LMFDB - Embedded newform 320.9.h.f.319.10

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 1394 x^{14} + 1332306 x^{12} - 657883370 x^{10} + 233333110300 x^{8} - 45644912663250 x^{6} + \cdots + 23\!\cdots\!00 $ x^16 - 1394x^14 + 1332306x^12 - 657883370x^10 + 233333110300x^8 - 45644912663250x^6 + 6418092533338125x^4 - 470478785036343750*x^2 + 23585385849201562500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{99}\cdot 3^{9}\cdot 5^{10} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.10
Root			$23.2011 + 13.3951i$ of defining polynomial
Character	$\chi$	$=$	320.319
Dual form			320.9.h.f.319.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.3781 q^{3} +(202.796 + 591.184i) q^{5} -1597.12 q^{7} -6060.22 q^{9} +O(q^{10})$	\(q+22.3781 q^{3} +(202.796 + 591.184i) q^{5} -1597.12 q^{7} -6060.22 q^{9} -9158.27i q^{11} +5553.53i q^{13} +(4538.19 + 13229.6i) q^{15} +10451.3i q^{17} +121116. i q^{19} -35740.4 q^{21} +239374. q^{23} +(-308372. + 239780. i) q^{25} -282439. q^{27} -301677. q^{29} +788407. i q^{31} -204945. i q^{33} +(-323889. - 944190. i) q^{35} -1.93051e6i q^{37} +124277. i q^{39} -1.28515e6 q^{41} -6.03743e6 q^{43} +(-1.22899e6 - 3.58271e6i) q^{45} +8.00316e6 q^{47} -3.21402e6 q^{49} +233880. i q^{51} -6.59254e6i q^{53} +(5.41422e6 - 1.85726e6i) q^{55} +2.71034e6i q^{57} -1.67773e7i q^{59} +5.53062e6 q^{61} +9.67888e6 q^{63} +(-3.28316e6 + 1.12624e6i) q^{65} -1.45418e7 q^{67} +5.35674e6 q^{69} +1.26273e7i q^{71} +2.28159e7i q^{73} +(-6.90078e6 + 5.36581e6i) q^{75} +1.46268e7i q^{77} -3.51394e7i q^{79} +3.34407e7 q^{81} +3.74104e7 q^{83} +(-6.17864e6 + 2.11948e6i) q^{85} -6.75096e6 q^{87} +2.75481e7 q^{89} -8.86964e6i q^{91} +1.76430e7i q^{93} +(-7.16018e7 + 2.45619e7i) q^{95} -9.97232e7i q^{97} +5.55011e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 600 q^{5} + 42176 q^{9}+O(q^{10}) $ 16 * q - 600 * q^5 + 42176 * q^9	$ 16 q - 600 q^{5} + 42176 q^{9} - 619344 q^{21} + 1137040 q^{25} + 3497568 q^{29} - 2169168 q^{41} + 1930760 q^{45} + 26174912 q^{49} - 22772656 q^{61} + 12524160 q^{65} - 8461392 q^{69} - 224999456 q^{81} + 18124800 q^{85} - 94250976 q^{89}+O(q^{100}) $ 16 * q - 600 * q^5 + 42176 * q^9 - 619344 * q^21 + 1137040 * q^25 + 3497568 * q^29 - 2169168 * q^41 + 1930760 * q^45 + 26174912 * q^49 - 22772656 * q^61 + 12524160 * q^65 - 8461392 * q^69 - 224999456 * q^81 + 18124800 * q^85 - 94250976 * 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
$3$	22.3781			0.276273			0.138136	−	0.990413i	$-0.455889\pi$
$3$	22.3781			0.276273			0.138136	+	0.990413i	$0.455889\pi$
$4$		0			0
$5$	202.796	+	591.184i	0.324474	+	0.945895i
$5$	202.796	+	591.184i	0.324474	+	0.945895i
$6$		0			0
$7$	−1597.12			−0.665188			−0.332594	−	0.943070i	$-0.607924\pi$
$7$	−1597.12			−0.665188			−0.332594	+	0.943070i	$0.607924\pi$
$8$		0			0
$9$	−6060.22			−0.923673
$10$		0			0
$11$		−	9158.27i		−	0.625522i	−0.949832	−	0.312761i	$-0.898746\pi$
$11$		−	9158.27i		−	0.625522i	0.949832	−	0.312761i	$-0.101254\pi$
$12$		0			0
$13$			5553.53i			0.194445i	0.995263	+	0.0972223i	$0.0309958\pi$
$13$			5553.53i			0.194445i	−0.995263	+	0.0972223i	$0.969004\pi$
$14$		0			0
$15$	4538.19	+	13229.6i	0.0896433	+	0.261325i
$16$		0			0
$17$			10451.3i			0.125134i	0.998041	+	0.0625669i	$0.0199287\pi$
$17$			10451.3i			0.125134i	−0.998041	+	0.0625669i	$0.980071\pi$
$18$		0			0
$19$			121116.i			0.929366i	0.885477	+	0.464683i	$0.153832\pi$
$19$			121116.i			0.929366i	−0.885477	+	0.464683i	$0.846168\pi$
$20$		0			0
$21$	−35740.4			−0.183773
$22$		0			0
$23$	239374.			0.855394			0.427697	−	0.903922i	$-0.359325\pi$
$23$	239374.			0.855394			0.427697	+	0.903922i	$0.359325\pi$
$24$		0			0
$25$	−308372.	+	239780.i	−0.789433	+	0.613836i
$26$		0			0
$27$	−282439.			−0.531458
$28$		0			0
$29$	−301677.			−0.426531			−0.213266	−	0.976994i	$-0.568410\pi$
$29$	−301677.			−0.426531			−0.213266	+	0.976994i	$0.568410\pi$
$30$		0			0
$31$			788407.i			0.853697i	0.904323	+	0.426849i	$0.140376\pi$
$31$			788407.i			0.853697i	−0.904323	+	0.426849i	$0.859624\pi$
$32$		0			0
$33$		−	204945.i		−	0.172815i
$34$		0			0
$35$	−323889.	−	944190.i	−0.215836	−	0.629198i
$36$		0			0
$37$		−	1.93051e6i		−	1.03006i	−0.857171	−	0.515032i	$-0.827780\pi$
$37$		−	1.93051e6i		−	1.03006i	0.857171	−	0.515032i	$-0.172220\pi$
$38$		0			0
$39$			124277.i			0.0537197i
$40$		0			0
$41$	−1.28515e6			−0.454799			−0.227400	−	0.973802i	$-0.573022\pi$
$41$	−1.28515e6			−0.454799			−0.227400	+	0.973802i	$0.573022\pi$
$42$		0			0
$43$	−6.03743e6			−1.76595			−0.882975	−	0.469419i	$-0.844463\pi$
$43$	−6.03743e6			−1.76595			−0.882975	+	0.469419i	$0.844463\pi$
$44$		0			0
$45$	−1.22899e6	−	3.58271e6i	−0.299708	−	0.873698i
$46$		0			0
$47$	8.00316e6			1.64010			0.820049	−	0.572293i	$-0.193946\pi$
$47$	8.00316e6			1.64010			0.820049	+	0.572293i	$0.193946\pi$
$48$		0			0
$49$	−3.21402e6			−0.557525
$50$		0			0
$51$			233880.i			0.0345711i
$52$		0			0
$53$		−	6.59254e6i		−	0.835506i	−0.908561	−	0.417753i	$-0.862818\pi$
$53$		−	6.59254e6i		−	0.835506i	0.908561	−	0.417753i	$-0.137182\pi$
$54$		0			0
$55$	5.41422e6	−	1.85726e6i	0.591678	−	0.202966i
$56$		0			0
$57$			2.71034e6i			0.256759i
$58$		0			0
$59$		−	1.67773e7i		−	1.38456i	−0.721627	−	0.692282i	$-0.756605\pi$
$59$		−	1.67773e7i		−	1.38456i	0.721627	−	0.692282i	$-0.243395\pi$
$60$		0			0
$61$	5.53062e6			0.399443			0.199722	−	0.979853i	$-0.435996\pi$
$61$	5.53062e6			0.399443			0.199722	+	0.979853i	$0.435996\pi$
$62$		0			0
$63$	9.67888e6			0.614416
$64$		0			0
$65$	−3.28316e6	+	1.12624e6i	−0.183924	+	0.0630922i
$66$		0			0
$67$	−1.45418e7			−0.721639			−0.360820	−	0.932636i	$-0.617503\pi$
$67$	−1.45418e7			−0.721639			−0.360820	+	0.932636i	$0.617503\pi$
$68$		0			0
$69$	5.35674e6			0.236322
$70$		0			0
$71$			1.26273e7i			0.496910i	0.968643	+	0.248455i	$0.0799229\pi$
