LMFDB - Embedded newform 81.9.d.g.26.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(26,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("81.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	81.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.9976674150$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.15
Character	$\chi$	$=$	81.26
Dual form			81.9.d.g.53.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(22.2403 + 12.8404i) q^{2} +(201.753 + 349.447i) q^{4} +(-176.989 + 102.185i) q^{5} +(1851.21 - 3206.39i) q^{7} +3788.08i q^{8} -5248.38 q^{10} +(13951.7 + 8054.99i) q^{11} +(18463.8 + 31980.2i) q^{13} +(82342.9 - 47540.7i) q^{14} +(3008.19 - 5210.35i) q^{16} +47526.9i q^{17} +213962. q^{19} +(-71416.2 - 41232.2i) q^{20} +(206859. + 358290. i) q^{22} +(-143902. + 83082.0i) q^{23} +(-174429. + 302120. i) q^{25} +948331. i q^{26} +1.49395e6 q^{28} +(-1.17848e6 - 680393. i) q^{29} +(109298. + 189309. i) q^{31} +(973633. - 562128. i) q^{32} +(-610265. + 1.05701e6i) q^{34} +756662. i q^{35} +1.15091e6 q^{37} +(4.75857e6 + 2.74736e6i) q^{38} +(-387084. - 670449. i) q^{40} +(643528. - 371541. i) q^{41} +(1.46517e6 - 2.53774e6i) q^{43} +6.50048e6i q^{44} -4.26723e6 q^{46} +(6.31706e6 + 3.64716e6i) q^{47} +(-3.97156e6 - 6.87895e6i) q^{49} +(-7.75870e6 + 4.47949e6i) q^{50} +(-7.45025e6 + 1.29042e7i) q^{52} -2.55324e6i q^{53} -3.29239e6 q^{55} +(1.21461e7 + 7.01254e6i) q^{56} +(-1.74731e7 - 3.02643e7i) q^{58} +(-4.44279e6 + 2.56505e6i) q^{59} +(1.34768e6 - 2.33425e6i) q^{61} +5.61372e6i q^{62} +2.73316e7 q^{64} +(-6.53578e6 - 3.77343e6i) q^{65} +(-319308. - 553057. i) q^{67} +(-1.66081e7 + 9.58869e6i) q^{68} +(-9.71586e6 + 1.68284e7i) q^{70} +3.72932e7i q^{71} -2.13898e7 q^{73} +(2.55965e7 + 1.47782e7i) q^{74} +(4.31675e7 + 7.47683e7i) q^{76} +(5.16549e7 - 2.98230e7i) q^{77} +(2.50309e7 - 4.33548e7i) q^{79} +1.22957e6i q^{80} +1.90830e7 q^{82} +(-2.83851e7 - 1.63882e7i) q^{83} +(-4.85652e6 - 8.41174e6i) q^{85} +(6.51714e7 - 3.76267e7i) q^{86} +(-3.05130e7 + 5.28500e7i) q^{88} +6.85873e6i q^{89} +1.36721e8 q^{91} +(-5.80654e7 - 3.35241e7i) q^{92} +(9.36621e7 + 1.62228e8i) q^{94} +(-3.78689e7 + 2.18636e7i) q^{95} +(4.14058e7 - 7.17170e7i) q^{97} -2.03986e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40}+ \cdots + 133878688 q^{97}+O(q^{100}) $ 32 * q + 2048 * q^4 - 3692 * q^7 + 21504 * q^10 - 63860 * q^13 - 95116 * q^16 + 370216 * q^19 + 691980 * q^22 + 541712 * q^25 - 1994264 * q^28 - 571136 * q^31 - 1027656 * q^34 + 8708536 * q^37 + 2973768 * q^40 - 5453084 * q^43 - 8468088 * q^46 - 14602560 * q^49 + 25384960 * q^52 - 28905144 * q^55 - 1213068 * q^58 - 31037996 * q^61 + 48506000 * q^64 + 62356828 * q^67 - 11075124 * q^70 - 42547328 * q^73 - 69593876 * q^76 + 151045036 * q^79 - 80402280 * q^82 - 160995564 * q^85 - 366976068 * q^88 + 747361592 * q^91 + 544741584 * q^94 + 133878688 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$	22.2403	+	12.8404i	1.39002	+	0.802527i	0.993317	−	0.115422i	$-0.0368219\pi$
$2$	22.2403	+	12.8404i	1.39002	+	0.802527i	0.396700	+	0.917948i	$0.370155\pi$
$3$		0			0
$3$		0			0
$4$	201.753	+	349.447i	0.788098	+	1.36503i
$5$	−176.989	+	102.185i	−0.283183	+	0.163496i	−0.634863	−	0.772624i	$-0.718944\pi$
$5$	−176.989	+	102.185i	−0.283183	+	0.163496i	0.351681	+	0.936120i	$0.385610\pi$
$6$		0			0
$7$	1851.21	−	3206.39i	0.771017	−	1.33544i	−0.165990	−	0.986127i	$-0.553082\pi$
$7$	1851.21	−	3206.39i	0.771017	−	1.33544i	0.937006	−	0.349312i	$-0.113585\pi$
$8$			3788.08i			0.924825i
$9$		0			0
$10$	−5248.38			−0.524838
$11$	13951.7	+	8054.99i	0.952917	+	0.550167i	0.893986	−	0.448095i	$-0.147897\pi$
$11$	13951.7	+	8054.99i	0.952917	+	0.550167i	0.0589310	+	0.998262i	$0.481231\pi$
$12$		0			0
$13$	18463.8	+	31980.2i	0.646468	+	1.11972i	0.983960	+	0.178387i	$0.0570880\pi$
$13$	18463.8	+	31980.2i	0.646468	+	1.11972i	−0.337492	+	0.941328i	$0.609579\pi$
$14$	82342.9	−	47540.7i	2.14345	−	1.23752i
$15$		0			0
$16$	3008.19	−	5210.35i	0.0459014	−	0.0795036i
$17$			47526.9i			0.569041i	0.958670	+	0.284520i	$0.0918344\pi$
$17$			47526.9i			0.569041i	−0.958670	+	0.284520i	$0.908166\pi$
$18$		0			0
$19$	213962.			1.64181			0.820904	−	0.571066i	$-0.193470\pi$
$19$	213962.			1.64181			0.820904	+	0.571066i	$0.193470\pi$
$20$	−71416.2	−	41232.2i	−0.446351	−	0.257701i
$21$		0			0
$22$	206859.	+	358290.i	0.883047	+	1.52948i
$23$	−143902.	+	83082.0i	−0.514229	+	0.296890i	−0.734570	−	0.678533i	$-0.762616\pi$
$23$	−143902.	+	83082.0i	−0.514229	+	0.296890i	0.220342	+	0.975423i	$0.429283\pi$
$24$		0			0
$25$	−174429.	+	302120.i	−0.446538	+	0.773427i
$26$			948331.i			2.07523i
$27$		0			0
$28$	1.49395e6			2.43055
$29$	−1.17848e6	−	680393.i	−1.66621	−	0.961984i	−0.969655	−	0.244476i	$-0.921384\pi$
$29$	−1.17848e6	−	680393.i	−1.66621	−	0.961984i	−0.696550	−	0.717508i	$-0.745283\pi$
$30$		0			0
$31$	109298.	+	189309.i	0.118349	+	0.204986i	0.919114	−	0.393993i	$-0.128907\pi$
$31$	109298.	+	189309.i	0.118349	+	0.204986i	−0.800765	+	0.598979i	$0.795573\pi$
$32$	973633.	−	562128.i	0.928529	−	0.536087i
$33$		0			0
$34$	−610265.	+	1.05701e6i	−0.456670	+	0.790976i
$35$			756662.i			0.504231i
$36$		0			0
$37$	1.15091e6			0.614093			0.307046	−	0.951695i	$-0.400659\pi$
$37$	1.15091e6			0.614093			0.307046	+	0.951695i	$0.400659\pi$
$38$	4.75857e6	+	2.74736e6i	2.28214	+	1.31759i
$39$		0			0
$40$	−387084.	−	670449.i	−0.151205	−	0.261894i
$41$	643528.	−	371541.i	0.227736	−	0.131484i	−0.381791	−	0.924249i	$-0.624693\pi$
$41$	643528.	−	371541.i	0.227736	−	0.131484i	0.609527	+	0.792765i	$0.291359\pi$
$42$		0			0
$43$	1.46517e6	−	2.53774e6i	0.428562	−	0.742291i	−0.568184	−	0.822902i	$-0.692354\pi$
$43$	1.46517e6	−	2.53774e6i	0.428562	−	0.742291i	0.996746	+	0.0806110i	$0.0256871\pi$
$44$			6.50048e6i			1.73434i
$45$		0			0
$46$	−4.26723e6			−0.953048
$47$	6.31706e6	+	3.64716e6i	1.29456	+	0.747417i	0.979460	−	0.201640i	$-0.0646270\pi$
$47$	6.31706e6	+	3.64716e6i	1.29456	+	0.747417i	0.315105	+	0.949057i	$0.397960\pi$
$48$		0			0
$49$	−3.97156e6	−	6.87895e6i	−0.688933	−	1.19327i
$50$	−7.75870e6	+	4.47949e6i	−1.24139	+	0.716718i
$51$		0			0
$52$	−7.45025e6	+	1.29042e7i	−1.01896	+	1.76489i
$53$		−	2.55324e6i		−	0.323585i	−0.986825	−	0.161792i	$-0.948273\pi$
$53$		−	2.55324e6i		−	0.323585i	0.986825	−	0.161792i	$-0.0517275\pi$
$54$		0			0
$55$	−3.29239e6			−0.359799
$56$	1.21461e7	+	7.01254e6i	1.23505	+	0.713055i
$57$		0			0
$58$	−1.74731e7	−	3.02643e7i	−1.54404	−	2.67435i
$59$	−4.44279e6	+	2.56505e6i	−0.366647	+	0.211684i	−0.671993	−	0.740558i	$-0.734561\pi$
$59$	−4.44279e6	+	2.56505e6i	−0.366647	+	0.211684i	0.305346	+	0.952242i	$0.401228\pi$
$60$		0			0
$61$	1.34768e6	−	2.33425e6i	0.0973346	−	0.168588i	−0.813246	−	0.581920i	$-0.802302\pi$
$61$	1.34768e6	−	2.33425e6i	0.0973346	−	0.168588i	0.910581	+	0.413332i	$0.135635\pi$
$62$			5.61372e6i			0.379913i
$63$		0			0
$64$	2.73316e7			1.62909
$65$	−6.53578e6	−	3.77343e6i	−0.366137	−	0.211389i
$66$		0			0
$67$	−319308.	−	553057.i	−0.0158456	−	0.0274455i	0.857994	−	0.513660i	$-0.171711\pi$
$67$	−319308.	−	553057.i	−0.0158456	−	0.0274455i	−0.873839	+	0.486214i	$0.838377\pi$
$68$	−1.66081e7	+	9.58869e6i	−0.776755	+	0.448460i
$69$		0			0
$70$	−9.71586e6	+	1.68284e7i	−0.404659	+	0.700890i
$71$			3.72932e7i			1.46756i	0.679387	+	0.733780i	$0.262246\pi$
$71$			3.72932e7i			1.46756i	−0.679387	+	0.733780i	$0.737754\pi$
$72$		0			0
$73$	−2.13898e7			−0.753209			−0.376604	−	0.926374i	$-0.622908\pi$
$73$	−2.13898e7			−0.753209			−0.376604	+	0.926374i	$0.622908\pi$
