LMFDB - Embedded newform 5292.2.bm.a.2285.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(2285,5292)]

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, 1]))

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

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

chi := DirichletCharacter("5292.2285");

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.bm (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			2285.3
Root			$-0.268067 + 1.71118i$ of defining polynomial
Character	$\chi$	$=$	5292.2285
Dual form			5292.2.bm.a.4625.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68574 q^{5} +3.90538i q^{11} +(-5.24391 - 3.02757i) q^{13} +(0.201244 - 0.348565i) q^{17} +(0.145617 - 0.0840718i) q^{19} -8.88395i q^{23} -2.15829 q^{25} +(6.15380 - 3.55290i) q^{29} +(-5.44527 + 3.14383i) q^{31} +(3.13257 + 5.42578i) q^{37} +(-1.64707 + 2.85281i) q^{41} +(1.80474 + 3.12590i) q^{43} +(4.38482 - 7.59474i) q^{47} +(4.94628 + 2.85574i) q^{53} -6.58345i q^{55} +(2.25163 + 3.89994i) q^{59} +(-4.43678 - 2.56157i) q^{61} +(8.83986 + 5.10369i) q^{65} +(2.95521 + 5.11857i) q^{67} -11.4308i q^{71} +(6.05559 + 3.49620i) q^{73} +(-0.603968 + 1.04610i) q^{79} +(-0.181350 - 0.314108i) q^{83} +(-0.339244 + 0.587588i) q^{85} +(1.38526 + 2.39934i) q^{89} +(-0.245471 + 0.141723i) q^{95} +(-0.508914 + 0.293821i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 3 q^{13} - 9 q^{17} + 16 q^{25} - 6 q^{29} - 6 q^{31} + q^{37} + 6 q^{41} - 2 q^{43} - 18 q^{47} - 15 q^{59} - 3 q^{61} + 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} - 21 q^{89} - 6 q^{95} - 3 q^{97}+O(q^{100})$ 16 * q - 3 * q^13 - 9 * q^17 + 16 * q^25 - 6 * q^29 - 6 * q^31 + 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 - 21 * q^89 - 6 * q^95 - 3 * q^97

Character values

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

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

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
$5$	−1.68574			−0.753885			−0.376942	−	0.926237i	$-0.623025\pi$
$5$	−1.68574			−0.753885			−0.376942	+	0.926237i	$0.623025\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.90538i			1.17752i	0.808309	+	0.588758i	$0.200383\pi$
$11$			3.90538i			1.17752i	−0.808309	+	0.588758i	$0.799617\pi$
$12$		0			0
$13$	−5.24391	−	3.02757i	−1.45440	−	0.839698i	−0.455673	−	0.890147i	$-0.650601\pi$
$13$	−5.24391	−	3.02757i	−1.45440	−	0.839698i	−0.998727	+	0.0504496i	$0.983935\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.201244	−	0.348565i	0.0488088	−	0.0845393i	−0.840589	−	0.541674i	$-0.817791\pi$
$17$	0.201244	−	0.348565i	0.0488088	−	0.0845393i	0.889398	+	0.457134i	$0.151124\pi$
$18$		0			0
$19$	0.145617	−	0.0840718i	0.0334067	−	0.0192874i	−0.483204	−	0.875508i	$-0.660527\pi$
$19$	0.145617	−	0.0840718i	0.0334067	−	0.0192874i	0.516610	+	0.856221i	$0.327194\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.88395i		−	1.85243i	−0.376993	−	0.926216i	$-0.623042\pi$
$23$		−	8.88395i		−	1.85243i	0.376993	−	0.926216i	$-0.376958\pi$
$24$		0			0
$25$	−2.15829			−0.431658
$26$		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	−0.901666	−	0.432434i	$-0.857655\pi$
$31$	−5.44527	+	3.14383i	−0.978000	+	0.564649i	−0.0763342	+	0.997082i	$0.524322\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.13257	+	5.42578i	0.514992	+	0.891992i	0.999849	+	0.0173987i	$0.00553846\pi$
$37$	3.13257	+	5.42578i	0.514992	+	0.891992i	−0.484857	+	0.874594i	$0.661128\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.64707	+	2.85281i	−0.257229	+	0.445534i	−0.965499	−	0.260408i	$-0.916143\pi$
$41$	−1.64707	+	2.85281i	−0.257229	+	0.445534i	0.708269	+	0.705942i	$0.249476\pi$
$42$		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	−0.345930	−	0.938260i	$-0.612437\pi$
$47$	4.38482	−	7.59474i	0.639592	−	1.10781i	0.985522	−	0.169546i	$-0.0542301\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.94628	+	2.85574i	0.679424	+	0.392266i	0.799638	−	0.600482i	$-0.205025\pi$
$53$	4.94628	+	2.85574i	0.679424	+	0.392266i	−0.120214	+	0.992748i	$0.538358\pi$
$54$		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	$-0.0719673\pi$
$59$	2.25163	+	3.89994i	0.293138	+	0.507729i	−0.681412	+	0.731900i	$0.738634\pi$
$60$		0			0
$61$	−4.43678	−	2.56157i	−0.568071	−	0.327976i	0.188308	−	0.982110i	$-0.439700\pi$
$61$	−4.43678	−	2.56157i	−0.568071	−	0.327976i	−0.756379	+	0.654134i	$0.773033\pi$
