LMFDB - Embedded newform 45.9.g.b.28.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,9,Mod(28,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, 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:	$18.3320374528$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3006 x^{14} + 3660359 x^{12} + 2360769624 x^{10} + 888292333775 x^{8} + 201214811046486 x^{6} + \cdots + 60\!\cdots\!84 $ x^16 + 3006x^14 + 3660359x^12 + 2360769624x^10 + 888292333775x^8 + 201214811046486x^6 + 26986219156502761x^4 + 1971352965206494164*x^2 + 60370178541549816384
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{12}\cdot 5^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			28.8
Root			$19.4315i$ of defining polynomial
Character	$\chi$	$=$	45.28
Dual form			45.9.g.b.37.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(21.5835 - 21.5835i) q^{2} -675.695i q^{4} +(-47.4620 - 623.195i) q^{5} +(-2752.21 + 2752.21i) q^{7} +(-9058.50 - 9058.50i) q^{8} +O(q^{10})$	\(q+(21.5835 - 21.5835i) q^{2} -675.695i q^{4} +(-47.4620 - 623.195i) q^{5} +(-2752.21 + 2752.21i) q^{7} +(-9058.50 - 9058.50i) q^{8} +(-14475.1 - 12426.3i) q^{10} +13148.8 q^{11} +(-5875.06 - 5875.06i) q^{13} +118804. i q^{14} -218050. q^{16} +(59203.2 - 59203.2i) q^{17} -36141.3i q^{19} +(-421090. + 32069.8i) q^{20} +(283798. - 283798. i) q^{22} +(-166503. - 166503. i) q^{23} +(-386120. + 59156.1i) q^{25} -253609. q^{26} +(1.85965e6 + 1.85965e6i) q^{28} +87622.9i q^{29} +175190. q^{31} +(-2.38731e6 + 2.38731e6i) q^{32} -2.55562e6i q^{34} +(1.84579e6 + 1.58454e6i) q^{35} +(2.42934e6 - 2.42934e6i) q^{37} +(-780055. - 780055. i) q^{38} +(-5.21528e6 + 6.07515e6i) q^{40} -1.94648e6 q^{41} +(-388178. - 388178. i) q^{43} -8.88460e6i q^{44} -7.18744e6 q^{46} +(2.33832e6 - 2.33832e6i) q^{47} -9.38447e6i q^{49} +(-7.05702e6 + 9.61061e6i) q^{50} +(-3.96975e6 + 3.96975e6i) q^{52} +(-4.90768e6 - 4.90768e6i) q^{53} +(-624069. - 8.19429e6i) q^{55} +4.98617e7 q^{56} +(1.89121e6 + 1.89121e6i) q^{58} +1.26334e7i q^{59} +1.65637e7 q^{61} +(3.78121e6 - 3.78121e6i) q^{62} +4.72323e7i q^{64} +(-3.38247e6 + 3.94015e6i) q^{65} +(1.57973e7 - 1.57973e7i) q^{67} +(-4.00033e7 - 4.00033e7i) q^{68} +(7.40384e7 - 5.63869e6i) q^{70} +3.42203e7 q^{71} +(-3.22549e7 - 3.22549e7i) q^{73} -1.04867e8i q^{74} -2.44205e7 q^{76} +(-3.61883e7 + 3.61883e7i) q^{77} +4.43397e7i q^{79} +(1.03491e7 + 1.35888e8i) q^{80} +(-4.20118e7 + 4.20118e7i) q^{82} +(4.14392e7 + 4.14392e7i) q^{83} +(-3.97050e7 - 3.40852e7i) q^{85} -1.67565e7 q^{86} +(-1.19109e8 - 1.19109e8i) q^{88} -2.07571e7i q^{89} +3.23387e7 q^{91} +(-1.12505e8 + 1.12505e8i) q^{92} -1.00938e8i q^{94} +(-2.25231e7 + 1.71534e6i) q^{95} +(-4.79885e6 + 4.79885e6i) q^{97} +(-2.02550e8 - 2.02550e8i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4220 q^{7}+O(q^{10}) $ 16 * q - 4220 * q^7	$ 16 q - 4220 q^{7} - 47000 q^{10} - 37940 q^{13} - 508916 q^{16} + 844700 q^{22} - 1664300 q^{25} + 6009380 q^{28} - 944752 q^{31} + 10616140 q^{37} - 17493600 q^{40} + 4050760 q^{43} - 34233160 q^{46} + 7180240 q^{52} + 17430500 q^{55} - 27842100 q^{58} + 32032232 q^{61} + 75463480 q^{67} + 272773500 q^{70} - 198258320 q^{73} - 196046088 q^{76} + 172139600 q^{82} + 183614500 q^{85} - 624395100 q^{88} - 477271600 q^{91} + 662476480 q^{97}+O(q^{100}) $ 16 * q - 4220 * q^7 - 47000 * q^10 - 37940 * q^13 - 508916 * q^16 + 844700 * q^22 - 1664300 * q^25 + 6009380 * q^28 - 944752 * q^31 + 10616140 * q^37 - 17493600 * q^40 + 4050760 * q^43 - 34233160 * q^46 + 7180240 * q^52 + 17430500 * q^55 - 27842100 * q^58 + 32032232 * q^61 + 75463480 * q^67 + 272773500 * q^70 - 198258320 * q^73 - 196046088 * q^76 + 172139600 * q^82 + 183614500 * q^85 - 624395100 * q^88 - 477271600 * q^91 + 662476480 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	21.5835	−	21.5835i	1.34897	−	1.34897i	0.462186	−	0.886783i	$-0.347065\pi$
$2$	21.5835	−	21.5835i	1.34897	−	1.34897i	0.886783	−	0.462186i	$-0.152935\pi$
$3$		0			0
$3$		0			0
$4$		−	675.695i		−	2.63943i
$5$	−47.4620	−	623.195i	−0.0759391	−	0.997112i
$5$	−47.4620	−	623.195i	−0.0759391	−	0.997112i
$6$		0			0
$7$	−2752.21	+	2752.21i	−1.14627	+	1.14627i	−0.158995	+	0.987279i	$0.550826\pi$
$7$	−2752.21	+	2752.21i	−1.14627	+	1.14627i	−0.987279	+	0.158995i	$0.949174\pi$
$8$	−9058.50	−	9058.50i	−2.21155	−	2.21155i
$9$		0			0
$10$	−14475.1	−	12426.3i	−1.44751	−	1.24263i
$11$	13148.8			0.898083			0.449041	−	0.893511i	$-0.351766\pi$
$11$	13148.8			0.898083			0.449041	+	0.893511i	$0.351766\pi$
$12$		0			0
$13$	−5875.06	−	5875.06i	−0.205702	−	0.205702i	0.596736	−	0.802438i	$-0.296464\pi$
$13$	−5875.06	−	5875.06i	−0.205702	−	0.205702i	−0.802438	+	0.596736i	$0.796464\pi$
$14$			118804.i			3.09258i
$15$		0			0
$16$	−218050.			−3.32718
$17$	59203.2	−	59203.2i	0.708842	−	0.708842i	−0.257450	−	0.966292i	$-0.582882\pi$
$17$	59203.2	−	59203.2i	0.708842	−	0.708842i	0.966292	+	0.257450i	$0.0828822\pi$
$18$		0			0
$19$		−	36141.3i		−	0.277325i	−0.990340	−	0.138662i	$-0.955720\pi$
$19$		−	36141.3i		−	0.277325i	0.990340	−	0.138662i	$-0.0442803\pi$
$20$	−421090.	+	32069.8i	−2.63181	+	0.200436i
$21$		0			0
$22$	283798.	−	283798.i	1.21149	−	1.21149i
$23$	−166503.	−	166503.i	−0.594991	−	0.594991i	0.343984	−	0.938975i	$-0.388223\pi$
$23$	−166503.	−	166503.i	−0.594991	−	0.594991i	−0.938975	+	0.343984i	$0.888223\pi$
$24$		0			0
$25$	−386120.	+	59156.1i	−0.988466	+	0.151440i
$26$	−253609.			−0.554972
$27$		0			0
$28$	1.85965e6	+	1.85965e6i	3.02552	+	3.02552i
$29$			87622.9i			0.123887i	0.998080	+	0.0619435i	$0.0197299\pi$
$29$			87622.9i			0.123887i	−0.998080	+	0.0619435i	$0.980270\pi$
$30$		0			0
$31$	175190.			0.189698			0.0948489	−	0.995492i	$-0.469763\pi$
$31$	175190.			0.189698			0.0948489	+	0.995492i	$0.469763\pi$
$32$	−2.38731e6	+	2.38731e6i	−2.27672	+	2.27672i
$33$		0			0
$34$		−	2.55562e6i		−	1.91241i
$35$	1.84579e6	+	1.58454e6i	1.23001	+	1.05592i
$36$		0			0
$37$	2.42934e6	−	2.42934e6i	1.29623	−	1.29623i	0.365364	−	0.930865i	$-0.380944\pi$
$37$	2.42934e6	−	2.42934e6i	1.29623	−	1.29623i	0.930865	−	0.365364i	$-0.119056\pi$
$38$	−780055.	−	780055.i	−0.374103	−	0.374103i
$39$		0			0
$40$	−5.21528e6	+	6.07515e6i	−2.03722	+	2.37310i
$41$	−1.94648e6			−0.688833			−0.344416	−	0.938817i	$-0.611923\pi$
$41$	−1.94648e6			−0.688833			−0.344416	+	0.938817i	$0.611923\pi$
$42$		0			0
$43$	−388178.	−	388178.i	−0.113542	−	0.113542i	0.648053	−	0.761595i	$-0.275584\pi$
$43$	−388178.	−	388178.i	−0.113542	−	0.113542i	−0.761595	+	0.648053i	$0.775584\pi$
$44$		−	8.88460e6i		−	2.37043i
$45$		0			0
$46$	−7.18744e6			−1.60525
$47$	2.33832e6	−	2.33832e6i	0.479194	−	0.479194i	−0.425680	−	0.904874i	$-0.639965\pi$
$47$	2.33832e6	−	2.33832e6i	0.479194	−	0.479194i	0.904874	+	0.425680i	$0.139965\pi$
$48$		0			0
$49$		−	9.38447e6i		−	1.62789i
$50$	−7.05702e6	+	9.61061e6i	−1.12912	+	1.53770i
$51$		0			0
$52$	−3.96975e6	+	3.96975e6i	−0.542937	+	0.542937i
$53$	−4.90768e6	−	4.90768e6i	−0.621974	−	0.621974i	0.324062	−	0.946036i	$-0.394951\pi$
$53$	−4.90768e6	−	4.90768e6i	−0.621974	−	0.621974i	−0.946036	+	0.324062i	$0.894951\pi$
$54$		0			0
$55$	−624069.	−	8.19429e6i	−0.0681996	−	0.895489i
$56$	4.98617e7			5.07008
$57$		0			0
$58$	1.89121e6	+	1.89121e6i	0.167120	+	0.167120i
$59$			1.26334e7i			1.04259i	0.853376	+	0.521295i	$0.174551\pi$
$59$			1.26334e7i			1.04259i	−0.853376	+	0.521295i	$0.825449\pi$
$60$		0			0
$61$	1.65637e7			1.19629			0.598147	−	0.801387i	$-0.295904\pi$
$61$	1.65637e7			1.19629			0.598147	+	0.801387i	$0.295904\pi$
