LMFDB - Embedded newform 1620.4.i.r.1081.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.2
Root			$1.39564 + 0.228425i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.r.541.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{5} +(7.37386 + 12.7719i) q^{7} +O(q^{10})$	$q+(2.50000 - 4.33013i) q^{5} +(7.37386 + 12.7719i) q^{7} +(-3.62614 - 6.28065i) q^{11} +(-30.8693 + 53.4672i) q^{13} +108.234 q^{17} -56.2523 q^{19} +(23.4909 - 40.6874i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(107.112 + 185.524i) q^{29} +(130.856 - 226.649i) q^{31} +73.7386 q^{35} -286.000 q^{37} +(-127.851 + 221.445i) q^{41} +(180.617 + 312.838i) q^{43} +(2.76591 + 4.79070i) q^{47} +(62.7523 - 108.690i) q^{49} -595.693 q^{53} -36.2614 q^{55} +(-157.617 + 273.001i) q^{59} +(-138.202 - 239.373i) q^{61} +(154.347 + 267.336i) q^{65} +(-58.7023 + 101.675i) q^{67} +192.784 q^{71} -756.189 q^{73} +(53.4773 - 92.6254i) q^{77} +(575.351 + 996.537i) q^{79} +(-70.9682 - 122.920i) q^{83} +(270.585 - 468.667i) q^{85} +719.107 q^{89} -910.505 q^{91} +(-140.631 + 243.579i) q^{95} +(521.909 + 903.973i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q + 10 * q^5 + 2 * q^7	$ 4 q + 10 q^{5} + 2 q^{7} - 42 q^{11} + 14 q^{13} + 48 q^{17} - 280 q^{19} - 126 q^{23} - 50 q^{25} + 126 q^{29} + 56 q^{31} + 20 q^{35} - 1144 q^{37} + 66 q^{41} + 530 q^{43} + 396 q^{47} + 306 q^{49} - 1008 q^{53} - 420 q^{55} - 438 q^{59} + 602 q^{61} - 70 q^{65} + 920 q^{67} + 1596 q^{71} - 1540 q^{73} - 336 q^{77} + 1724 q^{79} + 486 q^{83} + 120 q^{85} - 588 q^{89} - 3752 q^{91} - 700 q^{95} - 112 q^{97}+O(q^{100}) $ 4 * q + 10 * q^5 + 2 * q^7 - 42 * q^11 + 14 * q^13 + 48 * q^17 - 280 * q^19 - 126 * q^23 - 50 * q^25 + 126 * q^29 + 56 * q^31 + 20 * q^35 - 1144 * q^37 + 66 * q^41 + 530 * q^43 + 396 * q^47 + 306 * q^49 - 1008 * q^53 - 420 * q^55 - 438 * q^59 + 602 * q^61 - 70 * q^65 + 920 * q^67 + 1596 * q^71 - 1540 * q^73 - 336 * q^77 + 1724 * q^79 + 486 * q^83 + 120 * q^85 - 588 * q^89 - 3752 * q^91 - 700 * q^95 - 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	7.37386	+	12.7719i	0.398151	+	0.689618i	0.993498	−	0.113851i	$-0.0363186\pi$
$7$	7.37386	+	12.7719i	0.398151	+	0.689618i	−0.595347	+	0.803469i	$0.702985\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.62614	−	6.28065i	−0.0993928	−	0.172153i	0.812041	−	0.583601i	$-0.198357\pi$
$11$	−3.62614	−	6.28065i	−0.0993928	−	0.172153i	−0.911434	+	0.411447i	$0.865023\pi$
$12$		0			0
$13$	−30.8693	+	53.4672i	−0.658585	+	1.14070i	0.322397	+	0.946605i	$0.395511\pi$
$13$	−30.8693	+	53.4672i	−0.658585	+	1.14070i	−0.980982	+	0.194098i	$0.937822\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	108.234			1.54415			0.772077	−	0.635529i	$-0.219218\pi$
$17$	108.234			1.54415			0.772077	+	0.635529i	$0.219218\pi$
$18$		0			0
$19$	−56.2523			−0.679219			−0.339609	−	0.940567i	$-0.610295\pi$
$19$	−56.2523			−0.679219			−0.339609	+	0.940567i	$0.610295\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	23.4909	−	40.6874i	0.212965	−	0.368866i	−0.739676	−	0.672963i	$-0.765021\pi$
$23$	23.4909	−	40.6874i	0.212965	−	0.368866i	0.952641	+	0.304097i	$0.0983547\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	107.112	+	185.524i	0.685872	+	1.18797i	0.973162	+	0.230122i	$0.0739124\pi$
$29$	107.112	+	185.524i	0.685872	+	1.18797i	−0.287290	+	0.957844i	$0.592754\pi$
$30$		0			0
$31$	130.856	−	226.649i	0.758141	−	1.31314i	−0.185656	−	0.982615i	$-0.559441\pi$
$31$	130.856	−	226.649i	0.758141	−	1.31314i	0.943798	−	0.330524i	$-0.107226\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	73.7386			0.356117
$36$		0			0
$37$	−286.000			−1.27076			−0.635380	−	0.772200i	$-0.719156\pi$
$37$	−286.000			−1.27076			−0.635380	+	0.772200i	$0.719156\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−127.851	+	221.445i	−0.487000	+	0.843508i	−0.999888	−	0.0149468i	$-0.995242\pi$
$41$	−127.851	+	221.445i	−0.487000	+	0.843508i	0.512888	+	0.858455i	$0.328575\pi$
$42$		0			0
$43$	180.617	+	312.838i	0.640554	+	1.10947i	0.985309	+	0.170780i	$0.0546288\pi$
$43$	180.617	+	312.838i	0.640554	+	1.10947i	−0.344755	+	0.938693i	$0.612038\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.76591	+	4.79070i	0.00858403	+	0.0148680i	0.870286	−	0.492548i	$-0.163934\pi$
$47$	2.76591	+	4.79070i	0.00858403	+	0.0148680i	−0.861701	+	0.507416i	$0.830601\pi$
$48$		0			0
$49$	62.7523	−	108.690i	0.182951	−	0.316881i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−595.693			−1.54386			−0.771932	−	0.635706i	$-0.780709\pi$
$53$	−595.693			−1.54386			−0.771932	+	0.635706i	$0.780709\pi$
$54$		0			0
$55$	−36.2614			−0.0888997
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−157.617	+	273.001i	−0.347796	+	0.602401i	−0.985858	−	0.167584i	$-0.946403\pi$
$59$	−157.617	+	273.001i	−0.347796	+	0.602401i	0.638061	+	0.769986i	$0.279737\pi$
$60$		0			0
$61$	−138.202	−	239.373i	−0.290082	−	0.502436i	0.683747	−	0.729719i	$-0.260349\pi$
$61$	−138.202	−	239.373i	−0.290082	−	0.502436i	−0.973829	+	0.227283i	$0.927016\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	154.347	+	267.336i	0.294528	+	0.510138i
$66$		0			0
$67$	−58.7023	+	101.675i	−0.107039	+	0.185397i	−0.914569	−	0.404429i	$-0.867470\pi$
$67$	−58.7023	+	101.675i	−0.107039	+	0.185397i	0.807530	+	0.589826i	$0.200804\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	192.784			0.322243			0.161122	−	0.986935i	$-0.448489\pi$
$71$	192.784			0.322243			0.161122	+	0.986935i	$0.448489\pi$
$72$		0			0
$73$	−756.189			−1.21240			−0.606200	−	0.795312i	$-0.707307\pi$
$73$	−756.189			−1.21240			−0.606200	+	0.795312i	$0.707307\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	53.4773	−	92.6254i	0.0791468	−	0.137086i
$78$		0			0
$79$	575.351	+	996.537i	0.819393	+	1.41923i	0.906130	+	0.422999i	$0.139023\pi$