$62$		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	0.988132	−	0.153610i	$-0.0490899\pi$
$67$	2.95521	+	5.11857i	0.361036	+	0.625332i	−0.627096	+	0.778942i	$0.715757\pi$
$68$		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
$73$	6.05559	+	3.49620i	0.708753	+	0.409199i	0.810599	−	0.585601i	$-0.199142\pi$
$73$	6.05559	+	3.49620i	0.708753	+	0.409199i	−0.101846	+	0.994800i	$0.532475\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.603968	+	1.04610i	−0.0679517	+	0.117696i	−0.898000	−	0.439996i	$-0.854980\pi$
$79$	−0.603968	+	1.04610i	−0.0679517	+	0.117696i	0.830048	+	0.557692i	$0.188313\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.181350	−	0.314108i	−0.0199058	−	0.0344779i	0.855901	−	0.517140i	$-0.173003\pi$
$83$	−0.181350	−	0.314108i	−0.0199058	−	0.0344779i	−0.875807	+	0.482662i	$0.839670\pi$
$84$		0			0
$85$	−0.339244	+	0.587588i	−0.0367962	+	0.0637329i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.38526	+	2.39934i	0.146837	+	0.254329i	0.930057	−	0.367416i	$-0.119757\pi$
$89$	1.38526	+	2.39934i	0.146837	+	0.254329i	−0.783220	+	0.621745i	$0.786424\pi$
$90$		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	−0.525614	−	0.850723i	$-0.676164\pi$
$97$	−0.508914	+	0.293821i	−0.0516723	+	0.0298330i	0.473941	+	0.880556i	$0.342831\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.8466			1.37779			0.688893	−	0.724863i	$-0.258097\pi$
$101$	13.8466			1.37779			0.688893	+	0.724863i	$0.258097\pi$
$102$		0			0
$103$			12.0793i			1.19021i	0.803647	+	0.595106i	$0.202890\pi$
$103$			12.0793i			1.19021i	−0.803647	+	0.595106i	$0.797110\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.9299	+	9.19711i	−1.54000	+	0.889118i	−0.541159	+	0.840920i	$0.682014\pi$
$107$	−15.9299	+	9.19711i	−1.54000	+	0.889118i	−0.998838	+	0.0481978i	$0.984652\pi$
$108$		0			0
$109$	−5.51036	+	9.54422i	−0.527796	+	0.914170i	0.471679	+	0.881771i	$0.343648\pi$
$109$	−5.51036	+	9.54422i	−0.527796	+	0.914170i	−0.999475	+	0.0323997i	$0.989685\pi$
$110$		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
$115$			14.9760i			1.39652i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.25200			−0.386545
$122$		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
$131$	6.33480			0.553474			0.276737	−	0.960946i	$-0.410747\pi$
$131$	6.33480			0.553474			0.276737	+	0.960946i	$0.410747\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			16.6459i			1.42216i	0.703113	+	0.711078i	$0.251793\pi$
$137$			16.6459i			1.42216i	−0.703113	+	0.711078i	$0.748207\pi$
$138$		0			0
$139$	4.24007	+	2.44800i	0.359638	+	0.207637i	0.668922	−	0.743333i	$-0.266756\pi$
$139$	4.24007	+	2.44800i	0.359638	+	0.207637i	−0.309284	+	0.950970i	$0.600089\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.8238	−	20.4795i	0.988758	−	1.71258i
$144$		0			0
$145$	−10.3737	+	5.98926i	−0.861489	+	0.497381i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.28696i			0.433125i	0.976269	+	0.216562i	$0.0694845\pi$
$149$			5.28696i			0.433125i	−0.976269	+	0.216562i	$0.930516\pi$
$150$		0			0
$151$	−14.5833			−1.18677			−0.593385	−	0.804919i	$-0.702209\pi$
$151$	−14.5833			−1.18677			−0.593385	+	0.804919i	$0.702209\pi$
$152$		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	0.263232	−	0.964732i	$-0.415211\pi$
$157$	15.4160	−	8.90044i	1.23033	−	0.710332i	0.967099	+	0.254400i	$0.0818781\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.0482228	+	0.0835243i	0.00377710	+	0.00654213i	0.867908	−	0.496725i	$-0.165464\pi$
$163$	0.0482228	+	0.0835243i	0.00377710	+	0.00654213i	−0.864131	+	0.503267i	$0.832131\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.47872	+	4.29327i	−0.191809	+	0.332224i	−0.945850	−	0.324604i	$-0.894769\pi$
$167$	−2.47872	+	4.29327i	−0.191809	+	0.332224i	0.754041	+	0.656828i	$0.228102\pi$
$168$		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	0.434629	+	0.900610i	$0.356880\pi$
$173$	−7.40033	+	12.8177i	−0.562637	+	0.974515i	−0.997265	+	0.0739055i	$0.976454\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.592751	−	0.342225i	−0.0443043	−	0.0255791i	0.477684	−	0.878532i	$-0.341476\pi$
$179$	−0.592751	−	0.342225i	−0.0443043	−	0.0255791i	−0.521989	+	0.852952i	$0.674810\pi$
$180$		0			0
$181$			7.84745i			0.583297i	0.956526	+	0.291648i	$0.0942037\pi$
$181$			7.84745i			0.583297i	−0.956526	+	0.291648i	$0.905796\pi$
$182$		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	$-0.0825219\pi$
