LMFDB - Embedded newform 10.8.b.a.9.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10,8,Mod(9,10)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("10.9");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 10 = 2 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	10.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.12385025484$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{31})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 15x^{2} + 64 $ x^4 - 15*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.4
Root			$-2.78388 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	10.9
Dual form			10.8.b.a.9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{2} +27.6776i q^{3} -64.0000 q^{4} +(-207.711 + 187.033i) q^{5} -221.421 q^{6} +593.744i q^{7} -512.000i q^{8} +1420.95 q^{9} +O(q^{10})$	\(q+8.00000i q^{2} +27.6776i q^{3} -64.0000 q^{4} +(-207.711 + 187.033i) q^{5} -221.421 q^{6} +593.744i q^{7} -512.000i q^{8} +1420.95 q^{9} +(-1496.26 - 1661.68i) q^{10} +5342.84 q^{11} -1771.37i q^{12} -2590.86i q^{13} -4749.95 q^{14} +(-5176.63 - 5748.94i) q^{15} +4096.00 q^{16} +31177.8i q^{17} +11367.6i q^{18} -18842.8 q^{19} +(13293.5 - 11970.1i) q^{20} -16433.4 q^{21} +42742.7i q^{22} -83098.3i q^{23} +14171.0 q^{24} +(8162.37 - 77697.4i) q^{25} +20726.8 q^{26} +99859.5i q^{27} -37999.6i q^{28} +186706. q^{29} +(45991.5 - 41413.0i) q^{30} -168014. q^{31} +32768.0i q^{32} +147877. i q^{33} -249423. q^{34} +(-111050. - 123327. i) q^{35} -90940.7 q^{36} +344801. i q^{37} -150743. i q^{38} +71708.8 q^{39} +(95760.9 + 106348. i) q^{40} +490626. q^{41} -131467. i q^{42} -101510. i q^{43} -341942. q^{44} +(-295146. + 265764. i) q^{45} +664787. q^{46} -1.39947e6i q^{47} +113368. i q^{48} +471012. q^{49} +(621579. + 65298.9i) q^{50} -862929. q^{51} +165815. i q^{52} +294530. i q^{53} -798876. q^{54} +(-1.10976e6 + 999287. i) q^{55} +303997. q^{56} -521526. i q^{57} +1.49365e6i q^{58} +1.18171e6 q^{59} +(331304. + 367932. i) q^{60} -1.23135e6 q^{61} -1.34411e6i q^{62} +843679. i q^{63} -262144. q^{64} +(484575. + 538148. i) q^{65} -1.18302e6 q^{66} -928902. i q^{67} -1.99538e6i q^{68} +2.29997e6 q^{69} +(986614. - 888397. i) q^{70} +1.00307e6 q^{71} -727525. i q^{72} -2.48740e6i q^{73} -2.75841e6 q^{74} +(2.15048e6 + 225915. i) q^{75} +1.20594e6 q^{76} +3.17228e6i q^{77} +573670. i q^{78} -842323. q^{79} +(-850783. + 766087. i) q^{80} +343738. q^{81} +3.92501e6i q^{82} +4.01528e6i q^{83} +1.05174e6 q^{84} +(-5.83128e6 - 6.47597e6i) q^{85} +812084. q^{86} +5.16759e6i q^{87} -2.73554e6i q^{88} -1.03006e7 q^{89} +(-2.12611e6 - 2.36117e6i) q^{90} +1.53830e6 q^{91} +5.31829e6i q^{92} -4.65022e6i q^{93} +1.11958e7 q^{94} +(3.91386e6 - 3.52423e6i) q^{95} -906941. q^{96} -93943.7i q^{97} +3.76809e6i q^{98} +7.59190e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9}+O(q^{10}) $ 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9	$ 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9} - 640 q^{10} + 17808 q^{11} - 6528 q^{14} - 34960 q^{15} + 16384 q^{16} + 63600 q^{19} - 3840 q^{20} - 63952 q^{21} - 57344 q^{24} + 86100 q^{25} - 56064 q^{26} + 169560 q^{29} + 410240 q^{30} - 394112 q^{31} - 698368 q^{34} - 276720 q^{35} + 434432 q^{36} + 1163424 q^{39} + 40960 q^{40} + 232488 q^{41} - 1139712 q^{44} - 2879420 q^{45} + 1146496 q^{46} + 2520108 q^{49} + 2361600 q^{50} + 361088 q^{51} - 5116160 q^{54} - 526480 q^{55} + 417792 q^{56} + 2093520 q^{59} + 2237440 q^{60} - 5251432 q^{61} - 1048576 q^{64} + 2761440 q^{65} + 2401792 q^{66} + 6514864 q^{69} + 2679680 q^{70} - 7832352 q^{71} - 3126528 q^{74} - 7397600 q^{75} - 4070400 q^{76} + 7727040 q^{79} + 245760 q^{80} + 16428404 q^{81} + 4092928 q^{84} - 7995520 q^{85} + 4411776 q^{86} - 33470040 q^{89} - 15579520 q^{90} + 5340768 q^{91} + 15540352 q^{94} + 31904400 q^{95} + 3670016 q^{96} - 19109776 q^{99}+O(q^{100}) $ 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9 - 640 * q^10 + 17808 * q^11 - 6528 * q^14 - 34960 * q^15 + 16384 * q^16 + 63600 * q^19 - 3840 * q^20 - 63952 * q^21 - 57344 * q^24 + 86100 * q^25 - 56064 * q^26 + 169560 * q^29 + 410240 * q^30 - 394112 * q^31 - 698368 * q^34 - 276720 * q^35 + 434432 * q^36 + 1163424 * q^39 + 40960 * q^40 + 232488 * q^41 - 1139712 * q^44 - 2879420 * q^45 + 1146496 * q^46 + 2520108 * q^49 + 2361600 * q^50 + 361088 * q^51 - 5116160 * q^54 - 526480 * q^55 + 417792 * q^56 + 2093520 * q^59 + 2237440 * q^60 - 5251432 * q^61 - 1048576 * q^64 + 2761440 * q^65 + 2401792 * q^66 + 6514864 * q^69 + 2679680 * q^70 - 7832352 * q^71 - 3126528 * q^74 - 7397600 * q^75 - 4070400 * q^76 + 7727040 * q^79 + 245760 * q^80 + 16428404 * q^81 + 4092928 * q^84 - 7995520 * q^85 + 4411776 * q^86 - 33470040 * q^89 - 15579520 * q^90 + 5340768 * q^91 + 15540352 * q^94 + 31904400 * q^95 + 3670016 * q^96 - 19109776 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$			8.00000i			0.707107i
$2$			8.00000i			0.707107i
$3$			27.6776i			0.591841i	0.955213	+	0.295920i	$0.0956263\pi$
$3$			27.6776i			0.591841i	−0.955213	+	0.295920i	$0.904374\pi$
$4$	−64.0000			−0.500000
$5$	−207.711	+	187.033i	−0.743128	+	0.669149i
$5$	−207.711	+	187.033i	−0.743128	+	0.669149i
$6$	−221.421			−0.418494
$7$			593.744i			0.654268i	0.944978	+	0.327134i	$0.106083\pi$
$7$			593.744i			0.654268i	−0.944978	+	0.327134i	$0.893917\pi$
$8$		−	512.000i		−	0.353553i
$9$	1420.95			0.649725
$10$	−1496.26	−	1661.68i	−0.473160	−	0.525471i
$11$	5342.84			1.21031			0.605157	−	0.796106i	$-0.293110\pi$
$11$	5342.84			1.21031			0.605157	+	0.796106i	$0.293110\pi$
$12$		−	1771.37i		−	0.295920i
$13$		−	2590.86i		−	0.327071i	−0.986537	−	0.163535i	$-0.947710\pi$
$13$		−	2590.86i		−	0.327071i	0.986537	−	0.163535i	$-0.0522898\pi$
$14$	−4749.95			−0.462637
$15$	−5176.63	−	5748.94i	−0.396030	−	0.439813i
$16$	4096.00			0.250000
$17$			31177.8i			1.53913i	0.638569	+	0.769564i	$0.279527\pi$
$17$			31177.8i			1.53913i	−0.638569	+	0.769564i	$0.720473\pi$
$18$			11367.6i			0.459425i
$19$	−18842.8			−0.630244			−0.315122	−	0.949051i	$-0.602046\pi$
$19$	−18842.8			−0.630244			−0.315122	+	0.949051i	$0.602046\pi$
$20$	13293.5	−	11970.1i	0.371564	−	0.334575i
$21$	−16433.4			−0.387222
$22$			42742.7i			0.855821i
$23$		−	83098.3i		−	1.42411i	−0.702122	−	0.712057i	$-0.747764\pi$
$23$		−	83098.3i		−	1.42411i	0.702122	−	0.712057i	$-0.252236\pi$
$24$	14171.0			0.209247
$25$	8162.37	−	77697.4i	0.104478	−	0.994527i
$26$	20726.8			0.231274
$27$			99859.5i			0.976374i
$28$		−	37999.6i		−	0.327134i
$29$	186706.			1.42156			0.710782	−	0.703412i	$-0.248341\pi$
$29$	186706.			1.42156			0.710782	+	0.703412i	$0.248341\pi$
$30$	45991.5	−	41413.0i	0.310995	−	0.280035i
$31$	−168014.			−1.01293			−0.506464	−	0.862261i	$-0.669048\pi$
$31$	−168014.			−1.01293			−0.506464	+	0.862261i	$0.669048\pi$
$32$			32768.0i			0.176777i
$33$			147877.i			0.716313i
$34$	−249423.			−1.08833
$35$	−111050.	−	123327.i	−0.437803	−	0.486205i
$36$	−90940.7			−0.324862
$37$			344801.i			1.11908i	0.828802	+	0.559542i	$0.189023\pi$
$37$			344801.i			1.11908i	−0.828802	+	0.559542i	$0.810977\pi$
$38$		−	150743.i		−	0.445650i
$39$	71708.8			0.193574
$40$	95760.9	+	106348.i	0.236580	+	0.262735i
$41$	490626.			1.11175			0.555875	−	0.831266i	$-0.312383\pi$
$41$	490626.			1.11175			0.555875	+	0.831266i	$0.312383\pi$
$42$		−	131467.i		−	0.273808i
$43$		−	101510.i		−	0.194702i	−0.995250	−	0.0973512i	$-0.968963\pi$
$43$		−	101510.i		−	0.194702i	0.995250	−	0.0973512i	$-0.0310370\pi$
$44$	−341942.			−0.605157
$45$	−295146.	+	265764.i	−0.482829	+	0.434763i
$46$	664787.			1.00700
$47$		−	1.39947e6i		−	1.96617i	−0.183138	−	0.983087i	$-0.558625\pi$
$47$		−	1.39947e6i		−	1.96617i	0.183138	−	0.983087i	$-0.441375\pi$
$48$			113368.i			0.147960i
$49$	471012.			0.571933
$50$	621579.	+	65298.9i	0.703237	+	0.0738773i
