LMFDB - Embedded newform 5292.2.i.f.2125.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1549,5292)]

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, 2, 2]))

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

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

chi := DirichletCharacter("5292.1549");

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.i (of order $3$, 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:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2125.3
Root			$0.500000 + 2.05195i$ of defining polynomial
Character	$\chi$	$=$	5292.2125
Dual form			5292.2.i.f.1549.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.23025 - 2.13086i) q^{5} +O(q^{10})$	$q+(1.23025 - 2.13086i) q^{5} +(2.32383 + 4.02499i) q^{11} +(-3.55408 - 6.15585i) q^{13} +(-2.25729 + 3.90975i) q^{17} +(-2.16372 - 3.74766i) q^{19} +(2.93346 - 5.08091i) q^{23} +(-0.527042 - 0.912864i) q^{25} +(-3.48755 + 6.04061i) q^{29} -7.38151 q^{31} +(0.363327 + 0.629301i) q^{37} +(0.136673 + 0.236725i) q^{41} +(2.41741 - 4.18708i) q^{43} -3.67257 q^{47} +(2.52704 - 4.37697i) q^{53} +11.4356 q^{55} -9.13307 q^{59} -13.8171 q^{61} -17.4897 q^{65} -1.32743 q^{67} -13.5218 q^{71} +(2.16372 - 3.74766i) q^{73} +6.43560 q^{79} +(0.742705 - 1.28640i) q^{83} +(5.55408 + 9.61996i) q^{85} +(4.91741 + 8.51721i) q^{89} -10.6477 q^{95} +(0.246304 - 0.426611i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5}+O(q^{10}) $ 6 * q + q^5	$ 6 q + q^{5} + 2 q^{11} - 3 q^{13} + 2 q^{17} - 3 q^{19} + 14 q^{23} + 6 q^{25} + q^{29} - 6 q^{31} + 3 q^{37} - 3 q^{43} - 42 q^{47} + 6 q^{53} + 12 q^{55} - 62 q^{59} + 12 q^{61} - 30 q^{65} + 12 q^{67} - 34 q^{71} + 3 q^{73} - 18 q^{79} + 20 q^{83} + 15 q^{85} + 12 q^{89} - 40 q^{95} + 9 q^{97}+O(q^{100}) $ 6 * q + q^5 + 2 * q^11 - 3 * q^13 + 2 * q^17 - 3 * q^19 + 14 * q^23 + 6 * q^25 + q^29 - 6 * q^31 + 3 * q^37 - 3 * q^43 - 42 * q^47 + 6 * q^53 + 12 * q^55 - 62 * q^59 + 12 * q^61 - 30 * q^65 + 12 * q^67 - 34 * q^71 + 3 * q^73 - 18 * q^79 + 20 * q^83 + 15 * q^85 + 12 * q^89 - 40 * q^95 + 9 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	1.23025	−	2.13086i	0.550186	−	0.952949i	−0.448075	−	0.893996i	$-0.647890\pi$
$5$	1.23025	−	2.13086i	0.550186	−	0.952949i	0.998261	−	0.0589535i	$-0.0187764\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.32383	+	4.02499i	0.700662	+	1.21358i	0.968234	+	0.250044i	$0.0804451\pi$
$11$	2.32383	+	4.02499i	0.700662	+	1.21358i	−0.267573	+	0.963538i	$0.586222\pi$
$12$		0			0
$13$	−3.55408	−	6.15585i	−0.985726	−	1.70733i	−0.638667	−	0.769484i	$-0.720514\pi$
$13$	−3.55408	−	6.15585i	−0.985726	−	1.70733i	−0.347059	−	0.937843i	$-0.612820\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.25729	+	3.90975i	−0.547474	+	0.948253i	0.450972	+	0.892538i	$0.351077\pi$
$17$	−2.25729	+	3.90975i	−0.547474	+	0.948253i	−0.998447	+	0.0557155i	$0.982256\pi$
$18$		0			0
$19$	−2.16372	−	3.74766i	−0.496390	−	0.859773i	0.503601	−	0.863936i	$-0.332008\pi$
$19$	−2.16372	−	3.74766i	−0.496390	−	0.859773i	−0.999991	+	0.00416311i	$0.998675\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.93346	−	5.08091i	0.611669	−	1.05944i	−0.379290	−	0.925278i	$-0.623832\pi$
$23$	2.93346	−	5.08091i	0.611669	−	1.05944i	0.990959	−	0.134164i	$-0.0428350\pi$
$24$		0			0
$25$	−0.527042	−	0.912864i	−0.105408	−	0.182573i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.48755	+	6.04061i	−0.647621	+	1.12171i	0.336068	+	0.941838i	$0.390903\pi$
$29$	−3.48755	+	6.04061i	−0.647621	+	1.12171i	−0.983689	+	0.179875i	$0.942431\pi$
$30$		0			0
$31$	−7.38151			−1.32576			−0.662880	−	0.748726i	$-0.730666\pi$
$31$	−7.38151			−1.32576			−0.662880	+	0.748726i	$0.730666\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.363327	+	0.629301i	0.0597306	+	0.103456i	0.894344	−	0.447379i	$-0.147643\pi$
$37$	0.363327	+	0.629301i	0.0597306	+	0.103456i	−0.834614	+	0.550835i	$0.814309\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.136673	+	0.236725i	0.0213448	+	0.0369702i	0.876500	−	0.481401i	$-0.159872\pi$
$41$	0.136673	+	0.236725i	0.0213448	+	0.0369702i	−0.855156	+	0.518371i	$0.826539\pi$
$42$		0			0
$43$	2.41741	−	4.18708i	0.368652	−	0.638524i	−0.620703	−	0.784046i	$-0.713153\pi$
$43$	2.41741	−	4.18708i	0.368652	−	0.638524i	0.989355	+	0.145522i	$0.0464862\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.67257			−0.535699			−0.267850	−	0.963461i	$-0.586313\pi$
$47$	−3.67257			−0.535699			−0.267850	+	0.963461i	$0.586313\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.52704	−	4.37697i	0.347116	−	0.601222i	−0.638620	−	0.769522i	$-0.720494\pi$
$53$	2.52704	−	4.37697i	0.347116	−	0.601222i	0.985736	+	0.168300i	$0.0538277\pi$
$54$		0			0
$55$	11.4356			1.54198
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.13307			−1.18903			−0.594513	−	0.804086i	$-0.702655\pi$
$59$	−9.13307			−1.18903			−0.594513	+	0.804086i	$0.702655\pi$
$60$		0			0
$61$	−13.8171			−1.76910			−0.884550	−	0.466445i	$-0.845534\pi$
$61$	−13.8171			−1.76910			−0.884550	+	0.466445i	$0.845534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−17.4897			−2.16933
$66$		0			0
$67$	−1.32743			−0.162171			−0.0810857	−	0.996707i	$-0.525839\pi$
$67$	−1.32743			−0.162171			−0.0810857	+	0.996707i	$0.525839\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13.5218			−1.60474			−0.802370	−	0.596826i	$-0.796428\pi$
$71$	−13.5218			−1.60474			−0.802370	+	0.596826i	$0.796428\pi$
$72$		0			0
$73$	2.16372	−	3.74766i	0.253244	−	0.438631i	−0.711173	−	0.703017i	$-0.751836\pi$
$73$	2.16372	−	3.74766i	0.253244	−	0.438631i	0.964417	+	0.264386i	$0.0851692\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.43560			0.724061			0.362031	−	0.932166i	$-0.382084\pi$