$191$	16.9694	+	9.79729i	1.22786	+	0.708907i	0.261281	+	0.965263i	$0.415855\pi$
$192$		0			0
$193$	−9.18116	−	15.9022i	−0.660875	−	1.14467i	−0.980386	−	0.197086i	$-0.936852\pi$
$193$	−9.18116	−	15.9022i	−0.660875	−	1.14467i	0.319512	−	0.947582i	$-0.396481\pi$
$194$		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
$199$	13.6268	+	7.86741i	0.965975	+	0.557706i	0.898007	−	0.439982i	$-0.145015\pi$
$199$	13.6268	+	7.86741i	0.965975	+	0.557706i	0.0679681	+	0.997687i	$0.478348\pi$
$200$		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
$215$	−3.04231	−	5.26944i	−0.207484	−	0.359373i
$216$		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	0.0207437	−	0.999785i	$-0.493397\pi$
$223$	13.3944	−	7.73325i	0.896955	−	0.517857i	0.876211	+	0.481928i	$0.160063\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	28.0719			1.86320			0.931600	−	0.363486i	$-0.118414\pi$
$227$	28.0719			1.86320			0.931600	+	0.363486i	$0.118414\pi$
$228$		0			0
$229$		−	17.0264i		−	1.12514i	−0.826751	−	0.562568i	$-0.809814\pi$
$229$		−	17.0264i		−	1.12514i	0.826751	−	0.562568i	$-0.190186\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.0015	−	9.23847i	1.04829	−	0.605233i	0.126122	−	0.992015i	$-0.459747\pi$
$233$	16.0015	−	9.23847i	1.04829	−	0.605233i	0.922171	+	0.386782i	$0.126413\pi$
$234$		0			0
$235$	−7.39166	+	12.8027i	−0.482179	+	0.835158i
$236$		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
$241$		−	6.21759i		−	0.400510i	−0.979744	−	0.200255i	$-0.935823\pi$
$241$		−	6.21759i		−	0.400510i	0.979744	−	0.200255i	$-0.0641771\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.01813			−0.0647823
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.81844			−0.619734			−0.309867	−	0.950780i	$-0.600285\pi$
$251$	−9.81844			−0.619734			−0.309867	+	0.950780i	$0.600285\pi$
$252$		0			0
$253$	34.6952			2.18127
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.33581			−0.0833254			−0.0416627	−	0.999132i	$-0.513265\pi$
$257$	−1.33581			−0.0833254			−0.0416627	+	0.999132i	$0.513265\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			20.3502i			1.25485i	0.778678	+	0.627424i	$0.215891\pi$
$263$			20.3502i			1.25485i	−0.778678	+	0.627424i	$0.784109\pi$
$264$		0			0
$265$	−8.33814	−	4.81402i	−0.512208	−	0.295723i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.3614	+	23.1426i	−0.814659	+	1.41103i	0.0949131	+	0.995486i	$0.469743\pi$
$269$	−13.3614	+	23.1426i	−0.814659	+	1.41103i	−0.909572	+	0.415546i	$0.863591\pi$
$270$		0			0
$271$	−3.76517	+	2.17382i	−0.228718	+	0.132050i	−0.609980	−	0.792417i	$-0.708823\pi$
$271$	−3.76517	+	2.17382i	−0.228718	+	0.132050i	0.381263	+	0.924467i	$0.375489\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	8.42894i		−	0.508284i
$276$		0			0
$277$	−4.39803			−0.264252			−0.132126	−	0.991233i	$-0.542180\pi$
$277$	−4.39803			−0.264252			−0.132126	+	0.991233i	$0.542180\pi$
$278$		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	0.0390505	−	0.999237i	$-0.487567\pi$
$283$	15.5431	−	8.97381i	0.923941	−	0.533437i	0.884890	+	0.465800i	$0.154233\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.41900	+	14.5821i	0.495235	+	0.857773i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.1126	−	22.7117i	0.766048	−	1.32683i	−0.173642	−	0.984809i	$-0.555554\pi$
$293$	13.1126	−	22.7117i	0.766048	−	1.32683i	0.939691	−	0.342026i	$-0.111113\pi$
$294$		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
$305$	7.47924	+	4.31814i	0.428260	+	0.247256i
$306$		0			0
$307$			7.19520i			0.410652i	0.978694	+	0.205326i	$0.0658254\pi$
$307$			7.19520i			0.410652i	−0.978694	+	0.205326i	$0.934175\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.08721	+	1.88311i	0.0616503	+	0.106781i	0.895203	−	0.445658i	$-0.147030\pi$
$311$	1.08721	+	1.88311i	0.0616503	+	0.106781i	−0.833553	+	0.552440i	$0.813697\pi$
$312$		0			0
$313$	10.2870	+	5.93922i	0.581457	+	0.335704i	0.761712	−	0.647916i	$-0.224359\pi$
$313$	10.2870	+	5.93922i	0.581457	+	0.335704i	−0.180255	+	0.983620i	$0.557692\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.09969	−	4.09901i	−0.398758	−	0.230223i	0.287190	−	0.957874i	$-0.407279\pi$
$317$	−7.09969	−	4.09901i	−0.398758	−	0.230223i	−0.685948	+	0.727651i	$0.740612\pi$
$318$		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
$325$	11.3179	+	6.53438i	0.627803	+	0.362462i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.58540	+	14.8704i	−0.471897	+	0.817349i	−0.999483	−	0.0321526i	$-0.989764\pi$