$51$	−862929.			−0.910919
$52$			165815.i			0.163535i
$53$			294530.i			0.271747i	0.990726	+	0.135873i	$0.0433840\pi$
$53$			294530.i			0.271747i	−0.990726	+	0.135873i	$0.956616\pi$
$54$	−798876.			−0.690401
$55$	−1.10976e6	+	999287.i	−0.899418	+	0.809881i
$56$	303997.			0.231319
$57$		−	521526.i		−	0.373004i
$58$			1.49365e6i			1.00520i
$59$	1.18171e6			0.749083			0.374541	−	0.927210i	$-0.377800\pi$
$59$	1.18171e6			0.749083			0.374541	+	0.927210i	$0.377800\pi$
$60$	331304.	+	367932.i	0.198015	+	0.219907i
$61$	−1.23135e6			−0.694585			−0.347293	−	0.937757i	$-0.612899\pi$
$61$	−1.23135e6			−0.694585			−0.347293	+	0.937757i	$0.612899\pi$
$62$		−	1.34411e6i		−	0.716249i
$63$			843679.i			0.425094i
$64$	−262144.			−0.125000
$65$	484575.	+	538148.i	0.218859	+	0.243055i
$66$	−1.18302e6			−0.506510
$67$		−	928902.i		−	0.377319i	−0.982043	−	0.188659i	$-0.939586\pi$
$67$		−	928902.i		−	0.377319i	0.982043	−	0.188659i	$-0.0604142\pi$
$68$		−	1.99538e6i		−	0.769564i
$69$	2.29997e6			0.842849
$70$	986614.	−	888397.i	0.343799	−	0.309574i
$71$	1.00307e6			0.332604			0.166302	−	0.986075i	$-0.446817\pi$
$71$	1.00307e6			0.332604			0.166302	+	0.986075i	$0.446817\pi$
$72$		−	727525.i		−	0.229712i
$73$		−	2.48740e6i		−	0.748369i	−0.927354	−	0.374184i	$-0.877923\pi$
$73$		−	2.48740e6i		−	0.748369i	0.927354	−	0.374184i	$-0.122077\pi$
$74$	−2.75841e6			−0.791312
$75$	2.15048e6	+	225915.i	0.588602	+	0.0618345i
$76$	1.20594e6			0.315122
$77$			3.17228e6i			0.791870i
$78$			573670.i			0.136877i
$79$	−842323.			−0.192213			−0.0961067	−	0.995371i	$-0.530639\pi$
$79$	−842323.			−0.192213			−0.0961067	+	0.995371i	$0.530639\pi$
$80$	−850783.	+	766087.i	−0.185782	+	0.167287i
$81$	343738.			0.0718670
$82$			3.92501e6i			0.786126i
$83$			4.01528e6i			0.770801i	0.922749	+	0.385401i	$0.125937\pi$
$83$			4.01528e6i			0.770801i	−0.922749	+	0.385401i	$0.874063\pi$
$84$	1.05174e6			0.193611
$85$	−5.83128e6	−	6.47597e6i	−1.02991	−	1.14377i
$86$	812084.			0.137675
$87$			5.16759e6i			0.841339i
$88$		−	2.73554e6i		−	0.427911i
$89$	−1.03006e7			−1.54881			−0.774407	−	0.632688i	$-0.781952\pi$
$89$	−1.03006e7			−1.54881			−0.774407	+	0.632688i	$0.781952\pi$
$90$	−2.12611e6	−	2.36117e6i	−0.307424	−	0.341411i
$91$	1.53830e6			0.213992
$92$			5.31829e6i			0.712057i
$93$		−	4.65022e6i		−	0.599492i
$94$	1.11958e7			1.39030
$95$	3.91386e6	−	3.52423e6i	0.468352	−	0.421728i
$96$	−906941.			−0.104624
$97$		−	93943.7i		−	0.0104512i	−0.999986	−	0.00522561i	$-0.998337\pi$
$97$		−	93943.7i		−	0.0104512i	0.999986	−	0.00522561i	$-0.00166337\pi$
$98$			3.76809e6i			0.404418i
$99$	7.59190e6			0.786371
$100$	−522391.	+	4.97264e6i	−0.0522391	+	0.497264i
$101$	5.45339e6			0.526674			0.263337	−	0.964704i	$-0.415177\pi$
$101$	5.45339e6			0.526674			0.263337	+	0.964704i	$0.415177\pi$
$102$		−	6.90343e6i		−	0.644117i
$103$		−	1.06239e7i		−	0.957976i	−0.877821	−	0.478988i	$-0.841004\pi$
$103$		−	1.06239e7i		−	0.957976i	0.877821	−	0.478988i	$-0.158996\pi$
$104$	−1.32652e6			−0.115637
$105$	3.41340e6	−	3.07359e6i	0.287756	−	0.259110i
$106$	−2.35624e6			−0.192154
$107$			7.62614e6i			0.601813i	0.953654	+	0.300906i	$0.0972892\pi$
$107$			7.62614e6i			0.601813i	−0.953654	+	0.300906i	$0.902711\pi$
$108$		−	6.39101e6i		−	0.488187i
$109$	−8.04427e6			−0.594968			−0.297484	−	0.954727i	$-0.596148\pi$
$109$	−8.04427e6			−0.594968			−0.297484	+	0.954727i	$0.596148\pi$
$110$	−7.99430e6	−	8.87812e6i	−0.572672	−	0.635985i
$111$	−9.54329e6			−0.662320
$112$			2.43197e6i			0.163567i
$113$			1.36799e7i			0.891885i	0.895062	+	0.445942i	$0.147131\pi$
$113$			1.36799e7i			0.891885i	−0.895062	+	0.445942i	$0.852869\pi$
$114$	4.17221e6			0.263754
$115$	1.55421e7	+	1.72604e7i	0.952945	+	1.05830i
$116$	−1.19492e7			−0.710782
$117$		−	3.68147e6i		−	0.212506i
$118$			9.45370e6i			0.529681i
$119$	−1.85116e7			−1.00700
$120$	−2.94346e6	+	2.65043e6i	−0.155497	+	0.140018i
$121$	9.05879e6			0.464859
$122$		−	9.85077e6i		−	0.491146i
$123$			1.35794e7i			0.657979i
$124$	1.07529e7			0.506464
$125$	1.28366e7	+	1.76652e7i	0.587846	+	0.808973i
$126$	−6.74943e6			−0.300587
$127$		−	3.18598e7i		−	1.38016i	−0.723732	−	0.690081i	$-0.757575\pi$
$127$		−	3.18598e7i		−	1.38016i	0.723732	−	0.690081i	$-0.242425\pi$
$128$		−	2.09715e6i		−	0.0883883i
$129$	2.80957e6			0.115233
$130$	−4.30519e6	+	3.87660e6i	−0.171866	+	0.154757i
$131$	−3.68319e6			−0.143144			−0.0715722	−	0.997435i	$-0.522802\pi$
$131$	−3.68319e6			−0.143144			−0.0715722	+	0.997435i	$0.522802\pi$
$132$		−	9.46415e6i		−	0.358156i
$133$		−	1.11878e7i		−	0.412349i
$134$	7.43122e6			0.266805
$135$	−1.86770e7	−	2.07419e7i	−0.653340	−	0.725571i
$136$	1.59631e7			0.544164
$137$		−	3.40447e7i		−	1.13117i	−0.824690	−	0.565585i	$-0.808650\pi$
$137$		−	3.40447e7i		−	1.13117i	0.824690	−	0.565585i	$-0.191350\pi$
$138$			1.83997e7i			0.595984i
$139$	6.44053e6			0.203409			0.101704	−	0.994815i	$-0.467570\pi$
$139$	6.44053e6			0.203409			0.101704	+	0.994815i	$0.467570\pi$
$140$	7.10717e6	+	7.89292e6i	0.218902	+	0.243102i
$141$	3.87341e7			1.16366
$142$			8.02457e6i			0.235187i
$143$		−	1.38425e7i		−	0.395858i
$144$	5.82020e6			0.162431
$145$	−3.87809e7	+	3.49203e7i	−1.05640	+	0.951238i
$146$	1.98992e7			0.529177
$147$			1.30365e7i			0.338493i
$148$		−	2.20673e7i		−	0.559542i
$149$	−6.94584e6			−0.172018			−0.0860088	−	0.996294i	$-0.527411\pi$
$149$	−6.94584e6			−0.172018			−0.0860088	+	0.996294i	$0.527411\pi$
$150$	−1.80732e6	+	1.72039e7i	−0.0437236	+	0.416204i
$151$	−6.85713e7			−1.62078			−0.810388	−	0.585894i	$-0.800744\pi$
$151$	−6.85713e7			−1.62078			−0.810388	+	0.585894i	$0.800744\pi$
$152$			9.64754e6i			0.222825i
$153$			4.43021e7i			1.00001i
$154$	−2.53782e7			−0.559936
$155$	3.48982e7	−	3.14241e7i	0.752735	−	0.677800i
$156$	−4.58936e6			−0.0967868
$157$			4.59334e7i			0.947283i	0.880718	+	0.473641i	$0.157061\pi$
$157$			4.59334e7i			0.947283i	−0.880718	+	0.473641i	$0.842939\pi$
$158$		−	6.73858e6i		−	0.135915i
$159$	−8.15190e6			−0.160831
$160$	−6.12870e6	−	6.80626e6i	−0.118290	−	0.131368i
$161$	4.93391e7			0.931752
$162$			2.74990e6i			0.0508177i
$163$		−	1.61919e7i		−	0.292847i	−0.989222	−	0.146424i	$-0.953224\pi$
$163$		−	1.61919e7i		−	0.292847i	0.989222	−	0.146424i	$-0.0467762\pi$
$164$	−3.14001e7			−0.555875
$165$	−2.76579e7	−	3.07157e7i	−0.479320	−	0.532312i
$166$	−3.21222e7			−0.545039
$167$			1.47618e6i			0.0245262i	0.999925	+	0.0122631i	$0.00390357\pi$
$167$			1.47618e6i			0.0245262i	−0.999925	+	0.0122631i	$0.996096\pi$
$168$			8.41391e6i			0.136904i
$169$	5.60360e7			0.893025
$170$	5.18077e7	−	4.66503e7i	0.808767	−	0.728254i
$171$	−2.67747e7			−0.409485
$172$			6.49667e6i			0.0973512i
$173$		−	7.62099e6i		−	0.111905i	−0.998433	−	0.0559526i	$-0.982180\pi$
$173$		−	7.62099e6i		−	0.111905i	0.998433	−	0.0559526i	$-0.0178196\pi$
$174$	−4.13408e7			−0.594917
$175$	4.61323e7	+	4.84635e6i	0.650687	+	0.0683568i
$176$	2.18843e7			0.302578
$177$			3.27070e7i			0.443338i
$178$		−	8.24051e7i		−	1.09518i
$179$	1.23954e8			1.61538			0.807688	−	0.589610i	$-0.200719\pi$
$179$	1.23954e8			1.61538			0.807688	+	0.589610i	$0.200719\pi$
$180$	1.88893e7	−	1.70089e7i	0.241414	−	0.217381i
$181$	2.44651e7			0.306670			0.153335	−	0.988174i	$-0.450999\pi$
$181$	2.44651e7			0.306670			0.153335	+	0.988174i	$0.450999\pi$
$182$			1.23064e7i			0.151315i
$183$		−	3.40808e7i		−	0.411084i
$184$	−4.25463e7			−0.503500
$185$	−6.44892e7	−	7.16189e7i	−0.748835	−	0.831623i
$186$	3.72018e7			0.423905
$187$			1.66578e8i			1.86283i
$188$			8.95663e7i			0.983087i
$189$	−5.92909e7			−0.638810
$190$	2.81939e7	+	3.13109e7i	0.298206	+	0.331175i
$191$	−1.86266e8			−1.93427			−0.967137	−	0.254257i	$-0.918169\pi$
$191$	−1.86266e8			−1.93427			−0.967137	+	0.254257i	$0.918169\pi$
$192$		−	7.25553e6i		−	0.0739801i
