LMFDB - Embedded newform 20.11.f.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,11,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, 11, names="a")

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

chi := DirichletCharacter("20.13");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 11 $
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:	$12.7071450535$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 75402 x^{8} + 1918432665 x^{6} + 20025190470928 x^{4} + \cdots + 13\!\cdots\!00 $ x^10 + 75402x^8 + 1918432665x^6 + 20025190470928x^4 + 87673968468747264x^2 + 130532036219814297600
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.1
Root			$-149.896i$ of defining polynomial
Character	$\chi$	$=$	20.17
Dual form			20.11.f.a.13.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-190.862 + 190.862i) q^{3} +(2756.18 - 1472.78i) q^{5} +(14849.7 + 14849.7i) q^{7} -13807.6i q^{9} +O(q^{10})$	\(q+(-190.862 + 190.862i) q^{3} +(2756.18 - 1472.78i) q^{5} +(14849.7 + 14849.7i) q^{7} -13807.6i q^{9} -319087. q^{11} +(-76905.8 + 76905.8i) q^{13} +(-244952. + 807149. i) q^{15} +(-797011. - 797011. i) q^{17} +3.23229e6i q^{19} -5.66849e6 q^{21} +(-5.27364e6 + 5.27364e6i) q^{23} +(5.42745e6 - 8.11851e6i) q^{25} +(-8.63486e6 - 8.63486e6i) q^{27} +2.58390e7i q^{29} +1.45717e7 q^{31} +(6.09017e7 - 6.09017e7i) q^{33} +(6.27989e7 + 1.90581e7i) q^{35} +(-2.45616e7 - 2.45616e7i) q^{37} -2.93568e7i q^{39} -6.80482e7 q^{41} +(-5.22255e6 + 5.22255e6i) q^{43} +(-2.03356e7 - 3.80563e7i) q^{45} +(4.11962e7 + 4.11962e7i) q^{47} +1.58553e8i q^{49} +3.04238e8 q^{51} +(-2.99133e8 + 2.99133e8i) q^{53} +(-8.79463e8 + 4.69947e8i) q^{55} +(-6.16922e8 - 6.16922e8i) q^{57} -4.68335e7i q^{59} +2.29974e8 q^{61} +(2.05039e8 - 2.05039e8i) q^{63} +(-9.87008e7 + 3.25232e8i) q^{65} +(1.40597e9 + 1.40597e9i) q^{67} -2.01307e9i q^{69} +1.91787e9 q^{71} +(2.58703e9 - 2.58703e9i) q^{73} +(5.13623e8 + 2.58541e9i) q^{75} +(-4.73836e9 - 4.73836e9i) q^{77} +5.01453e9i q^{79} +4.11146e9 q^{81} +(2.23915e9 - 2.23915e9i) q^{83} +(-3.37053e9 - 1.02288e9i) q^{85} +(-4.93168e9 - 4.93168e9i) q^{87} +7.96413e9i q^{89} -2.28406e9 q^{91} +(-2.78119e9 + 2.78119e9i) q^{93} +(4.76046e9 + 8.90878e9i) q^{95} +(-4.52809e9 - 4.52809e9i) q^{97} +4.40584e9i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7}+O(q^{10}) $ 10 * q + 62 * q^3 + 894 * q^5 + 22286 * q^7	$ 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7} - 201700 q^{11} + 239298 q^{13} + 213662 q^{15} + 1045442 q^{17} + 4578860 q^{21} - 4097986 q^{23} - 4233934 q^{25} - 4817488 q^{27} + 23221660 q^{31} + 31816220 q^{33} - 55388242 q^{35} - 87811974 q^{37} + 29776460 q^{41} + 156325470 q^{43} - 144135236 q^{45} - 450750018 q^{47} + 1632585820 q^{51} + 701393866 q^{53} - 1301185140 q^{55} - 2564330416 q^{57} + 2991488220 q^{61} + 3352397678 q^{63} - 2867494182 q^{65} - 6990333394 q^{67} + 9915200380 q^{71} + 8401915018 q^{73} - 10170758642 q^{75} - 19825815140 q^{77} + 26071184290 q^{81} + 16998617454 q^{83} - 24280829854 q^{85} - 36065578576 q^{87} + 52347612540 q^{91} + 26277966572 q^{93} - 23431125296 q^{95} - 48945511254 q^{97}+O(q^{100}) $ 10 * q + 62 * q^3 + 894 * q^5 + 22286 * q^7 - 201700 * q^11 + 239298 * q^13 + 213662 * q^15 + 1045442 * q^17 + 4578860 * q^21 - 4097986 * q^23 - 4233934 * q^25 - 4817488 * q^27 + 23221660 * q^31 + 31816220 * q^33 - 55388242 * q^35 - 87811974 * q^37 + 29776460 * q^41 + 156325470 * q^43 - 144135236 * q^45 - 450750018 * q^47 + 1632585820 * q^51 + 701393866 * q^53 - 1301185140 * q^55 - 2564330416 * q^57 + 2991488220 * q^61 + 3352397678 * q^63 - 2867494182 * q^65 - 6990333394 * q^67 + 9915200380 * q^71 + 8401915018 * q^73 - 10170758642 * q^75 - 19825815140 * q^77 + 26071184290 * q^81 + 16998617454 * q^83 - 24280829854 * q^85 - 36065578576 * q^87 + 52347612540 * q^91 + 26277966572 * q^93 - 23431125296 * q^95 - 48945511254 * 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
$3$	−190.862	+	190.862i	−0.785440	+	0.785440i	−0.980743	−	0.195303i	$-0.937431\pi$
$3$	−190.862	+	190.862i	−0.785440	+	0.785440i	0.195303	+	0.980743i	$0.437431\pi$
$4$		0			0
$5$	2756.18	−	1472.78i	0.881978	−	0.471291i
$5$	2756.18	−	1472.78i	0.881978	−	0.471291i
$6$		0			0
$7$	14849.7	+	14849.7i	0.883543	+	0.883543i	0.993893	−	0.110349i	$-0.0351970\pi$
$7$	14849.7	+	14849.7i	0.883543	+	0.883543i	−0.110349	+	0.993893i	$0.535197\pi$
$8$		0			0
$9$		−	13807.6i		−	0.233833i
$10$		0			0
$11$	−319087.			−1.98128			−0.990641	−	0.136493i	$-0.956417\pi$
$11$	−319087.			−1.98128			−0.990641	+	0.136493i	$0.956417\pi$
$12$		0			0
$13$	−76905.8	+	76905.8i	−0.207130	+	0.207130i	−0.803046	−	0.595917i	$-0.796789\pi$
$13$	−76905.8	+	76905.8i	−0.207130	+	0.207130i	0.595917	+	0.803046i	$0.296789\pi$
$14$		0			0
$15$	−244952.	+	807149.i	−0.322570	+	1.06291i
$16$		0			0
$17$	−797011.	−	797011.i	−0.561332	−	0.561332i	0.368354	−	0.929686i	$-0.379922\pi$
$17$	−797011.	−	797011.i	−0.561332	−	0.561332i	−0.929686	+	0.368354i	$0.879922\pi$
$18$		0			0
$19$			3.23229e6i			1.30540i	0.757618	+	0.652698i	$0.226363\pi$
$19$			3.23229e6i			1.30540i	−0.757618	+	0.652698i	$0.773637\pi$
$20$		0			0
$21$	−5.66849e6			−1.38794
$22$		0			0
$23$	−5.27364e6	+	5.27364e6i	−0.819353	+	0.819353i	−0.986014	−	0.166661i	$-0.946701\pi$
$23$	−5.27364e6	+	5.27364e6i	−0.819353	+	0.819353i	0.166661	+	0.986014i	$0.446701\pi$
$24$		0			0
$25$	5.42745e6	−	8.11851e6i	0.555770	−	0.831336i
$26$		0			0
$27$	−8.63486e6	−	8.63486e6i	−0.601778	−	0.601778i
$28$		0			0
$29$			2.58390e7i			1.25975i	0.776696	+	0.629876i	$0.216894\pi$
$29$			2.58390e7i			1.25975i	−0.776696	+	0.629876i	$0.783106\pi$
$30$		0			0
$31$	1.45717e7			0.508982			0.254491	−	0.967075i	$-0.418092\pi$
$31$	1.45717e7			0.508982			0.254491	+	0.967075i	$0.418092\pi$
$32$		0			0
$33$	6.09017e7	−	6.09017e7i	1.55618	−	1.55618i
$34$		0			0
$35$	6.27989e7	+	1.90581e7i	1.19567	+	0.362860i
$36$		0			0
$37$	−2.45616e7	−	2.45616e7i	−0.354199	−	0.354199i	0.507470	−	0.861669i	$-0.330581\pi$
$37$	−2.45616e7	−	2.45616e7i	−0.354199	−	0.354199i	−0.861669	+	0.507470i	$0.830581\pi$
$38$		0			0
$39$		−	2.93568e7i		−	0.325376i
$40$		0			0
$41$	−6.80482e7			−0.587350			−0.293675	−	0.955905i	$-0.594878\pi$
$41$	−6.80482e7			−0.587350			−0.293675	+	0.955905i	$0.594878\pi$
$42$		0			0
$43$	−5.22255e6	+	5.22255e6i	−0.0355255	+	0.0355255i	−0.724646	−	0.689121i	$-0.757997\pi$
$43$	−5.22255e6	+	5.22255e6i	−0.0355255	+	0.0355255i	0.689121	+	0.724646i	$0.257997\pi$
$44$		0			0
$45$	−2.03356e7	−	3.80563e7i	−0.110203	−	0.206236i
$46$		0			0
$47$	4.11962e7	+	4.11962e7i	0.179625	+	0.179625i	0.791193	−	0.611567i	$-0.209461\pi$
$47$	4.11962e7	+	4.11962e7i	0.179625	+	0.179625i	−0.611567	+	0.791193i	$0.709461\pi$
$48$		0			0
$49$			1.58553e8i			0.561298i
$50$		0			0
$51$	3.04238e8			0.881786
$52$		0			0
$53$	−2.99133e8	+	2.99133e8i	−0.715294	+	0.715294i	−0.967638	−	0.252344i	$-0.918799\pi$
$53$	−2.99133e8	+	2.99133e8i	−0.715294	+	0.715294i	0.252344	+	0.967638i	$0.418799\pi$