$79$	575.351	+	996.537i	0.819393	+	1.41923i	−0.0867368	+	0.996231i	$0.527644\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−70.9682	−	122.920i	−0.0938526	−	0.162558i	0.815277	−	0.579072i	$-0.196585\pi$
$83$	−70.9682	−	122.920i	−0.0938526	−	0.162558i	−0.909129	+	0.416514i	$0.863252\pi$
$84$		0			0
$85$	270.585	−	468.667i	0.345283	−	0.598048i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	719.107			0.856463			0.428231	−	0.903669i	$-0.359137\pi$
$89$	719.107			0.856463			0.428231	+	0.903669i	$0.359137\pi$
$90$		0			0
$91$	−910.505			−1.04887
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−140.631	+	243.579i	−0.151878	+	0.263060i
$96$		0			0
$97$	521.909	+	903.973i	0.546308	+	0.946233i	0.998523	+	0.0543240i	$0.0173004\pi$
$97$	521.909	+	903.973i	0.546308	+	0.946233i	−0.452216	+	0.891909i	$0.649366\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−342.936	−	593.983i	−0.337856	−	0.585184i	0.646173	−	0.763191i	$-0.276368\pi$
$101$	−342.936	−	593.983i	−0.337856	−	0.585184i	−0.984029	+	0.178007i	$0.943035\pi$
$102$		0			0
$103$	185.027	−	320.477i	0.177003	−	0.306578i	−0.763850	−	0.645394i	$-0.776693\pi$
$103$	185.027	−	320.477i	0.177003	−	0.306578i	0.940853	+	0.338816i	$0.110027\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1354.85			−1.22409			−0.612046	−	0.790822i	$-0.709653\pi$
$107$	−1354.85			−1.22409			−0.612046	+	0.790822i	$0.709653\pi$
$108$		0			0
$109$	882.873			0.775815			0.387908	−	0.921698i	$-0.373198\pi$
$109$	882.873			0.775815			0.387908	+	0.921698i	$0.373198\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−493.280	+	854.385i	−0.410653	+	0.711273i	−0.994961	−	0.100259i	$-0.968033\pi$
$113$	−493.280	+	854.385i	−0.410653	+	0.711273i	0.584308	+	0.811532i	$0.301366\pi$
$114$		0			0
$115$	−117.455	−	203.437i	−0.0952408	−	0.164962i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	798.103	+	1382.36i	0.614807	+	1.06488i
$120$		0			0
$121$	639.202	−	1107.13i	0.480242	−	0.831804i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−809.405			−0.565536			−0.282768	−	0.959188i	$-0.591253\pi$
$127$	−809.405			−0.565536			−0.282768	+	0.959188i	$0.591253\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−644.064	+	1115.55i	−0.429558	+	0.744016i	−0.996834	−	0.0795115i	$-0.974664\pi$
$131$	−644.064	+	1115.55i	−0.429558	+	0.744016i	0.567276	+	0.823528i	$0.307997\pi$
$132$		0			0
$133$	−414.797	−	718.449i	−0.270432	−	0.468402i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1150.08	+	1992.00i	0.717212	+	1.24225i	0.962100	+	0.272696i	$0.0879153\pi$
$137$	1150.08	+	1992.00i	0.717212	+	1.24225i	−0.244888	+	0.969551i	$0.578751\pi$
$138$		0			0
$139$	679.945	−	1177.70i	0.414908	−	0.718642i	−0.580511	−	0.814253i	$-0.697147\pi$
$139$	679.945	−	1177.70i	0.414908	−	0.718642i	0.995419	+	0.0956108i	$0.0304804\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	447.745			0.261835
$144$		0			0
$145$	1071.12			0.613463
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−116.635	+	202.018i	−0.0641284	+	0.111074i	−0.896307	−	0.443434i	$-0.853760\pi$
$149$	−116.635	+	202.018i	−0.0641284	+	0.111074i	0.832179	+	0.554508i	$0.187093\pi$
$150$		0			0
$151$	38.0000	+	65.8179i	0.0204794	+	0.0354714i	0.876083	−	0.482159i	$-0.160147\pi$
$151$	38.0000	+	65.8179i	0.0204794	+	0.0354714i	−0.855604	+	0.517631i	$0.826814\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−654.278	−	1133.24i	−0.339051	−	0.587254i
$156$		0			0
$157$	−674.301	+	1167.92i	−0.342771	+	0.593697i	−0.984946	−	0.172861i	$-0.944699\pi$
$157$	−674.301	+	1167.92i	−0.342771	+	0.593697i	0.642175	+	0.766558i	$0.278032\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	692.875			0.339169
$162$		0			0
$163$	216.127			0.103855			0.0519276	−	0.998651i	$-0.483463\pi$
$163$	216.127			0.103855			0.0519276	+	0.998651i	$0.483463\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1300.78	+	2253.01i	−0.602738	+	1.04397i	0.389667	+	0.920956i	$0.372590\pi$
$167$	−1300.78	+	2253.01i	−0.602738	+	1.04397i	−0.992405	+	0.123016i	$0.960743\pi$
$168$		0			0
$169$	−807.330	−	1398.34i	−0.367469	−	0.636475i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1281.82	+	2220.18i	0.563323	+	0.975704i	0.997204	+	0.0747339i	$0.0238107\pi$
$173$	1281.82	+	2220.18i	0.563323	+	0.975704i	−0.433880	+	0.900971i	$0.642856\pi$
$174$		0			0
$175$	184.347	−	319.298i	0.0796302	−	0.137924i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3791.98			−1.58338			−0.791692	−	0.610921i	$-0.790799\pi$
$179$	−3791.98			−1.58338			−0.791692	+	0.610921i	$0.790799\pi$
$180$		0			0
$181$	1245.18			0.511346			0.255673	−	0.966763i	$-0.417703\pi$
$181$	1245.18			0.511346			0.255673	+	0.966763i	$0.417703\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−715.000	+	1238.42i	−0.284151	+	0.492163i
$186$		0			0
$187$	−392.472	−	679.781i	−0.153478	−	0.265831i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1733.29	+	3002.14i	0.656629	+	1.13732i	0.981483	+	0.191551i	$0.0613517\pi$
$191$	1733.29	+	3002.14i	0.656629	+	1.13732i	−0.324853	+	0.945764i	$0.605315\pi$
$192$		0			0
$193$	−1786.26	+	3093.89i	−0.666205	+	1.15390i	0.312753	+	0.949835i	$0.398749\pi$
$193$	−1786.26	+	3093.89i	−0.666205	+	1.15390i	−0.978957	+	0.204066i	$0.934584\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−761.048			−0.275241			−0.137620	−	0.990485i	$-0.543945\pi$
$197$	−761.048			−0.275241			−0.137620	+	0.990485i	$0.543945\pi$
$198$		0			0
$199$	764.195			0.272223			0.136112	−	0.990694i	$-0.456539\pi$
$199$	764.195			0.272223			0.136112	+	0.990694i	$0.456539\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1579.67	+	2736.06i	−0.546162	+	0.945980i
$204$		0			0
$205$	639.256	+	1107.22i	0.217793	+	0.377228i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	203.978	+	353.301i	0.0675095	+	0.116930i
$210$		0			0