$193$		−	1.26292e8i		−	1.26452i	−0.774758	−	0.632258i	$-0.782128\pi$
$193$		−	1.26292e8i		−	1.26452i	0.774758	−	0.632258i	$-0.217872\pi$
$194$	751550.			0.00739012
$195$	−1.48947e7	+	1.34119e7i	−0.143850	+	0.129530i
$196$	−3.01447e7			−0.285967
$197$		−	5.43973e7i		−	0.506927i	−0.967345	−	0.253463i	$-0.918430\pi$
$197$		−	5.43973e7i		−	0.506927i	0.967345	−	0.253463i	$-0.0815697\pi$
$198$			6.07352e7i			0.556048i
$199$	5.56142e7			0.500265			0.250132	−	0.968212i	$-0.419526\pi$
$199$	5.56142e7			0.500265			0.250132	+	0.968212i	$0.419526\pi$
$200$	−3.97811e7	−	4.17913e6i	−0.351618	−	0.0369386i
$201$	2.57098e7			0.223312
$202$			4.36271e7i			0.372414i
$203$			1.10856e8i			0.930084i
$204$	5.52275e7			0.455459
$205$	−1.01908e8	+	9.17632e7i	−0.826172	+	0.743927i
$206$	8.49914e7			0.677392
$207$		−	1.18078e8i		−	0.925282i
$208$		−	1.06121e7i		−	0.0817677i
$209$	−1.00674e8			−0.762793
$210$	2.45887e7	+	2.73072e7i	0.183218	+	0.203474i
$211$	1.75227e8			1.28414			0.642071	−	0.766645i	$-0.278076\pi$
$211$	1.75227e8			1.28414			0.642071	+	0.766645i	$0.278076\pi$
$212$		−	1.88499e7i		−	0.135873i
$213$			2.77627e7i			0.196849i
$214$	−6.10091e7			−0.425546
$215$	1.89858e7	+	2.10848e7i	0.130285	+	0.144689i
$216$	5.11281e7			0.345200
$217$		−	9.97570e7i		−	0.662727i
$218$		−	6.43542e7i		−	0.420706i
$219$	6.88454e7			0.442915
$220$	7.10249e7	−	6.39544e7i	0.449709	−	0.404940i
$221$	8.07773e7			0.503404
$222$		−	7.63463e7i		−	0.468331i
$223$		−	7.18570e7i		−	0.433913i	−0.976181	−	0.216956i	$-0.930387\pi$
$223$		−	7.18570e7i		−	0.433913i	0.976181	−	0.216956i	$-0.0696129\pi$
$224$	−1.94558e7			−0.115659
$225$	1.15983e7	−	1.10404e8i	0.0678821	−	0.646169i
$226$	−1.09439e8			−0.630658
$227$		−	1.75914e8i		−	0.998185i	−0.866549	−	0.499093i	$-0.833667\pi$
$227$		−	1.75914e8i		−	0.998185i	0.866549	−	0.499093i	$-0.166333\pi$
$228$			3.33776e7i			0.186502i
$229$	−1.04748e8			−0.576395			−0.288198	−	0.957571i	$-0.593056\pi$
$229$	−1.04748e8			−0.576395			−0.288198	+	0.957571i	$0.593056\pi$
$230$	−1.38083e8	+	1.24337e8i	−0.748330	+	0.673834i
$231$	−8.78012e7			−0.468661
$232$		−	9.55937e7i		−	0.502599i
$233$			2.98119e8i			1.54399i	0.635631	+	0.771993i	$0.280740\pi$
$233$			2.98119e8i			1.54399i	−0.635631	+	0.771993i	$0.719260\pi$
$234$	2.94518e7			0.150264
$235$	2.61748e8	+	2.90685e8i	1.31566	+	1.46112i
$236$	−7.56296e7			−0.374541
$237$		−	2.33135e7i		−	0.113760i
$238$		−	1.48093e8i		−	0.712058i
$239$	2.28183e7			0.108116			0.0540580	−	0.998538i	$-0.482784\pi$
$239$	2.28183e7			0.108116			0.0540580	+	0.998538i	$0.482784\pi$
$240$	−2.12035e7	−	2.35477e7i	−0.0990074	−	0.109953i
$241$	1.60830e8			0.740127			0.370063	−	0.929006i	$-0.379336\pi$
$241$	1.60830e8			0.740127			0.370063	+	0.929006i	$0.379336\pi$
$242$			7.24703e7i			0.328705i
$243$			2.27907e8i			1.01891i
$244$	7.88061e7			0.347293
$245$	−9.78341e7	+	8.80947e7i	−0.425020	+	0.382709i
$246$	−1.08635e8			−0.465261
$247$			4.88191e7i			0.206134i
$248$			8.60230e7i			0.358124i
$249$	−1.11133e8			−0.456191
$250$	−1.41322e8	+	1.02693e8i	−0.572030	+	0.415670i
$251$	−4.62567e8			−1.84636			−0.923180	−	0.384367i	$-0.874420\pi$
$251$	−4.62567e8			−1.84636			−0.923180	+	0.384367i	$0.874420\pi$
$252$		−	5.39954e7i		−	0.212547i
$253$		−	4.43981e8i		−	1.72362i
$254$	2.54879e8			0.975922
$255$	1.79240e8	−	1.61396e8i	0.676929	−	0.609541i
$256$	1.67772e7			0.0625000
$257$			1.68347e8i			0.618642i	0.950958	+	0.309321i	$0.100102\pi$
$257$			1.68347e8i			0.618642i	−0.950958	+	0.309321i	$0.899898\pi$
$258$			2.24766e7i			0.0814819i
$259$	−2.04724e8			−0.732181
$260$	−3.10128e7	−	3.44415e7i	−0.109430	−	0.121528i
$261$	2.65300e8			0.923625
$262$		−	2.94655e7i		−	0.101218i
$263$			1.23997e7i			0.0420308i	0.999779	+	0.0210154i	$0.00668989\pi$
$263$			1.23997e7i			0.0420308i	−0.999779	+	0.0210154i	$0.993310\pi$
$264$	7.57132e7			0.253255
$265$	−5.50868e7	−	6.11770e7i	−0.181839	−	0.201943i
$266$	8.95026e7			0.291575
$267$		−	2.85097e8i		−	0.916651i
$268$			5.94497e7i			0.188659i
$269$	2.68614e8			0.841385			0.420693	−	0.907203i	$-0.361787\pi$
$269$	2.68614e8			0.841385			0.420693	+	0.907203i	$0.361787\pi$
$270$	1.65935e8	−	1.49416e8i	0.513056	−	0.461981i
$271$	−8.57975e7			−0.261868			−0.130934	−	0.991391i	$-0.541798\pi$
$271$	−8.57975e7			−0.261868			−0.130934	+	0.991391i	$0.541798\pi$
$272$			1.27704e8i			0.384782i
$273$			4.25766e7i			0.126649i
$274$	2.72358e8			0.799858
$275$	4.36102e7	−	4.15125e8i	0.126451	−	1.20369i
$276$	−1.47198e8			−0.421424
$277$			2.82124e8i			0.797556i	0.917047	+	0.398778i	$0.130566\pi$
$277$			2.82124e8i			0.797556i	−0.917047	+	0.398778i	$0.869434\pi$
$278$			5.15243e7i			0.143832i
$279$	−2.38739e8			−0.658125
$280$	−6.31433e7	+	5.68574e7i	−0.171899	+	0.154787i
$281$	3.48129e8			0.935982			0.467991	−	0.883733i	$-0.344978\pi$
$281$	3.48129e8			0.935982			0.467991	+	0.883733i	$0.344978\pi$
$282$			3.09873e8i			0.822833i
$283$		−	4.89087e8i		−	1.28273i	−0.767238	−	0.641363i	$-0.778369\pi$
$283$		−	4.89087e8i		−	1.28273i	0.767238	−	0.641363i	$-0.221631\pi$
$284$	−6.41966e7			−0.166302
$285$	9.75425e7	+	1.08326e8i	0.249596	+	0.277190i
$286$	1.10740e8			0.279914
$287$			2.91306e8i			0.727382i
$288$			4.65616e7i			0.114856i
$289$	−5.61719e8			−1.36892
$290$	−2.79362e8	−	3.10247e8i	−0.672627	−	0.746990i
$291$	2.60014e6			0.00618545
$292$			1.59194e8i			0.374184i
$293$			1.16120e8i			0.269693i	0.990867	+	0.134846i	$0.0430541\pi$
$293$			1.16120e8i			0.269693i	−0.990867	+	0.134846i	$0.956946\pi$
$294$	−1.04292e8			−0.239351
$295$	−2.45454e8	+	2.21019e8i	−0.556664	+	0.501248i
$296$	1.76538e8			0.395656
$297$			5.33534e8i			1.18172i
$298$		−	5.55667e7i		−	0.121635i
$299$	−2.15296e8			−0.465786
$300$	−1.37631e8	−	1.44586e7i	−0.294301	−	0.0309172i
$301$	6.02712e7			0.127388
$302$		−	5.48570e8i		−	1.14606i
$303$			1.50937e8i			0.311707i
$304$	−7.71803e7			−0.157561
$305$	2.55764e8	−	2.30302e8i	0.516166	−	0.464781i
$306$	−3.54417e8			−0.707114
$307$			3.91725e8i			0.772676i	0.922357	+	0.386338i	$0.126260\pi$
$307$			3.91725e8i			0.772676i	−0.922357	+	0.386338i	$0.873740\pi$
$308$		−	2.03026e8i		−	0.395935i
$309$	2.94045e8			0.566969
$310$	2.51393e8	+	2.79186e8i	0.479277	+	0.532264i
$311$	3.01811e8			0.568949			0.284475	−	0.958684i	$-0.408181\pi$
$311$	3.01811e8			0.568949			0.284475	+	0.958684i	$0.408181\pi$
$312$		−	3.67149e7i		−	0.0684386i
$313$			1.48081e8i			0.272958i	0.990643	+	0.136479i	$0.0435786\pi$
$313$			1.48081e8i			0.272958i	−0.990643	+	0.136479i	$0.956421\pi$
$314$	−3.67467e8			−0.669830
$315$	−1.57796e8	−	1.75241e8i	−0.284451	−	0.315899i
$316$	5.39087e7			0.0961067
$317$		−	3.26518e8i		−	0.575704i	−0.957675	−	0.287852i	$-0.907059\pi$
$317$		−	3.26518e8i		−	0.575704i	0.957675	−	0.287852i	$-0.0929411\pi$
$318$		−	6.52152e7i		−	0.113724i
$319$	9.97543e8			1.72054
$320$	5.44501e7	−	4.90296e7i	0.0928910	−	0.0836437i
$321$	−2.11074e8			−0.356177
$322$			3.94713e8i			0.658848i
$323$		−	5.87479e8i		−	0.970027i
$324$	−2.19992e7			−0.0359335
$325$	−2.01303e8	−	2.11475e7i	−0.325281	−	0.0341718i
$326$	1.29535e8			0.207074
$327$		−	2.22646e8i		−	0.352126i
$328$		−	2.51200e8i		−	0.393063i
$329$	8.30928e8			1.28641
$330$	2.45725e8	−	2.21263e8i	0.376401	−	0.338931i
$331$	−3.83566e8			−0.581355			−0.290678	−	0.956821i	$-0.593881\pi$
$331$	−3.83566e8			−0.581355			−0.290678	+	0.956821i	$0.593881\pi$
$332$		−	2.56978e8i		−	0.385401i
$333$			4.89945e8i			0.727097i
$334$	−1.18094e7			−0.0173427
$335$	1.73735e8	+	1.92943e8i	0.252483	+	0.280396i
$336$	−6.73113e7			−0.0968056
$337$		−	1.28992e9i		−	1.83594i	−0.396645	−	0.917972i	$-0.629826\pi$
$337$		−	1.28992e9i		−	1.83594i	0.396645	−	0.917972i	$-0.370174\pi$
$338$			4.48288e8i			0.631464i
$339$	−3.78628e8			−0.527854
$340$	3.73202e8	+	4.14462e8i	0.514953	+	0.571885i
$341$	−8.97671e8			−1.22596
