LMFDB - Embedded newform 20.9.f.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,9,Mod(13,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("20.13");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	20.f (of order $4$, degree $2$, 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:	$8.14757220122$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4 x^{7} + 8 x^{6} + 22254 x^{5} + 4820745 x^{4} + 50131374 x^{3} + 307615702 x^{2} + \cdots + 2405464244 $ x^8 - 4x^7 + 8x^6 + 22254x^5 + 4820745x^4 + 50131374x^3 + 307615702x^2 - 1770757924*x + 2405464244
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{13}\cdot 5^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.2
Root			$36.4975 - 35.4975i$ of defining polynomial
Character	$\chi$	$=$	20.17
Dual form			20.9.f.a.13.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-29.4437 + 29.4437i) q^{3} +(98.2101 - 617.236i) q^{5} +(1846.83 + 1846.83i) q^{7} +4827.14i q^{9} +O(q^{10})$	\(q+(-29.4437 + 29.4437i) q^{3} +(98.2101 - 617.236i) q^{5} +(1846.83 + 1846.83i) q^{7} +4827.14i q^{9} +27269.8 q^{11} +(3770.73 - 3770.73i) q^{13} +(15282.0 + 21065.4i) q^{15} +(75231.2 + 75231.2i) q^{17} -107151. i q^{19} -108755. q^{21} +(-66095.9 + 66095.9i) q^{23} +(-371335. - 121237. i) q^{25} +(-335309. - 335309. i) q^{27} -269781. i q^{29} -200724. q^{31} +(-802924. + 802924. i) q^{33} +(1.32130e6 - 958551. i) q^{35} +(2.11986e6 + 2.11986e6i) q^{37} +222048. i q^{39} -1.25140e6 q^{41} +(2.52813e6 - 2.52813e6i) q^{43} +(2.97948e6 + 474074. i) q^{45} +(2.48633e6 + 2.48633e6i) q^{47} +1.05674e6i q^{49} -4.43017e6 q^{51} +(-1.04167e7 + 1.04167e7i) q^{53} +(2.67817e6 - 1.68319e7i) q^{55} +(3.15491e6 + 3.15491e6i) q^{57} -1.75660e7i q^{59} -2.19031e7 q^{61} +(-8.91489e6 + 8.91489e6i) q^{63} +(-1.95711e6 - 2.69775e6i) q^{65} +(1.53701e7 + 1.53701e7i) q^{67} -3.89222e6i q^{69} +4.03078e6 q^{71} +(8.92029e6 - 8.92029e6i) q^{73} +(1.45031e7 - 7.36378e6i) q^{75} +(5.03627e7 + 5.03627e7i) q^{77} -1.80828e7i q^{79} -1.19254e7 q^{81} +(-3.87237e7 + 3.87237e7i) q^{83} +(5.38238e7 - 3.90469e7i) q^{85} +(7.94334e6 + 7.94334e6i) q^{87} -5.94178e6i q^{89} +1.39278e7 q^{91} +(5.91006e6 - 5.91006e6i) q^{93} +(-6.61373e7 - 1.05233e7i) q^{95} +(-1.06523e8 - 1.06523e8i) q^{97} +1.31635e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7}+O(q^{10}) $ 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7	$ 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7} - 420 q^{11} + 33180 q^{13} - 48478 q^{15} + 43620 q^{17} + 108668 q^{21} - 663270 q^{23} + 163396 q^{25} + 1576040 q^{27} - 3178492 q^{31} - 944020 q^{33} + 2571618 q^{35} + 5344080 q^{37} - 10185252 q^{41} - 10342710 q^{43} + 20284834 q^{45} + 19232250 q^{47} - 47126684 q^{51} - 24320640 q^{53} + 21483180 q^{55} + 88218320 q^{57} - 82515684 q^{61} - 77441350 q^{63} + 72045768 q^{65} + 100675930 q^{67} - 99290076 q^{71} - 93528520 q^{73} + 76524178 q^{75} + 134199660 q^{77} - 161920268 q^{81} - 10450350 q^{83} + 51676156 q^{85} + 164801600 q^{87} - 130681068 q^{91} - 50183620 q^{93} + 84367944 q^{95} - 179570760 q^{97}+O(q^{100}) $ 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7 - 420 * q^11 + 33180 * q^13 - 48478 * q^15 + 43620 * q^17 + 108668 * q^21 - 663270 * q^23 + 163396 * q^25 + 1576040 * q^27 - 3178492 * q^31 - 944020 * q^33 + 2571618 * q^35 + 5344080 * q^37 - 10185252 * q^41 - 10342710 * q^43 + 20284834 * q^45 + 19232250 * q^47 - 47126684 * q^51 - 24320640 * q^53 + 21483180 * q^55 + 88218320 * q^57 - 82515684 * q^61 - 77441350 * q^63 + 72045768 * q^65 + 100675930 * q^67 - 99290076 * q^71 - 93528520 * q^73 + 76524178 * q^75 + 134199660 * q^77 - 161920268 * q^81 - 10450350 * q^83 + 51676156 * q^85 + 164801600 * q^87 - 130681068 * q^91 - 50183620 * q^93 + 84367944 * q^95 - 179570760 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−29.4437	+	29.4437i	−0.363502	+	0.363502i	−0.865101	−	0.501598i	$-0.832746\pi$
$3$	−29.4437	+	29.4437i	−0.363502	+	0.363502i	0.501598	+	0.865101i	$0.332746\pi$
$4$		0			0
$5$	98.2101	−	617.236i	0.157136	−	0.987577i
$5$	98.2101	−	617.236i	0.157136	−	0.987577i
$6$		0			0
$7$	1846.83	+	1846.83i	0.769191	+	0.769191i	0.977964	−	0.208773i	$-0.0669470\pi$
$7$	1846.83	+	1846.83i	0.769191	+	0.769191i	−0.208773	+	0.977964i	$0.566947\pi$
$8$		0			0
$9$			4827.14i			0.735732i
$10$		0			0
$11$	27269.8			1.86257			0.931283	−	0.364297i	$-0.118691\pi$
$11$	27269.8			1.86257			0.931283	+	0.364297i	$0.118691\pi$
$12$		0			0
$13$	3770.73	−	3770.73i	0.132024	−	0.132024i	−0.638007	−	0.770031i	$-0.720241\pi$
$13$	3770.73	−	3770.73i	0.132024	−	0.132024i	0.770031	+	0.638007i	$0.220241\pi$
$14$		0			0
$15$	15282.0	+	21065.4i	0.301867	+	0.416106i
$16$		0			0
$17$	75231.2	+	75231.2i	0.900746	+	0.900746i	0.995501	−	0.0947550i	$-0.0302068\pi$
$17$	75231.2	+	75231.2i	0.900746	+	0.900746i	−0.0947550	+	0.995501i	$0.530207\pi$
$18$		0			0
$19$		−	107151.i		−	0.822207i	−0.911589	−	0.411103i	$-0.865144\pi$
$19$		−	107151.i		−	0.822207i	0.911589	−	0.411103i	$-0.134856\pi$
$20$		0			0
$21$	−108755.			−0.559205
$22$		0			0
$23$	−66095.9	+	66095.9i	−0.236191	+	0.236191i	−0.815271	−	0.579080i	$-0.803412\pi$
$23$	−66095.9	+	66095.9i	−0.236191	+	0.236191i	0.579080	+	0.815271i	$0.303412\pi$
$24$		0			0
$25$	−371335.	−	121237.i	−0.950616	−	0.310368i
$26$		0			0
$27$	−335309.	−	335309.i	−0.630943	−	0.630943i
$28$		0			0
$29$		−	269781.i		−	0.381434i	−0.981645	−	0.190717i	$-0.938919\pi$
$29$		−	269781.i		−	0.381434i	0.981645	−	0.190717i	$-0.0610812\pi$
$30$		0			0
$31$	−200724.			−0.217347			−0.108673	−	0.994078i	$-0.534660\pi$
$31$	−200724.			−0.217347			−0.108673	+	0.994078i	$0.534660\pi$
$32$		0			0
$33$	−802924.	+	802924.i	−0.677047	+	0.677047i
$34$		0			0
$35$	1.32130e6	−	958551.i	0.880503	−	0.638768i
$36$		0			0
$37$	2.11986e6	+	2.11986e6i	1.13110	+	1.13110i	0.989995	+	0.141106i	$0.0450657\pi$
$37$	2.11986e6	+	2.11986e6i	1.13110	+	1.13110i	0.141106	+	0.989995i	$0.454934\pi$
$38$		0			0
$39$			222048.i			0.0959819i
$40$		0			0
$41$	−1.25140e6			−0.442854			−0.221427	−	0.975177i	$-0.571071\pi$
$41$	−1.25140e6			−0.442854			−0.221427	+	0.975177i	$0.571071\pi$
$42$		0			0
$43$	2.52813e6	−	2.52813e6i	0.739478	−	0.739478i	−0.232999	−	0.972477i	$-0.574854\pi$
$43$	2.52813e6	−	2.52813e6i	0.739478	−	0.739478i	0.972477	+	0.232999i	$0.0748538\pi$
$44$		0			0
$45$	2.97948e6	+	474074.i	0.726592	+	0.115610i
$46$		0			0
$47$	2.48633e6	+	2.48633e6i	0.509526	+	0.509526i	0.914381	−	0.404855i	$-0.132678\pi$
$47$	2.48633e6	+	2.48633e6i	0.509526	+	0.509526i	−0.404855	+	0.914381i	$0.632678\pi$
$48$		0			0
$49$			1.05674e6i			0.183310i
$50$		0			0
$51$	−4.43017e6			−0.654846
$52$		0			0
$53$	−1.04167e7	+	1.04167e7i	−1.32016	+	1.32016i	−0.406524	+	0.913640i	$0.633259\pi$
$53$	−1.04167e7	+	1.04167e7i	−1.32016	+	1.32016i	−0.913640	+	0.406524i	$0.866741\pi$
$54$		0			0
$55$	2.67817e6	−	1.68319e7i	0.292676	−	1.83943i
$56$		0			0