$211$	−2106.69	+	3648.90i	−0.687349	+	1.19052i	0.285343	+	0.958425i	$0.407892\pi$
$211$	−2106.69	+	3648.90i	−0.687349	+	1.19052i	−0.972692	+	0.232098i	$0.925441\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1806.17			0.572929
$216$		0			0
$217$	3859.65			1.20742
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3341.11	+	5786.98i	−1.01696	+	1.76142i
$222$		0			0
$223$	1703.77	+	2951.02i	0.511628	+	0.886166i	0.999909	+	0.0134794i	$0.00429075\pi$
$223$	1703.77	+	2951.02i	0.511628	+	0.886166i	−0.488281	+	0.872686i	$0.662376\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2819.20	+	4883.00i	0.824304	+	1.42774i	0.902450	+	0.430795i	$0.141767\pi$
$227$	2819.20	+	4883.00i	0.824304	+	1.42774i	−0.0781459	+	0.996942i	$0.524900\pi$
$228$		0			0
$229$	36.5432	−	63.2946i	0.0105452	−	0.0182647i	−0.860705	−	0.509105i	$-0.829977\pi$
$229$	36.5432	−	63.2946i	0.0105452	−	0.0182647i	0.871250	+	0.490840i	$0.163310\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1865.78			0.524598			0.262299	−	0.964987i	$-0.415519\pi$
$233$	1865.78			0.524598			0.262299	+	0.964987i	$0.415519\pi$
$234$		0			0
$235$	27.6591			0.00767779
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1573.80	+	2725.91i	−0.425945	+	0.737758i	−0.996508	−	0.0834956i	$-0.973392\pi$
$239$	−1573.80	+	2725.91i	−0.425945	+	0.737758i	0.570563	+	0.821254i	$0.306725\pi$
$240$		0			0
$241$	−2549.29	−	4415.51i	−0.681388	−	1.18020i	−0.974558	−	0.224138i	$-0.928044\pi$
$241$	−2549.29	−	4415.51i	−0.681388	−	1.18020i	0.293170	−	0.956060i	$-0.405290\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−313.761	−	543.451i	−0.0818183	−	0.141713i
$246$		0			0
$247$	1736.47	−	3007.65i	0.447323	−	0.774787i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6566.90			1.65139			0.825695	−	0.564117i	$-0.190783\pi$
$251$	6566.90			1.65139			0.825695	+	0.564117i	$0.190783\pi$
$252$		0			0
$253$	−340.725			−0.0846688
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3059.31	−	5298.88i	0.742547	−	1.28613i	−0.208785	−	0.977961i	$-0.566951\pi$
$257$	3059.31	−	5298.88i	0.742547	−	1.28613i	0.951332	−	0.308167i	$-0.0997157\pi$
$258$		0			0
$259$	−2108.92	−	3652.77i	−0.505955	−	0.876339i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3634.01	+	6294.28i	0.852025	+	1.47575i	0.879378	+	0.476124i	$0.157959\pi$
$263$	3634.01	+	6294.28i	0.852025	+	1.47575i	−0.0273536	+	0.999626i	$0.508708\pi$
$264$		0			0
$265$	−1489.23	+	2579.43i	−0.345218	+	0.597936i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4292.72			−0.972981			−0.486491	−	0.873686i	$-0.661723\pi$
$269$	−4292.72			−0.972981			−0.486491	+	0.873686i	$0.661723\pi$
$270$		0			0
$271$	−3847.73			−0.862483			−0.431242	−	0.902237i	$-0.641924\pi$
$271$	−3847.73			−0.862483			−0.431242	+	0.902237i	$0.641924\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−90.6534	+	157.016i	−0.0198786	+	0.0344307i
$276$		0			0
$277$	−3885.13	−	6729.24i	−0.842725	−	1.45964i	−0.887583	−	0.460649i	$-0.847617\pi$
$277$	−3885.13	−	6729.24i	−0.842725	−	1.45964i	0.0448579	−	0.998993i	$-0.485716\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3162.52	−	5477.64i	−0.671388	−	1.16288i	−0.977511	−	0.210886i	$-0.932365\pi$
$281$	−3162.52	−	5477.64i	−0.671388	−	1.16288i	0.306123	−	0.951992i	$-0.400968\pi$
$282$		0			0
$283$	−1331.48	+	2306.19i	−0.279676	+	0.484413i	−0.971304	−	0.237841i	$-0.923560\pi$
$283$	−1331.48	+	2306.19i	−0.279676	+	0.484413i	0.691628	+	0.722254i	$0.256894\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3771.03			−0.775598
$288$		0			0
$289$	6801.62			1.38441
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1755.16	+	3040.03i	−0.349958	+	0.606144i	−0.986242	−	0.165311i	$-0.947137\pi$
$293$	−1755.16	+	3040.03i	−0.349958	+	0.606144i	0.636284	+	0.771455i	$0.280471\pi$
$294$		0			0
$295$	788.085	+	1365.00i	0.155539	+	0.269402i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1450.30	+	2511.99i	0.280511	+	0.485859i
$300$		0			0
$301$	−2663.69	+	4613.65i	−0.510075	+	0.883476i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1382.02			−0.259457
$306$		0			0
$307$	−3451.19			−0.641596			−0.320798	−	0.947148i	$-0.603951\pi$
$307$	−3451.19			−0.641596			−0.320798	+	0.947148i	$0.603951\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4695.02	+	8132.02i	−0.856046	+	1.48271i	0.0196256	+	0.999807i	$0.493753\pi$
$311$	−4695.02	+	8132.02i	−0.856046	+	1.48271i	−0.875671	+	0.482907i	$0.839581\pi$
$312$		0			0
$313$	−3558.33	−	6163.21i	−0.642584	−	1.11299i	−0.984854	−	0.173386i	$-0.944529\pi$
$313$	−3558.33	−	6163.21i	−0.642584	−	1.11299i	0.342270	−	0.939602i	$-0.388804\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3109.77	+	5386.27i	0.550984	+	0.954332i	0.998204	+	0.0599082i	$0.0190808\pi$
$317$	3109.77	+	5386.27i	0.550984	+	0.954332i	−0.447220	+	0.894424i	$0.647586\pi$
$318$		0			0
$319$	776.809	−	1345.47i	0.136342	−	0.236151i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6088.41			−1.04882
$324$		0			0
$325$	1543.47			0.263434
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−40.7909	+	70.6519i	−0.00683549	+	0.0118394i
$330$		0			0
$331$	4221.53	+	7311.91i	0.701017	+	1.21420i	0.968110	+	0.250526i	$0.0806035\pi$
$331$	4221.53	+	7311.91i	0.701017	+	1.21420i	−0.267093	+	0.963671i	$0.586063\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	293.511	+	508.377i	0.0478694	+	0.0829122i
$336$		0			0
$337$	−823.915	+	1427.06i	−0.133180	+	0.230674i	−0.924901	−	0.380209i	$-0.875852\pi$
$337$	−823.915	+	1427.06i	−0.133180	+	0.230674i	0.791721	+	0.610883i	$0.209185\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1898.00			−0.301415
$342$		0			0
$343$	6909.38			1.08767
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1231.24	+	2132.57i	−0.190480	+	0.329921i	−0.945409	−	0.325885i	$-0.894338\pi$
