LMFDB - Embedded newform 54.8.c.b.19.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,8,Mod(19,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("54.19");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	54.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.8687913761$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
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} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 $ x^8 + 1336x^6 + 633664x^4 + 125389995*x^2 + 8783438400
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			19.2
Root			$-13.4379i$ of defining polynomial
Character	$\chi$	$=$	54.19
Dual form			54.8.c.b.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.00000 + 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(-103.252 - 178.837i) q^{5} +(-808.604 + 1400.54i) q^{7} +512.000 q^{8} +1652.03 q^{10} +(2338.01 - 4049.55i) q^{11} +(-3336.65 - 5779.25i) q^{13} +(-6468.83 - 11204.3i) q^{14} +(-2048.00 + 3547.24i) q^{16} +25731.9 q^{17} +22384.5 q^{19} +(-6608.11 + 11445.6i) q^{20} +(18704.1 + 32396.4i) q^{22} +(11710.7 + 20283.5i) q^{23} +(17740.6 - 30727.7i) q^{25} +53386.5 q^{26} +103501. q^{28} +(81488.4 - 141142. i) q^{29} +(116733. + 202187. i) q^{31} +(-16384.0 - 28377.9i) q^{32} +(-102928. + 178276. i) q^{34} +333959. q^{35} +308867. q^{37} +(-89537.8 + 155084. i) q^{38} +(-52864.9 - 91564.7i) q^{40} +(-157555. - 272893. i) q^{41} +(61173.2 - 105955. i) q^{43} -299265. q^{44} -187371. q^{46} +(-97801.6 + 169397. i) q^{47} +(-895910. - 1.55176e6i) q^{49} +(141925. + 245822. i) q^{50} +(-213546. + 369872. i) q^{52} +369367. q^{53} -965615. q^{55} +(-414005. + 717078. i) q^{56} +(651907. + 1.12914e6i) q^{58} +(-560669. - 971107. i) q^{59} +(-45155.7 + 78212.0i) q^{61} -1.86772e6 q^{62} +262144. q^{64} +(-689031. + 1.19344e6i) q^{65} +(-1.76399e6 - 3.05533e6i) q^{67} +(-823420. - 1.42621e6i) q^{68} +(-1.33584e6 + 2.31374e6i) q^{70} +703989. q^{71} +4.06113e6 q^{73} +(-1.23547e6 + 2.13990e6i) q^{74} +(-716302. - 1.24067e6i) q^{76} +(3.78105e6 + 6.54897e6i) q^{77} +(-2.33651e6 + 4.04696e6i) q^{79} +845839. q^{80} +2.52088e6 q^{82} +(4.38570e6 - 7.59625e6i) q^{83} +(-2.65686e6 - 4.60182e6i) q^{85} +(489386. + 847641. i) q^{86} +(1.19706e6 - 2.07337e6i) q^{88} -1.07368e7 q^{89} +1.07921e7 q^{91} +(749485. - 1.29815e6i) q^{92} +(-782413. - 1.35518e6i) q^{94} +(-2.31123e6 - 4.00317e6i) q^{95} +(-3.72029e6 + 6.44373e6i) q^{97} +1.43346e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{2} - 256 q^{4} - 54 q^{5} - 44 q^{7} + 4096 q^{8} + 864 q^{10} - 2172 q^{11} - 6398 q^{13} - 352 q^{14} - 16384 q^{16} + 51972 q^{17} + 90712 q^{19} - 3456 q^{20} - 17376 q^{22} + 2028 q^{23}+ \cdots + 20368608 q^{98}+O(q^{100}) $ 8 * q - 32 * q^2 - 256 * q^4 - 54 * q^5 - 44 * q^7 + 4096 * q^8 + 864 * q^10 - 2172 * q^11 - 6398 * q^13 - 352 * q^14 - 16384 * q^16 + 51972 * q^17 + 90712 * q^19 - 3456 * q^20 - 17376 * q^22 + 2028 * q^23 - 173446 * q^25 + 102368 * q^26 + 5632 * q^28 - 283098 * q^29 - 812 * q^31 - 131072 * q^32 - 207888 * q^34 + 1492992 * q^35 + 441688 * q^37 - 362848 * q^38 - 27648 * q^40 - 610704 * q^41 - 84380 * q^43 + 278016 * q^44 - 32448 * q^46 - 855708 * q^47 - 1273038 * q^49 - 1387568 * q^50 - 409472 * q^52 + 1842600 * q^53 + 7016760 * q^55 - 22528 * q^56 - 2264784 * q^58 - 380796 * q^59 - 3151130 * q^61 + 12992 * q^62 + 2097152 * q^64 + 326538 * q^65 - 9961580 * q^67 - 1663104 * q^68 - 5971968 * q^70 - 8970384 * q^71 + 29234932 * q^73 - 1766752 * q^74 - 2902784 * q^76 + 5954826 * q^77 - 17309396 * q^79 + 442368 * q^80 + 9771264 * q^82 + 11520192 * q^83 - 14675148 * q^85 - 675040 * q^86 - 1112064 * q^88 - 26066064 * q^89 + 26231936 * q^91 + 129792 * q^92 - 6845664 * q^94 + 31016952 * q^95 - 22003112 * q^97 + 20368608 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−4.00000	+	6.92820i	−0.353553	+	0.612372i
$2$	−4.00000	+	6.92820i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−32.0000	−	55.4256i	−0.250000	−	0.433013i
$5$	−103.252	−	178.837i	−0.369405	−	0.639828i	0.620068	−	0.784548i	$-0.287105\pi$
$5$	−103.252	−	178.837i	−0.369405	−	0.639828i	−0.989473	+	0.144720i	$0.953772\pi$
$6$		0			0
$7$	−808.604	+	1400.54i	−0.891031	+	1.54331i	−0.0523898	+	0.998627i	$0.516684\pi$
$7$	−808.604	+	1400.54i	−0.891031	+	1.54331i	−0.838641	+	0.544684i	$0.816650\pi$
$8$	512.000			0.353553
$9$		0			0
$10$	1652.03			0.522417
$11$	2338.01	−	4049.55i	0.529629	−	0.917345i	−0.469774	−	0.882787i	$-0.655664\pi$
$11$	2338.01	−	4049.55i	0.529629	−	0.917345i	0.999403	−	0.0345577i	$-0.0110022\pi$
$12$		0			0
$13$	−3336.65	−	5779.25i	−0.421220	−	0.729575i	0.574839	−	0.818267i	$-0.305065\pi$
$13$	−3336.65	−	5779.25i	−0.421220	−	0.729575i	−0.996059	+	0.0886915i	$0.971731\pi$
$14$	−6468.83	−	11204.3i	−0.630054	−	1.09129i
$15$		0			0
$16$	−2048.00	+	3547.24i	−0.125000	+	0.216506i
$17$	25731.9			1.27028			0.635141	−	0.772396i	$-0.280942\pi$
$17$	25731.9			1.27028			0.635141	+	0.772396i	$0.280942\pi$
$18$		0			0
$19$	22384.5			0.748702			0.374351	−	0.927287i	$-0.377866\pi$
$19$	22384.5			0.748702			0.374351	+	0.927287i	$0.377866\pi$
$20$	−6608.11	+	11445.6i	−0.184702	+	0.319914i
$21$		0			0
$22$	18704.1	+	32396.4i	0.374504	+	0.648661i
$23$	11710.7	+	20283.5i	0.200695	+	0.347613i	0.948752	−	0.316020i	$-0.102347\pi$
$23$	11710.7	+	20283.5i	0.200695	+	0.347613i	−0.748058	+	0.663634i	$0.769013\pi$
$24$		0			0
$25$	17740.6	−	30727.7i	0.227080	−	0.393315i
$26$	53386.5			0.595696
$27$		0			0
$28$	103501.			0.891031
$29$	81488.4	−	141142.i	0.620444	−	1.07464i	−0.368959	−	0.929446i	$-0.620286\pi$
$29$	81488.4	−	141142.i	0.620444	−	1.07464i	0.989403	−	0.145195i	$-0.0463811\pi$
$30$		0			0
$31$	116733.	+	202187.i	0.703763	+	1.21895i	0.967136	+	0.254260i	$0.0818318\pi$
$31$	116733.	+	202187.i	0.703763	+	1.21895i	−0.263373	+	0.964694i	$0.584835\pi$
$32$	−16384.0	−	28377.9i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−102928.	+	178276.i	−0.449113	+	0.777886i
$35$	333959.			1.31660
$36$		0			0
$37$	308867.			1.00246			0.501229	−	0.865315i	$-0.332882\pi$
$37$	308867.			1.00246			0.501229	+	0.865315i	$0.332882\pi$
$38$	−89537.8	+	155084.i	−0.264706	+	0.458484i
$39$		0			0
$40$	−52864.9	−	91564.7i	−0.130604	−	0.226213i
$41$	−157555.	−	272893.i	−0.357017	−	0.618372i	0.630444	−	0.776235i	$-0.282873\pi$
$41$	−157555.	−	272893.i	−0.357017	−	0.618372i	−0.987461	+	0.157863i	$0.949539\pi$
$42$		0			0
$43$	61173.2	−	105955.i	0.117333	−	0.203228i	−0.801377	−	0.598160i	$-0.795899\pi$
$43$	61173.2	−	105955.i	0.117333	−	0.203228i	0.918710	+	0.394933i	$0.129232\pi$
$44$	−299265.			−0.529629
$45$		0			0
$46$	−187371.			−0.283825
$47$	−97801.6	+	169397.i	−0.137405	+	0.237993i	−0.926514	−	0.376261i	$-0.877210\pi$
$47$	−97801.6	+	169397.i	−0.137405	+	0.237993i	0.789108	+	0.614254i	$0.210543\pi$
$48$		0			0
$49$	−895910.	−	1.55176e6i	−1.08787	−	1.88425i
$50$	141925.	+	245822.i	0.160570	+	0.278115i
$51$		0			0
$52$	−213546.	+	369872.i	−0.210610	+	0.364788i
$53$	369367.			0.340794			0.170397	−	0.985375i	$-0.445495\pi$
$53$	369367.			0.340794			0.170397	+	0.985375i	$0.445495\pi$
$54$		0			0
$55$	−965615.			−0.782590
$56$	−414005.	+	717078.i	−0.315027	+	0.545643i
$57$		0			0
$58$	651907.	+	1.12914e6i	0.438720	+	0.759886i
$59$	−560669.	−	971107.i	−0.355406	−	0.615581i	0.631781	−	0.775147i	$-0.282324\pi$
$59$	−560669.	−	971107.i	−0.355406	−	0.615581i	−0.987187	+	0.159565i	$0.948991\pi$
$60$		0			0
$61$	−45155.7	+	78212.0i	−0.0254717	+	0.0441183i	−0.878480	−	0.477779i	$-0.841442\pi$
$61$	−45155.7	+	78212.0i	−0.0254717	+	0.0441183i	0.853009	+	0.521897i	$0.174775\pi$
$62$	−1.86772e6			−0.995272