$57$	3.15491e6	+	3.15491e6i	0.298874	+	0.298874i
$58$		0			0
$59$		−	1.75660e7i		−	1.44966i	−0.688928	−	0.724829i	$-0.741919\pi$
$59$		−	1.75660e7i		−	1.44966i	0.688928	−	0.724829i	$-0.258081\pi$
$60$		0			0
$61$	−2.19031e7			−1.58192			−0.790962	−	0.611866i	$-0.790419\pi$
$61$	−2.19031e7			−1.58192			−0.790962	+	0.611866i	$0.790419\pi$
$62$		0			0
$63$	−8.91489e6	+	8.91489e6i	−0.565919	+	0.565919i
$64$		0			0
$65$	−1.95711e6	−	2.69775e6i	−0.109638	−	0.151129i
$66$		0			0
$67$	1.53701e7	+	1.53701e7i	0.762743	+	0.762743i	0.976817	−	0.214075i	$-0.0686736\pi$
$67$	1.53701e7	+	1.53701e7i	0.762743	+	0.762743i	−0.214075	+	0.976817i	$0.568674\pi$
$68$		0			0
$69$		−	3.89222e6i		−	0.171712i
$70$		0			0
$71$	4.03078e6			0.158619			0.0793097	−	0.996850i	$-0.474728\pi$
$71$	4.03078e6			0.158619			0.0793097	+	0.996850i	$0.474728\pi$
$72$		0			0
$73$	8.92029e6	−	8.92029e6i	0.314114	−	0.314114i	−0.532387	−	0.846501i	$-0.678705\pi$
$73$	8.92029e6	−	8.92029e6i	0.314114	−	0.314114i	0.846501	+	0.532387i	$0.178705\pi$
$74$		0			0
$75$	1.45031e7	−	7.36378e6i	0.458371	−	0.232732i
$76$		0			0
$77$	5.03627e7	+	5.03627e7i	1.43267	+	1.43267i
$78$		0			0
$79$		−	1.80828e7i		−	0.464256i	−0.972685	−	0.232128i	$-0.925431\pi$
$79$		−	1.80828e7i		−	0.464256i	0.972685	−	0.232128i	$-0.0745688\pi$
$80$		0			0
$81$	−1.19254e7			−0.277034
$82$		0			0
$83$	−3.87237e7	+	3.87237e7i	−0.815952	+	0.815952i	−0.985519	−	0.169567i	$-0.945763\pi$
$83$	−3.87237e7	+	3.87237e7i	−0.815952	+	0.815952i	0.169567	+	0.985519i	$0.445763\pi$
$84$		0			0
$85$	5.38238e7	−	3.90469e7i	1.03110	−	0.748016i
$86$		0			0
$87$	7.94334e6	+	7.94334e6i	0.138652	+	0.138652i
$88$		0			0
$89$		−	5.94178e6i		−	0.0947014i	−0.998878	−	0.0473507i	$-0.984922\pi$
$89$		−	5.94178e6i		−	0.0947014i	0.998878	−	0.0473507i	$-0.0150778\pi$
$90$		0			0
$91$	1.39278e7			0.203103
$92$		0			0
$93$	5.91006e6	−	5.91006e6i	0.0790060	−	0.0790060i
$94$		0			0
$95$	−6.61373e7	−	1.05233e7i	−0.811993	−	0.129198i
$96$		0			0
$97$	−1.06523e8	−	1.06523e8i	−1.20326	−	1.20326i	−0.973170	−	0.230087i	$-0.926099\pi$
$97$	−1.06523e8	−	1.06523e8i	−1.20326	−	1.20326i	−0.230087	−	0.973170i	$-0.573901\pi$
$98$		0			0
$99$			1.31635e8i			1.37035i
$100$		0			0
$101$	−8.08731e7			−0.777175			−0.388587	−	0.921412i	$-0.627037\pi$
$101$	−8.08731e7			−0.777175			−0.388587	+	0.921412i	$0.627037\pi$
$102$		0			0
$103$	4.63044e7	−	4.63044e7i	0.411409	−	0.411409i	−0.470820	−	0.882229i	$-0.656042\pi$
$103$	4.63044e7	−	4.63044e7i	0.411409	−	0.411409i	0.882229	+	0.470820i	$0.156042\pi$
$104$		0			0
$105$	−1.06808e7	+	6.71274e7i	−0.0878714	+	0.552258i
$106$		0			0
$107$	−1.87307e7	−	1.87307e7i	−0.142895	−	0.142895i	0.632040	−	0.774936i	$-0.282218\pi$
$107$	−1.87307e7	−	1.87307e7i	−0.142895	−	0.142895i	−0.774936	+	0.632040i	$0.782218\pi$
$108$		0			0
$109$		−	2.35934e8i		−	1.67142i	−0.549172	−	0.835709i	$-0.685057\pi$
$109$		−	2.35934e8i		−	1.67142i	0.549172	−	0.835709i	$-0.314943\pi$
$110$		0			0
$111$	−1.24833e8			−0.822315
$112$		0			0
$113$	2.29118e7	−	2.29118e7i	0.140522	−	0.140522i	−0.633346	−	0.773869i	$-0.718319\pi$
$113$	2.29118e7	−	2.29118e7i	0.140522	−	0.140522i	0.773869	+	0.633346i	$0.218319\pi$
$114$		0			0
$115$	3.43055e7	+	4.72881e7i	0.196143	+	0.270371i
$116$		0			0
$117$	1.82018e7	+	1.82018e7i	0.0971341	+	0.0971341i
$118$		0			0
$119$			2.77878e8i			1.38569i
$120$		0			0
$121$	5.29284e8			2.46915
$122$		0			0
$123$	3.68458e7	−	3.68458e7i	0.160978	−	0.160978i
$124$		0			0
$125$	−1.11301e8	+	2.17294e8i	−0.455888	+	0.890037i
$126$		0			0
$127$	−1.23765e8	−	1.23765e8i	−0.475753	−	0.475753i	0.428017	−	0.903770i	$-0.359212\pi$
$127$	−1.23765e8	−	1.23765e8i	−0.475753	−	0.475753i	−0.903770	+	0.428017i	$0.859212\pi$
$128$		0			0
$129$			1.48875e8i			0.537604i
$130$		0			0
$131$	4.41499e8			1.49915			0.749575	−	0.661920i	$-0.230258\pi$
$131$	4.41499e8			1.49915			0.749575	+	0.661920i	$0.230258\pi$
$132$		0			0
$133$	1.97889e8	−	1.97889e8i	0.632434	−	0.632434i
$134$		0			0
$135$	−2.39895e8	+	1.74034e8i	−0.722248	+	0.523961i
$136$		0			0
$137$	−1.97677e8	−	1.97677e8i	−0.561143	−	0.561143i	0.368489	−	0.929632i	$-0.379875\pi$
$137$	−1.97677e8	−	1.97677e8i	−0.561143	−	0.561143i	−0.929632	+	0.368489i	$0.879875\pi$
$138$		0			0
$139$		−	1.53083e8i		−	0.410078i	−0.978754	−	0.205039i	$-0.934268\pi$
$139$		−	1.53083e8i		−	0.410078i	0.978754	−	0.205039i	$-0.0657322\pi$
$140$		0			0
$141$	−1.46413e8			−0.370428
$142$		0			0
$143$	1.02827e8	−	1.02827e8i	0.245903	−	0.245903i
$144$		0			0
$145$	−1.66518e8	−	2.64952e7i	−0.376695	−	0.0599370i
$146$		0			0
$147$	−3.11145e7	−	3.11145e7i	−0.0666335	−	0.0666335i
$148$		0			0
$149$			2.39143e8i			0.485190i	0.970128	+	0.242595i	$0.0779986\pi$
$149$			2.39143e8i			0.485190i	−0.970128	+	0.242595i	$0.922001\pi$
$150$		0			0
$151$	−4.48626e8			−0.862933			−0.431466	−	0.902129i	$-0.642004\pi$
$151$	−4.48626e8			−0.862933			−0.431466	+	0.902129i	$0.642004\pi$
$152$		0			0
$153$	−3.63151e8	+	3.63151e8i	−0.662708	+	0.662708i
$154$		0			0
$155$	−1.97131e7	+	1.23894e8i	−0.0341530	+	0.214646i
$156$		0			0
$157$	−4.83810e8	−	4.83810e8i	−0.796300	−	0.796300i	0.186210	−	0.982510i	$-0.440379\pi$
$157$	−4.83810e8	−	4.83810e8i	−0.796300	−	0.796300i	−0.982510	+	0.186210i	$0.940379\pi$
$158$		0			0
$159$		−	6.13414e8i		−	0.959765i
$160$		0			0
$161$	−2.44136e8			−0.363352
$162$		0			0
$163$	7.10719e8	−	7.10719e8i	1.00681	−	1.00681i	0.00683400	−	0.999977i	$-0.497825\pi$
$163$	7.10719e8	−	7.10719e8i	1.00681	−	1.00681i	0.999977	−	0.00683400i	$-0.00217535\pi$
$164$		0			0
$165$	4.16738e8	+	5.74449e8i	0.562247	+	0.775024i
$166$		0			0
$167$	−2.93696e8	−	2.93696e8i	−0.377600	−	0.377600i	0.492635	−	0.870236i	$-0.336034\pi$
$167$	−2.93696e8	−	2.93696e8i	−0.377600	−	0.377600i	−0.870236	+	0.492635i	$0.836034\pi$
$168$		0			0
$169$			7.87294e8i			0.965139i
$170$		0			0
$171$	5.17232e8			0.604924
$172$		0			0
$173$	−6.53255e8	+	6.53255e8i	−0.729287	+	0.729287i	−0.970478	−	0.241191i	$-0.922462\pi$
$173$	−6.53255e8	+	6.53255e8i	−0.729287	+	0.729287i	0.241191	+	0.970478i	$0.422462\pi$
$174$		0			0
$175$	−4.61886e8	−	9.09696e8i	−0.492473	−	0.969938i
$176$		0			0
$177$	5.17209e8	+	5.17209e8i	0.526954	+	0.526954i
$178$		0			0
$179$			8.87937e8i			0.864908i	0.901656	+	0.432454i	$0.142352\pi$
$179$			8.87937e8i			0.864908i	−0.901656	+	0.432454i	$0.857648\pi$
$180$		0			0
$181$	7.89992e8			0.736052			0.368026	−	0.929816i	$-0.380034\pi$
$181$	7.89992e8			0.736052			0.368026	+	0.929816i	$0.380034\pi$
$182$		0			0
$183$	6.44907e8	−	6.44907e8i	0.575033	−	0.575033i
$184$		0			0
$185$	1.51665e9	−	1.10026e9i	1.29479	−	0.939312i
$186$		0			0
$187$	2.05154e9	+	2.05154e9i	1.67770	+	1.67770i
$188$		0			0
$189$		−	1.23852e9i		−	0.970631i
$190$		0			0
$191$	−1.33172e9			−1.00065			−0.500323	−	0.865839i	$-0.666786\pi$
$191$	−1.33172e9			−1.00065			−0.500323	+	0.865839i	$0.666786\pi$
$192$		0			0
$193$	−1.69278e9	+	1.69278e9i	−1.22003	+	1.22003i	−0.252414	+	0.967619i	$0.581224\pi$