$79$	6.43560			0.724061			0.362031	+	0.932166i	$0.382084\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.742705	−	1.28640i	0.0815225	−	0.141201i	−0.822382	−	0.568936i	$-0.807355\pi$
$83$	0.742705	−	1.28640i	0.0815225	−	0.141201i	0.903904	+	0.427735i	$0.140688\pi$
$84$		0			0
$85$	5.55408	+	9.61996i	0.602425	+	1.04343i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.91741	+	8.51721i	0.521245	+	0.902822i	0.999695	+	0.0247073i	$0.00786539\pi$
$89$	4.91741	+	8.51721i	0.521245	+	0.902822i	−0.478450	+	0.878115i	$0.658801\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.6477			−1.09243
$96$		0			0
$97$	0.246304	−	0.426611i	0.0250084	−	0.0433158i	−0.853250	−	0.521502i	$-0.825372\pi$
$97$	0.246304	−	0.426611i	0.0250084	−	0.0433158i	0.878259	+	0.478186i	$0.158705\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.70321	+	2.95005i	0.169476	+	0.293541i	0.938236	−	0.345997i	$-0.112459\pi$
$101$	1.70321	+	2.95005i	0.169476	+	0.293541i	−0.768760	+	0.639537i	$0.779126\pi$
$102$		0			0
$103$	−2.58113	+	4.47064i	−0.254326	+	0.440505i	−0.964712	−	0.263307i	$-0.915187\pi$
$103$	−2.58113	+	4.47064i	−0.254326	+	0.440505i	0.710386	+	0.703812i	$0.248520\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.88151	−	4.99093i	−0.278567	−	0.482491i	0.692462	−	0.721454i	$-0.256526\pi$
$107$	−2.88151	−	4.99093i	−0.278567	−	0.482491i	−0.971029	+	0.238963i	$0.923193\pi$
$108$		0			0
$109$	4.49115	−	7.77889i	0.430174	−	0.745083i	−0.566714	−	0.823914i	$-0.691786\pi$
$109$	4.49115	−	7.77889i	0.430174	−	0.745083i	0.996888	+	0.0788317i	$0.0251190\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.679767	−	1.17739i	−0.0639471	−	0.110760i	0.832279	−	0.554356i	$-0.187036\pi$
$113$	−0.679767	−	1.17739i	−0.0639471	−	0.110760i	−0.896226	+	0.443597i	$0.853702\pi$
$114$		0			0
$115$	−7.21780	−	12.5016i	−0.673063	−	1.16578i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.30039	+	9.18054i	−0.481853	+	0.834595i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.70895			0.868394
$126$		0			0
$127$	−0.820039			−0.0727667			−0.0363833	−	0.999338i	$-0.511584\pi$
$127$	−0.820039			−0.0727667			−0.0363833	+	0.999338i	$0.511584\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.89397	−	6.74455i	0.340218	−	0.589274i	−0.644255	−	0.764810i	$-0.722833\pi$
$131$	3.89397	−	6.74455i	0.340218	−	0.589274i	0.984473	+	0.175536i	$0.0561660\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.49640	−	2.59184i	−0.127846	−	0.221436i	0.794996	−	0.606615i	$-0.207473\pi$
$137$	−1.49640	−	2.59184i	−0.127846	−	0.221436i	−0.922842	+	0.385179i	$0.874140\pi$
$138$		0			0
$139$	−3.16372	−	5.47972i	−0.268343	−	0.464783i	0.700091	−	0.714053i	$-0.253143\pi$
$139$	−3.16372	−	5.47972i	−0.268343	−	0.464783i	−0.968434	+	0.249270i	$0.919809\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.5182	−	28.6103i	1.38132	−	2.39252i
$144$		0			0
$145$	8.58113	+	14.8629i	0.712624	+	1.23430i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.19076	+	3.79450i	−0.179474	+	0.310858i	−0.941700	−	0.336452i	$-0.890773\pi$
$149$	−2.19076	+	3.79450i	−0.179474	+	0.310858i	0.762227	+	0.647310i	$0.224106\pi$
$150$		0			0
$151$	−3.30039	−	5.71644i	−0.268582	−	0.465197i	0.699914	−	0.714227i	$-0.253222\pi$
$151$	−3.30039	−	5.71644i	−0.268582	−	0.465197i	−0.968496	+	0.249030i	$0.919888\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.08113	+	15.7290i	−0.729414	+	1.26338i
$156$		0			0
$157$	−5.78074			−0.461353			−0.230677	−	0.973030i	$-0.574094\pi$
$157$	−5.78074			−0.461353			−0.230677	+	0.973030i	$0.574094\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.66372	−	6.34574i	−0.286964	−	0.497037i	0.686119	−	0.727489i	$-0.259313\pi$
$163$	−3.66372	−	6.34574i	−0.286964	−	0.497037i	−0.973084	+	0.230452i	$0.925979\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.01459	−	10.4176i	−0.465423	−	0.806136i	0.533798	−	0.845612i	$-0.320764\pi$
$167$	−6.01459	−	10.4176i	−0.465423	−	0.806136i	−0.999221	+	0.0394762i	$0.987431\pi$
$168$		0			0
$169$	−18.7630	+	32.4985i	−1.44331	+	2.49989i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.89903			−0.372466			−0.186233	−	0.982506i	$-0.559628\pi$
$173$	−4.89903			−0.372466			−0.186233	+	0.982506i	$0.559628\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.890369	+	1.54216i	−0.0665493	+	0.115267i	−0.897380	−	0.441258i	$-0.854532\pi$
$179$	−0.890369	+	1.54216i	−0.0665493	+	0.115267i	0.830831	+	0.556525i	$0.187866\pi$
$180$		0			0
$181$	−16.9430			−1.25936			−0.629681	−	0.776854i	$-0.716815\pi$
$181$	−16.9430			−1.25936			−0.629681	+	0.776854i	$0.716815\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.78794			0.131452
$186$		0			0
$187$	−20.9823			−1.53438
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.48968			0.397220			0.198610	−	0.980079i	$-0.436357\pi$
$191$	5.48968			0.397220			0.198610	+	0.980079i	$0.436357\pi$
$192$		0			0
$193$	−5.50739			−0.396431			−0.198215	−	0.980158i	$-0.563515\pi$
$193$	−5.50739			−0.396431			−0.198215	+	0.980158i	$0.563515\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.6300			−0.828600			−0.414300	−	0.910140i	$-0.635974\pi$
$197$	−11.6300			−0.828600			−0.414300	+	0.910140i	$0.635974\pi$
$198$		0			0
$199$	2.07373	−	3.59181i	0.147003	−	0.254617i	−0.783115	−	0.621876i	$-0.786371\pi$
$199$	2.07373	−	3.59181i	0.147003	−	0.254617i	0.930118	+	0.367260i	$0.119704\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.672570			0.0469743
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.0562	−	17.4179i	0.695603	−	1.20482i
$210$		0			0
$211$	13.6082	+	23.5700i	0.936825	+	1.62263i	0.771347	+	0.636415i	$0.219583\pi$
