LMFDB - Embedded newform 72.8.f.a.35.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(35,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.f (of order $2$, degree $1$, 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:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.11
Character	$\chi$	$=$	72.35
Dual form			72.8.f.a.35.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.16822 - 10.8610i) q^{2} +(-107.925 + 68.8204i) q^{4} -238.250 q^{5} +737.398i q^{7} +(1089.39 + 954.138i) q^{8} +O(q^{10})$	\(q+(-3.16822 - 10.8610i) q^{2} +(-107.925 + 68.8204i) q^{4} -238.250 q^{5} +737.398i q^{7} +(1089.39 + 954.138i) q^{8} +(754.827 + 2587.64i) q^{10} -1701.82i q^{11} +4482.93i q^{13} +(8008.92 - 2336.24i) q^{14} +(6911.51 - 14854.8i) q^{16} -19959.6i q^{17} -8090.28 q^{19} +(25713.0 - 16396.4i) q^{20} +(-18483.5 + 5391.73i) q^{22} +110629. q^{23} -21362.2 q^{25} +(48689.3 - 14202.9i) q^{26} +(-50748.0 - 79583.6i) q^{28} -19262.3 q^{29} -70291.5i q^{31} +(-183236. - 28002.9i) q^{32} +(-216782. + 63236.4i) q^{34} -175685. i q^{35} -441575. i q^{37} +(25631.8 + 87868.9i) q^{38} +(-259547. - 227323. i) q^{40} -382570. i q^{41} -329066. q^{43} +(117120. + 183668. i) q^{44} +(-350497. - 1.20155e6i) q^{46} +224714. q^{47} +279787. q^{49} +(67680.0 + 232016. i) q^{50} +(-308517. - 483819. i) q^{52} +1.27611e6 q^{53} +405457. i q^{55} +(-703580. + 803315. i) q^{56} +(61027.1 + 209209. i) q^{58} -2.48468e6i q^{59} +2.93707e6i q^{61} +(-763440. + 222699. i) q^{62} +(276392. + 2.07886e6i) q^{64} -1.06805e6i q^{65} +2.94806e6 q^{67} +(1.37363e6 + 2.15413e6i) q^{68} +(-1.90812e6 + 556608. i) q^{70} +1.63172e6 q^{71} +6.26023e6 q^{73} +(-4.79597e6 + 1.39901e6i) q^{74} +(873141. - 556776. i) q^{76} +1.25492e6 q^{77} +448216. i q^{79} +(-1.64667e6 + 3.53916e6i) q^{80} +(-4.15512e6 + 1.21207e6i) q^{82} -3.63088e6i q^{83} +4.75536e6i q^{85} +(1.04255e6 + 3.57400e6i) q^{86} +(1.62377e6 - 1.85394e6i) q^{88} +1.41664e6i q^{89} -3.30570e6 q^{91} +(-1.19396e7 + 7.61352e6i) q^{92} +(-711942. - 2.44063e6i) q^{94} +1.92750e6 q^{95} +1.10470e7 q^{97} +(-886425. - 3.03878e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 52 q^{4}+O(q^{10}) $ 28 * q + 52 * q^4	$ 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−3.16822	−	10.8610i	−0.280034	−	0.959990i
$2$	−3.16822	−	10.8610i	−0.280034	−	0.959990i
$3$		0			0
$3$		0			0
$4$	−107.925	+	68.8204i	−0.843162	+	0.537659i
$5$	−238.250			−0.852387			−0.426194	−	0.904632i	$-0.640146\pi$
$5$	−238.250			−0.852387			−0.426194	+	0.904632i	$0.640146\pi$
$6$		0			0
$7$			737.398i			0.812567i	0.913747	+	0.406283i	$0.133175\pi$
$7$			737.398i			0.812567i	−0.913747	+	0.406283i	$0.866825\pi$
$8$	1089.39	+	954.138i	0.752261	+	0.658865i
$9$		0			0
$10$	754.827	+	2587.64i	0.238697	+	0.818283i
$11$		−	1701.82i		−	0.385512i	−0.981247	−	0.192756i	$-0.938257\pi$
$11$		−	1701.82i		−	0.385512i	0.981247	−	0.192756i	$-0.0617426\pi$
$12$		0			0
$13$			4482.93i			0.565926i	0.959131	+	0.282963i	$0.0913174\pi$
$13$			4482.93i			0.565926i	−0.959131	+	0.282963i	$0.908683\pi$
$14$	8008.92	−	2336.24i	0.780056	−	0.227546i
$15$		0			0
$16$	6911.51	−	14854.8i	0.421845	−	0.906668i
$17$		−	19959.6i		−	0.985327i	−0.870220	−	0.492664i	$-0.836023\pi$
$17$		−	19959.6i		−	0.985327i	0.870220	−	0.492664i	$-0.163977\pi$
$18$		0			0
$19$	−8090.28			−0.270599			−0.135299	−	0.990805i	$-0.543200\pi$
$19$	−8090.28			−0.270599			−0.135299	+	0.990805i	$0.543200\pi$
$20$	25713.0	−	16396.4i	0.718701	−	0.458294i
$21$		0			0
$22$	−18483.5	+	5391.73i	−0.370088	+	0.107956i
$23$	110629.			1.89593			0.947963	−	0.318381i	$-0.103139\pi$
$23$	110629.			1.89593			0.947963	+	0.318381i	$0.103139\pi$
$24$		0			0
$25$	−21362.2			−0.273436
$26$	48689.3	−	14202.9i	0.543284	−	0.158478i
$27$		0			0
$28$	−50748.0	−	79583.6i	−0.436884	−	0.685126i
$29$	−19262.3			−0.146661			−0.0733305	−	0.997308i	$-0.523363\pi$
$29$	−19262.3			−0.146661			−0.0733305	+	0.997308i	$0.523363\pi$
$30$		0			0
$31$		−	70291.5i		−	0.423777i	−0.977294	−	0.211888i	$-0.932039\pi$
$31$		−	70291.5i		−	0.423777i	0.977294	−	0.211888i	$-0.0679613\pi$
$32$	−183236.	−	28002.9i	−0.988523	−	0.151070i
$33$		0			0
$34$	−216782.	+	63236.4i	−0.945904	+	0.275925i
$35$		−	175685.i		−	0.692622i
$36$		0			0
$37$		−	441575.i		−	1.43317i	−0.697499	−	0.716586i	$-0.745704\pi$
$37$		−	441575.i		−	1.43317i	0.697499	−	0.716586i	$-0.254296\pi$
$38$	25631.8	+	87868.9i	0.0757767	+	0.259772i
$39$		0			0
$40$	−259547.	−	227323.i	−0.641218	−	0.561608i
$41$		−	382570.i		−	0.866898i	−0.901178	−	0.433449i	$-0.857297\pi$
$41$		−	382570.i		−	0.866898i	0.901178	−	0.433449i	$-0.142703\pi$
$42$		0			0
$43$	−329066.			−0.631166			−0.315583	−	0.948898i	$-0.602200\pi$
$43$	−329066.			−0.631166			−0.315583	+	0.948898i	$0.602200\pi$
$44$	117120.	+	183668.i	0.207274	+	0.325049i
$45$		0			0
$46$	−350497.	−	1.20155e6i	−0.530923	−	1.82007i
$47$	224714.			0.315709			0.157854	−	0.987462i	$-0.449542\pi$
$47$	224714.			0.315709			0.157854	+	0.987462i	$0.449542\pi$
$48$		0			0
$49$	279787.			0.339735
$50$	67680.0	+	232016.i	0.0765712	+	0.262496i
$51$		0			0
$52$	−308517.	−	483819.i	−0.304275	−	0.477168i
$53$	1.27611e6			1.17740			0.588698	−	0.808353i	$-0.299641\pi$
$53$	1.27611e6			1.17740			0.588698	+	0.808353i	$0.299641\pi$
$54$		0			0
$55$			405457.i			0.328606i
$56$	−703580.	+	803315.i	−0.535372	+	0.611263i
$57$		0			0
$58$	61027.1	+	209209.i	0.0410700	+	0.140793i
$59$		−	2.48468e6i		−	1.57503i	−0.616298	−	0.787513i	$-0.711368\pi$
$59$		−	2.48468e6i		−	1.57503i	0.616298	−	0.787513i	$-0.288632\pi$
$60$		0			0
$61$			2.93707e6i			1.65676i	0.560166	+	0.828380i	$0.310737\pi$
$61$			2.93707e6i			1.65676i	−0.560166	+	0.828380i	$0.689263\pi$
$62$	−763440.	+	222699.i	−0.406821	+	0.118672i
$63$		0			0
$64$	276392.	+	2.07886e6i	0.131794	+	0.991277i
$65$		−	1.06805e6i		−	0.482388i
$66$		0			0
$67$	2.94806e6			1.19750			0.598749	−	0.800937i	$-0.295665\pi$
$67$	2.94806e6			1.19750			0.598749	+	0.800937i	$0.295665\pi$
$68$	1.37363e6	+	2.15413e6i	0.529770	+	0.830791i
$69$		0			0
$70$	−1.90812e6	+	556608.i	−0.664910	+	0.193957i
$71$	1.63172e6			0.541057			0.270528	−	0.962712i	$-0.412802\pi$
$71$	1.63172e6			0.541057			0.270528	+	0.962712i	$0.412802\pi$
$72$		0			0
$73$	6.26023e6			1.88348			0.941738	−	0.336346i	$-0.109191\pi$
$73$	6.26023e6			1.88348			0.941738	+	0.336346i	$0.109191\pi$
$74$	−4.79597e6	+	1.39901e6i	−1.37583	+	0.401336i
$75$		0			0
$76$	873141.	−	556776.i	0.228159	−	0.145490i
$77$	1.25492e6			0.313254
$78$		0			0
$79$			448216.i			0.102280i	0.998691	+	0.0511402i	$0.0162855\pi$
$79$			448216.i			0.102280i	−0.998691	+	0.0511402i	$0.983714\pi$
$80$	−1.64667e6	+	3.53916e6i	−0.359576	+	0.772832i
$81$		0			0
$82$	−4.15512e6	+	1.21207e6i	−0.832213	+	0.242760i
