LMFDB - Embedded newform 80.9.p.c.33.3

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			33.3
Root			$3.70505 - 3.70505i$ of defining polynomial
Character	$\chi$	$=$	80.33
Dual form			80.9.p.c.17.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(90.9660 + 90.9660i) q^{3} +(-434.373 + 449.383i) q^{5} +(-508.219 + 508.219i) q^{7} +9988.61i q^{9} +O(q^{10})$	\(q+(90.9660 + 90.9660i) q^{3} +(-434.373 + 449.383i) q^{5} +(-508.219 + 508.219i) q^{7} +9988.61i q^{9} +7021.93 q^{11} +(-8073.75 - 8073.75i) q^{13} +(-80391.7 + 1365.33i) q^{15} +(-102958. + 102958. i) q^{17} +59599.0i q^{19} -92461.2 q^{21} +(-132866. - 132866. i) q^{23} +(-13264.5 - 390400. i) q^{25} +(-311796. + 311796. i) q^{27} -392047. i q^{29} +507883. q^{31} +(638757. + 638757. i) q^{33} +(-7628.00 - 449141. i) q^{35} +(-61012.9 + 61012.9i) q^{37} -1.46887e6i q^{39} -1.81765e6 q^{41} +(-1.47723e6 - 1.47723e6i) q^{43} +(-4.48871e6 - 4.33879e6i) q^{45} +(1.79287e6 - 1.79287e6i) q^{47} +5.24823e6i q^{49} -1.87314e7 q^{51} +(-5.66320e6 - 5.66320e6i) q^{53} +(-3.05014e6 + 3.15553e6i) q^{55} +(-5.42148e6 + 5.42148e6i) q^{57} +1.74855e7i q^{59} -1.96459e7 q^{61} +(-5.07640e6 - 5.07640e6i) q^{63} +(7.13522e6 - 121181. i) q^{65} +(1.12859e7 - 1.12859e7i) q^{67} -2.41725e7i q^{69} -3.01001e7 q^{71} +(2.52645e7 + 2.52645e7i) q^{73} +(3.43065e7 - 3.67197e7i) q^{75} +(-3.56868e6 + 3.56868e6i) q^{77} +8.14025e6i q^{79} +8.80962e6 q^{81} +(1.99635e7 + 1.99635e7i) q^{83} +(-1.54533e6 - 9.09902e7i) q^{85} +(3.56630e7 - 3.56630e7i) q^{87} +8.20905e7i q^{89} +8.20646e6 q^{91} +(4.62000e7 + 4.62000e7i) q^{93} +(-2.67827e7 - 2.58882e7i) q^{95} +(3.37379e7 - 3.37379e7i) q^{97} +7.01393e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	90.9660	+	90.9660i	1.12304	+	1.12304i	0.991282	+	0.131754i	$0.0420609\pi$
$3$	90.9660	+	90.9660i	1.12304	+	1.12304i	0.131754	+	0.991282i	$0.457939\pi$
$4$		0			0
$5$	−434.373	+	449.383i	−0.694997	+	0.719012i
$5$	−434.373	+	449.383i	−0.694997	+	0.719012i
$6$		0			0
$7$	−508.219	+	508.219i	−0.211670	+	0.211670i	−0.804976	−	0.593307i	$-0.797822\pi$
$7$	−508.219	+	508.219i	−0.211670	+	0.211670i	0.593307	+	0.804976i	$0.297822\pi$
$8$		0			0
$9$			9988.61i			1.52242i
$10$		0			0
$11$	7021.93			0.479607			0.239804	−	0.970821i	$-0.422917\pi$
$11$	7021.93			0.479607			0.239804	+	0.970821i	$0.422917\pi$
$12$		0			0
$13$	−8073.75	−	8073.75i	−0.282684	−	0.282684i	0.551494	−	0.834179i	$-0.314058\pi$
$13$	−8073.75	−	8073.75i	−0.282684	−	0.282684i	−0.834179	+	0.551494i	$0.814058\pi$
$14$		0			0
$15$	−80391.7	+	1365.33i	−1.58798	+	0.0269695i
$16$		0			0
$17$	−102958.	+	102958.i	−1.23273	+	1.23273i	−0.269813	+	0.962913i	$0.586962\pi$
$17$	−102958.	+	102958.i	−1.23273	+	1.23273i	−0.962913	+	0.269813i	$0.913038\pi$
$18$		0			0
$19$			59599.0i			0.457324i	0.973506	+	0.228662i	$0.0734351\pi$
$19$			59599.0i			0.457324i	−0.973506	+	0.228662i	$0.926565\pi$
$20$		0			0
$21$	−92461.2			−0.475425
$22$		0			0
$23$	−132866.	−	132866.i	−0.474790	−	0.474790i	0.428671	−	0.903461i	$-0.358982\pi$
$23$	−132866.	−	132866.i	−0.474790	−	0.474790i	−0.903461	+	0.428671i	$0.858982\pi$
$24$		0			0
$25$	−13264.5	−	390400.i	−0.0339572	−	0.999423i
$26$		0			0
$27$	−311796.	+	311796.i	−0.586699	+	0.586699i
$28$		0			0
$29$		−	392047.i		−	0.554302i	−0.960826	−	0.277151i	$-0.910610\pi$
$29$		−	392047.i		−	0.554302i	0.960826	−	0.277151i	$-0.0893902\pi$
$30$		0			0
$31$	507883.			0.549942			0.274971	−	0.961453i	$-0.411332\pi$
$31$	507883.			0.549942			0.274971	+	0.961453i	$0.411332\pi$
$32$		0			0
$33$	638757.	+	638757.i	0.538617	+	0.538617i
$34$		0			0
$35$	−7628.00	−	449141.i	−0.00508321	−	0.299303i
$36$		0			0
$37$	−61012.9	+	61012.9i	−0.0325548	+	0.0325548i	−0.723197	−	0.690642i	$-0.757328\pi$
$37$	−61012.9	+	61012.9i	−0.0325548	+	0.0325548i	0.690642	+	0.723197i	$0.257328\pi$
$38$		0			0
$39$		−	1.46887e6i		−	0.634930i
$40$		0			0
$41$	−1.81765e6			−0.643242			−0.321621	−	0.946868i	$-0.604228\pi$
$41$	−1.81765e6			−0.643242			−0.321621	+	0.946868i	$0.604228\pi$
$42$		0			0
$43$	−1.47723e6	−	1.47723e6i	−0.432090	−	0.432090i	0.457249	−	0.889339i	$-0.348835\pi$
$43$	−1.47723e6	−	1.47723e6i	−0.432090	−	0.432090i	−0.889339	+	0.457249i	$0.848835\pi$
$44$		0			0
$45$	−4.48871e6	−	4.33879e6i	−1.09464	−	1.05808i
$46$		0			0
$47$	1.79287e6	−	1.79287e6i	0.367416	−	0.367416i	−0.499118	−	0.866534i	$-0.666343\pi$
$47$	1.79287e6	−	1.79287e6i	0.367416	−	0.367416i	0.866534	+	0.499118i	$0.166343\pi$
$48$		0			0
$49$			5.24823e6i			0.910392i
$50$		0			0
$51$	−1.87314e7			−2.76879
$52$		0			0
$53$	−5.66320e6	−	5.66320e6i	−0.717725	−	0.717725i	0.250414	−	0.968139i	$-0.419433\pi$
$53$	−5.66320e6	−	5.66320e6i	−0.717725	−	0.717725i	−0.968139	+	0.250414i	$0.919433\pi$
$54$		0			0
$55$	−3.05014e6	+	3.15553e6i	−0.333326	+	0.344844i
$56$		0			0
$57$	−5.42148e6	+	5.42148e6i	−0.513592	+	0.513592i
$58$		0			0
$59$			1.74855e7i			1.44301i	0.692410	+	0.721504i	$0.256549\pi$
$59$			1.74855e7i			1.44301i	−0.692410	+	0.721504i	$0.743451\pi$
$60$		0			0
$61$	−1.96459e7			−1.41890			−0.709452	−	0.704754i	$-0.751057\pi$
$61$	−1.96459e7			−1.41890			−0.709452	+	0.704754i	$0.751057\pi$
$62$		0			0
$63$	−5.07640e6	−	5.07640e6i	−0.322251	−	0.322251i
$64$		0			0
$65$	7.13522e6	−	121181.i	0.399718	−	0.00678862i
$66$		0			0
$67$	1.12859e7	−	1.12859e7i	0.560061	−	0.560061i	−0.369264	−	0.929325i	$-0.620390\pi$
$67$	1.12859e7	−	1.12859e7i	0.560061	−	0.560061i	0.929325	+	0.369264i	$0.120390\pi$
$68$		0			0
$69$		−	2.41725e7i		−	1.06641i
$70$		0			0
$71$	−3.01001e7			−1.18450			−0.592249	−	0.805755i	$-0.701760\pi$
$71$	−3.01001e7			−1.18450			−0.592249	+	0.805755i	$0.701760\pi$
$72$		0			0
$73$	2.52645e7	+	2.52645e7i	0.889649	+	0.889649i	0.994489	−	0.104840i	$-0.0334331\pi$
$73$	2.52645e7	+	2.52645e7i	0.889649	+	0.889649i	−0.104840	+	0.994489i	$0.533433\pi$
$74$		0			0
$75$	3.43065e7	−	3.67197e7i	1.08425	−	1.16052i
$76$		0			0
$77$	−3.56868e6	+	3.56868e6i	−0.101518	+	0.101518i
$78$		0			0
$79$			8.14025e6i			0.208992i	0.994525	+	0.104496i	$0.0333229\pi$
$79$			8.14025e6i			0.208992i	−0.994525	+	0.104496i	$0.966677\pi$
$80$		0			0
$81$	8.80962e6			0.204653
$82$		0			0
$83$	1.99635e7	+	1.99635e7i	0.420652	+	0.420652i	0.885428	−	0.464776i	$-0.153865\pi$
$83$	1.99635e7	+	1.99635e7i	0.420652	+	0.420652i	−0.464776	+	0.885428i	$0.653865\pi$
$84$		0			0
$85$	−1.54533e6	−	9.09902e7i	−0.0296037	−	1.74309i
$86$		0			0
$87$	3.56630e7	−	3.56630e7i	0.622501	−	0.622501i
$88$		0			0