$54$		0			0
$55$	−8.79463e8	+	4.69947e8i	−1.74745	+	0.933760i
$56$		0			0
$57$	−6.16922e8	−	6.16922e8i	−1.02531	−	1.02531i
$58$		0			0
$59$		−	4.68335e7i		−	0.0655083i	−0.999463	−	0.0327541i	$-0.989572\pi$
$59$		−	4.68335e7i		−	0.0655083i	0.999463	−	0.0327541i	$-0.0104278\pi$
$60$		0			0
$61$	2.29974e8			0.272289			0.136144	−	0.990689i	$-0.456529\pi$
$61$	2.29974e8			0.272289			0.136144	+	0.990689i	$0.456529\pi$
$62$		0			0
$63$	2.05039e8	−	2.05039e8i	0.206602	−	0.206602i
$64$		0			0
$65$	−9.87008e7	+	3.25232e8i	−0.0850656	+	0.280302i
$66$		0			0
$67$	1.40597e9	+	1.40597e9i	1.04136	+	1.04136i	0.999107	+	0.0422576i	$0.0134550\pi$
$67$	1.40597e9	+	1.40597e9i	1.04136	+	1.04136i	0.0422576	+	0.999107i	$0.486545\pi$
$68$		0			0
$69$		−	2.01307e9i		−	1.28711i
$70$		0			0
$71$	1.91787e9			1.06299			0.531493	−	0.847063i	$-0.321631\pi$
$71$	1.91787e9			1.06299			0.531493	+	0.847063i	$0.321631\pi$
$72$		0			0
$73$	2.58703e9	−	2.58703e9i	1.24792	−	1.24792i	0.291283	−	0.956637i	$-0.405918\pi$
$73$	2.58703e9	−	2.58703e9i	1.24792	−	1.24792i	0.956637	−	0.291283i	$-0.0940822\pi$
$74$		0			0
$75$	5.13623e8	+	2.58541e9i	0.216440	+	1.08949i
$76$		0			0
$77$	−4.73836e9	−	4.73836e9i	−1.75055	−	1.75055i
$78$		0			0
$79$			5.01453e9i			1.62965i	0.579705	+	0.814826i	$0.303168\pi$
$79$			5.01453e9i			1.62965i	−0.579705	+	0.814826i	$0.696832\pi$
$80$		0			0
$81$	4.11146e9			1.17916
$82$		0			0
$83$	2.23915e9	−	2.23915e9i	0.568452	−	0.568452i	−0.363243	−	0.931694i	$-0.618330\pi$
$83$	2.23915e9	−	2.23915e9i	0.568452	−	0.568452i	0.931694	+	0.363243i	$0.118330\pi$
$84$		0			0
$85$	−3.37053e9	−	1.02288e9i	−0.759633	−	0.230532i
$86$		0			0
$87$	−4.93168e9	−	4.93168e9i	−0.989460	−	0.989460i
$88$		0			0
$89$			7.96413e9i			1.42623i	0.701049	+	0.713113i	$0.252716\pi$
$89$			7.96413e9i			1.42623i	−0.701049	+	0.713113i	$0.747284\pi$
$90$		0			0
$91$	−2.28406e9			−0.366016
$92$		0			0
$93$	−2.78119e9	+	2.78119e9i	−0.399775	+	0.399775i
$94$		0			0
$95$	4.76046e9	+	8.90878e9i	0.615221	+	1.15133i
$96$		0			0
$97$	−4.52809e9	−	4.52809e9i	−0.527299	−	0.527299i	0.392467	−	0.919766i	$-0.371622\pi$
$97$	−4.52809e9	−	4.52809e9i	−0.527299	−	0.527299i	−0.919766	+	0.392467i	$0.871622\pi$
$98$		0			0
$99$			4.40584e9i			0.463289i
$100$		0			0
$101$	−1.88425e9			−0.179280			−0.0896401	−	0.995974i	$-0.528572\pi$
$101$	−1.88425e9			−0.179280			−0.0896401	+	0.995974i	$0.528572\pi$
$102$		0			0
$103$	6.87289e9	−	6.87289e9i	0.592861	−	0.592861i	−0.345542	−	0.938403i	$-0.612305\pi$
$103$	6.87289e9	−	6.87289e9i	0.592861	−	0.592861i	0.938403	+	0.345542i	$0.112305\pi$
$104$		0			0
$105$	−1.56234e10	+	8.34846e9i	−1.22413	+	0.654124i
$106$		0			0
$107$	−1.21847e10	−	1.21847e10i	−0.868754	−	0.868754i	0.123580	−	0.992335i	$-0.460562\pi$
$107$	−1.21847e10	−	1.21847e10i	−0.868754	−	0.868754i	−0.992335	+	0.123580i	$0.960562\pi$
$108$		0			0
$109$			1.41780e9i			0.0921472i	0.998938	+	0.0460736i	$0.0146709\pi$
$109$			1.41780e9i			0.0921472i	−0.998938	+	0.0460736i	$0.985329\pi$
$110$		0			0
$111$	9.37575e9			0.556405
$112$		0			0
$113$	1.20980e10	−	1.20980e10i	0.656632	−	0.656632i	−0.297950	−	0.954582i	$-0.596303\pi$
$113$	1.20980e10	−	1.20980e10i	0.656632	−	0.656632i	0.954582	+	0.297950i	$0.0963028\pi$
$114$		0			0
$115$	−6.76818e9	+	2.23020e10i	−0.336498	+	1.10880i
$116$		0			0
$117$	1.06189e9	+	1.06189e9i	0.0484338	+	0.0484338i
$118$		0			0
$119$		−	2.36708e10i		−	0.991923i
$120$		0			0
$121$	7.58794e10			2.92548
$122$		0			0
$123$	1.29878e10	−	1.29878e10i	0.461329	−	0.461329i
$124$		0			0
$125$	3.00221e9	−	3.03695e10i	0.0983765	−	0.995149i
$126$		0			0
$127$	2.41897e10	+	2.41897e10i	0.732171	+	0.732171i	0.971050	−	0.238878i	$-0.0767797\pi$
$127$	2.41897e10	+	2.41897e10i	0.732171	+	0.732171i	−0.238878	+	0.971050i	$0.576780\pi$
$128$		0			0
$129$		−	1.99357e9i		−	0.0558063i
$130$		0			0
$131$	−3.74132e10			−0.969768			−0.484884	−	0.874578i	$-0.661138\pi$
$131$	−3.74132e10			−0.969768			−0.484884	+	0.874578i	$0.661138\pi$
$132$		0			0
$133$	−4.79986e10	+	4.79986e10i	−1.15337	+	1.15337i
$134$		0			0
$135$	−3.65165e10	−	1.10820e10i	−0.814368	−	0.247143i
$136$		0			0
$137$	−2.09683e10	−	2.09683e10i	−0.434471	−	0.434471i	0.455675	−	0.890146i	$-0.349398\pi$
$137$	−2.09683e10	−	2.09683e10i	−0.434471	−	0.434471i	−0.890146	+	0.455675i	$0.849398\pi$
$138$		0			0
$139$		−	1.84411e10i		−	0.355397i	−0.984085	−	0.177698i	$-0.943135\pi$
$139$		−	1.84411e10i		−	0.355397i	0.984085	−	0.177698i	$-0.0568651\pi$
$140$		0			0
$141$	−1.57256e10			−0.282170
$142$		0			0
$143$	2.45397e10	−	2.45397e10i	0.410382	−	0.410382i
$144$		0			0
$145$	3.80552e10	+	7.12169e10i	0.593709	+	1.11107i
$146$		0			0
$147$	−3.02617e10	−	3.02617e10i	−0.440866	−	0.440866i
$148$		0			0
$149$			1.14977e11i			1.56560i	0.622274	+	0.782800i	$0.286209\pi$
$149$			1.14977e11i			1.56560i	−0.622274	+	0.782800i	$0.713791\pi$
$150$		0			0
$151$	−1.03720e11			−1.32122			−0.660612	−	0.750728i	$-0.729703\pi$
$151$	−1.03720e11			−1.32122			−0.660612	+	0.750728i	$0.729703\pi$
$152$		0			0
$153$	−1.10048e10	+	1.10048e10i	−0.131258	+	0.131258i
$154$		0			0
$155$	4.01623e10	−	2.14610e10i	0.448911	−	0.239878i
$156$		0			0
$157$	1.05929e11	+	1.05929e11i	1.11050	+	1.11050i	0.993083	+	0.117416i	$0.0374612\pi$
$157$	1.05929e11	+	1.05929e11i	1.11050	+	1.11050i	0.117416	+	0.993083i	$0.462539\pi$
$158$		0			0
$159$		−	1.14186e11i		−	1.12364i
$160$		0			0
$161$	−1.56624e11			−1.44787
$162$		0			0
$163$	−7.88198e10	+	7.88198e10i	−0.685010	+	0.685010i	−0.961125	−	0.276114i	$-0.910953\pi$
$163$	−7.88198e10	+	7.88198e10i	−0.685010	+	0.685010i	0.276114	+	0.961125i	$0.410953\pi$
$164$		0			0
$165$	7.81611e10	−	2.57551e11i	0.639103	−	2.10593i
$166$		0			0
$167$	1.18284e10	+	1.18284e10i	0.0910634	+	0.0910634i	0.751171	−	0.660108i	$-0.229489\pi$
$167$	1.18284e10	+	1.18284e10i	0.0910634	+	0.0910634i	−0.660108	+	0.751171i	$0.729489\pi$
$168$		0			0
$169$			1.26029e11i			0.914195i
$170$		0			0
$171$	4.46302e10			0.305245
$172$		0			0
$173$	4.17464e10	−	4.17464e10i	0.269394	−	0.269394i	−0.559462	−	0.828856i	$-0.688992\pi$
$173$	4.17464e10	−	4.17464e10i	0.269394	−	0.269394i	0.828856	+	0.559462i	$0.188992\pi$
$174$		0			0
$175$	2.01154e11	−	3.99616e10i	1.22557	−	0.243474i
$176$		0			0
$177$	8.93873e9	+	8.93873e9i	0.0514528	+	0.0514528i
$178$		0			0
$179$		−	1.89753e11i		−	1.03258i	−0.856415	−	0.516289i	$-0.827313\pi$
$179$		−	1.89753e11i		−	1.03258i	0.856415	−	0.516289i	$-0.172687\pi$
$180$		0			0
$181$	1.75239e11			0.902067			0.451033	−	0.892507i	$-0.351056\pi$
$181$	1.75239e11			0.902067			0.451033	+	0.892507i	$0.351056\pi$
$182$		0			0
$183$	−4.38933e10	+	4.38933e10i	−0.213867	+	0.213867i
$184$		0			0
$185$	−1.03870e11	−	3.15223e10i	−0.479327	−	0.145465i
$186$		0			0
$187$	2.54316e11	+	2.54316e11i	1.11216	+	1.11216i
$188$		0			0
$189$		−	2.56450e11i		−	1.06339i
$190$		0			0
$191$	1.09224e10			0.0429684			0.0214842	−	0.999769i	$-0.493161\pi$
$191$	1.09224e10			0.0429684			0.0214842	+	0.999769i	$0.493161\pi$
$192$		0			0