$63$		0			0
$64$	262144.			0.125000
$65$	−689031.	+	1.19344e6i	−0.311202	+	0.539017i
$66$		0			0
$67$	−1.76399e6	−	3.05533e6i	−0.716532	−	1.24107i	−0.962366	−	0.271758i	$-0.912395\pi$
$67$	−1.76399e6	−	3.05533e6i	−0.716532	−	1.24107i	0.245834	−	0.969312i	$-0.420938\pi$
$68$	−823420.	−	1.42621e6i	−0.317571	−	0.550048i
$69$		0			0
$70$	−1.33584e6	+	2.31374e6i	−0.465490	+	0.806252i
$71$	703989.			0.233433			0.116716	−	0.993165i	$-0.462763\pi$
$71$	703989.			0.233433			0.116716	+	0.993165i	$0.462763\pi$
$72$		0			0
$73$	4.06113e6			1.22185			0.610923	−	0.791690i	$-0.290799\pi$
$73$	4.06113e6			1.22185			0.610923	+	0.791690i	$0.290799\pi$
$74$	−1.23547e6	+	2.13990e6i	−0.354422	+	0.613877i
$75$		0			0
$76$	−716302.	−	1.24067e6i	−0.187175	−	0.324197i
$77$	3.78105e6	+	6.54897e6i	0.943832	+	1.63477i
$78$		0			0
$79$	−2.33651e6	+	4.04696e6i	−0.533180	+	0.923494i	0.466069	+	0.884748i	$0.345670\pi$
$79$	−2.33651e6	+	4.04696e6i	−0.533180	+	0.923494i	−0.999249	+	0.0387462i	$0.987664\pi$
$80$	845839.			0.184702
$81$		0			0
$82$	2.52088e6			0.504898
$83$	4.38570e6	−	7.59625e6i	0.841910	−	1.45823i	−0.0463686	−	0.998924i	$-0.514765\pi$
$83$	4.38570e6	−	7.59625e6i	0.841910	−	1.45823i	0.888278	−	0.459306i	$-0.151902\pi$
$84$		0			0
$85$	−2.65686e6	−	4.60182e6i	−0.469248	−	0.812762i
$86$	489386.	+	847641.i	0.0829673	+	0.143704i
$87$		0			0
$88$	1.19706e6	−	2.07337e6i	0.187252	−	0.324330i
$89$	−1.07368e7			−1.61439			−0.807194	−	0.590286i	$-0.799015\pi$
$89$	−1.07368e7			−1.61439			−0.807194	+	0.590286i	$0.799015\pi$
$90$		0			0
$91$	1.07921e7			1.50128
$92$	749485.	−	1.29815e6i	0.100347	−	0.173807i
$93$		0			0
$94$	−782413.	−	1.35518e6i	−0.0971602	−	0.168286i
$95$	−2.31123e6	−	4.00317e6i	−0.276574	−	0.479040i
$96$		0			0
$97$	−3.72029e6	+	6.44373e6i	−0.413881	+	0.716863i	−0.995310	−	0.0967340i	$-0.969160\pi$
$97$	−3.72029e6	+	6.44373e6i	−0.413881	+	0.716863i	0.581429	+	0.813597i	$0.302494\pi$
$98$	1.43346e7			1.53848
$99$		0			0
$100$	−2.27080e6			−0.227080
$101$	7.40397e6	−	1.28241e7i	0.715056	−	1.23851i	−0.247882	−	0.968790i	$-0.579735\pi$
$101$	7.40397e6	−	1.28241e7i	0.715056	−	1.23851i	0.962938	−	0.269723i	$-0.0869321\pi$
$102$		0			0
$103$	8.62804e6	+	1.49442e7i	0.778004	+	1.34754i	0.933090	+	0.359642i	$0.117101\pi$
$103$	8.62804e6	+	1.49442e7i	0.778004	+	1.34754i	−0.155086	+	0.987901i	$0.549565\pi$
$104$	−1.70837e6	−	2.95898e6i	−0.148924	−	0.257944i
$105$		0			0
$106$	−1.47747e6	+	2.55905e6i	−0.120489	+	0.208693i
$107$	1.31928e7			1.04110			0.520552	−	0.853830i	$-0.325726\pi$
$107$	1.31928e7			1.04110			0.520552	+	0.853830i	$0.325726\pi$
$108$		0			0
$109$	−2.46882e6			−0.182598			−0.0912990	−	0.995824i	$-0.529102\pi$
$109$	−2.46882e6			−0.182598			−0.0912990	+	0.995824i	$0.529102\pi$
$110$	3.86246e6	−	6.68997e6i	0.276687	−	0.479237i
$111$		0			0
$112$	−3.31204e6	−	5.73663e6i	−0.222758	−	0.385828i
$113$	1.72194e6	+	2.98249e6i	0.112265	+	0.194448i	0.916683	−	0.399615i	$-0.130856\pi$
$113$	1.72194e6	+	2.98249e6i	0.112265	+	0.194448i	−0.804418	+	0.594063i	$0.797523\pi$
$114$		0			0
$115$	2.41830e6	−	4.18862e6i	0.148275	−	0.256820i
$116$	−1.04305e7			−0.620444
$117$		0			0
$118$	8.97071e6			0.502620
$119$	−2.08069e7	+	3.60386e7i	−1.13186	+	1.96044i
$120$		0			0
$121$	−1.18899e6	−	2.05940e6i	−0.0610141	−	0.105680i
$122$	−361246.	−	625696.i	−0.0180112	−	0.0311964i
$123$		0			0
$124$	7.47089e6	−	1.29400e7i	0.351882	−	0.609477i
$125$	−2.34601e7			−1.07435
$126$		0			0
$127$	−119100.			−0.00515940			−0.00257970	−	0.999997i	$-0.500821\pi$
$127$	−119100.			−0.00515940			−0.00257970	+	0.999997i	$0.500821\pi$
$128$	−1.04858e6	+	1.81619e6i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−5.51225e6	−	9.54749e6i	−0.220053	−	0.381143i
$131$	1.10622e7	+	1.91602e7i	0.429923	+	0.744648i	0.996866	−	0.0791089i	$-0.0252075\pi$
$131$	1.10622e7	+	1.91602e7i	0.429923	+	0.744648i	−0.566943	+	0.823757i	$0.691874\pi$
$132$		0			0
$133$	−1.81002e7	+	3.13504e7i	−0.667116	+	1.15548i
$134$	2.82239e7			1.01333
$135$		0			0
$136$	1.31747e7			0.449113
$137$	−2.94797e6	+	5.10604e6i	−0.0979493	+	0.169653i	−0.910836	−	0.412769i	$-0.864562\pi$
$137$	−2.94797e6	+	5.10604e6i	−0.0979493	+	0.169653i	0.812886	+	0.582422i	$0.197895\pi$
$138$		0			0
$139$	−1.77894e7	−	3.08121e7i	−0.561834	−	0.973126i	−0.997336	−	0.0729381i	$-0.976762\pi$
$139$	−1.77894e7	−	3.08121e7i	−0.561834	−	0.973126i	0.435502	−	0.900188i	$-0.356571\pi$
$140$	−1.06867e7	−	1.85099e7i	−0.329151	−	0.570106i
$141$		0			0
$142$	−2.81596e6	+	4.87738e6i	−0.0825310	+	0.142948i
$143$	−3.12045e7			−0.892363
$144$		0			0
$145$	−3.36553e7			−0.916780
$146$	−1.62445e7	+	2.81363e7i	−0.431988	+	0.748225i
$147$		0			0
$148$	−9.88376e6	−	1.71192e7i	−0.250614	−	0.434077i
$149$	−1.67110e7	−	2.89443e7i	−0.413857	−	0.716821i	0.581451	−	0.813582i	$-0.302485\pi$
$149$	−1.67110e7	−	2.89443e7i	−0.413857	−	0.716821i	−0.995308	+	0.0967603i	$0.969152\pi$
$150$		0			0
$151$	2.49541e7	−	4.32217e7i	0.589823	−	1.02160i	−0.404432	−	0.914568i	$-0.632531\pi$
$151$	2.49541e7	−	4.32217e7i	0.589823	−	1.02160i	0.994255	−	0.107036i	$-0.0341359\pi$
$152$	1.14608e7			0.264706
$153$		0			0
$154$	−6.04968e7			−1.33478
$155$	2.41057e7	−	4.17523e7i	0.519947	−	0.900575i
$156$		0			0
$157$	2.13156e7	+	3.69197e7i	0.439590	+	0.761393i	0.997658	−	0.0684026i	$-0.0217902\pi$
$157$	2.13156e7	+	3.69197e7i	0.439590	+	0.761393i	−0.558067	+	0.829796i	$0.688457\pi$
$158$	−1.86921e7	−	3.23757e7i	−0.377015	−	0.653009i
$159$		0			0
$160$	−3.38335e6	+	5.86014e6i	−0.0653022	+	0.113107i
$161$	−3.78773e7			−0.715300
$162$		0			0
$163$	−4.06125e7			−0.734519			−0.367259	−	0.930119i	$-0.619704\pi$
$163$	−4.06125e7			−0.734519			−0.367259	+	0.930119i	$0.619704\pi$
$164$	−1.00835e7	+	1.74652e7i	−0.178508	+	0.309186i
$165$		0			0
$166$	3.50856e7	+	6.07700e7i	0.595320	+	1.03112i
$167$	3.62955e7	+	6.28656e7i	0.603038	+	1.04449i	0.992358	+	0.123390i	$0.0393766\pi$
$167$	3.62955e7	+	6.28656e7i	0.603038	+	1.04449i	−0.389320	+	0.921102i	$0.627290\pi$
$168$		0			0
$169$	9.10774e6	−	1.57751e7i	0.145147	−	0.251401i
$170$	4.25098e7			0.663617
$171$		0			0
$172$	−7.83017e6			−0.117333
$173$	1.61254e7	−	2.79301e7i	0.236783	−	0.410120i	−0.723006	−	0.690841i	$-0.757240\pi$
$173$	1.61254e7	−	2.79301e7i	0.236783	−	0.410120i	0.959789	+	0.280721i	$0.0905736\pi$
$174$		0			0
$175$	2.86903e7	+	4.96931e7i	0.404671	+	0.700911i
$176$	9.57649e6	+	1.65870e7i	0.132407	+	0.229336i
$177$		0			0
$178$	4.29470e7	−	7.43864e7i	0.570772	−	0.988607i
$179$	5.74576e7			0.748793			0.374396	−	0.927269i	$-0.377850\pi$
$179$	5.74576e7			0.748793			0.374396	+	0.927269i	$0.377850\pi$
$180$		0			0
$181$	2.89626e7			0.363047			0.181524	−	0.983387i	$-0.441897\pi$
$181$	2.89626e7			0.363047			0.181524	+	0.983387i	$0.441897\pi$
$182$	−4.31685e7	+	7.47701e7i	−0.530783	+	0.919344i
$183$		0			0
$184$	5.99588e6	+	1.03852e7i	0.0709563	+	0.122900i
$185$	−3.18911e7	−	5.52370e7i	−0.370313	−	0.641400i
$186$		0			0
$187$	6.01614e7	−	1.04203e8i	0.672779	−	1.16529i
$188$	1.25186e7			0.137405
$189$		0			0
$190$	3.69797e7			0.391135
$191$	4.94718e7	−	8.56877e7i	0.513737	−	0.889819i	−0.486136	−	0.873883i	$-0.661594\pi$
$191$	4.94718e7	−	8.56877e7i	0.513737	−	0.889819i	0.999873	−	0.0159358i	$-0.00507275\pi$
$192$		0			0
$193$	7.72389e6	+	1.33782e7i	0.0773367	+	0.133951i	0.902100	−	0.431527i	$-0.142025\pi$
$193$	7.72389e6	+	1.33782e7i	0.0773367	+	0.133951i	−0.824763	+	0.565478i	$0.808692\pi$
$194$	−2.97623e7	−	5.15498e7i	−0.292658	−	0.506899i
$195$		0			0
$196$	−5.73382e7	+	9.93127e7i	−0.543936	+	0.942125i
$197$	9.85117e7			0.918028			0.459014	−	0.888429i	$-0.348203\pi$