$211$	13.6082	+	23.5700i	0.936825	+	1.62263i	0.165478	+	0.986213i	$0.447083\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.94805	−	10.3023i	−0.405654	−	0.702613i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	32.0905			2.15864
$222$		0			0
$223$	1.60817	−	2.78543i	0.107691	−	0.186526i	−0.807144	−	0.590355i	$-0.798988\pi$
$223$	1.60817	−	2.78543i	0.107691	−	0.186526i	0.914834	+	0.403829i	$0.132321\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.97296	+	13.8096i	0.529184	+	0.916573i	0.999421	+	0.0340330i	$0.0108351\pi$
$227$	7.97296	+	13.8096i	0.529184	+	0.916573i	−0.470237	+	0.882540i	$0.655832\pi$
$228$		0			0
$229$	0.608168	−	1.05338i	0.0401889	−	0.0696092i	−0.845231	−	0.534401i	$-0.820537\pi$
$229$	0.608168	−	1.05338i	0.0401889	−	0.0696092i	0.885420	+	0.464791i	$0.153871\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.98608	+	17.2964i	0.654210	+	1.13313i	0.982091	+	0.188406i	$0.0603321\pi$
$233$	9.98608	+	17.2964i	0.654210	+	1.13313i	−0.327881	+	0.944719i	$0.606335\pi$
$234$		0			0
$235$	−4.51819	+	7.82573i	−0.294734	+	0.510494i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00739	+	5.20896i	0.194532	+	0.336939i	0.946747	−	0.321978i	$-0.104348\pi$
$239$	3.00739	+	5.20896i	0.194532	+	0.336939i	−0.752215	+	0.658918i	$0.771015\pi$
$240$		0			0
$241$	9.30778	+	16.1215i	0.599567	+	1.03848i	0.992885	+	0.119078i	$0.0379938\pi$
$241$	9.30778	+	16.1215i	0.599567	+	1.03848i	−0.393318	+	0.919402i	$0.628673\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−15.3801	+	26.6390i	−0.978609	+	1.69500i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.99707			−0.441651			−0.220826	−	0.975313i	$-0.570875\pi$
$251$	−6.99707			−0.441651			−0.220826	+	0.975313i	$0.570875\pi$
$252$		0			0
$253$	27.2675			1.71429
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.88891	+	15.3960i	−0.554475	+	0.960378i	0.443469	+	0.896289i	$0.353747\pi$
$257$	−8.88891	+	15.3960i	−0.554475	+	0.960378i	−0.997944	+	0.0640889i	$0.979586\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.5993	−	23.5547i	−0.838570	−	1.45245i	−0.891091	−	0.453825i	$-0.850059\pi$
$263$	−13.5993	−	23.5547i	−0.838570	−	1.45245i	0.0525210	−	0.998620i	$-0.483274\pi$
$264$		0			0
$265$	−6.21780	−	10.7695i	−0.381956	−	0.661568i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.9481	−	20.6946i	0.728486	−	1.26177i	−0.229038	−	0.973418i	$-0.573558\pi$
$269$	11.9481	−	20.6946i	0.728486	−	1.26177i	0.957523	−	0.288356i	$-0.0931088\pi$
$270$		0			0
$271$	−6.13667	−	10.6290i	−0.372776	−	0.645668i	0.617215	−	0.786794i	$-0.288261\pi$
$271$	−6.13667	−	10.6290i	−0.372776	−	0.645668i	−0.989992	+	0.141127i	$0.954927\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.44951	−	4.24268i	0.147711	−	0.255843i
$276$		0			0
$277$	−6.39037	−	11.0684i	−0.383960	−	0.665038i	0.607664	−	0.794194i	$-0.292107\pi$
$277$	−6.39037	−	11.0684i	−0.383960	−	0.665038i	−0.991624	+	0.129156i	$0.958773\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.2573	+	24.6944i	−0.850519	+	1.47314i	0.0302219	+	0.999543i	$0.490379\pi$
$281$	−14.2573	+	24.6944i	−0.850519	+	1.47314i	−0.880741	+	0.473599i	$0.842955\pi$
$282$		0			0
$283$	0.726654			0.0431951			0.0215975	−	0.999767i	$-0.493125\pi$
$283$	0.726654			0.0431951			0.0215975	+	0.999767i	$0.493125\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.69076	−	2.92848i	−0.0994563	−	0.172263i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.7901	+	22.1531i	0.747204	+	1.29420i	0.949158	+	0.314800i	$0.101938\pi$
$293$	12.7901	+	22.1531i	0.747204	+	1.29420i	−0.201954	+	0.979395i	$0.564729\pi$
$294$		0			0
$295$	−11.2360	+	19.4613i	−0.654184	+	1.13308i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−41.7031			−2.41175
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.9985	+	29.4423i	−0.973333	+	1.68586i
$306$		0			0
$307$	−6.23405			−0.355796			−0.177898	−	0.984049i	$-0.556930\pi$
$307$	−6.23405			−0.355796			−0.177898	+	0.984049i	$0.556930\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−29.2383			−1.65795			−0.828976	−	0.559284i	$-0.811076\pi$
$311$	−29.2383			−1.65795			−0.828976	+	0.559284i	$0.811076\pi$
$312$		0			0
$313$	28.4868			1.61017			0.805083	−	0.593162i	$-0.202121\pi$
$313$	28.4868			1.61017			0.805083	+	0.593162i	$0.202121\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.61849			−0.0909032			−0.0454516	−	0.998967i	$-0.514473\pi$
$317$	−1.61849			−0.0909032			−0.0454516	+	0.998967i	$0.514473\pi$
$318$		0			0
$319$	−32.4179			−1.81505
$320$		0			0
$321$		0			0
$322$		0			0
$323$	19.5366			1.08704
$324$		0			0
$325$	−3.74630	+	6.48879i	−0.207808	+	0.359933i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−13.9823			−0.768536			−0.384268	−	0.923222i	$-0.625546\pi$
$331$	−13.9823			−0.768536			−0.384268	+	0.923222i	$0.625546\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.63307	+	2.82857i	−0.0892244	+	0.154541i
$336$		0			0
$337$	−13.8619	−	24.0095i	−0.755104	−	1.30788i	−0.945323	−	0.326137i	$-0.894253\pi$
$337$	−13.8619	−	24.0095i	−0.755104	−	1.30788i	0.190219	−	0.981742i	$-0.439080\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.1534	−	29.7106i	−0.928909	−	1.60892i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.52898			−0.404177			−0.202089	−	0.979367i	$-0.564773\pi$
$347$	−7.52898			−0.404177			−0.202089	+	0.979367i	$0.564773\pi$
$348$		0			0
$349$	15.0541	−	26.0744i	0.805827	−	1.39573i	−0.109905	−	0.993942i	$-0.535055\pi$
$349$	15.0541	−	26.0744i	0.805827	−	1.39573i	0.915732	−	0.401791i	$-0.131612\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.1819	−	17.6356i	−0.541928	−	0.938647i	−0.998793	−	0.0491110i	$-0.984361\pi$
