615	+	0.915588i	−0.0217812	+	0.0377261i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.26071			0.339227			0.169613	−	0.985511i	$-0.445748\pi$
$593$	8.26071			0.339227			0.169613	+	0.985511i	$0.445748\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	30.7618	+	17.7603i	1.25689	+	0.725667i	0.972469	−	0.233033i	$-0.0748649\pi$
$599$	30.7618	+	17.7603i	1.25689	+	0.725667i	0.284422	+	0.958699i	$0.408198\pi$
$600$		0			0
$601$	35.8981	−	20.7258i	1.46432	−	0.845423i	0.465109	−	0.885254i	$-0.346015\pi$
$601$	35.8981	−	20.7258i	1.46432	−	0.845423i	0.999206	+	0.0398308i	$0.0126819\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.58388	+	6.20746i	0.145705	+	0.252369i
$606$		0			0
$607$	−2.09569	−	1.20995i	−0.0850616	−	0.0491103i	0.456866	−	0.889536i	$-0.348972\pi$
$607$	−2.09569	−	1.20995i	−0.0850616	−	0.0491103i	−0.541927	+	0.840425i	$0.682305\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	53.1015i		−	2.14826i
$612$		0			0
$613$	−42.6456			−1.72244			−0.861219	−	0.508234i	$-0.830298\pi$
$613$	−42.6456			−1.72244			−0.861219	+	0.508234i	$0.830298\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.2535	−	7.65193i	−0.533567	−	0.308055i	0.208901	−	0.977937i	$-0.433011\pi$
$617$	−13.2535	−	7.65193i	−0.533567	−	0.308055i	−0.742468	+	0.669882i	$0.766345\pi$
$618$		0			0
$619$	23.9177	−	13.8089i	0.961334	−	0.555026i	0.0647505	−	0.997901i	$-0.479375\pi$
$619$	23.9177	−	13.8089i	0.961334	−	0.555026i	0.896583	+	0.442875i	$0.146042\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.77517	−	8.27084i	0.191007	−	0.330834i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.52164			−0.100545
$630$		0			0
$631$	8.28775			0.329930			0.164965	−	0.986299i	$-0.447249\pi$
$631$	8.28775			0.329930			0.164965	+	0.986299i	$0.447249\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.96069	−	15.5204i	0.355594	−	0.615907i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.58307	+	4.95544i	−0.339011	+	0.195728i	−0.659835	−	0.751411i	$-0.729374\pi$
$641$	−8.58307	+	4.95544i	−0.339011	+	0.195728i	0.320824	+	0.947139i	$0.396040\pi$
$642$		0			0
$643$	6.83668	+	3.94716i	0.269612	+	0.155661i	0.628711	−	0.777639i	$-0.283583\pi$
$643$	6.83668	+	3.94716i	0.269612	+	0.155661i	−0.359099	+	0.933299i	$0.616916\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.31931			0.169810			0.0849049	−	0.996389i	$-0.472941\pi$
$647$	4.31931			0.169810			0.0849049	+	0.996389i	$0.472941\pi$
$648$		0			0
$649$			17.5870i			0.690349i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	37.5853	+	21.6999i	1.47082	+	0.849181i	0.999463	−	0.0327591i	$-0.0104294\pi$
$653$	37.5853	+	21.6999i	1.47082	+	0.849181i	0.471361	+	0.881940i	$0.343763\pi$
$654$		0			0
$655$	5.33940	+	9.24812i	0.208628	+	0.361354i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.34894	−	5.39761i	0.364183	−	0.210261i	−0.306731	−	0.951796i	$-0.599235\pi$
$659$	9.34894	−	5.39761i	0.364183	−	0.210261i	0.670914	+	0.741535i	$0.265902\pi$
$660$		0			0
$661$	−3.39495	−	1.96008i	−0.132048	−	0.0762381i	0.432521	−	0.901624i	$-0.357624\pi$
