LMFDB - Embedded newform 5.18.b.a.4.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5,18,Mod(4,5)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("5.4");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	5.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.16110436723$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 203459x^{6} + 12362849196x^{4} + 237701205446144x^{2} + 1320400799499206656 $ x^8 + 203459x^6 + 12362849196x^4 + 237701205446144*x^2 + 1320400799499206656
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{8}\cdot 5^{12}\cdot 11 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4.5
Root			$98.5951i$ of defining polynomial
Character	$\chi$	$=$	5.4
Dual form			5.18.b.a.4.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+197.190i q^{2} +12817.1i q^{3} +92188.0 q^{4} +(620811. + 614437. i) q^{5} -2.52741e6 q^{6} -4.49140e6i q^{7} +4.40247e7i q^{8} -3.51381e7 q^{9} +O(q^{10})$	\(q+197.190i q^{2} +12817.1i q^{3} +92188.0 q^{4} +(620811. + 614437. i) q^{5} -2.52741e6 q^{6} -4.49140e6i q^{7} +4.40247e7i q^{8} -3.51381e7 q^{9} +(-1.21161e8 + 1.22418e8i) q^{10} -1.08959e9 q^{11} +1.18158e9i q^{12} +2.20399e9i q^{13} +8.85659e8 q^{14} +(-7.87531e9 + 7.95700e9i) q^{15} +3.40203e9 q^{16} -4.96836e10i q^{17} -6.92889e9i q^{18} +6.53775e10 q^{19} +(5.72314e10 + 5.66438e10i) q^{20} +5.75667e10 q^{21} -2.14857e11i q^{22} -6.03064e10i q^{23} -5.64269e11 q^{24} +(7.87347e9 + 7.62899e11i) q^{25} -4.34604e11 q^{26} +1.20483e12i q^{27} -4.14053e11i q^{28} -2.63186e10 q^{29} +(-1.56904e12 - 1.55293e12i) q^{30} +3.03326e12 q^{31} +6.44125e12i q^{32} -1.39654e13i q^{33} +9.79712e12 q^{34} +(2.75968e12 - 2.78831e12i) q^{35} -3.23931e12 q^{36} -8.44655e12i q^{37} +1.28918e13i q^{38} -2.82487e13 q^{39} +(-2.70504e13 + 2.73310e13i) q^{40} +5.70631e13 q^{41} +1.13516e13i q^{42} +6.40476e13i q^{43} -1.00447e14 q^{44} +(-2.18141e13 - 2.15901e13i) q^{45} +1.18918e13 q^{46} -2.44975e14i q^{47} +4.36042e13i q^{48} +2.12458e14 q^{49} +(-1.50436e14 + 1.55257e12i) q^{50} +6.36800e14 q^{51} +2.03181e14i q^{52} -1.01305e14i q^{53} -2.37582e14 q^{54} +(-6.76432e14 - 6.69487e14i) q^{55} +1.97732e14 q^{56} +8.37950e14i q^{57} -5.18977e12i q^{58} -1.41965e15 q^{59} +(-7.26009e14 + 7.33540e14i) q^{60} +2.31838e15 q^{61} +5.98128e14i q^{62} +1.57819e14i q^{63} -8.24240e14 q^{64} +(-1.35421e15 + 1.36826e15i) q^{65} +2.75385e15 q^{66} -4.06306e15i q^{67} -4.58023e15i q^{68} +7.72954e14 q^{69} +(5.49827e14 + 5.44182e14i) q^{70} +2.68034e15 q^{71} -1.54694e15i q^{72} +6.95024e15i q^{73} +1.66558e15 q^{74} +(-9.77816e15 + 1.00915e14i) q^{75} +6.02702e15 q^{76} +4.89380e15i q^{77} -5.57037e15i q^{78} -8.28477e15 q^{79} +(2.11202e15 + 2.09034e15i) q^{80} -1.99802e16 q^{81} +1.12523e16i q^{82} +2.66711e16i q^{83} +5.30696e15 q^{84} +(3.05275e16 - 3.08441e16i) q^{85} -1.26296e16 q^{86} -3.37328e14i q^{87} -4.79690e16i q^{88} +6.55891e15 q^{89} +(4.25736e15 - 4.30153e15i) q^{90} +9.89897e15 q^{91} -5.55953e15i q^{92} +3.88776e16i q^{93} +4.83067e16 q^{94} +(4.05871e16 + 4.01704e16i) q^{95} -8.25582e16 q^{96} -1.11864e17i q^{97} +4.18946e16i q^{98} +3.82862e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 579096 q^{4} + 379200 q^{5} + 357816 q^{6} - 234916344 q^{9}+O(q^{10}) $ 8 * q - 579096 * q^4 + 379200 * q^5 + 357816 * q^6 - 234916344 * q^9	$ 8 q - 579096 q^{4} + 379200 q^{5} + 357816 q^{6} - 234916344 q^{9} + 329570200 q^{10} + 463296576 q^{11} - 29937907992 q^{14} + 30646226400 q^{15} + 30848001568 q^{16} - 20615713280 q^{19} - 47558579400 q^{20} - 75039699024 q^{21} + 1768741136160 q^{24} - 1789249435000 q^{25} - 838901194224 q^{26} - 4079017824720 q^{29} + 2416984007400 q^{30} + 11329328658496 q^{31} - 36406243632832 q^{34} + 4019663899200 q^{35} + 59729752432728 q^{36} + 40318460422272 q^{39} - 209747532172000 q^{40} + 97217252847456 q^{41} - 116357853210912 q^{44} - 366841998003600 q^{45} + 10\!\cdots\!36 q^{46}+ \cdots - 56\!\cdots\!68 q^{99}+O(q^{100}) $ 8 * q - 579096 * q^4 + 379200 * q^5 + 357816 * q^6 - 234916344 * q^9 + 329570200 * q^10 + 463296576 * q^11 - 29937907992 * q^14 + 30646226400 * q^15 + 30848001568 * q^16 - 20615713280 * q^19 - 47558579400 * q^20 - 75039699024 * q^21 + 1768741136160 * q^24 - 1789249435000 * q^25 - 838901194224 * q^26 - 4079017824720 * q^29 + 2416984007400 * q^30 + 11329328658496 * q^31 - 36406243632832 * q^34 + 4019663899200 * q^35 + 59729752432728 * q^36 + 40318460422272 * q^39 - 209747532172000 * q^40 + 97217252847456 * q^41 - 116357853210912 * q^44 - 366841998003600 * q^45 + 1081224261700136 * q^46 - 856574357621656 * q^49 - 1283266301910000 * q^50 + 2468309514424896 * q^51 - 3408409774777680 * q^54 - 2042713226757600 * q^55 + 8363016326678880 * q^56 - 1091409512240640 * q^59 - 11479379108104800 * q^60 + 8064731010774976 * q^61 - 3616160324265856 * q^64 - 11989509557901600 * q^65 + 27318846906958752 * q^66 - 12078989597365008 * q^69 - 13190931213697800 * q^70 + 25241492058140736 * q^71 - 29902523510328912 * q^74 + 3839235716880000 * q^75 + 20767634734678560 * q^76 + 3229852337730880 * q^79 + 1263407265391200 * q^80 - 49353541005202632 * q^81 + 101439947332382688 * q^84 + 25693702369787200 * q^85 - 165112838769552744 * q^86 + 92963987535626640 * q^89 + 206315814421823400 * q^90 - 225591670236809664 * q^91 + 162612564681867848 * q^94 + 204319715505252000 * q^95 - 654303222993538944 * q^96 - 56327331239952768 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$			197.190i			0.544666i	0.962203	+	0.272333i	$0.0877952\pi$
$2$			197.190i			0.544666i	−0.962203	+	0.272333i	$0.912205\pi$
$3$			12817.1i			1.12787i	0.825819	+	0.563935i	$0.190713\pi$
$3$			12817.1i			1.12787i	−0.825819	+	0.563935i	$0.809287\pi$
$4$	92188.0			0.703339
$5$	620811.	+	614437.i	0.710746	+	0.703449i
$5$	620811.	+	614437.i	0.710746	+	0.703449i
$6$	−2.52741e6			−0.614313
$7$		−	4.49140e6i		−	0.294475i	−0.989101	−	0.147237i	$-0.952962\pi$
$7$		−	4.49140e6i		−	0.294475i	0.989101	−	0.147237i	$-0.0470381\pi$
$8$			4.40247e7i			0.927751i
$9$	−3.51381e7			−0.272093
$10$	−1.21161e8	+	1.22418e8i	−0.383145	+	0.387119i
$11$	−1.08959e9			−1.53259			−0.766296	−	0.642487i	$-0.777903\pi$
$11$	−1.08959e9			−1.53259			−0.766296	+	0.642487i	$0.777903\pi$
$12$			1.18158e9i			0.793275i
$13$			2.20399e9i			0.749359i	0.927154	+	0.374680i	$0.122247\pi$
$13$			2.20399e9i			0.749359i	−0.927154	+	0.374680i	$0.877753\pi$
$14$	8.85659e8			0.160390
$15$	−7.87531e9	+	7.95700e9i	−0.793399	+	0.801630i
$16$	3.40203e9			0.198024
$17$		−	4.96836e10i		−	1.72742i	−0.503991	−	0.863709i	$-0.668136\pi$
$17$		−	4.96836e10i		−	1.72742i	0.503991	−	0.863709i	$-0.331864\pi$
$18$		−	6.92889e9i		−	0.148200i
$19$	6.53775e10			0.883126			0.441563	−	0.897230i	$-0.354424\pi$
$19$	6.53775e10			0.883126			0.441563	+	0.897230i	$0.354424\pi$
$20$	5.72314e10	+	5.66438e10i	0.499895	+	0.494763i
$21$	5.75667e10			0.332129
$22$		−	2.14857e11i		−	0.834751i
$23$		−	6.03064e10i		−	0.160575i	−0.996772	−	0.0802873i	$-0.974416\pi$
$23$		−	6.03064e10i		−	0.160575i	0.996772	−	0.0802873i	$-0.0255838\pi$
$24$	−5.64269e11			−1.04638
$25$	7.87347e9	+	7.62899e11i	0.0103199	+	0.999947i
$26$	−4.34604e11			−0.408151
$27$			1.20483e12i			0.820986i
$28$		−	4.14053e11i		−	0.207115i
$29$	−2.63186e10			−0.00976967			−0.00488483	−	0.999988i	$-0.501555\pi$
$29$	−2.63186e10			−0.00976967			−0.00488483	+	0.999988i	$0.501555\pi$
$30$	−1.56904e12	−	1.55293e12i	−0.436621	−	0.432138i
$31$	3.03326e12			0.638756			0.319378	−	0.947627i	$-0.396526\pi$
$31$	3.03326e12			0.638756			0.319378	+	0.947627i	$0.396526\pi$
$32$			6.44125e12i			1.03561i
$33$		−	1.39654e13i		−	1.72857i
$34$	9.79712e12			0.940866
$35$	2.75968e12	−	2.78831e12i	0.207148	−	0.209297i
$36$	−3.23931e12			−0.191373
$37$		−	8.44655e12i		−	0.395334i	−0.980269	−	0.197667i	$-0.936663\pi$
$37$		−	8.44655e12i		−	0.395334i	0.980269	−	0.197667i	$-0.0633365\pi$
$38$			1.28918e13i			0.481009i
$39$	−2.82487e13			−0.845181
$40$	−2.70504e13	+	2.73310e13i	−0.652625	+	0.659395i
$41$	5.70631e13			1.11607			0.558036	−	0.829817i	$-0.311555\pi$
