LMFDB - Embedded newform 45.14.f.a.8.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,14,Mod(8,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.8"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	45.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$48.2539180284$
Analytic rank:	$0$
Dimension:	$52$
Relative dimension:	$26$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.2
Character	$\chi$	$=$	45.8
Dual form			45.14.f.a.17.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-120.296 - 120.296i) q^{2} +20750.4i q^{4} +(33048.8 + 11335.0i) q^{5} +(306864. - 306864. i) q^{7} +(1.51073e6 - 1.51073e6i) q^{8} +(-2.61208e6 - 5.33921e6i) q^{10} +4.98599e6i q^{11} +(-8.44751e6 - 8.44751e6i) q^{13} -7.38292e7 q^{14} -1.93484e8 q^{16} +(9.51433e7 + 9.51433e7i) q^{17} -2.88041e8i q^{19} +(-2.35207e8 + 6.85776e8i) q^{20} +(5.99796e8 - 5.99796e8i) q^{22} +(4.76777e8 - 4.76777e8i) q^{23} +(9.63737e8 + 7.49217e8i) q^{25} +2.03241e9i q^{26} +(6.36755e9 + 6.36755e9i) q^{28} -2.75947e9 q^{29} +5.75777e9 q^{31} +(1.08995e10 + 1.08995e10i) q^{32} -2.28908e10i q^{34} +(1.36198e10 - 6.66316e9i) q^{35} +(-1.26910e10 + 1.26910e10i) q^{37} +(-3.46503e10 + 3.46503e10i) q^{38} +(6.70520e10 - 3.28036e10i) q^{40} +3.49369e10i q^{41} +(1.01073e10 + 1.01073e10i) q^{43} -1.03461e11 q^{44} -1.14709e11 q^{46} +(7.14406e10 + 7.14406e10i) q^{47} -9.14417e10i q^{49} +(-2.58060e10 - 2.06062e11i) q^{50} +(1.75290e11 - 1.75290e11i) q^{52} +(9.55368e10 - 9.55368e10i) q^{53} +(-5.65163e10 + 1.64781e11i) q^{55} -9.27178e11i q^{56} +(3.31954e11 + 3.31954e11i) q^{58} +3.45406e10 q^{59} -3.52010e11 q^{61} +(-6.92639e11 - 6.92639e11i) q^{62} -1.03731e12i q^{64} +(-1.83427e11 - 3.74933e11i) q^{65} +(9.27329e11 - 9.27329e11i) q^{67} +(-1.97426e12 + 1.97426e12i) q^{68} +(-2.43996e12 - 8.36856e11i) q^{70} -2.71549e11i q^{71} +(-8.54773e11 - 8.54773e11i) q^{73} +3.05337e12 q^{74} +5.97697e12 q^{76} +(1.53002e12 + 1.53002e12i) q^{77} +4.01411e11i q^{79} +(-6.39440e12 - 2.19314e12i) q^{80} +(4.20278e12 - 4.20278e12i) q^{82} +(-2.65734e12 + 2.65734e12i) q^{83} +(2.06591e12 + 4.22282e12i) q^{85} -2.43174e12i q^{86} +(7.53249e12 + 7.53249e12i) q^{88} +7.50253e12 q^{89} -5.18447e12 q^{91} +(9.89332e12 + 9.89332e12i) q^{92} -1.71881e13i q^{94} +(3.26495e12 - 9.51939e12i) q^{95} +(2.18614e11 - 2.18614e11i) q^{97} +(-1.10001e13 + 1.10001e13i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q - 325928 q^{7} - 1271408 q^{10} - 14021276 q^{13} - 1225161608 q^{16} + 2938227248 q^{22} + 2536151056 q^{25} + 1169060768 q^{28} + 16189871792 q^{31} - 37236688556 q^{37} - 55693852632 q^{40} + 75180385888 q^{43}+ \cdots + 8822313963316 q^{97}+O(q^{100}) $ 52 * q - 325928 * q^7 - 1271408 * q^10 - 14021276 * q^13 - 1225161608 * q^16 + 2938227248 * q^22 + 2536151056 * q^25 + 1169060768 * q^28 + 16189871792 * q^31 - 37236688556 * q^37 - 55693852632 * q^40 + 75180385888 * q^43 - 154908345088 * q^46 + 1397910700024 * q^52 - 495232924312 * q^55 + 2026205726832 * q^58 + 394585459616 * q^61 + 4547875952656 * q^67 - 13248903926736 * q^70 + 12810003310564 * q^73 - 10698307157280 * q^76 + 37147387647776 * q^82 - 18754575936752 * q^85 + 50209944877536 * q^88 - 44737049270608 * q^91 + 8822313963316 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−120.296	−	120.296i	−1.32910	−	1.32910i	−0.906156	−	0.422943i	$-0.860997\pi$
$2$	−120.296	−	120.296i	−1.32910	−	1.32910i	−0.422943	−	0.906156i	$-0.639003\pi$
$3$		0			0
$3$		0			0
$4$			20750.4i			2.53301i
$5$	33048.8	+	11335.0i	0.945911	+	0.324427i
$5$	33048.8	+	11335.0i	0.945911	+	0.324427i
$6$		0			0
$7$	306864.	−	306864.i	0.985844	−	0.985844i	−0.0140569	−	0.999901i	$-0.504475\pi$
$7$	306864.	−	306864.i	0.985844	−	0.985844i	0.999901	+	0.0140569i	$0.00447458\pi$
$8$	1.51073e6	−	1.51073e6i	2.03752	−	2.03752i
$9$		0			0
$10$	−2.61208e6	−	5.33921e6i	−0.826013	−	1.68841i
$11$			4.98599e6i			0.848591i	0.905524	+	0.424296i	$0.139478\pi$
$11$			4.98599e6i			0.848591i	−0.905524	+	0.424296i	$0.860522\pi$
$12$		0			0
$13$	−8.44751e6	−	8.44751e6i	−0.485397	−	0.485397i	0.421453	−	0.906850i	$-0.361520\pi$
$13$	−8.44751e6	−	8.44751e6i	−0.485397	−	0.485397i	−0.906850	+	0.421453i	$0.861520\pi$
$14$	−7.38292e7			−2.62057
$15$		0			0
$16$	−1.93484e8			−2.88313
$17$	9.51433e7	+	9.51433e7i	0.956005	+	0.956005i	0.999072	−	0.0430675i	$-0.0137131\pi$
$17$	9.51433e7	+	9.51433e7i	0.956005	+	0.956005i	−0.0430675	+	0.999072i	$0.513713\pi$
$18$		0			0
$19$		−	2.88041e8i		−	1.40461i	−0.711876	−	0.702305i	$-0.752154\pi$
$19$		−	2.88041e8i		−	1.40461i	0.711876	−	0.702305i	$-0.247846\pi$
$20$	−2.35207e8	+	6.85776e8i	−0.821778	+	2.39600i
$21$		0			0
$22$	5.99796e8	−	5.99796e8i	1.12786	−	1.12786i
$23$	4.76777e8	−	4.76777e8i	0.671559	−	0.671559i	−0.286516	−	0.958075i	$-0.592497\pi$
$23$	4.76777e8	−	4.76777e8i	0.671559	−	0.671559i	0.958075	+	0.286516i	$0.0924973\pi$
$24$		0			0
$25$	9.63737e8	+	7.49217e8i	0.789494	+	0.613759i
$26$			2.03241e9i			1.29028i
$27$		0			0
$28$	6.36755e9	+	6.36755e9i	2.49715	+	2.49715i
$29$	−2.75947e9			−0.861467			−0.430734	−	0.902479i	$-0.641745\pi$
$29$	−2.75947e9			−0.861467			−0.430734	+	0.902479i	$0.641745\pi$
$30$		0			0
$31$	5.75777e9			1.16521			0.582604	−	0.812756i	$-0.302034\pi$
$31$	5.75777e9			1.16521			0.582604	+	0.812756i	$0.302034\pi$
$32$	1.08995e10	+	1.08995e10i	1.79445	+	1.79445i
$33$		0			0
$34$		−	2.28908e10i		−	2.54125i
$35$	1.36198e10	−	6.66316e9i	1.25236	−	0.612686i
$36$		0			0
$37$	−1.26910e10	+	1.26910e10i	−0.813179	+	0.813179i	−0.985109	−	0.171930i	$-0.945000\pi$
$37$	−1.26910e10	+	1.26910e10i	−0.813179	+	0.813179i	0.171930	+	0.985109i	$0.445000\pi$
$38$	−3.46503e10	+	3.46503e10i	−1.86687	+	1.86687i
$39$		0			0
$40$	6.70520e10	−	3.28036e10i	2.58834	−	1.26629i
$41$			3.49369e10i			1.14865i	0.818626	+	0.574327i	$0.194736\pi$
$41$			3.49369e10i			1.14865i	−0.818626	+	0.574327i	$0.805264\pi$
$42$		0			0
$43$	1.01073e10	+	1.01073e10i	0.243831	+	0.243831i	0.818433	−	0.574602i	$-0.194843\pi$
$43$	1.01073e10	+	1.01073e10i	0.243831	+	0.243831i	−0.574602	+	0.818433i	$0.694843\pi$
$44$	−1.03461e11			−2.14949
$45$		0			0
$46$	−1.14709e11			−1.78514
$47$	7.14406e10	+	7.14406e10i	0.966739	+	0.966739i	0.999464	−	0.0327256i	$-0.0104187\pi$
$47$	7.14406e10	+	7.14406e10i	0.966739	+	0.966739i	−0.0327256	+	0.999464i	$0.510419\pi$
$48$		0			0
$49$		−	9.14417e10i		−	0.943778i
$50$	−2.58060e10	−	2.06062e11i	−0.233569	−	1.86506i
$51$		0			0
$52$	1.75290e11	−	1.75290e11i	1.22952	−	1.22952i
$53$	9.55368e10	−	9.55368e10i	0.592076	−	0.592076i	−0.346115	−	0.938192i	$-0.612499\pi$
$53$	9.55368e10	−	9.55368e10i	0.592076	−	0.592076i	0.938192	+	0.346115i	$0.112499\pi$
$54$		0			0
$55$	−5.65163e10	+	1.64781e11i	−0.275306	+	0.802692i
$56$		−	9.27178e11i		−	4.01736i
$57$		0			0
$58$	3.31954e11	+	3.31954e11i	1.14498	+	1.14498i
$59$	3.45406e10			0.106608			0.0533042	−	0.998578i	$-0.483025\pi$
$59$	3.45406e10			0.106608			0.0533042	+	0.998578i	$0.483025\pi$
$60$		0			0
$61$	−3.52010e11			−0.874805			−0.437403	−	0.899266i	$-0.644102\pi$
$61$	−3.52010e11			−0.874805			−0.437403	+	0.899266i	$0.644102\pi$
$62$	−6.92639e11	−	6.92639e11i	−1.54868	−	1.54868i
$63$		0			0
$64$		−	1.03731e12i		−	1.88686i
$65$	−1.83427e11	−	3.74933e11i	−0.301666	−	0.616618i
$66$		0			0
$67$	9.27329e11	−	9.27329e11i	1.25241	−	1.25241i	0.297778	−	0.954635i	$-0.403754\pi$
$67$	9.27329e11	−	9.27329e11i	1.25241	−	1.25241i	0.954635	−	0.297778i	$-0.0962456\pi$
$68$	−1.97426e12	+	1.97426e12i	−2.42157	+	2.42157i
