LMFDB - Embedded newform 80.9.p.c.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,9,Mod(17,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("80.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	80.p (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.5902888049$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 30x^{3} + 1089x^{2} - 3168x + 4608 $ x^6 - 2x^5 + 2x^4 - 30x^3 + 1089x^2 - 3168*x + 4608
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.1
Root			$-4.23471 - 4.23471i$ of defining polynomial
Character	$\chi$	$=$	80.17
Dual form			80.9.p.c.33.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-75.2981 + 75.2981i) q^{3} +(-14.1685 + 624.839i) q^{5} +(-730.992 - 730.992i) q^{7} -4778.60i q^{9} +O(q^{10})$	\(q+(-75.2981 + 75.2981i) q^{3} +(-14.1685 + 624.839i) q^{5} +(-730.992 - 730.992i) q^{7} -4778.60i q^{9} -19599.8 q^{11} +(-24915.1 + 24915.1i) q^{13} +(-45982.3 - 48116.1i) q^{15} +(-11288.6 - 11288.6i) q^{17} +171525. i q^{19} +110085. q^{21} +(132074. - 132074. i) q^{23} +(-390224. - 17706.1i) q^{25} +(-134211. - 134211. i) q^{27} -127019. i q^{29} +960715. q^{31} +(1.47583e6 - 1.47583e6i) q^{33} +(467110. - 446396. i) q^{35} +(-243873. - 243873. i) q^{37} -3.75212e6i q^{39} +2.50747e6 q^{41} +(6763.95 - 6763.95i) q^{43} +(2.98586e6 + 67705.6i) q^{45} +(1.79394e6 + 1.79394e6i) q^{47} -4.69610e6i q^{49} +1.70003e6 q^{51} +(-2.97161e6 + 2.97161e6i) q^{53} +(277700. - 1.22467e7i) q^{55} +(-1.29155e7 - 1.29155e7i) q^{57} -313805. i q^{59} +1.76977e7 q^{61} +(-3.49312e6 + 3.49312e6i) q^{63} +(-1.52149e7 - 1.59209e7i) q^{65} +(-4.41349e6 - 4.41349e6i) q^{67} +1.98899e7i q^{69} +8.89315e6 q^{71} +(1.95076e7 - 1.95076e7i) q^{73} +(3.07163e7 - 2.80498e7i) q^{75} +(1.43273e7 + 1.43273e7i) q^{77} +1.11272e7i q^{79} +5.15641e7 q^{81} +(-1.58712e7 + 1.58712e7i) q^{83} +(7.21353e6 - 6.89365e6i) q^{85} +(9.56429e6 + 9.56429e6i) q^{87} +4.85032e7i q^{89} +3.64255e7 q^{91} +(-7.23400e7 + 7.23400e7i) q^{93} +(-1.07176e8 - 2.43025e6i) q^{95} +(-1.07411e8 - 1.07411e8i) q^{97} +9.36596e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 72 q^{3} + 220 q^{5} + 2352 q^{7}+O(q^{10}) $ 6 * q + 72 * q^3 + 220 * q^5 + 2352 * q^7	$ 6 q + 72 q^{3} + 220 q^{5} + 2352 q^{7} - 23192 q^{11} - 119142 q^{13} - 241440 q^{15} - 265502 q^{17} + 231672 q^{21} - 28888 q^{23} - 340350 q^{25} - 392040 q^{27} + 747648 q^{31} + 4269096 q^{33} + 4971680 q^{35} - 454002 q^{37} + 2489432 q^{41} - 792648 q^{43} + 210690 q^{45} + 15313352 q^{47} - 35567712 q^{51} - 13509122 q^{53} - 4448040 q^{55} - 34625520 q^{57} + 24111192 q^{61} - 44837688 q^{63} - 30943610 q^{65} + 32827752 q^{67} + 13992928 q^{71} + 111859638 q^{73} + 126793200 q^{75} + 26260136 q^{77} + 65834226 q^{81} + 14768432 q^{83} - 19713030 q^{85} + 133207680 q^{87} - 167542032 q^{91} - 96798024 q^{93} - 239661000 q^{95} - 186656202 q^{97}+O(q^{100}) $ 6 * q + 72 * q^3 + 220 * q^5 + 2352 * q^7 - 23192 * q^11 - 119142 * q^13 - 241440 * q^15 - 265502 * q^17 + 231672 * q^21 - 28888 * q^23 - 340350 * q^25 - 392040 * q^27 + 747648 * q^31 + 4269096 * q^33 + 4971680 * q^35 - 454002 * q^37 + 2489432 * q^41 - 792648 * q^43 + 210690 * q^45 + 15313352 * q^47 - 35567712 * q^51 - 13509122 * q^53 - 4448040 * q^55 - 34625520 * q^57 + 24111192 * q^61 - 44837688 * q^63 - 30943610 * q^65 + 32827752 * q^67 + 13992928 * q^71 + 111859638 * q^73 + 126793200 * q^75 + 26260136 * q^77 + 65834226 * q^81 + 14768432 * q^83 - 19713030 * q^85 + 133207680 * q^87 - 167542032 * q^91 - 96798024 * q^93 - 239661000 * q^95 - 186656202 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−75.2981	+	75.2981i	−0.929606	+	0.929606i	−0.997680	−	0.0680745i	$-0.978314\pi$
$3$	−75.2981	+	75.2981i	−0.929606	+	0.929606i	0.0680745	+	0.997680i	$0.478314\pi$
$4$		0			0
$5$	−14.1685	+	624.839i	−0.0226696	+	0.999743i
$5$	−14.1685	+	624.839i	−0.0226696	+	0.999743i
$6$		0			0
$7$	−730.992	−	730.992i	−0.304453	−	0.304453i	0.538300	−	0.842753i	$-0.319067\pi$
$7$	−730.992	−	730.992i	−0.304453	−	0.304453i	−0.842753	+	0.538300i	$0.819067\pi$
$8$		0			0
$9$		−	4778.60i		−	0.728334i
$10$		0			0
$11$	−19599.8			−1.33869			−0.669347	−	0.742950i	$-0.733426\pi$
$11$	−19599.8			−1.33869			−0.669347	+	0.742950i	$0.733426\pi$
$12$		0			0
$13$	−24915.1	+	24915.1i	−0.872347	+	0.872347i	−0.992728	−	0.120381i	$-0.961588\pi$
$13$	−24915.1	+	24915.1i	−0.872347	+	0.872347i	0.120381	+	0.992728i	$0.461588\pi$
$14$		0			0
$15$	−45982.3	−	48116.1i	−0.908293	−	0.950441i
$16$		0			0
$17$	−11288.6	−	11288.6i	−0.135159	−	0.135159i	0.636290	−	0.771450i	$-0.280468\pi$
$17$	−11288.6	−	11288.6i	−0.135159	−	0.135159i	−0.771450	+	0.636290i	$0.780468\pi$
$18$		0			0
$19$			171525.i			1.31617i	0.752943	+	0.658086i	$0.228634\pi$
$19$			171525.i			1.31617i	−0.752943	+	0.658086i	$0.771366\pi$
$20$		0			0
$21$	110085.			0.566043
$22$		0			0
$23$	132074.	−	132074.i	0.471961	−	0.471961i	−0.430587	−	0.902549i	$-0.641694\pi$
$23$	132074.	−	132074.i	0.471961	−	0.471961i	0.902549	+	0.430587i	$0.141694\pi$
$24$		0			0
$25$	−390224.	−	17706.1i	−0.998972	−	0.0453275i
$26$		0			0
$27$	−134211.	−	134211.i	−0.252542	−	0.252542i
$28$		0			0
$29$		−	127019.i		−	0.179588i	−0.995960	−	0.0897939i	$-0.971379\pi$
$29$		−	127019.i		−	0.179588i	0.995960	−	0.0897939i	$-0.0286208\pi$
$30$		0			0
$31$	960715.			1.04027			0.520137	−	0.854083i	$-0.325881\pi$
$31$	960715.			1.04027			0.520137	+	0.854083i	$0.325881\pi$
$32$		0			0
$33$	1.47583e6	−	1.47583e6i	1.24446	−	1.24446i
$34$		0			0
$35$	467110.	−	446396.i	0.311277	−	0.297473i
$36$		0			0
$37$	−243873.	−	243873.i	−0.130124	−	0.130124i	0.639045	−	0.769169i	$-0.279330\pi$
$37$	−243873.	−	243873.i	−0.130124	−	0.130124i	−0.769169	+	0.639045i	$0.779330\pi$
$38$		0			0
$39$		−	3.75212e6i		−	1.62188i
$40$		0			0
$41$	2.50747e6			0.887360			0.443680	−	0.896185i	$-0.353673\pi$
$41$	2.50747e6			0.887360			0.443680	+	0.896185i	$0.353673\pi$
$42$		0			0
$43$	6763.95	−	6763.95i	0.00197846	−	0.00197846i	−0.706117	−	0.708095i	$-0.749555\pi$
$43$	6763.95	−	6763.95i	0.00197846	−	0.00197846i	0.708095	+	0.706117i	$0.249555\pi$
$44$		0			0
$45$	2.98586e6	+	67705.6i	0.728147	+	0.0165110i
$46$		0			0
$47$	1.79394e6	+	1.79394e6i	0.367635	+	0.367635i	0.866614	−	0.498979i	$-0.166291\pi$
$47$	1.79394e6	+	1.79394e6i	0.367635	+	0.367635i	−0.498979	+	0.866614i	$0.666291\pi$
$48$		0			0
$49$		−	4.69610e6i		−	0.814617i
$50$		0			0
$51$	1.70003e6			0.251290
$52$		0			0
$53$	−2.97161e6	+	2.97161e6i	−0.376608	+	0.376608i	−0.869877	−	0.493269i	$-0.835802\pi$
$53$	−2.97161e6	+	2.97161e6i	−0.376608	+	0.376608i	0.493269	+	0.869877i	$0.335802\pi$
$54$		0			0
$55$	277700.	−	1.22467e7i	0.0303476	−	1.33835i
$56$		0			0
$57$	−1.29155e7	−	1.29155e7i	−1.22352	−	1.22352i
$58$		0			0
$59$		−	313805.i		−	0.0258972i	−0.999916	−	0.0129486i	$-0.995878\pi$
$59$		−	313805.i		−	0.0258972i	0.999916	−	0.0129486i	$-0.00412178\pi$