$197$	9.85117e7			0.918028			0.459014	+	0.888429i	$0.348203\pi$
$198$		0			0
$199$	1.40825e8			1.26676			0.633380	−	0.773841i	$-0.281667\pi$
$199$	1.40825e8			1.26676			0.633380	+	0.773841i	$0.281667\pi$
$200$	9.08321e6	−	1.57326e7i	0.0802850	−	0.139058i
$201$		0			0
$202$	5.92318e7	+	1.02592e8i	0.505621	+	0.875761i
$203$	1.31784e8	+	2.28256e8i	1.10567	+	1.91508i
$204$		0			0
$205$	−3.25357e7	+	5.63534e7i	−0.263768	+	0.456859i
$206$	−1.38049e8			−1.10026
$207$		0			0
$208$	2.73339e7			0.210610
$209$	5.23351e7	−	9.06470e7i	0.396534	−	0.686818i
$210$		0			0
$211$	3.02133e7	+	5.23309e7i	0.221416	+	0.383504i	0.955238	−	0.295838i	$-0.0955987\pi$
$211$	3.02133e7	+	5.23309e7i	0.221416	+	0.383504i	−0.733822	+	0.679342i	$0.762265\pi$
$212$	−1.18197e7	−	2.04724e7i	−0.0851986	−	0.147568i
$213$		0			0
$214$	−5.27713e7	+	9.14025e7i	−0.368086	+	0.637544i
$215$	−2.52650e7			−0.173374
$216$		0			0
$217$	−3.77562e8			−2.50830
$218$	9.87527e6	−	1.71045e7i	0.0645582	−	0.111818i
$219$		0			0
$220$	3.08997e7	+	5.35198e7i	0.195648	+	0.338871i
$221$	−8.58584e7	−	1.48711e8i	−0.535069	−	0.926767i
$222$		0			0
$223$	1.42446e8	−	2.46723e8i	0.860165	−	1.48985i	−0.0116033	−	0.999933i	$-0.503694\pi$
$223$	1.42446e8	−	2.46723e8i	0.860165	−	1.48985i	0.871769	−	0.489918i	$-0.162973\pi$
$224$	5.29927e7			0.315027
$225$		0			0
$226$	−2.75510e7			−0.158766
$227$	7.13578e7	−	1.23595e8i	0.404903	−	0.701312i	−0.589407	−	0.807836i	$-0.700639\pi$
$227$	7.13578e7	−	1.23595e8i	0.404903	−	0.701312i	0.994310	+	0.106524i	$0.0339720\pi$
$228$		0			0
$229$	−6.39437e7	−	1.10754e8i	−0.351863	−	0.609445i	0.634713	−	0.772748i	$-0.281118\pi$
$229$	−6.39437e7	−	1.10754e8i	−0.351863	−	0.609445i	−0.986576	+	0.163303i	$0.947785\pi$
$230$	1.93464e7	+	3.35090e7i	0.104846	+	0.181599i
$231$		0			0
$232$	4.17221e7	−	7.22647e7i	0.219360	−	0.379943i
$233$	−2.85503e8			−1.47865			−0.739324	−	0.673350i	$-0.764855\pi$
$233$	−2.85503e8			−1.47865			−0.739324	+	0.673350i	$0.764855\pi$
$234$		0			0
$235$	4.03928e7			0.203033
$236$	−3.58828e7	+	6.21509e7i	−0.177703	+	0.307791i
$237$		0			0
$238$	−1.66455e8	−	2.88309e8i	−0.800347	−	1.38624i
$239$	5.90868e6	+	1.02341e7i	0.0279961	+	0.0484907i	0.879684	−	0.475559i	$-0.157754\pi$
$239$	5.90868e6	+	1.02341e7i	0.0279961	+	0.0484907i	−0.851688	+	0.524049i	$0.824421\pi$
$240$		0			0
$241$	−7.41195e7	+	1.28379e8i	−0.341093	+	0.590791i	−0.984636	−	0.174619i	$-0.944130\pi$
$241$	−7.41195e7	+	1.28379e8i	−0.341093	+	0.590791i	0.643543	+	0.765410i	$0.277464\pi$
$242$	1.90239e7			0.0862870
$243$		0			0
$244$	5.77993e6			0.0254717
$245$	−1.85009e8	+	3.20444e8i	−0.803731	+	1.39210i
$246$		0			0
$247$	−7.46892e7	−	1.29365e8i	−0.315368	−	0.546234i
$248$	5.97672e7	+	1.03520e8i	0.248818	+	0.430965i
$249$		0			0
$250$	9.38404e7	−	1.62536e8i	0.379839	−	0.657901i
$251$	−6.92645e7			−0.276473			−0.138237	−	0.990399i	$-0.544143\pi$
$251$	−6.92645e7			−0.276473			−0.138237	+	0.990399i	$0.544143\pi$
$252$		0			0
$253$	1.09519e8			0.425175
$254$	476400.	−	825149.i	0.00182412	−	0.00315947i
$255$		0			0
$256$	−8.38861e6	−	1.45295e7i	−0.0312500	−	0.0541266i
$257$	8.27289e7	+	1.43291e8i	0.304013	+	0.526565i	0.977041	−	0.213051i	$-0.0683402\pi$
$257$	8.27289e7	+	1.43291e8i	0.304013	+	0.526565i	−0.673028	+	0.739617i	$0.735007\pi$
$258$		0			0
$259$	−2.49751e8	+	4.32582e8i	−0.893221	+	1.54710i
$260$	8.81960e7			0.311202
$261$		0			0
$262$	−1.76995e8			−0.608003
$263$	−8.54750e7	+	1.48047e8i	−0.289730	+	0.501828i	−0.973745	−	0.227640i	$-0.926899\pi$
$263$	−8.54750e7	+	1.48047e8i	−0.289730	+	0.501828i	0.684015	+	0.729468i	$0.260232\pi$
$264$		0			0
$265$	−3.81378e7	−	6.60566e7i	−0.125891	−	0.218050i
$266$	−1.44801e8	−	2.50803e8i	−0.471723	−	0.817047i
$267$		0			0
$268$	−1.12896e8	+	1.95541e8i	−0.358266	+	0.620535i
$269$	−2.40056e8			−0.751932			−0.375966	−	0.926633i	$-0.622689\pi$
$269$	−2.40056e8			−0.751932			−0.375966	+	0.926633i	$0.622689\pi$
$270$		0			0
$271$	2.92612e8			0.893098			0.446549	−	0.894759i	$-0.352653\pi$
$271$	2.92612e8			0.893098			0.446549	+	0.894759i	$0.352653\pi$
$272$	−5.26989e7	+	9.12772e7i	−0.158785	+	0.275024i
$273$		0			0
$274$	−2.35838e7	−	4.08483e7i	−0.0692606	−	0.119963i
$275$	−8.29556e7	−	1.43683e8i	−0.240537	−	0.416622i
$276$		0			0
$277$	−4.35557e7	+	7.54407e7i	−0.123131	+	0.213269i	−0.921001	−	0.389561i	$-0.872627\pi$
$277$	−4.35557e7	+	7.54407e7i	−0.123131	+	0.213269i	0.797870	+	0.602829i	$0.205960\pi$
$278$	2.84630e8			0.794554
$279$		0			0
$280$	1.70987e8			0.465490
$281$	−7.68148e7	+	1.33047e8i	−0.206525	+	0.357712i	−0.950618	−	0.310365i	$-0.899549\pi$
$281$	−7.68148e7	+	1.33047e8i	−0.206525	+	0.357712i	0.744092	+	0.668077i	$0.232882\pi$
$282$		0			0
$283$	1.09494e8	+	1.89650e8i	0.287170	+	0.497393i	0.973133	−	0.230243i	$-0.0739522\pi$
$283$	1.09494e8	+	1.89650e8i	0.287170	+	0.497393i	−0.685963	+	0.727636i	$0.740619\pi$
$284$	−2.25277e7	−	3.90191e7i	−0.0583582	−	0.101079i
$285$		0			0
$286$	1.24818e8	−	2.16191e8i	0.315498	−	0.546458i
$287$	5.09599e8			1.27245
$288$		0			0
$289$	2.51791e8			0.613617
$290$	1.34621e8	−	2.33171e8i	0.324131	−	0.561411i
$291$		0			0
$292$	−1.29956e8	−	2.25090e8i	−0.305461	−	0.529075i
$293$	−1.45004e8	−	2.51154e8i	−0.336777	−	0.583315i	0.647048	−	0.762450i	$-0.276004\pi$
$293$	−1.45004e8	−	2.51154e8i	−0.336777	−	0.583315i	−0.983825	+	0.179135i	$0.942670\pi$
$294$		0			0
$295$	−1.15780e8	+	2.00537e8i	−0.262577	+	0.454797i
$296$	1.58140e8			0.354422
$297$		0			0
$298$	2.67376e8			0.585282
$299$	7.81492e7	−	1.35358e8i	0.169073	−	0.292844i
$300$		0			0
$301$	9.89299e7	+	1.71352e8i	0.209095	+	0.362164i
$302$	1.99632e8	+	3.45774e8i	0.417068	+	0.722383i
$303$		0			0
$304$	−4.58434e7	+	7.94030e7i	−0.0935877	+	0.162099i
$305$	1.86496e7			0.0376375
$306$		0			0
$307$	2.99914e7			0.0591579			0.0295789	−	0.999562i	$-0.490583\pi$
$307$	2.99914e7			0.0591579			0.0295789	+	0.999562i	$0.490583\pi$
$308$	2.41987e8	−	4.19134e8i	0.471916	−	0.817383i
$309$		0			0
$310$	1.92846e8	+	3.34019e8i	0.367658	+	0.636803i
$311$	−7.31419e6	−	1.26686e7i	−0.0137881	−	0.0238817i	0.859049	−	0.511893i	$-0.171056\pi$
$311$	−7.31419e6	−	1.26686e7i	−0.0137881	−	0.0238817i	−0.872837	+	0.488012i	$0.837722\pi$
$312$		0			0
$313$	2.52793e7	−	4.37851e7i	0.0465973	−	0.0807089i	−0.841786	−	0.539811i	$-0.818496\pi$
$313$	2.52793e7	−	4.37851e7i	0.0465973	−	0.0807089i	0.888383	+	0.459102i	$0.151829\pi$
$314$	−3.41049e8			−0.621675
$315$		0			0
$316$	2.99074e8			0.533180
$317$	−3.24798e7	+	5.62566e7i	−0.0572671	+	0.0991896i	−0.893238	−	0.449585i	$-0.851572\pi$
$317$	−3.24798e7	+	5.62566e7i	−0.0572671	+	0.0991896i	0.835971	+	0.548774i	$0.184905\pi$
$318$		0			0
$319$	−3.81041e8	−	6.59983e8i	−0.657211	−	1.13832i
$320$	−2.70668e7	−	4.68811e7i	−0.0461756	−	0.0799785i
$321$		0			0
$322$	1.51509e8	−	2.62422e8i	0.252897	−	0.438030i
$323$	5.75994e8			0.951063
$324$		0			0
$325$	−2.36778e8			−0.382603
$326$	1.62450e8	−	2.81371e8i	0.259692	−	0.449799i
$327$		0			0
$328$	−8.06682e7	−	1.39721e8i	−0.126225	−	0.218627i
$329$	−1.58166e8	−	2.73951e8i	−0.244865	−	0.424118i
$330$		0			0
$331$	−4.42586e7	+	7.66582e7i	−0.0670811	+	0.116188i	−0.897615	−	0.440780i	$-0.854702\pi$
$331$	−4.42586e7	+	7.66582e7i	−0.0670811	+	0.116188i	0.830534	+	0.556968i	$0.188035\pi$
$332$	−5.61369e8			−0.841910
$333$		0			0
$334$	−5.80727e8			−0.852824
$335$	−3.64271e8	+	6.30936e8i	−0.529381	+	0.916914i
$336$		0			0
$337$	2.27637e8	+	3.94279e8i	0.323995	+	0.561176i	0.981309	−	0.192441i	$-0.0616404\pi$
$337$	2.27637e8	+	3.94279e8i	0.323995	+	0.561176i	−0.657313	+	0.753618i	$0.728307\pi$
$338$	7.28619e7	+	1.26201e8i	0.102634	+	0.177768i
$339$		0			0
$340$	−1.70039e8	+	2.94516e8i	−0.234624	+	0.406381i