$193$	6.23768e10	−	6.23768e10i	0.232936	−	0.232936i	−0.580981	−	0.813917i	$-0.697331\pi$
$193$	6.23768e10	−	6.23768e10i	0.232936	−	0.232936i	0.813917	+	0.580981i	$0.197331\pi$
$194$		0			0
$195$	−4.32362e10	−	8.09126e10i	−0.153347	−	0.286975i
$196$		0			0
$197$	1.51443e11	+	1.51443e11i	0.510408	+	0.510408i	0.914652	−	0.404243i	$-0.132465\pi$
$197$	1.51443e11	+	1.51443e11i	0.510408	+	0.510408i	−0.404243	+	0.914652i	$0.632465\pi$
$198$		0			0
$199$			2.08856e11i			0.669241i	0.942353	+	0.334620i	$0.108608\pi$
$199$			2.08856e11i			0.669241i	−0.942353	+	0.334620i	$0.891392\pi$
$200$		0			0
$201$	−5.36693e11			−1.63586
$202$		0			0
$203$	−3.83701e11	+	3.83701e11i	−1.11305	+	1.11305i
$204$		0			0
$205$	−1.87553e11	+	1.00220e11i	−0.518030	+	0.276813i
$206$		0			0
$207$	7.28163e10	+	7.28163e10i	0.191592	+	0.191592i
$208$		0			0
$209$		−	1.03138e12i		−	2.58636i
$210$		0			0
$211$	−3.67671e11			−0.879118			−0.439559	−	0.898214i	$-0.644865\pi$
$211$	−3.67671e11			−0.879118			−0.439559	+	0.898214i	$0.644865\pi$
$212$		0			0
$213$	−3.66048e11	+	3.66048e11i	−0.834911	+	0.834911i
$214$		0			0
$215$	−6.70261e9	+	2.20860e10i	−0.0145899	+	0.0480755i
$216$		0			0
$217$	2.16386e11	+	2.16386e11i	0.449707	+	0.449707i
$218$		0			0
$219$			9.87531e11i			1.96033i
$220$		0			0
$221$	1.22590e11			0.232537
$222$		0			0
$223$	3.04430e11	−	3.04430e11i	0.552030	−	0.552030i	−0.374997	−	0.927026i	$-0.622356\pi$
$223$	3.04430e11	−	3.04430e11i	0.552030	−	0.552030i	0.927026	+	0.374997i	$0.122356\pi$
$224$		0			0
$225$	−1.12097e11	−	7.49400e10i	−0.194394	−	0.129958i
$226$		0			0
$227$	−4.23326e11	−	4.23326e11i	−0.702337	−	0.702337i	0.262575	−	0.964912i	$-0.415428\pi$
$227$	−4.23326e11	−	4.23326e11i	−0.702337	−	0.702337i	−0.964912	+	0.262575i	$0.915428\pi$
$228$		0			0
$229$			5.45518e11i			0.866227i	0.901339	+	0.433114i	$0.142585\pi$
$229$			5.45518e11i			0.866227i	−0.901339	+	0.433114i	$0.857415\pi$
$230$		0			0
$231$	1.80875e12			2.74990
$232$		0			0
$233$	5.88340e11	−	5.88340e11i	0.856739	−	0.856739i	−0.134213	−	0.990952i	$-0.542851\pi$
$233$	5.88340e11	−	5.88340e11i	0.856739	−	0.856739i	0.990952	+	0.134213i	$0.0428507\pi$
$234$		0			0
$235$	1.74217e11	+	5.28711e10i	0.243081	+	0.0737699i
$236$		0			0
$237$	−9.57084e11	−	9.57084e11i	−1.28000	−	1.28000i
$238$		0			0
$239$		−	4.02141e11i		−	0.515691i	−0.966186	−	0.257845i	$-0.916988\pi$
$239$		−	4.02141e11i		−	0.515691i	0.966186	−	0.257845i	$-0.0830125\pi$
$240$		0			0
$241$	−7.74825e11			−0.953056			−0.476528	−	0.879159i	$-0.658105\pi$
$241$	−7.74825e11			−0.953056			−0.476528	+	0.879159i	$0.658105\pi$
$242$		0			0
$243$	−2.74841e11	+	2.74841e11i	−0.324378	+	0.324378i
$244$		0			0
$245$	2.33514e11	+	4.37000e11i	0.264535	+	0.495053i
$246$		0			0
$247$	−2.48582e11	−	2.48582e11i	−0.270386	−	0.270386i
$248$		0			0
$249$			8.54739e11i			0.892970i
$250$		0			0
$251$	−2.69007e11			−0.270019			−0.135010	−	0.990844i	$-0.543107\pi$
$251$	−2.69007e11			−0.270019			−0.135010	+	0.990844i	$0.543107\pi$
$252$		0			0
$253$	1.68275e12	−	1.68275e12i	1.62337	−	1.62337i
$254$		0			0
$255$	8.38536e11	−	4.48077e11i	0.777716	−	0.415577i
$256$		0			0
$257$	9.77764e11	+	9.77764e11i	0.872104	+	0.872104i	0.992701	−	0.120597i	$-0.0384810\pi$
$257$	9.77764e11	+	9.77764e11i	0.872104	+	0.872104i	−0.120597	+	0.992701i	$0.538481\pi$
$258$		0			0
$259$		−	7.29465e11i		−	0.625901i
$260$		0			0
$261$	3.56774e11			0.294572
$262$		0			0
$263$	1.33478e11	−	1.33478e11i	0.106079	−	0.106079i	−0.652075	−	0.758154i	$-0.726101\pi$
$263$	1.33478e11	−	1.33478e11i	0.106079	−	0.106079i	0.758154	+	0.652075i	$0.226101\pi$
$264$		0			0
$265$	−3.83906e11	+	1.26502e12i	−0.293762	+	0.967984i
$266$		0			0
$267$	−1.52005e12	−	1.52005e12i	−1.12022	−	1.12022i
$268$		0			0
$269$			2.23255e12i			1.58504i	0.609848	+	0.792518i	$0.291231\pi$
$269$			2.23255e12i			1.58504i	−0.609848	+	0.792518i	$0.708769\pi$
$270$		0			0
$271$	9.38370e11			0.641989			0.320994	−	0.947081i	$-0.395983\pi$
$271$	9.38370e11			0.641989			0.320994	+	0.947081i	$0.395983\pi$
$272$		0			0
$273$	4.35940e11	−	4.35940e11i	0.287484	−	0.287484i
$274$		0			0
$275$	−1.73183e12	+	2.59052e12i	−1.10114	+	1.64711i
$276$		0			0
$277$	−1.70517e12	−	1.70517e12i	−1.04561	−	1.04561i	−0.998909	−	0.0466965i	$-0.985131\pi$
$277$	−1.70517e12	−	1.70517e12i	−1.04561	−	1.04561i	−0.0466965	−	0.998909i	$-0.514869\pi$
$278$		0			0
$279$		−	2.01201e11i		−	0.119017i
$280$		0			0
$281$	6.23468e10			0.0355863			0.0177931	−	0.999842i	$-0.494336\pi$
$281$	6.23468e10			0.0355863			0.0177931	+	0.999842i	$0.494336\pi$
$282$		0			0
$283$	−2.52509e12	+	2.52509e12i	−1.39106	+	1.39106i	−0.568094	+	0.822963i	$0.692319\pi$
$283$	−2.52509e12	+	2.52509e12i	−1.39106	+	1.39106i	−0.822963	+	0.568094i	$0.807681\pi$
$284$		0			0
$285$	−2.60894e12	−	7.91756e11i	−1.38752	−	0.421082i
$286$		0			0
$287$	−1.01050e12	−	1.01050e12i	−0.518950	−	0.518950i
$288$		0			0
$289$		−	7.45539e11i		−	0.369812i
$290$		0			0
$291$	1.72848e12			0.828324
$292$		0			0
$293$	2.59186e11	−	2.59186e11i	0.120025	−	0.120025i	−0.644543	−	0.764568i	$-0.722952\pi$
$293$	2.59186e11	−	2.59186e11i	0.120025	−	0.120025i	0.764568	+	0.644543i	$0.222952\pi$
$294$		0			0
$295$	−6.89755e10	−	1.29081e11i	−0.0308734	−	0.0577769i
$296$		0			0
$297$	2.75528e12	+	2.75528e12i	1.19229	+	1.19229i
$298$		0			0
$299$		−	8.11147e11i		−	0.339425i
$300$		0			0
$301$	−1.55107e11			−0.0627766
$302$		0			0
$303$	3.59632e11	−	3.59632e11i	0.140814	−	0.140814i
$304$		0			0
$305$	6.33850e11	−	3.38702e11i	0.240153	−	0.128327i
$306$		0			0
$307$	−5.15760e11	−	5.15760e11i	−0.189128	−	0.189128i	0.606191	−	0.795319i	$-0.292697\pi$
$307$	−5.15760e11	−	5.15760e11i	−0.189128	−	0.189128i	−0.795319	+	0.606191i	$0.792697\pi$
$308$		0			0
$309$			2.62355e12i			0.931315i
$310$		0			0
$311$	7.79126e11			0.267797			0.133899	−	0.990995i	$-0.457250\pi$
$311$	7.79126e11			0.267797			0.133899	+	0.990995i	$0.457250\pi$
$312$		0			0
$313$	4.98495e11	−	4.98495e11i	0.165935	−	0.165935i	−0.619255	−	0.785190i	$-0.712565\pi$
$313$	4.98495e11	−	4.98495e11i	0.165935	−	0.165935i	0.785190	+	0.619255i	$0.212565\pi$
$314$		0			0
$315$	2.63147e11	−	8.67103e11i	0.0848487	−	0.279588i
$316$		0			0
$317$	2.12435e12	+	2.12435e12i	0.663637	+	0.663637i	0.956235	−	0.292598i	$-0.0945199\pi$
$317$	2.12435e12	+	2.12435e12i	0.663637	+	0.663637i	−0.292598	+	0.956235i	$0.594520\pi$
$318$		0			0
$319$		−	8.24489e12i		−	2.49593i
$320$		0			0
$321$	4.65120e12			1.36471
$322$		0			0
$323$	2.57617e12	−	2.57617e12i	0.732761	−	0.732761i
$324$		0			0
$325$	2.06959e11	+	1.04176e12i	0.0570778	+	0.287311i
$326$		0			0
$327$	−2.70604e11	−	2.70604e11i	−0.0723761	−	0.0723761i
$328$		0			0
$329$			1.22350e12i			0.317414i
$330$		0			0
$331$	−6.93944e12			−1.74656			−0.873282	−	0.487214i	$-0.838013\pi$
$331$	−6.93944e12			−1.74656			−0.873282	+	0.487214i	$0.838013\pi$
$332$		0			0
$333$	−3.39137e11	+	3.39137e11i	−0.0828235	+	0.0828235i
$334$		0			0
$335$	5.94581e12	+	1.80442e12i	1.40925	+	0.427675i
$336$		0			0
