LMFDB - Embedded newform 5292.2.w.b.1097.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(521,5292)]

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, 1, 1]))

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

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

chi := DirichletCharacter("5292.521");

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.w (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			1097.3
Root			$-0.268067 + 1.71118i$ of defining polynomial
Character	$\chi$	$=$	5292.1097
Dual form			5292.2.w.b.521.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.842869 - 1.45989i) q^{5} +(-3.38216 - 1.95269i) q^{11} +(5.24391 + 3.02757i) q^{13} +(-0.201244 - 0.348565i) q^{17} +(0.145617 + 0.0840718i) q^{19} +(-7.69373 + 4.44198i) q^{23} +(1.07914 - 1.86913i) q^{25} +(6.15380 - 3.55290i) q^{29} +6.28766i q^{31} +(3.13257 - 5.42578i) q^{37} +(1.64707 - 2.85281i) q^{41} +(1.80474 + 3.12590i) q^{43} +8.76965 q^{47} +(-4.94628 + 2.85574i) q^{53} +6.58345i q^{55} +4.50326 q^{59} -5.12315i q^{61} -10.2074i q^{65} -5.91041 q^{67} -11.4308i q^{71} +(6.05559 - 3.49620i) q^{73} +1.20794 q^{79} +(0.181350 + 0.314108i) q^{83} +(-0.339244 + 0.587588i) q^{85} +(-1.38526 + 2.39934i) q^{89} -0.283446i q^{95} +(0.508914 - 0.293821i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 6 q^{11} + 3 q^{13} + 9 q^{17} - 21 q^{23} - 8 q^{25} - 6 q^{29} + q^{37} - 6 q^{41} - 2 q^{43} - 36 q^{47} - 30 q^{59} + 14 q^{67} + 2 q^{79} + 6 q^{85} + 21 q^{89} + 3 q^{97}+O(q^{100})$ 16 * q + 6 * q^11 + 3 * q^13 + 9 * q^17 - 21 * q^23 - 8 * q^25 - 6 * q^29 + q^37 - 6 * q^41 - 2 * q^43 - 36 * q^47 - 30 * q^59 + 14 * q^67 + 2 * q^79 + 6 * q^85 + 21 * q^89 + 3 * q^97

Character values

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

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

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
$5$	−0.842869	−	1.45989i	−0.376942	−	0.652883i	0.613673	−	0.789560i	$-0.289691\pi$
$5$	−0.842869	−	1.45989i	−0.376942	−	0.652883i	−0.990616	+	0.136677i	$0.956358\pi$
$6$		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	$-0.533716\pi$
$11$	−3.38216	−	1.95269i	−1.01976	−	0.588758i	−0.914034	+	0.405637i	$0.867050\pi$
$12$		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
$17$	−0.201244	−	0.348565i	−0.0488088	−	0.0845393i	0.840589	−	0.541674i	$-0.182209\pi$
$17$	−0.201244	−	0.348565i	−0.0488088	−	0.0845393i	−0.889398	+	0.457134i	$0.848876\pi$
$18$		0			0
$19$	0.145617	+	0.0840718i	0.0334067	+	0.0192874i	0.516610	−	0.856221i	$-0.327194\pi$
$19$	0.145617	+	0.0840718i	0.0334067	+	0.0192874i	−0.483204	+	0.875508i	$0.660527\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.69373	+	4.44198i	−1.60425	+	0.926216i	−0.613630	+	0.789593i	$0.710291\pi$
$23$	−7.69373	+	4.44198i	−1.60425	+	0.926216i	−0.990623	+	0.136623i	$0.956375\pi$
$24$		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
$31$			6.28766i			1.12930i	0.825331	+	0.564649i	$0.190988\pi$
$31$			6.28766i			1.12930i	−0.825331	+	0.564649i	$0.809012\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.13257	−	5.42578i	0.514992	−	0.891992i	−0.484857	−	0.874594i	$-0.661128\pi$
$37$	3.13257	−	5.42578i	0.514992	−	0.891992i	0.999849	−	0.0173987i	$-0.00553846\pi$
$38$		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
$47$	8.76965			1.27918			0.639592	−	0.768714i	$-0.279103\pi$
$47$	8.76965			1.27918			0.639592	+	0.768714i	$0.279103\pi$
$48$		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	$-0.794975\pi$
$53$	−4.94628	+	2.85574i	−0.679424	+	0.392266i	0.120214	+	0.992748i	$0.461642\pi$