$69$		0			0
$70$	−2.43996e12	−	8.36856e11i	−2.47883	−	0.850185i
$71$		−	2.71549e11i		−	0.251576i	−0.992057	−	0.125788i	$-0.959854\pi$
$71$		−	2.71549e11i		−	0.251576i	0.992057	−	0.125788i	$-0.0401459\pi$
$72$		0			0
$73$	−8.54773e11	−	8.54773e11i	−0.661077	−	0.661077i	0.294557	−	0.955634i	$-0.404828\pi$
$73$	−8.54773e11	−	8.54773e11i	−0.661077	−	0.661077i	−0.955634	+	0.294557i	$0.904828\pi$
$74$	3.05337e12			2.16159
$75$		0			0
$76$	5.97697e12			3.55789
$77$	1.53002e12	+	1.53002e12i	0.836579	+	0.836579i
$78$		0			0
$79$			4.01411e11i			0.185786i	0.995676	+	0.0928931i	$0.0296115\pi$
$79$			4.01411e11i			0.185786i	−0.995676	+	0.0928931i	$0.970389\pi$
$80$	−6.39440e12	−	2.19314e12i	−2.72719	−	0.935367i
$81$		0			0
$82$	4.20278e12	−	4.20278e12i	1.52668	−	1.52668i
$83$	−2.65734e12	+	2.65734e12i	−0.892152	+	0.892152i	−0.994725	−	0.102574i	$-0.967292\pi$
$83$	−2.65734e12	+	2.65734e12i	−0.892152	+	0.892152i	0.102574	+	0.994725i	$0.467292\pi$
$84$		0			0
$85$	2.06591e12	+	4.22282e12i	0.594141	+	1.21445i
$86$		−	2.43174e12i		−	0.648152i
$87$		0			0
$88$	7.53249e12	+	7.53249e12i	1.72903	+	1.72903i
$89$	7.50253e12			1.60019			0.800097	−	0.599871i	$-0.204782\pi$
$89$	7.50253e12			1.60019			0.800097	+	0.599871i	$0.204782\pi$
$90$		0			0
$91$	−5.18447e12			−0.957052
$92$	9.89332e12	+	9.89332e12i	1.70107	+	1.70107i
$93$		0			0
$94$		−	1.71881e13i		−	2.56978i
$95$	3.26495e12	−	9.51939e12i	0.455694	−	1.32864i
$96$		0			0
$97$	2.18614e11	−	2.18614e11i	0.0266478	−	0.0266478i	−0.693657	−	0.720305i	$-0.744002\pi$
$97$	2.18614e11	−	2.18614e11i	0.0266478	−	0.0266478i	0.720305	+	0.693657i	$0.244002\pi$
$98$	−1.10001e13	+	1.10001e13i	−1.25438	+	1.25438i
$99$		0			0
$100$	−1.55466e13	+	1.99980e13i	−1.55466	+	1.99980i
$101$			3.96282e12i			0.371463i	0.982601	+	0.185731i	$0.0594654\pi$
$101$			3.96282e12i			0.371463i	−0.982601	+	0.185731i	$0.940535\pi$
$102$		0			0
$103$	−3.43968e12	−	3.43968e12i	−0.283842	−	0.283842i	0.550797	−	0.834639i	$-0.314324\pi$
$103$	−3.43968e12	−	3.43968e12i	−0.283842	−	0.283842i	−0.834639	+	0.550797i	$0.814324\pi$
$104$	−2.55239e13			−1.97802
$105$		0			0
$106$	−2.29855e13			−1.57386
$107$	−9.91100e12	−	9.91100e12i	−0.638444	−	0.638444i	0.311727	−	0.950172i	$-0.399092\pi$
$107$	−9.91100e12	−	9.91100e12i	−0.638444	−	0.638444i	−0.950172	+	0.311727i	$0.899092\pi$
$108$		0			0
$109$		−	2.46980e13i		−	1.41055i	−0.708933	−	0.705276i	$-0.750823\pi$
$109$		−	2.46980e13i		−	1.41055i	0.708933	−	0.705276i	$-0.249177\pi$
$110$	2.66212e13	−	1.30238e13i	1.43277	−	0.700948i
$111$		0			0
$112$	−5.93732e13	+	5.93732e13i	−2.84232	+	2.84232i
$113$	1.20812e13	−	1.20812e13i	0.545883	−	0.545883i	−0.379364	−	0.925247i	$-0.623857\pi$
$113$	1.20812e13	−	1.20812e13i	0.545883	−	0.545883i	0.925247	+	0.379364i	$0.123857\pi$
$114$		0			0
$115$	2.11612e13	−	1.03526e13i	0.853107	−	0.417363i
$116$		−	5.72602e13i		−	2.18211i
$117$		0			0
$118$	−4.15511e12	−	4.15511e12i	−0.141693	−	0.141693i
$119$	5.83920e13			1.88494
$120$		0			0
$121$	9.66265e12			0.279893
$122$	4.23456e13	+	4.23456e13i	1.16270	+	1.16270i
$123$		0			0
$124$			1.19476e14i			2.95148i
$125$	2.33579e13	+	3.56847e13i	0.547670	+	0.836694i
$126$		0			0
$127$	3.34357e13	−	3.34357e13i	0.707109	−	0.707109i	−0.258817	−	0.965926i	$-0.583333\pi$
$127$	3.34357e13	−	3.34357e13i	0.707109	−	0.707109i	0.965926	+	0.258817i	$0.0833326\pi$
$128$	−3.54967e13	+	3.54967e13i	−0.713383	+	0.713383i
$129$		0			0
$130$	−2.30374e13	+	6.71686e13i	−0.418603	+	1.22049i
$131$		−	2.81783e13i		−	0.487138i	−0.969884	−	0.243569i	$-0.921682\pi$
$131$		−	2.81783e13i		−	0.487138i	0.969884	−	0.243569i	$-0.0783182\pi$
$132$		0			0
$133$	−8.83893e13	−	8.83893e13i	−1.38473	−	1.38473i
$134$	−2.23109e14			−3.32916
$135$		0			0
$136$	2.87472e14			3.89576
$137$	4.83509e13	+	4.83509e13i	0.624771	+	0.624771i	0.946748	−	0.321976i	$-0.104347\pi$
$137$	4.83509e13	+	4.83509e13i	0.624771	+	0.624771i	−0.321976	+	0.946748i	$0.604347\pi$
$138$		0			0
$139$		−	2.18903e13i		−	0.257428i	−0.991682	−	0.128714i	$-0.958915\pi$
$139$		−	2.18903e13i		−	0.257428i	0.991682	−	0.128714i	$-0.0410849\pi$
$140$	1.38263e14	+	2.82616e14i	1.55194	+	3.17223i
$141$		0			0
$142$	−3.26664e13	+	3.26664e13i	−0.334370	+	0.334370i
$143$	4.21192e13	−	4.21192e13i	0.411904	−	0.411904i
$144$		0			0
$145$	−9.11971e13	−	3.12787e13i	−0.814871	−	0.279484i
$146$			2.05652e14i			1.75727i
$147$		0			0
$148$	−2.63345e14	−	2.63345e14i	−2.05979	−	2.05979i
$149$	1.42089e14			1.06378			0.531888	−	0.846814i	$-0.321483\pi$
$149$	1.42089e14			1.06378			0.531888	+	0.846814i	$0.321483\pi$
$150$		0			0
$151$	2.67833e14			1.83871			0.919355	−	0.393429i	$-0.128711\pi$
$151$	2.67833e14			1.83871			0.919355	+	0.393429i	$0.128711\pi$
$152$	−4.35153e14	−	4.35153e14i	−2.86193	−	2.86193i
$153$		0			0
$154$		−	3.68111e14i		−	2.22379i
$155$	1.90287e14	+	6.52645e13i	1.10218	+	0.378025i
$156$		0			0
$157$	−1.46838e14	+	1.46838e14i	−0.782513	+	0.782513i	−0.980254	−	0.197741i	$-0.936640\pi$
$157$	−1.46838e14	+	1.46838e14i	−0.782513	+	0.782513i	0.197741	+	0.980254i	$0.436640\pi$
$158$	4.82883e13	−	4.82883e13i	0.246928	−	0.246928i
$159$		0			0
$160$	2.36668e14	+	4.83760e14i	1.11522	+	2.27955i
$161$		−	2.92611e14i		−	1.32410i
$162$		0			0
$163$	7.88309e13	+	7.88309e13i	0.329213	+	0.329213i	0.852287	−	0.523074i	$-0.175215\pi$
$163$	7.88309e13	+	7.88309e13i	0.329213	+	0.329213i	−0.523074	+	0.852287i	$0.675215\pi$
$164$	−7.24955e14			−2.90955
$165$		0			0
$166$	6.39336e14			2.37152
$167$	1.40854e14	+	1.40854e14i	0.502473	+	0.502473i	0.912206	−	0.409732i	$-0.134378\pi$
$167$	1.40854e14	+	1.40854e14i	0.502473	+	0.502473i	−0.409732	+	0.912206i	$0.634378\pi$
$168$		0			0
$169$		−	1.60154e14i		−	0.528779i
$170$	2.59468e14	−	7.56512e14i	0.824451	−	2.40380i
$171$		0			0
$172$	−2.09731e14	+	2.09731e14i	−0.617628	+	0.617628i
$173$	2.44231e14	−	2.44231e14i	0.692629	−	0.692629i	−0.270181	−	0.962810i	$-0.587083\pi$
$173$	2.44231e14	−	2.44231e14i	0.692629	−	0.692629i	0.962810	+	0.270181i	$0.0870834\pi$
$174$		0			0
$175$	5.25644e14	−	6.58285e13i	1.38339	−	0.173247i
$176$		−	9.64708e14i		−	2.44660i
$177$		0			0
$178$	−9.02526e14	−	9.02526e14i	−2.12682	−	2.12682i
$179$	−2.48826e14			−0.565395			−0.282697	−	0.959209i	$-0.591229\pi$
$179$	−2.48826e14			−0.565395			−0.282697	+	0.959209i	$0.591229\pi$
$180$		0			0
$181$	−6.48348e14			−1.37056			−0.685279	−	0.728280i	$-0.740320\pi$
$181$	−6.48348e14			−1.37056			−0.685279	+	0.728280i	$0.740320\pi$
$182$	6.23673e14	+	6.23673e14i	1.27202	+	1.27202i
$183$		0			0
$184$		−	1.44056e15i		−	2.73663i
$185$	−5.63276e14	+	2.75570e14i	−1.03301	+	0.505377i
$186$		0			0
$187$	−4.74383e14	+	4.74383e14i	−0.811257	+	0.811257i
$188$	−1.48242e15	+	1.48242e15i	−2.44876	+	2.44876i
$189$		0			0
$190$	−1.53791e15	+	7.52387e14i	−2.37155	+	1.16023i
$191$		−	5.00359e14i		−	0.745702i	−0.927891	−	0.372851i	$-0.878380\pi$
$191$		−	5.00359e14i		−	0.745702i	0.927891	−	0.372851i	$-0.121620\pi$
$192$		0			0
$193$	6.90126e14	+	6.90126e14i	0.961183	+	0.961183i	0.999274	−	0.0380916i	$-0.0121279\pi$
$193$	6.90126e14	+	6.90126e14i	0.961183	+	0.961183i	−0.0380916	+	0.999274i	$0.512128\pi$
$194$	−5.25969e13			−0.0708351
$195$		0			0
$196$	1.89745e15			2.39060
$197$	−2.41177e14	−	2.41177e14i	−0.293971	−	0.293971i	0.544676	−	0.838647i	$-0.316653\pi$
$197$	−2.41177e14	−	2.41177e14i	−0.293971	−	0.293971i	−0.838647	+	0.544676i	$0.816653\pi$
$198$		0			0
$199$		−	7.76308e14i		−	0.886112i	−0.896494	−	0.443056i	$-0.853894\pi$