$347$	−1231.24	+	2132.57i	−0.190480	+	0.329921i	0.754930	+	0.655806i	$0.227671\pi$
$348$		0			0
$349$	4063.50	+	7038.18i	0.623249	+	1.07950i	0.988877	+	0.148738i	$0.0475210\pi$
$349$	4063.50	+	7038.18i	0.623249	+	1.07950i	−0.365628	+	0.930761i	$0.619146\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2688.32	−	4656.31i	−0.405340	−	0.702069i	0.589021	−	0.808117i	$-0.299513\pi$
$353$	−2688.32	−	4656.31i	−0.405340	−	0.702069i	−0.994361	+	0.106049i	$0.966180\pi$
$354$		0			0
$355$	481.960	−	834.780i	0.0720558	−	0.124804i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4860.14			0.714508			0.357254	−	0.934007i	$-0.383713\pi$
$359$	4860.14			0.714508			0.357254	+	0.934007i	$0.383713\pi$
$360$		0			0
$361$	−3694.68			−0.538662
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1890.47	+	3274.39i	−0.271101	+	0.469560i
$366$		0			0
$367$	482.147	+	835.102i	0.0685772	+	0.118779i	0.898275	−	0.439433i	$-0.144821\pi$
$367$	482.147	+	835.102i	0.0685772	+	0.118779i	−0.829698	+	0.558213i	$0.811487\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4392.56	−	7608.14i	−0.614691	−	1.06468i
$372$		0			0
$373$	3057.07	−	5294.99i	0.424367	−	0.735025i	−0.571994	−	0.820258i	$-0.693830\pi$
$373$	3057.07	−	5294.99i	0.424367	−	0.735025i	0.996361	+	0.0852326i	$0.0271633\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13226.0			−1.80682
$378$		0			0
$379$	−12063.6			−1.63501			−0.817504	−	0.575924i	$-0.804643\pi$
$379$	−12063.6			−1.63501			−0.817504	+	0.575924i	$0.804643\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4637.57	−	8032.50i	0.618717	−	1.07165i	−0.371003	−	0.928632i	$-0.620986\pi$
$383$	4637.57	−	8032.50i	0.618717	−	1.07165i	0.989720	−	0.143017i	$-0.0456805\pi$
$384$		0			0
$385$	−267.386	−	463.127i	−0.0353955	−	0.0613068i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5654.70	+	9794.23i	0.737030	+	1.27657i	0.953827	+	0.300356i	$0.0971056\pi$
$389$	5654.70	+	9794.23i	0.737030	+	1.27657i	−0.216797	+	0.976217i	$0.569561\pi$
$390$		0			0
$391$	2542.52	−	4403.77i	0.328851	−	0.569586i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5753.51			0.732888
$396$		0			0
$397$	−6235.73			−0.788319			−0.394159	−	0.919042i	$-0.628964\pi$
$397$	−6235.73			−0.788319			−0.394159	+	0.919042i	$0.628964\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6998.67	+	12122.1i	−0.871564	+	1.50959i	−0.0111849	+	0.999937i	$0.503560\pi$
$401$	−6998.67	+	12122.1i	−0.871564	+	1.50959i	−0.860379	+	0.509655i	$0.829773\pi$
$402$		0			0
$403$	8078.85	+	13993.0i	0.998601	+	1.72963i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1037.08	+	1796.27i	0.126304	+	0.218766i
$408$		0			0
$409$	−3386.26	+	5865.17i	−0.409388	+	0.709081i	−0.994821	−	0.101640i	$-0.967591\pi$
$409$	−3386.26	+	5865.17i	−0.409388	+	0.709081i	0.585433	+	0.810721i	$0.300924\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4648.99			−0.553902
$414$		0			0
$415$	−709.682			−0.0839444
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2318.41	−	4015.60i	0.270314	−	0.468197i	−0.698628	−	0.715485i	$-0.746206\pi$
$419$	2318.41	−	4015.60i	0.270314	−	0.468197i	0.968942	+	0.247287i	$0.0795392\pi$
$420$		0			0
$421$	−4588.74	−	7947.92i	−0.531215	−	0.920091i	−0.999336	−	0.0364266i	$-0.988402\pi$
$421$	−4588.74	−	7947.92i	−0.531215	−	0.920091i	0.468122	−	0.883664i	$-0.344931\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1352.93	−	2343.34i	−0.154415	−	0.267455i
$426$		0			0
$427$	2038.17	−	3530.21i	0.230993	−	0.400091i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16939.0			1.89310			0.946549	−	0.322561i	$-0.104544\pi$
$431$	16939.0			1.89310			0.946549	+	0.322561i	$0.104544\pi$
$432$		0			0
$433$	7368.61			0.817812			0.408906	−	0.912576i	$-0.365910\pi$
$433$	7368.61			0.817812			0.408906	+	0.912576i	$0.365910\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1321.42	+	2288.76i	−0.144650	+	0.250541i
$438$		0			0
$439$	−3949.88	−	6841.40i	−0.429425	−	0.743786i	0.567397	−	0.823444i	$-0.307950\pi$
$439$	−3949.88	−	6841.40i	−0.429425	−	0.743786i	−0.996822	+	0.0796582i	$0.974617\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1865.06	+	3230.38i	0.200027	+	0.346456i	0.948537	−	0.316667i	$-0.102564\pi$
$443$	1865.06	+	3230.38i	0.200027	+	0.346456i	−0.748510	+	0.663123i	$0.769231\pi$
$444$		0			0
$445$	1797.77	−	3113.82i	0.191511	−	0.331707i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	742.811			0.0780745			0.0390372	−	0.999238i	$-0.487571\pi$
$449$	742.811			0.0780745			0.0390372	+	0.999238i	$0.487571\pi$
$450$		0			0
$451$	1854.42			0.193617
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2276.26	+	3942.60i	−0.234534	+	0.406224i
$456$		0			0
$457$	1640.51	+	2841.45i	0.167921	+	0.290848i	0.937689	−	0.347476i	$-0.112961\pi$
$457$	1640.51	+	2841.45i	0.167921	+	0.290848i	−0.769768	+	0.638324i	$0.779628\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1354.28	−	2345.67i	−0.136822	−	0.236982i	0.789470	−	0.613789i	$-0.210355\pi$
$461$	−1354.28	−	2345.67i	−0.136822	−	0.236982i	−0.926292	+	0.376807i	$0.877022\pi$
$462$		0			0
$463$	6148.11	−	10648.8i	0.617120	−	1.06888i	−0.372888	−	0.927876i	$-0.621632\pi$
$463$	6148.11	−	10648.8i	0.617120	−	1.06888i	0.990009	−	0.141008i	$-0.0450342\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−347.168			−0.0344005			−0.0172002	−	0.999852i	$-0.505475\pi$
$467$	−347.168			−0.0344005			−0.0172002	+	0.999852i	$0.505475\pi$
$468$		0			0
$469$	−1731.45			−0.170471
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1309.88	−	2268.79i	0.127333	−	0.220547i
$474$		0			0
$475$	703.153	+	1217.90i	0.0679219	+	0.117644i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5984.02	+	10364.6i	0.570808	+	0.988668i	0.996483	+	0.0837924i	$0.0267033\pi$
$479$	5984.02	+	10364.6i	0.570808	+	0.988668i	−0.425675	+	0.904876i	$0.639963\pi$