$83$		−	3.63088e6i		−	0.697009i	−0.937307	−	0.348505i	$-0.886690\pi$
$83$		−	3.63088e6i		−	0.697009i	0.937307	−	0.348505i	$-0.113310\pi$
$84$		0			0
$85$			4.75536e6i			0.839880i
$86$	1.04255e6	+	3.57400e6i	0.176748	+	0.605913i
$87$		0			0
$88$	1.62377e6	−	1.85394e6i	0.254001	−	0.290006i
$89$			1.41664e6i			0.213008i	0.994312	+	0.106504i	$0.0339657\pi$
$89$			1.41664e6i			0.213008i	−0.994312	+	0.106504i	$0.966034\pi$
$90$		0			0
$91$	−3.30570e6			−0.459853
$92$	−1.19396e7	+	7.61352e6i	−1.59857	+	1.01936i
$93$		0			0
$94$	−711942.	−	2.44063e6i	−0.0884091	−	0.303077i
$95$	1.92750e6			0.230655
$96$		0			0
$97$	1.10470e7			1.22898			0.614488	−	0.788927i	$-0.289363\pi$
$97$	1.10470e7			1.22898			0.614488	+	0.788927i	$0.289363\pi$
$98$	−886425.	−	3.03878e6i	−0.0951373	−	0.326142i
$99$		0			0
$100$	2.30551e6	−	1.47015e6i	0.230551	−	0.147015i
$101$	−1.13145e7			−1.09273			−0.546364	−	0.837548i	$-0.683988\pi$
$101$	−1.13145e7			−1.09273			−0.546364	+	0.837548i	$0.683988\pi$
$102$		0			0
$103$			1.00479e7i			0.906034i	0.891502	+	0.453017i	$0.149652\pi$
$103$			1.00479e7i			0.906034i	−0.891502	+	0.453017i	$0.850348\pi$
$104$	−4.27733e6	+	4.88366e6i	−0.372869	+	0.425724i
$105$		0			0
$106$	−4.04300e6	−	1.38599e7i	−0.329711	−	1.13029i
$107$		−	1.33501e7i		−	1.05352i	−0.850014	−	0.526760i	$-0.823407\pi$
$107$		−	1.33501e7i		−	1.05352i	0.850014	−	0.526760i	$-0.176593\pi$
$108$		0			0
$109$		−	5.57713e6i		−	0.412494i	−0.978500	−	0.206247i	$-0.933875\pi$
$109$		−	5.57713e6i		−	0.412494i	0.978500	−	0.206247i	$-0.0661251\pi$
$110$	4.40369e6	−	1.28458e6i	0.315458	−	0.0920207i
$111$		0			0
$112$	1.09539e7	+	5.09654e6i	0.736728	+	0.342778i
$113$			4.54059e6i			0.296031i	0.988985	+	0.148016i	$0.0472886\pi$
$113$			4.54059e6i			0.296031i	−0.988985	+	0.148016i	$0.952711\pi$
$114$		0			0
$115$	−2.63573e7			−1.61606
$116$	2.07888e6	−	1.32564e6i	0.123659	−	0.0788536i
$117$		0			0
$118$	−2.69862e7	+	7.87200e6i	−1.51201	+	0.441060i
$119$	1.47182e7			0.800644
$120$		0			0
$121$	1.65910e7			0.851380
$122$	3.18996e7	−	9.30528e6i	1.59047	−	0.463949i
$123$		0			0
$124$	4.83749e6	+	7.58620e6i	0.227847	+	0.357313i
$125$	2.37028e7			1.08546
$126$		0			0
$127$		−	3.64412e7i		−	1.57863i	−0.613990	−	0.789314i	$-0.710436\pi$
$127$		−	3.64412e7i		−	1.57863i	0.613990	−	0.789314i	$-0.289564\pi$
$128$	2.17029e7	−	9.58819e6i	0.914710	−	0.404112i
$129$		0			0
$130$	−1.16002e7	+	3.38383e6i	−0.463088	+	0.135085i
$131$		−	2.32094e7i		−	0.902017i	−0.892520	−	0.451009i	$-0.851064\pi$
$131$		−	2.32094e7i		−	0.902017i	0.892520	−	0.451009i	$-0.148936\pi$
$132$		0			0
$133$		−	5.96576e6i		−	0.219880i
$134$	−9.34011e6	−	3.20191e7i	−0.335340	−	1.14959i
$135$		0			0
$136$	1.90442e7	−	2.17438e7i	0.649197	−	0.741223i
$137$			4.20151e7i			1.39599i	0.716101	+	0.697996i	$0.245925\pi$
$137$			4.20151e7i			1.39599i	−0.716101	+	0.697996i	$0.754075\pi$
$138$		0			0
$139$	−4.66398e7			−1.47301			−0.736504	−	0.676433i	$-0.763525\pi$
$139$	−4.66398e7			−1.47301			−0.736504	+	0.676433i	$0.763525\pi$
$140$	1.20907e7	+	1.89607e7i	0.372394	+	0.583992i
$141$		0			0
$142$	−5.16966e6	−	1.77222e7i	−0.151514	−	0.519409i
$143$	7.62911e6			0.218171
$144$		0			0
$145$	4.58923e6			0.125012
$146$	−1.98338e7	−	6.79927e7i	−0.527437	−	1.80812i
$147$		0			0
$148$	3.03893e7	+	4.76569e7i	0.770558	+	1.20840i
$149$	−4.42435e7			−1.09571			−0.547857	−	0.836572i	$-0.684556\pi$
$149$	−4.42435e7			−1.09571			−0.547857	+	0.836572i	$0.684556\pi$
$150$		0			0
$151$		−	4.69078e7i		−	1.10873i	−0.832274	−	0.554365i	$-0.812961\pi$
$151$		−	4.69078e7i		−	1.10873i	0.832274	−	0.554365i	$-0.187039\pi$
$152$	−8.81347e6	−	7.71924e6i	−0.203561	−	0.178288i
$153$		0			0
$154$	−3.97585e6	−	1.36297e7i	−0.0877218	−	0.300721i
$155$			1.67469e7i			0.361222i
$156$		0			0
$157$			6.21526e7i			1.28177i	0.767636	+	0.640886i	$0.221433\pi$
$157$			6.21526e7i			1.28177i	−0.767636	+	0.640886i	$0.778567\pi$
$158$	4.86809e6	−	1.42005e6i	0.0981881	−	0.0286419i
$159$		0			0
$160$	4.36560e7	+	6.67168e6i	0.842605	+	0.128770i
$161$			8.15776e7i			1.54057i
$162$		0			0
$163$	−9.54624e7			−1.72654			−0.863269	−	0.504745i	$-0.831587\pi$
$163$	−9.54624e7			−1.72654			−0.863269	+	0.504745i	$0.831587\pi$
$164$	2.63286e7	+	4.12888e7i	0.466095	+	0.730935i
$165$		0			0
$166$	−3.94352e7	+	1.15034e7i	−0.669122	+	0.195186i
$167$	−1.14834e8			−1.90793			−0.953963	−	0.299923i	$-0.903039\pi$
$167$	−1.14834e8			−1.90793			−0.953963	+	0.299923i	$0.903039\pi$
$168$		0			0
$169$	4.26519e7			0.679728
$170$	5.16482e7	−	1.50660e7i	0.806277	−	0.235195i
$171$		0			0
$172$	3.55144e7	−	2.26465e7i	0.532175	−	0.339352i
$173$	−5.67601e7			−0.833454			−0.416727	−	0.909032i	$-0.636823\pi$
$173$	−5.67601e7			−0.833454			−0.416727	+	0.909032i	$0.636823\pi$
$174$		0			0
$175$		−	1.57524e7i		−	0.222185i
$176$	−2.52802e7	−	1.17621e7i	−0.349532	−	0.162627i
$177$		0			0
$178$	1.53862e7	−	4.48824e6i	0.204486	−	0.0596494i
$179$		−	1.12835e7i		−	0.147048i	−0.997293	−	0.0735241i	$-0.976575\pi$
$179$		−	1.12835e7i		−	0.147048i	0.997293	−	0.0735241i	$-0.0234246\pi$
$180$		0			0
$181$		−	8.60787e7i		−	1.07900i	−0.841986	−	0.539499i	$-0.818614\pi$
$181$		−	8.60787e7i		−	1.07900i	0.841986	−	0.539499i	$-0.181386\pi$
$182$	1.04732e7	+	3.59034e7i	0.128774	+	0.441454i
$183$		0			0
$184$	1.20518e8	+	1.05555e8i	1.42623	+	1.24916i
$185$			1.05205e8i			1.22162i
$186$		0			0
$187$	−3.39676e7			−0.379856
$188$	−2.42522e7	+	1.54649e7i	−0.266194	+	0.169744i
$189$		0			0
$190$	−6.10676e6	−	2.09347e7i	−0.0645911	−	0.221426i
$191$	9.04440e7			0.939210			0.469605	−	0.882877i	$-0.344396\pi$
$191$	9.04440e7			0.939210			0.469605	+	0.882877i	$0.344396\pi$
$192$		0			0
$193$	1.25636e8			1.25795			0.628977	−	0.777424i	$-0.283474\pi$
$193$	1.25636e8			1.25795			0.628977	+	0.777424i	$0.283474\pi$
$194$	−3.49993e7	−	1.19982e8i	−0.344154	−	1.17980i
$195$		0			0
$196$	−3.01959e7	+	1.92550e7i	−0.286452	+	0.182662i
$197$	−3.98131e7			−0.371017			−0.185509	−	0.982643i	$-0.559393\pi$
$197$	−3.98131e7			−0.371017			−0.185509	+	0.982643i	$0.559393\pi$
$198$		0			0
$199$			1.82217e8i			1.63910i	0.573011	+	0.819548i	$0.305775\pi$
$199$			1.82217e8i			1.63910i	−0.573011	+	0.819548i	$0.694225\pi$
$200$	−2.32718e7	−	2.03825e7i	−0.205695	−	0.180157i
$201$		0			0
$202$	3.58469e7	+	1.22888e8i	0.306000	+	1.04901i
$203$		−	1.42040e7i		−	0.119172i
$204$		0			0
$205$			9.11472e7i			0.738933i
$206$	1.09131e8	−	3.18339e7i	0.869783	−	0.253720i
$207$		0			0
$208$	6.65932e7	+	3.09838e7i	0.513107	+	0.238733i
$209$			1.37682e7i			0.104319i
$210$		0			0
$211$	3.29195e7			0.241249			0.120624	−	0.992698i	$-0.461510\pi$
$211$	3.29195e7			0.241249			0.120624	+	0.992698i	$0.461510\pi$
$212$	−1.37724e8	+	8.78224e7i	−0.992736	+	0.633038i
$213$		0			0