$41$	5.70631e13			1.11607			0.558036	+	0.829817i	$0.311555\pi$
$42$			1.13516e13i			0.180900i
$43$			6.40476e13i			0.835644i	0.908529	+	0.417822i	$0.137206\pi$
$43$			6.40476e13i			0.835644i	−0.908529	+	0.417822i	$0.862794\pi$
$44$	−1.00447e14			−1.07793
$45$	−2.18141e13	−	2.15901e13i	−0.193389	−	0.191403i
$46$	1.18918e13			0.0874596
$47$		−	2.44975e14i		−	1.50069i	−0.661048	−	0.750344i	$-0.729888\pi$
$47$		−	2.44975e14i		−	1.50069i	0.661048	−	0.750344i	$-0.270112\pi$
$48$			4.36042e13i			0.223346i
$49$	2.12458e14			0.913285
$50$	−1.50436e14	+	1.55257e12i	−0.544637	+	0.00562090i
$51$	6.36800e14			1.94830
$52$			2.03181e14i			0.527054i
$53$		−	1.01305e14i		−	0.223504i	−0.993736	−	0.111752i	$-0.964354\pi$
$53$		−	1.01305e14i		−	0.223504i	0.993736	−	0.111752i	$-0.0356463\pi$
$54$	−2.37582e14			−0.447163
$55$	−6.76432e14	−	6.69487e14i	−1.08928	−	1.07810i
$56$	1.97732e14			0.273199
$57$			8.37950e14i			0.996052i
$58$		−	5.18977e12i		−	0.00532121i
$59$	−1.41965e15			−1.25875			−0.629373	−	0.777103i	$-0.716688\pi$
$59$	−1.41965e15			−1.25875			−0.629373	+	0.777103i	$0.716688\pi$
$60$	−7.26009e14	+	7.33540e14i	−0.558029	+	0.563817i
$61$	2.31838e15			1.54839			0.774195	−	0.632947i	$-0.218155\pi$
$61$	2.31838e15			1.54839			0.774195	+	0.632947i	$0.218155\pi$
$62$			5.98128e14i			0.347908i
$63$			1.57819e14i			0.0801244i
$64$	−8.24240e14			−0.366036
$65$	−1.35421e15	+	1.36826e15i	−0.527136	+	0.532604i
$66$	2.75385e15			0.941492
$67$		−	4.06306e15i		−	1.22241i	−0.791472	−	0.611206i	$-0.790685\pi$
$67$		−	4.06306e15i		−	1.22241i	0.791472	−	0.611206i	$-0.209315\pi$
$68$		−	4.58023e15i		−	1.21496i
$69$	7.72954e14			0.181107
$70$	5.49827e14	+	5.44182e14i	0.113997	+	0.112826i
$71$	2.68034e15			0.492599			0.246299	−	0.969194i	$-0.420785\pi$
$71$	2.68034e15			0.492599			0.246299	+	0.969194i	$0.420785\pi$
$72$		−	1.54694e15i		−	0.252434i
$73$			6.95024e15i			1.00869i	0.863504	+	0.504343i	$0.168265\pi$
$73$			6.95024e15i			1.00869i	−0.863504	+	0.504343i	$0.831735\pi$
$74$	1.66558e15			0.215325
$75$	−9.77816e15	+	1.00915e14i	−1.12781	+	0.0116395i
$76$	6.02702e15			0.621137
$77$			4.89380e15i			0.451310i
$78$		−	5.57037e15i		−	0.460341i
$79$	−8.28477e15			−0.614399			−0.307199	−	0.951645i	$-0.599392\pi$
$79$	−8.28477e15			−0.614399			−0.307199	+	0.951645i	$0.599392\pi$
$80$	2.11202e15	+	2.09034e15i	0.140745	+	0.139300i
$81$	−1.99802e16			−1.19806
$82$			1.12523e16i			0.607887i
$83$			2.66711e16i			1.29980i	0.760018	+	0.649902i	$0.225190\pi$
$83$			2.66711e16i			1.29980i	−0.760018	+	0.649902i	$0.774810\pi$
$84$	5.30696e15			0.233600
$85$	3.05275e16	−	3.08441e16i	1.21515	−	1.22776i
$86$	−1.26296e16			−0.455147
$87$		−	3.37328e14i		−	0.0110189i
$88$		−	4.79690e16i		−	1.42186i
$89$	6.55891e15			0.176611			0.0883053	−	0.996093i	$-0.471855\pi$
$89$	6.55891e15			0.176611			0.0883053	+	0.996093i	$0.471855\pi$
$90$	4.25736e15	−	4.30153e15i	0.104251	−	0.105332i
$91$	9.89897e15			0.220667
$92$		−	5.55953e15i		−	0.112938i
$93$			3.88776e16i			0.720434i
$94$	4.83067e16			0.817373
$95$	4.05871e16	+	4.01704e16i	0.627678	+	0.621234i
$96$	−8.25582e16			−1.16803
$97$		−	1.11864e17i		−	1.44920i	−0.689169	−	0.724601i	$-0.742024\pi$
$97$		−	1.11864e17i		−	1.44920i	0.689169	−	0.724601i	$-0.257976\pi$
$98$			4.18946e16i			0.497435i
$99$	3.82862e16			0.417007
$100$	7.25839e14	+	7.03301e16i	0.00725839	+	0.703301i
$101$	−1.99528e17			−1.83346			−0.916732	−	0.399503i	$-0.869183\pi$
$101$	−1.99528e17			−1.83346			−0.916732	+	0.399503i	$0.869183\pi$
$102$			1.25571e17i			1.06117i
$103$		−	1.36484e17i		−	1.06161i	−0.847495	−	0.530804i	$-0.821890\pi$
$103$		−	1.36484e17i		−	1.06161i	0.847495	−	0.530804i	$-0.178110\pi$
$104$	−9.70298e16			−0.695219
$105$	3.57380e16	+	3.53711e16i	0.236060	+	0.233636i
$106$	1.99764e16			0.121735
$107$			4.55554e16i			0.256317i	0.991754	+	0.128158i	$0.0409066\pi$
$107$			4.55554e16i			0.256317i	−0.991754	+	0.128158i	$0.959093\pi$
$108$			1.11071e17i			0.577431i
$109$	2.16082e17			1.03871			0.519355	−	0.854559i	$-0.326172\pi$
$109$	2.16082e17			1.03871			0.519355	+	0.854559i	$0.326172\pi$
$110$	1.32016e17	−	1.33386e17i	0.587205	−	0.593296i
$111$	1.08260e17			0.445886
$112$		−	1.52799e16i		−	0.0583132i
$113$		−	2.03188e17i		−	0.719005i	−0.933144	−	0.359502i	$-0.882946\pi$
$113$		−	2.03188e17i		−	0.719005i	0.933144	−	0.359502i	$-0.117054\pi$
$114$	−1.65236e17			−0.542516
$115$	3.70545e16	−	3.74389e16i	0.112956	−	0.114128i
$116$	−2.42626e15			−0.00687139
$117$		−	7.74438e16i		−	0.203895i
$118$		−	2.79941e17i		−	0.685597i
$119$	−2.23149e17			−0.508681
$120$	−3.50305e17	−	3.46708e17i	−0.743713	−	0.736077i
$121$	6.81767e17			1.34884
$122$			4.57161e17i			0.843355i
$123$			7.31384e17i			1.25879i
$124$	2.79630e17			0.449262
$125$	−4.63865e17	+	4.78454e17i	−0.696076	+	0.717968i
$126$	−3.11204e16			−0.0436410
$127$		−	1.49219e17i		−	0.195655i	−0.995203	−	0.0978277i	$-0.968811\pi$
$127$		−	1.49219e17i		−	0.195655i	0.995203	−	0.0978277i	$-0.0311894\pi$
$128$			6.81736e17i			0.836241i
$129$	−8.20905e17			−0.942498
$130$	−2.69807e17	−	2.67037e17i	−0.290091	−	0.287113i
$131$	−4.10281e17			−0.413309			−0.206655	−	0.978414i	$-0.566258\pi$
$131$	−4.10281e17			−0.413309			−0.206655	+	0.978414i	$0.566258\pi$
$132$		−	1.28745e18i		−	1.21577i
$133$		−	2.93636e17i		−	0.260058i
$134$	8.01196e17			0.665806
$135$	−7.40295e17	+	7.47975e17i	−0.577521	+	0.583512i
$136$	2.18731e18			1.60261
$137$		−	6.17653e17i		−	0.425226i	−0.977137	−	0.212613i	$-0.931803\pi$
$137$		−	6.17653e17i		−	0.425226i	0.977137	−	0.212613i	$-0.0681973\pi$
$138$			1.52419e17i			0.0986431i
$139$	−1.59873e18			−0.973080			−0.486540	−	0.873658i	$-0.661741\pi$
$139$	−1.59873e18			−0.973080			−0.486540	+	0.873658i	$0.661741\pi$
$140$	2.54409e17	−	2.57049e17i	0.145695	−	0.147207i
$141$	3.13987e18			1.69258
$142$			5.28536e17i			0.268302i
$143$		−	2.40145e18i		−	1.14846i
$144$	−1.19541e17			−0.0538810
$145$	−1.63389e16	−	1.61711e16i	−0.00694375	−	0.00687246i
$146$	−1.37052e18			−0.549397
$147$			2.72310e18i			1.03007i
$148$		−	7.78671e17i		−	0.278054i
$149$	1.08941e18			0.367373			0.183687	−	0.982985i	$-0.441197\pi$
$149$	1.08941e18			0.367373			0.183687	+	0.982985i	$0.441197\pi$
$150$	−1.98995e16	−	1.92816e18i	−0.00633965	−	0.614280i
$151$	−4.15139e18			−1.24994			−0.624970	−	0.780649i	$-0.714889\pi$
$151$	−4.15139e18			−1.24994			−0.624970	+	0.780649i	$0.714889\pi$
$152$			2.87822e18i			0.819321i
$153$			1.74579e18i			0.470018i
$154$	−9.65009e17			−0.245813
$155$	1.88308e18	+	1.86375e18i	0.453993	+	0.449332i
$156$	−2.60419e18			−0.594448
$157$		−	8.05540e16i		−	0.0174157i	−0.999962	−	0.00870783i	$-0.997228\pi$
$157$		−	8.05540e16i		−	0.0174157i	0.999962	−	0.00870783i	$-0.00277182\pi$
$158$		−	1.63368e18i		−	0.334642i
$159$	1.29844e18			0.252084
$160$	−3.95774e18	+	3.99880e18i	−0.728497	+	0.736054i
$161$	−2.70860e17			−0.0472852
$162$		−	3.93991e18i		−	0.652542i
$163$			2.71154e18i			0.426207i	0.977030	+	0.213104i	$0.0683572\pi$
$163$			2.71154e18i			0.426207i	−0.977030	+	0.213104i	$0.931643\pi$
$164$	5.26053e18			0.784977
$165$	8.58088e18	−	8.66990e18i	1.21596	−	1.22857i
$166$	−5.25929e18			−0.707959
$167$		−	7.86788e18i		−	1.00639i	−0.864172	−	0.503197i	$-0.832157\pi$
$167$		−	7.86788e18i		−	1.00639i	0.864172	−	0.503197i	$-0.167843\pi$
$168$			2.53436e18i			0.308133i
$169$	3.79286e18			0.438460
$170$	6.08216e18	+	6.01971e18i	0.668717	+	0.661851i
$171$	−2.29724e18			−0.240292
$172$			5.90443e18i			0.587741i
$173$		−	3.01075e18i		−	0.285288i	−0.989774	−	0.142644i	$-0.954440\pi$
$173$		−	3.01075e18i		−	0.285288i	0.989774	−	0.142644i	$-0.0455604\pi$
$174$	6.65178e16			0.00600163
$175$	3.42648e18	−	3.53628e16i	0.294459	−	0.00303895i
$176$	−3.70683e18			−0.303491
$177$		−	1.81958e19i		−	1.41970i
$178$			1.29335e18i			0.0961938i
$179$	−1.11507e19			−0.790774			−0.395387	−	0.918515i	$-0.629390\pi$
$179$	−1.11507e19			−0.790774			−0.395387	+	0.918515i	$0.629390\pi$
$180$	−2.01100e18	−	1.99035e18i	−0.136018	−	0.134621i
$181$	−1.16284e19			−0.750332			−0.375166	−	0.926958i	$-0.622414\pi$
