LMFDB - Embedded newform 320.9.h.g.319.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,9,Mod(319,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("320.319");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	320.h (of order $2$, degree $1$, 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:	$130.361155220$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 94 x^{18} + 5343 x^{16} + 172772 x^{14} + 36131456 x^{12} + 3044563968 x^{10} + \cdots + 11\!\cdots\!76 $ x^20 + 94x^18 + 5343x^16 + 172772x^14 + 36131456x^12 + 3044563968x^10 + 147994443776x^8 + 2898633162752x^6 + 367168164200448x^4 + 26458647810801664*x^2 + 1152921504606846976
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{140}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.20
Root			$3.53165 - 7.17826i$ of defining polynomial
Character	$\chi$	$=$	320.319
Dual form			320.9.h.g.319.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+134.970 q^{3} +(416.235 + 466.234i) q^{5} -1863.96 q^{7} +11655.8 q^{9} +O(q^{10})$	\(q+134.970 q^{3} +(416.235 + 466.234i) q^{5} -1863.96 q^{7} +11655.8 q^{9} +5278.32i q^{11} +33338.9i q^{13} +(56179.0 + 62927.3i) q^{15} +142617. i q^{17} -70346.9i q^{19} -251578. q^{21} -255957. q^{23} +(-44122.6 + 388125. i) q^{25} +687639. q^{27} -534050. q^{29} -1.24954e6i q^{31} +712412. i q^{33} +(-775844. - 869041. i) q^{35} -169861. i q^{37} +4.49973e6i q^{39} +2.36651e6 q^{41} -4.03853e6 q^{43} +(4.85153e6 + 5.43431e6i) q^{45} -1.65928e6 q^{47} -2.29046e6 q^{49} +1.92489e7i q^{51} -981179. i q^{53} +(-2.46093e6 + 2.19702e6i) q^{55} -9.49469e6i q^{57} -7.47526e6i q^{59} +1.41028e7 q^{61} -2.17259e7 q^{63} +(-1.55437e7 + 1.38768e7i) q^{65} -3.75634e7 q^{67} -3.45463e7 q^{69} +3.57630e7i q^{71} +7.26944e6i q^{73} +(-5.95521e6 + 5.23851e7i) q^{75} -9.83857e6i q^{77} +3.72731e7i q^{79} +1.63368e7 q^{81} +2.09461e7 q^{83} +(-6.64928e7 + 5.93621e7i) q^{85} -7.20805e7 q^{87} +2.52812e7 q^{89} -6.21423e7i q^{91} -1.68650e8i q^{93} +(3.27981e7 - 2.92808e7i) q^{95} -1.35709e6i q^{97} +6.15229e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 1420 q^{5} + 2556 q^{9}+O(q^{10}) $ 20 * q + 1420 * q^5 + 2556 * q^9	$ 20 q + 1420 q^{5} + 2556 q^{9} - 410256 q^{21} - 657260 q^{25} - 660136 q^{29} + 7068520 q^{41} + 18729060 q^{45} + 11719036 q^{49} + 17660440 q^{61} - 44202240 q^{65} - 111747216 q^{69} - 154212444 q^{81} - 68800000 q^{85} + 105006376 q^{89}+O(q^{100}) $ 20 * q + 1420 * q^5 + 2556 * q^9 - 410256 * q^21 - 657260 * q^25 - 660136 * q^29 + 7068520 * q^41 + 18729060 * q^45 + 11719036 * q^49 + 17660440 * q^61 - 44202240 * q^65 - 111747216 * q^69 - 154212444 * q^81 - 68800000 * q^85 + 105006376 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	134.970			1.66629			0.833145	−	0.553054i	$-0.186538\pi$
$3$	134.970			1.66629			0.833145	+	0.553054i	$0.186538\pi$
$4$		0			0
$5$	416.235	+	466.234i	0.665975	+	0.745974i
$5$	416.235	+	466.234i	0.665975	+	0.745974i
$6$		0			0
$7$	−1863.96			−0.776326			−0.388163	−	0.921591i	$-0.626890\pi$
$7$	−1863.96			−0.776326			−0.388163	+	0.921591i	$0.626890\pi$
$8$		0			0
$9$	11655.8			1.77652
$10$		0			0
$11$			5278.32i			0.360516i	0.983619	+	0.180258i	$0.0576933\pi$
$11$			5278.32i			0.360516i	−0.983619	+	0.180258i	$0.942307\pi$
$12$		0			0
$13$			33338.9i			1.16729i	0.812010	+	0.583643i	$0.198373\pi$
$13$			33338.9i			1.16729i	−0.812010	+	0.583643i	$0.801627\pi$
$14$		0			0
$15$	56179.0	+	62927.3i	1.10971	+	1.24301i
$16$		0			0
$17$			142617.i			1.70756i	0.520636	+	0.853779i	$0.325695\pi$
$17$			142617.i			1.70756i	−0.520636	+	0.853779i	$0.674305\pi$
$18$		0			0
$19$		−	70346.9i		−	0.539797i	−0.962889	−	0.269899i	$-0.913010\pi$
$19$		−	70346.9i		−	0.539797i	0.962889	−	0.269899i	$-0.0869902\pi$
$20$		0			0
$21$	−251578.			−1.29359
$22$		0			0
$23$	−255957.			−0.914650			−0.457325	−	0.889300i	$-0.651192\pi$
$23$	−255957.			−0.914650			−0.457325	+	0.889300i	$0.651192\pi$
$24$		0			0
$25$	−44122.6	+	388125.i	−0.112954	+	0.993600i
$26$		0			0
$27$	687639.			1.29391
$28$		0			0
$29$	−534050.			−0.755075			−0.377537	−	0.925994i	$-0.623229\pi$
$29$	−534050.			−0.755075			−0.377537	+	0.925994i	$0.623229\pi$
$30$		0			0
$31$		−	1.24954e6i		−	1.35302i	−0.736433	−	0.676511i	$-0.763491\pi$
$31$		−	1.24954e6i		−	1.35302i	0.736433	−	0.676511i	$-0.236509\pi$
$32$		0			0
$33$			712412.i			0.600725i
$34$		0			0
$35$	−775844.	−	869041.i	−0.517014	−	0.579119i
$36$		0			0
$37$		−	169861.i		−	0.0906332i	−0.998973	−	0.0453166i	$-0.985570\pi$
$37$		−	169861.i		−	0.0906332i	0.998973	−	0.0453166i	$-0.0144297\pi$
$38$		0			0
$39$			4.49973e6i			1.94504i
$40$		0			0
$41$	2.36651e6			0.837479			0.418739	−	0.908106i	$-0.362472\pi$
$41$	2.36651e6			0.837479			0.418739	+	0.908106i	$0.362472\pi$
$42$		0			0
$43$	−4.03853e6			−1.18127			−0.590635	−	0.806939i	$-0.701123\pi$
$43$	−4.03853e6			−1.18127			−0.590635	+	0.806939i	$0.701123\pi$
$44$		0			0
$45$	4.85153e6	+	5.43431e6i	1.18312	+	1.32524i
$46$		0			0
$47$	−1.65928e6			−0.340038			−0.170019	−	0.985441i	$-0.554383\pi$
$47$	−1.65928e6			−0.340038			−0.170019	+	0.985441i	$0.554383\pi$
$48$		0			0
$49$	−2.29046e6			−0.397317
$50$		0			0
$51$			1.92489e7i			2.84529i
$52$		0			0
$53$		−	981179.i		−	0.124350i	−0.998065	−	0.0621748i	$-0.980196\pi$
$53$		−	981179.i		−	0.124350i	0.998065	−	0.0621748i	$-0.0198036\pi$
$54$		0			0
$55$	−2.46093e6	+	2.19702e6i	−0.268936	+	0.240095i
$56$		0			0
$57$		−	9.49469e6i		−	0.899459i
$58$		0			0
$59$		−	7.47526e6i		−	0.616905i	−0.951240	−	0.308453i	$-0.900189\pi$
$59$		−	7.47526e6i		−	0.616905i	0.951240	−	0.308453i	$-0.0998111\pi$
$60$		0			0
$61$	1.41028e7			1.01856			0.509280	−	0.860601i	$-0.329912\pi$
$61$	1.41028e7			1.01856			0.509280	+	0.860601i	$0.329912\pi$
$62$		0			0
$63$	−2.17259e7			−1.37916
$64$		0			0
$65$	−1.55437e7	+	1.38768e7i	−0.870765	+	0.777384i
$66$		0			0
$67$	−3.75634e7			−1.86409			−0.932043	−	0.362347i	$-0.881976\pi$