$331$	−8.58540	+	14.8704i	−0.471897	+	0.817349i	0.527586	+	0.849501i	$0.323097\pi$
$332$		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
$341$	−12.2779	−	21.2659i	−0.664883	−	1.15161i
$342$		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	$-0.662288\pi$
$347$	0.443850	−	0.256257i	0.0238271	−	0.0137566i	0.511866	+	0.859065i	$0.328954\pi$
$348$		0			0
$349$	5.74612	−	3.31752i	0.307583	−	0.177583i	−0.338262	−	0.941052i	$-0.609839\pi$
$349$	5.74612	−	3.31752i	0.307583	−	0.177583i	0.645844	+	0.763469i	$0.276506\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0687			0.961702			0.480851	−	0.876802i	$-0.340328\pi$
$353$	18.0687			0.961702			0.480851	+	0.876802i	$0.340328\pi$
$354$		0			0
$355$			19.2693i			1.02271i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.52677	−	0.881479i	0.0805796	−	0.0465227i	−0.459169	−	0.888349i	$-0.651853\pi$
$359$	1.52677	−	0.881479i	0.0805796	−	0.0465227i	0.539748	+	0.841826i	$0.318519\pi$
$360$		0			0
$361$	−9.48586	+	16.4300i	−0.499256	+	0.864737i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.2081	−	5.89367i	−0.534318	−	0.308489i
$366$		0			0
$367$			33.4417i			1.74564i	0.488038	+	0.872822i	$0.337713\pi$
$367$			33.4417i			1.74564i	−0.488038	+	0.872822i	$0.662287\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	25.5688			1.32390			0.661952	−	0.749546i	$-0.269728\pi$
$373$	25.5688			1.32390			0.661952	+	0.749546i	$0.269728\pi$
$374$		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
$383$	32.8316			1.67762			0.838808	−	0.544427i	$-0.183253\pi$
$383$	32.8316			1.67762			0.838808	+	0.544427i	$0.183253\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	20.1544i		−	1.02187i	−0.859619	−	0.510935i	$-0.829299\pi$
$389$		−	20.1544i		−	1.02187i	0.859619	−	0.510935i	$-0.170701\pi$
$390$		0			0
$391$	−3.09663	−	1.78784i	−0.156603	−	0.0904150i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.01813	−	1.76346i	0.0512278	−	0.0887291i
$396$		0			0
$397$	30.2125	−	17.4432i	1.51632	−	0.875449i	0.516506	−	0.856284i	$-0.327232\pi$
$397$	30.2125	−	17.4432i	1.51632	−	0.875449i	0.999816	−	0.0191652i	$-0.00610086\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	9.66245i		−	0.482520i	−0.970461	−	0.241260i	$-0.922439\pi$
$401$		−	9.66245i		−	0.482520i	0.970461	−	0.241260i	$-0.0775606\pi$
$402$		0			0
$403$	38.0727			1.89654
$404$		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	−0.594647	+	0.803987i	$0.702708\pi$
$409$	−32.1202	+	18.5446i	−1.58824	+	0.916973i	−0.993597	+	0.112986i	$0.963958\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.305709	+	0.529504i	0.0150067	+	0.0259923i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.84193	+	3.19031i	−0.0899841	+	0.155857i	−0.907504	−	0.420043i	$-0.862015\pi$
$419$	−1.84193	+	3.19031i	−0.0899841	+	0.155857i	0.817520	+	0.575900i	$0.195348\pi$
$420$		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
$431$	27.3242	+	15.7756i	1.31616	+	0.759885i	0.983108	−	0.183024i	$-0.0585887\pi$
$431$	27.3242	+	15.7756i	1.31616	+	0.759885i	0.333051	+	0.942909i	$0.391922\pi$
$432$		0			0
$433$			10.0692i			0.483893i	0.970290	+	0.241947i	$0.0777859\pi$
$433$			10.0692i			0.483893i	−0.970290	+	0.241947i	$0.922214\pi$
$434$		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	−0.204779	−	0.978808i	$-0.565648\pi$
$439$	−24.1966	−	13.9699i	−1.15484	−	0.666748i	−0.950062	+	0.312060i	$0.898981\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.1930	+	17.4319i	1.43451	+	0.828215i	0.997460	−	0.0712223i	$-0.0226900\pi$
$443$	30.1930	+	17.4319i	1.43451	+	0.828215i	0.437050	+	0.899437i	$0.356023\pi$
$444$		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
$451$	−11.1413	−	6.43244i	−0.524624	−	0.302892i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.10938	−	5.38560i	0.145451	−	0.251928i	−0.784090	−	0.620647i	$-0.786870\pi$
$457$	3.10938	−	5.38560i	0.145451	−	0.251928i	0.929541	+	0.368719i	$0.120203\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.17165	+	3.76140i	0.101144	+	0.175186i	0.912156	−	0.409843i	$-0.134416\pi$
$461$	2.17165	+	3.76140i	0.101144	+	0.175186i	−0.811012	+	0.585029i	$0.801083\pi$
$462$		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
$467$	−0.944451	−	1.63584i	−0.0437040	−	0.0756975i	0.843346	−	0.537371i	$-0.180583\pi$
$467$	−0.944451	−	1.63584i	−0.0437040	−	0.0756975i	−0.887050	+	0.461673i	$0.847249\pi$