$341$	1.09169e9			1.49093
$342$		0			0
$343$	1.56590e9			2.09525
$344$	3.13207e7	−	5.42490e7i	0.0414836	−	0.0718518i
$345$		0			0
$346$	1.29004e8	+	2.23441e8i	0.167431	+	0.289999i
$347$	1.48016e8	+	2.56372e8i	0.190176	+	0.329395i	0.945309	−	0.326178i	$-0.105761\pi$
$347$	1.48016e8	+	2.56372e8i	0.190176	+	0.329395i	−0.755132	+	0.655572i	$0.772427\pi$
$348$		0			0
$349$	−3.80328e8	+	6.58748e8i	−0.478927	+	0.829526i	−0.999708	−	0.0241640i	$-0.992308\pi$
$349$	−3.80328e8	+	6.58748e8i	−0.478927	+	0.829526i	0.520781	+	0.853691i	$0.325641\pi$
$350$	−4.59045e8			−0.572291
$351$		0			0
$352$	−1.53224e8			−0.187252
$353$	7.00254e8	−	1.21288e9i	0.847313	−	1.46759i	−0.0362837	−	0.999342i	$-0.511552\pi$
$353$	7.00254e8	−	1.21288e9i	0.847313	−	1.46759i	0.883597	−	0.468248i	$-0.155115\pi$
$354$		0			0
$355$	−7.26882e7	−	1.25900e8i	−0.0862312	−	0.149357i
$356$	3.43576e8	+	5.95091e8i	0.403597	+	0.699051i
$357$		0			0
$358$	−2.29830e8	+	3.98078e8i	−0.264738	+	0.458540i
$359$	−1.32132e9			−1.50722			−0.753612	−	0.657320i	$-0.771690\pi$
$359$	−1.32132e9			−1.50722			−0.753612	+	0.657320i	$0.771690\pi$
$360$		0			0
$361$	−3.92808e8			−0.439446
$362$	−1.15851e8	+	2.00659e8i	−0.128357	+	0.222320i
$363$		0			0
$364$	−3.45348e8	−	5.98160e8i	−0.375320	−	0.650074i
$365$	−4.19318e8	−	7.26281e8i	−0.451356	−	0.781771i
$366$		0			0
$367$	3.70961e8	−	6.42523e8i	0.391739	−	0.678511i	−0.600940	−	0.799294i	$-0.705207\pi$
$367$	3.70961e8	−	6.42523e8i	0.391739	−	0.678511i	0.992679	+	0.120783i	$0.0385404\pi$
$368$	−9.59341e7			−0.100347
$369$		0			0
$370$	5.10258e8			0.523701
$371$	−2.98672e8	+	5.17314e8i	−0.303658	+	0.525952i
$372$		0			0
$373$	−4.39730e8	−	7.61635e8i	−0.438738	−	0.759916i	0.558855	−	0.829266i	$-0.311241\pi$
$373$	−4.39730e8	−	7.61635e8i	−0.438738	−	0.759916i	−0.997592	+	0.0693494i	$0.977908\pi$
$374$	4.81291e8	+	8.33621e8i	0.475726	+	0.823982i
$375$		0			0
$376$	−5.00744e7	+	8.67314e7i	−0.0485801	+	0.0841432i
$377$	−1.08759e9			−1.04538
$378$		0			0
$379$	−1.99736e9			−1.88460			−0.942301	−	0.334767i	$-0.891342\pi$
$379$	−1.99736e9			−1.88460			−0.942301	+	0.334767i	$0.891342\pi$
$380$	−1.47919e8	+	2.56203e8i	−0.138287	+	0.239520i
$381$		0			0
$382$	3.95775e8	+	6.85502e8i	0.363267	+	0.629197i
$383$	1.54585e8	+	2.67750e8i	0.140596	+	0.243519i	0.927721	−	0.373274i	$-0.121765\pi$
$383$	1.54585e8	+	2.67750e8i	0.140596	+	0.243519i	−0.787125	+	0.616793i	$0.788432\pi$
$384$		0			0
$385$	7.80800e8	−	1.35239e9i	0.697312	−	1.20778i
$386$	−1.23582e8			−0.109371
$387$		0			0
$388$	4.76197e8			0.413881
$389$	−9.64068e8	+	1.66982e9i	−0.830394	+	1.43828i	0.0673322	+	0.997731i	$0.478551\pi$
$389$	−9.64068e8	+	1.66982e9i	−0.830394	+	1.43828i	−0.897726	+	0.440554i	$0.854782\pi$
$390$		0			0
$391$	3.01338e8	+	5.21934e8i	0.254939	+	0.441567i
$392$	−4.58706e8	−	7.94502e8i	−0.384621	−	0.666183i
$393$		0			0
$394$	−3.94047e8	+	6.82509e8i	−0.324572	+	0.562175i
$395$	9.64997e8			0.787837
$396$		0			0
$397$	−1.08397e9			−0.869460			−0.434730	−	0.900561i	$-0.643156\pi$
$397$	−1.08397e9			−0.869460			−0.434730	+	0.900561i	$0.643156\pi$
$398$	−5.63300e8	+	9.75665e8i	−0.447867	+	0.775729i
$399$		0			0
$400$	7.26657e7	+	1.25861e8i	0.0567701	+	0.0983286i
$401$	4.81857e8	+	8.34600e8i	0.373175	+	0.646358i	0.990052	−	0.140701i	$-0.0449357\pi$
$401$	4.81857e8	+	8.34600e8i	0.373175	+	0.646358i	−0.616877	+	0.787060i	$0.711602\pi$
$402$		0			0
$403$	7.78993e8	−	1.34926e9i	0.592879	−	1.02690i
$404$	−9.47708e8			−0.715056
$405$		0			0
$406$	−2.10854e9			−1.56365
$407$	7.22135e8	−	1.25077e9i	0.530931	−	0.919599i
$408$		0			0
$409$	6.07460e7	+	1.05215e8i	0.0439022	+	0.0760408i	0.887141	−	0.461498i	$-0.152688\pi$
$409$	6.07460e7	+	1.05215e8i	0.0439022	+	0.0760408i	−0.843239	+	0.537538i	$0.819354\pi$
$410$	−2.60285e8	−	4.50828e8i	−0.186512	−	0.323048i
$411$		0			0
$412$	5.52195e8	−	9.56429e8i	0.389002	−	0.673772i
$413$	1.81344e9			1.26671
$414$		0			0
$415$	−1.81132e9			−1.24402
$416$	−1.09335e8	+	1.89375e8i	−0.0744620	+	0.128972i
$417$		0			0
$418$	4.18680e8	+	7.25176e8i	0.280392	+	0.485653i
$419$	−4.52373e8	−	7.83532e8i	−0.300433	−	0.520365i	0.675801	−	0.737084i	$-0.263798\pi$
$419$	−4.52373e8	−	7.83532e8i	−0.300433	−	0.520365i	−0.976234	+	0.216719i	$0.930464\pi$
$420$		0			0
$421$	5.94491e8	−	1.02969e9i	0.388292	−	0.672541i	−0.603928	−	0.797039i	$-0.706399\pi$
$421$	5.94491e8	−	1.02969e9i	0.388292	−	0.672541i	0.992220	+	0.124498i	$0.0397320\pi$
$422$	−4.83413e8			−0.313130
$423$		0			0
$424$	1.89116e8			0.120489
$425$	4.56500e8	−	7.90681e8i	0.288456	−	0.499621i
$426$		0			0
$427$	−7.30262e7	−	1.26485e8i	−0.0453922	−	0.0786216i
$428$	−4.22170e8	−	7.31220e8i	−0.260276	−	0.450812i
$429$		0			0
$430$	1.01060e8	−	1.75041e8i	0.0612970	−	0.106170i
$431$	−1.77750e9			−1.06940			−0.534700	−	0.845042i	$-0.679575\pi$
$431$	−1.77750e9			−1.06940			−0.534700	+	0.845042i	$0.679575\pi$
$432$		0			0
$433$	1.98626e9			1.17579			0.587894	−	0.808938i	$-0.299957\pi$
$433$	1.98626e9			1.17579			0.587894	+	0.808938i	$0.299957\pi$
$434$	1.51025e9	−	2.61583e9i	0.886818	−	1.53601i
$435$		0			0
$436$	7.90021e7	+	1.36836e8i	0.0456495	+	0.0790673i
$437$	2.62138e8	+	4.54036e8i	0.150260	+	0.260259i
$438$		0			0
$439$	−6.54719e8	+	1.13401e9i	−0.369342	+	0.639719i	−0.989463	−	0.144787i	$-0.953750\pi$
$439$	−6.54719e8	+	1.13401e9i	−0.369342	+	0.639719i	0.620121	+	0.784506i	$0.287084\pi$
$440$	−4.94395e8			−0.276687
$441$		0			0
$442$	1.37373e9			0.756702
$443$	9.12222e8	−	1.58002e9i	0.498526	−	0.863472i	−0.501473	−	0.865173i	$-0.667208\pi$
$443$	9.12222e8	−	1.58002e9i	0.498526	−	0.863472i	0.999999	+	0.00170136i	$0.000541559\pi$
$444$		0			0
$445$	1.10859e9	+	1.92013e9i	0.596363	+	1.03293i
$446$	1.13956e9	+	1.97378e9i	0.608229	+	1.05348i
$447$		0			0
$448$	−2.11971e8	+	3.67144e8i	−0.111379	+	0.192914i
$449$	3.58855e9			1.87093			0.935464	−	0.353423i	$-0.114982\pi$
$449$	3.58855e9			1.87093			0.935464	+	0.353423i	$0.114982\pi$
$450$		0			0
$451$	−1.47346e9			−0.756346
$452$	1.10204e8	−	1.90879e8i	0.0561324	−	0.0972241i
$453$		0			0
$454$	5.70862e8	+	9.88762e8i	0.286310	+	0.495903i
$455$	−1.11431e9	−	1.93004e9i	−0.554581	−	0.960562i
$456$		0			0
$457$	−1.59845e9	+	2.76860e9i	−0.783417	+	1.35692i	0.146522	+	0.989207i	$0.453192\pi$
$457$	−1.59845e9	+	2.76860e9i	−0.783417	+	1.35692i	−0.929940	+	0.367711i	$0.880141\pi$
$458$	1.02310e9			0.497610
$459$		0			0
$460$	−3.09543e8			−0.148275
$461$	−1.87364e9	+	3.24524e9i	−0.890704	+	1.54275i	−0.0516717	+	0.998664i	$0.516455\pi$
$461$	−1.87364e9	+	3.24524e9i	−0.890704	+	1.54275i	−0.839033	+	0.544081i	$0.816878\pi$
$462$		0			0
$463$	1.09312e9	+	1.89333e9i	0.511838	+	0.886530i	0.999906	+	0.0137240i	$0.00436861\pi$
$463$	1.09312e9	+	1.89333e9i	0.511838	+	0.886530i	−0.488068	+	0.872806i	$0.662298\pi$
$464$	3.33777e8	+	5.78118e8i	0.155111	+	0.268660i
$465$		0			0
$466$	1.14201e9	−	1.97802e9i	0.522781	−	0.905483i
$467$	−9.63843e8			−0.437922			−0.218961	−	0.975734i	$-0.570267\pi$
$467$	−9.63843e8			−0.437922			−0.218961	+	0.975734i	$0.570267\pi$
$468$		0			0
$469$	5.70549e9			2.55381
$470$	−1.61571e8	+	2.79849e8i	−0.0717829	+	0.124332i
$471$		0			0
$472$	−2.87063e8	−	4.97207e8i	−0.125655	−	0.217641i
$473$	−2.86047e8	−	4.95448e8i	−0.124286	−	0.215270i
$474$		0			0
$475$	3.97115e8	−	6.87823e8i	0.170015	−	0.294475i
$476$	2.66328e9			1.13186
$477$		0			0
$478$	−9.45388e7			−0.0395925
$479$	−8.61804e8	+	1.49269e9i	−0.358290	+	0.620576i	−0.987675	−	0.156517i	$-0.949973\pi$
$479$	−8.61804e8	+	1.49269e9i	−0.358290	+	0.620576i	0.629386	+	0.777093i	$0.283307\pi$
$480$		0			0
$481$	−1.03058e9	−	1.78502e9i	−0.422256	−	0.731368i