$353$	−10.1819	−	17.6356i	−0.541928	−	0.938647i	0.456865	−	0.889536i	$-0.348972\pi$
$354$		0			0
$355$	−16.6352	+	28.8130i	−0.882905	+	1.52924i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.01313	−	13.8791i	−0.422917	−	0.732513i	0.573307	−	0.819341i	$-0.305660\pi$
$359$	−8.01313	−	13.8791i	−0.422917	−	0.732513i	−0.996223	+	0.0868277i	$0.972327\pi$
$360$		0			0
$361$	0.136673	−	0.236725i	0.00719332	−	0.0124592i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.32383	−	9.22115i	−0.278662	−	0.482657i
$366$		0			0
$367$	6.79893	+	11.7761i	0.354901	+	0.614707i	0.987101	−	0.160099i	$-0.0511812\pi$
$367$	6.79893	+	11.7761i	0.354901	+	0.614707i	−0.632200	+	0.774805i	$0.717848\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.9641	−	18.9904i	0.567700	−	0.983285i	−0.429093	−	0.903260i	$-0.641167\pi$
$373$	10.9641	−	18.9904i	0.567700	−	0.983285i	0.996793	−	0.0800246i	$-0.0254999\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	49.5801			2.55351
$378$		0			0
$379$	−29.7965			−1.53054			−0.765271	−	0.643708i	$-0.777395\pi$
$379$	−29.7965			−1.53054			−0.765271	+	0.643708i	$0.777395\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.0109905	+	0.0190361i	−0.000561587	+	0.000972697i	−0.866306	−	0.499514i	$-0.833512\pi$
$383$	−0.0109905	+	0.0190361i	−0.000561587	+	0.000972697i	0.865744	+	0.500486i	$0.166845\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.6783	−	30.6197i	−0.896326	−	1.55248i	−0.832155	−	0.554543i	$-0.812893\pi$
$389$	−17.6783	−	30.6197i	−0.896326	−	1.55248i	−0.0641702	−	0.997939i	$-0.520440\pi$
$390$		0			0
$391$	13.2434	+	22.9382i	0.669746	+	1.16003i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.91741	−	13.7134i	0.398368	−	0.689994i
$396$		0			0
$397$	8.47150	+	14.6731i	0.425172	+	0.736420i	0.996436	−	0.0843464i	$-0.0268802\pi$
$397$	8.47150	+	14.6731i	0.425172	+	0.736420i	−0.571264	+	0.820766i	$0.693547\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.48181	+	2.56657i	−0.0739982	+	0.128169i	−0.900650	−	0.434545i	$-0.856909\pi$
$401$	−1.48181	+	2.56657i	−0.0739982	+	0.128169i	0.826652	+	0.562713i	$0.190243\pi$
$402$		0			0
$403$	26.2345	+	45.4395i	1.30683	+	2.26350i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.68862	+	2.92478i	−0.0837018	+	0.144976i
$408$		0			0
$409$	−14.6549			−0.724636			−0.362318	−	0.932054i	$-0.618015\pi$
$409$	−14.6549			−0.724636			−0.362318	+	0.932054i	$0.618015\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.82743	−	3.16520i	−0.0897050	−	0.155374i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.6352	−	21.8848i	−0.617270	−	1.06914i	−0.989982	−	0.141196i	$-0.954905\pi$
$419$	−12.6352	−	21.8848i	−0.617270	−	1.06914i	0.372711	−	0.927947i	$-0.378428\pi$
$420$		0			0
$421$	7.99854	−	13.8539i	0.389825	−	0.675196i	−0.602601	−	0.798043i	$-0.705869\pi$
$421$	7.99854	−	13.8539i	0.389825	−	0.675196i	0.992426	+	0.122846i	$0.0392022\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.75876			0.230834
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.51673	−	11.2873i	0.313900	−	0.543690i	−0.665303	−	0.746573i	$-0.731698\pi$
$431$	6.51673	−	11.2873i	0.313900	−	0.543690i	0.979203	+	0.202883i	$0.0650311\pi$
$432$		0			0
$433$	23.5467			1.13158			0.565791	−	0.824549i	$-0.308571\pi$
$433$	23.5467			1.13158			0.565791	+	0.824549i	$0.308571\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−25.3887			−1.21451
$438$		0			0
$439$	6.70895			0.320200			0.160100	−	0.987101i	$-0.448818\pi$
$439$	6.70895			0.320200			0.160100	+	0.987101i	$0.448818\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−35.2455			−1.67456			−0.837282	−	0.546771i	$-0.815857\pi$
$443$	−35.2455			−1.67456			−0.837282	+	0.546771i	$0.815857\pi$
$444$		0			0
$445$	24.1986			1.14712
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.9387			0.610616			0.305308	−	0.952254i	$-0.401241\pi$
$449$	12.9387			0.610616			0.305308	+	0.952254i	$0.401241\pi$
$450$		0			0
$451$	−0.635211	+	1.10022i	−0.0299109	+	0.0518072i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−29.1986			−1.36585			−0.682927	−	0.730487i	$-0.739293\pi$
$457$	−29.1986			−1.36585			−0.682927	+	0.730487i	$0.739293\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.34348	+	16.1834i	−0.435169	+	0.753735i	−0.997309	−	0.0733066i	$-0.976645\pi$
$461$	−9.34348	+	16.1834i	−0.435169	+	0.753735i	0.562140	+	0.827042i	$0.309978\pi$
$462$		0			0
$463$	19.1249	+	33.1253i	0.888809	+	1.53946i	0.841285	+	0.540593i	$0.181800\pi$
$463$	19.1249	+	33.1253i	0.888809	+	1.53946i	0.0475247	+	0.998870i	$0.484867\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.64387	−	13.2396i	−0.353716	−	0.612654i	0.633181	−	0.774004i	$-0.281749\pi$
$467$	−7.64387	−	13.2396i	−0.353716	−	0.612654i	−0.986897	+	0.161349i	$0.948415\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	22.4706			1.03320
$474$		0			0
$475$	−2.28074	+	3.95035i	−0.104647	+	0.181255i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.51605	+	9.55408i	0.252035	+	0.436537i	0.964086	−	0.265591i	$-0.0855669\pi$
$479$	5.51605	+	9.55408i	0.252035	+	0.436537i	−0.712051	+	0.702128i	$0.752234\pi$
$480$		0			0
$481$	2.58259	−	4.47318i	0.117756	−	0.203959i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.606032	−	1.04968i	−0.0275185	−	0.0476635i
$486$		0			0
$487$	−8.30039	+	14.3767i	−0.376126	+	0.651470i	−0.990495	−	0.137549i	$-0.956078\pi$
$487$	−8.30039	+	14.3767i	−0.376126	+	0.651470i	0.614368	+	0.789019i	$0.289411\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.3633	−	23.1460i	−0.603079	−	1.04456i	−0.992352	−	0.123440i	$-0.960607\pi$
$491$	−13.3633	−	23.1460i	−0.603079	−	1.04456i	0.389273	−	0.921122i	$-0.372726\pi$