$193$	−1.69278e9	+	1.69278e9i	−1.22003	+	1.22003i	−0.967619	+	0.252414i	$0.918776\pi$
$194$		0			0
$195$	1.37056e8	+	2.18074e7i	0.0947895	+	0.0150822i
$196$		0			0
$197$	2.16650e8	+	2.16650e8i	0.143844	+	0.143844i	0.775362	−	0.631517i	$-0.217568\pi$
$197$	2.16650e8	+	2.16650e8i	0.143844	+	0.143844i	−0.631517	+	0.775362i	$0.717568\pi$
$198$		0			0
$199$		−	5.35280e8i		−	0.341325i	−0.985330	−	0.170663i	$-0.945409\pi$
$199$		−	5.35280e8i		−	0.341325i	0.985330	−	0.170663i	$-0.0545908\pi$
$200$		0			0
$201$	−9.05106e8			−0.554517
$202$		0			0
$203$	4.98239e8	−	4.98239e8i	0.293395	−	0.293395i
$204$		0			0
$205$	−1.22900e8	+	7.72409e8i	−0.0695884	+	0.437353i
$206$		0			0
$207$	−3.19054e8	−	3.19054e8i	−0.173773	−	0.173773i
$208$		0			0
$209$		−	2.92198e9i		−	1.53141i
$210$		0			0
$211$	−2.19737e9			−1.10860			−0.554298	−	0.832318i	$-0.687013\pi$
$211$	−2.19737e9			−1.10860			−0.554298	+	0.832318i	$0.687013\pi$
$212$		0			0
$213$	−1.18681e8	+	1.18681e8i	−0.0576585	+	0.0576585i
$214$		0			0
$215$	−1.31216e9	−	1.80874e9i	−0.614093	−	0.846491i
$216$		0			0
$217$	−3.70703e8	−	3.70703e8i	−0.167181	−	0.167181i
$218$		0			0
$219$			5.25292e8i			0.228362i
$220$		0			0
$221$	5.67353e8			0.237840
$222$		0			0
$223$	−3.32535e9	+	3.32535e9i	−1.34468	+	1.34468i	−0.453339	+	0.891338i	$0.649767\pi$
$223$	−3.32535e9	+	3.32535e9i	−1.34468	+	1.34468i	−0.891338	+	0.453339i	$0.850233\pi$
$224$		0			0
$225$	5.85230e8	−	1.79248e9i	0.228348	−	0.699399i
$226$		0			0
$227$	2.74827e9	+	2.74827e9i	1.03504	+	1.03504i	0.999363	+	0.0356750i	$0.0113581\pi$
$227$	2.74827e9	+	2.74827e9i	1.03504	+	1.03504i	0.0356750	+	0.999363i	$0.488642\pi$
$228$		0			0
$229$		−	7.50119e8i		−	0.272765i	−0.990656	−	0.136382i	$-0.956452\pi$
$229$		−	7.50119e8i		−	0.272765i	0.990656	−	0.136382i	$-0.0435476\pi$
$230$		0			0
$231$	−2.96573e9			−1.04156
$232$		0			0
$233$	7.57232e8	−	7.57232e8i	0.256924	−	0.256924i	−0.566878	−	0.823802i	$-0.691849\pi$
$233$	7.57232e8	−	7.57232e8i	0.256924	−	0.256924i	0.823802	+	0.566878i	$0.191849\pi$
$234$		0			0
$235$	1.77883e9	−	1.29047e9i	0.583262	−	0.423132i
$236$		0			0
$237$	5.32425e8	+	5.32425e8i	0.168758	+	0.168758i
$238$		0			0
$239$			3.00002e9i			0.919459i	0.888059	+	0.459730i	$0.152054\pi$
$239$			3.00002e9i			0.919459i	−0.888059	+	0.459730i	$0.847946\pi$
$240$		0			0
$241$	2.25204e9			0.667587			0.333794	−	0.942646i	$-0.391671\pi$
$241$	2.25204e9			0.667587			0.333794	+	0.942646i	$0.391671\pi$
$242$		0			0
$243$	2.55109e9	−	2.55109e9i	0.731645	−	0.731645i
$244$		0			0
$245$	6.52260e8	+	1.03783e8i	0.181033	+	0.0288046i
$246$		0			0
$247$	−4.04037e8	−	4.04037e8i	−0.108551	−	0.108551i
$248$		0			0
$249$		−	2.28034e9i		−	0.593201i
$250$		0			0
$251$	4.65138e9			1.17189			0.585945	−	0.810351i	$-0.300723\pi$
$251$	4.65138e9			1.17189			0.585945	+	0.810351i	$0.300723\pi$
$252$		0			0
$253$	−1.80242e9	+	1.80242e9i	−0.439921	+	0.439921i
$254$		0			0
$255$	−4.35087e8	+	2.73446e9i	−0.102900	+	0.646711i
$256$		0			0
$257$	−2.11705e9	−	2.11705e9i	−0.485287	−	0.485287i	0.421529	−	0.906815i	$-0.361494\pi$
$257$	−2.11705e9	−	2.11705e9i	−0.485287	−	0.485287i	−0.906815	+	0.421529i	$0.861494\pi$
$258$		0			0
$259$			7.83005e9i			1.74006i
$260$		0			0
$261$	1.30227e9			0.280633
$262$		0			0
$263$	4.61007e9	−	4.61007e9i	0.963572	−	0.963572i	−0.0357872	−	0.999359i	$-0.511394\pi$
$263$	4.61007e9	−	4.61007e9i	0.963572	−	0.963572i	0.999359	+	0.0357872i	$0.0113939\pi$
$264$		0			0
$265$	5.40655e9	+	7.45260e9i	1.09632	+	1.51121i
$266$		0			0
$267$	1.74948e8	+	1.74948e8i	0.0344242	+	0.0344242i
$268$		0			0
$269$		−	4.82025e9i		−	0.920577i	−0.887769	−	0.460289i	$-0.847746\pi$
$269$		−	4.82025e9i		−	0.920577i	0.887769	−	0.460289i	$-0.152254\pi$
$270$		0			0
$271$	−4.27766e9			−0.793101			−0.396551	−	0.918013i	$-0.629793\pi$
$271$	−4.27766e9			−0.793101			−0.396551	+	0.918013i	$0.629793\pi$
$272$		0			0
$273$	−4.10085e8	+	4.10085e8i	−0.0738284	+	0.0738284i
$274$		0			0
$275$	−1.01262e10	−	3.30613e9i	−1.77059	−	0.578081i
$276$		0			0
$277$	−3.86181e9	−	3.86181e9i	−0.655952	−	0.655952i	0.298468	−	0.954420i	$-0.403524\pi$
$277$	−3.86181e9	−	3.86181e9i	−0.655952	−	0.655952i	−0.954420	+	0.298468i	$0.903524\pi$
$278$		0			0
$279$		−	9.68923e8i		−	0.159909i
$280$		0			0
$281$	−3.80378e9			−0.610085			−0.305042	−	0.952339i	$-0.598671\pi$
$281$	−3.80378e9			−0.610085			−0.305042	+	0.952339i	$0.598671\pi$
$282$		0			0
$283$	2.29828e9	−	2.29828e9i	0.358309	−	0.358309i	−0.504880	−	0.863189i	$-0.668463\pi$
$283$	2.29828e9	−	2.29828e9i	0.358309	−	0.358309i	0.863189	+	0.504880i	$0.168463\pi$
$284$		0			0
$285$	2.25717e9	−	1.63748e9i	0.342125	−	0.248197i
$286$		0			0
$287$	−2.31112e9	−	2.31112e9i	−0.340639	−	0.340639i
$288$		0			0
$289$			4.34370e9i			0.622685i
$290$		0			0
$291$	6.27289e9			0.874773
$292$		0			0
$293$	−4.73797e9	+	4.73797e9i	−0.642868	+	0.642868i	−0.951259	−	0.308392i	$-0.900209\pi$
$293$	−4.73797e9	+	4.73797e9i	−0.642868	+	0.642868i	0.308392	+	0.951259i	$0.400209\pi$
$294$		0			0
$295$	−1.08424e10	−	1.72516e9i	−1.43165	−	0.227794i
$296$		0			0
$297$	−9.14381e9	−	9.14381e9i	−1.17517	−	1.17517i
$298$		0			0
$299$			4.98460e8i			0.0623657i
$300$		0			0
$301$	9.33804e9			1.13760
$302$		0			0
$303$	2.38120e9	−	2.38120e9i	0.282505	−	0.282505i
$304$		0			0
$305$	−2.15110e9	+	1.35193e10i	−0.248577	+	1.56227i
$306$		0			0
$307$	−3.69814e9	−	3.69814e9i	−0.416322	−	0.416322i	0.467612	−	0.883934i	$-0.345114\pi$
$307$	−3.69814e9	−	3.69814e9i	−0.416322	−	0.416322i	−0.883934	+	0.467612i	$0.845114\pi$
$308$		0			0
$309$			2.72675e9i			0.299096i
$310$		0			0
$311$	8.98153e9			0.960083			0.480041	−	0.877246i	$-0.340622\pi$
$311$	8.98153e9			0.960083			0.480041	+	0.877246i	$0.340622\pi$
$312$		0			0
$313$	4.28730e9	−	4.28730e9i	0.446691	−	0.446691i	−0.447562	−	0.894253i	$-0.647708\pi$
$313$	4.28730e9	−	4.28730e9i	0.446691	−	0.446691i	0.894253	+	0.447562i	$0.147708\pi$
$314$		0			0
$315$	4.62706e9	+	6.37812e9i	0.469962	+	0.647814i
$316$		0			0
$317$	5.30286e9	+	5.30286e9i	0.525138	+	0.525138i	0.919119	−	0.393981i	$-0.128902\pi$
$317$	5.30286e9	+	5.30286e9i	0.525138	+	0.525138i	−0.393981	+	0.919119i	$0.628902\pi$
$318$		0			0
$319$		−	7.35688e9i		−	0.710446i
$320$		0			0
$321$	1.10300e9			0.103886
$322$		0			0
$323$	8.06108e9	−	8.06108e9i	0.740599	−	0.740599i
$324$		0			0
$325$	−1.85736e9	+	9.43049e8i	−0.166480	+	0.0845280i
$326$		0			0
$327$	6.94678e9	+	6.94678e9i	0.607564	+	0.607564i
$328$		0			0
$329$			9.18363e9i			0.783846i
$330$		0			0
$331$	−3.22662e9			−0.268804			−0.134402	−	0.990927i	$-0.542911\pi$
$331$	−3.22662e9			−0.268804			−0.134402	+	0.990927i	$0.542911\pi$
$332$		0			0
$333$	−1.02329e10	+	1.02329e10i	−0.832187	+	0.832187i
$334$		0			0
$335$	1.09965e10	−	7.97748e9i	0.873121	−	0.633413i
$336$		0			0
$337$	5.78115e9	+	5.78115e9i	0.448224	+	0.448224i	0.894764	−	0.446540i	$-0.147344\pi$
$337$	5.78115e9	+	5.78115e9i	0.448224	+	0.448224i	−0.446540	+	0.894764i	$0.647344\pi$