$199$		−	7.76308e14i		−	0.886112i	0.896494	−	0.443056i	$-0.146106\pi$
$200$	2.58782e15	−	3.24083e14i	2.85916	−	0.358064i
$201$		0			0
$202$	4.76713e14	−	4.76713e14i	0.493711	−	0.493711i
$203$	−8.46782e14	+	8.46782e14i	−0.849273	+	0.849273i
$204$		0			0
$205$	−3.96011e14	+	1.15462e15i	−0.372655	+	1.08652i
$206$			8.27562e14i			0.754507i
$207$		0			0
$208$	1.63446e15	+	1.63446e15i	1.39946	+	1.39946i
$209$	1.43617e15			1.19194
$210$		0			0
$211$	1.31315e15			1.02442			0.512211	−	0.858860i	$-0.328827\pi$
$211$	1.31315e15			1.02442			0.512211	+	0.858860i	$0.328827\pi$
$212$	1.98243e15	+	1.98243e15i	1.49974	+	1.49974i
$213$		0			0
$214$			2.38451e15i			1.69711i
$215$	2.19467e14	+	4.48600e14i	0.151537	+	0.309748i
$216$		0			0
$217$	1.76685e15	−	1.76685e15i	1.14871	−	1.14871i
$218$	−2.97108e15	+	2.97108e15i	−1.87476	+	1.87476i
$219$		0			0
$220$	−3.41927e15	−	1.17274e15i	−2.03323	−	0.697354i
$221$		−	1.60745e15i		−	0.928084i
$222$		0			0
$223$	1.32367e15	+	1.32367e15i	0.720771	+	0.720771i	0.968762	−	0.247991i	$-0.0797705\pi$
$223$	1.32367e15	+	1.32367e15i	0.720771	+	0.720771i	−0.247991	+	0.968762i	$0.579770\pi$
$224$	6.68931e15			3.53809
$225$		0			0
$226$	−2.90665e15			−1.45107
$227$	−1.57648e15	−	1.57648e15i	−0.764754	−	0.764754i	0.212424	−	0.977178i	$-0.431864\pi$
$227$	−1.57648e15	−	1.57648e15i	−0.764754	−	0.764754i	−0.977178	+	0.212424i	$0.931864\pi$
$228$		0			0
$229$			1.92458e15i			0.881871i	0.897539	+	0.440935i	$0.145353\pi$
$229$			1.92458e15i			0.881871i	−0.897539	+	0.440935i	$0.854647\pi$
$230$	−3.79099e15	−	1.30023e15i	−1.68858	−	0.579147i
$231$		0			0
$232$	−4.16882e15	+	4.16882e15i	−1.75526	+	1.75526i
$233$	−8.52667e14	+	8.52667e14i	−0.349113	+	0.349113i	−0.859779	−	0.510666i	$-0.829399\pi$
$233$	−8.52667e14	+	8.52667e14i	−0.349113	+	0.349113i	0.510666	+	0.859779i	$0.329399\pi$
$234$		0			0
$235$	1.55124e15	+	3.17080e15i	0.600812	+	1.22809i
$236$			7.16732e14i			0.270040i
$237$		0			0
$238$	−7.02435e15	−	7.02435e15i	−2.50528	−	2.50528i
$239$	−2.38339e15			−0.827195			−0.413598	−	0.910460i	$-0.635728\pi$
$239$	−2.38339e15			−0.827195			−0.413598	+	0.910460i	$0.635728\pi$
$240$		0			0
$241$	−1.02353e15			−0.336504			−0.168252	−	0.985744i	$-0.553812\pi$
$241$	−1.02353e15			−0.336504			−0.168252	+	0.985744i	$0.553812\pi$
$242$	−1.16238e15	−	1.16238e15i	−0.372005	−	0.372005i
$243$		0			0
$244$		−	7.30436e15i		−	2.21589i
$245$	1.03649e15	−	3.02204e15i	0.306188	−	0.892730i
$246$		0			0
$247$	−2.43323e15	+	2.43323e15i	−0.681793	+	0.681793i
$248$	8.69845e15	−	8.69845e15i	2.37414	−	2.37414i
$249$		0			0
$250$	1.48286e15	−	7.10261e15i	0.384141	−	1.83996i
$251$			4.94075e15i			1.24714i	0.781769	+	0.623568i	$0.214317\pi$
$251$			4.94075e15i			1.24714i	−0.781769	+	0.623568i	$0.785683\pi$
$252$		0			0
$253$	2.37720e15	+	2.37720e15i	0.569879	+	0.569879i
$254$	−8.04439e15			−1.87964
$255$		0			0
$256$	4.25572e13			0.00944961
$257$	−4.24600e15	−	4.24600e15i	−0.919210	−	0.919210i	0.0777622	−	0.996972i	$-0.475223\pi$
$257$	−4.24600e15	−	4.24600e15i	−0.919210	−	0.919210i	−0.996972	+	0.0777622i	$0.975223\pi$
$258$		0			0
$259$			7.78884e15i			1.60334i
$260$	7.78001e15	−	3.80619e15i	1.56190	−	0.764123i
$261$		0			0
$262$	−3.38975e15	+	3.38975e15i	−0.647455	+	0.647455i
$263$	4.10690e15	−	4.10690e15i	0.765248	−	0.765248i	−0.212018	−	0.977266i	$-0.568004\pi$
$263$	4.10690e15	−	4.10690e15i	0.765248	−	0.765248i	0.977266	+	0.212018i	$0.0680036\pi$
$264$		0			0
$265$	4.24029e15	−	2.07446e15i	0.752137	−	0.367966i
$266$			2.12658e16i			3.68088i
$267$		0			0
$268$	1.92425e16	+	1.92425e16i	3.17238	+	3.17238i
$269$	1.70453e15			0.274293			0.137146	−	0.990551i	$-0.456207\pi$
$269$	1.70453e15			0.274293			0.137146	+	0.990551i	$0.456207\pi$
$270$		0			0
$271$	3.73538e15			0.572842			0.286421	−	0.958104i	$-0.407534\pi$
$271$	3.73538e15			0.572842			0.286421	+	0.958104i	$0.407534\pi$
$272$	−1.84087e16	−	1.84087e16i	−2.75629	−	2.75629i
$273$		0			0
$274$		−	1.16329e16i		−	1.66077i
$275$	−3.73559e15	+	4.80518e15i	−0.520830	+	0.669958i
$276$		0			0
$277$	2.10618e15	−	2.10618e15i	0.280141	−	0.280141i	−0.553024	−	0.833165i	$-0.686526\pi$
$277$	2.10618e15	−	2.10618e15i	0.280141	−	0.280141i	0.833165	+	0.553024i	$0.186526\pi$
$278$	−2.63333e15	+	2.63333e15i	−0.342148	+	0.342148i
$279$		0			0
$280$	1.05096e16	−	3.06421e16i	1.30334	−	3.80007i
$281$		−	7.65332e15i		−	0.927382i	−0.885997	−	0.463691i	$-0.846525\pi$
$281$		−	7.65332e15i		−	0.927382i	0.885997	−	0.463691i	$-0.153475\pi$
$282$		0			0
$283$	−5.28958e15	−	5.28958e15i	−0.612082	−	0.612082i	0.331406	−	0.943488i	$-0.392477\pi$
$283$	−5.28958e15	−	5.28958e15i	−0.612082	−	0.612082i	−0.943488	+	0.331406i	$0.892477\pi$
$284$	5.63476e15			0.637245
$285$		0			0
$286$	−1.01336e16			−1.09492
$287$	1.07209e16	+	1.07209e16i	1.13239	+	1.13239i
$288$		0			0
$289$			8.19990e15i			0.827890i
$290$	7.20797e15	+	1.47334e16i	0.711583	+	1.45451i
$291$		0			0
$292$	1.77369e16	−	1.77369e16i	1.67452	−	1.67452i
$293$	7.64264e15	−	7.64264e15i	0.705674	−	0.705674i	−0.259948	−	0.965622i	$-0.583706\pi$
$293$	7.64264e15	−	7.64264e15i	0.705674	−	0.705674i	0.965622	+	0.259948i	$0.0837056\pi$
$294$		0			0
$295$	1.14152e15	+	3.91518e14i	0.100842	+	0.0345867i
$296$			3.83455e16i			3.31374i
$297$		0			0
$298$	−1.70928e16	−	1.70928e16i	−1.41387	−	1.41387i
$299$	−8.05515e15			−0.651945
$300$		0			0
$301$	6.20312e15			0.480760
$302$	−3.22193e16	−	3.22193e16i	−2.44383	−	2.44383i
$303$		0			0
$304$			5.57312e16i			4.04968i
$305$	−1.16335e16	−	3.99005e15i	−0.827488	−	0.283811i
$306$		0			0
$307$	2.66109e15	−	2.66109e15i	0.181410	−	0.181410i	−0.610560	−	0.791970i	$-0.709056\pi$
$307$	2.66109e15	−	2.66109e15i	0.181410	−	0.181410i	0.791970	+	0.610560i	$0.209056\pi$
$308$	−3.17485e16	+	3.17485e16i	−2.11906	+	2.11906i
$309$		0			0
$310$	−1.50398e16	−	3.07419e16i	−0.962477	−	1.96734i
$311$		−	1.40245e16i		−	0.878910i	−0.898265	−	0.439455i	$-0.855172\pi$
$311$		−	1.40245e16i		−	0.878910i	0.898265	−	0.439455i	$-0.144828\pi$
$312$		0			0
$313$	1.65598e16	+	1.65598e16i	0.995442	+	0.995442i	0.999990	−	0.00454812i	$-0.00144772\pi$
$313$	1.65598e16	+	1.65598e16i	0.995442	+	0.995442i	−0.00454812	+	0.999990i	$0.501448\pi$
$314$	3.53282e16			2.08008
$315$		0			0
$316$	−8.32946e15			−0.470599
$317$	1.41761e16	+	1.41761e16i	0.784644	+	0.784644i	0.980611	−	0.195967i	$-0.0627845\pi$
$317$	1.41761e16	+	1.41761e16i	0.784644	+	0.784644i	−0.195967	+	0.980611i	$0.562784\pi$
$318$		0			0
$319$		−	1.37587e16i		−	0.731034i
$320$	1.17580e16	−	3.42820e16i	0.612150	−	1.78480i
$321$		0			0
$322$	−3.52000e16	+	3.52000e16i	−1.75987	+	1.75987i
$323$	2.74051e16	−	2.74051e16i	1.34281	−	1.34281i
$324$		0			0
$325$	−1.81216e15	−	1.44702e16i	−0.0853013	−	0.681135i
$326$		−	1.89661e16i		−	0.875114i
$327$		0			0
$328$	5.27803e16	+	5.27803e16i	2.34041	+	2.34041i
$329$	4.38451e16			1.90611
$330$		0			0
$331$	8.39338e14			0.0350796			0.0175398	−	0.999846i	$-0.494417\pi$
$331$	8.39338e14			0.0350796			0.0175398	+	0.999846i	$0.494417\pi$
$332$	−5.51408e16	−	5.51408e16i	−2.25983	−	2.25983i
$333$		0			0
$334$		−	3.38885e16i		−	1.33567i
$335$	4.11584e16	−	2.01358e16i	1.59099	−	0.778354i
$336$		0			0
$337$	1.73454e15	−	1.73454e15i	0.0645044	−	0.0645044i	−0.674119	−	0.738623i	$-0.735476\pi$
$337$	1.73454e15	−	1.73454e15i	0.0645044	−	0.0645044i	0.738623	+	0.674119i	$0.235476\pi$
$338$	−1.92660e16	+	1.92660e16i	−0.702801	+	0.702801i
$339$		0			0
$340$	−8.76253e16	+	4.28686e16i	−3.07621	+	1.50497i