$337$	−1.87434e12	−	1.87434e12i	−0.431221	−	0.431221i	0.457823	−	0.889043i	$-0.348629\pi$
$337$	−1.87434e12	−	1.87434e12i	−0.431221	−	0.431221i	−0.889043	+	0.457823i	$0.848629\pi$
$338$		0			0
$339$			4.61810e12i			1.03149i
$340$		0			0
$341$	−4.64965e12			−1.00844
$342$		0			0
$343$	1.84021e12	−	1.84021e12i	0.387612	−	0.387612i
$344$		0			0
$345$	−2.96482e12	−	5.54840e12i	−0.606601	−	1.13520i
$346$		0			0
$347$	8.74787e11	+	8.74787e11i	0.173882	+	0.173882i	0.788683	−	0.614801i	$-0.210764\pi$
$347$	8.74787e11	+	8.74787e11i	0.173882	+	0.173882i	−0.614801	+	0.788683i	$0.710764\pi$
$348$		0			0
$349$			2.13024e11i			0.0411436i	0.999788	+	0.0205718i	$0.00654866\pi$
$349$			2.13024e11i			0.0411436i	−0.999788	+	0.0205718i	$0.993451\pi$
$350$		0			0
$351$	1.32814e12			0.249292
$352$		0			0
$353$	−3.66682e12	+	3.66682e12i	−0.668985	+	0.668985i	−0.957481	−	0.288496i	$-0.906845\pi$
$353$	−3.66682e12	+	3.66682e12i	−0.668985	+	0.668985i	0.288496	+	0.957481i	$0.406845\pi$
$354$		0			0
$355$	5.28599e12	−	2.82460e12i	0.937529	−	0.500975i
$356$		0			0
$357$	4.51785e12	+	4.51785e12i	0.779096	+	0.779096i
$358$		0			0
$359$			3.38682e12i			0.567962i	0.958830	+	0.283981i	$0.0916553\pi$
$359$			3.38682e12i			0.567962i	−0.958830	+	0.283981i	$0.908345\pi$
$360$		0			0
$361$	−4.31664e12			−0.704061
$362$		0			0
$363$	−1.44825e13	+	1.44825e13i	−2.29779	+	2.29779i
$364$		0			0
$365$	3.32019e12	−	1.09404e13i	0.512505	−	1.68877i
$366$		0			0
$367$	2.90521e12	+	2.90521e12i	0.436362	+	0.436362i	0.890786	−	0.454424i	$-0.150155\pi$
$367$	2.90521e12	+	2.90521e12i	0.436362	+	0.436362i	−0.454424	+	0.890786i	$0.650155\pi$
$368$		0			0
$369$			9.39583e11i			0.137342i
$370$		0			0
$371$	−8.88407e12			−1.26399
$372$		0			0
$373$	5.85230e12	−	5.85230e12i	0.810555	−	0.810555i	−0.174162	−	0.984717i	$-0.555722\pi$
$373$	5.85230e12	−	5.85230e12i	0.810555	−	0.810555i	0.984717	+	0.174162i	$0.0557218\pi$
$374$		0			0
$375$	5.22338e12	+	6.36940e12i	0.704362	+	0.858899i
$376$		0			0
$377$	−1.98717e12	−	1.98717e12i	−0.260932	−	0.260932i
$378$		0			0
$379$		−	3.40305e12i		−	0.435183i	−0.976040	−	0.217592i	$-0.930180\pi$
$379$		−	3.40305e12i		−	0.435183i	0.976040	−	0.217592i	$-0.0698201\pi$
$380$		0			0
$381$	−9.23381e12			−1.15015
$382$		0			0
$383$	−3.07744e12	+	3.07744e12i	−0.373418	+	0.373418i	−0.868721	−	0.495302i	$-0.835057\pi$
$383$	−3.07744e12	+	3.07744e12i	−0.373418	+	0.373418i	0.495302	+	0.868721i	$0.335057\pi$
$384$		0			0
$385$	−2.00383e13	−	6.08120e12i	−2.36896	−	0.718928i
$386$		0			0
$387$	7.21109e10	+	7.21109e10i	0.00830704	+	0.00830704i
$388$		0			0
$389$		−	1.92474e12i		−	0.216084i	−0.994146	−	0.108042i	$-0.965542\pi$
$389$		−	1.92474e12i		−	0.216084i	0.994146	−	0.108042i	$-0.0344582\pi$
$390$		0			0
$391$	8.40630e12			0.919858
$392$		0			0
$393$	7.14075e12	−	7.14075e12i	0.761695	−	0.761695i
$394$		0			0
$395$	7.38532e12	+	1.38210e13i	0.768040	+	1.43732i
$396$		0			0
$397$	8.74684e12	+	8.74684e12i	0.886949	+	0.886949i	0.994229	−	0.107280i	$-0.0342142\pi$
$397$	8.74684e12	+	8.74684e12i	0.886949	+	0.886949i	−0.107280	+	0.994229i	$0.534214\pi$
$398$		0			0
$399$		−	1.83222e13i		−	1.81181i
$400$		0			0
$401$	1.24070e13			1.19659			0.598295	−	0.801276i	$-0.295845\pi$
$401$	1.24070e13			1.19659			0.598295	+	0.801276i	$0.295845\pi$
$402$		0			0
$403$	−1.12065e12	+	1.12065e12i	−0.105425	+	0.105425i
$404$		0			0
$405$	1.13319e13	−	6.05529e12i	1.03999	−	0.555725i
$406$		0			0
$407$	7.83729e12	+	7.83729e12i	0.701769	+	0.701769i
$408$		0			0
$409$			1.81003e12i			0.158150i	0.996869	+	0.0790750i	$0.0251967\pi$
$409$			1.81003e12i			0.158150i	−0.996869	+	0.0790750i	$0.974803\pi$
$410$		0			0
$411$	8.00412e12			0.682503
$412$		0			0
$413$	6.95463e11	−	6.95463e11i	0.0578794	−	0.0578794i
$414$		0			0
$415$	2.87373e12	−	9.46930e12i	0.233456	−	0.769268i
$416$		0			0
$417$	3.51971e12	+	3.51971e12i	0.279143	+	0.279143i
$418$		0			0
$419$		−	2.08744e13i		−	1.61638i	−0.588921	−	0.808190i	$-0.700447\pi$
$419$		−	2.08744e13i		−	1.61638i	0.588921	−	0.808190i	$-0.299553\pi$
$420$		0			0
$421$	−1.44273e13			−1.09087			−0.545437	−	0.838152i	$-0.683636\pi$
$421$	−1.44273e13			−1.09087			−0.545437	+	0.838152i	$0.683636\pi$
$422$		0			0
$423$	5.68821e11	−	5.68821e11i	0.0420023	−	0.0420023i
$424$		0			0
$425$	−1.07963e13	+	2.14481e12i	−0.778627	+	0.154684i
$426$		0			0
$427$	3.41505e12	+	3.41505e12i	0.240579	+	0.240579i
$428$		0			0
$429$			9.36739e12i			0.644662i
$430$		0			0
$431$	8.62278e12			0.579777			0.289888	−	0.957060i	$-0.406382\pi$
$431$	8.62278e12			0.579777			0.289888	+	0.957060i	$0.406382\pi$
$432$		0			0
$433$	−4.88088e12	+	4.88088e12i	−0.320670	+	0.320670i	−0.849024	−	0.528354i	$-0.822809\pi$
$433$	−4.88088e12	+	4.88088e12i	−0.320670	+	0.320670i	0.528354	+	0.849024i	$0.322809\pi$
$434$		0			0
$435$	−2.08559e13	−	6.32931e12i	−1.33901	−	0.406359i
$436$		0			0
$437$	−1.70459e13	−	1.70459e13i	−1.06958	−	1.06958i
$438$		0			0
$439$			2.52662e13i			1.54959i	0.632212	+	0.774796i	$0.282147\pi$
$439$			2.52662e13i			1.54959i	−0.632212	+	0.774796i	$0.717853\pi$
$440$		0			0
$441$	2.18924e12			0.131250
$442$		0			0
$443$	−6.36035e12	+	6.36035e12i	−0.372789	+	0.372789i	−0.868492	−	0.495703i	$-0.834910\pi$
$443$	−6.36035e12	+	6.36035e12i	−0.372789	+	0.372789i	0.495703	+	0.868492i	$0.334910\pi$
$444$		0			0
$445$	1.17294e13	+	2.19506e13i	0.672167	+	1.25790i
$446$		0			0
$447$	−2.19448e13	−	2.19448e13i	−1.22968	−	1.22968i
$448$		0			0
$449$			4.59384e12i			0.251736i	0.992047	+	0.125868i	$0.0401715\pi$
$449$			4.59384e12i			0.251736i	−0.992047	+	0.125868i	$0.959828\pi$
$450$		0			0
$451$	2.17133e13			1.16371
$452$		0			0
$453$	1.97961e13	−	1.97961e13i	1.03774	−	1.03774i
$454$		0			0
$455$	−6.29528e12	+	3.36392e12i	−0.322818	+	0.172500i
$456$		0			0
$457$	2.85709e12	+	2.85709e12i	0.143332	+	0.143332i	0.775132	−	0.631800i	$-0.217684\pi$
$457$	2.85709e12	+	2.85709e12i	0.143332	+	0.143332i	−0.631800	+	0.775132i	$0.717684\pi$
$458$		0			0
$459$			1.37642e13i			0.675595i
$460$		0			0
$461$	−7.01154e12			−0.336751			−0.168375	−	0.985723i	$-0.553852\pi$
$461$	−7.01154e12			−0.336751			−0.168375	+	0.985723i	$0.553852\pi$
$462$		0			0
$463$	−1.02937e13	+	1.02937e13i	−0.483802	+	0.483802i	−0.906343	−	0.422542i	$-0.861138\pi$
$463$	−1.02937e13	+	1.02937e13i	−0.483802	+	0.483802i	0.422542	+	0.906343i	$0.361138\pi$
$464$		0			0
$465$	−3.56937e12	+	1.17615e13i	−0.164182	+	0.541003i
$466$		0			0
$467$	−2.34972e13	−	2.34972e13i	−1.05787	−	1.05787i	−0.998219	−	0.0596513i	$-0.981001\pi$
$467$	−2.34972e13	−	2.34972e13i	−1.05787	−	1.05787i	−0.0596513	−	0.998219i	$-0.518999\pi$
$468$		0			0
$469$			4.17566e13i			1.84018i
$470$		0			0
$471$	−4.04358e13			−1.74446
$472$		0			0
$473$	1.66645e12	−	1.66645e12i	0.0703860	−	0.0703860i
$474$		0			0
$475$	2.62414e13	+	1.75431e13i	1.08522	+	0.725501i
$476$		0			0
$477$	4.13031e12	+	4.13031e12i	0.167259	+	0.167259i
$478$		0			0
$479$		−	1.29636e13i		−	0.514102i	−0.966398	−	0.257051i	$-0.917249\pi$