$67$	−3.75634e7			−1.86409			−0.932043	+	0.362347i	$0.881976\pi$
$68$		0			0
$69$	−3.45463e7			−1.52407
$70$		0			0
$71$			3.57630e7i			1.40734i	0.710524	+	0.703672i	$0.248458\pi$
$71$			3.57630e7i			1.40734i	−0.710524	+	0.703672i	$0.751542\pi$
$72$		0			0
$73$			7.26944e6i			0.255982i	0.991775	+	0.127991i	$0.0408529\pi$
$73$			7.26944e6i			0.255982i	−0.991775	+	0.127991i	$0.959147\pi$
$74$		0			0
$75$	−5.95521e6	+	5.23851e7i	−0.188214	+	1.65563i
$76$		0			0
$77$		−	9.83857e6i		−	0.279878i
$78$		0			0
$79$			3.72731e7i			0.956945i	0.878102	+	0.478473i	$0.158809\pi$
$79$			3.72731e7i			0.956945i	−0.878102	+	0.478473i	$0.841191\pi$
$80$		0			0
$81$	1.63368e7			0.379512
$82$		0			0
$83$	2.09461e7			0.441357			0.220679	−	0.975347i	$-0.429173\pi$
$83$	2.09461e7			0.441357			0.220679	+	0.975347i	$0.429173\pi$
$84$		0			0
$85$	−6.64928e7	+	5.93621e7i	−1.27379	+	1.13719i
$86$		0			0
$87$	−7.20805e7			−1.25817
$88$		0			0
$89$	2.52812e7			0.402938			0.201469	−	0.979495i	$-0.435429\pi$
$89$	2.52812e7			0.402938			0.201469	+	0.979495i	$0.435429\pi$
$90$		0			0
$91$		−	6.21423e7i		−	0.906195i
$92$		0			0
$93$		−	1.68650e8i		−	2.25453i
$94$		0			0
$95$	3.27981e7	−	2.92808e7i	0.402674	−	0.359492i
$96$		0			0
$97$		−	1.35709e6i		−	0.0153293i	−0.999971	−	0.00766465i	$-0.997560\pi$
$97$		−	1.35709e6i		−	0.0153293i	0.999971	−	0.00766465i	$-0.00243976\pi$
$98$		0			0
$99$			6.15229e7i			0.640465i
$100$		0			0
$101$	1.25091e8			1.20210			0.601052	−	0.799210i	$-0.294749\pi$
$101$	1.25091e8			1.20210			0.601052	+	0.799210i	$0.294749\pi$
$102$		0			0
$103$	4.55443e7			0.404655			0.202327	−	0.979318i	$-0.435149\pi$
$103$	4.55443e7			0.404655			0.202327	+	0.979318i	$0.435149\pi$
$104$		0			0
$105$	−1.04715e8	−	1.17294e8i	−0.861496	−	0.964981i
$106$		0			0
$107$	−1.11806e7			−0.0852959			−0.0426480	−	0.999090i	$-0.513579\pi$
$107$	−1.11806e7			−0.0852959			−0.0426480	+	0.999090i	$0.513579\pi$
$108$		0			0
$109$	3.48944e7			0.247201			0.123600	−	0.992332i	$-0.460556\pi$
$109$	3.48944e7			0.247201			0.123600	+	0.992332i	$0.460556\pi$
$110$		0			0
$111$		−	2.29261e7i		−	0.151021i
$112$		0			0
$113$			1.55089e8i			0.951187i	0.879665	+	0.475594i	$0.157767\pi$
$113$			1.55089e8i			0.951187i	−0.879665	+	0.475594i	$0.842233\pi$
$114$		0			0
$115$	−1.06538e8	−	1.19336e8i	−0.609135	−	0.682305i
$116$		0			0
$117$			3.88590e8i			2.07371i
$118$		0			0
$119$		−	2.65832e8i		−	1.32562i
$120$		0			0
$121$	1.86498e8			0.870028
$122$		0			0
$123$	3.19407e8			1.39548
$124$		0			0
$125$	−1.99322e8	+	1.40980e8i	−0.816424	+	0.577453i
$126$		0			0
$127$	2.11442e8			0.812786			0.406393	−	0.913698i	$-0.366786\pi$
$127$	2.11442e8			0.812786			0.406393	+	0.913698i	$0.366786\pi$
$128$		0			0
$129$	−5.45078e8			−1.96834
$130$		0			0
$131$			1.14013e8i			0.387140i	0.981086	+	0.193570i	$0.0620067\pi$
$131$			1.14013e8i			0.387140i	−0.981086	+	0.193570i	$0.937993\pi$
$132$		0			0
$133$			1.31124e8i			0.419059i
$134$		0			0
$135$	2.86219e8	+	3.20600e8i	0.861714	+	0.965226i
$136$		0			0
$137$			3.96731e8i			1.12620i	0.826390	+	0.563098i	$0.190391\pi$
$137$			3.96731e8i			1.12620i	−0.826390	+	0.563098i	$0.809609\pi$
$138$		0			0
$139$		−	2.81838e8i		−	0.754988i	−0.926012	−	0.377494i	$-0.876786\pi$
$139$		−	2.81838e8i		−	0.754988i	0.926012	−	0.377494i	$-0.123214\pi$
$140$		0			0
$141$	−2.23952e8			−0.566601
$142$		0			0
$143$	−1.75973e8			−0.420825
$144$		0			0
$145$	−2.22290e8	−	2.48992e8i	−0.502861	−	0.563266i
$146$		0			0
$147$	−3.09142e8			−0.662046
$148$		0			0
$149$	−4.38854e8			−0.890379			−0.445190	−	0.895436i	$-0.646864\pi$
$149$	−4.38854e8			−0.890379			−0.445190	+	0.895436i	$0.646864\pi$
$150$		0			0
$151$		−	7.34766e8i		−	1.41332i	−0.707552	−	0.706662i	$-0.750200\pi$
$151$		−	7.34766e8i		−	1.41332i	0.707552	−	0.706662i	$-0.249800\pi$
$152$		0			0
$153$			1.66231e9i			3.03352i
$154$		0			0
$155$	5.82579e8	−	5.20103e8i	1.00932	−	0.901079i
$156$		0			0
$157$			3.92920e8i			0.646704i	0.946279	+	0.323352i	$0.104810\pi$
$157$			3.92920e8i			0.646704i	−0.946279	+	0.323352i	$0.895190\pi$
$158$		0			0
$159$		−	1.32429e8i		−	0.207203i
$160$		0			0
$161$	4.77093e8			0.710067
$162$		0			0
$163$	8.78730e8			1.24482			0.622408	−	0.782693i	$-0.286155\pi$
$163$	8.78730e8			1.24482			0.622408	+	0.782693i	$0.286155\pi$
$164$		0			0
$165$	−3.32150e8	+	2.96530e8i	−0.448125	+	0.400068i
$166$		0			0
$167$	−4.28526e8			−0.550948			−0.275474	−	0.961308i	$-0.588835\pi$
$167$	−4.28526e8			−0.550948			−0.275474	+	0.961308i	$0.588835\pi$
$168$		0			0
$169$	−2.95749e8			−0.362557
$170$		0			0
$171$		−	8.19947e8i		−	0.958962i
$172$		0			0
$173$		−	3.08766e8i		−	0.344703i	−0.985035	−	0.172352i	$-0.944863\pi$
$173$		−	3.08766e8i		−	0.344703i	0.985035	−	0.172352i	$-0.0551366\pi$
$174$		0			0
$175$	8.22427e7	−	7.23450e8i	0.0876891	−	0.771358i
$176$		0			0
$177$		−	1.00893e9i		−	1.02794i
$178$		0			0
$179$			2.34312e8i			0.228235i	0.993467	+	0.114117i	$0.0364040\pi$
$179$			2.34312e8i			0.228235i	−0.993467	+	0.114117i	$0.963596\pi$
$180$		0			0
$181$	1.98225e8			0.184691			0.0923454	−	0.995727i	$-0.470564\pi$
$181$	1.98225e8			0.184691			0.0923454	+	0.995727i	$0.470564\pi$
$182$		0			0
$183$	1.90345e9			1.69722
$184$		0			0
$185$	7.91950e7	−	7.07021e7i	0.0676100	−	0.0603595i
$186$		0			0
$187$	−7.52778e8			−0.615602
$188$		0			0
$189$	−1.28173e9			−1.00450
$190$		0			0
$191$			2.29845e9i			1.72704i	0.504317	+	0.863518i	$0.331744\pi$
$191$			2.29845e9i			1.72704i	−0.504317	+	0.863518i	$0.668256\pi$
$192$		0			0
$193$		−	6.50744e8i		−	0.469009i	−0.972115	−	0.234505i	$-0.924653\pi$
$193$		−	6.50744e8i		−	0.469009i	0.972115	−	0.234505i	$-0.0753468\pi$
$194$		0			0
$195$	−2.09793e9	+	1.87294e9i	−1.45095	+	1.29535i
$196$		0			0
$197$		−	1.18078e9i		−	0.783981i	−0.919969	−	0.391990i	$-0.871787\pi$
$197$		−	1.18078e9i		−	0.783981i	0.919969	−	0.391990i	$-0.128213\pi$
