LMFDB - Embedded newform 320.9.h.f.319.4

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			319.4
Root			$8.60649 + 4.96896i$ of defining polynomial
Character	$\chi$	$=$	320.319
Dual form			320.9.h.f.319.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-120.512 q^{3} +(622.222 + 58.8612i) q^{5} -1534.67 q^{7} +7962.26 q^{9} +O(q^{10})$	\(q-120.512 q^{3} +(622.222 + 58.8612i) q^{5} -1534.67 q^{7} +7962.26 q^{9} +16188.1i q^{11} -37582.9i q^{13} +(-74985.5 - 7093.51i) q^{15} +139612. i q^{17} -224741. i q^{19} +184948. q^{21} -32765.1 q^{23} +(383696. + 73249.5i) q^{25} -168869. q^{27} +852404. q^{29} +220128. i q^{31} -1.95087e6i q^{33} +(-954909. - 90332.8i) q^{35} +147461. i q^{37} +4.52921e6i q^{39} -1.12962e6 q^{41} -4.66121e6 q^{43} +(4.95430e6 + 468668. i) q^{45} -6.39030e6 q^{47} -3.40957e6 q^{49} -1.68250e7i q^{51} -4.05503e6i q^{53} +(-952850. + 1.00726e7i) q^{55} +2.70841e7i q^{57} +3.00137e6i q^{59} +9.13574e6 q^{61} -1.22195e7 q^{63} +(2.21218e6 - 2.33849e7i) q^{65} -3.89358e6 q^{67} +3.94860e6 q^{69} -2.31111e6i q^{71} -3.84831e7i q^{73} +(-4.62401e7 - 8.82748e6i) q^{75} -2.48434e7i q^{77} +3.64057e7i q^{79} -3.18895e7 q^{81} -5.60465e7 q^{83} +(-8.21774e6 + 8.68697e7i) q^{85} -1.02725e8 q^{87} +4.14743e7 q^{89} +5.76776e7i q^{91} -2.65281e7i q^{93} +(1.32285e7 - 1.39839e8i) q^{95} +1.54653e8i q^{97} +1.28894e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−120.512			−1.48781			−0.743904	−	0.668286i	$-0.767028\pi$
$3$	−120.512			−1.48781			−0.743904	+	0.668286i	$0.767028\pi$
$4$		0			0
$5$	622.222	+	58.8612i	0.995555	+	0.0941779i
$5$	622.222	+	58.8612i	0.995555	+	0.0941779i
$6$		0			0
$7$	−1534.67			−0.639182			−0.319591	−	0.947556i	$-0.603545\pi$
$7$	−1534.67			−0.639182			−0.319591	+	0.947556i	$0.603545\pi$
$8$		0			0
$9$	7962.26			1.21357
$10$		0			0
$11$			16188.1i			1.10567i	0.833291	+	0.552834i	$0.186454\pi$
$11$			16188.1i			1.10567i	−0.833291	+	0.552834i	$0.813546\pi$
$12$		0			0
$13$		−	37582.9i		−	1.31588i	−0.753069	−	0.657941i	$-0.771427\pi$
$13$		−	37582.9i		−	1.31588i	0.753069	−	0.657941i	$-0.228573\pi$
$14$		0			0
$15$	−74985.5	−	7093.51i	−1.48120	−	0.140119i
$16$		0			0
$17$			139612.i			1.67158i	0.549049	+	0.835790i	$0.314990\pi$
$17$			139612.i			1.67158i	−0.549049	+	0.835790i	$0.685010\pi$
$18$		0			0
$19$		−	224741.i		−	1.72452i	−0.506468	−	0.862259i	$-0.669049\pi$
$19$		−	224741.i		−	1.72452i	0.506468	−	0.862259i	$-0.330951\pi$
$20$		0			0
$21$	184948.			0.950980
$22$		0			0
$23$	−32765.1			−0.117085			−0.0585424	−	0.998285i	$-0.518645\pi$
$23$	−32765.1			−0.117085			−0.0585424	+	0.998285i	$0.518645\pi$
$24$		0			0
$25$	383696.	+	73249.5i	0.982261	+	0.187519i
$26$		0			0
$27$	−168869.			−0.317758
$28$		0			0
$29$	852404.			1.20518			0.602592	−	0.798049i	$-0.294135\pi$
$29$	852404.			1.20518			0.602592	+	0.798049i	$0.294135\pi$
$30$		0			0
$31$			220128.i			0.238357i	0.992873	+	0.119179i	$0.0380261\pi$
$31$			220128.i			0.238357i	−0.992873	+	0.119179i	$0.961974\pi$
$32$		0			0
$33$		−	1.95087e6i		−	1.64502i
$34$		0			0
$35$	−954909.	−	90332.8i	−0.636341	−	0.0601968i
$36$		0			0
$37$			147461.i			0.0786809i	0.999226	+	0.0393405i	$0.0125257\pi$
$37$			147461.i			0.0786809i	−0.999226	+	0.0393405i	$0.987474\pi$
$38$		0			0
$39$			4.52921e6i			1.95778i
$40$		0			0
$41$	−1.12962e6			−0.399758			−0.199879	−	0.979821i	$-0.564055\pi$
$41$	−1.12962e6			−0.399758			−0.199879	+	0.979821i	$0.564055\pi$
$42$		0			0
$43$	−4.66121e6			−1.36341			−0.681703	−	0.731629i	$-0.738760\pi$
$43$	−4.66121e6			−1.36341			−0.681703	+	0.731629i	$0.738760\pi$
$44$		0			0
$45$	4.95430e6	+	468668.i	1.20818	+	0.114292i
$46$		0			0
$47$	−6.39030e6			−1.30957			−0.654786	−	0.755814i	$-0.727241\pi$
$47$	−6.39030e6			−1.30957			−0.654786	+	0.755814i	$0.727241\pi$
$48$		0			0
$49$	−3.40957e6			−0.591447
$50$		0			0
$51$		−	1.68250e7i		−	2.48699i
$52$		0			0
$53$		−	4.05503e6i		−	0.513914i	−0.966423	−	0.256957i	$-0.917280\pi$
$53$		−	4.05503e6i		−	0.513914i	0.966423	−	0.256957i	$-0.0827199\pi$
$54$		0			0
$55$	−952850.	+	1.00726e7i	−0.104130	+	1.10075i
$56$		0			0
$57$			2.70841e7i			2.56575i
$58$		0			0
$59$			3.00137e6i			0.247692i	0.992301	+	0.123846i	$0.0395228\pi$
$59$			3.00137e6i			0.247692i	−0.992301	+	0.123846i	$0.960477\pi$
$60$		0			0
$61$	9.13574e6			0.659818			0.329909	−	0.944013i	$-0.392982\pi$
$61$	9.13574e6			0.659818			0.329909	+	0.944013i	$0.392982\pi$
$62$		0			0
$63$	−1.22195e7			−0.775694
$64$		0			0
$65$	2.21218e6	−	2.33849e7i	0.123927	−	1.31003i
$66$		0			0
$67$	−3.89358e6			−0.193219			−0.0966096	−	0.995322i	$-0.530800\pi$
$67$	−3.89358e6			−0.193219			−0.0966096	+	0.995322i	$0.530800\pi$
$68$		0			0
$69$	3.94860e6			0.174200
$70$		0			0
$71$		−	2.31111e6i		−	0.0909468i	−0.998966	−	0.0454734i	$-0.985520\pi$
$71$		−	2.31111e6i		−	0.0909468i	0.998966	−	0.0454734i	$-0.0144796\pi$
$72$		0			0
$73$		−	3.84831e7i		−	1.35512i	−0.735467	−	0.677561i	$-0.763037\pi$
$73$		−	3.84831e7i		−	1.35512i	0.735467	−	0.677561i	$-0.236963\pi$
$74$		0			0
$75$	−4.62401e7	−	8.82748e6i	−1.46142	−	0.278992i
$76$		0			0
$77$		−	2.48434e7i		−	0.706723i
$78$		0			0
$79$			3.64057e7i			0.934677i	0.884078	+	0.467338i	$0.154787\pi$
$79$			3.64057e7i			0.934677i	−0.884078	+	0.467338i	$0.845213\pi$
$80$		0			0
$81$	−3.18895e7			−0.740812
$82$		0			0
$83$	−5.60465e7			−1.18096			−0.590481	−	0.807051i	$-0.701062\pi$
$83$	−5.60465e7			−1.18096			−0.590481	+	0.807051i	$0.701062\pi$
$84$		0			0
$85$	−8.21774e6	+	8.68697e7i	−0.157426	+	1.66415i
$86$		0			0
$87$	−1.02725e8			−1.79308
$88$		0			0
$89$	4.14743e7			0.661026			0.330513	−	0.943801i	$-0.392778\pi$