$338$		0			0
$339$			1.34922e9i			0.102160i
$340$		0			0
$341$	−5.47371e9			−0.404822
$342$		0			0
$343$	8.69497e9	−	8.69497e9i	0.628191	−	0.628191i
$344$		0			0
$345$	−2.40241e9	−	3.82255e8i	−0.169579	−	0.0269821i
$346$		0			0
$347$	−3.25108e9	−	3.25108e9i	−0.224239	−	0.224239i	0.586042	−	0.810281i	$-0.300685\pi$
$347$	−3.25108e9	−	3.25108e9i	−0.224239	−	0.224239i	−0.810281	+	0.586042i	$0.800685\pi$
$348$		0			0
$349$		−	1.43278e10i		−	0.965777i	−0.875682	−	0.482889i	$-0.839588\pi$
$349$		−	1.43278e10i		−	0.965777i	0.875682	−	0.482889i	$-0.160412\pi$
$350$		0			0
$351$	−2.52872e9			−0.166599
$352$		0			0
$353$	2.51690e9	−	2.51690e9i	0.162094	−	0.162094i	−0.621400	−	0.783494i	$-0.713436\pi$
$353$	2.51690e9	−	2.51690e9i	0.162094	−	0.162094i	0.783494	+	0.621400i	$0.213436\pi$
$354$		0			0
$355$	3.95864e8	−	2.48794e9i	0.0249248	−	0.156649i
$356$		0			0
$357$	−8.18175e9	−	8.18175e9i	−0.503702	−	0.503702i
$358$		0			0
$359$			2.77305e10i			1.66948i	0.550646	+	0.834739i	$0.314381\pi$
$359$			2.77305e10i			1.66948i	−0.550646	+	0.834739i	$0.685619\pi$
$360$		0			0
$361$	5.50227e9			0.323976
$362$		0			0
$363$	−1.55841e10	+	1.55841e10i	−0.897542	+	0.897542i
$364$		0			0
$365$	−4.62986e9	−	6.38198e9i	−0.260853	−	0.359571i
$366$		0			0
$367$	−3.81266e9	−	3.81266e9i	−0.210167	−	0.210167i	0.594172	−	0.804338i	$-0.297480\pi$
$367$	−3.81266e9	−	3.81266e9i	−0.210167	−	0.210167i	−0.804338	+	0.594172i	$0.797480\pi$
$368$		0			0
$369$		−	6.04068e9i		−	0.325822i
$370$		0			0
$371$	−3.84758e10			−2.03092
$372$		0			0
$373$	−8.58377e9	+	8.58377e9i	−0.443448	+	0.443448i	−0.893169	−	0.449721i	$-0.851523\pi$
$373$	−8.58377e9	+	8.58377e9i	−0.443448	+	0.443448i	0.449721	+	0.893169i	$0.351523\pi$
$374$		0			0
$375$	−3.12083e9	−	9.67505e9i	−0.157814	−	0.489247i
$376$		0			0
$377$	−1.01727e9	−	1.01727e9i	−0.0503583	−	0.0503583i
$378$		0			0
$379$		−	3.12745e10i		−	1.51577i	−0.652387	−	0.757886i	$-0.726232\pi$
$379$		−	3.12745e10i		−	1.51577i	0.652387	−	0.757886i	$-0.273768\pi$
$380$		0			0
$381$	7.28817e9			0.345875
$382$		0			0
$383$	−3.13695e9	+	3.13695e9i	−0.145785	+	0.145785i	−0.776232	−	0.630447i	$-0.782872\pi$
$383$	−3.13695e9	+	3.13695e9i	−0.145785	+	0.145785i	0.630447	+	0.776232i	$0.282872\pi$
$384$		0			0
$385$	3.60318e10	−	2.61395e10i	1.63999	−	1.18975i
$386$		0			0
$387$	1.22036e10	+	1.22036e10i	0.544058	+	0.544058i
$388$		0			0
$389$		−	8.87869e9i		−	0.387749i	−0.981026	−	0.193874i	$-0.937895\pi$
$389$		−	8.87869e9i		−	0.387749i	0.981026	−	0.193874i	$-0.0621054\pi$
$390$		0			0
$391$	−9.94495e9			−0.425496
$392$		0			0
$393$	−1.29994e10	+	1.29994e10i	−0.544944	+	0.544944i
$394$		0			0
$395$	−1.11614e10	−	1.77591e9i	−0.458489	−	0.0729514i
$396$		0			0
$397$	1.95333e10	+	1.95333e10i	0.786347	+	0.786347i	0.980893	−	0.194546i	$-0.0623235\pi$
$397$	1.95333e10	+	1.95333e10i	0.786347	+	0.786347i	−0.194546	+	0.980893i	$0.562323\pi$
$398$		0			0
$399$			1.16532e10i			0.459783i
$400$		0			0
$401$	−3.02617e10			−1.17035			−0.585176	−	0.810907i	$-0.698974\pi$
$401$	−3.02617e10			−1.17035			−0.585176	+	0.810907i	$0.698974\pi$
$402$		0			0
$403$	−7.56876e8	+	7.56876e8i	−0.0286949	+	0.0286949i
$404$		0			0
$405$	−1.17120e9	+	7.36079e9i	−0.0435321	+	0.273592i
$406$		0			0
$407$	5.78083e10	+	5.78083e10i	2.10675	+	2.10675i
$408$		0			0
$409$		−	2.76095e10i		−	0.986655i	−0.869844	−	0.493328i	$-0.835780\pi$
$409$		−	2.76095e10i		−	0.986655i	0.869844	−	0.493328i	$-0.164220\pi$
$410$		0			0
$411$	1.16407e10			0.407953
$412$		0			0
$413$	3.24414e10	−	3.24414e10i	1.11506	−	1.11506i
$414$		0			0
$415$	2.00986e10	+	2.77047e10i	0.677600	+	0.934031i
$416$		0			0
$417$	4.50732e9	+	4.50732e9i	0.149064	+	0.149064i
$418$		0			0
$419$			2.03500e10i			0.660250i	0.943937	+	0.330125i	$0.107091\pi$
$419$			2.03500e10i			0.660250i	−0.943937	+	0.330125i	$0.892909\pi$
$420$		0			0
$421$	3.20774e10			1.02111			0.510553	−	0.859846i	$-0.329441\pi$
$421$	3.20774e10			1.02111			0.510553	+	0.859846i	$0.329441\pi$
$422$		0			0
$423$	−1.20018e10	+	1.20018e10i	−0.374875	+	0.374875i
$424$		0			0
$425$	−1.88151e10	−	3.70568e10i	−0.576701	−	1.13583i
$426$		0			0
$427$	−4.04512e10	−	4.04512e10i	−1.21680	−	1.21680i
$428$		0			0
$429$			6.05522e9i			0.178773i
$430$		0			0
$431$	2.99257e10			0.867232			0.433616	−	0.901098i	$-0.357237\pi$
$431$	2.99257e10			0.867232			0.433616	+	0.901098i	$0.357237\pi$
$432$		0			0
$433$	−2.28016e10	+	2.28016e10i	−0.648655	+	0.648655i	−0.952668	−	0.304013i	$-0.901674\pi$
$433$	−2.28016e10	+	2.28016e10i	−0.648655	+	0.648655i	0.304013	+	0.952668i	$0.401674\pi$
$434$		0			0
$435$	5.68303e9	−	4.12280e9i	0.158717	−	0.115142i
$436$		0			0
$437$	7.08223e9	+	7.08223e9i	0.194198	+	0.194198i
$438$		0			0
$439$		−	3.89963e10i		−	1.04994i	−0.851120	−	0.524971i	$-0.824076\pi$
$439$		−	3.89963e10i		−	1.04994i	0.851120	−	0.524971i	$-0.175924\pi$
$440$		0			0
$441$	−5.10105e9			−0.134867
$442$		0			0
$443$	−1.95575e10	+	1.95575e10i	−0.507808	+	0.507808i	−0.913853	−	0.406045i	$-0.866908\pi$
$443$	−1.95575e10	+	1.95575e10i	−0.507808	+	0.507808i	0.406045	+	0.913853i	$0.366908\pi$
$444$		0			0
$445$	−3.66748e9	−	5.83542e8i	−0.0935249	−	0.0148810i
$446$		0			0
$447$	−7.04124e9	−	7.04124e9i	−0.176368	−	0.176368i
$448$		0			0
$449$			4.35112e10i			1.07057i	0.844671	+	0.535286i	$0.179796\pi$
$449$			4.35112e10i			1.07057i	−0.844671	+	0.535286i	$0.820204\pi$
$450$		0			0
$451$	−3.41255e10			−0.824845
$452$		0			0
$453$	1.32092e10	−	1.32092e10i	0.313678	−	0.313678i
$454$		0			0
$455$	1.36785e9	−	8.59672e9i	0.0319148	−	0.200580i
$456$		0			0
$457$	2.46845e9	+	2.46845e9i	0.0565925	+	0.0565925i	0.734837	−	0.678244i	$-0.237259\pi$
$457$	2.46845e9	+	2.46845e9i	0.0565925	+	0.0565925i	−0.678244	+	0.734837i	$0.737259\pi$
$458$		0			0
$459$		−	5.04513e10i		−	1.13664i
$460$		0			0
$461$	−6.81683e10			−1.50931			−0.754656	−	0.656121i	$-0.772196\pi$
$461$	−6.81683e10			−1.50931			−0.754656	+	0.656121i	$0.772196\pi$
$462$		0			0
$463$	−2.38984e10	+	2.38984e10i	−0.520049	+	0.520049i	−0.917586	−	0.397537i	$-0.869865\pi$
$463$	−2.38984e10	+	2.38984e10i	−0.520049	+	0.520049i	0.397537	+	0.917586i	$0.369865\pi$
$464$		0			0
$465$	−3.06747e9	−	4.22832e9i	−0.0656098	−	0.0904392i
$466$		0			0
$467$	−3.02391e10	−	3.02391e10i	−0.635772	−	0.635772i	0.313738	−	0.949510i	$-0.398419\pi$
$467$	−3.02391e10	−	3.02391e10i	−0.635772	−	0.635772i	−0.949510	+	0.313738i	$0.898419\pi$
$468$		0			0
$469$			5.67719e10i			1.17339i
$470$		0			0
$471$	2.84903e10			0.578914
$472$		0			0
$473$	6.89416e10	−	6.89416e10i	1.37733	−	1.37733i
$474$		0			0
$475$	−1.29907e10	+	3.97888e10i	−0.255187	+	0.781603i
$476$		0			0
$477$	−5.02830e10	−	5.02830e10i	−0.971287	−	0.971287i
$478$		0			0
$479$		−	1.43401e10i		−	0.272403i	−0.990681	−	0.136201i	$-0.956511\pi$
$479$		−	1.43401e10i		−	0.272403i	0.990681	−	0.136201i	$-0.0434894\pi$
$480$		0			0
$481$	1.59869e10			0.298664
$482$		0			0
$483$	7.18825e9	−	7.18825e9i	0.132079	−	0.132079i