$492$		0			0
$493$	−15.7448	−	27.2709i	−0.709112	−	1.22822i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.618485	+	1.07125i	−0.0276872	+	0.0479557i	−0.879537	−	0.475830i	$-0.842148\pi$
$499$	−0.618485	+	1.07125i	−0.0276872	+	0.0479557i	0.851850	+	0.523786i	$0.175481\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.07179			0.0477889			0.0238944	−	0.999714i	$-0.492393\pi$
$503$	1.07179			0.0477889			0.0238944	+	0.999714i	$0.492393\pi$
$504$		0			0
$505$	8.38151			0.372973
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.0344	−	17.3801i	0.444768	−	0.770362i	−0.553268	−	0.833004i	$-0.686619\pi$
$509$	10.0344	−	17.3801i	0.444768	−	0.770362i	0.998036	+	0.0626420i	$0.0199526\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.35087	+	11.0000i	0.279853	+	0.484720i
$516$		0			0
$517$	−8.53443	−	14.7821i	−0.375344	−	0.650115i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.4430	+	26.7480i	−0.676570	+	1.17185i	0.299438	+	0.954116i	$0.403201\pi$
$521$	−15.4430	+	26.7480i	−0.676570	+	1.17185i	−0.976007	+	0.217737i	$0.930132\pi$
$522$		0			0
$523$	3.69961	+	6.40792i	0.161773	+	0.280199i	0.935505	−	0.353315i	$-0.114946\pi$
$523$	3.69961	+	6.40792i	0.161773	+	0.280199i	−0.773732	+	0.633513i	$0.781612\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.6623	−	28.8599i	0.725819	−	1.25716i
$528$		0			0
$529$	−5.71041	−	9.89072i	−0.248279	−	0.430031i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.971495	−	1.68268i	0.0420801	−	0.0728849i
$534$		0			0
$535$	−14.1800			−0.613053
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−11.3348	−	19.6325i	−0.487322	−	0.844067i	0.512572	−	0.858644i	$-0.328693\pi$
$541$	−11.3348	−	19.6325i	−0.487322	−	0.844067i	−0.999894	+	0.0145779i	$0.995360\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.0505	−	19.1400i	−0.473351	−	0.819868i
$546$		0			0
$547$	3.07373	−	5.32386i	0.131423	−	0.227632i	−0.792802	−	0.609479i	$-0.791379\pi$
$547$	3.07373	−	5.32386i	0.131423	−	0.227632i	0.924225	+	0.381847i	$0.124712\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	30.1842			1.28589
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.8370	−	25.6984i	0.628662	−	1.08887i	−0.359158	−	0.933277i	$-0.616936\pi$
$557$	14.8370	−	25.6984i	0.628662	−	1.08887i	0.987820	−	0.155598i	$-0.0497305\pi$
$558$		0			0
$559$	−34.3667			−1.45356
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.3111			1.23531			0.617657	−	0.786447i	$-0.288082\pi$
$563$	29.3111			1.23531			0.617657	+	0.786447i	$0.288082\pi$
$564$		0			0
$565$	−3.34514			−0.140731
$566$		0			0
$567$		0			0
$568$		0			0
$569$	36.8860			1.54634			0.773170	−	0.634198i	$-0.218670\pi$
$569$	36.8860			1.54634			0.773170	+	0.634198i	$0.218670\pi$
$570$		0			0
$571$	32.3786			1.35500			0.677501	−	0.735522i	$-0.263063\pi$
$571$	32.3786			1.35500			0.677501	+	0.735522i	$0.263063\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.18423			−0.257900
$576$		0			0
$577$	−11.5093	+	19.9348i	−0.479140	+	0.829895i	−0.999714	−	0.0239220i	$-0.992385\pi$
$577$	−11.5093	+	19.9348i	−0.479140	+	0.829895i	0.520574	+	0.853817i	$0.325718\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	23.4897			0.972843
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.87052	−	4.97189i	0.118479	−	0.205212i	−0.800686	−	0.599084i	$-0.795531\pi$
$587$	2.87052	−	4.97189i	0.118479	−	0.205212i	0.919165	+	0.393872i	$0.128865\pi$
$588$		0			0
$589$	15.9715	+	27.6634i	0.658094	+	1.13985i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.8727	+	24.0282i	0.569682	+	0.986718i	0.996597	+	0.0824263i	$0.0262669\pi$
$593$	13.8727	+	24.0282i	0.569682	+	0.986718i	−0.426915	+	0.904292i	$0.640400\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.10817			−0.167855			−0.0839276	−	0.996472i	$-0.526746\pi$
$599$	−4.10817			−0.167855			−0.0839276	+	0.996472i	$0.526746\pi$
$600$		0			0
$601$	−7.80924	+	13.5260i	−0.318546	+	0.551737i	−0.980185	−	0.198085i	$-0.936528\pi$
$601$	−7.80924	+	13.5260i	−0.318546	+	0.551737i	0.661639	+	0.749822i	$0.269861\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.0416	+	22.5888i	0.530218	+	0.918364i
$606$		0			0
$607$	0.280738	−	0.486253i	0.0113948	−	0.0197364i	−0.860272	−	0.509836i	$-0.829706\pi$
$607$	0.280738	−	0.486253i	0.0113948	−	0.0197364i	0.871667	+	0.490099i	$0.163040\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.0526	+	22.6078i	0.528053	+	0.914614i
$612$		0			0
$613$	10.1008	−	17.4951i	0.407967	−	0.706619i	−0.586695	−	0.809808i	$-0.699571\pi$
$613$	10.1008	−	17.4951i	0.407967	−	0.706619i	0.994662	+	0.103189i	$0.0329047\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.4569	−	19.8439i	−0.461238	−	0.798887i	0.537785	−	0.843082i	$-0.319261\pi$
$617$	−11.4569	−	19.8439i	−0.461238	−	0.798887i	−0.999023	+	0.0441948i	$0.985928\pi$
$618$		0			0
$619$	−19.8515	−	34.3839i	−0.797901	−	1.38201i	−0.920981	−	0.389608i	$-0.872610\pi$
$619$	−19.8515	−	34.3839i	−0.797901	−	1.38201i	0.123080	−	0.992397i	$-0.460723\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.5797	−	25.2527i	0.583187	−	1.01011i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.28054			−0.130804
$630$		0			0
$631$	−31.0364			−1.23554			−0.617769	−	0.786359i	$-0.711963\pi$
$631$	−31.0364			−1.23554			−0.617769	+	0.786359i	$0.711963\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.00885	+	1.74739i	−0.0400352	+	0.0693429i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.7932	−	25.6226i	−0.584296	−	1.01203i	−0.994963	−	0.100245i	$-0.968037\pi$
$641$	−14.7932	−	25.6226i	−0.584296	−	1.01203i	0.410667	−	0.911785i	$-0.365296\pi$
$642$		0			0