$89$	4.14743e7			0.661026			0.330513	+	0.943801i	$0.392778\pi$
$90$		0			0
$91$			5.76776e7i			0.841088i
$92$		0			0
$93$		−	2.65281e7i		−	0.354630i
$94$		0			0
$95$	1.32285e7	−	1.39839e8i	0.162412	−	1.71685i
$96$		0			0
$97$			1.54653e8i			1.74691i	0.486906	+	0.873454i	$0.338125\pi$
$97$			1.54653e8i			1.74691i	−0.486906	+	0.873454i	$0.661875\pi$
$98$		0			0
$99$			1.28894e8i			1.34181i
$100$		0			0
$101$	−1.82375e8			−1.75259			−0.876295	−	0.481775i	$-0.839992\pi$
$101$	−1.82375e8			−1.75259			−0.876295	+	0.481775i	$0.839992\pi$
$102$		0			0
$103$	−4.33848e6			−0.0385468			−0.0192734	−	0.999814i	$-0.506135\pi$
$103$	−4.33848e6			−0.0385468			−0.0192734	+	0.999814i	$0.506135\pi$
$104$		0			0
$105$	1.15078e8	+	1.08862e7i	0.946753	+	0.0895613i
$106$		0			0
$107$	1.65463e8			1.26231			0.631157	−	0.775656i	$-0.282581\pi$
$107$	1.65463e8			1.26231			0.631157	+	0.775656i	$0.282581\pi$
$108$		0			0
$109$	8.40764e7			0.595619			0.297809	−	0.954625i	$-0.403744\pi$
$109$	8.40764e7			0.595619			0.297809	+	0.954625i	$0.403744\pi$
$110$		0			0
$111$		−	1.77709e7i		−	0.117062i
$112$		0			0
$113$		−	1.50847e8i		−	0.925173i	−0.886574	−	0.462587i	$-0.846921\pi$
$113$		−	1.50847e8i		−	0.925173i	0.886574	−	0.462587i	$-0.153079\pi$
$114$		0			0
$115$	−2.03872e7	−	1.92859e6i	−0.116564	−	0.0110268i
$116$		0			0
$117$		−	2.99245e8i		−	1.59692i
$118$		0			0
$119$		−	2.14259e8i		−	1.06844i
$120$		0			0
$121$	−4.76951e7			−0.222501
$122$		0			0
$123$	1.36133e8			0.594764
$124$		0			0
$125$	2.34432e8	+	6.81622e7i	0.960235	+	0.279193i
$126$		0			0
$127$	4.34976e8			1.67205			0.836026	−	0.548689i	$-0.184873\pi$
$127$	4.34976e8			1.67205			0.836026	+	0.548689i	$0.184873\pi$
$128$		0			0
$129$	5.61735e8			2.02849
$130$		0			0
$131$		−	1.49354e8i		−	0.507144i	−0.967317	−	0.253572i	$-0.918395\pi$
$131$		−	1.49354e8i		−	0.507144i	0.967317	−	0.253572i	$-0.0816055\pi$
$132$		0			0
$133$			3.44904e8i			1.10228i
$134$		0			0
$135$	−1.05074e8	−	9.93986e6i	−0.316345	−	0.0299258i
$136$		0			0
$137$		−	6.68094e8i		−	1.89651i	−0.317508	−	0.948256i	$-0.602846\pi$
$137$		−	6.68094e8i		−	1.89651i	0.317508	−	0.948256i	$-0.397154\pi$
$138$		0			0
$139$			8.41936e7i			0.225538i	0.993621	+	0.112769i	$0.0359720\pi$
$139$			8.41936e7i			0.225538i	−0.993621	+	0.112769i	$0.964028\pi$
$140$		0			0
$141$	7.70110e8			1.94839
$142$		0			0
$143$	6.08396e8			1.45493
$144$		0			0
$145$	5.30384e8	+	5.01735e7i	1.19983	+	0.113502i
$146$		0			0
$147$	4.10896e8			0.879960
$148$		0			0
$149$	4.99074e8			1.01256			0.506278	−	0.862370i	$-0.331021\pi$
$149$	4.99074e8			1.01256			0.506278	+	0.862370i	$0.331021\pi$
$150$		0			0
$151$			7.00966e8i			1.34831i	0.738591	+	0.674154i	$0.235491\pi$
$151$			7.00966e8i			1.34831i	−0.738591	+	0.674154i	$0.764509\pi$
$152$		0			0
$153$			1.11163e9i			2.02859i
$154$		0			0
$155$	−1.29570e7	+	1.36968e8i	−0.0224480	+	0.237298i
$156$		0			0
$157$			6.03846e8i			0.993866i	0.867789	+	0.496933i	$0.165541\pi$
$157$			6.03846e8i			0.993866i	−0.867789	+	0.496933i	$0.834459\pi$
$158$		0			0
$159$			4.88682e8i			0.764606i
$160$		0			0
$161$	5.02838e7			0.0748384
$162$		0			0
$163$	9.77331e8			1.38449			0.692247	−	0.721660i	$-0.256621\pi$
$163$	9.77331e8			1.38449			0.692247	+	0.721660i	$0.256621\pi$
$164$		0			0
$165$	1.14830e8	−	1.21387e9i	0.154925	−	1.63771i
$166$		0			0
$167$	6.22691e8			0.800583			0.400292	−	0.916388i	$-0.368909\pi$
$167$	6.22691e8			0.800583			0.400292	+	0.916388i	$0.368909\pi$
$168$		0			0
$169$	−5.96746e8			−0.731548
$170$		0			0
$171$		−	1.78945e9i		−	2.09283i
$172$		0			0
$173$		−	9.79732e8i		−	1.09376i	−0.837210	−	0.546881i	$-0.815815\pi$
$173$		−	9.79732e8i		−	1.09376i	0.837210	−	0.546881i	$-0.184185\pi$
$174$		0			0
$175$	−5.88848e8	−	1.12414e8i	−0.627843	−	0.119858i
$176$		0			0
$177$		−	3.61702e8i		−	0.368518i
$178$		0			0
$179$		−	1.38679e9i		−	1.35082i	−0.737443	−	0.675409i	$-0.763967\pi$
$179$		−	1.38679e9i		−	1.35082i	0.737443	−	0.675409i	$-0.236033\pi$
$180$		0			0
$181$	1.12770e9			1.05070			0.525349	−	0.850887i	$-0.323935\pi$
$181$	1.12770e9			1.05070			0.525349	+	0.850887i	$0.323935\pi$
$182$		0			0
$183$	−1.10097e9			−0.981683
$184$		0			0
$185$	−8.67972e6	+	9.17533e7i	−0.00741001	+	0.0783312i
$186$		0			0
$187$	−2.26005e9			−1.84821
$188$		0			0
$189$	2.59160e8			0.203105
$190$		0			0
$191$			1.28581e9i			0.966147i	0.875580	+	0.483073i	$0.160480\pi$
$191$			1.28581e9i			0.966147i	−0.875580	+	0.483073i	$0.839520\pi$
$192$		0			0
$193$		−	1.07613e9i		−	0.775595i	−0.921745	−	0.387798i	$-0.873236\pi$
$193$		−	1.07613e9i		−	0.775595i	0.921745	−	0.387798i	$-0.126764\pi$
$194$		0			0
$195$	−2.66595e8	+	2.81818e9i	−0.184380	+	1.94908i
$196$		0			0
$197$			2.59964e9i			1.72603i	0.505179	+	0.863015i	$0.331426\pi$
$197$			2.59964e9i			1.72603i	−0.505179	+	0.863015i	$0.668574\pi$
$198$		0			0
$199$		−	1.92276e9i		−	1.22606i	−0.790059	−	0.613031i	$-0.789950\pi$
$199$		−	1.92276e9i		−	1.22606i	0.790059	−	0.613031i	$-0.210050\pi$
$200$		0			0
$201$	4.69225e8			0.287473
$202$		0			0
$203$	−1.30816e9			−0.770331
$204$		0			0
$205$	−7.02875e8	−	6.64908e7i	−0.397981	−	0.0376484i
$206$		0			0
$207$	−2.60884e8			−0.142091
$208$		0			0
$209$	3.63812e9			1.90674
$210$		0			0
$211$		−	1.37284e9i		−	0.692612i	−0.938122	−	0.346306i	$-0.887436\pi$
$211$		−	1.37284e9i		−	0.692612i	0.938122	−	0.346306i	$-0.112564\pi$
$212$		0			0
$213$			2.78518e8i			0.135311i
$214$		0			0
$215$	−2.90031e9	−	2.74365e8i	−1.35735	−	0.128403i
$216$		0			0
$217$		−	3.37825e8i		−	0.152353i
$218$		0			0
$219$			4.63769e9i			2.01616i
$220$		0			0
$221$	5.24703e9			2.19960
$222$		0			0
$223$	2.68536e9			1.08588			0.542942	−	0.839770i	$-0.317310\pi$