$661$	−3.39495	−	1.96008i	−0.132048	−	0.0762381i	−0.564569	+	0.825386i	$0.690958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	63.1276			2.44431
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.0039	+	17.3273i	−0.386197	+	0.668913i
$672$		0			0
$673$	−12.3404	−	21.3742i	−0.475687	−	0.823915i	0.523925	−	0.851765i	$-0.324467\pi$
$673$	−12.3404	−	21.3742i	−0.475687	−	0.823915i	−0.999612	+	0.0278497i	$0.991134\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.36327	−	12.7536i	−0.282994	−	0.490159i	0.689127	−	0.724641i	$-0.257994\pi$
$677$	−7.36327	−	12.7536i	−0.282994	−	0.490159i	−0.972121	+	0.234481i	$0.924661\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.84900i		−	0.0707499i	−0.999374	−	0.0353750i	$-0.988737\pi$
$683$		−	1.84900i		−	0.0707499i	0.999374	−	0.0353750i	$-0.0112625\pi$
$684$		0			0
$685$			28.0606i			1.07214i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17.2919	−	29.9505i	0.658769	−	1.14102i
$690$		0			0
$691$	33.7613	−	19.4921i	1.28434	−	0.741514i	0.306701	−	0.951806i	$-0.400775\pi$
$691$	33.7613	−	19.4921i	1.28434	−	0.741514i	0.977639	+	0.210292i	$0.0674415\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.14764	−	4.12669i	0.271126	−	0.156534i
$696$		0			0
$697$	0.662926	−	1.14822i	0.0251101	−	0.0434920i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			25.4389i			0.960813i	0.877046	+	0.480406i	$0.159511\pi$
$701$			25.4389i			0.960813i	−0.877046	+	0.480406i	$0.840489\pi$
$702$		0			0
$703$			1.05344i			0.0397314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.14517	+	12.3758i	0.268342	+	0.464783i	0.968434	−	0.249270i	$-0.0801908\pi$
$709$	7.14517	+	12.3758i	0.268342	+	0.464783i	−0.700092	+	0.714053i	$0.746857\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−27.9296	−	48.3756i	−1.04597	−	1.81168i
$714$		0			0
$715$	−19.9319	+	34.5230i	−0.745410	+	1.29109i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33.4688			−1.24818			−0.624088	−	0.781354i	$-0.714529\pi$
$719$	−33.4688			−1.24818			−0.624088	+	0.781354i	$0.714529\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.2817	+	7.66819i	0.493269	+	0.284789i
$726$		0			0
$727$	12.1354	−	7.00636i	0.450076	−	0.259851i	−0.257786	−	0.966202i	$-0.582993\pi$
$727$	12.1354	−	7.00636i	0.450076	−	0.259851i	0.707862	+	0.706350i	$0.249660\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.726384	+	1.25813i	0.0268663	+	0.0465338i
$732$		0			0
$733$	23.6491	+	13.6538i	0.873501	+	0.504316i	0.868510	−	0.495672i	$-0.165078\pi$
$733$	23.6491	+	13.6538i	0.873501	+	0.504316i	0.00499085	+	0.999988i	$0.498411\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	23.0824i		−	0.850251i
$738$		0			0
$739$	−52.6314			−1.93608			−0.968039	−	0.250801i	$-0.919306\pi$
$739$	−52.6314			−1.93608			−0.968039	+	0.250801i	$0.919306\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	30.9523	+	17.8703i	1.13553	+	0.655599i	0.945320	−	0.326144i	$-0.105750\pi$
$743$	30.9523	+	17.8703i	1.13553	+	0.655599i	0.190211	+	0.981743i	$0.439083\pi$
$744$		0			0