$89$			8.20905e7i			1.30838i	0.756332	+	0.654189i	$0.226990\pi$
$89$			8.20905e7i			1.30838i	−0.756332	+	0.654189i	$0.773010\pi$
$90$		0			0
$91$	8.20646e6			0.119671
$92$		0			0
$93$	4.62000e7	+	4.62000e7i	0.617605	+	0.617605i
$94$		0			0
$95$	−2.67827e7	−	2.58882e7i	−0.328822	−	0.317839i
$96$		0			0
$97$	3.37379e7	−	3.37379e7i	0.381093	−	0.381093i	−0.490403	−	0.871496i	$-0.663150\pi$
$97$	3.37379e7	−	3.37379e7i	0.381093	−	0.381093i	0.871496	+	0.490403i	$0.163150\pi$
$98$		0			0
$99$			7.01393e7i			0.730165i
$100$		0			0
$101$	−1.91592e7			−0.184116			−0.0920582	−	0.995754i	$-0.529345\pi$
$101$	−1.91592e7			−0.184116			−0.0920582	+	0.995754i	$0.529345\pi$
$102$		0			0
$103$	1.35177e8	+	1.35177e8i	1.20103	+	1.20103i	0.973854	+	0.227173i	$0.0729483\pi$
$103$	1.35177e8	+	1.35177e8i	1.20103	+	1.20103i	0.227173	+	0.973854i	$0.427052\pi$
$104$		0			0
$105$	4.01627e7	−	4.15505e7i	0.330419	−	0.341837i
$106$		0			0
$107$	−1.22825e8	+	1.22825e8i	−0.937029	+	0.937029i	−0.998132	−	0.0611024i	$-0.980538\pi$
$107$	−1.22825e8	+	1.22825e8i	−0.937029	+	0.937029i	0.0611024	+	0.998132i	$0.480538\pi$
$108$		0			0
$109$			1.66780e8i			1.18151i	0.806850	+	0.590757i	$0.201171\pi$
$109$			1.66780e8i			1.18151i	−0.806850	+	0.590757i	$0.798829\pi$
$110$		0			0
$111$	−1.11002e7			−0.0731204
$112$		0			0
$113$	3.23356e7	+	3.23356e7i	0.198320	+	0.198320i	0.799280	−	0.600959i	$-0.205215\pi$
$113$	3.23356e7	+	3.23356e7i	0.198320	+	0.198320i	−0.600959	+	0.799280i	$0.705215\pi$
$114$		0			0
$115$	1.17421e8	−	1.99422e6i	0.671357	−	0.0114020i
$116$		0			0
$117$	8.06456e7	−	8.06456e7i	0.430365	−	0.430365i
$118$		0			0
$119$		−	1.04651e8i		−	0.521861i
$120$		0			0
$121$	−1.65051e8			−0.769977
$122$		0			0
$123$	−1.65344e8	−	1.65344e8i	−0.722385	−	0.722385i
$124$		0			0
$125$	1.81201e8	+	1.63618e8i	0.742198	+	0.670181i
$126$		0			0
$127$	2.93412e8	−	2.93412e8i	1.12788	−	1.12788i	0.137357	−	0.990522i	$-0.456139\pi$
$127$	2.93412e8	−	2.93412e8i	1.12788	−	1.12788i	0.990522	−	0.137357i	$-0.0438608\pi$
$128$		0			0
$129$		−	2.68756e8i		−	0.970507i
$130$		0			0
$131$	3.37671e8			1.14659			0.573295	−	0.819349i	$-0.305665\pi$
$131$	3.37671e8			1.14659			0.573295	+	0.819349i	$0.305665\pi$
$132$		0			0
$133$	−3.02893e7	−	3.02893e7i	−0.0968017	−	0.0968017i
$134$		0			0
$135$	−4.67983e6	−	2.75552e8i	−0.0140895	−	0.829599i
$136$		0			0
$137$	1.06123e8	−	1.06123e8i	0.301249	−	0.301249i	−0.540253	−	0.841502i	$-0.681672\pi$
$137$	1.06123e8	−	1.06123e8i	0.301249	−	0.301249i	0.841502	+	0.540253i	$0.181672\pi$
$138$		0			0
$139$			5.86652e8i			1.57153i	0.618528	+	0.785763i	$0.287729\pi$
$139$			5.86652e8i			1.57153i	−0.618528	+	0.785763i	$0.712271\pi$
$140$		0			0
$141$	3.26181e8			0.825243
$142$		0			0
$143$	−5.66933e7	−	5.66933e7i	−0.135577	−	0.135577i
$144$		0			0
$145$	1.76179e8	+	1.70295e8i	0.398550	+	0.385238i
$146$		0			0
$147$	−4.77410e8	+	4.77410e8i	−1.02240	+	1.02240i
$148$		0			0
$149$			5.45945e8i			1.10765i	0.832632	+	0.553827i	$0.186833\pi$
$149$			5.45945e8i			1.10765i	−0.832632	+	0.553827i	$0.813167\pi$
$150$		0			0
$151$	−8.88293e7			−0.170863			−0.0854316	−	0.996344i	$-0.527227\pi$
$151$	−8.88293e7			−0.170863			−0.0854316	+	0.996344i	$0.527227\pi$
$152$		0			0
$153$	−1.02841e9	−	1.02841e9i	−1.87673	−	1.87673i
$154$		0			0
$155$	−2.20611e8	+	2.28234e8i	−0.382208	+	0.395415i
$156$		0			0
$157$	−3.21072e8	+	3.21072e8i	−0.528450	+	0.528450i	−0.920110	−	0.391660i	$-0.871901\pi$
$157$	−3.21072e8	+	3.21072e8i	−0.528450	+	0.528450i	0.391660	+	0.920110i	$0.371901\pi$
$158$		0			0
$159$		−	1.03032e9i		−	1.61206i
$160$		0			0
$161$	1.35050e8			0.200997
$162$		0			0
$163$	−2.14053e8	−	2.14053e8i	−0.303229	−	0.303229i	0.539047	−	0.842276i	$-0.318785\pi$
$163$	−2.14053e8	−	2.14053e8i	−0.303229	−	0.303229i	−0.842276	+	0.539047i	$0.818785\pi$
$164$		0			0
$165$	−5.64505e8	+	9.58728e6i	−0.761609	+	0.0129348i
$166$		0			0
$167$	−6.91005e8	+	6.91005e8i	−0.888414	+	0.888414i	−0.994371	−	0.105957i	$-0.966209\pi$
$167$	−6.91005e8	+	6.91005e8i	−0.888414	+	0.888414i	0.105957	+	0.994371i	$0.466209\pi$
$168$		0			0
$169$		−	6.85360e8i		−	0.840179i
$170$		0			0
$171$	−5.95311e8			−0.696241
$172$		0			0
$173$	−3.40367e8	−	3.40367e8i	−0.379982	−	0.379982i	0.491114	−	0.871095i	$-0.336590\pi$
$173$	−3.40367e8	−	3.40367e8i	−0.379982	−	0.379982i	−0.871095	+	0.491114i	$0.836590\pi$
$174$		0			0
$175$	2.05150e8	+	1.91667e8i	0.218735	+	0.204360i
$176$		0			0
$177$	−1.59058e9	+	1.59058e9i	−1.62055	+	1.62055i
$178$		0			0
$179$			7.51587e7i			0.0732094i	0.999330	+	0.0366047i	$0.0116542\pi$
$179$			7.51587e7i			0.0732094i	−0.999330	+	0.0366047i	$0.988346\pi$
$180$		0			0
$181$	1.46153e8			0.136174			0.0680870	−	0.997679i	$-0.478310\pi$
$181$	1.46153e8			0.136174			0.0680870	+	0.997679i	$0.478310\pi$
$182$		0			0
$183$	−1.78711e9	−	1.78711e9i	−1.59348	−	1.59348i
$184$		0			0
$185$	−915760.	−	5.39205e7i	−0.000781798	−	0.0460328i
$186$		0			0
$187$	−7.22967e8	+	7.22967e8i	−0.591224	+	0.591224i
$188$		0			0
$189$		−	3.16921e8i		−	0.248373i
$190$		0			0
$191$	2.54450e9			1.91191			0.955957	−	0.293506i	$-0.0948222\pi$
$191$	2.54450e9			1.91191			0.955957	+	0.293506i	$0.0948222\pi$
$192$		0			0
$193$	9.46964e8	+	9.46964e8i	0.682503	+	0.682503i	0.960563	−	0.278061i	$-0.0896917\pi$
$193$	9.46964e8	+	9.46964e8i	0.682503	+	0.682503i	−0.278061	+	0.960563i	$0.589692\pi$
$194$		0			0
$195$	6.60086e8	+	6.38039e8i	0.456522	+	0.441275i
$196$		0			0
$197$	6.42356e8	−	6.42356e8i	0.426492	−	0.426492i	−0.460940	−	0.887432i	$-0.652487\pi$
$197$	6.42356e8	−	6.42356e8i	0.426492	−	0.426492i	0.887432	+	0.460940i	$0.152487\pi$
$198$		0			0
$199$		−	3.47565e8i		−	0.221628i	−0.993841	−	0.110814i	$-0.964654\pi$
$199$		−	3.47565e8i		−	0.221628i	0.993841	−	0.110814i	$-0.0353457\pi$
$200$		0			0
$201$	2.05326e9			1.25794
$202$		0			0
$203$	1.99246e8	+	1.99246e8i	0.117329	+	0.117329i
$204$		0			0
$205$	7.89538e8	−	8.16820e8i	0.447052	−	0.462499i
$206$		0			0
$207$	1.32714e9	−	1.32714e9i	0.722831	−	0.722831i
$208$		0			0
$209$			4.18500e8i			0.219336i
$210$		0			0
$211$	1.32439e9			0.668168			0.334084	−	0.942543i	$-0.391573\pi$
$211$	1.32439e9			0.668168			0.334084	+	0.942543i	$0.391573\pi$
$212$		0			0
$213$	−2.73808e9	−	2.73808e9i	−1.33023	−	1.33023i
$214$		0			0
$215$	1.30551e9	−	2.21722e7i	0.610980	−	0.0103766i
$216$		0			0
$217$	−2.58115e8	+	2.58115e8i	−0.116406	+	0.116406i
$218$		0			0
$219$			4.59641e9i			1.99822i
$220$		0			0
$221$	1.66252e9			0.696945
$222$		0			0