$74$	2.55965e7	+	1.47782e7i	0.853599	+	0.492826i
$75$		0			0
$76$	4.31675e7	+	7.47683e7i	1.29391	+	2.24111i
$77$	5.16549e7	−	2.98230e7i	1.46943	−	0.848375i
$78$		0			0
$79$	2.50309e7	−	4.33548e7i	0.642641	−	1.11309i	−0.342200	−	0.939627i	$-0.611172\pi$
$79$	2.50309e7	−	4.33548e7i	0.642641	−	1.11309i	0.984841	−	0.173460i	$-0.0554946\pi$
$80$			1.22957e6i			0.0300187i
$81$		0			0
$82$	1.90830e7			0.422076
$83$	−2.83851e7	−	1.63882e7i	−0.598106	−	0.345317i	0.170190	−	0.985411i	$-0.445562\pi$
$83$	−2.83851e7	−	1.63882e7i	−0.598106	−	0.345317i	−0.768296	+	0.640094i	$0.778895\pi$
$84$		0			0
$85$	−4.85652e6	−	8.41174e6i	−0.0930357	−	0.161142i
$86$	6.51714e7	−	3.76267e7i	1.19142	−	0.687864i
$87$		0			0
$88$	−3.05130e7	+	5.28500e7i	−0.508808	+	0.881281i
$89$			6.85873e6i			0.109316i	0.998505	+	0.0546580i	$0.0174069\pi$
$89$			6.85873e6i			0.109316i	−0.998505	+	0.0546580i	$0.982593\pi$
$90$		0			0
$91$	1.36721e8			1.99375
$92$	−5.80654e7	−	3.35241e7i	−0.810525	−	0.467957i
$93$		0			0
$94$	9.36621e7	+	1.62228e8i	1.19964	+	2.07785i
$95$	−3.78689e7	+	2.18636e7i	−0.464931	+	0.268428i
$96$		0			0
$97$	4.14058e7	−	7.17170e7i	0.467708	−	0.810093i	−0.531612	−	0.846988i	$-0.678413\pi$
$97$	4.14058e7	−	7.17170e7i	0.467708	−	0.810093i	0.999319	+	0.0368950i	$0.0117467\pi$
$98$		−	2.03986e8i		−	2.21155i
$99$		0			0
$100$	−1.40766e8			−1.40766
$101$	−1.71673e8	−	9.91154e7i	−1.64974	−	0.952480i	−0.977173	−	0.212447i	$-0.931857\pi$
$101$	−1.71673e8	−	9.91154e7i	−1.64974	−	0.952480i	−0.672571	−	0.740033i	$-0.734810\pi$
$102$		0			0
$103$	−1.05983e8	−	1.83567e8i	−0.941641	−	1.63097i	−0.762340	−	0.647177i	$-0.775950\pi$
$103$	−1.05983e8	−	1.83567e8i	−0.941641	−	1.63097i	−0.179302	−	0.983794i	$-0.557384\pi$
$104$	−1.21144e8	+	6.99423e7i	−1.03554	+	0.597870i
$105$		0			0
$106$	3.27847e7	−	5.67847e7i	0.259685	−	0.449788i
$107$			1.63763e8i			1.24934i	0.780887	+	0.624672i	$0.214767\pi$
$107$			1.63763e8i			1.24934i	−0.780887	+	0.624672i	$0.785233\pi$
$108$		0			0
$109$	−2.21458e8			−1.56887			−0.784434	−	0.620213i	$-0.787046\pi$
$109$	−2.21458e8			−1.56887			−0.784434	+	0.620213i	$0.787046\pi$
$110$	−7.32236e7	−	4.22757e7i	−0.500127	−	0.288748i
$111$		0			0
$112$	−1.11376e7	−	1.92909e7i	−0.0707815	−	0.122597i
$113$	−6.14338e7	+	3.54688e7i	−0.376785	+	0.217537i	−0.676418	−	0.736518i	$-0.736469\pi$
$113$	−6.14338e7	+	3.54688e7i	−0.376785	+	0.217537i	0.299634	+	0.954054i	$0.403136\pi$
$114$		0			0
$115$	1.69794e7	−	2.94092e7i	0.0970804	−	0.168148i
$116$		−	5.49086e8i		−	3.03255i
$117$		0			0
$118$	−1.31745e8			−0.679527
$119$	1.52390e8	+	8.79822e7i	0.759920	+	0.438740i
$120$		0			0
$121$	2.25863e7	+	3.91207e7i	0.105367	+	0.182501i
$122$	5.99455e7	−	3.46096e7i	0.270593	−	0.156227i
$123$		0			0
$124$	−4.41023e7	+	7.63875e7i	−0.186541	+	0.323099i
$125$		−	1.51128e8i		−	0.619019i
$126$		0			0
$127$	−2.24298e8			−0.862205			−0.431102	−	0.902303i	$-0.641875\pi$
$127$	−2.24298e8			−0.862205			−0.431102	+	0.902303i	$0.641875\pi$
$128$	3.58613e8	+	2.07045e8i	1.33594	+	0.771303i
$129$		0			0
$130$	−9.69049e7	−	1.67844e8i	−0.339291	−	0.587669i
$131$	3.07378e8	−	1.77465e8i	1.04373	−	0.602596i	0.122841	−	0.992426i	$-0.460800\pi$
$131$	3.07378e8	−	1.77465e8i	1.04373	−	0.602596i	0.920887	+	0.389830i	$0.127466\pi$
$132$		0			0
$133$	3.96089e8	−	6.86046e8i	1.26586	−	2.19254i
$134$		−	1.64002e7i		−	0.0508662i
$135$		0			0
$136$	−1.80036e8			−0.526263
$137$	−6.45816e7	−	3.72862e7i	−0.183327	−	0.105844i	0.405528	−	0.914083i	$-0.367088\pi$
$137$	−6.45816e7	−	3.72862e7i	−0.183327	−	0.105844i	−0.588855	+	0.808239i	$0.700421\pi$
$138$		0			0
$139$	−4.16583e6	−	7.21542e6i	−0.0111594	−	0.0193287i	0.860392	−	0.509633i	$-0.170219\pi$
$139$	−4.16583e6	−	7.21542e6i	−0.0111594	−	0.0193287i	−0.871551	+	0.490305i	$0.836886\pi$
$140$	−2.64413e8	+	1.52659e8i	−0.688288	+	0.397383i
$141$		0			0
$142$	−4.78860e8	+	8.29410e8i	−1.17776	+	2.03993i
$143$			5.94902e8i			1.42266i
$144$		0			0
$145$	2.78103e8			0.629121
$146$	−4.75715e8	−	2.74654e8i	−1.04697	−	0.604470i
$147$		0			0
$148$	2.32199e8	+	4.02181e8i	0.483965	+	0.838252i
$149$	−5.91209e8	+	3.41334e8i	−1.19949	+	0.692524i	−0.960441	−	0.278484i	$-0.910168\pi$
$149$	−5.91209e8	+	3.41334e8i	−1.19949	+	0.692524i	−0.239046	+	0.971008i	$0.576835\pi$
$150$		0			0
$151$	−6.08390e7	+	1.05376e8i	−0.117024	+	0.202691i	−0.918587	−	0.395219i	$-0.870669\pi$
$151$	−6.08390e7	+	1.05376e8i	−0.117024	+	0.202691i	0.801563	+	0.597910i	$0.204002\pi$
$152$			8.10506e8i			1.51838i
$153$		0			0
$154$	1.53176e9			2.72338
$155$	−3.86890e7	−	2.23371e7i	−0.0670288	−	0.0386991i
$156$		0			0
$157$	3.73425e7	+	6.46792e7i	0.0614618	+	0.106455i	0.895119	−	0.445827i	$-0.147090\pi$
$157$	3.73425e7	+	6.46792e7i	0.0614618	+	0.106455i	−0.833657	+	0.552282i	$0.813757\pi$
$158$	1.11339e9	−	6.42815e8i	1.78656	−	1.03147i
$159$		0			0
$160$	−1.14882e8	+	1.98981e8i	−0.175296	+	0.303621i
$161$			6.15209e8i			0.915628i
$162$		0			0
$163$	5.65546e8			0.801157			0.400578	−	0.916262i	$-0.368809\pi$
$163$	5.65546e8			0.801157			0.400578	+	0.916262i	$0.368809\pi$
$164$	2.59667e8	+	1.49919e8i	0.358957	+	0.207244i
$165$		0			0
$166$	−4.20862e8	−	7.28954e8i	−0.554252	−	0.959992i
$167$	−2.18287e8	+	1.26028e8i	−0.280648	+	0.162032i	−0.633717	−	0.773565i	$-0.718471\pi$
$167$	−2.18287e8	+	1.26028e8i	−0.280648	+	0.162032i	0.353069	+	0.935597i	$0.385138\pi$
$168$		0			0
$169$	−2.73957e8	+	4.74507e8i	−0.335842	+	0.581696i
$170$		−	2.49439e8i		−	0.298654i
$171$		0			0
$172$	1.18241e9			1.35099
$173$	−2.42772e8	−	1.40164e8i	−0.271028	−	0.156478i	0.358327	−	0.933596i	$-0.383347\pi$
$173$	−2.42772e8	−	1.40164e8i	−0.271028	−	0.156478i	−0.629355	+	0.777118i	$0.716681\pi$
$174$		0			0
$175$	6.45810e8	+	1.11858e9i	0.688577	+	1.19265i
$176$	8.39386e7	−	4.84620e7i	0.0874804	−	0.0505069i
$177$		0			0
$178$	−8.80690e7	+	1.52540e8i	−0.0877290	+	0.151951i
$179$			4.01384e8i			0.390974i	0.980706	+	0.195487i	$0.0626287\pi$
$179$			4.01384e8i			0.390974i	−0.980706	+	0.195487i	$0.937371\pi$
$180$		0			0
$181$	−1.39158e9			−1.29657			−0.648284	−	0.761399i	$-0.724513\pi$
$181$	−1.39158e9			−1.29657			−0.648284	+	0.761399i	$0.724513\pi$
$182$	3.04072e9	+	1.75556e9i	2.77135	+	1.60004i
$183$		0			0
$184$	−3.14721e8	−	5.45113e8i	−0.274571	−	0.475571i
$185$	−2.03698e8	+	1.17605e8i	−0.173900	+	0.100401i
$186$		0			0
$187$	−3.82829e8	+	6.63078e8i	−0.313067	+	0.542249i
$188$			2.94330e9i			2.35615i
$189$		0			0
$190$	−1.12295e9			−0.861683
$191$	−1.47287e8	−	8.50360e7i	−0.110670	−	0.0638953i	0.443643	−	0.896203i	$-0.353686\pi$
$191$	−1.47287e8	−	8.50360e7i	−0.110670	−	0.0638953i	−0.554313	+	0.832308i	$0.687019\pi$
$192$		0			0
$193$	2.90945e8	+	5.03932e8i	0.209692	+	0.363198i	0.951618	−	0.307285i	$-0.0994205\pi$
$193$	2.90945e8	+	5.03932e8i	0.209692	+	0.363198i	−0.741925	+	0.670483i	$0.766087\pi$
$194$	1.84175e9	−	1.06334e9i	1.30024	−	0.750695i
$195$		0			0
$196$	1.60255e9	−	2.77570e9i	1.08589	−	1.88082i
$197$		−	1.19043e9i		−	0.790385i	−0.918598	−	0.395193i	$-0.870678\pi$
$197$		−	1.19043e9i		−	0.790385i	0.918598	−	0.395193i	$-0.129322\pi$
$198$		0			0
$199$	−1.88525e9			−1.20214			−0.601071	−	0.799196i	$-0.705259\pi$
$199$	−1.88525e9			−1.20214			−0.601071	+	0.799196i	$0.705259\pi$
$200$	−1.14446e9	−	6.60752e8i	−0.715285	−	0.412970i
$201$		0			0
$202$	−2.54537e9	−	4.40871e9i	−1.52878	−	2.64793i
$203$	−4.36321e9	+	2.51910e9i	−2.56934	+	1.48341i
$204$		0			0
$205$	−7.59316e7	+	1.31517e8i	−0.0429939	+	0.0744677i
$206$		−	5.44345e9i		−	3.02277i
$207$		0			0
$208$	2.22171e8			0.118695
$209$	2.98512e9	+	1.72346e9i	1.56451	+	0.903268i