$214$	−1.44996e8	+	4.22962e7i	−1.01137	+	0.295021i
$215$	7.83998e7			0.537998
$216$		0			0
$217$	5.18329e7			0.344347
$218$	−6.05735e7	+	1.76696e7i	−0.395990	+	0.115512i
$219$		0			0
$220$	−2.79037e7	−	4.37588e7i	−0.176678	−	0.277068i
$221$	8.94774e7			0.557622
$222$		0			0
$223$		−	1.71254e8i		−	1.03412i	−0.855948	−	0.517062i	$-0.827026\pi$
$223$		−	1.71254e8i		−	1.03412i	0.855948	−	0.517062i	$-0.172974\pi$
$224$	2.06493e7	−	1.35118e8i	0.122754	−	0.803241i
$225$		0			0
$226$	4.93156e7	−	1.43856e7i	0.284187	−	0.0828988i
$227$		−	2.45746e7i		−	0.139443i	−0.997566	−	0.0697214i	$-0.977789\pi$
$227$		−	2.45746e7i		−	0.139443i	0.997566	−	0.0697214i	$-0.0222110\pi$
$228$		0			0
$229$			7.85931e7i			0.432474i	0.976341	+	0.216237i	$0.0693784\pi$
$229$			7.85931e7i			0.432474i	−0.976341	+	0.216237i	$0.930622\pi$
$230$	8.35057e7	+	2.86268e8i	0.452552	+	1.55140i
$231$		0			0
$232$	−2.09842e7	−	1.83789e7i	−0.110327	−	0.0966298i
$233$		−	3.01184e8i		−	1.55986i	−0.625866	−	0.779931i	$-0.715254\pi$
$233$		−	3.01184e8i		−	1.55986i	0.625866	−	0.779931i	$-0.284746\pi$
$234$		0			0
$235$	−5.35379e7			−0.269106
$236$	1.70996e8	+	2.68158e8i	0.846827	+	1.32800i
$237$		0			0
$238$	−4.66304e7	−	1.59855e8i	−0.224207	−	0.768610i
$239$	1.18967e8			0.563681			0.281840	−	0.959461i	$-0.409055\pi$
$239$	1.18967e8			0.563681			0.281840	+	0.959461i	$0.409055\pi$
$240$		0			0
$241$	−9.68834e7			−0.445851			−0.222925	−	0.974835i	$-0.571561\pi$
$241$	−9.68834e7			−0.445851			−0.222925	+	0.974835i	$0.571561\pi$
$242$	−5.25639e7	−	1.80196e8i	−0.238415	−	0.817317i
$243$		0			0
$244$	−2.02130e8	−	3.16982e8i	−0.890772	−	1.39692i
$245$	−6.66590e7			−0.289586
$246$		0			0
$247$		−	3.62681e7i		−	0.153139i
$248$	6.70678e7	−	7.65749e7i	0.279212	−	0.318791i
$249$		0			0
$250$	−7.50956e7	−	2.57437e8i	−0.303965	−	1.04203i
$251$		−	3.13129e8i		−	1.24987i	−0.780677	−	0.624935i	$-0.785126\pi$
$251$		−	3.13129e8i		−	1.24987i	0.780677	−	0.624935i	$-0.214874\pi$
$252$		0			0
$253$		−	1.88270e8i		−	0.730903i
$254$	−3.95790e8	+	1.15454e8i	−1.51547	+	0.442069i
$255$		0			0
$256$	−1.72897e8	−	2.05339e8i	−0.644093	−	0.764947i
$257$		−	1.31292e8i		−	0.482472i	−0.970466	−	0.241236i	$-0.922447\pi$
$257$		−	1.31292e8i		−	0.482472i	0.970466	−	0.241236i	$-0.0775528\pi$
$258$		0			0
$259$	3.25617e8			1.16455
$260$	7.35039e7	+	1.15270e8i	0.259360	+	0.406732i
$261$		0			0
$262$	−2.52079e8	+	7.35325e7i	−0.865927	+	0.252595i
$263$	−2.72319e8			−0.923068			−0.461534	−	0.887123i	$-0.652701\pi$
$263$	−2.72319e8			−0.923068			−0.461534	+	0.887123i	$0.652701\pi$
$264$		0			0
$265$	−3.04033e8			−1.00360
$266$	−6.47944e7	+	1.89008e7i	−0.211082	+	0.0615737i
$267$		0			0
$268$	−3.18169e8	+	2.02887e8i	−1.00969	+	0.643846i
$269$	5.42080e8			1.69797			0.848985	−	0.528416i	$-0.177214\pi$
$269$	5.42080e8			1.69797			0.848985	+	0.528416i	$0.177214\pi$
$270$		0			0
$271$		−	4.54506e7i		−	0.138723i	−0.997592	−	0.0693613i	$-0.977904\pi$
$271$		−	4.54506e7i		−	0.138723i	0.997592	−	0.0693613i	$-0.0220961\pi$
$272$	−2.96497e8	−	1.37951e8i	−0.893364	−	0.415656i
$273$		0			0
$274$	4.56328e8	−	1.33113e8i	1.34014	−	0.390925i
$275$			3.63545e7i			0.105413i
$276$		0			0
$277$			1.51529e8i			0.428368i	0.976793	+	0.214184i	$0.0687092\pi$
$277$			1.51529e8i			0.428368i	−0.976793	+	0.214184i	$0.931291\pi$
$278$	1.47765e8	+	5.06557e8i	0.412492	+	1.41407i
$279$		0			0
$280$	1.67628e8	−	1.91389e8i	0.456344	−	0.521032i
$281$		−	2.01196e8i		−	0.540938i	−0.962729	−	0.270469i	$-0.912821\pi$
$281$		−	2.01196e8i		−	0.540938i	0.962729	−	0.270469i	$-0.0871788\pi$
$282$		0			0
$283$	3.07402e8			0.806222			0.403111	−	0.915151i	$-0.367929\pi$
$283$	3.07402e8			0.806222			0.403111	+	0.915151i	$0.367929\pi$
$284$	−1.76104e8	+	1.12296e8i	−0.456199	+	0.290904i
$285$		0			0
$286$	−2.41707e7	−	8.28602e7i	−0.0610954	−	0.209442i
$287$	2.82107e8			0.704412
$288$		0			0
$289$	1.19534e7			0.0291306
$290$	−1.45397e7	−	4.98438e7i	−0.0350076	−	0.120010i
$291$		0			0
$292$	−6.75634e8	+	4.30831e8i	−1.58808	+	1.01267i
$293$	4.38990e8			1.01957			0.509786	−	0.860301i	$-0.329724\pi$
$293$	4.38990e8			1.01957			0.509786	+	0.860301i	$0.329724\pi$
$294$		0			0
$295$			5.91973e8i			1.34253i
$296$	4.21324e8	−	4.81048e8i	0.944267	−	1.07812i
$297$		0			0
$298$	1.40173e8	+	4.80531e8i	0.306837	+	1.05188i
$299$			4.95941e8i			1.07295i
$300$		0			0
$301$		−	2.42653e8i		−	0.512865i
$302$	−5.09468e8	+	1.48614e8i	−1.06437	+	0.310482i
$303$		0			0
$304$	−5.59161e7	+	1.20180e8i	−0.114151	+	0.245343i
$305$		−	6.99755e8i		−	1.41220i
$306$		0			0
$307$	4.94522e8			0.975442			0.487721	−	0.873000i	$-0.337828\pi$
$307$	4.94522e8			0.975442			0.487721	+	0.873000i	$0.337828\pi$
$308$	−1.35437e8	+	8.63638e7i	−0.264124	+	0.168424i
$309$		0			0
$310$	1.81889e8	−	5.30579e7i	0.346769	−	0.101154i
$311$	3.88359e7			0.0732103			0.0366051	−	0.999330i	$-0.488346\pi$
$311$	3.88359e7			0.0732103			0.0366051	+	0.999330i	$0.488346\pi$
$312$		0			0
$313$	−4.59451e7			−0.0846905			−0.0423452	−	0.999103i	$-0.513483\pi$
$313$	−4.59451e7			−0.0846905			−0.0423452	+	0.999103i	$0.513483\pi$
$314$	6.75043e8	−	1.96913e8i	1.23049	−	0.358939i
$315$		0			0
$316$	−3.08464e7	−	4.83736e7i	−0.0549920	−	0.0862389i
$317$	−8.40916e8			−1.48267			−0.741336	−	0.671134i	$-0.765808\pi$
$317$	−8.40916e8			−1.48267			−0.741336	+	0.671134i	$0.765808\pi$
$318$		0			0
$319$			3.27809e7i			0.0565396i
$320$	−6.58503e7	−	4.95287e8i	−0.112340	−	0.844952i
$321$		0			0
$322$	8.86018e8	−	2.58456e8i	1.47893	−	0.431410i
$323$			1.61479e8i			0.266628i
$324$		0			0
$325$		−	9.57650e7i		−	0.154744i
$326$	3.02446e8	+	1.03682e9i	0.483488	+	1.65746i
$327$		0			0
$328$	3.65025e8	−	4.16769e8i	0.571168	−	0.652133i
$329$			1.65703e8i			0.256535i
$330$		0			0
$331$	8.63353e8			1.30855			0.654275	−	0.756257i	$-0.272974\pi$
$331$	8.63353e8			1.30855			0.654275	+	0.756257i	$0.272974\pi$
$332$	2.49878e8	+	3.91862e8i	0.374753	+	0.587692i
$333$		0			0
$334$	3.63818e8	+	1.24721e9i	0.534284	+	1.83159i
$335$	−7.02375e8			−1.02073
$336$		0			0
$337$	−1.07596e9			−1.53141			−0.765703	−	0.643194i	$-0.777609\pi$
$337$	−1.07596e9			−1.53141			−0.765703	+	0.643194i	$0.777609\pi$
$338$	−1.35131e8	−	4.63244e8i	−0.190347	−	0.652532i
$339$		0			0
$340$	−3.27266e8	−	5.13221e8i	−0.451569	−	0.708155i
$341$	−1.19623e8			−0.163371
$342$		0			0
$343$			8.13593e8i			1.08862i
$344$	−3.58482e8	−	3.13975e8i	−0.474802	−	0.415853i
$345$		0			0
$346$	1.79828e8	+	6.16474e8i	0.233395	+	0.800108i
$347$			1.40774e9i			1.80871i	0.426785	+	0.904353i	$0.359646\pi$
$347$			1.40774e9i			1.80871i	−0.426785	+	0.904353i	$0.640354\pi$
$348$		0			0
$349$			4.13124e8i			0.520225i	0.965578	+	0.260112i	$0.0837596\pi$
$349$			4.13124e8i			0.520225i	−0.965578	+	0.260112i	$0.916240\pi$