$484$		0			0
$485$	−7.62118e10	+	5.52884e10i	−1.37738	+	0.999234i
$486$		0			0
$487$	−2.89344e10	−	2.89344e10i	−0.514397	−	0.514397i	0.401474	−	0.915870i	$-0.368498\pi$
$487$	−2.89344e10	−	2.89344e10i	−0.514397	−	0.514397i	−0.915870	+	0.401474i	$0.868498\pi$
$488$		0			0
$489$			4.18524e10i			0.731956i
$490$		0			0
$491$	5.86208e10			1.00862			0.504308	−	0.863524i	$-0.331748\pi$
$491$	5.86208e10			1.00862			0.504308	+	0.863524i	$0.331748\pi$
$492$		0			0
$493$	2.02959e10	−	2.02959e10i	0.343575	−	0.343575i
$494$		0			0
$495$	8.12500e10	+	1.29279e10i	1.35333	+	0.215331i
$496$		0			0
$497$	7.44416e9	+	7.44416e9i	0.122009	+	0.122009i
$498$		0			0
$499$			6.42826e10i			1.03679i	0.855141	+	0.518396i	$0.173470\pi$
$499$			6.42826e10i			1.03679i	−0.855141	+	0.518396i	$0.826530\pi$
$500$		0			0
$501$	1.72950e10			0.274517
$502$		0			0
$503$	6.02745e10	−	6.02745e10i	0.941590	−	0.941590i	−0.0567955	−	0.998386i	$-0.518088\pi$
$503$	6.02745e10	−	6.02745e10i	0.941590	−	0.941590i	0.998386	+	0.0567955i	$0.0180883\pi$
$504$		0			0
$505$	−7.94255e9	+	4.99178e10i	−0.122122	+	0.767520i
$506$		0			0
$507$	−2.31808e10	−	2.31808e10i	−0.350830	−	0.350830i
$508$		0			0
$509$		−	1.72000e10i		−	0.256246i	−0.991758	−	0.128123i	$-0.959105\pi$
$509$		−	1.72000e10i		−	0.256246i	0.991758	−	0.128123i	$-0.0408952\pi$
$510$		0			0
$511$	3.29485e10			0.483228
$512$		0			0
$513$	−3.59286e10	+	3.59286e10i	−0.518765	+	0.518765i
$514$		0			0
$515$	−2.40332e10	−	3.31283e10i	−0.341651	−	0.470945i
$516$		0			0
$517$	6.78017e10	+	6.78017e10i	0.949027	+	0.949027i
$518$		0			0
$519$		−	3.84685e10i		−	0.530195i
$520$		0			0
$521$	8.58625e10			1.16534			0.582670	−	0.812709i	$-0.302008\pi$
$521$	8.58625e10			1.16534			0.582670	+	0.812709i	$0.302008\pi$
$522$		0			0
$523$	−1.96034e10	+	1.96034e10i	−0.262014	+	0.262014i	−0.825872	−	0.563858i	$-0.809317\pi$
$523$	−1.96034e10	+	1.96034e10i	−0.262014	+	0.262014i	0.563858	+	0.825872i	$0.309317\pi$
$524$		0			0
$525$	4.03844e10	+	1.31852e10i	0.531590	+	0.173559i
$526$		0			0
$527$	−1.51007e10	−	1.51007e10i	−0.195774	−	0.195774i
$528$		0			0
$529$			6.95736e10i			0.888428i
$530$		0			0
$531$	8.47937e10			1.06656
$532$		0			0
$533$	−4.71869e9	+	4.71869e9i	−0.0584673	+	0.0584673i
$534$		0			0
$535$	−1.34008e10	+	9.72170e9i	−0.163574	+	0.118666i
$536$		0			0
$537$	−2.61441e10	−	2.61441e10i	−0.314396	−	0.314396i
$538$		0			0
$539$			2.88172e10i			0.341427i
$540$		0			0
$541$	−3.03881e10			−0.354744			−0.177372	−	0.984144i	$-0.556760\pi$
$541$	−3.03881e10			−0.354744			−0.177372	+	0.984144i	$0.556760\pi$
$542$		0			0
$543$	−2.32603e10	+	2.32603e10i	−0.267557	+	0.267557i
$544$		0			0
$545$	−1.45627e11	−	2.31711e10i	−1.65065	−	0.262640i
$546$		0			0
$547$	−1.07672e11	−	1.07672e11i	−1.20268	−	1.20268i	−0.973347	−	0.229338i	$-0.926344\pi$
$547$	−1.07672e11	−	1.07672e11i	−1.20268	−	1.20268i	−0.229338	−	0.973347i	$-0.573656\pi$
$548$		0			0
$549$		−	1.05729e11i		−	1.16387i
$550$		0			0
$551$	−2.89072e10			−0.313617
$552$		0			0
$553$	3.33958e10	−	3.33958e10i	0.357102	−	0.357102i
$554$		0			0
$555$	−1.22599e10	+	7.70515e10i	−0.129215	+	0.812099i
$556$		0			0
$557$	3.79612e10	+	3.79612e10i	0.394384	+	0.394384i	0.876247	−	0.481863i	$-0.160040\pi$
$557$	3.79612e10	+	3.79612e10i	0.394384	+	0.394384i	−0.481863	+	0.876247i	$0.660040\pi$
$558$		0			0
$559$		−	1.90658e10i		−	0.195257i
$560$		0			0
$561$	−1.20810e11			−1.21969
$562$		0			0
$563$	9.04448e10	−	9.04448e10i	0.900223	−	0.900223i	−0.0952324	−	0.995455i	$-0.530359\pi$
$563$	9.04448e10	−	9.04448e10i	0.900223	−	0.900223i	0.995455	+	0.0952324i	$0.0303594\pi$
$564$		0			0
$565$	−1.18918e10	−	1.63922e10i	−0.116696	−	0.160858i
$566$		0			0
$567$	−2.20242e10	−	2.20242e10i	−0.213092	−	0.213092i
$568$		0			0
$569$			2.33053e10i			0.222334i	0.993802	+	0.111167i	$0.0354589\pi$
$569$			2.33053e10i			0.222334i	−0.993802	+	0.111167i	$0.964541\pi$
$570$		0			0
$571$	−1.63516e10			−0.153821			−0.0769103	−	0.997038i	$-0.524506\pi$
$571$	−1.63516e10			−0.153821			−0.0769103	+	0.997038i	$0.524506\pi$
$572$		0			0
$573$	3.92109e10	−	3.92109e10i	0.363737	−	0.363737i
$574$		0			0
$575$	3.25570e10	−	1.65304e10i	0.297833	−	0.151221i
$576$		0			0
$577$	5.69386e9	+	5.69386e9i	0.0513692	+	0.0513692i	0.732325	−	0.680956i	$-0.238435\pi$
$577$	5.69386e9	+	5.69386e9i	0.0513692	+	0.0513692i	−0.680956	+	0.732325i	$0.738435\pi$
$578$		0			0
$579$		−	9.96834e10i		−	0.886970i
$580$		0			0
$581$	−1.43032e11			−1.25525
$582$		0			0
$583$	−2.84062e11	+	2.84062e11i	−2.45889	+	2.45889i
$584$		0			0
$585$	1.30224e10	−	9.44722e9i	0.111191	−	0.0806641i
$586$		0			0
$587$	1.29582e11	+	1.29582e11i	1.09142	+	1.09142i	0.995377	+	0.0960465i	$0.0306197\pi$
$587$	1.29582e11	+	1.29582e11i	1.09142	+	1.09142i	0.0960465	+	0.995377i	$0.469380\pi$
$588$		0			0
$589$			2.15078e10i			0.178704i
$590$		0			0
$591$	−1.27579e10			−0.104576
$592$		0			0
$593$	9.52038e10	−	9.52038e10i	0.769902	−	0.769902i	−0.208187	−	0.978089i	$-0.566756\pi$
$593$	9.52038e10	−	9.52038e10i	0.769902	−	0.769902i	0.978089	+	0.208187i	$0.0667563\pi$
$594$		0			0
$595$	1.71516e11	+	2.72904e10i	1.36848	+	0.217742i
$596$		0			0
$597$	1.57606e10	+	1.57606e10i	0.124073	+	0.124073i
$598$		0			0
$599$		−	9.76852e9i		−	0.0758790i	−0.999280	−	0.0379395i	$-0.987921\pi$
$599$		−	9.76852e9i		−	0.0758790i	0.999280	−	0.0379395i	$-0.0120794\pi$
$600$		0			0
$601$	1.16960e11			0.896474			0.448237	−	0.893915i	$-0.352052\pi$
$601$	1.16960e11			0.896474			0.448237	+	0.893915i	$0.352052\pi$
$602$		0			0
$603$	−7.41937e10	+	7.41937e10i	−0.561174	+	0.561174i
$604$		0			0
$605$	5.19811e10	−	3.26693e11i	0.387993	−	2.43848i
$606$		0			0
$607$	2.56369e10	+	2.56369e10i	0.188847	+	0.188847i	0.795198	−	0.606350i	$-0.207367\pi$
$607$	2.56369e10	+	2.56369e10i	0.188847	+	0.188847i	−0.606350	+	0.795198i	$0.707367\pi$
$608$		0			0
$609$			2.93400e10i			0.213300i
$610$		0			0
$611$	1.87505e10			0.134539
$612$		0			0
$613$	3.57093e10	−	3.57093e10i	0.252894	−	0.252894i	−0.569262	−	0.822156i	$-0.692771\pi$
$613$	3.57093e10	−	3.57093e10i	0.252894	−	0.252894i	0.822156	+	0.569262i	$0.192771\pi$
$614$		0			0
$615$	−1.91239e10	−	2.63612e10i	−0.133683	−	0.184274i
$616$		0			0
$617$	6.31791e10	+	6.31791e10i	0.435946	+	0.435946i	0.890645	−	0.454699i	$-0.150253\pi$
$617$	6.31791e10	+	6.31791e10i	0.435946	+	0.435946i	−0.454699	+	0.890645i	$0.650253\pi$
$618$		0			0
$619$		−	5.89874e10i		−	0.401788i	−0.979613	−	0.200894i	$-0.935615\pi$
$619$		−	5.89874e10i		−	0.401788i	0.979613	−	0.200894i	$-0.0643846\pi$
$620$		0			0
$621$	4.43251e10			0.298046
$622$		0			0
$623$	1.09734e10	−	1.09734e10i	0.0728435	−	0.0728435i
$624$		0			0
$625$	1.23191e11	+	9.00393e10i	0.807343	+	0.590082i
$626$		0			0
$627$	8.60340e10	+	8.60340e10i	0.556673	+	0.556673i
$628$		0			0
$629$			3.18960e11i			2.03767i
$630$		0			0
$631$	−8.68315e10			−0.547721			−0.273861	−	0.961769i	$-0.588301\pi$