$181$	−1.16284e19			−0.750332			−0.375166	+	0.926958i	$0.622414\pi$
$182$			1.95198e18i			0.120190i
$183$			2.97149e19i			1.74638i
$184$	2.65497e18			0.148973
$185$	5.18987e18	−	5.24371e18i	0.278097	−	0.280982i
$186$	−7.66627e18			−0.392396
$187$			5.41349e19i			2.64743i
$188$		−	2.25838e19i		−	1.05549i
$189$	5.41139e18			0.241759
$190$	−7.92120e18	+	8.00337e18i	−0.338365	+	0.341875i
$191$	1.23347e19			0.503901			0.251951	−	0.967740i	$-0.418928\pi$
$191$	1.23347e19			0.503901			0.251951	+	0.967740i	$0.418928\pi$
$192$		−	1.05644e19i		−	0.412842i
$193$		−	4.31622e19i		−	1.61386i	−0.590645	−	0.806931i	$-0.701127\pi$
$193$		−	4.31622e19i		−	1.61386i	0.590645	−	0.806931i	$-0.298873\pi$
$194$	2.20584e19			0.789331
$195$	−1.75371e19	−	1.73571e19i	−0.600709	−	0.594541i
$196$	1.95861e19			0.642349
$197$		−	2.74515e18i		−	0.0862191i	−0.999070	−	0.0431096i	$-0.986274\pi$
$197$		−	2.74515e18i		−	0.0862191i	0.999070	−	0.0431096i	$-0.0137265\pi$
$198$			7.54967e18i			0.227130i
$199$	−5.45248e19			−1.57160			−0.785802	−	0.618478i	$-0.787750\pi$
$199$	−5.45248e19			−1.57160			−0.785802	+	0.618478i	$0.787750\pi$
$200$	−3.35864e19	+	3.46627e17i	−0.927701	+	0.00957430i
$201$	5.20767e19			1.37872
$202$		−	3.93450e19i		−	0.998625i
$203$			1.18207e17i			0.00287692i
$204$	5.87053e19			1.37032
$205$	3.54254e19	+	3.50617e19i	0.793244	+	0.785100i
$206$	2.69132e19			0.578222
$207$			2.11905e18i			0.0436912i
$208$			7.49803e18i			0.148391i
$209$	−7.12349e19			−1.35347
$210$	−6.97484e18	+	7.04719e18i	−0.127254	+	0.128574i
$211$	8.75529e18			0.153416			0.0767078	−	0.997054i	$-0.475559\pi$
$211$	8.75529e18			0.153416			0.0767078	+	0.997054i	$0.475559\pi$
$212$		−	9.33911e18i		−	0.157199i
$213$			3.43541e19i			0.555588i
$214$	−8.98307e18			−0.139607
$215$	−3.93532e19	+	3.97615e19i	−0.587833	+	0.593931i
$216$	−5.30425e19			−0.761670
$217$		−	1.36236e19i		−	0.188097i
$218$			4.26093e19i			0.565750i
$219$	−8.90820e19			−1.13767
$220$	−6.23589e19	−	6.17187e19i	−0.766136	−	0.758270i
$221$	1.09502e20			1.29446
$222$			2.13479e19i			0.242859i
$223$		−	4.58536e19i		−	0.502091i	−0.967975	−	0.251045i	$-0.919226\pi$
$223$		−	4.58536e19i		−	0.502091i	0.967975	−	0.251045i	$-0.0807743\pi$
$224$	2.89302e19			0.304960
$225$	−2.76659e17	−	2.68068e19i	−0.00280797	−	0.272078i
$226$	4.00667e19			0.391617
$227$			4.24431e19i			0.399565i	0.979840	+	0.199782i	$0.0640235\pi$
$227$			4.24431e19i			0.399565i	−0.979840	+	0.199782i	$0.935977\pi$
$228$			7.72490e19i			0.700562i
$229$	1.66894e20			1.45828			0.729138	−	0.684367i	$-0.239921\pi$
$229$	1.66894e20			1.45828			0.729138	+	0.684367i	$0.239921\pi$
$230$	7.38258e18	+	7.30679e18i	0.0621615	+	0.0615233i
$231$	−6.27243e19			−0.509019
$232$		−	1.15867e18i		−	0.00906382i
$233$		−	6.95294e19i		−	0.524377i	−0.965017	−	0.262188i	$-0.915556\pi$
$233$		−	6.95294e19i		−	0.524377i	0.965017	−	0.262188i	$-0.0844442\pi$
$234$	1.52712e19			0.111055
$235$	1.50522e20	−	1.52083e20i	1.05566	−	1.06661i
$236$	−1.30875e20			−0.885326
$237$		−	1.06187e20i		−	0.692962i
$238$		−	4.40027e19i		−	0.277061i
$239$	−4.98454e19			−0.302861			−0.151430	−	0.988468i	$-0.548388\pi$
$239$	−4.98454e19			−0.302861			−0.151430	+	0.988468i	$0.548388\pi$
$240$	−2.67921e19	+	2.70700e19i	−0.157112	+	0.158742i
$241$	−2.34973e19			−0.133007			−0.0665033	−	0.997786i	$-0.521184\pi$
$241$	−2.34973e19			−0.133007			−0.0665033	+	0.997786i	$0.521184\pi$
$242$			1.34438e20i			0.734668i
$243$		−	1.00496e20i		−	0.530269i
$244$	2.13727e20			1.08904
$245$	1.31896e20	+	1.30542e20i	0.649113	+	0.642449i
$246$	−1.44222e20			−0.685618
$247$			1.44091e20i			0.661779i
$248$			1.33538e20i			0.592606i
$249$	−3.41847e20			−1.46601
$250$	−9.43464e19	−	9.14697e19i	−0.391053	−	0.379129i
$251$	−1.73046e20			−0.693322			−0.346661	−	0.937990i	$-0.612685\pi$
$251$	−1.73046e20			−0.693322			−0.346661	+	0.937990i	$0.612685\pi$
$252$			1.45490e19i			0.0563546i
$253$			6.57095e19i			0.246096i
$254$	2.94245e19			0.106567
$255$	3.95333e20	+	3.91274e20i	1.38475	+	1.37053i
$256$	−2.42466e20			−0.821508
$257$		−	4.12299e20i		−	1.35139i	−0.737181	−	0.675695i	$-0.763844\pi$
$257$		−	4.12299e20i		−	1.35139i	0.737181	−	0.675695i	$-0.236156\pi$
$258$		−	1.61875e20i		−	0.513347i
$259$	−3.79368e19			−0.116416
$260$	−1.24842e20	+	1.26137e20i	−0.370755	+	0.374601i
$261$	9.24785e17			0.00265825
$262$		−	8.09033e19i		−	0.225116i
$263$			2.09492e20i			0.564343i	0.959364	+	0.282171i	$0.0910546\pi$
$263$			2.09492e20i			0.564343i	−0.959364	+	0.282171i	$0.908945\pi$
$264$	6.14824e20			1.60368
$265$	6.22456e19	−	6.28913e19i	0.157224	−	0.158855i
$266$	5.79022e19			0.141645
$267$			8.40663e19i			0.199194i
$268$		−	3.74566e20i		−	0.859770i
$269$	−1.23146e20			−0.273859			−0.136930	−	0.990581i	$-0.543723\pi$
$269$	−1.23146e20			−0.273859			−0.136930	+	0.990581i	$0.543723\pi$
$270$	−1.47493e20	−	1.45979e20i	−0.317819	−	0.314556i
$271$	1.71222e20			0.357536			0.178768	−	0.983891i	$-0.442789\pi$
$271$	1.71222e20			0.357536			0.178768	+	0.983891i	$0.442789\pi$
$272$		−	1.69025e20i		−	0.342071i
$273$			1.26876e20i			0.248884i
$274$	1.21795e20			0.231606
$275$	−8.57888e18	−	8.31250e20i	−0.0158162	−	1.53251i
$276$	7.12571e19			0.127380
$277$			4.23825e20i			0.734698i	0.930083	+	0.367349i	$0.119734\pi$
$277$			4.23825e20i			0.734698i	−0.930083	+	0.367349i	$0.880266\pi$
$278$		−	3.15253e20i		−	0.530004i
$279$	−1.06583e20			−0.173801
$280$	1.22754e20	+	1.21494e20i	0.194175	+	0.192182i
$281$	−3.64626e20			−0.559556			−0.279778	−	0.960065i	$-0.590261\pi$
$281$	−3.64626e20			−0.559556			−0.279778	+	0.960065i	$0.590261\pi$
$282$			6.19152e20i			0.921892i
$283$			5.68589e20i			0.821511i	0.911745	+	0.410756i	$0.134735\pi$
$283$			5.68589e20i			0.821511i	−0.911745	+	0.410756i	$0.865265\pi$
$284$	2.47095e20			0.346464
$285$	−5.14868e20	+	5.20209e20i	−0.700671	+	0.707940i
$286$	4.73542e20			0.625529
$287$		−	2.56293e20i		−	0.328655i
$288$		−	2.26333e20i		−	0.281781i
$289$	−1.64122e21			−1.98397
$290$	3.18879e18	−	3.22187e18i	0.00374320	−	0.00378203i
$291$	1.43377e21			1.63451
$292$			6.40729e20i			0.709448i
$293$			1.46631e21i			1.57707i	0.614991	+	0.788534i	$0.289159\pi$
$293$			1.46631e21i			1.57707i	−0.614991	+	0.788534i	$0.710841\pi$
$294$	−5.36968e20			−0.561043
$295$	−8.81333e20	−	8.72284e20i	−0.894649	−	0.885464i
$296$	3.71857e20			0.366772
$297$		−	1.31278e21i		−	1.25824i
$298$			2.14821e20i			0.200096i
$299$	1.32914e20			0.120328
$300$	−9.01429e20	+	9.30316e18i	−0.793233	+	0.00818653i
$301$	2.87663e20			0.246076
$302$		−	8.18613e20i		−	0.680800i
$303$		−	2.55737e21i		−	2.06791i
$304$	2.22416e20			0.174880
$305$	1.43927e21	+	1.42450e21i	1.10051	+	1.08921i
$306$	−3.44252e20			−0.256003
$307$			1.00902e21i			0.729833i	0.931040	+	0.364916i	$0.118902\pi$
$307$			1.00902e21i			0.729833i	−0.931040	+	0.364916i	$0.881098\pi$
$308$			4.51149e20i			0.317424i
$309$	1.74933e21			1.19736
$310$	−3.67512e20	+	3.71325e20i	−0.244736	+	0.247275i
$311$	1.79361e21			1.16216			0.581078	−	0.813848i	$-0.302631\pi$
$311$	1.79361e21			1.16216			0.581078	+	0.813848i	$0.302631\pi$
$312$		−	1.24364e21i		−	0.784117i
$313$			6.24766e20i			0.383345i	0.981459	+	0.191673i	$0.0613912\pi$
$313$			6.24766e20i			0.383345i	−0.981459	+	0.191673i	$0.938609\pi$
$314$	1.58845e19			0.00948572
$315$	−9.69699e19	+	9.79758e19i	−0.0563634	+	0.0569481i
$316$	−7.63757e20			−0.432130
$317$		−	3.52085e20i		−	0.193930i	−0.995288	−	0.0969648i	$-0.969087\pi$
$317$		−	3.52085e20i		−	0.193930i	0.995288	−	0.0969648i	$-0.0309134\pi$
$318$			2.56039e20i			0.137302i
$319$	2.86766e19			0.0149729
$320$	−5.11697e20	−	5.06444e20i	−0.260159	−	0.257488i
$321$	−5.83888e20			−0.289092
$322$		−	5.34109e19i		−	0.0257546i
$323$		−	3.24819e21i		−	1.52553i
$324$	−1.84194e21			−0.842641
$325$	−1.68142e21	+	1.73530e19i	−0.749320	+	0.00773332i
$326$	−5.34689e20			−0.232141
$327$			2.76955e21i			1.17153i
$328$			2.51218e21i			1.03544i
$329$	−1.10028e21			−0.441914
$330$	1.70962e21	+	1.69207e21i	0.669161	+	0.662291i
$331$	−2.87953e21			−1.09846			−0.549229	−	0.835672i	$-0.685079\pi$