$341$			2.87082e16i			0.988785i
$342$		0			0
$343$	1.67157e15	+	1.67157e15i	0.0554260	+	0.0554260i
$344$	3.05388e16			0.993625
$345$		0			0
$346$	−5.87601e16			−1.84115
$347$	2.20036e16	+	2.20036e16i	0.676631	+	0.676631i	0.959236	−	0.282605i	$-0.0911986\pi$
$347$	2.20036e16	+	2.20036e16i	0.676631	+	0.676631i	−0.282605	+	0.959236i	$0.591199\pi$
$348$		0			0
$349$		−	2.92039e16i		−	0.865120i	−0.901605	−	0.432560i	$-0.857610\pi$
$349$		−	2.92039e16i		−	0.865120i	0.901605	−	0.432560i	$-0.142390\pi$
$350$	−7.11519e16	−	5.53141e16i	−2.06892	−	1.60840i
$351$		0			0
$352$	−5.43446e16	+	5.43446e16i	−1.52275	+	1.52275i
$353$	−1.74437e16	+	1.74437e16i	−0.479847	+	0.479847i	−0.905083	−	0.425236i	$-0.860191\pi$
$353$	−1.74437e16	+	1.74437e16i	−0.479847	+	0.479847i	0.425236	+	0.905083i	$0.360191\pi$
$354$		0			0
$355$	3.07802e15	−	8.97436e15i	0.0816181	−	0.237968i
$356$			1.55681e17i			4.05331i
$357$		0			0
$358$	2.99329e16	+	2.99329e16i	0.751466	+	0.751466i
$359$	−4.73668e16			−1.16778			−0.583888	−	0.811834i	$-0.698469\pi$
$359$	−4.73668e16			−1.16778			−0.583888	+	0.811834i	$0.698469\pi$
$360$		0			0
$361$	−4.09146e16			−0.972929
$362$	7.79939e16	+	7.79939e16i	1.82161	+	1.82161i
$363$		0			0
$364$		−	1.07580e17i		−	2.42422i
$365$	−1.85603e16	−	3.79380e16i	−0.410848	−	0.839791i
$366$		0			0
$367$	1.80637e16	−	1.80637e16i	0.385902	−	0.385902i	−0.487321	−	0.873223i	$-0.662026\pi$
$367$	1.80637e16	−	1.80637e16i	0.385902	−	0.385902i	0.873223	+	0.487321i	$0.162026\pi$
$368$	−9.22486e16	+	9.22486e16i	−1.93619	+	1.93619i
$369$		0			0
$370$	1.00910e17	+	3.46101e16i	2.04467	+	0.701279i
$371$		−	5.86336e16i		−	1.16739i
$372$		0			0
$373$	−2.34824e16	−	2.34824e16i	−0.451476	−	0.451476i	0.444368	−	0.895844i	$-0.353428\pi$
$373$	−2.34824e16	−	2.34824e16i	−0.451476	−	0.451476i	−0.895844	+	0.444368i	$0.853428\pi$
$374$	1.14133e17			2.15648
$375$		0			0
$376$	2.15855e17			3.93951
$377$	2.33107e16	+	2.33107e16i	0.418154	+	0.418154i
$378$		0			0
$379$			3.92402e15i			0.0680106i	0.999422	+	0.0340053i	$0.0108263\pi$
$379$			3.92402e15i			0.0680106i	−0.999422	+	0.0340053i	$0.989174\pi$
$380$	1.97531e17	+	6.77491e16i	3.36545	+	1.15428i
$381$		0			0
$382$	−6.01914e16	+	6.01914e16i	−0.991112	+	0.991112i
$383$	−5.70569e15	+	5.70569e15i	−0.0923669	+	0.0923669i	−0.751780	−	0.659414i	$-0.770805\pi$
$383$	−5.70569e15	+	5.70569e15i	−0.0923669	+	0.0923669i	0.659414	+	0.751780i	$0.270805\pi$
$384$		0			0
$385$	3.32224e16	+	6.79080e16i	0.519920	+	1.06274i
$386$		−	1.66039e17i		−	2.55501i
$387$		0			0
$388$	4.53633e15	+	4.53633e15i	0.0674991	+	0.0674991i
$389$	−6.89206e16			−1.00850			−0.504251	−	0.863557i	$-0.668231\pi$
$389$	−6.89206e16			−1.00850			−0.504251	+	0.863557i	$0.668231\pi$
$390$		0			0
$391$	9.07242e16			1.28403
$392$	−1.38144e17	−	1.38144e17i	−1.92297	−	1.92297i
$393$		0			0
$394$			5.80254e16i			0.781434i
$395$	−4.55001e15	+	1.32661e16i	−0.0602742	+	0.175737i
$396$		0			0
$397$	3.16814e16	−	3.16814e16i	0.406131	−	0.406131i	−0.474256	−	0.880387i	$-0.657283\pi$
$397$	3.16814e16	−	3.16814e16i	0.406131	−	0.406131i	0.880387	+	0.474256i	$0.157283\pi$
$398$	−9.33870e16	+	9.33870e16i	−1.17773	+	1.17773i
$399$		0			0
$400$	−1.86468e17	−	1.44961e17i	−2.27622	−	1.76955i
$401$			1.16198e17i			1.39560i	0.716295	+	0.697798i	$0.245837\pi$
$401$			1.16198e17i			1.39560i	−0.716295	+	0.697798i	$0.754163\pi$
$402$		0			0
$403$	−4.86388e16	−	4.86388e16i	−0.565588	−	0.565588i
$404$	−8.22302e16			−0.940919
$405$		0			0
$406$	2.03730e17			2.25754
$407$	−6.32774e16	−	6.32774e16i	−0.690057	−	0.690057i
$408$		0			0
$409$			6.16286e16i			0.650999i	0.945542	+	0.325500i	$0.105532\pi$
$409$			6.16286e16i			0.650999i	−0.945542	+	0.325500i	$0.894468\pi$
$410$	1.86535e17	−	9.12580e16i	1.93939	−	0.948803i
$411$		0			0
$412$	7.13748e16	−	7.13748e16i	0.718974	−	0.718974i
$413$	1.05993e16	−	1.05993e16i	0.105099	−	0.105099i
$414$		0			0
$415$	−1.17943e17	+	5.77007e16i	−1.13333	+	0.554457i
$416$		−	1.84147e17i		−	1.74204i
$417$		0			0
$418$	−1.72766e17	−	1.72766e17i	−1.58421	−	1.58421i
$419$	−8.71397e15			−0.0786728			−0.0393364	−	0.999226i	$-0.512524\pi$
$419$	−8.71397e15			−0.0786728			−0.0393364	+	0.999226i	$0.512524\pi$
$420$		0			0
$421$	−1.99837e17			−1.74921			−0.874607	−	0.484833i	$-0.838881\pi$
$421$	−1.99837e17			−1.74921			−0.874607	+	0.484833i	$0.838881\pi$
$422$	−1.57967e17	−	1.57967e17i	−1.36156	−	1.36156i
$423$		0			0
$424$		−	2.88661e17i		−	2.41274i
$425$	2.04102e16	+	1.62976e17i	0.168004	+	1.34152i
$426$		0			0
$427$	−1.08019e17	+	1.08019e17i	−0.862422	+	0.862422i
$428$	2.05658e17	−	2.05658e17i	1.61719	−	1.61719i
$429$		0			0
$430$	2.75638e16	−	8.03660e16i	0.210278	−	0.613094i
$431$			8.38140e15i			0.0629817i	0.999504	+	0.0314909i	$0.0100255\pi$
$431$			8.38140e15i			0.0629817i	−0.999504	+	0.0314909i	$0.989974\pi$
$432$		0			0
$433$	4.03528e15	+	4.03528e15i	0.0294240	+	0.0294240i	0.721666	−	0.692242i	$-0.243377\pi$
$433$	4.03528e15	+	4.03528e15i	0.0294240	+	0.0294240i	−0.692242	+	0.721666i	$0.743377\pi$
$434$	−4.25092e17			−3.05351
$435$		0			0
$436$	5.12494e17			3.57295
$437$	−1.37331e17	−	1.37331e17i	−0.943278	−	0.943278i
$438$		0			0
$439$		−	1.80474e16i		−	0.120336i	−0.998188	−	0.0601679i	$-0.980836\pi$
$439$		−	1.80474e16i		−	0.120336i	0.998188	−	0.0601679i	$-0.0191636\pi$
$440$	1.63559e17	+	3.34321e17i	1.07456	+	2.19645i
$441$		0			0
$442$	−1.93370e17	+	1.93370e17i	−1.23352	+	1.23352i
$443$	−8.49326e16	+	8.49326e16i	−0.533888	+	0.533888i	−0.921727	−	0.387839i	$-0.873222\pi$
$443$	−8.49326e16	+	8.49326e16i	−0.533888	+	0.533888i	0.387839	+	0.921727i	$0.373222\pi$
$444$		0			0
$445$	2.47949e17	+	8.50413e16i	1.51364	+	0.519147i
$446$		−	3.18465e17i		−	1.91595i
$447$		0			0
$448$	−3.18314e17	−	3.18314e17i	−1.86015	−	1.86015i
$449$	−1.40506e17			−0.809272			−0.404636	−	0.914478i	$-0.632602\pi$
$449$	−1.40506e17			−0.809272			−0.404636	+	0.914478i	$0.632602\pi$
$450$		0			0
$451$	−1.74195e17			−0.974738
$452$	2.50690e17	+	2.50690e17i	1.38273	+	1.38273i
$453$		0			0
$454$			3.79291e17i			2.03287i
$455$	−1.71340e17	−	5.87661e16i	−0.905285	−	0.310494i
$456$		0			0
$457$	−1.00107e17	+	1.00107e17i	−0.514056	+	0.514056i	−0.915767	−	0.401711i	$-0.868416\pi$
$457$	−1.00107e17	+	1.00107e17i	−0.514056	+	0.514056i	0.401711	+	0.915767i	$0.368416\pi$
$458$	2.31520e17	−	2.31520e17i	1.17209	−	1.17209i
$459$		0			0
$460$	2.14821e17	+	4.39103e17i	1.05718	+	2.16093i
$461$			1.23603e16i			0.0599753i	0.999550	+	0.0299876i	$0.00954679\pi$
$461$			1.23603e16i			0.0599753i	−0.999550	+	0.0299876i	$0.990453\pi$
$462$		0			0
$463$	1.64681e17	+	1.64681e17i	0.776904	+	0.776904i	0.979303	−	0.202399i	$-0.0648738\pi$
$463$	1.64681e17	+	1.64681e17i	0.776904	+	0.776904i	−0.202399	+	0.979303i	$0.564874\pi$
$464$	5.33913e17			2.48373
$465$		0			0
$466$	2.05145e17			0.928012
$467$	2.43285e17	+	2.43285e17i	1.08531	+	1.08531i	0.996004	+	0.0893099i	$0.0284661\pi$
$467$	2.43285e17	+	2.43285e17i	1.08531	+	1.08531i	0.0893099	+	0.996004i	$0.471534\pi$
$468$		0			0
$469$		−	5.69127e17i		−	2.46937i
$470$	1.94827e17	−	5.68045e17i	0.833708	−	2.43079i
$471$		0			0
$472$	5.21816e16	−	5.21816e16i	0.217217	−	0.217217i
$473$	−5.03948e16	+	5.03948e16i	−0.206913	+	0.206913i
$474$		0			0
$475$	2.15805e17	−	2.77596e17i	0.862091	−	1.10893i
$476$			1.21166e18i			4.77458i
$477$		0			0
$478$	2.86713e17	+	2.86713e17i	1.09943	+	1.09943i
$479$	−2.65388e17			−1.00392			−0.501962	−	0.864890i	$-0.667388\pi$