$631$	−8.68315e10			−0.547721			−0.273861	+	0.961769i	$0.588301\pi$
$632$		0			0
$633$	6.46987e10	−	6.46987e10i	0.402977	−	0.402977i
$634$		0			0
$635$	−8.85468e10	+	6.42370e10i	−0.544601	+	0.395085i
$636$		0			0
$637$	3.98470e9	+	3.98470e9i	0.0242013	+	0.0242013i
$638$		0			0
$639$			1.94572e10i			0.116701i
$640$		0			0
$641$	−1.04888e11			−0.621290			−0.310645	−	0.950526i	$-0.600545\pi$
$641$	−1.04888e11			−0.621290			−0.310645	+	0.950526i	$0.600545\pi$
$642$		0			0
$643$	−6.21828e10	+	6.21828e10i	−0.363770	+	0.363770i	−0.865199	−	0.501429i	$-0.832808\pi$
$643$	−6.21828e10	+	6.21828e10i	−0.363770	+	0.363770i	0.501429	+	0.865199i	$0.332808\pi$
$644$		0			0
$645$	9.18909e10	+	1.46210e10i	0.530925	+	0.0844770i
$646$		0			0
$647$	6.55391e10	+	6.55391e10i	0.374010	+	0.374010i	0.868936	−	0.494925i	$-0.164805\pi$
$647$	6.55391e10	+	6.55391e10i	0.374010	+	0.374010i	−0.494925	+	0.868936i	$0.664805\pi$
$648$		0			0
$649$		−	4.79023e11i		−	2.70008i
$650$		0			0
$651$	2.18297e10			0.121541
$652$		0			0
$653$	−6.93583e10	+	6.93583e10i	−0.381457	+	0.381457i	−0.871627	−	0.490170i	$-0.836935\pi$
$653$	−6.93583e10	+	6.93583e10i	−0.381457	+	0.381457i	0.490170	+	0.871627i	$0.336935\pi$
$654$		0			0
$655$	4.33597e10	−	2.72509e11i	0.235571	−	1.48053i
$656$		0			0
$657$	4.30595e10	+	4.30595e10i	0.231104	+	0.231104i
$658$		0			0
$659$			2.40343e11i			1.27435i	0.770718	+	0.637176i	$0.219898\pi$
$659$			2.40343e11i			1.27435i	−0.770718	+	0.637176i	$0.780102\pi$
$660$		0			0
$661$	1.14988e10			0.0602348			0.0301174	−	0.999546i	$-0.490412\pi$
$661$	1.14988e10			0.0602348			0.0301174	+	0.999546i	$0.490412\pi$
$662$		0			0
$663$	−1.67050e10	+	1.67050e10i	−0.0864552	+	0.0864552i
$664$		0			0
$665$	−1.02710e11	−	1.41579e11i	−0.525199	−	0.723956i
$666$		0			0
$667$	1.78314e10	+	1.78314e10i	0.0900913	+	0.0900913i
$668$		0			0
$669$		−	1.95821e11i		−	0.977586i
$670$		0			0
$671$	−5.97293e11			−2.94644
$672$		0			0
$673$	−2.17942e11	+	2.17942e11i	−1.06238	+	1.06238i	−0.0644605	+	0.997920i	$0.520533\pi$
$673$	−2.17942e11	+	2.17942e11i	−1.06238	+	1.06238i	−0.997920	+	0.0644605i	$0.979467\pi$
$674$		0			0
$675$	8.38597e10	+	1.65164e11i	0.403960	+	0.795609i
$676$		0			0
$677$	−4.78802e10	−	4.78802e10i	−0.227930	−	0.227930i	0.583898	−	0.811827i	$-0.301527\pi$
$677$	−4.78802e10	−	4.78802e10i	−0.227930	−	0.227930i	−0.811827	+	0.583898i	$0.801527\pi$
$678$		0			0
$679$		−	3.93461e11i		−	1.85107i
$680$		0			0
$681$	−1.61839e11			−0.752478
$682$		0			0
$683$	2.08426e11	−	2.08426e11i	0.957787	−	0.957787i	−0.0413575	−	0.999144i	$-0.513168\pi$
$683$	2.08426e11	−	2.08426e11i	0.957787	−	0.957787i	0.999144	+	0.0413575i	$0.0131683\pi$
$684$		0			0
$685$	−1.41427e11	+	1.02599e11i	−0.642348	+	0.465996i
$686$		0			0
$687$	2.20863e10	+	2.20863e10i	0.0991507	+	0.0991507i
$688$		0			0
$689$			7.85573e10i			0.348586i
$690$		0			0
$691$	3.71711e11			1.63040			0.815198	−	0.579183i	$-0.196628\pi$
$691$	3.71711e11			1.63040			0.815198	+	0.579183i	$0.196628\pi$
$692$		0			0
$693$	−2.43108e11	+	2.43108e11i	−1.05406	+	1.05406i
$694$		0			0
$695$	−9.44881e10	−	1.50343e10i	−0.404984	−	0.0644381i
$696$		0			0
$697$	−9.41443e10	−	9.41443e10i	−0.398899	−	0.398899i
$698$		0			0
$699$			4.45914e10i			0.186785i
$700$		0			0
$701$	3.27111e11			1.35464			0.677318	−	0.735690i	$-0.263142\pi$
$701$	3.27111e11			1.35464			0.677318	+	0.735690i	$0.263142\pi$
$702$		0			0
$703$	2.27145e11	−	2.27145e11i	0.929998	−	0.929998i
$704$		0			0
$705$	−1.43793e10	+	9.03715e10i	−0.0582076	+	0.365826i
$706$		0			0
$707$	−1.49359e11	−	1.49359e11i	−0.597796	−	0.597796i
$708$		0			0
$709$			3.56319e11i			1.41011i	0.709151	+	0.705056i	$0.249078\pi$
$709$			3.56319e11i			1.41011i	−0.709151	+	0.705056i	$0.750922\pi$
$710$		0			0
$711$	8.72882e10			0.341568
$712$		0			0
$713$	1.32670e10	−	1.32670e10i	0.0513353	−	0.0513353i
$714$		0			0
$715$	−5.33699e10	−	7.35672e10i	−0.204208	−	0.281488i
$716$		0			0
$717$	−8.83316e10	−	8.83316e10i	−0.334225	−	0.334225i
$718$		0			0
$719$			1.32561e11i			0.496022i	0.968757	+	0.248011i	$0.0797769\pi$
$719$			1.32561e11i			0.496022i	−0.968757	+	0.248011i	$0.920223\pi$
$720$		0			0
$721$	1.71033e11			0.632904
$722$		0			0
$723$	−6.63084e10	+	6.63084e10i	−0.242669	+	0.242669i
$724$		0			0
$725$	−3.27076e10	+	1.00179e11i	−0.118385	+	0.362597i
$726$		0			0
$727$	3.61153e10	+	3.61153e10i	0.129287	+	0.129287i	0.768789	−	0.639503i	$-0.220860\pi$
$727$	3.61153e10	+	3.61153e10i	0.129287	+	0.129287i	−0.639503	+	0.768789i	$0.720860\pi$
$728$		0			0
$729$			7.19843e10i			0.254875i
$730$		0			0
$731$	3.80388e11			1.33216
$732$		0			0
$733$	2.93274e11	−	2.93274e11i	1.01591	−	1.01591i	0.0160430	−	0.999871i	$-0.494893\pi$
$733$	2.93274e11	−	2.93274e11i	1.01591	−	1.01591i	0.999871	−	0.0160430i	$-0.00510688\pi$
$734$		0			0
$735$	−2.22607e10	+	1.61492e10i	−0.0762763	+	0.0553352i
$736$		0			0
$737$	4.19140e11	+	4.19140e11i	1.42066	+	1.42066i
$738$		0			0
$739$			1.49433e11i			0.501036i	0.968112	+	0.250518i	$0.0806010\pi$
$739$			1.49433e11i			0.501036i	−0.968112	+	0.250518i	$0.919399\pi$
$740$		0			0
$741$	2.37927e10			0.0789169
$742$		0			0
$743$	5.62819e10	−	5.62819e10i	0.184677	−	0.184677i	−0.608713	−	0.793390i	$-0.708314\pi$
$743$	5.62819e10	−	5.62819e10i	0.184677	−	0.184677i	0.793390	+	0.608713i	$0.208314\pi$
$744$		0			0
$745$	1.47607e11	+	2.34862e10i	0.479162	+	0.0762408i
$746$		0			0
$747$	−1.86925e11	−	1.86925e11i	−0.600322	−	0.600322i
$748$		0			0
$749$		−	6.91846e10i		−	0.219828i
$750$		0			0
$751$	−2.89364e11			−0.909671			−0.454836	−	0.890575i	$-0.650302\pi$
$751$	−2.89364e11			−0.909671			−0.454836	+	0.890575i	$0.650302\pi$
$752$		0			0
$753$	−1.36954e11	+	1.36954e11i	−0.425985	+	0.425985i
$754$		0			0
$755$	−4.40596e10	+	2.76908e11i	−0.135598	+	0.852212i
$756$		0			0
$757$	−2.89205e11	−	2.89205e11i	−0.880688	−	0.880688i	0.112916	−	0.993604i	$-0.463981\pi$
$757$	−2.89205e11	−	2.89205e11i	−0.880688	−	0.880688i	−0.993604	+	0.112916i	$0.963981\pi$
$758$		0			0
$759$		−	1.06140e11i		−	0.319825i
$760$		0			0
$761$	−9.87295e10			−0.294380			−0.147190	−	0.989108i	$-0.547023\pi$
$761$	−9.87295e10			−0.294380			−0.147190	+	0.989108i	$0.547023\pi$
$762$		0			0
$763$	4.35730e11	−	4.35730e11i	1.28564	−	1.28564i
$764$		0			0
$765$	1.88485e11	+	2.59815e11i	0.550339	+	0.758610i
$766$		0			0
$767$	−6.62368e10	−	6.62368e10i	−0.191389	−	0.191389i
$768$		0			0
$769$		−	1.60740e10i		−	0.0459640i	−0.999736	−	0.0229820i	$-0.992684\pi$
$769$		−	1.60740e10i		−	0.0459640i	0.999736	−	0.0229820i	$-0.00731603\pi$
$770$		0			0
$771$	1.24667e11			0.352805
$772$		0			0
$773$	−5.26755e10	+	5.26755e10i	−0.147533	+	0.147533i	−0.777015	−	0.629482i	$-0.783267\pi$
$773$	−5.26755e10	+	5.26755e10i	−0.147533	+	0.147533i	0.629482	+	0.777015i	$0.283267\pi$
$774$		0			0
$775$	7.45358e10	+	2.43353e10i	0.206613	+	0.0674574i
$776$		0			0
$777$	−2.30545e11	−	2.30545e11i	−0.632517	−	0.632517i
$778$		0			0
$779$			1.34089e11i			0.364118i
$780$		0			0