$331$	−2.87953e21			−1.09846			−0.549229	+	0.835672i	$0.685079\pi$
$332$			2.45876e21i			0.914202i
$333$			2.96795e20i			0.107568i
$334$	1.55147e21			0.548148
$335$	2.49650e21	−	2.52239e21i	0.859904	−	0.868824i
$336$	1.95844e20			0.0657697
$337$			6.69424e20i			0.219203i	0.993976	+	0.109602i	$0.0349575\pi$
$337$			6.69424e20i			0.219203i	−0.993976	+	0.109602i	$0.965042\pi$
$338$			7.47916e20i			0.238814i
$339$	2.60428e21			0.810944
$340$	2.81427e21	−	2.84346e21i	0.854662	−	0.863528i
$341$	−3.30502e21			−0.978952
$342$		−	4.52993e20i		−	0.130879i
$343$		−	1.99907e21i		−	0.563414i
$344$	−2.81968e21			−0.775269
$345$	4.79858e20	+	4.74932e20i	0.128721	+	0.127400i
$346$	5.93691e20			0.155387
$347$		−	8.92234e20i		−	0.227865i	−0.993488	−	0.113933i	$-0.963655\pi$
$347$		−	8.92234e20i		−	0.227865i	0.993488	−	0.113933i	$-0.0363448\pi$
$348$		−	3.10976e19i		−	0.00775004i
$349$	2.36533e21			0.575274			0.287637	−	0.957739i	$-0.407130\pi$
$349$	2.36533e21			0.575274			0.287637	+	0.957739i	$0.407130\pi$
$350$	6.97321e18	+	6.75668e20i	0.00165521	+	0.160382i
$351$	−2.65544e21			−0.615213
$352$		−	7.01835e21i		−	1.58717i
$353$			2.51330e21i			0.554829i	0.960750	+	0.277414i	$0.0894774\pi$
$353$			2.51330e21i			0.554829i	−0.960750	+	0.277414i	$0.910523\pi$
$354$	3.58803e21			0.773265
$355$	1.66398e21	+	1.64690e21i	0.350113	+	0.346518i
$356$	6.04653e20			0.124217
$357$		−	2.86012e21i		−	0.573726i
$358$		−	2.19881e21i		−	0.430708i
$359$	2.10167e21			0.402033			0.201017	−	0.979588i	$-0.435576\pi$
$359$	2.10167e21			0.402033			0.201017	+	0.979588i	$0.435576\pi$
$360$	9.50499e20	−	9.60360e20i	0.177574	−	0.179417i
$361$	−1.20617e21			−0.220089
$362$		−	2.29301e21i		−	0.408680i
$363$			8.73829e21i			1.52132i
$364$	9.12566e20			0.155204
$365$	−4.27049e21	+	4.31479e21i	−0.709558	+	0.716919i
$366$	−5.85948e21			−0.951196
$367$			7.00996e21i			1.11187i	0.831226	+	0.555935i	$0.187640\pi$
$367$			7.00996e21i			1.11187i	−0.831226	+	0.555935i	$0.812360\pi$
$368$		−	2.05165e20i		−	0.0317977i
$369$	−2.00509e21			−0.303675
$370$	1.03401e21	+	1.02339e21i	0.153042	+	0.151470i
$371$	−4.55001e20			−0.0658164
$372$			3.58405e21i			0.506709i
$373$		−	5.69623e21i		−	0.787158i	−0.919291	−	0.393579i	$-0.871237\pi$
$373$		−	5.69623e21i		−	0.787158i	0.919291	−	0.393579i	$-0.128763\pi$
$374$	−1.06749e22			−1.44196
$375$	−6.13239e21	−	5.94541e21i	−0.809775	−	0.785084i
$376$	1.07850e22			1.39226
$377$		−	5.80058e19i		−	0.00732099i
$378$			1.06707e21i			0.131678i
$379$	4.63351e21			0.559083			0.279542	−	0.960134i	$-0.409817\pi$
$379$	4.63351e21			0.559083			0.279542	+	0.960134i	$0.409817\pi$
$380$	3.74164e21	+	3.70323e21i	0.441470	+	0.436938i
$381$	1.91255e21			0.220674
$382$			2.43228e21i			0.274458i
$383$		−	8.22870e20i		−	0.0908117i	−0.998969	−	0.0454058i	$-0.985542\pi$
$383$		−	8.22870e20i		−	0.0908117i	0.998969	−	0.0454058i	$-0.0144581\pi$
$384$	−8.73788e21			−0.943171
$385$	−3.00693e21	+	3.03812e21i	−0.317473	+	0.320767i
$386$	8.51117e21			0.879016
$387$		−	2.25051e21i		−	0.227373i
$388$		−	1.03125e22i		−	1.01928i
$389$	6.11949e21			0.591757			0.295879	−	0.955226i	$-0.404388\pi$
$389$	6.11949e21			0.591757			0.295879	+	0.955226i	$0.404388\pi$
$390$	3.42264e21	−	3.45815e21i	0.323826	−	0.327186i
$391$	−2.99624e21			−0.277379
$392$			9.35339e21i			0.847301i
$393$		−	5.25861e21i		−	0.466159i
$394$	5.41317e20			0.0469606
$395$	−5.14328e21	−	5.09047e21i	−0.436681	−	0.432198i
$396$	3.52953e21			0.293297
$397$		−	1.23524e22i		−	1.00469i	−0.864668	−	0.502344i	$-0.832471\pi$
$397$		−	1.23524e22i		−	1.00469i	0.864668	−	0.502344i	$-0.167529\pi$
$398$		−	1.07518e22i		−	0.855999i
$399$	3.76357e21			0.293312
$400$	2.67858e19	+	2.59541e21i	0.00204359	+	0.198014i
$401$	2.56078e22			1.91269			0.956346	−	0.292237i	$-0.0943996\pi$
$401$	2.56078e22			1.91269			0.956346	+	0.292237i	$0.0943996\pi$
$402$			1.02690e22i			0.750943i
$403$			6.68525e21i			0.478657i
$404$	−1.83941e22			−1.28955
$405$	−1.24040e22	−	1.22766e22i	−0.851515	−	0.842772i
$406$	−2.33093e19			−0.00156696
$407$			9.20330e21i			0.605887i
$408$			2.80349e22i			1.80754i
$409$	1.54083e22			0.972986			0.486493	−	0.873684i	$-0.338276\pi$
$409$	1.54083e22			0.972986			0.486493	+	0.873684i	$0.338276\pi$
$410$	−6.91382e21	+	6.98554e21i	−0.427617	+	0.432053i
$411$	7.91652e21			0.479600
$412$		−	1.25822e22i		−	0.746670i
$413$			6.37620e21i			0.370669i
$414$	−4.17856e20			−0.0237971
$415$	−1.63877e22	+	1.65577e22i	−0.914345	+	0.923830i
$416$	−1.41964e22			−0.776043
$417$		−	2.04911e22i		−	1.09751i
$418$		−	1.40468e22i		−	0.737190i
$419$	−2.60959e22			−1.34200			−0.671001	−	0.741456i	$-0.734136\pi$
$419$	−2.60959e22			−1.34200			−0.671001	+	0.741456i	$0.734136\pi$
$420$	3.29462e21	+	3.26079e21i	0.166030	+	0.164325i
$421$	−2.31572e22			−1.14364			−0.571818	−	0.820380i	$-0.693762\pi$
$421$	−2.31572e22			−1.14364			−0.571818	+	0.820380i	$0.693762\pi$
$422$			1.72646e21i			0.0835603i
$423$			8.60796e21i			0.408326i
$424$	4.45992e21			0.207356
$425$	3.79036e22	−	3.91182e20i	1.72733	−	0.0178268i
$426$	−6.77430e21			−0.302610
$427$		−	1.04127e22i		−	0.455962i
$428$			4.19966e21i			0.180278i
$429$	3.07796e22			1.29532
$430$	−7.84057e21	−	7.76007e21i	−0.323494	−	0.320172i
$431$	6.11457e20			0.0247348			0.0123674	−	0.999924i	$-0.496063\pi$
$431$	6.11457e20			0.0247348			0.0123674	+	0.999924i	$0.496063\pi$
$432$			4.09889e21i			0.162575i
$433$			3.10026e22i			1.20573i	0.797842	+	0.602867i	$0.205975\pi$
$433$			3.10026e22i			1.20573i	−0.797842	+	0.602867i	$0.794025\pi$
$434$	2.68643e21			0.102450
$435$	2.07267e20	−	2.09417e20i	0.00775125	−	0.00783166i
$436$	1.99202e22			0.730565
$437$		−	3.94268e21i		−	0.141808i
$438$		−	1.75661e22i		−	0.619649i
$439$	−1.11795e22			−0.386787			−0.193394	−	0.981121i	$-0.561949\pi$
$439$	−1.11795e22			−0.386787			−0.193394	+	0.981121i	$0.561949\pi$
$440$	2.94739e22	−	2.97797e22i	1.00021	−	1.01058i
$441$	−7.46536e21			−0.248498
$442$			2.15927e22i			0.705047i
$443$		−	4.87194e22i		−	1.56052i	−0.625454	−	0.780261i	$-0.715086\pi$
$443$		−	4.87194e22i		−	1.56052i	0.625454	−	0.780261i	$-0.284914\pi$
$444$	9.98030e21			0.313609
$445$	4.07185e21	+	4.03004e21i	0.125525	+	0.124237i
$446$	9.04188e21			0.273472
$447$			1.39631e22i			0.414349i
$448$			3.70199e21i			0.107788i
$449$	−8.32387e21			−0.237811			−0.118905	−	0.992906i	$-0.537938\pi$
$449$	−8.32387e21			−0.237811			−0.118905	+	0.992906i	$0.537938\pi$
$450$	5.28604e21	−	5.45543e19i	0.148192	−	0.00152941i
$451$	−6.21756e22			−1.71048
$452$		−	1.87315e22i		−	0.505704i
$453$		−	5.32088e22i		−	1.40977i
$454$	−8.36936e21			−0.217629
$455$	6.14539e21	+	6.08229e21i	0.156838	+	0.155228i
$456$	−3.68905e22			−0.924088
$457$			6.51348e22i			1.60149i	0.599002	+	0.800747i	$0.295564\pi$
$457$			6.51348e22i			1.60149i	−0.599002	+	0.800747i	$0.704436\pi$
$458$			3.29099e22i			0.794273i
$459$	5.98605e22			1.41818
$460$	3.41598e21	−	3.45142e21i	0.0794464	−	0.0802705i
$461$	−3.15500e22			−0.720346			−0.360173	−	0.932886i	$-0.617282\pi$
$461$	−3.15500e22			−0.720346			−0.360173	+	0.932886i	$0.617282\pi$
$462$		−	1.23686e22i		−	0.277245i
$463$			7.28815e22i			1.60391i	0.597388	+	0.801953i	$0.296205\pi$
$463$			7.28815e22i			1.60391i	−0.597388	+	0.801953i	$0.703795\pi$
$464$	−8.95368e19			−0.00193463
$465$	−2.38878e22	+	2.41356e22i	−0.506788	+	0.512045i
$466$	1.37105e22			0.285610
$467$			2.24397e22i			0.459011i	0.973307	+	0.229506i	$0.0737109\pi$
$467$			2.24397e22i			0.459011i	−0.973307	+	0.229506i	$0.926289\pi$
$468$		−	7.13939e21i		−	0.143407i
$469$	−1.82488e22			−0.359969
$470$	2.99893e22	+	2.96814e22i	0.580945	+	0.574980i
$471$	1.03247e21			0.0196426
$472$		−	6.24995e22i		−	1.16780i
$473$		−	6.97859e22i		−	1.28070i
$474$	2.09390e22			0.377433
$475$	5.14747e20	+	4.98764e22i	0.00911378	+	0.883079i
$476$	−2.05716e22			−0.357775
$477$			3.55966e21i			0.0608139i
$478$		−	9.82902e21i		−	0.164958i
$479$	−6.47401e22			−1.06739			−0.533693	−	0.845678i	$-0.679196\pi$
$479$	−6.47401e22			−1.06739			−0.533693	+	0.845678i	$0.679196\pi$