$479$	−2.65388e17			−1.00392			−0.501962	+	0.864890i	$0.667388\pi$
$480$		0			0
$481$	2.14416e17			0.789429
$482$	1.23127e17	+	1.23127e17i	0.447248	+	0.447248i
$483$		0			0
$484$			2.00504e17i			0.708971i
$485$	9.70290e15	−	4.74692e15i	0.0338517	−	0.0165611i
$486$		0			0
$487$	−2.38709e17	+	2.38709e17i	−0.810830	+	0.810830i	−0.984758	−	0.173929i	$-0.944354\pi$
$487$	−2.38709e17	+	2.38709e17i	−0.810830	+	0.810830i	0.173929	+	0.984758i	$0.444354\pi$
$488$	−5.31793e17	+	5.31793e17i	−1.78244	+	1.78244i
$489$		0			0
$490$	−4.88226e17	+	2.38853e17i	−1.59348	+	0.779573i
$491$			1.93370e17i			0.622815i	0.950277	+	0.311407i	$0.100800\pi$
$491$			1.93370e17i			0.622815i	−0.950277	+	0.311407i	$0.899200\pi$
$492$		0			0
$493$	−2.62545e17	−	2.62545e17i	−0.823567	−	0.823567i
$494$	5.85417e17			1.81234
$495$		0			0
$496$	−1.11404e18			−3.35945
$497$	−8.33286e16	−	8.33286e16i	−0.248015	−	0.248015i
$498$		0			0
$499$		−	4.22682e17i		−	1.22563i	−0.790225	−	0.612816i	$-0.790037\pi$
$499$		−	4.22682e17i		−	1.22563i	0.790225	−	0.612816i	$-0.209963\pi$
$500$	−7.40472e17	+	4.84687e17i	−2.11936	+	1.38725i
$501$		0			0
$502$	5.94354e17	−	5.94354e17i	1.65757	−	1.65757i
$503$	−1.93804e17	+	1.93804e17i	−0.533544	+	0.533544i	−0.921625	−	0.388081i	$-0.873138\pi$
$503$	−1.93804e17	+	1.93804e17i	−0.533544	+	0.533544i	0.388081	+	0.921625i	$0.373138\pi$
$504$		0			0
$505$	−4.49187e16	+	1.30966e17i	−0.120513	+	0.351371i
$506$		−	5.71937e17i		−	1.51485i
$507$		0			0
$508$	6.93806e17	+	6.93806e17i	1.79112	+	1.79112i
$509$	−1.68004e17			−0.428208			−0.214104	−	0.976811i	$-0.568683\pi$
$509$	−1.68004e17			−0.428208			−0.214104	+	0.976811i	$0.568683\pi$
$510$		0			0
$511$	−5.24598e17			−1.30344
$512$	2.85669e17	+	2.85669e17i	0.700823	+	0.700823i
$513$		0			0
$514$			1.02156e18i			2.44344i
$515$	−7.46883e16	−	1.52666e17i	−0.176403	−	0.360575i
$516$		0			0
$517$	−3.56202e17	+	3.56202e17i	−0.820366	+	0.820366i
$518$	9.36969e17	−	9.36969e17i	2.13099	−	2.13099i
$519$		0			0
$520$	−8.43532e17	−	2.89314e17i	−1.87103	−	0.641723i
$521$		−	4.71176e17i		−	1.03214i	−0.856547	−	0.516069i	$-0.827395\pi$
$521$		−	4.71176e17i		−	1.03214i	0.856547	−	0.516069i	$-0.172605\pi$
$522$		0			0
$523$	2.57345e17	+	2.57345e17i	0.549863	+	0.549863i	0.926401	−	0.376538i	$-0.122886\pi$
$523$	2.57345e17	+	2.57345e17i	0.549863	+	0.549863i	−0.376538	+	0.926401i	$0.622886\pi$
$524$	5.84712e17			1.23393
$525$		0			0
$526$	−9.88091e17			−2.03418
$527$	5.47813e17	+	5.47813e17i	1.11394	+	1.11394i
$528$		0			0
$529$			4.94044e16i			0.0980175i
$530$	−7.59641e17	−	2.60541e17i	−1.48873	−	0.510602i
$531$		0			0
$532$	1.83412e18	−	1.83412e18i	3.50753	−	3.50753i
$533$	2.95130e17	−	2.95130e17i	0.557553	−	0.557553i
$534$		0			0
$535$	−2.15205e17	−	4.39888e17i	−0.396782	−	0.811040i
$536$		−	2.80189e18i		−	5.10364i
$537$		0			0
$538$	−2.05048e17	−	2.05048e17i	−0.364562	−	0.364562i
$539$	4.55927e17			0.800882
$540$		0			0
$541$	2.24095e15			0.00384282			0.00192141	−	0.999998i	$-0.499388\pi$
$541$	2.24095e15			0.00384282			0.00192141	+	0.999998i	$0.499388\pi$
$542$	−4.49353e17	−	4.49353e17i	−0.761364	−	0.761364i
$543$		0			0
$544$			2.07402e18i			3.43100i
$545$	2.79952e17	−	8.16238e17i	0.457622	−	1.33426i
$546$		0			0
$547$	1.83143e17	−	1.83143e17i	0.292330	−	0.292330i	−0.545670	−	0.838000i	$-0.683725\pi$
$547$	1.83143e17	−	1.83143e17i	0.292330	−	0.292330i	0.838000	+	0.545670i	$0.183725\pi$
$548$	−1.00330e18	+	1.00330e18i	−1.58255	+	1.58255i
$549$		0			0
$550$	1.02742e18	−	1.28668e17i	1.58268	−	0.198205i
$551$			7.94841e17i			1.21003i
$552$		0			0
$553$	1.23179e17	+	1.23179e17i	0.183156	+	0.183156i
$554$	−5.06731e17			−0.744670
$555$		0			0
$556$	4.54234e17			0.652068
$557$	−3.71457e17	−	3.71457e17i	−0.527048	−	0.527048i	0.392643	−	0.919691i	$-0.371561\pi$
$557$	−3.71457e17	−	3.71457e17i	−0.527048	−	0.527048i	−0.919691	+	0.392643i	$0.871561\pi$
$558$		0			0
$559$		−	1.70763e17i		−	0.236710i
$560$	−2.63521e18	+	1.28921e18i	−3.61071	+	1.76645i
$561$		0			0
$562$	−9.20666e17	+	9.20666e17i	−1.23258	+	1.23258i
$563$	−4.60055e16	+	4.60055e16i	−0.0608843	+	0.0608843i	−0.736893	−	0.676009i	$-0.763708\pi$
$563$	−4.60055e16	+	4.60055e16i	−0.0608843	+	0.0608843i	0.676009	+	0.736893i	$0.263708\pi$
$564$		0			0
$565$	5.36209e17	−	2.62328e17i	0.693456	−	0.339257i
$566$			1.27264e18i			1.62704i
$567$		0			0
$568$	−4.10238e17	−	4.10238e17i	−0.512592	−	0.512592i
$569$	1.33270e18			1.64628			0.823139	−	0.567839i	$-0.192221\pi$
$569$	1.33270e18			1.64628			0.823139	+	0.567839i	$0.192221\pi$
$570$		0			0
$571$	−1.00154e18			−1.20930			−0.604650	−	0.796491i	$-0.706687\pi$
$571$	−1.00154e18			−1.20930			−0.604650	+	0.796491i	$0.706687\pi$
$572$	8.73991e17	+	8.73991e17i	1.04336	+	1.04336i
$573$		0			0
$574$		−	2.57936e18i		−	3.01013i
$575$	8.16697e17	−	1.02278e17i	0.942367	−	0.118016i
$576$		0			0
$577$	5.62634e17	−	5.62634e17i	0.634721	−	0.634721i	−0.314527	−	0.949248i	$-0.601846\pi$
$577$	5.62634e17	−	5.62634e17i	0.634721	−	0.634721i	0.949248	+	0.314527i	$0.101846\pi$
$578$	9.86418e17	−	9.86418e17i	1.10035	−	1.10035i
$579$		0			0
$580$	6.49046e17	−	1.89238e18i	0.707935	−	2.06408i
$581$			1.63088e18i			1.75905i
$582$		0			0
$583$	4.76345e17	+	4.76345e17i	0.502431	+	0.502431i
$584$	−2.58267e18			−2.69392
$585$		0			0
$586$	−1.83876e18			−1.87582
$587$	9.91204e16	+	9.91204e16i	0.100004	+	0.100004i	0.755338	−	0.655335i	$-0.227472\pi$
$587$	9.91204e16	+	9.91204e16i	0.100004	+	0.100004i	−0.655335	+	0.755338i	$0.727472\pi$
$588$		0			0
$589$		−	1.65847e18i		−	1.63666i
$590$	−9.02229e16	−	1.84419e17i	−0.0880600	−	0.179998i
$591$		0			0
$592$	2.45551e18	−	2.45551e18i	2.34450	−	2.34450i
$593$	8.74534e17	−	8.74534e17i	0.825888	−	0.825888i	−0.161057	−	0.986945i	$-0.551490\pi$
$593$	8.74534e17	−	8.74534e17i	0.825888	−	0.825888i	0.986945	+	0.161057i	$0.0514904\pi$
$594$		0			0
$595$	1.92978e18	+	6.61875e17i	1.78299	+	0.611527i
$596$			2.94841e18i			2.69456i
$597$		0			0
$598$	9.69006e17	+	9.69006e17i	0.866500	+	0.866500i
$599$	−2.04520e18			−1.80909			−0.904546	−	0.426376i	$-0.859790\pi$
$599$	−2.04520e18			−1.80909			−0.904546	+	0.426376i	$0.859790\pi$
$600$		0			0
$601$	−6.40263e17			−0.554210			−0.277105	−	0.960840i	$-0.589375\pi$
$601$	−6.40263e17			−0.554210			−0.277105	+	0.960840i	$0.589375\pi$
$602$	−7.46213e17	−	7.46213e17i	−0.638977	−	0.638977i
$603$		0			0
$604$			5.55764e18i			4.65747i
$605$	3.19339e17	+	1.09526e17i	0.264753	+	0.0908048i
$606$		0			0
$607$	−5.40554e17	+	5.40554e17i	−0.438644	+	0.438644i	−0.891556	−	0.452911i	$-0.850385\pi$
$607$	−5.40554e17	+	5.40554e17i	−0.438644	+	0.438644i	0.452911	+	0.891556i	$0.350385\pi$
$608$	3.13949e18	−	3.13949e18i	2.52050	−	2.52050i
$609$		0			0
$610$	9.19480e17	+	1.87946e18i	0.722601	+	1.47703i
$611$		−	1.20699e18i		−	0.938504i
$612$		0			0
$613$	−1.32862e18	−	1.32862e18i	−1.01136	−	1.01136i	−0.999935	−	0.0114274i	$-0.996362\pi$
$613$	−1.32862e18	−	1.32862e18i	−1.01136	−	1.01136i	−0.0114274	−	0.999935i	$-0.503638\pi$
$614$	−6.40239e17			−0.482223
$615$		0			0
$616$	4.62290e18			3.40910
$617$	7.49374e17	+	7.49374e17i	0.546821	+	0.546821i	0.925520	−	0.378699i	$-0.123629\pi$
$617$	7.49374e17	+	7.49374e17i	0.546821	+	0.546821i	−0.378699	+	0.925520i	$0.623629\pi$
$618$		0			0
$619$			2.39852e17i			0.171378i	0.996322	+	0.0856889i	$0.0273091\pi$
$619$			2.39852e17i			0.171378i	−0.996322	+	0.0856889i	$0.972691\pi$
$620$	−1.35427e18	+	3.94854e18i	−0.957542	+	2.79184i
$621$		0			0
$622$	−1.68709e18	+	1.68709e18i	−1.16816	+	1.16816i
