LMFDB - Embedded newform 76.8.e.a.49.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(45,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, 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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.6
Character	$\chi$	$=$	76.49
Dual form			76.8.e.a.45.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.69911 - 2.94294i) q^{3} +(105.538 - 182.797i) q^{5} +1469.14 q^{7} +(1087.73 + 1884.00i) q^{9} +O(q^{10})$	\(q+(1.69911 - 2.94294i) q^{3} +(105.538 - 182.797i) q^{5} +1469.14 q^{7} +(1087.73 + 1884.00i) q^{9} -7030.02 q^{11} +(512.187 + 887.134i) q^{13} +(-358.641 - 621.184i) q^{15} +(10115.7 - 17520.9i) q^{17} +(27108.5 - 12609.5i) q^{19} +(2496.22 - 4323.58i) q^{21} +(16084.9 + 27859.9i) q^{23} +(16785.9 + 29074.0i) q^{25} +14824.5 q^{27} +(-65841.6 - 114041. i) q^{29} +299767. q^{31} +(-11944.7 + 20688.9i) q^{33} +(155050. - 268555. i) q^{35} +348230. q^{37} +3481.04 q^{39} +(-83105.4 + 143943. i) q^{41} +(165120. - 285996. i) q^{43} +459186. q^{45} +(-391947. - 678872. i) q^{47} +1.33482e6 q^{49} +(-34375.2 - 59539.6i) q^{51} +(189084. + 327503. i) q^{53} +(-741935. + 1.28507e6i) q^{55} +(8951.38 - 101204. i) q^{57} +(699619. - 1.21178e6i) q^{59} +(-373370. - 646696. i) q^{61} +(1.59802e6 + 2.76785e6i) q^{63} +216221. q^{65} +(-2.45164e6 - 4.24636e6i) q^{67} +109320. q^{69} +(-1.76532e6 + 3.05763e6i) q^{71} +(1.66909e6 - 2.89095e6i) q^{73} +114084. q^{75} -1.03281e7 q^{77} +(-3.74009e6 + 6.47802e6i) q^{79} +(-2.35367e6 + 4.07667e6i) q^{81} +3.62399e6 q^{83} +(-2.13518e6 - 3.69824e6i) q^{85} -447487. q^{87} +(5.14127e6 + 8.90494e6i) q^{89} +(752473. + 1.30332e6i) q^{91} +(509336. - 882196. i) q^{93} +(556006. - 6.28615e6i) q^{95} +(1.88121e6 - 3.25836e6i) q^{97} +(-7.64673e6 - 1.32445e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.69911	−	2.94294i	0.0363326	−	0.0629299i	−0.847287	−	0.531135i	$-0.821766\pi$
$3$	1.69911	−	2.94294i	0.0363326	−	0.0629299i	0.883620	+	0.468205i	$0.155099\pi$
$4$		0			0
$5$	105.538	−	182.797i	0.377585	−	0.653996i	−0.613126	−	0.789986i	$-0.710088\pi$
$5$	105.538	−	182.797i	0.377585	−	0.653996i	0.990710	+	0.135990i	$0.0434213\pi$
$6$		0			0
$7$	1469.14			1.61890			0.809449	−	0.587190i	$-0.199766\pi$
$7$	1469.14			1.61890			0.809449	+	0.587190i	$0.199766\pi$
$8$		0			0
$9$	1087.73	+	1884.00i	0.497360	+	0.861453i
$10$		0			0
$11$	−7030.02			−1.59251			−0.796254	−	0.604962i	$-0.793188\pi$
$11$	−7030.02			−1.59251			−0.796254	+	0.604962i	$0.793188\pi$
$12$		0			0
$13$	512.187	+	887.134i	0.0646587	+	0.111992i	0.896543	−	0.442958i	$-0.146071\pi$
$13$	512.187	+	887.134i	0.0646587	+	0.111992i	−0.831884	+	0.554950i	$0.812737\pi$
$14$		0			0
$15$	−358.641	−	621.184i	−0.0274372	−	0.0475227i
$16$		0			0
$17$	10115.7	−	17520.9i	0.499372	−	0.864937i	−0.500628	−	0.865662i	$-0.666898\pi$
$17$	10115.7	−	17520.9i	0.499372	−	0.864937i	1.00000		0.000725555i	$0.000230951\pi$
$18$		0			0
$19$	27108.5	−	12609.5i	0.906710	−	0.421754i
$19$	27108.5	−	12609.5i	0.906710	−	0.421754i
$20$		0			0
$21$	2496.22	−	4323.58i	0.0588187	−	0.101877i
$22$		0			0
$23$	16084.9	+	27859.9i	0.275658	+	0.477454i	0.970301	−	0.241901i	$-0.0777708\pi$
$23$	16084.9	+	27859.9i	0.275658	+	0.477454i	−0.694643	+	0.719355i	$0.744437\pi$
$24$		0			0