$480$		0			0
$481$	8828.62	−	15291.6i	0.836904	−	1.44956i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5219.09			0.488632
$486$		0			0
$487$	−827.189			−0.0769682			−0.0384841	−	0.999259i	$-0.512253\pi$
$487$	−827.189			−0.0769682			−0.0384841	+	0.999259i	$0.512253\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5134.46	−	8893.14i	0.471924	−	0.817397i	−0.527560	−	0.849518i	$-0.676893\pi$
$491$	5134.46	−	8893.14i	0.471924	−	0.817397i	0.999484	+	0.0321212i	$0.0102263\pi$
$492$		0			0
$493$	11593.2	+	20080.1i	1.05909	+	1.83440i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1421.56	+	2462.22i	0.128301	+	0.222225i
$498$		0			0
$499$	4156.65	−	7199.53i	0.372900	−	0.645882i	−0.617110	−	0.786877i	$-0.711697\pi$
$499$	4156.65	−	7199.53i	0.372900	−	0.645882i	0.990010	+	0.140995i	$0.0450301\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7575.17			0.671492			0.335746	−	0.941953i	$-0.391012\pi$
$503$	7575.17			0.671492			0.335746	+	0.941953i	$0.391012\pi$
$504$		0			0
$505$	−3429.36			−0.302187
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5319.19	−	9213.11i	0.463200	−	0.802287i	−0.535918	−	0.844270i	$-0.680034\pi$
$509$	5319.19	−	9213.11i	0.463200	−	0.802287i	0.999118	+	0.0419835i	$0.0133677\pi$
$510$		0			0
$511$	−5576.03	−	9657.97i	−0.482718	−	0.836093i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−925.136	−	1602.38i	−0.0791580	−	0.137106i
$516$		0			0
$517$	20.0591	−	34.7434i	0.00170638	−	0.00295554i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−289.152			−0.0243148			−0.0121574	−	0.999926i	$-0.503870\pi$
$521$	−289.152			−0.0243148			−0.0121574	+	0.999926i	$0.503870\pi$
$522$		0			0
$523$	5708.40			0.477268			0.238634	−	0.971110i	$-0.423300\pi$
$523$	5708.40			0.477268			0.238634	+	0.971110i	$0.423300\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14163.0	−	24531.1i	1.17069	−	2.02769i
$528$		0			0
$529$	4979.85	+	8625.36i	0.409292	+	0.708914i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7893.35	−	13671.7i	−0.641462	−	1.11104i
$534$		0			0
$535$	−3387.11	+	5866.65i	−0.273715	+	0.474089i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−910.193			−0.0727362
$540$		0			0
$541$	−7904.89			−0.628203			−0.314101	−	0.949389i	$-0.601703\pi$
$541$	−7904.89			−0.628203			−0.314101	+	0.949389i	$0.601703\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2207.18	−	3822.95i	0.173478	−	0.300472i
$546$		0			0
$547$	2281.55	+	3951.77i	0.178340	+	0.308895i	0.941312	−	0.337537i	$-0.109594\pi$
$547$	2281.55	+	3951.77i	0.178340	+	0.308895i	−0.762972	+	0.646432i	$0.776261\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6025.32	−	10436.2i	−0.465857	−	0.806888i
$552$		0			0
$553$	−8485.12	+	14696.7i	−0.652485	+	1.13014i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1729.19			0.131540			0.0657702	−	0.997835i	$-0.479050\pi$
$557$	1729.19			0.131540			0.0657702	+	0.997835i	$0.479050\pi$
$558$		0			0
$559$	−22302.1			−1.68744
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7828.00	+	13558.5i	−0.585988	+	1.01496i	0.408764	+	0.912640i	$0.365960\pi$
$563$	−7828.00	+	13558.5i	−0.585988	+	1.01496i	−0.994752	+	0.102320i	$0.967373\pi$
$564$		0			0
$565$	2466.40	+	4271.93i	0.183650	+	0.318091i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7934.24	−	13742.5i	−0.584570	−	1.01251i	−0.994929	−	0.100581i	$-0.967930\pi$
$569$	−7934.24	−	13742.5i	−0.584570	−	1.01251i	0.410359	−	0.911924i	$-0.365404\pi$
$570$		0			0
$571$	6494.57	−	11248.9i	0.475988	−	0.824435i	−0.523634	−	0.851944i	$-0.675424\pi$
$571$	6494.57	−	11248.9i	0.475988	−	0.824435i	0.999622	+	0.0275081i	$0.00875722\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1174.55			−0.0851860
$576$		0			0
$577$	−6404.62			−0.462093			−0.231046	−	0.972943i	$-0.574215\pi$
$577$	−6404.62			−0.462093			−0.231046	+	0.972943i	$0.574215\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1046.62	−	1812.80i	0.0747351	−	0.129445i
$582$		0			0
$583$	2160.06	+	3741.34i	0.153449	+	0.265781i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10540.8	+	18257.1i	0.741165	+	1.28373i	0.951965	+	0.306206i	$0.0990596\pi$
$587$	10540.8	+	18257.1i	0.741165	+	1.28373i	−0.210801	+	0.977529i	$0.567607\pi$
$588$		0			0
$589$	−7360.93	+	12749.5i	−0.514944	+	0.891909i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1379.98			−0.0955629			−0.0477814	−	0.998858i	$-0.515215\pi$
$593$	−1379.98			−0.0955629			−0.0477814	+	0.998858i	$0.515215\pi$
$594$		0			0
$595$	7981.03			0.549900
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6476.03	+	11216.8i	−0.441742	+	0.765119i	−0.997819	−	0.0660115i	$-0.978973\pi$
$599$	−6476.03	+	11216.8i	−0.441742	+	0.765119i	0.556077	+	0.831131i	$0.312306\pi$
$600$		0			0
$601$	−11570.3	−	20040.3i	−0.785293	−	1.36017i	−0.928824	−	0.370522i	$-0.879179\pi$
$601$	−11570.3	−	20040.3i	−0.785293	−	1.36017i	0.143531	−	0.989646i	$-0.454154\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3196.01	−	5535.65i	−0.214771	−	0.371994i
$606$		0			0
$607$	739.826	−	1281.42i	0.0494705	−	0.0856855i	−0.840230	−	0.542231i	$-0.817580\pi$
$607$	739.826	−	1281.42i	0.0494705	−	0.0856855i	0.889700	+	0.456545i	$0.150913\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−341.527			−0.0226133
$612$		0			0
$613$	20327.3			1.33934			0.669668	−	0.742661i	$-0.266436\pi$
$613$	20327.3			1.33934			0.669668	+	0.742661i	$0.266436\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7181.42	−	12438.6i	0.468579	−	0.811603i	−0.530776	−	0.847512i	$-0.678099\pi$
$617$	7181.42	−	12438.6i	0.468579	−	0.811603i	0.999355	+	0.0359095i	$0.0114328\pi$
$618$		0			0
$619$	−1557.60	−	2697.84i	−0.101139	−	0.175178i	0.811015	−	0.585025i	$-0.198915\pi$