$223$	−2.96426e9	−	2.96426e9i	−1.19866	−	1.19866i	−0.974567	−	0.224096i	$-0.928057\pi$
$223$	−2.96426e9	−	2.96426e9i	−1.19866	−	1.19866i	−0.224096	−	0.974567i	$-0.571943\pi$
$224$		0			0
$225$	3.89955e9	−	1.32494e8i	1.52154	−	0.0516972i
$226$		0			0
$227$	8.37535e8	−	8.37535e8i	0.315428	−	0.315428i	−0.531580	−	0.847008i	$-0.678402\pi$
$227$	8.37535e8	−	8.37535e8i	0.315428	−	0.315428i	0.847008	+	0.531580i	$0.178402\pi$
$228$		0			0
$229$			2.66676e9i			0.969709i	0.874595	+	0.484854i	$0.161127\pi$
$229$			2.66676e9i			0.969709i	−0.874595	+	0.484854i	$0.838873\pi$
$230$		0			0
$231$	−6.49256e8			−0.228017
$232$		0			0
$233$	9.91964e8	+	9.91964e8i	0.336568	+	0.336568i	0.855074	−	0.518506i	$-0.173512\pi$
$233$	9.91964e8	+	9.91964e8i	0.336568	+	0.336568i	−0.518506	+	0.855074i	$0.673512\pi$
$234$		0			0
$235$	2.69097e7	+	1.58446e9i	0.00882343	+	0.519529i
$236$		0			0
$237$	−7.40485e8	+	7.40485e8i	−0.234705	+	0.234705i
$238$		0			0
$239$			1.26088e9i			0.386440i	0.981155	+	0.193220i	$0.0618931\pi$
$239$			1.26088e9i			0.386440i	−0.981155	+	0.193220i	$0.938107\pi$
$240$		0			0
$241$	−1.40173e9			−0.415524			−0.207762	−	0.978179i	$-0.566618\pi$
$241$	−1.40173e9			−0.415524			−0.207762	+	0.978179i	$0.566618\pi$
$242$		0			0
$243$	2.84707e9	+	2.84707e9i	0.816532	+	0.816532i
$244$		0			0
$245$	−2.35846e9	−	2.27969e9i	−0.654583	−	0.632720i
$246$		0			0
$247$	4.81187e8	−	4.81187e8i	0.129278	−	0.129278i
$248$		0			0
$249$			3.63199e9i			0.944816i
$250$		0			0
$251$	−5.24024e9			−1.32025			−0.660126	−	0.751155i	$-0.729497\pi$
$251$	−5.24024e9			−1.32025			−0.660126	+	0.751155i	$0.729497\pi$
$252$		0			0
$253$	−9.32973e8	−	9.32973e8i	−0.227713	−	0.227713i
$254$		0			0
$255$	8.13644e9	−	8.41758e9i	1.92430	−	1.99080i
$256$		0			0
$257$	−2.38954e9	+	2.38954e9i	−0.547749	+	0.547749i	−0.925789	−	0.378040i	$-0.876598\pi$
$257$	−2.38954e9	+	2.38954e9i	−0.547749	+	0.547749i	0.378040	+	0.925789i	$0.376598\pi$
$258$		0			0
$259$		−	6.20158e7i		−	0.0137817i
$260$		0			0
$261$	3.91601e9			0.843882
$262$		0			0
$263$	1.29445e9	+	1.29445e9i	0.270559	+	0.270559i	0.829325	−	0.558766i	$-0.188725\pi$
$263$	1.29445e9	+	1.29445e9i	0.270559	+	0.270559i	−0.558766	+	0.829325i	$0.688725\pi$
$264$		0			0
$265$	5.00488e9	−	8.50005e7i	1.01487	−	0.0172361i
$266$		0			0
$267$	−7.46744e9	+	7.46744e9i	−1.46936	+	1.46936i
$268$		0			0
$269$			6.14346e9i			1.17329i	0.809845	+	0.586643i	$0.199551\pi$
$269$			6.14346e9i			1.17329i	−0.809845	+	0.586643i	$0.800449\pi$
$270$		0			0
$271$	5.21208e9			0.966349			0.483174	−	0.875524i	$-0.339484\pi$
$271$	5.21208e9			0.966349			0.483174	+	0.875524i	$0.339484\pi$
$272$		0			0
$273$	7.46509e8	+	7.46509e8i	0.134395	+	0.134395i
$274$		0			0
$275$	−9.31427e7	−	2.74136e9i	−0.0162861	−	0.479331i
$276$		0			0
$277$	5.42598e9	−	5.42598e9i	0.921635	−	0.921635i	−0.0755097	−	0.997145i	$-0.524058\pi$
$277$	5.42598e9	−	5.42598e9i	0.921635	−	0.921635i	0.997145	+	0.0755097i	$0.0240584\pi$
$278$		0			0
$279$			5.07304e9i			0.837243i
$280$		0			0
$281$	−7.13397e9			−1.14421			−0.572106	−	0.820180i	$-0.693873\pi$
$281$	−7.13397e9			−1.14421			−0.572106	+	0.820180i	$0.693873\pi$
$282$		0			0
$283$	−2.63941e9	−	2.63941e9i	−0.411492	−	0.411492i	0.470766	−	0.882258i	$-0.343978\pi$
$283$	−2.63941e9	−	2.63941e9i	−0.411492	−	0.411492i	−0.882258	+	0.470766i	$0.843978\pi$
$284$		0			0
$285$	−8.13725e7	−	4.79126e9i	−0.0123338	−	0.726224i
$286$		0			0
$287$	9.23763e8	−	9.23763e8i	0.136155	−	0.136155i
$288$		0			0
$289$		−	1.42251e10i		−	2.03923i
$290$		0			0
$291$	6.13800e9			0.855963
$292$		0			0
$293$	−2.26873e9	−	2.26873e9i	−0.307831	−	0.307831i	0.536237	−	0.844068i	$-0.319845\pi$
$293$	−2.26873e9	−	2.26873e9i	−0.307831	−	0.307831i	−0.844068	+	0.536237i	$0.819845\pi$
$294$		0			0
$295$	−7.85766e9	−	7.59522e9i	−1.03754	−	1.00289i
$296$		0			0
$297$	−2.18941e9	+	2.18941e9i	−0.281385	+	0.281385i
$298$		0			0
$299$			2.14545e9i			0.268431i
$300$		0			0
$301$	1.50151e9			0.182921
$302$		0			0
$303$	−1.74284e9	−	1.74284e9i	−0.206769	−	0.206769i
$304$		0			0
$305$	8.53366e9	−	8.82853e9i	0.986134	−	1.02021i
$306$		0			0
$307$	5.16937e9	−	5.16937e9i	0.581948	−	0.581948i	−0.353490	−	0.935438i	$-0.615005\pi$
$307$	5.16937e9	−	5.16937e9i	0.581948	−	0.581948i	0.935438	+	0.353490i	$0.115005\pi$
$308$		0			0
$309$			2.45930e10i			2.69760i
$310$		0			0
$311$	−1.32155e10			−1.41268			−0.706340	−	0.707873i	$-0.749655\pi$
$311$	−1.32155e10			−1.41268			−0.706340	+	0.707873i	$0.749655\pi$
$312$		0			0
$313$	−9.02814e9	−	9.02814e9i	−0.940635	−	0.940635i	0.0576992	−	0.998334i	$-0.481624\pi$
$313$	−9.02814e9	−	9.02814e9i	−0.940635	−	0.940635i	−0.998334	+	0.0576992i	$0.981624\pi$
$314$		0			0
$315$	4.48630e9	−	7.61931e7i	0.455665	−	0.00773880i
$316$		0			0
$317$	−4.91807e9	+	4.91807e9i	−0.487032	+	0.487032i	−0.907369	−	0.420336i	$-0.861912\pi$
$317$	−4.91807e9	+	4.91807e9i	−0.487032	+	0.487032i	0.420336	+	0.907369i	$0.361912\pi$
$318$		0			0
$319$		−	2.75293e9i		−	0.265847i
$320$		0			0
$321$	−2.23459e10			−2.10464
$322$		0			0
$323$	−6.13622e9	−	6.13622e9i	−0.563756	−	0.563756i
$324$		0			0
$325$	−3.04489e9	+	3.25908e9i	−0.272922	+	0.292121i
$326$		0			0
$327$	−1.51713e10	+	1.51713e10i	−1.32688	+	1.32688i
$328$		0			0
$329$			1.82234e9i			0.155541i
$330$		0			0
$331$	−1.72045e9			−0.143327			−0.0716637	−	0.997429i	$-0.522831\pi$
$331$	−1.72045e9			−0.143327			−0.0716637	+	0.997429i	$0.522831\pi$
$332$		0			0
$333$	−6.09434e8	−	6.09434e8i	−0.0495621	−	0.0495621i
$334$		0			0
$335$	1.69393e8	+	9.97394e9i	0.0134498	+	0.791932i
$336$		0			0
$337$	−5.92808e9	+	5.92808e9i	−0.459615	+	0.459615i	−0.898529	−	0.438914i	$-0.855363\pi$
$337$	−5.92808e9	+	5.92808e9i	−0.459615	+	0.459615i	0.438914	+	0.898529i	$0.355363\pi$
$338$		0			0
$339$			5.88287e9i			0.445442i
$340$		0			0
$341$	3.56632e9			0.263756
$342$		0			0
$343$	−5.59703e9	−	5.59703e9i	−0.404372	−	0.404372i
$344$		0			0
$345$	1.08627e10	+	1.04999e10i	0.766764	+	0.741154i
$346$		0			0
$347$	−4.12838e9	+	4.12838e9i	−0.284749	+	0.284749i	−0.834999	−	0.550251i	$-0.814532\pi$
$347$	−4.12838e9	+	4.12838e9i	−0.284749	+	0.284749i	0.550251	+	0.834999i	$0.314532\pi$
$348$		0			0
$349$		−	3.79762e9i		−	0.255983i	−0.991775	−	0.127991i	$-0.959147\pi$
$349$		−	3.79762e9i		−	0.255983i	0.991775	−	0.127991i	$-0.0408530\pi$
$350$		0			0
$351$	5.03473e9			0.331702
$352$		0			0
$353$	−4.46363e9	−	4.46363e9i	−0.287468	−	0.287468i	0.548610	−	0.836078i	$-0.315157\pi$