$71$			1.26273e7i			0.496910i	−0.968643	+	0.248455i	$0.920077\pi$
$72$		0			0
$73$			2.28159e7i			0.803425i	0.915766	+	0.401713i	$0.131585\pi$
$73$			2.28159e7i			0.803425i	−0.915766	+	0.401713i	$0.868415\pi$
$74$		0			0
$75$	−6.90078e6	+	5.36581e6i	−0.218099	+	0.169586i
$76$		0			0
$77$			1.46268e7i			0.416090i
$78$		0			0
$79$		−	3.51394e7i		−	0.902165i	−0.892482	−	0.451083i	$-0.851038\pi$
$79$		−	3.51394e7i		−	0.902165i	0.892482	−	0.451083i	$-0.148962\pi$
$80$		0			0
$81$	3.34407e7			0.776846
$82$		0			0
$83$	3.74104e7			0.788279			0.394140	−	0.919051i	$-0.371043\pi$
$83$	3.74104e7			0.788279			0.394140	+	0.919051i	$0.371043\pi$
$84$		0			0
$85$	−6.17864e6	+	2.11948e6i	−0.118363	+	0.0406027i
$86$		0			0
$87$	−6.75096e6			−0.117839
$88$		0			0
$89$	2.75481e7			0.439068			0.219534	−	0.975605i	$-0.429546\pi$
$89$	2.75481e7			0.439068			0.219534	+	0.975605i	$0.429546\pi$
$90$		0			0
$91$		−	8.86964e6i		−	0.129342i
$92$		0			0
$93$			1.76430e7i			0.235853i
$94$		0			0
$95$	−7.16018e7	+	2.45619e7i	−0.879083	+	0.301555i
$96$		0			0
$97$		−	9.97232e7i		−	1.12644i	−0.826306	−	0.563222i	$-0.809562\pi$
$97$		−	9.97232e7i		−	1.12644i	0.826306	−	0.563222i	$-0.190438\pi$
$98$		0			0
$99$			5.55011e7i			0.577778i
$100$		0			0
$101$	1.23920e8			1.19085			0.595424	−	0.803411i	$-0.296984\pi$
$101$	1.23920e8			1.19085			0.595424	+	0.803411i	$0.296984\pi$
$102$		0			0
$103$	−1.14815e8			−1.02012			−0.510060	−	0.860139i	$-0.670377\pi$
$103$	−1.14815e8			−1.02012			−0.510060	+	0.860139i	$0.670377\pi$
$104$		0			0
$105$	−7.24802e6	−	2.11292e7i	−0.0596297	−	0.173830i
$106$		0			0
$107$	1.84428e8			1.40699			0.703497	−	0.710698i	$-0.251621\pi$
$107$	1.84428e8			1.40699			0.703497	+	0.710698i	$0.251621\pi$
$108$		0			0
$109$	−8.34305e7			−0.591043			−0.295521	−	0.955336i	$-0.595493\pi$
$109$	−8.34305e7			−0.591043			−0.295521	+	0.955336i	$0.595493\pi$
$110$		0			0
$111$		−	4.32010e7i		−	0.284579i
$112$		0			0
$113$		−	2.27463e8i		−	1.39508i	−0.716548	−	0.697538i	$-0.754279\pi$
$113$		−	2.27463e8i		−	1.39508i	0.716548	−	0.697538i	$-0.245721\pi$
$114$		0			0
$115$	4.85442e7	+	1.41514e8i	0.277553	+	0.809112i
$116$		0			0
$117$		−	3.36556e7i		−	0.179603i
$118$		0			0
$119$		−	1.66919e7i		−	0.0832375i
$120$		0			0
$121$	1.30485e8			0.608722
$122$		0			0
$123$	−2.87593e7			−0.125649
$124$		0			0
$125$	−2.04291e8	−	1.33678e8i	−0.836775	−	0.547547i
$126$		0			0
$127$	5.00525e7			0.192403			0.0962013	−	0.995362i	$-0.469331\pi$
$127$	5.00525e7			0.192403			0.0962013	+	0.995362i	$0.469331\pi$
$128$		0			0
$129$	−1.35106e8			−0.487884
$130$		0			0
$131$		−	1.67481e8i		−	0.568695i	−0.958721	−	0.284347i	$-0.908223\pi$
$131$		−	1.67481e8i		−	0.568695i	0.958721	−	0.284347i	$-0.0917769\pi$
$132$		0			0
$133$		−	1.93436e8i		−	0.618203i
$134$		0			0
$135$	−5.72775e7	−	1.66973e8i	−0.172444	−	0.502704i
$136$		0			0
$137$		−	1.49477e8i		−	0.424319i	−0.977235	−	0.212160i	$-0.931950\pi$
$137$		−	1.49477e8i		−	0.424319i	0.977235	−	0.212160i	$-0.0680497\pi$
$138$		0			0
$139$		−	1.15806e8i		−	0.310221i	−0.987897	−	0.155111i	$-0.950427\pi$
$139$		−	1.15806e8i		−	0.310221i	0.987897	−	0.155111i	$-0.0495734\pi$
$140$		0			0
$141$	1.79095e8			0.453114
$142$		0			0
$143$	5.08607e7			0.121629
$144$		0			0
$145$	−6.11790e7	−	1.78347e8i	−0.138398	−	0.403453i
$146$		0			0
$147$	−7.19236e7			−0.154029
$148$		0			0
$149$	7.15334e7			0.145132			0.0725661	−	0.997364i	$-0.476881\pi$
$149$	7.15334e7			0.145132			0.0725661	+	0.997364i	$0.476881\pi$
$150$		0			0
$151$			7.42099e8i			1.42743i	0.700438	+	0.713714i	$0.252988\pi$
$151$			7.42099e8i			1.42743i	−0.700438	+	0.713714i	$0.747012\pi$
$152$		0			0
$153$		−	6.33372e7i		−	0.115583i
$154$		0			0
$155$	−4.66094e8	+	1.59886e8i	−0.807508	+	0.277003i
$156$		0			0
$157$		−	9.42107e8i		−	1.55061i	−0.631589	−	0.775303i	$-0.717597\pi$
$157$		−	9.42107e8i		−	1.55061i	0.631589	−	0.775303i	$-0.282403\pi$
$158$		0			0
$159$		−	1.47529e8i		−	0.230827i
$160$		0			0
$161$	−3.82309e8			−0.568998
$162$		0			0
$163$	−8.06537e8			−1.14255			−0.571273	−	0.820760i	$-0.693550\pi$
$163$	−8.06537e8			−1.14255			−0.571273	+	0.820760i	$0.693550\pi$
$164$		0			0
$165$	1.21160e8	−	4.15620e7i	0.163465	−	0.0560739i
$166$		0			0
$167$	1.05441e9			1.35564			0.677818	−	0.735230i	$-0.262926\pi$
$167$	1.05441e9			1.35564			0.677818	+	0.735230i	$0.262926\pi$
$168$		0			0
$169$	7.84889e8			0.962191
$170$		0			0
$171$		−	7.33990e8i		−	0.858431i
$172$		0			0
$173$		−	1.44631e9i		−	1.61465i	−0.590108	−	0.807324i	$-0.700915\pi$
$173$		−	1.44631e9i		−	1.61465i	0.590108	−	0.807324i	$-0.299085\pi$
$174$		0			0
$175$	4.92507e8	−	3.82956e8i	0.525122	−	0.408317i
$176$		0			0
$177$		−	3.75443e8i		−	0.382518i
$178$		0			0
$179$			1.10783e9i			1.07909i	0.841955	+	0.539547i	$0.181405\pi$
$179$			1.10783e9i			1.07909i	−0.841955	+	0.539547i	$0.818595\pi$
$180$		0			0
$181$	1.66421e9			1.55058			0.775291	−	0.631605i	$-0.217603\pi$
$181$	1.66421e9			1.55058			0.775291	+	0.631605i	$0.217603\pi$
$182$		0			0
$183$	1.23765e8			0.110355
$184$		0			0
$185$	1.14128e9	−	3.91499e8i	0.974332	−	0.334229i
$186$		0			0
$187$	9.57159e7			0.0782740
$188$		0			0
$189$	4.51088e8			0.353520
$190$		0			0
$191$			1.75551e9i			1.31908i	0.751670	+	0.659539i	$0.229248\pi$
$191$			1.75551e9i			1.31908i	−0.751670	+	0.659539i	$0.770752\pi$
$192$		0			0
$193$		−	1.19976e9i		−	0.864699i	−0.901706	−	0.432349i	$-0.857685\pi$
$193$		−	1.19976e9i		−	0.864699i	0.901706	−	0.432349i	$-0.142315\pi$
$194$		0			0
$195$	−7.34708e7	+	2.52030e7i	−0.0508132	+	0.0174307i
$196$		0			0
$197$		−	9.30981e8i		−	0.618125i	−0.951042	−	0.309062i	$-0.899985\pi$
$197$		−	9.30981e8i		−	0.618125i	0.951042	−	0.309062i	$-0.100015\pi$
$198$		0			0
$199$			6.75378e7i			0.0430660i	0.999768	+	0.0215330i	$0.00685470\pi$
$199$			6.75378e7i			0.0430660i	−0.999768	+	0.0215330i	$0.993145\pi$
$200$		0			0
$201$	−3.25419e8			−0.199369