$468$		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
$479$	−10.4498			−0.477465			−0.238732	−	0.971085i	$-0.576732\pi$
$479$	−10.4498			−0.477465			−0.238732	+	0.971085i	$0.576732\pi$
$480$		0			0
$481$		−	37.9364i		−	1.72975i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.857895	−	0.495306i	0.0389550	−	0.0224907i
$486$		0			0
$487$	−11.8298	+	20.4898i	−0.536060	+	0.928483i	0.463052	+	0.886331i	$0.346754\pi$
$487$	−11.8298	+	20.4898i	−0.536060	+	0.928483i	−0.999111	+	0.0421513i	$0.986579\pi$
$488$		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
$493$		−	2.86000i		−	0.128808i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−12.0807			−0.540807			−0.270403	−	0.962747i	$-0.587157\pi$
$499$	−12.0807			−0.540807			−0.270403	+	0.962747i	$0.587157\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20.5283			−0.915310			−0.457655	−	0.889130i	$-0.651310\pi$
$503$	−20.5283			−0.915310			−0.457655	+	0.889130i	$0.651310\pi$
$504$		0			0
$505$	−23.3417			−1.03869
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.18085			0.362610			0.181305	−	0.983427i	$-0.441968\pi$
$509$	8.18085			0.362610			0.181305	+	0.983427i	$0.441968\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	20.3626i		−	0.897283i
$516$		0			0
$517$	29.6603	+	17.1244i	1.30446	+	0.753131i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.8746	+	24.0314i	−0.607856	+	1.05284i	0.383738	+	0.923442i	$0.374637\pi$
$521$	−13.8746	+	24.0314i	−0.607856	+	1.05284i	−0.991593	+	0.129395i	$0.958697\pi$
$522$		0			0
$523$	−19.8843	+	11.4802i	−0.869478	+	0.501993i	−0.867175	−	0.498004i	$-0.834066\pi$
$523$	−19.8843	+	11.4802i	−0.869478	+	0.501993i	−0.00230311	+	0.999997i	$0.500733\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			2.53071i			0.110239i
$528$		0			0
$529$	−55.9246			−2.43151
$530$		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
$541$	−2.60405	−	4.51035i	−0.111957	−	0.193915i	0.804602	−	0.593814i	$-0.202379\pi$
$541$	−2.60405	−	4.51035i	−0.111957	−	0.193915i	−0.916559	+	0.399899i	$0.869045\pi$
$542$		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
$557$	11.0945	+	6.40543i	0.470090	+	0.271407i	0.716277	−	0.697816i	$-0.245844\pi$
$557$	11.0945	+	6.40543i	0.470090	+	0.271407i	−0.246187	+	0.969222i	$0.579178\pi$
$558$		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	−0.926101	−	0.377277i	$-0.876861\pi$
$563$	−18.7396	−	32.4580i	−0.789781	−	1.36794i	0.136319	−	0.990665i	$-0.456473\pi$
$564$		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	$-0.379327\pi$
$569$	−5.94906	−	3.43469i	−0.249397	−	0.143990i	−0.619488	+	0.785006i	$0.712660\pi$
$570$		0			0
$571$	−0.0847909	−	0.146862i	−0.00354839	−	0.00614599i	0.864246	−	0.503070i	$-0.167796\pi$
$571$	−0.0847909	−	0.146862i	−0.00354839	−	0.00614599i	−0.867794	+	0.496924i	$0.834463\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			19.1741i			0.799617i
$576$		0			0
$577$	5.41193	+	3.12458i	0.225302	+	0.130078i	0.608403	−	0.793628i	$-0.291811\pi$
$577$	5.41193	+	3.12458i	0.225302	+	0.130078i	−0.383101	+	0.923706i	$0.625144\pi$
$578$		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	0.998071	−	0.0620801i	$-0.0197734\pi$
$587$	10.7881	+	18.6855i	0.445273	+	0.771235i	−0.552799	+	0.833315i	$0.686440\pi$
$588$		0			0
$589$	−0.528615	+	0.915588i	−0.0217812	+	0.0377261i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.13036	+	7.15399i	0.169613	+	0.293779i	0.938284	−	0.345866i	$-0.112415\pi$
$593$	4.13036	+	7.15399i	0.169613	+	0.293779i	−0.768671	+	0.639645i	$0.779081\pi$
$594$		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	$-0.925135\pi$
$599$	−30.7618	+	17.7603i	−1.25689	+	0.725667i	−0.284422	+	0.958699i	$0.591802\pi$
$600$		0			0
$601$	−35.8981	+	20.7258i	−1.46432	+	0.845423i	−0.999206	−	0.0398308i	$-0.987318\pi$
$601$	−35.8981	+	20.7258i	−1.46432	+	0.845423i	−0.465109	+	0.885254i	$0.653985\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.16776			0.291411
$606$		0			0
$607$		−	2.41990i		−	0.0982206i	−0.998793	−	0.0491103i	$-0.984361\pi$
$607$		−	2.41990i		−	0.0982206i	0.998793	−	0.0491103i	$-0.0156386\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−45.9873	+	26.5508i	−1.86045	+	1.07413i
$612$		0			0
$613$	21.3228	−	36.9321i	0.861219	−	1.49168i	−0.00953416	−	0.999955i	$-0.503035\pi$
$613$	21.3228	−	36.9321i	0.861219	−	1.49168i	0.870753	−	0.491720i	$-0.163632\pi$