$342$		−	2.14198e8i		−	0.289550i
$343$			7.68633e8i			1.02847i
$344$	−5.19734e7			−0.0688377
$345$	−4.77727e8	+	4.30169e8i	−0.626344	+	0.563992i
$346$	6.09679e7			0.0791289
$347$		−	1.81664e8i		−	0.233408i	−0.993167	−	0.116704i	$-0.962767\pi$
$347$		−	1.81664e8i		−	0.233408i	0.993167	−	0.116704i	$-0.0372329\pi$
$348$		−	3.30726e8i		−	0.420670i
$349$	−9.16337e8			−1.15390			−0.576948	−	0.816781i	$-0.695756\pi$
$349$	−9.16337e8			−1.15390			−0.576948	+	0.816781i	$0.695756\pi$
$350$	−3.87708e7	+	3.69059e8i	−0.0483356	+	0.460105i
$351$	2.58722e8			0.319343
$352$			1.75074e8i			0.213955i
$353$			2.23842e8i			0.270851i	0.990788	+	0.135426i	$0.0432402\pi$
$353$			2.23842e8i			0.270851i	−0.990788	+	0.135426i	$0.956760\pi$
$354$	−2.61656e8			−0.313487
$355$	−2.08349e8	+	1.87607e8i	−0.247168	+	0.222562i
$356$	6.59241e8			0.774407
$357$		−	5.12359e8i		−	0.595985i
$358$			9.91629e8i			1.14224i
$359$	5.30502e8			0.605141			0.302570	−	0.953127i	$-0.402155\pi$
$359$	5.30502e8			0.605141			0.302570	+	0.953127i	$0.402155\pi$
$360$	1.36071e8	+	1.51115e8i	0.153712	+	0.170706i
$361$	−5.38819e8			−0.602792
$362$			1.95721e8i			0.216849i
$363$			2.50726e8i			0.275123i
$364$	−9.84515e7			−0.106996
$365$	4.65226e8	+	5.16659e8i	0.500770	+	0.556134i
$366$	2.72646e8			0.290680
$367$			2.40908e8i			0.254402i	0.991877	+	0.127201i	$0.0405993\pi$
$367$			2.40908e8i			0.254402i	−0.991877	+	0.127201i	$0.959401\pi$
$368$		−	3.40371e8i		−	0.356029i
$369$	6.97154e8			0.722331
$370$	5.72951e8	−	5.15914e8i	0.588046	−	0.529506i
$371$	−1.74875e8			−0.177795
$372$			2.97614e8i			0.299746i
$373$		−	1.58549e9i		−	1.58191i	−0.611874	−	0.790955i	$-0.709584\pi$
$373$		−	1.58549e9i		−	1.58191i	0.611874	−	0.790955i	$-0.290416\pi$
$374$	−1.33263e9			−1.31722
$375$	−4.88931e8	+	3.55286e8i	−0.478783	+	0.347911i
$376$	−7.16530e8			−0.695148
$377$		−	4.83730e8i		−	0.464952i
$378$		−	4.74327e8i		−	0.451707i
$379$	2.12379e8			0.200389			0.100195	−	0.994968i	$-0.468053\pi$
$379$	2.12379e8			0.200389			0.100195	+	0.994968i	$0.468053\pi$
$380$	−2.50487e8	+	2.25551e8i	−0.234176	+	0.210864i
$381$	8.81805e8			0.816836
$382$		−	1.49013e9i		−	1.36774i
$383$			3.98208e8i			0.362172i	0.983467	+	0.181086i	$0.0579612\pi$
$383$			3.98208e8i			0.362172i	−0.983467	+	0.181086i	$0.942039\pi$
$384$	5.80442e7			0.0523118
$385$	−5.93320e8	−	6.58916e8i	−0.529879	−	0.588460i
$386$	1.01033e9			0.894148
$387$		−	1.44241e8i		−	0.126503i
$388$			6.01240e6i			0.00522561i
$389$	−2.10695e7			−0.0181480			−0.00907402	−	0.999959i	$-0.502888\pi$
$389$	−2.10695e7			−0.0181480			−0.00907402	+	0.999959i	$0.502888\pi$
$390$	−1.07295e8	−	1.19157e8i	−0.0915913	−	0.101717i
$391$	2.59083e9			2.19189
$392$		−	2.41158e8i		−	0.202209i
$393$		−	1.01942e8i		−	0.0847187i
$394$	4.35178e8			0.358451
$395$	1.74959e8	−	1.57542e8i	0.142839	−	0.128620i
$396$	−4.85882e8			−0.393185
$397$			1.41010e9i			1.13105i	0.824731	+	0.565526i	$0.191327\pi$
$397$			1.41010e9i			1.13105i	−0.824731	+	0.565526i	$0.808673\pi$
$398$			4.44914e8i			0.353741i
$399$	3.09652e8			0.244045
$400$	3.34330e7	−	3.18249e8i	0.0261196	−	0.248632i
$401$	−1.78639e9			−1.38347			−0.691737	−	0.722149i	$-0.743154\pi$
$401$	−1.78639e9			−1.38347			−0.691737	+	0.722149i	$0.743154\pi$
$402$			2.05679e8i			0.157906i
$403$			4.35299e8i			0.331299i
$404$	−3.49017e8			−0.263337
$405$	−7.13980e7	+	6.42903e7i	−0.0534064	+	0.0480898i
$406$	−8.86846e8			−0.657669
$407$			1.84222e9i			1.35444i
$408$			4.41820e8i			0.322058i
$409$	6.99574e8			0.505594			0.252797	−	0.967519i	$-0.418650\pi$
$409$	6.99574e8			0.505594			0.252797	+	0.967519i	$0.418650\pi$
$410$	−7.34106e8	−	8.15266e8i	−0.526036	−	0.584192i
$411$	9.42278e8			0.669473
$412$			6.79931e8i			0.478988i
$413$			7.01634e8i			0.490101i
$414$	9.44627e8			0.654273
$415$	−7.50990e8	−	8.34016e8i	−0.515781	−	0.572804i
$416$	8.48972e7			0.0578185
$417$			1.78259e8i			0.120386i
$418$		−	8.05395e8i		−	0.539376i
$419$	−2.51332e9			−1.66916			−0.834582	−	0.550884i	$-0.814291\pi$
$419$	−2.51332e9			−1.66916			−0.834582	+	0.550884i	$0.814291\pi$
$420$	−2.18457e8	+	1.96710e8i	−0.143878	+	0.129555i
$421$	−2.24308e9			−1.46507			−0.732533	−	0.680732i	$-0.761662\pi$
$421$	−2.24308e9			−1.46507			−0.732533	+	0.680732i	$0.761662\pi$
$422$			1.40182e9i			0.908025i
$423$		−	1.98858e9i		−	1.27747i
$424$	1.50799e8			0.0960770
$425$	2.42244e9	+	2.54485e8i	1.53071	+	0.160805i
$426$	−2.22101e8			−0.139193
$427$		−	7.31104e8i		−	0.454445i
$428$		−	4.88073e8i		−	0.300906i
$429$	3.83129e8			0.234285
$430$	−1.68678e8	+	1.51886e8i	−0.102310	+	0.0921254i
$431$	−1.52879e9			−0.919766			−0.459883	−	0.887980i	$-0.652109\pi$
$431$	−1.52879e9			−0.919766			−0.459883	+	0.887980i	$0.652109\pi$
$432$			4.09025e8i			0.244094i
$433$			8.91266e8i			0.527594i	0.964578	+	0.263797i	$0.0849749\pi$
$433$			8.91266e8i			0.527594i	−0.964578	+	0.263797i	$0.915025\pi$
$434$	7.98056e8			0.468619
$435$	−9.66510e8	−	1.07336e9i	−0.562981	−	0.625223i
$436$	5.14833e8			0.297484
$437$			1.56581e9i			0.897540i
$438$			5.50763e8i			0.313188i
$439$	6.50171e8			0.366777			0.183388	−	0.983041i	$-0.441293\pi$
$439$	6.50171e8			0.366777			0.183388	+	0.983041i	$0.441293\pi$
$440$	5.11635e8	+	5.68200e8i	0.286336	+	0.317992i
$441$	6.69283e8			0.371599
$442$			6.46218e8i			0.355960i
$443$		−	1.60107e9i		−	0.874981i	−0.899223	−	0.437490i	$-0.855867\pi$
$443$		−	1.60107e9i		−	0.874981i	0.899223	−	0.437490i	$-0.144133\pi$
$444$	6.10771e8			0.331160
$445$	2.13955e9	−	1.92656e9i	1.15097	−	1.03639i
$446$	5.74856e8			0.306823
$447$		−	1.92244e8i		−	0.101807i
$448$		−	1.55646e8i		−	0.0817835i
$449$	−1.33678e9			−0.696945			−0.348473	−	0.937319i	$-0.613300\pi$
$449$	−1.33678e9			−0.696945			−0.348473	+	0.937319i	$0.613300\pi$
$450$	8.83232e8	+	9.27864e7i	0.456910	+	0.0479999i
$451$	2.62134e9			1.34557
$452$		−	8.75515e8i		−	0.445942i
$453$		−	1.89789e9i		−	0.959241i
$454$	1.40732e9			0.705824
$455$	−3.19522e8	+	2.87714e8i	−0.159023	+	0.143193i
$456$	−2.67021e8			−0.131877
$457$			6.52558e8i			0.319825i	0.987131	+	0.159912i	$0.0511212\pi$
$457$			6.52558e8i			0.319825i	−0.987131	+	0.159912i	$0.948879\pi$
$458$		−	8.37982e8i		−	0.407573i
$459$	−3.11340e9			−1.50276
$460$	−9.94696e8	−	1.10467e9i	−0.476473	−	0.529150i
$461$	2.80778e8			0.133478			0.0667389	−	0.997770i	$-0.478741\pi$
$461$	2.80778e8			0.133478			0.0667389	+	0.997770i	$0.478741\pi$
$462$		−	7.02409e8i		−	0.331393i
$463$			3.47632e9i			1.62774i	0.581044	+	0.813872i	$0.302645\pi$
$463$			3.47632e9i			1.62774i	−0.581044	+	0.813872i	$0.697355\pi$
$464$	7.64750e8			0.355391
$465$	8.69745e8	+	9.65901e8i	0.401150	+	0.445499i
$466$	−2.38495e9			−1.09176
$467$		−	2.84726e9i		−	1.29365i	−0.762637	−	0.646827i	$-0.776095\pi$
$467$		−	2.84726e9i		−	1.29365i	0.762637	−	0.646827i	$-0.223905\pi$
$468$			2.35614e8i			0.106253i
$469$	5.51530e8			0.246868
$470$	−2.32548e9	+	2.09398e9i	−1.03317	+	0.930315i
$471$	−1.27133e9			−0.560640
$472$		−	6.05037e8i		−	0.264841i
$473$		−	5.42355e8i		−	0.235651i
$474$	1.86508e8			0.0804403
$475$	−1.53802e8	+	1.46404e9i	−0.0658468	+	0.626795i
$476$	1.18475e9			0.503501
$477$			4.18512e8i			0.176561i
$478$			1.82546e8i			0.0764496i
$479$	−1.48737e8			−0.0618366			−0.0309183	−	0.999522i	$-0.509843\pi$
$479$	−1.48737e8			−0.0618366			−0.0309183	+	0.999522i	$0.509843\pi$
$480$	1.88381e8	−	1.69628e8i	0.0777487	−	0.0700088i
$481$	8.93331e8			0.366020
$482$			1.28664e9i			0.523349i
$483$			1.36559e9i			0.551449i
$484$	−5.79763e8			−0.232430
$485$	1.75706e7	+	1.95131e7i	0.00699342	+	0.00776659i
$486$	−1.82325e9			−0.720477
$487$		−	1.32115e9i		−	0.518325i	−0.965834	−	0.259163i	$-0.916553\pi$
$487$		−	1.32115e9i		−	0.518325i	0.965834	−	0.259163i	$-0.0834465\pi$
$488$			6.30449e8i			0.245573i
$489$	4.48153e8			0.173319
$490$	−7.04758e8	−	7.82673e8i	−0.270616	−	0.300534i