$350$	−1.71088e8	+	4.99072e7i	−0.213295	+	0.0622192i
$351$		0			0
$352$	−4.76558e7	+	3.11835e8i	−0.0582393	+	0.381088i
$353$			5.35837e8i			0.648367i	0.945994	+	0.324183i	$0.105090\pi$
$353$			5.35837e8i			0.648367i	−0.945994	+	0.324183i	$0.894910\pi$
$354$		0			0
$355$	−3.88758e8			−0.461190
$356$	−9.74940e7	−	1.52891e8i	−0.114526	−	0.179600i
$357$		0			0
$358$	−1.22551e8	+	3.57487e7i	−0.141165	+	0.0411784i
$359$	−1.32439e7			−0.0151072			−0.00755362	−	0.999971i	$-0.502404\pi$
$359$	−1.32439e7			−0.0151072			−0.00755362	+	0.999971i	$0.502404\pi$
$360$		0			0
$361$	−8.28419e8			−0.926776
$362$	−9.34905e8	+	2.72716e8i	−1.03583	+	0.302156i
$363$		0			0
$364$	3.56767e8	−	2.27500e8i	0.387731	−	0.247244i
$365$	−1.49150e9			−1.60545
$366$		0			0
$367$		−	5.85755e8i		−	0.618565i	−0.950970	−	0.309282i	$-0.899911\pi$
$367$		−	5.85755e8i		−	0.618565i	0.950970	−	0.309282i	$-0.100089\pi$
$368$	7.64614e8	−	1.64338e9i	0.799788	−	1.71897i
$369$		0			0
$370$	1.14264e9	−	3.33312e8i	1.17274	−	0.342094i
$371$			9.41002e8i			0.956713i
$372$		0			0
$373$		−	8.81595e8i		−	0.879606i	−0.898094	−	0.439803i	$-0.855048\pi$
$373$		−	8.81595e8i		−	0.879606i	0.898094	−	0.439803i	$-0.144952\pi$
$374$	1.07617e8	+	3.68923e8i	0.106372	+	0.364658i
$375$		0			0
$376$	2.44801e8	+	2.14408e8i	0.237496	+	0.208010i
$377$		−	8.63514e7i		−	0.0829993i
$378$		0			0
$379$	−1.38059e8			−0.130265			−0.0651325	−	0.997877i	$-0.520747\pi$
$379$	−1.38059e8			−0.130265			−0.0651325	+	0.997877i	$0.520747\pi$
$380$	−2.08025e8	+	1.32652e8i	−0.194480	+	0.124014i
$381$		0			0
$382$	−2.86546e8	−	9.82317e8i	−0.263011	−	0.901633i
$383$	−5.81141e8			−0.528550			−0.264275	−	0.964447i	$-0.585133\pi$
$383$	−5.81141e8			−0.528550			−0.264275	+	0.964447i	$0.585133\pi$
$384$		0			0
$385$	−2.98983e8			−0.267014
$386$	−3.98044e8	−	1.36454e9i	−0.352270	−	1.20762i
$387$		0			0
$388$	−1.19224e9	+	7.60258e8i	−1.03623	+	0.660770i
$389$	1.53201e9			1.31959			0.659796	−	0.751445i	$-0.270643\pi$
$389$	1.53201e9			1.31959			0.659796	+	0.751445i	$0.270643\pi$
$390$		0			0
$391$		−	2.20811e9i		−	1.86811i
$392$	3.04797e8	+	2.66955e8i	0.255570	+	0.223840i
$393$		0			0
$394$	1.26137e8	+	4.32412e8i	0.103897	+	0.356173i
$395$		−	1.06787e8i		−	0.0871825i
$396$		0			0
$397$		−	1.41811e9i		−	1.13748i	−0.822518	−	0.568740i	$-0.807431\pi$
$397$		−	1.41811e9i		−	1.13748i	0.822518	−	0.568740i	$-0.192569\pi$
$398$	1.97907e9	−	5.77305e8i	1.57352	−	0.459002i
$399$		0			0
$400$	−1.47645e8	+	3.17332e8i	−0.115348	+	0.247915i
$401$			2.25678e8i			0.174777i	0.996174	+	0.0873884i	$0.0278521\pi$
$401$			2.25678e8i			0.174777i	−0.996174	+	0.0873884i	$0.972148\pi$
$402$		0			0
$403$	3.15112e8			0.239826
$404$	1.22112e9	−	7.78670e8i	0.921347	−	0.587515i
$405$		0			0
$406$	−1.54270e8	+	4.50013e7i	−0.114404	+	0.0333721i
$407$	−7.51479e8			−0.552505
$408$		0			0
$409$	−6.63525e8			−0.479541			−0.239770	−	0.970830i	$-0.577072\pi$
$409$	−6.63525e8			−0.479541			−0.239770	+	0.970830i	$0.577072\pi$
$410$	9.89954e8	−	2.88774e8i	0.709368	−	0.206926i
$411$		0			0
$412$	−6.91499e8	−	1.08442e9i	−0.487137	−	0.763933i
$413$	1.83220e9			1.27981
$414$		0			0
$415$			8.65055e8i			0.594122i
$416$	1.25535e8	−	8.21435e8i	0.0854944	−	0.559431i
$417$		0			0
$418$	1.49537e8	−	4.36206e7i	0.100145	−	0.0292129i
$419$		−	1.25579e9i		−	0.834006i	−0.908905	−	0.417003i	$-0.863080\pi$
$419$		−	1.25579e9i		−	0.834006i	0.908905	−	0.417003i	$-0.136920\pi$
$420$		0			0
$421$			1.15612e9i			0.755118i	0.925986	+	0.377559i	$0.123236\pi$
$421$			1.15612e9i			0.755118i	−0.925986	+	0.377559i	$0.876764\pi$
$422$	−1.04296e8	−	3.57540e8i	−0.0675577	−	0.231596i
$423$		0			0
$424$	1.39018e9	+	1.21759e9i	0.885710	+	0.775745i
$425$			4.26380e8i			0.269424i
$426$		0			0
$427$	−2.16579e9			−1.34623
$428$	9.18761e8	+	1.44081e9i	0.566434	+	0.888288i
$429$		0			0
$430$	−2.48388e8	−	8.51505e8i	−0.150658	−	0.516473i
$431$	9.49566e8			0.571287			0.285644	−	0.958336i	$-0.407793\pi$
$431$	9.49566e8			0.571287			0.285644	+	0.958336i	$0.407793\pi$
$432$		0			0
$433$	−3.02435e8			−0.179030			−0.0895148	−	0.995985i	$-0.528532\pi$
$433$	−3.02435e8			−0.179030			−0.0895148	+	0.995985i	$0.528532\pi$
$434$	−1.64218e8	−	5.62959e8i	−0.0964287	−	0.330570i
$435$		0			0
$436$	3.83820e8	+	6.01910e8i	0.221781	+	0.347800i
$437$	−8.95019e8			−0.513035
$438$		0			0
$439$			8.77959e8i			0.495277i	0.968852	+	0.247639i	$0.0796546\pi$
$439$			8.77959e8i			0.495277i	−0.968852	+	0.247639i	$0.920345\pi$
$440$	−3.86862e8	+	4.41701e8i	−0.216507	+	0.247197i
$441$		0			0
$442$	−2.83484e8	−	9.71818e8i	−0.156153	−	0.535312i
$443$		−	7.35244e8i		−	0.401808i	−0.979611	−	0.200904i	$-0.935612\pi$
$443$		−	7.35244e8i		−	0.401808i	0.979611	−	0.200904i	$-0.0643879\pi$
$444$		0			0
$445$		−	3.37515e8i		−	0.181565i
$446$	−1.85999e9	+	5.42569e8i	−0.992749	+	0.289590i
$447$		0			0
$448$	−1.53295e9	+	2.03811e8i	−0.805479	+	0.107091i
$449$		−	5.48949e8i		−	0.286200i	−0.989708	−	0.143100i	$-0.954293\pi$
$449$		−	5.48949e8i		−	0.286200i	0.989708	−	0.143100i	$-0.0457071\pi$
$450$		0			0
$451$	−6.51064e8			−0.334200
$452$	−3.12485e8	−	4.90042e8i	−0.159164	−	0.249603i
$453$		0			0
$454$	−2.66906e8	+	7.78578e7i	−0.133864	+	0.0390487i
$455$	7.87582e8			0.391973
$456$		0			0
$457$	3.04791e9			1.49381			0.746905	−	0.664930i	$-0.231539\pi$
$457$	3.04791e9			1.49381			0.746905	+	0.664930i	$0.231539\pi$
$458$	8.53604e8	−	2.49000e8i	0.415171	−	0.121107i
$459$		0			0
$460$	2.84460e9	−	1.81392e9i	1.36260	−	0.868891i
$461$	−4.10410e8			−0.195103			−0.0975516	−	0.995230i	$-0.531101\pi$
$461$	−4.10410e8			−0.195103			−0.0975516	+	0.995230i	$0.531101\pi$
$462$		0			0
$463$			6.43419e8i			0.301273i	0.988589	+	0.150636i	$0.0481323\pi$
$463$			6.43419e8i			0.301273i	−0.988589	+	0.150636i	$0.951868\pi$
$464$	−1.33132e8	+	2.86138e8i	−0.0618683	+	0.132973i
$465$		0			0
$466$	−3.27117e9	+	9.54216e8i	−1.49745	+	0.436814i
$467$		−	9.09110e8i		−	0.413054i	−0.978441	−	0.206527i	$-0.933784\pi$
$467$		−	9.09110e8i		−	0.413054i	0.978441	−	0.206527i	$-0.0662162\pi$
$468$		0			0
$469$			2.17390e9i			0.973047i
$470$	1.69620e8	+	5.81478e8i	0.0753588	+	0.258339i
$471$		0			0
$472$	2.37072e9	−	2.70678e9i	1.03773	−	1.18483i
$473$			5.60010e8i			0.243322i
$474$		0			0
$475$	1.72826e8			0.0739914
$476$	−1.58846e9	+	1.01291e9i	−0.675073	+	0.430474i
$477$		0			0
$478$	−3.76913e8	−	1.29210e9i	−0.157850	−	0.541128i
$479$	5.75601e8			0.239302			0.119651	−	0.992816i	$-0.461822\pi$
$479$	5.75601e8			0.239302			0.119651	+	0.992816i	$0.461822\pi$
$480$		0			0
$481$	1.97955e9			0.811070
$482$	3.06948e8	+	1.05226e9i	0.124853	+	0.428012i
$483$		0			0
$484$	−1.79058e9	+	1.14180e9i	−0.717852	+	0.457752i
$485$	−2.63194e9			−1.04756
$486$		0			0
$487$		−	4.63509e9i		−	1.81847i	−0.416279	−	0.909237i	$-0.636666\pi$