$619$	−1557.60	−	2697.84i	−0.101139	−	0.175178i	−0.912154	+	0.409847i	$0.865582\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5302.60	+	9184.36i	0.341002	+	0.590632i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−30954.9			−1.96225
$630$		0			0
$631$	−23024.2			−1.45258			−0.726292	−	0.687386i	$-0.758758\pi$
$631$	−23024.2			−1.45258			−0.726292	+	0.687386i	$0.758758\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2023.51	+	3504.82i	−0.126458	+	0.219031i
$636$		0			0
$637$	3874.24	+	6710.38i	0.240978	+	0.417386i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13617.1	−	23585.5i	−0.839068	−	1.45331i	−0.890675	−	0.454640i	$-0.849768\pi$
$641$	−13617.1	−	23585.5i	−0.839068	−	1.45331i	0.0516074	−	0.998667i	$-0.483566\pi$
$642$		0			0
$643$	−1122.75	+	1944.66i	−0.0688600	+	0.119269i	−0.898400	−	0.439179i	$-0.855270\pi$
$643$	−1122.75	+	1944.66i	−0.0688600	+	0.119269i	0.829540	+	0.558448i	$0.188603\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7245.60			−0.440269			−0.220135	−	0.975470i	$-0.570650\pi$
$647$	−7245.60			−0.440269			−0.220135	+	0.975470i	$0.570650\pi$
$648$		0			0
$649$	2286.16			0.138274
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6030.52	−	10445.2i	0.361398	−	0.625959i	−0.626794	−	0.779185i	$-0.715633\pi$
$653$	6030.52	−	10445.2i	0.361398	−	0.625959i	0.988191	+	0.153226i	$0.0489664\pi$
$654$		0			0
$655$	3220.32	+	5577.75i	0.192104	+	0.332734i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15816.4	−	27394.7i	−0.934929	−	1.61934i	−0.774761	−	0.632254i	$-0.782130\pi$
$659$	−15816.4	−	27394.7i	−0.934929	−	1.61934i	−0.160168	−	0.987090i	$-0.551203\pi$
$660$		0			0
$661$	4656.13	−	8064.66i	0.273983	−	0.474552i	−0.695895	−	0.718143i	$-0.744992\pi$
$661$	4656.13	−	8064.66i	0.273983	−	0.474552i	0.969878	+	0.243591i	$0.0783256\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4147.97			−0.241882
$666$		0			0
$667$	10064.7			0.584267
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1002.28	+	1736.00i	−0.0576641	+	0.0998772i
$672$		0			0
$673$	1880.51	+	3257.14i	0.107709	+	0.186558i	0.914842	−	0.403812i	$-0.132315\pi$
$673$	1880.51	+	3257.14i	0.107709	+	0.186558i	−0.807133	+	0.590370i	$0.798982\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14948.8	−	25892.1i	−0.848642	−	1.46989i	−0.882421	−	0.470461i	$-0.844088\pi$
$677$	−14948.8	−	25892.1i	−0.848642	−	1.46989i	0.0337792	−	0.999429i	$-0.489246\pi$
$678$		0			0
$679$	−7696.97	+	13331.5i	−0.435026	+	0.753487i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7931.11			0.444327			0.222164	−	0.975009i	$-0.428688\pi$
$683$	7931.11			0.444327			0.222164	+	0.975009i	$0.428688\pi$
$684$		0			0
$685$	11500.8			0.641494
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18388.6	−	31850.1i	1.01677	−	1.76109i
$690$		0			0
$691$	−9718.46	−	16832.9i	−0.535033	−	0.926704i	−0.999162	−	0.0409363i	$-0.986966\pi$
$691$	−9718.46	−	16832.9i	−0.535033	−	0.926704i	0.464129	−	0.885768i	$-0.346367\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3399.73	−	5888.50i	−0.185553	−	0.321386i
$696$		0			0
$697$	−13837.9	+	23967.9i	−0.752003	+	1.30251i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26442.4			1.42470			0.712351	−	0.701823i	$-0.247630\pi$
$701$	26442.4			1.42470			0.712351	+	0.701823i	$0.247630\pi$
$702$		0			0
$703$	16088.2			0.863124
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5057.53	−	8759.90i	0.269035	−	0.465983i
$708$		0			0
$709$	−11198.7	−	19396.8i	−0.593199	−	1.02745i	−0.993798	−	0.111197i	$-0.964532\pi$
$709$	−11198.7	−	19396.8i	−0.593199	−	1.02745i	0.400600	−	0.916253i	$-0.368802\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6147.84	−	10648.4i	−0.322915	−	0.559305i
$714$		0			0
$715$	1119.36	−	1938.79i	0.0585480	−	0.101408i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	29828.0			1.54715			0.773573	−	0.633707i	$-0.218467\pi$
$719$	29828.0			1.54715			0.773573	+	0.633707i	$0.218467\pi$
$720$		0			0
$721$	5457.46			0.281895
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2677.81	−	4638.11i	0.137174	−	0.237593i
$726$		0			0
$727$	−18650.5	−	32303.5i	−0.951454	−	1.64797i	−0.742282	−	0.670088i	$-0.766257\pi$
$727$	−18650.5	−	32303.5i	−0.951454	−	1.64797i	−0.209172	−	0.977879i	$-0.567077\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	19548.9	+	33859.7i	0.989115	+	1.71320i
$732$		0			0
$733$	7917.94	−	13714.3i	0.398985	−	0.691061i	−0.594616	−	0.804010i	$-0.702696\pi$
$733$	7917.94	−	13714.3i	0.398985	−	0.691061i	0.993601	+	0.112948i	$0.0360294\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	851.450			0.0425557
$738$		0			0
$739$	26946.7			1.34134			0.670669	−	0.741756i	$-0.266007\pi$
$739$	26946.7			1.34134			0.670669	+	0.741756i	$0.266007\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18690.0	−	32372.0i	0.922839	−	1.59840i	0.127838	−	0.991795i	$-0.459196\pi$
$743$	18690.0	−	32372.0i	0.922839	−	1.59840i	0.795001	−	0.606608i	$-0.207470\pi$
$744$		0			0
$745$	583.176	+	1010.09i	0.0286791	+	0.0496736i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9990.45	−	17304.0i	−0.487374	−	0.844156i
$750$		0			0
$751$	−19826.2	+	34340.0i	−0.963339	+	1.66855i	−0.249328	+	0.968419i	$0.580210\pi$
$751$	−19826.2	+	34340.0i	−0.963339	+	1.66855i	−0.714012	+	0.700134i	$0.753124\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	380.000			0.0183174
$756$		0			0
$757$	−27952.6			−1.34208			−0.671041	−	0.741420i	$-0.734153\pi$
$757$	−27952.6			−1.34208			−0.671041	+	0.741420i	$0.734153\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12809.8	−	22187.2i	0.610190	−	1.05688i	−0.381018	−	0.924567i	$-0.624427\pi$
$761$	12809.8	−	22187.2i	0.610190	−	1.05688i	0.991208	−	0.132312i	$-0.0422402\pi$
$762$		0			0
$763$	6510.18	+	11276.0i	0.308892	+	0.535016i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9731.06	−	16854.7i	−0.458107	−	0.793465i
$768$		0			0