$480$	−5.12531e22	−	5.07268e22i	−0.830174	−	0.821651i
$481$	1.86161e22			0.296248
$482$		−	4.63344e21i		−	0.0724442i
$483$		−	3.47164e21i		−	0.0533316i
$484$	6.28508e22			0.948692
$485$	6.87331e22	−	6.94461e22i	1.01944	−	1.03001i
$486$	1.98169e22			0.288820
$487$		−	1.32206e23i		−	1.89346i	−0.322030	−	0.946730i	$-0.604365\pi$
$487$		−	1.32206e23i		−	1.89346i	0.322030	−	0.946730i	$-0.395635\pi$
$488$			1.02066e23i			1.43652i
$489$	−3.47541e22			−0.480707
$490$	−2.57416e22	+	2.60086e22i	−0.349920	+	0.353550i
$491$	1.24365e23			1.66151			0.830757	−	0.556636i	$-0.187908\pi$
$491$	1.24365e23			1.66151			0.830757	+	0.556636i	$0.187908\pi$
$492$			6.74248e22i			0.885353i
$493$			1.30760e21i			0.0168763i
$494$	−2.84133e22			−0.360448
$495$	2.37685e22	+	2.35245e22i	0.296386	+	0.293343i
$496$	1.03192e22			0.126489
$497$		−	1.20384e22i		−	0.145058i
$498$		−	6.74088e22i		−	0.798486i
$499$	1.71668e22			0.199910			0.0999550	−	0.994992i	$-0.468130\pi$
$499$	1.71668e22			0.199910			0.0999550	+	0.994992i	$0.468130\pi$
$500$	−4.27628e22	+	4.41077e22i	−0.489578	+	0.504975i
$501$	1.00844e23			1.13508
$502$		−	3.41230e22i		−	0.377629i
$503$			6.39694e22i			0.696057i	0.937484	+	0.348028i	$0.113149\pi$
$503$			6.39694e22i			0.696057i	−0.937484	+	0.348028i	$0.886851\pi$
$504$	−6.94793e21			−0.0743355
$505$	−1.23869e23	−	1.22597e23i	−1.30313	−	1.28975i
$506$	−1.29573e22			−0.134040
$507$			4.86136e22i			0.494527i
$508$		−	1.37562e22i		−	0.137612i
$509$	−1.87713e23			−1.84669			−0.923346	−	0.383970i	$-0.874557\pi$
$509$	−1.87713e23			−1.84669			−0.923346	+	0.383970i	$0.874557\pi$
$510$	−7.71553e22	+	7.79557e22i	−0.746482	+	0.754226i
$511$	3.12163e22			0.297032
$512$			4.15445e22i			0.388793i
$513$			7.87691e22i			0.725034i
$514$	8.13013e22			0.736057
$515$	8.38606e22	−	8.47306e22i	0.746787	−	0.754534i
$516$	−7.56777e22			−0.662896
$517$			2.66923e23i			2.29994i
$518$		−	7.48076e21i		−	0.0634078i
$519$	3.85891e22			0.321768
$520$	−6.02372e22	−	5.96187e22i	−0.494124	−	0.489051i
$521$	2.51826e22			0.203226			0.101613	−	0.994824i	$-0.467600\pi$
$521$	2.51826e22			0.203226			0.101613	+	0.994824i	$0.467600\pi$
$522$			1.82359e20i			0.00144786i
$523$		−	1.41643e23i		−	1.10645i	−0.833033	−	0.553223i	$-0.813398\pi$
$523$		−	1.41643e23i		−	1.10645i	0.833033	−	0.553223i	$-0.186602\pi$
$524$	−3.78230e22			−0.290696
$525$	4.53249e20	+	4.39176e22i	0.00342754	+	0.332112i
$526$	−4.13097e22			−0.307378
$527$		−	1.50703e23i		−	1.10340i
$528$		−	4.75109e22i		−	0.342298i
$529$	1.37413e23			0.974216
$530$	1.24015e22	+	1.22742e22i	0.0865229	+	0.0856345i
$531$	4.98837e22			0.342496
$532$		−	2.70697e22i		−	0.182909i
$533$			1.25766e23i			0.836340i
$534$	−1.65771e22			−0.108494
$535$	−2.79909e22	+	2.82813e22i	−0.180306	+	0.182176i
$536$	1.78875e23			1.13409
$537$		−	1.42920e23i		−	0.891891i
$538$		−	2.42833e22i		−	0.149162i
$539$	−2.31493e23			−1.39969
$540$	−6.82464e22	+	6.89543e22i	−0.406193	+	0.410407i
$541$	4.04123e22			0.236776			0.118388	−	0.992967i	$-0.462227\pi$
$541$	4.04123e22			0.236776			0.118388	+	0.992967i	$0.462227\pi$
$542$			3.37632e22i			0.194738i
$543$		−	1.49043e23i		−	0.846278i
$544$	3.20025e23			1.78893
$545$	1.34146e23	+	1.32769e23i	0.738259	+	0.730679i
$546$	−2.50187e22			−0.135559
$547$		−	2.57609e23i		−	1.37426i	−0.726535	−	0.687130i	$-0.758870\pi$
$547$		−	2.57609e23i		−	1.37426i	0.726535	−	0.687130i	$-0.241130\pi$
$548$		−	5.69402e22i		−	0.299078i
$549$	−8.14633e22			−0.421305
$550$	1.63914e23	−	1.69167e21i	0.834707	−	0.00861456i
$551$	−1.72064e21			−0.00862785
$552$			3.40291e22i			0.168023i
$553$			3.72102e22i			0.180925i
$554$	−8.35742e22			−0.400165
$555$	6.72092e22	+	6.65191e22i	0.316912	+	0.313658i
$556$	−1.47383e23			−0.684405
$557$			2.94142e23i			1.34520i	0.740004	+	0.672602i	$0.234824\pi$
$557$			2.94142e23i			1.34520i	−0.740004	+	0.672602i	$0.765176\pi$
$558$		−	2.10171e22i		−	0.0946633i
$559$	−1.41160e23			−0.626198
$560$	9.38852e21	−	9.48592e21i	0.0410203	−	0.0414459i
$561$	−6.93853e23			−2.98596
$562$		−	7.19007e22i		−	0.304771i
$563$			2.02557e23i			0.845719i	0.906195	+	0.422859i	$0.138974\pi$
$563$			2.02557e23i			0.845719i	−0.906195	+	0.422859i	$0.861026\pi$
$564$	2.89459e23			1.19046
$565$	1.24846e23	−	1.26141e23i	0.505783	−	0.511030i
$566$	−1.12120e23			−0.447449
$567$			8.97391e22i			0.352798i
$568$			1.18001e23i			0.457009i
$569$	−1.65130e23			−0.630045			−0.315022	−	0.949084i	$-0.602012\pi$
$569$	−1.65130e23			−0.630045			−0.315022	+	0.949084i	$0.602012\pi$
$570$	−1.02580e23	−	1.01527e23i	−0.385591	−	0.381632i
$571$	9.39406e22			0.347894			0.173947	−	0.984755i	$-0.444348\pi$
$571$	9.39406e22			0.347894			0.173947	+	0.984755i	$0.444348\pi$
$572$		−	2.21385e23i		−	0.807759i
$573$			1.58095e23i			0.568336i
$574$	5.05384e22			0.179007
$575$	4.60077e22	−	4.74821e20i	0.160566	−	0.00165712i
$576$	2.89622e22			0.0995957
$577$			3.22752e23i			1.09364i	0.837250	+	0.546820i	$0.184162\pi$
$577$			3.22752e23i			1.09364i	−0.837250	+	0.546820i	$0.815838\pi$
$578$		−	3.23633e23i		−	1.08060i
$579$	5.53215e23			1.82023
$580$	−1.50625e21	−	1.49078e21i	−0.00488381	−	0.00483367i
$581$	1.19791e23			0.382759
$582$			2.82725e23i			0.890263i
$583$			1.10381e23i			0.342541i
$584$	−3.05982e23			−0.935809
$585$	4.75844e22	−	4.80780e22i	0.143430	−	0.144918i
$586$	−2.89141e23			−0.858975
$587$		−	1.54906e23i		−	0.453570i	−0.973945	−	0.226785i	$-0.927178\pi$
$587$		−	1.54906e23i		−	0.453570i	0.973945	−	0.226785i	$-0.0728215\pi$
$588$			2.51037e23i			0.724486i
$589$	1.98307e23			0.564102
$590$	1.72006e23	−	1.73790e23i	0.482282	−	0.487285i
$591$	3.51849e22			0.0972440
$592$		−	2.87354e22i		−	0.0782859i
$593$		−	6.15543e23i		−	1.65308i	−0.562879	−	0.826539i	$-0.690306\pi$
$593$		−	6.15543e23i		−	1.65308i	0.562879	−	0.826539i	$-0.309694\pi$
$594$	2.58867e23			0.685319
$595$	−1.38533e23	−	1.37111e23i	−0.361543	−	0.357831i
$596$	1.00430e23			0.258388
$597$		−	6.98851e23i		−	1.77257i
$598$			2.62094e22i			0.0655387i
$599$	−2.44389e23			−0.602496			−0.301248	−	0.953546i	$-0.597403\pi$
$599$	−2.44389e23			−0.602496			−0.301248	+	0.953546i	$0.597403\pi$
$600$	−4.44275e21	−	4.30480e23i	−0.0107986	−	1.04633i
$601$	−2.29017e23			−0.548825			−0.274413	−	0.961612i	$-0.588483\pi$
$601$	−2.29017e23			−0.548825			−0.274413	+	0.961612i	$0.588483\pi$
$602$			5.67244e22i			0.134029i
$603$			1.42768e23i			0.332609i
$604$	−3.82708e23			−0.879131
$605$	4.23249e23	+	4.18903e23i	0.958683	+	0.948840i
$606$	5.04289e23			1.12632
$607$			5.08927e23i			1.12086i	0.828202	+	0.560430i	$0.189364\pi$
$607$			5.08927e23i			1.12086i	−0.828202	+	0.560430i	$0.810636\pi$
$608$			4.21113e23i			0.914572i
$609$	−1.51507e21			−0.00324479
$610$	−2.80897e23	+	2.83811e23i	−0.593257	+	0.599412i
$611$	5.39922e23			1.12455
$612$			1.60941e23i			0.330582i
$613$		−	3.80505e23i		−	0.770808i	−0.922748	−	0.385404i	$-0.874062\pi$
$613$		−	3.80505e23i		−	0.770808i	0.922748	−	0.385404i	$-0.125938\pi$
$614$	−1.98969e23			−0.397515
$615$	−4.49389e23	+	4.54051e23i	−0.885491	+	0.894677i
$616$	−2.15448e23			−0.418703
$617$		−	3.08703e23i		−	0.591721i	−0.955231	−	0.295860i	$-0.904394\pi$
$617$		−	3.08703e23i		−	0.591721i	0.955231	−	0.295860i	$-0.0956063\pi$
$618$			3.44950e23i			0.652160i
$619$	−2.71878e22			−0.0506995			−0.0253497	−	0.999679i	$-0.508070\pi$
$619$	−2.71878e22			−0.0506995			−0.0253497	+	0.999679i	$0.508070\pi$
$620$	1.73597e23	+	1.71815e23i	0.319311	+	0.316032i
$621$	7.26593e22			0.131829
$622$			3.53682e23i			0.632987i
$623$		−	2.94587e22i		−	0.0520074i
$624$	−9.61031e22			−0.167366
$625$	−5.81953e23	+	1.20133e22i	−0.999787	+	0.0206387i
$626$	−1.23198e23			−0.208795
$627$		−	9.13025e23i		−	1.52654i
$628$		−	7.42611e21i		−	0.0122491i
$629$	−4.19655e23			−0.682907
$630$	−1.93199e22	−	1.91215e22i	−0.0310177	−	0.0306992i
$631$	1.08541e22			0.0171927			0.00859636	−	0.999963i	$-0.497264\pi$
$631$	1.08541e22			0.0171927			0.00859636	+	0.999963i	$0.497264\pi$
$632$		−	3.64734e23i		−	0.570009i
$633$			1.12217e23i			0.173033i
$634$	6.94278e22			0.105627