$353$	−4.46363e9	−	4.46363e9i	−0.287468	−	0.287468i	−0.836078	+	0.548610i	$0.815157\pi$
$354$		0			0
$355$	1.30747e10	−	1.35265e10i	0.823223	−	0.851668i
$356$		0			0
$357$	9.51967e9	−	9.51967e9i	0.586069	−	0.586069i
$358$		0			0
$359$		−	9.04307e8i		−	0.0544425i	−0.999629	−	0.0272213i	$-0.991334\pi$
$359$		−	9.04307e8i		−	0.0544425i	0.999629	−	0.0272213i	$-0.00866586\pi$
$360$		0			0
$361$	1.34315e10			0.790854
$362$		0			0
$363$	−1.50141e10	−	1.50141e10i	−0.864712	−	0.864712i
$364$		0			0
$365$	−2.23276e10	+	3.79201e8i	−1.25797	+	0.0213648i
$366$		0			0
$367$	9.88793e9	−	9.88793e9i	0.545056	−	0.545056i	−0.379951	−	0.925007i	$-0.624059\pi$
$367$	9.88793e9	−	9.88793e9i	0.545056	−	0.545056i	0.925007	+	0.379951i	$0.124059\pi$
$368$		0			0
$369$		−	1.81558e10i		−	0.979287i
$370$		0			0
$371$	5.75629e9			0.303841
$372$		0			0
$373$	−1.56902e10	−	1.56902e10i	−0.810575	−	0.810575i	0.174145	−	0.984720i	$-0.444284\pi$
$373$	−1.56902e10	−	1.56902e10i	−0.810575	−	0.810575i	−0.984720	+	0.174145i	$0.944284\pi$
$374$		0			0
$375$	1.59938e9	+	3.13668e10i	0.0808775	+	1.58615i
$376$		0			0
$377$	−3.16529e9	+	3.16529e9i	−0.156692	+	0.156692i
$378$		0			0
$379$		−	4.03797e8i		−	0.0195707i	−0.999952	−	0.00978534i	$-0.996885\pi$
$379$		−	4.03797e8i		−	0.0195707i	0.999952	−	0.00978534i	$-0.00311482\pi$
$380$		0			0
$381$	5.33809e10			2.53330
$382$		0			0
$383$	1.94253e10	+	1.94253e10i	0.902761	+	0.902761i	0.995674	−	0.0929130i	$-0.0296178\pi$
$383$	1.94253e10	+	1.94253e10i	0.902761	+	0.902761i	−0.0929130	+	0.995674i	$0.529618\pi$
$384$		0			0
$385$	−5.35633e7	−	3.15384e9i	−0.00243795	−	0.143548i
$386$		0			0
$387$	1.47555e10	−	1.47555e10i	0.657824	−	0.657824i
$388$		0			0
$389$			1.00547e10i			0.439109i	0.975600	+	0.219555i	$0.0704604\pi$
$389$			1.00547e10i			0.439109i	−0.975600	+	0.219555i	$0.929540\pi$
$390$		0			0
$391$	2.73593e10			1.17057
$392$		0			0
$393$	3.07165e10	+	3.07165e10i	1.28766	+	1.28766i
$394$		0			0
$395$	−3.65809e9	−	3.53591e9i	−0.150268	−	0.145249i
$396$		0			0
$397$	4.59921e9	−	4.59921e9i	0.185149	−	0.185149i	−0.608446	−	0.793595i	$-0.708207\pi$
$397$	4.59921e9	−	4.59921e9i	0.185149	−	0.185149i	0.793595	+	0.608446i	$0.208207\pi$
$398$		0			0
$399$		−	5.51059e9i		−	0.217424i
$400$		0			0
$401$	3.93282e10			1.52099			0.760496	−	0.649343i	$-0.224956\pi$
$401$	3.93282e10			1.52099			0.760496	+	0.649343i	$0.224956\pi$
$402$		0			0
$403$	−4.10052e9	−	4.10052e9i	−0.155460	−	0.155460i
$404$		0			0
$405$	−3.82667e9	+	3.95889e9i	−0.142233	+	0.147148i
$406$		0			0
$407$	−4.28428e8	+	4.28428e8i	−0.0156135	+	0.0156135i
$408$		0			0
$409$		−	3.53396e10i		−	1.26290i	−0.775417	−	0.631450i	$-0.782460\pi$
$409$		−	3.53396e10i		−	1.26290i	0.775417	−	0.631450i	$-0.217540\pi$
$410$		0			0
$411$	1.93071e10			0.676628
$412$		0			0
$413$	−8.88644e9	−	8.88644e9i	−0.305441	−	0.305441i
$414$		0			0
$415$	−1.76428e10	+	2.99637e8i	−0.594806	+	0.0101019i
$416$		0			0
$417$	−5.33654e10	+	5.33654e10i	−1.76488	+	1.76488i
$418$		0			0
$419$		−	1.65282e10i		−	0.536251i	−0.963384	−	0.268126i	$-0.913596\pi$
$419$		−	1.65282e10i		−	0.536251i	0.963384	−	0.268126i	$-0.0864042\pi$
$420$		0			0
$421$	−2.05277e10			−0.653449			−0.326725	−	0.945120i	$-0.605945\pi$
$421$	−2.05277e10			−0.653449			−0.326725	+	0.945120i	$0.605945\pi$
$422$		0			0
$423$	1.79083e10	+	1.79083e10i	0.559362	+	0.559362i
$424$		0			0
$425$	4.15607e10	+	3.88293e10i	1.27387	+	1.19015i
$426$		0			0
$427$	9.98442e9	−	9.98442e9i	0.300339	−	0.300339i
$428$		0			0
$429$		−	1.03143e10i		−	0.304517i
$430$		0			0
$431$	−2.06955e9			−0.0599745			−0.0299873	−	0.999550i	$-0.509547\pi$
$431$	−2.06955e9			−0.0599745			−0.0299873	+	0.999550i	$0.509547\pi$
$432$		0			0
$433$	2.90398e10	+	2.90398e10i	0.826118	+	0.826118i	0.986977	−	0.160860i	$-0.0514266\pi$
$433$	2.90398e10	+	2.90398e10i	0.826118	+	0.826118i	−0.160860	+	0.986977i	$0.551427\pi$
$434$		0			0
$435$	5.35275e8	+	3.15173e10i	0.0149493	+	0.880223i
$436$		0			0
$437$	7.91866e9	−	7.91866e9i	0.217133	−	0.217133i
$438$		0			0
$439$			1.56289e10i			0.420795i	0.977616	+	0.210398i	$0.0674758\pi$
$439$			1.56289e10i			0.420795i	−0.977616	+	0.210398i	$0.932524\pi$
$440$		0			0
$441$	−5.24225e10			−1.38600
$442$		0			0
$443$	−7.72628e9	−	7.72628e9i	−0.200611	−	0.200611i	0.599651	−	0.800262i	$-0.295306\pi$
$443$	−7.72628e9	−	7.72628e9i	−0.200611	−	0.200611i	−0.800262	+	0.599651i	$0.795306\pi$
$444$		0			0
$445$	−3.68901e10	−	3.56579e10i	−0.940739	−	0.909319i
$446$		0			0
$447$	−4.96625e10	+	4.96625e10i	−1.24394	+	1.24394i
$448$		0			0
$449$			1.21836e10i			0.299771i	0.988703	+	0.149886i	$0.0478905\pi$
$449$			1.21836e10i			0.299771i	−0.988703	+	0.149886i	$0.952109\pi$
$450$		0			0
$451$	−1.27634e10			−0.308504
$452$		0			0
$453$	−8.08045e9	−	8.08045e9i	−0.191886	−	0.191886i
$454$		0			0
$455$	−3.56467e9	+	3.68784e9i	−0.0831713	+	0.0860452i
$456$		0			0
$457$	4.53311e10	−	4.53311e10i	1.03928	−	1.03928i	0.0400801	−	0.999196i	$-0.487239\pi$
$457$	4.53311e10	−	4.53311e10i	1.03928	−	1.03928i	0.999196	−	0.0400801i	$-0.0127613\pi$
$458$		0			0
$459$		−	6.42041e10i		−	1.44648i
$460$		0			0
$461$	−3.53353e9			−0.0782357			−0.0391178	−	0.999235i	$-0.512455\pi$
$461$	−3.53353e9			−0.0782357			−0.0391178	+	0.999235i	$0.512455\pi$
$462$		0			0
$463$	−1.72557e10	−	1.72557e10i	−0.375499	−	0.375499i	0.493976	−	0.869475i	$-0.335543\pi$
$463$	−1.72557e10	−	1.72557e10i	−0.375499	−	0.375499i	−0.869475	+	0.493976i	$0.835543\pi$
$464$		0			0
$465$	−4.08296e10	+	6.93429e8i	−0.873299	+	0.0148317i
$466$		0			0
$467$	2.45996e10	−	2.45996e10i	0.517203	−	0.517203i	−0.399521	−	0.916724i	$-0.630823\pi$
$467$	2.45996e10	−	2.45996e10i	0.517203	−	0.517203i	0.916724	+	0.399521i	$0.130823\pi$
$468$		0			0
$469$			1.14714e10i			0.237096i
$470$		0			0
$471$	−5.84133e10			−1.18694
$472$		0			0
$473$	−1.03730e10	−	1.03730e10i	−0.207234	−	0.207234i
$474$		0			0
$475$	2.32674e10	−	7.90553e8i	0.457061	−	0.0155295i
$476$		0			0
$477$	5.65675e10	−	5.65675e10i	1.09268	−	1.09268i
$478$		0			0
$479$			7.37614e10i			1.40116i	0.713575	+	0.700579i	$0.247075\pi$
$479$			7.37614e10i			1.40116i	−0.713575	+	0.700579i	$0.752925\pi$
$480$		0			0
$481$	9.85206e8			0.0184055
$482$		0			0
$483$	1.22849e10	+	1.22849e10i	0.225727	+	0.225727i
$484$		0			0
$485$	5.06382e8	+	2.98161e10i	0.00915189	+	0.538869i
$486$		0			0
$487$	2.80441e10	−	2.80441e10i	0.498569	−	0.498569i	−0.412423	−	0.910992i	$-0.635318\pi$
$487$	2.80441e10	−	2.80441e10i	0.498569	−	0.498569i	0.910992	+	0.412423i	$0.135318\pi$