$210$		0			0
$211$	1.98576e8	+	3.43944e8i	0.100184	+	0.173524i	0.911760	−	0.410723i	$-0.134724\pi$
$211$	1.98576e8	+	3.43944e8i	0.100184	+	0.173524i	−0.811576	+	0.584246i	$0.801390\pi$
$212$	8.92220e8	−	5.15123e8i	0.441701	−	0.255016i
$213$		0			0
$214$	−2.10279e9	+	3.64214e9i	−1.00263	+	1.73661i
$215$			5.98871e8i			0.280272i
$216$		0			0
$217$	8.09333e8			0.364996
$218$	−4.92530e9	−	2.84362e9i	−2.18075	−	1.25906i
$219$		0			0
$220$	−6.64249e8	−	1.15051e9i	−0.283557	−	0.491135i
$221$	−1.51992e9	+	8.77526e8i	−0.637164	+	0.367867i
$222$		0			0
$223$	9.30529e8	−	1.61172e9i	0.376279	−	0.651735i	−0.614238	−	0.789121i	$-0.710537\pi$
$223$	9.30529e8	−	1.61172e9i	0.376279	−	0.651735i	0.990518	+	0.137386i	$0.0438699\pi$
$224$		−	4.16247e9i		−	1.65333i
$225$		0			0
$226$	−1.82174e9			−0.698316
$227$	−1.89595e9	−	1.09463e9i	−0.714041	−	0.412252i	0.0985143	−	0.995136i	$-0.468591\pi$
$227$	−1.89595e9	−	1.09463e9i	−0.714041	−	0.412252i	−0.812556	+	0.582884i	$0.801924\pi$
$228$		0			0
$229$	1.07525e9	+	1.86238e9i	0.390990	+	0.677215i	0.992580	−	0.121590i	$-0.0387993\pi$
$229$	1.07525e9	+	1.86238e9i	0.390990	+	0.677215i	−0.601590	+	0.798805i	$0.705466\pi$
$230$	7.55254e8	−	4.36046e8i	0.269887	−	0.155819i
$231$		0			0
$232$	2.57739e9	−	4.46416e9i	0.889667	−	1.54095i
$233$			3.41313e9i			1.15805i	0.815308	+	0.579027i	$0.196567\pi$
$233$			3.41313e9i			1.15805i	−0.815308	+	0.579027i	$0.803433\pi$
$234$		0			0
$235$	−1.49074e9			−0.488798
$236$	−1.79269e9	−	1.03501e9i	−0.577907	−	0.333655i
$237$		0			0
$238$	2.25946e9	+	3.91350e9i	0.704201	+	1.21971i
$239$	2.84547e9	−	1.64283e9i	0.872091	−	0.503502i	0.00404838	−	0.999992i	$-0.498711\pi$
$239$	2.84547e9	−	1.64283e9i	0.872091	−	0.503502i	0.868042	+	0.496490i	$0.165378\pi$
$240$		0			0
$241$	−2.01741e9	+	3.49426e9i	−0.598035	+	1.03583i	0.395076	+	0.918648i	$0.370718\pi$
$241$	−2.01741e9	+	3.49426e9i	−0.598035	+	1.03583i	−0.993111	+	0.117178i	$0.962615\pi$
$242$			1.16007e9i			0.338239i
$243$		0			0
$244$	1.08759e9			0.306837
$245$	1.40585e9	+	8.11666e8i	0.390188	+	0.225275i
$246$		0			0
$247$	3.95055e9	+	6.84255e9i	1.06138	+	1.83836i
$248$	−7.17119e8	+	4.14029e8i	−0.189577	+	0.109452i
$249$		0			0
$250$	1.94054e9	−	3.36112e9i	0.496779	−	0.860447i
$251$		−	4.69477e9i		−	1.18282i	−0.806371	−	0.591411i	$-0.798571\pi$
$251$		−	4.69477e9i		−	1.18282i	0.806371	−	0.591411i	$-0.201429\pi$
$252$		0			0
$253$	−2.67690e9			−0.653356
$254$	−4.98845e9	−	2.88008e9i	−1.19848	−	0.691942i
$255$		0			0
$256$	1.81865e9	+	3.14999e9i	0.423437	+	0.733414i
$257$	1.10426e8	−	6.37546e7i	0.0253128	−	0.0146143i	−0.487290	−	0.873240i	$-0.662015\pi$
$257$	1.10426e8	−	6.37546e7i	0.0253128	−	0.0146143i	0.512603	+	0.858626i	$0.328681\pi$
$258$		0			0
$259$	2.13057e9	−	3.69026e9i	0.473476	−	0.820084i
$260$		−	3.04521e9i		−	0.666382i
$261$		0			0
$262$	9.11488e9			1.93440
$263$	2.01039e9	+	1.16070e9i	0.420201	+	0.242603i	0.695163	−	0.718852i	$-0.255332\pi$
$263$	2.01039e9	+	1.16070e9i	0.420201	+	0.242603i	−0.274962	+	0.961455i	$0.588665\pi$
$264$		0			0
$265$	2.60902e8	+	4.51895e8i	0.0529046	+	0.0916335i
$266$	1.76182e10	−	1.01719e10i	3.51914	−	2.03177i
$267$		0			0
$268$	1.28843e8	−	2.23162e8i	0.0249758	−	0.0432594i
$269$			7.86019e9i			1.50115i	0.660785	+	0.750575i	$0.270223\pi$
$269$			7.86019e9i			1.50115i	−0.660785	+	0.750575i	$0.729777\pi$
$270$		0			0
$271$	2.14381e9			0.397474			0.198737	−	0.980053i	$-0.436316\pi$
$271$	2.14381e9			0.397474			0.198737	+	0.980053i	$0.436316\pi$
$272$	2.47631e8	+	1.42970e8i	0.0452408	+	0.0261198i
$273$		0			0
$274$	−9.57541e8	−	1.65851e9i	−0.169885	−	0.294249i
$275$	−4.86715e9	+	2.81005e9i	−0.851028	+	0.491341i
$276$		0			0
$277$	−3.87519e8	+	6.71202e8i	−0.0658224	+	0.114008i	−0.897059	−	0.441912i	$-0.854300\pi$
$277$	−3.87519e8	+	6.71202e8i	−0.0658224	+	0.114008i	0.831236	+	0.555920i	$0.187634\pi$
$278$		−	2.13964e8i		−	0.0358229i
$279$		0			0
$280$	−2.86630e9			−0.466325
$281$	−6.89145e9	−	3.97878e9i	−1.10531	−	0.638153i	−0.167702	−	0.985838i	$-0.553634\pi$
$281$	−6.89145e9	−	3.97878e9i	−1.10531	−	0.638153i	−0.937611	+	0.347685i	$0.886968\pi$
$282$		0			0
$283$	−9.79004e8	−	1.69568e9i	−0.152630	−	0.264362i	0.779564	−	0.626323i	$-0.215441\pi$
$283$	−9.79004e8	−	1.69568e9i	−0.152630	−	0.264362i	−0.932193	+	0.361961i	$0.882107\pi$
$284$	−1.30320e10	+	7.52401e9i	−2.00326	+	1.15658i
$285$		0			0
$286$	−7.63880e9	+	1.32308e10i	−1.14172	+	1.97752i
$287$		−	2.75120e9i		−	0.405504i
$288$		0			0
$289$	4.71695e9			0.676192
$290$	6.18509e9	+	3.57096e9i	0.874488	+	0.504886i
$291$		0			0
$292$	−4.31546e9	−	7.47459e9i	−0.593602	−	1.02815i
$293$	−1.07304e10	+	6.19523e9i	−1.45595	+	0.840595i	−0.998809	−	0.0487979i	$-0.984461\pi$
$293$	−1.07304e10	+	6.19523e9i	−1.45595	+	0.840595i	−0.457144	+	0.889393i	$0.651128\pi$
$294$		0			0
$295$	5.24217e8	−	9.07971e8i	0.0692187	−	0.119890i
$296$			4.35974e9i			0.567928i
$297$		0			0
$298$	−1.75315e10			−2.22308
$299$	−5.31396e9	−	3.06801e9i	−0.664865	−	0.383860i
$300$		0			0
$301$	−5.42467e9	−	9.39580e9i	−0.660856	−	1.14464i
$302$	−2.70615e9	+	1.56240e9i	−0.325330	+	0.187829i
$303$		0			0
$304$	6.43639e8	−	1.11482e9i	0.0753613	−	0.130530i
$305$			5.50849e8i			0.0636551i
$306$		0			0
$307$	−3.61263e8			−0.0406696			−0.0203348	−	0.999793i	$-0.506473\pi$
$307$	−3.61263e8			−0.0406696			−0.0203348	+	0.999793i	$0.506473\pi$
$308$	2.08431e10	+	1.20338e10i	2.31611	+	1.33721i
$309$		0			0
$310$	−5.73636e8	−	9.93567e8i	−0.0621141	−	0.107585i
$311$	4.51507e9	−	2.60677e9i	0.482639	−	0.278652i	−0.238876	−	0.971050i	$-0.576779\pi$
$311$	4.51507e9	−	2.60677e9i	0.482639	−	0.278652i	0.721516	+	0.692398i	$0.243446\pi$
$312$		0			0
$313$	7.25733e9	−	1.25701e10i	0.756136	−	1.30967i	−0.188672	−	0.982040i	$-0.560418\pi$
$313$	7.25733e9	−	1.25701e10i	0.756136	−	1.30967i	0.944808	−	0.327625i	$-0.106248\pi$
$314$			1.91798e9i			0.197299i
$315$		0			0
$316$	2.02003e10			2.02586
$317$	9.91318e9	+	5.72338e9i	0.981694	+	0.566781i	0.902781	−	0.430100i	$-0.141522\pi$
$317$	9.91318e9	+	5.72338e9i	0.981694	+	0.566781i	0.0789126	+	0.996882i	$0.474855\pi$
$318$		0			0
$319$	−1.09611e10	−	1.89852e10i	−1.05850	−	1.83338i
$320$	−4.83740e9	+	2.79287e9i	−0.461331	+	0.266349i
$321$		0			0
$322$	−7.89955e9	+	1.36824e10i	−0.734816	+	1.27274i
$323$			1.01689e10i			0.934256i
$324$		0			0
$325$	−1.28825e10			−1.15469
$326$	1.25779e10	+	7.26185e9i	1.11362	+	0.642950i
$327$		0			0
$328$	1.40743e9	+	2.43774e9i	0.121599	+	0.210616i
$329$	2.33884e10	−	1.35033e10i	1.99626	−	1.15254i
$330$		0			0
$331$	−6.60298e9	+	1.14367e10i	−0.550083	+	0.952771i	0.448185	+	0.893941i	$0.352071\pi$
$331$	−6.60298e9	+	1.14367e10i	−0.550083	+	0.952771i	−0.998268	+	0.0588306i	$0.981263\pi$
$332$		−	1.32254e10i		−	1.08857i
$333$		0			0
$334$	−6.47301e9			−0.520140
$335$	1.13028e8	+	6.52567e7i	0.00897442	+	0.00518139i
$336$		0			0
$337$	5.96753e9	+	1.03361e10i	0.462674	+	0.801375i	0.999093	−	0.0425770i	$-0.0135568\pi$
$337$	5.96753e9	+	1.03361e10i	0.462674	+	0.801375i	−0.536419	+	0.843952i	$0.680223\pi$
$338$	−1.21857e10	+	7.03544e9i	−0.933653	+	0.539045i
$339$		0			0
$340$	1.95964e9	−	3.39419e9i	0.146642	−	0.253992i
$341$			3.52157e9i			0.260447i
$342$		0			0
$343$	−8.06507e9			−0.582682
$344$	9.61318e9	+	5.55017e9i	0.686489	+	0.396344i
$345$		0			0
$346$	−3.59954e9	−	6.23458e9i	−0.251155	−	0.435014i
$347$	7.95396e9	−	4.59222e9i	0.548612	−	0.316741i	−0.199950	−	0.979806i	$-0.564078\pi$
$347$	7.95396e9	−	4.59222e9i	0.548612	−	0.316741i	0.748562	+	0.663065i	$0.230745\pi$
$348$		0			0