$482$	−5.92956e8	−	1.02703e9i	−0.241189	−	0.417752i
$483$		0			0
$484$	−7.60956e7	+	1.31801e8i	−0.0305071	+	0.0528398i
$485$	1.53651e9			0.611558
$486$		0			0
$487$	−6.57115e7			−0.0257804			−0.0128902	−	0.999917i	$-0.504103\pi$
$487$	−6.57115e7			−0.0257804			−0.0128902	+	0.999917i	$0.504103\pi$
$488$	−2.31197e7	+	4.00446e7i	−0.00900562	+	0.0155982i
$489$		0			0
$490$	−1.48007e9	−	2.56355e9i	−0.568323	−	0.984365i
$491$	9.79411e8	+	1.69639e9i	0.373405	+	0.646756i	0.990087	−	0.140456i	$-0.0448570\pi$
$491$	9.79411e8	+	1.69639e9i	0.373405	+	0.646756i	−0.616682	+	0.787212i	$0.711524\pi$
$492$		0			0
$493$	2.09685e9	−	3.63185e9i	0.788140	−	1.36510i
$494$	1.19503e9			0.445998
$495$		0			0
$496$	−9.56274e8			−0.351882
$497$	−5.69249e8	+	9.85968e8i	−0.207996	+	0.360259i
$498$		0			0
$499$	1.54210e9	+	2.67100e9i	0.555599	+	0.962325i	0.997857	+	0.0654372i	$0.0208442\pi$
$499$	1.54210e9	+	2.67100e9i	0.555599	+	0.962325i	−0.442258	+	0.896888i	$0.645822\pi$
$500$	7.50723e8	+	1.30029e9i	0.268587	+	0.465206i
$501$		0			0
$502$	2.77058e8	−	4.79879e8i	0.0977480	−	0.169304i
$503$	−3.95457e9			−1.38552			−0.692758	−	0.721170i	$-0.743605\pi$
$503$	−3.95457e9			−1.38552			−0.692758	+	0.721170i	$0.743605\pi$
$504$		0			0
$505$	−3.05789e9			−1.05658
$506$	−4.38076e8	+	7.58770e8i	−0.150322	+	0.260365i
$507$		0			0
$508$	3.81120e6	+	6.60119e6i	0.00128985	+	0.00223408i
$509$	−1.92422e9	−	3.33285e9i	−0.646760	−	1.12022i	−0.983892	−	0.178764i	$-0.942790\pi$
$509$	−1.92422e9	−	3.33285e9i	−0.646760	−	1.12022i	0.337132	−	0.941457i	$-0.390543\pi$
$510$		0			0
$511$	−3.28384e9	+	5.68778e9i	−1.08870	+	1.88569i
$512$	1.34218e8			0.0441942
$513$		0			0
$514$	−1.32366e9			−0.429939
$515$	1.78172e9	−	3.08603e9i	0.574797	−	0.995578i
$516$		0			0
$517$	4.57322e8	+	7.92105e8i	0.145548	+	0.252096i
$518$	−1.99801e9	−	3.46066e9i	−0.631602	−	1.09397i
$519$		0			0
$520$	−3.52784e8	+	6.11039e8i	−0.110026	+	0.190571i
$521$	1.90239e8			0.0589344			0.0294672	−	0.999566i	$-0.490619\pi$
$521$	1.90239e8			0.0589344			0.0294672	+	0.999566i	$0.490619\pi$
$522$		0			0
$523$	−9.89767e8			−0.302536			−0.151268	−	0.988493i	$-0.548336\pi$
$523$	−9.89767e8			−0.302536			−0.151268	+	0.988493i	$0.548336\pi$
$524$	7.07978e8	−	1.22625e9i	0.214961	−	0.372324i
$525$		0			0
$526$	−6.83800e8	−	1.18438e9i	−0.204870	−	0.354846i
$527$	3.00375e9	+	5.20265e9i	0.893978	+	1.54842i
$528$		0			0
$529$	1.42813e9	−	2.47360e9i	0.419443	−	0.726497i
$530$	6.10205e8			0.178037
$531$		0			0
$532$	2.31682e9			0.667116
$533$	−1.05141e9	+	1.82110e9i	−0.300766	+	0.520941i
$534$		0			0
$535$	−1.36218e9	−	2.35937e9i	−0.384589	−	0.666128i
$536$	−9.03165e8	−	1.56433e9i	−0.253332	−	0.438784i
$537$		0			0
$538$	9.60222e8	−	1.66315e9i	0.265848	−	0.460463i
$539$	−8.37858e9			−2.30468
$540$		0			0
$541$	−3.06000e9			−0.830865			−0.415433	−	0.909624i	$-0.636370\pi$
$541$	−3.06000e9			−0.830865			−0.415433	+	0.909624i	$0.636370\pi$
$542$	−1.17045e9	+	2.02727e9i	−0.315758	+	0.546909i
$543$		0			0
$544$	−4.21591e8	−	7.30217e8i	−0.112278	−	0.194471i
$545$	2.54910e8	+	4.41517e8i	0.0674526	+	0.116831i
$546$		0			0
$547$	−2.69716e9	+	4.67161e9i	−0.704612	+	1.22042i	0.262219	+	0.965008i	$0.415546\pi$
$547$	−2.69716e9	+	4.67161e9i	−0.704612	+	1.22042i	−0.966831	+	0.255416i	$0.917788\pi$
$548$	3.77340e8			0.0979493
$549$		0			0
$550$	1.32729e9			0.340170
$551$	1.82407e9	−	3.15939e9i	0.464528	−	0.804586i
$552$		0			0
$553$	−3.77863e9	−	6.54478e9i	−0.950159	−	1.64572i
$554$	−3.48446e8	−	6.03526e8i	−0.0870665	−	0.150804i
$555$		0			0
$556$	−1.13852e9	+	1.97197e9i	−0.280917	+	0.486563i
$557$	704537.			0.000172747			8.63736e−5	−	1.00000i	$-0.499973\pi$
$557$	704537.			0.000172747			8.63736e−5		1.00000i	$0.499973\pi$
$558$		0			0
$559$	−8.16456e8			−0.197693
$560$	−6.83948e8	+	1.18463e9i	−0.164576	+	0.285053i
$561$		0			0
$562$	−6.14518e8	−	1.06438e9i	−0.146035	−	0.252941i
$563$	−1.58230e9	−	2.74063e9i	−0.373688	−	0.647247i	0.616441	−	0.787401i	$-0.288574\pi$
$563$	−1.58230e9	−	2.74063e9i	−0.373688	−	0.647247i	−0.990130	+	0.140153i	$0.955240\pi$
$564$		0			0
$565$	3.55587e8	−	6.15894e8i	0.0829423	−	0.143660i
$566$	−1.75191e9			−0.406120
$567$		0			0
$568$	3.60443e8			0.0825310
$569$	6.94504e8	−	1.20292e9i	0.158045	−	0.273742i	−0.776118	−	0.630587i	$-0.782814\pi$
$569$	6.94504e8	−	1.20292e9i	0.158045	−	0.273742i	0.934164	+	0.356845i	$0.116147\pi$
$570$		0			0
$571$	−2.83085e9	−	4.90317e9i	−0.636341	−	1.10217i	−0.986229	−	0.165383i	$-0.947114\pi$
$571$	−2.83085e9	−	4.90317e9i	−0.636341	−	1.10217i	0.349889	−	0.936791i	$-0.386219\pi$
$572$	9.98545e8	+	1.72953e9i	0.223091	+	0.386404i
$573$		0			0
$574$	−2.03839e9	+	3.53060e9i	−0.449880	+	0.779215i
$575$	8.31022e8			0.182295
$576$		0			0
$577$	3.72547e9			0.807358			0.403679	−	0.914901i	$-0.367731\pi$
$577$	3.72547e9			0.807358			0.403679	+	0.914901i	$0.367731\pi$
$578$	−1.00716e9	+	1.74446e9i	−0.216946	+	0.375762i
$579$		0			0
$580$	1.07697e9	+	1.86537e9i	0.229195	+	0.396978i
$581$	7.09259e9	+	1.22847e10i	1.50034	+	2.59866i
$582$		0			0
$583$	8.63583e8	−	1.49577e9i	0.180495	−	0.312626i
$584$	2.07930e9			0.431988
$585$		0			0
$586$	2.32006e9			0.476275
$587$	−6.23155e8	+	1.07934e9i	−0.127164	+	0.220254i	−0.922577	−	0.385814i	$-0.873921\pi$
$587$	−6.23155e8	+	1.07934e9i	−0.127164	+	0.220254i	0.795413	+	0.606068i	$0.207254\pi$
$588$		0			0
$589$	2.61300e9	+	4.52584e9i	0.526909	+	0.912633i
$590$	−9.26241e8	−	1.60430e9i	−0.185670	−	0.321590i
$591$		0			0
$592$	−6.32560e8	+	1.09563e9i	−0.125307	+	0.217038i
$593$	−2.28414e9			−0.449813			−0.224906	−	0.974380i	$-0.572208\pi$
$593$	−2.28414e9			−0.449813			−0.224906	+	0.974380i	$0.572208\pi$
$594$		0			0
$595$	8.59340e9			1.67246
$596$	−1.06950e9	+	1.85243e9i	−0.206929	+	0.358411i
$597$		0			0
$598$	6.25193e8	+	1.08287e9i	0.119553	+	0.207072i
$599$	9.17038e8	+	1.58836e9i	0.174339	+	0.301963i	0.939932	−	0.341361i	$-0.110888\pi$
$599$	9.17038e8	+	1.58836e9i	0.174339	+	0.301963i	−0.765594	+	0.643325i	$0.777555\pi$
$600$		0			0
$601$	−1.04820e9	+	1.81554e9i	−0.196962	+	0.341149i	−0.947542	−	0.319631i	$-0.896441\pi$
$601$	−1.04820e9	+	1.81554e9i	−0.196962	+	0.341149i	0.750580	+	0.660780i	$0.229774\pi$
$602$	−1.58288e9			−0.295706
$603$		0			0
$604$	−3.19412e9			−0.589823
$605$	−2.45531e8	+	4.25273e8i	−0.0450778	+	0.0780771i
$606$		0			0
$607$	−1.96750e9	−	3.40781e9i	−0.357070	−	0.618464i	0.630400	−	0.776271i	$-0.282891\pi$
$607$	−1.96750e9	−	3.40781e9i	−0.357070	−	0.618464i	−0.987470	+	0.157807i	$0.949558\pi$
$608$	−3.66747e8	−	6.35224e8i	−0.0661765	−	0.114621i
$609$		0			0
$610$	−7.45985e7	+	1.29208e8i	−0.0133069	+	0.0230482i
$611$	1.30532e9			0.231512
$612$		0			0
$613$	8.35535e9			1.46505			0.732526	−	0.680739i	$-0.238341\pi$
$613$	8.35535e9			1.46505			0.732526	+	0.680739i	$0.238341\pi$
$614$	−1.19966e8	+	2.07787e8i	−0.0209155	+	0.0362266i
$615$		0			0
$616$	1.93590e9	+	3.35307e9i	0.333695	+	0.577977i
$617$	−5.33262e9	−	9.23638e9i	−0.913993	−	1.58308i	−0.808369	−	0.588676i	$-0.799649\pi$
$617$	−5.33262e9	−	9.23638e9i	−0.913993	−	1.58308i	−0.105624	−	0.994406i	$-0.533684\pi$
$618$		0			0
$619$	3.59538e9	−	6.22739e9i	0.609295	−	1.05533i	−0.382061	−	0.924137i	$-0.624786\pi$
$619$	3.59538e9	−	6.22739e9i	0.609295	−	1.05533i	0.991357	−	0.131194i	$-0.0418809\pi$
$620$	−3.08553e9			−0.519947
$621$		0			0
$622$	1.17027e8			0.0194994
$623$	8.68178e9	−	1.50373e10i	1.43847	−	2.49150i
$624$		0			0
$625$	1.03631e9	+	1.79494e9i	0.169789	+	0.294083i
$626$	2.02235e8	+	3.50281e8i	0.0329493	+	0.0570698i
$627$		0			0
$628$	1.36420e9	−	2.36286e9i	0.219795	−	0.380697i
$629$	7.94774e9			1.27340
$630$		0			0