$479$		−	1.29636e13i		−	0.514102i	0.966398	−	0.257051i	$-0.0827508\pi$
$480$		0			0
$481$	3.77786e12			0.146730
$482$		0			0
$483$	2.98936e13	−	2.98936e13i	1.13721	−	1.13721i
$484$		0			0
$485$	−1.91492e13	−	5.81135e12i	−0.713577	−	0.216555i
$486$		0			0
$487$	−2.45596e13	−	2.45596e13i	−0.896553	−	0.896553i	0.0985765	−	0.995129i	$-0.468571\pi$
$487$	−2.45596e13	−	2.45596e13i	−0.896553	−	0.896553i	−0.995129	+	0.0985765i	$0.968571\pi$
$488$		0			0
$489$		−	3.00874e13i		−	1.07607i
$490$		0			0
$491$	2.26671e13			0.794308			0.397154	−	0.917752i	$-0.369998\pi$
$491$	2.26671e13			0.794308			0.397154	+	0.917752i	$0.369998\pi$
$492$		0			0
$493$	2.05940e13	−	2.05940e13i	0.707140	−	0.707140i
$494$		0			0
$495$	6.48884e12	+	1.21433e13i	0.218344	+	0.408611i
$496$		0			0
$497$	2.84798e13	+	2.84798e13i	0.939194	+	0.939194i
$498$		0			0
$499$			5.67102e12i			0.183298i	0.995791	+	0.0916492i	$0.0292139\pi$
$499$			5.67102e12i			0.183298i	−0.995791	+	0.0916492i	$0.970786\pi$
$500$		0			0
$501$	−4.51519e12			−0.143050
$502$		0			0
$503$	−2.21600e13	+	2.21600e13i	−0.688225	+	0.688225i	−0.961840	−	0.273614i	$-0.911781\pi$
$503$	−2.21600e13	+	2.21600e13i	−0.688225	+	0.688225i	0.273614	+	0.961840i	$0.411781\pi$
$504$		0			0
$505$	−5.19335e12	+	2.77510e12i	−0.158121	+	0.0844931i
$506$		0			0
$507$	−2.40542e13	−	2.40542e13i	−0.718045	−	0.718045i
$508$		0			0
$509$			1.36304e13i			0.398950i	0.979903	+	0.199475i	$0.0639236\pi$
$509$			1.36304e13i			0.398950i	−0.979903	+	0.199475i	$0.936076\pi$
$510$		0			0
$511$	7.68333e13			2.20518
$512$		0			0
$513$	2.79104e13	−	2.79104e13i	0.785560	−	0.785560i
$514$		0			0
$515$	8.82065e12	−	2.90652e13i	0.243481	−	0.802301i
$516$		0			0
$517$	−1.31452e13	−	1.31452e13i	−0.355888	−	0.355888i
$518$		0			0
$519$			1.59356e13i			0.423187i
$520$		0			0
$521$	7.15320e13			1.86342			0.931712	−	0.363199i	$-0.118315\pi$
$521$	7.15320e13			1.86342			0.931712	+	0.363199i	$0.118315\pi$
$522$		0			0
$523$	3.53933e12	−	3.53933e12i	0.0904508	−	0.0904508i	−0.660434	−	0.750884i	$-0.729628\pi$
$523$	3.53933e12	−	3.53933e12i	0.0904508	−	0.0904508i	0.750884	+	0.660434i	$0.229628\pi$
$524$		0			0
$525$	−3.07654e13	+	4.60197e13i	−0.771377	+	1.15385i
$526$		0			0
$527$	−1.16138e13	−	1.16138e13i	−0.285708	−	0.285708i
$528$		0			0
$529$		−	1.41960e13i		−	0.342679i
$530$		0			0
$531$	−6.46658e11			−0.0153180
$532$		0			0
$533$	5.23330e12	−	5.23330e12i	0.121658	−	0.121658i
$534$		0			0
$535$	−5.15288e13	−	1.56379e13i	−1.17566	−	0.356786i
$536$		0			0
$537$	3.62166e13	+	3.62166e13i	0.811028	+	0.811028i
$538$		0			0
$539$		−	5.05922e13i		−	1.11209i
$540$		0			0
$541$	−4.89956e13			−1.05723			−0.528616	−	0.848861i	$-0.677289\pi$
$541$	−4.89956e13			−1.05723			−0.528616	+	0.848861i	$0.677289\pi$
$542$		0			0
$543$	−3.34465e13	+	3.34465e13i	−0.708520	+	0.708520i
$544$		0			0
$545$	2.08811e12	+	3.90771e12i	0.0434281	+	0.0812718i
$546$		0			0
$547$	5.59543e13	+	5.59543e13i	1.14261	+	1.14261i	0.987972	+	0.154636i	$0.0494203\pi$
$547$	5.59543e13	+	5.59543e13i	1.14261	+	1.14261i	0.154636	+	0.987972i	$0.450580\pi$
$548$		0			0
$549$		−	3.17539e12i		−	0.0636701i
$550$		0			0
$551$	−8.35191e13			−1.64448
$552$		0			0
$553$	−7.44644e13	+	7.44644e13i	−1.43987	+	1.43987i
$554$		0			0
$555$	2.58413e13	−	1.38084e13i	0.490737	−	0.262228i
$556$		0			0
$557$	−6.33595e13	−	6.33595e13i	−1.18178	−	1.18178i	−0.979284	−	0.202493i	$-0.935096\pi$
$557$	−6.33595e13	−	6.33595e13i	−1.18178	−	1.18178i	−0.202493	−	0.979284i	$-0.564904\pi$
$558$		0			0
$559$		−	8.03289e11i		−	0.0147168i
$560$		0			0
$561$	−9.70787e13			−1.74707
$562$		0			0
$563$	−2.81203e13	+	2.81203e13i	−0.497139	+	0.497139i	−0.910546	−	0.413407i	$-0.864339\pi$
$563$	−2.81203e13	+	2.81203e13i	−0.497139	+	0.497139i	0.413407	+	0.910546i	$0.364339\pi$
$564$		0			0
$565$	1.55266e13	−	5.11621e13i	0.269670	−	0.888599i
$566$		0			0
$567$	6.10540e13	+	6.10540e13i	1.04183	+	1.04183i
$568$		0			0
$569$		−	5.38796e13i		−	0.903364i	−0.892179	−	0.451682i	$-0.850824\pi$
$569$		−	5.38796e13i		−	0.903364i	0.892179	−	0.451682i	$-0.149176\pi$
$570$		0			0
$571$	3.12311e13			0.514526			0.257263	−	0.966341i	$-0.417179\pi$
$571$	3.12311e13			0.514526			0.257263	+	0.966341i	$0.417179\pi$
$572$		0			0
$573$	−2.08466e12	+	2.08466e12i	−0.0337491	+	0.0337491i
$574$		0			0
$575$	1.41917e13	+	7.14365e13i	0.225785	+	1.13653i
$576$		0			0
$577$	4.42054e13	+	4.42054e13i	0.691188	+	0.691188i	0.962493	−	0.271305i	$-0.0874554\pi$
$577$	4.42054e13	+	4.42054e13i	0.691188	+	0.691188i	−0.271305	+	0.962493i	$0.587455\pi$
$578$		0			0
$579$			2.38107e13i			0.365915i
$580$		0			0
$581$	6.65016e13			1.00450
$582$		0			0
$583$	9.54494e13	−	9.54494e13i	1.41720	−	1.41720i
$584$		0			0
$585$	4.49068e12	+	1.36282e12i	0.0655439	+	0.0198911i
$586$		0			0
$587$	1.14912e11	+	1.14912e11i	0.00164882	+	0.00164882i	0.707931	−	0.706282i	$-0.249629\pi$
$587$	1.14912e11	+	1.14912e11i	0.00164882	+	0.00164882i	−0.706282	+	0.707931i	$0.749629\pi$
$588$		0			0
$589$			4.71000e13i			0.664423i
$590$		0			0
$591$	−5.78094e13			−0.801791
$592$		0			0
$593$	−1.66944e13	+	1.66944e13i	−0.227666	+	0.227666i	−0.811717	−	0.584051i	$-0.801467\pi$
$593$	−1.66944e13	+	1.66944e13i	−0.227666	+	0.227666i	0.584051	+	0.811717i	$0.301467\pi$
$594$		0			0
$595$	−3.48619e13	−	6.52410e13i	−0.467484	−	0.874854i
$596$		0			0
$597$	−3.98628e13	−	3.98628e13i	−0.525649	−	0.525649i
$598$		0			0
$599$			1.09809e14i			1.42398i	0.702190	+	0.711990i	$0.252206\pi$
$599$			1.09809e14i			1.42398i	−0.702190	+	0.711990i	$0.747794\pi$
$600$		0			0
$601$	−7.86667e13			−1.00327			−0.501636	−	0.865079i	$-0.667268\pi$
$601$	−7.86667e13			−1.00327			−0.501636	+	0.865079i	$0.667268\pi$
$602$		0			0
$603$	1.94131e13	−	1.94131e13i	0.243505	−	0.243505i
$604$		0			0
$605$	2.09137e14	−	1.11754e14i	2.58021	−	1.37875i
$606$		0			0
$607$	5.76042e12	+	5.76042e12i	0.0699054	+	0.0699054i	0.741195	−	0.671290i	$-0.234259\pi$
$607$	5.76042e12	+	5.76042e12i	0.0699054	+	0.0699054i	−0.671290	+	0.741195i	$0.734259\pi$
$608$		0			0
$609$		−	1.46468e14i		−	1.74846i
$610$		0			0
$611$	−6.33645e12			−0.0744115
$612$		0			0
$613$	−4.00113e13	+	4.00113e13i	−0.462254	+	0.462254i	−0.899394	−	0.437140i	$-0.855991\pi$
$613$	−4.00113e13	+	4.00113e13i	−0.462254	+	0.462254i	0.437140	+	0.899394i	$0.355991\pi$
$614$		0			0
$615$	1.66685e13	−	5.49250e13i	0.189462	−	0.624302i
$616$		0			0
$617$	−4.85097e13	−	4.85097e13i	−0.542503	−	0.542503i	0.381759	−	0.924262i	$-0.375319\pi$
$617$	−4.85097e13	−	4.85097e13i	−0.542503	−	0.542503i	−0.924262	+	0.381759i	$0.875319\pi$
$618$		0			0
$619$		−	2.11162e13i		−	0.232360i	−0.993228	−	0.116180i	$-0.962935\pi$
$619$		−	2.11162e13i		−	0.232360i	0.993228	−	0.116180i	$-0.0370650\pi$
$620$		0			0
$621$	9.10743e13			0.986138
$622$		0			0
$623$	−1.18265e14	+	1.18265e14i	−1.26013	+	1.26013i
$624$		0			0
$625$	−3.64531e13	−	8.81256e13i	−0.382239	−	0.924064i
$626$		0			0
$627$	1.96852e14	+	1.96852e14i	2.03143	+	2.03143i
$628$		0			0
$629$			3.91517e13i			0.397647i
$630$		0			0