$54$		0			0
$55$			6.58345i			0.887712i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.50326			0.586275			0.293138	−	0.956070i	$-0.405301\pi$
$59$	4.50326			0.586275			0.293138	+	0.956070i	$0.405301\pi$
$60$		0			0
$61$		−	5.12315i		−	0.655952i	−0.944686	−	0.327976i	$-0.893634\pi$
$61$		−	5.12315i		−	0.655952i	0.944686	−	0.327976i	$-0.106366\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	10.2074i		−	1.26607i
$66$		0			0
$67$	−5.91041			−0.722072			−0.361036	−	0.932552i	$-0.617577\pi$
$67$	−5.91041			−0.722072			−0.361036	+	0.932552i	$0.617577\pi$
$68$		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	−0.101846	−	0.994800i	$-0.532475\pi$
$73$	6.05559	−	3.49620i	0.708753	−	0.409199i	0.810599	+	0.585601i	$0.199142\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.20794			0.135903			0.0679517	−	0.997689i	$-0.478354\pi$
$79$	1.20794			0.135903			0.0679517	+	0.997689i	$0.478354\pi$
$80$		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
$89$	−1.38526	+	2.39934i	−0.146837	+	0.254329i	−0.930057	−	0.367416i	$-0.880243\pi$
$89$	−1.38526	+	2.39934i	−0.146837	+	0.254329i	0.783220	+	0.621745i	$0.213576\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	0.283446i		−	0.0290809i
$96$		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	−0.283303	−	0.959030i	$-0.591430\pi$
$101$	6.92329	−	11.9915i	0.688893	−	1.19320i	0.972196	−	0.234167i	$-0.0752364\pi$
$102$		0			0
$103$	−10.4610	+	6.03967i	−1.03075	+	0.595106i	−0.917201	−	0.398425i	$-0.869557\pi$
$103$	−10.4610	+	6.03967i	−1.03075	+	0.595106i	−0.113554	+	0.993532i	$0.536223\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	15.9299	+	9.19711i	1.54000	+	0.889118i	0.998838	+	0.0481978i	$0.0153478\pi$
$107$	15.9299	+	9.19711i	1.54000	+	0.889118i	0.541159	+	0.840920i	$0.317986\pi$
$108$		0			0
$109$	−5.51036	−	9.54422i	−0.527796	−	0.914170i	−0.999475	−	0.0323997i	$-0.989685\pi$
$109$	−5.51036	−	9.54422i	−0.527796	−	0.914170i	0.471679	−	0.881771i	$-0.343648\pi$
$110$		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	0.970572	−	0.240812i	$-0.0774136\pi$
$131$	3.16740	+	5.48610i	0.276737	+	0.479322i	−0.693835	+	0.720134i	$0.744080\pi$
$132$		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	$-0.585126\pi$
$137$	−14.4158	−	8.32296i	−1.23162	−	0.711078i	−0.967368	+	0.253375i	$0.918459\pi$
$138$		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
$143$	−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
$149$	4.57864	−	2.64348i	0.375097	−	0.216562i	−0.300586	−	0.953755i	$-0.597182\pi$
$149$	4.57864	−	2.64348i	0.375097	−	0.216562i	0.675683	+	0.737192i	$0.263849\pi$
$150$		0			0
$151$	7.29163	−	12.6295i	0.593385	−	1.02777i	−0.400388	−	0.916346i	$-0.631125\pi$
$151$	7.29163	−	12.6295i	0.593385	−	1.02777i	0.993773	−	0.111427i	$-0.0355421\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.17930	−	5.29967i	0.737299	−	0.425680i
$156$		0			0
$157$		−	17.8009i		−	1.42066i	−0.703867	−	0.710332i	$-0.748545\pi$
$157$		−	17.8009i		−	1.42066i	0.703867	−	0.710332i	$-0.251455\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.0482228	−	0.0835243i	0.00377710	−	0.00654213i	−0.864131	−	0.503267i	$-0.832131\pi$
$163$	0.0482228	−	0.0835243i	0.00377710	−	0.00654213i	0.867908	+	0.496725i	$0.165464\pi$
$164$		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
$173$	−14.8007			−1.12527			−0.562637	−	0.826704i	$-0.690213\pi$
$173$	−14.8007			−1.12527			−0.562637	+	0.826704i	$0.690213\pi$
$174$		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	$-0.658524\pi$