$198$		0			0
$199$			8.61830e8i			0.549553i	0.961508	+	0.274776i	$0.0886038\pi$
$199$			8.61830e8i			0.549553i	−0.961508	+	0.274776i	$0.911396\pi$
$200$		0			0
$201$	−5.06992e9			−3.10611
$202$		0			0
$203$	9.95448e8			0.586185
$204$		0			0
$205$	9.85025e8	+	1.10335e9i	0.557740	+	0.624737i
$206$		0			0
$207$	−2.98337e9			−1.62490
$208$		0			0
$209$	3.71313e8			0.194606
$210$		0			0
$211$			1.94443e9i			0.980985i	0.871445	+	0.490493i	$0.163183\pi$
$211$			1.94443e9i			0.980985i	−0.871445	+	0.490493i	$0.836817\pi$
$212$		0			0
$213$			4.82691e9i			2.34505i
$214$		0			0
$215$	−1.68097e9	−	1.88290e9i	−0.786697	−	0.881197i
$216$		0			0
$217$			2.32910e9i			1.05039i
$218$		0			0
$219$			9.81153e8i			0.426541i
$220$		0			0
$221$	−4.75469e9			−1.99321
$222$		0			0
$223$	−5.65989e8			−0.228870			−0.114435	−	0.993431i	$-0.536506\pi$
$223$	−5.65989e8			−0.228870			−0.114435	+	0.993431i	$0.536506\pi$
$224$		0			0
$225$	−5.14283e8	+	4.52390e9i	−0.200665	+	1.76515i
$226$		0			0
$227$	−2.24952e9			−0.847200			−0.423600	−	0.905849i	$-0.639234\pi$
$227$	−2.24952e9			−0.847200			−0.423600	+	0.905849i	$0.639234\pi$
$228$		0			0
$229$	1.02108e9			0.371295			0.185648	−	0.982616i	$-0.440562\pi$
$229$	1.02108e9			0.371295			0.185648	+	0.982616i	$0.440562\pi$
$230$		0			0
$231$		−	1.32791e9i		−	0.466358i
$232$		0			0
$233$			1.60926e9i			0.546013i	0.962012	+	0.273007i	$0.0880181\pi$
$233$			1.60926e9i			0.546013i	−0.962012	+	0.273007i	$0.911982\pi$
$234$		0			0
$235$	−6.90648e8	−	7.73610e8i	−0.226457	−	0.253659i
$236$		0			0
$237$			5.03073e9i			1.59455i
$238$		0			0
$239$			6.14150e9i			1.88227i	0.338027	+	0.941136i	$0.390240\pi$
$239$			6.14150e9i			1.88227i	−0.338027	+	0.941136i	$0.609760\pi$
$240$		0			0
$241$	−1.57617e9			−0.467234			−0.233617	−	0.972329i	$-0.575056\pi$
$241$	−1.57617e9			−0.467234			−0.233617	+	0.972329i	$0.575056\pi$
$242$		0			0
$243$	−2.30663e9			−0.661536
$244$		0			0
$245$	−9.53367e8	−	1.06789e9i	−0.264603	−	0.296388i
$246$		0			0
$247$	2.34528e9			0.630098
$248$		0			0
$249$	2.82708e9			0.735429
$250$		0			0
$251$			6.05238e8i			0.152486i	0.997089	+	0.0762432i	$0.0242925\pi$
$251$			6.05238e8i			0.152486i	−0.997089	+	0.0762432i	$0.975707\pi$
$252$		0			0
$253$		−	1.35102e9i		−	0.329746i
$254$		0			0
$255$	−8.97450e9	+	8.01207e9i	−2.12251	+	1.89489i
$256$		0			0
$257$			7.33383e9i			1.68112i	0.541720	+	0.840559i	$0.317773\pi$
$257$			7.33383e9i			1.68112i	−0.541720	+	0.840559i	$0.682227\pi$
$258$		0			0
$259$			3.16614e8i			0.0703609i
$260$		0			0
$261$	−6.22476e9			−1.34141
$262$		0			0
$263$	8.11216e9			1.69556			0.847780	−	0.530348i	$-0.177939\pi$
$263$	8.11216e9			1.69556			0.847780	+	0.530348i	$0.177939\pi$
$264$		0			0
$265$	4.57458e8	−	4.08400e8i	0.0927616	−	0.0828138i
$266$		0			0
$267$	3.41219e9			0.671411
$268$		0			0
$269$	−1.31912e9			−0.251927			−0.125964	−	0.992035i	$-0.540202\pi$
$269$	−1.31912e9			−0.251927			−0.125964	+	0.992035i	$0.540202\pi$
$270$		0			0
$271$		−	3.01624e9i		−	0.559228i	−0.960113	−	0.279614i	$-0.909794\pi$
$271$		−	3.01624e9i		−	0.559228i	0.960113	−	0.279614i	$-0.0902064\pi$
$272$		0			0
$273$		−	8.38731e9i		−	1.50998i
$274$		0			0
$275$	−2.04865e9	−	2.32893e8i	−0.358209	−	0.0407217i
$276$		0			0
$277$		−	1.19710e9i		−	0.203334i	−0.994818	−	0.101667i	$-0.967582\pi$
$277$		−	1.19710e9i		−	0.203334i	0.994818	−	0.101667i	$-0.0324177\pi$
$278$		0			0
$279$		−	1.45644e10i		−	2.40367i
$280$		0			0
$281$	2.05582e9			0.329732			0.164866	−	0.986316i	$-0.447281\pi$
$281$	2.05582e9			0.329732			0.164866	+	0.986316i	$0.447281\pi$
$282$		0			0
$283$	1.21382e10			1.89238			0.946189	−	0.323616i	$-0.104898\pi$
$283$	1.21382e10			1.89238			0.946189	+	0.323616i	$0.104898\pi$
$284$		0			0
$285$	4.42674e9	−	3.95202e9i	0.670973	−	0.599017i
$286$		0			0
$287$	−4.41109e9			−0.650157
$288$		0			0
$289$	−1.33638e10			−1.91575
$290$		0			0
$291$		−	1.83166e8i		−	0.0255431i
$292$		0			0
$293$			8.85830e9i			1.20193i	0.799275	+	0.600966i	$0.205217\pi$
$293$			8.85830e9i			1.20193i	−0.799275	+	0.600966i	$0.794783\pi$
$294$		0			0
$295$	3.48522e9	−	3.11146e9i	0.460195	−	0.410844i
$296$		0			0
$297$			3.62958e9i			0.466477i
$298$		0			0
$299$		−	8.53330e9i		−	1.06766i
$300$		0			0
$301$	7.52765e9			0.917051
$302$		0			0
$303$	1.68835e10			2.00305
$304$		0			0
$305$	5.87008e9	+	6.57520e9i	0.678335	+	0.759818i
$306$		0			0
$307$	−2.27115e9			−0.255678			−0.127839	−	0.991795i	$-0.540804\pi$
$307$	−2.27115e9			−0.255678			−0.127839	+	0.991795i	$0.540804\pi$
$308$		0			0
$309$	6.14709e9			0.674273
$310$		0			0
$311$		−	9.95031e8i		−	0.106364i	−0.998585	−	0.0531820i	$-0.983064\pi$
$311$		−	9.95031e8i		−	0.106364i	0.998585	−	0.0531820i	$-0.0169364\pi$
$312$		0			0
$313$		−	9.67741e9i		−	1.00828i	−0.863621	−	0.504141i	$-0.831809\pi$
$313$		−	9.67741e9i		−	1.00828i	0.863621	−	0.504141i	$-0.168191\pi$
$314$		0			0
$315$	−9.04306e9	−	1.01293e10i	−0.918488	−	1.02882i
$316$		0			0
$317$			5.41589e9i			0.536330i	0.963373	+	0.268165i	$0.0864173\pi$
$317$			5.41589e9i			0.536330i	−0.963373	+	0.268165i	$0.913583\pi$
$318$		0			0
$319$		−	2.81889e9i		−	0.272217i
$320$		0			0
$321$	−1.50903e9			−0.142128
$322$		0			0
$323$	1.00327e10			0.921735
$324$		0			0
$325$	−1.29396e10	−	1.47100e9i	−1.15982	−	0.131849i
$326$		0			0
$327$	4.70968e9			0.411908
$328$		0			0
$329$	3.09282e9			0.263980
$330$		0			0
$331$		−	1.61383e10i		−	1.34445i	−0.740345	−	0.672227i	$-0.765338\pi$
$331$		−	1.61383e10i		−	1.34445i	0.740345	−	0.672227i	$-0.234662\pi$
$332$		0			0
$333$		−	1.97986e9i		−	0.161012i
$334$		0			0
$335$	−1.56352e10	−	1.75133e10i	−1.24144	−	1.39056i
$336$		0			0
$337$		−	1.60080e10i		−	1.24113i	−0.784154	−	0.620566i	$-0.786903\pi$
$337$		−	1.60080e10i		−	1.24113i	0.784154	−	0.620566i	$-0.213097\pi$
$338$		0			0
$339$			2.09322e10i			1.58495i
$340$		0			0
$341$	6.59549e9			0.487786
$342$		0			0
$343$	1.50147e10			1.08477