$643$	−12.8442	−	22.2467i	−0.506524	−	0.877325i	−0.999972	−	0.00754978i	$-0.997597\pi$
$643$	−12.8442	−	22.2467i	−0.506524	−	0.877325i	0.493447	−	0.869776i	$-0.335737\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.50885	−	14.7378i	0.334518	−	0.579401i	−0.648874	−	0.760895i	$-0.724760\pi$
$647$	8.50885	−	14.7378i	0.334518	−	0.579401i	0.983392	+	0.181494i	$0.0580933\pi$
$648$		0			0
$649$	−21.2237	−	36.7606i	−0.833104	−	1.44298i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.735508	−	1.27394i	0.0287827	−	0.0498530i	−0.851275	−	0.524719i	$-0.824170\pi$
$653$	0.735508	−	1.27394i	0.0287827	−	0.0498530i	0.880058	+	0.474866i	$0.157504\pi$
$654$		0			0
$655$	−9.58113	−	16.5950i	−0.374366	−	0.648420i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.7003	+	35.8539i	−0.806369	+	1.39667i	0.108995	+	0.994042i	$0.465237\pi$
$659$	−20.7003	+	35.8539i	−0.806369	+	1.39667i	−0.915363	+	0.402629i	$0.868096\pi$
$660$		0			0
$661$	38.2704			1.48855			0.744273	−	0.667875i	$-0.232796\pi$
$661$	38.2704			1.48855			0.744273	+	0.667875i	$0.232796\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	20.4612	+	35.4398i	0.792260	+	1.37223i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−32.1086	−	55.6138i	−1.23954	−	2.14695i
$672$		0			0
$673$	15.2448	−	26.4048i	0.587645	−	1.01783i	−0.406894	−	0.913475i	$-0.633388\pi$
$673$	15.2448	−	26.4048i	0.587645	−	1.01783i	0.994540	−	0.104357i	$-0.0332783\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−44.9253			−1.72662			−0.863309	−	0.504675i	$-0.831612\pi$
$677$	−44.9253			−1.72662			−0.863309	+	0.504675i	$0.831612\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.1986	+	41.9133i	−0.925935	+	1.60377i	−0.135884	+	0.990725i	$0.543387\pi$
$683$	−24.1986	+	41.9133i	−0.925935	+	1.60377i	−0.790051	+	0.613041i	$0.789946\pi$
$684$		0			0
$685$	−7.36381			−0.281357
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−35.9253			−1.36864
$690$		0			0
$691$	18.3815			0.699266			0.349633	−	0.936887i	$-0.386306\pi$
$691$	18.3815			0.699266			0.349633	+	0.936887i	$0.386306\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−15.5687			−0.590553
$696$		0			0
$697$	−1.23405			−0.0467428
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.0292			1.02088			0.510439	−	0.859914i	$-0.329483\pi$
$701$	27.0292			1.02088			0.510439	+	0.859914i	$0.329483\pi$
$702$		0			0
$703$	1.57227	−	2.72325i	0.0592994	−	0.102710i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.98522			0.187224			0.0936119	−	0.995609i	$-0.470159\pi$
$709$	4.98522			0.187224			0.0936119	+	0.995609i	$0.470159\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−21.6534	+	37.5048i	−0.810926	+	1.40457i
$714$		0			0
$715$	−40.6431	−	70.3959i	−1.51997	−	2.63266i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.84708	+	13.5915i	0.292647	+	0.506879i	0.974435	−	0.224671i	$-0.0721307\pi$
$719$	7.84708	+	13.5915i	0.292647	+	0.506879i	−0.681788	+	0.731550i	$0.738797\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.35234			0.273059
$726$		0			0
$727$	−10.9071	+	18.8916i	−0.404522	+	0.700652i	−0.994266	−	0.106938i	$-0.965895\pi$
$727$	−10.9071	+	18.8916i	−0.404522	+	0.700652i	0.589744	+	0.807590i	$0.299229\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.9136	+	18.9029i	0.403655	+	0.699151i
$732$		0			0
$733$	12.0074	−	20.7974i	0.443503	−	0.768170i	−0.554443	−	0.832221i	$-0.687069\pi$
$733$	12.0074	−	20.7974i	0.443503	−	0.768170i	0.997947	+	0.0640514i	$0.0204022\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.08472	−	5.34290i	−0.113627	−	0.196808i
$738$		0			0
$739$	−9.35447	+	16.2024i	−0.344110	+	0.596016i	−0.985192	−	0.171457i	$-0.945153\pi$
$739$	−9.35447	+	16.2024i	−0.344110	+	0.596016i	0.641082	+	0.767473i	$0.278486\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.1534	−	34.9067i	−0.739356	−	1.28060i	−0.952785	−	0.303644i	$-0.901797\pi$
$743$	−20.1534	−	34.9067i	−0.739356	−	1.28060i	0.213429	−	0.976959i	$-0.431537\pi$
$744$		0			0
$745$	5.39037	+	9.33639i	0.197488	+	0.342059i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	10.5629	−	18.2955i	0.385447	−	0.667614i	−0.606384	−	0.795172i	$-0.707381\pi$
$751$	10.5629	−	18.2955i	0.385447	−	0.667614i	0.991831	+	0.127558i	$0.0407139\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.2412			−0.591079
$756$		0			0
$757$	8.85934			0.321998			0.160999	−	0.986955i	$-0.448528\pi$
$757$	8.85934			0.321998			0.160999	+	0.986955i	$0.448528\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.694551	−	1.20300i	0.0251774	−	0.0436086i	−0.853162	−	0.521646i	$-0.825318\pi$
$761$	0.694551	−	1.20300i	0.0251774	−	0.0436086i	0.878340	+	0.478037i	$0.158652\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32.4597	+	56.2219i	1.17205	+	2.03005i
$768$		0			0
$769$	18.9626	+	32.8443i	0.683810	+	1.18439i	0.973809	+	0.227367i	$0.0730118\pi$
$769$	18.9626	+	32.8443i	0.683810	+	1.18439i	−0.289999	+	0.957027i	$0.593655\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.657981	−	1.13966i	0.0236659	−	0.0409906i	−0.853950	−	0.520355i	$-0.825800\pi$
$773$	0.657981	−	1.13966i	0.0236659	−	0.0409906i	0.877616	+	0.479365i	$0.159133\pi$
$774$		0			0
$775$	3.89037	+	6.73832i	0.139746	+	0.242047i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.591443	−	1.02441i	0.0211907	−	0.0367033i
$780$		0			0
$781$	−31.4224	−	54.4251i	−1.12438	−	1.94748i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.11177	+	12.3179i	−0.253830	+	0.439646i
$786$		0			0
$787$	12.2586			0.436971			0.218485	−	0.975840i	$-0.429888\pi$
$787$	12.2586			0.436971			0.218485	+	0.975840i	$0.429888\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	49.1072	+	85.0561i	1.74385	+	3.02043i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.7178	+	18.5638i	0.379644	+	0.657563i	0.991010	−	0.133785i	$-0.0427132\pi$