$179$	0.592751	−	0.342225i	0.0443043	−	0.0255791i	0.521989	+	0.852952i	$0.325190\pi$
$180$		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
$185$	−10.5614			−0.776489
$186$		0			0
$187$			1.57187i			0.114946i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	19.5946i		−	1.41781i	−0.705302	−	0.708907i	$-0.749189\pi$
$191$		−	19.5946i		−	1.41781i	0.705302	−	0.708907i	$-0.250811\pi$
$192$		0			0
$193$	18.3623			1.32175			0.660875	−	0.750496i	$-0.270186\pi$
$193$	18.3623			1.32175			0.660875	+	0.750496i	$0.270186\pi$
$194$		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	0.0679681	−	0.997687i	$-0.478348\pi$
$199$	13.6268	−	7.86741i	0.965975	−	0.557706i	0.898007	+	0.439982i	$0.145015\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−5.55306			−0.387842
$206$		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
$221$		−	2.43712i		−	0.163939i
$222$		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	0.151011	−	0.988532i	$-0.451747\pi$
$227$	14.0360	−	24.3110i	0.931600	−	1.61358i	0.780588	−	0.625046i	$-0.214920\pi$
$228$		0			0
$229$	14.7453	−	8.51319i	0.974396	−	0.562568i	0.0738222	−	0.997271i	$-0.476480\pi$
$229$	14.7453	−	8.51319i	0.974396	−	0.562568i	0.900573	+	0.434704i	$0.143147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.0015	−	9.23847i	−1.04829	−	0.605233i	−0.126122	−	0.992015i	$-0.540253\pi$
$233$	−16.0015	−	9.23847i	−1.04829	−	0.605233i	−0.922171	+	0.386782i	$0.873587\pi$
$234$		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
$241$	−5.38459	−	3.10879i	−0.346852	−	0.200255i	0.316446	−	0.948611i	$-0.397510\pi$
$241$	−5.38459	−	3.10879i	−0.346852	−	0.200255i	−0.663298	+	0.748355i	$0.730844\pi$
$242$		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	0.844442	−	0.535647i	$-0.179932\pi$
$257$	−0.667904	−	1.15684i	−0.0416627	−	0.0721619i	−0.886105	+	0.463485i	$0.846599\pi$
$258$		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	$-0.549224\pi$
$263$	−17.6238	−	10.1751i	−1.08673	−	0.627424i	−0.932704	+	0.360643i	$0.882557\pi$
$264$		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	0.909572	+	0.415546i	$0.136409\pi$
$269$	13.3614	+	23.1426i	0.814659	+	1.41103i	−0.0949131	+	0.995486i	$0.530257\pi$
$270$		0			0
$271$	−3.76517	−	2.17382i	−0.228718	−	0.132050i	0.381263	−	0.924467i	$-0.375489\pi$
$271$	−3.76517	−	2.17382i	−0.228718	−	0.132050i	−0.609980	+	0.792417i	$0.708823\pi$
$272$		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	−0.792370	−	0.610041i	$-0.791153\pi$
$277$	2.19901	−	3.80880i	0.132126	−	0.228849i	0.924496	+	0.381192i	$0.124486\pi$
$278$		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
$283$		−	17.9476i		−	1.06687i	−0.845840	−	0.533437i	$-0.820900\pi$
$283$		−	17.9476i		−	1.06687i	0.845840	−	0.533437i	$-0.179100\pi$
$284$		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	$0.444446\pi$
$293$	−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
$299$	−53.7936			−3.11097
$300$		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	$-0.934175\pi$
$307$		−	7.19520i		−	0.410652i	0.978694	−	0.205326i	$-0.0658254\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.17443			0.123301			0.0616503	−	0.998098i	$-0.480364\pi$
$311$	2.17443			0.123301			0.0616503	+	0.998098i	$0.480364\pi$
$312$		0			0
$313$			11.8784i			0.671409i	0.941967	+	0.335704i	$0.108974\pi$
$313$			11.8784i			0.671409i	−0.941967	+	0.335704i	$0.891026\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			8.19801i			0.460446i	0.973138	+	0.230223i	$0.0739456\pi$
$317$			8.19801i			0.460446i	−0.973138	+	0.230223i	$0.926054\pi$
$318$		0			0
$319$	−27.7509			−1.55375