$344$		0			0
$345$	−1.43794e10	−	1.61067e10i	−1.01499	−	1.13692i
$346$		0			0
$347$	1.34569e10			0.928168			0.464084	−	0.885791i	$-0.346384\pi$
$347$	1.34569e10			0.928168			0.464084	+	0.885791i	$0.346384\pi$
$348$		0			0
$349$	−1.78901e10			−1.20590			−0.602951	−	0.797779i	$-0.706008\pi$
$349$	−1.78901e10			−1.20590			−0.602951	+	0.797779i	$0.706008\pi$
$350$		0			0
$351$			2.29251e10i			1.51037i
$352$		0			0
$353$			2.70557e10i			1.74245i	0.490886	+	0.871224i	$0.336673\pi$
$353$			2.70557e10i			1.74245i	−0.490886	+	0.871224i	$0.663327\pi$
$354$		0			0
$355$	−1.66739e10	+	1.48858e10i	−1.04984	+	0.937257i
$356$		0			0
$357$		−	3.58793e10i		−	2.20887i
$358$		0			0
$359$			5.50402e9i			0.331361i	0.986179	+	0.165681i	$0.0529821\pi$
$359$			5.50402e9i			0.331361i	−0.986179	+	0.165681i	$0.947018\pi$
$360$		0			0
$361$	1.20349e10			0.708619
$362$		0			0
$363$	2.51716e10			1.44972
$364$		0			0
$365$	−3.38926e9	+	3.02579e9i	−0.190956	+	0.170478i
$366$		0			0
$367$	−9.67406e9			−0.533267			−0.266633	−	0.963798i	$-0.585911\pi$
$367$	−9.67406e9			−0.533267			−0.266633	+	0.963798i	$0.585911\pi$
$368$		0			0
$369$	2.75835e10			1.48780
$370$		0			0
$371$			1.82888e9i			0.0965359i
$372$		0			0
$373$		−	2.57338e10i		−	1.32944i	−0.747093	−	0.664720i	$-0.768551\pi$
$373$		−	2.57338e10i		−	1.32944i	0.747093	−	0.664720i	$-0.231449\pi$
$374$		0			0
$375$	−2.69024e10	+	1.90280e10i	−1.36040	+	0.962204i
$376$		0			0
$377$		−	1.78046e10i		−	0.881388i
$378$		0			0
$379$		−	3.39099e10i		−	1.64350i	−0.569849	−	0.821749i	$-0.692998\pi$
$379$		−	3.39099e10i		−	1.64350i	0.569849	−	0.821749i	$-0.307002\pi$
$380$		0			0
$381$	2.85382e10			1.35434
$382$		0			0
$383$	1.07466e10			0.499433			0.249717	−	0.968319i	$-0.419663\pi$
$383$	1.07466e10			0.499433			0.249717	+	0.968319i	$0.419663\pi$
$384$		0			0
$385$	4.58707e9	−	4.09515e9i	0.208782	−	0.186392i
$386$		0			0
$387$	−4.70722e10			−2.09855
$388$		0			0
$389$	4.17407e10			1.82289			0.911446	−	0.411419i	$-0.134967\pi$
$389$	4.17407e10			1.82289			0.911446	+	0.411419i	$0.134967\pi$
$390$		0			0
$391$		−	3.65038e10i		−	1.56182i
$392$		0			0
$393$			1.53882e10i			0.645088i
$394$		0			0
$395$	−1.73780e10	+	1.55144e10i	−0.713856	+	0.637302i
$396$		0			0
$397$			1.28145e10i			0.515871i	0.966162	+	0.257936i	$0.0830422\pi$
$397$			1.28145e10i			0.515871i	−0.966162	+	0.257936i	$0.916958\pi$
$398$		0			0
$399$			1.76977e10i			0.698274i
$400$		0			0
$401$	4.75527e10			1.83907			0.919533	−	0.393013i	$-0.128567\pi$
$401$	4.75527e10			1.83907			0.919533	+	0.393013i	$0.128567\pi$
$402$		0			0
$403$	4.16584e10			1.57936
$404$		0			0
$405$	6.79992e9	+	7.61674e9i	0.252746	+	0.283106i
$406$		0			0
$407$	8.96581e8			0.0326747
$408$		0			0
$409$	2.44716e10			0.874518			0.437259	−	0.899336i	$-0.355949\pi$
$409$	2.44716e10			0.874518			0.437259	+	0.899336i	$0.355949\pi$
$410$		0			0
$411$			5.35466e10i			1.87657i
$412$		0			0
$413$			1.39336e10i			0.478920i
$414$		0			0
$415$	8.71848e9	+	9.76576e9i	0.293933	+	0.329241i
$416$		0			0
$417$		−	3.80395e10i		−	1.25803i
$418$		0			0
$419$		−	3.99144e9i		−	0.129501i	−0.997901	−	0.0647505i	$-0.979375\pi$
$419$		−	3.99144e9i		−	0.129501i	0.997901	−	0.0647505i	$-0.0206252\pi$
$420$		0			0
$421$	8.14500e9			0.259276			0.129638	−	0.991561i	$-0.458618\pi$
$421$	8.14500e9			0.259276			0.129638	+	0.991561i	$0.458618\pi$
$422$		0			0
$423$	−1.93401e10			−0.604085
$424$		0			0
$425$	−5.53532e10	−	6.29263e9i	−1.69663	−	0.192875i
$426$		0			0
$427$	−2.62871e10			−0.790734
$428$		0			0
$429$	−2.37510e10			−0.701217
$430$		0			0
$431$		−	4.14023e9i		−	0.119982i	−0.998199	−	0.0599909i	$-0.980893\pi$
$431$		−	4.14023e9i		−	0.119982i	0.998199	−	0.0599909i	$-0.0191072\pi$
$432$		0			0
$433$		−	2.45166e10i		−	0.697443i	−0.937226	−	0.348722i	$-0.886616\pi$
$433$		−	2.45166e10i		−	0.697443i	0.937226	−	0.348722i	$-0.113384\pi$
$434$		0			0
$435$	−3.00024e10	−	3.36063e10i	−0.837913	−	0.938565i
$436$		0			0
$437$			1.80058e10i			0.493726i
$438$		0			0
$439$			5.66902e10i			1.52633i	0.646201	+	0.763167i	$0.276357\pi$
$439$			5.66902e10i			1.52633i	−0.646201	+	0.763167i	$0.723643\pi$
$440$		0			0
$441$	−2.66970e10			−0.705843
$442$		0			0
$443$	−2.12919e10			−0.552840			−0.276420	−	0.961037i	$-0.589148\pi$
$443$	−2.12919e10			−0.552840			−0.276420	+	0.961037i	$0.589148\pi$
$444$		0			0
$445$	1.05229e10	+	1.17870e10i	0.268347	+	0.300581i
$446$		0			0
$447$	−5.92319e10			−1.48363
$448$		0			0
$449$	−5.16478e10			−1.27077			−0.635384	−	0.772196i	$-0.719158\pi$
$449$	−5.16478e10			−1.27077			−0.635384	+	0.772196i	$0.719158\pi$
$450$		0			0
$451$			1.24912e10i			0.301925i
$452$		0			0
$453$		−	9.91711e10i		−	2.35501i
$454$		0			0
$455$	2.89728e10	−	2.58658e10i	0.675998	−	0.603503i
$456$		0			0
$457$			6.18862e10i			1.41883i	0.704793	+	0.709413i	$0.251040\pi$
$457$			6.18862e10i			1.41883i	−0.704793	+	0.709413i	$0.748960\pi$
$458$		0			0
$459$			9.80689e10i			2.20943i
$460$		0			0
$461$	−1.84997e9			−0.0409601			−0.0204800	−	0.999790i	$-0.506519\pi$
$461$	−1.84997e9			−0.0409601			−0.0204800	+	0.999790i	$0.506519\pi$
$462$		0			0
$463$	4.89801e10			1.06585			0.532924	−	0.846163i	$-0.321093\pi$
$463$	4.89801e10			1.06585			0.532924	+	0.846163i	$0.321093\pi$
$464$		0			0
$465$	7.86304e10	−	7.01981e10i	1.68182	−	1.50146i
$466$		0			0
$467$	2.58985e9			0.0544511			0.0272255	−	0.999629i	$-0.491333\pi$
$467$	2.58985e9			0.0544511			0.0272255	+	0.999629i	$0.491333\pi$
$468$		0			0
$469$	7.00167e10			1.44714
$470$		0			0
$471$			5.30322e10i			1.07760i
$472$		0			0
$473$		−	2.13166e10i		−	0.425867i
$474$		0			0
$475$	2.73034e10	+	3.10389e9i	0.536342	+	0.0609722i
$476$		0			0
$477$		−	1.14364e10i		−	0.220910i
$478$		0			0
$479$		−	7.24642e10i		−	1.37652i	−0.725466	−	0.688258i	$-0.758376\pi$
$479$		−	7.24642e10i		−	1.37652i	0.725466	−	0.688258i	$-0.241624\pi$
$480$		0			0
$481$	5.66298e9			0.105795