$769$	8332.98	−	14433.1i	0.390761	−	0.676817i	−0.601790	−	0.798655i	$-0.705545\pi$
$769$	8332.98	−	14433.1i	0.390761	−	0.676817i	0.992550	+	0.121838i	$0.0388787\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4266.62			0.198525			0.0992623	−	0.995061i	$-0.468352\pi$
$773$	4266.62			0.198525			0.0992623	+	0.995061i	$0.468352\pi$
$774$		0			0
$775$	−6542.78			−0.303256
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7191.92	−	12456.8i	0.330779	−	0.572927i
$780$		0			0
$781$	−699.061	−	1210.81i	−0.0320287	−	0.0554753i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3371.51	+	5839.62i	0.153292	+	0.265509i
$786$		0			0
$787$	6047.51	−	10474.6i	0.273914	−	0.474434i	−0.695946	−	0.718094i	$-0.745015\pi$
$787$	6047.51	−	10474.6i	0.273914	−	0.474434i	0.969861	+	0.243660i	$0.0783482\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14549.5			−0.654009
$792$		0			0
$793$	17064.8			0.764174
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7541.00	+	13061.4i	−0.335152	+	0.580500i	−0.983514	−	0.180832i	$-0.942121\pi$
$797$	−7541.00	+	13061.4i	−0.335152	+	0.580500i	0.648362	+	0.761332i	$0.275454\pi$
$798$		0			0
$799$	299.366	+	518.517i	0.0132551	+	0.0229585i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2742.04	+	4749.36i	0.120504	+	0.208719i
$804$		0			0
$805$	1732.19	−	3000.24i	0.0758405	−	0.131360i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−29078.6			−1.26372			−0.631859	−	0.775083i	$-0.717708\pi$
$809$	−29078.6			−1.26372			−0.631859	+	0.775083i	$0.717708\pi$
$810$		0			0
$811$	10900.8			0.471985			0.235992	−	0.971755i	$-0.424166\pi$
$811$	10900.8			0.471985			0.235992	+	0.971755i	$0.424166\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	540.318	−	935.859i	0.0232227	−	0.0402229i
$816$		0			0
$817$	−10160.1	−	17597.8i	−0.435077	−	0.753575i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12288.5	−	21284.4i	−0.522378	−	0.904786i	−0.999661	−	0.0260359i	$-0.991712\pi$
$821$	−12288.5	−	21284.4i	−0.522378	−	0.904786i	0.477283	−	0.878750i	$-0.341622\pi$
$822$		0			0
$823$	−18000.4	+	31177.6i	−0.762399	+	1.32051i	0.179211	+	0.983811i	$0.442645\pi$
$823$	−18000.4	+	31177.6i	−0.762399	+	1.32051i	−0.941611	+	0.336704i	$0.890688\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31061.1			1.30605			0.653024	−	0.757337i	$-0.273500\pi$
$827$	31061.1			1.30605			0.653024	+	0.757337i	$0.273500\pi$
$828$		0			0
$829$	−30816.6			−1.29108			−0.645540	−	0.763727i	$-0.723368\pi$
$829$	−30816.6			−1.29108			−0.645540	+	0.763727i	$0.723368\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6791.94	−	11764.0i	0.282505	−	0.489313i
$834$		0			0
$835$	6503.89	+	11265.1i	0.269552	+	0.466878i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13889.0	+	24056.4i	0.571515	+	0.989893i	0.996411	+	0.0846505i	$0.0269774\pi$
$839$	13889.0	+	24056.4i	0.571515	+	0.989893i	−0.424896	+	0.905242i	$0.639689\pi$
$840$		0			0
$841$	−10751.7	+	18622.4i	−0.440841	+	0.763559i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8073.30			−0.328674
$846$		0			0
$847$	18853.6			0.764836
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6718.40	+	11636.6i	−0.270627	+	0.468740i
$852$		0			0
$853$	23350.3	+	40443.9i	0.937279	+	1.62341i	0.770519	+	0.637417i	$0.219997\pi$
$853$	23350.3	+	40443.9i	0.937279	+	1.62341i	0.166760	+	0.985997i	$0.446669\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12872.9	−	22296.5i	−0.513103	−	0.888721i	−0.999885	−	0.0151968i	$-0.995163\pi$
$857$	−12872.9	−	22296.5i	−0.513103	−	0.888721i	0.486781	−	0.873524i	$-0.338171\pi$
$858$		0			0
$859$	−4987.40	+	8638.43i	−0.198100	+	0.343119i	−0.947912	−	0.318531i	$-0.896810\pi$
$859$	−4987.40	+	8638.43i	−0.198100	+	0.343119i	0.749812	+	0.661650i	$0.230144\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9723.99			−0.383555			−0.191778	−	0.981438i	$-0.561425\pi$
$863$	−9723.99			−0.383555			−0.191778	+	0.981438i	$0.561425\pi$
$864$		0			0
$865$	12818.2			0.503852
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4172.60	−	7227.16i	0.162884	−	0.282123i
$870$		0			0
$871$	−3624.20	−	6277.30i	−0.140989	−	0.244200i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−921.733	−	1596.49i	−0.0356117	−	0.0616813i
$876$		0			0
$877$	6827.46	−	11825.5i	0.262882	−	0.455324i	−0.704125	−	0.710076i	$-0.748660\pi$
$877$	6827.46	−	11825.5i	0.262882	−	0.455324i	0.967006	+	0.254752i	$0.0819938\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9893.62			0.378348			0.189174	−	0.981944i	$-0.439419\pi$
$881$	9893.62			0.378348			0.189174	+	0.981944i	$0.439419\pi$
$882$		0			0
$883$	41598.9			1.58541			0.792703	−	0.609607i	$-0.208673\pi$
$883$	41598.9			1.58541			0.792703	+	0.609607i	$0.208673\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8184.82	+	14176.5i	−0.309830	+	0.536641i	−0.978325	−	0.207075i	$-0.933605\pi$
$887$	−8184.82	+	14176.5i	−0.309830	+	0.536641i	0.668495	+	0.743717i	$0.266939\pi$
$888$		0			0
$889$	−5968.44	−	10337.6i	−0.225169	−	0.390004i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−155.589	−	269.488i	−0.00583044	−	0.0100986i
$894$		0			0
$895$	−9479.94	+	16419.7i	−0.354055	+	0.613242i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	56065.1			2.07995
$900$		0			0
$901$	−64474.3			−2.38396
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3112.95	−	5391.80i	0.114340	−	0.198043i
$906$		0			0
$907$	−19605.5	−	33957.7i	−0.717740	−	1.24316i	−0.961893	−	0.273425i	$-0.911844\pi$
$907$	−19605.5	−	33957.7i	−0.717740	−	1.24316i	0.244154	−	0.969737i	$-0.421490\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5429.42	−	9404.04i	−0.197459	−	0.342008i	0.750245	−	0.661160i	$-0.229935\pi$
$911$	−5429.42	−	9404.04i	−0.197459	−	0.342008i	−0.947704	+	0.319151i	$0.896602\pi$
$912$		0			0
$913$	−514.681	+	891.453i	−0.0186566	+	0.0323141i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18996.9			−0.684116