$635$	9.16855e22	−	9.26367e22i	0.137634	−	0.139061i
$636$	1.19700e23			0.177301
$637$			4.68254e23i			0.684379i
$638$			5.65474e21i			0.00815524i
$639$	−9.41819e22			−0.134032
$640$	−4.18884e23	+	4.23229e23i	−0.588252	+	0.594355i
$641$	1.29824e24			1.79913			0.899563	−	0.436792i	$-0.143885\pi$
$641$	1.29824e24			1.79913			0.899563	+	0.436792i	$0.143885\pi$
$642$		−	1.15137e23i		−	0.157459i
$643$		−	9.64078e23i		−	1.30113i	−0.759453	−	0.650563i	$-0.774533\pi$
$643$		−	9.64078e23i		−	1.30113i	0.759453	−	0.650563i	$-0.225467\pi$
$644$	−2.49701e22			−0.0332575
$645$	−5.09627e23	−	5.04395e23i	−0.669877	−	0.662999i
$646$	6.40511e23			0.830903
$647$			1.43576e24i			1.83821i	0.394017	+	0.919103i	$0.371085\pi$
$647$			1.43576e24i			1.83821i	−0.394017	+	0.919103i	$0.628915\pi$
$648$		−	8.79624e23i		−	1.11150i
$649$	1.54684e24			1.92915
$650$	−3.42184e21	−	3.31559e23i	−0.00421208	−	0.408129i
$651$	1.74615e23			0.212149
$652$			2.49971e23i			0.299768i
$653$			4.34101e23i			0.513841i	0.966433	+	0.256920i	$0.0827078\pi$
$653$			4.34101e23i			0.513841i	−0.966433	+	0.256920i	$0.917292\pi$
$654$	−5.46128e23			−0.638093
$655$	−2.54707e23	−	2.52092e23i	−0.293758	−	0.290742i
$656$	1.94131e23			0.221010
$657$		−	2.44218e23i		−	0.274456i
$658$		−	2.16964e23i		−	0.240696i
$659$	−2.96685e22			−0.0324915			−0.0162457	−	0.999868i	$-0.505171\pi$
$659$	−2.96685e22			−0.0324915			−0.0162457	+	0.999868i	$0.505171\pi$
$660$	7.91055e23	−	7.99261e23i	0.855230	−	0.864102i
$661$	−3.46155e23			−0.369453			−0.184726	−	0.982790i	$-0.559140\pi$
$661$	−3.46155e23			−0.369453			−0.184726	+	0.982790i	$0.559140\pi$
$662$		−	5.67815e23i		−	0.598293i
$663$			1.40350e24i			1.45998i
$664$	−1.17419e24			−1.20589
$665$	1.80421e23	−	1.82293e23i	0.182938	−	0.184835i
$666$	−5.85252e22			−0.0585884
$667$			1.58718e21i			0.00156876i
$668$		−	7.25325e23i		−	0.707836i
$669$	5.87711e23			0.566294
$670$	4.97391e23	+	4.92284e23i	0.473219	+	0.468360i
$671$	−2.52609e24			−2.37305
$672$			3.70802e23i			0.343956i
$673$			1.41988e24i			1.30054i	0.759703	+	0.650270i	$0.225344\pi$
$673$			1.41988e24i			1.30054i	−0.759703	+	0.650270i	$0.774656\pi$
$674$	−1.32004e23			−0.119393
$675$	−9.19167e23	+	9.48622e21i	−0.820942	+	0.00847250i
$676$	3.49657e23			0.308386
$677$		−	8.46838e23i		−	0.737559i	−0.929517	−	0.368779i	$-0.879776\pi$
$677$		−	8.46838e23i		−	0.737559i	0.929517	−	0.368779i	$-0.120224\pi$
$678$			5.13539e23i			0.441694i
$679$	−5.02423e23			−0.426753
$680$	1.35790e24	+	1.34396e24i	1.13905	+	1.12736i
$681$	−5.43997e23			−0.450657
$682$		−	6.51717e23i		−	0.533202i
$683$		−	3.07314e23i		−	0.248317i	−0.992262	−	0.124159i	$-0.960377\pi$
$683$		−	3.07314e23i		−	0.248317i	0.992262	−	0.124159i	$-0.0396232\pi$
$684$	−2.11778e23			−0.169007
$685$	3.79509e23	−	3.83446e23i	0.299124	−	0.302227i
$686$	3.94197e23			0.306872
$687$			2.13910e24i			1.64475i
$688$			2.17892e23i			0.165478i
$689$	2.23275e23			0.167485
$690$	−9.36519e22	+	9.46234e22i	−0.0693904	+	0.0701102i
$691$	2.97612e23			0.217815			0.108907	−	0.994052i	$-0.465265\pi$
$691$	2.97612e23			0.217815			0.108907	+	0.994052i	$0.465265\pi$
$692$		−	2.77555e23i		−	0.200654i
$693$		−	1.71959e23i		−	0.122798i
$694$	1.75940e23			0.124111
$695$	−9.92507e23	−	9.82317e23i	−0.691613	−	0.684512i
$696$	1.48508e22			0.0102228
$697$		−	2.83510e24i		−	1.92792i
$698$			4.66419e23i			0.313332i
$699$	8.91166e23			0.591429
$700$	3.15880e23	−	3.26003e21i	0.207104	−	0.00213741i
$701$	−2.35616e24			−1.52617			−0.763084	−	0.646299i	$-0.776316\pi$
$701$	−2.35616e24			−1.52617			−0.763084	+	0.646299i	$0.776316\pi$
$702$		−	5.23626e23i		−	0.335086i
$703$		−	5.52214e23i		−	0.349130i
$704$	8.98087e23			0.560984
$705$	1.94927e24	+	1.92925e24i	1.20300	+	1.19064i
$706$	−4.95597e23			−0.302196
$707$			8.96159e23i			0.539909i
$708$		−	1.67743e24i		−	0.998533i
$709$	2.77907e24			1.63458			0.817289	−	0.576227i	$-0.195476\pi$
$709$	2.77907e24			1.63458			0.817289	+	0.576227i	$0.195476\pi$
$710$	−3.24752e23	+	3.28121e23i	−0.188737	+	0.190694i
$711$	2.91111e23			0.167173
$712$			2.88754e23i			0.163851i
$713$		−	1.82925e23i		−	0.102568i
$714$	5.63988e23			0.312489
$715$	1.47554e24	−	1.49085e24i	0.807885	−	0.816265i
$716$	−1.02796e24			−0.556182
$717$		−	6.38874e23i		−	0.341588i
$718$			4.14429e23i			0.218974i
$719$	−9.31893e23			−0.486598			−0.243299	−	0.969951i	$-0.578230\pi$
$719$	−9.31893e23			−0.486598			−0.243299	+	0.969951i	$0.578230\pi$
$720$	−7.42124e22	−	7.34504e22i	−0.0382957	−	0.0379025i
$721$	−6.13002e23			−0.312617
$722$		−	2.37845e23i		−	0.119875i
$723$		−	3.01167e23i		−	0.150014i
$724$	−1.07200e24			−0.527738
$725$	−2.07219e20	−	2.00784e22i	−0.000100822	−	0.00976915i
$726$	−1.72310e24			−0.828610
$727$			7.77376e23i			0.369478i	0.982788	+	0.184739i	$0.0591440\pi$
$727$			7.77376e23i			0.369478i	−0.982788	+	0.184739i	$0.940856\pi$
$728$			4.35799e23i			0.204724i
$729$	−1.29218e24			−0.599983
$730$	−8.50834e23	−	8.42098e23i	−0.390482	−	0.386472i
$731$	3.18212e24			1.44351
$732$			2.73936e24i			1.22830i
$733$		−	2.62599e24i		−	1.16388i	−0.813230	−	0.581942i	$-0.802293\pi$
$733$		−	2.62599e24i		−	1.16388i	0.813230	−	0.581942i	$-0.197707\pi$
$734$	−1.38230e24			−0.605598
$735$	−1.67317e24	+	1.69053e24i	−0.724599	+	0.732116i
$736$	3.88449e23			0.166292
$737$			4.42708e24i			1.87346i
$738$		−	3.95384e23i		−	0.165402i
$739$	5.55636e23			0.229780			0.114890	−	0.993378i	$-0.463348\pi$
$739$	5.55636e23			0.229780			0.114890	+	0.993378i	$0.463348\pi$
$740$	4.78444e23	−	4.83407e23i	0.195597	−	0.197626i
$741$	−1.84683e24			−0.746401
$742$		−	8.97217e22i		−	0.0358479i
$743$			5.15655e23i			0.203683i	0.994801	+	0.101841i	$0.0324734\pi$
$743$			5.15655e23i			0.203683i	−0.994801	+	0.101841i	$0.967527\pi$
$744$	−1.71157e24			−0.668383
$745$	6.76317e23	+	6.69373e23i	0.261109	+	0.258428i
$746$	1.12324e24			0.428739
$747$		−	9.37173e23i		−	0.353667i
$748$			4.99059e24i			1.86204i
$749$	2.04607e23			0.0754788
$750$	1.17238e24	−	1.20925e24i	0.427609	−	0.441057i
$751$	3.82210e24			1.37836			0.689180	−	0.724590i	$-0.257971\pi$
$751$	3.82210e24			1.37836			0.689180	+	0.724590i	$0.257971\pi$
$752$		−	8.33414e23i		−	0.297173i
$753$		−	2.21795e24i		−	0.781978i
$754$	1.14382e22			0.00398750
$755$	−2.57723e24	−	2.55077e24i	−0.888390	−	0.879269i
$756$	4.98865e23			0.170039
$757$		−	3.94460e24i		−	1.32950i	−0.747067	−	0.664749i	$-0.768538\pi$
$757$		−	3.94460e24i		−	1.32950i	0.747067	−	0.664749i	$-0.231462\pi$
$758$			9.13682e23i			0.304514i
$759$	−8.42206e23			−0.277564
$760$	−1.76849e24	+	1.78683e24i	−0.576350	+	0.582329i
$761$	−3.70043e23			−0.119257			−0.0596284	−	0.998221i	$-0.518992\pi$
$761$	−3.70043e23			−0.119257			−0.0596284	+	0.998221i	$0.518992\pi$
$762$			3.77137e23i			0.120194i
$763$		−	9.70512e23i		−	0.305874i
$764$	1.13711e24			0.354413
$765$	−1.07268e24	+	1.08380e24i	−0.330633	+	0.334063i
$766$	1.62262e23			0.0494620
$767$		−	3.12888e24i		−	0.943254i
$768$		−	3.10772e24i		−	0.926555i
$769$	−1.11643e24			−0.329197			−0.164599	−	0.986361i	$-0.552633\pi$
$769$	−1.11643e24			−0.329197			−0.164599	+	0.986361i	$0.552633\pi$
$770$	−5.99088e23	−	5.92937e23i	−0.174711	−	0.172917i
$771$	5.28448e24			1.52419
$772$		−	3.97904e24i		−	1.13509i
$773$			4.91642e24i			1.38715i	0.720385	+	0.693574i	$0.243965\pi$
$773$			4.91642e24i			1.38715i	−0.720385	+	0.693574i	$0.756035\pi$
$774$	4.43779e23			0.123842
$775$	2.38822e22	+	2.31407e24i	0.00659190	+	0.638722i
$776$	4.92476e24			1.34450
$777$		−	4.86240e23i		−	0.131302i
$778$			1.20670e24i			0.322310i
$779$	3.73064e24			0.985633
$780$	−1.61671e24	−	1.60011e24i	−0.422502	−	0.418164i
$781$	−2.92048e24			−0.754953
$782$		−	5.90829e23i		−	0.151079i
$783$		−	3.17096e22i		−	0.00802076i
$784$	7.22789e23			0.180853
$785$	4.94953e22	−	5.00088e22i	0.0122510	−	0.0123781i
$786$	1.03695e24			0.253901
$787$			3.08300e24i			0.746772i	0.927676	+	0.373386i	$0.121803\pi$
$787$			3.08300e24i			0.746772i	−0.927676	+	0.373386i	$0.878197\pi$
$788$		−	2.53070e23i		−	0.0606412i
$789$	−2.68508e24			−0.636505
$790$	1.00379e24	−	1.01420e24i	0.235404	−	0.237846i