$614$		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
$619$		−	27.6178i		−	1.11005i	−0.831833	−	0.555026i	$-0.812708\pi$
$619$		−	27.6178i		−	1.11005i	0.831833	−	0.555026i	$-0.187292\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−9.55034			−0.382014
$626$		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
$635$	17.9214			0.711188
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	9.91088i		−	0.391456i	−0.980658	−	0.195728i	$-0.937293\pi$
$641$		−	9.91088i		−	0.391456i	0.980658	−	0.195728i	$-0.0627070\pi$
$642$		0			0
$643$	−6.83668	−	3.94716i	−0.269612	−	0.155661i	0.359099	−	0.933299i	$-0.383084\pi$
$643$	−6.83668	−	3.94716i	−0.269612	−	0.155661i	−0.628711	+	0.777639i	$0.716417\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.15966	−	3.74063i	0.0849049	−	0.147060i	−0.820446	−	0.571724i	$-0.806275\pi$
$647$	2.15966	−	3.74063i	0.0849049	−	0.147060i	0.905351	+	0.424665i	$0.139608\pi$
$648$		0			0
$649$	−15.2308	+	8.79348i	−0.597859	+	0.345174i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	43.3997i		−	1.69836i	−0.528102	−	0.849181i	$-0.677096\pi$
$653$		−	43.3997i		−	1.69836i	0.528102	−	0.849181i	$-0.322904\pi$
$654$		0			0
$655$	−10.6788			−0.417256
$656$		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	−0.564569	−	0.825386i	$-0.690958\pi$
$661$	−3.39495	+	1.96008i	−0.132048	+	0.0762381i	0.432521	+	0.901624i	$0.357624\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−31.5638	−	54.6701i	−1.22216	−	2.11683i
$668$		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	$-0.742006\pi$
$677$	7.36327	−	12.7536i	0.282994	−	0.490159i	0.972121	+	0.234481i	$0.0753392\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.60128	+	0.924499i	0.0612712	+	0.0353750i	0.530323	−	0.847796i	$-0.322071\pi$
$683$	1.60128	+	0.924499i	0.0612712	+	0.0353750i	−0.469051	+	0.883171i	$0.655404\pi$
$684$		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	0.977639	−	0.210292i	$-0.0674415\pi$
$691$	33.7613	+	19.4921i	1.28434	+	0.741514i	0.306701	+	0.951806i	$0.400775\pi$
$692$		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
$703$	0.912310	+	0.526722i	0.0344084	+	0.0198657i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.14517	−	12.3758i	0.268342	−	0.464783i	−0.700092	−	0.714053i	$-0.746857\pi$
$709$	7.14517	−	12.3758i	0.268342	−	0.464783i	0.968434	+	0.249270i	$0.0801908\pi$
$710$		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
$719$	−16.7344	−	28.9848i	−0.624088	−	1.08095i	−0.988716	−	0.149799i	$-0.952137\pi$
$719$	−16.7344	−	28.9848i	−0.624088	−	1.08095i	0.364629	−	0.931153i	$-0.381196\pi$
$720$		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	−0.707862	−	0.706350i	$-0.750340\pi$
$727$	−12.1354	+	7.00636i	−0.450076	+	0.259851i	0.257786	+	0.966202i	$0.417007\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.45277			0.0537326
$732$		0			0
$733$			27.3077i			1.00863i	0.863519	+	0.504316i	$0.168255\pi$
$733$			27.3077i			1.00863i	−0.863519	+	0.504316i	$0.831745\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.9899	+	11.5412i	−0.736339	+	0.425126i
$738$		0			0
$739$	26.3157	−	45.5801i	0.968039	−	1.67669i	0.266819	−	0.963747i	$-0.414027\pi$
$739$	26.3157	−	45.5801i	0.968039	−	1.67669i	0.701220	−	0.712945i	$-0.252639\pi$
$740$		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
$745$		−	8.91243i		−	0.326526i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−33.1282			−1.20887			−0.604433	−	0.796656i	$-0.706600\pi$
$751$	−33.1282			−1.20887			−0.604433	+	0.796656i	$0.706600\pi$
$752$		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
$761$	−13.0347			−0.472509			−0.236255	−	0.971691i	$-0.575920\pi$
$761$	−13.0347			−0.472509			−0.236255	+	0.971691i	$0.575920\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	27.2679i		−	0.984588i
$768$		0			0
$769$	18.4866	+	10.6732i	0.666642	+	0.384886i	0.794803	−	0.606867i	$-0.207574\pi$
$769$	18.4866	+	10.6732i	0.666642	+	0.384886i	−0.128161	+	0.991753i	$0.540907\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.73940	+	9.94093i	−0.206432	+	0.357550i	−0.950588	−	0.310455i	$-0.899518\pi$
$773$	−5.73940	+	9.94093i	−0.206432	+	0.357550i	0.744156	+	0.668006i	$0.232852\pi$
$774$		0			0
$775$	11.7525	−	6.78529i	0.422161	−	0.243735i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			0.553889i			0.0198451i
$780$		0			0
$781$	44.6416			1.59740
$782$		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	−0.975342	−	0.220698i	$-0.929166\pi$