$488$		0			0
$489$		−	3.89430e10i		−	0.681074i
$490$		0			0
$491$	1.16769e10			0.200911			0.100455	−	0.994942i	$-0.467970\pi$
$491$	1.16769e10			0.200911			0.100455	+	0.994942i	$0.467970\pi$
$492$		0			0
$493$	4.03646e10	+	4.03646e10i	0.683302	+	0.683302i
$494$		0			0
$495$	−3.15194e10	−	3.04667e10i	−0.524997	−	0.507463i
$496$		0			0
$497$	1.52974e10	−	1.52974e10i	0.250722	−	0.250722i
$498$		0			0
$499$		−	7.08522e10i		−	1.14275i	−0.820689	−	0.571375i	$-0.806410\pi$
$499$		−	7.08522e10i		−	1.14275i	0.820689	−	0.571375i	$-0.193590\pi$
$500$		0			0
$501$	−1.25716e11			−1.99544
$502$		0			0
$503$	7.32217e10	+	7.32217e10i	1.14385	+	1.14385i	0.987740	+	0.156107i	$0.0498944\pi$
$503$	7.32217e10	+	7.32217e10i	1.14385	+	1.14385i	0.156107	+	0.987740i	$0.450106\pi$
$504$		0			0
$505$	8.32225e9	−	8.60982e9i	0.127960	−	0.132382i
$506$		0			0
$507$	6.23444e10	−	6.23444e10i	0.943552	−	0.943552i
$508$		0			0
$509$			1.10598e11i			1.64769i	0.566813	+	0.823846i	$0.308176\pi$
$509$			1.10598e11i			1.64769i	−0.566813	+	0.823846i	$0.691824\pi$
$510$		0			0
$511$	−2.56797e10			−0.376623
$512$		0			0
$513$	−1.85827e10	−	1.85827e10i	−0.268312	−	0.268312i
$514$		0			0
$515$	−1.19463e11	+	2.02890e9i	−1.69826	+	0.0288425i
$516$		0			0
$517$	1.25894e10	−	1.25894e10i	0.176215	−	0.176215i
$518$		0			0
$519$		−	6.19236e10i		−	0.853467i
$520$		0			0
$521$	−8.07216e10			−1.09557			−0.547783	−	0.836620i	$-0.684528\pi$
$521$	−8.07216e10			−1.09557			−0.547783	+	0.836620i	$0.684528\pi$
$522$		0			0
$523$	3.77792e10	+	3.77792e10i	0.504948	+	0.504948i	0.912971	−	0.408024i	$-0.133782\pi$
$523$	3.77792e10	+	3.77792e10i	0.504948	+	0.504948i	−0.408024	+	0.912971i	$0.633782\pi$
$524$		0			0
$525$	1.22646e9	+	3.60968e10i	0.0161441	+	0.475151i
$526$		0			0
$527$	−5.22908e10	+	5.22908e10i	−0.677927	+	0.677927i
$528$		0			0
$529$		−	4.30044e10i		−	0.549149i
$530$		0			0
$531$	−1.74655e11			−2.19687
$532$		0			0
$533$	1.46752e10	+	1.46752e10i	0.181835	+	0.181835i
$534$		0			0
$535$	−1.84352e9	−	1.08548e11i	−0.0225026	−	1.32497i
$536$		0			0
$537$	−6.83688e9	+	6.83688e9i	−0.0822169	+	0.0822169i
$538$		0			0
$539$			3.68527e10i			0.436631i
$540$		0			0
$541$	7.53948e10			0.880141			0.440071	−	0.897963i	$-0.354953\pi$
$541$	7.53948e10			0.880141			0.440071	+	0.897963i	$0.354953\pi$
$542$		0			0
$543$	1.32950e10	+	1.32950e10i	0.152928	+	0.152928i
$544$		0			0
$545$	−7.49481e10	−	7.24449e10i	−0.849522	−	0.821149i
$546$		0			0
$547$	−2.20178e10	+	2.20178e10i	−0.245937	+	0.245937i	−0.819301	−	0.573364i	$-0.805638\pi$
$547$	−2.20178e10	+	2.20178e10i	−0.245937	+	0.245937i	0.573364	+	0.819301i	$0.305638\pi$
$548$		0			0
$549$		−	1.96235e11i		−	2.16017i
$550$		0			0
$551$	2.33656e10			0.253496
$552$		0			0
$553$	−4.13703e9	−	4.13703e9i	−0.0442372	−	0.0442372i
$554$		0			0
$555$	4.82163e9	−	4.98823e9i	0.0508185	−	0.0525745i
$556$		0			0
$557$	−1.03428e11	+	1.03428e11i	−1.07453	+	1.07453i	−0.0775357	+	0.996990i	$0.524705\pi$
$557$	−1.03428e11	+	1.03428e11i	−1.07453	+	1.07453i	−0.996990	+	0.0775357i	$0.975295\pi$
$558$		0			0
$559$			2.38536e10i			0.244290i
$560$		0			0
$561$	−1.31531e11			−1.32793
$562$		0			0
$563$	7.70545e10	+	7.70545e10i	0.766945	+	0.766945i	0.977567	−	0.210623i	$-0.0675491\pi$
$563$	7.70545e10	+	7.70545e10i	0.766945	+	0.766945i	−0.210623	+	0.977567i	$0.567549\pi$
$564$		0			0
$565$	−2.85768e10	+	4.85334e8i	−0.280427	+	0.00476263i
$566$		0			0
$567$	−4.47721e9	+	4.47721e9i	−0.0433187	+	0.0433187i
$568$		0			0
$569$			6.19669e10i			0.591168i	0.955317	+	0.295584i	$0.0955142\pi$
$569$			6.19669e10i			0.591168i	−0.955317	+	0.295584i	$0.904486\pi$
$570$		0			0
$571$	−2.38428e10			−0.224292			−0.112146	−	0.993692i	$-0.535772\pi$
$571$	−2.38428e10			−0.224292			−0.112146	+	0.993692i	$0.535772\pi$
$572$		0			0
$573$	2.31463e11	+	2.31463e11i	2.14715	+	2.14715i
$574$		0			0
$575$	−5.01083e10	+	5.36331e10i	−0.458393	+	0.490638i
$576$		0			0
$577$	1.08821e11	−	1.08821e11i	0.981772	−	0.981772i	−0.0180652	−	0.999837i	$-0.505751\pi$
$577$	1.08821e11	−	1.08821e11i	0.981772	−	0.981772i	0.999837	+	0.0180652i	$0.00575065\pi$
$578$		0			0
$579$			1.72283e11i			1.53295i
$580$		0			0
$581$	−2.02916e10			−0.178079
$582$		0			0
$583$	−3.97666e10	−	3.97666e10i	−0.344226	−	0.344226i
$584$		0			0
$585$	1.21043e9	+	7.12710e10i	0.0103351	+	0.608540i
$586$		0			0
$587$	9.61187e10	−	9.61187e10i	0.809572	−	0.809572i	−0.174997	−	0.984569i	$-0.555992\pi$
$587$	9.61187e10	−	9.61187e10i	0.809572	−	0.809572i	0.984569	+	0.174997i	$0.0559916\pi$
$588$		0			0
$589$			3.02693e10i			0.251502i
$590$		0			0
$591$	1.16865e11			0.957932
$592$		0			0
$593$	1.01154e10	+	1.01154e10i	0.0818021	+	0.0818021i	0.746824	−	0.665022i	$-0.231578\pi$
$593$	1.01154e10	+	1.01154e10i	0.0818021	+	0.0818021i	−0.665022	+	0.746824i	$0.731578\pi$
$594$		0			0
$595$	4.70283e10	+	4.54575e10i	0.375225	+	0.362692i
$596$		0			0
$597$	3.16166e10	−	3.16166e10i	0.248896	−	0.248896i
$598$		0			0
$599$		−	1.61079e11i		−	1.25121i	−0.780139	−	0.625606i	$-0.784852\pi$
$599$		−	1.61079e11i		−	1.25121i	0.780139	−	0.625606i	$-0.215148\pi$
$600$		0			0
$601$	2.16088e11			1.65627			0.828137	−	0.560526i	$-0.189401\pi$
$601$	2.16088e11			1.65627			0.828137	+	0.560526i	$0.189401\pi$
$602$		0			0
$603$	1.12730e11	+	1.12730e11i	0.852649	+	0.852649i
$604$		0			0
$605$	7.16939e10	−	7.41712e10i	0.535132	−	0.553623i
$606$		0			0
$607$	8.94773e10	−	8.94773e10i	0.659111	−	0.659111i	−0.296059	−	0.955170i	$-0.595673\pi$
$607$	8.94773e10	−	8.94773e10i	0.659111	−	0.659111i	0.955170	+	0.296059i	$0.0956725\pi$
$608$		0			0
$609$			3.62492e10i			0.263529i
$610$		0			0
$611$	−2.89504e10			−0.207725
$612$		0			0
$613$	−1.23726e11	−	1.23726e11i	−0.876229	−	0.876229i	0.116913	−	0.993142i	$-0.462700\pi$
$613$	−1.23726e11	−	1.23726e11i	−0.876229	−	0.876229i	−0.993142	+	0.116913i	$0.962700\pi$
$614$		0			0
$615$	1.46124e11	−	2.48170e9i	1.02146	−	0.0173480i
$616$		0			0
$617$	1.10854e11	−	1.10854e11i	0.764908	−	0.764908i	−0.212297	−	0.977205i	$-0.568094\pi$
$617$	1.10854e11	−	1.10854e11i	0.764908	−	0.764908i	0.977205	+	0.212297i	$0.0680945\pi$
$618$		0			0
$619$		−	1.30648e11i		−	0.889897i	−0.895556	−	0.444949i	$-0.853222\pi$
$619$		−	1.30648e11i		−	0.889897i	0.895556	−	0.444949i	$-0.146778\pi$
$620$		0			0
$621$	8.28540e10			0.557118
$622$		0			0
$623$	−4.17199e10	−	4.17199e10i	−0.276944	−	0.276944i
$624$		0			0
$625$	−1.52236e11	+	1.03569e10i	−0.997694	+	0.0678753i
$626$		0			0