$223$	2.68536e9			1.08588			0.542942	+	0.839770i	$0.317310\pi$
$224$		0			0
$225$	3.05509e9	+	5.83232e8i	1.19205	+	0.227568i
$226$		0			0
$227$	9.29784e8			0.350170			0.175085	−	0.984553i	$-0.443980\pi$
$227$	9.29784e8			0.350170			0.175085	+	0.984553i	$0.443980\pi$
$228$		0			0
$229$	1.30243e9			0.473599			0.236800	−	0.971558i	$-0.423901\pi$
$229$	1.30243e9			0.473599			0.236800	+	0.971558i	$0.423901\pi$
$230$		0			0
$231$			2.99395e9i			1.05147i
$232$		0			0
$233$		−	1.31356e8i		−	0.0445681i	−0.999752	−	0.0222841i	$-0.992906\pi$
$233$		−	1.31356e8i		−	0.0445681i	0.999752	−	0.0222841i	$-0.00709383\pi$
$234$		0			0
$235$	−3.97618e9	−	3.76140e8i	−1.30375	−	0.123333i
$236$		0			0
$237$		−	4.38735e9i		−	1.39062i
$238$		0			0
$239$		−	4.66096e9i		−	1.42851i	−0.699884	−	0.714256i	$-0.746765\pi$
$239$		−	4.66096e9i		−	1.42851i	0.699884	−	0.714256i	$-0.253235\pi$
$240$		0			0
$241$	3.50875e9			1.04012			0.520062	−	0.854129i	$-0.325909\pi$
$241$	3.50875e9			1.04012			0.520062	+	0.854129i	$0.325909\pi$
$242$		0			0
$243$	4.95104e9			1.41994
$244$		0			0
$245$	−2.12151e9	−	2.00692e8i	−0.588818	−	0.0557012i
$246$		0			0
$247$	−8.44642e9			−2.26926
$248$		0			0
$249$	6.75431e9			1.75705
$250$		0			0
$251$			1.77120e9i			0.446245i	0.974790	+	0.223123i	$0.0716250\pi$
$251$			1.77120e9i			0.446245i	−0.974790	+	0.223123i	$0.928375\pi$
$252$		0			0
$253$		−	5.30404e8i		−	0.129457i
$254$		0			0
$255$	9.90340e8	−	1.04689e10i	0.234220	−	2.47594i
$256$		0			0
$257$			5.05381e9i			1.15848i	0.815159	+	0.579238i	$0.196650\pi$
$257$			5.05381e9i			1.15848i	−0.815159	+	0.579238i	$0.803350\pi$
$258$		0			0
$259$		−	2.26304e8i		−	0.0502914i
$260$		0			0
$261$	6.78706e9			1.46258
$262$		0			0
$263$	2.78242e9			0.581567			0.290783	−	0.956789i	$-0.406084\pi$
$263$	2.78242e9			0.581567			0.290783	+	0.956789i	$0.406084\pi$
$264$		0			0
$265$	2.38684e8	−	2.52313e9i	0.0483994	−	0.511630i
$266$		0			0
$267$	−4.99817e9			−0.983481
$268$		0			0
$269$	−1.37946e9			−0.263451			−0.131725	−	0.991286i	$-0.542052\pi$
$269$	−1.37946e9			−0.263451			−0.131725	+	0.991286i	$0.542052\pi$
$270$		0			0
$271$			1.00479e10i			1.86293i	0.363831	+	0.931465i	$0.381469\pi$
$271$			1.00479e10i			1.86293i	−0.363831	+	0.931465i	$0.618531\pi$
$272$		0			0
$273$		−	6.95087e9i		−	1.25138i
$274$		0			0
$275$	−1.18577e9	+	6.21130e9i	−0.207333	+	1.08605i
$276$		0			0
$277$		−	6.27722e9i		−	1.06622i	−0.846045	−	0.533111i	$-0.821023\pi$
$277$		−	6.27722e9i		−	1.06622i	0.846045	−	0.533111i	$-0.178977\pi$
$278$		0			0
$279$			1.75272e9i			0.289264i
$280$		0			0
$281$	1.54217e9			0.247347			0.123673	−	0.992323i	$-0.460532\pi$
$281$	1.54217e9			0.247347			0.123673	+	0.992323i	$0.460532\pi$
$282$		0			0
$283$	2.45734e9			0.383106			0.191553	−	0.981482i	$-0.438648\pi$
$283$	2.45734e9			0.383106			0.191553	+	0.981482i	$0.438648\pi$
$284$		0			0
$285$	−1.59420e9	+	1.68523e10i	−0.241637	+	2.55435i
$286$		0			0
$287$	1.73360e9			0.255518
$288$		0			0
$289$	−1.25158e10			−1.79418
$290$		0			0
$291$		−	1.86376e10i		−	2.59907i
$292$		0			0
$293$			7.18324e8i			0.0974652i	0.998812	+	0.0487326i	$0.0155182\pi$
$293$			7.18324e8i			0.0974652i	−0.998812	+	0.0487326i	$0.984482\pi$
$294$		0			0
$295$	−1.76664e8	+	1.86752e9i	−0.0233271	+	0.246591i
$296$		0			0
$297$		−	2.73367e9i		−	0.351335i
$298$		0			0
$299$			1.23141e9i			0.154070i
$300$		0			0
$301$	7.15345e9			0.871464
$302$		0			0
$303$	2.19785e10			2.60752
$304$		0			0
$305$	5.68446e9	+	5.37740e8i	0.656885	+	0.0621403i
$306$		0			0
$307$	8.02642e9			0.903584			0.451792	−	0.892123i	$-0.350785\pi$
$307$	8.02642e9			0.903584			0.451792	+	0.892123i	$0.350785\pi$
$308$		0			0
$309$	5.22841e8			0.0573503
$310$		0			0
$311$			7.25842e8i			0.0775891i	0.999247	+	0.0387945i	$0.0123518\pi$
$311$			7.25842e8i			0.0775891i	−0.999247	+	0.0387945i	$0.987648\pi$
$312$		0			0
$313$			8.02136e9i			0.835739i	0.908507	+	0.417870i	$0.137223\pi$
$313$			8.02136e9i			0.835739i	−0.908507	+	0.417870i	$0.862777\pi$
$314$		0			0
$315$	−7.60323e9	−	7.19253e8i	−0.772247	−	0.0730533i
$316$		0			0
$317$		−	6.01792e9i		−	0.595950i	−0.954574	−	0.297975i	$-0.903689\pi$
$317$		−	6.01792e9i		−	0.595950i	0.954574	−	0.297975i	$-0.0963111\pi$
$318$		0			0
$319$			1.37988e10i			1.33253i
$320$		0			0
$321$	−1.99404e10			−1.87808
$322$		0			0
$323$	3.13765e10			2.88267
$324$		0			0
$325$	2.75293e9	−	1.44204e10i	0.246753	−	1.29254i
$326$		0			0
$327$	−1.01323e10			−0.886166
$328$		0			0
$329$	9.80703e9			0.837055
$330$		0			0
$331$		−	2.10140e10i		−	1.75064i	−0.483547	−	0.875318i	$-0.660652\pi$
$331$		−	2.10140e10i		−	1.75064i	0.483547	−	0.875318i	$-0.339348\pi$
$332$		0			0
$333$			1.17412e9i			0.0954852i
$334$		0			0
$335$	−2.42267e9	−	2.29181e8i	−0.192360	−	0.0181970i
$336$		0			0
$337$			1.25477e10i			0.972845i	0.873724	+	0.486422i	$0.161698\pi$
$337$			1.25477e10i			0.972845i	−0.873724	+	0.486422i	$0.838302\pi$
$338$		0			0
$339$			1.81790e10i			1.37648i
$340$		0			0
$341$	−3.56345e9			−0.263544
$342$		0			0
$343$	1.40797e10			1.01722
$344$		0			0
$345$	2.45691e9	+	2.32420e8i	0.173425	+	0.0164058i
$346$		0			0
$347$	−1.14689e10			−0.791052			−0.395526	−	0.918455i	$-0.629438\pi$
$347$	−1.14689e10			−0.791052			−0.395526	+	0.918455i	$0.629438\pi$
$348$		0			0
$349$	−1.79740e10			−1.21156			−0.605778	−	0.795634i	$-0.707138\pi$
$349$	−1.79740e10			−1.21156			−0.605778	+	0.795634i	$0.707138\pi$
$350$		0			0
$351$			6.34661e9i			0.418132i
$352$		0			0
$353$			3.47840e8i			0.0224017i	0.999937	+	0.0112008i	$0.00356541\pi$
$353$			3.47840e8i			0.0224017i	−0.999937	+	0.0112008i	$0.996435\pi$
$354$		0			0
$355$	1.36035e8	−	1.43802e9i	0.00856518	−	0.0905426i