$60$		0			0
$61$	1.76977e7			1.27820			0.639098	−	0.769125i	$-0.279308\pi$
$61$	1.76977e7			1.27820			0.639098	+	0.769125i	$0.279308\pi$
$62$		0			0
$63$	−3.49312e6	+	3.49312e6i	−0.221744	+	0.221744i
$64$		0			0
$65$	−1.52149e7	−	1.59209e7i	−0.852347	−	0.891899i
$66$		0			0
$67$	−4.41349e6	−	4.41349e6i	−0.219020	−	0.219020i	0.589066	−	0.808085i	$-0.299496\pi$
$67$	−4.41349e6	−	4.41349e6i	−0.219020	−	0.219020i	−0.808085	+	0.589066i	$0.799496\pi$
$68$		0			0
$69$			1.98899e7i			0.877476i
$70$		0			0
$71$	8.89315e6			0.349963			0.174982	−	0.984572i	$-0.444013\pi$
$71$	8.89315e6			0.349963			0.174982	+	0.984572i	$0.444013\pi$
$72$		0			0
$73$	1.95076e7	−	1.95076e7i	0.686928	−	0.686928i	−0.274624	−	0.961552i	$-0.588553\pi$
$73$	1.95076e7	−	1.95076e7i	0.686928	−	0.686928i	0.961552	+	0.274624i	$0.0885532\pi$
$74$		0			0
$75$	3.07163e7	−	2.80498e7i	0.970787	−	0.886514i
$76$		0			0
$77$	1.43273e7	+	1.43273e7i	0.407569	+	0.407569i
$78$		0			0
$79$			1.11272e7i			0.285680i	0.989746	+	0.142840i	$0.0456234\pi$
$79$			1.11272e7i			0.285680i	−0.989746	+	0.142840i	$0.954377\pi$
$80$		0			0
$81$	5.15641e7			1.19786
$82$		0			0
$83$	−1.58712e7	+	1.58712e7i	−0.334423	+	0.334423i	−0.854264	−	0.519840i	$-0.825992\pi$
$83$	−1.58712e7	+	1.58712e7i	−0.334423	+	0.334423i	0.519840	+	0.854264i	$0.325992\pi$
$84$		0			0
$85$	7.21353e6	−	6.89365e6i	0.138189	−	0.132061i
$86$		0			0
$87$	9.56429e6	+	9.56429e6i	0.166946	+	0.166946i
$88$		0			0
$89$			4.85032e7i			0.773056i	0.922278	+	0.386528i	$0.126326\pi$
$89$			4.85032e7i			0.773056i	−0.922278	+	0.386528i	$0.873674\pi$
$90$		0			0
$91$	3.64255e7			0.531178
$92$		0			0
$93$	−7.23400e7	+	7.23400e7i	−0.967045	+	0.967045i
$94$		0			0
$95$	−1.07176e8	−	2.43025e6i	−1.31583	−	0.0298371i
$96$		0			0
$97$	−1.07411e8	−	1.07411e8i	−1.21329	−	1.21329i	−0.969939	−	0.243348i	$-0.921754\pi$
$97$	−1.07411e8	−	1.07411e8i	−1.21329	−	1.21329i	−0.243348	−	0.969939i	$-0.578246\pi$
$98$		0			0
$99$			9.36596e7i			0.975016i
$100$		0			0
$101$	−7.16023e7			−0.688084			−0.344042	−	0.938954i	$-0.611796\pi$
$101$	−7.16023e7			−0.688084			−0.344042	+	0.938954i	$0.611796\pi$
$102$		0			0
$103$	−9.89158e7	+	9.89158e7i	−0.878854	+	0.878854i	−0.993416	−	0.114562i	$-0.963454\pi$
$103$	−9.89158e7	+	9.89158e7i	−0.878854	+	0.878854i	0.114562	+	0.993416i	$0.463454\pi$
$104$		0			0
$105$	−1.55973e6	+	6.87852e7i	−0.0128320	+	0.565897i
$106$		0			0
$107$	−4.73497e7	−	4.73497e7i	−0.361228	−	0.361228i	0.503037	−	0.864265i	$-0.332216\pi$
$107$	−4.73497e7	−	4.73497e7i	−0.361228	−	0.361228i	−0.864265	+	0.503037i	$0.832216\pi$
$108$		0			0
$109$			2.29072e7i			0.162280i	0.996703	+	0.0811401i	$0.0258561\pi$
$109$			2.29072e7i			0.162280i	−0.996703	+	0.0811401i	$0.974144\pi$
$110$		0			0
$111$	3.67264e7			0.241928
$112$		0			0
$113$	−1.95361e8	+	1.95361e8i	−1.19819	+	1.19819i	−0.223480	+	0.974709i	$0.571742\pi$
$113$	−1.95361e8	+	1.95361e8i	−1.19819	+	1.19819i	−0.974709	+	0.223480i	$0.928258\pi$
$114$		0			0
$115$	8.06538e7	+	8.43964e7i	0.461141	+	0.482539i
$116$		0			0
$117$	1.19059e8	+	1.19059e8i	0.635360	+	0.635360i
$118$		0			0
$119$			1.65038e7i			0.0822994i
$120$		0			0
$121$	1.69794e8			0.792100
$122$		0			0
$123$	−1.88808e8	+	1.88808e8i	−0.824895	+	0.824895i
$124$		0			0
$125$	1.65923e7	−	2.43576e8i	0.0679622	−	0.997688i
$126$		0			0
$127$	−1.84792e8	−	1.84792e8i	−0.710343	−	0.710343i	0.256264	−	0.966607i	$-0.417508\pi$
$127$	−1.84792e8	−	1.84792e8i	−0.710343	−	0.710343i	−0.966607	+	0.256264i	$0.917508\pi$
$128$		0			0
$129$			1.01862e6i			0.00367837i
$130$		0			0
$131$	3.05076e8			1.03591			0.517956	−	0.855407i	$-0.326693\pi$
$131$	3.05076e8			1.03591			0.517956	+	0.855407i	$0.326693\pi$
$132$		0			0
$133$	1.25383e8	−	1.25383e8i	0.400713	−	0.400713i
$134$		0			0
$135$	8.57622e7	−	8.19590e7i	0.258203	−	0.246753i
$136$		0			0
$137$	3.75735e8	+	3.75735e8i	1.06659	+	1.06659i	0.997618	+	0.0689766i	$0.0219734\pi$
$137$	3.75735e8	+	3.75735e8i	1.06659	+	1.06659i	0.0689766	+	0.997618i	$0.478027\pi$
$138$		0			0
$139$		−	1.10575e8i		−	0.296208i	−0.988972	−	0.148104i	$-0.952683\pi$
$139$		−	1.10575e8i		−	0.296208i	0.988972	−	0.148104i	$-0.0473171\pi$
$140$		0			0
$141$	−2.70161e8			−0.683512
$142$		0			0
$143$	4.88331e8	−	4.88331e8i	1.16781	−	1.16781i
$144$		0			0
$145$	7.93665e7	+	1.79967e6i	0.179542	+	0.00407118i
$146$		0			0
$147$	3.53607e8	+	3.53607e8i	0.757272	+	0.757272i
$148$		0			0
$149$		−	7.23744e8i		−	1.46838i	−0.678942	−	0.734192i	$-0.737561\pi$
$149$		−	7.23744e8i		−	1.46838i	0.678942	−	0.734192i	$-0.262439\pi$
$150$		0			0
$151$	−7.67246e8			−1.47580			−0.737899	−	0.674911i	$-0.764182\pi$
$151$	−7.67246e8			−1.47580			−0.737899	+	0.674911i	$0.764182\pi$
$152$		0			0
$153$	−5.39439e7	+	5.39439e7i	−0.0984411	+	0.0984411i
$154$		0			0
$155$	−1.36119e7	+	6.00293e8i	−0.0235826	+	1.04001i
$156$		0			0
$157$	−8.03264e8	−	8.03264e8i	−1.32209	−	1.32209i	−0.912085	−	0.410001i	$-0.865528\pi$
$157$	−8.03264e8	−	8.03264e8i	−1.32209	−	1.32209i	−0.410001	−	0.912085i	$-0.634472\pi$
$158$		0			0
$159$		−	4.47514e8i		−	0.700193i
$160$		0			0
$161$	−1.93090e8			−0.287380
$162$		0			0
$163$	4.34915e8	−	4.34915e8i	0.616103	−	0.616103i	−0.328426	−	0.944530i	$-0.606518\pi$
$163$	4.34915e8	−	4.34915e8i	0.616103	−	0.616103i	0.944530	+	0.328426i	$0.106518\pi$
$164$		0			0
$165$	9.01245e8	+	9.43066e8i	1.21593	+	1.27235i
$166$		0			0
$167$	−7.16803e8	−	7.16803e8i	−0.921582	−	0.921582i	0.0755590	−	0.997141i	$-0.475926\pi$
$167$	−7.16803e8	−	7.16803e8i	−0.921582	−	0.921582i	−0.997141	+	0.0755590i	$0.975926\pi$
$168$		0			0
$169$		−	4.25794e8i		−	0.521979i
$170$		0			0
$171$	8.19649e8			0.958613
$172$		0			0
$173$	3.40359e8	−	3.40359e8i	0.379973	−	0.379973i	−0.491119	−	0.871092i	$-0.663412\pi$
$173$	3.40359e8	−	3.40359e8i	0.379973	−	0.379973i	0.871092	+	0.491119i	$0.163412\pi$
$174$		0			0
$175$	2.72307e8	+	2.98193e8i	0.290340	+	0.317940i
$176$		0			0
$177$	2.36289e7	+	2.36289e7i	0.0240741	+	0.0240741i
$178$		0			0
$179$		−	1.29181e9i		−	1.25830i	−0.777283	−	0.629151i	$-0.783403\pi$
$179$		−	1.29181e9i		−	1.25830i	0.777283	−	0.629151i	$-0.216597\pi$
$180$		0			0
$181$	1.92912e8			0.179740			0.0898700	−	0.995954i	$-0.471355\pi$
$181$	1.92912e8			0.179740			0.0898700	+	0.995954i	$0.471355\pi$
$182$		0			0
$183$	−1.33260e9	+	1.33260e9i	−1.18822	+	1.18822i
$184$		0			0
$185$	1.55837e8	−	1.48926e8i	0.133040	−	0.127141i
$186$		0			0
$187$	2.21255e8	+	2.21255e8i	0.180937	+	0.180937i
$188$		0			0
$189$			1.96215e8i			0.153775i
$190$		0			0
$191$	−7.64646e8			−0.574549			−0.287274	−	0.957848i	$-0.592749\pi$
$191$	−7.64646e8			−0.574549			−0.287274	+	0.957848i	$0.592749\pi$
$192$		0			0
$193$	1.15389e8	−	1.15389e8i	0.0831641	−	0.0831641i	−0.664301	−	0.747465i	$-0.731271\pi$
$193$	1.15389e8	−	1.15389e8i	0.0831641	−	0.0831641i	0.747465	+	0.664301i	$0.231271\pi$
$194$		0			0
$195$	2.34447e9	+	5.31619e7i	1.62146	+	0.0367673i
$196$		0			0