$918$		0			0
$919$	50169.6			1.80081			0.900403	−	0.435057i	$-0.143272\pi$
$919$	50169.6			1.80081			0.900403	+	0.435057i	$0.143272\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5951.11	+	10307.6i	−0.212225	+	0.367584i
$924$		0			0
$925$	3575.00	+	6192.08i	0.127076	+	0.220102i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6333.18	+	10969.4i	0.223665	+	0.387399i	0.955918	−	0.293633i	$-0.0948645\pi$
$929$	6333.18	+	10969.4i	0.223665	+	0.387399i	−0.732253	+	0.681033i	$0.761531\pi$
$930$		0			0
$931$	−3529.96	+	6114.07i	−0.124264	+	0.215231i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3924.72			−0.137275
$936$		0			0
$937$	6348.26			0.221333			0.110666	−	0.993858i	$-0.464702\pi$
$937$	6348.26			0.221333			0.110666	+	0.993858i	$0.464702\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19570.7	−	33897.5i	0.677988	−	1.17431i	−0.297597	−	0.954691i	$-0.596185\pi$
$941$	19570.7	−	33897.5i	0.677988	−	1.17431i	0.975586	−	0.219619i	$-0.0704814\pi$
$942$		0			0
$943$	6006.68	+	10403.9i	0.207428	+	0.359275i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13290.1	+	23019.2i	0.456041	+	0.789886i	0.998747	−	0.0500365i	$-0.0159338\pi$
$947$	13290.1	+	23019.2i	0.456041	+	0.789886i	−0.542707	+	0.839922i	$0.682600\pi$
$948$		0			0
$949$	23343.0	−	40431.3i	0.798469	−	1.38299i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−406.982			−0.0138336			−0.00691680	−	0.999976i	$-0.502202\pi$
$953$	−406.982			−0.0138336			−0.00691680	+	0.999976i	$0.502202\pi$
$954$		0			0
$955$	17332.9			0.587307
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16961.1	+	29377.4i	−0.571118	+	0.989205i
$960$		0			0
$961$	−19350.9	−	33516.8i	−0.649556	−	1.12506i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8931.28	+	15469.4i	0.297936	+	0.516040i
$966$		0			0
$967$	29436.8	−	50986.0i	0.978928	−	1.69555i	0.312618	−	0.949879i	$-0.398794\pi$
$967$	29436.8	−	50986.0i	0.978928	−	1.69555i	0.666310	−	0.745675i	$-0.267873\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−52987.1			−1.75122			−0.875611	−	0.483016i	$-0.839541\pi$
$971$	−52987.1			−1.75122			−0.875611	+	0.483016i	$0.839541\pi$
$972$		0			0
$973$	20055.3			0.660785
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3191.24	−	5527.40i	0.104500	−	0.181000i	−0.809034	−	0.587762i	$-0.800009\pi$
$977$	3191.24	−	5527.40i	0.104500	−	0.181000i	0.913534	+	0.406762i	$0.133342\pi$
$978$		0			0
$979$	−2607.58	−	4516.46i	−0.0851263	−	0.147443i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12831.0	−	22223.9i	−0.416323	−	0.721092i	0.579244	−	0.815155i	$-0.303348\pi$
$983$	−12831.0	−	22223.9i	−0.416323	−	0.721092i	−0.995566	+	0.0940624i	$0.970015\pi$
$984$		0			0
$985$	−1902.62	+	3295.43i	−0.0615457	+	0.106600i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16971.4			0.545662
$990$		0			0
$991$	−49058.5			−1.57255			−0.786275	−	0.617877i	$-0.787993\pi$
$991$	−49058.5			−1.57255			−0.786275	+	0.617877i	$0.787993\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1910.49	−	3309.06i	0.0608709	−	0.105432i
$996$		0			0
$997$	2365.90	+	4097.86i	0.0751543	+	0.130171i	0.901153	−	0.433500i	$-0.142722\pi$
$997$	2365.90	+	4097.86i	0.0751543	+	0.130171i	−0.825999	+	0.563671i	$0.809388\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.r.1081.2		4
3.2	odd	2		1620.4.i.o.1081.2		4
9.2	odd	6		1620.4.i.o.541.2		4
9.4	even	3		540.4.a.e.1.1	✓	2
9.5	odd	6		540.4.a.h.1.1	yes	2
9.7	even	3	inner	1620.4.i.r.541.2		4
36.23	even	6		2160.4.a.bc.1.2		2
36.31	odd	6		2160.4.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.e.1.1	✓	2	9.4	even	3
540.4.a.h.1.1	yes	2	9.5	odd	6
1620.4.i.o.541.2		4	9.2	odd	6
1620.4.i.o.1081.2		4	3.2	odd	2
1620.4.i.r.541.2		4	9.7	even	3	inner
1620.4.i.r.1081.2		4	1.1	even	1	trivial
2160.4.a.x.1.2		2	36.31	odd	6
2160.4.a.bc.1.2		2	36.23	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 - 4.33013i) q^{5} +(7.37386 + 12.7719i) q^{7} +O(q^{10})\)	\(q+(2.50000 - 4.33013i) q^{5} +(7.37386 + 12.7719i) q^{7} +(-3.62614 - 6.28065i) q^{11} +(-30.8693 + 53.4672i) q^{13} +108.234 q^{17} -56.2523 q^{19} +(23.4909 - 40.6874i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(107.112 + 185.524i) q^{29} +(130.856 - 226.649i) q^{31} +73.7386 q^{35} -286.000 q^{37} +(-127.851 + 221.445i) q^{41} +(180.617 + 312.838i) q^{43} +(2.76591 + 4.79070i) q^{47} +(62.7523 - 108.690i) q^{49} -595.693 q^{53} -36.2614 q^{55} +(-157.617 + 273.001i) q^{59} +(-138.202 - 239.373i) q^{61} +(154.347 + 267.336i) q^{65} +(-58.7023 + 101.675i) q^{67} +192.784 q^{71} -756.189 q^{73} +(53.4773 - 92.6254i) q^{77} +(575.351 + 996.537i) q^{79} +(-70.9682 - 122.920i) q^{83} +(270.585 - 468.667i) q^{85} +719.107 q^{89} -910.505 q^{91} +(-140.631 + 243.579i) q^{95} +(521.909 + 903.973i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q + 10 * q^5 + 2 * q^7	\( 4 q + 10 q^{5} + 2 q^{7} - 42 q^{11} + 14 q^{13} + 48 q^{17} - 280 q^{19} - 126 q^{23} - 50 q^{25} + 126 q^{29} + 56 q^{31} + 20 q^{35} - 1144 q^{37} + 66 q^{41} + 530 q^{43} + 396 q^{47} + 306 q^{49} - 1008 q^{53} - 420 q^{55} - 438 q^{59} + 602 q^{61} - 70 q^{65} + 920 q^{67} + 1596 q^{71} - 1540 q^{73} - 336 q^{77} + 1724 q^{79} + 486 q^{83} + 120 q^{85} - 588 q^{89} - 3752 q^{91} - 700 q^{95} - 112 q^{97}+O(q^{100}) \) 4 * q + 10 * q^5 + 2 * q^7 - 42 * q^11 + 14 * q^13 + 48 * q^17 - 280 * q^19 - 126 * q^23 - 50 * q^25 + 126 * q^29 + 56 * q^31 + 20 * q^35 - 1144 * q^37 + 66 * q^41 + 530 * q^43 + 396 * q^47 + 306 * q^49 - 1008 * q^53 - 420 * q^55 - 438 * q^59 + 602 * q^61 - 70 * q^65 + 920 * q^67 + 1596 * q^71 - 1540 * q^73 - 336 * q^77 + 1724 * q^79 + 486 * q^83 + 120 * q^85 - 588 * q^89 - 3752 * q^91 - 700 * q^95 - 112 * q^97

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