$320$		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
$331$	17.1708			0.943793			0.471897	−	0.881654i	$-0.343570\pi$
$331$	17.1708			0.943793			0.471897	+	0.881654i	$0.343570\pi$
$332$		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
$347$			0.512514i			0.0275132i	0.999905	+	0.0137566i	$0.00437900\pi$
$347$			0.512514i			0.0275132i	−0.999905	+	0.0137566i	$0.995621\pi$
$348$		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	−0.518908	−	0.854830i	$-0.673661\pi$
$353$	9.03437	−	15.6480i	0.480851	−	0.832858i	0.999759	+	0.0219721i	$0.00699449\pi$
$354$		0			0
$355$	−16.6877	+	9.63465i	−0.885691	+	0.511354i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.52677	−	0.881479i	−0.0805796	−	0.0465227i	0.459169	−	0.888349i	$-0.348147\pi$
$359$	−1.52677	−	0.881479i	−0.0805796	−	0.0465227i	−0.539748	+	0.841826i	$0.681481\pi$
$360$		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
$367$	28.9614	+	16.7209i	1.51177	+	0.872822i	0.999905	+	0.0137576i	$0.00437931\pi$
$367$	28.9614	+	16.7209i	1.51177	+	0.872822i	0.511867	+	0.859065i	$0.328954\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−12.7844	−	22.1433i	−0.661952	−	1.14653i	−0.980102	−	0.198494i	$-0.936395\pi$
$373$	−12.7844	−	22.1433i	−0.661952	−	1.14653i	0.318150	−	0.948040i	$-0.396938\pi$
$374$		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	0.890892	+	0.454215i	$0.150080\pi$
$383$	16.4158	+	28.4330i	0.838808	+	1.45286i	−0.0520838	+	0.998643i	$0.516586\pi$
$384$		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	$-0.162633\pi$
$389$	17.4542	+	10.0772i	0.884965	+	0.510935i	0.0126730	+	0.999920i	$0.495966\pi$
$390$		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	0.999816	+	0.0191652i	$0.00610086\pi$
$397$	30.2125	+	17.4432i	1.51632	+	0.875449i	0.516506	+	0.856284i	$0.327232\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.36793	+	4.83122i	−0.417874	+	0.241260i	−0.694167	−	0.719814i	$-0.744227\pi$
$401$	−8.36793	+	4.83122i	−0.417874	+	0.241260i	0.276293	+	0.961073i	$0.410894\pi$
$402$		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
$409$			37.0893i			1.83395i	0.398949	+	0.916973i	$0.369375\pi$
$409$			37.0893i			1.83395i	−0.398949	+	0.916973i	$0.630625\pi$
$410$		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	−0.817520	−	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
$425$	−0.868685			−0.0421374
$426$		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	$-0.941411\pi$
$431$	−27.3242	+	15.7756i	−1.31616	+	0.759885i	−0.333051	+	0.942909i	$0.608078\pi$
$432$		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
$437$	−1.49378			−0.0714572
$438$		0			0
$439$		−	27.9398i		−	1.33350i	−0.745283	−	0.666748i	$-0.767686\pi$
$439$		−	27.9398i		−	1.33350i	0.745283	−	0.666748i	$-0.232314\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	34.8638i		−	1.65643i	−0.560411	−	0.828215i	$-0.689357\pi$
$443$		−	34.8638i		−	1.65643i	0.560411	−	0.828215i	$-0.310643\pi$
$444$		0			0
$445$	4.67037			0.221397
$446$		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
$457$	−6.21876			−0.290901			−0.145451	−	0.989366i	$-0.546463\pi$
$457$	−6.21876			−0.290901			−0.145451	+	0.989366i	$0.546463\pi$
$458$		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
$467$	0.944451	−	1.63584i	0.0437040	−	0.0756975i	−0.843346	−	0.537371i	$-0.819417\pi$
$467$	0.944451	−	1.63584i	0.0437040	−	0.0756975i	0.887050	+	0.461673i	$0.152751\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	14.0964i		−	0.648152i
$474$		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	−0.960351	−	0.278794i	$-0.910065\pi$