$197$	−9.95415e7	−	9.95415e7i	−0.0660906	−	0.0660906i	0.673289	−	0.739379i	$-0.264881\pi$
$197$	−9.95415e7	−	9.95415e7i	−0.0660906	−	0.0660906i	−0.739379	+	0.673289i	$0.764881\pi$
$198$		0			0
$199$		−	1.71187e9i		−	1.09159i	−0.837919	−	0.545795i	$-0.816228\pi$
$199$		−	1.71187e9i		−	1.09159i	0.837919	−	0.545795i	$-0.183772\pi$
$200$		0			0
$201$	6.64655e8			0.407204
$202$		0			0
$203$	−9.28499e7	+	9.28499e7i	−0.0546761	+	0.0546761i
$204$		0			0
$205$	−3.55271e7	+	1.56676e9i	−0.0201161	+	0.887132i
$206$		0			0
$207$	−6.31129e8	−	6.31129e8i	−0.343745	−	0.343745i
$208$		0			0
$209$		−	3.36185e9i		−	1.76195i
$210$		0			0
$211$	6.81230e8			0.343688			0.171844	−	0.985124i	$-0.445028\pi$
$211$	6.81230e8			0.343688			0.171844	+	0.985124i	$0.445028\pi$
$212$		0			0
$213$	−6.69637e8	+	6.69637e8i	−0.325328	+	0.325328i
$214$		0			0
$215$	4.13055e6	+	4.32222e6i	0.00193310	+	0.00202280i
$216$		0			0
$217$	−7.02275e8	−	7.02275e8i	−0.316715	−	0.316715i
$218$		0			0
$219$			2.93776e9i			1.27715i
$220$		0			0
$221$	5.62515e8			0.235812
$222$		0			0
$223$	−9.37608e8	+	9.37608e8i	−0.379142	+	0.379142i	−0.870793	−	0.491651i	$-0.836394\pi$
$223$	−9.37608e8	+	9.37608e8i	−0.379142	+	0.379142i	0.491651	+	0.870793i	$0.336394\pi$
$224$		0			0
$225$	−8.46102e7	+	1.86472e9i	−0.0330136	+	0.727585i
$226$		0			0
$227$	3.47956e9	+	3.47956e9i	1.31045	+	1.31045i	0.921082	+	0.389368i	$0.127307\pi$
$227$	3.47956e9	+	3.47956e9i	1.31045	+	1.31045i	0.389368	+	0.921082i	$0.372693\pi$
$228$		0			0
$229$			3.67273e9i			1.33551i	0.744382	+	0.667754i	$0.232744\pi$
$229$			3.67273e9i			1.33551i	−0.744382	+	0.667754i	$0.767256\pi$
$230$		0			0
$231$	−2.15764e9			−0.757758
$232$		0			0
$233$	−3.62893e9	+	3.62893e9i	−1.23128	+	1.23128i	−0.267801	+	0.963474i	$0.586297\pi$
$233$	−3.62893e9	+	3.62893e9i	−1.23128	+	1.23128i	−0.963474	+	0.267801i	$0.913703\pi$
$234$		0			0
$235$	−1.14634e9	+	1.09551e9i	−0.375875	+	0.359207i
$236$		0			0
$237$	−8.37860e8	−	8.37860e8i	−0.265569	−	0.265569i
$238$		0			0
$239$		−	2.64336e9i		−	0.810148i	−0.914284	−	0.405074i	$-0.867246\pi$
$239$		−	2.64336e9i		−	0.810148i	0.914284	−	0.405074i	$-0.132754\pi$
$240$		0			0
$241$	−7.20231e7			−0.0213503			−0.0106751	−	0.999943i	$-0.503398\pi$
$241$	−7.20231e7			−0.0213503			−0.0106751	+	0.999943i	$0.503398\pi$
$242$		0			0
$243$	−3.00212e9	+	3.00212e9i	−0.860999	+	0.860999i
$244$		0			0
$245$	2.93431e9	+	6.65367e7i	0.814407	+	0.0184670i
$246$		0			0
$247$	−4.27356e9	−	4.27356e9i	−1.14816	−	1.14816i
$248$		0			0
$249$		−	2.39014e9i		−	0.621764i
$250$		0			0
$251$	−5.78911e7			−0.0145854			−0.00729268	−	0.999973i	$-0.502321\pi$
$251$	−5.78911e7			−0.0145854			−0.00729268	+	0.999973i	$0.502321\pi$
$252$		0			0
$253$	−2.58863e9	+	2.58863e9i	−0.631811	+	0.631811i
$254$		0			0
$255$	−2.40868e7	+	1.06224e9i	−0.00569664	+	0.251225i
$256$		0			0
$257$	−4.73425e8	−	4.73425e8i	−0.108522	−	0.108522i	0.650761	−	0.759283i	$-0.274450\pi$
$257$	−4.73425e8	−	4.73425e8i	−0.108522	−	0.108522i	−0.759283	+	0.650761i	$0.774450\pi$
$258$		0			0
$259$			3.56539e8i			0.0792333i
$260$		0			0
$261$	−6.06973e8			−0.130800
$262$		0			0
$263$	−6.55957e9	+	6.55957e9i	−1.37105	+	1.37105i	−0.512155	+	0.858893i	$0.671153\pi$
$263$	−6.55957e9	+	6.55957e9i	−1.37105	+	1.37105i	−0.858893	+	0.512155i	$0.828847\pi$
$264$		0			0
$265$	−1.81468e9	−	1.89889e9i	−0.367973	−	0.385048i
$266$		0			0
$267$	−3.65220e9	−	3.65220e9i	−0.718637	−	0.718637i
$268$		0			0
$269$			5.38818e9i			1.02904i	0.857478	+	0.514521i	$0.172030\pi$
$269$			5.38818e9i			1.02904i	−0.857478	+	0.514521i	$0.827970\pi$
$270$		0			0
$271$	8.79497e9			1.63064			0.815318	−	0.579013i	$-0.196562\pi$
$271$	8.79497e9			1.63064			0.815318	+	0.579013i	$0.196562\pi$
$272$		0			0
$273$	−2.74277e9	+	2.74277e9i	−0.493786	+	0.493786i
$274$		0			0
$275$	7.64831e9	+	3.47036e8i	1.33732	+	0.0606797i
$276$		0			0
$277$	−4.78151e9	−	4.78151e9i	−0.812168	−	0.812168i	0.172790	−	0.984959i	$-0.444722\pi$
$277$	−4.78151e9	−	4.78151e9i	−0.812168	−	0.812168i	−0.984959	+	0.172790i	$0.944722\pi$
$278$		0			0
$279$		−	4.59087e9i		−	0.757667i
$280$		0			0
$281$	1.22255e10			1.96083			0.980416	−	0.196936i	$-0.0630990\pi$
$281$	1.22255e10			1.96083			0.980416	+	0.196936i	$0.0630990\pi$
$282$		0			0
$283$	2.86857e8	−	2.86857e8i	0.0447219	−	0.0447219i	−0.684392	−	0.729114i	$-0.739932\pi$
$283$	2.86857e8	−	2.86857e8i	0.0447219	−	0.0447219i	0.729114	+	0.684392i	$0.239932\pi$
$284$		0			0
$285$	8.25310e9	−	7.88712e9i	1.25094	−	1.19547i
$286$		0			0
$287$	−1.83294e9	−	1.83294e9i	−0.270160	−	0.270160i
$288$		0			0
$289$		−	6.72089e9i		−	0.963464i
$290$		0			0
$291$	1.61757e10			2.25576
$292$		0			0
$293$	−1.02814e9	+	1.02814e9i	−0.139503	+	0.139503i	−0.773410	−	0.633907i	$-0.781450\pi$
$293$	−1.02814e9	+	1.02814e9i	−0.139503	+	0.139503i	0.633907	+	0.773410i	$0.281450\pi$
$294$		0			0
$295$	1.96078e8	+	4.44615e6i	0.0258905	+	0.000587078i
$296$		0			0
$297$	2.63052e9	+	2.63052e9i	0.338077	+	0.338077i
$298$		0			0
$299$			6.58128e9i			0.823428i
$300$		0			0
$301$	−9.88879e6			−0.00120469
$302$		0			0
$303$	5.39151e9	−	5.39151e9i	0.639647	−	0.639647i
$304$		0			0
$305$	−2.50750e8	+	1.10582e10i	−0.0289762	+	1.27787i
$306$		0			0
$307$	−7.92159e9	−	7.92159e9i	−0.891783	−	0.891783i	0.102908	−	0.994691i	$-0.467185\pi$
$307$	−7.92159e9	−	7.92159e9i	−0.891783	−	0.891783i	−0.994691	+	0.102908i	$0.967185\pi$
$308$		0			0
$309$		−	1.48963e10i		−	1.63398i
$310$		0			0
$311$	−3.74059e9			−0.399851			−0.199926	−	0.979811i	$-0.564070\pi$
$311$	−3.74059e9			−0.399851			−0.199926	+	0.979811i	$0.564070\pi$
$312$		0			0
$313$	−4.79633e9	+	4.79633e9i	−0.499725	+	0.499725i	−0.911352	−	0.411627i	$-0.864961\pi$
$313$	−4.79633e9	+	4.79633e9i	−0.499725	+	0.499725i	0.411627	+	0.911352i	$0.364961\pi$
$314$		0			0
$315$	−2.13314e9	−	2.23213e9i	−0.216660	−	0.226713i
$316$		0			0
$317$	−5.54903e9	−	5.54903e9i	−0.549516	−	0.549516i	0.376785	−	0.926301i	$-0.377030\pi$
$317$	−5.54903e9	−	5.54903e9i	−0.549516	−	0.549516i	−0.926301	+	0.376785i	$0.877030\pi$
$318$		0			0
$319$			2.48955e9i			0.240413i
$320$		0			0
$321$	7.13068e9			0.671600
$322$		0			0
$323$	1.93628e9	−	1.93628e9i	0.177893	−	0.177893i
$324$		0			0
$325$	1.01636e10	−	9.28131e9i	0.910992	−	0.831909i
$326$		0			0
$327$	−1.72487e9	−	1.72487e9i	−0.150857	−	0.150857i
$328$		0			0
$329$		−	2.62272e9i		−	0.223855i
$330$		0			0
$331$	−1.49782e10			−1.24780			−0.623902	−	0.781503i	$-0.714454\pi$
$331$	−1.49782e10			−1.24780			−0.623902	+	0.781503i	$0.714454\pi$
$332$		0			0
$333$	−1.16537e9	+	1.16537e9i	−0.0947737	+	0.0947737i
$334$		0			0
$335$	2.82026e9	−	2.69519e9i	0.223929	−	0.213998i
$336$		0			0
$337$	6.29203e9	+	6.29203e9i	0.487833	+	0.487833i	0.907622	−	0.419789i	$-0.137896\pi$
$337$	6.29203e9	+	6.29203e9i	0.487833	+	0.487833i	−0.419789	+	0.907622i	$0.637896\pi$
$338$		0			0
$339$		−	2.94207e10i		−	2.22769i
$340$		0			0