$62$	3.78121e6	−	3.78121e6i	0.255896	−	0.255896i
$63$		0			0
$64$			4.72323e7i			2.81526i
$65$	−3.38247e6	+	3.94015e6i	−0.189487	+	0.220729i
$66$		0			0
$67$	1.57973e7	−	1.57973e7i	0.783943	−	0.783943i	−0.196551	−	0.980494i	$-0.562974\pi$
$67$	1.57973e7	−	1.57973e7i	0.783943	−	0.783943i	0.980494	+	0.196551i	$0.0629741\pi$
$68$	−4.00033e7	−	4.00033e7i	−1.87094	−	1.87094i
$69$		0			0
$70$	7.40384e7	−	5.63869e6i	3.08365	−	0.234848i
$71$	3.42203e7			1.34664			0.673319	−	0.739353i	$-0.264868\pi$
$71$	3.42203e7			1.34664			0.673319	+	0.739353i	$0.264868\pi$
$72$		0			0
$73$	−3.22549e7	−	3.22549e7i	−1.13581	−	1.13581i	−0.989194	−	0.146613i	$-0.953163\pi$
$73$	−3.22549e7	−	3.22549e7i	−1.13581	−	1.13581i	−0.146613	−	0.989194i	$-0.546837\pi$
$74$		−	1.04867e8i		−	3.49715i
$75$		0			0
$76$	−2.44205e7			−0.731981
$77$	−3.61883e7	+	3.61883e7i	−1.02945	+	1.02945i
$78$		0			0
$79$			4.43397e7i			1.13837i	0.822208	+	0.569186i	$0.192742\pi$
$79$			4.43397e7i			1.13837i	−0.822208	+	0.569186i	$0.807258\pi$
$80$	1.03491e7	+	1.35888e8i	0.252663	+	3.31757i
$81$		0			0
$82$	−4.20118e7	+	4.20118e7i	−0.929214	+	0.929214i
$83$	4.14392e7	+	4.14392e7i	0.873171	+	0.873171i	0.992817	−	0.119646i	$-0.0381758\pi$
$83$	4.14392e7	+	4.14392e7i	0.873171	+	0.873171i	−0.119646	+	0.992817i	$0.538176\pi$
$84$		0			0
$85$	−3.97050e7	−	3.40852e7i	−0.760624	−	0.652966i
$86$	−1.67565e7			−0.306330
$87$		0			0
$88$	−1.19109e8	−	1.19109e8i	−1.98615	−	1.98615i
$89$		−	2.07571e7i		−	0.330831i	−0.986224	−	0.165416i	$-0.947103\pi$
$89$		−	2.07571e7i		−	0.330831i	0.986224	−	0.165416i	$-0.0528965\pi$
$90$		0			0
$91$	3.23387e7			0.471582
$92$	−1.12505e8	+	1.12505e8i	−1.57044	+	1.57044i
$93$		0			0
$94$		−	1.00938e8i		−	1.29284i
$95$	−2.25231e7	+	1.71534e6i	−0.276524	+	0.0210598i
$96$		0			0
$97$	−4.79885e6	+	4.79885e6i	−0.0542063	+	0.0542063i	−0.733690	−	0.679484i	$-0.762204\pi$
$97$	−4.79885e6	+	4.79885e6i	−0.0542063	+	0.0542063i	0.679484	+	0.733690i	$0.262204\pi$
$98$	−2.02550e8	−	2.02550e8i	−2.19598	−	2.19598i
$99$		0			0
$100$	3.99715e7	+	2.60899e8i	0.399715	+	2.60899i
$101$	9.15239e7			0.879527			0.439763	−	0.898114i	$-0.355062\pi$
$101$	9.15239e7			0.879527			0.439763	+	0.898114i	$0.355062\pi$
$102$		0			0
$103$	9.69070e7	+	9.69070e7i	0.861006	+	0.861006i	0.991455	−	0.130449i	$-0.0416419\pi$
$103$	9.69070e7	+	9.69070e7i	0.861006	+	0.861006i	−0.130449	+	0.991455i	$0.541642\pi$
$104$			1.06438e8i			0.909840i
$105$		0			0
$106$	−2.11850e8			−1.67805
$107$	5.01152e7	−	5.01152e7i	0.382327	−	0.382327i	−0.489613	−	0.871940i	$-0.662862\pi$
$107$	5.01152e7	−	5.01152e7i	0.382327	−	0.382327i	0.871940	+	0.489613i	$0.162862\pi$
$108$		0			0
$109$			9.70946e7i			0.687843i	0.938998	+	0.343921i	$0.111755\pi$
$109$			9.70946e7i			0.687843i	−0.938998	+	0.343921i	$0.888245\pi$
$110$	−1.90331e8	−	1.63392e8i	−1.29999	−	1.11599i
$111$		0			0
$112$	6.00119e8	−	6.00119e8i	3.81386	−	3.81386i
$113$	1.14708e8	+	1.14708e8i	0.703526	+	0.703526i	0.965166	−	0.261640i	$-0.0842632\pi$
$113$	1.14708e8	+	1.14708e8i	0.703526	+	0.703526i	−0.261640	+	0.965166i	$0.584263\pi$
$114$		0			0
$115$	−9.58613e7	+	1.11666e8i	−0.548090	+	0.638456i
$116$	5.92064e7			0.326992
$117$		0			0
$118$	2.72674e8	+	2.72674e8i	1.40642	+	1.40642i
$119$			3.25879e8i			1.62506i
$120$		0			0
$121$	−4.14672e7			−0.193448
$122$	3.57502e8	−	3.57502e8i	1.61376	−	1.61376i
$123$		0			0
$124$		−	1.18375e8i		−	0.500695i
$125$	5.51918e7	+	2.37820e8i	0.226066	+	0.974112i
$126$		0			0
$127$	−2.59623e8	+	2.59623e8i	−0.997993	+	0.997993i	−0.999998	−	0.00200471i	$-0.999362\pi$
$127$	−2.59623e8	+	2.59623e8i	−0.997993	+	0.997993i	0.00200471	+	0.999998i	$0.499362\pi$
$128$	4.08286e8	+	4.08286e8i	1.52098	+	1.52098i
$129$		0			0
$130$	1.20368e7	+	1.58048e8i	0.0421441	+	0.553369i
$131$	−3.05242e8			−1.03648			−0.518238	−	0.855237i	$-0.673412\pi$
$131$	−3.05242e8			−1.03648			−0.518238	+	0.855237i	$0.673412\pi$
$132$		0			0
$133$	9.94682e7	+	9.94682e7i	0.317891	+	0.317891i
$134$		−	6.81923e8i		−	2.11503i
$135$		0			0
$136$	−1.07258e9			−3.13527
$137$	6.96635e7	−	6.96635e7i	0.197753	−	0.197753i	−0.601283	−	0.799036i	$-0.705344\pi$
$137$	6.96635e7	−	6.96635e7i	0.197753	−	0.197753i	0.799036	+	0.601283i	$0.205344\pi$
$138$		0			0
$139$		−	4.11944e7i		−	0.110352i	−0.998477	−	0.0551759i	$-0.982428\pi$
$139$		−	4.11944e7i		−	0.110352i	0.998477	−	0.0551759i	$-0.0175720\pi$
$140$	1.07066e9	−	1.24719e9i	2.78703	−	3.24654i
$141$		0			0
$142$	7.38594e8	−	7.38594e8i	1.81657	−	1.81657i
$143$	−7.72501e7	−	7.72501e7i	−0.184737	−	0.184737i
$144$		0			0
$145$	5.46062e7	−	4.15876e6i	0.123529	−	0.00940787i
$146$	−1.39235e9			−3.06434
$147$		0			0
$148$	−1.64149e9	−	1.64149e9i	−3.42131	−	3.42131i
$149$			5.66140e8i			1.14863i	0.818636	+	0.574313i	$0.194731\pi$
$149$			5.66140e8i			1.14863i	−0.818636	+	0.574313i	$0.805269\pi$
$150$		0			0
$151$	−9.66981e7			−0.185999			−0.0929994	−	0.995666i	$-0.529645\pi$
$151$	−9.66981e7			−0.185999			−0.0929994	+	0.995666i	$0.529645\pi$
$152$	−3.27385e8	+	3.27385e8i	−0.613317	+	0.613317i
$153$		0			0
$154$			1.56214e9i			2.77739i
$155$	−8.31486e6	−	1.09178e8i	−0.0144055	−	0.189150i
$156$		0			0
$157$	−2.84498e8	+	2.84498e8i	−0.468253	+	0.468253i	−0.901348	−	0.433095i	$-0.857421\pi$
$157$	−2.84498e8	+	2.84498e8i	−0.468253	+	0.468253i	0.433095	+	0.901348i	$0.357421\pi$
$158$	9.57006e8	+	9.57006e8i	1.53563	+	1.53563i
$159$		0			0
$160$	1.60107e9	+	1.37446e9i	2.44304	+	2.09725i
$161$	9.16501e8			1.36405
$162$		0			0
$163$	−1.19600e8	−	1.19600e8i	−0.169426	−	0.169426i	0.617301	−	0.786727i	$-0.288226\pi$
$163$	−1.19600e8	−	1.19600e8i	−0.169426	−	0.169426i	−0.786727	+	0.617301i	$0.788226\pi$
$164$			1.31523e9i			1.81813i
$165$		0			0
$166$	1.78881e9			2.35576
$167$	5.01138e8	−	5.01138e8i	0.644305	−	0.644305i	−0.307305	−	0.951611i	$-0.599427\pi$
$167$	5.01138e8	−	5.01138e8i	0.644305	−	0.644305i	0.951611	+	0.307305i	$0.0994273\pi$
$168$		0			0
$169$		−	7.46698e8i		−	0.915373i
$170$	−1.59265e9	+	1.21295e8i	−1.90689	+	0.145227i
$171$		0			0
$172$	−2.62290e8	+	2.62290e8i	−0.299687	+	0.299687i
$173$	−8.62681e7	−	8.62681e7i	−0.0963087	−	0.0963087i	0.657311	−	0.753620i	$-0.271694\pi$
$173$	−8.62681e7	−	8.62681e7i	−0.0963087	−	0.0963087i	−0.753620	+	0.657311i	$0.771694\pi$
$174$		0			0
$175$	8.99871e8	−	1.22549e9i	0.959463	−	1.30665i
$176$	−2.86710e9			−2.98808
$177$		0			0
$178$	−4.48011e8	−	4.48011e8i	−0.446281	−	0.446281i
$179$			6.55883e8i			0.638872i	0.947608	+	0.319436i	$0.103494\pi$
$179$			6.55883e8i			0.638872i	−0.947608	+	0.319436i	$0.896506\pi$
$180$		0			0
$181$	1.22810e9			1.14424			0.572122	−	0.820168i	$-0.306120\pi$
$181$	1.22810e9			1.14424			0.572122	+	0.820168i	$0.306120\pi$
$182$	6.97983e8	−	6.97983e8i	0.636150	−	0.636150i
$183$		0			0
$184$			3.01653e9i			2.63170i
$185$	−1.62926e9	−	1.39865e9i	−1.39092	−	1.19405i
$186$		0			0
$187$	7.78452e8	−	7.78452e8i	0.636599	−	0.636599i
$188$	−1.57999e9	−	1.57999e9i	−1.26480	−	1.26480i
$189$		0			0
$190$	−4.49104e8	+	5.23150e8i	−0.344613	+	0.401431i
$191$	9.07068e8			0.681564			0.340782	−	0.940142i	$-0.389308\pi$
$191$	9.07068e8			0.681564			0.340782	+	0.940142i	$0.389308\pi$
$192$		0			0
$193$	−4.25882e8	−	4.25882e8i	−0.306945	−	0.306945i	0.536778	−	0.843723i	$-0.319641\pi$
$193$	−4.25882e8	−	4.25882e8i	−0.306945	−	0.306945i	−0.843723	+	0.536778i	$0.819641\pi$
$194$			2.07152e8i			0.146245i
$195$		0			0
$196$	−6.34104e9			−4.29671
$197$	−1.12753e9	+	1.12753e9i	−0.748626	+	0.748626i	−0.974221	−	0.225596i	$-0.927567\pi$