$627$	−3.80692e10	+	3.80692e10i	−0.246323	+	0.246323i
$628$		0			0
$629$		−	1.25636e10i		−	0.0802622i
$630$		0			0
$631$	2.21106e11			1.39470			0.697352	−	0.716729i	$-0.254361\pi$
$631$	2.21106e11			1.39470			0.697352	+	0.716729i	$0.254361\pi$
$632$		0			0
$633$	1.20474e11	+	1.20474e11i	0.750377	+	0.750377i
$634$		0			0
$635$	4.40390e9	+	2.59304e11i	0.0270858	+	1.59483i
$636$		0			0
$637$	4.23729e10	−	4.23729e10i	0.257354	−	0.257354i
$638$		0			0
$639$		−	3.00658e11i		−	1.80331i
$640$		0			0
$641$	2.04377e11			1.21060			0.605298	−	0.795999i	$-0.293054\pi$
$641$	2.04377e11			1.21060			0.605298	+	0.795999i	$0.293054\pi$
$642$		0			0
$643$	−1.72443e11	−	1.72443e11i	−1.00879	−	1.00879i	−0.999961	−	0.00882849i	$-0.997190\pi$
$643$	−1.72443e11	−	1.72443e11i	−1.00879	−	1.00879i	−0.00882849	−	0.999961i	$-0.502810\pi$
$644$		0			0
$645$	1.20774e11	+	1.16740e11i	0.697806	+	0.674500i
$646$		0			0
$647$	1.33246e11	−	1.33246e11i	0.760392	−	0.760392i	−0.216001	−	0.976393i	$-0.569301\pi$
$647$	1.33246e11	−	1.33246e11i	0.760392	−	0.760392i	0.976393	+	0.216001i	$0.0693014\pi$
$648$		0			0
$649$			1.22782e11i			0.692078i
$650$		0			0
$651$	−4.69594e10			−0.261456
$652$		0			0
$653$	−7.89023e10	−	7.89023e10i	−0.433947	−	0.433947i	0.456022	−	0.889969i	$-0.349274\pi$
$653$	−7.89023e10	−	7.89023e10i	−0.433947	−	0.433947i	−0.889969	+	0.456022i	$0.849274\pi$
$654$		0			0
$655$	−1.46675e11	+	1.51743e11i	−0.796877	+	0.824412i
$656$		0			0
$657$	−2.52357e11	+	2.52357e11i	−1.35442	+	1.35442i
$658$		0			0
$659$			3.33275e11i			1.76710i	0.468339	+	0.883549i	$0.344853\pi$
$659$			3.33275e11i			1.76710i	−0.468339	+	0.883549i	$0.655147\pi$
$660$		0			0
$661$	−2.67623e11			−1.40190			−0.700951	−	0.713209i	$-0.747241\pi$
$661$	−2.67623e11			−1.40190			−0.700951	+	0.713209i	$0.747241\pi$
$662$		0			0
$663$	1.51233e11	+	1.51233e11i	0.782694	+	0.782694i
$664$		0			0
$665$	2.67684e10	−	4.54621e8i	0.136879	−	0.00232468i
$666$		0			0
$667$	−5.20896e10	+	5.20896e10i	−0.263177	+	0.263177i
$668$		0			0
$669$		−	5.39294e11i		−	2.69228i
$670$		0			0
$671$	−1.37952e11			−0.680517
$672$		0			0
$673$	−4.65870e10	−	4.65870e10i	−0.227093	−	0.227093i	0.584384	−	0.811477i	$-0.301336\pi$
$673$	−4.65870e10	−	4.65870e10i	−0.227093	−	0.227093i	−0.811477	+	0.584384i	$0.801336\pi$
$674$		0			0
$675$	1.25861e11	+	1.17589e11i	0.606284	+	0.566438i
$676$		0			0
$677$	−2.04905e11	+	2.04905e11i	−0.975432	+	0.975432i	−0.999705	−	0.0242736i	$-0.992273\pi$
$677$	−2.04905e11	+	2.04905e11i	−0.975432	+	0.975432i	0.0242736	+	0.999705i	$0.492273\pi$
$678$		0			0
$679$			3.42925e10i			0.161332i
$680$		0			0
$681$	1.52374e11			0.708473
$682$		0			0
$683$	−2.26392e11	−	2.26392e11i	−1.04035	−	1.04035i	−0.999151	−	0.0411961i	$-0.986883\pi$
$683$	−2.26392e11	−	2.26392e11i	−1.04035	−	1.04035i	−0.0411961	−	0.999151i	$-0.513117\pi$
$684$		0			0
$685$	1.59283e9	+	9.37866e10i	0.00723445	+	0.425969i
$686$		0			0
$687$	−2.42584e11	+	2.42584e11i	−1.08902	+	1.08902i
$688$		0			0
$689$			9.14465e10i			0.405779i
$690$		0			0
$691$	1.15111e11			0.504900			0.252450	−	0.967610i	$-0.418764\pi$
$691$	1.15111e11			0.504900			0.252450	+	0.967610i	$0.418764\pi$
$692$		0			0
$693$	−3.56461e10	−	3.56461e10i	−0.154554	−	0.154554i
$694$		0			0
$695$	−2.63631e11	−	2.54826e11i	−1.12995	−	1.09221i
$696$		0			0
$697$	1.87142e11	−	1.87142e11i	0.792941	−	0.792941i
$698$		0			0
$699$			1.80470e11i			0.755956i
$700$		0			0
$701$	3.26439e10			0.135185			0.0675927	−	0.997713i	$-0.478468\pi$
$701$	3.26439e10			0.135185			0.0675927	+	0.997713i	$0.478468\pi$
$702$		0			0
$703$	−3.63631e9	−	3.63631e9i	−0.0148881	−	0.0148881i
$704$		0			0
$705$	−1.41684e11	+	1.46580e11i	−0.573542	+	0.593360i
$706$		0			0
$707$	9.73707e9	−	9.73707e9i	0.0389718	−	0.0389718i
$708$		0			0
$709$		−	1.10106e11i		−	0.435737i	−0.975978	−	0.217869i	$-0.930090\pi$
$709$		−	1.10106e11i		−	0.435737i	0.975978	−	0.217869i	$-0.0699104\pi$
$710$		0			0
$711$	−8.13098e10			−0.318174
$712$		0			0
$713$	−6.74802e10	−	6.74802e10i	−0.261107	−	0.261107i
$714$		0			0
$715$	5.01030e10	−	8.50926e8i	0.191708	−	0.00325587i
$716$		0			0
$717$	−1.14697e11	+	1.14697e11i	−0.433986	+	0.433986i
$718$		0			0
$719$		−	2.80995e11i		−	1.05144i	−0.850658	−	0.525719i	$-0.823796\pi$
$719$		−	2.80995e11i		−	1.05144i	0.850658	−	0.525719i	$-0.176204\pi$
$720$		0			0
$721$	−1.37399e11			−0.508442
$722$		0			0
$723$	−1.27510e11	−	1.27510e11i	−0.466649	−	0.466649i
$724$		0			0
$725$	−1.53055e11	+	5.20033e9i	−0.553982	+	0.0188226i
$726$		0			0
$727$	−7.91155e10	+	7.91155e10i	−0.283220	+	0.283220i	−0.834392	−	0.551172i	$-0.814181\pi$
$727$	−7.91155e10	+	7.91155e10i	−0.283220	+	0.283220i	0.551172	+	0.834392i	$0.314181\pi$
$728$		0			0
$729$			4.60173e11i			1.62934i
$730$		0			0
$731$	3.04187e11			1.06530
$732$		0			0
$733$	−1.00588e11	−	1.00588e11i	−0.348441	−	0.348441i	0.511088	−	0.859529i	$-0.329243\pi$
$733$	−1.00588e11	−	1.00588e11i	−0.348441	−	0.348441i	−0.859529	+	0.511088i	$0.829243\pi$
$734$		0			0
$735$	−7.16558e9	−	4.21914e11i	−0.0245529	−	1.44569i
$736$		0			0
$737$	7.92485e10	−	7.92485e10i	0.268609	−	0.268609i
$738$		0			0
$739$			1.06947e11i			0.358584i	0.983796	+	0.179292i	$0.0573806\pi$
$739$			1.06947e11i			0.358584i	−0.983796	+	0.179292i	$0.942619\pi$
$740$		0			0
$741$	8.75433e10			0.290369
$742$		0			0
$743$	−5.96783e10	−	5.96783e10i	−0.195822	−	0.195822i	0.602384	−	0.798206i	$-0.294218\pi$
$743$	−5.96783e10	−	5.96783e10i	−0.195822	−	0.195822i	−0.798206	+	0.602384i	$0.794218\pi$
$744$		0			0
$745$	−2.45338e11	−	2.37144e11i	−0.796417	−	0.769817i
$746$		0			0
$747$	−1.99407e11	+	1.99407e11i	−0.640410	+	0.640410i
$748$		0			0
$749$		−	1.24844e11i		−	0.396681i
$750$		0			0
$751$	2.90374e11			0.912846			0.456423	−	0.889763i	$-0.349130\pi$
$751$	2.90374e11			0.912846			0.456423	+	0.889763i	$0.349130\pi$
$752$		0			0
$753$	−4.76684e11	−	4.76684e11i	−1.48269	−	1.48269i
$754$		0			0
$755$	3.85851e10	−	3.99184e10i	0.118750	−	0.122853i
$756$		0			0
$757$	−2.22407e11	+	2.22407e11i	−0.677274	+	0.677274i	−0.959383	−	0.282108i	$-0.908966\pi$
$757$	−2.22407e11	+	2.22407e11i	−0.677274	+	0.677274i	0.282108	+	0.959383i	$0.408966\pi$
$758$		0			0
$759$		−	1.69738e11i		−	0.511459i
$760$		0			0
$761$	5.59405e11			1.66797			0.833984	−	0.551788i	$-0.186054\pi$
$761$	5.59405e11			1.66797			0.833984	+	0.551788i	$0.186054\pi$
$762$		0			0
$763$	−8.47608e10	−	8.47608e10i	−0.250090	−	0.250090i
$764$		0			0
$765$	9.08866e11	−	1.54357e10i	2.65371	−	0.0450694i