$356$		0			0
$357$			2.58209e10i			1.58964i
$358$		0			0
$359$		−	1.23147e10i		−	0.741388i	−0.928755	−	0.370694i	$-0.879120\pi$
$359$		−	1.23147e10i		−	0.741388i	0.928755	−	0.370694i	$-0.120880\pi$
$360$		0			0
$361$	−3.35249e10			−1.97396
$362$		0			0
$363$	5.74786e9			0.331039
$364$		0			0
$365$	2.26516e9	−	2.39450e10i	0.127623	−	1.34910i
$366$		0			0
$367$	8.49274e9			0.468148			0.234074	−	0.972219i	$-0.424794\pi$
$367$	8.49274e9			0.468148			0.234074	+	0.972219i	$0.424794\pi$
$368$		0			0
$369$	−8.99434e9			−0.485136
$370$		0			0
$371$			6.22316e9i			0.328485i
$372$		0			0
$373$			1.22212e10i			0.631361i	0.948866	+	0.315680i	$0.102233\pi$
$373$			1.22212e10i			0.631361i	−0.948866	+	0.315680i	$0.897767\pi$
$374$		0			0
$375$	−2.82520e10	−	8.21440e9i	−1.42865	−	0.415385i
$376$		0			0
$377$		−	3.20358e10i		−	1.58588i
$378$		0			0
$379$		−	2.22220e10i		−	1.07703i	−0.842617	−	0.538514i	$-0.818986\pi$
$379$		−	2.22220e10i		−	1.07703i	0.842617	−	0.538514i	$-0.181014\pi$
$380$		0			0
$381$	−5.24200e10			−2.48769
$382$		0			0
$383$	1.91425e10			0.889617			0.444808	−	0.895626i	$-0.353272\pi$
$383$	1.91425e10			0.889617			0.444808	+	0.895626i	$0.353272\pi$
$384$		0			0
$385$	1.46232e9	−	1.54581e10i	0.0665577	−	0.703581i
$386$		0			0
$387$	−3.71138e10			−1.65459
$388$		0			0
$389$	2.08109e10			0.908852			0.454426	−	0.890785i	$-0.349844\pi$
$389$	2.08109e10			0.908852			0.454426	+	0.890785i	$0.349844\pi$
$390$		0			0
$391$		−	4.57440e9i		−	0.195717i
$392$		0			0
$393$			1.79990e10i			0.754533i
$394$		0			0
$395$	−2.14289e9	+	2.26525e10i	−0.0880259	+	0.930523i
$396$		0			0
$397$			2.06428e10i			0.831010i	0.909591	+	0.415505i	$0.136395\pi$
$397$			2.06428e10i			0.831010i	−0.909591	+	0.415505i	$0.863605\pi$
$398$		0			0
$399$		−	4.15653e10i		−	1.63998i
$400$		0			0
$401$	3.88938e10			1.50419			0.752096	−	0.659054i	$-0.229043\pi$
$401$	3.88938e10			1.50419			0.752096	+	0.659054i	$0.229043\pi$
$402$		0			0
$403$	8.27305e9			0.313650
$404$		0			0
$405$	−1.98424e10	−	1.87706e9i	−0.737519	−	0.0697681i
$406$		0			0
$407$	−2.38711e9			−0.0869950
$408$		0			0
$409$	−1.06496e10			−0.380576			−0.190288	−	0.981728i	$-0.560942\pi$
$409$	−1.06496e10			−0.380576			−0.190288	+	0.981728i	$0.560942\pi$
$410$		0			0
$411$			8.05137e10i			2.82165i
$412$		0			0
$413$		−	4.60612e9i		−	0.158320i
$414$		0			0
$415$	−3.48734e10	−	3.29897e9i	−1.17571	−	0.111221i
$416$		0			0
$417$		−	1.01464e10i		−	0.335557i
$418$		0			0
$419$			5.51889e9i			0.179059i	0.995984	+	0.0895294i	$0.0285363\pi$
$419$			5.51889e9i			0.179059i	−0.995984	+	0.0895294i	$0.971464\pi$
$420$		0			0
$421$	6.08601e9			0.193733			0.0968667	−	0.995297i	$-0.469118\pi$
$421$	6.08601e9			0.193733			0.0968667	+	0.995297i	$0.469118\pi$
$422$		0			0
$423$	−5.08812e10			−1.58926
$424$		0			0
$425$	−1.02265e10	+	5.35686e10i	−0.313453	+	1.64193i
$426$		0			0
$427$	−1.40204e10			−0.421744
$428$		0			0
$429$	−7.33193e10			−2.16466
$430$		0			0
$431$		−	2.77495e9i		−	0.0804166i	−0.999191	−	0.0402083i	$-0.987198\pi$
$431$		−	2.77495e9i		−	0.0804166i	0.999191	−	0.0402083i	$-0.0128022\pi$
$432$		0			0
$433$			1.60701e10i			0.457159i	0.973525	+	0.228579i	$0.0734081\pi$
$433$			1.60701e10i			0.457159i	−0.973525	+	0.228579i	$0.926592\pi$
$434$		0			0
$435$	−6.39179e10	−	6.04653e9i	−1.78511	−	0.168869i
$436$		0			0
$437$			7.36366e9i			0.201915i
$438$		0			0
$439$			5.34609e10i			1.43939i	0.694291	+	0.719694i	$0.255718\pi$
$439$			5.34609e10i			1.43939i	−0.694291	+	0.719694i	$0.744282\pi$
$440$		0			0
$441$	−2.71479e10			−0.717765
$442$		0			0
$443$	1.96255e10			0.509572			0.254786	−	0.966998i	$-0.417995\pi$
$443$	1.96255e10			0.509572			0.254786	+	0.966998i	$0.417995\pi$
$444$		0			0
$445$	2.58062e10	+	2.44123e9i	0.658088	+	0.0622541i
$446$		0			0
$447$	−6.01446e10			−1.50649
$448$		0			0
$449$	−2.90598e10			−0.715002			−0.357501	−	0.933913i	$-0.616371\pi$
$449$	−2.90598e10			−0.715002			−0.357501	+	0.933913i	$0.616371\pi$
$450$		0			0
$451$		−	1.82864e10i		−	0.442000i
$452$		0			0
$453$		−	8.44751e10i		−	2.00602i
$454$		0			0
$455$	−3.39497e9	+	3.58883e10i	−0.0792119	+	0.837350i
$456$		0			0
$457$			1.13331e10i			0.259828i	0.991525	+	0.129914i	$0.0414701\pi$
$457$			1.13331e10i			0.259828i	−0.991525	+	0.129914i	$0.958530\pi$
$458$		0			0
$459$		−	2.35762e10i		−	0.531158i
$460$		0			0
$461$	−1.09720e10			−0.242930			−0.121465	−	0.992596i	$-0.538759\pi$
$461$	−1.09720e10			−0.242930			−0.121465	+	0.992596i	$0.538759\pi$
$462$		0			0
$463$	4.86558e10			1.05879			0.529396	−	0.848375i	$-0.322419\pi$
$463$	4.86558e10			1.05879			0.529396	+	0.848375i	$0.322419\pi$
$464$		0			0
$465$	1.56148e9	−	1.65064e10i	0.0333983	−	0.353054i
$466$		0			0
$467$	7.74081e9			0.162749			0.0813746	−	0.996684i	$-0.474069\pi$
$467$	7.74081e9			0.162749			0.0813746	+	0.996684i	$0.474069\pi$
$468$		0			0
$469$	5.97538e9			0.123502
$470$		0			0
$471$		−	7.27710e10i		−	1.47868i
$472$		0			0
$473$		−	7.54561e10i		−	1.50747i
$474$		0			0
$475$	1.64622e10	−	8.62321e10i	0.323379	−	1.69393i
$476$		0			0
$477$		−	3.22872e10i		−	0.623673i
$478$		0			0
$479$		−	5.28605e10i		−	1.00413i	−0.864830	−	0.502064i	$-0.832574\pi$
$479$		−	5.28605e10i		−	1.00413i	0.864830	−	0.502064i	$-0.167426\pi$
$480$		0			0
$481$	5.54201e9			0.103535
$482$		0			0
$483$	−6.05982e9			−0.111345
$484$		0			0
$485$	−9.10304e9	+	9.62283e10i	−0.164520	+	1.73914i
$486$		0			0
$487$	8.64235e10			1.53644			0.768221	−	0.640185i	$-0.221142\pi$
$487$	8.64235e10			1.53644			0.768221	+	0.640185i	$0.221142\pi$
$488$		0			0
$489$	−1.17781e11			−2.05986
$490$		0			0
$491$			2.62564e10i			0.451761i	0.974155	+	0.225880i	$0.0725259\pi$