$341$	−1.88298e10			−1.39261
$342$		0			0
$343$	−7.64684e9	+	7.64684e9i	−0.552466	+	0.552466i
$344$		0			0
$345$	−1.24280e10	−	2.81809e8i	−0.877251	−	0.0198920i
$346$		0			0
$347$	8.28230e9	+	8.28230e9i	0.571259	+	0.571259i	0.932480	−	0.361221i	$-0.117640\pi$
$347$	8.28230e9	+	8.28230e9i	0.571259	+	0.571259i	−0.361221	+	0.932480i	$0.617640\pi$
$348$		0			0
$349$			1.11611e10i			0.752328i	0.926553	+	0.376164i	$0.122757\pi$
$349$			1.11611e10i			0.752328i	−0.926553	+	0.376164i	$0.877243\pi$
$350$		0			0
$351$	6.68778e9			0.440609
$352$		0			0
$353$	1.05904e10	−	1.05904e10i	0.682047	−	0.682047i	−0.278414	−	0.960461i	$-0.589809\pi$
$353$	1.05904e10	−	1.05904e10i	0.682047	−	0.682047i	0.960461	+	0.278414i	$0.0898088\pi$
$354$		0			0
$355$	−1.26003e8	+	5.55679e9i	−0.00793352	+	0.349873i
$356$		0			0
$357$	−1.24271e9	−	1.24271e9i	−0.0765060	−	0.0765060i
$358$		0			0
$359$			9.48530e9i			0.571049i	0.958371	+	0.285524i	$0.0921677\pi$
$359$			9.48530e9i			0.571049i	−0.958371	+	0.285524i	$0.907832\pi$
$360$		0			0
$361$	−1.24372e10			−0.732309
$362$		0			0
$363$	−1.27851e10	+	1.27851e10i	−0.736340	+	0.736340i
$364$		0			0
$365$	1.19127e10	+	1.24655e10i	0.671179	+	0.702324i
$366$		0			0
$367$	9.59996e9	+	9.59996e9i	0.529182	+	0.529182i	0.920329	−	0.391146i	$-0.127921\pi$
$367$	9.59996e9	+	9.59996e9i	0.529182	+	0.529182i	−0.391146	+	0.920329i	$0.627921\pi$
$368$		0			0
$369$		−	1.19822e10i		−	0.646295i
$370$		0			0
$371$	4.34445e9			0.229319
$372$		0			0
$373$	8.48382e9	−	8.48382e9i	0.438285	−	0.438285i	−0.453150	−	0.891434i	$-0.649700\pi$
$373$	8.48382e9	−	8.48382e9i	0.438285	−	0.438285i	0.891434	+	0.453150i	$0.149700\pi$
$374$		0			0
$375$	1.70914e10	+	1.95902e10i	0.864278	+	0.990634i
$376$		0			0
$377$	3.16469e9	+	3.16469e9i	0.156663	+	0.156663i
$378$		0			0
$379$		−	2.19075e10i		−	1.06178i	−0.847440	−	0.530891i	$-0.821857\pi$
$379$		−	2.19075e10i		−	1.06178i	0.847440	−	0.530891i	$-0.178143\pi$
$380$		0			0
$381$	2.78289e10			1.32068
$382$		0			0
$383$	2.31285e10	−	2.31285e10i	1.07486	−	1.07486i	0.0778987	−	0.996961i	$-0.475179\pi$
$383$	2.31285e10	−	2.31285e10i	1.07486	−	1.07486i	0.996961	−	0.0778987i	$-0.0248211\pi$
$384$		0			0
$385$	−9.15526e9	+	8.74927e9i	−0.416704	+	0.398225i
$386$		0			0
$387$	−3.23222e7	−	3.23222e7i	−0.00144098	−	0.00144098i
$388$		0			0
$389$			2.23175e10i			0.974647i	0.873221	+	0.487324i	$0.162027\pi$
$389$			2.23175e10i			0.974647i	−0.873221	+	0.487324i	$0.837973\pi$
$390$		0			0
$391$	−2.98188e9			−0.127580
$392$		0			0
$393$	−2.29716e10	+	2.29716e10i	−0.962989	+	0.962989i
$394$		0			0
$395$	−6.95274e9	−	1.57656e8i	−0.285606	−	0.00647624i
$396$		0			0
$397$	−2.26036e10	−	2.26036e10i	−0.909947	−	0.909947i	0.0863200	−	0.996267i	$-0.472489\pi$
$397$	−2.26036e10	−	2.26036e10i	−0.909947	−	0.909947i	−0.996267	+	0.0863200i	$0.972489\pi$
$398$		0			0
$399$			1.88822e10i			0.745010i
$400$		0			0
$401$	1.77913e10			0.688064			0.344032	−	0.938958i	$-0.388207\pi$
$401$	1.77913e10			0.688064			0.344032	+	0.938958i	$0.388207\pi$
$402$		0			0
$403$	−2.39363e10	+	2.39363e10i	−0.907480	+	0.907480i
$404$		0			0
$405$	−7.30586e8	+	3.22193e10i	−0.0271551	+	1.19756i
$406$		0			0
$407$	4.77987e9	+	4.77987e9i	0.174196	+	0.174196i
$408$		0			0
$409$		−	4.37112e10i		−	1.56206i	−0.624490	−	0.781032i	$-0.714693\pi$
$409$		−	4.37112e10i		−	1.56206i	0.624490	−	0.781032i	$-0.285307\pi$
$410$		0			0
$411$	−5.65843e10			−1.98303
$412$		0			0
$413$	−2.29389e8	+	2.29389e8i	−0.00788447	+	0.00788447i
$414$		0			0
$415$	−9.69206e9	−	1.01418e10i	−0.326756	−	0.341919i
$416$		0			0
$417$	8.32607e9	+	8.32607e9i	0.275357	+	0.275357i
$418$		0			0
$419$			2.98593e10i			0.968776i	0.874853	+	0.484388i	$0.160958\pi$
$419$			2.98593e10i			0.968776i	−0.874853	+	0.484388i	$0.839042\pi$
$420$		0			0
$421$	−5.33816e10			−1.69927			−0.849637	−	0.527368i	$-0.823179\pi$
$421$	−5.33816e10			−1.69927			−0.849637	+	0.527368i	$0.823179\pi$
$422$		0			0
$423$	8.57253e9	−	8.57253e9i	0.267761	−	0.267761i
$424$		0			0
$425$	4.20522e9	+	4.60497e9i	0.128894	+	0.141147i
$426$		0			0
$427$	−1.29369e10	−	1.29369e10i	−0.389151	−	0.389151i
$428$		0			0
$429$			7.35408e10i			2.17120i
$430$		0			0
$431$	−1.68739e10			−0.488997			−0.244499	−	0.969650i	$-0.578623\pi$
$431$	−1.68739e10			−0.488997			−0.244499	+	0.969650i	$0.578623\pi$
$432$		0			0
$433$	7.15187e9	−	7.15187e9i	0.203455	−	0.203455i	−0.598024	−	0.801478i	$-0.704047\pi$
$433$	7.15187e9	−	7.15187e9i	0.203455	−	0.203455i	0.801478	+	0.598024i	$0.204047\pi$
$434$		0			0
$435$	−6.11165e9	+	5.84063e9i	−0.170688	+	0.163118i
$436$		0			0
$437$	2.26540e10	+	2.26540e10i	0.621182	+	0.621182i
$438$		0			0
$439$		−	2.62124e9i		−	0.0705747i	−0.999377	−	0.0352873i	$-0.988765\pi$
$439$		−	2.62124e9i		−	0.0705747i	0.999377	−	0.0352873i	$-0.0112346\pi$
$440$		0			0
$441$	−2.24408e10			−0.593313
$442$		0			0
$443$	3.30317e10	−	3.30317e10i	0.857663	−	0.857663i	−0.133400	−	0.991062i	$-0.542589\pi$
$443$	3.30317e10	−	3.30317e10i	0.857663	−	0.857663i	0.991062	+	0.133400i	$0.0425894\pi$
$444$		0			0
$445$	−3.03067e10	−	6.87218e8i	−0.772857	−	0.0175249i
$446$		0			0
$447$	5.44965e10	+	5.44965e10i	1.36502	+	1.36502i
$448$		0			0
$449$		−	2.56081e10i		−	0.630074i	−0.949079	−	0.315037i	$-0.897983\pi$
$449$		−	2.56081e10i		−	0.630074i	0.949079	−	0.315037i	$-0.102017\pi$
$450$		0			0
$451$	−4.91459e10			−1.18790
$452$		0			0
$453$	5.77721e10	−	5.77721e10i	1.37191	−	1.37191i
$454$		0			0
$455$	−5.16094e8	+	2.27601e10i	−0.0120416	+	0.531041i
$456$		0			0
$457$	1.31162e10	+	1.31162e10i	0.300707	+	0.300707i	0.841290	−	0.540583i	$-0.181796\pi$
$457$	1.31162e10	+	1.31162e10i	0.300707	+	0.300707i	−0.540583	+	0.841290i	$0.681796\pi$
$458$		0			0
$459$			3.03013e9i			0.0682669i
$460$		0			0
$461$	−3.99207e10			−0.883883			−0.441942	−	0.897044i	$-0.645710\pi$
$461$	−3.99207e10			−0.883883			−0.441942	+	0.897044i	$0.645710\pi$
$462$		0			0
$463$	−1.43482e10	+	1.43482e10i	−0.312229	+	0.312229i	−0.845772	−	0.533544i	$-0.820860\pi$
$463$	−1.43482e10	+	1.43482e10i	−0.312229	+	0.312229i	0.533544	+	0.845772i	$0.320860\pi$
$464$		0			0
$465$	−4.41759e10	−	4.62258e10i	−0.944874	−	0.988719i
$466$		0			0
$467$	−2.44200e10	−	2.44200e10i	−0.513427	−	0.513427i	0.402148	−	0.915575i	$-0.368264\pi$
$467$	−2.44200e10	−	2.44200e10i	−0.513427	−	0.513427i	−0.915575	+	0.402148i	$0.868264\pi$
$468$		0			0
$469$			6.45246e9i			0.133363i
$470$		0			0
$471$	1.20968e11			2.45804
$472$		0			0
$473$	−1.32572e8	+	1.32572e8i	−0.00264855	+	0.00264855i
$474$		0			0
$475$	3.03703e9	−	6.69330e10i	0.0596588	−	1.31482i
$476$		0			0
$477$	1.42002e10	+	1.42002e10i	0.274296	+	0.274296i
$478$		0			0
$479$			9.03857e9i			0.171695i	0.996308	+	0.0858475i	$0.0273598\pi$
$479$			9.03857e9i			0.171695i	−0.996308	+	0.0858475i	$0.972640\pi$
$480$		0			0
$481$	1.21523e10			0.227026
$482$		0			0
$483$	1.45393e10	−	1.45393e10i	0.267150	−	0.267150i
$484$		0			0