$482$		0			0
$483$	6.43930e10			1.18318
$484$		0			0
$485$	6.32722e8	−	5.64869e8i	0.0114353	−	0.0102089i
$486$		0			0
$487$	−6.78628e10			−1.20647			−0.603234	−	0.797564i	$-0.706121\pi$
$487$	−6.78628e10			−1.20647			−0.603234	+	0.797564i	$0.706121\pi$
$488$		0			0
$489$	1.18602e11			2.07422
$490$		0			0
$491$		−	1.04720e11i		−	1.80179i	−0.434032	−	0.900897i	$-0.642910\pi$
$491$		−	1.04720e11i		−	1.80179i	0.434032	−	0.900897i	$-0.357090\pi$
$492$		0			0
$493$		−	7.61646e10i		−	1.28933i
$494$		0			0
$495$	−2.86840e10	+	2.56079e10i	−0.477770	+	0.426534i
$496$		0			0
$497$		−	6.66608e10i		−	1.09256i
$498$		0			0
$499$		−	5.70756e10i		−	0.920552i	−0.887776	−	0.460276i	$-0.847750\pi$
$499$		−	5.70756e10i		−	0.920552i	0.887776	−	0.460276i	$-0.152250\pi$
$500$		0			0
$501$	−5.78379e10			−0.918040
$502$		0			0
$503$	1.65834e10			0.259061			0.129531	−	0.991575i	$-0.458653\pi$
$503$	1.65834e10			0.259061			0.129531	+	0.991575i	$0.458653\pi$
$504$		0			0
$505$	5.20673e10	+	5.83218e10i	0.800571	+	0.896738i
$506$		0			0
$507$	−3.99171e10			−0.604125
$508$		0			0
$509$	−3.82746e9			−0.0570216			−0.0285108	−	0.999593i	$-0.509076\pi$
$509$	−3.82746e9			−0.0570216			−0.0285108	+	0.999593i	$0.509076\pi$
$510$		0			0
$511$		−	1.35499e10i		−	0.198726i
$512$		0			0
$513$		−	4.83732e10i		−	0.698451i
$514$		0			0
$515$	1.89571e10	+	2.12343e10i	0.269490	+	0.301862i
$516$		0			0
$517$		−	8.75818e9i		−	0.122589i
$518$		0			0
$519$		−	4.16740e10i		−	0.574376i
$520$		0			0
$521$	−7.15272e10			−0.970779			−0.485389	−	0.874298i	$-0.661322\pi$
$521$	−7.15272e10			−0.970779			−0.485389	+	0.874298i	$0.661322\pi$
$522$		0			0
$523$	−2.30293e10			−0.307803			−0.153902	−	0.988086i	$-0.549184\pi$
$523$	−2.30293e10			−0.307803			−0.153902	+	0.988086i	$0.549184\pi$
$524$		0			0
$525$	1.11003e10	−	9.76436e10i	0.146115	−	1.28531i
$526$		0			0
$527$	1.78206e11			2.31036
$528$		0			0
$529$	−1.27972e10			−0.163415
$530$		0			0
$531$		−	8.71300e10i		−	1.09595i
$532$		0			0
$533$			7.88969e10i			0.977577i
$534$		0			0
$535$	−4.65374e9	−	5.21275e9i	−0.0568050	−	0.0636285i
$536$		0			0
$537$			3.16250e10i			0.380306i
$538$		0			0
$539$		−	1.20897e10i		−	0.143239i
$540$		0			0
$541$	−5.52906e10			−0.645450			−0.322725	−	0.946493i	$-0.604599\pi$
$541$	−5.52906e10			−0.645450			−0.322725	+	0.946493i	$0.604599\pi$
$542$		0			0
$543$	2.67544e10			0.307748
$544$		0			0
$545$	1.45242e10	+	1.62689e10i	0.164630	+	0.184405i
$546$		0			0
$547$	8.25380e10			0.921945			0.460972	−	0.887415i	$-0.347501\pi$
$547$	8.25380e10			0.921945			0.460972	+	0.887415i	$0.347501\pi$
$548$		0			0
$549$	1.64379e11			1.80949
$550$		0			0
$551$			3.75688e10i			0.407587i
$552$		0			0
$553$		−	6.94755e10i		−	0.742902i
$554$		0			0
$555$	1.06889e10	−	9.54263e9i	0.112658	−	0.100576i
$556$		0			0
$557$		−	6.76291e10i		−	0.702608i	−0.936261	−	0.351304i	$-0.885738\pi$
$557$		−	6.76291e10i		−	0.702608i	0.936261	−	0.351304i	$-0.114262\pi$
$558$		0			0
$559$		−	1.34640e11i		−	1.37888i
$560$		0			0
$561$	−1.01602e11			−1.02577
$562$		0			0
$563$	1.16824e11			1.16278			0.581391	−	0.813624i	$-0.302509\pi$
$563$	1.16824e11			1.16278			0.581391	+	0.813624i	$0.302509\pi$
$564$		0			0
$565$	−7.23075e10	+	6.45532e10i	−0.709561	+	0.633467i
$566$		0			0
$567$	−3.04510e10			−0.294625
$568$		0			0
$569$	−1.83290e9			−0.0174860			−0.00874298	−	0.999962i	$-0.502783\pi$
$569$	−1.83290e9			−0.0174860			−0.00874298	+	0.999962i	$0.502783\pi$
$570$		0			0
$571$			1.83754e11i			1.72859i	0.502984	+	0.864296i	$0.332235\pi$
$571$			1.83754e11i			1.72859i	−0.502984	+	0.864296i	$0.667765\pi$
$572$		0			0
$573$			3.10221e11i			2.87774i
$574$		0			0
$575$	1.12935e10	−	9.93432e10i	0.103313	−	0.908797i
$576$		0			0
$577$			1.07447e11i			0.969374i	0.874688	+	0.484687i	$0.161066\pi$
$577$			1.07447e11i			0.969374i	−0.874688	+	0.484687i	$0.838934\pi$
$578$		0			0
$579$		−	8.78307e10i		−	0.781505i
$580$		0			0
$581$	−3.90426e10			−0.342637
$582$		0			0
$583$	5.17897e9			0.0448301
$584$		0			0
$585$	−1.81174e11	+	1.61745e11i	−1.54693	+	1.38104i
$586$		0			0
$587$	8.33835e10			0.702308			0.351154	−	0.936318i	$-0.385789\pi$
$587$	8.33835e10			0.702308			0.351154	+	0.936318i	$0.385789\pi$
$588$		0			0
$589$	−8.79015e10			−0.730357
$590$		0			0
$591$		−	1.59370e11i		−	1.30634i
$592$		0			0
$593$			8.33128e10i			0.673741i	0.941551	+	0.336871i	$0.109369\pi$
$593$			8.33128e10i			0.673741i	−0.941551	+	0.336871i	$0.890631\pi$
$594$		0			0
$595$	1.23940e11	−	1.10649e11i	0.988880	−	0.882832i
$596$		0			0
$597$			1.16321e11i			0.915714i
$598$		0			0
$599$			7.46618e10i			0.579951i	0.957034	+	0.289976i	$0.0936472\pi$
$599$			7.46618e10i			0.579951i	−0.957034	+	0.289976i	$0.906353\pi$
$600$		0			0
$601$	−1.14915e11			−0.880803			−0.440401	−	0.897801i	$-0.645164\pi$
$601$	−1.14915e11			−0.880803			−0.440401	+	0.897801i	$0.645164\pi$
$602$		0			0
$603$	−4.37831e11			−3.31159
$604$		0			0
$605$	7.76270e10	+	8.69518e10i	0.579417	+	0.649018i
$606$		0			0
$607$	3.63432e10			0.267712			0.133856	−	0.991001i	$-0.457264\pi$
$607$	3.63432e10			0.267712			0.133856	+	0.991001i	$0.457264\pi$
$608$		0			0
$609$	1.34355e11			0.976754
$610$		0			0
$611$		−	5.53183e10i		−	0.396921i
$612$		0			0
$613$			4.54903e10i			0.322164i	0.986941	+	0.161082i	$0.0514983\pi$
$613$			4.54903e10i			0.322164i	−0.986941	+	0.161082i	$0.948502\pi$
$614$		0			0
$615$	1.32948e11	+	1.48918e11i	0.929357	+	1.04099i
$616$		0			0
$617$		−	9.39914e10i		−	0.648556i	−0.945962	−	0.324278i	$-0.894879\pi$
$617$		−	9.39914e10i		−	0.648556i	0.945962	−	0.324278i	$-0.105121\pi$
$618$		0			0
$619$		−	6.86980e10i		−	0.467931i	−0.972245	−	0.233965i	$-0.924830\pi$
$619$		−	6.86980e10i		−	0.467931i	0.972245	−	0.233965i	$-0.0751702\pi$
$620$		0			0
$621$	−1.76006e11			−1.18348
$622$		0			0
$623$	−4.71232e10			−0.312811
$624$		0			0
$625$	−1.48694e11	−	3.42502e10i	−0.974483	−	0.224462i