$25$	16785.9	+	29074.0i	0.214859	+	0.372148i
$26$		0			0
$27$	14824.5			0.144947
$28$		0			0
$29$	−65841.6	−	114041.i	−0.501311	−	0.868296i	−0.999999	−	0.00151461i	$-0.999518\pi$
$29$	−65841.6	−	114041.i	−0.501311	−	0.868296i	0.498688	−	0.866782i	$-0.333815\pi$
$30$		0			0
$31$	299767.			1.80725			0.903624	−	0.428326i	$-0.140897\pi$
$31$	299767.			1.80725			0.903624	+	0.428326i	$0.140897\pi$
$32$		0			0
$33$	−11944.7	+	20688.9i	−0.0578599	+	0.100216i
$34$		0			0
$35$	155050.	−	268555.i	0.611271	−	1.05875i
$36$		0			0
$37$	348230.			1.13021			0.565106	−	0.825018i	$-0.308835\pi$
$37$	348230.			1.13021			0.565106	+	0.825018i	$0.308835\pi$
$38$		0			0
$39$	3481.04			0.00939686
$40$		0			0
$41$	−83105.4	+	143943.i	−0.188315	+	0.326172i	−0.944689	−	0.327969i	$-0.893636\pi$
$41$	−83105.4	+	143943.i	−0.188315	+	0.326172i	0.756373	+	0.654140i	$0.226969\pi$
$42$		0			0
$43$	165120.	−	285996.i	0.316709	−	0.548556i	−0.663091	−	0.748539i	$-0.730756\pi$
$43$	165120.	−	285996.i	0.316709	−	0.548556i	0.979799	+	0.199984i	$0.0640889\pi$
$44$		0			0
$45$	459186.			0.751182
$46$		0			0
$47$	−391947.	−	678872.i	−0.550662	−	0.953774i	−0.998227	−	0.0595228i	$-0.981042\pi$
$47$	−391947.	−	678872.i	−0.550662	−	0.953774i	0.447565	−	0.894251i	$-0.352291\pi$
$48$		0			0
$49$	1.33482e6			1.62083
$50$		0			0
$51$	−34375.2	−	59539.6i	−0.0362869	−	0.0628508i
$52$		0			0
$53$	189084.	+	327503.i	0.174457	+	0.302169i	0.939973	−	0.341248i	$-0.110850\pi$
$53$	189084.	+	327503.i	0.174457	+	0.302169i	−0.765516	+	0.643417i	$0.777516\pi$
$54$		0			0
$55$	−741935.	+	1.28507e6i	−0.601307	+	1.04149i
$56$		0			0
$57$	8951.38	−	101204.i	0.00640218	−	0.0723826i
$58$		0			0
$59$	699619.	−	1.21178e6i	0.443486	−	0.768140i	−0.554460	−	0.832211i	$-0.687075\pi$
$59$	699619.	−	1.21178e6i	0.443486	−	0.768140i	0.997945	+	0.0640709i	$0.0204084\pi$
$60$		0			0
$61$	−373370.	−	646696.i	−0.210613	−	0.364793i	0.741293	−	0.671181i	$-0.234213\pi$
$61$	−373370.	−	646696.i	−0.210613	−	0.364793i	−0.951907	+	0.306388i	$0.900879\pi$
$62$		0			0
$63$	1.59802e6	+	2.76785e6i	0.805175	+	1.39460i
$64$		0			0
$65$	216221.			0.0976565
$66$		0			0
$67$	−2.45164e6	−	4.24636e6i	−0.995851	−	1.72487i	−0.576730	−	0.816935i	$-0.695671\pi$
$67$	−2.45164e6	−	4.24636e6i	−0.995851	−	1.72487i	−0.419121	−	0.907930i	$-0.637662\pi$
$68$		0			0
$69$	109320.			0.0400615
$70$		0			0
$71$	−1.76532e6	+	3.05763e6i	−0.585355	+	1.01386i	0.409476	+	0.912321i	$0.365712\pi$
$71$	−1.76532e6	+	3.05763e6i	−0.585355	+	1.01386i	−0.994831	+	0.101544i	$0.967622\pi$
$72$		0			0
$73$	1.66909e6	−	2.89095e6i	0.502168	−	0.869781i	−0.497829	−	0.867275i	$-0.665869\pi$
$73$	1.66909e6	−	2.89095e6i	0.502168	−	0.869781i	0.999997	−	0.00250543i	$-0.000797504\pi$
$74$		0			0
$75$	114084.			0.0312256
$76$		0			0
$77$	−1.03281e7			−2.57811
$78$		0			0
$79$	−3.74009e6	+	6.47802e6i	−0.853467	+	1.47825i	0.0245927	+	0.999698i	$0.492171\pi$
$79$	−3.74009e6	+	6.47802e6i	−0.853467	+	1.47825i	−0.878060	+	0.478551i	$0.841162\pi$
$80$		0			0
$81$	−2.35367e6	+	4.07667e6i	−0.492094	+	0.852331i
$82$		0			0
$83$	3.62399e6			0.695687			0.347843	−	0.937553i	$-0.386914\pi$
$83$	3.62399e6			0.695687			0.347843	+	0.937553i	$0.386914\pi$
$84$		0			0
$85$	−2.13518e6	−	3.69824e6i	−0.377110	−	0.653174i
$86$		0			0
$87$	−447487.			−0.0728557