$745$	7.71839	−	4.45621i	0.282780	−	0.163263i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	16.5641	−	28.6899i	0.604433	−	1.04691i	−0.387708	−	0.921782i	$-0.626733\pi$
$751$	16.5641	−	28.6899i	0.604433	−	1.04691i	0.992141	−	0.125126i	$-0.0399336\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.5836			−0.894687
$756$		0			0
$757$	−13.6903			−0.497584			−0.248792	−	0.968557i	$-0.580034\pi$
$757$	−13.6903			−0.497584			−0.248792	+	0.968557i	$0.580034\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.51737	+	11.2884i	−0.236255	+	0.409205i	−0.959637	−	0.281243i	$-0.909253\pi$
$761$	−6.51737	+	11.2884i	−0.236255	+	0.409205i	0.723382	+	0.690448i	$0.242587\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−23.6147	+	13.6340i	−0.852678	+	0.492294i
$768$		0			0
$769$	−18.4866	−	10.6732i	−0.666642	−	0.384886i	0.128161	−	0.991753i	$-0.459093\pi$
$769$	−18.4866	−	10.6732i	−0.666642	−	0.384886i	−0.794803	+	0.606867i	$0.792426\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11.4788			−0.412864			−0.206432	−	0.978461i	$-0.566185\pi$
$773$	−11.4788			−0.412864			−0.206432	+	0.978461i	$0.566185\pi$
$774$		0			0
$775$		−	13.5706i		−	0.487470i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.479682	−	0.276944i	−0.0171864	−	0.00992256i
$780$		0			0
$781$	−22.3208	−	38.6607i	−0.798701	−	1.38339i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−25.9873	+	15.0038i	−0.927528	+	0.535509i
$786$		0			0
$787$	−35.6808	−	20.6003i	−1.27188	−	0.734322i	−0.296541	−	0.955020i	$-0.595833\pi$
$787$	−35.6808	−	20.6003i	−1.27188	−	0.734322i	−0.975342	+	0.220698i	$0.929166\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−31.0214			−1.10160
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.0066	+	43.3127i	−0.885779	+	1.53421i	−0.0409600	+	0.999161i	$0.513042\pi$
$797$	−25.0066	+	43.3127i	−0.885779	+	1.53421i	−0.844819	+	0.535053i	$0.820292\pi$
$798$		0			0
$799$	−1.76484	−	3.05679i	−0.0624355	−	0.108141i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.6540	+	23.6494i	0.481838	+	0.834568i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			50.6908i			1.78219i	0.453812	+	0.891097i	$0.350064\pi$
$809$			50.6908i			1.78219i	−0.453812	+	0.891097i	$0.649936\pi$
$810$		0			0
$811$		−	8.96566i		−	0.314827i	−0.987533	−	0.157413i	$-0.949684\pi$
$811$		−	8.96566i		−	0.314827i	0.987533	−	0.157413i	$-0.0503155\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.0812910	−	0.140800i	0.00284750	−	0.00493201i
$816$		0			0
$817$	0.525599	−	0.303455i	0.0183884	−	0.0106165i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.2190	+	16.2922i	−0.984849	+	0.568603i	−0.903731	−	0.428102i	$-0.859183\pi$
$821$	−28.2190	+	16.2922i	−0.984849	+	0.568603i	−0.0811184	+	0.996704i	$0.525849\pi$
$822$		0			0
$823$	10.0877	−	17.4724i	0.351636	−	0.609051i	−0.634901	−	0.772594i	$-0.718959\pi$
$823$	10.0877	−	17.4724i	0.351636	−	0.609051i	0.986536	+	0.163543i	$0.0522923\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			0.253288i			0.00880770i	0.999990	+	0.00440385i	$0.00140179\pi$
$827$			0.253288i			0.00880770i	−0.999990	+	0.00440385i	$0.998598\pi$
$828$		0			0