$485$	6.86368e10	−	6.55930e10i	1.24048	−	1.18547i
$486$		0			0
$487$	−3.53950e10	−	3.53950e10i	−0.629255	−	0.629255i	0.318626	−	0.947881i	$-0.396779\pi$
$487$	−3.53950e10	−	3.53950e10i	−0.629255	−	0.629255i	−0.947881	+	0.318626i	$0.896779\pi$
$488$		0			0
$489$			6.54965e10i			1.14547i
$490$		0			0
$491$	5.87159e10			1.01025			0.505126	−	0.863045i	$-0.331446\pi$
$491$	5.87159e10			1.01025			0.505126	+	0.863045i	$0.331446\pi$
$492$		0			0
$493$	−1.43387e9	+	1.43387e9i	−0.0242730	+	0.0242730i
$494$		0			0
$495$	−5.85222e10	−	1.32702e9i	−0.974765	−	0.0221032i
$496$		0			0
$497$	−6.50082e9	−	6.50082e9i	−0.106547	−	0.106547i
$498$		0			0
$499$		−	2.57394e10i		−	0.415141i	−0.978220	−	0.207571i	$-0.933444\pi$
$499$		−	2.57394e10i		−	0.415141i	0.978220	−	0.207571i	$-0.0665557\pi$
$500$		0			0
$501$	1.07948e11			1.71342
$502$		0			0
$503$	4.75499e10	−	4.75499e10i	0.742810	−	0.742810i	−0.230308	−	0.973118i	$-0.573973\pi$
$503$	4.75499e10	−	4.75499e10i	0.742810	−	0.742810i	0.973118	+	0.230308i	$0.0739733\pi$
$504$		0			0
$505$	1.01450e9	−	4.47399e10i	0.0155986	−	0.687907i
$506$		0			0
$507$	3.20615e10	+	3.20615e10i	0.485235	+	0.485235i
$508$		0			0
$509$		−	8.69280e10i		−	1.29506i	−0.762042	−	0.647528i	$-0.775803\pi$
$509$		−	8.69280e10i		−	1.29506i	0.762042	−	0.647528i	$-0.224197\pi$
$510$		0			0
$511$	−2.85197e10			−0.418275
$512$		0			0
$513$	2.30206e10	−	2.30206e10i	0.332389	−	0.332389i
$514$		0			0
$515$	−6.04050e10	−	6.32080e10i	−0.858705	−	0.898552i
$516$		0			0
$517$	−3.51609e10	−	3.51609e10i	−0.492151	−	0.492151i
$518$		0			0
$519$			5.12568e10i			0.706451i
$520$		0			0
$521$	6.44123e10			0.874214			0.437107	−	0.899409i	$-0.356003\pi$
$521$	6.44123e10			0.874214			0.437107	+	0.899409i	$0.356003\pi$
$522$		0			0
$523$	5.99903e10	−	5.99903e10i	0.801815	−	0.801815i	−0.181564	−	0.983379i	$-0.558116\pi$
$523$	5.99903e10	−	5.99903e10i	0.801815	−	0.801815i	0.983379	+	0.181564i	$0.0581160\pi$
$524$		0			0
$525$	−4.29576e10	−	1.94917e9i	−0.565461	−	0.0256573i
$526$		0			0
$527$	−1.08452e10	−	1.08452e10i	−0.140603	−	0.140603i
$528$		0			0
$529$			4.34238e10i			0.554505i
$530$		0			0
$531$	−1.49955e9			−0.0188618
$532$		0			0
$533$	−6.24738e10	+	6.24738e10i	−0.774086	+	0.774086i
$534$		0			0
$535$	3.02568e10	−	2.89151e10i	0.369324	−	0.352947i
$536$		0			0
$537$	9.72705e10	+	9.72705e10i	1.16973	+	1.16973i
$538$		0			0
$539$			9.20427e10i			1.09052i
$540$		0			0
$541$	−9.38341e10			−1.09540			−0.547698	−	0.836676i	$-0.684496\pi$
$541$	−9.38341e10			−1.09540			−0.547698	+	0.836676i	$0.684496\pi$
$542$		0			0
$543$	−1.45259e10	+	1.45259e10i	−0.167087	+	0.167087i
$544$		0			0
$545$	−1.43133e10	−	3.24560e8i	−0.162238	−	0.00367883i
$546$		0			0
$547$	−1.86813e10	−	1.86813e10i	−0.208669	−	0.208669i	0.595032	−	0.803702i	$-0.297139\pi$
$547$	−1.86813e10	−	1.86813e10i	−0.208669	−	0.208669i	−0.803702	+	0.595032i	$0.797139\pi$
$548$		0			0
$549$		−	8.45702e10i		−	0.930954i
$550$		0			0
$551$	2.17869e10			0.236368
$552$		0			0
$553$	8.13393e9	−	8.13393e9i	0.0869761	−	0.0869761i
$554$		0			0
$555$	−5.20357e8	+	2.29481e10i	−0.00548441	+	0.241866i
$556$		0			0
$557$	−4.32331e9	−	4.32331e9i	−0.0449154	−	0.0449154i	0.684292	−	0.729208i	$-0.260111\pi$
$557$	−4.32331e9	−	4.32331e9i	−0.0449154	−	0.0449154i	−0.729208	+	0.684292i	$0.760111\pi$
$558$		0			0
$559$			3.37049e8i			0.00345180i
$560$		0			0
$561$	−3.33202e10			−0.336400
$562$		0			0
$563$	−5.85471e10	+	5.85471e10i	−0.582736	+	0.582736i	−0.935654	−	0.352918i	$-0.885189\pi$
$563$	−5.85471e10	+	5.85471e10i	−0.582736	+	0.582736i	0.352918	+	0.935654i	$0.385189\pi$
$564$		0			0
$565$	−1.19302e11	−	1.24837e11i	−1.17072	−	1.22504i
$566$		0			0
$567$	−3.76929e10	−	3.76929e10i	−0.364693	−	0.364693i
$568$		0			0
$569$			1.98793e11i			1.89650i	0.317524	+	0.948250i	$0.397149\pi$
$569$			1.98793e11i			1.89650i	−0.317524	+	0.948250i	$0.602851\pi$
$570$		0			0
$571$	−6.21842e10			−0.584973			−0.292487	−	0.956270i	$-0.594483\pi$
$571$	−6.21842e10			−0.584973			−0.292487	+	0.956270i	$0.594483\pi$
$572$		0			0
$573$	5.75764e10	−	5.75764e10i	0.534104	−	0.534104i
$574$		0			0
$575$	−5.38769e10	+	4.91999e10i	−0.492869	+	0.450083i
$576$		0			0
$577$	−1.02056e11	−	1.02056e11i	−0.920739	−	0.920739i	0.0763427	−	0.997082i	$-0.475676\pi$
$577$	−1.02056e11	−	1.02056e11i	−0.920739	−	0.920739i	−0.997082	+	0.0763427i	$0.975676\pi$
$578$		0			0
$579$			1.73772e10i			0.154620i
$580$		0			0
$581$	2.32034e10			0.203633
$582$		0			0
$583$	5.82431e10	−	5.82431e10i	0.504162	−	0.504162i
$584$		0			0
$585$	−7.60798e10	+	7.27060e10i	−0.649600	+	0.620793i
$586$		0			0
$587$	1.52761e11	+	1.52761e11i	1.28665	+	1.28665i	0.936811	+	0.349837i	$0.113763\pi$
$587$	1.52761e11	+	1.52761e11i	1.28665	+	1.28665i	0.349837	+	0.936811i	$0.386237\pi$
$588$		0			0
$589$			1.64787e11i			1.36918i
$590$		0			0
$591$	1.49906e10			0.122876
$592$		0			0
$593$	9.35153e10	−	9.35153e10i	0.756247	−	0.756247i	−0.219390	−	0.975637i	$-0.570407\pi$
$593$	9.35153e10	−	9.35153e10i	0.756247	−	0.756247i	0.975637	+	0.219390i	$0.0704067\pi$
$594$		0			0
$595$	−1.03122e10	−	2.33834e8i	−0.0822782	−	0.00186569i
$596$		0			0
$597$	1.28901e11	+	1.28901e11i	1.01475	+	1.01475i
$598$		0			0
$599$			5.65120e10i			0.438968i	0.975616	+	0.219484i	$0.0704374\pi$
$599$			5.65120e10i			0.438968i	−0.975616	+	0.219484i	$0.929563\pi$
$600$		0			0
$601$	−9.99220e10			−0.765884			−0.382942	−	0.923772i	$-0.625089\pi$
$601$	−9.99220e10			−0.765884			−0.382942	+	0.923772i	$0.625089\pi$
$602$		0			0
$603$	−2.10903e10	+	2.10903e10i	−0.159520	+	0.159520i
$604$		0			0
$605$	−2.40572e9	+	1.06094e11i	−0.0179566	+	0.791896i
$606$		0			0
$607$	−1.27776e11	−	1.27776e11i	−0.941229	−	0.941229i	0.0571370	−	0.998366i	$-0.481803\pi$
$607$	−1.27776e11	−	1.27776e11i	−0.941229	−	0.941229i	−0.998366	+	0.0571370i	$0.981803\pi$
$608$		0			0
$609$		−	1.39828e10i		−	0.101654i
$610$		0			0
$611$	−8.93925e10			−0.641411
$612$		0			0
$613$	6.34636e10	−	6.34636e10i	0.449452	−	0.449452i	−0.445720	−	0.895172i	$-0.647052\pi$
$613$	6.34636e10	−	6.34636e10i	0.449452	−	0.449452i	0.895172	+	0.445720i	$0.147052\pi$
$614$		0			0
$615$	−1.15299e11	−	1.20649e11i	−0.805983	−	0.843383i
$616$		0			0
$617$	−6.39126e10	−	6.39126e10i	−0.441007	−	0.441007i	0.451343	−	0.892350i	$-0.350945\pi$
$617$	−6.39126e10	−	6.39126e10i	−0.441007	−	0.441007i	−0.892350	+	0.451343i	$0.850945\pi$
$618$		0			0
$619$			5.76225e10i			0.392491i	0.980555	+	0.196245i	$0.0628749\pi$
$619$			5.76225e10i			0.392491i	−0.980555	+	0.196245i	$0.937125\pi$
$620$		0			0
$621$	−3.54517e10			−0.238381
$622$		0			0
$623$	3.54555e10	−	3.54555e10i	0.235359	−	0.235359i
$624$		0			0
$625$	1.51961e11	+	1.38186e10i	0.995891	+	0.0905619i
$626$		0			0
$627$	2.53141e11	+	2.53141e11i	1.63792	+	1.63792i
$628$		0			0
$629$			5.50600e9i			0.0351749i
$630$		0			0
$631$	−1.53061e11			−0.965486			−0.482743	−	0.875762i	$-0.660359\pi$