$88$		0			0
$89$	5.14127e6	+	8.90494e6i	0.773046	+	1.33895i	0.935886	+	0.352302i	$0.114601\pi$
$89$	5.14127e6	+	8.90494e6i	0.773046	+	1.33895i	−0.162841	+	0.986652i	$0.552066\pi$
$90$		0			0
$91$	752473.	+	1.30332e6i	0.104676	+	0.181304i
$92$		0			0
$93$	509336.	−	882196.i	0.0656620	−	0.113730i
$94$		0			0
$95$	556006.	−	6.28615e6i	0.0665344	−	0.752233i
$96$		0			0
$97$	1.88121e6	−	3.25836e6i	0.209284	−	0.362491i	−0.742205	−	0.670173i	$-0.766220\pi$
$97$	1.88121e6	−	3.25836e6i	0.209284	−	0.362491i	0.951489	+	0.307682i	$0.0995532\pi$
$98$		0			0
$99$	−7.64673e6	−	1.32445e7i	−0.792050	−	1.37187i
$100$		0			0
$101$	8.38014e6	+	1.45148e7i	0.809331	+	1.40180i	0.913328	+	0.407225i	$0.133504\pi$
$101$	8.38014e6	+	1.45148e7i	0.809331	+	1.40180i	−0.103997	+	0.994578i	$0.533163\pi$
$102$		0			0
$103$	−1.29287e7			−1.16580			−0.582902	−	0.812542i	$-0.698083\pi$
$103$	−1.29287e7			−1.16580			−0.582902	+	0.812542i	$0.698083\pi$
$104$		0			0
$105$	−526893.	−	912606.i	−0.0444181	−	0.0769344i
$106$		0			0
$107$	−2.19417e7			−1.73152			−0.865758	−	0.500463i	$-0.833163\pi$
$107$	−2.19417e7			−1.73152			−0.865758	+	0.500463i	$0.833163\pi$
$108$		0			0
$109$	−7.86107e6	+	1.36158e7i	−0.581419	+	1.00705i	0.413893	+	0.910326i	$0.364169\pi$
$109$	−7.86107e6	+	1.36158e7i	−0.581419	+	1.00705i	−0.995312	+	0.0967213i	$0.969164\pi$
$110$		0			0
$111$	591680.	−	1.02482e6i	0.0410635	−	0.0711241i
$112$		0			0
$113$	4.90046e6			0.319493			0.159747	−	0.987158i	$-0.448932\pi$
$113$	4.90046e6			0.319493			0.159747	+	0.987158i	$0.448932\pi$
$114$		0			0
$115$	6.79029e6			0.416338
$116$		0			0
$117$	−1.11424e6	+	1.92992e6i	−0.0643172	+	0.111401i
$118$		0			0
$119$	1.48613e7	−	2.57406e7i	0.808432	−	1.40024i
$120$		0			0
$121$	2.99339e7			1.53608
$122$		0			0
$123$	282410.	+	489148.i	0.0136840	+	0.0237013i
$124$		0			0
$125$	2.35765e7			1.07968
$126$		0			0
$127$	−209212.	−	362366.i	−0.00906303	−	0.0156976i	0.861458	−	0.507828i	$-0.169552\pi$
$127$	−209212.	−	362366.i	−0.00906303	−	0.0156976i	−0.870521	+	0.492131i	$0.836218\pi$
$128$		0			0
$129$	−561113.	−	971876.i	−0.0230137	−	0.0398609i
$130$		0			0
$131$	1.30448e6	−	2.25942e6i	0.0506976	−	0.0878108i	−0.839563	−	0.543262i	$-0.817189\pi$
$131$	1.30448e6	−	2.25942e6i	0.0506976	−	0.0878108i	0.890261	+	0.455452i	$0.150522\pi$
$132$		0			0
$133$	3.98262e7	−	1.85251e7i	1.46787	−	0.682777i
$134$		0			0
$135$	1.56455e6	−	2.70989e6i	0.0547296	−	0.0947945i
$136$		0			0
$137$	−2.43397e6	−	4.21576e6i	−0.0808711	−	0.140073i	0.822753	−	0.568399i	$-0.192437\pi$
$137$	−2.43397e6	−	4.21576e6i	−0.0808711	−	0.140073i	−0.903624	+	0.428326i	$0.859104\pi$
$138$		0			0
$139$	1.12781e7	+	1.95342e7i	0.356191	+	0.616942i	0.987321	−	0.158736i	$-0.0507418\pi$
$139$	1.12781e7	+	1.95342e7i	0.356191	+	0.616942i	−0.631130	+	0.775677i	$0.717408\pi$
$140$		0			0
$141$	−2.66384e6			−0.0800278
$142$		0			0
$143$	−3.60068e6	−	6.23656e6i	−0.102969	−	0.178348i
$144$		0			0
$145$	−2.77952e7			−0.757150
$146$		0			0
$147$	2.26801e6	−	3.92830e6i	0.0588889	−	0.101999i
$148$		0			0
$149$	−1.12939e7	+	1.95617e7i	−0.279701	+	0.484456i	−0.971310	−	0.237816i	$-0.923569\pi$
$149$	−1.12939e7	+	1.95617e7i	−0.279701	+	0.484456i	0.691609	+	0.722272i	$0.256902\pi$
$150$		0			0
$151$	−1.33693e7			−0.316002			−0.158001	−	0.987439i	$-0.550505\pi$