$202$		0			0
$203$	4.81814e8			0.283723
$204$		0			0
$205$	−2.60624e8	−	7.59762e8i	−0.147570	−	0.430192i
$206$		0			0
$207$	−1.45066e9			−0.790105
$208$		0			0
$209$	1.10921e9			0.581339
$210$		0			0
$211$		−	2.69449e9i		−	1.35940i	−0.733490	−	0.679700i	$-0.762110\pi$
$211$		−	2.69449e9i		−	1.35940i	0.733490	−	0.679700i	$-0.237890\pi$
$212$		0			0
$213$			2.82575e8i			0.137283i
$214$		0			0
$215$	−1.22437e9	−	3.56924e9i	−0.573005	−	1.67040i
$216$		0			0
$217$		−	1.25918e9i		−	0.567869i
$218$		0			0
$219$			5.10575e8i			0.221964i
$220$		0			0
$221$	−5.80416e7			−0.0243316
$222$		0			0
$223$	−4.89597e8			−0.197979			−0.0989895	−	0.995088i	$-0.531561\pi$
$223$	−4.89597e8			−0.197979			−0.0989895	+	0.995088i	$0.531561\pi$
$224$		0			0
$225$	1.86880e9	−	1.45312e9i	0.729178	−	0.566984i
$226$		0			0
$227$	1.86765e9			0.703383			0.351691	−	0.936116i	$-0.385607\pi$
$227$	1.86765e9			0.703383			0.351691	+	0.936116i	$0.385607\pi$
$228$		0			0
$229$	−4.57230e8			−0.166262			−0.0831309	−	0.996539i	$-0.526492\pi$
$229$	−4.57230e8			−0.166262			−0.0831309	+	0.996539i	$0.526492\pi$
$230$		0			0
$231$			3.27320e8i			0.114954i
$232$		0			0
$233$		−	4.05778e9i		−	1.37678i	−0.725341	−	0.688390i	$-0.758318\pi$
$233$		−	4.05778e9i		−	1.37678i	0.725341	−	0.688390i	$-0.241682\pi$
$234$		0			0
$235$	1.62301e9	+	4.73134e9i	0.532169	+	1.55136i
$236$		0			0
$237$		−	7.86353e8i		−	0.249244i
$238$		0			0
$239$		−	2.83485e9i		−	0.868837i	−0.900711	−	0.434418i	$-0.856954\pi$
$239$		−	2.83485e9i		−	0.868837i	0.900711	−	0.434418i	$-0.143046\pi$
$240$		0			0
$241$	−3.85082e9			−1.14152			−0.570762	−	0.821116i	$-0.693352\pi$
$241$	−3.85082e9			−1.14152			−0.570762	+	0.821116i	$0.693352\pi$
$242$		0			0
$243$	2.60142e9			0.746080
$244$		0			0
$245$	−6.51791e8	−	1.90008e9i	−0.180902	−	0.527360i
$246$		0			0
$247$	−6.72621e8			−0.180710
$248$		0			0
$249$	8.37174e8			0.217780
$250$		0			0
$251$		−	6.59447e9i		−	1.66144i	−0.556689	−	0.830721i	$-0.687929\pi$
$251$		−	6.59447e9i		−	1.66144i	0.556689	−	0.830721i	$-0.312071\pi$
$252$		0			0
$253$		−	2.19225e9i		−	0.535068i
$254$		0			0
$255$	−1.38266e8	+	4.74300e7i	−0.0327006	+	0.0112174i
$256$		0			0
$257$		−	7.47455e9i		−	1.71338i	−0.515835	−	0.856688i	$-0.672518\pi$
$257$		−	7.47455e9i		−	1.71338i	0.515835	−	0.856688i	$-0.327482\pi$
$258$		0			0
$259$			3.08324e9i			0.685186i
$260$		0			0
$261$	1.82823e9			0.393975
$262$		0			0
$263$	−4.26079e9			−0.890568			−0.445284	−	0.895389i	$-0.646897\pi$
$263$	−4.26079e9			−0.890568			−0.445284	+	0.895389i	$0.646897\pi$
$264$		0			0
$265$	3.89741e9	−	1.33694e9i	0.790301	−	0.271100i
$266$		0			0
$267$	6.16474e8			0.121303
$268$		0			0
$269$	−1.11794e9			−0.213505			−0.106753	−	0.994286i	$-0.534045\pi$
$269$	−1.11794e9			−0.213505			−0.106753	+	0.994286i	$0.534045\pi$
$270$		0			0
$271$			8.89723e9i			1.64960i	0.565427	+	0.824798i	$0.308711\pi$
$271$			8.89723e9i			1.64960i	−0.565427	+	0.824798i	$0.691289\pi$
$272$		0			0
$273$		−	1.98486e8i		−	0.0357337i
$274$		0			0
$275$	2.19597e9	+	2.82416e9i	0.383968	+	0.493808i
$276$		0			0
$277$		−	4.87529e9i		−	0.828098i	−0.910255	−	0.414049i	$-0.864114\pi$
$277$		−	4.87529e9i		−	0.828098i	0.910255	−	0.414049i	$-0.135886\pi$
$278$		0			0
$279$		−	4.77792e9i		−	0.788537i
$280$		0			0
$281$	−6.51378e9			−1.04474			−0.522369	−	0.852719i	$-0.674952\pi$
$281$	−6.51378e9			−1.04474			−0.522369	+	0.852719i	$0.674952\pi$
$282$		0			0
$283$	−6.99985e9			−1.09130			−0.545648	−	0.838014i	$-0.683717\pi$
$283$	−6.99985e9			−1.09130			−0.545648	+	0.838014i	$0.683717\pi$
$284$		0			0
$285$	−1.60231e9	+	5.49648e8i	−0.242867	+	0.0833115i
$286$		0			0
$287$	2.05254e9			0.302527
$288$		0			0
$289$	6.86653e9			0.984342
$290$		0			0
$291$		−	2.23162e9i		−	0.311206i
$292$		0			0
$293$		−	7.35332e9i		−	0.997729i	−0.866680	−	0.498865i	$-0.833750\pi$
$293$		−	7.35332e9i		−	0.997729i	0.866680	−	0.498865i	$-0.166250\pi$
$294$		0			0
$295$	9.91846e9	−	3.40237e9i	1.30965	−	0.449255i
$296$		0			0
$297$			2.58665e9i			0.332439i
$298$		0			0
$299$			1.32937e9i			0.166327i
$300$		0			0
$301$	9.64249e9			1.17469
$302$		0			0
$303$	2.77310e9			0.328999
$304$		0			0
$305$	1.12159e9	+	3.26962e9i	0.129609	+	0.377831i
$306$		0			0
$307$	−1.62237e10			−1.82640			−0.913202	−	0.407508i	$-0.866398\pi$
$307$	−1.62237e10			−1.82640			−0.913202	+	0.407508i	$0.866398\pi$
$308$		0			0
$309$	−2.56935e9			−0.281831
$310$		0			0
$311$			1.53086e10i			1.63641i	0.574925	+	0.818206i	$0.305031\pi$
$311$			1.53086e10i			1.63641i	−0.574925	+	0.818206i	$0.694969\pi$
$312$		0			0
$313$			1.04251e10i			1.08618i	0.839675	+	0.543089i	$0.182745\pi$
$313$			1.04251e10i			1.08618i	−0.839675	+	0.543089i	$0.817255\pi$
$314$		0			0
$315$	1.96284e9	+	5.72200e9i	0.199362	+	0.581173i
$316$		0			0
$317$			7.88064e9i			0.780413i	0.920727	+	0.390207i	$0.127596\pi$
$317$			7.88064e9i			0.780413i	−0.920727	+	0.390207i	$0.872404\pi$
$318$		0			0
$319$			2.76284e9i			0.266805i
$320$		0			0
$321$	4.12715e9			0.388714
$322$		0			0
$323$	−1.26582e9			−0.116295
$324$		0			0
$325$	−1.33162e9	−	1.71256e9i	−0.119357	−	0.153501i
$326$		0			0
$327$	−1.86702e9			−0.163289
$328$		0			0
$329$	−1.27820e10			−1.09097
$330$		0			0
$331$		−	1.29603e10i		−	1.07970i	−0.841761	−	0.539851i	$-0.818481\pi$
$331$		−	1.29603e10i		−	1.07970i	0.841761	−	0.539851i	$-0.181519\pi$
$332$		0			0
$333$			1.16993e10i			0.951443i
$334$		0			0
$335$	−2.94903e9	−	8.59690e9i	−0.234153	−	0.682595i
$336$		0			0
$337$			1.72887e10i			1.34043i	0.742168	+	0.670214i	$0.233798\pi$
$337$			1.72887e10i			1.34043i	−0.742168	+	0.670214i	$0.766202\pi$
$338$		0			0
$339$		−	5.09020e9i		−	0.385421i
$340$		0			0
$341$	7.22045e9			0.534007
$342$		0			0
$343$	1.43402e10			1.03605
$344$		0			0
$345$	1.08633e9	+	3.16682e9i	0.0766803	+	0.223536i
$346$		0			0
$347$	2.19389e9			0.151320			0.0756599	−	0.997134i	$-0.475894\pi$