$197$	−1.12753e9	+	1.12753e9i	−0.748626	+	0.748626i	0.225596	+	0.974221i	$0.427567\pi$
$198$		0			0
$199$		−	2.18387e9i		−	1.39256i	−0.717771	−	0.696279i	$-0.754837\pi$
$199$		−	2.18387e9i		−	1.39256i	0.717771	−	0.696279i	$-0.245163\pi$
$200$	4.03353e9	+	2.96180e9i	2.52096	+	1.85112i
$201$		0			0
$202$	1.97541e9	−	1.97541e9i	1.18645	−	1.18645i
$203$	−2.41156e8	−	2.41156e8i	−0.142009	−	0.142009i
$204$		0			0
$205$	9.23836e7	+	1.21304e9i	0.0523094	+	0.686844i
$206$	4.18319e9			2.32294
$207$		0			0
$208$	1.28106e9	+	1.28106e9i	0.684408	+	0.684408i
$209$		−	4.75215e8i		−	0.249061i
$210$		0			0
$211$	−2.69138e9			−1.35783			−0.678915	−	0.734217i	$-0.737550\pi$
$211$	−2.69138e9			−1.35783			−0.678915	+	0.734217i	$0.737550\pi$
$212$	−3.31609e9	+	3.31609e9i	−1.64166	+	1.64166i
$213$		0			0
$214$		−	2.16332e9i		−	1.03149i
$215$	−2.23487e8	+	2.60335e8i	−0.104592	+	0.121837i
$216$		0			0
$217$	−4.82159e8	+	4.82159e8i	−0.217446	+	0.217446i
$218$	2.09564e9	+	2.09564e9i	0.927878	+	0.927878i
$219$		0			0
$220$	−5.53684e9	+	4.21681e8i	−2.36359	+	0.180008i
$221$	−6.95644e8			−0.291621
$222$		0			0
$223$	1.40724e9	+	1.40724e9i	0.569049	+	0.569049i	0.931862	−	0.362813i	$-0.118184\pi$
$223$	1.40724e9	+	1.40724e9i	0.569049	+	0.569049i	−0.362813	+	0.931862i	$0.618184\pi$
$224$		−	1.31407e10i		−	5.21949i
$225$		0			0
$226$	4.95160e9			1.89807
$227$	3.36682e9	−	3.36682e9i	1.26799	−	1.26799i	0.320868	−	0.947124i	$-0.396025\pi$
$227$	3.36682e9	−	3.36682e9i	1.26799	−	1.26799i	0.947124	−	0.320868i	$-0.103975\pi$
$228$		0			0
$229$			8.58581e8i			0.312205i	0.987741	+	0.156102i	$0.0498930\pi$
$229$			8.58581e8i			0.312205i	−0.987741	+	0.156102i	$0.950107\pi$
$230$	3.41130e8	+	4.47918e9i	0.121901	+	1.60061i
$231$		0			0
$232$	7.93732e8	−	7.93732e8i	0.273982	−	0.273982i
$233$	−3.47469e9	−	3.47469e9i	−1.17894	−	1.17894i	−0.980015	−	0.198926i	$-0.936255\pi$
$233$	−3.47469e9	−	3.47469e9i	−1.17894	−	1.17894i	−0.198926	−	0.980015i	$-0.563745\pi$
$234$		0			0
$235$	−1.56821e9	−	1.34625e9i	−0.514200	−	0.441421i
$236$	8.53636e9			2.75185
$237$		0			0
$238$	7.03360e9	+	7.03360e9i	2.19215	+	2.19215i
$239$			8.14065e8i			0.249498i	0.992188	+	0.124749i	$0.0398126\pi$
$239$			8.14065e8i			0.249498i	−0.992188	+	0.124749i	$0.960187\pi$
$240$		0			0
$241$	2.96616e9			0.879279			0.439639	−	0.898174i	$-0.355106\pi$
$241$	2.96616e9			0.879279			0.439639	+	0.898174i	$0.355106\pi$
$242$	−8.95008e8	+	8.95008e8i	−0.260955	+	0.260955i
$243$		0			0
$244$		−	1.11920e10i		−	3.15754i
$245$	−5.84836e9	+	4.45405e8i	−1.62319	+	0.123621i
$246$		0			0
$247$	−2.12332e8	+	2.12332e8i	−0.0570463	+	0.0570463i
$248$	−1.58696e9	−	1.58696e9i	−0.419526	−	0.419526i
$249$		0			0
$250$	6.32423e9	+	3.94176e9i	1.61900	+	1.00909i
$251$	2.86118e9			0.720859			0.360430	−	0.932786i	$-0.382630\pi$
$251$	2.86118e9			0.720859			0.360430	+	0.932786i	$0.382630\pi$
$252$		0			0
$253$	−2.18932e9	−	2.18932e9i	−0.534351	−	0.534351i
$254$			1.12071e10i			2.69252i
$255$		0			0
$256$	5.53303e9			1.28826
$257$	4.87604e9	−	4.87604e9i	1.11773	−	1.11773i	0.125651	−	0.992075i	$-0.459898\pi$
$257$	4.87604e9	−	4.87604e9i	1.11773	−	1.11773i	0.992075	−	0.125651i	$-0.0401019\pi$
$258$		0			0
$259$			1.33721e10i			2.97167i
$260$	2.66234e9	+	2.28552e9i	0.582600	+	0.500139i
$261$		0			0
$262$	−6.58819e9	+	6.58819e9i	−1.39817	+	1.39817i
$263$	−5.25726e9	−	5.25726e9i	−1.09884	−	1.09884i	−0.994546	−	0.104299i	$-0.966740\pi$
$263$	−5.25726e9	−	5.25726e9i	−1.09884	−	1.09884i	−0.104299	−	0.994546i	$-0.533260\pi$
$264$		0			0
$265$	−2.82551e9	+	3.29137e9i	−0.572946	+	0.667410i
$266$	4.29374e9			0.857649
$267$		0			0
$268$	−1.06742e10	−	1.06742e10i	−2.06917	−	2.06917i
$269$			9.59406e9i			1.83229i	0.400851	+	0.916143i	$0.368714\pi$
$269$			9.59406e9i			1.83229i	−0.400851	+	0.916143i	$0.631286\pi$
$270$		0			0
$271$	2.85777e8			0.0529847			0.0264924	−	0.999649i	$-0.491566\pi$
$271$	2.85777e8			0.0529847			0.0264924	+	0.999649i	$0.491566\pi$
$272$	−1.29093e10	+	1.29093e10i	−2.35845	+	2.35845i
$273$		0			0
$274$		−	3.00717e9i		−	0.533525i
$275$	−5.07702e9	+	7.77834e8i	−0.887725	+	0.136005i
$276$		0			0
$277$	−3.13452e8	+	3.13452e8i	−0.0532417	+	0.0532417i	−0.733226	−	0.679985i	$-0.761986\pi$
$277$	−3.13452e8	+	3.13452e8i	−0.0532417	+	0.0532417i	0.679985	+	0.733226i	$0.261986\pi$
$278$	−8.89120e8	−	8.89120e8i	−0.148861	−	0.148861i
$279$		0			0
$280$	−2.36653e9	−	3.10736e10i	−0.385018	−	5.05544i
$281$	4.58192e8			0.0734889			0.0367445	−	0.999325i	$-0.488301\pi$
$281$	4.58192e8			0.0734889			0.0367445	+	0.999325i	$0.488301\pi$
$282$		0			0
$283$	−1.99336e9	−	1.99336e9i	−0.310770	−	0.310770i	0.534438	−	0.845208i	$-0.320523\pi$
$283$	−1.99336e9	−	1.99336e9i	−0.310770	−	0.310770i	−0.845208	+	0.534438i	$0.820523\pi$
$284$		−	2.31225e10i		−	3.55436i
$285$		0			0
$286$	−3.33466e9			−0.498410
$287$	5.35711e9	−	5.35711e9i	0.789592	−	0.789592i
$288$		0			0
$289$		−	3.42755e7i		−	0.00491351i
$290$	1.08883e9	−	1.26835e9i	0.153946	−	0.179328i
$291$		0			0
$292$	−2.17945e10	+	2.17945e10i	−2.99789	+	2.99789i
$293$	3.43513e9	+	3.43513e9i	0.466093	+	0.466093i	0.900646	−	0.434553i	$-0.143094\pi$
$293$	3.43513e9	+	3.43513e9i	0.466093	+	0.466093i	−0.434553	+	0.900646i	$0.643094\pi$
$294$		0			0
$295$	7.87311e9	−	5.99608e8i	1.03958	−	0.0791735i
$296$	−4.40124e10			−5.73334
$297$		0			0
$298$	1.22193e10	+	1.22193e10i	1.54946	+	1.54946i
$299$			1.95643e9i			0.244782i
$300$		0			0
$301$	2.13669e9			0.260301
$302$	−2.08708e9	+	2.08708e9i	−0.250907	+	0.250907i
$303$		0			0
$304$			7.88061e9i			0.922710i
$305$	−7.86145e8	−	1.03224e10i	−0.0908455	−	1.19284i
$306$		0			0
$307$	−3.57429e8	+	3.57429e8i	−0.0402380	+	0.0402380i	−0.726939	−	0.686702i	$-0.759058\pi$
$307$	−3.57429e8	+	3.57429e8i	−0.0402380	+	0.0402380i	0.686702	+	0.726939i	$0.259058\pi$
$308$	2.44522e10	+	2.44522e10i	2.71716	+	2.71716i
$309$		0			0
$310$	−2.53590e9	−	2.17697e9i	−0.274590	−	0.235725i
$311$	1.29610e10			1.38547			0.692736	−	0.721191i	$-0.256405\pi$
$311$	1.29610e10			1.38547			0.692736	+	0.721191i	$0.256405\pi$
$312$		0			0
$313$	−7.76022e9	−	7.76022e9i	−0.808531	−	0.808531i	0.175880	−	0.984412i	$-0.443723\pi$
$313$	−7.76022e9	−	7.76022e9i	−0.808531	−	0.808531i	−0.984412	+	0.175880i	$0.943723\pi$
$314$			1.22809e10i			1.26332i
$315$		0			0
$316$	2.99601e10			3.00466
$317$	−8.96390e9	+	8.96390e9i	−0.887688	+	0.887688i	−0.994301	−	0.106613i	$-0.965999\pi$
$317$	−8.96390e9	+	8.96390e9i	−0.887688	+	0.887688i	0.106613	+	0.994301i	$0.465999\pi$
$318$		0			0
$319$			1.15214e9i			0.111261i
$320$	2.94349e10	−	2.24174e9i	2.80713	−	0.213789i
$321$		0			0
$322$	1.97813e10	−	1.97813e10i	1.84006	−	1.84006i
$323$	−2.13968e9	−	2.13968e9i	−0.196579	−	0.196579i
$324$		0			0
$325$	2.61602e9	+	1.92093e9i	0.234481	+	0.172178i
$326$	−5.16278e9			−0.457102
$327$		0			0
$328$	1.76322e10	+	1.76322e10i	1.52339	+	1.52339i
$329$			1.28711e10i			1.09858i
$330$		0			0
$331$	−1.02611e10			−0.854832			−0.427416	−	0.904055i	$-0.640576\pi$
$331$	−1.02611e10			−0.854832			−0.427416	+	0.904055i	$0.640576\pi$
$332$	2.80003e10	−	2.80003e10i	2.30468	−	2.30468i
$333$		0			0
$334$		−	2.16326e10i		−	1.73830i
$335$	−1.05946e10	−	9.09505e9i	−0.841211	−	0.722147i
$336$		0			0
$337$	−1.22083e9	+	1.22083e9i	−0.0946531	+	0.0946531i	−0.752848	−	0.658195i	$-0.771320\pi$
$337$	−1.22083e9	+	1.22083e9i	−0.0946531	+	0.0946531i	0.658195	+	0.752848i	$0.271320\pi$
$338$	−1.61164e10	−	1.61164e10i	−1.23481	−	1.23481i
$339$		0			0