$626$		0			0
$627$	5.01160e10			0.324269
$628$		0			0
$629$	2.42251e10			0.154761
$630$		0			0
$631$		−	3.95536e10i		−	0.249499i	−0.992188	−	0.124750i	$-0.960187\pi$
$631$		−	3.95536e10i		−	0.249499i	0.992188	−	0.124750i	$-0.0398128\pi$
$632$		0			0
$633$			2.62439e11i			1.63461i
$634$		0			0
$635$	8.80094e10	+	9.85813e10i	0.541295	+	0.606317i
$636$		0			0
$637$		−	7.63612e10i		−	0.463783i
$638$		0			0
$639$			4.16845e11i			2.50018i
$640$		0			0
$641$	3.16726e11			1.87608			0.938040	−	0.346526i	$-0.112639\pi$
$641$	3.16726e11			1.87608			0.938040	+	0.346526i	$0.112639\pi$
$642$		0			0
$643$	−2.41563e11			−1.41315			−0.706574	−	0.707640i	$-0.749760\pi$
$643$	−2.41563e11			−1.41315			−0.706574	+	0.707640i	$0.749760\pi$
$644$		0			0
$645$	−2.26880e11	−	2.54134e11i	−1.31087	−	1.46833i
$646$		0			0
$647$	−1.42487e11			−0.813124			−0.406562	−	0.913623i	$-0.633272\pi$
$647$	−1.42487e11			−0.813124			−0.406562	+	0.913623i	$0.633272\pi$
$648$		0			0
$649$	3.94568e10			0.222404
$650$		0			0
$651$			3.14357e11i			1.75025i
$652$		0			0
$653$			1.22267e11i			0.672447i	0.941782	+	0.336224i	$0.109150\pi$
$653$			1.22267e11i			0.672447i	−0.941782	+	0.336224i	$0.890850\pi$
$654$		0			0
$655$	−5.31566e10	+	4.74560e10i	−0.288796	+	0.257826i
$656$		0			0
$657$			8.47310e10i			0.454758i
$658$		0			0
$659$			1.05381e11i			0.558753i	0.960182	+	0.279377i	$0.0901278\pi$
$659$			1.05381e11i			0.558753i	−0.960182	+	0.279377i	$0.909872\pi$
$660$		0			0
$661$	2.12360e11			1.11242			0.556209	−	0.831043i	$-0.312255\pi$
$661$	2.12360e11			1.11242			0.556209	+	0.831043i	$0.312255\pi$
$662$		0			0
$663$	−6.41738e11			−3.32126
$664$		0			0
$665$	−6.11343e10	+	5.45782e10i	−0.312607	+	0.279083i
$666$		0			0
$667$	1.36694e11			0.690629
$668$		0			0
$669$	−7.63913e10			−0.381364
$670$		0			0
$671$			7.44391e10i			0.367207i
$672$		0			0
$673$		−	3.80972e11i		−	1.85709i	−0.371219	−	0.928545i	$-0.621060\pi$
$673$		−	3.80972e11i		−	1.85709i	0.371219	−	0.928545i	$-0.378940\pi$
$674$		0			0
$675$	−3.03404e10	+	2.66890e11i	−0.146152	+	1.28563i
$676$		0			0
$677$		−	2.80516e11i		−	1.33538i	−0.744441	−	0.667688i	$-0.767284\pi$
$677$		−	2.80516e11i		−	1.33538i	0.744441	−	0.667688i	$-0.232716\pi$
$678$		0			0
$679$			2.52957e9i			0.0119005i
$680$		0			0
$681$	−3.03616e11			−1.41168
$682$		0			0
$683$	−5.79360e10			−0.266235			−0.133118	−	0.991100i	$-0.542499\pi$
$683$	−5.79360e10			−0.266235			−0.133118	+	0.991100i	$0.542499\pi$
$684$		0			0
$685$	−1.84969e11	+	1.65133e11i	−0.840112	+	0.750018i
$686$		0			0
$687$	1.37815e11			0.618686
$688$		0			0
$689$	3.27114e10			0.145152
$690$		0			0
$691$		−	2.55640e11i		−	1.12129i	−0.828057	−	0.560644i	$-0.810554\pi$
$691$		−	2.55640e11i		−	1.12129i	0.828057	−	0.560644i	$-0.189446\pi$
$692$		0			0
$693$		−	1.14676e11i		−	0.497210i
$694$		0			0
$695$	1.31402e11	−	1.17311e11i	0.563201	−	0.502803i
$696$		0			0
$697$			3.37505e11i			1.43004i
$698$		0			0
$699$			2.17201e11i			0.909817i
$700$		0			0
$701$	−1.52264e11			−0.630556			−0.315278	−	0.948999i	$-0.602098\pi$
$701$	−1.52264e11			−0.630556			−0.315278	+	0.948999i	$0.602098\pi$
$702$		0			0
$703$	−1.19492e10			−0.0489235
$704$		0			0
$705$	−9.32164e10	−	1.04414e11i	−0.377343	−	0.422670i
$706$		0			0
$707$	−2.33165e11			−0.933225
$708$		0			0
$709$	−8.99510e10			−0.355976			−0.177988	−	0.984033i	$-0.556959\pi$
$709$	−8.99510e10			−0.355976			−0.177988	+	0.984033i	$0.556959\pi$
$710$		0			0
$711$			4.34447e11i			1.70004i
$712$		0			0
$713$			3.19829e11i			1.23754i
$714$		0			0
$715$	−7.32461e10	−	8.20446e10i	−0.280259	−	0.313925i
$716$		0			0
$717$			8.28915e11i			3.13641i
$718$		0			0
$719$		−	3.34871e11i		−	1.25303i	−0.779409	−	0.626515i	$-0.784481\pi$
$719$		−	3.34871e11i		−	1.25303i	0.779409	−	0.626515i	$-0.215519\pi$
$720$		0			0
$721$	−8.48927e10			−0.314144
$722$		0			0
$723$	−2.12735e11			−0.778547
$724$		0			0
$725$	2.35637e10	−	2.07278e11i	0.0852886	−	0.750243i
$726$		0			0
$727$	4.68905e11			1.67860			0.839300	−	0.543669i	$-0.182965\pi$
$727$	4.68905e11			1.67860			0.839300	+	0.543669i	$0.182965\pi$
$728$		0			0
$729$	−4.18511e11			−1.48182
$730$		0			0
$731$		−	5.75963e11i		−	2.01709i
$732$		0			0
$733$			6.30452e10i			0.218392i	0.994020	+	0.109196i	$0.0348276\pi$
$733$			6.30452e10i			0.218392i	−0.994020	+	0.109196i	$0.965172\pi$
$734$		0			0
$735$	−1.28675e11	−	1.44132e11i	−0.440906	−	0.493869i
$736$		0			0
$737$		−	1.98272e11i		−	0.672033i
$738$		0			0
$739$			2.09954e11i			0.703959i	0.936008	+	0.351979i	$0.114491\pi$
$739$			2.09954e11i			0.703959i	−0.936008	+	0.351979i	$0.885509\pi$
$740$		0			0
$741$	3.16542e11			1.04993
$742$		0			0
$743$	−2.18225e11			−0.716060			−0.358030	−	0.933710i	$-0.616552\pi$
$743$	−2.18225e11			−0.716060			−0.358030	+	0.933710i	$0.616552\pi$
$744$		0			0
$745$	−1.82666e11	−	2.04608e11i	−0.592970	−	0.664199i
$746$		0			0
$747$	2.44143e11			0.784081
$748$		0			0
$749$	2.08401e10			0.0662175
$750$		0			0
$751$		−	5.33771e10i		−	0.167801i	−0.996474	−	0.0839006i	$-0.973262\pi$
$751$		−	5.33771e10i		−	0.167801i	0.996474	−	0.0839006i	$-0.0267378\pi$
$752$		0			0
$753$			8.16886e10i			0.254087i
$754$		0			0
$755$	3.42573e11	−	3.05835e11i	1.05430	−	0.941238i
$756$		0			0
$757$			2.91321e11i			0.887132i	0.896242	+	0.443566i	$0.146287\pi$
$757$			2.91321e11i			0.887132i	−0.896242	+	0.443566i	$0.853713\pi$
$758$		0			0
$759$		−	1.82347e11i		−	0.549453i
$760$		0			0
$761$	1.92349e10			0.0573523			0.0286762	−	0.999589i	$-0.490871\pi$
$761$	1.92349e10			0.0573523			0.0286762	+	0.999589i	$0.490871\pi$
$762$		0			0
$763$	−6.50417e10			−0.191908
$764$		0			0
$765$	−7.75025e11	+	6.91911e11i	−2.26292	+	2.02025i
$766$		0			0
$767$	2.49217e11			0.720105
$768$		0			0
$769$	3.49604e11			0.999703			0.499852	−	0.866111i	$-0.333388\pi$
$769$	3.49604e11			0.999703			0.499852	+	0.866111i	$0.333388\pi$
$770$		0			0
$771$			9.89843e11i			2.80123i