$829$			7.05416i			0.245001i	0.992468	+	0.122501i	$0.0390913\pi$
$829$			7.05416i			0.245001i	−0.992468	+	0.122501i	$0.960909\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	4.17848	+	7.23733i	0.144602	+	0.250458i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.0936	+	29.6069i	0.590136	+	1.02215i	0.994214	+	0.107420i	$0.0342591\pi$
$839$	17.0936	+	29.6069i	0.590136	+	1.02215i	−0.404078	+	0.914725i	$0.632408\pi$
$840$		0			0
$841$	10.7462	−	18.6130i	0.370558	−	0.641826i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−39.8926			−1.37235
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−48.2024	−	27.8296i	−1.65236	−	0.953988i
$852$		0			0
$853$	−21.7586	+	12.5623i	−0.745000	+	0.430126i	−0.823884	−	0.566758i	$-0.808198\pi$
$853$	−21.7586	+	12.5623i	−0.745000	+	0.430126i	0.0788844	+	0.996884i	$0.474864\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.0954	+	36.5383i	0.720604	+	1.24812i	0.960758	+	0.277388i	$0.0894688\pi$
$857$	21.0954	+	36.5383i	0.720604	+	1.24812i	−0.240154	+	0.970735i	$0.577198\pi$
$858$		0			0
$859$	4.08139	+	2.35639i	0.139255	+	0.0803990i	0.568009	−	0.823022i	$-0.307714\pi$
$859$	4.08139	+	2.35639i	0.139255	+	0.0803990i	−0.428754	+	0.903421i	$0.641047\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	35.6120i		−	1.21225i	−0.795371	−	0.606123i	$-0.792724\pi$
$863$		−	35.6120i		−	1.21225i	0.795371	−	0.606123i	$-0.207276\pi$
$864$		0			0
$865$	−24.9500			−0.848326
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.08543	−	2.35873i	−0.138589	−	0.0800143i
$870$		0			0
$871$	30.9937	−	17.8942i	1.05018	−	0.606322i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−20.4532	+	35.4260i	−0.690655	+	1.19625i	0.280969	+	0.959717i	$0.409344\pi$
$877$	−20.4532	+	35.4260i	−0.690655	+	1.19625i	−0.971624	+	0.236532i	$0.923989\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37.4443			−1.26153			−0.630765	−	0.775974i	$-0.717259\pi$
$881$	−37.4443			−1.26153			−0.630765	+	0.775974i	$0.717259\pi$
$882$		0			0
$883$	−49.8357			−1.67711			−0.838553	−	0.544821i	$-0.816598\pi$
$883$	−49.8357			−1.67711			−0.838553	+	0.544821i	$0.816598\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.4482	−	25.0251i	0.485124	−	0.840260i	−0.514730	−	0.857352i	$-0.672108\pi$
$887$	14.4482	−	25.0251i	0.485124	−	0.840260i	0.999854	+	0.0170929i	$0.00544110\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.27701	+	0.737280i	−0.0427334	+	0.0246721i
$894$		0			0
$895$	−0.999223	−	0.576902i	−0.0334003	−	0.0192837i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−44.6789			−1.49012
$900$		0			0
$901$		−	2.29880i		−	0.0765841i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	11.4564	+	6.61437i	0.380825	+	0.219869i
$906$		0			0
$907$	7.43498	+	12.8778i	0.246874	+	0.427599i	0.962657	−	0.270724i	$-0.0872632\pi$
$907$	7.43498	+	12.8778i	0.246874	+	0.427599i	−0.715783	+	0.698323i	$0.753930\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.81616	+	4.51266i	−0.258961	+	0.149511i	−0.623861	−	0.781536i	$-0.714437\pi$
$911$	−7.81616	+	4.51266i	−0.258961	+	0.149511i	0.364899	+	0.931047i	$0.381103\pi$