$349$	1.04434e10	−	1.80886e10i	0.703950	−	1.21928i	−0.263119	−	0.964763i	$-0.584751\pi$
$349$	1.04434e10	−	1.80886e10i	0.703950	−	1.21928i	0.967069	−	0.254514i	$-0.0819155\pi$
$350$			3.31699e10i			2.21041i
$351$		0			0
$352$	1.81117e10			1.17975
$353$	−9.50403e9	−	5.48716e9i	−0.612081	−	0.353385i	0.161698	−	0.986840i	$-0.448303\pi$
$353$	−9.50403e9	−	5.48716e9i	−0.612081	−	0.353385i	−0.773780	+	0.633455i	$0.781636\pi$
$354$		0			0
$355$	−3.81079e9	−	6.60049e9i	−0.239940	−	0.415588i
$356$	−2.39676e9	+	1.38377e9i	−0.149219	+	0.0861517i
$357$		0			0
$358$	−5.15394e9	+	8.92688e9i	−0.313767	+	0.543460i
$359$			1.19761e10i			0.721001i	0.932759	+	0.360501i	$0.117394\pi$
$359$			1.19761e10i			0.721001i	−0.932759	+	0.360501i	$0.882606\pi$
$360$		0			0
$361$	2.87962e10			1.69553
$362$	−3.09492e10	−	1.78685e10i	−1.80225	−	1.04053i
$363$		0			0
$364$	2.75840e10	+	4.77768e10i	1.57127	+	2.72152i
$365$	3.78576e9	−	2.18571e9i	0.213296	−	0.123146i
$366$		0			0
$367$	3.11074e9	−	5.38796e9i	0.171474	−	0.297002i	−0.767461	−	0.641096i	$-0.778480\pi$
$367$	3.11074e9	−	5.38796e9i	0.171474	−	0.297002i	0.938936	+	0.344093i	$0.111814\pi$
$368$			9.99707e8i			0.0545107i
$369$		0			0
$370$	−6.04041e9			−0.322299
$371$	−8.18668e9	−	4.72658e9i	−0.432128	−	0.249489i
$372$		0			0
$373$	6.15140e9	+	1.06545e10i	0.317789	+	0.550427i	0.980026	−	0.198867i	$-0.0637263\pi$
$373$	6.15140e9	+	1.06545e10i	0.317789	+	0.550427i	−0.662237	+	0.749294i	$0.730393\pi$
$374$	−1.70284e10	+	9.83136e9i	−0.870338	+	0.502490i
$375$		0			0
$376$	−1.38157e10	+	2.39296e10i	−0.691230	+	1.19725i
$377$		−	5.02505e10i		−	2.48757i
$378$		0			0
$379$	2.26268e10			1.09665			0.548324	−	0.836266i	$-0.315266\pi$
$379$	2.26268e10			1.09665			0.548324	+	0.836266i	$0.315266\pi$
$380$	−1.52804e10	−	8.82212e9i	−0.732823	−	0.423095i
$381$		0			0
$382$	−2.18380e9	−	3.78245e9i	−0.102555	−	0.177631i
$383$	−9.94555e9	+	5.74206e9i	−0.462204	+	0.266854i	−0.712971	−	0.701194i	$-0.752651\pi$
$383$	−9.94555e9	+	5.74206e9i	−0.462204	+	0.266854i	0.250767	+	0.968048i	$0.419317\pi$
$384$		0			0
$385$	−6.09490e9	+	1.05567e10i	−0.277411	+	0.480490i
$386$			1.49435e10i			0.673134i
$387$		0			0
$388$	3.34150e10			1.47440
$389$	−6.51932e8	−	3.76393e8i	−0.0284711	−	0.0164378i	0.485697	−	0.874127i	$-0.338566\pi$
$389$	−6.51932e8	−	3.76393e8i	−0.0284711	−	0.0164378i	−0.514168	+	0.857690i	$0.671899\pi$
$390$		0			0
$391$	−3.94863e9	−	6.83922e9i	−0.168943	−	0.292617i
$392$	2.60580e10	−	1.50446e10i	1.10356	−	0.637142i
$393$		0			0
$394$	1.52856e10	−	2.64755e10i	0.634305	−	1.09865i
$395$			1.02311e10i			0.420276i
$396$		0			0
$397$	−5.67666e9			−0.228524			−0.114262	−	0.993451i	$-0.536450\pi$
$397$	−5.67666e9			−0.228524			−0.114262	+	0.993451i	$0.536450\pi$
$398$	−4.19284e10	−	2.42074e10i	−1.67100	−	0.964751i
$399$		0			0
$400$	1.04943e9	+	1.81767e9i	0.0409935	+	0.0710028i
$401$	3.63092e10	−	2.09631e10i	1.40423	−	0.810735i	0.409410	−	0.912350i	$-0.365734\pi$
$401$	3.63092e10	−	2.09631e10i	1.40423	−	0.810735i	0.994824	+	0.101615i	$0.0324011\pi$
$402$		0			0
$403$	−4.03610e9	+	6.99073e9i	−0.153018	+	0.265034i
$404$		−	7.99874e10i		−	3.00259i
$405$		0			0
$406$	−1.29385e11			−4.76191
$407$	1.60571e10	+	9.27056e9i	0.585179	+	0.337853i
$408$		0			0
$409$	−1.25068e10	−	2.16625e10i	−0.446946	−	0.774133i	0.551240	−	0.834347i	$-0.314155\pi$
$409$	−1.25068e10	−	2.16625e10i	−0.446946	−	0.774133i	−0.998185	+	0.0602143i	$0.980822\pi$
$410$	−3.37748e9	+	1.94999e9i	−0.119525	+	0.0690075i
$411$		0			0
$412$	4.27646e10	−	7.40705e10i	1.48421	−	2.57073i
$413$			1.89938e10i			0.652846i
$414$		0			0
$415$	6.69848e9			0.225831
$416$	3.59539e10	+	2.07580e10i	1.20053	+	0.693126i
$417$		0			0
$418$	4.42600e10	+	7.66605e10i	1.44979	+	2.51112i
$419$	4.37377e9	−	2.52520e9i	0.141906	−	0.0819294i	−0.427366	−	0.904079i	$-0.640558\pi$
$419$	4.37377e9	−	2.52520e9i	0.141906	−	0.0819294i	0.569272	+	0.822149i	$0.307225\pi$
$420$		0			0
$421$	−7.11594e9	+	1.23252e10i	−0.226519	+	0.392342i	−0.956774	−	0.290832i	$-0.906068\pi$
$421$	−7.11594e9	+	1.23252e10i	−0.226519	+	0.392342i	0.730255	+	0.683174i	$0.239401\pi$
$422$			1.01992e10i			0.321601i
$423$		0			0
$424$	9.67187e9			0.299259
$425$	−1.43588e10	−	8.29007e9i	−0.440112	−	0.254099i
$426$		0			0
$427$	−4.98968e9	−	8.64237e9i	−0.150093	−	0.259969i
$428$	−5.72266e10	+	3.30398e10i	−1.70539	+	0.984605i
$429$		0			0
$430$	−7.68975e9	+	1.33190e10i	−0.224925	+	0.389582i
$431$		−	3.72124e10i		−	1.07840i	−0.842179	−	0.539199i	$-0.818727\pi$
$431$		−	3.72124e10i		−	1.07840i	0.842179	−	0.539199i	$-0.181273\pi$
$432$		0			0
$433$	−3.76284e10			−1.07044			−0.535222	−	0.844711i	$-0.679772\pi$
$433$	−3.76284e10			−1.07044			−0.535222	+	0.844711i	$0.679772\pi$
$434$	1.79998e10	+	1.03922e10i	0.507351	+	0.292919i
$435$		0			0
$436$	−4.46799e10	−	7.73879e10i	−1.23642	−	2.14154i
$437$	−3.07896e10	+	1.77764e10i	−0.844264	+	0.487436i
$438$		0			0
$439$	−2.68277e9	+	4.64669e9i	−0.0722312	+	0.125108i	−0.899879	−	0.436140i	$-0.856345\pi$
$439$	−2.68277e9	+	4.64669e9i	−0.0722312	+	0.125108i	0.827648	+	0.561248i	$0.189679\pi$
$440$		−	1.24718e10i		−	0.332751i
$441$		0			0
$442$	−4.50712e10			−1.18089
$443$	2.08109e10	+	1.20152e10i	0.540350	+	0.311971i	0.745221	−	0.666818i	$-0.232344\pi$
$443$	2.08109e10	+	1.20152e10i	0.540350	+	0.311971i	−0.204871	+	0.978789i	$0.565677\pi$
$444$		0			0
$445$	−7.00857e8	−	1.21392e9i	−0.0178727	−	0.0309564i
$446$	4.13904e10	−	2.38968e10i	1.04607	−	0.603949i
$447$		0			0
$448$	5.05966e10	−	8.76359e10i	1.25606	−	2.17555i
$449$		−	7.28050e10i		−	1.79133i	−0.444729	−	0.895665i	$-0.646700\pi$
$449$		−	7.28050e10i		−	1.79133i	0.444729	−	0.895665i	$-0.353300\pi$
$450$		0			0
$451$	1.19710e10			0.289351
$452$	−2.47889e10	−	1.43119e10i	−0.593886	−	0.342881i
$453$		0			0
$454$	−2.81110e10	−	4.86896e10i	−0.661686	−	1.14607i
$455$	−2.41982e10	+	1.39708e10i	−0.564595	+	0.325969i
$456$		0			0
$457$	−3.09877e10	+	5.36723e10i	−0.710436	+	1.23051i	0.254258	+	0.967137i	$0.418169\pi$
$457$	−3.09877e10	+	5.36723e10i	−0.710436	+	1.23051i	−0.964694	+	0.263375i	$0.915164\pi$
$458$			5.52265e10i			1.25512i
$459$		0			0
$460$	1.37026e10			0.306035
$461$	1.04284e10	+	6.02087e9i	0.230896	+	0.133308i	0.610985	−	0.791642i	$-0.290773\pi$
$461$	1.04284e10	+	6.02087e9i	0.230896	+	0.133308i	−0.380090	+	0.924950i	$0.624107\pi$
$462$		0			0
$463$	2.96311e9	+	5.13225e9i	0.0644797	+	0.111682i	0.896463	−	0.443118i	$-0.146128\pi$
$463$	2.96311e9	+	5.13225e9i	0.0644797	+	0.111682i	−0.831983	+	0.554801i	$0.812795\pi$
$464$	−7.09017e9	+	4.09351e9i	−0.152962	+	0.0883129i
$465$		0			0
$466$	−4.38260e10	+	7.59089e10i	−0.929369	+	1.60971i
$467$		−	5.80782e10i		−	1.22109i	−0.791983	−	0.610543i	$-0.790951\pi$
$467$		−	5.80782e10i		−	1.22109i	0.791983	−	0.610543i	$-0.209049\pi$
$468$		0			0
$469$	−2.36442e9			−0.0488690
$470$	−3.31544e10	−	1.91417e10i	−0.679437	−	0.392273i
$471$		0			0
$472$	−9.71661e9	−	1.68297e10i	−0.195770	−	0.339084i
$473$	4.08830e10	−	2.36038e10i	0.816767	−	0.471561i
$474$		0			0
$475$	−3.73212e10	+	6.46422e10i	−0.733130	+	1.26982i
$476$			7.10027e10i			1.38308i
$477$		0			0
$478$	8.43786e10			1.61629
$479$	−3.99551e10	−	2.30681e10i	−0.758980	−	0.438197i	0.0699495	−	0.997551i	$-0.477716\pi$
$479$	−3.99551e10	−	2.30681e10i	−0.758980	−	0.438197i	−0.828929	+	0.559353i	$0.811050\pi$
$480$		0			0
$481$	2.12501e10	+	3.68063e10i	0.396991	+	0.687609i
$482$	−8.97356e10	+	5.18088e10i	−1.66256	+	0.959877i
$483$		0			0
$484$	−9.11372e9	+	1.57854e10i	−0.166079	+	0.287657i
$485$			1.69242e10i			0.305872i
$486$		0			0