$491$			2.62564e10i			0.451761i	−0.974155	+	0.225880i	$0.927474\pi$
$492$		0			0
$493$			1.19006e11i			2.01456i
$494$		0			0
$495$	−7.58684e9	+	8.02005e10i	−0.126369	+	1.33585i
$496$		0			0
$497$			3.54681e9i			0.0581315i
$498$		0			0
$499$		−	5.13854e10i		−	0.828776i	−0.910100	−	0.414388i	$-0.863996\pi$
$499$		−	5.13854e10i		−	0.828776i	0.910100	−	0.414388i	$-0.136004\pi$
$500$		0			0
$501$	−7.50420e10			−1.19111
$502$		0			0
$503$	1.11774e11			1.74609			0.873047	−	0.487636i	$-0.162141\pi$
$503$	1.11774e11			1.74609			0.873047	+	0.487636i	$0.162141\pi$
$504$		0			0
$505$	−1.13478e11	−	1.07348e10i	−1.74480	−	0.165055i
$506$		0			0
$507$	7.19153e10			1.08840
$508$		0			0
$509$	−7.73223e10			−1.15195			−0.575975	−	0.817467i	$-0.695378\pi$
$509$	−7.73223e10			−1.15195			−0.575975	+	0.817467i	$0.695378\pi$
$510$		0			0
$511$			5.90590e10i			0.866168i
$512$		0			0
$513$			3.79519e10i			0.547979i
$514$		0			0
$515$	−2.69950e9	−	2.55368e8i	−0.0383755	−	0.00363026i
$516$		0			0
$517$		−	1.03447e11i		−	1.44795i
$518$		0			0
$519$			1.18070e11i			1.62731i
$520$		0			0
$521$	1.71598e10			0.232895			0.116448	−	0.993197i	$-0.462849\pi$
$521$	1.71598e10			0.232895			0.116448	+	0.993197i	$0.462849\pi$
$522$		0			0
$523$	2.66205e10			0.355803			0.177902	−	0.984048i	$-0.443069\pi$
$523$	2.66205e10			0.355803			0.177902	+	0.984048i	$0.443069\pi$
$524$		0			0
$525$	7.09636e10	+	1.35473e10i	0.934110	+	0.178326i
$526$		0			0
$527$	−3.07325e10			−0.398433
$528$		0			0
$529$	−7.72374e10			−0.986291
$530$		0			0
$531$			2.38977e10i			0.300592i
$532$		0			0
$533$			4.24545e10i			0.526035i
$534$		0			0
$535$	1.02955e11	+	9.73938e9i	1.25670	+	0.118882i
$536$		0			0
$537$			1.67125e11i			2.00976i
$538$		0			0
$539$		−	5.51945e10i		−	0.653944i
$540$		0			0
$541$	6.85541e10			0.800284			0.400142	−	0.916453i	$-0.368961\pi$
$541$	6.85541e10			0.800284			0.400142	+	0.916453i	$0.368961\pi$
$542$		0			0
$543$	−1.35901e11			−1.56324
$544$		0			0
$545$	5.23142e10	+	4.94884e9i	0.592971	+	0.0560941i
$546$		0			0
$547$	1.30693e11			1.45983			0.729915	−	0.683538i	$-0.239559\pi$
$547$	1.30693e11			1.45983			0.729915	+	0.683538i	$0.239559\pi$
$548$		0			0
$549$	7.27411e10			0.800738
$550$		0			0
$551$		−	1.91570e11i		−	2.07836i
$552$		0			0
$553$		−	5.58710e10i		−	0.597428i
$554$		0			0
$555$	1.04601e9	−	1.10574e10i	0.0110247	−	0.116542i
$556$		0			0
$557$		−	6.80108e10i		−	0.706573i	−0.935515	−	0.353287i	$-0.885064\pi$
$557$		−	6.80108e10i		−	0.706573i	0.935515	−	0.353287i	$-0.114936\pi$
$558$		0			0
$559$			1.75182e11i			1.79408i
$560$		0			0
$561$	2.72365e11			2.74979
$562$		0			0
$563$	−7.51567e10			−0.748056			−0.374028	−	0.927417i	$-0.622024\pi$
$563$	−7.51567e10			−0.748056			−0.374028	+	0.927417i	$0.622024\pi$
$564$		0			0
$565$	8.87904e9	−	9.38604e10i	0.0871309	−	0.921061i
$566$		0			0
$567$	4.89400e10			0.473513
$568$		0			0
$569$	5.25389e10			0.501224			0.250612	−	0.968088i	$-0.419368\pi$
$569$	5.25389e10			0.501224			0.250612	+	0.968088i	$0.419368\pi$
$570$		0			0
$571$			2.18843e10i			0.205868i	0.994688	+	0.102934i	$0.0328230\pi$
$571$			2.18843e10i			0.205868i	−0.994688	+	0.102934i	$0.967177\pi$
$572$		0			0
$573$		−	1.54956e11i		−	1.43744i
$574$		0			0
$575$	−1.25718e10	−	2.40003e9i	−0.115008	−	0.0219556i
$576$		0			0
$577$		−	4.43477e10i		−	0.400100i	−0.979786	−	0.200050i	$-0.935890\pi$
$577$		−	4.43477e10i		−	0.400100i	0.979786	−	0.200050i	$-0.0641104\pi$
$578$		0			0
$579$			1.29687e11i			1.15394i
$580$		0			0
$581$	8.60132e10			0.754850
$582$		0			0
$583$	6.56432e10			0.568219
$584$		0			0
$585$	1.76139e10	−	1.86197e11i	0.150395	−	1.58982i
$586$		0			0
$587$	1.03443e11			0.871265			0.435633	−	0.900125i	$-0.356525\pi$
$587$	1.03443e11			0.871265			0.435633	+	0.900125i	$0.356525\pi$
$588$		0			0
$589$	4.94717e10			0.411051
$590$		0			0
$591$		−	3.13289e11i		−	2.56800i
$592$		0			0
$593$		−	3.73406e10i		−	0.301969i	−0.988536	−	0.150985i	$-0.951756\pi$
$593$		−	3.73406e10i		−	0.301969i	0.988536	−	0.150985i	$-0.0482444\pi$
$594$		0			0
$595$	1.26116e10	−	1.33317e11i	0.100624	−	1.06369i
$596$		0			0
$597$			2.31717e11i			1.82415i
$598$		0			0
$599$		−	1.84560e11i		−	1.43361i	−0.697276	−	0.716803i	$-0.745605\pi$
$599$		−	1.84560e11i		−	1.43361i	0.697276	−	0.716803i	$-0.254395\pi$
$600$		0			0
$601$	8.41016e10			0.644624			0.322312	−	0.946634i	$-0.395540\pi$
$601$	8.41016e10			0.644624			0.322312	+	0.946634i	$0.395540\pi$
$602$		0			0
$603$	−3.10017e10			−0.234486
$604$		0			0
$605$	−2.96770e10	−	2.80739e9i	−0.221512	−	0.0209547i
$606$		0			0
$607$	−6.99218e10			−0.515060			−0.257530	−	0.966270i	$-0.582909\pi$
$607$	−6.99218e10			−0.515060			−0.257530	+	0.966270i	$0.582909\pi$
$608$		0			0
$609$	1.57650e11			1.14611
$610$		0			0
$611$			2.40166e11i			1.72324i
$612$		0			0
$613$			7.92935e10i			0.561559i	0.959772	+	0.280780i	$0.0905930\pi$
$613$			7.92935e10i			0.561559i	−0.959772	+	0.280780i	$0.909407\pi$
$614$		0			0
$615$	8.47052e10	+	8.01298e9i	0.592120	+	0.0560136i
$616$		0			0
$617$		−	7.71667e9i		−	0.0532463i	−0.999646	−	0.0266231i	$-0.991525\pi$
$617$		−	7.71667e9i		−	0.0532463i	0.999646	−	0.0266231i	$-0.00847541\pi$
$618$		0			0
$619$			1.03066e11i			0.702026i	0.936371	+	0.351013i	$0.114163\pi$
$619$			1.03066e11i			0.702026i	−0.936371	+	0.351013i	$0.885837\pi$
$620$		0			0
$621$	5.53303e9			0.0372046
$622$		0			0
$623$	−6.36495e10			−0.422516
$624$		0			0
$625$	1.41857e11	+	5.62110e10i	0.929673	+	0.368385i
$626$		0			0
$627$	−4.38439e11			−2.83687
$628$		0			0
$629$	−2.05873e10			−0.131522
$630$		0			0
$631$			2.53349e11i			1.59809i	0.601269	+	0.799047i	$0.294662\pi$
$631$			2.53349e11i			1.59809i	−0.601269	+	0.799047i	$0.705338\pi$