$631$	−1.53061e11			−0.965486			−0.482743	+	0.875762i	$0.660359\pi$
$632$		0			0
$633$	−5.12953e10	+	5.12953e10i	−0.319494	+	0.319494i
$634$		0			0
$635$	1.18083e11	−	1.12847e11i	0.726263	−	0.694057i
$636$		0			0
$637$	1.17004e11	+	1.17004e11i	0.710628	+	0.710628i
$638$		0			0
$639$		−	4.24968e10i		−	0.254890i
$640$		0			0
$641$	1.10388e11			0.653867			0.326934	−	0.945047i	$-0.393985\pi$
$641$	1.10388e11			0.653867			0.326934	+	0.945047i	$0.393985\pi$
$642$		0			0
$643$	−4.62452e10	+	4.62452e10i	−0.270535	+	0.270535i	−0.829315	−	0.558781i	$-0.811269\pi$
$643$	−4.62452e10	+	4.62452e10i	−0.270535	+	0.270535i	0.558781	+	0.829315i	$0.311269\pi$
$644$		0			0
$645$	−6.36477e8	−	1.44324e7i	−0.00367743	−	8.33872e-5i
$646$		0			0
$647$	9.64071e10	+	9.64071e10i	0.550164	+	0.550164i	0.926488	−	0.376324i	$-0.122812\pi$
$647$	9.64071e10	+	9.64071e10i	0.550164	+	0.550164i	−0.376324	+	0.926488i	$0.622812\pi$
$648$		0			0
$649$			6.15052e9i			0.0346683i
$650$		0			0
$651$	1.05760e11			0.588840
$652$		0			0
$653$	−2.02786e11	+	2.02786e11i	−1.11528	+	1.11528i	−0.122857	+	0.992424i	$0.539206\pi$
$653$	−2.02786e11	+	2.02786e11i	−1.11528	+	1.11528i	−0.992424	+	0.122857i	$0.960794\pi$
$654$		0			0
$655$	−4.32247e9	+	1.90623e11i	−0.0234837	+	1.03565i
$656$		0			0
$657$	−9.32188e10	−	9.32188e10i	−0.500313	−	0.500313i
$658$		0			0
$659$		−	5.78139e10i		−	0.306542i	−0.988184	−	0.153271i	$-0.951019\pi$
$659$		−	5.78139e10i		−	0.306542i	0.988184	−	0.153271i	$-0.0489808\pi$
$660$		0			0
$661$	3.80376e10			0.199254			0.0996271	−	0.995025i	$-0.468235\pi$
$661$	3.80376e10			0.199254			0.0996271	+	0.995025i	$0.468235\pi$
$662$		0			0
$663$	−4.23563e10	+	4.23563e10i	−0.219212	+	0.219212i
$664$		0			0
$665$	7.65679e10	+	8.01209e10i	0.391526	+	0.409694i
$666$		0			0
$667$	−1.67759e10	−	1.67759e10i	−0.0847585	−	0.0847585i
$668$		0			0
$669$		−	1.41200e11i		−	0.704905i
$670$		0			0
$671$	−3.46872e11			−1.71111
$672$		0			0
$673$	1.23735e11	−	1.23735e11i	0.603158	−	0.603158i	−0.337991	−	0.941149i	$-0.609747\pi$
$673$	1.23735e11	−	1.23735e11i	0.603158	−	0.603158i	0.941149	+	0.337991i	$0.109747\pi$
$674$		0			0
$675$	4.99961e10	+	5.47488e10i	0.240836	+	0.263730i
$676$		0			0
$677$	−8.44279e10	−	8.44279e10i	−0.401912	−	0.401912i	0.476994	−	0.878906i	$-0.341726\pi$
$677$	−8.44279e10	−	8.44279e10i	−0.401912	−	0.401912i	−0.878906	+	0.476994i	$0.841726\pi$
$678$		0			0
$679$			1.57034e11i			0.738778i
$680$		0			0
$681$	−5.24008e11			−2.43640
$682$		0			0
$683$	2.87370e10	−	2.87370e10i	0.132056	−	0.132056i	−0.637989	−	0.770045i	$-0.720234\pi$
$683$	2.87370e10	−	2.87370e10i	0.132056	−	0.132056i	0.770045	+	0.637989i	$0.220234\pi$
$684$		0			0
$685$	−2.40098e11	+	2.29450e11i	−1.09050	+	1.04214i
$686$		0			0
$687$	−2.76549e11	−	2.76549e11i	−1.24150	−	1.24150i
$688$		0			0
$689$		−	1.48076e11i		−	0.657065i
$690$		0			0
$691$	8.87703e10			0.389364			0.194682	−	0.980866i	$-0.437633\pi$
$691$	8.87703e10			0.389364			0.194682	+	0.980866i	$0.437633\pi$
$692$		0			0
$693$	6.84644e10	−	6.84644e10i	0.296847	−	0.296847i
$694$		0			0
$695$	6.90915e10	+	1.56668e9i	0.296132	+	0.00671492i
$696$		0			0
$697$	−2.83059e10	−	2.83059e10i	−0.119935	−	0.119935i
$698$		0			0
$699$		−	5.46503e11i		−	2.28920i
$700$		0			0
$701$	1.22790e11			0.508499			0.254249	−	0.967139i	$-0.418172\pi$
$701$	1.22790e11			0.508499			0.254249	+	0.967139i	$0.418172\pi$
$702$		0			0
$703$	4.18303e10	−	4.18303e10i	0.171266	−	0.171266i
$704$		0			0
$705$	3.82777e9	−	1.68807e11i	0.0154949	−	0.683336i
$706$		0			0
$707$	5.23407e10	+	5.23407e10i	0.209489	+	0.209489i
$708$		0			0
$709$		−	2.07670e11i		−	0.821842i	−0.911671	−	0.410921i	$-0.865207\pi$
$709$		−	2.07670e11i		−	0.821842i	0.911671	−	0.410921i	$-0.134793\pi$
$710$		0			0
$711$	5.31726e10			0.208070
$712$		0			0
$713$	1.26886e11	−	1.26886e11i	0.490969	−	0.490969i
$714$		0			0
$715$	2.98210e11	+	3.12048e11i	1.14103	+	1.19398i
$716$		0			0
$717$	1.99040e11	+	1.99040e11i	0.753119	+	0.753119i
$718$		0			0
$719$		−	1.66571e11i		−	0.623281i	−0.950200	−	0.311640i	$-0.899122\pi$
$719$		−	1.66571e11i		−	0.623281i	0.950200	−	0.311640i	$-0.100878\pi$
$720$		0			0
$721$	1.44613e11			0.535140
$722$		0			0
$723$	5.42320e9	−	5.42320e9i	0.0198474	−	0.0198474i
$724$		0			0
$725$	−2.24901e9	+	4.95658e10i	−0.00814027	+	0.179403i
$726$		0			0
$727$	−1.61369e11	−	1.61369e11i	−0.577673	−	0.577673i	0.356588	−	0.934262i	$-0.383940\pi$
$727$	−1.61369e11	−	1.61369e11i	−0.577673	−	0.577673i	−0.934262	+	0.356588i	$0.883940\pi$
$728$		0			0
$729$		−	1.13795e11i		−	0.402915i
$730$		0			0
$731$	−1.52712e8			−0.000534814
$732$		0			0
$733$	4.72057e10	−	4.72057e10i	0.163523	−	0.163523i	−0.620603	−	0.784125i	$-0.713112\pi$
$733$	4.72057e10	−	4.72057e10i	0.163523	−	0.163523i	0.784125	+	0.620603i	$0.213112\pi$
$734$		0			0
$735$	−2.25958e11	+	2.15938e11i	−0.774245	+	0.739911i
$736$		0			0
$737$	8.65036e10	+	8.65036e10i	0.293200	+	0.293200i
$738$		0			0
$739$			2.13693e11i			0.716494i	0.933627	+	0.358247i	$0.116625\pi$
$739$			2.13693e11i			0.716494i	−0.933627	+	0.358247i	$0.883375\pi$
$740$		0			0
$741$	6.43582e11			2.13467
$742$		0			0
$743$	3.10741e11	−	3.10741e11i	1.01963	−	1.01963i	0.0198284	−	0.999803i	$-0.493688\pi$
$743$	3.10741e11	−	3.10741e11i	1.01963	−	1.01963i	0.999803	−	0.0198284i	$-0.00631198\pi$
$744$		0			0
$745$	4.52224e11	+	1.02544e10i	1.46801	+	0.0332877i
$746$		0			0
$747$	7.58420e10	+	7.58420e10i	0.243572	+	0.243572i
$748$		0			0
$749$			6.92245e10i			0.219954i
$750$		0			0
$751$	5.29510e11			1.66462			0.832308	−	0.554313i	$-0.187019\pi$
$751$	5.29510e11			1.66462			0.832308	+	0.554313i	$0.187019\pi$
$752$		0			0
$753$	4.35909e9	−	4.35909e9i	0.0135586	−	0.0135586i
$754$		0			0
$755$	1.08707e10	−	4.79405e11i	0.0334557	−	1.47542i
$756$		0			0
$757$	−1.66172e11	−	1.66172e11i	−0.506029	−	0.506029i	0.407276	−	0.913305i	$-0.366479\pi$
$757$	−1.66172e11	−	1.66172e11i	−0.506029	−	0.506029i	−0.913305	+	0.407276i	$0.866479\pi$
$758$		0			0
$759$		−	3.89837e11i		−	1.17467i
$760$		0			0
$761$	−3.61095e11			−1.07667			−0.538335	−	0.842731i	$-0.680946\pi$
$761$	−3.61095e11			−1.07667			−0.538335	+	0.842731i	$0.680946\pi$
$762$		0			0
$763$	1.67450e10	−	1.67450e10i	0.0494067	−	0.0494067i
$764$		0			0
$765$	−3.29420e10	−	3.44706e10i	−0.0961842	−	0.100647i
$766$		0			0
$767$	7.81849e9	+	7.81849e9i	0.0225913	+	0.0225913i
$768$		0			0
$769$			2.04700e11i			0.585345i	0.956213	+	0.292672i	$0.0945446\pi$
$769$			2.04700e11i			0.585345i	−0.956213	+	0.292672i	$0.905455\pi$
$770$		0			0
$771$	7.12960e10			0.201766
$772$		0			0
$773$	−8.07889e10	+	8.07889e10i	−0.226274	+	0.226274i	−0.811134	−	0.584860i	$-0.801149\pi$
$773$	−8.07889e10	+	8.07889e10i	−0.226274	+	0.226274i	0.584860	+	0.811134i	$0.301149\pi$
$774$		0			0
$775$	−3.74894e11	−	1.70105e10i	−1.03921	−	0.0471531i
$776$		0			0
$777$	−2.68467e10	−	2.68467e10i	−0.0736557	−	0.0736557i
$778$		0			0