$479$	−5.22491	+	9.04981i	−0.238732	+	0.413497i	0.721618	+	0.692291i	$0.243399\pi$
$480$		0			0
$481$	32.8539	−	18.9682i	1.49801	−	0.864875i
$482$		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	−0.999111	−	0.0421513i	$-0.986579\pi$
$487$	−11.8298	−	20.4898i	−0.536060	−	0.928483i	0.463052	−	0.886331i	$-0.346754\pi$
$488$		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	0.968965	−	0.247197i	$-0.0795096\pi$
$499$	6.04035	+	10.4622i	0.270403	+	0.468352i	−0.698562	+	0.715550i	$0.746176\pi$
$500$		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	0.942325	−	0.334699i	$-0.108635\pi$
$509$	4.09043	+	7.08483i	0.181305	+	0.314029i	−0.761020	+	0.648728i	$0.775301\pi$
$510$		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
$521$	13.8746	+	24.0314i	0.607856	+	1.05284i	0.991593	+	0.129395i	$0.0413034\pi$
$521$	13.8746	+	24.0314i	0.607856	+	1.05284i	−0.383738	+	0.923442i	$0.625363\pi$
$522$		0			0
$523$	−19.8843	−	11.4802i	−0.869478	−	0.501993i	−0.00230311	−	0.999997i	$-0.500733\pi$
$523$	−19.8843	−	11.4802i	−0.869478	−	0.501993i	−0.867175	+	0.498004i	$0.834066\pi$
$524$		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
$535$		−	31.0078i		−	1.34058i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.60405	+	4.51035i	−0.111957	+	0.193915i	−0.916559	−	0.399899i	$-0.869045\pi$
$541$	−2.60405	+	4.51035i	−0.111957	+	0.193915i	0.804602	+	0.593814i	$0.202379\pi$
$542$		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
$551$	1.19479			0.0509000
$552$		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	$-0.754156\pi$
$557$	−11.0945	+	6.40543i	−0.470090	+	0.271407i	0.246187	+	0.969222i	$0.420822\pi$
$558$		0			0
$559$			21.8559i			0.924406i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−37.4793			−1.57956			−0.789781	−	0.613388i	$-0.789806\pi$
$563$	−37.4793			−1.57956			−0.789781	+	0.613388i	$0.789806\pi$
$564$		0			0
$565$		−	14.3422i		−	0.603380i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			6.86938i			0.287979i	0.989579	+	0.143990i	$0.0459932\pi$
$569$			6.86938i			0.287979i	−0.989579	+	0.143990i	$0.954007\pi$
$570$		0			0
$571$	0.169582			0.00709678			0.00354839	−	0.999994i	$-0.498871\pi$
$571$	0.169582			0.00709678			0.00354839	+	0.999994i	$0.498871\pi$
$572$		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	−0.383101	−	0.923706i	$-0.625144\pi$
$577$	5.41193	−	3.12458i	0.225302	−	0.130078i	0.608403	+	0.793628i	$0.291811\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	22.3055			0.923799
$584$		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
$593$	−4.13036	+	7.15399i	−0.169613	+	0.293779i	−0.938284	−	0.345866i	$-0.887585\pi$
$593$	−4.13036	+	7.15399i	−0.169613	+	0.293779i	0.768671	+	0.639645i	$0.220919\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	35.5206i		−	1.45133i	−0.688047	−	0.725667i	$-0.741532\pi$
$599$		−	35.5206i		−	1.45133i	0.688047	−	0.725667i	$-0.258468\pi$
$600$		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	$-0.651028\pi$
$607$	2.09569	−	1.20995i	0.0850616	−	0.0491103i	0.541927	+	0.840425i	$0.317695\pi$
$608$		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	0.870753	+	0.491720i	$0.163632\pi$
$613$	21.3228	+	36.9321i	0.861219	+	1.49168i	−0.00953416	+	0.999955i	$0.503035\pi$
$614$		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	$-0.520625\pi$
$619$	−23.9177	−	13.8089i	−0.961334	−	0.555026i	−0.896583	+	0.442875i	$0.853958\pi$
$620$		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	$-0.270626\pi$
$641$	8.58307	+	4.95544i	0.339011	+	0.195728i	−0.320824	+	0.947139i	$0.603960\pi$