$151$	−1.33693e7			−0.316002			−0.158001	+	0.987439i	$0.550505\pi$
$152$		0			0
$153$	4.40123e7			0.993469
$154$		0			0
$155$	3.16369e7	−	5.47966e7i	0.682389	−	1.18193i
$156$		0			0
$157$	3.96189e6	−	6.86219e6i	0.0817059	−	0.141519i	−0.822277	−	0.569088i	$-0.807297\pi$
$157$	3.96189e6	−	6.86219e6i	0.0817059	−	0.141519i	0.903983	+	0.427569i	$0.140630\pi$
$158$		0			0
$159$	1.28509e6			0.0253539
$160$		0			0
$161$	2.36310e7	+	4.09300e7i	0.446263	+	0.772950i
$162$		0			0
$163$	−3.91903e7			−0.708797			−0.354399	−	0.935094i	$-0.615314\pi$
$163$	−3.91903e7			−0.708797			−0.354399	+	0.935094i	$0.615314\pi$
$164$		0			0
$165$	2.52125e6	+	4.36694e6i	0.0436941	+	0.0756803i
$166$		0			0
$167$	−4.04121e7	−	6.99959e7i	−0.671435	−	1.16296i	−0.977497	−	0.210949i	$-0.932345\pi$
$167$	−4.04121e7	−	6.99959e7i	−0.671435	−	1.16296i	0.306062	−	0.952012i	$-0.400989\pi$
$168$		0			0
$169$	3.08496e7	−	5.34331e7i	0.491639	−	0.851543i
$170$		0			0
$171$	5.32429e7	+	3.73567e7i	0.814283	+	0.571324i
$172$		0			0
$173$	−2.15166e7	+	3.72679e7i	−0.315946	+	0.547234i	−0.979638	−	0.200771i	$-0.935655\pi$
$173$	−2.15166e7	+	3.72679e7i	−0.315946	+	0.547234i	0.663692	+	0.748006i	$0.268988\pi$
$174$		0			0
$175$	2.46608e7	+	4.27138e7i	0.347836	+	0.602469i
$176$		0			0
$177$	−2.37745e6	−	4.11787e6i	−0.0322259	−	0.0558170i
$178$		0			0
$179$	−1.45981e8			−1.90243			−0.951217	−	0.308522i	$-0.900166\pi$
$179$	−1.45981e8			−1.90243			−0.951217	+	0.308522i	$0.900166\pi$
$180$		0			0
$181$	3.31815e7	+	5.74721e7i	0.415931	+	0.720413i	0.995526	−	0.0944914i	$-0.0301225\pi$
$181$	3.31815e7	+	5.74721e7i	0.415931	+	0.720413i	−0.579595	+	0.814905i	$0.696789\pi$
$182$		0			0
$183$	−2.53758e6			−0.0306085
$184$		0			0
$185$	3.67516e7	−	6.36556e7i	0.426751	−	0.739155i
$186$		0			0
$187$	−7.11134e7	+	1.23172e8i	−0.795254	+	1.37742i
$188$		0			0
$189$	2.17793e7			0.234654
$190$		0			0
$191$	−6.54259e7			−0.679411			−0.339706	−	0.940532i	$-0.610327\pi$
$191$	−6.54259e7			−0.679411			−0.339706	+	0.940532i	$0.610327\pi$
$192$		0			0
$193$	−7.70291e7	+	1.33418e8i	−0.771266	+	1.33587i	0.165603	+	0.986193i	$0.447043\pi$
$193$	−7.70291e7	+	1.33418e8i	−0.771266	+	1.33587i	−0.936869	+	0.349680i	$0.886290\pi$
$194$		0			0
$195$	367382.	−	636325.i	0.00354811	−	0.00614551i
$196$		0			0
$197$	−5.27524e7			−0.491598			−0.245799	−	0.969321i	$-0.579050\pi$
$197$	−5.27524e7			−0.491598			−0.245799	+	0.969321i	$0.579050\pi$
$198$		0			0
$199$	−6.33010e7	−	1.09641e8i	−0.569410	−	0.986247i	−0.996624	−	0.0820964i	$-0.973838\pi$
$199$	−6.33010e7	−	1.09641e8i	−0.569410	−	0.986247i	0.427215	−	0.904150i	$-0.359495\pi$
$200$		0			0
$201$	−1.66624e7			−0.144727
$202$		0			0
$203$	−9.67304e7	−	1.67542e8i	−0.811572	−	1.40568i
$204$		0			0
$205$	1.75416e7	+	3.03829e7i	0.142210	+	0.246315i
$206$		0			0
$207$	−3.49920e7	+	6.06079e7i	−0.274203	+	0.474933i
$208$		0			0
$209$	−1.90573e8	+	8.86448e7i	−1.44394	+	0.671647i
$210$		0			0
$211$	7.48370e6	−	1.29622e7i	0.0548439	−	0.0949923i	−0.837300	−	0.546744i	$-0.815867\pi$
$211$	7.48370e6	−	1.29622e7i	0.0548439	−	0.0949923i	0.892144	+	0.451751i	$0.149201\pi$
$212$		0			0
$213$	5.99894e6	+	1.03905e7i	0.0425349	+	0.0736726i
$214$		0			0
$215$	−3.48529e7	−	6.03670e7i	−0.239169	−	0.414252i
$216$		0			0