$787$	−35.6808	+	20.6003i	−1.27188	+	0.734322i	−0.296541	+	0.955020i	$0.595833\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	15.5107	+	26.8653i	0.550801	+	0.954016i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	25.0066	−	43.3127i	0.885779	−	1.53421i	0.0409600	−	0.999161i	$-0.486958\pi$
$797$	25.0066	−	43.3127i	0.885779	−	1.53421i	0.844819	−	0.535053i	$-0.179708\pi$
$798$		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
$809$	−43.8995	−	25.3454i	−1.54343	−	0.891097i	−0.998619	−	0.0525356i	$-0.983270\pi$
$809$	−43.8995	−	25.3454i	−1.54343	−	0.891097i	−0.544807	−	0.838562i	$-0.683397\pi$
$810$		0			0
$811$			8.96566i			0.314827i	0.987533	+	0.157413i	$0.0503155\pi$
$811$			8.96566i			0.314827i	−0.987533	+	0.157413i	$0.949684\pi$
$812$		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	$-0.140817\pi$
$821$	28.2190	+	16.2922i	0.984849	+	0.568603i	0.0811184	+	0.996704i	$0.474151\pi$
$822$		0			0
$823$	10.0877	+	17.4724i	0.351636	+	0.609051i	0.986536	−	0.163543i	$-0.0522923\pi$
$823$	10.0877	+	17.4724i	0.351636	+	0.609051i	−0.634901	+	0.772594i	$0.718959\pi$
$824$		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
$829$	6.10909	+	3.52708i	0.212177	+	0.122501i	0.602323	−	0.798253i	$-0.294242\pi$
$829$	6.10909	+	3.52708i	0.212177	+	0.122501i	−0.390146	+	0.920753i	$0.627575\pi$
$830$		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	$-0.965741\pi$
$839$	−17.0936	−	29.6069i	−0.590136	−	1.02215i	0.404078	−	0.914725i	$-0.367592\pi$
$840$		0			0
$841$	10.7462	−	18.6130i	0.370558	−	0.641826i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.9463	−	34.5480i	−0.686174	−	1.18849i
$846$		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	−0.0788844	−	0.996884i	$-0.525136\pi$
$853$	21.7586	−	12.5623i	0.745000	−	0.430126i	0.823884	+	0.566758i	$0.191802\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	42.1907			1.44121			0.720604	−	0.693347i	$-0.243865\pi$
$857$	42.1907			1.44121			0.720604	+	0.693347i	$0.243865\pi$
$858$		0			0
$859$			4.71278i			0.160798i	0.996763	+	0.0803990i	$0.0256195\pi$
$859$			4.71278i			0.160798i	−0.996763	+	0.0803990i	$0.974381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−30.8409	+	17.8060i	−1.04984	+	0.606123i	−0.922603	−	0.385750i	$-0.873943\pi$
$863$	−30.8409	+	17.8060i	−1.04984	+	0.606123i	−0.127232	+	0.991873i	$0.540609\pi$
$864$		0			0
$865$	12.4750	−	21.6074i	0.424163	−	0.734672i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.08543	−	2.35873i	−0.138589	−	0.0800143i
$870$		0			0
$871$		−	35.7884i		−	1.21264i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	40.9064			1.38131			0.690655	−	0.723184i	$-0.257322\pi$
$877$	40.9064			1.38131			0.690655	+	0.723184i	$0.257322\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	37.4443			1.26153			0.630765	−	0.775974i	$-0.282741\pi$
$881$	37.4443			1.26153			0.630765	+	0.775974i	$0.282741\pi$
$882$		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
$887$	28.8965			0.970248			0.485124	−	0.874445i	$-0.338774\pi$
$887$	28.8965			0.970248			0.485124	+	0.874445i	$0.338774\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1.47456i		−	0.0493443i
$894$		0			0
$895$	0.999223	+	0.576902i	0.0334003	+	0.0192837i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.3394	+	38.6930i	−0.745062	+	1.29048i
$900$		0			0
$901$	1.99082	−	1.14940i	0.0663238	−	0.0382920i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	13.2287i		−	0.439738i
$906$		0			0
$907$	−14.8700			−0.493749			−0.246874	−	0.969048i	$-0.579403\pi$
$907$	−14.8700			−0.493749			−0.246874	+	0.969048i	$0.579403\pi$
$908$		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
$919$	13.2083	+	22.8774i	0.435702	+	0.754657i	0.997353	−	0.0727170i	$-0.0231670\pi$
$919$	13.2083	+	22.8774i	0.435702	+	0.754657i	−0.561651	+	0.827374i	$0.689834\pi$
$920$		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	$-0.714392\pi$
$929$	11.1259	−	19.2706i	0.365029	−	0.632249i	0.988781	+	0.149373i	$0.0477257\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.29476	−	1.32488i	−0.0750465	−	0.0433281i
$936$		0			0
$937$		−	14.6822i		−	0.479647i	−0.970817	−	0.239823i	$-0.922910\pi$
$937$		−	14.6822i		−	0.479647i	0.970817	−	0.239823i	$-0.0770896\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.0396	+	39.9058i	0.751070	+	1.30089i	0.947305	+	0.320334i	$0.103795\pi$