$487$	8.55813e10			1.52147			0.760735	−	0.649063i	$-0.224839\pi$
$487$	8.55813e10			1.52147			0.760735	+	0.649063i	$0.224839\pi$
$488$	8.84233e9	+	5.10512e9i	0.155915	+	0.0900174i
$489$		0			0
$490$	2.08443e10	+	3.61033e10i	0.361578	+	0.626272i
$491$	2.58098e10	−	1.49013e10i	0.444077	−	0.256388i	−0.261248	−	0.965272i	$-0.584134\pi$
$491$	2.58098e10	−	1.49013e10i	0.444077	−	0.256388i	0.705326	+	0.708884i	$0.250801\pi$
$492$		0			0
$493$	3.23370e10	−	5.60093e10i	0.547408	−	0.948139i
$494$			2.02907e11i			3.40713i
$495$		0			0
$496$	1.31516e9			0.0217295
$497$	1.19576e11	+	6.90375e10i	1.95984	+	1.13151i
$498$		0			0
$499$	2.57266e10	+	4.45598e10i	0.414936	+	0.718690i	0.995422	−	0.0955806i	$-0.0304708\pi$
$499$	2.57266e10	+	4.45598e10i	0.414936	+	0.718690i	−0.580486	+	0.814270i	$0.697137\pi$
$500$	5.28111e10	−	3.04905e10i	0.844977	−	0.487848i
$501$		0			0
$502$	6.02828e10	−	1.04413e11i	0.949246	−	1.64414i
$503$			1.97425e10i			0.308412i	0.988039	+	0.154206i	$0.0492819\pi$
$503$			1.97425e10i			0.308412i	−0.988039	+	0.154206i	$0.950718\pi$
$504$		0			0
$505$	4.05123e10			0.622905
$506$	−5.95350e10	−	3.43725e10i	−0.908176	−	0.524336i
$507$		0			0
$508$	−4.52528e10	−	7.83801e10i	−0.679502	−	1.17693i
$509$	2.46734e10	−	1.42452e10i	0.367586	−	0.212226i	−0.304817	−	0.952411i	$-0.598595\pi$
$509$	2.46734e10	−	1.42452e10i	0.367586	−	0.212226i	0.672403	+	0.740185i	$0.265262\pi$
$510$		0			0
$511$	−3.95970e10	+	6.85841e10i	−0.580736	+	1.00586i
$512$		−	1.25983e10i		−	0.183330i
$513$		0			0
$514$	3.27455e9			0.0469136
$515$	3.75155e10	+	2.16596e10i	0.533313	+	0.307908i
$516$		0			0
$517$	5.87557e10	+	1.01768e11i	0.822408	+	1.42445i
$518$	9.47691e10	−	5.47150e10i	1.31628	−	0.759954i
$519$		0			0
$520$	1.42941e10	−	2.47581e10i	0.195498	−	0.338613i
$521$			1.19310e11i			1.61929i	0.586919	+	0.809646i	$0.300341\pi$
$521$			1.19310e11i			1.61929i	−0.586919	+	0.809646i	$0.699659\pi$
$522$		0			0
$523$	−8.33461e10			−1.11398			−0.556992	−	0.830518i	$-0.688044\pi$
$523$	−8.33461e10			−1.11398			−0.556992	+	0.830518i	$0.688044\pi$
$524$	1.24029e11	+	7.16081e10i	1.64512	+	0.949810i
$525$		0			0
$526$	2.98077e10	+	5.16285e10i	0.389391	+	0.674445i
$527$	−8.99728e9	+	5.19458e9i	−0.116646	+	0.0673454i
$528$		0			0
$529$	−2.53503e10	+	4.39079e10i	−0.323713	+	0.560687i
$530$			1.34004e10i			0.169829i
$531$		0			0
$532$	3.19648e11			3.99049
$533$	2.37639e10	+	1.37201e10i	0.294448	+	0.170000i
$534$		0			0
$535$	−1.67341e10	−	2.89843e10i	−0.204262	−	0.353792i
$536$	2.09503e9	−	1.20956e9i	0.0253822	−	0.0146544i
$537$		0			0
$538$	−1.00928e11	+	1.74813e11i	−1.20471	+	2.08662i
$539$		−	1.27964e11i		−	1.51611i
$540$		0			0
$541$	−1.38210e11			−1.61343			−0.806717	−	0.590937i	$-0.798758\pi$
$541$	−1.38210e11			−1.61343			−0.806717	+	0.590937i	$0.798758\pi$
$542$	4.76789e10	+	2.75274e10i	0.552495	+	0.318983i
$543$		0			0
$544$	2.67162e10	+	4.62737e10i	0.305055	+	0.528371i
$545$	3.91957e10	−	2.26297e10i	0.444276	−	0.256503i
$546$		0			0
$547$	3.95682e10	−	6.85341e10i	0.441974	−	0.765522i	−0.555862	−	0.831275i	$-0.687612\pi$
$547$	3.95682e10	−	6.85341e10i	0.441974	−	0.765522i	0.997836	+	0.0657529i	$0.0209449\pi$
$548$		−	3.00904e10i		−	0.333661i
$549$		0			0
$550$	−1.44329e11			−1.57726
$551$	−2.52149e11	−	1.45578e11i	−2.73559	−	1.57939i
$552$		0			0
$553$	−9.26750e10	−	1.60518e11i	−0.990974	−	1.71642i
$554$	−1.72370e10	+	9.95182e9i	−0.182989	+	0.105648i
$555$		0			0
$556$	1.68094e9	−	2.91147e9i	0.0175894	−	0.0304658i
$557$			1.12395e11i			1.16769i	0.811865	+	0.583846i	$0.198453\pi$
$557$			1.12395e11i			1.16769i	−0.811865	+	0.583846i	$0.801547\pi$
$558$		0			0
$559$	1.08210e11			1.10821
$560$	3.94247e9	+	2.27619e9i	0.0400882	+	0.0231449i
$561$		0			0
$562$	−1.02178e11	−	1.76978e11i	−1.02427	−	1.77409i
$563$	−1.03943e11	+	6.00113e10i	−1.03457	+	0.597310i	−0.918291	−	0.395906i	$-0.870430\pi$
$563$	−1.03943e11	+	6.00113e10i	−1.03457	+	0.597310i	−0.116280	+	0.993216i	$0.537097\pi$
$564$		0			0
$565$	7.24874e9	−	1.25552e10i	0.0711326	−	0.123205i
$566$		−	5.02833e10i		−	0.489957i
$567$		0			0
$568$	−1.41270e11			−1.35724
$569$	1.32890e11	+	7.67240e10i	1.26778	+	0.731951i	0.974567	−	0.224097i	$-0.0719432\pi$
$569$	1.32890e11	+	7.67240e10i	1.26778	+	0.731951i	0.293210	+	0.956048i	$0.405277\pi$
$570$		0			0
$571$	−2.33391e10	−	4.04245e10i	−0.219553	−	0.380277i	0.735118	−	0.677939i	$-0.237127\pi$
$571$	−2.33391e10	−	4.04245e10i	−0.219553	−	0.380277i	−0.954671	+	0.297662i	$0.903793\pi$
$572$	−2.07887e11	+	1.20023e11i	−1.94197	+	1.12120i
$573$		0			0
$574$	3.53266e10	−	6.11875e10i	0.325428	−	0.563657i
$575$		−	5.79677e10i		−	0.530291i
$576$		0			0
$577$	9.25710e10			0.835164			0.417582	−	0.908639i	$-0.362878\pi$
$577$	9.25710e10			0.835164			0.417582	+	0.908639i	$0.362878\pi$
$578$	1.04906e11	+	6.05677e10i	0.939919	+	0.542662i
$579$		0			0
$580$	5.61082e10	+	9.71822e10i	0.495809	+	0.858766i
$581$	−1.05094e11	+	6.06759e10i	−0.922300	+	0.532490i
$582$		0			0
$583$	2.05663e10	−	3.56219e10i	0.178025	−	0.308349i
$584$		−	8.10263e10i		−	0.696586i
$585$		0			0
$586$	−3.18197e11			−2.69840
$587$	−6.72211e10	−	3.88101e10i	−0.566178	−	0.326883i	0.189443	−	0.981892i	$-0.439332\pi$
$587$	−6.72211e10	−	3.88101e10i	−0.566178	−	0.326883i	−0.755621	+	0.655009i	$0.772665\pi$
$588$		0			0
$589$	2.33856e10	+	4.05050e10i	0.194306	+	0.336548i
$590$	2.33175e10	−	1.34623e10i	0.192430	−	0.111100i
$591$		0			0
$592$	3.46216e9	−	5.99663e9i	0.0281877	−	0.0488226i
$593$		−	2.09190e11i		−	1.69169i	−0.533427	−	0.845846i	$-0.679096\pi$
$593$		−	2.09190e11i		−	1.69169i	0.533427	−	0.845846i	$-0.320904\pi$
$594$		0			0
$595$	−3.59618e10			−0.286928
$596$	−2.38556e11	−	1.37731e11i	−1.89063	−	1.09155i
$597$		0			0
$598$	−7.87892e10	−	1.36467e11i	−0.616116	−	1.06714i
$599$	−8.07861e10	+	4.66419e10i	−0.627523	+	0.362300i	−0.779792	−	0.626039i	$-0.784675\pi$
$599$	−8.07861e10	+	4.66419e10i	−0.627523	+	0.362300i	0.152269	+	0.988339i	$0.451342\pi$
$600$		0			0
$601$	4.06269e10	−	7.03678e10i	0.311398	−	0.539357i	−0.667267	−	0.744818i	$-0.732536\pi$
$601$	4.06269e10	−	7.03678e10i	0.311398	−	0.539357i	0.978665	+	0.205461i	$0.0658695\pi$
$602$		−	2.78620e11i		−	2.12142i
$603$		0			0
$604$	−4.90978e10			−0.368905
$605$	−7.99507e9	−	4.61596e9i	−0.0596762	−	0.0344540i
$606$		0			0
$607$	−2.06162e10	−	3.57084e10i	−0.151864	−	0.263036i	0.780049	−	0.625719i	$-0.215194\pi$
$607$	−2.06162e10	−	3.57084e10i	−0.151864	−	0.263036i	−0.931913	+	0.362683i	$0.881861\pi$
$608$	2.08321e11	−	1.20274e11i	1.52447	−	0.880151i
$609$		0			0
$610$	−7.07313e9	+	1.22510e10i	−0.0510849	+	0.0884816i
$611$			2.69361e11i			1.93273i
$612$		0			0
$613$	2.01264e10			0.142536			0.0712681	−	0.997457i	$-0.477295\pi$
$613$	2.01264e10			0.142536			0.0712681	+	0.997457i	$0.477295\pi$
$614$	−8.03459e9	−	4.63877e9i	−0.0565315	−	0.0326385i
$615$		0			0
$616$	1.12972e11	+	1.95673e11i	0.784599	+	1.35896i
$617$	1.18902e11	−	6.86484e10i	0.820446	−	0.473685i	−0.0301240	−	0.999546i	$-0.509590\pi$
$617$	1.18902e11	−	6.86484e10i	0.820446	−	0.473685i	0.850570	+	0.525861i	$0.176257\pi$
$618$		0			0
$619$	4.85200e10	−	8.40391e10i	0.330490	−	0.572425i	−0.652118	−	0.758117i	$-0.726119\pi$
$619$	4.85200e10	−	8.40391e10i	0.330490	−	0.572425i	0.982608	+	0.185692i	$0.0594527\pi$
$620$		−	1.80263e10i		−	0.121995i
$621$		0			0
$622$	1.33888e11			0.894502
$623$	2.19918e10	+	1.26970e10i	0.145985	+	0.0842845i
$624$		0			0
$625$	−5.26934e10	−	9.12677e10i	−0.345332	−	0.598132i
$626$	3.22810e11	−	1.86375e11i	2.10208	−	1.21364i
$627$		0			0
$628$	−1.50679e10	+	2.60984e10i	−0.0968758	+	0.167794i