$642$		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
$647$	−2.15966	−	3.74063i	−0.0849049	−	0.147060i	0.820446	−	0.571724i	$-0.193725\pi$
$647$	−2.15966	−	3.74063i	−0.0849049	−	0.147060i	−0.905351	+	0.424665i	$0.860392\pi$
$648$		0			0
$649$	−15.2308	−	8.79348i	−0.597859	−	0.345174i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−37.5853	+	21.6999i	−1.47082	+	0.849181i	−0.999463	−	0.0327591i	$-0.989571\pi$
$653$	−37.5853	+	21.6999i	−1.47082	+	0.849181i	−0.471361	+	0.881940i	$0.656237\pi$
$654$		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
$661$			3.92015i			0.152476i	0.997090	+	0.0762381i	$0.0242909\pi$
$661$			3.92015i			0.152476i	−0.997090	+	0.0762381i	$0.975709\pi$
$662$		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
$677$	14.7265			0.565987			0.282994	−	0.959122i	$-0.408672\pi$
$677$	14.7265			0.565987			0.282994	+	0.959122i	$0.408672\pi$
$678$		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	$-0.677929\pi$
$683$	−1.60128	+	0.924499i	−0.0612712	+	0.0353750i	0.469051	+	0.883171i	$0.344596\pi$
$684$		0			0
$685$			28.0606i			1.07214i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−34.5838			−1.31754
$690$		0			0
$691$			38.9842i			1.48303i	0.670938	+	0.741514i	$0.265892\pi$
$691$			38.9842i			1.48303i	−0.670938	+	0.741514i	$0.734108\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			8.25339i			0.313069i
$696$		0			0
$697$	−1.32585			−0.0502202
$698$		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
$709$	−14.2903			−0.536685			−0.268342	−	0.963324i	$-0.586476\pi$
$709$	−14.2903			−0.536685			−0.268342	+	0.963324i	$0.586476\pi$
$710$		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	−0.364629	−	0.931153i	$-0.618804\pi$
$719$	16.7344	−	28.9848i	0.624088	−	1.08095i	0.988716	−	0.149799i	$-0.0478626\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	15.3364i		−	0.569579i
$726$		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	$-0.834922\pi$
$733$	−23.6491	+	13.6538i	−0.873501	+	0.504316i	−0.00499085	+	0.999988i	$0.501589\pi$
$734$		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	0.701220	+	0.712945i	$0.252639\pi$
$739$	26.3157	+	45.5801i	0.968039	+	1.67669i	0.266819	+	0.963747i	$0.414027\pi$
$740$		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	0.992141	+	0.125126i	$0.0399336\pi$
$751$	16.5641	+	28.6899i	0.604433	+	1.04691i	−0.387708	+	0.921782i	$0.626733\pi$
$752$		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	0.723382	−	0.690448i	$-0.242587\pi$
$761$	−6.51737	−	11.2884i	−0.236255	−	0.409205i	−0.959637	+	0.281243i	$0.909253\pi$
$762$		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
$773$	5.73940	+	9.94093i	0.206432	+	0.357550i	0.950588	−	0.310455i	$-0.100482\pi$
$773$	5.73940	+	9.94093i	0.206432	+	0.357550i	−0.744156	+	0.668006i	$0.767148\pi$
$774$		0			0
$775$	11.7525	+	6.78529i	0.422161	+	0.243735i
$776$		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
$787$			41.2006i			1.46864i	0.678802	+	0.734322i	$0.262500\pi$
$787$			41.2006i			1.46864i	−0.678802	+	0.734322i	$0.737500\pi$
$788$		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	$0.513042\pi$
$797$	−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
$803$	−27.3079			−0.963677
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	43.8995	−	25.3454i	1.54343	−	0.891097i	0.544807	−	0.838562i	$-0.316603\pi$
$809$	43.8995	−	25.3454i	1.54343	−	0.891097i	0.998619	−	0.0525356i	$-0.0167303\pi$
$810$		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
$815$	−0.162582			−0.00569500
$816$		0			0