$347$	2.19389e9			0.151320			0.0756599	+	0.997134i	$0.475894\pi$
$348$		0			0
$349$	6.68047e9			0.450303			0.225152	−	0.974324i	$-0.427712\pi$
$349$	6.68047e9			0.450303			0.225152	+	0.974324i	$0.427712\pi$
$350$		0			0
$351$		−	1.56853e9i		−	0.103339i
$352$		0			0
$353$		−	4.05658e9i		−	0.261253i	−0.991432	−	0.130626i	$-0.958301\pi$
$353$		−	4.05658e9i		−	0.261253i	0.991432	−	0.130626i	$-0.0416988\pi$
$354$		0			0
$355$	−7.46507e9	+	2.56077e9i	−0.470025	+	0.161234i
$356$		0			0
$357$		−	3.73534e8i		−	0.0229963i
$358$		0			0
$359$		−	6.13114e9i		−	0.369116i	−0.982822	−	0.184558i	$-0.940915\pi$
$359$		−	6.13114e9i		−	0.369116i	0.982822	−	0.184558i	$-0.0590854\pi$
$360$		0			0
$361$	2.31449e9			0.136278
$362$		0			0
$363$	2.92000e9			0.168173
$364$		0			0
$365$	−1.34884e10	+	4.62697e9i	−0.759955	+	0.260691i
$366$		0			0
$367$	2.49628e9			0.137603			0.0688017	−	0.997630i	$-0.478082\pi$
$367$	2.49628e9			0.137603			0.0688017	+	0.997630i	$0.478082\pi$
$368$		0			0
$369$	7.78831e9			0.420086
$370$		0			0
$371$			1.05291e10i			0.555769i
$372$		0			0
$373$			6.35142e9i			0.328122i	0.986450	+	0.164061i	$0.0524594\pi$
$373$			6.35142e9i			0.328122i	−0.986450	+	0.164061i	$0.947541\pi$
$374$		0			0
$375$	−4.57164e9	−	2.99147e9i	−0.231178	−	0.151272i
$376$		0			0
$377$		−	1.67537e9i		−	0.0829367i
$378$		0			0
$379$			5.45573e9i			0.264421i	0.991222	+	0.132210i	$0.0422075\pi$
$379$			5.45573e9i			0.264421i	−0.991222	+	0.132210i	$0.957793\pi$
$380$		0			0
$381$	1.12008e9			0.0531556
$382$		0			0
$383$	−2.27276e10			−1.05623			−0.528114	−	0.849173i	$-0.677101\pi$
$383$	−2.27276e10			−1.05623			−0.528114	+	0.849173i	$0.677101\pi$
$384$		0			0
$385$	−8.64715e9	+	2.96627e9i	−0.393577	+	0.135010i
$386$		0			0
$387$	3.65882e10			1.63116
$388$		0			0
$389$	−4.42468e10			−1.93234			−0.966169	−	0.257909i	$-0.916966\pi$
$389$	−4.42468e10			−1.93234			−0.966169	+	0.257909i	$0.916966\pi$
$390$		0			0
$391$			2.50177e9i			0.107039i
$392$		0			0
$393$		−	3.74790e9i		−	0.157115i
$394$		0			0
$395$	2.07739e10	−	7.12614e9i	0.853353	−	0.292729i
$396$		0			0
$397$			1.27954e10i			0.515100i	0.966265	+	0.257550i	$0.0829152\pi$
$397$			1.27954e10i			0.515100i	−0.966265	+	0.257550i	$0.917085\pi$
$398$		0			0
$399$		−	4.32873e9i		−	0.170793i
$400$		0			0
$401$	−5.64673e9			−0.218384			−0.109192	−	0.994021i	$-0.534826\pi$
$401$	−5.64673e9			−0.218384			−0.109192	+	0.994021i	$0.534826\pi$
$402$		0			0
$403$	−4.37844e9			−0.165997
$404$		0			0
$405$	6.78164e9	+	1.97696e10i	0.252066	+	0.734814i
$406$		0			0
$407$	−1.76801e10			−0.644328
$408$		0			0
$409$	−3.38375e10			−1.20922			−0.604610	−	0.796522i	$-0.706671\pi$
$409$	−3.38375e10			−1.20922			−0.604610	+	0.796522i	$0.706671\pi$
$410$		0			0
$411$		−	3.34502e9i		−	0.117228i
$412$		0			0
$413$			2.67953e10i			0.920996i
$414$		0			0
$415$	7.58669e9	+	2.21164e10i	0.255776	+	0.745629i
$416$		0			0
$417$		−	2.59151e9i		−	0.0857056i
$418$		0			0
$419$			1.74509e10i			0.566188i	0.959092	+	0.283094i	$0.0913609\pi$
$419$			1.74509e10i			0.566188i	−0.959092	+	0.283094i	$0.908639\pi$
$420$		0			0
$421$	−1.05702e10			−0.336476			−0.168238	−	0.985746i	$-0.553808\pi$
$421$	−1.05702e10			−0.336476			−0.168238	+	0.985746i	$0.553808\pi$
$422$		0			0
$423$	−4.85009e10			−1.51492
$424$		0			0
$425$	−2.50601e9	−	3.22289e9i	−0.0768117	−	0.0987848i
$426$		0			0
$427$	−8.83305e9			−0.265705
$428$		0			0
$429$	1.13817e9			0.0336029
$430$		0			0
$431$		−	5.85670e10i		−	1.69724i	−0.529000	−	0.848622i	$-0.677433\pi$
$431$		−	5.85670e10i		−	1.69724i	0.529000	−	0.848622i	$-0.322567\pi$
$432$		0			0
$433$			3.38808e10i			0.963833i	0.876217	+	0.481917i	$0.160059\pi$
$433$			3.38808e10i			0.963833i	−0.876217	+	0.481917i	$0.839941\pi$
$434$		0			0
$435$	−1.36907e9	−	3.99106e9i	−0.0382357	−	0.111463i
$436$		0			0
$437$			2.89920e10i			0.794974i
$438$		0			0
$439$			1.23627e10i			0.332855i	0.986054	+	0.166427i	$0.0532232\pi$
$439$			1.23627e10i			0.332855i	−0.986054	+	0.166427i	$0.946777\pi$
$440$		0			0
$441$	1.94777e10			0.514971
$442$		0			0
$443$	4.72275e9			0.122625			0.0613126	−	0.998119i	$-0.480471\pi$
$443$	4.72275e9			0.122625			0.0613126	+	0.998119i	$0.480471\pi$
$444$		0			0
$445$	5.58666e9	+	1.62860e10i	0.142466	+	0.415312i
$446$		0			0
$447$	1.60078e9			0.0400961
$448$		0			0
$449$	−9.72921e9			−0.239382			−0.119691	−	0.992811i	$-0.538190\pi$
$449$	−9.72921e9			−0.239382			−0.119691	+	0.992811i	$0.538190\pi$
$450$		0			0
$451$			1.17698e10i			0.284487i
$452$		0			0
$453$			1.66068e10i			0.394359i
$454$		0			0
$455$	5.24359e9	−	1.79873e9i	0.122344	−	0.0419682i
$456$		0			0
$457$			6.95026e10i			1.59344i	0.604348	+	0.796721i	$0.293434\pi$
$457$			6.95026e10i			1.59344i	−0.604348	+	0.796721i	$0.706566\pi$
$458$		0			0
$459$		−	2.95185e9i		−	0.0665034i
$460$		0			0
$461$	−6.11928e10			−1.35487			−0.677433	−	0.735584i	$-0.736908\pi$
$461$	−6.11928e10			−1.35487			−0.677433	+	0.735584i	$0.736908\pi$
$462$		0			0
$463$	6.02944e10			1.31206			0.656029	−	0.754735i	$-0.272235\pi$
$463$	6.02944e10			1.31206			0.656029	+	0.754735i	$0.272235\pi$
$464$		0			0
$465$	−1.04303e10	+	3.57794e9i	−0.223092	+	0.0765282i
$466$		0			0
$467$	2.66514e9			0.0560341			0.0280170	−	0.999607i	$-0.491081\pi$
$467$	2.66514e9			0.0560341			0.0280170	+	0.999607i	$0.491081\pi$
$468$		0			0
$469$	2.32250e10			0.480026
$470$		0			0
$471$		−	2.10826e10i		−	0.428390i
$472$		0			0
$473$			5.52925e10i			1.10464i
$474$		0			0
$475$	−2.90412e10	−	3.73488e10i	−0.570479	−	0.733673i
$476$		0			0
$477$			3.99523e10i			0.771735i
$478$		0			0
$479$			7.94312e10i			1.50886i	0.656380	+	0.754431i	$0.272087\pi$
$479$			7.94312e10i			1.50886i	−0.656380	+	0.754431i	$0.727913\pi$
$480$		0			0
$481$	1.07211e10			0.200290
$482$		0			0
$483$	−8.55534e9			−0.157199
$484$		0			0
$485$	5.89548e10	−	2.02235e10i	1.06550	−	0.365502i