$631$	−6.37278e13			−0.637063			−0.318531	−	0.947912i	$-0.603190\pi$
$631$	−6.37278e13			−0.637063			−0.318531	+	0.947912i	$0.603190\pi$
$632$		0			0
$633$	7.01745e13	−	7.01745e13i	0.690495	−	0.690495i
$634$		0			0
$635$	1.02298e14	+	3.10451e13i	0.990824	+	0.300694i
$636$		0			0
$637$	−1.21936e13	−	1.21936e13i	−0.116262	−	0.116262i
$638$		0			0
$639$		−	2.64812e13i		−	0.248561i
$640$		0			0
$641$	2.92687e13			0.270467			0.135233	−	0.990814i	$-0.456822\pi$
$641$	2.92687e13			0.270467			0.135233	+	0.990814i	$0.456822\pi$
$642$		0			0
$643$	1.39211e14	−	1.39211e14i	1.26654	−	1.26654i	0.318670	−	0.947866i	$-0.396764\pi$
$643$	1.39211e14	−	1.39211e14i	1.26654	−	1.26654i	0.947866	−	0.318670i	$-0.103236\pi$
$644$		0			0
$645$	−2.93610e12	−	5.49465e12i	−0.0263010	−	0.0492199i
$646$		0			0
$647$	4.48473e13	+	4.48473e13i	0.395562	+	0.395562i	0.876664	−	0.481102i	$-0.159763\pi$
$647$	4.48473e13	+	4.48473e13i	0.395562	+	0.395562i	−0.481102	+	0.876664i	$0.659763\pi$
$648$		0			0
$649$			1.49440e13i			0.129790i
$650$		0			0
$651$	−8.25996e13			−0.706437
$652$		0			0
$653$	−1.43170e14	+	1.43170e14i	−1.20583	+	1.20583i	−0.233470	+	0.972364i	$0.575008\pi$
$653$	−1.43170e14	+	1.43170e14i	−1.20583	+	1.20583i	−0.972364	+	0.233470i	$0.924992\pi$
$654$		0			0
$655$	−1.03117e14	+	5.51015e13i	−0.855314	+	0.457043i
$656$		0			0
$657$	−3.57207e13	−	3.57207e13i	−0.291805	−	0.291805i
$658$		0			0
$659$		−	1.09354e14i		−	0.879851i	−0.898034	−	0.439925i	$-0.855005\pi$
$659$		−	1.09354e14i		−	0.879851i	0.898034	−	0.439925i	$-0.144995\pi$
$660$		0			0
$661$	−1.18122e14			−0.936099			−0.468049	−	0.883702i	$-0.655043\pi$
$661$	−1.18122e14			−0.936099			−0.468049	+	0.883702i	$0.655043\pi$
$662$		0			0
$663$	−2.33977e13	+	2.33977e13i	−0.182644	+	0.182644i
$664$		0			0
$665$	−6.16013e13	+	2.02984e14i	−0.473676	+	1.56083i
$666$		0			0
$667$	−1.36265e14	−	1.36265e14i	−1.03218	−	1.03218i
$668$		0			0
$669$			1.16208e14i			0.867173i
$670$		0			0
$671$	−7.33819e13			−0.539481
$672$		0			0
$673$	−8.66086e13	+	8.66086e13i	−0.627315	+	0.627315i	−0.947392	−	0.320076i	$-0.896291\pi$
$673$	−8.66086e13	+	8.66086e13i	−0.627315	+	0.627315i	0.320076	+	0.947392i	$0.396291\pi$
$674$		0			0
$675$	−1.16967e14	+	2.32370e13i	−0.834731	+	0.165829i
$676$		0			0
$677$	1.26223e14	+	1.26223e14i	0.887557	+	0.887557i	0.994288	−	0.106731i	$-0.0340383\pi$
$677$	1.26223e14	+	1.26223e14i	0.887557	+	0.887557i	−0.106731	+	0.994288i	$0.534038\pi$
$678$		0			0
$679$		−	1.34482e14i		−	0.931783i
$680$		0			0
$681$	1.61594e14			1.10329
$682$		0			0
$683$	6.22898e13	−	6.22898e13i	0.419096	−	0.419096i	−0.465796	−	0.884892i	$-0.654232\pi$
$683$	6.22898e13	−	6.22898e13i	0.419096	−	0.419096i	0.884892	+	0.465796i	$0.154232\pi$
$684$		0			0
$685$	−8.86744e13	−	2.69107e13i	−0.587957	−	0.178432i
$686$		0			0
$687$	−1.04119e14	−	1.04119e14i	−0.680370	−	0.680370i
$688$		0			0
$689$		−	4.60101e13i		−	0.296317i
$690$		0			0
$691$	2.99647e12			0.0190204			0.00951022	−	0.999955i	$-0.496973\pi$
$691$	2.99647e12			0.0190204			0.00951022	+	0.999955i	$0.496973\pi$
$692$		0			0
$693$	−6.54254e13	+	6.54254e13i	−0.409336	+	0.409336i
$694$		0			0
$695$	−2.71598e13	−	5.08271e13i	−0.167495	−	0.313452i
$696$		0			0
$697$	5.42352e13	+	5.42352e13i	0.329699	+	0.329699i
$698$		0			0
$699$			2.24583e14i			1.34584i
$700$		0			0
$701$	2.46559e14			1.45657			0.728283	−	0.685277i	$-0.240319\pi$
$701$	2.46559e14			1.45657			0.728283	+	0.685277i	$0.240319\pi$
$702$		0			0
$703$	7.93902e13	−	7.93902e13i	0.462371	−	0.462371i
$704$		0			0
$705$	−4.33425e13	+	2.31603e13i	−0.248868	+	0.132984i
$706$		0			0
$707$	−2.79806e13	−	2.79806e13i	−0.158402	−	0.158402i
$708$		0			0
$709$		−	1.85832e14i		−	1.03726i	−0.854998	−	0.518632i	$-0.826441\pi$
$709$		−	1.85832e14i		−	1.03726i	0.854998	−	0.518632i	$-0.173559\pi$
$710$		0			0
$711$	6.92387e13			0.381067
$712$		0			0
$713$	−7.68459e13	+	7.68459e13i	−0.417036	+	0.417036i
$714$		0			0
$715$	3.14942e13	−	1.03777e14i	0.168539	−	0.555358i
$716$		0			0
$717$	7.67535e13	+	7.67535e13i	0.405044	+	0.405044i
$718$		0			0
$719$			2.85619e13i			0.148642i	0.997234	+	0.0743211i	$0.0236790\pi$
$719$			2.85619e13i			0.148642i	−0.997234	+	0.0743211i	$0.976321\pi$
$720$		0			0
$721$	2.04121e14			1.04764
$722$		0			0
$723$	1.47885e14	−	1.47885e14i	0.748569	−	0.748569i
$724$		0			0
$725$	2.09774e14	+	1.40240e14i	1.04728	+	0.700133i
$726$		0			0
$727$	−3.34110e13	−	3.34110e13i	−0.164520	−	0.164520i	0.620046	−	0.784566i	$-0.287114\pi$
$727$	−3.34110e13	−	3.34110e13i	−0.164520	−	0.164520i	−0.784566	+	0.620046i	$0.787114\pi$
$728$		0			0
$729$			1.37864e14i			0.669597i
$730$		0			0
$731$	8.32486e12			0.0398832
$732$		0			0
$733$	1.67919e14	−	1.67919e14i	0.793561	−	0.793561i	−0.188510	−	0.982071i	$-0.560366\pi$
$733$	1.67919e14	−	1.67919e14i	0.793561	−	0.793561i	0.982071	+	0.188510i	$0.0603658\pi$
$734$		0			0
$735$	−1.27976e14	−	3.88378e13i	−0.596610	−	0.181058i
$736$		0			0
$737$	−4.48628e14	−	4.48628e14i	−2.06324	−	2.06324i
$738$		0			0
$739$			1.45713e14i			0.661112i	0.943786	+	0.330556i	$0.107236\pi$
$739$			1.45713e14i			0.661112i	−0.943786	+	0.330556i	$0.892764\pi$
$740$		0			0
$741$	9.48897e13			0.424745
$742$		0			0
$743$	2.55814e14	−	2.55814e14i	1.12974	−	1.12974i	0.139526	−	0.990218i	$-0.455442\pi$
$743$	2.55814e14	−	2.55814e14i	1.12974	−	1.12974i	0.990218	−	0.139526i	$-0.0445578\pi$
$744$		0			0
$745$	1.69337e14	+	3.16898e14i	0.737852	+	1.38082i
$746$		0			0
$747$	−3.09174e13	−	3.09174e13i	−0.132923	−	0.132923i
$748$		0			0
$749$		−	3.61879e14i		−	1.53516i
$750$		0			0
$751$	6.70445e13			0.280649			0.140325	−	0.990106i	$-0.455185\pi$
$751$	6.70445e13			0.280649			0.140325	+	0.990106i	$0.455185\pi$
$752$		0			0
$753$	5.13432e13	−	5.13432e13i	0.212084	−	0.212084i
$754$		0			0
$755$	−2.85870e14	+	1.52757e14i	−1.16529	+	0.622680i
$756$		0			0
$757$	−2.60082e14	−	2.60082e14i	−1.04624	−	1.04624i	−0.998878	−	0.0473609i	$-0.984919\pi$
$757$	−2.60082e14	−	2.60082e14i	−1.04624	−	1.04624i	−0.0473609	−	0.998878i	$-0.515081\pi$
$758$		0			0
$759$			6.42347e14i			2.55012i
$760$		0			0
$761$	9.47162e13			0.371109			0.185554	−	0.982634i	$-0.440592\pi$
$761$	9.47162e13			0.371109			0.185554	+	0.982634i	$0.440592\pi$
$762$		0			0
$763$	−2.10539e13	+	2.10539e13i	−0.0814160	+	0.0814160i
$764$		0			0
$765$	−1.41236e13	+	4.65390e13i	−0.0539060	+	0.177627i
$766$		0			0
$767$	3.60176e12	+	3.60176e12i	0.0135687	+	0.0135687i
$768$		0			0
$769$			2.60425e14i			0.968391i	0.874960	+	0.484195i	$0.160888\pi$
$769$			2.60425e14i			0.968391i	−0.874960	+	0.484195i	$0.839112\pi$
$770$		0			0
$771$	−3.73236e14			−1.36997
$772$		0			0
$773$	−5.98900e13	+	5.98900e13i	−0.216999	+	0.216999i	−0.807232	−	0.590234i	$-0.799036\pi$
$773$	−5.98900e13	+	5.98900e13i	−0.216999	+	0.216999i	0.590234	+	0.807232i	$0.299036\pi$
$774$		0			0
$775$	7.90872e13	−	1.18301e14i	0.282877	−	0.423135i
$776$		0			0
$777$	1.39227e14	+	1.39227e14i	0.491608	+	0.491608i
$778$		0			0
$779$		−	2.19952e14i		−	0.766725i