$491$	2.02552e9			0.772238			0.386119	−	0.922449i	$-0.373815\pi$
$491$	2.02552e9			0.772238			0.386119	+	0.922449i	$0.373815\pi$
$492$		−	8.69080e8i		−	0.328989i
$493$			5.82110e9i			2.18797i
$494$	−3.90553e8			−0.145759
$495$	−1.57692e9	+	1.41994e9i	−0.584374	+	0.526200i
$496$	−6.88184e8			−0.253232
$497$			5.95567e8i			0.217612i
$498$		−	8.89068e8i		−	0.322576i
$499$	−1.18422e9			−0.426658			−0.213329	−	0.976980i	$-0.568431\pi$
$499$	−1.18422e9			−0.426658			−0.213329	+	0.976980i	$0.568431\pi$
$500$	−8.21540e8	−	1.13057e9i	−0.293923	−	0.404486i
$501$	−4.08571e7			−0.0145156
$502$		−	3.70054e9i		−	1.30557i
$503$		−	4.51308e9i		−	1.58120i	−0.612336	−	0.790598i	$-0.709770\pi$
$503$		−	4.51308e9i		−	1.58120i	0.612336	−	0.790598i	$-0.290230\pi$
$504$	4.31963e8			0.150293
$505$	−1.13273e9	+	1.01996e9i	−0.391386	+	0.352423i
$506$	3.55185e9			1.21879
$507$			1.55094e9i			0.528528i
$508$			2.03903e9i			0.690081i
$509$	3.64968e9			1.22671			0.613355	−	0.789807i	$-0.289819\pi$
$509$	3.64968e9			1.22671			0.613355	+	0.789807i	$0.289819\pi$
$510$	1.29117e9	+	1.43392e9i	0.431010	+	0.478661i
$511$	1.47688e9			0.489634
$512$			1.34218e8i			0.0441942i
$513$		−	1.88164e9i		−	0.615354i
$514$	−1.34678e9			−0.437446
$515$	1.98702e9	+	2.20670e9i	0.641029	+	0.711899i
$516$	−1.79813e8			−0.0576164
$517$		−	7.47716e9i		−	2.37969i
$518$		−	1.63779e9i		−	0.517730i
$519$	2.10931e8			0.0662300
$520$	2.75532e8	−	2.48103e8i	0.0859330	−	0.0773784i
$521$	−2.44041e9			−0.756017			−0.378008	−	0.925802i	$-0.623391\pi$
$521$	−2.44041e9			−0.756017			−0.378008	+	0.925802i	$0.623391\pi$
$522$			2.12240e9i			0.653102i
$523$		−	5.40105e8i		−	0.165090i	−0.996587	−	0.0825452i	$-0.973695\pi$
$523$		−	5.40105e8i		−	0.165090i	0.996587	−	0.0825452i	$-0.0263049\pi$
$524$	2.35724e8			0.0715722
$525$	−1.34136e8	+	1.27683e9i	−0.0404563	+	0.385103i
$526$	−9.91979e7			−0.0297202
$527$		−	5.23830e9i		−	1.55903i
$528$			6.05705e8i			0.179078i
$529$	−3.50051e9			−1.02810
$530$	4.89416e8	−	4.40695e8i	0.142795	−	0.128580i
$531$	1.67915e9			0.486698
$532$			7.16020e8i			0.206174i
$533$		−	1.27114e9i		−	0.363621i
$534$	2.28078e9			0.648170
$535$	−1.42634e9	−	1.58403e9i	−0.402703	−	0.447224i
$536$	−4.75598e8			−0.133402
$537$			3.43074e9i			0.956045i
$538$			2.14891e9i			0.594949i
$539$	2.51654e9			0.692219
$540$	1.19533e9	+	1.32748e9i	0.326670	+	0.362785i
$541$	−3.22245e9			−0.874975			−0.437487	−	0.899225i	$-0.644132\pi$
$541$	−3.22245e9			−0.874975			−0.437487	+	0.899225i	$0.644132\pi$
$542$		−	6.86380e8i		−	0.185168i
$543$			6.77136e8i			0.181500i
$544$	−1.02164e9			−0.272082
$545$	1.67088e9	−	1.50454e9i	0.442138	−	0.398123i
$546$	−3.40613e8			−0.0895544
$547$			6.94609e9i			1.81462i	0.420468	+	0.907308i	$0.361866\pi$
$547$			6.94609e9i			1.81462i	−0.420468	+	0.907308i	$0.638134\pi$
$548$			2.17886e9i			0.565585i
$549$	−1.74968e9			−0.451289
$550$	3.32100e9	+	3.48882e8i	0.851137	+	0.0894147i
$551$	−3.51808e9			−0.895933
$552$		−	1.17758e9i		−	0.297992i
$553$		−	5.00124e8i		−	0.125759i
$554$	−2.25699e9			−0.563958
$555$	1.98224e9	−	1.78491e9i	0.492188	−	0.443191i
$556$	−4.12194e8			−0.101704
$557$			3.27363e9i			0.802669i	0.915932	+	0.401335i	$0.131454\pi$
$557$			3.27363e9i			0.802669i	−0.915932	+	0.401335i	$0.868546\pi$
$558$		−	1.90991e9i		−	0.465364i
$559$	−2.62999e8			−0.0636814
$560$	−4.54859e8	−	5.05147e8i	−0.109451	−	0.121551i
$561$	−4.61049e9			−1.10250
$562$			2.78503e9i			0.661839i
$563$			6.64466e9i			1.56925i	0.619968	+	0.784627i	$0.287146\pi$
$563$			6.64466e9i			1.56925i	−0.619968	+	0.784627i	$0.712854\pi$
$564$	−2.47898e9			−0.581831
$565$	−2.55859e9	−	2.84146e9i	−0.596804	−	0.662785i
$566$	3.91270e9			0.907024
$567$			2.04092e8i			0.0470203i
$568$		−	5.13573e8i		−	0.117593i
$569$	4.72242e9			1.07466			0.537331	−	0.843372i	$-0.319433\pi$
$569$	4.72242e9			1.07466			0.537331	+	0.843372i	$0.319433\pi$
$570$	−8.66611e8	+	7.80340e8i	−0.196003	+	0.176491i
$571$	4.68329e9			1.05275			0.526374	−	0.850253i	$-0.323551\pi$
$571$	4.68329e9			1.05275			0.526374	+	0.850253i	$0.323551\pi$
$572$			8.85922e8i			0.197929i
$573$		−	5.15542e9i		−	1.14478i
$574$	−2.33045e9			−0.514337
$575$	−6.45653e9	−	6.78279e8i	−1.41632	−	0.148789i
$576$	−3.72493e8			−0.0812156
$577$		−	3.85243e9i		−	0.834872i	−0.908706	−	0.417436i	$-0.862929\pi$
$577$		−	3.85243e9i		−	0.834872i	0.908706	−	0.417436i	$-0.137071\pi$
$578$		−	4.49375e9i		−	0.967970i
$579$	3.49546e9			0.748392
$580$	2.48198e9	−	2.23490e9i	0.528202	−	0.475619i
$581$	−2.38405e9			−0.504311
$582$			2.08011e7i			0.00437377i
$583$			1.57363e9i			0.328899i
$584$	−1.27355e9			−0.264588
$585$	6.88557e8	+	7.64681e8i	0.142198	+	0.157919i
$586$	−9.28957e8			−0.190702
$587$			6.20959e9i			1.26715i	0.773680	+	0.633577i	$0.218414\pi$
$587$			6.20959e9i			1.26715i	−0.773680	+	0.633577i	$0.781586\pi$
$588$		−	8.34336e8i		−	0.169247i
$589$	3.16586e9			0.638392
$590$	−1.76815e9	−	1.96363e9i	−0.354436	−	0.393621i
$591$	1.50559e9			0.300020
$592$			1.41231e9i			0.279771i
$593$		−	5.38056e9i		−	1.05958i	−0.848127	−	0.529792i	$-0.822270\pi$
$593$		−	5.38056e9i		−	1.05958i	0.848127	−	0.529792i	$-0.177730\pi$
$594$	−4.26827e9			−0.835601
$595$	3.84506e9	−	3.46229e9i	0.748332	−	0.673835i
$596$	4.44534e8			0.0860088
$597$			1.53927e9i			0.296077i
$598$		−	1.72237e9i		−	0.329360i
$599$	9.48876e9			1.80391			0.901957	−	0.431826i	$-0.142131\pi$
$599$	9.48876e9			1.80391			0.901957	+	0.431826i	$0.142131\pi$
$600$	1.15669e8	−	1.10105e9i	0.0218618	−	0.208102i
$601$	8.26392e9			1.55284			0.776418	−	0.630218i	$-0.217035\pi$
$601$	8.26392e9			1.55284			0.776418	+	0.630218i	$0.217035\pi$
$602$			4.82170e8i			0.0900766i
$603$		−	1.31992e9i		−	0.245153i
$604$	4.38856e9			0.810388
$605$	−1.88161e9	+	1.69429e9i	−0.345450	+	0.311060i
$606$	−1.20750e9			−0.220410
$607$			2.68979e9i			0.488156i	0.969756	+	0.244078i	$0.0784853\pi$
$607$			2.68979e9i			0.488156i	−0.969756	+	0.244078i	$0.921515\pi$
$608$		−	6.17442e8i		−	0.111413i
$609$	−3.06823e9			−0.550461
$610$	1.84242e9	+	2.04611e9i	0.328650	+	0.364984i
$611$	−3.62583e9			−0.643078
$612$		−	2.83533e9i		−	0.500005i
$613$			7.96983e9i			1.39745i	0.715389	+	0.698726i	$0.246250\pi$
$613$			7.96983e9i			1.39745i	−0.715389	+	0.698726i	$0.753750\pi$
$614$	−3.13380e9			−0.546364
$615$	−2.53979e9	−	2.82058e9i	−0.440286	−	0.488962i
$616$	1.62421e9			0.279968
$617$			1.27485e9i			0.218506i	0.994014	+	0.109253i	$0.0348458\pi$
$617$			1.27485e9i			0.218506i	−0.994014	+	0.109253i	$0.965154\pi$
$618$			2.35236e9i			0.400908i
$619$	−7.29379e9			−1.23605			−0.618025	−	0.786158i	$-0.712067\pi$
$619$	−7.29379e9			−1.23605			−0.618025	+	0.786158i	$0.712067\pi$
$620$	−2.23349e9	+	2.01114e9i	−0.376368	+	0.338900i
$621$	8.29816e9			1.39047
$622$			2.41449e9i			0.402308i
$623$		−	6.11594e9i		−	1.01334i
$624$	2.93719e8			0.0483934
$625$	−5.97027e9	−	1.26839e9i	−0.978169	−	0.207813i
$626$	−1.18465e9			−0.193010
$627$		−	2.78643e9i		−	0.451452i
$628$		−	2.93974e9i		−	0.473641i
$629$	−1.07502e10			−1.72242
$630$	1.40193e9	−	1.26237e9i	0.223375	−	0.201138i
$631$	−1.14564e10			−1.81529			−0.907644	−	0.419741i	$-0.862121\pi$
$631$	−1.14564e10			−1.81529			−0.907644	+	0.419741i	$0.862121\pi$
$632$			4.31269e8i			0.0679577i
$633$			4.84988e9i			0.760007i
$634$	2.61214e9			0.407084
$635$	5.95883e9	+	6.61762e9i	0.923535	+	1.02564i
$636$	5.21722e8			0.0804154
$637$		−	1.22032e9i		−	0.187063i
$638$			7.98035e9i			1.21660i
$639$	1.42531e9			0.216101
$640$	3.92236e8	+	4.35601e8i	0.0591450	+	0.0656839i
$641$	−1.37149e9			−0.205680			−0.102840	−	0.994698i	$-0.532793\pi$
$641$	−1.37149e9			−0.205680			−0.102840	+	0.994698i	$0.532793\pi$
$642$		−	1.68859e9i		−	0.251855i
$643$		−	7.34501e9i		−	1.08957i	−0.838577	−	0.544784i	$-0.816612\pi$
$643$		−	7.34501e9i		−	1.08957i	0.838577	−	0.544784i	$-0.183388\pi$