$486$		0			0
$487$	1.01535e11			1.80510			0.902552	−	0.430582i	$-0.141692\pi$
$487$	1.01535e11			1.80510			0.902552	+	0.430582i	$0.141692\pi$
$488$		0			0
$489$	−1.80488e10			−0.315654
$490$		0			0
$491$			6.25226e10i			1.07575i	0.843025	+	0.537874i	$0.180773\pi$
$491$			6.25226e10i			1.07575i	−0.843025	+	0.537874i	$0.819227\pi$
$492$		0			0
$493$		−	3.15292e9i		−	0.0533735i
$494$		0			0
$495$	−3.28114e10	+	1.12554e10i	−0.546517	+	0.187474i
$496$		0			0
$497$		−	2.01673e10i		−	0.330539i
$498$		0			0
$499$		−	1.02766e11i		−	1.65748i	−0.559635	−	0.828739i	$-0.689059\pi$
$499$		−	1.02766e11i		−	1.65748i	0.559635	−	0.828739i	$-0.310941\pi$
$500$		0			0
$501$	2.35956e10			0.374525
$502$		0			0
$503$	−8.14362e9			−0.127217			−0.0636085	−	0.997975i	$-0.520261\pi$
$503$	−8.14362e9			−0.127217			−0.0636085	+	0.997975i	$0.520261\pi$
$504$		0			0
$505$	2.51306e10	+	7.32597e10i	0.386399	+	1.12642i
$506$		0			0
$507$	1.75643e10			0.265827
$508$		0			0
$509$	1.19816e10			0.178502			0.0892509	−	0.996009i	$-0.471553\pi$
$509$	1.19816e10			0.178502			0.0892509	+	0.996009i	$0.471553\pi$
$510$		0			0
$511$		−	3.64396e10i		−	0.534429i
$512$		0			0
$513$		−	3.42078e10i		−	0.493920i
$514$		0			0
$515$	−2.32841e10	−	6.78770e10i	−0.331002	−	0.964926i
$516$		0			0
$517$		−	7.32951e10i		−	1.02592i
$518$		0			0
$519$		−	3.23657e10i		−	0.446083i
$520$		0			0
$521$	3.58216e10			0.486177			0.243088	−	0.970004i	$-0.421839\pi$
$521$	3.58216e10			0.486177			0.243088	+	0.970004i	$0.421839\pi$
$522$		0			0
$523$	−1.25596e11			−1.67868			−0.839341	−	0.543605i	$-0.817059\pi$
$523$	−1.25596e11			−1.67868			−0.839341	+	0.543605i	$0.817059\pi$
$524$		0			0
$525$	1.10214e10	−	8.56983e9i	0.145077	−	0.112807i
$526$		0			0
$527$	−8.23988e9			−0.106826
$528$		0			0
$529$	−2.10109e10			−0.268301
$530$		0			0
$531$			1.01674e11i			1.27889i
$532$		0			0
$533$		−	7.13714e9i		−	0.0884332i
$534$		0			0
$535$	3.74014e10	+	1.09031e11i	0.456533	+	1.33087i
$536$		0			0
$537$			2.47910e10i			0.298124i
$538$		0			0
$539$			2.94349e10i			0.348744i
$540$		0			0
$541$	−7.43291e10			−0.867701			−0.433850	−	0.900985i	$-0.642845\pi$
$541$	−7.43291e10			−0.867701			−0.433850	+	0.900985i	$0.642845\pi$
$542$		0			0
$543$	3.72419e10			0.428383
$544$		0			0
$545$	−1.69194e10	−	4.93228e10i	−0.191778	−	0.559064i
$546$		0			0
$547$	−1.08968e11			−1.21717			−0.608583	−	0.793490i	$-0.708262\pi$
$547$	−1.08968e11			−1.21717			−0.608583	+	0.793490i	$0.708262\pi$
$548$		0			0
$549$	−3.35168e10			−0.368955
$550$		0			0
$551$		−	3.65379e10i		−	0.396404i
$552$		0			0
$553$			5.61217e10i			0.600110i
$554$		0			0
$555$	2.55398e10	−	8.76101e9i	0.269181	−	0.0923384i
$556$		0			0
$557$			1.83899e10i			0.191055i	0.995427	+	0.0955275i	$0.0304538\pi$
$557$			1.83899e10i			0.191055i	−0.995427	+	0.0955275i	$0.969546\pi$
$558$		0			0
$559$		−	3.35291e10i		−	0.343380i
$560$		0			0
$561$	2.14194e9			0.0216250
$562$		0			0
$563$	1.42955e11			1.42287			0.711434	−	0.702753i	$-0.248046\pi$
$563$	1.42955e11			1.42287			0.711434	+	0.702753i	$0.248046\pi$
$564$		0			0
$565$	1.34473e11	−	4.61287e10i	1.31960	−	0.452666i
$566$		0			0
$567$	−5.34086e10			−0.516749
$568$		0			0
$569$	−9.98523e10			−0.952596			−0.476298	−	0.879284i	$-0.658022\pi$
$569$	−9.98523e10			−0.952596			−0.476298	+	0.879284i	$0.658022\pi$
$570$		0			0
$571$			1.00938e11i			0.949531i	0.880112	+	0.474766i	$0.157467\pi$
$571$			1.00938e11i			0.949531i	−0.880112	+	0.474766i	$0.842533\pi$
$572$		0			0
$573$			3.92850e10i			0.364425i
$574$		0			0
$575$	−7.38164e10	+	5.73971e10i	−0.675276	+	0.525072i
$576$		0			0
$577$		−	1.49635e11i		−	1.34999i	−0.737823	−	0.674994i	$-0.764146\pi$
$577$		−	1.49635e11i		−	1.34999i	0.737823	−	0.674994i	$-0.235854\pi$
$578$		0			0
$579$		−	2.68483e10i		−	0.238893i
$580$		0			0
$581$	−5.97488e10			−0.524354
$582$		0			0
$583$	−6.03763e10			−0.522628
$584$		0			0
$585$	1.98967e10	−	6.82523e9i	0.169886	−	0.0582766i
$586$		0			0
$587$	1.28684e11			1.08385			0.541927	−	0.840426i	$-0.317695\pi$
$587$	1.28684e11			1.08385			0.541927	+	0.840426i	$0.317695\pi$
$588$		0			0
$589$	−9.54887e10			−0.793398
$590$		0			0
$591$		−	2.08336e10i		−	0.170771i
$592$		0			0
$593$		−	1.92463e11i		−	1.55642i	−0.628002	−	0.778212i	$-0.716127\pi$
$593$		−	1.92463e11i		−	1.55642i	0.628002	−	0.778212i	$-0.283873\pi$
$594$		0			0
$595$	9.86801e9	−	3.38506e9i	0.0787339	−	0.0270084i
$596$		0			0
$597$			1.51137e9i			0.0118980i
$598$		0			0
$599$			8.17954e10i			0.635362i	0.948198	+	0.317681i	$0.102904\pi$
$599$			8.17954e10i			0.635362i	−0.948198	+	0.317681i	$0.897096\pi$
$600$		0			0
$601$	6.26688e10			0.480345			0.240173	−	0.970730i	$-0.422796\pi$
$601$	6.26688e10			0.480345			0.240173	+	0.970730i	$0.422796\pi$
$602$		0			0
$603$	8.81267e10			0.666559
$604$		0			0
$605$	2.64619e10	+	7.71406e10i	0.197514	+	0.575787i
$606$		0			0
$607$	−6.90116e10			−0.508355			−0.254178	−	0.967158i	$-0.581805\pi$
$607$	−6.90116e10			−0.508355			−0.254178	+	0.967158i	$0.581805\pi$
$608$		0			0
$609$	1.07821e10			0.0783850
$610$		0			0
$611$			4.44458e10i			0.318908i
$612$		0			0
$613$		−	2.61992e11i		−	1.85544i	−0.373282	−	0.927718i	$-0.621768\pi$
$613$		−	2.61992e11i		−	1.85544i	0.373282	−	0.927718i	$-0.378232\pi$
$614$		0			0
$615$	−5.83227e9	−	1.70020e10i	−0.0407697	−	0.118850i
$616$		0			0
$617$			2.25547e9i			0.0155631i	0.999970	+	0.00778154i	$0.00247697\pi$
$617$			2.25547e9i			0.0155631i	−0.999970	+	0.00778154i	$0.997523\pi$
$618$		0			0
$619$		−	6.50054e10i		−	0.442779i	−0.975185	−	0.221389i	$-0.928941\pi$
$619$		−	6.50054e10i		−	0.442779i	0.975185	−	0.221389i	$-0.0710592\pi$
$620$		0			0
$621$	−6.76086e10			−0.454606
$622$		0			0
$623$	−4.39976e10			−0.292063
$624$		0			0
$625$	3.75991e10	−	1.47883e11i	0.246410	−	0.969166i
$626$		0			0
$627$	2.48221e10			0.160608
$628$		0			0
$629$	2.01763e10			0.128896
$630$		0			0