$797$	10.7178	+	18.5638i	0.379644	+	0.657563i	−0.611366	+	0.791348i	$0.709380\pi$
$798$		0			0
$799$	8.29007	−	14.3588i	0.293282	−	0.507979i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	20.1124			0.709753
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13.3478	−	23.1190i	0.469282	−	0.812820i	−0.530101	−	0.847934i	$-0.677846\pi$
$809$	13.3478	−	23.1190i	0.469282	−	0.812820i	0.999383	+	0.0351140i	$0.0111794\pi$
$810$		0			0
$811$	38.2852			1.34438			0.672188	−	0.740381i	$-0.265355\pi$
$811$	38.2852			1.34438			0.672188	+	0.740381i	$0.265355\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.0292			−0.631535
$816$		0			0
$817$	−20.9224			−0.731981
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.4998			0.366446			0.183223	−	0.983071i	$-0.441347\pi$
$821$	10.4998			0.366446			0.183223	+	0.983071i	$0.441347\pi$
$822$		0			0
$823$	16.0000			0.557725			0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000			0.557725			0.278862	+	0.960331i	$0.410043\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	48.7817			1.69631			0.848153	−	0.529752i	$-0.177715\pi$
$827$	48.7817			1.69631			0.848153	+	0.529752i	$0.177715\pi$
$828$		0			0
$829$	3.10963	−	5.38604i	0.108002	−	0.187065i	−0.806959	−	0.590608i	$-0.798888\pi$
$829$	3.10963	−	5.38604i	0.108002	−	0.187065i	0.914961	+	0.403543i	$0.132221\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−29.5979			−1.02428
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.0366	+	36.4364i	−0.726263	+	1.25792i	0.232189	+	0.972671i	$0.425411\pi$
$839$	−21.0366	+	36.4364i	−0.726263	+	1.25792i	−0.958452	+	0.285254i	$0.907922\pi$
$840$		0			0
$841$	−9.82597	−	17.0191i	−0.338826	−	0.586865i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	46.1665	+	79.9628i	1.58818	+	2.75080i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.26322			0.146141
$852$		0			0
$853$	6.72519	−	11.6484i	0.230266	−	0.398833i	−0.727620	−	0.685980i	$-0.759374\pi$
$853$	6.72519	−	11.6484i	0.230266	−	0.398833i	0.957886	+	0.287147i	$0.0927070\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.6893	+	35.8349i	0.706733	+	1.22410i	0.966063	+	0.258308i	$0.0831650\pi$
$857$	20.6893	+	35.8349i	0.706733	+	1.22410i	−0.259330	+	0.965789i	$0.583502\pi$
$858$		0			0
$859$	19.8815	−	34.4358i	0.678349	−	1.17493i	−0.297129	−	0.954837i	$-0.596029\pi$
$859$	19.8815	−	34.4358i	0.678349	−	1.17493i	0.975478	−	0.220097i	$-0.0706374\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.6929	+	46.2334i	0.908637	+	1.57380i	0.815960	+	0.578109i	$0.196209\pi$
$863$	26.6929	+	46.2334i	0.908637	+	1.57380i	0.0926768	+	0.995696i	$0.470458\pi$
$864$		0			0
$865$	−6.02704	+	10.4391i	−0.204926	+	0.354942i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.9552	+	25.9033i	0.507322	+	0.878708i
$870$		0			0
$871$	4.71780	+	8.17147i	0.159857	+	0.276880i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.42674	−	5.93530i	0.115713	−	0.200421i	−0.802352	−	0.596852i	$-0.796418\pi$
$877$	3.42674	−	5.93530i	0.115713	−	0.200421i	0.918065	+	0.396431i	$0.129751\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.5103			0.421483			0.210742	−	0.977542i	$-0.432412\pi$
$881$	12.5103			0.421483			0.210742	+	0.977542i	$0.432412\pi$
$882$		0			0
$883$	6.69124			0.225178			0.112589	−	0.993642i	$-0.464086\pi$
$883$	6.69124			0.225178			0.112589	+	0.993642i	$0.464086\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.0708	−	27.8355i	0.539605	−	0.934623i	−0.459320	−	0.888271i	$-0.651907\pi$
$887$	16.0708	−	27.8355i	0.539605	−	0.934623i	0.998925	−	0.0463524i	$-0.0147597\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.94639	+	13.7636i	0.265916	+	0.460580i
$894$		0			0
$895$	2.19076	+	3.79450i	0.0732289	+	0.126836i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	25.7434	−	44.5888i	0.858590	−	1.48712i
$900$		0			0
$901$	11.4086	+	19.7602i	0.380074	+	0.658308i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.8442	+	36.1031i	−0.692883	+	1.20011i
$906$		0			0
$907$	15.7016	+	27.1959i	0.521362	+	0.903025i	0.999691	+	0.0248444i	$0.00790902\pi$
$907$	15.7016	+	27.1959i	0.521362	+	0.903025i	−0.478330	+	0.878180i	$0.658758\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.8982	−	39.6609i	0.758653	−	1.31402i	−0.184885	−	0.982760i	$-0.559191\pi$
$911$	22.8982	−	39.6609i	0.758653	−	1.31402i	0.943538	−	0.331265i	$-0.107475\pi$
$912$		0			0
$913$	6.90369			0.228479
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−13.5900	−	23.5385i	−0.448292	−	0.776465i	0.549983	−	0.835176i	$-0.314634\pi$
$919$	−13.5900	−	23.5385i	−0.448292	−	0.776465i	−0.998275	+	0.0587112i	$0.981301\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	48.0576	+	83.2381i	1.58183	+	2.73982i
$924$		0			0
$925$	0.382977	−	0.663336i	0.0125922	−	0.0218104i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	40.6677			1.33426			0.667132	−	0.744940i	$-0.267522\pi$
$929$	40.6677			1.33426			0.667132	+	0.744940i	$0.267522\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−25.8135	+	44.7103i	−0.844192	+	1.46218i
$936$		0			0
$937$	16.4150			0.536254			0.268127	−	0.963384i	$-0.413595\pi$
$937$	16.4150			0.536254			0.268127	+	0.963384i	$0.413595\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.33755			−0.239197			−0.119599	−	0.992822i	$-0.538161\pi$
$941$	−7.33755			−0.239197			−0.119599	+	0.992822i	$0.538161\pi$
$942$		0			0
$943$	1.60370			0.0522237
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−59.1124			−1.92090			−0.960448	−	0.278459i	$-0.910176\pi$