$644$	−3.15770e9			−0.465876
$645$	−5.83578e8	+	5.25482e8i	−0.0856327	+	0.0771079i
$646$	4.69984e9			0.685913
$647$			2.96720e9i			0.430708i	0.976536	+	0.215354i	$0.0690905\pi$
$647$			2.96720e9i			0.430708i	−0.976536	+	0.215354i	$0.930910\pi$
$648$		−	1.75994e8i		−	0.0254088i
$649$	6.31370e9			0.906625
$650$	1.69180e8	−	1.61042e9i	0.0241631	−	0.230008i
$651$	2.76104e9			0.392229
$652$			1.03628e9i			0.146424i
$653$			6.34555e9i			0.891812i	0.895080	+	0.445906i	$0.147118\pi$
$653$			6.34555e9i			0.891812i	−0.895080	+	0.445906i	$0.852882\pi$
$654$	1.78117e9			0.248991
$655$	7.65037e8	−	6.88878e8i	0.106375	−	0.0957850i
$656$	2.00960e9			0.277937
$657$		−	3.53447e9i		−	0.486234i
$658$			6.64742e9i			0.909626i
$659$	−3.94753e9			−0.537312			−0.268656	−	0.963236i	$-0.586580\pi$
$659$	−3.94753e9			−0.537312			−0.268656	+	0.963236i	$0.586580\pi$
$660$	1.77011e9	+	1.96580e9i	0.239660	+	0.266156i
$661$	−1.36470e7			−0.00183794			−0.000918969	−	1.00000i	$-0.500293\pi$
$661$	−1.36470e7			−0.00183794			−0.000918969		1.00000i	$0.500293\pi$
$662$		−	3.06852e9i		−	0.411080i
$663$			2.23573e9i			0.297935i
$664$	2.05582e9			0.272519
$665$	2.09249e9	+	2.32383e9i	0.275923	+	0.306428i
$666$	−3.91956e9			−0.514135
$667$		−	1.55150e10i		−	2.02447i
$668$		−	9.44753e7i		−	0.0122631i
$669$	1.98883e9			0.256807
$670$	−1.54354e9	+	1.38988e9i	−0.198270	+	0.178532i
$671$	−6.57889e9			−0.840666
$672$		−	5.38490e8i		−	0.0684519i
$673$		−	1.11700e10i		−	1.41253i	−0.707946	−	0.706267i	$-0.750378\pi$
$673$		−	1.11700e10i		−	1.41253i	0.707946	−	0.706267i	$-0.249622\pi$
$674$	1.03194e10			1.29821
$675$	7.75883e9	+	8.15090e8i	0.971030	+	0.102010i
$676$	−3.58630e9			−0.446512
$677$		−	1.19620e10i		−	1.48165i	−0.671701	−	0.740823i	$-0.734436\pi$
$677$		−	1.19620e10i		−	1.48165i	0.671701	−	0.740823i	$-0.265564\pi$
$678$		−	3.02902e9i		−	0.373249i
$679$	5.57785e7			0.00683789
$680$	−3.31570e9	+	2.98562e9i	−0.404384	+	0.364127i
$681$	4.86890e9			0.590766
$682$		−	7.18137e9i		−	0.866885i
$683$			7.37957e9i			0.886255i	0.896459	+	0.443128i	$0.146131\pi$
$683$			7.37957e9i			0.886255i	−0.896459	+	0.443128i	$0.853869\pi$
$684$	1.71358e9			0.204743
$685$	6.36749e9	+	7.07145e9i	0.756922	+	0.840604i
$686$	−6.14907e9			−0.727235
$687$		−	2.89917e9i		−	0.341134i
$688$		−	4.15787e8i		−	0.0486756i
$689$	7.63085e8			0.0888804
$690$	−3.44135e9	−	3.82182e9i	−0.398802	−	0.442892i
$691$	−7.44734e9			−0.858674			−0.429337	−	0.903144i	$-0.641253\pi$
$691$	−7.44734e9			−0.858674			−0.429337	+	0.903144i	$0.641253\pi$
$692$			4.87744e8i			0.0559526i
$693$			4.50764e9i			0.514497i
$694$	1.45331e9			0.165044
$695$	−1.33777e9	+	1.20459e9i	−0.151159	+	0.136111i
$696$	2.64581e9			0.297458
$697$			1.52967e10i			1.71113i
$698$		−	7.33070e9i		−	0.815927i
$699$	−8.25122e9			−0.913794
$700$	−2.95247e9	−	3.10166e8i	−0.325344	−	0.0341784i
$701$	1.28712e10			1.41125			0.705627	−	0.708583i	$-0.250665\pi$
$701$	1.28712e10			1.41125			0.705627	+	0.708583i	$0.250665\pi$
$702$			2.06977e9i			0.225810i
$703$		−	6.49704e9i		−	0.705297i
$704$	−1.40059e9			−0.151289
$705$	−8.04549e9	+	7.24456e9i	−0.864750	+	0.778664i
$706$	−1.79074e9			−0.191521
$707$			3.23791e9i			0.344586i
$708$		−	2.09325e9i		−	0.221669i
$709$	−3.39607e9			−0.357861			−0.178931	−	0.983862i	$-0.557264\pi$
$709$	−3.39607e9			−0.357861			−0.178931	+	0.983862i	$0.557264\pi$
$710$	−1.50086e9	−	1.66679e9i	−0.157375	−	0.174774i
$711$	−1.19690e9			−0.124886
$712$			5.27393e9i			0.547588i
$713$			1.39617e10i			1.44253i
$714$	4.09887e9			0.421425
$715$	2.58901e9	+	2.87524e9i	0.264888	+	0.294173i
$716$	−7.93303e9			−0.807688
$717$			6.31556e8i			0.0639875i
$718$			4.24402e9i			0.427899i
$719$	7.48932e9			0.751434			0.375717	−	0.926734i	$-0.377396\pi$
$719$	7.48932e9			0.751434			0.375717	+	0.926734i	$0.377396\pi$
$720$	−1.20892e9	+	1.08857e9i	−0.120707	+	0.108691i
$721$	6.30789e9			0.626773
$722$		−	4.31055e9i		−	0.426238i
$723$			4.45138e9i			0.438037i
$724$	−1.56577e9			−0.153335
$725$	1.52397e9	−	1.45066e10i	0.148523	−	1.41378i
$726$	−2.00581e9			−0.194541
$727$		−	1.40205e9i		−	0.135330i	−0.997708	−	0.0676648i	$-0.978445\pi$
$727$		−	1.40205e9i		−	0.135330i	0.997708	−	0.0676648i	$-0.0215548\pi$
$728$		−	7.87612e8i		−	0.0756576i
$729$	−5.55616e9			−0.531164
$730$	−4.13327e9	+	3.72181e9i	−0.393246	+	0.354098i
$731$	3.16488e9			0.299672
$732$			2.18117e9i			0.205542i
$733$		−	1.75214e9i		−	0.164326i	−0.996619	−	0.0821628i	$-0.973817\pi$
$733$		−	1.75214e9i		−	0.164326i	0.996619	−	0.0821628i	$-0.0261828\pi$
$734$	−1.92727e9			−0.179889
$735$	−2.43825e9	−	2.70782e9i	−0.226503	−	0.251544i
$736$	2.72297e9			0.251750
$737$		−	4.96298e9i		−	0.456674i
$738$			5.57723e9i			0.510765i
$739$	−1.38739e10			−1.26457			−0.632287	−	0.774734i	$-0.717884\pi$
$739$	−1.38739e10			−1.26457			−0.632287	+	0.774734i	$0.717884\pi$
$740$	4.12731e9	+	4.58361e9i	0.374417	+	0.415812i
$741$	−1.35120e9			−0.121999
$742$		−	1.39900e9i		−	0.125720i
$743$			2.95000e8i			0.0263852i	0.999913	+	0.0131926i	$0.00419946\pi$
$743$			2.95000e8i			0.0263852i	−0.999913	+	0.0131926i	$0.995801\pi$
$744$	−2.38091e9			−0.211952
$745$	1.44272e9	−	1.29910e9i	0.127831	−	0.115105i
$746$	1.26839e10			1.11858
$747$			5.70550e9i			0.500809i
$748$		−	1.06610e10i		−	0.931414i
$749$	−4.52797e9			−0.393747
$750$	−2.84229e9	−	3.91145e9i	−0.246011	−	0.338551i
$751$	−3.41467e7			−0.00294177			−0.00147089	−	0.999999i	$-0.500468\pi$
$751$	−3.41467e7			−0.00294177			−0.00147089	+	0.999999i	$0.500468\pi$
$752$		−	5.73224e9i		−	0.491544i
$753$		−	1.28028e10i		−	1.09275i
$754$	3.86984e9			0.328771
$755$	1.42430e10	−	1.28251e10i	1.20444	−	1.08454i
$756$	3.79462e9			0.319405
$757$			1.60197e10i			1.34220i	0.741366	+	0.671101i	$0.234178\pi$
$757$			1.60197e10i			1.34220i	−0.741366	+	0.671101i	$0.765822\pi$
$758$			1.69903e9i			0.141697i
$759$	1.22884e10			1.02011
$760$	−1.80441e9	−	2.00390e9i	−0.149103	−	0.165588i
$761$	7.40377e9			0.608985			0.304493	−	0.952515i	$-0.401513\pi$
$761$	7.40377e9			0.608985			0.304493	+	0.952515i	$0.401513\pi$
$762$			7.05444e9i			0.577590i
$763$		−	4.77623e9i		−	0.389269i
$764$	1.19211e10			0.967137
$765$	−8.28595e9	−	9.20201e9i	−0.669156	−	0.743135i
$766$	−3.18567e9			−0.256094
$767$		−	3.06165e9i		−	0.245003i
$768$			4.64354e8i			0.0369900i
$769$	−9.29033e9			−0.736697			−0.368348	−	0.929688i	$-0.620077\pi$
$769$	−9.29033e9			−0.736697			−0.368348	+	0.929688i	$0.620077\pi$
$770$	5.27133e9	−	4.74656e9i	0.416104	−	0.374681i
$771$	−4.65945e9			−0.366137
$772$			8.08267e9i			0.632258i
$773$		−	2.05039e10i		−	1.59664i	−0.602233	−	0.798321i	$-0.705722\pi$
$773$		−	2.05039e10i		−	1.59664i	0.602233	−	0.798321i	$-0.294278\pi$
$774$	1.15393e9			0.0894511
$775$	−1.37139e9	+	1.30542e10i	−0.105829	+	1.00738i
$776$	−4.80992e7			−0.00369506
$777$		−	5.66627e9i		−	0.433335i
$778$		−	1.68556e8i		−	0.0128326i
$779$	−9.24479e9			−0.700674
$780$	9.53259e8	−	8.58362e8i	0.0719250	−	0.0647649i
$781$	5.35925e9			0.402555
$782$			2.07266e10i			1.54990i
$783$			1.86444e10i			1.38798i
$784$	1.92926e9			0.142983
$785$	−8.59105e9	−	9.54085e9i	−0.633874	−	0.703952i
$786$	8.15536e8			0.0599052
$787$			8.34278e9i			0.610098i	0.952337	+	0.305049i	$0.0986728\pi$
$787$			8.34278e9i			0.610098i	−0.952337	+	0.305049i	$0.901327\pi$
$788$			3.48142e9i			0.253463i
$789$	−3.43195e8			−0.0248755
$790$	1.26034e9	+	1.39968e9i	0.0909477	+	0.101003i
$791$	−8.12236e9			−0.583532
$792$		−	3.88705e9i		−	0.278024i
$793$			3.19024e9i			0.227179i
$794$	−1.12808e10			−0.799774
$795$	1.69324e9	−	1.52467e9i	0.119518	−	0.107620i
$796$	−3.55931e9			−0.250132
$797$			1.71990e10i			1.20337i	0.798733	+	0.601686i	$0.205504\pi$
$797$			1.71990e10i			1.20337i	−0.798733	+	0.601686i	$0.794496\pi$
$798$			2.47722e9i			0.172566i