$779$			4.30093e11i			1.16792i
$780$		0			0
$781$	−1.74304e11			−0.468493
$782$		0			0
$783$	−1.70474e10	+	1.70474e10i	−0.0453535	+	0.0453535i
$784$		0			0
$785$	5.13292e11	−	4.90530e11i	1.35172	−	1.29177i
$786$		0			0
$787$	1.41727e11	+	1.41727e11i	0.369448	+	0.369448i	0.867276	−	0.497828i	$-0.165869\pi$
$787$	1.41727e11	+	1.41727e11i	0.369448	+	0.369448i	−0.497828	+	0.867276i	$0.665869\pi$
$788$		0			0
$789$		−	9.87846e11i		−	2.54907i
$790$		0			0
$791$	2.85615e11			0.729584
$792$		0			0
$793$	−4.40940e11	+	4.40940e11i	−1.11503	+	1.11503i
$794$		0			0
$795$	2.79624e11	+	6.34060e9i	0.700013	+	0.0158731i
$796$		0			0
$797$	−2.66505e11	−	2.66505e11i	−0.660499	−	0.660499i	0.294998	−	0.955498i	$-0.404681\pi$
$797$	−2.66505e11	−	2.66505e11i	−0.660499	−	0.660499i	−0.955498	+	0.294998i	$0.904681\pi$
$798$		0			0
$799$		−	4.05024e10i		−	0.0993787i
$800$		0			0
$801$	2.31778e11			0.563043
$802$		0			0
$803$	−3.82344e11	+	3.82344e11i	−0.919586	+	0.919586i
$804$		0			0
$805$	2.73580e9	−	1.20650e11i	0.00651479	−	0.287306i
$806$		0			0
$807$	−4.05720e11	−	4.05720e11i	−0.956604	−	0.956604i
$808$		0			0
$809$			6.78011e11i			1.58286i	0.611259	+	0.791431i	$0.290663\pi$
$809$			6.78011e11i			1.58286i	−0.611259	+	0.791431i	$0.709337\pi$
$810$		0			0
$811$	−4.62623e11			−1.06941			−0.534704	−	0.845039i	$-0.679577\pi$
$811$	−4.62623e11			−1.06941			−0.534704	+	0.845039i	$0.679577\pi$
$812$		0			0
$813$	−6.62244e11	+	6.62244e11i	−1.51585	+	1.51585i
$814$		0			0
$815$	2.65590e11	+	2.77914e11i	0.601978	+	0.629912i
$816$		0			0
$817$	1.16019e9	+	1.16019e9i	0.00260399	+	0.00260399i
$818$		0			0
$819$		−	1.74063e11i		−	0.386875i
$820$		0			0
$821$	−2.02486e11			−0.445678			−0.222839	−	0.974855i	$-0.571532\pi$
$821$	−2.02486e11			−0.445678			−0.222839	+	0.974855i	$0.571532\pi$
$822$		0			0
$823$	−5.67731e11	+	5.67731e11i	−1.23749	+	1.23749i	−0.276473	+	0.961022i	$0.589166\pi$
$823$	−5.67731e11	+	5.67731e11i	−1.23749	+	1.23749i	−0.961022	+	0.276473i	$0.910834\pi$
$824$		0			0
$825$	−6.02034e11	+	5.49772e11i	−1.29959	+	1.18677i
$826$		0			0
$827$	−5.76065e11	−	5.76065e11i	−1.23154	−	1.23154i	−0.963371	−	0.268173i	$-0.913580\pi$
$827$	−5.76065e11	−	5.76065e11i	−1.23154	−	1.23154i	−0.268173	−	0.963371i	$-0.586420\pi$
$828$		0			0
$829$			2.12691e11i			0.450330i	0.974321	+	0.225165i	$0.0722922\pi$
$829$			2.12691e11i			0.450330i	−0.974321	+	0.225165i	$0.927708\pi$
$830$		0			0
$831$	7.20077e11			1.50999
$832$		0			0
$833$	−5.30126e10	+	5.30126e10i	−0.110103	+	0.110103i
$834$		0			0
$835$	4.58043e11	−	4.37731e11i	0.942237	−	0.900454i
$836$		0			0
$837$	−1.28939e11	−	1.28939e11i	−0.262713	−	0.262713i
$838$		0			0
$839$			8.36816e11i			1.68881i	0.535701	+	0.844407i	$0.320047\pi$
$839$			8.36816e11i			1.68881i	−0.535701	+	0.844407i	$0.679953\pi$
$840$		0			0
$841$	4.84113e11			0.967748
$842$		0			0
$843$	−9.20555e11	+	9.20555e11i	−1.82280	+	1.82280i
$844$		0			0
$845$	2.66053e11	+	6.03286e9i	0.521845	+	0.0118331i
$846$		0			0
$847$	−1.24118e11	−	1.24118e11i	−0.241157	−	0.241157i
$848$		0			0
$849$			4.31996e10i			0.0831475i
$850$		0			0
$851$	−6.44187e10			−0.122827
$852$		0			0
$853$	2.74662e11	−	2.74662e11i	0.518803	−	0.518803i	−0.398406	−	0.917209i	$-0.630436\pi$
$853$	2.74662e11	−	2.74662e11i	0.518803	−	0.518803i	0.917209	+	0.398406i	$0.130436\pi$
$854$		0			0
$855$	−1.16132e10	+	5.12149e11i	−0.0217314	+	0.958366i
$856$		0			0
$857$	−9.65564e10	−	9.65564e10i	−0.179002	−	0.179002i	0.611919	−	0.790921i	$-0.290398\pi$
$857$	−9.65564e10	−	9.65564e10i	−0.179002	−	0.179002i	−0.790921	+	0.611919i	$0.790398\pi$
$858$		0			0
$859$		−	7.22383e11i		−	1.32677i	−0.748280	−	0.663383i	$-0.769120\pi$
$859$		−	7.22383e11i		−	1.32677i	0.748280	−	0.663383i	$-0.230880\pi$
$860$		0			0
$861$	2.76034e11			0.502284
$862$		0			0
$863$	9.33124e10	−	9.33124e10i	0.168227	−	0.168227i	−0.617973	−	0.786200i	$-0.712046\pi$
$863$	9.33124e10	−	9.33124e10i	0.168227	−	0.168227i	0.786200	+	0.617973i	$0.212046\pi$
$864$		0			0
$865$	2.07848e11	+	2.17492e11i	0.371262	+	0.388490i
$866$		0			0
$867$	5.06070e11	+	5.06070e11i	0.895642	+	0.895642i
$868$		0			0
$869$		−	2.18092e11i		−	0.382437i
$870$		0			0
$871$	2.19925e11			0.382123
$872$		0			0
$873$	−5.13276e11	+	5.13276e11i	−0.883678	+	0.883678i
$874$		0			0
$875$	−1.90181e11	+	1.65923e11i	−0.324441	+	0.283058i
$876$		0			0
$877$	5.23388e11	+	5.23388e11i	0.884760	+	0.884760i	0.994014	−	0.109254i	$-0.0348463\pi$
$877$	5.23388e11	+	5.23388e11i	0.884760	+	0.884760i	−0.109254	+	0.994014i	$0.534846\pi$
$878$		0			0
$879$		−	1.54834e11i		−	0.259365i
$880$		0			0
$881$	−8.15621e10			−0.135389			−0.0676947	−	0.997706i	$-0.521564\pi$
$881$	−8.15621e10			−0.135389			−0.0676947	+	0.997706i	$0.521564\pi$
$882$		0			0
$883$	3.22895e11	−	3.22895e11i	0.531151	−	0.531151i	−0.389764	−	0.920915i	$-0.627443\pi$
$883$	3.22895e11	−	3.22895e11i	0.531151	−	0.531151i	0.920915	+	0.389764i	$0.127443\pi$
$884$		0			0
$885$	−1.50991e10	+	1.44295e10i	−0.0246137	+	0.0235222i
$886$		0			0
$887$	−4.59932e11	−	4.59932e11i	−0.743018	−	0.743018i	0.230140	−	0.973158i	$-0.426082\pi$
$887$	−4.59932e11	−	4.59932e11i	−0.743018	−	0.743018i	−0.973158	+	0.230140i	$0.926082\pi$
$888$		0			0
$889$			2.70163e11i			0.432532i
$890$		0			0
$891$	−1.01065e12			−1.60357
$892$		0			0
$893$	−3.07706e11	+	3.07706e11i	−0.483871	+	0.483871i
$894$		0			0
$895$	8.07171e11	+	1.83029e10i	1.25798	+	0.0285252i
$896$		0			0
$897$	−4.95558e11	−	4.95558e11i	−0.765464	−	0.765464i
$898$		0			0
$899$		−	1.22029e11i		−	0.186821i
$900$		0			0
$901$	6.70910e10			0.101804
$902$		0			0
$903$	7.44607e8	−	7.44607e8i	0.00111989	−	0.00111989i
$904$		0			0
$905$	−2.73327e9	+	1.20539e11i	−0.00407463	+	0.179694i
$906$		0			0
$907$	7.98076e11	+	7.98076e11i	1.17928	+	1.17928i	0.979929	+	0.199347i	$0.0638820\pi$
$907$	7.98076e11	+	7.98076e11i	1.17928	+	1.17928i	0.199347	+	0.979929i	$0.436118\pi$
$908$		0			0
$909$			3.42159e11i			0.501155i
$910$		0			0
$911$	1.50243e11			0.218132			0.109066	−	0.994034i	$-0.465214\pi$
$911$	1.50243e11			0.218132			0.109066	+	0.994034i	$0.465214\pi$
$912$		0			0
$913$	3.11072e11	−	3.11072e11i	0.447690	−	0.447690i
$914$		0			0
$915$	−8.13782e11	−	8.51544e11i	−1.16098	−	1.21485i
$916$		0			0
$917$	−2.23008e11	−	2.23008e11i	−0.315387	−	0.315387i
$918$		0			0
$919$			1.02471e12i			1.43662i	0.695725	+	0.718308i	$0.255083\pi$
$919$			1.02471e12i			1.43662i	−0.695725	+	0.718308i	$0.744917\pi$
$920$		0			0
$921$	1.19296e12			1.65801
$922$		0			0
$923$	−2.21574e11	+	2.21574e11i	−0.305289	+	0.305289i
$924$		0			0
$925$	9.08470e10	+	9.94831e10i	0.124092	+	0.135888i
$926$		0			0
$927$	4.72679e11	+	4.72679e11i	0.640099	+	0.640099i
$928$		0			0
$929$		−	1.38553e12i		−	1.86017i	−0.367346	−	0.930084i	$-0.619734\pi$
$929$		−	1.38553e12i		−	1.86017i	0.367346	−	0.930084i	$-0.380266\pi$
$930$		0			0
$931$	8.05498e11			1.07218
$932$		0			0