$623$	2.30225e18	−	2.30225e18i	1.57754	−	1.57754i
$624$		0			0
$625$	3.67464e17	+	1.44410e18i	0.246601	+	0.969117i
$626$		−	3.98416e18i		−	2.64608i
$627$		0			0
$628$	−3.04696e18	−	3.04696e18i	−1.98211	−	1.98211i
$629$	−2.41493e18			−1.55481
$630$		0			0
$631$	−1.55806e17			−0.0982637			−0.0491319	−	0.998792i	$-0.515645\pi$
$631$	−1.55806e17			−0.0982637			−0.0491319	+	0.998792i	$0.515645\pi$
$632$	6.06425e17	+	6.06425e17i	0.378544	+	0.378544i
$633$		0			0
$634$		−	3.41067e18i		−	2.08574i
$635$	1.48400e18	−	7.26014e17i	0.898268	−	0.439457i
$636$		0			0
$637$	−7.72455e17	+	7.72455e17i	−0.458107	+	0.458107i
$638$	−1.65512e18	+	1.65512e18i	−0.971617	+	0.971617i
$639$		0			0
$640$	−1.57548e18	+	7.70765e17i	−0.906237	+	0.443355i
$641$			2.75120e18i			1.56655i	0.621674	+	0.783276i	$0.286453\pi$
$641$			2.75120e18i			1.56655i	−0.621674	+	0.783276i	$0.713547\pi$
$642$		0			0
$643$	1.59479e18	+	1.59479e18i	0.889882	+	0.889882i	0.994511	−	0.104629i	$-0.0333656\pi$
$643$	1.59479e18	+	1.59479e18i	0.889882	+	0.889882i	−0.104629	+	0.994511i	$0.533366\pi$
$644$	6.07180e18			3.35397
$645$		0			0
$646$	−6.59348e18			−3.56947
$647$	2.22956e17	+	2.22956e17i	0.119492	+	0.119492i	0.764324	−	0.644832i	$-0.223073\pi$
$647$	2.22956e17	+	2.22956e17i	0.119492	+	0.119492i	−0.644832	+	0.764324i	$0.723073\pi$
$648$		0			0
$649$			1.72219e17i			0.0904670i
$650$	−1.52272e18	+	1.95871e18i	−0.791922	+	1.01867i
$651$		0			0
$652$	−1.63578e18	+	1.63578e18i	−0.833901	+	0.833901i
$653$	−1.12879e18	+	1.12879e18i	−0.569743	+	0.569743i	−0.932056	−	0.362314i	$-0.881987\pi$
$653$	−1.12879e18	+	1.12879e18i	−0.569743	+	0.569743i	0.362314	+	0.932056i	$0.381987\pi$
$654$		0			0
$655$	3.19402e17	−	9.31259e17i	0.158041	−	0.460789i
$656$		−	6.75972e18i		−	3.31172i
$657$		0			0
$658$	−5.27440e18	−	5.27440e18i	−2.53341	−	2.53341i
$659$	−1.71902e18			−0.817573			−0.408786	−	0.912630i	$-0.634048\pi$
$659$	−1.71902e18			−0.817573			−0.408786	+	0.912630i	$0.634048\pi$
$660$		0			0
$661$	4.27733e18			1.99464			0.997318	−	0.0731921i	$-0.0233186\pi$
$661$	4.27733e18			1.99464			0.997318	+	0.0731921i	$0.0233186\pi$
$662$	−1.00969e17	−	1.00969e17i	−0.0466243	−	0.0466243i
$663$		0			0
$664$			8.02905e18i			3.63556i
$665$	−1.91926e18	−	3.92305e18i	−0.860584	−	1.75907i
$666$		0			0
$667$	−1.31565e18	+	1.31565e18i	−0.578526	+	0.578526i
$668$	−2.92279e18	+	2.92279e18i	−1.27277	+	1.27277i
$669$		0			0
$670$	−7.37346e18	−	2.52894e18i	−3.14909	−	1.08007i
$671$		−	1.75512e18i		−	0.742352i
$672$		0			0
$673$	2.63256e18	+	2.63256e18i	1.09215	+	1.09215i	0.995299	+	0.0968470i	$0.0308757\pi$
$673$	2.63256e18	+	2.63256e18i	1.09215	+	1.09215i	0.0968470	+	0.995299i	$0.469124\pi$
$674$	−4.17317e17			−0.171465
$675$		0			0
$676$	3.32327e18			1.33940
$677$	−1.12099e18	−	1.12099e18i	−0.447480	−	0.447480i	0.447036	−	0.894516i	$-0.352480\pi$
$677$	−1.12099e18	−	1.12099e18i	−0.447480	−	0.447480i	−0.894516	+	0.447036i	$0.852480\pi$
$678$		0			0
$679$		−	1.34169e17i		−	0.0525411i
$680$	9.50059e18	+	3.25850e18i	3.68505	+	1.26389i
$681$		0			0
$682$	3.45349e18	−	3.45349e18i	1.31419	−	1.31419i
$683$	−1.35194e18	+	1.35194e18i	−0.509593	+	0.509593i	−0.914401	−	0.404809i	$-0.867338\pi$
$683$	−1.35194e18	+	1.35194e18i	−0.509593	+	0.509593i	0.404809	+	0.914401i	$0.367338\pi$
$684$		0			0
$685$	1.04988e18	+	2.14600e18i	0.388285	+	0.793671i
$686$		−	4.02168e17i		−	0.147333i
$687$		0			0
$688$	−1.95560e18	−	1.95560e18i	−0.702999	−	0.702999i
$689$	−1.61410e18			−0.574784
$690$		0			0
$691$	−6.79917e17			−0.237601			−0.118801	−	0.992918i	$-0.537905\pi$
$691$	−6.79917e17			−0.237601			−0.118801	+	0.992918i	$0.537905\pi$
$692$	5.06789e18	+	5.06789e18i	1.75444	+	1.75444i
$693$		0			0
$694$		−	5.29390e18i		−	1.79862i
$695$	2.48127e17	−	7.23448e17i	0.0835168	−	0.243504i
$696$		0			0
$697$	−3.32401e18	+	3.32401e18i	−1.09812	+	1.09812i
$698$	−3.51313e18	+	3.51313e18i	−1.14983	+	1.14983i
$699$		0			0
$700$	1.36597e18	+	1.09073e19i	0.438838	+	3.50414i
$701$			7.86024e17i			0.250189i	0.992145	+	0.125095i	$0.0399234\pi$
$701$			7.86024e17i			0.250189i	−0.992145	+	0.125095i	$0.960077\pi$
$702$		0			0
$703$	3.65554e18	+	3.65554e18i	1.14220	+	1.14220i
$704$	5.17204e18			1.60118
$705$		0			0
$706$	4.19682e18			1.27553
$707$	1.21605e18	+	1.21605e18i	0.366204	+	0.366204i
$708$		0			0
$709$		−	3.21891e18i		−	0.951717i	−0.879522	−	0.475859i	$-0.842137\pi$
$709$		−	3.21891e18i		−	0.951717i	0.879522	−	0.475859i	$-0.157863\pi$
$710$	−1.44986e18	+	7.09309e17i	−0.424762	+	0.207805i
$711$		0			0
$712$	1.13343e19	−	1.13343e19i	3.26043	−	3.26043i
$713$	2.74517e18	−	2.74517e18i	0.782506	−	0.782506i
$714$		0			0
$715$	1.86941e18	−	9.14565e17i	0.523257	−	0.255991i
$716$		−	5.16325e18i		−	1.43215i
$717$		0			0
$718$	5.69805e18	+	5.69805e18i	1.55209	+	1.55209i
$719$	−3.40579e18			−0.919349			−0.459675	−	0.888087i	$-0.652034\pi$
$719$	−3.40579e18			−0.919349			−0.459675	+	0.888087i	$0.652034\pi$
$720$		0			0
$721$	−2.11103e18			−0.559647
$722$	4.92187e18	+	4.92187e18i	1.29312	+	1.29312i
$723$		0			0
$724$		−	1.34535e19i		−	3.47164i
$725$	−2.65941e18	−	2.06744e18i	−0.680123	−	0.528733i
$726$		0			0
$727$	−5.02908e18	+	5.02908e18i	−1.26332	+	1.26332i	−0.313851	+	0.949472i	$0.601619\pi$
$727$	−5.02908e18	+	5.02908e18i	−1.26332	+	1.26332i	−0.949472	+	0.313851i	$0.898381\pi$
$728$	−7.83235e18	+	7.83235e18i	−1.95002	+	1.95002i
$729$		0			0
$730$	−2.33107e18	+	6.79655e18i	−0.570108	+	1.66222i
$731$			1.92328e18i			0.466208i
$732$		0			0
$733$	1.91229e18	+	1.91229e18i	0.455384	+	0.455384i	0.897137	−	0.441753i	$-0.145643\pi$
$733$	1.91229e18	+	1.91229e18i	0.455384	+	0.455384i	−0.441753	+	0.897137i	$0.645643\pi$
$734$	−4.34599e18			−1.02580
$735$		0			0
$736$	1.03932e19			2.41015
$737$	4.62365e18	+	4.62365e18i	1.06279	+	1.06279i
$738$		0			0
$739$		−	3.99079e18i		−	0.901301i	−0.892700	−	0.450651i	$-0.851192\pi$
$739$		−	3.99079e18i		−	0.901301i	0.892700	−	0.450651i	$-0.148808\pi$
$740$	−5.71819e18	−	1.16882e19i	−1.28013	−	2.61663i
$741$		0			0
$742$	−7.05341e18	+	7.05341e18i	−1.55158	+	1.55158i
$743$	−2.52718e18	+	2.52718e18i	−0.551072	+	0.551072i	−0.926750	−	0.375678i	$-0.877410\pi$
$743$	−2.52718e18	+	2.52718e18i	−0.551072	+	0.551072i	0.375678	+	0.926750i	$0.377410\pi$
$744$		0			0
$745$	4.69587e18	+	1.61059e18i	1.00624	+	0.345118i
$746$			5.64969e18i			1.20011i
$747$		0			0
$748$	−9.84365e18	−	9.84365e18i	−2.05492	−	2.05492i
$749$	−6.08266e18			−1.25881
$750$		0			0
$751$	−5.37324e18			−1.09289			−0.546445	−	0.837495i	$-0.684019\pi$
$751$	−5.37324e18			−1.09289			−0.546445	+	0.837495i	$0.684019\pi$
$752$	−1.38226e19	−	1.38226e19i	−2.78724	−	2.78724i
$753$		0			0
$754$		−	5.60838e18i		−	1.11154i
$755$	8.85153e18	+	3.03589e18i	1.73926	+	0.596528i
$756$		0			0
$757$	5.96574e18	−	5.96574e18i	1.15224	−	1.15224i	0.166132	−	0.986104i	$-0.446872\pi$
$757$	5.96574e18	−	5.96574e18i	1.15224	−	1.15224i	0.986104	−	0.166132i	$-0.0531278\pi$
$758$	4.72045e17	−	4.72045e17i	0.0903928	−	0.0903928i
$759$		0			0
$760$	−9.44879e18	−	1.93137e19i	−1.77864	−	3.63561i
$761$		−	2.63450e18i		−	0.491697i	−0.969308	−	0.245848i	$-0.920933\pi$
$761$		−	2.63450e18i		−	0.491697i	0.969308	−	0.245848i	$-0.0790665\pi$
$762$		0			0
$763$	−7.57892e18	−	7.57892e18i	−1.39059	−	1.39059i
$764$	1.03827e19			1.88887
$765$		0			0
$766$	1.37275e18			0.245530
$767$	−2.91782e17	−	2.91782e17i	−0.0517474	−	0.0517474i
$768$		0			0
$769$		−	6.98812e18i		−	1.21854i	−0.792964	−	0.609269i	$-0.791463\pi$