$632$		0			0
$633$			1.65444e11i			1.03047i
$634$		0			0
$635$	2.70651e11	+	2.56032e10i	1.66462	+	0.157470i
$636$		0			0
$637$			1.28142e11i			0.778275i
$638$		0			0
$639$		−	1.84017e10i		−	0.110371i
$640$		0			0
$641$	9.60782e10			0.569105			0.284553	−	0.958660i	$-0.408155\pi$
$641$	9.60782e10			0.569105			0.284553	+	0.958660i	$0.408155\pi$
$642$		0			0
$643$	−6.04152e10			−0.353429			−0.176714	−	0.984262i	$-0.556547\pi$
$643$	−6.04152e10			−0.353429			−0.176714	+	0.984262i	$0.556547\pi$
$644$		0			0
$645$	3.49524e11	+	3.30644e10i	2.01947	+	0.191039i
$646$		0			0
$647$	1.53200e11			0.874263			0.437132	−	0.899398i	$-0.355994\pi$
$647$	1.53200e11			0.874263			0.437132	+	0.899398i	$0.355994\pi$
$648$		0			0
$649$	−4.85864e10			−0.273865
$650$		0			0
$651$			4.07121e10i			0.226673i
$652$		0			0
$653$			7.80246e10i			0.429120i	0.976711	+	0.214560i	$0.0688317\pi$
$653$			7.80246e10i			0.429120i	−0.976711	+	0.214560i	$0.931168\pi$
$654$		0			0
$655$	8.79115e9	−	9.29313e10i	0.0477618	−	0.504890i
$656$		0			0
$657$		−	3.06412e11i		−	1.64454i
$658$		0			0
$659$		−	1.55912e11i		−	0.826682i	−0.910576	−	0.413341i	$-0.864362\pi$
$659$		−	1.55912e11i		−	0.826682i	0.910576	−	0.413341i	$-0.135638\pi$
$660$		0			0
$661$	2.82658e10			0.148066			0.0740329	−	0.997256i	$-0.476413\pi$
$661$	2.82658e10			0.148066			0.0740329	+	0.997256i	$0.476413\pi$
$662$		0			0
$663$	−6.32333e11			−3.27259
$664$		0			0
$665$	−2.03015e10	+	2.14607e11i	−0.103810	+	1.09738i
$666$		0			0
$667$	−2.79291e10			−0.141109
$668$		0			0
$669$	−3.23620e11			−1.61559
$670$		0			0
$671$			1.47890e11i			0.729540i
$672$		0			0
$673$			1.49713e11i			0.729793i	0.931048	+	0.364897i	$0.118896\pi$
$673$			1.49713e11i			0.729793i	−0.931048	+	0.364897i	$0.881104\pi$
$674$		0			0
$675$	−6.47945e10	−	1.23696e10i	−0.312121	−	0.0595855i
$676$		0			0
$677$		−	3.52288e11i		−	1.67704i	−0.544871	−	0.838520i	$-0.683421\pi$
$677$		−	3.52288e11i		−	1.67704i	0.544871	−	0.838520i	$-0.316579\pi$
$678$		0			0
$679$		−	2.37341e11i		−	1.11659i
$680$		0			0
$681$	−1.12051e11			−0.520986
$682$		0			0
$683$	−2.94121e11			−1.35159			−0.675793	−	0.737092i	$-0.736199\pi$
$683$	−2.94121e11			−1.35159			−0.675793	+	0.737092i	$0.736199\pi$
$684$		0			0
$685$	3.93248e10	−	4.15703e11i	0.178609	−	1.88808i
$686$		0			0
$687$	−1.56959e11			−0.704625
$688$		0			0
$689$	−1.52400e11			−0.676251
$690$		0			0
$691$			1.01323e11i			0.444423i	0.974999	+	0.222211i	$0.0713275\pi$
$691$			1.01323e11i			0.444423i	−0.974999	+	0.222211i	$0.928673\pi$
$692$		0			0
$693$		−	1.97810e11i		−	0.857660i
$694$		0			0
$695$	−4.95574e9	+	5.23871e10i	−0.0212407	+	0.224536i
$696$		0			0
$697$		−	1.57709e11i		−	0.668228i
$698$		0			0
$699$			1.58300e10i			0.0663089i
$700$		0			0
$701$	−2.70771e11			−1.12132			−0.560661	−	0.828045i	$-0.689453\pi$
$701$	−2.70771e11			−1.12132			−0.560661	+	0.828045i	$0.689453\pi$
$702$		0			0
$703$	3.31405e10			0.135687
$704$		0			0
$705$	4.79180e11	+	4.53296e10i	1.93973	+	0.183496i
$706$		0			0
$707$	2.79887e11			1.12022
$708$		0			0
$709$	4.53746e11			1.79568			0.897838	−	0.440325i	$-0.145137\pi$
$709$	4.53746e11			1.79568			0.897838	+	0.440325i	$0.145137\pi$
$710$		0			0
$711$			2.89872e11i			1.13430i
$712$		0			0
$713$		−	7.21251e9i		−	0.0279080i
$714$		0			0
$715$	3.78557e11	+	3.58109e10i	1.44846	+	0.137022i
$716$		0			0
$717$			5.61704e11i			2.12535i
$718$		0			0
$719$			4.53815e11i			1.69810i	0.528314	+	0.849049i	$0.322825\pi$
$719$			4.53815e11i			1.69810i	−0.528314	+	0.849049i	$0.677175\pi$
$720$		0			0
$721$	6.65816e9			0.0246384
$722$		0			0
$723$	−4.22849e11			−1.54750
$724$		0			0
$725$	3.27064e11	+	6.24381e10i	1.18381	+	0.225994i
$726$		0			0
$727$	1.74630e11			0.625146			0.312573	−	0.949894i	$-0.398809\pi$
$727$	1.74630e11			0.625146			0.312573	+	0.949894i	$0.398809\pi$
$728$		0			0
$729$	−3.87435e11			−1.37179
$730$		0			0
$731$		−	6.50762e11i		−	2.27904i
$732$		0			0
$733$			8.01917e10i			0.277788i	0.990307	+	0.138894i	$0.0443547\pi$
$733$			8.01917e10i			0.277788i	−0.990307	+	0.138894i	$0.955645\pi$
$734$		0			0
$735$	2.55669e11	+	2.41858e10i	0.876049	+	0.0828728i
$736$		0			0
$737$		−	6.30296e10i		−	0.213636i
$738$		0			0
$739$		−	4.64897e11i		−	1.55876i	−0.626552	−	0.779379i	$-0.715535\pi$
$739$		−	4.64897e11i		−	1.55876i	0.626552	−	0.779379i	$-0.284465\pi$
$740$		0			0
$741$	1.01790e12			3.37623
$742$		0			0
$743$	−5.15335e10			−0.169097			−0.0845483	−	0.996419i	$-0.526945\pi$
$743$	−5.15335e10			−0.169097			−0.0845483	+	0.996419i	$0.526945\pi$
$744$		0			0
$745$	3.10535e11	+	2.93761e10i	1.00806	+	0.0953605i
$746$		0			0
$747$	−4.46257e11			−1.43319
$748$		0			0
$749$	−2.53933e11			−0.806847
$750$		0			0
$751$		−	1.91297e11i		−	0.601380i	−0.953722	−	0.300690i	$-0.902783\pi$
$751$		−	1.91297e11i		−	0.601380i	0.953722	−	0.300690i	$-0.0972169\pi$
$752$		0			0
$753$		−	2.13452e11i		−	0.663928i
$754$		0			0
$755$	−4.12597e10	+	4.36156e11i	−0.126981	+	1.34231i
$756$		0			0
$757$		−	3.22051e11i		−	0.980712i	−0.871522	−	0.490356i	$-0.836867\pi$
$757$		−	3.22051e11i		−	0.980712i	0.871522	−	0.490356i	$-0.163133\pi$
$758$		0			0
$759$			6.39203e10i			0.192607i
$760$		0			0
$761$	−3.30820e11			−0.986399			−0.493199	−	0.869916i	$-0.664173\pi$
$761$	−3.30820e11			−0.986399			−0.493199	+	0.869916i	$0.664173\pi$
$762$		0			0
$763$	−1.29030e11			−0.380708
$764$		0			0
$765$	−6.54318e10	+	6.91679e11i	−0.191048	+	2.01957i
$766$		0			0
$767$	1.12800e11			0.325933
$768$		0			0
$769$	5.78247e11			1.65351			0.826757	−	0.562559i	$-0.190183\pi$
$769$	5.78247e11			1.65351			0.826757	+	0.562559i	$0.190183\pi$
$770$		0			0