$340$	−2.30312e10	+	2.68285e10i	−1.72346	+	2.00762i
$341$	2.30354e9			0.170364
$342$		0			0
$343$	9.96208e9	+	9.96208e9i	0.719736	+	0.719736i
$344$			7.03262e9i			0.502208i
$345$		0			0
$346$	−3.72393e9			−0.259835
$347$	8.09716e9	−	8.09716e9i	0.558489	−	0.558489i	−0.370388	−	0.928877i	$-0.620775\pi$
$347$	8.09716e9	−	8.09716e9i	0.558489	−	0.558489i	0.928877	+	0.370388i	$0.120775\pi$
$348$		0			0
$349$		−	2.20331e10i		−	1.48516i	−0.669755	−	0.742582i	$-0.733601\pi$
$349$		−	2.20331e10i		−	1.48516i	0.669755	−	0.742582i	$-0.266399\pi$
$350$	−7.02802e9	−	4.58728e10i	−0.468339	−	3.05691i
$351$		0			0
$352$	−3.13904e10	+	3.13904e10i	−2.04468	+	2.04468i
$353$	−4.00135e8	−	4.00135e8i	−0.0257696	−	0.0257696i	0.694105	−	0.719874i	$-0.255801\pi$
$353$	−4.00135e8	−	4.00135e8i	−0.0257696	−	0.0257696i	−0.719874	+	0.694105i	$0.755801\pi$
$354$		0			0
$355$	−1.62416e9	−	2.13259e10i	−0.102262	−	1.34275i
$356$	−1.40255e10			−0.873207
$357$		0			0
$358$	1.41562e10	+	1.41562e10i	0.861819	+	0.861819i
$359$			2.82827e10i			1.70272i	0.524580	+	0.851361i	$0.324222\pi$
$359$			2.82827e10i			1.70272i	−0.524580	+	0.851361i	$0.675778\pi$
$360$		0			0
$361$	1.56774e10			0.923091
$362$	2.65067e10	−	2.65067e10i	1.54355	−	1.54355i
$363$		0			0
$364$		−	2.18511e10i		−	1.24471i
$365$	−1.85702e10	+	2.16320e10i	−1.04628	+	1.21878i
$366$		0			0
$367$	2.11716e10	−	2.11716e10i	1.16705	−	1.16705i	0.184154	−	0.982897i	$-0.441046\pi$
$367$	2.11716e10	−	2.11716e10i	1.16705	−	1.16705i	0.982897	−	0.184154i	$-0.0589544\pi$
$368$	3.63060e10	+	3.63060e10i	1.97964	+	1.97964i
$369$		0			0
$370$	−6.53529e10	+	4.97721e9i	−3.48705	+	0.265570i
$371$	2.70139e10			1.42591
$372$		0			0
$373$	−1.95645e10	−	1.95645e10i	−1.01073	−	1.01073i	−0.999942	−	0.0107842i	$-0.996567\pi$
$373$	−1.95645e10	−	1.95645e10i	−1.01073	−	1.01073i	−0.0107842	−	0.999942i	$-0.503433\pi$
$374$		−	3.36035e10i		−	1.71750i
$375$		0			0
$376$	−4.23632e10			−2.11952
$377$	5.14790e8	−	5.14790e8i	0.0254838	−	0.0254838i
$378$		0			0
$379$		−	1.24124e9i		−	0.0601587i	−0.999548	−	0.0300793i	$-0.990424\pi$
$379$		−	1.24124e9i		−	0.0601587i	0.999548	−	0.0300793i	$-0.00957599\pi$
$380$	1.15904e9	+	1.52187e10i	0.0555860	+	0.729867i
$381$		0			0
$382$	1.95777e10	−	1.95777e10i	0.919408	−	0.919408i
$383$	2.10965e10	+	2.10965e10i	0.980428	+	0.980428i	0.999812	−	0.0193839i	$-0.00617048\pi$
$383$	2.10965e10	+	2.10965e10i	0.980428	+	0.980428i	−0.0193839	+	0.999812i	$0.506170\pi$
$384$		0			0
$385$	2.42699e10	+	2.08348e10i	1.10465	+	0.948301i
$386$	−1.83841e10			−0.828118
$387$		0			0
$388$	3.24256e9	+	3.24256e9i	0.143074	+	0.143074i
$389$			2.26757e10i			0.990289i	0.868811	+	0.495144i	$0.164885\pi$
$389$			2.26757e10i			0.990289i	−0.868811	+	0.495144i	$0.835115\pi$
$390$		0			0
$391$	−1.97150e10			−0.843510
$392$	−8.50092e10	+	8.50092e10i	−3.60016	+	3.60016i
$393$		0			0
$394$			4.86723e10i			2.01975i
$395$	2.76323e10	−	2.10445e9i	1.13509	−	0.0864471i
$396$		0			0
$397$	−2.35251e10	+	2.35251e10i	−0.947042	+	0.947042i	−0.998667	−	0.0516245i	$-0.983560\pi$
$397$	−2.35251e10	+	2.35251e10i	−0.947042	+	0.947042i	0.0516245	+	0.998667i	$0.483560\pi$
$398$	−4.71355e10	−	4.71355e10i	−1.87852	−	1.87852i
$399$		0			0
$400$	8.41935e10	−	1.28990e10i	3.28881	−	0.503867i
$401$	−4.19224e10			−1.62132			−0.810661	−	0.585516i	$-0.800892\pi$
$401$	−4.19224e10			−1.62132			−0.810661	+	0.585516i	$0.800892\pi$
$402$		0			0
$403$	−1.02925e9	−	1.02925e9i	−0.0390212	−	0.0390212i
$404$		−	6.18423e10i		−	2.32145i
$405$		0			0
$406$	−1.04100e10			−0.383130
$407$	3.19430e10	−	3.19430e10i	1.16412	−	1.16412i
$408$		0			0
$409$			2.58166e9i			0.0922583i	0.998935	+	0.0461291i	$0.0146886\pi$
$409$			2.58166e9i			0.0922583i	−0.998935	+	0.0461291i	$0.985311\pi$
$410$	2.81755e10	+	2.41876e10i	0.997095	+	0.855967i
$411$		0			0
$412$	6.54796e10	−	6.54796e10i	2.27257	−	2.27257i
$413$	−3.47699e10	−	3.47699e10i	−1.19510	−	1.19510i
$414$		0			0
$415$	2.38579e10	−	2.77915e10i	0.804342	−	0.936958i
$416$	2.80512e10			0.936651
$417$		0			0
$418$	−1.02568e10	−	1.02568e10i	−0.335975	−	0.335975i
$419$		−	2.80524e10i		−	0.910152i	−0.890453	−	0.455076i	$-0.849612\pi$
$419$		−	2.80524e10i		−	0.910152i	0.890453	−	0.455076i	$-0.150388\pi$
$420$		0			0
$421$	4.95314e10			1.57671			0.788356	−	0.615220i	$-0.210933\pi$
$421$	4.95314e10			1.57671			0.788356	+	0.615220i	$0.210933\pi$
$422$	−5.80894e10	+	5.80894e10i	−1.83167	+	1.83167i
$423$		0			0
$424$			8.89123e10i			2.75105i
$425$	−1.93573e10	+	2.63617e10i	−0.593320	+	0.808013i
$426$		0			0
$427$	−4.55867e10	+	4.55867e10i	−1.37128	+	1.37128i
$428$	−3.38626e10	−	3.38626e10i	−1.00913	−	1.00913i
$429$		0			0
$430$	7.95296e8	+	1.04426e10i	0.0232624	+	0.305445i
$431$	4.47627e10			1.29720			0.648600	−	0.761130i	$-0.275355\pi$
$431$	4.47627e10			1.29720			0.648600	+	0.761130i	$0.275355\pi$
$432$		0			0
$433$	6.18826e9	+	6.18826e9i	0.176042	+	0.176042i	0.789628	−	0.613586i	$-0.210274\pi$
$433$	6.18826e9	+	6.18826e9i	0.176042	+	0.176042i	−0.613586	+	0.789628i	$0.710274\pi$
$434$			2.08133e10i			0.586655i
$435$		0			0
$436$	6.56064e10			1.81552
$437$	−6.01763e9	+	6.01763e9i	−0.165006	+	0.165006i
$438$		0			0
$439$			4.22019e10i			1.13625i	0.822942	+	0.568125i	$0.192331\pi$
$439$			4.22019e10i			1.13625i	−0.822942	+	0.568125i	$0.807669\pi$
$440$	−6.85748e10	+	7.98810e10i	−1.82959	+	2.13124i
$441$		0			0
$442$	−1.50144e10	+	1.50144e10i	−0.393387	+	0.393387i
$443$	2.82723e10	+	2.82723e10i	0.734084	+	0.734084i	0.971426	−	0.237342i	$-0.0762763\pi$
$443$	2.82723e10	+	2.82723e10i	0.734084	+	0.734084i	−0.237342	+	0.971426i	$0.576276\pi$
$444$		0			0
$445$	−1.29357e10	+	9.85172e8i	−0.329876	+	0.0251230i
$446$	6.07465e10			1.53526
$447$		0			0
$448$	−1.29993e11	−	1.29993e11i	−3.22706	−	3.22706i
$449$			3.34411e10i			0.822803i	0.911454	+	0.411401i	$0.134961\pi$
$449$			3.34411e10i			0.822803i	−0.911454	+	0.411401i	$0.865039\pi$
$450$		0			0
$451$	−2.55939e10			−0.618629
$452$	7.75077e10	−	7.75077e10i	1.85691	−	1.85691i
$453$		0			0
$454$		−	1.45336e11i		−	3.42096i
$455$	−1.53486e9	−	2.01533e10i	−0.0358116	−	0.470221i
$456$		0			0
$457$	−3.18371e10	+	3.18371e10i	−0.729909	+	0.729909i	−0.970601	−	0.240693i	$-0.922625\pi$
$457$	−3.18371e10	+	3.18371e10i	−0.729909	+	0.729909i	0.240693	+	0.970601i	$0.422625\pi$
$458$	1.85312e10	+	1.85312e10i	0.421155	+	0.421155i
$459$		0			0
$460$	7.54525e10	+	6.47730e10i	1.68516	+	1.44665i
$461$	−6.00943e9			−0.133055			−0.0665273	−	0.997785i	$-0.521192\pi$
$461$	−6.00943e9			−0.133055			−0.0665273	+	0.997785i	$0.521192\pi$
$462$		0			0
$463$	−4.06901e10	−	4.06901e10i	−0.885452	−	0.885452i	0.108630	−	0.994082i	$-0.465354\pi$
$463$	−4.06901e10	−	4.06901e10i	−0.885452	−	0.885452i	−0.994082	+	0.108630i	$0.965354\pi$
$464$		−	1.91062e10i		−	0.412195i
$465$		0			0
$466$	−1.49992e11			−3.18071
$467$	−2.22625e10	+	2.22625e10i	−0.468065	+	0.468065i	−0.901287	−	0.433222i	$-0.857377\pi$
$467$	−2.22625e10	+	2.22625e10i	−0.468065	+	0.468065i	0.433222	+	0.901287i	$0.357377\pi$
$468$		0			0
$469$			8.69550e10i			1.79723i
$470$	−6.29041e10	+	4.79072e9i	−1.28910	+	0.0981769i
$471$		0			0
$472$	1.14440e11	−	1.14440e11i	2.30574	−	2.30574i
$473$	−5.10409e9	−	5.10409e9i	−0.101970	−	0.101970i
$474$		0			0
$475$	2.13798e9	+	1.39549e10i	0.0419980	+	0.274126i
$476$	2.20195e11			4.28923
$477$		0			0
$478$	1.75704e10	+	1.75704e10i	0.336565	+	0.336565i
$479$		−	7.26744e10i		−	1.38051i	−0.723567	−	0.690255i	$-0.757498\pi$
$479$		−	7.26744e10i		−	1.38051i	0.723567	−	0.690255i	$-0.242502\pi$