$781$	1.09919e11			0.295439
$782$		0			0
$783$	−9.04599e10	+	9.04599e10i	−0.240663	+	0.240663i
$784$		0			0
$785$	−3.46140e11	+	2.51110e11i	−0.911535	+	0.661280i
$786$		0			0
$787$	−3.52484e11	−	3.52484e11i	−0.918841	−	0.918841i	0.0781038	−	0.996945i	$-0.475113\pi$
$787$	−3.52484e11	−	3.52484e11i	−0.918841	−	0.918841i	−0.996945	+	0.0781038i	$0.975113\pi$
$788$		0			0
$789$			2.71475e11i			0.700521i
$790$		0			0
$791$	8.46283e10			0.216177
$792$		0			0
$793$	−8.25905e10	+	8.25905e10i	−0.208851	+	0.208851i
$794$		0			0
$795$	−3.78621e11	−	6.02434e10i	−0.947842	−	0.150814i
$796$		0			0
$797$	−3.84209e11	−	3.84209e11i	−0.952214	−	0.952214i	0.0466953	−	0.998909i	$-0.485131\pi$
$797$	−3.84209e11	−	3.84209e11i	−0.952214	−	0.952214i	−0.998909	+	0.0466953i	$0.985131\pi$
$798$		0			0
$799$			3.74099e11i			0.917907i
$800$		0			0
$801$	2.86818e10			0.0696749
$802$		0			0
$803$	2.43255e11	−	2.43255e11i	0.585058	−	0.585058i
$804$		0			0
$805$	−2.39766e10	+	1.50689e11i	−0.0570957	+	0.358838i
$806$		0			0
$807$	1.41926e11	+	1.41926e11i	0.334632	+	0.334632i
$808$		0			0
$809$		−	2.83223e11i		−	0.661202i	−0.943771	−	0.330601i	$-0.892749\pi$
$809$		−	2.83223e11i		−	0.661202i	0.943771	−	0.330601i	$-0.107251\pi$
$810$		0			0
$811$	−2.20272e11			−0.509185			−0.254593	−	0.967048i	$-0.581941\pi$
$811$	−2.20272e11			−0.509185			−0.254593	+	0.967048i	$0.581941\pi$
$812$		0			0
$813$	1.25950e11	−	1.25950e11i	0.288294	−	0.288294i
$814$		0			0
$815$	−3.68882e11	−	5.08481e11i	−0.836097	−	1.15251i
$816$		0			0
$817$	−2.70891e11	−	2.70891e11i	−0.608004	−	0.608004i
$818$		0			0
$819$			6.72313e10i			0.149429i
$820$		0			0
$821$	−1.46718e11			−0.322931			−0.161466	−	0.986878i	$-0.551622\pi$
$821$	−1.46718e11			−0.322931			−0.161466	+	0.986878i	$0.551622\pi$
$822$		0			0
$823$	−3.16433e11	+	3.16433e11i	−0.689736	+	0.689736i	−0.962173	−	0.272438i	$-0.912170\pi$
$823$	−3.16433e11	+	3.16433e11i	−0.689736	+	0.689736i	0.272438	+	0.962173i	$0.412170\pi$
$824$		0			0
$825$	3.95498e11	−	2.00809e11i	0.853746	−	0.433478i
$826$		0			0
$827$	3.32131e11	+	3.32131e11i	0.710047	+	0.710047i	0.966545	−	0.256498i	$-0.0825686\pi$
$827$	3.32131e11	+	3.32131e11i	0.710047	+	0.710047i	−0.256498	+	0.966545i	$0.582569\pi$
$828$		0			0
$829$		−	6.96417e11i		−	1.47452i	−0.675608	−	0.737261i	$-0.736119\pi$
$829$		−	6.96417e11i		−	1.47452i	0.675608	−	0.737261i	$-0.263881\pi$
$830$		0			0
$831$	2.27412e11			0.476880
$832$		0			0
$833$	−7.95001e10	+	7.95001e10i	−0.165116	+	0.165116i
$834$		0			0
$835$	−2.10124e11	+	1.52436e11i	−0.432244	+	0.313575i
$836$		0			0
$837$	6.73046e10	+	6.73046e10i	0.137133	+	0.137133i
$838$		0			0
$839$			8.69937e10i			0.175566i	0.996140	+	0.0877829i	$0.0279782\pi$
$839$			8.69937e10i			0.175566i	−0.996140	+	0.0877829i	$0.972022\pi$
$840$		0			0
$841$	4.27465e11			0.854508
$842$		0			0
$843$	1.11997e11	−	1.11997e11i	0.221767	−	0.221767i
$844$		0			0
$845$	4.85946e11	+	7.73202e10i	0.953149	+	0.151658i
$846$		0			0
$847$	9.77497e11	+	9.77497e11i	1.89925	+	1.89925i
$848$		0			0
$849$			1.35340e11i			0.260492i
$850$		0			0
$851$	−2.80229e11			−0.534311
$852$		0			0
$853$	−4.88735e11	+	4.88735e11i	−0.923162	+	0.923162i	−0.997252	−	0.0740899i	$-0.976395\pi$
$853$	−4.88735e11	+	4.88735e11i	−0.923162	+	0.923162i	0.0740899	+	0.997252i	$0.476395\pi$
$854$		0			0
$855$	5.07974e10	−	3.19254e11i	0.0950554	−	0.597409i
$856$		0			0
$857$	−4.47848e9	−	4.47848e9i	−0.00830246	−	0.00830246i	0.702943	−	0.711246i	$-0.251869\pi$
$857$	−4.47848e9	−	4.47848e9i	−0.00830246	−	0.00830246i	−0.711246	+	0.702943i	$0.751869\pi$
$858$		0			0
$859$			4.94570e11i			0.908354i	0.890911	+	0.454177i	$0.150067\pi$
$859$			4.94570e11i			0.908354i	−0.890911	+	0.454177i	$0.849933\pi$
$860$		0			0
$861$	1.36096e11			0.247646
$862$		0			0
$863$	3.85362e10	−	3.85362e10i	0.0694746	−	0.0694746i	−0.671516	−	0.740990i	$-0.734356\pi$
$863$	3.85362e10	−	3.85362e10i	0.0694746	−	0.0694746i	0.740990	+	0.671516i	$0.234356\pi$
$864$		0			0
$865$	3.39056e11	+	4.67369e11i	0.605630	+	0.834825i
$866$		0			0
$867$	−1.27895e11	−	1.27895e11i	−0.226347	−	0.226347i
$868$		0			0
$869$		−	4.93115e11i		−	0.864707i
$870$		0			0
$871$	1.15913e11			0.201400
$872$		0			0
$873$	5.14204e11	−	5.14204e11i	0.885275	−	0.885275i
$874$		0			0
$875$	−6.06858e11	+	1.95751e11i	−1.03527	+	0.333943i
$876$		0			0
$877$	2.96877e11	+	2.96877e11i	0.501856	+	0.501856i	0.912014	−	0.410159i	$-0.134527\pi$
$877$	2.96877e11	+	2.96877e11i	0.501856	+	0.501856i	−0.410159	+	0.912014i	$0.634527\pi$
$878$		0			0
$879$		−	2.79006e11i		−	0.467368i
$880$		0			0
$881$	5.89117e11			0.977907			0.488954	−	0.872310i	$-0.337379\pi$
$881$	5.89117e11			0.977907			0.488954	+	0.872310i	$0.337379\pi$
$882$		0			0
$883$	−1.14084e11	+	1.14084e11i	−0.187665	+	0.187665i	−0.794686	−	0.607021i	$-0.792364\pi$
$883$	−1.14084e11	+	1.14084e11i	−0.187665	+	0.187665i	0.607021	+	0.794686i	$0.292364\pi$
$884$		0			0
$885$	3.70035e11	−	2.68445e11i	0.603211	−	0.437604i
$886$		0			0
$887$	2.01818e11	+	2.01818e11i	0.326035	+	0.326035i	0.851077	−	0.525041i	$-0.175950\pi$
$887$	2.01818e11	+	2.01818e11i	0.326035	+	0.326035i	−0.525041	+	0.851077i	$0.675950\pi$
$888$		0			0
$889$		−	4.57144e11i		−	0.731890i
$890$		0			0
$891$	−3.25204e11			−0.515994
$892$		0			0
$893$	2.66412e11	−	2.66412e11i	0.418936	−	0.418936i
$894$		0			0
$895$	5.48066e11	+	8.72044e10i	0.854164	+	0.135908i
$896$		0			0
$897$	−1.46765e10	−	1.46765e10i	−0.0226701	−	0.0226701i
$898$		0			0
$899$			5.41515e10i			0.0829033i
$900$		0			0
$901$	−1.56733e12			−2.37826
$902$		0			0
$903$	−2.74946e11	+	2.74946e11i	−0.413520	+	0.413520i
$904$		0			0
$905$	7.75852e10	−	4.87611e11i	0.115660	−	0.726908i
$906$		0			0
$907$	−2.20776e11	−	2.20776e11i	−0.326229	−	0.326229i	0.524922	−	0.851150i	$-0.324095\pi$
$907$	−2.20776e11	−	2.20776e11i	−0.326229	−	0.326229i	−0.851150	+	0.524922i	$0.824095\pi$
$908$		0			0
$909$		−	3.90386e11i		−	0.571792i
$910$		0			0
$911$	−9.59264e11			−1.39272			−0.696361	−	0.717692i	$-0.745199\pi$
$911$	−9.59264e11			−1.39272			−0.696361	+	0.717692i	$0.745199\pi$
$912$		0			0
$913$	−1.05599e12	+	1.05599e12i	−1.51976	+	1.51976i
$914$		0			0
$915$	−3.34723e11	−	4.61396e11i	−0.477531	−	0.658247i
$916$		0			0
$917$	8.15373e11	+	8.15373e11i	1.15313	+	1.15313i
$918$		0			0
$919$			6.19847e11i			0.869005i	0.900671	+	0.434502i	$0.143076\pi$
$919$			6.19847e11i			0.869005i	−0.900671	+	0.434502i	$0.856924\pi$
$920$		0			0
$921$	2.17773e11			0.302668
$922$		0			0
$923$	1.51990e10	−	1.51990e10i	0.0209415	−	0.0209415i
$924$		0			0
$925$	−5.30172e11	−	1.04419e12i	−0.724185	−	1.42630i
$926$		0			0
$927$	2.23518e11	+	2.23518e11i	0.302687	+	0.302687i
$928$		0			0
$929$		−	1.15962e11i		−	0.155687i	−0.996966	−	0.0778434i	$-0.975197\pi$
$929$		−	1.15962e11i		−	0.155687i	0.996966	−	0.0778434i	$-0.0248034\pi$
$930$		0			0
$931$	1.13231e11			0.150719
$932$		0			0
$933$	−2.64449e11	+	2.64449e11i	−0.348992	+	0.348992i
$934$		0			0