$631$		−	2.17489e11i		−	1.37189i	−0.727652	−	0.685947i	$-0.759388\pi$
$631$		−	2.17489e11i		−	1.37189i	0.727652	−	0.685947i	$-0.240612\pi$
$632$		0			0
$633$		−	6.02976e10i		−	0.375565i
$634$		0			0
$635$	1.01505e10	+	2.95902e10i	0.0624296	+	0.181993i
$636$		0			0
$637$		−	1.78492e10i		−	0.108408i
$638$		0			0
$639$		−	7.65244e10i		−	0.458983i
$640$		0			0
$641$	2.32148e11			1.37510			0.687548	−	0.726139i	$-0.258687\pi$
$641$	2.32148e11			1.37510			0.687548	+	0.726139i	$0.258687\pi$
$642$		0			0
$643$	7.12806e10			0.416992			0.208496	−	0.978023i	$-0.433143\pi$
$643$	7.12806e10			0.416992			0.208496	+	0.978023i	$0.433143\pi$
$644$		0			0
$645$	−2.73990e10	−	7.98727e10i	−0.158306	−	0.461487i
$646$		0			0
$647$	−1.15500e11			−0.659122			−0.329561	−	0.944134i	$-0.606901\pi$
$647$	−1.15500e11			−0.659122			−0.329561	+	0.944134i	$0.606901\pi$
$648$		0			0
$649$	−1.53651e11			−0.866076
$650$		0			0
$651$		−	2.81780e10i		−	0.156887i
$652$		0			0
$653$			7.20788e10i			0.396419i	0.980160	+	0.198210i	$0.0635127\pi$
$653$			7.20788e10i			0.396419i	−0.980160	+	0.198210i	$0.936487\pi$
$654$		0			0
$655$	9.90119e10	−	3.39644e10i	0.537926	−	0.184527i
$656$		0			0
$657$		−	1.38269e11i		−	0.742102i
$658$		0			0
$659$			6.92396e10i			0.367124i	0.983008	+	0.183562i	$0.0587628\pi$
$659$			6.92396e10i			0.367124i	−0.983008	+	0.183562i	$0.941237\pi$
$660$		0			0
$661$	3.21734e11			1.68535			0.842676	−	0.538421i	$-0.180979\pi$
$661$	3.21734e11			1.68535			0.842676	+	0.538421i	$0.180979\pi$
$662$		0			0
$663$	−1.29886e9			−0.00672216
$664$		0			0
$665$	1.14356e11	−	3.92282e10i	0.584755	−	0.200591i
$666$		0			0
$667$	−7.22138e10			−0.364852
$668$		0			0
$669$	−1.09562e10			−0.0546962
$670$		0			0
$671$		−	5.06510e10i		−	0.249860i
$672$		0			0
$673$		−	1.71274e11i		−	0.834891i	−0.908702	−	0.417446i	$-0.862925\pi$
$673$		−	1.71274e11i		−	0.834891i	0.908702	−	0.417446i	$-0.137075\pi$
$674$		0			0
$675$	8.70963e10	−	6.77231e10i	0.419551	−	0.326229i
$676$		0			0
$677$		−	3.00702e11i		−	1.43147i	−0.698374	−	0.715733i	$-0.746093\pi$
$677$		−	3.00702e11i		−	1.43147i	0.698374	−	0.715733i	$-0.253907\pi$
$678$		0			0
$679$			1.59270e11i			0.749297i
$680$		0			0
$681$	4.17944e10			0.194325
$682$		0			0
$683$	−1.27871e11			−0.587611			−0.293806	−	0.955865i	$-0.594922\pi$
$683$	−1.27871e11			−0.587611			−0.293806	+	0.955865i	$0.594922\pi$
$684$		0			0
$685$	8.83686e10	−	3.03134e10i	0.401361	−	0.137681i
$686$		0			0
$687$	−1.02319e10			−0.0459336
$688$		0			0
$689$	3.66119e10			0.162460
$690$		0			0
$691$		−	2.65656e11i		−	1.16522i	−0.812753	−	0.582609i	$-0.802032\pi$
$691$		−	2.65656e11i		−	1.16522i	0.812753	−	0.582609i	$-0.197968\pi$
$692$		0			0
$693$		−	8.86418e10i		−	0.384331i
$694$		0			0
$695$	6.84626e10	−	2.34850e10i	0.293436	−	0.100659i
$696$		0			0
$697$		−	1.34315e10i		−	0.0569107i
$698$		0			0
$699$		−	9.08053e10i		−	0.380367i
$700$		0			0
$701$	4.18657e11			1.73375			0.866874	−	0.498527i	$-0.166126\pi$
$701$	4.18657e11			1.73375			0.866874	+	0.498527i	$0.166126\pi$
$702$		0			0
$703$	2.33815e11			0.957307
$704$		0			0
$705$	3.63199e10	+	1.05878e11i	0.147024	+	0.428598i
$706$		0			0
$707$	−1.97915e11			−0.792138
$708$		0			0
$709$	−2.62253e11			−1.03785			−0.518925	−	0.854820i	$-0.673668\pi$
$709$	−2.62253e11			−1.03785			−0.518925	+	0.854820i	$0.673668\pi$
$710$		0			0
$711$			2.12953e11i			0.833306i
$712$		0			0
$713$			1.88724e11i			0.730247i
$714$		0			0
$715$	1.03144e10	+	3.00681e10i	0.0394656	+	0.115049i
$716$		0			0
$717$		−	6.34385e10i		−	0.240036i
$718$		0			0
$719$		−	1.76165e11i		−	0.659181i	−0.944124	−	0.329590i	$-0.893089\pi$
$719$		−	1.76165e11i		−	0.659181i	0.944124	−	0.329590i	$-0.106911\pi$
$720$		0			0
$721$	1.83374e11			0.678572
$722$		0			0
$723$	−8.61740e10			−0.315372
$724$		0			0
$725$	9.30290e10	−	7.23361e10i	0.336718	−	0.261820i
$726$		0			0
$727$	9.70320e10			0.347358			0.173679	−	0.984802i	$-0.444434\pi$
$727$	9.70320e10			0.347358			0.173679	+	0.984802i	$0.444434\pi$
$728$		0			0
$729$	−1.61189e11			−0.570724
$730$		0			0
$731$		−	6.30991e10i		−	0.220980i
$732$		0			0
$733$			1.09468e11i			0.379202i	0.981861	+	0.189601i	$0.0607194\pi$
$733$			1.09468e11i			0.379202i	−0.981861	+	0.189601i	$0.939281\pi$
$734$		0			0
$735$	−1.45858e10	−	4.25201e10i	−0.0499784	−	0.145695i
$736$		0			0
$737$			1.33178e11i			0.451401i
$738$		0			0
$739$			3.98006e11i			1.33448i	0.744843	+	0.667240i	$0.232525\pi$
$739$			3.98006e11i			1.33448i	−0.744843	+	0.667240i	$0.767475\pi$
$740$		0			0
$741$	−1.50520e10			−0.0499253
$742$		0			0
$743$	3.73650e11			1.22605			0.613027	−	0.790062i	$-0.289952\pi$
$743$	3.73650e11			1.22605			0.613027	+	0.790062i	$0.289952\pi$
$744$		0			0
$745$	1.45067e10	+	4.22894e10i	0.0470916	+	0.137280i
$746$		0			0
$747$	−2.26715e11			−0.728113
$748$		0			0
$749$	−2.94553e11			−0.935916
$750$		0			0
$751$			5.64243e11i			1.77381i	0.461953	+	0.886904i	$0.347149\pi$
$751$			5.64243e11i			1.77381i	−0.461953	+	0.886904i	$0.652851\pi$
$752$		0			0
$753$		−	1.47572e11i		−	0.459011i
$754$		0			0
$755$	−4.38717e11	+	1.50495e11i	−1.35020	+	0.463163i
$756$		0			0
$757$		−	3.11081e11i		−	0.947305i	−0.880712	−	0.473653i	$-0.842935\pi$
$757$		−	3.11081e11i		−	0.947305i	0.880712	−	0.473653i	$-0.157065\pi$
$758$		0			0
$759$		−	4.90585e10i		−	0.147825i
$760$		0			0
$761$	−1.85990e11			−0.554563			−0.277282	−	0.960789i	$-0.589433\pi$
$761$	−1.85990e11			−0.554563			−0.277282	+	0.960789i	$0.589433\pi$
$762$		0			0
$763$	1.33248e11			0.393155
$764$		0			0
$765$	3.74440e10	−	1.28445e10i	0.109329	−	0.0375036i
$766$		0			0
$767$	9.31731e10			0.269221
$768$		0			0
$769$	−2.85199e11			−0.815535			−0.407768	−	0.913086i	$-0.633693\pi$
$769$	−2.85199e11			−0.815535			−0.407768	+	0.913086i	$0.633693\pi$
$770$		0			0
$771$		−	1.67266e11i		−	0.473359i
$772$		0			0
$773$		−	4.88863e11i		−	1.36921i	−0.728916	−	0.684604i	$-0.759975\pi$