$766$		0			0
$767$	1.41173e11	−	1.41173e11i	0.407916	−	0.407916i
$768$		0			0
$769$			3.07549e10i			0.0879446i	0.999033	+	0.0439723i	$0.0140013\pi$
$769$			3.07549e10i			0.0879446i	−0.999033	+	0.0439723i	$0.985999\pi$
$770$		0			0
$771$	−4.34734e11			−1.23029
$772$		0			0
$773$	−1.21896e11	−	1.21896e11i	−0.341405	−	0.341405i	0.515490	−	0.856896i	$-0.327610\pi$
$773$	−1.21896e11	−	1.21896e11i	−0.341405	−	0.341405i	−0.856896	+	0.515490i	$0.827610\pi$
$774$		0			0
$775$	−6.73683e9	−	1.98277e11i	−0.0186745	−	0.549625i
$776$		0			0
$777$	5.64133e9	−	5.64133e9i	0.0154774	−	0.0154774i
$778$		0			0
$779$		−	1.08330e11i		−	0.294170i
$780$		0			0
$781$	−2.11361e11			−0.568094
$782$		0			0
$783$	1.22239e11	+	1.22239e11i	0.325209	+	0.325209i
$784$		0			0
$785$	−4.81906e9	−	2.83749e11i	−0.0126906	−	0.747234i
$786$		0			0
$787$	2.26235e9	−	2.26235e9i	0.00589741	−	0.00589741i	−0.704152	−	0.710049i	$-0.748673\pi$
$787$	2.26235e9	−	2.26235e9i	0.00589741	−	0.00589741i	0.710049	+	0.704152i	$0.248673\pi$
$788$		0			0
$789$			2.35501e11i			0.607695i
$790$		0			0
$791$	−3.28671e10			−0.0839567
$792$		0			0
$793$	1.58616e11	+	1.58616e11i	0.401102	+	0.401102i
$794$		0			0
$795$	4.63006e11	+	4.47542e11i	1.15909	+	1.12038i
$796$		0			0
$797$	3.08554e11	−	3.08554e11i	0.764711	−	0.764711i	−0.212459	−	0.977170i	$-0.568147\pi$
$797$	3.08554e11	−	3.08554e11i	0.764711	−	0.764711i	0.977170	+	0.212459i	$0.0681471\pi$
$798$		0			0
$799$			3.69183e11i			0.905846i
$800$		0			0
$801$	−8.19970e11			−1.99190
$802$		0			0
$803$	1.77405e11	+	1.77405e11i	0.426682	+	0.426682i
$804$		0			0
$805$	−5.86619e10	+	6.06889e10i	−0.139692	+	0.144519i
$806$		0			0
$807$	−5.58846e11	+	5.58846e11i	−1.31764	+	1.31764i
$808$		0			0
$809$			3.71836e11i			0.868076i	0.900895	+	0.434038i	$0.142912\pi$
$809$			3.71836e11i			0.868076i	−0.900895	+	0.434038i	$0.857088\pi$
$810$		0			0
$811$	−7.35802e11			−1.70090			−0.850448	−	0.526060i	$-0.823669\pi$
$811$	−7.35802e11			−1.70090			−0.850448	+	0.526060i	$0.823669\pi$
$812$		0			0
$813$	4.74122e11	+	4.74122e11i	1.08525	+	1.08525i
$814$		0			0
$815$	1.89170e11	−	3.21278e9i	0.428768	−	0.00728199i
$816$		0			0
$817$	8.80415e10	−	8.80415e10i	0.197606	−	0.197606i
$818$		0			0
$819$			8.19712e10i			0.182190i
$820$		0			0
$821$	−5.89174e11			−1.29679			−0.648397	−	0.761303i	$-0.724560\pi$
$821$	−5.89174e11			−1.29679			−0.648397	+	0.761303i	$0.724560\pi$
$822$		0			0
$823$	6.93799e10	+	6.93799e10i	0.151229	+	0.151229i	0.778667	−	0.627438i	$-0.215896\pi$
$823$	6.93799e10	+	6.93799e10i	0.151229	+	0.151229i	−0.627438	+	0.778667i	$0.715896\pi$
$824$		0			0
$825$	2.40898e11	−	2.57843e11i	0.520016	−	0.556596i
$826$		0			0
$827$	6.01415e10	−	6.01415e10i	0.128574	−	0.128574i	−0.639891	−	0.768465i	$-0.721021\pi$
$827$	6.01415e10	−	6.01415e10i	0.128574	−	0.128574i	0.768465	+	0.639891i	$0.221021\pi$
$828$		0			0
$829$			8.00261e11i			1.69439i	0.531282	+	0.847195i	$0.321711\pi$
$829$			8.00261e11i			1.69439i	−0.531282	+	0.847195i	$0.678289\pi$
$830$		0			0
$831$	9.87159e11			2.07006
$832$		0			0
$833$	−5.40350e11	−	5.40350e11i	−1.12226	−	1.12226i
$834$		0			0
$835$	−1.03715e10	−	6.10680e11i	−0.0213351	−	1.25623i
$836$		0			0
$837$	−1.58356e11	+	1.58356e11i	−0.322650	+	0.322650i
$838$		0			0
$839$			6.08689e11i			1.22842i	0.789142	+	0.614211i	$0.210526\pi$
$839$			6.08689e11i			1.22842i	−0.789142	+	0.614211i	$0.789474\pi$
$840$		0			0
$841$	3.46545e11			0.692749
$842$		0			0
$843$	−6.48949e11	−	6.48949e11i	−1.28499	−	1.28499i
$844$		0			0
$845$	3.07989e11	+	2.97702e11i	0.604099	+	0.583922i
$846$		0			0
$847$	8.38822e10	−	8.38822e10i	0.162981	−	0.162981i
$848$		0			0
$849$		−	4.80193e11i		−	0.924241i
$850$		0			0
$851$	1.62130e10			0.0309134
$852$		0			0
$853$	3.37705e11	+	3.37705e11i	0.637883	+	0.637883i	0.950033	−	0.312150i	$-0.101049\pi$
$853$	3.37705e11	+	3.37705e11i	0.637883	+	0.637883i	−0.312150	+	0.950033i	$0.601049\pi$
$854$		0			0
$855$	2.58587e11	−	2.67522e11i	0.483886	−	0.500606i
$856$		0			0
$857$	2.05118e11	−	2.05118e11i	0.380260	−	0.380260i	−0.490935	−	0.871196i	$-0.663345\pi$
$857$	2.05118e11	−	2.05118e11i	0.380260	−	0.380260i	0.871196	+	0.490935i	$0.163345\pi$
$858$		0			0
$859$			1.00757e12i			1.85056i	0.379281	+	0.925282i	$0.376171\pi$
$859$			1.00757e12i			1.85056i	−0.379281	+	0.925282i	$0.623829\pi$
$860$		0			0
$861$	1.68062e11			0.305814
$862$		0			0
$863$	−1.70651e11	−	1.70651e11i	−0.307656	−	0.307656i	0.536344	−	0.844000i	$-0.319805\pi$
$863$	−1.70651e11	−	1.70651e11i	−0.307656	−	0.307656i	−0.844000	+	0.536344i	$0.819805\pi$
$864$		0			0
$865$	3.00801e11	−	5.10866e9i	0.537298	−	0.00912520i
$866$		0			0
$867$	1.29400e12	−	1.29400e12i	2.29013	−	2.29013i
$868$		0			0
$869$			5.71602e10i			0.100234i
$870$		0			0
$871$	−1.82238e11			−0.316641
$872$		0			0
$873$	3.36995e11	+	3.36995e11i	0.580185	+	0.580185i
$874$		0			0
$875$	−1.75243e11	+	8.93562e9i	−0.298958	+	0.0152438i
$876$		0			0
$877$	−1.79019e11	+	1.79019e11i	−0.302623	+	0.302623i	−0.842039	−	0.539416i	$-0.818645\pi$
$877$	−1.79019e11	+	1.79019e11i	−0.302623	+	0.302623i	0.539416	+	0.842039i	$0.318645\pi$
$878$		0			0
$879$		−	4.12754e11i		−	0.691411i
$880$		0			0
$881$	−4.12168e10			−0.0684180			−0.0342090	−	0.999415i	$-0.510891\pi$
$881$	−4.12168e10			−0.0684180			−0.0342090	+	0.999415i	$0.510891\pi$
$882$		0			0
$883$	4.90367e11	+	4.90367e11i	0.806638	+	0.806638i	0.984123	−	0.177486i	$-0.0567964\pi$
$883$	4.90367e11	+	4.90367e11i	0.806638	+	0.806638i	−0.177486	+	0.984123i	$0.556796\pi$
$884$		0			0
$885$	−2.38735e10	−	1.40569e12i	−0.0389173	−	2.29148i
$886$		0			0
$887$	−5.99973e11	+	5.99973e11i	−0.969253	+	0.969253i	−0.999541	−	0.0302885i	$-0.990357\pi$
$887$	−5.99973e11	+	5.99973e11i	−0.969253	+	0.969253i	0.0302885	+	0.999541i	$0.490357\pi$
$888$		0			0
$889$			2.98235e11i			0.477475i
$890$		0			0
$891$	6.18606e10			0.0981529
$892$		0			0
$893$	1.06853e11	+	1.06853e11i	0.168028	+	0.168028i
$894$		0			0
$895$	−3.37750e10	−	3.26469e10i	−0.0526385	−	0.0508804i
$896$		0			0
$897$	−1.95163e11	+	1.95163e11i	−0.301458	+	0.301458i
$898$		0			0
$899$		−	1.99114e11i		−	0.304834i
$900$		0			0
$901$	1.16615e12			1.76952
$902$		0			0
$903$	1.36587e11	+	1.36587e11i	0.205427	+	0.205427i
$904$		0			0
$905$	−6.34851e10	+	6.56787e10i	−0.0946406	+	0.0979108i
$906$		0			0
$907$	5.29755e11	−	5.29755e11i	0.782791	−	0.782791i	−0.197510	−	0.980301i	$-0.563285\pi$
$907$	5.29755e11	−	5.29755e11i	0.782791	−	0.782791i	0.980301	+	0.197510i	$0.0632854\pi$