$217$	4.40399e8			2.92575
$218$		0			0
$219$	−5.67191e6	−	9.82404e6i	−0.0364901	−	0.0632027i
$220$		0			0
$221$	2.07245e7			0.129155
$222$		0			0
$223$	1.98782e7	−	3.44300e7i	0.120035	−	0.207907i	−0.799746	−	0.600339i	$-0.795033\pi$
$223$	1.98782e7	−	3.44300e7i	0.120035	−	0.207907i	0.919781	+	0.392431i	$0.128366\pi$
$224$		0			0
$225$	−3.65169e7	+	6.32492e7i	−0.213725	+	0.370183i
$226$		0			0
$227$	8.80190e7			0.499443			0.249722	−	0.968318i	$-0.419661\pi$
$227$	8.80190e7			0.499443			0.249722	+	0.968318i	$0.419661\pi$
$228$		0			0
$229$	−8.55511e7			−0.470762			−0.235381	−	0.971903i	$-0.575634\pi$
$229$	−8.55511e7			−0.470762			−0.235381	+	0.971903i	$0.575634\pi$
$230$		0			0
$231$	−1.75485e7	+	3.03949e7i	−0.0936693	+	0.162240i
$232$		0			0
$233$	−1.70641e8	+	2.95559e8i	−0.883768	+	1.53073i	−0.0366490	+	0.999328i	$0.511668\pi$
$233$	−1.70641e8	+	2.95559e8i	−0.883768	+	1.53073i	−0.847119	+	0.531403i	$0.821665\pi$
$234$		0			0
$235$	−1.65461e8			−0.831686
$236$		0			0
$237$	1.27096e7	+	2.20137e7i	0.0620173	+	0.107417i
$238$		0			0
$239$	1.59033e8			0.753517			0.376759	−	0.926311i	$-0.377039\pi$
$239$	1.59033e8			0.753517			0.376759	+	0.926311i	$0.377039\pi$
$240$		0			0
$241$	−1.02181e8	−	1.76982e8i	−0.470228	−	0.814459i	0.529192	−	0.848502i	$-0.322495\pi$
$241$	−1.02181e8	−	1.76982e8i	−0.470228	−	0.814459i	−0.999420	+	0.0340430i	$0.989162\pi$
$242$		0			0
$243$	2.42089e7	+	4.19310e7i	0.108231	+	0.187462i
$244$		0			0
$245$	1.40875e8	−	2.44002e8i	0.612001	−	1.06002i
$246$		0			0
$247$	2.50709e7	+	1.75905e7i	0.105860	+	0.0742743i
$248$		0			0
$249$	6.15754e6	−	1.06652e7i	0.0252761	−	0.0437795i
$250$		0			0
$251$	1.33073e8	+	2.30488e8i	0.531166	+	0.920007i	0.999338	+	0.0363695i	$0.0115793\pi$
$251$	1.33073e8	+	2.30488e8i	0.531166	+	0.920007i	−0.468172	+	0.883637i	$0.655087\pi$
$252$		0			0
$253$	−1.13077e8	−	1.95855e8i	−0.438989	−	0.760350i
$254$		0			0
$255$	−1.45116e7			−0.0548055
$256$		0			0
$257$	−2.15172e6	−	3.72690e6i	−0.00790716	−	0.0136956i	0.862045	−	0.506832i	$-0.169184\pi$
$257$	−2.15172e6	−	3.72690e6i	−0.00790716	−	0.0136956i	−0.869952	+	0.493137i	$0.835850\pi$
$258$		0			0
$259$	5.11598e8			1.82970
$260$		0			0
$261$	1.43235e8	−	2.48091e8i	0.498664	−	0.863712i
$262$		0			0
$263$	1.91705e8	−	3.32042e8i	0.649812	−	1.12551i	−0.333356	−	0.942801i	$-0.608181\pi$
$263$	1.91705e8	−	3.32042e8i	0.649812	−	1.12551i	0.983168	−	0.182706i	$-0.0584855\pi$
$264$		0			0
$265$	7.98223e7			0.263490
$266$		0			0
$267$	3.49422e7			0.112347
$268$		0			0
$269$	2.42799e7	−	4.20539e7i	0.0760524	−	0.131727i	−0.825491	−	0.564415i	$-0.809102\pi$
$269$	2.42799e7	−	4.20539e7i	0.0760524	−	0.131727i	0.901543	+	0.432689i	$0.142435\pi$
$270$		0			0
$271$	7.67017e7	−	1.32851e8i	0.234106	−	0.405483i	−0.724907	−	0.688847i	$-0.758117\pi$
$271$	7.67017e7	−	1.32851e8i	0.234106	−	0.405483i	0.959012	+	0.283364i	$0.0914504\pi$
$272$		0			0
$273$	5.11413e6			0.0152126
$274$		0			0
$275$	−1.18005e8	−	2.04391e8i	−0.342166	−	0.592648i
$276$		0			0
$277$	1.10507e8			0.312399			0.156199	−	0.987726i	$-0.450076\pi$
$277$	1.10507e8			0.312399			0.156199	+	0.987726i	$0.450076\pi$
$278$		0			0
$279$	3.26064e8	+	5.64760e8i	0.898853	+	1.55686i
$280$		0			0
$281$	−8.07878e7	−	1.39929e8i	−0.217207	−	0.376213i	0.736746	−	0.676170i	$-0.236361\pi$