$817$			0.606910i			0.0212331i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	32.5845i		−	1.13721i	−0.822612	−	0.568603i	$-0.807484\pi$
$821$		−	32.5845i		−	1.13721i	0.822612	−	0.568603i	$-0.192516\pi$
$822$		0			0
$823$	−20.1754			−0.703271			−0.351636	−	0.936137i	$-0.614374\pi$
$823$	−20.1754			−0.703271			−0.351636	+	0.936137i	$0.614374\pi$
$824$		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	−0.390146	−	0.920753i	$-0.627575\pi$
$829$	6.10909	−	3.52708i	0.212177	−	0.122501i	0.602323	+	0.798253i	$0.294242\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−8.35695			−0.289204
$836$		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
$845$	19.9463	−	34.5480i	0.686174	−	1.18849i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$			55.6593i			1.90798i
$852$		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	−0.240154	−	0.970735i	$-0.577198\pi$
$857$	21.0954	−	36.5383i	0.720604	−	1.24812i	0.960758	−	0.277388i	$-0.0894688\pi$
$858$		0			0
$859$	−4.08139	+	2.35639i	−0.139255	+	0.0803990i	−0.568009	−	0.823022i	$-0.692286\pi$
$859$	−4.08139	+	2.35639i	−0.139255	+	0.0803990i	0.428754	+	0.903421i	$0.358953\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.8409	+	17.8060i	1.04984	+	0.606123i	0.922603	−	0.385750i	$-0.126057\pi$
$863$	30.8409	+	17.8060i	1.04984	+	0.606123i	0.127232	+	0.991873i	$0.459391\pi$
$864$		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
$871$	−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	−0.971624	−	0.236532i	$-0.923989\pi$
$877$	−20.4532	−	35.4260i	−0.690655	−	1.19625i	0.280969	−	0.959717i	$-0.409344\pi$
$878$		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	0.999854	−	0.0170929i	$-0.00544110\pi$
$887$	14.4482	+	25.0251i	0.485124	+	0.840260i	−0.514730	+	0.857352i	$0.672108\pi$
$888$		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
$899$	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
$905$	−11.4564	+	6.61437i	−0.380825	+	0.219869i
$906$		0			0
$907$	7.43498	−	12.8778i	0.246874	−	0.427599i	−0.715783	−	0.698323i	$-0.753930\pi$
$907$	7.43498	−	12.8778i	0.246874	−	0.427599i	0.962657	+	0.270724i	$0.0872632\pi$
$908$		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
$913$		−	1.41649i		−	0.0468788i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	13.2083	−	22.8774i	0.435702	−	0.754657i	−0.561651	−	0.827374i	$-0.689834\pi$
$919$	13.2083	−	22.8774i	0.435702	−	0.754657i	0.997353	+	0.0727170i	$0.0231670\pi$
$920$		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
$929$	22.2518			0.730058			0.365029	−	0.930996i	$-0.381059\pi$
$929$	22.2518			0.730058			0.365029	+	0.930996i	$0.381059\pi$
$930$		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	$0.0770896\pi$
$937$			14.6822i			0.479647i	−0.970817	+	0.239823i	$0.922910\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	46.0792			1.50214			0.751070	−	0.660223i	$-0.229538\pi$
$941$	46.0792			1.50214			0.751070	+	0.660223i	$0.229538\pi$
$942$		0			0
$943$			29.2650i			0.952999i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			8.03805i			0.261202i	0.991435	+	0.130601i	$0.0416906\pi$
$947$			8.03805i			0.261202i	−0.991435	+	0.130601i	$0.958309\pi$
$948$		0			0
$949$	42.3400			1.37441
$950$		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
$961$	−8.53466			−0.275312
$962$		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	$-0.911907\pi$
$971$	−7.61403	+	13.1879i	−0.244346	+	0.423219i	0.717602	+	0.696454i	$0.245240\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.72533i		−	0.0551983i	−0.999619	−	0.0275992i	$-0.991214\pi$