$769$		−	6.98812e18i		−	1.21854i	0.792964	−	0.609269i	$-0.208537\pi$
$770$	4.17255e18	−	1.21656e19i	0.721460	−	2.10351i
$771$		0			0
$772$	−1.43204e19	+	1.43204e19i	−2.43469	+	2.43469i
$773$	2.13967e18	−	2.13967e18i	0.360729	−	0.360729i	−0.503353	−	0.864081i	$-0.667900\pi$
$773$	2.13967e18	−	2.13967e18i	0.360729	−	0.360729i	0.864081	+	0.503353i	$0.167900\pi$
$774$		0			0
$775$	5.54898e18	+	4.31382e18i	0.919924	+	0.715156i
$776$		−	6.60534e17i		−	0.108591i
$777$		0			0
$778$	8.29090e18	+	8.29090e18i	1.34040	+	1.34040i
$779$	1.00633e19			1.61341
$780$		0			0
$781$	1.35394e18			0.213485
$782$	−1.09138e19	−	1.09138e19i	−1.70660	−	1.70660i
$783$		0			0
$784$			1.76925e19i			2.72104i
$785$	−6.51724e18	+	3.18841e18i	−0.994056	+	0.486319i
$786$		0			0
$787$	−3.76195e18	+	3.76195e18i	−0.564388	+	0.564388i	−0.930551	−	0.366163i	$-0.880671\pi$
$787$	−3.76195e18	+	3.76195e18i	−0.564388	+	0.564388i	0.366163	+	0.930551i	$0.380671\pi$
$788$	5.00452e18	−	5.00452e18i	0.744633	−	0.744633i
$789$		0			0
$790$	2.14322e18	−	1.04852e18i	0.313683	−	0.153462i
$791$		−	7.41456e18i		−	1.07631i
$792$		0			0
$793$	2.97361e18	+	2.97361e18i	0.424628	+	0.424628i
$794$	−7.62232e18			−1.07958
$795$		0			0
$796$	1.61087e19			2.24453
$797$	−4.87617e18	−	4.87617e18i	−0.673907	−	0.673907i	0.284707	−	0.958614i	$-0.408104\pi$
$797$	−4.87617e18	−	4.87617e18i	−0.673907	−	0.673907i	−0.958614	+	0.284707i	$0.908104\pi$
$798$		0			0
$799$			1.35942e19i			1.84841i
$800$	2.33816e18	+	1.86703e19i	0.315347	+	2.51806i
$801$		0			0
$802$	1.39782e19	−	1.39782e19i	1.85488	−	1.85488i
$803$	4.26189e18	−	4.26189e18i	0.560984	−	0.560984i
$804$		0			0
$805$	3.31675e18	−	9.67043e18i	0.429576	−	1.25248i
$806$			1.17022e19i			1.50345i
$807$		0			0
$808$	5.98676e18	+	5.98676e18i	0.756864	+	0.756864i
$809$	8.11892e18			1.01820			0.509100	−	0.860708i	$-0.329979\pi$
$809$	8.11892e18			1.01820			0.509100	+	0.860708i	$0.329979\pi$
$810$		0			0
$811$	−1.05843e19			−1.30626			−0.653128	−	0.757248i	$-0.726544\pi$
$811$	−1.05843e19			−1.30626			−0.653128	+	0.757248i	$0.726544\pi$
$812$	−1.75711e19	−	1.75711e19i	−2.15122	−	2.15122i
$813$		0			0
$814$			1.52241e19i			1.83431i
$815$	1.71171e18	+	3.49881e18i	0.204600	+	0.418212i
$816$		0			0
$817$	2.91131e18	−	2.91131e18i	0.342488	−	0.342488i
$818$	7.41369e18	−	7.41369e18i	0.865243	−	0.865243i
$819$		0			0
$820$	−2.39589e19	−	8.21739e18i	−2.75218	−	0.943939i
$821$			1.05993e19i			1.20794i	0.797006	+	0.603971i	$0.206416\pi$
$821$			1.05993e19i			1.20794i	−0.797006	+	0.603971i	$0.793584\pi$
$822$		0			0
$823$	−8.12712e18	−	8.12712e18i	−0.911670	−	0.911670i	0.0847338	−	0.996404i	$-0.472996\pi$
$823$	−8.12712e18	−	8.12712e18i	−0.911670	−	0.911670i	−0.996404	+	0.0847338i	$0.972996\pi$
$824$	−1.03929e19			−1.15667
$825$		0			0
$826$	−2.55010e18			−0.279375
$827$	7.23309e18	+	7.23309e18i	0.786209	+	0.786209i	0.980871	−	0.194661i	$-0.0623607\pi$
$827$	7.23309e18	+	7.23309e18i	0.786209	+	0.786209i	−0.194661	+	0.980871i	$0.562361\pi$
$828$		0			0
$829$		−	1.32329e19i		−	1.41595i	−0.706236	−	0.707977i	$-0.749608\pi$
$829$		−	1.32329e19i		−	1.41595i	0.706236	−	0.707977i	$-0.250392\pi$
$830$	2.11292e19	+	7.24689e18i	2.24324	+	0.769385i
$831$		0			0
$832$	−8.76273e18	+	8.76273e18i	−0.915878	+	0.915878i
$833$	8.70006e18	−	8.70006e18i	0.902256	−	0.902256i
$834$		0			0
$835$	3.05847e18	+	6.25165e18i	0.312279	+	0.638311i
$836$			2.98011e19i			3.01920i
$837$		0			0
$838$	1.04826e18	+	1.04826e18i	0.104564	+	0.104564i
$839$	−1.19461e19			−1.18242			−0.591211	−	0.806517i	$-0.701350\pi$
$839$	−1.19461e19			−1.18242			−0.591211	+	0.806517i	$0.701350\pi$
$840$		0			0
$841$	−2.64595e18			−0.257874
$842$	2.40397e19	+	2.40397e19i	2.32488	+	2.32488i
$843$		0			0
$844$			2.72484e19i			2.59487i
$845$	1.81535e18	−	5.29289e18i	0.171551	−	0.500178i
$846$		0			0
$847$	2.96512e18	−	2.96512e18i	0.275931	−	0.275931i
$848$	−1.84848e19	+	1.84848e19i	−1.70704	+	1.70704i
$849$		0			0
$850$	1.71502e19	−	2.20607e19i	1.55971	−	2.00630i
$851$			1.21016e19i			1.09219i
$852$		0			0
$853$	−1.11168e19	−	1.11168e19i	−0.988127	−	0.988127i	0.0118036	−	0.999930i	$-0.496243\pi$
$853$	−1.11168e19	−	1.11168e19i	−0.988127	−	0.988127i	−0.999930	+	0.0118036i	$0.996243\pi$
$854$	2.59886e19			2.29249
$855$		0			0
$856$	−2.99458e19			−2.60169
$857$	−8.13050e18	−	8.13050e18i	−0.701039	−	0.701039i	0.263595	−	0.964634i	$-0.415092\pi$
$857$	−8.13050e18	−	8.13050e18i	−0.701039	−	0.701039i	−0.964634	+	0.263595i	$0.915092\pi$
$858$		0			0
$859$		−	1.29817e19i		−	1.10249i	−0.834343	−	0.551246i	$-0.814152\pi$
$859$		−	1.29817e19i		−	1.10249i	0.834343	−	0.551246i	$-0.185848\pi$
$860$	−9.30863e18	+	4.55403e18i	−0.784596	+	0.383845i
$861$		0			0
$862$	1.00825e18	−	1.00825e18i	0.0837090	−	0.0837090i
$863$	−1.25247e19	+	1.25247e19i	−1.03204	+	1.03204i	−0.0325751	+	0.999469i	$0.510371\pi$
$863$	−1.25247e19	+	1.25247e19i	−1.03204	+	1.03204i	−0.999469	+	0.0325751i	$0.989629\pi$
$864$		0			0
$865$	1.08399e19	−	5.30316e18i	0.879873	−	0.430457i
$866$		−	9.70858e17i		−	0.0782149i
$867$		0			0
$868$	3.66629e19	+	3.66629e19i	2.90970	+	2.90970i
$869$	−2.00143e18			−0.157657
$870$		0			0
$871$	−1.56673e19			−1.21584
$872$	−3.73121e19	−	3.73121e19i	−2.87403	−	2.87403i
$873$		0			0
$874$			3.30409e19i			2.50742i
$875$	1.81180e19	+	3.78263e18i	1.36477	+	0.284933i
$876$		0			0
$877$	−1.72283e19	+	1.72283e19i	−1.27863	+	1.27863i	−0.337191	+	0.941436i	$0.609477\pi$
$877$	−1.72283e19	+	1.72283e19i	−1.27863	+	1.27863i	−0.941436	+	0.337191i	$0.890523\pi$
$878$	−2.17104e18	+	2.17104e18i	−0.159938	+	0.159938i
$879$		0			0
$880$	1.09350e19	−	3.18824e19i	0.793745	−	2.31427i
$881$			1.46159e19i			1.05313i	0.850135	+	0.526565i	$0.176520\pi$
$881$			1.46159e19i			1.05313i	−0.850135	+	0.526565i	$0.823480\pi$
$882$		0			0
$883$	1.43336e19	+	1.43336e19i	1.01768	+	1.01768i	0.999841	+	0.0178372i	$0.00567807\pi$
$883$	1.43336e19	+	1.43336e19i	1.01768	+	1.01768i	0.0178372	+	0.999841i	$0.494322\pi$
$884$	3.33552e19			2.35085
$885$		0			0
$886$	2.04342e19			1.41918
$887$	6.19080e18	+	6.19080e18i	0.426819	+	0.426819i	0.887543	−	0.460725i	$-0.152410\pi$
$887$	6.19080e18	+	6.19080e18i	0.426819	+	0.426819i	−0.460725	+	0.887543i	$0.652410\pi$
$888$		0			0
$889$		−	2.05204e19i		−	1.39420i
$890$	−1.95972e19	−	4.00575e19i	−1.32178	−	2.70178i
$891$		0			0
$892$	−2.74667e19	+	2.74667e19i	−1.82572	+	1.82572i
$893$	2.05778e19	−	2.05778e19i	1.35789	−	1.35789i
$894$		0			0
$895$	−8.22340e18	−	2.82045e18i	−0.534813	−	0.183430i
$896$			2.17853e19i			1.40657i
$897$		0			0
$898$	1.69024e19	+	1.69024e19i	1.07560	+	1.07560i
$899$	−1.58884e19			−1.00379
$900$		0			0
$901$	1.81794e19			1.13206
$902$	2.09550e19	+	2.09550e19i	1.29552	+	1.29552i
$903$		0			0
$904$		−	3.65029e19i		−	2.22450i
$905$	−2.14271e19	−	7.34905e18i	−1.29643	−	0.444647i
$906$		0			0
$907$	1.07470e19	−	1.07470e19i	0.640976	−	0.640976i	−0.309820	−	0.950795i	$-0.600269\pi$
$907$	1.07470e19	−	1.07470e19i	0.640976	−	0.640976i	0.950795	+	0.309820i	$0.100269\pi$
$908$	3.27127e19	−	3.27127e19i	1.93713	−	1.93713i
$909$		0			0
$910$	1.35423e19	+	2.76810e19i	0.790537	+	1.61589i
$911$		−	7.14808e18i		−	0.414305i	−0.978309	−	0.207152i	$-0.933580\pi$
$911$		−	7.14808e18i		−	0.414305i	0.978309	−	0.207152i	$-0.0664196\pi$
$912$		0			0
$913$	−1.32494e19	−	1.32494e19i	−0.757072	−	0.757072i
$914$	2.40851e19			1.36646
$915$		0			0
$916$	−3.99358e19			−2.23379
$917$	−8.64691e18	−	8.64691e18i	−0.480242	−	0.480242i