$281$	−8.07878e7	−	1.39929e8i	−0.217207	−	0.376213i	−0.953953	+	0.299956i	$0.903028\pi$
$282$		0			0
$283$	−1.48045e8	+	2.56421e8i	−0.388276	+	0.672513i	−0.992218	−	0.124515i	$-0.960262\pi$
$283$	−1.48045e8	+	2.56421e8i	−0.388276	+	0.672513i	0.603942	+	0.797028i	$0.293596\pi$
$284$		0			0
$285$	−1.75550e7	−	1.23171e7i	−0.0449205	−	0.0315176i
$286$		0			0
$287$	−1.22093e8	+	2.11472e8i	−0.304863	+	0.528039i
$288$		0			0
$289$	515455.	+	892795.i	0.00125617	+	0.00217575i
$290$		0			0
$291$	−6.39276e6	−	1.10726e7i	−0.0152077	−	0.0263405i
$292$		0			0
$293$	1.50276e8			0.349023			0.174512	−	0.984655i	$-0.444165\pi$
$293$	1.50276e8			0.349023			0.174512	+	0.984655i	$0.444165\pi$
$294$		0			0
$295$	−1.47673e8	−	2.55777e8i	−0.334907	−	0.580076i
$296$		0			0
$297$	−1.04217e8			−0.230829
$298$		0			0
$299$	−1.64770e7	+	2.85389e7i	−0.0356474	+	0.0617431i
$300$		0			0
$301$	2.42584e8	−	4.20168e8i	0.512719	−	0.888056i
$302$		0			0
$303$	5.69550e7			0.117620
$304$		0			0
$305$	−1.57619e8			−0.318097
$306$		0			0
$307$	1.49923e7	−	2.59675e7i	0.0295723	−	0.0512207i	−0.850860	−	0.525392i	$-0.823919\pi$
$307$	1.49923e7	−	2.59675e7i	0.0295723	−	0.0512207i	0.880433	+	0.474171i	$0.157252\pi$
$308$		0			0
$309$	−2.19673e7	+	3.80484e7i	−0.0423567	+	0.0733639i
$310$		0			0
$311$	−2.97838e8			−0.561461			−0.280730	−	0.959787i	$-0.590577\pi$
$311$	−2.97838e8			−0.561461			−0.280730	+	0.959787i	$0.590577\pi$
$312$		0			0
$313$	2.95691e8	+	5.12152e8i	0.545046	+	0.944047i	0.998604	+	0.0528201i	$0.0168210\pi$
$313$	2.95691e8	+	5.12152e8i	0.545046	+	0.944047i	−0.453559	+	0.891227i	$0.649846\pi$
$314$		0			0
$315$	6.74608e8			1.21609
$316$		0			0
$317$	−1.16112e8	−	2.01112e8i	−0.204725	−	0.354594i	0.745320	−	0.666707i	$-0.232297\pi$
$317$	−1.16112e8	−	2.01112e8i	−0.204725	−	0.354594i	−0.950045	+	0.312113i	$0.898963\pi$
$318$		0			0
$319$	4.62867e8	+	8.01710e8i	0.798342	+	1.38277i
$320$		0			0
$321$	−3.72812e7	+	6.45730e7i	−0.0629104	+	0.108964i
$322$		0			0
$323$	5.32923e7	−	6.02519e8i	0.0879946	−	0.994859i
$324$		0			0
$325$	−1.71950e7	+	2.97827e7i	−0.0277851	+	0.0481251i
$326$		0			0
$327$	2.67136e7	+	4.62693e7i	0.0422489	+	0.0731772i
$328$		0			0
$329$	−5.75824e8	−	9.97357e8i	−0.891465	−	1.54406i
$330$		0			0
$331$	−1.10867e9			−1.68037			−0.840185	−	0.542300i	$-0.817553\pi$
$331$	−1.10867e9			−1.68037			−0.840185	+	0.542300i	$0.817553\pi$
$332$		0			0
$333$	3.78779e8	+	6.56064e8i	0.562122	+	0.973625i
$334$		0			0
$335$	−1.03497e9			−1.50407
$336$		0			0
$337$	−5.97146e8	+	1.03429e9i	−0.849916	+	1.47210i	0.0313664	+	0.999508i	$0.490014\pi$
$337$	−5.97146e8	+	1.03429e9i	−0.849916	+	1.47210i	−0.881282	+	0.472590i	$0.843319\pi$
$338$		0			0
$339$	8.32639e6	−	1.44217e7i	0.0116080	−	0.0201057i
$340$		0			0
$341$	−2.10737e9			−2.87806
$342$		0			0
$343$	7.51142e8			1.00506
$344$		0			0
$345$	1.15374e7	−	1.99834e7i	0.0151266	−	0.0262001i
$346$		0			0
$347$	4.33934e8	−	7.51596e8i	0.557533	−	0.965676i	−0.440168	−	0.897915i	$-0.645081\pi$
$347$	4.33934e8	−	7.51596e8i	0.557533	−	0.965676i	0.997702	−	0.0677606i	$-0.0215854\pi$
$348$		0			0
$349$	4.29984e8			0.541457			0.270728	−	0.962656i	$-0.412735\pi$
$349$	4.29984e8			0.541457			0.270728	+	0.962656i	$0.412735\pi$
$350$		0			0
$351$	7.59293e6	+	1.31513e7i	0.00937205	+	0.0162329i