$771$		−	6.09048e11i		−	1.72359i
$772$		0			0
$773$			3.06841e11i			0.859400i	0.902972	+	0.429700i	$0.141381\pi$
$773$			3.06841e11i			0.859400i	−0.902972	+	0.429700i	$0.858619\pi$
$774$		0			0
$775$	−1.61242e10	+	8.44621e10i	−0.0446964	+	0.234129i
$776$		0			0
$777$			2.72725e10i			0.0748240i
$778$		0			0
$779$			2.53872e11i			0.689390i
$780$		0			0
$781$	3.74125e10			0.100557
$782$		0			0
$783$	−1.43945e11			−0.382957
$784$		0			0
$785$	−3.55431e10	+	3.75727e11i	−0.0936002	+	0.989449i
$786$		0			0
$787$	5.75892e11			1.50121			0.750607	−	0.660749i	$-0.229761\pi$
$787$	5.75892e11			1.50121			0.750607	+	0.660749i	$0.229761\pi$
$788$		0			0
$789$	−3.35316e11			−0.865260
$790$		0			0
$791$			2.31501e11i			0.591354i
$792$		0			0
$793$		−	3.43348e11i		−	0.868243i
$794$		0			0
$795$	−2.87644e10	+	3.04069e11i	−0.0720090	+	0.761208i
$796$		0			0
$797$		−	4.47788e11i		−	1.10979i	−0.831922	−	0.554893i	$-0.812759\pi$
$797$		−	4.47788e11i		−	1.10979i	0.831922	−	0.554893i	$-0.187241\pi$
$798$		0			0
$799$		−	8.92163e11i		−	2.18906i
$800$		0			0
$801$	3.30229e11			0.802205
$802$		0			0
$803$	6.22967e11			1.49831
$804$		0			0
$805$	3.12877e10	+	2.95976e9i	0.0745058	+	0.00704812i
$806$		0			0
$807$	1.66242e11			0.391964
$808$		0			0
$809$	−6.31321e11			−1.47386			−0.736930	−	0.675969i	$-0.763725\pi$
$809$	−6.31321e11			−1.47386			−0.736930	+	0.675969i	$0.763725\pi$
$810$		0			0
$811$			1.38612e11i			0.320418i	0.987083	+	0.160209i	$0.0512168\pi$
$811$			1.38612e11i			0.320418i	−0.987083	+	0.160209i	$0.948783\pi$
$812$		0			0
$813$		−	1.21089e12i		−	2.77168i
$814$		0			0
$815$	6.08117e11	+	5.75269e10i	1.37834	+	0.130389i
$816$		0			0
$817$			1.04757e12i			2.35122i
$818$		0			0
$819$			4.59244e11i			1.02072i
$820$		0			0
$821$	2.20489e11			0.485305			0.242652	−	0.970113i	$-0.421983\pi$
$821$	2.20489e11			0.485305			0.242652	+	0.970113i	$0.421983\pi$
$822$		0			0
$823$	−1.32102e11			−0.287944			−0.143972	−	0.989582i	$-0.545988\pi$
$823$	−1.32102e11			−0.287944			−0.143972	+	0.989582i	$0.545988\pi$
$824$		0			0
$825$	1.42900e11	−	7.48539e11i	0.308472	−	1.61584i
$826$		0			0
$827$	4.68442e11			1.00146			0.500730	−	0.865604i	$-0.333065\pi$
$827$	4.68442e11			1.00146			0.500730	+	0.865604i	$0.333065\pi$
$828$		0			0
$829$	−7.83488e11			−1.65888			−0.829439	−	0.558598i	$-0.811339\pi$
$829$	−7.83488e11			−1.65888			−0.829439	+	0.558598i	$0.811339\pi$
$830$		0			0
$831$			7.56483e11i			1.58634i
$832$		0			0
$833$		−	4.76018e11i		−	0.988651i
$834$		0			0
$835$	3.87452e11	+	3.66523e10i	0.797025	+	0.0753973i
$836$		0			0
$837$		−	3.71729e10i		−	0.0757398i
$838$		0			0
$839$		−	1.53524e11i		−	0.309834i	−0.987927	−	0.154917i	$-0.950489\pi$
$839$		−	1.53524e11i		−	0.309834i	0.987927	−	0.154917i	$-0.0495111\pi$
$840$		0			0
$841$	2.26346e11			0.452468
$842$		0			0
$843$	−1.85851e11			−0.368005
$844$		0			0
$845$	−3.71308e11	−	3.51252e10i	−0.728296	−	0.0688956i
$846$		0			0
$847$	7.31965e10			0.142219
$848$		0			0
$849$	−2.96140e11			−0.569989
$850$		0			0
$851$		−	4.83157e9i		−	0.00921233i
$852$		0			0
$853$			4.80988e11i			0.908528i	0.890867	+	0.454264i	$0.150098\pi$
$853$			4.80988e11i			0.908528i	−0.890867	+	0.454264i	$0.849902\pi$
$854$		0			0
$855$	1.05329e11	−	1.11343e12i	0.197098	−	2.08353i
$856$		0			0
$857$			3.12043e11i			0.578483i	0.957256	+	0.289242i	$0.0934031\pi$
$857$			3.12043e11i			0.578483i	−0.957256	+	0.289242i	$0.906597\pi$
$858$		0			0
$859$			7.51163e11i			1.37963i	0.723987	+	0.689813i	$0.242307\pi$
$859$			7.51163e11i			1.37963i	−0.723987	+	0.689813i	$0.757693\pi$
$860$		0			0
$861$	−2.08921e11			−0.380162
$862$		0			0
$863$	9.04684e11			1.63100			0.815499	−	0.578758i	$-0.196462\pi$
$863$	9.04684e11			1.63100			0.815499	+	0.578758i	$0.196462\pi$
$864$		0			0
$865$	5.76682e10	−	6.09611e11i	0.103008	−	1.08890i
$866$		0			0
$867$	1.50831e12			2.66940
$868$		0			0
$869$	−5.89339e11			−1.03344
$870$		0			0
$871$			1.46332e11i			0.254254i
$872$		0			0
$873$			1.23138e12i			2.12000i
$874$		0			0
$875$	−3.59778e11	−	1.04607e11i	−0.613765	−	0.178455i
$876$		0			0
$877$			4.30463e11i			0.727676i	0.931462	+	0.363838i	$0.118534\pi$
$877$			4.30463e11i			0.727676i	−0.931462	+	0.363838i	$0.881466\pi$
$878$		0			0
$879$		−	8.65670e10i		−	0.145010i
$880$		0			0
$881$	−1.96937e11			−0.326906			−0.163453	−	0.986551i	$-0.552263\pi$
$881$	−1.96937e11			−0.326906			−0.163453	+	0.986551i	$0.552263\pi$
$882$		0			0
$883$	−7.38394e11			−1.21463			−0.607316	−	0.794460i	$-0.707754\pi$
$883$	−7.38394e11			−1.21463			−0.607316	+	0.794460i	$0.707754\pi$
$884$		0			0
$885$	2.12902e10	−	2.25059e11i	0.0347062	−	0.366880i
$886$		0			0
$887$	−1.16055e11			−0.187487			−0.0937435	−	0.995596i	$-0.529883\pi$
$887$	−1.16055e11			−0.187487			−0.0937435	+	0.995596i	$0.529883\pi$
$888$		0			0
$889$	−6.67546e11			−1.06875
$890$		0			0
$891$		−	5.16230e11i		−	0.819092i
$892$		0			0
$893$			1.43616e12i			2.25838i
$894$		0			0
$895$	8.16278e10	−	8.62888e11i	0.127217	−	1.34481i
$896$		0			0
$897$		−	1.48400e11i		−	0.229226i
$898$		0			0
$899$			1.87638e11i			0.287264i
$900$		0			0
$901$	5.66131e11			0.859049
$902$		0			0
$903$	−8.62080e11			−1.29657
$904$		0			0
$905$	7.01677e11	+	6.63775e10i	1.04603	+	0.0989525i
$906$		0			0
$907$	−1.17401e12			−1.73478			−0.867389	−	0.497632i	$-0.834203\pi$
$907$	−1.17401e12			−1.73478			−0.867389	+	0.497632i	$0.834203\pi$
$908$		0			0
$909$	−1.45212e12			−2.12690
$910$		0			0
$911$		−	5.87783e11i		−	0.853383i	−0.904397	−	0.426691i	$-0.859679\pi$
$911$		−	5.87783e11i		−	0.853383i	0.904397	−	0.426691i	$-0.140321\pi$
$912$		0			0
$913$		−	9.07286e11i		−	1.30575i
$914$		0			0
$915$	−6.85048e11	−	6.48044e10i	−0.977320	−	0.0924529i