$487$		−	4.63509e9i		−	1.81847i	0.416279	−	0.909237i	$-0.363334\pi$
$488$	−2.80237e9	+	3.19962e9i	−1.09158	+	1.24632i
$489$		0			0
$490$	2.11190e8	+	7.23987e8i	0.0810938	+	0.278000i
$491$		−	1.75783e9i		−	0.670181i	−0.942186	−	0.335091i	$-0.891233\pi$
$491$		−	1.75783e9i		−	0.670181i	0.942186	−	0.335091i	$-0.108767\pi$
$492$		0			0
$493$			3.84467e8i			0.144509i
$494$	−3.93910e8	+	1.14905e8i	−0.147012	+	0.0428840i
$495$		0			0
$496$	−1.04417e9	−	4.85821e8i	−0.384225	−	0.178768i
$497$			1.20323e9i			0.439645i
$498$		0			0
$499$	9.64456e8			0.347481			0.173740	−	0.984791i	$-0.444415\pi$
$499$	9.64456e8			0.347481			0.173740	+	0.984791i	$0.444415\pi$
$500$	−2.55812e9	+	1.63123e9i	−0.915219	+	0.583608i
$501$		0			0
$502$	−3.40091e9	+	9.92060e8i	−1.19986	+	0.350006i
$503$	3.53727e9			1.23931			0.619656	−	0.784874i	$-0.287272\pi$
$503$	3.53727e9			1.23931			0.619656	+	0.784874i	$0.287272\pi$
$504$		0			0
$505$	2.69568e9			0.931427
$506$	−2.04481e9	+	5.96481e8i	−0.701659	+	0.204677i
$507$		0			0
$508$	2.50790e9	+	3.93291e9i	0.848764	+	1.33104i
$509$	−1.22175e9			−0.410649			−0.205324	−	0.978694i	$-0.565825\pi$
$509$	−1.22175e9			−0.410649			−0.205324	+	0.978694i	$0.565825\pi$
$510$		0			0
$511$			4.61628e9i			1.53045i
$512$	−1.68242e9	+	2.52841e9i	−0.553974	+	0.832534i
$513$		0			0
$514$	−1.42597e9	+	4.15962e8i	−0.463169	+	0.135108i
$515$		−	2.39390e9i		−	0.772292i
$516$		0			0
$517$		−	3.82421e8i		−	0.121710i
$518$	−1.03162e9	−	3.53654e9i	−0.326113	−	1.11795i
$519$		0			0
$520$	1.01907e9	−	1.16353e9i	0.317829	−	0.362882i
$521$		−	2.68053e8i		−	0.0830404i	−0.999138	−	0.0415202i	$-0.986780\pi$
$521$		−	2.68053e8i		−	0.0830404i	0.999138	−	0.0415202i	$-0.0132201\pi$
$522$		0			0
$523$	−3.14297e9			−0.960691			−0.480346	−	0.877079i	$-0.659489\pi$
$523$	−3.14297e9			−0.960691			−0.480346	+	0.877079i	$0.659489\pi$
$524$	1.59728e9	+	2.50487e9i	0.484978	+	0.760547i
$525$		0			0
$526$	8.62768e8	+	2.95767e9i	0.258490	+	0.886136i
$527$	−1.40299e9			−0.417559
$528$		0			0
$529$	8.83394e9			2.59454
$530$	9.63242e8	+	3.30211e9i	0.281041	+	0.963444i
$531$		0			0
$532$	4.10566e8	+	6.43853e8i	0.118220	+	0.185394i
$533$	1.71503e9			0.490600
$534$		0			0
$535$			3.18066e9i			0.898007i
$536$	3.21159e9	+	2.81286e9i	0.900832	+	0.788990i
$537$		0			0
$538$	−1.71743e9	−	5.88755e9i	−0.475489	−	1.63004i
$539$		−	4.76145e8i		−	0.130972i
$540$		0			0
$541$			6.12302e9i			1.66255i	0.555859	+	0.831276i	$0.312389\pi$
$541$			6.12302e9i			1.66255i	−0.555859	+	0.831276i	$0.687611\pi$
$542$	−4.93641e8	+	1.43997e8i	−0.133172	+	0.0388470i
$543$		0			0
$544$	−5.58926e8	+	3.65732e9i	−0.148853	+	0.974019i
$545$			1.32875e9i			0.351605i
$546$		0			0
$547$	1.33232e9			0.348058			0.174029	−	0.984740i	$-0.444321\pi$
$547$	1.33232e9			0.348058			0.174029	+	0.984740i	$0.444321\pi$
$548$	−2.89149e9	−	4.53447e9i	−0.750568	−	1.17705i
$549$		0			0
$550$	3.94848e8	−	1.15179e8i	0.101195	−	0.0295191i
$551$	1.55837e8			0.0396863
$552$		0			0
$553$	−3.30513e8			−0.0831096
$554$	1.64576e9	−	4.80077e8i	0.411229	−	0.119957i
$555$		0			0
$556$	5.03359e9	−	3.20977e9i	1.24198	−	0.791976i
$557$	4.96837e9			1.21821			0.609103	−	0.793091i	$-0.291530\pi$
$557$	4.96837e9			1.21821			0.609103	+	0.793091i	$0.291530\pi$
$558$		0			0
$559$		−	1.47518e9i		−	0.357193i
$560$	−2.60977e9	−	1.21425e9i	−0.627978	−	0.292179i
$561$		0			0
$562$	−2.18520e9	+	6.37434e8i	−0.519295	+	0.151481i
$563$		−	7.63988e9i		−	1.80429i	−0.431428	−	0.902147i	$-0.641990\pi$
$563$		−	7.63988e9i		−	1.80429i	0.431428	−	0.902147i	$-0.358010\pi$
$564$		0			0
$565$		−	1.08179e9i		−	0.252333i
$566$	−9.73918e8	−	3.33871e9i	−0.225769	−	0.773965i
$567$		0			0
$568$	1.77759e9	+	1.55689e9i	0.407016	+	0.356483i
$569$		−	4.81831e8i		−	0.109648i	−0.998496	−	0.0548242i	$-0.982540\pi$
$569$		−	4.81831e8i		−	0.109648i	0.998496	−	0.0548242i	$-0.0174598\pi$
$570$		0			0
$571$	−4.95723e9			−1.11433			−0.557164	−	0.830403i	$-0.688110\pi$
$571$	−4.95723e9			−1.11433			−0.557164	+	0.830403i	$0.688110\pi$
$572$	−8.23370e8	+	5.25038e8i	−0.183954	+	0.117302i
$573$		0			0
$574$	−8.93776e8	−	3.06398e9i	−0.197259	−	0.676229i
$575$	−2.36327e9			−0.518414
$576$		0			0
$577$	−3.49388e9			−0.757168			−0.378584	−	0.925567i	$-0.623589\pi$
$577$	−3.49388e9			−0.757168			−0.378584	+	0.925567i	$0.623589\pi$
$578$	−3.78710e7	−	1.29827e8i	−0.00815754	−	0.0279651i
$579$		0			0
$580$	−4.95292e8	+	3.15832e8i	−0.105405	+	0.0672139i
$581$	2.67740e9			0.566366
$582$		0			0
$583$		−	2.17171e9i		−	0.453901i
$584$	6.81984e9	+	5.97313e9i	1.41687	+	1.24096i
$585$		0			0
$586$	−1.39082e9	−	4.76789e9i	−0.285515	−	0.978780i
$587$			7.05369e8i			0.143940i	0.997407	+	0.0719702i	$0.0229287\pi$
$587$			7.05369e8i			0.143940i	−0.997407	+	0.0719702i	$0.977071\pi$
$588$		0			0
$589$			5.68678e8i			0.114673i
$590$	6.42944e9	−	1.87550e9i	1.28882	−	0.375954i
$591$		0			0
$592$	−6.55953e9	−	3.05195e9i	−1.29941	−	0.604577i
$593$			2.36729e9i			0.466188i	0.972454	+	0.233094i	$0.0748850\pi$
$593$			2.36729e9i			0.466188i	−0.972454	+	0.233094i	$0.925115\pi$
$594$		0			0
$595$	−3.50660e9			−0.682459
$596$	4.77497e9	−	3.04485e9i	0.923865	−	0.589121i
$597$		0			0
$598$	5.38644e9	−	1.57125e9i	1.03003	−	0.300463i
$599$	4.23088e9			0.804334			0.402167	−	0.915566i	$-0.368257\pi$
$599$	4.23088e9			0.804334			0.402167	+	0.915566i	$0.368257\pi$
$600$		0			0
$601$	−2.07946e8			−0.0390742			−0.0195371	−	0.999809i	$-0.506219\pi$
$601$	−2.07946e8			−0.0390742			−0.0195371	+	0.999809i	$0.506219\pi$
$602$	−2.63546e9	+	7.68777e8i	−0.492345	+	0.143619i
$603$		0			0
$604$	3.22821e9	+	5.06251e9i	0.596118	+	0.934839i
$605$	−3.95280e9			−0.725706
$606$		0			0
$607$		−	5.95951e9i		−	1.08156i	−0.841165	−	0.540779i	$-0.818130\pi$
$607$		−	5.95951e9i		−	1.08156i	0.841165	−	0.540779i	$-0.181870\pi$
$608$	1.48243e9	+	2.26551e8i	0.267493	+	0.0408793i
$609$		0			0
$610$	−7.60007e9	+	2.21698e9i	−1.35570	+	0.395464i
$611$			1.00737e9i			0.178668i
$612$		0			0
$613$		−	8.16210e9i		−	1.43117i	−0.698527	−	0.715584i	$-0.746161\pi$
$613$		−	8.16210e9i		−	1.43117i	0.698527	−	0.715584i	$-0.253839\pi$
$614$	−1.56675e9	−	5.37103e9i	−0.273156	−	0.936414i
$615$		0			0
$616$	1.36709e9	+	1.19736e9i	0.235649	+	0.206392i
$617$			4.21778e9i			0.722912i	0.932389	+	0.361456i	$0.117720\pi$
$617$			4.21778e9i			0.722912i	−0.932389	+	0.361456i	$0.882280\pi$
$618$		0			0
$619$	3.34483e9			0.566835			0.283417	−	0.958997i	$-0.408532\pi$
$619$	3.34483e9			0.566835			0.283417	+	0.958997i	$0.408532\pi$
$620$	−1.15253e9	−	1.80741e9i	−0.194214	−	0.304569i
$621$		0			0
$622$	−1.23041e8	−	4.21799e8i	−0.0205013	−	0.0702811i
$623$	−1.04463e9			−0.173083
$624$		0			0
$625$	−3.97825e9			−0.651797
$626$	1.45564e8	+	4.99012e8i	0.0237162	+	0.0813020i
$627$		0			0