$791$	−9.12598e23			−0.211729
$792$			1.68554e24i			0.386879i
$793$			5.10967e24i			1.16030i
$794$	2.43577e24			0.547220
$795$	8.06084e23	+	7.97808e23i	0.179168	+	0.177328i
$796$	−5.02654e24			−1.10537
$797$			2.52033e24i			0.548356i	0.961679	+	0.274178i	$0.0884057\pi$
$797$			2.52033e24i			0.548356i	−0.961679	+	0.274178i	$0.911594\pi$
$798$			7.42138e23i			0.159757i
$799$	−1.21712e25			−2.59231
$800$	−4.91402e24	+	5.07150e22i	−1.03555	+	0.0106874i
$801$	−2.30468e23			−0.0480545
$802$			5.04961e24i			1.04178i
$803$		−	7.57294e24i		−	1.54590i
$804$	4.80085e24			0.969709
$805$	−1.68153e23	−	1.66426e23i	−0.0336077	−	0.0332627i
$806$	−1.31827e24			−0.260709
$807$		−	1.57838e24i		−	0.308878i
$808$		−	8.78416e24i		−	1.70100i
$809$	−1.85576e24			−0.355598			−0.177799	−	0.984067i	$-0.556898\pi$
$809$	−1.85576e24			−0.355598			−0.177799	+	0.984067i	$0.556898\pi$
$810$	2.42082e24	−	2.44594e24i	0.459030	−	0.463791i
$811$	6.23069e24			1.16912			0.584560	−	0.811351i	$-0.301267\pi$
$811$	6.23069e24			1.16912			0.584560	+	0.811351i	$0.301267\pi$
$812$			1.08973e22i			0.00202345i
$813$			2.19457e24i			0.403254i
$814$	−1.81480e24			−0.330006
$815$	−1.66607e24	+	1.68335e24i	−0.299815	+	0.302925i
$816$	2.16642e24			0.385812
$817$			4.18727e24i			0.737979i
$818$			3.03837e24i			0.529953i
$819$	−3.47831e23			−0.0600420
$820$	3.26580e24	+	3.23227e24i	0.557919	+	0.552191i
$821$	−3.16422e24			−0.534995			−0.267498	−	0.963559i	$-0.586197\pi$
$821$	−3.16422e24			−0.534995			−0.267498	+	0.963559i	$0.586197\pi$
$822$			1.56106e24i			0.261222i
$823$			6.98168e24i			1.15628i	0.815939	+	0.578138i	$0.196220\pi$
$823$			6.98168e24i			1.15628i	−0.815939	+	0.578138i	$0.803780\pi$
$824$	6.00865e24			0.984908
$825$	1.06542e25	−	1.09956e23i	1.72847	−	0.0178386i
$826$	−1.25732e24			−0.201891
$827$		−	5.87487e24i		−	0.933688i	−0.884340	−	0.466844i	$-0.845391\pi$
$827$		−	5.87487e24i		−	0.933688i	0.884340	−	0.466844i	$-0.154609\pi$
$828$			1.95351e23i			0.0307297i
$829$	1.14242e24			0.177873			0.0889366	−	0.996037i	$-0.471653\pi$
$829$	1.14242e24			0.177873			0.0889366	+	0.996037i	$0.471653\pi$
$830$	−3.26502e24	−	3.23150e24i	−0.503179	−	0.498013i
$831$	−5.43221e24			−0.828644
$832$		−	1.81661e24i		−	0.274293i
$833$		−	1.05557e25i		−	1.57762i
$834$	4.04063e24			0.597776
$835$	4.83432e24	−	4.88447e24i	0.707946	−	0.715290i
$836$	−6.56700e24			−0.951950
$837$			3.65457e24i			0.524409i
$838$		−	5.14586e24i		−	0.730943i
$839$	1.17074e25			1.64621			0.823105	−	0.567889i	$-0.192240\pi$
$839$	1.17074e25			1.64621			0.823105	+	0.567889i	$0.192240\pi$
$840$	−1.55720e24	+	1.57336e24i	−0.216756	+	0.219005i
$841$	−7.25646e24			−0.999905
$842$		−	4.56637e24i		−	0.622900i
$843$		−	4.67345e24i		−	0.631107i
$844$	8.07133e23			0.107903
$845$	2.35465e24	+	2.33048e24i	0.311634	+	0.308434i
$846$	−1.69740e24			−0.222401
$847$		−	3.06209e24i		−	0.397199i
$848$		−	3.44643e23i		−	0.0442593i
$849$	−7.28766e24			−0.926559
$850$	7.71373e22	+	7.47421e24i	0.00970965	+	0.940816i
$851$	−5.09381e23			−0.0634807
$852$			3.16704e24i			0.390766i
$853$			1.03109e25i			1.25959i	0.776763	+	0.629793i	$0.216860\pi$
$853$			1.03109e25i			1.25959i	−0.776763	+	0.629793i	$0.783140\pi$
$854$	2.05329e24			0.248347
$855$	−1.42615e24	−	1.41151e24i	−0.170787	−	0.169033i
$856$	−2.00556e24			−0.237798
$857$			1.60769e23i			0.0188740i	0.999955	+	0.00943701i	$0.00300394\pi$
$857$			1.60769e23i			0.0188740i	−0.999955	+	0.00943701i	$0.996996\pi$
$858$			6.06944e24i			0.705516i
$859$	−1.64738e24			−0.189605			−0.0948027	−	0.995496i	$-0.530222\pi$
$859$	−1.64738e24			−0.189605			−0.0948027	+	0.995496i	$0.530222\pi$
$860$	−3.62790e24	+	3.66553e24i	−0.413446	+	0.417734i
$861$	3.28493e24			0.370681
$862$			1.20573e23i			0.0134722i
$863$			1.49755e25i			1.65687i	0.560086	+	0.828434i	$0.310768\pi$
$863$			1.49755e25i			1.65687i	−0.560086	+	0.828434i	$0.689232\pi$
$864$	−7.76064e24			−0.850219
$865$	1.84992e24	−	1.86911e24i	0.200685	−	0.202767i
$866$	−6.11342e24			−0.656723
$867$		−	2.10357e25i		−	2.23766i
$868$		−	1.25593e24i		−	0.132296i
$869$	9.02703e24			0.941623
$870$	4.12950e22	+	4.08710e22i	0.00426564	+	0.00422184i
$871$	8.95493e24			0.916026
$872$			9.51296e24i			0.963664i
$873$			3.93067e24i			0.394317i
$874$	7.77458e23			0.0772378
$875$	2.14893e24	+	2.08340e24i	0.211423	+	0.204977i
$876$	−8.21230e24			−0.800165
$877$			1.81287e24i			0.174932i	0.996167	+	0.0874662i	$0.0278770\pi$
$877$			1.81287e24i			0.174932i	−0.996167	+	0.0874662i	$0.972123\pi$
$878$		−	2.20448e24i		−	0.210670i
$879$	−1.87938e25			−1.77873
$880$	−2.30124e24	−	2.27762e24i	−0.215705	−	0.213490i
$881$	−1.37084e25			−1.27260			−0.636299	−	0.771442i	$-0.719536\pi$
$881$	−1.37084e25			−1.27260			−0.636299	+	0.771442i	$0.719536\pi$
$882$		−	1.47210e24i		−	0.135348i
$883$		−	7.88163e24i		−	0.717712i	−0.933393	−	0.358856i	$-0.883167\pi$
$883$		−	7.88163e24i		−	0.717712i	0.933393	−	0.358856i	$-0.116833\pi$
$884$	1.00948e25			0.910442
$885$	1.11802e25	−	1.12961e25i	0.998689	−	1.00905i
$886$	9.60698e24			0.849964
$887$			2.04571e25i			1.79264i	0.443410	+	0.896319i	$0.353769\pi$
$887$			2.04571e25i			1.79264i	−0.443410	+	0.896319i	$0.646231\pi$
$888$			4.76613e24i			0.413671i
$889$	−6.70200e23			−0.0576155
$890$	−7.94684e23	+	8.02928e23i	−0.0676674	+	0.0683694i
$891$	2.17703e25			1.83614
$892$		−	4.22715e24i		−	0.353140i
$893$		−	1.60159e25i		−	1.32530i
$894$	−2.75338e24			−0.225682
$895$	−6.92250e24	−	6.85142e24i	−0.562040	−	0.556269i
$896$	3.06194e24			0.246252
$897$			1.70358e24i			0.135715i
$898$		−	1.64138e24i		−	0.129527i
$899$	−7.98310e22			−0.00624043
$900$	−2.55046e22	−	2.47127e24i	−0.00197496	−	0.191363i
$901$	−5.03320e24			−0.386085
$902$		−	1.22604e25i		−	0.931643i
$903$			3.68701e24i			0.277542i
$904$	8.94530e24			0.667057
$905$	−7.21906e24	−	7.14494e24i	−0.533296	−	0.527820i
$906$	1.04922e25			0.767854
$907$			1.09711e25i			0.795407i	0.917514	+	0.397704i	$0.130193\pi$
$907$			1.09711e25i			0.795407i	−0.917514	+	0.397704i	$0.869807\pi$
$908$			3.91274e24i			0.281029i
$909$	7.01103e24			0.498872
$910$	−1.19937e24	+	1.21181e24i	−0.0845475	+	0.0854246i
$911$	1.51157e25			1.05566			0.527829	−	0.849350i	$-0.323006\pi$
$911$	1.51157e25			1.05566			0.527829	+	0.849350i	$0.323006\pi$
$912$			2.85074e24i			0.197243i
$913$		−	2.90607e25i		−	1.99207i
$914$	−1.28439e25			−0.872280
$915$	−1.82579e25	+	1.84473e25i	−1.22849	+	1.24124i
$916$	1.53857e25			1.02566
$917$			1.84273e24i			0.121709i
$918$			1.18039e25i			0.772437i
$919$	−1.47234e25			−0.954612			−0.477306	−	0.878737i	$-0.658387\pi$
$919$	−1.47234e25			−0.954612			−0.477306	+	0.878737i	$0.658387\pi$
$920$	1.64824e24	+	1.63131e24i	0.105882	+	0.104795i
$921$	−1.29327e25			−0.823157
$922$		−	6.22134e24i		−	0.392348i
$923$			5.90742e24i			0.369133i
$924$	−5.78243e24			−0.358013
$925$	6.44386e24	−	6.65036e22i	0.395313	−	0.00407981i
$926$	−1.43715e25			−0.873593
$927$			4.79578e24i			0.288856i
$928$		−	1.69525e23i		−	0.0101175i
$929$	2.22221e25			1.31417			0.657084	−	0.753817i	$-0.271790\pi$
$929$	2.22221e25			1.31417			0.657084	+	0.753817i	$0.271790\pi$
$930$	−4.75931e24	−	4.71044e24i	−0.278894	−	0.276030i
$931$	1.38900e25			0.806545
$932$		−	6.40978e24i		−	0.368815i
$933$			2.29889e25i			1.31076i
$934$	−4.42489e24			−0.250008
$935$	−3.32625e25	+	3.36076e25i	−1.86233	+	1.88165i
$936$	3.40944e24			0.189164
$937$			8.73364e24i			0.480185i	0.970750	+	0.240093i	$0.0771778\pi$
$937$			8.73364e24i			0.480185i	−0.970750	+	0.240093i	$0.922822\pi$
$938$		−	3.59849e24i		−	0.196063i
$939$	−8.00769e24			−0.432364
$940$	1.38763e25	−	1.40203e25i	0.742484	−	0.750187i
$941$	−2.19322e23			−0.0116298			−0.00581489	−	0.999983i	$-0.501851\pi$
$941$	−2.19322e23			−0.0116298			−0.00581489	+	0.999983i	$0.501851\pi$
$942$			2.03593e23i			0.0106987i
$943$		−	3.44127e24i		−	0.179213i
$944$	−4.82969e24			−0.249263
$945$	3.35945e24	+	3.32496e24i	0.171830	+	0.170065i
$946$	1.37611e25			0.697555
$947$		−	2.00026e25i		−	1.00487i	−0.864614	−	0.502436i	$-0.832437\pi$