$352$		0			0
$353$	−1.30327e9			−1.57697			−0.788485	−	0.615054i	$-0.789134\pi$
$353$	−1.30327e9			−1.57697			−0.788485	+	0.615054i	$0.789134\pi$
$354$		0			0
$355$	3.72617e8	+	6.45392e8i	0.442042	+	0.765640i
$356$		0			0
$357$	−5.05019e7	−	8.74719e7i	−0.0587448	−	0.101749i
$358$		0			0
$359$	−4.21641e8	+	7.30304e8i	−0.480964	+	0.833054i	−0.999761	−	0.0218430i	$-0.993047\pi$
$359$	−4.21641e8	+	7.30304e8i	−0.480964	+	0.833054i	0.518797	+	0.854897i	$0.326380\pi$
$360$		0			0
$361$	5.75874e8	−	6.83649e8i	0.644247	−	0.764818i
$362$		0			0
$363$	5.08609e7	−	8.80937e7i	0.0558099	−	0.0966656i
$364$		0			0
$365$	−3.52305e8	−	6.10210e8i	−0.379222	−	0.656832i
$366$		0			0
$367$	8.23076e8	+	1.42561e9i	0.869178	+	1.50546i	0.862838	+	0.505481i	$0.168685\pi$
$367$	8.23076e8	+	1.42561e9i	0.869178	+	1.50546i	0.00634013	+	0.999980i	$0.497982\pi$
$368$		0			0
$369$	−3.61583e8			−0.374642
$370$		0			0
$371$	2.77790e8	+	4.81147e8i	0.282429	+	0.489181i
$372$		0			0
$373$	−3.71517e8			−0.370678			−0.185339	−	0.982675i	$-0.559338\pi$
$373$	−3.71517e8			−0.370678			−0.185339	+	0.982675i	$0.559338\pi$
$374$		0			0
$375$	4.00591e7	−	6.93843e7i	0.0392276	−	0.0679441i
$376$		0			0
$377$	6.74464e7	−	1.16821e8i	0.0648282	−	0.112286i
$378$		0			0
$379$	−4.29208e8			−0.404978			−0.202489	−	0.979285i	$-0.564903\pi$
$379$	−4.29208e8			−0.404978			−0.202489	+	0.979285i	$0.564903\pi$
$380$		0			0
$381$	−1.42189e6			−0.00131713
$382$		0			0
$383$	3.56023e8	−	6.16649e8i	0.323804	−	0.560845i	−0.657466	−	0.753484i	$-0.728372\pi$
$383$	3.56023e8	−	6.16649e8i	0.323804	−	0.560845i	0.981270	+	0.192640i	$0.0617049\pi$
$384$		0			0
$385$	−1.09000e9	+	1.88794e9i	−0.973455	+	1.68607i
$386$		0			0
$387$	7.18421e8			0.630073
$388$		0			0
$389$	−6.02526e8	−	1.04361e9i	−0.518982	−	0.898903i	−0.999757	−	0.0220589i	$-0.992978\pi$
$389$	−6.02526e8	−	1.04361e9i	−0.518982	−	0.898903i	0.480775	−	0.876844i	$-0.340355\pi$
$390$		0			0
$391$	6.50839e8			0.550624
$392$		0			0
$393$	−4.43289e6	−	7.67800e6i	−0.00368395	−	0.00638078i
$394$		0			0
$395$	7.89444e8	+	1.36736e9i	0.644512	+	1.11633i
$396$		0			0
$397$	1.13820e9	−	1.97142e9i	0.912959	−	1.58129i	0.103096	−	0.994671i	$-0.467125\pi$
$397$	1.13820e9	−	1.97142e9i	0.912959	−	1.58129i	0.809863	−	0.586619i	$-0.199541\pi$
$398$		0			0
$399$	1.31508e7	−	1.48682e8i	0.0103645	−	0.117180i
$400$		0			0
$401$	9.31376e8	−	1.61319e9i	0.721306	−	1.24934i	−0.239171	−	0.970978i	$-0.576876\pi$
$401$	9.31376e8	−	1.61319e9i	0.721306	−	1.24934i	0.960477	−	0.278361i	$-0.0897912\pi$
$402$		0			0
$403$	1.53537e8	+	2.65933e8i	0.116854	+	0.202398i
$404$		0			0
$405$	4.96804e8	+	8.60489e8i	0.371614	+	0.643654i
$406$		0			0
$407$	−2.44806e9			−1.79987
$408$		0			0
$409$	7.56446e7	+	1.31020e8i	0.0546697	+	0.0946906i	0.892065	−	0.451907i	$-0.149256\pi$
$409$	7.56446e7	+	1.31020e8i	0.0546697	+	0.0946906i	−0.837395	+	0.546598i	$0.815923\pi$
$410$		0			0
$411$	−1.65423e7			−0.0117530
$412$		0			0
$413$	1.02784e9	−	1.78027e9i	0.717958	−	1.24354i
$414$		0			0
$415$	3.82469e8	−	6.62456e8i	0.262681	−	0.454976i
$416$		0			0
$417$	7.66506e7			0.0517654
$418$		0			0
$419$	−1.26852e9			−0.842460			−0.421230	−	0.906954i	$-0.638401\pi$
$419$	−1.26852e9			−0.842460			−0.421230	+	0.906954i	$0.638401\pi$
$420$		0			0