$629$			5.46991e10i			0.349444i
$630$		0			0
$631$	−2.86142e11			−1.80495			−0.902473	−	0.430747i	$-0.858250\pi$
$631$	−2.86142e11			−1.80495			−0.902473	+	0.430747i	$0.858250\pi$
$632$	1.64232e11	+	9.48192e10i	1.02941	+	0.594330i
$633$		0			0
$634$	1.46981e11	+	2.54579e11i	0.909714	+	1.57567i
$635$	3.96983e10	−	2.29198e10i	0.244161	−	0.140967i
$636$		0			0
$637$	1.46660e11	−	2.54023e11i	0.890747	−	1.54282i
$638$		−	5.62982e11i		−	3.39791i
$639$		0			0
$640$	−8.46274e10			−0.504418
$641$	4.96090e10	+	2.86417e10i	0.293852	+	0.169655i	0.639678	−	0.768643i	$-0.279068\pi$
$641$	4.96090e10	+	2.86417e10i	0.293852	+	0.169655i	−0.345826	+	0.938299i	$0.612401\pi$
$642$		0			0
$643$	6.72328e10	+	1.16451e11i	0.393312	+	0.681236i	0.992884	−	0.119084i	$-0.0379959\pi$
$643$	6.72328e10	+	1.16451e11i	0.393312	+	0.681236i	−0.599572	+	0.800321i	$0.704663\pi$
$644$	−2.14983e11	+	1.24120e11i	−1.24986	+	0.721605i
$645$		0			0
$646$	−1.30574e11	+	2.26160e11i	−0.749765	+	1.29863i
$647$			2.98401e11i			1.70288i	0.524454	+	0.851439i	$0.324269\pi$
$647$			2.98401e11i			1.70288i	−0.524454	+	0.851439i	$0.675731\pi$
$648$		0			0
$649$	−8.26457e10			−0.465845
$650$	−2.86510e11	−	1.65417e11i	−1.60504	−	0.926671i
$651$		0			0
$652$	1.14101e11	+	1.97628e11i	0.631390	+	1.09360i
$653$	8.81880e10	−	5.09154e10i	0.485017	−	0.280025i	−0.237488	−	0.971390i	$-0.576324\pi$
$653$	8.81880e10	−	5.09154e10i	0.485017	−	0.280025i	0.722505	+	0.691366i	$0.242991\pi$
$654$		0			0
$655$	−3.62683e10	+	6.28186e10i	−0.197044	+	0.341290i
$656$		−	4.47067e9i		−	0.0241411i
$657$		0			0
$658$	6.93553e11			3.69978
$659$	−2.09636e11	−	1.21033e11i	−1.11154	−	0.641746i	−0.172310	−	0.985043i	$-0.555123\pi$
$659$	−2.09636e11	−	1.21033e11i	−1.11154	−	0.641746i	−0.939227	+	0.343296i	$0.888456\pi$
$660$		0			0
$661$	1.53401e11	+	2.65699e11i	0.803570	+	1.39182i	0.917252	+	0.398307i	$0.130402\pi$
$661$	1.53401e11	+	2.65699e11i	0.803570	+	1.39182i	−0.113682	+	0.993517i	$0.536265\pi$
$662$	−2.93704e11	+	1.69570e11i	−1.52925	+	0.882912i
$663$		0			0
$664$	6.20797e10	−	1.07525e11i	0.319358	−	0.553143i
$665$			1.61897e11i			0.827850i
$666$		0			0
$667$	2.26114e11			1.14241
$668$	−8.80800e10	−	5.08530e10i	−0.442356	−	0.255394i
$669$		0			0
$670$	1.67585e9	+	2.90265e9i	0.00831640	+	0.0144044i
$671$	3.76047e10	−	2.17111e10i	0.185504	−	0.107101i
$672$		0			0
$673$	8.97126e10	−	1.55387e11i	0.437314	−	0.757450i	−0.560168	−	0.828379i	$-0.689263\pi$
$673$	8.97126e10	−	1.55387e11i	0.437314	−	0.757450i	0.997481	+	0.0709297i	$0.0225966\pi$
$674$			3.06502e11i			1.48523i
$675$		0			0
$676$	−2.21086e11			−1.05871
$677$	−1.27199e11	−	7.34384e10i	−0.605521	−	0.349597i	0.165690	−	0.986178i	$-0.447015\pi$
$677$	−1.27199e11	−	7.34384e10i	−0.605521	−	0.349597i	−0.771210	+	0.636580i	$0.780348\pi$
$678$		0			0
$679$	−1.53302e11	−	2.65526e11i	−0.721220	−	1.24919i
$680$	3.18644e10	−	1.83969e10i	0.149029	−	0.0860417i
$681$		0			0
$682$	−4.52185e10	+	7.83207e10i	−0.209015	+	0.362025i
$683$			5.39354e10i			0.247851i	0.992291	+	0.123926i	$0.0395485\pi$
$683$			5.39354e10i			0.247851i	−0.992291	+	0.123926i	$0.960452\pi$
$684$		0			0
$685$	1.52403e10			0.0692200
$686$	−1.79369e11	−	1.03559e11i	−0.809938	−	0.467618i
$687$		0			0
$688$	−8.81501e9	−	1.52681e10i	−0.0393432	−	0.0681444i
$689$	8.16531e10	−	4.71424e10i	0.362323	−	0.209187i
$690$		0			0
$691$	−5.19287e10	+	8.99432e10i	−0.227769	+	0.394508i	−0.957147	−	0.289603i	$-0.906477\pi$
$691$	−5.19287e10	+	8.99432e10i	−0.227769	+	0.394508i	0.729377	+	0.684112i	$0.239810\pi$
$692$		−	1.13114e11i		−	0.493279i
$693$		0			0
$694$	2.35864e11			1.01677
$695$	1.47461e9	+	8.51367e8i	0.00632031	+	0.00364903i
$696$		0			0
$697$	1.76582e10	+	3.05849e10i	0.0748195	+	0.129591i
$698$	4.64530e11	−	2.68196e11i	1.95700	−	1.12988i
$699$		0			0
$700$	−2.60588e11	+	4.51352e11i	−1.08533	+	1.87985i
$701$			1.65918e11i			0.687103i	0.939134	+	0.343552i	$0.111630\pi$
$701$			1.65918e11i			0.687103i	−0.939134	+	0.343552i	$0.888370\pi$
$702$		0			0
$703$	2.46251e11			1.00822
$704$	3.81321e11	+	2.20156e11i	1.55239	+	0.896272i
$705$		0			0
$706$	−1.40915e11	−	2.44072e11i	−0.567202	−	0.982423i
$707$	−6.35606e11	+	3.66967e11i	−2.54396	+	1.46876i
$708$		0			0
$709$	7.63444e10	−	1.32232e11i	0.302129	−	0.523303i	−0.674489	−	0.738285i	$-0.735636\pi$
$709$	7.63444e10	−	1.32232e11i	0.302129	−	0.523303i	0.976618	+	0.214982i	$0.0689693\pi$
$710$		−	1.95729e11i		−	0.770232i
$711$		0			0
$712$	−2.59814e10			−0.101098
$713$	−3.14564e10	−	1.81614e10i	−0.121717	−	0.0702733i
$714$		0			0
$715$	−6.07899e10	−	1.05291e11i	−0.232599	−	0.402873i
$716$	−1.40262e11	+	8.09804e10i	−0.533689	+	0.308125i
$717$		0			0
$718$	−1.53778e11	+	2.66351e11i	−0.578623	+	1.00220i
$719$		−	2.64075e11i		−	0.988125i	−0.869427	−	0.494062i	$-0.835511\pi$
$719$		−	2.64075e11i		−	0.988125i	0.869427	−	0.494062i	$-0.164489\pi$
$720$		0			0
$721$	−7.84784e11			−2.90408
$722$	6.40435e11	+	3.69755e11i	2.35682	+	1.36071i
$723$		0			0
$724$	−2.80756e11	−	4.86284e11i	−1.02182	−	1.76985i
$725$	4.11121e11	−	2.37361e11i	1.48805	−	0.859126i
$726$		0			0
$727$	1.28330e11	−	2.22274e11i	0.459399	−	0.795703i	−0.539530	−	0.841966i	$-0.681398\pi$
$727$	1.28330e11	−	2.22274e11i	0.459399	−	0.795703i	0.998929	+	0.0462634i	$0.0147314\pi$
$728$			5.17912e11i			1.84387i
$729$		0			0
$730$	1.12262e11			0.395313
$731$	1.20611e11	+	6.96348e10i	0.422394	+	0.243869i
$732$		0			0
$733$	1.38206e11	+	2.39380e11i	0.478752	+	0.829223i	0.999703	−	0.0243635i	$-0.00775591\pi$
$733$	1.38206e11	+	2.39380e11i	0.478752	+	0.829223i	−0.520951	+	0.853587i	$0.674423\pi$
$734$	1.38367e11	−	7.98864e10i	0.476705	−	0.275226i
$735$		0			0
$736$	−9.34053e10	+	1.61783e11i	−0.318317	+	0.551342i
$737$		−	1.02881e10i		−	0.0348710i
$738$		0			0
$739$	3.69193e11			1.23787			0.618935	−	0.785442i	$-0.287564\pi$
$739$	3.69193e11			1.23787			0.618935	+	0.785442i	$0.287564\pi$
$740$	−8.21935e10	−	4.74545e10i	−0.274101	−	0.158252i
$741$		0			0
$742$	−1.21383e11	−	2.10241e11i	−0.400443	−	0.693588i
$743$	−4.52457e11	+	2.61226e11i	−1.48464	+	0.857160i	−0.999847	−	0.0174674i	$-0.994440\pi$
$743$	−4.52457e11	+	2.61226e11i	−1.48464	+	0.857160i	−0.484797	+	0.874627i	$0.661106\pi$
$744$		0			0
$745$	6.97583e10	−	1.20825e11i	0.226449	−	0.392222i
$746$			3.15947e11i			1.02014i
$747$		0			0
$748$	−3.08947e11			−0.986911
$749$	5.25090e11	+	3.03161e11i	1.66842	+	0.963265i
$750$		0			0
$751$	−1.11747e11	−	1.93552e11i	−0.351300	−	0.608469i	0.635178	−	0.772366i	$-0.280927\pi$
$751$	−1.11747e11	−	1.93552e11i	−0.351300	−	0.608469i	−0.986477	+	0.163897i	$0.947594\pi$
$752$	3.80059e10	−	2.19427e10i	0.118845	−	0.0686150i
$753$		0			0
$754$	6.45238e11	−	1.11758e12i	1.99634	−	3.45776i
$755$		−	2.48673e10i		−	0.0765315i
$756$		0			0
$757$	−2.44183e11			−0.743588			−0.371794	−	0.928315i	$-0.621257\pi$
$757$	−2.44183e11			−0.743588			−0.371794	+	0.928315i	$0.621257\pi$
$758$	5.03227e11	+	2.90538e11i	1.52436	+	0.880089i
$759$		0			0
$760$	−8.28213e10	−	1.43451e11i	−0.248249	−	0.429980i
$761$	−1.94881e11	+	1.12515e11i	−0.581073	+	0.335483i	−0.761560	−	0.648095i	$-0.775566\pi$
$761$	−1.94881e11	+	1.12515e11i	−0.581073	+	0.335483i	0.180487	+	0.983577i	$0.442233\pi$
$762$		0			0
$763$	−4.09966e11	+	7.10082e11i	−1.20962	+	2.09513i
$764$		−	6.86251e10i		−	0.201423i
$765$		0			0
$766$	−2.94922e11			−0.856628
$767$	−1.64061e11	−	9.47209e10i	−0.474051	−	0.273693i
$768$		0			0
$769$	2.41526e11	+	4.18336e11i	0.690652	+	1.19624i	0.971625	+	0.236528i	$0.0760095\pi$