$772$		0			0
$773$		−	6.52120e11i		−	1.82646i	−0.407447	−	0.913229i	$-0.633581\pi$
$773$		−	6.52120e11i		−	1.82646i	0.407447	−	0.913229i	$-0.366419\pi$
$774$		0			0
$775$	4.84979e11	+	5.51331e10i	1.34436	+	0.152829i
$776$		0			0
$777$			4.27333e10i			0.117242i
$778$		0			0
$779$		−	1.66477e11i		−	0.452068i
$780$		0			0
$781$	−1.88768e11			−0.507371
$782$		0			0
$783$	−3.67233e11			−0.977001
$784$		0			0
$785$	−1.83192e11	+	1.63547e11i	−0.482424	+	0.430689i
$786$		0			0
$787$	3.62094e11			0.943892			0.471946	−	0.881627i	$-0.343552\pi$
$787$	3.62094e11			0.943892			0.471946	+	0.881627i	$0.343552\pi$
$788$		0			0
$789$	1.09489e12			2.82530
$790$		0			0
$791$		−	2.89079e11i		−	0.738432i
$792$		0			0
$793$			4.70171e11i			1.18895i
$794$		0			0
$795$	6.17430e10	−	5.51216e10i	0.154568	−	0.137992i
$796$		0			0
$797$		−	1.33733e11i		−	0.331440i	−0.986173	−	0.165720i	$-0.947005\pi$
$797$		−	1.33733e11i		−	0.331440i	0.986173	−	0.165720i	$-0.0529948\pi$
$798$		0			0
$799$		−	2.36641e11i		−	0.580634i
$800$		0			0
$801$	2.94672e11			0.715828
$802$		0			0
$803$	−3.83704e10			−0.0922857
$804$		0			0
$805$	1.98583e11	+	2.22437e11i	0.472887	+	0.529692i
$806$		0			0
$807$	−1.78041e11			−0.419784
$808$		0			0
$809$	4.11621e11			0.960956			0.480478	−	0.877007i	$-0.340463\pi$
$809$	4.11621e11			0.960956			0.480478	+	0.877007i	$0.340463\pi$
$810$		0			0
$811$		−	7.23209e11i		−	1.67178i	−0.548894	−	0.835892i	$-0.684951\pi$
$811$		−	7.23209e11i		−	1.67178i	0.548894	−	0.835892i	$-0.315049\pi$
$812$		0			0
$813$		−	4.07100e11i		−	0.931836i
$814$		0			0
$815$	3.65758e11	+	4.09693e11i	0.829016	+	0.928600i
$816$		0			0
$817$			2.84098e11i			0.637646i
$818$		0			0
$819$		−	7.24316e11i		−	1.60988i
$820$		0			0
$821$	−6.38146e11			−1.40458			−0.702291	−	0.711890i	$-0.747840\pi$
$821$	−6.38146e11			−1.40458			−0.702291	+	0.711890i	$0.747840\pi$
$822$		0			0
$823$	2.86794e11			0.625130			0.312565	−	0.949896i	$-0.398812\pi$
$823$	2.86794e11			0.625130			0.312565	+	0.949896i	$0.398812\pi$
$824$		0			0
$825$	−2.76505e11	−	3.14335e10i	−0.596880	−	0.0678541i
$826$		0			0
$827$	6.21880e11			1.32949			0.664745	−	0.747071i	$-0.268540\pi$
$827$	6.21880e11			1.32949			0.664745	+	0.747071i	$0.268540\pi$
$828$		0			0
$829$	5.50205e11			1.16495			0.582474	−	0.812849i	$-0.302085\pi$
$829$	5.50205e11			1.16495			0.582474	+	0.812849i	$0.302085\pi$
$830$		0			0
$831$		−	1.61572e11i		−	0.338814i
$832$		0			0
$833$		−	3.26658e11i		−	0.678442i
$834$		0			0
$835$	−1.78367e11	−	1.99793e11i	−0.366918	−	0.410993i
$836$		0			0
$837$		−	8.59235e11i		−	1.75069i
$838$		0			0
$839$		−	9.29570e9i		−	0.0187601i	−0.999956	−	0.00938003i	$-0.997014\pi$
$839$		−	9.29570e9i		−	0.0187601i	0.999956	−	0.00938003i	$-0.00298580\pi$
$840$		0			0
$841$	−2.15037e11			−0.429862
$842$		0			0
$843$	2.77474e11			0.549429
$844$		0			0
$845$	−1.23101e11	−	1.37888e11i	−0.241454	−	0.270458i
$846$		0			0
$847$	−3.47625e11			−0.675426
$848$		0			0
$849$	1.63828e12			3.15325
$850$		0			0
$851$			4.34771e10i			0.0828977i
$852$		0			0
$853$			2.76680e11i			0.522615i	0.965256	+	0.261308i	$0.0841537\pi$
$853$			2.76680e11i			0.522615i	−0.965256	+	0.261308i	$0.915846\pi$
$854$		0			0
$855$	3.82287e11	−	3.41290e11i	0.715361	−	0.638645i
$856$		0			0
$857$		−	8.09328e11i		−	1.50038i	−0.661223	−	0.750190i	$-0.729962\pi$
$857$		−	8.09328e11i		−	1.50038i	0.661223	−	0.750190i	$-0.270038\pi$
$858$		0			0
$859$			2.89123e11i			0.531020i	0.964108	+	0.265510i	$0.0855403\pi$
$859$			2.89123e11i			0.531020i	−0.964108	+	0.265510i	$0.914460\pi$
$860$		0			0
$861$	−5.95362e11			−1.08335
$862$		0			0
$863$	−6.16481e11			−1.11142			−0.555708	−	0.831378i	$-0.687553\pi$
$863$	−6.16481e11			−1.11142			−0.555708	+	0.831378i	$0.687553\pi$
$864$		0			0
$865$	1.43957e11	−	1.28519e11i	0.257140	−	0.229564i
$866$		0			0
$867$	−1.80371e12			−3.19220
$868$		0			0
$869$	−1.96739e11			−0.344994
$870$		0			0
$871$		−	1.25232e12i		−	2.17592i
$872$		0			0
$873$		−	1.58180e10i		−	0.0272329i
$874$		0			0
$875$	3.71529e11	−	2.62780e11i	0.633812	−	0.448292i
$876$		0			0
$877$			6.46830e10i			0.109343i	0.998504	+	0.0546716i	$0.0174112\pi$
$877$			6.46830e10i			0.109343i	−0.998504	+	0.0546716i	$0.982589\pi$
$878$		0			0
$879$			1.19560e12i			2.00277i
$880$		0			0
$881$	−9.16075e11			−1.52064			−0.760322	−	0.649547i	$-0.774959\pi$
$881$	−9.16075e11			−1.52064			−0.760322	+	0.649547i	$0.774959\pi$
$882$		0			0
$883$	7.21518e11			1.18687			0.593437	−	0.804881i	$-0.297771\pi$
$883$	7.21518e11			1.18687			0.593437	+	0.804881i	$0.297771\pi$
$884$		0			0
$885$	4.70398e11	−	4.19953e11i	0.766819	−	0.684585i
$886$		0			0
$887$	−1.56512e10			−0.0252844			−0.0126422	−	0.999920i	$-0.504024\pi$
$887$	−1.56512e10			−0.0252844			−0.0126422	+	0.999920i	$0.504024\pi$
$888$		0			0
$889$	−3.94119e11			−0.630987
$890$		0			0
$891$			8.62306e10i			0.136820i
$892$		0			0
$893$			1.16725e11i			0.183551i
$894$		0			0
$895$	−1.09244e11	+	9.75287e10i	−0.170257	+	0.151999i
$896$		0			0
$897$		−	1.15174e12i		−	1.77903i
$898$		0			0
$899$			6.67319e11i			1.02163i
$900$		0			0
$901$	1.39933e11			0.212334
$902$		0			0
$903$	1.01600e12			1.52807
$904$		0			0
$905$	8.25083e10	+	9.24194e10i	0.122999	+	0.137774i
$906$		0			0
$907$	7.40442e11			1.09411			0.547056	−	0.837096i	$-0.315748\pi$
$907$	7.40442e11			1.09411			0.547056	+	0.837096i	$0.315748\pi$
$908$		0			0
$909$	1.45804e12			2.13556
$910$		0			0
$911$			1.11646e12i			1.62095i	0.585770	+	0.810477i	$0.300792\pi$
$911$			1.11646e12i			1.62095i	−0.585770	+	0.810477i	$0.699208\pi$
$912$		0			0
$913$			1.10560e11i			0.159116i
$914$		0			0
$915$	7.92281e11	+	8.87452e11i	1.13030	+	1.26608i
$916$		0			0
$917$		−	2.12515e11i		−	0.300547i
$918$		0			0
$919$		−	4.29970e11i		−	0.602804i	−0.953497	−	0.301402i	$-0.902545\pi$
$919$		−	4.29970e11i		−	0.602804i	0.953497	−	0.301402i	$-0.0974546\pi$
$920$		0			0
$921$	−3.06536e11			−0.426033