$935$	1.46777e12	−	1.06480e12i	1.92048	−	1.39323i
$936$		0			0
$937$	−2.76920e11	−	2.76920e11i	−0.359249	−	0.359249i	0.504287	−	0.863536i	$-0.331755\pi$
$937$	−2.76920e11	−	2.76920e11i	−0.359249	−	0.359249i	−0.863536	+	0.504287i	$0.831755\pi$
$938$		0			0
$939$			2.52468e11i			0.324746i
$940$		0			0
$941$	−1.25899e11			−0.160569			−0.0802847	−	0.996772i	$-0.525583\pi$
$941$	−1.25899e11			−0.160569			−0.0802847	+	0.996772i	$0.525583\pi$
$942$		0			0
$943$	8.27125e10	−	8.27125e10i	0.104598	−	0.104598i
$944$		0			0
$945$	−7.64456e11	−	1.21635e11i	−0.958573	−	0.152521i
$946$		0			0
$947$	−1.06723e12	−	1.06723e12i	−1.32696	−	1.32696i	−0.908005	−	0.418959i	$-0.862395\pi$
$947$	−1.06723e12	−	1.06723e12i	−1.32696	−	1.32696i	−0.418959	−	0.908005i	$-0.637605\pi$
$948$		0			0
$949$		−	6.72720e10i		−	0.0829410i
$950$		0			0
$951$	−3.12271e11			−0.381777
$952$		0			0
$953$	−1.25047e11	+	1.25047e11i	−0.151601	+	0.151601i	−0.778833	−	0.627232i	$-0.784188\pi$
$953$	−1.25047e11	+	1.25047e11i	−0.151601	+	0.151601i	0.627232	+	0.778833i	$0.284188\pi$
$954$		0			0
$955$	−1.30789e11	+	8.21988e11i	−0.157238	+	0.988216i
$956$		0			0
$957$	2.16614e11	+	2.16614e11i	0.258249	+	0.258249i
$958$		0			0
$959$		−	7.30150e11i		−	0.863252i
$960$		0			0
$961$	−8.12601e11			−0.952760
$962$		0			0
$963$	9.04155e10	−	9.04155e10i	0.105133	−	0.105133i
$964$		0			0
$965$	8.78597e11	+	1.21109e12i	1.01317	+	1.39659i
$966$		0			0
$967$	7.92466e11	+	7.92466e11i	0.906306	+	0.906306i	0.995972	−	0.0896656i	$-0.0285798\pi$
$967$	7.92466e11	+	7.92466e11i	0.906306	+	0.906306i	−0.0896656	+	0.995972i	$0.528580\pi$
$968$		0			0
$969$			4.74696e11i			0.538419i
$970$		0			0
$971$	1.24092e9			0.00139595			0.000697973	−	1.00000i	$-0.499778\pi$
$971$	1.24092e9			0.00139595			0.000697973		1.00000i	$0.499778\pi$
$972$		0			0
$973$	2.82717e11	−	2.82717e11i	0.315429	−	0.315429i
$974$		0			0
$975$	2.69206e10	−	8.24542e10i	0.0297897	−	0.0912419i
$976$		0			0
$977$	−3.89017e11	−	3.89017e11i	−0.426963	−	0.426963i	0.460629	−	0.887593i	$-0.347624\pi$
$977$	−3.89017e11	−	3.89017e11i	−0.426963	−	0.426963i	−0.887593	+	0.460629i	$0.847624\pi$
$978$		0			0
$979$		−	1.62031e11i		−	0.176388i
$980$		0			0
$981$	1.13889e12			1.22972
$982$		0			0
$983$	3.02595e10	−	3.02595e10i	0.0324077	−	0.0324077i	−0.690717	−	0.723125i	$-0.742705\pi$
$983$	3.02595e10	−	3.02595e10i	0.0324077	−	0.0324077i	0.723125	+	0.690717i	$0.242705\pi$
$984$		0			0
$985$	1.55001e11	−	1.12447e11i	0.164661	−	0.119454i
$986$		0			0
$987$	−2.70400e11	−	2.70400e11i	−0.284930	−	0.284930i
$988$		0			0
$989$			3.34198e11i			0.349316i
$990$		0			0
$991$	7.19692e11			0.746195			0.373097	−	0.927792i	$-0.378296\pi$
$991$	7.19692e11			0.746195			0.373097	+	0.927792i	$0.378296\pi$
$992$		0			0
$993$	9.50035e10	−	9.50035e10i	0.0977108	−	0.0977108i
$994$		0			0
$995$	−3.30394e11	−	5.25698e10i	−0.337085	−	0.0536345i
$996$		0			0
$997$	8.06297e11	+	8.06297e11i	0.816046	+	0.816046i	0.985532	−	0.169487i	$-0.0542110\pi$
$997$	8.06297e11	+	8.06297e11i	0.816046	+	0.816046i	−0.169487	+	0.985532i	$0.554211\pi$
$998$		0			0
$999$		−	1.42162e12i		−	1.42732i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.9.f.a.17.2	yes	8
3.2	odd	2		180.9.l.a.37.3		8
4.3	odd	2		80.9.p.d.17.3		8
5.2	odd	4		100.9.f.b.93.3		8
5.3	odd	4	inner	20.9.f.a.13.2	✓	8
5.4	even	2		100.9.f.b.57.3		8
15.8	even	4		180.9.l.a.73.3		8
20.3	even	4		80.9.p.d.33.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.f.a.13.2	✓	8	5.3	odd	4	inner
20.9.f.a.17.2	yes	8	1.1	even	1	trivial
80.9.p.d.17.3		8	4.3	odd	2
80.9.p.d.33.3		8	20.3	even	4
100.9.f.b.57.3		8	5.4	even	2
100.9.f.b.93.3		8	5.2	odd	4
180.9.l.a.37.3		8	3.2	odd	2
180.9.l.a.73.3		8	15.8	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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	20.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-29.4437 + 29.4437i) q^{3} +(98.2101 - 617.236i) q^{5} +(1846.83 + 1846.83i) q^{7} +4827.14i q^{9} +O(q^{10})\)	\(q+(-29.4437 + 29.4437i) q^{3} +(98.2101 - 617.236i) q^{5} +(1846.83 + 1846.83i) q^{7} +4827.14i q^{9} +27269.8 q^{11} +(3770.73 - 3770.73i) q^{13} +(15282.0 + 21065.4i) q^{15} +(75231.2 + 75231.2i) q^{17} -107151. i q^{19} -108755. q^{21} +(-66095.9 + 66095.9i) q^{23} +(-371335. - 121237. i) q^{25} +(-335309. - 335309. i) q^{27} -269781. i q^{29} -200724. q^{31} +(-802924. + 802924. i) q^{33} +(1.32130e6 - 958551. i) q^{35} +(2.11986e6 + 2.11986e6i) q^{37} +222048. i q^{39} -1.25140e6 q^{41} +(2.52813e6 - 2.52813e6i) q^{43} +(2.97948e6 + 474074. i) q^{45} +(2.48633e6 + 2.48633e6i) q^{47} +1.05674e6i q^{49} -4.43017e6 q^{51} +(-1.04167e7 + 1.04167e7i) q^{53} +(2.67817e6 - 1.68319e7i) q^{55} +(3.15491e6 + 3.15491e6i) q^{57} -1.75660e7i q^{59} -2.19031e7 q^{61} +(-8.91489e6 + 8.91489e6i) q^{63} +(-1.95711e6 - 2.69775e6i) q^{65} +(1.53701e7 + 1.53701e7i) q^{67} -3.89222e6i q^{69} +4.03078e6 q^{71} +(8.92029e6 - 8.92029e6i) q^{73} +(1.45031e7 - 7.36378e6i) q^{75} +(5.03627e7 + 5.03627e7i) q^{77} -1.80828e7i q^{79} -1.19254e7 q^{81} +(-3.87237e7 + 3.87237e7i) q^{83} +(5.38238e7 - 3.90469e7i) q^{85} +(7.94334e6 + 7.94334e6i) q^{87} -5.94178e6i q^{89} +1.39278e7 q^{91} +(5.91006e6 - 5.91006e6i) q^{93} +(-6.61373e7 - 1.05233e7i) q^{95} +(-1.06523e8 - 1.06523e8i) q^{97} +1.31635e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7}+O(q^{10}) \) 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7	\( 8 q - 70 q^{3} + 894 q^{5} - 2030 q^{7} - 420 q^{11} + 33180 q^{13} - 48478 q^{15} + 43620 q^{17} + 108668 q^{21} - 663270 q^{23} + 163396 q^{25} + 1576040 q^{27} - 3178492 q^{31} - 944020 q^{33} + 2571618 q^{35} + 5344080 q^{37} - 10185252 q^{41} - 10342710 q^{43} + 20284834 q^{45} + 19232250 q^{47} - 47126684 q^{51} - 24320640 q^{53} + 21483180 q^{55} + 88218320 q^{57} - 82515684 q^{61} - 77441350 q^{63} + 72045768 q^{65} + 100675930 q^{67} - 99290076 q^{71} - 93528520 q^{73} + 76524178 q^{75} + 134199660 q^{77} - 161920268 q^{81} - 10450350 q^{83} + 51676156 q^{85} + 164801600 q^{87} - 130681068 q^{91} - 50183620 q^{93} + 84367944 q^{95} - 179570760 q^{97}+O(q^{100}) \) 8 * q - 70 * q^3 + 894 * q^5 - 2030 * q^7 - 420 * q^11 + 33180 * q^13 - 48478 * q^15 + 43620 * q^17 + 108668 * q^21 - 663270 * q^23 + 163396 * q^25 + 1576040 * q^27 - 3178492 * q^31 - 944020 * q^33 + 2571618 * q^35 + 5344080 * q^37 - 10185252 * q^41 - 10342710 * q^43 + 20284834 * q^45 + 19232250 * q^47 - 47126684 * q^51 - 24320640 * q^53 + 21483180 * q^55 + 88218320 * q^57 - 82515684 * q^61 - 77441350 * q^63 + 72045768 * q^65 + 100675930 * q^67 - 99290076 * q^71 - 93528520 * q^73 + 76524178 * q^75 + 134199660 * q^77 - 161920268 * q^81 - 10450350 * q^83 + 51676156 * q^85 + 164801600 * q^87 - 130681068 * q^91 - 50183620 * q^93 + 84367944 * q^95 - 179570760 * q^97

Introduction

L-functions

Modular forms

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Database

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