$908$		0			0
$909$		−	1.91374e11i		−	0.280303i
$910$		0			0
$911$	4.06858e11			0.590703			0.295351	−	0.955389i	$-0.404563\pi$
$911$	4.06858e11			0.590703			0.295351	+	0.955389i	$0.404563\pi$
$912$		0			0
$913$	1.40182e11	+	1.40182e11i	0.201748	+	0.201748i
$914$		0			0
$915$	1.57937e12	−	2.68232e10i	2.25320	−	0.0382672i
$916$		0			0
$917$	−1.71611e11	+	1.71611e11i	−0.242698	+	0.242698i
$918$		0			0
$919$			1.03523e12i			1.45135i	0.688036	+	0.725676i	$0.258473\pi$
$919$			1.03523e12i			1.45135i	−0.688036	+	0.725676i	$0.741527\pi$
$920$		0			0
$921$	9.40473e11			1.30710
$922$		0			0
$923$	2.43020e11	+	2.43020e11i	0.334839	+	0.334839i
$924$		0			0
$925$	2.46287e10	+	2.30101e10i	0.0336415	+	0.0314305i
$926$		0			0
$927$	−1.35023e12	+	1.35023e12i	−1.82847	+	1.82847i
$928$		0			0
$929$			5.10944e11i			0.685978i	0.939339	+	0.342989i	$0.111439\pi$
$929$			5.10944e11i			0.685978i	−0.939339	+	0.342989i	$0.888561\pi$
$930$		0			0
$931$	−3.12789e11			−0.416345
$932$		0			0
$933$	−1.20217e12	−	1.20217e12i	−1.58649	−	1.58649i
$934$		0			0
$935$	−1.08512e10	−	6.38927e11i	−0.0141982	−	0.835997i
$936$		0			0
$937$	−6.26235e11	+	6.26235e11i	−0.812417	+	0.812417i	−0.984996	−	0.172579i	$-0.944790\pi$
$937$	−6.26235e11	+	6.26235e11i	−0.812417	+	0.812417i	0.172579	+	0.984996i	$0.444790\pi$
$938$		0			0
$939$		−	1.64251e12i		−	2.11273i
$940$		0			0
$941$	5.32097e11			0.678628			0.339314	−	0.940673i	$-0.389805\pi$
$941$	5.32097e11			0.678628			0.339314	+	0.940673i	$0.389805\pi$
$942$		0			0
$943$	2.41503e11	+	2.41503e11i	0.305405	+	0.305405i
$944$		0			0
$945$	1.42419e11	+	1.37662e11i	0.178583	+	0.172618i
$946$		0			0
$947$	6.94903e11	−	6.94903e11i	0.864021	−	0.864021i	−0.127781	−	0.991802i	$-0.540786\pi$
$947$	6.94903e11	−	6.94903e11i	0.864021	−	0.864021i	0.991802	+	0.127781i	$0.0407855\pi$
$948$		0			0
$949$		−	4.07958e11i		−	0.502980i
$950$		0			0
$951$	−8.94754e11			−1.09391
$952$		0			0
$953$	1.46903e11	+	1.46903e11i	0.178098	+	0.178098i	0.790526	−	0.612428i	$-0.209807\pi$
$953$	1.46903e11	+	1.46903e11i	0.178098	+	0.178098i	−0.612428	+	0.790526i	$0.709807\pi$
$954$		0			0
$955$	−1.10526e12	+	1.14345e12i	−1.32878	+	1.37469i
$956$		0			0
$957$	2.50423e11	−	2.50423e11i	0.298556	−	0.298556i
$958$		0			0
$959$			1.07867e11i			0.127531i
$960$		0			0
$961$	−5.94946e11			−0.697564
$962$		0			0
$963$	−1.22686e12	−	1.22686e12i	−1.42655	−	1.42655i
$964$		0			0
$965$	−8.36885e11	+	1.42132e10i	−0.965065	+	0.0163902i
$966$		0			0
$967$	−8.32464e11	+	8.32464e11i	−0.952050	+	0.952050i	−0.998902	−	0.0468522i	$-0.985081\pi$
$967$	−8.32464e11	+	8.32464e11i	−0.952050	+	0.952050i	0.0468522	+	0.998902i	$0.485081\pi$
$968$		0			0
$969$		−	1.11637e12i		−	1.26624i
$970$		0			0
$971$	−1.00239e11			−0.112761			−0.0563804	−	0.998409i	$-0.517956\pi$
$971$	−1.00239e11			−0.112761			−0.0563804	+	0.998409i	$0.517956\pi$
$972$		0			0
$973$	−2.98148e11	−	2.98148e11i	−0.332644	−	0.332644i
$974$		0			0
$975$	−5.73447e11	+	1.94839e10i	−0.634564	+	0.0215605i
$976$		0			0
$977$	−1.07451e11	+	1.07451e11i	−0.117932	+	0.117932i	−0.763610	−	0.645678i	$-0.776575\pi$
$977$	−1.07451e11	+	1.07451e11i	−0.117932	+	0.117932i	0.645678	+	0.763610i	$0.276575\pi$
$978$		0			0
$979$			5.76434e11i			0.627507i
$980$		0			0
$981$	−1.66590e12			−1.79876
$982$		0			0
$983$	−5.36756e10	−	5.36756e10i	−0.0574861	−	0.0574861i	0.677779	−	0.735265i	$-0.262942\pi$
$983$	−5.36756e10	−	5.36756e10i	−0.0574861	−	0.0574861i	−0.735265	+	0.677779i	$0.762942\pi$
$984$		0			0
$985$	9.64130e9	+	5.67686e11i	0.0102421	+	0.603064i
$986$		0			0
$987$	−1.65771e11	+	1.65771e11i	−0.174679	+	0.174679i
$988$		0			0
$989$			3.92547e11i			0.410304i
$990$		0			0
$991$	1.54321e12			1.60004			0.800019	−	0.599974i	$-0.204823\pi$
$991$	1.54321e12			1.60004			0.800019	+	0.599974i	$0.204823\pi$
$992$		0			0
$993$	−1.56502e11	−	1.56502e11i	−0.160962	−	0.160962i
$994$		0			0
$995$	1.56190e11	+	1.50973e11i	0.159353	+	0.154031i
$996$		0			0
$997$	−7.69594e11	+	7.69594e11i	−0.778898	+	0.778898i	−0.979644	−	0.200745i	$-0.935664\pi$
$997$	−7.69594e11	+	7.69594e11i	−0.778898	+	0.778898i	0.200745	+	0.979644i	$0.435664\pi$
$998$		0			0
$999$		−	3.80472e10i		−	0.0381997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.9.p.c.33.3		6
4.3	odd	2		5.9.c.a.3.1	yes	6
5.2	odd	4	inner	80.9.p.c.17.3		6
12.11	even	2		45.9.g.a.28.3		6
20.3	even	4		25.9.c.b.7.3		6
20.7	even	4		5.9.c.a.2.1	✓	6
20.19	odd	2		25.9.c.b.18.3		6
60.47	odd	4		45.9.g.a.37.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.9.c.a.2.1	✓	6	20.7	even	4
5.9.c.a.3.1	yes	6	4.3	odd	2
25.9.c.b.7.3		6	20.3	even	4
25.9.c.b.18.3		6	20.19	odd	2
45.9.g.a.28.3		6	12.11	even	2
45.9.g.a.37.3		6	60.47	odd	4
80.9.p.c.17.3		6	5.2	odd	4	inner
80.9.p.c.33.3		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(90.9660 + 90.9660i) q^{3} +(-434.373 + 449.383i) q^{5} +(-508.219 + 508.219i) q^{7} +9988.61i q^{9} +O(q^{10})\)	\(q+(90.9660 + 90.9660i) q^{3} +(-434.373 + 449.383i) q^{5} +(-508.219 + 508.219i) q^{7} +9988.61i q^{9} +7021.93 q^{11} +(-8073.75 - 8073.75i) q^{13} +(-80391.7 + 1365.33i) q^{15} +(-102958. + 102958. i) q^{17} +59599.0i q^{19} -92461.2 q^{21} +(-132866. - 132866. i) q^{23} +(-13264.5 - 390400. i) q^{25} +(-311796. + 311796. i) q^{27} -392047. i q^{29} +507883. q^{31} +(638757. + 638757. i) q^{33} +(-7628.00 - 449141. i) q^{35} +(-61012.9 + 61012.9i) q^{37} -1.46887e6i q^{39} -1.81765e6 q^{41} +(-1.47723e6 - 1.47723e6i) q^{43} +(-4.48871e6 - 4.33879e6i) q^{45} +(1.79287e6 - 1.79287e6i) q^{47} +5.24823e6i q^{49} -1.87314e7 q^{51} +(-5.66320e6 - 5.66320e6i) q^{53} +(-3.05014e6 + 3.15553e6i) q^{55} +(-5.42148e6 + 5.42148e6i) q^{57} +1.74855e7i q^{59} -1.96459e7 q^{61} +(-5.07640e6 - 5.07640e6i) q^{63} +(7.13522e6 - 121181. i) q^{65} +(1.12859e7 - 1.12859e7i) q^{67} -2.41725e7i q^{69} -3.01001e7 q^{71} +(2.52645e7 + 2.52645e7i) q^{73} +(3.43065e7 - 3.67197e7i) q^{75} +(-3.56868e6 + 3.56868e6i) q^{77} +8.14025e6i q^{79} +8.80962e6 q^{81} +(1.99635e7 + 1.99635e7i) q^{83} +(-1.54533e6 - 9.09902e7i) q^{85} +(3.56630e7 - 3.56630e7i) q^{87} +8.20905e7i q^{89} +8.20646e6 q^{91} +(4.62000e7 + 4.62000e7i) q^{93} +(-2.67827e7 - 2.58882e7i) q^{95} +(3.37379e7 - 3.37379e7i) q^{97} +7.01393e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 72 q^{3} + 220 q^{5} + 2352 q^{7}+O(q^{10}) \) 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

Representations

Groups

Database

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