$631$	−6.01287e9			−0.952750			−0.476375	−	0.879242i	$-0.658049\pi$
$631$	−6.01287e9			−0.952750			−0.476375	+	0.879242i	$0.658049\pi$
$632$	−1.19630e9	+	2.07204e9i	−0.188507	+	0.326505i
$633$		0			0
$634$	−2.59838e8	−	4.50053e8i	−0.0404940	−	0.0701376i
$635$	1.22973e7	+	2.12995e7i	0.00190591	+	0.00330113i
$636$		0			0
$637$	−5.97868e9	+	1.03554e10i	−0.916468	+	1.58737i
$638$	6.09666e9			0.929437
$639$		0			0
$640$	4.33069e8			0.0653022
$641$	1.26243e9	−	2.18659e9i	0.189323	−	0.327917i	−0.755702	−	0.654916i	$-0.772704\pi$
$641$	1.26243e9	−	2.18659e9i	0.189323	−	0.327917i	0.945025	+	0.326999i	$0.106037\pi$
$642$		0			0
$643$	4.05815e9	+	7.02892e9i	0.601990	+	1.04268i	0.992519	+	0.122087i	$0.0389587\pi$
$643$	4.05815e9	+	7.02892e9i	0.601990	+	1.04268i	−0.390529	+	0.920590i	$0.627708\pi$
$644$	1.21207e9	+	2.09937e9i	0.178825	+	0.309734i
$645$		0			0
$646$	−2.30398e9	+	3.99060e9i	−0.336251	+	0.582405i
$647$	6.70712e9			0.973578			0.486789	−	0.873519i	$-0.338168\pi$
$647$	6.70712e9			0.973578			0.486789	+	0.873519i	$0.338168\pi$
$648$		0			0
$649$	−5.24340e9			−0.752933
$650$	9.47110e8	−	1.64044e9i	0.135271	−	0.234296i
$651$		0			0
$652$	1.29960e9	+	2.25097e9i	0.183630	+	0.318056i
$653$	−3.33252e9	−	5.77210e9i	−0.468357	−	0.811218i	0.530989	−	0.847379i	$-0.321821\pi$
$653$	−3.33252e9	−	5.77210e9i	−0.468357	−	0.811218i	−0.999346	+	0.0361606i	$0.988487\pi$
$654$		0			0
$655$	2.28438e9	−	3.95665e9i	0.317631	−	0.550153i
$656$	1.29069e9			0.178508
$657$		0			0
$658$	2.53065e9			0.346291
$659$	−6.56243e9	+	1.13665e10i	−0.893235	+	1.54713i	−0.0572622	+	0.998359i	$0.518237\pi$
$659$	−6.56243e9	+	1.13665e10i	−0.893235	+	1.54713i	−0.835973	+	0.548770i	$0.815096\pi$
$660$		0			0
$661$	8.40716e8	+	1.45616e9i	0.113226	+	0.196112i	0.917069	−	0.398728i	$-0.130548\pi$
$661$	8.40716e8	+	1.45616e9i	0.113226	+	0.196112i	−0.803843	+	0.594841i	$0.797215\pi$
$662$	−3.54069e8	−	6.13266e8i	−0.0474335	−	0.0821572i
$663$		0			0
$664$	2.24548e9	−	3.88928e9i	0.297660	−	0.515562i
$665$	7.47549e9			0.985744
$666$		0			0
$667$	3.81715e9			0.498079
$668$	2.32291e9	−	4.02340e9i	0.301519	−	0.522246i
$669$		0			0
$670$	−2.91417e9	−	5.04749e9i	−0.374329	−	0.648356i
$671$	2.11149e8	+	3.65721e8i	0.0269811	+	0.0467327i
$672$		0			0
$673$	1.90889e9	−	3.30630e9i	0.241395	−	0.418109i	−0.719717	−	0.694268i	$-0.755728\pi$
$673$	1.90889e9	−	3.30630e9i	0.241395	−	0.418109i	0.961112	+	0.276159i	$0.0890616\pi$
$674$	−3.64220e9			−0.458199
$675$		0			0
$676$	−1.16579e9			−0.145147
$677$	4.08674e9	−	7.07845e9i	0.506194	−	0.876754i	−0.493780	−	0.869587i	$-0.664385\pi$
$677$	4.08674e9	−	7.07845e9i	0.506194	−	0.876754i	0.999974	−	0.00716716i	$-0.00228140\pi$
$678$		0			0
$679$	−6.01648e9	−	1.04209e10i	−0.737562	−	1.27749i
$680$	−1.36031e9	−	2.35613e9i	−0.165904	−	0.287355i
$681$		0			0
$682$	−4.36676e9	+	7.56344e9i	−0.527125	+	0.913007i
$683$	−1.11162e10			−1.33501			−0.667507	−	0.744604i	$-0.732638\pi$
$683$	−1.11162e10			−1.33501			−0.667507	+	0.744604i	$0.732638\pi$
$684$		0			0
$685$	1.21753e9			0.144732
$686$	−6.26362e9	+	1.08489e10i	−0.740783	+	1.28307i
$687$		0			0
$688$	2.50566e8	+	4.33992e8i	0.0293334	+	0.0508069i
$689$	−1.23245e9	−	2.13467e9i	−0.143550	−	0.248635i
$690$		0			0
$691$	1.06038e9	−	1.83664e9i	0.122262	−	0.211763i	−0.798398	−	0.602131i	$-0.794319\pi$
$691$	1.06038e9	−	1.83664e9i	0.122262	−	0.211763i	0.920659	+	0.390367i	$0.127652\pi$
$692$	−2.06406e9			−0.236783
$693$		0			0
$694$	−2.36826e9			−0.268950
$695$	−3.67356e9	+	6.36280e9i	−0.415089	+	0.718955i
$696$		0			0
$697$	−4.05419e9	−	7.02206e9i	−0.453512	−	0.785506i
$698$	−3.04263e9	−	5.26998e9i	−0.338653	−	0.586564i
$699$		0			0
$700$	1.83618e9	−	3.18036e9i	0.202336	−	0.350455i
$701$	−1.27398e10			−1.39684			−0.698422	−	0.715686i	$-0.746114\pi$
$701$	−1.27398e10			−1.39684			−0.698422	+	0.715686i	$0.746114\pi$
$702$		0			0
$703$	6.91383e9			0.750542
$704$	6.12895e8	−	1.06157e9i	0.0662036	−	0.114668i
$705$		0			0
$706$	5.60203e9	+	9.70300e9i	0.599141	+	1.03774i
$707$	1.19738e10	+	2.07392e10i	1.27427	+	2.20711i
$708$		0			0
$709$	3.84065e9	−	6.65221e9i	0.404710	−	0.700977i	−0.589578	−	0.807711i	$-0.700706\pi$
$709$	3.84065e9	−	6.65221e9i	0.404710	−	0.700977i	0.994288	+	0.106734i	$0.0340393\pi$
$710$	1.16301e9			0.121949
$711$		0			0
$712$	−5.49722e9			−0.570772
$713$	−2.73405e9	+	4.73551e9i	−0.282483	+	0.489275i
$714$		0			0
$715$	3.22192e9	+	5.58053e9i	0.329643	+	0.570958i
$716$	−1.83864e9	−	3.18462e9i	−0.187198	−	0.324237i
$717$		0			0
$718$	5.28528e9	−	9.15438e9i	0.532884	−	0.922982i
$719$	1.29899e10			1.30333			0.651666	−	0.758506i	$-0.274071\pi$
$719$	1.29899e10			1.30333			0.651666	+	0.758506i	$0.274071\pi$
$720$		0			0
$721$	−2.79067e10			−2.77290
$722$	1.57123e9	−	2.72145e9i	0.155368	−	0.269104i
$723$		0			0
$724$	−9.26805e8	−	1.60527e9i	−0.0907618	−	0.157204i
$725$	−2.89131e9	−	5.00790e9i	−0.281781	−	0.488060i
$726$		0			0
$727$	4.67747e8	−	8.10161e8i	0.0451482	−	0.0781990i	−0.842568	−	0.538590i	$-0.818957\pi$
$727$	4.67747e8	−	8.10161e8i	0.0451482	−	0.0781990i	0.887716	+	0.460391i	$0.152291\pi$
$728$	5.52557e9			0.530783
$729$		0			0
$730$	6.70910e9			0.638313
$731$	1.57410e9	−	2.72643e9i	0.149047	−	0.258156i
$732$		0			0
$733$	5.82375e9	+	1.00870e10i	0.546184	+	0.946019i	0.998531	+	0.0541767i	$0.0172534\pi$
$733$	5.82375e9	+	1.00870e10i	0.546184	+	0.946019i	−0.452347	+	0.891842i	$0.649413\pi$
$734$	2.96768e9	+	5.14018e9i	0.277001	+	0.479780i
$735$		0			0
$736$	3.83736e8	−	6.64651e8i	0.0354781	−	0.0614499i
$737$	−1.64969e10			−1.51798
$738$		0			0
$739$	−1.36168e10			−1.24114			−0.620570	−	0.784151i	$-0.713099\pi$
$739$	−1.36168e10			−1.24114			−0.620570	+	0.784151i	$0.713099\pi$
$740$	−2.04103e9	+	3.53517e9i	−0.185156	+	0.320700i
$741$		0			0
$742$	−2.38937e9	−	4.13852e9i	−0.214719	−	0.371904i
$743$	7.55834e8	+	1.30914e9i	0.0676029	+	0.117092i	0.897846	−	0.440310i	$-0.145132\pi$
$743$	7.55834e8	+	1.30914e9i	0.0676029	+	0.117092i	−0.830243	+	0.557402i	$0.811798\pi$
$744$		0			0
$745$	−3.45088e9	+	5.97710e9i	−0.305762	+	0.529594i
$746$	7.03568e9			0.620469
$747$		0			0
$748$	−7.70066e9			−0.672779
$749$	−1.06678e10	+	1.84771e10i	−0.927657	+	1.60675i
$750$		0			0
$751$	9.52029e9	+	1.64896e10i	0.820183	+	1.42060i	0.905546	+	0.424248i	$0.139462\pi$
$751$	9.52029e9	+	1.64896e10i	0.820183	+	1.42060i	−0.0853635	+	0.996350i	$0.527205\pi$
$752$	−4.00595e8	−	6.93852e8i	−0.0343513	−	0.0594982i
$753$		0			0
$754$	4.35038e9	−	7.53508e9i	0.369596	−	0.640159i
$755$	−1.03062e10			−0.871534
$756$		0			0
$757$	−6.00984e9			−0.503532			−0.251766	−	0.967788i	$-0.581011\pi$
$757$	−6.00984e9			−0.503532			−0.251766	+	0.967788i	$0.581011\pi$
$758$	7.98945e9	−	1.38381e10i	0.666307	−	1.15408i
$759$		0			0
$760$	−1.18335e9	−	2.04963e9i	−0.0977837	−	0.169366i
$761$	−1.12107e10	−	1.94175e10i	−0.922116	−	1.59715i	−0.796135	−	0.605119i	$-0.793126\pi$
$761$	−1.12107e10	−	1.94175e10i	−0.922116	−	1.59715i	−0.125981	−	0.992033i	$-0.540208\pi$
$762$		0			0
$763$	1.99630e9	−	3.45769e9i	0.162701	−	0.281806i
$764$	−6.33239e9			−0.513737
$765$		0			0
$766$	−2.47336e9			−0.198833
$767$	−3.74152e9	+	6.48050e9i	−0.299408	+	0.518591i
$768$		0			0
$769$	3.36877e9	+	5.83488e9i	0.267134	+	0.462690i	0.968121	−	0.250485i	$-0.0805900\pi$
$769$	3.36877e9	+	5.83488e9i	0.267134	+	0.462690i	−0.700987	+	0.713175i	$0.747257\pi$
$770$	6.24640e9	+	1.08191e10i	0.493074	+	0.854029i
$771$		0			0
$772$	4.94329e8	−	8.56203e8i	0.0386683	−	0.0669755i
$773$	9.36137e9			0.728973			0.364486	−	0.931209i	$-0.381245\pi$
$773$	9.36137e9			0.728973			0.364486	+	0.931209i	$0.381245\pi$
$774$		0			0
$775$	8.28365e9			0.639243