$933$	2.81659e11	−	2.81659e11i	0.371704	−	0.371704i
$934$		0			0
$935$	−1.41384e11	+	1.35114e11i	−0.184992	+	0.176789i
$936$		0			0
$937$	−7.72786e11	−	7.72786e11i	−1.00254	−	1.00254i	−0.999997	−	0.00254091i	$-0.999191\pi$
$937$	−7.72786e11	−	7.72786e11i	−1.00254	−	1.00254i	−0.00254091	−	0.999997i	$-0.500809\pi$
$938$		0			0
$939$		−	7.22308e11i		−	0.929095i
$940$		0			0
$941$	−6.91185e11			−0.881528			−0.440764	−	0.897623i	$-0.645292\pi$
$941$	−6.91185e11			−0.881528			−0.440764	+	0.897623i	$0.645292\pi$
$942$		0			0
$943$	3.31172e11	−	3.31172e11i	0.418800	−	0.418800i
$944$		0			0
$945$	−1.22603e11	−	2.78007e9i	−0.153735	−	0.00348601i
$946$		0			0
$947$	−6.05426e11	−	6.05426e11i	−0.752768	−	0.752768i	0.222227	−	0.974995i	$-0.428667\pi$
$947$	−6.05426e11	−	6.05426e11i	−0.752768	−	0.752768i	−0.974995	+	0.222227i	$0.928667\pi$
$948$		0			0
$949$			9.72066e11i			1.19848i
$950$		0			0
$951$	8.35663e11			1.02167
$952$		0			0
$953$	−3.45330e11	+	3.45330e11i	−0.418661	+	0.418661i	−0.884742	−	0.466081i	$-0.845665\pi$
$953$	−3.45330e11	+	3.45330e11i	−0.418661	+	0.418661i	0.466081	+	0.884742i	$0.345665\pi$
$954$		0			0
$955$	1.08339e10	−	4.77781e11i	0.0130248	−	0.574401i
$956$		0			0
$957$	−1.87458e11	−	1.87458e11i	−0.223489	−	0.223489i
$958$		0			0
$959$		−	5.49319e11i		−	0.649456i
$960$		0			0
$961$	7.00825e10			0.0821705
$962$		0			0
$963$	−2.26265e11	+	2.26265e11i	−0.263095	+	0.263095i
$964$		0			0
$965$	7.04648e10	+	7.37346e10i	0.0812575	+	0.0850281i
$966$		0			0
$967$	1.77273e10	+	1.77273e10i	0.0202739	+	0.0202739i	0.717171	−	0.696897i	$-0.245437\pi$
$967$	1.77273e10	+	1.77273e10i	0.0202739	+	0.0202739i	−0.696897	+	0.717171i	$0.745437\pi$
$968$		0			0
$969$			2.91597e11i			0.330741i
$970$		0			0
$971$	−1.52187e12			−1.71199			−0.855994	−	0.516986i	$-0.827054\pi$
$971$	−1.52187e12			−1.71199			−0.855994	+	0.516986i	$0.827054\pi$
$972$		0			0
$973$	−8.08294e10	+	8.08294e10i	−0.0901816	+	0.0901816i
$974$		0			0
$975$	−6.64353e10	+	1.46416e12i	−0.0735157	+	1.62021i
$976$		0			0
$977$	2.88131e11	+	2.88131e11i	0.316237	+	0.316237i	0.847320	−	0.531083i	$-0.178215\pi$
$977$	2.88131e11	+	2.88131e11i	0.316237	+	0.316237i	−0.531083	+	0.847320i	$0.678215\pi$
$978$		0			0
$979$		−	9.50654e11i		−	1.03488i
$980$		0			0
$981$	1.09464e11			0.118194
$982$		0			0
$983$	2.15222e9	−	2.15222e9i	0.00230501	−	0.00230501i	−0.705953	−	0.708258i	$-0.749481\pi$
$983$	2.15222e9	−	2.15222e9i	0.00230501	−	0.00230501i	0.708258	+	0.705953i	$0.249481\pi$
$984$		0			0
$985$	6.36078e10	−	6.07871e10i	0.0675718	−	0.0645753i
$986$		0			0
$987$	1.97485e11	+	1.97485e11i	0.208097	+	0.208097i
$988$		0			0
$989$		−	1.78669e9i		−	0.00186751i
$990$		0			0
$991$	−1.40286e10			−0.0145452			−0.00727261	−	0.999974i	$-0.502315\pi$
$991$	−1.40286e10			−0.0145452			−0.00727261	+	0.999974i	$0.502315\pi$
$992$		0			0
$993$	1.12783e12	−	1.12783e12i	1.15997	−	1.15997i
$994$		0			0
$995$	1.06965e12	+	2.42547e10i	1.09131	+	0.0247459i
$996$		0			0
$997$	7.09850e11	+	7.09850e11i	0.718433	+	0.718433i	0.968284	−	0.249851i	$-0.0803818\pi$
$997$	7.09850e11	+	7.09850e11i	0.718433	+	0.718433i	−0.249851	+	0.968284i	$0.580382\pi$
$998$		0			0
$999$			6.54611e10i			0.0657236i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.9.p.c.17.1		6
4.3	odd	2		5.9.c.a.2.2	✓	6
5.3	odd	4	inner	80.9.p.c.33.1		6
12.11	even	2		45.9.g.a.37.2		6
20.3	even	4		5.9.c.a.3.2	yes	6
20.7	even	4		25.9.c.b.18.2		6
20.19	odd	2		25.9.c.b.7.2		6
60.23	odd	4		45.9.g.a.28.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.9.c.a.2.2	✓	6	4.3	odd	2
5.9.c.a.3.2	yes	6	20.3	even	4
25.9.c.b.7.2		6	20.19	odd	2
25.9.c.b.18.2		6	20.7	even	4
45.9.g.a.28.2		6	60.23	odd	4
45.9.g.a.37.2		6	12.11	even	2
80.9.p.c.17.1		6	1.1	even	1	trivial
80.9.p.c.33.1		6	5.3	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	80.p (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-75.2981 + 75.2981i) q^{3} +(-14.1685 + 624.839i) q^{5} +(-730.992 - 730.992i) q^{7} -4778.60i q^{9} +O(q^{10})\)	\(q+(-75.2981 + 75.2981i) q^{3} +(-14.1685 + 624.839i) q^{5} +(-730.992 - 730.992i) q^{7} -4778.60i q^{9} -19599.8 q^{11} +(-24915.1 + 24915.1i) q^{13} +(-45982.3 - 48116.1i) q^{15} +(-11288.6 - 11288.6i) q^{17} +171525. i q^{19} +110085. q^{21} +(132074. - 132074. i) q^{23} +(-390224. - 17706.1i) q^{25} +(-134211. - 134211. i) q^{27} -127019. i q^{29} +960715. q^{31} +(1.47583e6 - 1.47583e6i) q^{33} +(467110. - 446396. i) q^{35} +(-243873. - 243873. i) q^{37} -3.75212e6i q^{39} +2.50747e6 q^{41} +(6763.95 - 6763.95i) q^{43} +(2.98586e6 + 67705.6i) q^{45} +(1.79394e6 + 1.79394e6i) q^{47} -4.69610e6i q^{49} +1.70003e6 q^{51} +(-2.97161e6 + 2.97161e6i) q^{53} +(277700. - 1.22467e7i) q^{55} +(-1.29155e7 - 1.29155e7i) q^{57} -313805. i q^{59} +1.76977e7 q^{61} +(-3.49312e6 + 3.49312e6i) q^{63} +(-1.52149e7 - 1.59209e7i) q^{65} +(-4.41349e6 - 4.41349e6i) q^{67} +1.98899e7i q^{69} +8.89315e6 q^{71} +(1.95076e7 - 1.95076e7i) q^{73} +(3.07163e7 - 2.80498e7i) q^{75} +(1.43273e7 + 1.43273e7i) q^{77} +1.11272e7i q^{79} +5.15641e7 q^{81} +(-1.58712e7 + 1.58712e7i) q^{83} +(7.21353e6 - 6.89365e6i) q^{85} +(9.56429e6 + 9.56429e6i) q^{87} +4.85032e7i q^{89} +3.64255e7 q^{91} +(-7.23400e7 + 7.23400e7i) q^{93} +(-1.07176e8 - 2.43025e6i) q^{95} +(-1.07411e8 - 1.07411e8i) q^{97} +9.36596e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 72 q^{3} + 220 q^{5} + 2352 q^{7}+O(q^{10}) \) 6 * q + 72 * q^3 + 220 * q^5 + 2352 * q^7	\( 6 q + 72 q^{3} + 220 q^{5} + 2352 q^{7} - 23192 q^{11} - 119142 q^{13} - 241440 q^{15} - 265502 q^{17} + 231672 q^{21} - 28888 q^{23} - 340350 q^{25} - 392040 q^{27} + 747648 q^{31} + 4269096 q^{33} + 4971680 q^{35} - 454002 q^{37} + 2489432 q^{41} - 792648 q^{43} + 210690 q^{45} + 15313352 q^{47} - 35567712 q^{51} - 13509122 q^{53} - 4448040 q^{55} - 34625520 q^{57} + 24111192 q^{61} - 44837688 q^{63} - 30943610 q^{65} + 32827752 q^{67} + 13992928 q^{71} + 111859638 q^{73} + 126793200 q^{75} + 26260136 q^{77} + 65834226 q^{81} + 14768432 q^{83} - 19713030 q^{85} + 133207680 q^{87} - 167542032 q^{91} - 96798024 q^{93} - 239661000 q^{95} - 186656202 q^{97}+O(q^{100}) \) 6 * q + 72 * q^3 + 220 * q^5 + 2352 * q^7 - 23192 * q^11 - 119142 * q^13 - 241440 * q^15 - 265502 * q^17 + 231672 * q^21 - 28888 * q^23 - 340350 * q^25 - 392040 * q^27 + 747648 * q^31 + 4269096 * q^33 + 4971680 * q^35 - 454002 * q^37 + 2489432 * q^41 - 792648 * q^43 + 210690 * q^45 + 15313352 * q^47 - 35567712 * q^51 - 13509122 * q^53 - 4448040 * q^55 - 34625520 * q^57 + 24111192 * q^61 - 44837688 * q^63 - 30943610 * q^65 + 32827752 * q^67 + 13992928 * q^71 + 111859638 * q^73 + 126793200 * q^75 + 26260136 * q^77 + 65834226 * q^81 + 14768432 * q^83 - 19713030 * q^85 + 133207680 * q^87 - 167542032 * q^91 - 96798024 * q^93 - 239661000 * q^95 - 186656202 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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