$480$		0			0
$481$	−2.85450e10			−0.533274
$482$	6.40201e10	−	6.40201e10i	1.18612	−	1.18612i
$483$		0			0
$484$			2.80192e10i			0.510593i
$485$	3.21838e9	+	2.76286e9i	0.0581662	+	0.0499334i
$486$		0			0
$487$	9.02091e9	−	9.02091e9i	0.160374	−	0.160374i	−0.622358	−	0.782732i	$-0.713825\pi$
$487$	9.02091e9	−	9.02091e9i	0.160374	−	0.160374i	0.782732	+	0.622358i	$0.213825\pi$
$488$	−1.50042e11	−	1.50042e11i	−2.64566	−	2.64566i
$489$		0			0
$490$	−1.16615e11	+	1.35841e11i	−2.02287	+	2.35639i
$491$	−2.74456e10			−0.472222			−0.236111	−	0.971726i	$-0.575873\pi$
$491$	−2.74456e10			−0.472222			−0.236111	+	0.971726i	$0.575873\pi$
$492$		0			0
$493$	5.18756e9	+	5.18756e9i	0.0878163	+	0.0878163i
$494$			9.16574e9i			0.153907i
$495$		0			0
$496$	−3.82002e10			−0.631159
$497$	−9.41813e10	+	9.41813e10i	−1.54362	+	1.54362i
$498$		0			0
$499$			1.68809e10i			0.272266i	0.990691	+	0.136133i	$0.0434674\pi$
$499$			1.68809e10i			0.272266i	−0.990691	+	0.136133i	$0.956533\pi$
$500$	1.60694e11	−	3.72929e10i	2.57111	−	0.596686i
$501$		0			0
$502$	6.17543e10	−	6.17543e10i	0.972417	−	0.972417i
$503$	−1.66635e10	−	1.66635e10i	−0.260311	−	0.260311i	0.564869	−	0.825181i	$-0.308927\pi$
$503$	−1.66635e10	−	1.66635e10i	−0.260311	−	0.260311i	−0.825181	+	0.564869i	$0.808927\pi$
$504$		0			0
$505$	−4.34390e9	−	5.70373e10i	−0.0667905	−	0.876987i
$506$	−9.45063e10			−1.44165
$507$		0			0
$508$	1.75426e11	+	1.75426e11i	2.63414	+	2.63414i
$509$		−	1.05991e11i		−	1.57906i	−0.613713	−	0.789529i	$-0.710325\pi$
$509$		−	1.05991e11i		−	1.57906i	0.613713	−	0.789529i	$-0.289675\pi$
$510$		0			0
$511$	1.77544e11			2.60389
$512$	1.49009e10	−	1.49009e10i	0.216836	−	0.216836i
$513$		0			0
$514$		−	2.10484e11i		−	3.01555i
$515$	5.57926e10	−	6.49914e10i	0.793136	−	0.923904i
$516$		0			0
$517$	3.07461e10	−	3.07461e10i	0.430356	−	0.430356i
$518$	2.88617e11	+	2.88617e11i	4.00869	+	4.00869i
$519$		0			0
$520$	6.63319e10	−	5.05177e9i	0.907213	−	0.0690924i
$521$	−5.55100e10			−0.753390			−0.376695	−	0.926337i	$-0.622940\pi$
$521$	−5.55100e10			−0.753390			−0.376695	+	0.926337i	$0.622940\pi$
$522$		0			0
$523$	5.02384e9	+	5.02384e9i	0.0671474	+	0.0671474i	0.739883	−	0.672736i	$-0.234881\pi$
$523$	5.02384e9	+	5.02384e9i	0.0671474	+	0.0671474i	−0.672736	+	0.739883i	$0.734881\pi$
$524$			2.06251e11i			2.73571i
$525$		0			0
$526$	−2.26940e11			−2.96461
$527$	1.03718e10	−	1.03718e10i	0.134466	−	0.134466i
$528$		0			0
$529$		−	2.28645e10i		−	0.291971i
$530$	1.00548e10	+	1.32024e11i	0.127430	+	1.67320i
$531$		0			0
$532$	6.72102e10	−	6.72102e10i	0.839051	−	0.839051i
$533$	1.14357e10	+	1.14357e10i	0.141694	+	0.141694i
$534$		0			0
$535$	−3.36101e10	−	2.88530e10i	−0.410256	−	0.352189i
$536$	−2.86200e11			−3.46745
$537$		0			0
$538$	2.07073e11	+	2.07073e11i	2.47170	+	2.47170i
$539$		−	1.23395e11i		−	1.46198i
$540$		0			0
$541$	−4.75814e8			−0.00555454			−0.00277727	−	0.999996i	$-0.500884\pi$
$541$	−4.75814e8			−0.00555454			−0.00277727	+	0.999996i	$0.500884\pi$
$542$	6.16808e9	−	6.16808e9i	0.0714747	−	0.0714747i
$543$		0			0
$544$			2.82673e11i			3.22767i
$545$	6.05089e10	−	4.60830e9i	0.685856	−	0.0522342i
$546$		0			0
$547$	2.94676e8	−	2.94676e8i	0.00329151	−	0.00329151i	−0.705459	−	0.708751i	$-0.749259\pi$
$547$	2.94676e8	−	2.94676e8i	0.00329151	−	0.00329151i	0.708751	+	0.705459i	$0.249259\pi$
$548$	−4.70713e10	−	4.70713e10i	−0.521956	−	0.521956i
$549$		0			0
$550$	−9.27915e10	+	1.26368e11i	−1.01405	+	1.38098i
$551$	3.16680e9			0.0343569
$552$		0			0
$553$	−1.22032e11	−	1.22032e11i	−1.30489	−	1.30489i
$554$			1.35308e10i			0.143643i
$555$		0			0
$556$	−2.78349e10			−0.291266
$557$	4.75093e10	−	4.75093e10i	0.493581	−	0.493581i	−0.415852	−	0.909432i	$-0.636516\pi$
$557$	4.75093e10	−	4.75093e10i	0.493581	−	0.493581i	0.909432	+	0.415852i	$0.136516\pi$
$558$		0			0
$559$			4.56114e9i			0.0467117i
$560$	−4.02474e11	−	3.45508e11i	−4.09247	−	3.51323i
$561$		0			0
$562$	9.88938e9	−	9.88938e9i	0.0991343	−	0.0991343i
$563$	−2.73644e10	−	2.73644e10i	−0.272366	−	0.272366i	0.557686	−	0.830052i	$-0.311689\pi$
$563$	−2.73644e10	−	2.73644e10i	−0.272366	−	0.272366i	−0.830052	+	0.557686i	$0.811689\pi$
$564$		0			0
$565$	6.60413e10	−	7.69298e10i	0.648069	−	0.754920i
$566$	−8.60473e10			−0.838439
$567$		0			0
$568$	−3.09985e11	−	3.09985e11i	−2.97815	−	2.97815i
$569$			7.91512e10i			0.755107i	0.925988	+	0.377554i	$0.123235\pi$
$569$			7.91512e10i			0.755107i	−0.925988	+	0.377554i	$0.876765\pi$
$570$		0			0
$571$	−5.92977e10			−0.557819			−0.278910	−	0.960317i	$-0.589973\pi$
$571$	−5.92977e10			−0.557819			−0.278910	+	0.960317i	$0.589973\pi$
$572$	−5.21975e10	+	5.21975e10i	−0.487603	+	0.487603i
$573$		0			0
$574$		−	2.31250e11i		−	2.13027i
$575$	7.41398e10	+	5.44404e10i	0.678234	+	0.498024i
$576$		0			0
$577$	4.73451e10	−	4.73451e10i	0.427142	−	0.427142i	−0.460512	−	0.887654i	$-0.652334\pi$
$577$	4.73451e10	−	4.73451e10i	0.427142	−	0.427142i	0.887654	+	0.460512i	$0.152334\pi$
$578$	−7.39784e8	−	7.39784e8i	−0.00662817	−	0.00662817i
$579$		0			0
$580$	−2.81005e9	−	3.68971e10i	−0.0248315	−	0.326047i
$581$	−2.28099e11			−2.00179
$582$		0			0
$583$	−6.45302e10	−	6.45302e10i	−0.558584	−	0.558584i
$584$			5.84362e11i			5.02378i
$585$		0			0
$586$	1.48284e11			1.25749
$587$	8.29317e10	−	8.29317e10i	0.698503	−	0.698503i	−0.265585	−	0.964088i	$-0.585565\pi$
$587$	8.29317e10	−	8.29317e10i	0.698503	−	0.698503i	0.964088	+	0.265585i	$0.0855650\pi$
$588$		0			0
$589$		−	6.33158e9i		−	0.0526079i
$590$	1.56988e11	−	1.82871e11i	1.29556	−	1.50916i
$591$		0			0
$592$	−5.29718e11	+	5.29718e11i	−4.31279	+	4.31279i
$593$	4.64477e9	+	4.64477e9i	0.0375617	+	0.0375617i	0.725638	−	0.688076i	$-0.241545\pi$
$593$	4.64477e9	+	4.64477e9i	0.0375617	+	0.0375617i	−0.688076	+	0.725638i	$0.741545\pi$
$594$		0			0
$595$	2.03086e11	−	1.54668e10i	1.62036	−	0.123405i
$596$	3.82538e11			3.03172
$597$		0			0
$598$	4.22266e10	+	4.22266e10i	0.330203	+	0.330203i
$599$		−	5.67206e9i		−	0.0440589i	−0.999757	−	0.0220294i	$-0.992987\pi$
$599$		−	5.67206e9i		−	0.0440589i	0.999757	−	0.0220294i	$-0.00701276\pi$
$600$		0			0
$601$	5.00895e10			0.383927			0.191964	−	0.981402i	$-0.438514\pi$
$601$	5.00895e10			0.383927			0.191964	+	0.981402i	$0.438514\pi$
$602$	4.61173e10	−	4.61173e10i	0.351138	−	0.351138i
$603$		0			0
$604$			6.53385e10i			0.490932i
$605$	1.96812e9	+	2.58422e10i	0.0146903	+	0.192889i
$606$		0			0
$607$	−1.49734e11	+	1.49734e11i	−1.10297	+	1.10297i	−0.108923	+	0.994050i	$0.534740\pi$
$607$	−1.49734e11	+	1.49734e11i	−1.10297	+	1.10297i	−0.994050	+	0.108923i	$0.965260\pi$
$608$	8.62805e10	+	8.62805e10i	0.631391	+	0.631391i
$609$		0			0
$610$	−2.39762e11	−	2.05826e11i	−1.73165	−	1.48656i
$611$	−2.74755e10			−0.197143
$612$		0			0
$613$	−4.45340e10	−	4.45340e10i	−0.315391	−	0.315391i	0.531603	−	0.846994i	$-0.321590\pi$
$613$	−4.45340e10	−	4.45340e10i	−0.315391	−	0.315391i	−0.846994	+	0.531603i	$0.821590\pi$
$614$			1.54291e10i			0.108560i
$615$		0			0
$616$	6.55623e11			4.55335
$617$	−2.00317e11	+	2.00317e11i	−1.38222	+	1.38222i	−0.541563	+	0.840660i	$0.682167\pi$
$617$	−2.00317e11	+	2.00317e11i	−1.38222	+	1.38222i	−0.840660	+	0.541563i	$0.817833\pi$
$618$		0			0
$619$			8.78558e9i			0.0598422i	0.999552	+	0.0299211i	$0.00952561\pi$
$619$			8.78558e9i			0.0598422i	−0.999552	+	0.0299211i	$0.990474\pi$
$620$	−7.37708e10	+	5.61831e9i	−0.499249	+	0.0380224i
$621$		0			0
$622$	2.79745e11	−	2.79745e11i	1.86896	−	1.86896i
$623$	5.71278e10	+	5.71278e10i	0.379223	+	0.379223i