$628$	−4.27737e9	−	6.70781e9i	−0.689156	−	1.08074i
$629$	−8.81365e9			−1.41214
$630$		0			0
$631$		−	9.88860e9i		−	1.56687i	−0.621475	−	0.783434i	$-0.713466\pi$
$631$		−	9.88860e9i		−	1.56687i	0.621475	−	0.783434i	$-0.286534\pi$
$632$	−4.27660e8	+	4.88282e8i	−0.0673889	+	0.0769415i
$633$		0			0
$634$	2.66421e9	+	9.13323e9i	0.415198	+	1.42335i
$635$			8.68210e9i			1.34560i
$636$		0			0
$637$			1.25426e9i			0.192265i
$638$	3.56035e8	−	1.03857e8i	0.0542775	−	0.0158330i
$639$		0			0
$640$	−5.17071e9	+	2.28438e9i	−0.779687	+	0.344460i
$641$		−	4.30111e9i		−	0.645026i	−0.946565	−	0.322513i	$-0.895472\pi$
$641$		−	4.30111e9i		−	0.645026i	0.946565	−	0.322513i	$-0.104528\pi$
$642$		0			0
$643$	−6.64317e9			−0.985455			−0.492728	−	0.870184i	$-0.664000\pi$
$643$	−6.64317e9			−0.985455			−0.492728	+	0.870184i	$0.664000\pi$
$644$	−5.61420e9	−	8.80425e9i	−0.828300	−	1.29895i
$645$		0			0
$646$	1.75383e9	−	5.11600e8i	0.255960	−	0.0746649i
$647$	−9.43332e9			−1.36930			−0.684651	−	0.728871i	$-0.740045\pi$
$647$	−9.43332e9			−1.36930			−0.684651	+	0.728871i	$0.740045\pi$
$648$		0			0
$649$	−4.22846e9			−0.607192
$650$	−1.04011e9	+	3.03405e8i	−0.148553	+	0.0433337i
$651$		0			0
$652$	1.03028e10	−	6.56976e9i	1.45575	−	0.928288i
$653$	5.73991e9			0.806694			0.403347	−	0.915047i	$-0.367847\pi$
$653$	5.73991e9			0.806694			0.403347	+	0.915047i	$0.367847\pi$
$654$		0			0
$655$			5.52963e9i			0.768868i
$656$	−5.68302e9	−	2.64414e9i	−0.785988	−	0.365697i
$657$		0			0
$658$	1.79971e9	−	5.24985e8i	0.246271	−	0.0718383i
$659$			3.23892e9i			0.440860i	0.975403	+	0.220430i	$0.0707461\pi$
$659$			3.23892e9i			0.440860i	−0.975403	+	0.220430i	$0.929254\pi$
$660$		0			0
$661$		−	1.77949e9i		−	0.239657i	−0.992795	−	0.119829i	$-0.961766\pi$
$661$		−	1.77949e9i		−	0.239657i	0.992795	−	0.119829i	$-0.0382345\pi$
$662$	−2.73529e9	−	9.37691e9i	−0.366438	−	1.25619i
$663$		0			0
$664$	3.46436e9	−	3.95545e9i	0.459235	−	0.524333i
$665$			1.42134e9i			0.187423i
$666$		0			0
$667$	−2.13097e9			−0.278058
$668$	1.23934e10	−	7.90290e9i	1.60869	−	1.02581i
$669$		0			0
$670$	2.22528e9	+	7.62853e9i	0.285839	+	0.979893i
$671$	4.99835e9			0.638701
$672$		0			0
$673$	−3.44495e9			−0.435642			−0.217821	−	0.975989i	$-0.569895\pi$
$673$	−3.44495e9			−0.435642			−0.217821	+	0.975989i	$0.569895\pi$
$674$	3.40887e9	+	1.16860e10i	0.428845	+	1.47014i
$675$		0			0
$676$	−4.60320e9	+	2.93532e9i	−0.573121	+	0.365462i
$677$	−5.11751e9			−0.633868			−0.316934	−	0.948448i	$-0.602653\pi$
$677$	−5.11751e9			−0.633868			−0.316934	+	0.948448i	$0.602653\pi$
$678$		0			0
$679$			8.14604e9i			0.998624i
$680$	−4.53727e9	+	5.18045e9i	−0.553368	+	0.631809i
$681$		0			0
$682$	3.78993e8	+	1.29923e9i	0.0457494	+	0.156835i
$683$			1.48675e10i			1.78552i	0.450530	+	0.892761i	$0.351235\pi$
$683$			1.48675e10i			1.78552i	−0.450530	+	0.892761i	$0.648765\pi$
$684$		0			0
$685$		−	1.00101e10i		−	1.18993i
$686$	8.83648e9	−	2.57764e9i	1.04507	−	0.304851i
$687$		0			0
$688$	−2.27435e9	+	4.88823e9i	−0.266254	+	0.572258i
$689$			5.72071e9i			0.666319i
$690$		0			0
$691$	1.52586e10			1.75931			0.879655	−	0.475612i	$-0.157773\pi$
$691$	1.52586e10			1.75931			0.879655	+	0.475612i	$0.157773\pi$
$692$	6.12582e9	−	3.90625e9i	0.702737	−	0.448114i
$693$		0			0
$694$	1.52895e10	−	4.46002e9i	1.73634	−	0.506499i
$695$	1.11119e10			1.25557
$696$		0			0
$697$	−7.63595e9			−0.854178
$698$	4.48696e9	−	1.30887e9i	0.499411	−	0.145680i
$699$		0			0
$700$	1.08409e9	+	1.70008e9i	0.119460	+	0.187338i
$701$	−4.98929e9			−0.547049			−0.273524	−	0.961865i	$-0.588189\pi$
$701$	−4.98929e9			−0.547049			−0.273524	+	0.961865i	$0.588189\pi$
$702$		0			0
$703$			3.57246e9i			0.387815i
$704$	3.53784e9	−	4.70369e8i	0.382149	−	0.0508082i
$705$		0			0
$706$	5.81975e9	−	1.69765e9i	0.622426	−	0.181565i
$707$		−	8.34332e9i		−	0.887914i
$708$		0			0
$709$			1.01729e10i			1.07197i	0.844226	+	0.535987i	$0.180060\pi$
$709$			1.01729e10i			1.07197i	−0.844226	+	0.535987i	$0.819940\pi$
$710$	1.23167e9	+	4.22232e9i	0.129149	+	0.442738i
$711$		0			0
$712$	−1.35167e9	+	1.54328e9i	−0.140343	+	0.160238i
$713$		−	7.77628e9i		−	0.803449i
$714$		0			0
$715$	−1.81763e9			−0.185967
$716$	7.76537e8	+	1.21777e9i	0.0790618	+	0.123985i
$717$		0			0
$718$	4.19596e7	+	1.43843e8i	0.00423054	+	0.0145028i
$719$	−1.76261e10			−1.76850			−0.884248	−	0.467017i	$-0.845329\pi$
$719$	−1.76261e10			−1.76850			−0.884248	+	0.467017i	$0.845329\pi$
$720$		0			0
$721$	−7.40929e9			−0.736213
$722$	2.62461e9	+	8.99750e9i	0.259529	+	0.889696i
$723$		0			0
$724$	5.92397e9	+	9.29002e9i	0.580133	+	0.909770i
$725$	4.11484e8			0.0401024
$726$		0			0
$727$			1.77014e10i			1.70859i	0.519790	+	0.854294i	$0.326010\pi$
$727$			1.77014e10i			1.70859i	−0.519790	+	0.854294i	$0.673990\pi$
$728$	−3.60120e9	−	3.15410e9i	−0.345929	−	0.302981i
$729$		0			0
$730$	4.72539e9	+	1.61992e10i	0.449581	+	1.54122i
$731$			6.56803e9i			0.621905i
$732$		0			0
$733$		−	3.93288e9i		−	0.368848i	−0.982847	−	0.184424i	$-0.940958\pi$
$733$		−	3.93288e9i		−	0.368848i	0.982847	−	0.184424i	$-0.0590419\pi$
$734$	−6.36192e9	+	1.85580e9i	−0.593816	+	0.173219i
$735$		0			0
$736$	−2.02712e10	−	3.09793e9i	−1.87417	−	0.286417i
$737$		−	5.01706e9i		−	0.461650i
$738$		0			0
$739$	2.06051e10			1.87810			0.939051	−	0.343777i	$-0.111706\pi$
$739$	2.06051e10			1.87810			0.939051	+	0.343777i	$0.111706\pi$
$740$	−7.24025e9	−	1.13542e10i	−0.656814	−	1.03002i
$741$		0			0
$742$	1.02203e10	−	2.98130e9i	0.918435	−	0.267912i
$743$	−1.69616e10			−1.51707			−0.758537	−	0.651630i	$-0.774085\pi$
$743$	−1.69616e10			−1.51707			−0.758537	+	0.651630i	$0.774085\pi$
$744$		0			0
$745$	1.05410e10			0.933973
$746$	−9.57504e9	+	2.79308e9i	−0.844413	+	0.246319i
$747$		0			0
$748$	3.66594e9	−	2.33766e9i	0.320280	−	0.204233i
$749$	9.84437e9			0.856055
$750$		0			0
$751$		−	1.23098e10i		−	1.06050i	−0.847841	−	0.530251i	$-0.822098\pi$
$751$		−	1.23098e10i		−	1.06050i	0.847841	−	0.530251i	$-0.177902\pi$
$752$	1.55311e9	−	3.33809e9i	0.133180	−	0.286243i
$753$		0			0
$754$	−9.37867e8	+	2.73580e8i	−0.0796785	+	0.0232426i
$755$			1.11758e10i			0.945067i
$756$		0			0
$757$			8.82480e9i			0.739383i	0.929155	+	0.369691i	$0.120537\pi$
$757$			8.82480e9i			0.739383i	−0.929155	+	0.369691i	$0.879463\pi$
$758$	4.37402e8	+	1.49947e9i	0.0364786	+	0.125053i
$759$		0			0
$760$	2.09981e9	+	1.83911e9i	0.173513	+	0.151970i
$761$		−	8.05695e8i		−	0.0662712i	−0.999451	−	0.0331356i	$-0.989451\pi$
$761$		−	8.05695e8i		−	0.0662712i	0.999451	−	0.0331356i	$-0.0105493\pi$
$762$		0			0
$763$	4.11257e9			0.335179
$764$	−9.76115e9	+	6.22439e9i	−0.791907	+	0.504975i
$765$		0			0
$766$	1.84118e9	+	6.31180e9i	0.148012	+	0.507403i
$767$	1.11386e10			0.891348
$768$		0			0
$769$	−1.00081e10			−0.793611			−0.396806	−	0.917903i	$-0.629881\pi$