$776$	−1.90479e9	+	3.29919e9i	−0.146329	+	0.253449i
$777$		0			0
$778$	−7.71255e9	−	1.33585e10i	−0.587177	−	1.01702i
$779$	−3.52678e9	−	6.10857e9i	−0.267299	−	0.462976i
$780$		0			0
$781$	1.64593e9	−	2.85084e9i	0.123633	−	0.214138i
$782$	−4.82142e9			−0.360538
$783$		0			0
$784$	7.33929e9			0.543936
$785$	4.40174e9	−	7.62404e9i	0.324774	−	0.562524i
$786$		0			0
$787$	−3.25442e9	−	5.63682e9i	−0.237992	−	0.412214i	0.722146	−	0.691740i	$-0.243156\pi$
$787$	−3.25442e9	−	5.63682e9i	−0.237992	−	0.412214i	−0.960138	+	0.279527i	$0.909822\pi$
$788$	−3.15237e9	−	5.46007e9i	−0.229507	−	0.397518i
$789$		0			0
$790$	−3.85999e9	+	6.68569e9i	−0.278542	+	0.482449i
$791$	−5.56947e9			−0.400125
$792$		0			0
$793$	6.02676e8			0.0429169
$794$	4.33587e9	−	7.50995e9i	0.307401	−	0.532434i
$795$		0			0
$796$	−4.50640e9	−	7.80532e9i	−0.316690	−	0.548523i
$797$	8.12967e9	+	1.40810e10i	0.568812	+	0.985211i	0.996684	+	0.0813716i	$0.0259300\pi$
$797$	8.12967e9	+	1.40810e10i	0.568812	+	0.985211i	−0.427872	+	0.903839i	$0.640737\pi$
$798$		0			0
$799$	−2.51662e9	+	4.35891e9i	−0.174544	+	0.302318i
$800$	−1.16265e9			−0.0802850
$801$		0			0
$802$	−7.70971e9			−0.527749
$803$	9.49495e9	−	1.64457e10i	0.647125	−	1.12085i
$804$		0			0
$805$	3.91090e9	+	6.77387e9i	0.264235	+	0.457669i
$806$	6.23195e9	+	1.07940e10i	0.419229	+	0.726126i
$807$		0			0
$808$	3.79083e9	−	6.56592e9i	0.252810	−	0.437880i
$809$	2.11275e10			1.40291			0.701454	−	0.712715i	$-0.252535\pi$
$809$	2.11275e10			1.40291			0.701454	+	0.712715i	$0.252535\pi$
$810$		0			0
$811$	1.05784e10			0.696383			0.348191	−	0.937423i	$-0.386796\pi$
$811$	1.05784e10			0.696383			0.348191	+	0.937423i	$0.386796\pi$
$812$	8.43416e9	−	1.46084e10i	0.552835	−	0.957539i
$813$		0			0
$814$	5.77708e9	+	1.00062e10i	0.375425	+	0.650255i
$815$	4.19331e9	+	7.26303e9i	0.271335	+	0.469965i
$816$		0			0
$817$	1.36933e9	−	2.37175e9i	0.0878478	−	0.152157i
$818$	−9.71936e8			−0.0620870
$819$		0			0
$820$	4.16457e9			0.263768
$821$	−1.46609e10	+	2.53934e10i	−0.924610	+	1.60147i	−0.132422	+	0.991193i	$0.542275\pi$
$821$	−1.46609e10	+	2.53934e10i	−0.924610	+	1.60147i	−0.792188	+	0.610277i	$0.791058\pi$
$822$		0			0
$823$	−4.13709e9	−	7.16565e9i	−0.258700	−	0.448081i	0.707194	−	0.707019i	$-0.249961\pi$
$823$	−4.13709e9	−	7.16565e9i	−0.258700	−	0.448081i	−0.965894	+	0.258939i	$0.916627\pi$
$824$	4.41756e9	+	7.65144e9i	0.275066	+	0.476428i
$825$		0			0
$826$	−7.25375e9	+	1.25639e10i	−0.447850	+	0.775699i
$827$	1.68712e10			1.03723			0.518616	−	0.855007i	$-0.326448\pi$
$827$	1.68712e10			1.03723			0.518616	+	0.855007i	$0.326448\pi$
$828$		0			0
$829$	1.62532e10			0.990830			0.495415	−	0.868657i	$-0.335016\pi$
$829$	1.62532e10			0.990830			0.495415	+	0.868657i	$0.335016\pi$
$830$	7.24530e9	−	1.25492e10i	0.439828	−	0.761805i
$831$		0			0
$832$	−8.74684e8	−	1.51500e9i	−0.0526526	−	0.0911969i
$833$	−2.30534e10	−	3.99297e10i	−1.38191	−	2.39353i
$834$		0			0
$835$	7.49514e9	−	1.29820e10i	0.445530	−	0.771681i
$836$	−6.69889e9			−0.396534
$837$		0			0
$838$	7.23796e9			0.424876
$839$	1.12134e10	−	1.94222e10i	0.655498	−	1.13536i	−0.326271	−	0.945276i	$-0.605792\pi$
$839$	1.12134e10	−	1.94222e10i	0.655498	−	1.13536i	0.981769	−	0.190079i	$-0.0608745\pi$
$840$		0			0
$841$	−4.65578e9	−	8.06405e9i	−0.269902	−	0.467485i
$842$	4.75593e9	+	8.23751e9i	0.274564	+	0.475558i
$843$		0			0
$844$	1.93365e9	−	3.34918e9i	0.110708	−	0.191752i
$845$	−3.76156e9			−0.214471
$846$		0			0
$847$	3.84570e9			0.217462
$848$	−7.56463e8	+	1.31023e9i	−0.0425993	+	0.0737842i
$849$		0			0
$850$	3.65200e9	+	6.32545e9i	0.203969	+	0.353285i
$851$	3.61706e9	+	6.26492e9i	0.201188	+	0.348468i
$852$		0			0
$853$	−1.11195e10	+	1.92595e10i	−0.613428	+	1.06249i	0.377230	+	0.926120i	$0.376877\pi$
$853$	−1.11195e10	+	1.92595e10i	−0.613428	+	1.06249i	−0.990658	+	0.136369i	$0.956457\pi$
$854$	1.16842e9			0.0641943
$855$		0			0
$856$	6.75472e9			0.368086
$857$	−8.25516e9	+	1.42984e10i	−0.448015	+	0.775985i	−0.998257	−	0.0590205i	$-0.981202\pi$
$857$	−8.25516e9	+	1.42984e10i	−0.448015	+	0.775985i	0.550242	+	0.835005i	$0.314536\pi$
$858$		0			0
$859$	1.01406e10	+	1.75640e10i	0.545866	+	0.945468i	0.998552	+	0.0537978i	$0.0171326\pi$
$859$	1.01406e10	+	1.75640e10i	0.545866	+	0.945468i	−0.452686	+	0.891670i	$0.649534\pi$
$860$	8.08479e8	+	1.40033e9i	0.0433435	+	0.0750732i
$861$		0			0
$862$	7.11002e9	−	1.23149e10i	0.378090	−	0.654871i
$863$	−2.39028e10			−1.26593			−0.632967	−	0.774179i	$-0.718163\pi$
$863$	−2.39028e10			−1.26593			−0.632967	+	0.774179i	$0.718163\pi$
$864$		0			0
$865$	−6.65992e9			−0.349875
$866$	−7.94505e9	+	1.37612e10i	−0.415704	+	0.720020i
$867$		0			0
$868$	1.20820e10	+	2.09266e10i	0.627075	+	1.08613i
$869$	1.09256e10	+	1.89237e10i	0.564775	+	0.978219i
$870$		0			0
$871$	−1.17717e10	+	2.03891e10i	−0.603636	+	1.04553i
$872$	−1.26403e9			−0.0645582
$873$		0			0
$874$	−4.19420e9			−0.212500
$875$	1.89699e10	−	3.28569e10i	0.957277	−	1.65805i
$876$		0			0
$877$	1.28617e9	+	2.22771e9i	0.0643873	+	0.111522i	0.896422	−	0.443201i	$-0.146157\pi$
$877$	1.28617e9	+	2.22771e9i	0.0643873	+	0.111522i	−0.832035	+	0.554724i	$0.812824\pi$
$878$	−5.23775e9	−	9.07205e9i	−0.261164	−	0.452350i
$879$		0			0
$880$	1.97758e9	−	3.42527e9i	0.0978238	−	0.169436i
$881$	3.40624e10			1.67826			0.839130	−	0.543931i	$-0.183065\pi$
$881$	3.40624e10			1.67826			0.839130	+	0.543931i	$0.183065\pi$
$882$		0			0
$883$	5.47993e9			0.267863			0.133931	−	0.990991i	$-0.457240\pi$
$883$	5.47993e9			0.267863			0.133931	+	0.990991i	$0.457240\pi$
$884$	−5.49494e9	+	9.51751e9i	−0.267534	+	0.463383i
$885$		0			0
$886$	7.29778e9	+	1.26401e10i	0.352511	+	0.610567i
$887$	1.08266e10	+	1.87522e10i	0.520904	+	0.902233i	0.999704	+	0.0243090i	$0.00773856\pi$
$887$	1.08266e10	+	1.87522e10i	0.520904	+	0.902233i	−0.478800	+	0.877924i	$0.658928\pi$
$888$		0			0
$889$	9.63048e7	−	1.66805e8i	0.00459718	−	0.00796255i
$890$	−1.77374e10			−0.843384
$891$		0			0
$892$	−1.82330e10			−0.860165
$893$	−2.18924e9	+	3.79187e9i	−0.102876	+	0.178186i
$894$		0			0
$895$	−5.93260e9	−	1.02756e10i	−0.276608	−	0.479098i
$896$	−1.69577e9	−	2.93715e9i	−0.0787568	−	0.136411i
$897$		0			0
$898$	−1.43542e10	+	2.48622e10i	−0.661473	+	1.14570i
$899$	3.80495e10			1.74658
$900$		0			0
$901$	9.50451e9			0.432905
$902$	5.89384e9	−	1.02084e10i	0.267409	−	0.463166i
$903$		0			0
$904$	8.81633e8	+	1.52703e9i	0.0396916	+	0.0687478i
$905$	−2.99044e9	−	5.17960e9i	−0.134111	−	0.232288i
$906$		0			0
$907$	1.64024e10	−	2.84098e10i	0.729931	−	1.26428i	−0.226982	−	0.973899i	$-0.572886\pi$
$907$	1.64024e10	−	2.84098e10i	0.729931	−	1.26428i	0.956912	−	0.290378i	$-0.0937810\pi$
$908$	−9.13379e9			−0.404903
$909$		0			0
$910$	1.78289e10			0.784296
$911$	−9.25174e9	+	1.60245e10i	−0.405423	+	0.702214i	−0.994371	−	0.105957i	$-0.966209\pi$
$911$	−9.25174e9	+	1.60245e10i	−0.405423	+	0.702214i	0.588947	+	0.808172i	$0.299543\pi$
$912$		0			0
$913$	−2.05076e10	−	3.55202e10i	−0.891800	−	1.54464i
$914$	−1.27876e10	−	2.21488e10i	−0.553960	−	0.959487i
$915$		0			0
$916$	−4.09240e9	+	7.08824e9i	−0.175932	+	0.304723i
$917$	−3.57796e10			−1.53230
$918$		0			0
$919$	1.47226e10			0.625720			0.312860	−	0.949799i	$-0.398713\pi$
$919$	1.47226e10			0.625720			0.312860	+	0.949799i	$0.398713\pi$
$920$	1.23817e9	−	2.14457e9i	0.0524232	−	0.0907996i
$921$		0			0
$922$	−1.49891e10	−	2.59620e10i	−0.629823	−	1.09089i
$923$	−2.34897e9	−	4.06853e9i	−0.0983267	−	0.170307i
$924$		0			0
$925$	5.47951e9	−	9.49078e9i	0.227638	−	0.394281i
$926$	−1.74899e10			−0.723849
$927$		0			0
$928$	−5.34042e9			−0.219360
$929$	−2.24590e9	+	3.89000e9i	−0.0919040	+	0.159182i	−0.908312	−	0.418293i	$-0.862629\pi$