$624$		0			0
$625$	1.45589e11	−	4.56827e10i	0.954132	−	0.299386i
$626$	−3.34986e11			−2.18137
$627$		0			0
$628$	1.92234e11	+	1.92234e11i	1.23592	+	1.23592i
$629$		−	2.87649e11i		−	1.83764i
$630$		0			0
$631$	1.29831e10			0.0818959			0.0409480	−	0.999161i	$-0.486962\pi$
$631$	1.29831e10			0.0818959			0.0409480	+	0.999161i	$0.486962\pi$
$632$	4.01651e11	−	4.01651e11i	2.51757	−	2.51757i
$633$		0			0
$634$			3.86945e11i			2.39493i
$635$	1.74118e11	+	1.49473e11i	1.07090	+	0.919325i
$636$		0			0
$637$	−5.51343e10	+	5.51343e10i	−0.334861	+	0.334861i
$638$	2.48672e10	+	2.48672e10i	0.150087	+	0.150087i
$639$		0			0
$640$	2.35064e11	−	2.73820e11i	1.40109	−	1.63209i
$641$	−1.39848e11			−0.828368			−0.414184	−	0.910193i	$-0.635933\pi$
$641$	−1.39848e11			−0.828368			−0.414184	+	0.910193i	$0.635933\pi$
$642$		0			0
$643$	−1.28438e11	−	1.28438e11i	−0.751366	−	0.751366i	0.223368	−	0.974734i	$-0.428295\pi$
$643$	−1.28438e11	−	1.28438e11i	−0.751366	−	0.751366i	−0.974734	+	0.223368i	$0.928295\pi$
$644$		−	6.19275e11i		−	3.60031i
$645$		0			0
$646$	−9.23635e10			−0.530359
$647$	2.14258e10	−	2.14258e10i	0.122270	−	0.122270i	−0.643324	−	0.765594i	$-0.722445\pi$
$647$	2.14258e10	−	2.14258e10i	0.122270	−	0.122270i	0.765594	+	0.643324i	$0.222445\pi$
$648$		0			0
$649$			1.66115e11i			0.936333i
$650$	9.79233e10	−	1.50025e10i	0.548571	−	0.0840447i
$651$		0			0
$652$	−8.08132e10	+	8.08132e10i	−0.447190	+	0.447190i
$653$	1.24928e11	+	1.24928e11i	0.687078	+	0.687078i	0.961585	−	0.274507i	$-0.0885147\pi$
$653$	1.24928e11	+	1.24928e11i	0.687078	+	0.687078i	−0.274507	+	0.961585i	$0.588515\pi$
$654$		0			0
$655$	1.44874e10	+	1.90225e11i	0.0787091	+	1.03348i
$656$	4.24430e11			2.29187
$657$		0			0
$658$	2.77802e11	+	2.77802e11i	1.48195	+	1.48195i
$659$		−	1.95671e11i		−	1.03749i	−0.854928	−	0.518747i	$-0.826399\pi$
$659$		−	1.95671e11i		−	1.03749i	0.854928	−	0.518747i	$-0.173601\pi$
$660$		0			0
$661$	−1.72257e11			−0.902344			−0.451172	−	0.892437i	$-0.648994\pi$
$661$	−1.72257e11			−0.902344			−0.451172	+	0.892437i	$0.648994\pi$
$662$	−2.21470e11	+	2.21470e11i	−1.15314	+	1.15314i
$663$		0			0
$664$		−	7.50754e11i		−	3.86212i
$665$	5.72671e10	−	6.67091e10i	0.292832	−	0.341113i
$666$		0			0
$667$	1.45895e10	−	1.45895e10i	0.0737117	−	0.0737117i
$668$	−3.38617e11	−	3.38617e11i	−1.70060	−	1.70060i
$669$		0			0
$670$	−4.24971e11	+	3.23654e10i	−2.10892	+	0.160613i
$671$	2.17793e11			1.07437
$672$		0			0
$673$	1.80619e11	+	1.80619e11i	0.880445	+	0.880445i	0.993580	−	0.113135i	$-0.0360893\pi$
$673$	1.80619e11	+	1.80619e11i	0.880445	+	0.880445i	−0.113135	+	0.993580i	$0.536089\pi$
$674$			5.26995e10i			0.255368i
$675$		0			0
$676$	−5.04540e11			−2.41607
$677$	−2.04251e11	+	2.04251e11i	−0.972323	+	0.972323i	−0.999627	−	0.0273046i	$-0.991308\pi$
$677$	−2.04251e11	+	2.04251e11i	−0.972323	+	0.972323i	0.0273046	+	0.999627i	$0.491308\pi$
$678$		0			0
$679$		−	2.64148e10i		−	0.124271i
$680$	5.09069e10	+	6.68429e11i	0.238090	+	3.12622i
$681$		0			0
$682$	4.97185e10	−	4.97185e10i	0.229816	−	0.229816i
$683$	2.10972e11	+	2.10972e11i	0.969486	+	0.969486i	0.999548	−	0.0300621i	$-0.00957051\pi$
$683$	2.10972e11	+	2.10972e11i	0.969486	+	0.969486i	−0.0300621	+	0.999548i	$0.509571\pi$
$684$		0			0
$685$	−4.67203e10	−	4.01076e10i	−0.212199	−	0.182165i
$686$	4.30033e11			1.94180
$687$		0			0
$688$	8.46423e10	+	8.46423e10i	0.377776	+	0.377776i
$689$			5.76658e10i			0.255883i
$690$		0			0
$691$	4.03399e11			1.76939			0.884693	−	0.466174i	$-0.154368\pi$
$691$	4.03399e11			1.76939			0.884693	+	0.466174i	$0.154368\pi$
$692$	−5.82909e10	+	5.82909e10i	−0.254201	+	0.254201i
$693$		0			0
$694$		−	3.49530e11i		−	1.50677i
$695$	−2.56722e10	+	1.95517e9i	−0.110033	+	0.00838002i
$696$		0			0
$697$	−1.15238e11	+	1.15238e11i	−0.488274	+	0.488274i
$698$	−4.75552e11	−	4.75552e11i	−2.00344	−	2.00344i
$699$		0			0
$700$	−8.28058e11	−	6.08039e11i	−3.44881	−	2.53244i
$701$	1.27175e11			0.526658			0.263329	−	0.964706i	$-0.415180\pi$
$701$	1.27175e11			0.526658			0.263329	+	0.964706i	$0.415180\pi$
$702$		0			0
$703$	−8.77995e10	−	8.77995e10i	−0.359477	−	0.359477i
$704$			6.21049e11i			2.52834i
$705$		0			0
$706$	−1.72726e10			−0.0695247
$707$	−2.51893e11	+	2.51893e11i	−1.00818	+	1.00818i
$708$		0			0
$709$			3.96460e11i			1.56897i	0.620147	+	0.784485i	$0.287073\pi$
$709$			3.96460e11i			1.56897i	−0.620147	+	0.784485i	$0.712927\pi$
$710$	−4.95344e11	−	4.25233e11i	−1.94928	−	1.67338i
$711$		0			0
$712$	−1.88028e11	+	1.88028e11i	−0.731648	+	0.731648i
$713$	−2.91696e10	−	2.91696e10i	−0.112869	−	0.112869i
$714$		0			0
$715$	−4.44755e10	+	5.18084e10i	−0.170175	+	0.198233i
$716$	4.43177e11			1.68626
$717$		0			0
$718$	6.10441e11	+	6.10441e11i	2.29692	+	2.29692i
$719$			2.47872e11i			0.927495i	0.885967	+	0.463748i	$0.153496\pi$
$719$			2.47872e11i			0.927495i	−0.885967	+	0.463748i	$0.846504\pi$
$720$		0			0
$721$	−5.33416e11			−1.97390
$722$	3.38373e11	−	3.38373e11i	1.24522	−	1.24522i
$723$		0			0
$724$		−	8.29820e11i		−	3.02016i
$725$	−5.18343e9	−	3.38329e10i	−0.0187614	−	0.122458i
$726$		0			0
$727$	−2.43200e11	+	2.43200e11i	−0.870616	+	0.870616i	−0.992540	−	0.121923i	$-0.961094\pi$
$727$	−2.43200e11	+	2.43200e11i	−0.870616	+	0.870616i	0.121923	+	0.992540i	$0.461094\pi$
$728$	−2.92940e11	−	2.92940e11i	−1.04293	−	1.04293i
$729$		0			0
$730$	6.60836e10	+	8.67705e11i	0.232703	+	3.05549i
$731$	−4.59628e10			−0.160967
$732$		0			0
$733$	5.23835e10	+	5.23835e10i	0.181459	+	0.181459i	0.791991	−	0.610532i	$-0.209044\pi$
$733$	5.23835e10	+	5.23835e10i	0.181459	+	0.181459i	−0.610532	+	0.791991i	$0.709044\pi$
$734$		−	9.13916e11i		−	3.14863i
$735$		0			0
$736$	7.94989e11			2.70926
$737$	2.07716e11	−	2.07716e11i	0.704045	−	0.704045i
$738$		0			0
$739$		−	8.87871e10i		−	0.297695i	−0.988860	−	0.148848i	$-0.952444\pi$
$739$		−	8.87871e10i		−	0.297695i	0.988860	−	0.148848i	$-0.0475564\pi$
$740$	−9.45063e11	+	1.10088e12i	−3.15162	+	3.67124i
$741$		0			0
$742$	5.83054e11	−	5.83054e11i	1.92350	−	1.92350i
$743$	−1.90402e11	−	1.90402e11i	−0.624764	−	0.624764i	0.321982	−	0.946746i	$-0.395651\pi$
$743$	−1.90402e11	−	1.90402e11i	−0.624764	−	0.624764i	−0.946746	+	0.321982i	$0.895651\pi$
$744$		0			0
$745$	3.52816e11	−	2.68701e10i	1.14531	−	0.0872257i
$746$	−8.44541e11			−2.72688
$747$		0			0
$748$	−5.25997e11	−	5.25997e11i	−1.68026	−	1.68026i
$749$			2.75855e11i			0.876503i
$750$		0			0
$751$	−1.51599e11			−0.476581			−0.238290	−	0.971194i	$-0.576587\pi$
$751$	−1.51599e11			−0.476581			−0.238290	+	0.971194i	$0.576587\pi$
$752$	−5.09870e11	+	5.09870e11i	−1.59437	+	1.59437i
$753$		0			0
$754$		−	2.22219e10i		−	0.0687538i
$755$	4.58948e9	+	6.02618e10i	0.0141246	+	0.185462i
$756$		0			0
$757$	1.98644e11	−	1.98644e11i	0.604913	−	0.604913i	−0.336699	−	0.941612i	$-0.609311\pi$
$757$	1.98644e11	−	1.98644e11i	0.604913	−	0.604913i	0.941612	+	0.336699i	$0.109311\pi$
$758$	−2.67903e10	−	2.67903e10i	−0.0811522	−	0.0811522i
$759$		0			0
$760$	2.19563e11	+	1.88487e11i	0.658121	+	0.564971i
$761$	5.37314e11			1.60210			0.801049	−	0.598599i	$-0.204276\pi$
$761$	5.37314e11			1.60210			0.801049	+	0.598599i	$0.204276\pi$
$762$		0			0
$763$	−2.67224e11	−	2.67224e11i	−0.788457	−	0.788457i
$764$		−	6.12902e11i		−	1.79894i
$765$		0			0
$766$	9.10674e11			2.64513
$767$	7.42223e10	−	7.42223e10i	0.214463	−	0.214463i
$768$		0			0
$769$			1.69574e11i			0.484902i	0.970164	+	0.242451i	$0.0779514\pi$
$769$			1.69574e11i			0.484902i	−0.970164	+	0.242451i	$0.922049\pi$