$773$		−	4.88863e11i		−	1.36921i	0.728916	−	0.684604i	$-0.240025\pi$
$774$		0			0
$775$	−1.89044e11	−	2.43123e11i	−0.524030	−	0.673937i
$776$		0			0
$777$			6.89971e10i			0.189298i
$778$		0			0
$779$		−	1.55653e11i		−	0.422675i
$780$		0			0
$781$	1.15644e11			0.310828
$782$		0			0
$783$	8.52054e10			0.226684
$784$		0			0
$785$	5.56959e11	−	1.91056e11i	1.46671	−	0.503132i
$786$		0			0
$787$	2.44090e11			0.636283			0.318142	−	0.948043i	$-0.396941\pi$
$787$	2.44090e11			0.636283			0.318142	+	0.948043i	$0.396941\pi$
$788$		0			0
$789$	−9.53483e10			−0.246040
$790$		0			0
$791$			3.63286e11i			0.927988i
$792$		0			0
$793$			3.07145e10i			0.0776695i
$794$		0			0
$795$	8.72165e10	−	2.99182e10i	0.218338	−	0.0748975i
$796$		0			0
$797$			5.89137e11i			1.46010i	0.683393	+	0.730051i	$0.260504\pi$
$797$			5.89137e11i			1.46010i	−0.683393	+	0.730051i	$0.739496\pi$
$798$		0			0
$799$			8.36434e10i			0.205232i
$800$		0			0
$801$	−1.66948e11			−0.405556
$802$		0			0
$803$	2.08954e11			0.502560
$804$		0			0
$805$	−7.75308e10	−	2.26015e11i	−0.184625	−	0.538212i
$806$		0			0
$807$	−2.50173e10			−0.0589857
$808$		0			0
$809$	−3.77151e11			−0.880483			−0.440241	−	0.897879i	$-0.645107\pi$
$809$	−3.77151e11			−0.880483			−0.440241	+	0.897879i	$0.645107\pi$
$810$		0			0
$811$		−	1.98883e11i		−	0.459742i	−0.973221	−	0.229871i	$-0.926170\pi$
$811$		−	1.98883e11i		−	0.459742i	0.973221	−	0.229871i	$-0.0738304\pi$
$812$		0			0
$813$			1.99103e11i			0.455739i
$814$		0			0
$815$	−1.63563e11	−	4.76812e11i	−0.370727	−	1.08073i
$816$		0			0
$817$		−	7.31230e11i		−	1.64122i
$818$		0			0
$819$			5.37520e10i			0.119470i
$820$		0			0
$821$	3.87163e11			0.852159			0.426080	−	0.904686i	$-0.359894\pi$
$821$	3.87163e11			0.852159			0.426080	+	0.904686i	$0.359894\pi$
$822$		0			0
$823$	−7.88136e11			−1.71792			−0.858958	−	0.512047i	$-0.828887\pi$
$823$	−7.88136e11			−1.71792			−0.858958	+	0.512047i	$0.828887\pi$
$824$		0			0
$825$	4.91416e10	+	6.31993e10i	0.106080	+	0.136426i
$826$		0			0
$827$	1.77430e10			0.0379319			0.0189660	−	0.999820i	$-0.493963\pi$
$827$	1.77430e10			0.0379319			0.0189660	+	0.999820i	$0.493963\pi$
$828$		0			0
$829$	−8.13711e11			−1.72287			−0.861434	−	0.507870i	$-0.830433\pi$
$829$	−8.13711e11			−1.72287			−0.861434	+	0.507870i	$0.830433\pi$
$830$		0			0
$831$		−	1.09100e11i		−	0.228781i
$832$		0			0
$833$		−	3.35907e10i		−	0.0697652i
$834$		0			0
$835$	2.13830e11	+	6.23349e11i	0.439868	+	1.28229i
$836$		0			0
$837$		−	2.22677e11i		−	0.453705i
$838$		0			0
$839$		−	1.76196e11i		−	0.355589i	−0.984068	−	0.177794i	$-0.943104\pi$
$839$		−	1.76196e11i		−	0.355589i	0.984068	−	0.177794i	$-0.0568962\pi$
$840$		0			0
$841$	−4.09237e11			−0.818071
$842$		0			0
$843$	−1.45766e11			−0.288633
$844$		0			0
$845$	1.59173e11	+	4.64014e11i	0.312206	+	0.910132i
$846$		0			0
$847$	−2.08400e11			−0.404915
$848$		0			0
$849$	−1.56643e11			−0.301495
$850$		0			0
$851$		−	4.62113e11i		−	0.881111i
$852$		0			0
$853$			1.22129e11i			0.230686i	0.993326	+	0.115343i	$0.0367967\pi$
$853$			1.22129e11i			0.230686i	−0.993326	+	0.115343i	$0.963203\pi$
$854$		0			0
$855$	4.33923e11	−	1.48850e11i	0.811985	−	0.278539i
$856$		0			0
$857$			5.89065e11i			1.09204i	0.837771	+	0.546022i	$0.183858\pi$
$857$			5.89065e11i			1.09204i	−0.837771	+	0.546022i	$0.816142\pi$
$858$		0			0
$859$			5.92618e11i			1.08843i	0.838944	+	0.544217i	$0.183173\pi$
$859$			5.92618e11i			1.08843i	−0.838944	+	0.544217i	$0.816827\pi$
$860$		0			0
$861$	4.59319e10			0.0835799
$862$		0			0
$863$	−2.76008e11			−0.497597			−0.248799	−	0.968555i	$-0.580036\pi$
$863$	−2.76008e11			−0.497597			−0.248799	+	0.968555i	$0.580036\pi$
$864$		0			0
$865$	8.55038e11	−	2.93307e11i	1.52729	−	0.523911i
$866$		0			0
$867$	1.53660e11			0.271947
$868$		0			0
$869$	−3.21816e11			−0.564324
$870$		0			0
$871$		−	8.07585e10i		−	0.140319i
$872$		0			0
$873$			6.04345e11i			1.04047i
$874$		0			0
$875$	3.26276e11	+	2.13500e11i	0.556613	+	0.364222i
$876$		0			0
$877$			9.47400e11i			1.60153i	0.598978	+	0.800765i	$0.295574\pi$
$877$			9.47400e11i			1.60153i	−0.598978	+	0.800765i	$0.704426\pi$
$878$		0			0
$879$		−	1.64553e11i		−	0.275645i
$880$		0			0
$881$	1.85781e11			0.308388			0.154194	−	0.988041i	$-0.450722\pi$
$881$	1.85781e11			0.308388			0.154194	+	0.988041i	$0.450722\pi$
$882$		0			0
$883$	3.50379e11			0.576362			0.288181	−	0.957576i	$-0.406950\pi$
$883$	3.50379e11			0.576362			0.288181	+	0.957576i	$0.406950\pi$
$884$		0			0
$885$	2.21956e11	−	7.61385e10i	0.361821	−	0.124117i
$886$		0			0
$887$	−1.31377e11			−0.212239			−0.106120	−	0.994353i	$-0.533843\pi$
$887$	−1.31377e11			−0.212239			−0.106120	+	0.994353i	$0.533843\pi$
$888$		0			0
$889$	−7.99397e10			−0.127984
$890$		0			0
$891$		−	3.06259e11i		−	0.485934i
$892$		0			0
$893$			9.69310e11i			1.52425i
$894$		0			0
$895$	−6.54929e11	+	2.24663e11i	−1.02071	+	0.350138i
$896$		0			0
$897$			2.97488e10i			0.0459515i
$898$		0			0
$899$		−	2.37845e11i		−	0.364128i
$900$		0			0
$901$	6.89007e10			0.104550
$902$		0			0
$903$	2.15780e11			0.324535
$904$		0			0
$905$	3.37496e11	+	9.83856e11i	0.503123	+	1.46669i
$906$		0			0
$907$	9.89559e11			1.46222			0.731110	−	0.682260i	$-0.239003\pi$
$907$	9.89559e11			1.46222			0.731110	+	0.682260i	$0.239003\pi$
$908$		0			0
$909$	−7.50984e11			−1.09996
$910$		0			0
$911$			1.12268e12i			1.62998i	0.579476	+	0.814989i	$0.303257\pi$
$911$			1.12268e12i			1.62998i	−0.579476	+	0.814989i	$0.696743\pi$
$912$		0			0
$913$		−	3.42615e11i		−	0.493086i
$914$		0			0
$915$	2.50990e10	+	7.31678e10i	0.0358074	+	0.104384i
$916$		0			0
$917$			2.67486e11i			0.378289i
$918$		0			0
$919$		−	9.46459e11i		−	1.32691i	−0.748218	−	0.663453i	$-0.769090\pi$
$919$		−	9.46459e11i		−	1.32691i	0.748218	−	0.663453i	$-0.230910\pi$
$920$		0			0
$921$	−3.63056e11			−0.504585
$922$		0			0