$780$		0			0
$781$	−6.11968e14			−2.10607
$782$		0			0
$783$	2.23116e14	−	2.23116e14i	0.758092	−	0.758092i
$784$		0			0
$785$	4.47972e14	+	1.35950e14i	1.50280	+	0.456068i
$786$		0			0
$787$	2.16758e14	+	2.16758e14i	0.717962	+	0.717962i	0.968188	−	0.250225i	$-0.0805047\pi$
$787$	2.16758e14	+	2.16758e14i	0.717962	+	0.717962i	−0.250225	+	0.968188i	$0.580505\pi$
$788$		0			0
$789$			5.09516e13i			0.166638i
$790$		0			0
$791$	3.59304e14			1.16033
$792$		0			0
$793$	−1.76863e13	+	1.76863e13i	−0.0563991	+	0.0563991i
$794$		0			0
$795$	−1.68171e14	−	3.14718e14i	−0.529561	−	0.991026i
$796$		0			0
$797$	3.14993e13	+	3.14993e13i	0.0979512	+	0.0979512i	0.754384	−	0.656433i	$-0.227936\pi$
$797$	3.14993e13	+	3.14993e13i	0.0979512	+	0.0979512i	−0.656433	+	0.754384i	$0.727936\pi$
$798$		0			0
$799$		−	6.56676e13i		−	0.201659i
$800$		0			0
$801$	1.09966e14			0.333499
$802$		0			0
$803$	−8.25488e14	+	8.25488e14i	−2.47248	+	2.47248i
$804$		0			0
$805$	−4.31684e14	+	2.30673e14i	−1.27699	+	0.682367i
$806$		0			0
$807$	−4.26109e14	−	4.26109e14i	−1.24495	−	1.24495i
$808$		0			0
$809$		−	2.30864e13i		−	0.0666214i	−0.999445	−	0.0333107i	$-0.989395\pi$
$809$		−	2.30864e13i		−	0.0666214i	0.999445	−	0.0333107i	$-0.0106051\pi$
$810$		0			0
$811$	1.56268e14			0.445417			0.222708	−	0.974885i	$-0.428510\pi$
$811$	1.56268e14			0.445417			0.222708	+	0.974885i	$0.428510\pi$
$812$		0			0
$813$	−1.79099e14	+	1.79099e14i	−0.504244	+	0.504244i
$814$		0			0
$815$	−1.01157e14	+	3.33326e14i	−0.281325	+	0.927003i
$816$		0			0
$817$	−1.68808e13	−	1.68808e13i	−0.0463749	−	0.0463749i
$818$		0			0
$819$			3.15374e13i			0.0855867i
$820$		0			0
$821$	−2.91291e14			−0.780927			−0.390464	−	0.920618i	$-0.627685\pi$
$821$	−2.91291e14			−0.780927			−0.390464	+	0.920618i	$0.627685\pi$
$822$		0			0
$823$	3.98092e14	−	3.98092e14i	1.05435	−	1.05435i	0.0559143	−	0.998436i	$-0.482193\pi$
$823$	3.98092e14	−	3.98092e14i	1.05435	−	1.05435i	0.998436	−	0.0559143i	$-0.0178074\pi$
$824$		0			0
$825$	−1.63891e14	−	8.24972e14i	−0.428829	−	2.15859i
$826$		0			0
$827$	−5.02893e13	−	5.02893e13i	−0.130002	−	0.130002i	0.639112	−	0.769114i	$-0.279302\pi$
$827$	−5.02893e13	−	5.02893e13i	−0.130002	−	0.130002i	−0.769114	+	0.639112i	$0.779302\pi$
$828$		0			0
$829$			4.45415e14i			1.13761i	0.822473	+	0.568804i	$0.192594\pi$
$829$			4.45415e14i			1.13761i	−0.822473	+	0.568804i	$0.807406\pi$
$830$		0			0
$831$	6.50903e14			1.64252
$832$		0			0
$833$	1.26368e14	−	1.26368e14i	0.315075	−	0.315075i
$834$		0			0
$835$	5.00219e13	+	1.51805e13i	0.123233	+	0.0373986i
$836$		0			0
$837$	−1.25825e14	−	1.25825e14i	−0.306294	−	0.306294i
$838$		0			0
$839$		−	6.00226e14i		−	1.44379i	−0.692001	−	0.721897i	$-0.743271\pi$
$839$		−	6.00226e14i		−	1.44379i	0.692001	−	0.721897i	$-0.256729\pi$
$840$		0			0
$841$	−2.46945e14			−0.586976
$842$		0			0
$843$	−1.18996e13	+	1.18996e13i	−0.0279509	+	0.0279509i
$844$		0			0
$845$	1.85614e14	+	3.47360e14i	0.430851	+	0.806299i
$846$		0			0
$847$	1.12679e15	+	1.12679e15i	2.58479	+	2.58479i
$848$		0			0
$849$		−	9.63889e14i		−	2.18519i
$850$		0			0
$851$	2.59058e14			0.580429
$852$		0			0
$853$	3.59725e14	−	3.59725e14i	0.796572	−	0.796572i	−0.185981	−	0.982553i	$-0.559546\pi$
$853$	3.59725e14	−	3.59725e14i	0.796572	−	0.796572i	0.982553	+	0.185981i	$0.0595464\pi$
$854$		0			0
$855$	1.23009e14	−	6.57306e13i	0.269219	−	0.143859i
$856$		0			0
$857$	3.54053e14	+	3.54053e14i	0.765885	+	0.765885i	0.977379	−	0.211494i	$-0.0678329\pi$
$857$	3.54053e14	+	3.54053e14i	0.765885	+	0.765885i	−0.211494	+	0.977379i	$0.567833\pi$
$858$		0			0
$859$			5.09001e14i			1.08831i	0.838985	+	0.544155i	$0.183150\pi$
$859$			5.09001e14i			1.08831i	−0.838985	+	0.544155i	$0.816850\pi$
$860$		0			0
$861$	3.85731e14			0.815208
$862$		0			0
$863$	−4.61464e14	+	4.61464e14i	−0.964015	+	0.964015i	−0.999375	−	0.0353594i	$-0.988742\pi$
$863$	−4.61464e14	+	4.61464e14i	−0.964015	+	0.964015i	0.0353594	+	0.999375i	$0.488742\pi$
$864$		0			0
$865$	5.35773e13	−	1.76544e14i	0.110637	−	0.364563i
$866$		0			0
$867$	1.42295e14	+	1.42295e14i	0.290466	+	0.290466i
$868$		0			0
$869$		−	1.60008e15i		−	3.22880i
$870$		0			0
$871$	−2.16255e14			−0.431395
$872$		0			0
$873$	−6.25222e13	+	6.25222e13i	−0.123300	+	0.123300i
$874$		0			0
$875$	4.95561e14	−	4.06397e14i	0.966178	−	0.792338i
$876$		0			0
$877$	3.59343e14	+	3.59343e14i	0.692645	+	0.692645i	0.962813	−	0.270168i	$-0.0870792\pi$
$877$	3.59343e14	+	3.59343e14i	0.692645	+	0.692645i	−0.270168	+	0.962813i	$0.587079\pi$
$878$		0			0
$879$			9.89374e13i			0.188546i
$880$		0			0
$881$	−9.71158e14			−1.82983			−0.914914	−	0.403648i	$-0.867742\pi$
$881$	−9.71158e14			−1.82983			−0.914914	+	0.403648i	$0.867742\pi$
$882$		0			0
$883$	5.87538e13	−	5.87538e13i	0.109454	−	0.109454i	−0.650259	−	0.759713i	$-0.725339\pi$
$883$	5.87538e13	−	5.87538e13i	0.109454	−	0.109454i	0.759713	+	0.650259i	$0.225339\pi$
$884$		0			0
$885$	3.78016e13	+	1.14719e13i	0.0696295	+	0.0211310i
$886$		0			0
$887$	−5.22917e14	−	5.22917e14i	−0.952389	−	0.952389i	0.0465275	−	0.998917i	$-0.485184\pi$
$887$	−5.22917e14	−	5.22917e14i	−0.952389	−	0.952389i	−0.998917	+	0.0465275i	$0.985184\pi$
$888$		0			0
$889$			7.18422e14i			1.29381i
$890$		0			0
$891$	−1.31192e15			−2.33624
$892$		0			0
$893$	−1.33158e14	+	1.33158e14i	−0.234482	+	0.234482i
$894$		0			0
$895$	−2.79464e14	−	5.22993e14i	−0.486644	−	0.910710i
$896$		0			0
$897$	1.54817e14	+	1.54817e14i	0.266598	+	0.266598i
$898$		0			0
$899$			3.76518e14i			0.641191i
$900$		0			0
$901$	4.76824e14			0.803035
$902$		0			0
$903$	2.96040e13	−	2.96040e13i	0.0493073	−	0.0493073i
$904$		0			0
$905$	4.82991e14	−	2.58090e14i	0.795603	−	0.425136i
$906$		0			0
$907$	5.34644e14	+	5.34644e14i	0.871020	+	0.871020i	0.992584	−	0.121563i	$-0.0387907\pi$
$907$	5.34644e14	+	5.34644e14i	0.871020	+	0.871020i	−0.121563	+	0.992584i	$0.538791\pi$
$908$		0			0
$909$			2.60170e13i			0.0419217i
$910$		0			0
$911$	2.20072e13			0.0350730			0.0175365	−	0.999846i	$-0.494418\pi$
$911$	2.20072e13			0.0350730			0.0175365	+	0.999846i	$0.494418\pi$
$912$		0			0
$913$	−7.14486e14	+	7.14486e14i	−1.12626	+	1.12626i
$914$		0			0
$915$	−5.63326e13	+	1.85623e14i	−0.0878323	+	0.289419i
$916$		0			0
$917$	−5.55575e14	−	5.55575e14i	−0.856832	−	0.856832i
$918$		0			0
$919$		−	1.65331e13i		−	0.0252219i	−0.999920	−	0.0126109i	$-0.995986\pi$
$919$		−	1.65331e13i		−	0.0252219i	0.999920	−	0.0126109i	$-0.00401430\pi$
$920$		0			0
$921$	1.96878e14			0.297098
$922$		0			0
$923$	−1.47495e14	+	1.47495e14i	−0.220176	+	0.220176i
$924$		0			0
$925$	−3.32710e14	+	6.60969e13i	−0.491312	+	0.0976051i
$926$		0			0
$927$	−9.48982e13	−	9.48982e13i	−0.138631	−	0.138631i
$928$		0			0
$929$			9.99452e14i			1.44439i	0.691691	+	0.722193i	$0.256866\pi$
$929$			9.99452e14i			1.44439i	−0.691691	+	0.722193i	$0.743134\pi$
$930$		0			0
$931$	−5.12489e14			−0.732717
$932$		0			0
$933$	−1.48706e14	+	1.48706e14i	−0.210339	+	0.210339i
$934$		0			0
$935$	1.07549e15	+	3.26389e14i	1.50505	+	0.456749i
$936$		0			0