$977$		−	1.72533i		−	0.0551983i	0.999619	−	0.0275992i	$-0.00878620\pi$
$978$		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	−0.239780	+	0.970827i	$0.577075\pi$
$983$	−30.1191	+	52.1679i	−0.960651	+	1.66390i	−0.720871	+	0.693070i	$0.756258\pi$
$984$		0			0
$985$	−8.64713	+	4.99242i	−0.275521	+	0.159072i
$986$		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	0.816777	−	0.576953i	$-0.195759\pi$
$991$	−2.87312	−	4.97639i	−0.0912676	−	0.158080i	−0.908045	+	0.418873i	$0.862425\pi$
$992$		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	$-0.333464\pi$
$997$	−0.0224508	−	0.0129620i	−0.000711024	−	0.000410510i	−0.500355	+	0.865820i	$0.666797\pi$
$998$		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.w.b.1097.3		16
3.2	odd	2		1764.2.w.b.509.5		16
7.2	even	3		5292.2.x.b.881.3		16
7.3	odd	6		5292.2.bm.a.2285.3		16
7.4	even	3		756.2.bm.a.17.6		16
7.5	odd	6		5292.2.x.a.881.6		16
7.6	odd	2		756.2.w.a.341.6		16
9.2	odd	6		5292.2.bm.a.4625.3		16
9.7	even	3		1764.2.bm.a.1685.2		16
21.2	odd	6		1764.2.x.b.293.1		16
21.5	even	6		1764.2.x.a.293.8		16
21.11	odd	6		252.2.bm.a.185.7	yes	16
21.17	even	6		1764.2.bm.a.1697.2		16
21.20	even	2		252.2.w.a.5.4	✓	16
28.11	odd	6		3024.2.df.d.17.6		16
28.27	even	2		3024.2.ca.d.2609.6		16
63.2	odd	6		5292.2.x.a.4409.6		16
63.4	even	3		2268.2.t.b.1781.3		16
63.11	odd	6		756.2.w.a.521.6		16
63.13	odd	6		2268.2.t.a.2105.6		16
63.16	even	3		1764.2.x.a.1469.8		16
63.20	even	6		756.2.bm.a.89.6		16
63.25	even	3		252.2.w.a.101.4	yes	16
63.32	odd	6		2268.2.t.a.1781.6		16
63.34	odd	6		252.2.bm.a.173.7	yes	16
63.38	even	6	inner	5292.2.w.b.521.3		16
63.41	even	6		2268.2.t.b.2105.3		16
63.47	even	6		5292.2.x.b.4409.3		16
63.52	odd	6		1764.2.w.b.1109.5		16
63.61	odd	6		1764.2.x.b.1469.1		16
84.11	even	6		1008.2.df.d.689.2		16
84.83	odd	2		1008.2.ca.d.257.5		16
252.11	even	6		3024.2.ca.d.2033.6		16
252.83	odd	6		3024.2.df.d.1601.6		16
252.151	odd	6		1008.2.ca.d.353.5		16
252.223	even	6		1008.2.df.d.929.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.w.a.5.4	✓	16	21.20	even	2
252.2.w.a.101.4	yes	16	63.25	even	3
252.2.bm.a.173.7	yes	16	63.34	odd	6
252.2.bm.a.185.7	yes	16	21.11	odd	6
756.2.w.a.341.6		16	7.6	odd	2
756.2.w.a.521.6		16	63.11	odd	6
756.2.bm.a.17.6		16	7.4	even	3
756.2.bm.a.89.6		16	63.20	even	6
1008.2.ca.d.257.5		16	84.83	odd	2
1008.2.ca.d.353.5		16	252.151	odd	6
1008.2.df.d.689.2		16	84.11	even	6
1008.2.df.d.929.2		16	252.223	even	6
1764.2.w.b.509.5		16	3.2	odd	2
1764.2.w.b.1109.5		16	63.52	odd	6
1764.2.x.a.293.8		16	21.5	even	6
1764.2.x.a.1469.8		16	63.16	even	3
1764.2.x.b.293.1		16	21.2	odd	6
1764.2.x.b.1469.1		16	63.61	odd	6
1764.2.bm.a.1685.2		16	9.7	even	3
1764.2.bm.a.1697.2		16	21.17	even	6
2268.2.t.a.1781.6		16	63.32	odd	6
2268.2.t.a.2105.6		16	63.13	odd	6
2268.2.t.b.1781.3		16	63.4	even	3
2268.2.t.b.2105.3		16	63.41	even	6
3024.2.ca.d.2033.6		16	252.11	even	6
3024.2.ca.d.2609.6		16	28.27	even	2
3024.2.df.d.17.6		16	28.11	odd	6
3024.2.df.d.1601.6		16	252.83	odd	6
5292.2.w.b.521.3		16	63.38	even	6	inner
5292.2.w.b.1097.3		16	1.1	even	1	trivial
5292.2.x.a.881.6		16	7.5	odd	6
5292.2.x.a.4409.6		16	63.2	odd	6
5292.2.x.b.881.3		16	7.2	even	3
5292.2.x.b.4409.3		16	63.47	even	6
5292.2.bm.a.2285.3		16	7.3	odd	6
5292.2.bm.a.4625.3		16	9.2	odd	6

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

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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.w.b.1097.3

Level	$5292$
Weight	$2$
Character	5292.1097
Analytic conductor	$42.257$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$