$922$		0			0
$923$	−1.19230e12			−1.64277
$924$		0			0
$925$	6.59274e10	+	7.49472e9i	0.0900532	+	0.0102374i
$926$		0			0
$927$	5.30854e11			0.718879
$928$		0			0
$929$	1.03621e12			1.39119			0.695594	−	0.718435i	$-0.255141\pi$
$929$	1.03621e12			1.39119			0.695594	+	0.718435i	$0.255141\pi$
$930$		0			0
$931$			1.61126e11i			0.214471i
$932$		0			0
$933$		−	1.34299e11i		−	0.177233i
$934$		0			0
$935$	−3.13332e11	−	3.50970e11i	−0.409976	−	0.459223i
$936$		0			0
$937$			9.79343e11i			1.27051i	0.772304	+	0.635253i	$0.219104\pi$
$937$			9.79343e11i			1.27051i	−0.772304	+	0.635253i	$0.780896\pi$
$938$		0			0
$939$		−	1.30616e12i		−	1.68009i
$940$		0			0
$941$	3.95736e11			0.504715			0.252358	−	0.967634i	$-0.418794\pi$
$941$	3.95736e11			0.504715			0.252358	+	0.967634i	$0.418794\pi$
$942$		0			0
$943$	−6.05725e11			−0.766000
$944$		0			0
$945$	−5.33501e11	−	5.97586e11i	−0.668972	−	0.749330i
$946$		0			0
$947$	−7.08577e11			−0.881022			−0.440511	−	0.897747i	$-0.645203\pi$
$947$	−7.08577e11			−0.881022			−0.440511	+	0.897747i	$0.645203\pi$
$948$		0			0
$949$	−2.42355e11			−0.298804
$950$		0			0
$951$			7.30979e11i			0.893682i
$952$		0			0
$953$		−	4.04943e11i		−	0.490932i	−0.969405	−	0.245466i	$-0.921059\pi$
$953$		−	4.04943e11i		−	0.490932i	0.969405	−	0.245466i	$-0.0789411\pi$
$954$		0			0
$955$	−1.07161e12	+	9.56694e11i	−1.28832	+	1.15016i
$956$		0			0
$957$		−	3.80464e11i		−	0.453592i
$958$		0			0
$959$		−	7.39490e11i		−	0.874295i
$960$		0			0
$961$	−7.08468e11			−0.830667
$962$		0			0
$963$	−1.30318e11			−0.151530
$964$		0			0
$965$	3.03399e11	−	2.70862e11i	0.349868	−	0.312348i
$966$		0			0
$967$	−1.36193e12			−1.55758			−0.778789	−	0.627286i	$-0.784166\pi$
$967$	−1.36193e12			−1.55758			−0.778789	+	0.627286i	$0.784166\pi$
$968$		0			0
$969$	1.35410e12			1.53588
$970$		0			0
$971$			1.12453e12i			1.26501i	0.774556	+	0.632505i	$0.217973\pi$
$971$			1.12453e12i			1.26501i	−0.774556	+	0.632505i	$0.782027\pi$
$972$		0			0
$973$			5.25334e11i			0.586117i
$974$		0			0
$975$	−1.74646e12	−	1.98540e11i	−1.93259	−	0.219699i
$976$		0			0
$977$		−	1.36587e12i		−	1.49910i	−0.661949	−	0.749549i	$-0.730270\pi$
$977$		−	1.36587e12i		−	1.49910i	0.661949	−	0.749549i	$-0.269730\pi$
$978$		0			0
$979$			1.33442e11i			0.145266i
$980$		0			0
$981$	4.06721e11			0.439158
$982$		0			0
$983$	1.23632e12			1.32409			0.662046	−	0.749463i	$-0.269688\pi$
$983$	1.23632e12			1.32409			0.662046	+	0.749463i	$0.269688\pi$
$984$		0			0
$985$	5.50521e11	−	4.91483e11i	0.584829	−	0.522112i
$986$		0			0
$987$	4.17437e11			0.439868
$988$		0			0
$989$	1.03369e12			1.08045
$990$		0			0
$991$			1.30551e12i			1.35359i	0.736172	+	0.676794i	$0.236631\pi$
$991$			1.30551e12i			1.35359i	−0.736172	+	0.676794i	$0.763369\pi$
$992$		0			0
$993$		−	2.17818e12i		−	2.24025i
$994$		0			0
$995$	−4.01814e11	+	3.58723e11i	−0.409952	+	0.365989i
$996$		0			0
$997$		−	1.69693e12i		−	1.71744i	−0.512441	−	0.858722i	$-0.671259\pi$
$997$		−	1.69693e12i		−	1.71744i	0.512441	−	0.858722i	$-0.328741\pi$
$998$		0			0
$999$		−	1.16803e11i		−	0.117272i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.h.g.319.20		20
4.3	odd	2	inner	320.9.h.g.319.2		20
5.4	even	2	inner	320.9.h.g.319.1		20
8.3	odd	2		20.9.d.c.19.13	yes	20
8.5	even	2		20.9.d.c.19.7	✓	20
20.19	odd	2	inner	320.9.h.g.319.19		20
40.3	even	4		100.9.b.g.51.3		20
40.13	odd	4		100.9.b.g.51.4		20
40.19	odd	2		20.9.d.c.19.8	yes	20
40.27	even	4		100.9.b.g.51.18		20
40.29	even	2		20.9.d.c.19.14	yes	20
40.37	odd	4		100.9.b.g.51.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.d.c.19.7	✓	20	8.5	even	2
20.9.d.c.19.8	yes	20	40.19	odd	2
20.9.d.c.19.13	yes	20	8.3	odd	2
20.9.d.c.19.14	yes	20	40.29	even	2
100.9.b.g.51.3		20	40.3	even	4
100.9.b.g.51.4		20	40.13	odd	4
100.9.b.g.51.17		20	40.37	odd	4
100.9.b.g.51.18		20	40.27	even	4
320.9.h.g.319.1		20	5.4	even	2	inner
320.9.h.g.319.2		20	4.3	odd	2	inner
320.9.h.g.319.19		20	20.19	odd	2	inner
320.9.h.g.319.20		20	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	320.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+134.970 q^{3} +(416.235 + 466.234i) q^{5} -1863.96 q^{7} +11655.8 q^{9} +O(q^{10})\)	\(q+134.970 q^{3} +(416.235 + 466.234i) q^{5} -1863.96 q^{7} +11655.8 q^{9} +5278.32i q^{11} +33338.9i q^{13} +(56179.0 + 62927.3i) q^{15} +142617. i q^{17} -70346.9i q^{19} -251578. q^{21} -255957. q^{23} +(-44122.6 + 388125. i) q^{25} +687639. q^{27} -534050. q^{29} -1.24954e6i q^{31} +712412. i q^{33} +(-775844. - 869041. i) q^{35} -169861. i q^{37} +4.49973e6i q^{39} +2.36651e6 q^{41} -4.03853e6 q^{43} +(4.85153e6 + 5.43431e6i) q^{45} -1.65928e6 q^{47} -2.29046e6 q^{49} +1.92489e7i q^{51} -981179. i q^{53} +(-2.46093e6 + 2.19702e6i) q^{55} -9.49469e6i q^{57} -7.47526e6i q^{59} +1.41028e7 q^{61} -2.17259e7 q^{63} +(-1.55437e7 + 1.38768e7i) q^{65} -3.75634e7 q^{67} -3.45463e7 q^{69} +3.57630e7i q^{71} +7.26944e6i q^{73} +(-5.95521e6 + 5.23851e7i) q^{75} -9.83857e6i q^{77} +3.72731e7i q^{79} +1.63368e7 q^{81} +2.09461e7 q^{83} +(-6.64928e7 + 5.93621e7i) q^{85} -7.20805e7 q^{87} +2.52812e7 q^{89} -6.21423e7i q^{91} -1.68650e8i q^{93} +(3.27981e7 - 2.92808e7i) q^{95} -1.35709e6i q^{97} +6.15229e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 1420 q^{5} + 2556 q^{9}+O(q^{10}) \) 20 * q + 1420 * q^5 + 2556 * q^9	\( 20 q + 1420 q^{5} + 2556 q^{9} - 410256 q^{21} - 657260 q^{25} - 660136 q^{29} + 7068520 q^{41} + 18729060 q^{45} + 11719036 q^{49} + 17660440 q^{61} - 44202240 q^{65} - 111747216 q^{69} - 154212444 q^{81} - 68800000 q^{85} + 105006376 q^{89}+O(q^{100}) \) 20 * q + 1420 * q^5 + 2556 * q^9 - 410256 * q^21 - 657260 * q^25 - 660136 * q^29 + 7068520 * q^41 + 18729060 * q^45 + 11719036 * q^49 + 17660440 * q^61 - 44202240 * q^65 - 111747216 * q^69 - 154212444 * q^81 - 68800000 * q^85 + 105006376 * q^89

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