$918$		0			0
$919$			1.48462e19i			0.812952i	0.913661	+	0.406476i	$0.133243\pi$
$919$			1.48462e19i			0.812952i	−0.913661	+	0.406476i	$0.866757\pi$
$920$	1.63288e19	−	4.76089e19i	0.887839	−	2.58861i
$921$		0			0
$922$	1.48690e18	−	1.48690e18i	0.0797131	−	0.0797131i
$923$	−2.29392e18	+	2.29392e18i	−0.122114	+	0.122114i
$924$		0			0
$925$	−2.17392e19	+	2.72249e18i	−1.14110	+	0.142904i
$926$		−	3.96211e19i		−	2.06517i
$927$		0			0
$928$	−3.00768e19	−	3.00768e19i	−1.54586	−	1.54586i
$929$	−3.08858e19			−1.57636			−0.788182	−	0.615442i	$-0.788978\pi$
$929$	−3.08858e19			−1.57636			−0.788182	+	0.615442i	$0.788978\pi$
$930$		0			0
$931$	−2.63390e19			−1.32564
$932$	−1.76932e19	−	1.76932e19i	−0.884307	−	0.884307i
$933$		0			0
$934$		−	5.85326e19i		−	2.88498i
$935$	−2.10549e19	+	1.03006e19i	−1.03057	+	0.504183i
$936$		0			0
$937$	1.26169e19	−	1.26169e19i	0.609041	−	0.609041i	−0.333655	−	0.942695i	$-0.608282\pi$
$937$	1.26169e19	−	1.26169e19i	0.609041	−	0.609041i	0.942695	+	0.333655i	$0.108282\pi$
$938$	−6.84640e19	+	6.84640e19i	−3.28204	+	3.28204i
$939$		0			0
$940$	−6.57956e19	+	3.21889e19i	−3.11075	+	1.52186i
$941$		−	7.19140e18i		−	0.337661i	−0.985645	−	0.168831i	$-0.946001\pi$
$941$		−	7.19140e18i		−	0.337661i	0.985645	−	0.168831i	$-0.0539991\pi$
$942$		0			0
$943$	1.66571e19	+	1.66571e19i	0.771389	+	0.771389i
$944$	−6.68304e18			−0.307366
$945$		0			0
$946$	1.21246e19			0.550017
$947$	2.67888e19	+	2.67888e19i	1.20692	+	1.20692i	0.972019	+	0.234901i	$0.0754764\pi$
$947$	2.67888e19	+	2.67888e19i	1.20692	+	1.20692i	0.234901	+	0.972019i	$0.424524\pi$
$948$		0			0
$949$			1.44414e19i			0.641770i
$950$	−5.93543e19	+	7.43319e18i	−2.61968	+	0.328074i
$951$		0			0
$952$	8.82148e19	−	8.82148e19i	3.84062	−	3.84062i
$953$	−9.23420e18	+	9.23420e18i	−0.399296	+	0.399296i	−0.877985	−	0.478688i	$-0.841112\pi$
$953$	−9.23420e18	+	9.23420e18i	−0.399296	+	0.399296i	0.478688	+	0.877985i	$0.341112\pi$
$954$		0			0
$955$	5.67159e18	−	1.65363e19i	0.241926	−	0.705367i
$956$		−	4.94563e19i		−	2.09530i
$957$		0			0
$958$	3.19252e19	+	3.19252e19i	1.33431	+	1.33431i
$959$	2.96743e19			1.23185
$960$		0			0
$961$	8.73438e18			0.357709
$962$	−2.57934e19	−	2.57934e19i	−1.04923	−	1.04923i
$963$		0			0
$964$		−	2.12387e19i		−	0.852369i
$965$	1.49852e19	+	3.06304e19i	0.597359	+	1.22103i
$966$		0			0
$967$	−2.90818e19	+	2.90818e19i	−1.14380	+	1.14380i	−0.156047	+	0.987750i	$0.549875\pi$
$967$	−2.90818e19	+	2.90818e19i	−1.14380	+	1.14380i	−0.987750	+	0.156047i	$0.950125\pi$
$968$	1.45977e19	−	1.45977e19i	0.570288	−	0.570288i
$969$		0			0
$970$	−1.73826e18	−	5.96187e17i	−0.0670037	−	0.0229808i
$971$			1.22242e19i			0.468052i	0.972230	+	0.234026i	$0.0751900\pi$
$971$			1.22242e19i			0.468052i	−0.972230	+	0.234026i	$0.924810\pi$
$972$		0			0
$973$	−6.71735e18	−	6.71735e18i	−0.253784	−	0.253784i
$974$	5.74315e19			2.15535
$975$		0			0
$976$	6.81083e19			2.52218
$977$	−1.40631e19	−	1.40631e19i	−0.517329	−	0.517329i	0.399433	−	0.916762i	$-0.369207\pi$
$977$	−1.40631e19	−	1.40631e19i	−0.517329	−	0.517329i	−0.916762	+	0.399433i	$0.869207\pi$
$978$		0			0
$979$			3.74075e19i			1.35791i
$980$	6.27085e19	+	2.15077e19i	2.26129	+	0.775576i
$981$		0			0
$982$	2.32617e19	−	2.32617e19i	0.827783	−	0.827783i
$983$	−1.99319e18	+	1.99319e18i	−0.0704613	+	0.0704613i	−0.741459	−	0.670998i	$-0.765866\pi$
$983$	−1.99319e18	+	1.99319e18i	−0.0704613	+	0.0704613i	0.670998	+	0.741459i	$0.265866\pi$
$984$		0			0
$985$	−5.23685e18	−	1.07043e19i	−0.182698	−	0.373443i
$986$			6.31664e19i			2.18920i
$987$		0			0
$988$	−5.04905e19	−	5.04905e19i	−1.72699	−	1.72699i
$989$	9.63784e18			0.327494
$990$		0			0
$991$	4.82498e19			1.61814			0.809071	−	0.587711i	$-0.199971\pi$
$991$	4.82498e19			1.61814			0.809071	+	0.587711i	$0.199971\pi$
$992$	6.27567e19	+	6.27567e19i	2.09090	+	2.09090i
$993$		0			0
$994$			2.00483e19i			0.659273i
$995$	8.79947e18	−	2.56560e19i	0.287479	−	0.838183i
$996$		0			0
$997$	−1.37567e19	+	1.37567e19i	−0.443603	+	0.443603i	−0.893221	−	0.449618i	$-0.851560\pi$
$997$	−1.37567e19	+	1.37567e19i	−0.443603	+	0.443603i	0.449618	+	0.893221i	$0.351560\pi$
$998$	−5.08471e19	+	5.08471e19i	−1.62899	+	1.62899i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.14.f.a.8.2	✓	52
3.2	odd	2	inner	45.14.f.a.8.25	yes	52
5.2	odd	4	inner	45.14.f.a.17.25	yes	52
15.2	even	4	inner	45.14.f.a.17.2	yes	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.14.f.a.8.2	✓	52	1.1	even	1	trivial
45.14.f.a.8.25	yes	52	3.2	odd	2	inner
45.14.f.a.17.2	yes	52	15.2	even	4	inner
45.14.f.a.17.25	yes	52	5.2	odd	4	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	45.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-120.296 - 120.296i) q^{2} +20750.4i q^{4} +(33048.8 + 11335.0i) q^{5} +(306864. - 306864. i) q^{7} +(1.51073e6 - 1.51073e6i) q^{8} +(-2.61208e6 - 5.33921e6i) q^{10} +4.98599e6i q^{11} +(-8.44751e6 - 8.44751e6i) q^{13} -7.38292e7 q^{14} -1.93484e8 q^{16} +(9.51433e7 + 9.51433e7i) q^{17} -2.88041e8i q^{19} +(-2.35207e8 + 6.85776e8i) q^{20} +(5.99796e8 - 5.99796e8i) q^{22} +(4.76777e8 - 4.76777e8i) q^{23} +(9.63737e8 + 7.49217e8i) q^{25} +2.03241e9i q^{26} +(6.36755e9 + 6.36755e9i) q^{28} -2.75947e9 q^{29} +5.75777e9 q^{31} +(1.08995e10 + 1.08995e10i) q^{32} -2.28908e10i q^{34} +(1.36198e10 - 6.66316e9i) q^{35} +(-1.26910e10 + 1.26910e10i) q^{37} +(-3.46503e10 + 3.46503e10i) q^{38} +(6.70520e10 - 3.28036e10i) q^{40} +3.49369e10i q^{41} +(1.01073e10 + 1.01073e10i) q^{43} -1.03461e11 q^{44} -1.14709e11 q^{46} +(7.14406e10 + 7.14406e10i) q^{47} -9.14417e10i q^{49} +(-2.58060e10 - 2.06062e11i) q^{50} +(1.75290e11 - 1.75290e11i) q^{52} +(9.55368e10 - 9.55368e10i) q^{53} +(-5.65163e10 + 1.64781e11i) q^{55} -9.27178e11i q^{56} +(3.31954e11 + 3.31954e11i) q^{58} +3.45406e10 q^{59} -3.52010e11 q^{61} +(-6.92639e11 - 6.92639e11i) q^{62} -1.03731e12i q^{64} +(-1.83427e11 - 3.74933e11i) q^{65} +(9.27329e11 - 9.27329e11i) q^{67} +(-1.97426e12 + 1.97426e12i) q^{68} +(-2.43996e12 - 8.36856e11i) q^{70} -2.71549e11i q^{71} +(-8.54773e11 - 8.54773e11i) q^{73} +3.05337e12 q^{74} +5.97697e12 q^{76} +(1.53002e12 + 1.53002e12i) q^{77} +4.01411e11i q^{79} +(-6.39440e12 - 2.19314e12i) q^{80} +(4.20278e12 - 4.20278e12i) q^{82} +(-2.65734e12 + 2.65734e12i) q^{83} +(2.06591e12 + 4.22282e12i) q^{85} -2.43174e12i q^{86} +(7.53249e12 + 7.53249e12i) q^{88} +7.50253e12 q^{89} -5.18447e12 q^{91} +(9.89332e12 + 9.89332e12i) q^{92} -1.71881e13i q^{94} +(3.26495e12 - 9.51939e12i) q^{95} +(2.18614e11 - 2.18614e11i) q^{97} +(-1.10001e13 + 1.10001e13i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 325928 q^{7} - 1271408 q^{10} - 14021276 q^{13} - 1225161608 q^{16} + 2938227248 q^{22} + 2536151056 q^{25} + 1169060768 q^{28} + 16189871792 q^{31} - 37236688556 q^{37} - 55693852632 q^{40} + 75180385888 q^{43}+ \cdots + 8822313963316 q^{97}+O(q^{100}) \) 52 * q - 325928 * q^7 - 1271408 * q^10 - 14021276 * q^13 - 1225161608 * q^16 + 2938227248 * q^22 + 2536151056 * q^25 + 1169060768 * q^28 + 16189871792 * q^31 - 37236688556 * q^37 - 55693852632 * q^40 + 75180385888 * q^43 - 154908345088 * q^46 + 1397910700024 * q^52 - 495232924312 * q^55 + 2026205726832 * q^58 + 394585459616 * q^61 + 4547875952656 * q^67 - 13248903926736 * q^70 + 12810003310564 * q^73 - 10698307157280 * q^76 + 37147387647776 * q^82 - 18754575936752 * q^85 + 50209944877536 * q^88 - 44737049270608 * q^91 + 8822313963316 * q^97

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