$769$	−1.00081e10			−0.793611			−0.396806	+	0.917903i	$0.629881\pi$
$770$	9.47244e8	+	3.24727e9i	0.0747729	+	0.256331i
$771$		0			0
$772$	−1.35593e10	+	8.64635e9i	−1.06066	+	0.676351i
$773$	1.09881e10			0.855648			0.427824	−	0.903862i	$-0.359280\pi$
$773$	1.09881e10			0.855648			0.427824	+	0.903862i	$0.359280\pi$
$774$		0			0
$775$			1.50158e9i			0.115876i
$776$	1.20345e10	+	1.05404e10i	0.924510	+	0.809729i
$777$		0			0
$778$	−4.85376e9	−	1.66393e10i	−0.369530	−	1.26679i
$779$			3.09510e9i			0.234581i
$780$		0			0
$781$		−	2.77690e9i		−	0.208584i
$782$	−2.39824e10	+	6.99577e9i	−1.79336	+	0.523133i
$783$		0			0
$784$	1.93375e9	−	4.15619e9i	0.143316	−	0.308027i
$785$		−	1.48078e10i		−	1.09257i
$786$		0			0
$787$	1.23147e10			0.900562			0.450281	−	0.892887i	$-0.351324\pi$
$787$	1.23147e10			0.900562			0.450281	+	0.892887i	$0.351324\pi$
$788$	4.29682e9	−	2.73995e9i	0.312828	−	0.199481i
$789$		0			0
$790$	−1.15982e9	+	3.38325e8i	−0.0836943	+	0.0244140i
$791$	−3.34823e9			−0.240545
$792$		0			0
$793$	−1.31667e10			−0.937604
$794$	−1.54022e10	+	4.49289e9i	−1.09197	+	0.318532i
$795$		0			0
$796$	−1.25403e10	−	1.96658e10i	−0.881274	−	1.38202i
$797$	−7.04586e9			−0.492981			−0.246490	−	0.969145i	$-0.579277\pi$
$797$	−7.04586e9			−0.492981			−0.246490	+	0.969145i	$0.579277\pi$
$798$		0			0
$799$		−	4.48519e9i		−	0.311076i
$800$	3.91433e9	+	5.98203e8i	0.270298	+	0.0413079i
$801$		0			0
$802$	2.45110e9	−	7.14998e8i	0.167784	−	0.0489434i
$803$		−	1.06538e10i		−	0.726103i
$804$		0			0
$805$		−	1.94358e10i		−	1.31316i
$806$	−9.98343e8	−	3.42244e9i	−0.0671594	−	0.230231i
$807$		0			0
$808$	−1.23259e10	−	1.07956e10i	−0.822016	−	0.719960i
$809$		−	7.50045e9i		−	0.498044i	−0.968498	−	0.249022i	$-0.919891\pi$
$809$		−	7.50045e9i		−	0.498044i	0.968498	−	0.249022i	$-0.0801091\pi$
$810$		0			0
$811$	−7.77055e9			−0.511539			−0.255770	−	0.966738i	$-0.582329\pi$
$811$	−7.77055e9			−0.511539			−0.255770	+	0.966738i	$0.582329\pi$
$812$	9.77523e8	+	1.53296e9i	0.0640739	+	0.100481i
$813$		0			0
$814$	2.38085e9	+	8.16185e9i	0.154720	+	0.530400i
$815$	2.27439e10			1.47168
$816$		0			0
$817$	2.66224e9			0.170793
$818$	2.10219e9	+	7.20658e9i	0.134288	+	0.460355i
$819$		0			0
$820$	−6.27278e9	−	9.83704e9i	−0.397294	−	0.623040i
$821$	2.32808e10			1.46824			0.734120	−	0.679019i	$-0.237595\pi$
$821$	2.32808e10			1.46824			0.734120	+	0.679019i	$0.237595\pi$
$822$		0			0
$823$		−	1.37296e10i		−	0.858537i	−0.903177	−	0.429268i	$-0.858771\pi$
$823$		−	1.37296e10i		−	0.858537i	0.903177	−	0.429268i	$-0.141229\pi$
$824$	−9.58707e9	+	1.09461e10i	−0.596954	+	0.681574i
$825$		0			0
$826$	−5.80480e9	−	1.98996e10i	−0.358391	−	1.22861i
$827$			6.49685e9i			0.399423i	0.979855	+	0.199712i	$0.0640006\pi$
$827$			6.49685e9i			0.399423i	−0.979855	+	0.199712i	$0.935999\pi$
$828$		0			0
$829$			1.86189e10i			1.13504i	0.823359	+	0.567521i	$0.192098\pi$
$829$			1.86189e10i			1.13504i	−0.823359	+	0.567521i	$0.807902\pi$
$830$	9.39541e9	−	2.74068e9i	0.570351	−	0.166374i
$831$		0			0
$832$	−9.31937e9	+	1.23905e9i	−0.560990	+	0.0745857i
$833$		−	5.58443e9i		−	0.334750i
$834$		0			0
$835$	2.73591e10			1.62629
$836$	−9.47530e8	−	1.48593e9i	−0.0560881	−	0.0879580i
$837$		0			0
$838$	−1.36392e10	+	3.97863e9i	−0.800638	+	0.233550i
$839$	−1.39216e10			−0.813811			−0.406906	−	0.913470i	$-0.633392\pi$
$839$	−1.39216e10			−0.813811			−0.406906	+	0.913470i	$0.633392\pi$
$840$		0			0
$841$	−1.68788e10			−0.978491
$842$	1.25566e10	−	3.66283e9i	0.724905	−	0.211458i
$843$		0			0
$844$	−3.55283e9	+	2.26553e9i	−0.203412	+	0.129710i
$845$	−1.01618e10			−0.579391
$846$		0			0
$847$			1.22342e10i			0.691803i
$848$	8.81986e9	−	1.89564e10i	0.496679	−	1.06751i
$849$		0			0
$850$	4.63094e9	−	1.35087e9i	0.258644	−	0.0754477i
$851$		−	4.88510e10i		−	2.71719i
$852$		0			0
$853$			1.40948e10i			0.777568i	0.921329	+	0.388784i	$0.127105\pi$
$853$			1.40948e10i			0.777568i	−0.921329	+	0.388784i	$0.872895\pi$
$854$	6.86170e9	+	2.35227e10i	0.376989	+	1.29237i
$855$		0			0
$856$	1.27379e10	−	1.45435e10i	0.694127	−	0.792522i
$857$		−	2.00414e10i		−	1.08766i	−0.839194	−	0.543832i	$-0.816973\pi$
$857$		−	2.00414e10i		−	1.08766i	0.839194	−	0.543832i	$-0.183027\pi$
$858$		0			0
$859$	−1.04263e10			−0.561245			−0.280623	−	0.959818i	$-0.590541\pi$
$859$	−1.04263e10			−0.561245			−0.280623	+	0.959818i	$0.590541\pi$
$860$	−8.46129e9	+	5.39551e9i	−0.453620	+	0.289259i
$861$		0			0
$862$	−3.00843e9	−	1.03133e10i	−0.159980	−	0.548430i
$863$	1.98495e10			1.05127			0.525633	−	0.850712i	$-0.323829\pi$
$863$	1.98495e10			1.05127			0.525633	+	0.850712i	$0.323829\pi$
$864$		0			0
$865$	1.35231e10			0.710426
$866$	9.58181e8	+	3.28476e9i	0.0501343	+	0.171867i
$867$		0			0
$868$	−5.59405e9	+	3.56716e9i	−0.290340	+	0.185141i
$869$	7.62781e8			0.0394303
$870$		0			0
$871$			1.32159e10i			0.677696i
$872$	5.32135e9	−	6.07567e9i	0.271778	−	0.310303i
$873$		0			0
$874$	2.83562e9	+	9.72084e9i	0.143667	+	0.492509i
$875$			1.74784e10i			0.882009i
$876$		0			0
$877$		−	1.64195e10i		−	0.821982i	−0.911639	−	0.410991i	$-0.865183\pi$
$877$		−	1.64195e10i		−	0.821982i	0.911639	−	0.410991i	$-0.134817\pi$
$878$	9.53556e9	−	2.78157e9i	0.475461	−	0.138694i
$879$		0			0
$880$	6.02300e9	+	2.80232e9i	0.297936	+	0.138621i
$881$		−	2.36845e10i		−	1.16694i	−0.812135	−	0.583470i	$-0.801695\pi$
$881$		−	2.36845e10i		−	1.16694i	0.812135	−	0.583470i	$-0.198305\pi$
$882$		0			0
$883$	−1.57236e10			−0.768580			−0.384290	−	0.923213i	$-0.625554\pi$
$883$	−1.57236e10			−0.768580			−0.384290	+	0.923213i	$0.625554\pi$
$884$	−9.65682e9	+	6.15786e9i	−0.470166	+	0.299811i
$885$		0			0
$886$	−7.98552e9	+	2.32941e9i	−0.385732	+	0.112520i
$887$	1.88503e10			0.906953			0.453477	−	0.891268i	$-0.350184\pi$
$887$	1.88503e10			0.906953			0.453477	+	0.891268i	$0.350184\pi$
$888$		0			0
$889$	2.68717e10			1.28274
$890$	−3.66577e9	+	1.06932e9i	−0.174301	+	0.0508444i
$891$		0			0
$892$	1.17857e10	+	1.84825e10i	0.556006	+	0.871935i
$893$	−1.81799e9			−0.0854304
$894$		0			0
$895$			2.68830e9i			0.125342i
$896$	7.07031e9	+	1.60037e10i	0.328368	+	0.743263i
$897$		0			0
$898$	−5.96216e9	+	1.73919e9i	−0.274749	+	0.0801457i
$899$			1.35398e9i			0.0621515i
$900$		0			0
$901$		−	2.54706e10i		−	1.16012i
$902$	2.06271e9	+	7.07124e9i	0.0935871	+	0.320828i
$903$		0			0
$904$	−4.33235e9	+	4.94648e9i	−0.195045	+	0.222693i
$905$			2.05082e10i			0.919724i
$906$		0			0
$907$	−5.59421e9			−0.248951			−0.124475	−	0.992223i	$-0.539725\pi$
$907$	−5.59421e9			−0.248951			−0.124475	+	0.992223i	$0.539725\pi$
$908$	1.69123e9	+	2.65221e9i	0.0749727	+	0.117573i
$909$		0			0
$910$	−2.49523e9	−	8.55397e9i	−0.109766	−	0.376290i
$911$	−5.38461e9			−0.235961			−0.117980	−	0.993016i	$-0.537642\pi$