$929$	−2.24590e9	+	3.89000e9i	−0.0919040	+	0.159182i	0.816408	+	0.577475i	$0.195962\pi$
$930$		0			0
$931$	−2.00544e10	−	3.47353e10i	−0.814492	−	1.41074i
$932$	9.13609e9	+	1.58242e10i	0.369662	+	0.640273i
$933$		0			0
$934$	3.85537e9	−	6.67770e9i	0.154829	−	0.268172i
$935$	−2.48471e10			−0.994111
$936$		0			0
$937$	−3.45477e10			−1.37193			−0.685963	−	0.727637i	$-0.740619\pi$
$937$	−3.45477e10			−1.37193			−0.685963	+	0.727637i	$0.740619\pi$
$938$	−2.28220e10	+	3.95288e10i	−0.902908	+	1.56388i
$939$		0			0
$940$	−1.29257e9	−	2.23879e9i	−0.0507582	−	0.0879157i
$941$	−1.06261e10	−	1.84050e10i	−0.415729	−	0.720065i	0.579775	−	0.814776i	$-0.303140\pi$
$941$	−1.06261e10	−	1.84050e10i	−0.415729	−	0.720065i	−0.995505	+	0.0947119i	$0.969807\pi$
$942$		0			0
$943$	3.69016e9	−	6.39155e9i	0.143303	−	0.248208i
$944$	4.59300e9			0.177703
$945$		0			0
$946$	4.57676e9			0.175768
$947$	−1.52119e10	+	2.63477e10i	−0.582046	+	1.00813i	0.413190	+	0.910645i	$0.364415\pi$
$947$	−1.52119e10	+	2.63477e10i	−0.582046	+	1.00813i	−0.995237	+	0.0974893i	$0.968919\pi$
$948$		0			0
$949$	−1.35506e10	−	2.34703e10i	−0.514666	−	0.891429i
$950$	3.17692e9	+	5.50258e9i	0.120219	+	0.208225i
$951$		0			0
$952$	−1.06531e10	+	1.84518e10i	−0.400173	+	0.693120i
$953$	3.86478e10			1.44644			0.723218	−	0.690620i	$-0.242662\pi$
$953$	3.86478e10			1.44644			0.723218	+	0.690620i	$0.242662\pi$
$954$		0			0
$955$	−2.04322e10			−0.759108
$956$	3.78155e8	−	6.54984e8i	0.0139981	−	0.0242453i
$957$		0			0
$958$	−6.89443e9	−	1.19415e10i	−0.253349	−	0.438813i
$959$	−4.76749e9	−	8.25753e9i	−0.174552	−	0.302333i
$960$		0			0
$961$	−1.34967e10	+	2.33771e10i	−0.490566	+	0.849685i
$962$	1.64893e10			0.597160
$963$		0			0
$964$	9.48730e9			0.341093
$965$	1.59501e9	−	2.76264e9i	0.0571371	−	0.0989643i
$966$		0			0
$967$	1.70024e10	+	2.94491e10i	0.604670	+	1.04732i	0.992104	+	0.125421i	$0.0400282\pi$
$967$	1.70024e10	+	2.94491e10i	0.604670	+	1.04732i	−0.387434	+	0.921898i	$0.626638\pi$
$968$	−6.08764e8	−	1.05441e9i	−0.0215718	−	0.0373634i
$969$		0			0
$970$	−6.14602e9	+	1.06452e10i	−0.216219	+	0.374502i
$971$	−2.08691e10			−0.731538			−0.365769	−	0.930706i	$-0.619194\pi$
$971$	−2.08691e10			−0.731538			−0.365769	+	0.930706i	$0.619194\pi$
$972$		0			0
$973$	5.75382e10			2.00245
$974$	2.62846e8	−	4.55263e8i	0.00911476	−	0.0157872i
$975$		0			0
$976$	−1.84958e8	−	3.20356e8i	−0.00636793	−	0.0110296i
$977$	−2.13334e10	−	3.69505e10i	−0.731862	−	1.26762i	−0.956087	−	0.293085i	$-0.905318\pi$
$977$	−2.13334e10	−	3.69505e10i	−0.731862	−	1.26762i	0.224225	−	0.974537i	$-0.428015\pi$
$978$		0			0
$979$	−2.51026e10	+	4.34790e10i	−0.855027	+	1.48095i
$980$	2.36811e10			0.803731
$981$		0			0
$982$	−1.56706e10			−0.528074
$983$	−9.99083e9	+	1.73046e10i	−0.335478	+	0.581065i	−0.983577	−	0.180491i	$-0.942231\pi$
$983$	−9.99083e9	+	1.73046e10i	−0.335478	+	0.581065i	0.648098	+	0.761557i	$0.275565\pi$
$984$		0			0
$985$	−1.01715e10	−	1.76176e10i	−0.339124	−	0.587380i
$986$	1.67748e10	+	2.90548e10i	0.557299	+	0.965270i
$987$		0			0
$988$	−4.78011e9	+	8.27939e9i	−0.157684	+	0.273117i
$989$	2.86553e9			0.0941928
$990$		0			0
$991$	−3.14380e10			−1.02612			−0.513059	−	0.858353i	$-0.671488\pi$
$991$	−3.14380e10			−1.02612			−0.513059	+	0.858353i	$0.671488\pi$
$992$	3.82510e9	−	6.62526e9i	0.124409	−	0.215483i
$993$		0			0
$994$	−4.55399e9	−	7.88774e9i	−0.147075	−	0.254742i
$995$	−1.45404e10	−	2.51848e10i	−0.467947	−	0.810508i
$996$		0			0
$997$	−1.75392e10	+	3.03787e10i	−0.560500	+	0.970814i	0.436953	+	0.899484i	$0.356058\pi$
$997$	−1.75392e10	+	3.03787e10i	−0.560500	+	0.970814i	−0.997453	+	0.0713299i	$0.977276\pi$
$998$	−2.46736e10			−0.785735
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.c.b.19.2		8
3.2	odd	2		18.8.c.b.7.1	✓	8
4.3	odd	2		432.8.i.b.289.2		8
9.2	odd	6		162.8.a.h.1.2		4
9.4	even	3	inner	54.8.c.b.37.2		8
9.5	odd	6		18.8.c.b.13.1	yes	8
9.7	even	3		162.8.a.i.1.3		4
12.11	even	2		144.8.i.b.97.4		8
36.23	even	6		144.8.i.b.49.4		8
36.31	odd	6		432.8.i.b.145.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.8.c.b.7.1	✓	8	3.2	odd	2
18.8.c.b.13.1	yes	8	9.5	odd	6
54.8.c.b.19.2		8	1.1	even	1	trivial
54.8.c.b.37.2		8	9.4	even	3	inner
144.8.i.b.49.4		8	36.23	even	6
144.8.i.b.97.4		8	12.11	even	2
162.8.a.h.1.2		4	9.2	odd	6
162.8.a.i.1.3		4	9.7	even	3
432.8.i.b.145.2		8	36.31	odd	6
432.8.i.b.289.2		8	4.3	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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.00000 + 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(-103.252 - 178.837i) q^{5} +(-808.604 + 1400.54i) q^{7} +512.000 q^{8} +1652.03 q^{10} +(2338.01 - 4049.55i) q^{11} +(-3336.65 - 5779.25i) q^{13} +(-6468.83 - 11204.3i) q^{14} +(-2048.00 + 3547.24i) q^{16} +25731.9 q^{17} +22384.5 q^{19} +(-6608.11 + 11445.6i) q^{20} +(18704.1 + 32396.4i) q^{22} +(11710.7 + 20283.5i) q^{23} +(17740.6 - 30727.7i) q^{25} +53386.5 q^{26} +103501. q^{28} +(81488.4 - 141142. i) q^{29} +(116733. + 202187. i) q^{31} +(-16384.0 - 28377.9i) q^{32} +(-102928. + 178276. i) q^{34} +333959. q^{35} +308867. q^{37} +(-89537.8 + 155084. i) q^{38} +(-52864.9 - 91564.7i) q^{40} +(-157555. - 272893. i) q^{41} +(61173.2 - 105955. i) q^{43} -299265. q^{44} -187371. q^{46} +(-97801.6 + 169397. i) q^{47} +(-895910. - 1.55176e6i) q^{49} +(141925. + 245822. i) q^{50} +(-213546. + 369872. i) q^{52} +369367. q^{53} -965615. q^{55} +(-414005. + 717078. i) q^{56} +(651907. + 1.12914e6i) q^{58} +(-560669. - 971107. i) q^{59} +(-45155.7 + 78212.0i) q^{61} -1.86772e6 q^{62} +262144. q^{64} +(-689031. + 1.19344e6i) q^{65} +(-1.76399e6 - 3.05533e6i) q^{67} +(-823420. - 1.42621e6i) q^{68} +(-1.33584e6 + 2.31374e6i) q^{70} +703989. q^{71} +4.06113e6 q^{73} +(-1.23547e6 + 2.13990e6i) q^{74} +(-716302. - 1.24067e6i) q^{76} +(3.78105e6 + 6.54897e6i) q^{77} +(-2.33651e6 + 4.04696e6i) q^{79} +845839. q^{80} +2.52088e6 q^{82} +(4.38570e6 - 7.59625e6i) q^{83} +(-2.65686e6 - 4.60182e6i) q^{85} +(489386. + 847641. i) q^{86} +(1.19706e6 - 2.07337e6i) q^{88} -1.07368e7 q^{89} +1.07921e7 q^{91} +(749485. - 1.29815e6i) q^{92} +(-782413. - 1.35518e6i) q^{94} +(-2.31123e6 - 4.00317e6i) q^{95} +(-3.72029e6 + 6.44373e6i) q^{97} +1.43346e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 256 q^{4} - 54 q^{5} - 44 q^{7} + 4096 q^{8} + 864 q^{10} - 2172 q^{11} - 6398 q^{13} - 352 q^{14} - 16384 q^{16} + 51972 q^{17} + 90712 q^{19} - 3456 q^{20} - 17376 q^{22} + 2028 q^{23}+ \cdots + 20368608 q^{98}+O(q^{100}) \) 8 * q - 32 * q^2 - 256 * q^4 - 54 * q^5 - 44 * q^7 + 4096 * q^8 + 864 * q^10 - 2172 * q^11 - 6398 * q^13 - 352 * q^14 - 16384 * q^16 + 51972 * q^17 + 90712 * q^19 - 3456 * q^20 - 17376 * q^22 + 2028 * q^23 - 173446 * q^25 + 102368 * q^26 + 5632 * q^28 - 283098 * q^29 - 812 * q^31 - 131072 * q^32 - 207888 * q^34 + 1492992 * q^35 + 441688 * q^37 - 362848 * q^38 - 27648 * q^40 - 610704 * q^41 - 84380 * q^43 + 278016 * q^44 - 32448 * q^46 - 855708 * q^47 - 1273038 * q^49 - 1387568 * q^50 - 409472 * q^52 + 1842600 * q^53 + 7016760 * q^55 - 22528 * q^56 - 2264784 * q^58 - 380796 * q^59 - 3151130 * q^61 + 12992 * q^62 + 2097152 * q^64 + 326538 * q^65 - 9961580 * q^67 - 1663104 * q^68 - 5971968 * q^70 - 8970384 * q^71 + 29234932 * q^73 - 1766752 * q^74 - 2902784 * q^76 + 5954826 * q^77 - 17309396 * q^79 + 442368 * q^80 + 9771264 * q^82 + 11520192 * q^83 - 14675148 * q^85 - 675040 * q^86 - 1112064 * q^88 - 26066064 * q^89 + 26231936 * q^91 + 129792 * q^92 - 6845664 * q^94 + 31016952 * q^95 - 22003112 * q^97 + 20368608 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more