$947$		−	2.00026e25i		−	1.00487i	0.864614	−	0.502436i	$-0.167563\pi$
$948$		−	9.78915e24i		−	0.487387i
$949$	−1.53182e25			−0.755868
$950$	−9.83514e24	+	1.01503e23i	−0.480983	+	0.00496397i
$951$	4.51272e24			0.218728
$952$		−	9.82405e24i		−	0.471929i
$953$		−	4.30467e24i		−	0.204951i	−0.994736	−	0.102475i	$-0.967324\pi$
$953$		−	4.30467e24i		−	0.204951i	0.994736	−	0.102475i	$-0.0326763\pi$
$954$	−7.01931e23			−0.0331233
$955$	7.65753e24	+	7.57891e24i	0.358146	+	0.354469i
$956$	−4.59515e24			−0.213014
$957$			3.67551e23i			0.0168875i
$958$		−	1.27661e25i		−	0.581369i
$959$	−2.77412e24			−0.125218
$960$	6.49114e24	−	6.55848e24i	0.290413	−	0.293425i
$961$	−1.33495e25			−0.591991
$962$			3.67091e24i			0.161356i
$963$		−	1.60073e24i		−	0.0697420i
$964$	−2.16617e24			−0.0935487
$965$	2.65205e25	−	2.67956e25i	1.13527	−	1.14705i
$966$	6.84574e23			0.0290479
$967$			3.71481e24i			0.156247i	0.996944	+	0.0781234i	$0.0248928\pi$
$967$			3.71481e24i			0.156247i	−0.996944	+	0.0781234i	$0.975107\pi$
$968$			3.00146e25i			1.25139i
$969$	4.16324e25			1.72060
$970$	1.36941e25	+	1.35535e25i	0.561014	+	0.555254i
$971$	−8.87377e24			−0.360367			−0.180183	−	0.983633i	$-0.557669\pi$
$971$	−8.87377e24			−0.360367			−0.180183	+	0.983633i	$0.557669\pi$
$972$		−	9.26455e24i		−	0.372959i
$973$			7.18051e24i			0.286547i
$974$	2.60698e25			1.03130
$975$	−2.22415e23	−	2.15509e25i	−0.00872219	−	0.845136i
$976$	7.88719e24			0.306619
$977$		−	4.31141e24i		−	0.166156i	−0.996543	−	0.0830780i	$-0.973525\pi$
$977$		−	4.31141e24i		−	0.166156i	0.996543	−	0.0830780i	$-0.0264751\pi$
$978$		−	6.85316e24i		−	0.261825i
$979$	−7.14655e24			−0.270672
$980$	1.21593e25	+	1.20344e25i	0.456547	+	0.451859i
$981$	−7.59272e24			−0.282625
$982$			2.45235e25i			0.904970i
$983$		−	1.30001e25i		−	0.475598i	−0.971314	−	0.237799i	$-0.923574\pi$
$983$		−	1.30001e25i		−	0.475598i	0.971314	−	0.237799i	$-0.0764260\pi$
$984$	−3.21989e25			−1.16784
$985$	1.68672e24	−	1.70422e24i	0.0606507	−	0.0612799i
$986$	−2.57846e23			−0.00919194
$987$		−	1.41024e25i		−	0.498422i
$988$			1.32835e25i			0.465455i
$989$	3.86248e24			0.134183
$990$	−4.63880e24	+	4.68692e24i	−0.159774	+	0.161432i
$991$	−3.28849e25			−1.12298			−0.561488	−	0.827485i	$-0.689771\pi$
$991$	−3.28849e25			−1.12298			−0.561488	+	0.827485i	$0.689771\pi$
$992$			1.95380e25i			0.661500i
$993$		−	3.69072e25i		−	1.23892i
$994$	2.37386e24			0.0790081
$995$	−3.38496e25	−	3.35021e25i	−1.11701	−	1.10554i
$996$	−3.15142e25			−1.03110
$997$			1.26903e25i			0.411685i	0.978585	+	0.205842i	$0.0659934\pi$
$997$			1.26903e25i			0.411685i	−0.978585	+	0.205842i	$0.934007\pi$
$998$			3.38512e24i			0.108884i
$999$	1.01767e25			0.324564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.18.b.a.4.5	yes	8
3.2	odd	2		45.18.b.b.19.4		8
4.3	odd	2		80.18.c.b.49.2		8
5.2	odd	4		25.18.a.f.1.4		8
5.3	odd	4		25.18.a.f.1.5		8
5.4	even	2	inner	5.18.b.a.4.4	✓	8
15.14	odd	2		45.18.b.b.19.5		8
20.19	odd	2		80.18.c.b.49.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.18.b.a.4.4	✓	8	5.4	even	2	inner
5.18.b.a.4.5	yes	8	1.1	even	1	trivial
25.18.a.f.1.4		8	5.2	odd	4
25.18.a.f.1.5		8	5.3	odd	4
45.18.b.b.19.4		8	3.2	odd	2
45.18.b.b.19.5		8	15.14	odd	2
80.18.c.b.49.2		8	4.3	odd	2
80.18.c.b.49.7		8	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	5.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+197.190i q^{2} +12817.1i q^{3} +92188.0 q^{4} +(620811. + 614437. i) q^{5} -2.52741e6 q^{6} -4.49140e6i q^{7} +4.40247e7i q^{8} -3.51381e7 q^{9} +O(q^{10})\)	\(q+197.190i q^{2} +12817.1i q^{3} +92188.0 q^{4} +(620811. + 614437. i) q^{5} -2.52741e6 q^{6} -4.49140e6i q^{7} +4.40247e7i q^{8} -3.51381e7 q^{9} +(-1.21161e8 + 1.22418e8i) q^{10} -1.08959e9 q^{11} +1.18158e9i q^{12} +2.20399e9i q^{13} +8.85659e8 q^{14} +(-7.87531e9 + 7.95700e9i) q^{15} +3.40203e9 q^{16} -4.96836e10i q^{17} -6.92889e9i q^{18} +6.53775e10 q^{19} +(5.72314e10 + 5.66438e10i) q^{20} +5.75667e10 q^{21} -2.14857e11i q^{22} -6.03064e10i q^{23} -5.64269e11 q^{24} +(7.87347e9 + 7.62899e11i) q^{25} -4.34604e11 q^{26} +1.20483e12i q^{27} -4.14053e11i q^{28} -2.63186e10 q^{29} +(-1.56904e12 - 1.55293e12i) q^{30} +3.03326e12 q^{31} +6.44125e12i q^{32} -1.39654e13i q^{33} +9.79712e12 q^{34} +(2.75968e12 - 2.78831e12i) q^{35} -3.23931e12 q^{36} -8.44655e12i q^{37} +1.28918e13i q^{38} -2.82487e13 q^{39} +(-2.70504e13 + 2.73310e13i) q^{40} +5.70631e13 q^{41} +1.13516e13i q^{42} +6.40476e13i q^{43} -1.00447e14 q^{44} +(-2.18141e13 - 2.15901e13i) q^{45} +1.18918e13 q^{46} -2.44975e14i q^{47} +4.36042e13i q^{48} +2.12458e14 q^{49} +(-1.50436e14 + 1.55257e12i) q^{50} +6.36800e14 q^{51} +2.03181e14i q^{52} -1.01305e14i q^{53} -2.37582e14 q^{54} +(-6.76432e14 - 6.69487e14i) q^{55} +1.97732e14 q^{56} +8.37950e14i q^{57} -5.18977e12i q^{58} -1.41965e15 q^{59} +(-7.26009e14 + 7.33540e14i) q^{60} +2.31838e15 q^{61} +5.98128e14i q^{62} +1.57819e14i q^{63} -8.24240e14 q^{64} +(-1.35421e15 + 1.36826e15i) q^{65} +2.75385e15 q^{66} -4.06306e15i q^{67} -4.58023e15i q^{68} +7.72954e14 q^{69} +(5.49827e14 + 5.44182e14i) q^{70} +2.68034e15 q^{71} -1.54694e15i q^{72} +6.95024e15i q^{73} +1.66558e15 q^{74} +(-9.77816e15 + 1.00915e14i) q^{75} +6.02702e15 q^{76} +4.89380e15i q^{77} -5.57037e15i q^{78} -8.28477e15 q^{79} +(2.11202e15 + 2.09034e15i) q^{80} -1.99802e16 q^{81} +1.12523e16i q^{82} +2.66711e16i q^{83} +5.30696e15 q^{84} +(3.05275e16 - 3.08441e16i) q^{85} -1.26296e16 q^{86} -3.37328e14i q^{87} -4.79690e16i q^{88} +6.55891e15 q^{89} +(4.25736e15 - 4.30153e15i) q^{90} +9.89897e15 q^{91} -5.55953e15i q^{92} +3.88776e16i q^{93} +4.83067e16 q^{94} +(4.05871e16 + 4.01704e16i) q^{95} -8.25582e16 q^{96} -1.11864e17i q^{97} +4.18946e16i q^{98} +3.82862e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 579096 q^{4} + 379200 q^{5} + 357816 q^{6} - 234916344 q^{9}+O(q^{10}) \) 8 * q - 579096 * q^4 + 379200 * q^5 + 357816 * q^6 - 234916344 * q^9	\( 8 q - 579096 q^{4} + 379200 q^{5} + 357816 q^{6} - 234916344 q^{9} + 329570200 q^{10} + 463296576 q^{11} - 29937907992 q^{14} + 30646226400 q^{15} + 30848001568 q^{16} - 20615713280 q^{19} - 47558579400 q^{20} - 75039699024 q^{21} + 1768741136160 q^{24} - 1789249435000 q^{25} - 838901194224 q^{26} - 4079017824720 q^{29} + 2416984007400 q^{30} + 11329328658496 q^{31} - 36406243632832 q^{34} + 4019663899200 q^{35} + 59729752432728 q^{36} + 40318460422272 q^{39} - 209747532172000 q^{40} + 97217252847456 q^{41} - 116357853210912 q^{44} - 366841998003600 q^{45} + 10\!\cdots\!36 q^{46}+ \cdots - 56\!\cdots\!68 q^{99}+O(q^{100}) \) 8 * q - 579096 * q^4 + 379200 * q^5 + 357816 * q^6 - 234916344 * q^9 + 329570200 * q^10 + 463296576 * q^11 - 29937907992 * q^14 + 30646226400 * q^15 + 30848001568 * q^16 - 20615713280 * q^19 - 47558579400 * q^20 - 75039699024 * q^21 + 1768741136160 * q^24 - 1789249435000 * q^25 - 838901194224 * q^26 - 4079017824720 * q^29 + 2416984007400 * q^30 + 11329328658496 * q^31 - 36406243632832 * q^34 + 4019663899200 * q^35 + 59729752432728 * q^36 + 40318460422272 * q^39 - 209747532172000 * q^40 + 97217252847456 * q^41 - 116357853210912 * q^44 - 366841998003600 * q^45 + 1081224261700136 * q^46 - 856574357621656 * q^49 - 1283266301910000 * q^50 + 2468309514424896 * q^51 - 3408409774777680 * q^54 - 2042713226757600 * q^55 + 8363016326678880 * q^56 - 1091409512240640 * q^59 - 11479379108104800 * q^60 + 8064731010774976 * q^61 - 3616160324265856 * q^64 - 11989509557901600 * q^65 + 27318846906958752 * q^66 - 12078989597365008 * q^69 - 13190931213697800 * q^70 + 25241492058140736 * q^71 - 29902523510328912 * q^74 + 3839235716880000 * q^75 + 20767634734678560 * q^76 + 3229852337730880 * q^79 + 1263407265391200 * q^80 - 49353541005202632 * q^81 + 101439947332382688 * q^84 + 25693702369787200 * q^85 - 165112838769552744 * q^86 + 92963987535626640 * q^89 + 206315814421823400 * q^90 - 225591670236809664 * q^91 + 162612564681867848 * q^94 + 204319715505252000 * q^95 - 654303222993538944 * q^96 - 56327331239952768 * q^99

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