$770$	9.73518e11	−	7.41422e10i	2.76937	−	0.210913i
$771$		0			0
$772$	−2.87767e11	+	2.87767e11i	−0.810161	+	0.810161i
$773$	3.58947e11	+	3.58947e11i	1.00534	+	1.00534i	0.999986	+	0.00535155i	$0.00170346\pi$
$773$	3.58947e11	+	3.58947e11i	1.00534	+	1.00534i	0.00535155	+	0.999986i	$0.498297\pi$
$774$		0			0
$775$	−6.76443e10	+	1.03636e10i	−0.187510	+	0.0287278i
$776$	8.69407e10			0.239760
$777$		0			0
$778$	4.89420e11	+	4.89420e11i	1.33587	+	1.33587i
$779$			7.03481e10i			0.191031i
$780$		0			0
$781$	4.49957e11			1.20939
$782$	−4.25519e11	+	4.25519e11i	−1.13787	+	1.13787i
$783$		0			0
$784$			2.04629e12i			5.41629i
$785$	1.90801e11	+	1.63795e11i	0.502460	+	0.431342i
$786$		0			0
$787$	2.17163e11	−	2.17163e11i	0.566093	−	0.566093i	−0.364939	−	0.931032i	$-0.618910\pi$
$787$	2.17163e11	−	2.17163e11i	0.566093	−	0.566093i	0.931032	+	0.364939i	$0.118910\pi$
$788$	7.61869e11	+	7.61869e11i	1.97595	+	1.97595i
$789$		0			0
$790$	5.50981e11	−	6.41823e11i	1.41458	−	1.64781i
$791$	−6.31400e11			−1.61287
$792$		0			0
$793$	−9.73126e10	−	9.73126e10i	−0.246080	−	0.246080i
$794$			1.01551e12i			2.55506i
$795$		0			0
$796$	−1.47563e12			−3.67557
$797$	1.95735e11	−	1.95735e11i	0.485104	−	0.485104i	−0.421653	−	0.906757i	$-0.638550\pi$
$797$	1.95735e11	−	1.95735e11i	0.485104	−	0.485104i	0.906757	+	0.421653i	$0.138550\pi$
$798$		0			0
$799$		−	2.76871e11i		−	0.679346i
$800$	7.80564e11	−	1.06301e12i	1.90567	−	2.59525i
$801$		0			0
$802$	−9.04833e11	+	9.04833e11i	−2.18711	+	2.18711i
$803$	−4.24114e11	−	4.24114e11i	−1.02005	−	1.02005i
$804$		0			0
$805$	−4.34989e10	−	5.71159e11i	−0.103585	−	1.36011i
$806$	−4.44297e10			−0.105277
$807$		0			0
$808$	−8.29069e11	−	8.29069e11i	−1.94511	−	1.94511i
$809$		−	2.86879e11i		−	0.669738i	−0.942265	−	0.334869i	$-0.891308\pi$
$809$		−	2.86879e11i		−	0.669738i	0.942265	−	0.334869i	$-0.108692\pi$
$810$		0			0
$811$	6.46329e11			1.49407			0.747034	−	0.664786i	$-0.231477\pi$
$811$	6.46329e11			1.49407			0.747034	+	0.664786i	$0.231477\pi$
$812$	−1.62948e11	+	1.62948e11i	−0.374822	+	0.374822i
$813$		0			0
$814$		−	1.37888e12i		−	3.14073i
$815$	−6.88577e10	+	8.02106e10i	−0.156071	+	0.181803i
$816$		0			0
$817$	−1.40293e10	+	1.40293e10i	−0.0314881	+	0.0314881i
$818$	5.57212e10	+	5.57212e10i	0.124454	+	0.124454i
$819$		0			0
$820$	8.19642e11	−	6.24232e10i	1.81288	−	0.138067i
$821$	2.10538e11			0.463401			0.231701	−	0.972787i	$-0.425571\pi$
$821$	2.10538e11			0.463401			0.231701	+	0.972787i	$0.425571\pi$
$822$		0			0
$823$	−2.04250e11	−	2.04250e11i	−0.445208	−	0.445208i	0.448550	−	0.893758i	$-0.351941\pi$
$823$	−2.04250e11	−	2.04250e11i	−0.445208	−	0.445208i	−0.893758	+	0.448550i	$0.851941\pi$
$824$		−	1.75566e12i		−	3.80831i
$825$		0			0
$826$	−1.50091e12			−3.22429
$827$	2.22889e11	−	2.22889e11i	0.476504	−	0.476504i	−0.427507	−	0.904012i	$-0.640608\pi$
$827$	2.22889e11	−	2.22889e11i	0.476504	−	0.476504i	0.904012	+	0.427507i	$0.140608\pi$
$828$		0			0
$829$			1.71733e11i			0.363609i	0.983335	+	0.181805i	$0.0581938\pi$
$829$			1.71733e11i			0.363609i	−0.983335	+	0.181805i	$0.941806\pi$
$830$	−8.49003e10	−	1.11478e12i	−0.178894	−	2.34896i
$831$		0			0
$832$	2.77492e11	−	2.77492e11i	0.579105	−	0.579105i
$833$	−5.55591e11	−	5.55591e11i	−1.15392	−	1.15392i
$834$		0			0
$835$	−3.36092e11	−	2.88522e11i	−0.691373	−	0.593517i
$836$	−3.21101e11			−0.657379
$837$		0			0
$838$	−6.05469e11	−	6.05469e11i	−1.22777	−	1.22777i
$839$			5.18727e11i			1.04687i	0.852067	+	0.523433i	$0.175349\pi$
$839$			5.18727e11i			1.04687i	−0.852067	+	0.523433i	$0.824651\pi$
$840$		0			0
$841$	4.92569e11			0.984652
$842$	1.06906e12	−	1.06906e12i	2.12693	−	2.12693i
$843$		0			0
$844$			1.81855e12i			3.58390i
$845$	−4.65339e11	+	3.54398e10i	−0.912730	+	0.0695127i
$846$		0			0
$847$	1.14126e11	−	1.14126e11i	0.221744	−	0.221744i
$848$	1.07012e12	+	1.07012e12i	2.06942	+	2.06942i
$849$		0			0
$850$	1.51181e11	+	9.86777e11i	0.289615	+	1.89035i
$851$	−8.08985e11			−1.54249
$852$		0			0
$853$	−5.76113e11	−	5.76113e11i	−1.08821	−	1.08821i	−0.995713	−	0.0924928i	$-0.970516\pi$
$853$	−5.76113e11	−	5.76113e11i	−1.08821	−	1.08821i	−0.0924928	−	0.995713i	$-0.529484\pi$
$854$			1.96784e12i			3.69963i
$855$		0			0
$856$	−9.07937e11			−1.69107
$857$	3.82245e11	−	3.82245e11i	0.708629	−	0.708629i	−0.257618	−	0.966247i	$-0.582938\pi$
$857$	3.82245e11	−	3.82245e11i	0.708629	−	0.708629i	0.966247	+	0.257618i	$0.0829377\pi$
$858$		0			0
$859$			4.88244e11i			0.896736i	0.893849	+	0.448368i	$0.147995\pi$
$859$			4.88244e11i			0.896736i	−0.893849	+	0.448368i	$0.852005\pi$
$860$	1.75907e11	+	1.51009e11i	0.321580	+	0.276064i
$861$		0			0
$862$	9.66135e11	−	9.66135e11i	1.74988	−	1.74988i
$863$	4.87256e11	+	4.87256e11i	0.878444	+	0.878444i	0.993374	−	0.114930i	$-0.0366643\pi$
$863$	4.87256e11	+	4.87256e11i	0.878444	+	0.878444i	−0.114930	+	0.993374i	$0.536664\pi$
$864$		0			0
$865$	−4.96674e10	+	5.78563e10i	−0.0887170	+	0.103344i
$866$	2.67129e11			0.474951
$867$		0			0
$868$	3.25792e11	+	3.25792e11i	0.573934	+	0.573934i
$869$			5.83015e11i			1.02235i
$870$		0			0
$871$	−1.85620e11			−0.322517
$872$	8.79531e11	−	8.79531e11i	1.52120	−	1.52120i
$873$		0			0
$874$			2.59763e11i			0.445176i
$875$	−8.06430e11	−	5.02631e11i	−1.37573	−	0.857467i
$876$		0			0
$877$	3.48945e11	−	3.48945e11i	0.589873	−	0.589873i	−0.347724	−	0.937597i	$-0.613045\pi$
$877$	3.48945e11	−	3.48945e11i	0.589873	−	0.589873i	0.937597	+	0.347724i	$0.113045\pi$
$878$	9.10865e11	+	9.10865e11i	1.53277	+	1.53277i
$879$		0			0
$880$	1.36078e11	+	1.78677e12i	0.226913	+	2.97946i
$881$	9.75114e11			1.61865			0.809323	−	0.587364i	$-0.199834\pi$
$881$	9.75114e11			1.61865			0.809323	+	0.587364i	$0.199834\pi$
$882$		0			0
$883$	7.39746e11	+	7.39746e11i	1.21686	+	1.21686i	0.968727	+	0.248130i	$0.0798162\pi$
$883$	7.39746e11	+	7.39746e11i	1.21686	+	1.21686i	0.248130	+	0.968727i	$0.420184\pi$
$884$			4.70044e11i			0.769713i
$885$		0			0
$886$	1.22043e12			1.98051
$887$	−4.98872e11	+	4.98872e11i	−0.805924	+	0.805924i	−0.984014	−	0.178090i	$-0.943008\pi$
$887$	−4.98872e11	+	4.98872e11i	−0.805924	+	0.805924i	0.178090	+	0.984014i	$0.443008\pi$
$888$		0			0
$889$		−	1.42907e12i		−	2.28795i
$890$	−2.57935e11	+	3.00462e11i	−0.411102	+	0.478882i
$891$		0			0
$892$	9.50868e11	−	9.50868e11i	1.50197	−	1.50197i
$893$	−8.45097e10	−	8.45097e10i	−0.132893	−	0.132893i
$894$		0			0
$895$	4.08743e11	−	3.11295e10i	0.637028	−	0.0485154i
$896$	−2.24737e12			−3.48693
$897$		0			0
$898$	7.21777e11	+	7.21777e11i	1.10994	+	1.10994i
$899$			1.53507e10i			0.0235011i
$900$		0			0
$901$	−5.81100e11			−0.881763
$902$	−5.52406e11	+	5.52406e11i	−0.834511	+	0.834511i
$903$		0			0
$904$		−	2.07817e12i		−	3.11176i
$905$	−5.82880e10	−	7.65345e11i	−0.0868930	−	1.14094i
$906$		0			0
$907$	−3.10116e11	+	3.10116e11i	−0.458242	+	0.458242i	−0.898078	−	0.439836i	$-0.855037\pi$
$907$	−3.10116e11	+	3.10116e11i	−0.458242	+	0.458242i	0.439836	+	0.898078i	$0.355037\pi$
$908$	−2.27495e12	−	2.27495e12i	−3.34678	−	3.34678i
$909$		0			0
$910$	−4.68108e11	−	4.01852e11i	−0.682622	−	0.586004i
$911$	7.33153e11			1.06444			0.532220	−	0.846606i	$-0.321358\pi$
$911$	7.33153e11			1.06444			0.532220	+	0.846606i	$0.321358\pi$
$912$		0			0
$913$	5.44877e11	+	5.44877e11i	0.784180	+	0.784180i
$914$			1.37431e12i			1.96925i
$915$		0			0
$916$	5.80139e11			0.824044
$917$	8.40089e11	−	8.40089e11i	1.18809	−	1.18809i
$918$		0			0
$919$			4.25211e11i			0.596132i	0.954545	+	0.298066i	$0.0963416\pi$