$937$	−5.99227e14	−	5.99227e14i	−0.829648	−	0.829648i	0.157820	−	0.987468i	$-0.449553\pi$
$937$	−5.99227e14	−	5.99227e14i	−0.829648	−	0.829648i	−0.987468	+	0.157820i	$0.949553\pi$
$938$		0			0
$939$			1.90287e14i			0.260665i
$940$		0			0
$941$	4.36336e14			0.591388			0.295694	−	0.955283i	$-0.404449\pi$
$941$	4.36336e14			0.591388			0.295694	+	0.955283i	$0.404449\pi$
$942$		0			0
$943$	3.58861e14	−	3.58861e14i	0.481247	−	0.481247i
$944$		0			0
$945$	−3.77696e14	−	7.06824e14i	−0.501168	−	0.937891i
$946$		0			0
$947$	−4.31092e14	−	4.31092e14i	−0.566005	−	0.566005i	0.365002	−	0.931007i	$-0.381068\pi$
$947$	−4.31092e14	−	4.31092e14i	−0.566005	−	0.566005i	−0.931007	+	0.365002i	$0.881068\pi$
$948$		0			0
$949$			3.97915e14i			0.516963i
$950$		0			0
$951$	−8.10917e14			−1.04249
$952$		0			0
$953$	−1.43542e14	+	1.43542e14i	−0.182605	+	0.182605i	−0.792490	−	0.609885i	$-0.791216\pi$
$953$	−1.43542e14	+	1.43542e14i	−0.182605	+	0.182605i	0.609885	+	0.792490i	$0.291216\pi$
$954$		0			0
$955$	3.01040e13	−	1.60863e13i	0.0378972	−	0.0202506i
$956$		0			0
$957$	1.57364e15	+	1.57364e15i	1.96040	+	1.96040i
$958$		0			0
$959$		−	6.22748e14i		−	0.767749i
$960$		0			0
$961$	−6.07293e14			−0.740938
$962$		0			0
$963$	−1.68242e14	+	1.68242e14i	−0.203144	+	0.203144i
$964$		0			0
$965$	8.00543e13	−	2.63789e14i	0.0956639	−	0.315225i
$966$		0			0
$967$	−2.96216e14	−	2.96216e14i	−0.350329	−	0.350329i	0.509903	−	0.860232i	$-0.329681\pi$
$967$	−2.96216e14	−	2.96216e14i	−0.350329	−	0.350329i	−0.860232	+	0.509903i	$0.829681\pi$
$968$		0			0
$969$			9.83387e14i			1.15108i
$970$		0			0
$971$	4.69824e13			0.0544301			0.0272151	−	0.999630i	$-0.491336\pi$
$971$	4.69824e13			0.0544301			0.0272151	+	0.999630i	$0.491336\pi$
$972$		0			0
$973$	2.73845e14	−	2.73845e14i	0.314008	−	0.314008i
$974$		0			0
$975$	−2.38334e14	−	1.59332e14i	−0.270497	−	0.180834i
$976$		0			0
$977$	1.47654e13	+	1.47654e13i	0.0165871	+	0.0165871i	0.715352	−	0.698765i	$-0.246266\pi$
$977$	1.47654e13	+	1.47654e13i	0.0165871	+	0.0165871i	−0.698765	+	0.715352i	$0.746266\pi$
$978$		0			0
$979$		−	2.54126e15i		−	2.82576i
$980$		0			0
$981$	1.95764e13			0.0215471
$982$		0			0
$983$	2.72740e14	−	2.72740e14i	0.297153	−	0.297153i	−0.542745	−	0.839898i	$-0.682615\pi$
$983$	2.72740e14	−	2.72740e14i	0.297153	−	0.297153i	0.839898	+	0.542745i	$0.182615\pi$
$984$		0			0
$985$	6.40447e14	+	1.94362e14i	0.690720	+	0.209618i
$986$		0			0
$987$	−2.33520e14	−	2.33520e14i	−0.249309	−	0.249309i
$988$		0			0
$989$		−	5.50836e13i		−	0.0582158i
$990$		0			0
$991$	1.08965e15			1.14004			0.570018	−	0.821632i	$-0.306936\pi$
$991$	1.08965e15			1.14004			0.570018	+	0.821632i	$0.306936\pi$
$992$		0			0
$993$	1.32448e15	−	1.32448e15i	1.37182	−	1.37182i
$994$		0			0
$995$	3.07600e14	+	5.75646e14i	0.315407	+	0.590256i
$996$		0			0
$997$	6.24498e14	+	6.24498e14i	0.633951	+	0.633951i	0.949056	−	0.315106i	$-0.102040\pi$
$997$	6.24498e14	+	6.24498e14i	0.633951	+	0.633951i	−0.315106	+	0.949056i	$0.602040\pi$
$998$		0			0
$999$			4.24172e14i			0.426299i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.11.f.a.17.1	yes	10
3.2	odd	2		180.11.l.a.37.1		10
4.3	odd	2		80.11.p.e.17.5		10
5.2	odd	4		100.11.f.b.93.5		10
5.3	odd	4	inner	20.11.f.a.13.1	✓	10
5.4	even	2		100.11.f.b.57.5		10
15.8	even	4		180.11.l.a.73.1		10
20.3	even	4		80.11.p.e.33.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.11.f.a.13.1	✓	10	5.3	odd	4	inner
20.11.f.a.17.1	yes	10	1.1	even	1	trivial
80.11.p.e.17.5		10	4.3	odd	2
80.11.p.e.33.5		10	20.3	even	4
100.11.f.b.57.5		10	5.4	even	2
100.11.f.b.93.5		10	5.2	odd	4
180.11.l.a.37.1		10	3.2	odd	2
180.11.l.a.73.1		10	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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	20.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-190.862 + 190.862i) q^{3} +(2756.18 - 1472.78i) q^{5} +(14849.7 + 14849.7i) q^{7} -13807.6i q^{9} +O(q^{10})\)	\(q+(-190.862 + 190.862i) q^{3} +(2756.18 - 1472.78i) q^{5} +(14849.7 + 14849.7i) q^{7} -13807.6i q^{9} -319087. q^{11} +(-76905.8 + 76905.8i) q^{13} +(-244952. + 807149. i) q^{15} +(-797011. - 797011. i) q^{17} +3.23229e6i q^{19} -5.66849e6 q^{21} +(-5.27364e6 + 5.27364e6i) q^{23} +(5.42745e6 - 8.11851e6i) q^{25} +(-8.63486e6 - 8.63486e6i) q^{27} +2.58390e7i q^{29} +1.45717e7 q^{31} +(6.09017e7 - 6.09017e7i) q^{33} +(6.27989e7 + 1.90581e7i) q^{35} +(-2.45616e7 - 2.45616e7i) q^{37} -2.93568e7i q^{39} -6.80482e7 q^{41} +(-5.22255e6 + 5.22255e6i) q^{43} +(-2.03356e7 - 3.80563e7i) q^{45} +(4.11962e7 + 4.11962e7i) q^{47} +1.58553e8i q^{49} +3.04238e8 q^{51} +(-2.99133e8 + 2.99133e8i) q^{53} +(-8.79463e8 + 4.69947e8i) q^{55} +(-6.16922e8 - 6.16922e8i) q^{57} -4.68335e7i q^{59} +2.29974e8 q^{61} +(2.05039e8 - 2.05039e8i) q^{63} +(-9.87008e7 + 3.25232e8i) q^{65} +(1.40597e9 + 1.40597e9i) q^{67} -2.01307e9i q^{69} +1.91787e9 q^{71} +(2.58703e9 - 2.58703e9i) q^{73} +(5.13623e8 + 2.58541e9i) q^{75} +(-4.73836e9 - 4.73836e9i) q^{77} +5.01453e9i q^{79} +4.11146e9 q^{81} +(2.23915e9 - 2.23915e9i) q^{83} +(-3.37053e9 - 1.02288e9i) q^{85} +(-4.93168e9 - 4.93168e9i) q^{87} +7.96413e9i q^{89} -2.28406e9 q^{91} +(-2.78119e9 + 2.78119e9i) q^{93} +(4.76046e9 + 8.90878e9i) q^{95} +(-4.52809e9 - 4.52809e9i) q^{97} +4.40584e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7}+O(q^{10}) \) 10 * q + 62 * q^3 + 894 * q^5 + 22286 * q^7	\( 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7} - 201700 q^{11} + 239298 q^{13} + 213662 q^{15} + 1045442 q^{17} + 4578860 q^{21} - 4097986 q^{23} - 4233934 q^{25} - 4817488 q^{27} + 23221660 q^{31} + 31816220 q^{33} - 55388242 q^{35} - 87811974 q^{37} + 29776460 q^{41} + 156325470 q^{43} - 144135236 q^{45} - 450750018 q^{47} + 1632585820 q^{51} + 701393866 q^{53} - 1301185140 q^{55} - 2564330416 q^{57} + 2991488220 q^{61} + 3352397678 q^{63} - 2867494182 q^{65} - 6990333394 q^{67} + 9915200380 q^{71} + 8401915018 q^{73} - 10170758642 q^{75} - 19825815140 q^{77} + 26071184290 q^{81} + 16998617454 q^{83} - 24280829854 q^{85} - 36065578576 q^{87} + 52347612540 q^{91} + 26277966572 q^{93} - 23431125296 q^{95} - 48945511254 q^{97}+O(q^{100}) \) 10 * q + 62 * q^3 + 894 * q^5 + 22286 * q^7 - 201700 * q^11 + 239298 * q^13 + 213662 * q^15 + 1045442 * q^17 + 4578860 * q^21 - 4097986 * q^23 - 4233934 * q^25 - 4817488 * q^27 + 23221660 * q^31 + 31816220 * q^33 - 55388242 * q^35 - 87811974 * q^37 + 29776460 * q^41 + 156325470 * q^43 - 144135236 * q^45 - 450750018 * q^47 + 1632585820 * q^51 + 701393866 * q^53 - 1301185140 * q^55 - 2564330416 * q^57 + 2991488220 * q^61 + 3352397678 * q^63 - 2867494182 * q^65 - 6990333394 * q^67 + 9915200380 * q^71 + 8401915018 * q^73 - 10170758642 * q^75 - 19825815140 * q^77 + 26071184290 * q^81 + 16998617454 * q^83 - 24280829854 * q^85 - 36065578576 * q^87 + 52347612540 * q^91 + 26277966572 * q^93 - 23431125296 * q^95 - 48945511254 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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