$941$	23.0396	+	39.9058i	0.751070	+	1.30089i	−0.196235	+	0.980557i	$0.562871\pi$
$942$		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	$-0.375024\pi$
$947$	−6.96116	−	4.01903i	−0.226207	−	0.130601i	−0.608821	+	0.793308i	$0.708357\pi$
$948$		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
$955$	−28.6060	−	16.5157i	−0.925667	−	0.534434i
$956$		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
$971$	7.61403	+	13.1879i	0.244346	+	0.423219i	0.961947	−	0.273234i	$-0.0880935\pi$
$971$	7.61403	+	13.1879i	0.244346	+	0.423219i	−0.717602	+	0.696454i	$0.754760\pi$
$972$		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	$-0.675453\pi$
$977$	−1.49418	+	0.862667i	−0.0478031	+	0.0275992i	0.475908	+	0.879495i	$0.342120\pi$
$978$		0			0
$979$	−9.37033	+	5.40997i	−0.299477	+	0.172903i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−60.2383			−1.92130			−0.960651	−	0.277758i	$-0.910409\pi$
$983$	−60.2383			−1.92130			−0.960651	+	0.277758i	$0.910409\pi$
$984$		0			0
$985$			9.98485i			0.318144i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	27.7703	−	16.0332i	0.883044	−	0.509826i
$990$		0			0
$991$	−2.87312	+	4.97639i	−0.0912676	+	0.158080i	−0.908045	−	0.418873i	$-0.862425\pi$
$991$	−2.87312	+	4.97639i	−0.0912676	+	0.158080i	0.816777	+	0.576953i	$0.195759\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.9711	−	13.2624i	−0.728234	−	0.420446i
$996$		0			0
$997$		−	0.0259240i		−	0.000821020i	−1.00000		0.000410510i	$-0.999869\pi$
$997$		−	0.0259240i		−	0.000821020i	1.00000		0.000410510i	$-0.000130669\pi$
$998$		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.bm.a.2285.3		16
3.2	odd	2		1764.2.bm.a.1697.2		16
7.2	even	3		756.2.w.a.341.6		16
7.3	odd	6		5292.2.x.b.881.3		16
7.4	even	3		5292.2.x.a.881.6		16
7.5	odd	6		5292.2.w.b.1097.3		16
7.6	odd	2		756.2.bm.a.17.6		16
9.2	odd	6		5292.2.w.b.521.3		16
9.7	even	3		1764.2.w.b.1109.5		16
21.2	odd	6		252.2.w.a.5.4	✓	16
21.5	even	6		1764.2.w.b.509.5		16
21.11	odd	6		1764.2.x.a.293.8		16
21.17	even	6		1764.2.x.b.293.1		16
21.20	even	2		252.2.bm.a.185.7	yes	16
28.23	odd	6		3024.2.ca.d.2609.6		16
28.27	even	2		3024.2.df.d.17.6		16
63.2	odd	6		756.2.bm.a.89.6		16
63.11	odd	6		5292.2.x.b.4409.3		16
63.13	odd	6		2268.2.t.b.1781.3		16
63.16	even	3		252.2.bm.a.173.7	yes	16
63.20	even	6		756.2.w.a.521.6		16
63.23	odd	6		2268.2.t.b.2105.3		16
63.25	even	3		1764.2.x.b.1469.1		16
63.34	odd	6		252.2.w.a.101.4	yes	16
63.38	even	6		5292.2.x.a.4409.6		16
63.41	even	6		2268.2.t.a.1781.6		16
63.47	even	6	inner	5292.2.bm.a.4625.3		16
63.52	odd	6		1764.2.x.a.1469.8		16
63.58	even	3		2268.2.t.a.2105.6		16
63.61	odd	6		1764.2.bm.a.1685.2		16
84.23	even	6		1008.2.ca.d.257.5		16
84.83	odd	2		1008.2.df.d.689.2		16
252.79	odd	6		1008.2.df.d.929.2		16
252.83	odd	6		3024.2.ca.d.2033.6		16
252.191	even	6		3024.2.df.d.1601.6		16
252.223	even	6		1008.2.ca.d.353.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.w.a.5.4	✓	16	21.2	odd	6
252.2.w.a.101.4	yes	16	63.34	odd	6
252.2.bm.a.173.7	yes	16	63.16	even	3
252.2.bm.a.185.7	yes	16	21.20	even	2
756.2.w.a.341.6		16	7.2	even	3
756.2.w.a.521.6		16	63.20	even	6
756.2.bm.a.17.6		16	7.6	odd	2
756.2.bm.a.89.6		16	63.2	odd	6
1008.2.ca.d.257.5		16	84.23	even	6
1008.2.ca.d.353.5		16	252.223	even	6
1008.2.df.d.689.2		16	84.83	odd	2
1008.2.df.d.929.2		16	252.79	odd	6
1764.2.w.b.509.5		16	21.5	even	6
1764.2.w.b.1109.5		16	9.7	even	3
1764.2.x.a.293.8		16	21.11	odd	6
1764.2.x.a.1469.8		16	63.52	odd	6
1764.2.x.b.293.1		16	21.17	even	6
1764.2.x.b.1469.1		16	63.25	even	3
1764.2.bm.a.1685.2		16	63.61	odd	6
1764.2.bm.a.1697.2		16	3.2	odd	2
2268.2.t.a.1781.6		16	63.41	even	6
2268.2.t.a.2105.6		16	63.58	even	3
2268.2.t.b.1781.3		16	63.13	odd	6
2268.2.t.b.2105.3		16	63.23	odd	6
3024.2.ca.d.2033.6		16	252.83	odd	6
3024.2.ca.d.2609.6		16	28.23	odd	6
3024.2.df.d.17.6		16	28.27	even	2
3024.2.df.d.1601.6		16	252.191	even	6
5292.2.w.b.521.3		16	9.2	odd	6
5292.2.w.b.1097.3		16	7.5	odd	6
5292.2.x.a.881.6		16	7.4	even	3
5292.2.x.a.4409.6		16	63.38	even	6
5292.2.x.b.881.3		16	7.3	odd	6
5292.2.x.b.4409.3		16	63.11	odd	6
5292.2.bm.a.2285.3		16	1.1	even	1	trivial
5292.2.bm.a.4625.3		16	63.47	even	6	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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.bm.a.2285.3

Level	$5292$
Weight	$2$
Character	5292.2285
Analytic conductor	$42.257$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$