$919$			4.25211e11i			0.596132i	−0.954545	+	0.298066i	$0.903658\pi$
$920$	1.87989e12	−	1.43171e11i	2.62410	−	0.199849i
$921$		0			0
$922$	−1.29705e11	+	1.29705e11i	−0.179486	+	0.179486i
$923$	−2.01046e11	−	2.01046e11i	−0.277006	−	0.277006i
$924$		0			0
$925$	−7.94306e11	+	1.08173e12i	−1.08498	+	1.47758i
$926$	−1.75647e12			−2.38889
$927$		0			0
$928$	−2.09183e11	−	2.09183e11i	−0.282056	−	0.282056i
$929$		−	9.00171e11i		−	1.20854i	−0.796779	−	0.604271i	$-0.793464\pi$
$929$		−	9.00171e11i		−	1.20854i	0.796779	−	0.604271i	$-0.206536\pi$
$930$		0			0
$931$	−3.39167e11			−0.451455
$932$	−2.34783e12	+	2.34783e12i	−3.11174	+	3.11174i
$933$		0			0
$934$			9.61005e11i			1.26281i
$935$	−5.22075e11	−	4.48181e11i	−0.683103	−	0.586418i
$936$		0			0
$937$	−7.70690e11	+	7.70690e11i	−0.999819	+	0.999819i	−1.00000		0.000181418i	$-0.999942\pi$
$937$	−7.70690e11	+	7.70690e11i	−0.999819	+	0.999819i	0.000181418		1.00000i	$0.499942\pi$
$938$	1.87679e12	+	1.87679e12i	2.42440	+	2.42440i
$939$		0			0
$940$	−9.09652e11	+	1.05963e12i	−1.16510	+	1.35720i
$941$	−9.46747e11			−1.20747			−0.603733	−	0.797186i	$-0.706321\pi$
$941$	−9.46747e11			−1.20747			−0.603733	+	0.797186i	$0.706321\pi$
$942$		0			0
$943$	3.24094e11	+	3.24094e11i	0.409850	+	0.409850i
$944$		−	2.75473e12i		−	3.46889i
$945$		0			0
$946$	−2.20328e11			−0.275110
$947$	−8.02306e10	+	8.02306e10i	−0.0997562	+	0.0997562i	−0.755224	−	0.655467i	$-0.772472\pi$
$947$	−8.02306e10	+	8.02306e10i	−0.0997562	+	0.0997562i	0.655467	+	0.755224i	$0.272472\pi$
$948$		0			0
$949$			3.78999e11i			0.467276i
$950$	3.47340e11	+	2.55050e11i	0.426442	+	0.313134i
$951$		0			0
$952$	2.95197e12	−	2.95197e12i	3.59389	−	3.59389i
$953$	−4.83892e11	−	4.83892e11i	−0.586647	−	0.586647i	0.350074	−	0.936722i	$-0.386156\pi$
$953$	−4.83892e11	−	4.83892e11i	−0.586647	−	0.586647i	−0.936722	+	0.350074i	$0.886156\pi$
$954$		0			0
$955$	−4.30512e10	−	5.65281e11i	−0.0517574	−	0.679596i
$956$	5.50060e11			0.658534
$957$		0			0
$958$	−1.56857e12	−	1.56857e12i	−1.86226	−	1.86226i
$959$			3.83457e11i			0.453359i
$960$		0			0
$961$	−8.22200e11			−0.964015
$962$	−6.16102e11	+	6.16102e11i	−0.719370	+	0.719370i
$963$		0			0
$964$		−	2.00422e12i		−	2.32080i
$965$	−2.45195e11	+	2.85621e11i	−0.282749	+	0.329368i
$966$		0			0
$967$	−6.13625e11	+	6.13625e11i	−0.701774	+	0.701774i	−0.964791	−	0.263017i	$-0.915282\pi$
$967$	−6.13625e11	+	6.13625e11i	−0.701774	+	0.701774i	0.263017	+	0.964791i	$0.415282\pi$
$968$	3.75631e11	+	3.75631e11i	0.427819	+	0.427819i
$969$		0			0
$970$	1.29096e11	−	9.83183e9i	0.145823	−	0.0111057i
$971$	1.58537e12			1.78342			0.891711	−	0.452605i	$-0.149505\pi$
$971$	1.58537e12			1.78342			0.891711	+	0.452605i	$0.149505\pi$
$972$		0			0
$973$	1.13376e11	+	1.13376e11i	0.126493	+	0.126493i
$974$		−	3.89405e11i		−	0.432679i
$975$		0			0
$976$	−3.61172e12			−3.98029
$977$	2.59654e11	−	2.59654e11i	0.284982	−	0.284982i	−0.550110	−	0.835092i	$-0.685414\pi$
$977$	2.59654e11	−	2.59654e11i	0.284982	−	0.284982i	0.835092	+	0.550110i	$0.185414\pi$
$978$		0			0
$979$		−	2.72931e11i		−	0.297114i
$980$	3.00958e11	+	3.95171e12i	0.326289	+	4.28431i
$981$		0			0
$982$	−5.92372e11	+	5.92372e11i	−0.637014	+	0.637014i
$983$	−4.25838e11	−	4.25838e11i	−0.456068	−	0.456068i	0.441294	−	0.897363i	$-0.354520\pi$
$983$	−4.25838e11	−	4.25838e11i	−0.456068	−	0.456068i	−0.897363	+	0.441294i	$0.854520\pi$
$984$		0			0
$985$	7.56189e11	+	6.49159e11i	0.803314	+	0.689614i
$986$	2.23931e11			0.236923
$987$		0			0
$988$	1.43472e11	+	1.43472e11i	0.150570	+	0.150570i
$989$			1.29266e11i			0.135113i
$990$		0			0
$991$	−2.07410e11			−0.215048			−0.107524	−	0.994202i	$-0.534292\pi$
$991$	−2.07410e11			−0.215048			−0.107524	+	0.994202i	$0.534292\pi$
$992$	−4.18233e11	+	4.18233e11i	−0.431888	+	0.431888i
$993$		0			0
$994$			4.06553e12i			4.16458i
$995$	−1.36097e12	+	1.03651e11i	−1.38854	+	0.105750i
$996$		0			0
$997$	1.98300e11	−	1.98300e11i	0.200698	−	0.200698i	−0.599601	−	0.800299i	$-0.704674\pi$
$997$	1.98300e11	−	1.98300e11i	0.200698	−	0.200698i	0.800299	+	0.599601i	$0.204674\pi$
$998$	3.64349e11	+	3.64349e11i	0.367278	+	0.367278i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.9.g.b.28.8	yes	16
3.2	odd	2	inner	45.9.g.b.28.1	✓	16
5.2	odd	4	inner	45.9.g.b.37.8	yes	16
15.2	even	4	inner	45.9.g.b.37.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.g.b.28.1	✓	16	3.2	odd	2	inner
45.9.g.b.28.8	yes	16	1.1	even	1	trivial
45.9.g.b.37.1	yes	16	15.2	even	4	inner
45.9.g.b.37.8	yes	16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(21.5835 - 21.5835i) q^{2} -675.695i q^{4} +(-47.4620 - 623.195i) q^{5} +(-2752.21 + 2752.21i) q^{7} +(-9058.50 - 9058.50i) q^{8} +O(q^{10})\)	\(q+(21.5835 - 21.5835i) q^{2} -675.695i q^{4} +(-47.4620 - 623.195i) q^{5} +(-2752.21 + 2752.21i) q^{7} +(-9058.50 - 9058.50i) q^{8} +(-14475.1 - 12426.3i) q^{10} +13148.8 q^{11} +(-5875.06 - 5875.06i) q^{13} +118804. i q^{14} -218050. q^{16} +(59203.2 - 59203.2i) q^{17} -36141.3i q^{19} +(-421090. + 32069.8i) q^{20} +(283798. - 283798. i) q^{22} +(-166503. - 166503. i) q^{23} +(-386120. + 59156.1i) q^{25} -253609. q^{26} +(1.85965e6 + 1.85965e6i) q^{28} +87622.9i q^{29} +175190. q^{31} +(-2.38731e6 + 2.38731e6i) q^{32} -2.55562e6i q^{34} +(1.84579e6 + 1.58454e6i) q^{35} +(2.42934e6 - 2.42934e6i) q^{37} +(-780055. - 780055. i) q^{38} +(-5.21528e6 + 6.07515e6i) q^{40} -1.94648e6 q^{41} +(-388178. - 388178. i) q^{43} -8.88460e6i q^{44} -7.18744e6 q^{46} +(2.33832e6 - 2.33832e6i) q^{47} -9.38447e6i q^{49} +(-7.05702e6 + 9.61061e6i) q^{50} +(-3.96975e6 + 3.96975e6i) q^{52} +(-4.90768e6 - 4.90768e6i) q^{53} +(-624069. - 8.19429e6i) q^{55} +4.98617e7 q^{56} +(1.89121e6 + 1.89121e6i) q^{58} +1.26334e7i q^{59} +1.65637e7 q^{61} +(3.78121e6 - 3.78121e6i) q^{62} +4.72323e7i q^{64} +(-3.38247e6 + 3.94015e6i) q^{65} +(1.57973e7 - 1.57973e7i) q^{67} +(-4.00033e7 - 4.00033e7i) q^{68} +(7.40384e7 - 5.63869e6i) q^{70} +3.42203e7 q^{71} +(-3.22549e7 - 3.22549e7i) q^{73} -1.04867e8i q^{74} -2.44205e7 q^{76} +(-3.61883e7 + 3.61883e7i) q^{77} +4.43397e7i q^{79} +(1.03491e7 + 1.35888e8i) q^{80} +(-4.20118e7 + 4.20118e7i) q^{82} +(4.14392e7 + 4.14392e7i) q^{83} +(-3.97050e7 - 3.40852e7i) q^{85} -1.67565e7 q^{86} +(-1.19109e8 - 1.19109e8i) q^{88} -2.07571e7i q^{89} +3.23387e7 q^{91} +(-1.12505e8 + 1.12505e8i) q^{92} -1.00938e8i q^{94} +(-2.25231e7 + 1.71534e6i) q^{95} +(-4.79885e6 + 4.79885e6i) q^{97} +(-2.02550e8 - 2.02550e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4220 q^{7}+O(q^{10}) \) 16 * q - 4220 * q^7	\( 16 q - 4220 q^{7} - 47000 q^{10} - 37940 q^{13} - 508916 q^{16} + 844700 q^{22} - 1664300 q^{25} + 6009380 q^{28} - 944752 q^{31} + 10616140 q^{37} - 17493600 q^{40} + 4050760 q^{43} - 34233160 q^{46} + 7180240 q^{52} + 17430500 q^{55} - 27842100 q^{58} + 32032232 q^{61} + 75463480 q^{67} + 272773500 q^{70} - 198258320 q^{73} - 196046088 q^{76} + 172139600 q^{82} + 183614500 q^{85} - 624395100 q^{88} - 477271600 q^{91} + 662476480 q^{97}+O(q^{100}) \) 16 * q - 4220 * q^7 - 47000 * q^10 - 37940 * q^13 - 508916 * q^16 + 844700 * q^22 - 1664300 * q^25 + 6009380 * q^28 - 944752 * q^31 + 10616140 * q^37 - 17493600 * q^40 + 4050760 * q^43 - 34233160 * q^46 + 7180240 * q^52 + 17430500 * q^55 - 27842100 * q^58 + 32032232 * q^61 + 75463480 * q^67 + 272773500 * q^70 - 198258320 * q^73 - 196046088 * q^76 + 172139600 * q^82 + 183614500 * q^85 - 624395100 * q^88 - 477271600 * q^91 + 662476480 * q^97

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