$799$	4.36326e10			3.02620
$800$	2.54599e9	+	2.67464e8i	0.175809	+	0.0184693i
$801$	−1.46367e10			−1.00630
$802$		−	1.42911e10i		−	0.978264i
$803$		−	1.32898e10i		−	0.905761i
$804$	−1.64543e9			−0.111656
$805$	−1.02483e10	+	9.22803e9i	−0.692411	+	0.623482i
$806$	−3.48239e9			−0.234264
$807$			7.43459e9i			0.497966i
$808$		−	2.79213e9i		−	0.186207i
$809$	−1.76602e10			−1.17267			−0.586335	−	0.810068i	$-0.699430\pi$
$809$	−1.76602e10			−1.17267			−0.586335	+	0.810068i	$0.699430\pi$
$810$	−5.14322e8	−	5.71184e8i	−0.0340046	−	0.0377640i
$811$	−1.29259e10			−0.850921			−0.425461	−	0.904977i	$-0.639888\pi$
$811$	−1.29259e10			−0.850921			−0.425461	+	0.904977i	$0.639888\pi$
$812$		−	7.09477e9i		−	0.465042i
$813$		−	2.37467e9i		−	0.154984i
$814$	−1.47378e10			−0.957736
$815$	3.02842e9	+	3.36323e9i	0.195958	+	0.217623i
$816$	−3.53456e9			−0.227730
$817$			1.91275e9i			0.122710i
$818$			5.59659e9i			0.357509i
$819$	2.18585e9			0.139036
$820$	6.52212e9	−	5.87285e9i	0.413086	−	0.371963i
$821$	−4.68261e9			−0.295316			−0.147658	−	0.989038i	$-0.547173\pi$
$821$	−4.68261e9			−0.295316			−0.147658	+	0.989038i	$0.547173\pi$
$822$			7.53823e9i			0.473389i
$823$			8.04444e8i			0.0503033i	0.999684	+	0.0251516i	$0.00800686\pi$
$823$			8.04444e8i			0.0503033i	−0.999684	+	0.0251516i	$0.991993\pi$
$824$	−5.43945e9			−0.338696
$825$	1.14897e10	+	1.20703e9i	0.712392	+	0.0748391i
$826$	−5.61307e9			−0.346554
$827$		−	9.35990e9i		−	0.575442i	−0.957714	−	0.287721i	$-0.907102\pi$
$827$		−	9.35990e9i		−	0.575442i	0.957714	−	0.287721i	$-0.0928976\pi$
$828$			7.55702e9i			0.462641i
$829$	2.14103e10			1.30521			0.652607	−	0.757697i	$-0.273675\pi$
$829$	2.14103e10			1.30521			0.652607	+	0.757697i	$0.273675\pi$
$830$	6.67213e9	−	6.00792e9i	0.405034	−	0.364712i
$831$	−7.80854e9			−0.472026
$832$			6.79177e8i			0.0408838i
$833$			1.46851e10i			0.880279i
$834$	−1.42607e9			−0.0851255
$835$	−2.76094e8	−	3.06617e8i	−0.0164117	−	0.0182261i
$836$	6.44316e9			0.381397
$837$		−	1.67778e10i		−	0.988997i
$838$		−	2.01066e10i		−	1.18028i
$839$	1.98675e10			1.16139			0.580693	−	0.814123i	$-0.302782\pi$
$839$	1.98675e10			1.16139			0.580693	+	0.814123i	$0.302782\pi$
$840$	−1.57368e9	−	1.74766e9i	−0.0916091	−	0.101737i
$841$	1.76094e10			1.02084
$842$		−	1.79446e10i		−	1.03596i
$843$			9.63538e9i			0.553952i
$844$	−1.12145e10			−0.642071
$845$	−1.16393e10	+	1.04806e10i	−0.663632	+	0.597567i
$846$	1.59086e10			0.903309
$847$			5.37860e9i			0.304143i
$848$			1.20640e9i			0.0679367i
$849$	1.35368e10			0.759169
$850$	−2.03588e9	+	1.93795e10i	−0.113707	+	1.08237i
$851$	2.86524e10			1.59370
$852$		−	1.77681e9i		−	0.0984243i
$853$		−	1.46955e10i		−	0.810704i	−0.914161	−	0.405352i	$-0.867149\pi$
$853$		−	1.46955e10i		−	0.810704i	0.914161	−	0.405352i	$-0.132851\pi$
$854$	5.84883e9			0.321341
$855$	5.56139e9	−	5.00775e9i	0.304300	−	0.274007i
$856$	3.90458e9			0.212773
$857$		−	2.40440e10i		−	1.30489i	−0.757835	−	0.652446i	$-0.773743\pi$
$857$		−	2.40440e10i		−	1.30489i	0.757835	−	0.652446i	$-0.226257\pi$
$858$			3.06503e9i			0.165664i
$859$	−3.43457e10			−1.84883			−0.924414	−	0.381391i	$-0.875445\pi$
$859$	−3.43457e10			−1.84883			−0.924414	+	0.381391i	$0.875445\pi$
$860$	−1.21509e9	−	1.34943e9i	−0.0651425	−	0.0723444i
$861$	−8.06266e9			−0.430494
$862$		−	1.22303e10i		−	0.650373i
$863$			2.30103e10i			1.21867i	0.792914	+	0.609334i	$0.208563\pi$
$863$			2.30103e10i			1.21867i	−0.792914	+	0.609334i	$0.791437\pi$
$864$	−3.27220e9			−0.172600
$865$	1.42538e9	+	1.58296e9i	0.0748813	+	0.0831599i
$866$	−7.13013e9			−0.373065
$867$		−	1.55471e10i		−	0.810180i
$868$			6.38445e9i			0.331363i
$869$	−4.50040e9			−0.232639
$870$	8.58691e9	−	7.73208e9i	0.442099	−	0.398088i
$871$	−2.40665e9			−0.123410
$872$			4.11867e9i			0.210353i
$873$		−	1.33489e8i		−	0.00679041i
$874$	−1.25265e10			−0.634657
$875$	−1.04886e10	+	7.62163e9i	−0.529285	+	0.384609i
$876$	−4.40610e9			−0.221457
$877$		−	4.20994e9i		−	0.210755i	−0.994432	−	0.105377i	$-0.966395\pi$
$877$		−	4.20994e9i		−	0.210755i	0.994432	−	0.105377i	$-0.0336051\pi$
$878$			5.20137e9i			0.259350i
$879$	−3.21392e9			−0.159615
$880$	−4.54560e9	+	4.09308e9i	−0.224854	+	0.202470i
$881$	−8.33612e9			−0.410722			−0.205361	−	0.978686i	$-0.565837\pi$
$881$	−8.33612e9			−0.410722			−0.205361	+	0.978686i	$0.565837\pi$
$882$			5.35426e9i			0.262760i
$883$		−	3.18818e10i		−	1.55840i	−0.626772	−	0.779202i	$-0.715624\pi$
$883$		−	3.18818e10i		−	1.55840i	0.626772	−	0.779202i	$-0.284376\pi$
$884$	−5.16975e9			−0.251702
$885$	−6.11729e9	−	6.79359e9i	−0.296659	−	0.329457i
$886$	1.28086e10			0.618705
$887$			1.35692e10i			0.652863i	0.945221	+	0.326432i	$0.105846\pi$
$887$			1.35692e10i			0.652863i	−0.945221	+	0.326432i	$0.894154\pi$
$888$			4.88616e9i			0.234165i
$889$	1.89166e10			0.902996
$890$	1.54125e10	+	1.71164e10i	0.732837	+	0.813856i
$891$	1.83654e9			0.0869816
$892$			4.59885e9i			0.216956i
$893$			2.63701e10i			1.23917i
$894$	1.53796e9			0.0719884
$895$	−2.57465e10	+	2.31834e10i	−1.20043	+	1.08093i
$896$	1.24517e9			0.0578297
$897$		−	5.95888e9i		−	0.275671i
$898$		−	1.06943e10i		−	0.492815i
$899$	−3.13692e10			−1.43994
$900$	−7.42291e8	+	7.06586e9i	−0.0339411	+	0.323084i
$901$	−9.18282e9			−0.418253
$902$			2.09707e10i			0.951459i
$903$			1.66816e9i			0.0753931i
$904$	7.00412e9			0.315329
$905$	−5.08166e9	+	4.57578e9i	−0.227895	+	0.205208i
$906$	1.51831e10			0.678286
$907$		−	2.74121e10i		−	1.21988i	−0.792448	−	0.609939i	$-0.791194\pi$
$907$		−	2.74121e10i		−	1.21988i	0.792448	−	0.609939i	$-0.208806\pi$
$908$			1.12585e10i			0.499093i
$909$	7.74898e9			0.342193
$910$	−2.30171e9	−	2.55618e9i	−0.101252	−	0.112446i
$911$	5.32528e9			0.233361			0.116680	−	0.993170i	$-0.462775\pi$
$911$	5.32528e9			0.233361			0.116680	+	0.993170i	$0.462775\pi$
$912$		−	2.13617e9i		−	0.0932510i
$913$			2.14530e10i			0.932911i
$914$	−5.22046e9			−0.226150
$915$	6.37422e9	+	7.07893e9i	0.275076	+	0.305488i
$916$	6.70386e9			0.288198
$917$		−	2.18687e9i		−	0.0936548i
$918$		−	2.49072e10i		−	1.06262i
$919$	2.89226e10			1.22923			0.614615	−	0.788827i	$-0.289311\pi$
$919$	2.89226e10			1.22923			0.614615	+	0.788827i	$0.289311\pi$
$920$	8.83732e9	−	7.95757e9i	0.374165	−	0.336917i
$921$	−1.08420e10			−0.457301
$922$			2.24622e9i			0.0943831i
$923$		−	2.59881e9i		−	0.108785i
$924$	5.61928e9			0.234330
$925$	2.67902e10	+	2.81439e9i	1.11296	+	0.116920i
$926$	−2.78106e10			−1.15099
$927$		−	1.50960e10i		−	0.622421i
$928$			6.11800e9i			0.251299i
$929$	−5.82377e9			−0.238314			−0.119157	−	0.992875i	$-0.538019\pi$
$929$	−5.82377e9			−0.238314			−0.119157	+	0.992875i	$0.538019\pi$
$930$	−7.72720e9	+	6.95796e9i	−0.315016	+	0.283656i
$931$	−8.87520e9			−0.360458
$932$		−	1.90796e10i		−	0.771993i
$933$			8.35341e9i			0.336727i
$934$	2.27781e10			0.914752
$935$	−3.11556e10	−	3.46001e10i	−1.24651	−	1.38432i
$936$	−1.88491e9			−0.0751322
$937$			1.28190e10i			0.509055i	0.967065	+	0.254528i	$0.0819200\pi$
$937$			1.28190e10i			0.509055i	−0.967065	+	0.254528i	$0.918080\pi$
$938$			4.41224e9i			0.174562i
$939$	−4.09854e9			−0.161547
$940$	−1.67518e10	−	1.86039e10i	−0.657832	−	0.730560i
$941$	1.82995e10			0.715937			0.357968	−	0.933734i	$-0.383470\pi$
$941$	1.82995e10			0.715937			0.357968	+	0.933734i	$0.383470\pi$
$942$		−	1.01706e10i		−	0.396433i
$943$		−	4.07702e10i		−	1.58326i
$944$	4.84029e9			0.187271
$945$	1.23154e10	−	1.10894e10i	0.474718	−	0.427460i
$946$	4.33884e9			0.166630
$947$			1.97030e10i			0.753889i	0.926236	+	0.376944i	$0.123025\pi$
$947$			1.97030e10i			0.753889i	−0.926236	+	0.376944i	$0.876975\pi$
$948$			1.49206e9i			0.0568799i
$949$	−6.44450e9			−0.244769
$950$	−1.17123e10	−	1.23042e9i	−0.443211	−	0.0465608i
$951$	9.03724e9			0.340725
$952$			9.47796e9i			0.356029i
$953$		−	1.68804e10i		−	0.631769i	−0.948798	−	0.315885i	$-0.897699\pi$