$912$		0			0
$913$	1.22671	+	0.708243i	0.0405982	+	0.0234394i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−26.4166			−0.871403			−0.435702	−	0.900091i	$-0.643500\pi$
$919$	−26.4166			−0.871403			−0.435702	+	0.900091i	$0.643500\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	34.6075	−	59.9420i	1.13912	−	1.97302i
$924$		0			0
$925$	−6.76100	−	11.7104i	−0.222300	−	0.385036i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.1259	−	19.2706i	−0.365029	−	0.632249i	0.623752	−	0.781623i	$-0.285608\pi$
$929$	−11.1259	−	19.2706i	−0.365029	−	0.632249i	−0.988781	+	0.149373i	$0.952274\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2.64976i			0.0866563i
$936$		0			0
$937$			14.6822i			0.479647i	0.970817	+	0.239823i	$0.0770896\pi$
$937$			14.6822i			0.479647i	−0.970817	+	0.239823i	$0.922910\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.0396	+	39.9058i	−0.751070	+	1.30089i	0.196235	+	0.980557i	$0.437129\pi$
$941$	−23.0396	+	39.9058i	−0.751070	+	1.30089i	−0.947305	+	0.320334i	$0.896205\pi$
$942$		0			0
$943$	25.3442	−	14.6325i	0.825322	−	0.476500i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.96116	−	4.01903i	0.226207	−	0.130601i	−0.382614	−	0.923908i	$-0.624976\pi$
$947$	6.96116	−	4.01903i	0.226207	−	0.130601i	0.608821	+	0.793308i	$0.291643\pi$
$948$		0			0
$949$	−21.1700	+	36.6675i	−0.687206	+	1.19028i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	54.9348i		−	1.77951i	−0.456437	−	0.889756i	$-0.650875\pi$
$953$		−	54.9348i		−	1.77951i	0.456437	−	0.889756i	$-0.349125\pi$
$954$		0			0
$955$		−	33.0313i		−	1.06887i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	4.26733	+	7.39124i	0.137656	+	0.238427i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−15.4770	−	26.8070i	−0.498223	−	0.862948i
$966$		0			0
$967$	−26.5917	+	46.0582i	−0.855132	+	1.48113i	0.0213900	+	0.999771i	$0.493191\pi$
$967$	−26.5917	+	46.0582i	−0.855132	+	1.48113i	−0.876522	+	0.481361i	$0.840143\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15.2281			0.488692			0.244346	−	0.969688i	$-0.421427\pi$
$971$	15.2281			0.488692			0.244346	+	0.969688i	$0.421427\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.49418	+	0.862667i	0.0478031	+	0.0275992i	0.523711	−	0.851896i	$-0.324547\pi$
$977$	1.49418	+	0.862667i	0.0478031	+	0.0275992i	−0.475908	+	0.879495i	$0.657880\pi$
$978$		0			0
$979$	9.37033	−	5.40997i	0.299477	−	0.172903i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−30.1191	−	52.1679i	−0.960651	−	1.66390i	−0.720871	−	0.693070i	$-0.756258\pi$
$983$	−30.1191	−	52.1679i	−0.960651	−	1.66390i	−0.239780	−	0.970827i	$-0.577075\pi$
$984$		0			0
$985$	8.64713	+	4.99242i	0.275521	+	0.159072i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			32.0664i			1.01965i
$990$		0			0
$991$	5.74624			0.182535			0.0912676	−	0.995826i	$-0.470908\pi$
$991$	5.74624			0.182535			0.0912676	+	0.995826i	$0.470908\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.9711	−	13.2624i	−0.728234	−	0.420446i
$996$		0			0
$997$	0.0224508	−	0.0129620i	0.000711024	−	0.000410510i	−0.499644	−	0.866231i	$-0.666536\pi$