$769$	2.41526e11	+	4.18336e11i	0.690652	+	1.19624i	−0.280973	+	0.959716i	$0.590657\pi$
$770$	−2.71105e11	+	1.56522e11i	−0.771212	+	0.445260i
$771$		0			0
$772$	−1.17398e11	+	2.03340e11i	−0.330516	+	0.572470i
$773$			1.32482e10i			0.0371056i	0.999828	+	0.0185528i	$0.00590588\pi$
$773$			1.32482e10i			0.0371056i	−0.999828	+	0.0185528i	$0.994094\pi$
$774$		0			0
$775$	−7.62588e10			−0.211390
$776$	2.71670e11	+	1.56849e11i	0.749194	+	0.432547i
$777$		0			0
$778$	−9.66609e9	−	1.67422e10i	−0.0263835	−	0.0456976i
$779$	1.37691e11	−	7.94957e10i	0.373899	−	0.215871i
$780$		0			0
$781$	−3.00396e11	+	5.20301e11i	−0.807403	+	1.39846i
$782$		−	2.02808e11i		−	0.542324i
$783$		0			0
$784$	−4.77889e10			−0.126492
$785$	−1.32184e10	−	7.63167e9i	−0.0348098	−	0.0200975i
$786$		0			0
$787$	−2.37300e11	−	4.11015e11i	−0.618584	−	1.07142i	−0.989744	−	0.142850i	$-0.954373\pi$
$787$	−2.37300e11	−	4.11015e11i	−0.618584	−	1.07142i	0.371161	−	0.928569i	$-0.378960\pi$
$788$	4.15992e11	−	2.40173e11i	1.07890	−	0.622901i
$789$		0			0
$790$	−1.31372e11	+	2.27543e11i	−0.337282	+	0.584190i
$791$			2.62641e11i			0.670898i
$792$		0			0
$793$	9.95330e10			0.251695
$794$	−1.26250e11	−	7.28907e10i	−0.317652	−	0.183396i
$795$		0			0
$796$	−3.80354e11	−	6.58792e11i	−0.947405	−	1.64095i
$797$	−2.58248e11	+	1.49100e11i	−0.640035	+	0.369524i	−0.784628	−	0.619967i	$-0.787146\pi$
$797$	−2.58248e11	+	1.49100e11i	−0.640035	+	0.369524i	0.144593	+	0.989491i	$0.453813\pi$
$798$		0			0
$799$	−1.73338e11	+	3.00230e11i	−0.425311	+	0.736660i
$800$			3.92206e11i			0.957533i
$801$		0			0
$802$	1.07670e12			2.60255
$803$	−2.98423e11	−	1.72295e11i	−0.717745	−	0.414390i
$804$		0			0
$805$	−6.28650e10	−	1.08885e11i	−0.149701	−	0.259290i
$806$	−1.79528e11	+	1.03650e11i	−0.425394	+	0.245602i
$807$		0			0
$808$	3.75457e11	−	6.50311e11i	0.880877	−	1.52572i
$809$		−	3.19136e11i		−	0.745044i	−0.928023	−	0.372522i	$-0.878493\pi$
$809$		−	3.19136e11i		−	0.745044i	0.928023	−	0.372522i	$-0.121507\pi$
$810$		0			0
$811$	2.94415e11			0.680575			0.340288	−	0.940321i	$-0.389476\pi$
$811$	2.94415e11			0.680575			0.340288	+	0.940321i	$0.389476\pi$
$812$	−1.76058e12	−	1.01647e12i	−4.04979	−	2.33815i
$813$		0			0
$814$	2.38076e11	+	4.12360e11i	0.542273	+	0.939244i
$815$	−1.00095e11	+	5.77902e10i	−0.226874	+	0.130986i
$816$		0			0
$817$	3.13490e11	−	5.42981e11i	0.703616	−	1.21870i
$818$		−	6.42373e11i		−	1.43474i
$819$		0			0
$820$	−6.12777e10			−0.135534
$821$	5.33500e10	+	3.08016e10i	0.117425	+	0.0677956i	0.557562	−	0.830135i	$-0.311737\pi$
$821$	5.33500e10	+	3.08016e10i	0.117425	+	0.0677956i	−0.440137	+	0.897931i	$0.645070\pi$
$822$		0			0
$823$	2.35526e11	+	4.07944e11i	0.513381	+	0.889203i	0.999880	+	0.0155212i	$0.00494074\pi$
$823$	2.35526e11	+	4.07944e11i	0.513381	+	0.889203i	−0.486498	+	0.873682i	$0.661726\pi$
$824$	6.95368e11	−	4.01471e11i	1.50836	−	0.870853i
$825$		0			0
$826$	−2.43888e11	+	4.22427e11i	−0.523927	+	0.907467i
$827$		−	4.35038e11i		−	0.930048i	−0.885298	−	0.465024i	$-0.846046\pi$
$827$		−	4.35038e11i		−	0.930048i	0.885298	−	0.465024i	$-0.153954\pi$
$828$		0			0
$829$	2.07438e11			0.439208			0.219604	−	0.975589i	$-0.429524\pi$
$829$	2.07438e11			0.439208			0.219604	+	0.975589i	$0.429524\pi$
$830$	1.48976e11	+	8.60113e10i	0.313909	+	0.181235i
$831$		0			0
$832$	5.04645e11	+	8.74071e11i	1.05316	+	1.82412i
$833$	3.26935e11	−	1.88756e11i	0.679018	−	0.392031i
$834$		0			0
$835$	2.57562e10	−	4.46111e10i	0.0529830	−	0.0917693i
$836$			1.39086e12i			2.84745i
$837$		0			0
$838$	1.29699e11			0.263002
$839$	2.06839e11	+	1.19419e11i	0.417431	+	0.241004i	0.693978	−	0.719996i	$-0.255857\pi$
$839$	2.06839e11	+	1.19419e11i	0.417431	+	0.241004i	−0.276546	+	0.961001i	$0.589190\pi$
$840$		0			0
$841$	6.75747e11	+	1.17043e12i	1.35083	+	2.33970i
$842$	−3.16521e11	+	1.82743e11i	−0.629730	+	0.363575i
$843$		0			0
$844$	−8.01268e10	+	1.38784e11i	−0.157909	+	0.273507i
$845$		−	1.11977e11i		−	0.219635i
$846$		0			0
$847$	1.67248e11			0.324959
$848$	−1.33033e10	−	7.68064e9i	−0.0257261	−	0.0148530i
$849$		0			0
$850$	−2.12896e11	−	3.68747e11i	−0.407842	−	0.706403i
$851$	−1.65618e11	+	9.56198e10i	−0.315784	+	0.182318i
$852$		0			0
$853$	1.57842e11	−	2.73391e11i	0.298144	−	0.516401i	−0.677567	−	0.735461i	$-0.736966\pi$
$853$	1.57842e11	−	2.73391e11i	0.298144	−	0.516401i	0.975711	+	0.219060i	$0.0702990\pi$
$854$		−	2.56278e11i		−	0.481815i
$855$		0			0
$856$	−6.20349e11			−1.15542
$857$	4.18908e11	+	2.41857e11i	0.776597	+	0.448369i	0.835223	−	0.549911i	$-0.185339\pi$
$857$	4.18908e11	+	2.41857e11i	0.776597	+	0.448369i	−0.0586257	+	0.998280i	$0.518672\pi$
$858$		0			0
$859$	2.84594e11	+	4.92930e11i	0.522700	+	0.905343i	0.999651	+	0.0264130i	$0.00840849\pi$
$859$	2.84594e11	+	4.92930e11i	0.522700	+	0.905343i	−0.476951	+	0.878930i	$0.658258\pi$
$860$	−2.09273e11	+	1.20824e11i	−0.382578	+	0.220882i
$861$		0			0
$862$	4.77823e11	−	8.27614e11i	0.865443	−	1.49899i
$863$		−	1.70442e11i		−	0.307279i	−0.988127	−	0.153639i	$-0.950901\pi$
$863$		−	1.70442e11i		−	0.307279i	0.988127	−	0.153639i	$-0.0490994\pi$
$864$		0			0
$865$	5.72906e10			0.102334
$866$	−8.36865e11	−	4.83164e11i	−1.48793	−	0.859060i
$867$		0			0
$868$	1.63285e11	+	2.82819e11i	0.287653	+	0.498229i
$869$	6.98445e11	−	4.03248e11i	1.22477	−	0.707119i
$870$		0			0
$871$	1.17912e10	−	2.04230e10i	0.0204874	−	0.0354852i
$872$		−	8.38903e11i		−	1.45093i
$873$		0			0
$874$	−9.13026e11			−1.56472
$875$	−4.84575e11	−	2.79769e11i	−0.826663	−	0.477274i
$876$		0			0
$877$	−3.49396e11	−	6.05172e11i	−0.590636	−	1.02301i	−0.994147	−	0.108037i	$-0.965544\pi$
$877$	−3.49396e11	−	6.05172e11i	−0.590636	−	1.02301i	0.403511	−	0.914975i	$-0.367790\pi$
$878$	−1.19331e11	+	6.88957e10i	−0.200805	+	0.115935i
$879$		0			0
$880$	−9.90414e9	+	1.71545e10i	−0.0165153	+	0.0286053i
$881$			2.05065e11i			0.340399i	0.985410	+	0.170199i	$0.0544412\pi$
$881$			2.05065e11i			0.340399i	−0.985410	+	0.170199i	$0.945559\pi$
$882$		0			0
$883$	−7.79882e11			−1.28288			−0.641440	−	0.767173i	$-0.721663\pi$
$883$	−7.79882e11			−1.28288			−0.641440	+	0.767173i	$0.721663\pi$
$884$	−6.13296e11	−	3.54087e11i	−1.00430	−	0.579830i
$885$		0			0
$886$	3.08559e11	+	5.34441e11i	0.500730	+	0.867290i
$887$	4.20319e11	−	2.42671e11i	0.679023	−	0.392034i	−0.120464	−	0.992718i	$-0.538438\pi$
$887$	4.20319e11	−	2.42671e11i	0.679023	−	0.392034i	0.799487	+	0.600684i	$0.205105\pi$
$888$		0			0
$889$	−4.15223e11	+	7.19187e11i	−0.664774	+	1.15142i
$890$		−	3.59972e10i		−	0.0573732i
$891$		0			0
$892$	7.50948e11			1.18618
$893$	1.35161e12	+	7.80353e11i	2.12543	+	1.22712i
$894$		0			0
$895$	−4.10153e10	−	7.10405e10i	−0.0639224	−	0.110717i
$896$	1.32774e12	−	7.66568e11i	2.06006	−	1.18938i
$897$		0			0
$898$	9.34847e11	−	1.61920e12i	1.43759	−	2.48998i
$899$		−	2.97462e11i		−	0.455400i
$900$		0			0
$901$	1.21347e11			0.184133
$902$	2.66239e11	+	1.53713e11i	0.402203	+	0.232212i
$903$		0			0
$904$	−1.34359e11	−	2.32716e11i	−0.201183	−	0.348460i
$905$	2.46295e11	−	1.42199e11i	0.367165	−	0.211983i
$906$		0			0
$907$	−3.68722e11	+	6.38645e11i	−0.544841	+	0.943693i	0.453776	+	0.891116i	$0.350077\pi$
$907$	−3.68722e11	+	6.38645e11i	−0.544841	+	0.943693i	−0.998617	+	0.0525767i	$0.983257\pi$
$908$		−	8.83377e11i		−	1.29958i
$909$		0			0
$910$	−7.17566e11			−1.04640
$911$	−5.44603e11	−	3.14427e11i	−0.790691	−	0.456506i	0.0495146	−	0.998773i	$-0.484233\pi$
$911$	−5.44603e11	−	3.14427e11i	−0.790691	−	0.456506i	−0.840206	+	0.542268i	$0.817566\pi$
$912$		0			0
$913$	−2.64013e11	−	4.57284e11i	−0.379964	−	0.658116i
$914$	−1.37835e12	+	7.95791e11i	−1.97504	+	1.14029i