$916$		0			0
$917$			2.29210e11i			0.324157i
$918$		0			0
$919$			2.35534e11i			0.330211i	0.986276	+	0.165105i	$0.0527964\pi$
$919$			2.35534e11i			0.330211i	−0.986276	+	0.165105i	$0.947204\pi$
$920$		0			0
$921$	−9.67284e11			−1.34436
$922$		0			0
$923$	−8.68583e10			−0.119675
$924$		0			0
$925$	−1.08014e10	+	5.65801e10i	−0.0147541	+	0.0772852i
$926$		0			0
$927$	−3.45441e10			−0.0467794
$928$		0			0
$929$	−6.07791e11			−0.816003			−0.408001	−	0.912981i	$-0.633774\pi$
$929$	−6.07791e11			−0.816003			−0.408001	+	0.912981i	$0.633774\pi$
$930$		0			0
$931$			7.66271e11i			1.01996i
$932$		0			0
$933$		−	8.74730e10i		−	0.115438i
$934$		0			0
$935$	−1.40625e12	−	1.33029e11i	−1.84000	−	0.174061i
$936$		0			0
$937$		−	4.25011e10i		−	0.0551368i	−0.999620	−	0.0275684i	$-0.991224\pi$
$937$		−	4.25011e10i		−	0.0551368i	0.999620	−	0.0275684i	$-0.00877641\pi$
$938$		0			0
$939$		−	9.66674e11i		−	1.24342i
$940$		0			0
$941$	−1.70120e11			−0.216968			−0.108484	−	0.994098i	$-0.534600\pi$
$941$	−1.70120e11			−0.216968			−0.108484	+	0.994098i	$0.534600\pi$
$942$		0			0
$943$	3.70121e10			0.0468056
$944$		0			0
$945$	1.61255e11	+	1.52545e10i	0.202202	+	0.0191280i
$946$		0			0
$947$	−1.01764e12			−1.26530			−0.632652	−	0.774436i	$-0.718034\pi$
$947$	−1.01764e12			−1.26530			−0.632652	+	0.774436i	$0.718034\pi$
$948$		0			0
$949$	−1.44631e12			−1.78318
$950$		0			0
$951$			7.25235e11i			0.886659i
$952$		0			0
$953$			3.24062e10i			0.0392877i	0.999807	+	0.0196439i	$0.00625324\pi$
$953$			3.24062e10i			0.0392877i	−0.999807	+	0.0196439i	$0.993747\pi$
$954$		0			0
$955$	−7.56843e10	+	8.00059e11i	−0.0909897	+	0.961852i
$956$		0			0
$957$		−	1.66293e12i		−	1.98255i
$958$		0			0
$959$			1.02531e12i			1.21222i
$960$		0			0
$961$	8.04435e11			0.943186
$962$		0			0
$963$	1.31746e12			1.53191
$964$		0			0
$965$	6.33422e10	−	6.69591e11i	0.0730439	−	0.772148i
$966$		0			0
$967$	−3.27368e11			−0.374396			−0.187198	−	0.982322i	$-0.559941\pi$
$967$	−3.27368e11			−0.374396			−0.187198	+	0.982322i	$0.559941\pi$
$968$		0			0
$969$	−3.78127e12			−4.28886
$970$		0			0
$971$		−	9.34077e11i		−	1.05077i	−0.850866	−	0.525383i	$-0.823922\pi$
$971$		−	9.34077e11i		−	1.05077i	0.850866	−	0.525383i	$-0.176078\pi$
$972$		0			0
$973$		−	1.29210e11i		−	0.144160i
$974$		0			0
$975$	−3.31762e11	+	1.73784e12i	−0.367121	+	1.92305i
$976$		0			0
$977$		−	9.79841e11i		−	1.07542i	−0.843130	−	0.537709i	$-0.819290\pi$
$977$		−	9.79841e11i		−	1.07542i	0.843130	−	0.537709i	$-0.180710\pi$
$978$		0			0
$979$			6.71389e11i			0.730876i
$980$		0			0
$981$	6.69438e11			0.722827
$982$		0			0
$983$	9.99101e10			0.107003			0.0535014	−	0.998568i	$-0.482962\pi$
$983$	9.99101e10			0.107003			0.0535014	+	0.998568i	$0.482962\pi$
$984$		0			0
$985$	−1.53018e11	+	1.61755e12i	−0.162554	+	1.71836i
$986$		0			0
$987$	−1.18187e12			−1.24538
$988$		0			0
$989$	1.52725e11			0.159634
$990$		0			0
$991$			1.05539e12i			1.09425i	0.837051	+	0.547125i	$0.184278\pi$
$991$			1.05539e12i			1.09425i	−0.837051	+	0.547125i	$0.815722\pi$
$992$		0			0
$993$			2.53244e12i			2.60461i
$994$		0			0
$995$	1.13176e11	−	1.19638e12i	0.115468	−	1.22061i
$996$		0			0
$997$			1.38748e12i			1.40425i	0.712051	+	0.702127i	$0.247766\pi$
$997$			1.38748e12i			1.40425i	−0.712051	+	0.702127i	$0.752234\pi$
$998$		0			0
$999$		−	2.49016e10i		−	0.0250015i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.h.f.319.4		16
4.3	odd	2	inner	320.9.h.f.319.14		16
5.4	even	2	inner	320.9.h.f.319.13		16
8.3	odd	2		80.9.h.d.79.3	✓	16
8.5	even	2		80.9.h.d.79.13	yes	16
20.19	odd	2	inner	320.9.h.f.319.3		16
40.3	even	4		400.9.b.m.351.4		16
40.13	odd	4		400.9.b.m.351.13		16
40.19	odd	2		80.9.h.d.79.14	yes	16
40.27	even	4		400.9.b.m.351.14		16
40.29	even	2		80.9.h.d.79.4	yes	16
40.37	odd	4		400.9.b.m.351.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.9.h.d.79.3	✓	16	8.3	odd	2
80.9.h.d.79.4	yes	16	40.29	even	2
80.9.h.d.79.13	yes	16	8.5	even	2
80.9.h.d.79.14	yes	16	40.19	odd	2
320.9.h.f.319.3		16	20.19	odd	2	inner
320.9.h.f.319.4		16	1.1	even	1	trivial
320.9.h.f.319.13		16	5.4	even	2	inner
320.9.h.f.319.14		16	4.3	odd	2	inner
400.9.b.m.351.3		16	40.37	odd	4
400.9.b.m.351.4		16	40.3	even	4
400.9.b.m.351.13		16	40.13	odd	4
400.9.b.m.351.14		16	40.27	even	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)

\(f(q)\)	\(=\)	\(q-120.512 q^{3} +(622.222 + 58.8612i) q^{5} -1534.67 q^{7} +7962.26 q^{9} +O(q^{10})\)	\(q-120.512 q^{3} +(622.222 + 58.8612i) q^{5} -1534.67 q^{7} +7962.26 q^{9} +16188.1i q^{11} -37582.9i q^{13} +(-74985.5 - 7093.51i) q^{15} +139612. i q^{17} -224741. i q^{19} +184948. q^{21} -32765.1 q^{23} +(383696. + 73249.5i) q^{25} -168869. q^{27} +852404. q^{29} +220128. i q^{31} -1.95087e6i q^{33} +(-954909. - 90332.8i) q^{35} +147461. i q^{37} +4.52921e6i q^{39} -1.12962e6 q^{41} -4.66121e6 q^{43} +(4.95430e6 + 468668. i) q^{45} -6.39030e6 q^{47} -3.40957e6 q^{49} -1.68250e7i q^{51} -4.05503e6i q^{53} +(-952850. + 1.00726e7i) q^{55} +2.70841e7i q^{57} +3.00137e6i q^{59} +9.13574e6 q^{61} -1.22195e7 q^{63} +(2.21218e6 - 2.33849e7i) q^{65} -3.89358e6 q^{67} +3.94860e6 q^{69} -2.31111e6i q^{71} -3.84831e7i q^{73} +(-4.62401e7 - 8.82748e6i) q^{75} -2.48434e7i q^{77} +3.64057e7i q^{79} -3.18895e7 q^{81} -5.60465e7 q^{83} +(-8.21774e6 + 8.68697e7i) q^{85} -1.02725e8 q^{87} +4.14743e7 q^{89} +5.76776e7i q^{91} -2.65281e7i q^{93} +(1.32285e7 - 1.39839e8i) q^{95} +1.54653e8i q^{97} +1.28894e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 600 q^{5} + 42176 q^{9}+O(q^{10}) \) 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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