$947$	−59.1124			−1.92090			−0.960448	+	0.278459i	$0.910176\pi$
$948$		0			0
$949$	−30.7601			−0.998515
$950$		0			0
$951$		0			0
$952$		0			0
$953$	16.9354			0.548592			0.274296	−	0.961645i	$-0.411555\pi$
$953$	16.9354			0.548592			0.274296	+	0.961645i	$0.411555\pi$
$954$		0			0
$955$	6.75370	−	11.6977i	0.218544	−	0.378530i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	23.4868			0.757637
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.77548	+	11.7355i	−0.218110	+	0.377778i
$966$		0			0
$967$	3.55555	+	6.15839i	0.114339	+	0.198040i	0.917515	−	0.397701i	$-0.130192\pi$
$967$	3.55555	+	6.15839i	0.114339	+	0.198040i	−0.803177	+	0.595741i	$0.796858\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.735508	−	1.27394i	−0.0236036	−	0.0408826i	0.853982	−	0.520302i	$-0.174181\pi$
$971$	−0.735508	−	1.27394i	−0.0236036	−	0.0408826i	−0.877586	+	0.479419i	$0.840847\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.4327			0.621706			0.310853	−	0.950458i	$-0.399385\pi$
$977$	19.4327			0.621706			0.310853	+	0.950458i	$0.399385\pi$
$978$		0			0
$979$	−22.8545	+	39.5851i	−0.730432	+	1.26515i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.87218	+	6.70681i	0.123503	+	0.213914i	0.921147	−	0.389215i	$-0.127254\pi$
$983$	3.87218	+	6.70681i	0.123503	+	0.213914i	−0.797644	+	0.603129i	$0.793920\pi$
$984$		0			0
$985$	−14.3078	+	24.7818i	−0.455884	+	0.789614i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.1828	−	24.5653i	−0.450986	−	0.781130i
$990$		0			0
$991$	7.23551	−	12.5323i	0.229843	−	0.398101i	−0.727918	−	0.685664i	$-0.759512\pi$
$991$	7.23551	−	12.5323i	0.229843	−	0.398101i	0.957762	+	0.287563i	$0.0928452\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.10243	−	8.83767i	−0.161758	−	0.280173i
$996$		0			0
$997$	27.6549	+	47.8996i	0.875838	+	1.51700i	0.855867	+	0.517196i	$0.173024\pi$
$997$	27.6549	+	47.8996i	0.875838	+	1.51700i	0.0199711	+	0.999801i	$0.493643\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.i.f.2125.3		6
3.2	odd	2		1764.2.i.g.1537.3		6
7.2	even	3		5292.2.l.e.3313.1		6
7.3	odd	6		5292.2.j.d.3529.1		6
7.4	even	3		756.2.j.b.505.3		6
7.5	odd	6		5292.2.l.f.3313.3		6
7.6	odd	2		5292.2.i.e.2125.1		6
9.4	even	3		5292.2.l.e.361.1		6
9.5	odd	6		1764.2.l.e.949.2		6
21.2	odd	6		1764.2.l.e.961.2		6
21.5	even	6		1764.2.l.f.961.2		6
21.11	odd	6		252.2.j.a.169.2	yes	6
21.17	even	6		1764.2.j.e.1177.2		6
21.20	even	2		1764.2.i.d.1537.1		6
28.11	odd	6		3024.2.r.j.2017.3		6
63.4	even	3		756.2.j.b.253.3		6
63.5	even	6		1764.2.i.d.373.1		6
63.11	odd	6		2268.2.a.i.1.3		3
63.13	odd	6		5292.2.l.f.361.3		6
63.23	odd	6		1764.2.i.g.373.3		6
63.25	even	3		2268.2.a.h.1.1		3
63.31	odd	6		5292.2.j.d.1765.1		6
63.32	odd	6		252.2.j.a.85.2	✓	6
63.40	odd	6		5292.2.i.e.1549.1		6
63.41	even	6		1764.2.l.f.949.2		6
63.58	even	3	inner	5292.2.i.f.1549.3		6
63.59	even	6		1764.2.j.e.589.2		6
84.11	even	6		1008.2.r.j.673.2		6
252.11	even	6		9072.2.a.by.1.3		3
252.67	odd	6		3024.2.r.j.1009.3		6
252.95	even	6		1008.2.r.j.337.2		6
252.151	odd	6		9072.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.a.85.2	✓	6	63.32	odd	6
252.2.j.a.169.2	yes	6	21.11	odd	6
756.2.j.b.253.3		6	63.4	even	3
756.2.j.b.505.3		6	7.4	even	3
1008.2.r.j.337.2		6	252.95	even	6
1008.2.r.j.673.2		6	84.11	even	6
1764.2.i.d.373.1		6	63.5	even	6
1764.2.i.d.1537.1		6	21.20	even	2
1764.2.i.g.373.3		6	63.23	odd	6
1764.2.i.g.1537.3		6	3.2	odd	2
1764.2.j.e.589.2		6	63.59	even	6
1764.2.j.e.1177.2		6	21.17	even	6
1764.2.l.e.949.2		6	9.5	odd	6
1764.2.l.e.961.2		6	21.2	odd	6
1764.2.l.f.949.2		6	63.41	even	6
1764.2.l.f.961.2		6	21.5	even	6
2268.2.a.h.1.1		3	63.25	even	3
2268.2.a.i.1.3		3	63.11	odd	6
3024.2.r.j.1009.3		6	252.67	odd	6
3024.2.r.j.2017.3		6	28.11	odd	6
5292.2.i.e.1549.1		6	63.40	odd	6
5292.2.i.e.2125.1		6	7.6	odd	2
5292.2.i.f.1549.3		6	63.58	even	3	inner
5292.2.i.f.2125.3		6	1.1	even	1	trivial
5292.2.j.d.1765.1		6	63.31	odd	6
5292.2.j.d.3529.1		6	7.3	odd	6
5292.2.l.e.361.1		6	9.4	even	3
5292.2.l.e.3313.1		6	7.2	even	3
5292.2.l.f.361.3		6	63.13	odd	6
5292.2.l.f.3313.3		6	7.5	odd	6
9072.2.a.bv.1.1		3	252.151	odd	6
9072.2.a.by.1.3		3	252.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.23025 - 2.13086i) q^{5} +O(q^{10})\)	\(q+(1.23025 - 2.13086i) q^{5} +(2.32383 + 4.02499i) q^{11} +(-3.55408 - 6.15585i) q^{13} +(-2.25729 + 3.90975i) q^{17} +(-2.16372 - 3.74766i) q^{19} +(2.93346 - 5.08091i) q^{23} +(-0.527042 - 0.912864i) q^{25} +(-3.48755 + 6.04061i) q^{29} -7.38151 q^{31} +(0.363327 + 0.629301i) q^{37} +(0.136673 + 0.236725i) q^{41} +(2.41741 - 4.18708i) q^{43} -3.67257 q^{47} +(2.52704 - 4.37697i) q^{53} +11.4356 q^{55} -9.13307 q^{59} -13.8171 q^{61} -17.4897 q^{65} -1.32743 q^{67} -13.5218 q^{71} +(2.16372 - 3.74766i) q^{73} +6.43560 q^{79} +(0.742705 - 1.28640i) q^{83} +(5.55408 + 9.61996i) q^{85} +(4.91741 + 8.51721i) q^{89} -10.6477 q^{95} +(0.246304 - 0.426611i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5}+O(q^{10}) \) 6 * q + q^5	\( 6 q + q^{5} + 2 q^{11} - 3 q^{13} + 2 q^{17} - 3 q^{19} + 14 q^{23} + 6 q^{25} + q^{29} - 6 q^{31} + 3 q^{37} - 3 q^{43} - 42 q^{47} + 6 q^{53} + 12 q^{55} - 62 q^{59} + 12 q^{61} - 30 q^{65} + 12 q^{67} - 34 q^{71} + 3 q^{73} - 18 q^{79} + 20 q^{83} + 15 q^{85} + 12 q^{89} - 40 q^{95} + 9 q^{97}+O(q^{100}) \) 6 * q + q^5 + 2 * q^11 - 3 * q^13 + 2 * q^17 - 3 * q^19 + 14 * q^23 + 6 * q^25 + q^29 - 6 * q^31 + 3 * q^37 - 3 * q^43 - 42 * q^47 + 6 * q^53 + 12 * q^55 - 62 * q^59 + 12 * q^61 - 30 * q^65 + 12 * q^67 - 34 * q^71 + 3 * q^73 - 18 * q^79 + 20 * q^83 + 15 * q^85 + 12 * q^89 - 40 * q^95 + 9 * q^97

Introduction

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