$953$		−	1.68804e10i		−	0.631769i	0.948798	−	0.315885i	$-0.102301\pi$
$954$	−3.34810e9			−0.124847
$955$	3.86895e10	−	3.48380e10i	1.43741	−	1.29432i
$956$	−1.46037e9			−0.0540580
$957$			2.76096e10i			1.01828i
$958$		−	1.18990e9i		−	0.0437251i
$959$	2.02138e10			0.740089
$960$	1.35702e9	+	1.50705e9i	0.0495037	+	0.0549767i
$961$	7.15989e8			0.0260240
$962$			7.14665e9i			0.258815i
$963$			1.08363e10i			0.391013i
$964$	−1.02931e10			−0.370063
$965$	2.36207e10	+	2.62321e10i	0.846150	+	0.939697i
$966$	−1.09247e10			−0.389933
$967$			3.70867e10i			1.31894i	0.751731	+	0.659470i	$0.229219\pi$
$967$			3.70867e10i			1.31894i	−0.751731	+	0.659470i	$0.770781\pi$
$968$		−	4.63810e9i		−	0.164353i
$969$	1.62600e10			0.574101
$970$	−1.56105e8	+	1.40565e8i	−0.00549181	+	0.00494510i
$971$	−8.44393e9			−0.295990			−0.147995	−	0.988988i	$-0.547282\pi$
$971$	−8.44393e9			−0.295990			−0.147995	+	0.988988i	$0.547282\pi$
$972$		−	1.45860e10i		−	0.509454i
$973$			3.82402e9i			0.133084i
$974$	1.05692e10			0.366511
$975$	5.85313e8	−	5.57159e9i	0.0202242	−	0.192514i
$976$	−5.04359e9			−0.173646
$977$			1.47527e10i			0.506106i	0.967452	+	0.253053i	$0.0814347\pi$
$977$			1.47527e10i			0.506106i	−0.967452	+	0.253053i	$0.918565\pi$
$978$			3.58523e9i			0.122555i
$979$	−5.50347e10			−1.87455
$980$	6.26138e9	−	5.63806e9i	0.212510	−	0.191354i
$981$	−1.14305e10			−0.386566
$982$			1.62042e10i			0.546055i
$983$			2.96936e9i			0.0997071i	0.998757	+	0.0498535i	$0.0158755\pi$
$983$			2.96936e9i			0.0997071i	−0.998757	+	0.0498535i	$0.984125\pi$
$984$	6.95264e9			0.232631
$985$	1.01741e10	+	1.12989e10i	0.339210	+	0.376711i
$986$	−4.65688e10			−1.54713
$987$			2.29981e10i			0.761347i
$988$		−	3.12442e9i		−	0.103067i
$989$	−8.43535e9			−0.277278
$990$	−1.13595e10	−	1.26153e10i	−0.372079	−	0.413215i
$991$	1.05803e9			0.0345336			0.0172668	−	0.999851i	$-0.494504\pi$
$991$	1.05803e9			0.0345336			0.0172668	+	0.999851i	$0.494504\pi$
$992$		−	5.50547e9i		−	0.179062i
$993$		−	1.06162e10i		−	0.344070i
$994$	−4.76454e9			−0.153875
$995$	−1.15517e10	+	1.04017e10i	−0.371761	+	0.334752i
$996$	7.11254e9			0.228096
$997$		−	2.55398e9i		−	0.0816176i	−0.999167	−	0.0408088i	$-0.987007\pi$
$997$		−	2.55398e9i		−	0.0816176i	0.999167	−	0.0408088i	$-0.0129934\pi$
$998$		−	9.47375e9i		−	0.301693i
$999$	−3.44317e10			−1.09265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.8.b.a.9.4	yes	4
3.2	odd	2		90.8.c.c.19.2		4
4.3	odd	2		80.8.c.d.49.2		4
5.2	odd	4		50.8.a.i.1.2		2
5.3	odd	4		50.8.a.j.1.1		2
5.4	even	2	inner	10.8.b.a.9.1	✓	4
8.3	odd	2		320.8.c.g.129.3		4
8.5	even	2		320.8.c.f.129.2		4
15.2	even	4		450.8.a.bi.1.1		2
15.8	even	4		450.8.a.bd.1.2		2
15.14	odd	2		90.8.c.c.19.4		4
20.3	even	4		400.8.a.t.1.2		2
20.7	even	4		400.8.a.bf.1.1		2
20.19	odd	2		80.8.c.d.49.3		4
40.19	odd	2		320.8.c.g.129.2		4
40.29	even	2		320.8.c.f.129.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.8.b.a.9.1	✓	4	5.4	even	2	inner
10.8.b.a.9.4	yes	4	1.1	even	1	trivial
50.8.a.i.1.2		2	5.2	odd	4
50.8.a.j.1.1		2	5.3	odd	4
80.8.c.d.49.2		4	4.3	odd	2
80.8.c.d.49.3		4	20.19	odd	2
90.8.c.c.19.2		4	3.2	odd	2
90.8.c.c.19.4		4	15.14	odd	2
320.8.c.f.129.2		4	8.5	even	2
320.8.c.f.129.3		4	40.29	even	2
320.8.c.g.129.2		4	40.19	odd	2
320.8.c.g.129.3		4	8.3	odd	2
400.8.a.t.1.2		2	20.3	even	4
400.8.a.bf.1.1		2	20.7	even	4
450.8.a.bd.1.2		2	15.8	even	4
450.8.a.bi.1.1		2	15.2	even	4

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:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	10.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} +27.6776i q^{3} -64.0000 q^{4} +(-207.711 + 187.033i) q^{5} -221.421 q^{6} +593.744i q^{7} -512.000i q^{8} +1420.95 q^{9} +O(q^{10})\)	\(q+8.00000i q^{2} +27.6776i q^{3} -64.0000 q^{4} +(-207.711 + 187.033i) q^{5} -221.421 q^{6} +593.744i q^{7} -512.000i q^{8} +1420.95 q^{9} +(-1496.26 - 1661.68i) q^{10} +5342.84 q^{11} -1771.37i q^{12} -2590.86i q^{13} -4749.95 q^{14} +(-5176.63 - 5748.94i) q^{15} +4096.00 q^{16} +31177.8i q^{17} +11367.6i q^{18} -18842.8 q^{19} +(13293.5 - 11970.1i) q^{20} -16433.4 q^{21} +42742.7i q^{22} -83098.3i q^{23} +14171.0 q^{24} +(8162.37 - 77697.4i) q^{25} +20726.8 q^{26} +99859.5i q^{27} -37999.6i q^{28} +186706. q^{29} +(45991.5 - 41413.0i) q^{30} -168014. q^{31} +32768.0i q^{32} +147877. i q^{33} -249423. q^{34} +(-111050. - 123327. i) q^{35} -90940.7 q^{36} +344801. i q^{37} -150743. i q^{38} +71708.8 q^{39} +(95760.9 + 106348. i) q^{40} +490626. q^{41} -131467. i q^{42} -101510. i q^{43} -341942. q^{44} +(-295146. + 265764. i) q^{45} +664787. q^{46} -1.39947e6i q^{47} +113368. i q^{48} +471012. q^{49} +(621579. + 65298.9i) q^{50} -862929. q^{51} +165815. i q^{52} +294530. i q^{53} -798876. q^{54} +(-1.10976e6 + 999287. i) q^{55} +303997. q^{56} -521526. i q^{57} +1.49365e6i q^{58} +1.18171e6 q^{59} +(331304. + 367932. i) q^{60} -1.23135e6 q^{61} -1.34411e6i q^{62} +843679. i q^{63} -262144. q^{64} +(484575. + 538148. i) q^{65} -1.18302e6 q^{66} -928902. i q^{67} -1.99538e6i q^{68} +2.29997e6 q^{69} +(986614. - 888397. i) q^{70} +1.00307e6 q^{71} -727525. i q^{72} -2.48740e6i q^{73} -2.75841e6 q^{74} +(2.15048e6 + 225915. i) q^{75} +1.20594e6 q^{76} +3.17228e6i q^{77} +573670. i q^{78} -842323. q^{79} +(-850783. + 766087. i) q^{80} +343738. q^{81} +3.92501e6i q^{82} +4.01528e6i q^{83} +1.05174e6 q^{84} +(-5.83128e6 - 6.47597e6i) q^{85} +812084. q^{86} +5.16759e6i q^{87} -2.73554e6i q^{88} -1.03006e7 q^{89} +(-2.12611e6 - 2.36117e6i) q^{90} +1.53830e6 q^{91} +5.31829e6i q^{92} -4.65022e6i q^{93} +1.11958e7 q^{94} +(3.91386e6 - 3.52423e6i) q^{95} -906941. q^{96} -93943.7i q^{97} +3.76809e6i q^{98} +7.59190e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9}+O(q^{10}) \) 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9	\( 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9} - 640 q^{10} + 17808 q^{11} - 6528 q^{14} - 34960 q^{15} + 16384 q^{16} + 63600 q^{19} - 3840 q^{20} - 63952 q^{21} - 57344 q^{24} + 86100 q^{25} - 56064 q^{26} + 169560 q^{29} + 410240 q^{30} - 394112 q^{31} - 698368 q^{34} - 276720 q^{35} + 434432 q^{36} + 1163424 q^{39} + 40960 q^{40} + 232488 q^{41} - 1139712 q^{44} - 2879420 q^{45} + 1146496 q^{46} + 2520108 q^{49} + 2361600 q^{50} + 361088 q^{51} - 5116160 q^{54} - 526480 q^{55} + 417792 q^{56} + 2093520 q^{59} + 2237440 q^{60} - 5251432 q^{61} - 1048576 q^{64} + 2761440 q^{65} + 2401792 q^{66} + 6514864 q^{69} + 2679680 q^{70} - 7832352 q^{71} - 3126528 q^{74} - 7397600 q^{75} - 4070400 q^{76} + 7727040 q^{79} + 245760 q^{80} + 16428404 q^{81} + 4092928 q^{84} - 7995520 q^{85} + 4411776 q^{86} - 33470040 q^{89} - 15579520 q^{90} + 5340768 q^{91} + 15540352 q^{94} + 31904400 q^{95} + 3670016 q^{96} - 19109776 q^{99}+O(q^{100}) \) 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9 - 640 * q^10 + 17808 * q^11 - 6528 * q^14 - 34960 * q^15 + 16384 * q^16 + 63600 * q^19 - 3840 * q^20 - 63952 * q^21 - 57344 * q^24 + 86100 * q^25 - 56064 * q^26 + 169560 * q^29 + 410240 * q^30 - 394112 * q^31 - 698368 * q^34 - 276720 * q^35 + 434432 * q^36 + 1163424 * q^39 + 40960 * q^40 + 232488 * q^41 - 1139712 * q^44 - 2879420 * q^45 + 1146496 * q^46 + 2520108 * q^49 + 2361600 * q^50 + 361088 * q^51 - 5116160 * q^54 - 526480 * q^55 + 417792 * q^56 + 2093520 * q^59 + 2237440 * q^60 - 5251432 * q^61 - 1048576 * q^64 + 2761440 * q^65 + 2401792 * q^66 + 6514864 * q^69 + 2679680 * q^70 - 7832352 * q^71 - 3126528 * q^74 - 7397600 * q^75 - 4070400 * q^76 + 7727040 * q^79 + 245760 * q^80 + 16428404 * q^81 + 4092928 * q^84 - 7995520 * q^85 + 4411776 * q^86 - 33470040 * q^89 - 15579520 * q^90 + 5340768 * q^91 + 15540352 * q^94 + 31904400 * q^95 + 3670016 * q^96 - 19109776 * q^99

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