$915$		0			0
$916$	−4.33868e11	+	7.51482e11i	−0.616277	+	1.06742i
$917$		−	1.31410e12i		−	1.85845i
$918$		0			0
$919$	3.48346e11			0.488370			0.244185	−	0.969729i	$-0.421480\pi$
$919$	3.48346e11			0.488370			0.244185	+	0.969729i	$0.421480\pi$
$920$	1.11405e11	+	6.43194e10i	0.155508	+	0.0897823i
$921$		0			0
$922$	1.54621e11	+	2.67811e11i	0.213966	+	0.370600i
$923$	−1.19264e12	+	6.88573e11i	−1.64325	+	0.948731i
$924$		0			0
$925$	−2.00752e11	+	3.47713e11i	−0.274216	+	0.474956i
$926$			1.52190e11i			0.206987i
$927$		0			0
$928$	−1.52987e12			−2.06283
$929$	−5.27095e8	−	3.04319e8i	−0.000707663	−	0.000408569i	0.499646	−	0.866230i	$-0.333463\pi$
$929$	−5.27095e8	−	3.04319e8i	−0.000707663	−	0.000408569i	−0.500354	+	0.865821i	$0.666797\pi$
$930$		0			0
$931$	−8.49763e11	−	1.47183e12i	−1.13110	−	1.95912i
$932$	−1.19271e12	+	6.88609e11i	−1.58077	+	0.912660i
$933$		0			0
$934$	7.45749e11	−	1.29168e12i	0.979953	−	1.69733i
$935$		−	1.56477e11i		−	0.204740i
$936$		0			0
$937$	5.98470e10			0.0776398			0.0388199	−	0.999246i	$-0.487640\pi$
$937$	5.98470e10			0.0776398			0.0388199	+	0.999246i	$0.487640\pi$
$938$	−5.25854e10	−	3.03602e10i	−0.0679288	−	0.0392187i
$939$		0			0
$940$	−3.00760e11	−	5.20932e11i	−0.385220	−	0.667221i
$941$	1.12529e12	−	6.49688e11i	1.43518	−	0.828603i	0.437673	−	0.899134i	$-0.355803\pi$
$941$	1.12529e12	−	6.49688e11i	1.43518	−	0.828603i	0.997510	+	0.0705309i	$0.0224694\pi$
$942$		0			0
$943$	−6.17367e10	+	1.06931e11i	−0.0780723	+	0.135225i
$944$			3.08646e10i			0.0388663i
$945$		0			0
$946$	1.21233e12			1.51376
$947$	−1.13710e12	−	6.56502e11i	−1.41383	−	0.816275i	−0.418083	−	0.908409i	$-0.637298\pi$
$947$	−1.13710e12	−	6.56502e11i	−1.41383	−	0.816275i	−0.995747	+	0.0921343i	$0.970631\pi$
$948$		0			0
$949$	−3.94937e11	−	6.84050e11i	−0.486925	−	0.843380i
$950$	−1.66007e12	+	9.58440e11i	−2.03813	+	1.17671i
$951$		0			0
$952$	−3.33284e11	+	5.77265e11i	−0.405758	+	0.702793i
$953$			1.49915e12i			1.81750i	0.417341	+	0.908750i	$0.362962\pi$
$953$			1.49915e12i			1.81750i	−0.417341	+	0.908750i	$0.637038\pi$
$954$		0			0
$955$	3.47575e10			0.0417864
$956$	1.14816e12	+	6.62892e11i	1.37459	+	0.793618i
$957$		0			0
$958$	−5.92408e11	−	1.02608e12i	−0.703330	−	1.21820i
$959$	−2.39108e11	+	1.38049e11i	−0.282696	+	0.163215i
$960$		0			0
$961$	4.02554e11	−	6.97243e11i	0.471987	−	0.817506i
$962$			1.09144e12i			1.27438i
$963$		0			0
$964$	−1.62808e12			−1.88524
$965$	−1.02988e11	−	5.94604e10i	−0.118762	−	0.0685675i
$966$		0			0
$967$	1.29701e11	+	2.24648e11i	0.148332	+	0.256919i	0.930611	−	0.366009i	$-0.119276\pi$
$967$	1.29701e11	+	2.24648e11i	0.148332	+	0.256919i	−0.782279	+	0.622928i	$0.785943\pi$
$968$	−1.48192e11	+	8.55589e10i	−0.168781	+	0.0974459i
$969$		0			0
$970$	−2.17313e11	+	3.76398e11i	−0.245471	+	0.425168i
$971$		−	1.15804e12i		−	1.30271i	−0.758772	−	0.651356i	$-0.774200\pi$
$971$		−	1.15804e12i		−	1.30271i	0.758772	−	0.651356i	$-0.225800\pi$
$972$		0			0
$973$	−3.08473e10			−0.0344164
$974$	1.90335e12	+	1.09890e12i	2.11487	+	1.22102i
$975$		0			0
$976$	−8.10816e9	−	1.40437e10i	−0.00893559	−	0.0154769i
$977$	−7.79343e10	+	4.49954e10i	−0.0855363	+	0.0493844i	−0.542158	−	0.840277i	$-0.682393\pi$
$977$	−7.79343e10	+	4.49954e10i	−0.0855363	+	0.0493844i	0.456622	+	0.889661i	$0.349059\pi$
$978$		0			0
$979$	−5.52470e10	+	9.56907e10i	−0.0601420	+	0.104169i
$980$			6.55024e11i			0.710155i
$981$		0			0
$982$	7.65356e11			0.823033
$983$	−1.42514e12	−	8.22805e11i	−1.52631	−	0.881217i	−0.999512	−	0.0312278i	$-0.990058\pi$
$983$	−1.42514e12	−	8.22805e11i	−1.52631	−	0.881217i	−0.526800	−	0.849989i	$-0.676608\pi$
$984$		0			0
$985$	1.21644e11	+	2.10693e11i	0.129224	+	0.223823i
$986$	1.43837e12	−	8.30441e11i	1.52181	−	0.878620i
$987$		0			0
$988$	−1.59407e12	+	2.76101e12i	−1.67294	+	2.89761i
$989$			4.86916e11i			0.508943i
$990$		0			0
$991$	6.39028e11			0.662560			0.331280	−	0.943533i	$-0.392520\pi$
$991$	6.39028e11			0.662560			0.331280	+	0.943533i	$0.392520\pi$
$992$	2.12832e11	+	1.22879e11i	0.219781	+	0.126891i
$993$		0			0
$994$	1.77294e12	+	3.07083e12i	1.81614	+	3.14564i
$995$	3.33668e11	−	1.92643e11i	0.340426	−	0.196545i
$996$		0			0
$997$	4.46173e11	−	7.72795e11i	0.451568	−	0.782139i	−0.546916	−	0.837188i	$-0.684198\pi$
$997$	4.46173e11	−	7.72795e11i	0.451568	−	0.782139i	0.998484	+	0.0550490i	$0.0175315\pi$
$998$			1.32136e12i			1.33199i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.9.d.g.26.15		32
3.2	odd	2	inner	81.9.d.g.26.2		32
9.2	odd	6		81.9.b.b.80.15	yes	16
9.4	even	3	inner	81.9.d.g.53.2		32
9.5	odd	6	inner	81.9.d.g.53.15		32
9.7	even	3		81.9.b.b.80.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.9.b.b.80.2	✓	16	9.7	even	3
81.9.b.b.80.15	yes	16	9.2	odd	6
81.9.d.g.26.2		32	3.2	odd	2	inner
81.9.d.g.26.15		32	1.1	even	1	trivial
81.9.d.g.53.2		32	9.4	even	3	inner
81.9.d.g.53.15		32	9.5	odd	6	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(22.2403 + 12.8404i) q^{2} +(201.753 + 349.447i) q^{4} +(-176.989 + 102.185i) q^{5} +(1851.21 - 3206.39i) q^{7} +3788.08i q^{8} -5248.38 q^{10} +(13951.7 + 8054.99i) q^{11} +(18463.8 + 31980.2i) q^{13} +(82342.9 - 47540.7i) q^{14} +(3008.19 - 5210.35i) q^{16} +47526.9i q^{17} +213962. q^{19} +(-71416.2 - 41232.2i) q^{20} +(206859. + 358290. i) q^{22} +(-143902. + 83082.0i) q^{23} +(-174429. + 302120. i) q^{25} +948331. i q^{26} +1.49395e6 q^{28} +(-1.17848e6 - 680393. i) q^{29} +(109298. + 189309. i) q^{31} +(973633. - 562128. i) q^{32} +(-610265. + 1.05701e6i) q^{34} +756662. i q^{35} +1.15091e6 q^{37} +(4.75857e6 + 2.74736e6i) q^{38} +(-387084. - 670449. i) q^{40} +(643528. - 371541. i) q^{41} +(1.46517e6 - 2.53774e6i) q^{43} +6.50048e6i q^{44} -4.26723e6 q^{46} +(6.31706e6 + 3.64716e6i) q^{47} +(-3.97156e6 - 6.87895e6i) q^{49} +(-7.75870e6 + 4.47949e6i) q^{50} +(-7.45025e6 + 1.29042e7i) q^{52} -2.55324e6i q^{53} -3.29239e6 q^{55} +(1.21461e7 + 7.01254e6i) q^{56} +(-1.74731e7 - 3.02643e7i) q^{58} +(-4.44279e6 + 2.56505e6i) q^{59} +(1.34768e6 - 2.33425e6i) q^{61} +5.61372e6i q^{62} +2.73316e7 q^{64} +(-6.53578e6 - 3.77343e6i) q^{65} +(-319308. - 553057. i) q^{67} +(-1.66081e7 + 9.58869e6i) q^{68} +(-9.71586e6 + 1.68284e7i) q^{70} +3.72932e7i q^{71} -2.13898e7 q^{73} +(2.55965e7 + 1.47782e7i) q^{74} +(4.31675e7 + 7.47683e7i) q^{76} +(5.16549e7 - 2.98230e7i) q^{77} +(2.50309e7 - 4.33548e7i) q^{79} +1.22957e6i q^{80} +1.90830e7 q^{82} +(-2.83851e7 - 1.63882e7i) q^{83} +(-4.85652e6 - 8.41174e6i) q^{85} +(6.51714e7 - 3.76267e7i) q^{86} +(-3.05130e7 + 5.28500e7i) q^{88} +6.85873e6i q^{89} +1.36721e8 q^{91} +(-5.80654e7 - 3.35241e7i) q^{92} +(9.36621e7 + 1.62228e8i) q^{94} +(-3.78689e7 + 2.18636e7i) q^{95} +(4.14058e7 - 7.17170e7i) q^{97} -2.03986e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40}+ \cdots + 133878688 q^{97}+O(q^{100}) \) 32 * q + 2048 * q^4 - 3692 * q^7 + 21504 * q^10 - 63860 * q^13 - 95116 * q^16 + 370216 * q^19 + 691980 * q^22 + 541712 * q^25 - 1994264 * q^28 - 571136 * q^31 - 1027656 * q^34 + 8708536 * q^37 + 2973768 * q^40 - 5453084 * q^43 - 8468088 * q^46 - 14602560 * q^49 + 25384960 * q^52 - 28905144 * q^55 - 1213068 * q^58 - 31037996 * q^61 + 48506000 * q^64 + 62356828 * q^67 - 11075124 * q^70 - 42547328 * q^73 - 69593876 * q^76 + 151045036 * q^79 - 80402280 * q^82 - 160995564 * q^85 - 366976068 * q^88 + 747361592 * q^91 + 544741584 * q^94 + 133878688 * q^97

Introduction

L-functions

Modular forms

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Database

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