$997$	0.0224508	−	0.0129620i	0.000711024	−	0.000410510i	0.500355	+	0.865820i	$0.333203\pi$
$998$		0			0
$999$		0			0

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.x.b.881.3		16
3.2	odd	2		1764.2.x.b.293.1		16
7.2	even	3		756.2.bm.a.17.6		16
7.3	odd	6		756.2.w.a.341.6		16
7.4	even	3		5292.2.w.b.1097.3		16
7.5	odd	6		5292.2.bm.a.2285.3		16
7.6	odd	2		5292.2.x.a.881.6		16
9.2	odd	6		5292.2.x.a.4409.6		16
9.7	even	3		1764.2.x.a.1469.8		16
21.2	odd	6		252.2.bm.a.185.7	yes	16
21.5	even	6		1764.2.bm.a.1697.2		16
21.11	odd	6		1764.2.w.b.509.5		16
21.17	even	6		252.2.w.a.5.4	✓	16
21.20	even	2		1764.2.x.a.293.8		16
28.3	even	6		3024.2.ca.d.2609.6		16
28.23	odd	6		3024.2.df.d.17.6		16
63.2	odd	6		756.2.w.a.521.6		16
63.11	odd	6		5292.2.bm.a.4625.3		16
63.16	even	3		252.2.w.a.101.4	yes	16
63.20	even	6	inner	5292.2.x.b.4409.3		16
63.23	odd	6		2268.2.t.a.1781.6		16
63.25	even	3		1764.2.bm.a.1685.2		16
63.31	odd	6		2268.2.t.a.2105.6		16
63.34	odd	6		1764.2.x.b.1469.1		16
63.38	even	6		756.2.bm.a.89.6		16
63.47	even	6		5292.2.w.b.521.3		16
63.52	odd	6		252.2.bm.a.173.7	yes	16
63.58	even	3		2268.2.t.b.1781.3		16
63.59	even	6		2268.2.t.b.2105.3		16
63.61	odd	6		1764.2.w.b.1109.5		16
84.23	even	6		1008.2.df.d.689.2		16
84.59	odd	6		1008.2.ca.d.257.5		16
252.79	odd	6		1008.2.ca.d.353.5		16
252.115	even	6		1008.2.df.d.929.2		16
252.191	even	6		3024.2.ca.d.2033.6		16
252.227	odd	6		3024.2.df.d.1601.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.w.a.5.4	✓	16	21.17	even	6
252.2.w.a.101.4	yes	16	63.16	even	3
252.2.bm.a.173.7	yes	16	63.52	odd	6
252.2.bm.a.185.7	yes	16	21.2	odd	6
756.2.w.a.341.6		16	7.3	odd	6
756.2.w.a.521.6		16	63.2	odd	6
756.2.bm.a.17.6		16	7.2	even	3
756.2.bm.a.89.6		16	63.38	even	6
1008.2.ca.d.257.5		16	84.59	odd	6
1008.2.ca.d.353.5		16	252.79	odd	6
1008.2.df.d.689.2		16	84.23	even	6
1008.2.df.d.929.2		16	252.115	even	6
1764.2.w.b.509.5		16	21.11	odd	6
1764.2.w.b.1109.5		16	63.61	odd	6
1764.2.x.a.293.8		16	21.20	even	2
1764.2.x.a.1469.8		16	9.7	even	3
1764.2.x.b.293.1		16	3.2	odd	2
1764.2.x.b.1469.1		16	63.34	odd	6
1764.2.bm.a.1685.2		16	63.25	even	3
1764.2.bm.a.1697.2		16	21.5	even	6
2268.2.t.a.1781.6		16	63.23	odd	6
2268.2.t.a.2105.6		16	63.31	odd	6
2268.2.t.b.1781.3		16	63.58	even	3
2268.2.t.b.2105.3		16	63.59	even	6
3024.2.ca.d.2033.6		16	252.191	even	6
3024.2.ca.d.2609.6		16	28.3	even	6
3024.2.df.d.17.6		16	28.23	odd	6
3024.2.df.d.1601.6		16	252.227	odd	6
5292.2.w.b.521.3		16	63.47	even	6
5292.2.w.b.1097.3		16	7.4	even	3
5292.2.x.a.881.6		16	7.6	odd	2
5292.2.x.a.4409.6		16	9.2	odd	6
5292.2.x.b.881.3		16	1.1	even	1	trivial
5292.2.x.b.4409.3		16	63.20	even	6	inner
5292.2.bm.a.2285.3		16	7.5	odd	6
5292.2.bm.a.4625.3		16	63.11	odd	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Modular forms

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

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

Label	5292.2.x.b.881.3

Level	$5292$
Weight	$2$
Character	5292.881
Analytic conductor	$42.257$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$