$923$	−7.01262e10			−0.0966215
$924$		0			0
$925$	4.62896e11	+	5.95315e11i	0.632291	+	0.813167i
$926$		0			0
$927$	6.95807e11			0.942258
$928$		0			0
$929$	−9.37679e11			−1.25890			−0.629450	−	0.777041i	$-0.716720\pi$
$929$	−9.37679e11			−1.25890			−0.629450	+	0.777041i	$0.716720\pi$
$930$		0			0
$931$		−	3.89269e11i		−	0.518145i
$932$		0			0
$933$			3.42576e11i			0.452096i
$934$		0			0
$935$	1.94108e10	+	5.65857e10i	0.0253979	+	0.0740390i
$936$		0			0
$937$			9.74959e11i			1.26482i	0.774635	+	0.632409i	$0.217934\pi$
$937$			9.74959e11i			1.26482i	−0.774635	+	0.632409i	$0.782066\pi$
$938$		0			0
$939$			2.33293e11i			0.300081i
$940$		0			0
$941$	3.47861e10			0.0443657			0.0221829	−	0.999754i	$-0.492938\pi$
$941$	3.47861e10			0.0443657			0.0221829	+	0.999754i	$0.492938\pi$
$942$		0			0
$943$	−3.07633e11			−0.389032
$944$		0			0
$945$	9.14789e10	+	2.66676e11i	0.114708	+	0.334392i
$946$		0			0
$947$	−6.70932e11			−0.834216			−0.417108	−	0.908857i	$-0.636956\pi$
$947$	−6.70932e11			−0.834216			−0.417108	+	0.908857i	$0.636956\pi$
$948$		0			0
$949$	−1.26709e11			−0.156222
$950$		0			0
$951$			1.76354e11i			0.215607i
$952$		0			0
$953$			1.15574e12i			1.40116i	0.713572	+	0.700582i	$0.247076\pi$
$953$			1.15574e12i			1.40116i	−0.713572	+	0.700582i	$0.752924\pi$
$954$		0			0
$955$	−1.03783e12	+	3.56011e11i	−1.24771	+	0.428007i
$956$		0			0
$957$			6.18271e10i			0.0737109i
$958$		0			0
$959$			2.38733e11i			0.282252i
$960$		0			0
$961$	2.31305e11			0.271201
$962$		0			0
$963$	−1.11768e12			−1.29960
$964$		0			0
$965$	7.09279e11	−	2.43307e11i	0.817914	−	0.280572i
$966$		0			0
$967$	−3.70816e10			−0.0424084			−0.0212042	−	0.999775i	$-0.506750\pi$
$967$	−3.70816e10			−0.0424084			−0.0212042	+	0.999775i	$0.506750\pi$
$968$		0			0
$969$	−2.83266e10			−0.0321292
$970$		0			0
$971$			2.32263e9i			0.00261278i	0.999999	+	0.00130639i	$0.000415837\pi$
$971$			2.32263e9i			0.00261278i	−0.999999	+	0.00130639i	$0.999584\pi$
$972$		0			0
$973$			1.84955e11i			0.206355i
$974$		0			0
$975$	−2.97992e10	−	3.83237e10i	−0.0329751	−	0.0424081i
$976$		0			0
$977$			2.20075e11i			0.241542i	0.992680	+	0.120771i	$0.0385367\pi$
$977$			2.20075e11i			0.241542i	−0.992680	+	0.120771i	$0.961463\pi$
$978$		0			0
$979$		−	2.52293e11i		−	0.274647i
$980$		0			0
$981$	5.05607e11			0.545930
$982$		0			0
$983$	1.35506e12			1.45126			0.725628	−	0.688087i	$-0.241549\pi$
$983$	1.35506e12			1.45126			0.725628	+	0.688087i	$0.241549\pi$
$984$		0			0
$985$	5.50381e11	−	1.88799e11i	0.584681	−	0.200565i
$986$		0			0
$987$	−2.86036e11			−0.301406
$988$		0			0
$989$	−1.44521e12			−1.51058
$990$		0			0
$991$		−	8.47013e11i		−	0.878204i	−0.898437	−	0.439102i	$-0.855297\pi$
$991$		−	8.47013e11i		−	0.878204i	0.898437	−	0.439102i	$-0.144703\pi$
$992$		0			0
$993$		−	2.90027e11i		−	0.298292i
$994$		0			0
$995$	−3.99273e10	+	1.36964e10i	−0.0407359	+	0.0139738i
$996$		0			0
$997$		−	3.89571e11i		−	0.394281i	−0.980375	−	0.197140i	$-0.936835\pi$
$997$		−	3.89571e11i		−	0.394281i	0.980375	−	0.197140i	$-0.0631655\pi$
$998$		0			0
$999$			5.45250e11i			0.547436i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.h.f.319.10		16
4.3	odd	2	inner	320.9.h.f.319.8		16
5.4	even	2	inner	320.9.h.f.319.7		16
8.3	odd	2		80.9.h.d.79.9	yes	16
8.5	even	2		80.9.h.d.79.7	✓	16
20.19	odd	2	inner	320.9.h.f.319.9		16
40.3	even	4		400.9.b.m.351.9		16
40.13	odd	4		400.9.b.m.351.8		16
40.19	odd	2		80.9.h.d.79.8	yes	16
40.27	even	4		400.9.b.m.351.7		16
40.29	even	2		80.9.h.d.79.10	yes	16
40.37	odd	4		400.9.b.m.351.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.9.h.d.79.7	✓	16	8.5	even	2
80.9.h.d.79.8	yes	16	40.19	odd	2
80.9.h.d.79.9	yes	16	8.3	odd	2
80.9.h.d.79.10	yes	16	40.29	even	2
320.9.h.f.319.7		16	5.4	even	2	inner
320.9.h.f.319.8		16	4.3	odd	2	inner
320.9.h.f.319.9		16	20.19	odd	2	inner
320.9.h.f.319.10		16	1.1	even	1	trivial
400.9.b.m.351.7		16	40.27	even	4
400.9.b.m.351.8		16	40.13	odd	4
400.9.b.m.351.9		16	40.3	even	4
400.9.b.m.351.10		16	40.37	odd	4

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

\(f(q)\)	\(=\)	\(q+22.3781 q^{3} +(202.796 + 591.184i) q^{5} -1597.12 q^{7} -6060.22 q^{9} +O(q^{10})\)	\(q+22.3781 q^{3} +(202.796 + 591.184i) q^{5} -1597.12 q^{7} -6060.22 q^{9} -9158.27i q^{11} +5553.53i q^{13} +(4538.19 + 13229.6i) q^{15} +10451.3i q^{17} +121116. i q^{19} -35740.4 q^{21} +239374. q^{23} +(-308372. + 239780. i) q^{25} -282439. q^{27} -301677. q^{29} +788407. i q^{31} -204945. i q^{33} +(-323889. - 944190. i) q^{35} -1.93051e6i q^{37} +124277. i q^{39} -1.28515e6 q^{41} -6.03743e6 q^{43} +(-1.22899e6 - 3.58271e6i) q^{45} +8.00316e6 q^{47} -3.21402e6 q^{49} +233880. i q^{51} -6.59254e6i q^{53} +(5.41422e6 - 1.85726e6i) q^{55} +2.71034e6i q^{57} -1.67773e7i q^{59} +5.53062e6 q^{61} +9.67888e6 q^{63} +(-3.28316e6 + 1.12624e6i) q^{65} -1.45418e7 q^{67} +5.35674e6 q^{69} +1.26273e7i q^{71} +2.28159e7i q^{73} +(-6.90078e6 + 5.36581e6i) q^{75} +1.46268e7i q^{77} -3.51394e7i q^{79} +3.34407e7 q^{81} +3.74104e7 q^{83} +(-6.17864e6 + 2.11948e6i) q^{85} -6.75096e6 q^{87} +2.75481e7 q^{89} -8.86964e6i q^{91} +1.76430e7i q^{93} +(-7.16018e7 + 2.45619e7i) q^{95} -9.97232e7i q^{97} +5.55011e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 600 q^{5} + 42176 q^{9}+O(q^{10}) \) 16 * q - 600 * q^5 + 42176 * q^9	\( 16 q - 600 q^{5} + 42176 q^{9} - 619344 q^{21} + 1137040 q^{25} + 3497568 q^{29} - 2169168 q^{41} + 1930760 q^{45} + 26174912 q^{49} - 22772656 q^{61} + 12524160 q^{65} - 8461392 q^{69} - 224999456 q^{81} + 18124800 q^{85} - 94250976 q^{89}+O(q^{100}) \) 16 * q - 600 * q^5 + 42176 * q^9 - 619344 * q^21 + 1137040 * q^25 + 3497568 * q^29 - 2169168 * q^41 + 1930760 * q^45 + 26174912 * q^49 - 22772656 * q^61 + 12524160 * q^65 - 8461392 * q^69 - 224999456 * q^81 + 18124800 * q^85 - 94250976 * q^89

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