$911$	−5.38461e9			−0.235961			−0.117980	+	0.993016i	$0.537642\pi$
$912$		0			0
$913$	−6.17909e9			−0.268706
$914$	−9.65645e9	−	3.31035e10i	−0.418317	−	1.43404i
$915$		0			0
$916$	−5.40881e9	−	8.48214e9i	−0.232524	−	0.364646i
$917$	1.71146e10			0.732949
$918$		0			0
$919$			3.73542e10i			1.58758i	0.608193	+	0.793789i	$0.291895\pi$
$919$			3.73542e10i			1.58758i	−0.608193	+	0.793789i	$0.708105\pi$
$920$	−2.87134e10	−	2.51485e10i	−1.21570	−	1.06477i
$921$		0			0
$922$	1.30027e9	+	4.45748e9i	0.0546355	+	0.187297i
$923$			7.31490e9i			0.306198i
$924$		0			0
$925$			9.43300e9i			0.391881i
$926$	6.98820e9	−	2.03849e9i	0.289219	−	0.0843666i
$927$		0			0
$928$	3.52955e9	+	5.39400e8i	0.144978	+	0.0221561i
$929$			3.91715e10i			1.60293i	0.598040	+	0.801467i	$0.295947\pi$
$929$			3.91715e10i			1.60293i	−0.598040	+	0.801467i	$0.704053\pi$
$930$		0			0
$931$	−2.26355e9			−0.0919319
$932$	2.07276e10	+	3.25052e10i	0.838674	+	1.31522i
$933$		0			0
$934$	−9.87389e9	+	2.88026e9i	−0.396528	+	0.115669i
$935$	8.09275e9			0.323784
$936$		0			0
$937$	1.34561e10			0.534355			0.267177	−	0.963647i	$-0.413909\pi$
$937$	1.34561e10			0.534355			0.267177	+	0.963647i	$0.413909\pi$
$938$	2.36108e10	−	6.88738e9i	0.934116	−	0.272486i
$939$		0			0
$940$	5.77807e9	−	3.68450e9i	0.226900	−	0.144687i
$941$	−5.29393e9			−0.207116			−0.103558	−	0.994623i	$-0.533023\pi$
$941$	−5.29393e9			−0.207116			−0.103558	+	0.994623i	$0.533023\pi$
$942$		0			0
$943$		−	4.23234e10i		−	1.64357i
$944$	−3.69095e10	−	1.71729e10i	−1.42802	−	0.664417i
$945$		0			0
$946$	6.08230e9	−	1.77423e9i	0.233587	−	0.0681384i
$947$		−	2.20048e10i		−	0.841962i	−0.907070	−	0.420981i	$-0.861686\pi$
$947$		−	2.20048e10i		−	0.841962i	0.907070	−	0.420981i	$-0.138314\pi$
$948$		0			0
$949$			2.80641e10i			1.06591i
$950$	−5.47550e8	−	1.87707e9i	−0.0207201	−	0.0710310i
$951$		0			0
$952$	1.60338e10	+	1.40432e10i	0.602293	+	0.527516i
$953$			1.42103e10i			0.531836i	0.963996	+	0.265918i	$0.0856750\pi$
$953$			1.42103e10i			0.531836i	−0.963996	+	0.265918i	$0.914325\pi$
$954$		0			0
$955$	−2.15482e10			−0.800571
$956$	−1.28395e10	+	8.18734e9i	−0.475274	+	0.303068i
$957$		0			0
$958$	−1.82363e9	−	6.25163e9i	−0.0670127	−	0.229728i
$959$	−3.09818e10			−1.13434
$960$		0			0
$961$	2.25717e10			0.820413
$962$	−6.27164e9	−	2.15000e10i	−0.227127	−	0.778619i
$963$		0			0
$964$	1.04561e10	−	6.66755e9i	0.375925	−	0.239716i
$965$	−2.99328e10			−1.07226
$966$		0			0
$967$		−	3.05306e10i		−	1.08578i	−0.839803	−	0.542892i	$-0.817329\pi$
$967$		−	3.05306e10i		−	1.08578i	0.839803	−	0.542892i	$-0.182671\pi$
$968$	1.80741e10	+	1.58301e10i	0.640460	+	0.560945i
$969$		0			0
$970$	8.33857e9	+	2.85856e10i	0.293353	+	1.00565i
$971$			4.10450e10i			1.43877i	0.694610	+	0.719387i	$0.255577\pi$
$971$			4.10450e10i			1.43877i	−0.694610	+	0.719387i	$0.744423\pi$
$972$		0			0
$973$		−	3.43921e10i		−	1.19692i
$974$	−5.03420e10	+	1.46850e10i	−1.74572	+	0.509234i
$975$		0			0
$976$	4.36297e10	+	2.02996e10i	1.50213	+	0.698897i
$977$			4.37480e10i			1.50082i	0.660975	+	0.750408i	$0.270143\pi$
$977$			4.37480e10i			1.50082i	−0.660975	+	0.750408i	$0.729857\pi$
$978$		0			0
$979$	2.41087e9			0.0821172
$980$	7.19416e9	−	4.58750e9i	0.244168	−	0.155699i
$981$		0			0
$982$	−1.90919e10	+	5.56920e9i	−0.643368	+	0.187673i
$983$	−4.01275e10			−1.34742			−0.673712	−	0.738994i	$-0.735301\pi$
$983$	−4.01275e10			−1.34742			−0.673712	+	0.738994i	$0.735301\pi$
$984$		0			0
$985$	9.48545e9			0.316250
$986$	4.17572e9	−	1.21808e9i	0.138727	−	0.0404674i
$987$		0			0
$988$	2.49598e9	+	3.91423e9i	0.0823365	+	0.129121i
$989$	−3.64042e10			−1.19664
$990$		0			0
$991$		−	8.59795e9i		−	0.280632i	−0.990107	−	0.140316i	$-0.955188\pi$
$991$		−	8.59795e9i		−	0.280632i	0.990107	−	0.140316i	$-0.0448118\pi$
$992$	−1.96837e9	+	1.28800e10i	−0.0640199	+	0.418913i
$993$		0			0
$994$	1.30684e10	−	3.81210e9i	0.422055	−	0.123115i
$995$		−	4.34132e10i		−	1.39714i
$996$		0			0
$997$			5.27262e10i			1.68497i	0.538716	+	0.842487i	$0.318909\pi$
$997$			5.27262e10i			1.68497i	−0.538716	+	0.842487i	$0.681091\pi$
$998$	−3.05561e9	−	1.04750e10i	−0.0973063	−	0.333578i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.f.a.35.11	✓	28
3.2	odd	2	inner	72.8.f.a.35.18	yes	28
4.3	odd	2		288.8.f.a.143.7		28
8.3	odd	2	inner	72.8.f.a.35.17	yes	28
8.5	even	2		288.8.f.a.143.22		28
12.11	even	2		288.8.f.a.143.21		28
24.5	odd	2		288.8.f.a.143.8		28
24.11	even	2	inner	72.8.f.a.35.12	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.11	✓	28	1.1	even	1	trivial
72.8.f.a.35.12	yes	28	24.11	even	2	inner
72.8.f.a.35.17	yes	28	8.3	odd	2	inner
72.8.f.a.35.18	yes	28	3.2	odd	2	inner
288.8.f.a.143.7		28	4.3	odd	2
288.8.f.a.143.8		28	24.5	odd	2
288.8.f.a.143.21		28	12.11	even	2
288.8.f.a.143.22		28	8.5	even	2

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.16822 - 10.8610i) q^{2} +(-107.925 + 68.8204i) q^{4} -238.250 q^{5} +737.398i q^{7} +(1089.39 + 954.138i) q^{8} +O(q^{10})\)	\(q+(-3.16822 - 10.8610i) q^{2} +(-107.925 + 68.8204i) q^{4} -238.250 q^{5} +737.398i q^{7} +(1089.39 + 954.138i) q^{8} +(754.827 + 2587.64i) q^{10} -1701.82i q^{11} +4482.93i q^{13} +(8008.92 - 2336.24i) q^{14} +(6911.51 - 14854.8i) q^{16} -19959.6i q^{17} -8090.28 q^{19} +(25713.0 - 16396.4i) q^{20} +(-18483.5 + 5391.73i) q^{22} +110629. q^{23} -21362.2 q^{25} +(48689.3 - 14202.9i) q^{26} +(-50748.0 - 79583.6i) q^{28} -19262.3 q^{29} -70291.5i q^{31} +(-183236. - 28002.9i) q^{32} +(-216782. + 63236.4i) q^{34} -175685. i q^{35} -441575. i q^{37} +(25631.8 + 87868.9i) q^{38} +(-259547. - 227323. i) q^{40} -382570. i q^{41} -329066. q^{43} +(117120. + 183668. i) q^{44} +(-350497. - 1.20155e6i) q^{46} +224714. q^{47} +279787. q^{49} +(67680.0 + 232016. i) q^{50} +(-308517. - 483819. i) q^{52} +1.27611e6 q^{53} +405457. i q^{55} +(-703580. + 803315. i) q^{56} +(61027.1 + 209209. i) q^{58} -2.48468e6i q^{59} +2.93707e6i q^{61} +(-763440. + 222699. i) q^{62} +(276392. + 2.07886e6i) q^{64} -1.06805e6i q^{65} +2.94806e6 q^{67} +(1.37363e6 + 2.15413e6i) q^{68} +(-1.90812e6 + 556608. i) q^{70} +1.63172e6 q^{71} +6.26023e6 q^{73} +(-4.79597e6 + 1.39901e6i) q^{74} +(873141. - 556776. i) q^{76} +1.25492e6 q^{77} +448216. i q^{79} +(-1.64667e6 + 3.53916e6i) q^{80} +(-4.15512e6 + 1.21207e6i) q^{82} -3.63088e6i q^{83} +4.75536e6i q^{85} +(1.04255e6 + 3.57400e6i) q^{86} +(1.62377e6 - 1.85394e6i) q^{88} +1.41664e6i q^{89} -3.30570e6 q^{91} +(-1.19396e7 + 7.61352e6i) q^{92} +(-711942. - 2.44063e6i) q^{94} +1.92750e6 q^{95} +1.10470e7 q^{97} +(-886425. - 3.03878e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 52 q^{4}+O(q^{10}) \) 28 * q + 52 * q^4	\( 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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