$421$	−1.13990e9	+	1.97437e9i	−0.744528	+	1.28956i	0.205888	+	0.978576i	$0.433992\pi$
$421$	−1.13990e9	+	1.97437e9i	−0.744528	+	1.28956i	−0.950415	+	0.310984i	$0.899341\pi$
$422$		0			0
$423$	8.52662e8	−	1.47685e9i	0.547754	−	0.948738i
$424$		0			0
$425$	6.79203e8			0.429179
$426$		0			0
$427$	−5.48533e8	−	9.50086e8i	−0.340961	−	0.590562i
$428$		0			0
$429$	−2.44718e7			−0.0149646
$430$		0			0
$431$	7.19166e8	+	1.24563e9i	0.432672	+	0.749410i	0.997102	−	0.0760706i	$-0.0242375\pi$
$431$	7.19166e8	+	1.24563e9i	0.432672	+	0.749410i	−0.564430	+	0.825481i	$0.690904\pi$
$432$		0			0
$433$	2.00855e8	+	3.47892e8i	0.118898	+	0.205938i	0.919331	−	0.393484i	$-0.128730\pi$
$433$	2.00855e8	+	3.47892e8i	0.118898	+	0.205938i	−0.800433	+	0.599422i	$0.795397\pi$
$434$		0			0
$435$	−4.72270e7	+	8.17996e7i	−0.0275092	+	0.0476473i
$436$		0			0
$437$	7.87337e8	+	5.52419e8i	0.451311	+	0.316653i
$438$		0			0
$439$	9.32173e8	−	1.61457e9i	0.525861	−	0.910818i	−0.473685	−	0.880694i	$-0.657077\pi$
$439$	9.32173e8	−	1.61457e9i	0.525861	−	0.910818i	0.999546	−	0.0301235i	$-0.00959007\pi$
$440$		0			0
$441$	1.45192e9	+	2.51480e9i	0.806136	+	1.39627i
$442$		0			0
$443$	−4.16484e8	−	7.21371e8i	−0.227607	−	0.394227i	0.729492	−	0.683990i	$-0.239757\pi$
$443$	−4.16484e8	−	7.21371e8i	−0.227607	−	0.394227i	−0.957098	+	0.289763i	$0.906423\pi$
$444$		0			0
$445$	2.17040e9			1.16756
$446$		0			0
$447$	3.83792e7	+	6.64748e7i	0.0203245	+	0.0352031i
$448$		0			0
$449$	1.61629e9			0.842668			0.421334	−	0.906905i	$-0.361562\pi$
$449$	1.61629e9			0.842668			0.421334	+	0.906905i	$0.361562\pi$
$450$		0			0
$451$	5.84232e8	−	1.01192e9i	0.299894	−	0.519431i
$452$		0			0
$453$	−2.27159e7	+	3.93450e7i	−0.0114812	+	0.0198859i
$454$		0			0
$455$	3.17659e8			0.158096
$456$		0			0
$457$	1.79441e9			0.879456			0.439728	−	0.898131i	$-0.355075\pi$
$457$	1.79441e9			0.879456			0.439728	+	0.898131i	$0.355075\pi$
$458$		0			0
$459$	1.49960e8	−	2.59739e8i	0.0723822	−	0.125370i
$460$		0			0
$461$	5.50373e8	−	9.53274e8i	0.261640	−	0.453174i	−0.705038	−	0.709170i	$-0.749070\pi$
$461$	5.50373e8	−	9.53274e8i	0.261640	−	0.453174i	0.966678	+	0.255996i	$0.0824034\pi$
$462$		0			0
$463$	5.85563e7			0.0274183			0.0137091	−	0.999906i	$-0.495636\pi$
$463$	5.85563e7			0.0274183			0.0137091	+	0.999906i	$0.495636\pi$
$464$		0			0
$465$	−1.07509e8	−	1.86211e8i	−0.0495859	−	0.0858853i
$466$		0			0
$467$	2.86396e9			1.30124			0.650622	−	0.759402i	$-0.274508\pi$
$467$	2.86396e9			1.30124			0.650622	+	0.759402i	$0.274508\pi$
$468$		0			0
$469$	−3.60179e9	−	6.23849e9i	−1.61218	−	2.79238i
$470$		0			0
$471$	−1.34633e7	−	2.33192e7i	−0.00593717	−	0.0102835i
$472$		0			0
$473$	−1.16080e9	+	2.01056e9i	−0.504361	+	0.873580i
$474$		0			0
$475$	8.21649e8	+	5.76493e8i	0.351770	+	0.246812i
$476$		0			0
$477$	−4.11343e8	+	7.12467e8i	−0.173536	+	0.300573i
$478$		0			0
$479$	8.34926e8	+	1.44613e9i	0.347115	+	0.601221i	0.985736	−	0.168301i	$-0.0538280\pi$
$479$	8.34926e8	+	1.44613e9i	0.347115	+	0.601221i	−0.638621	+	0.769522i	$0.720495\pi$
$480$		0			0
$481$	1.78359e8	+	3.08927e8i	0.0730780	+	0.126575i
$482$		0			0
$483$	1.60606e8			0.0648555
$484$		0			0
$485$	−3.97079e8	−	6.87762e8i	−0.158045	−	0.273742i
$486$		0			0
$487$	−1.46933e9			−0.576460			−0.288230	−	0.957561i	$-0.593067\pi$
$487$	−1.46933e9			−0.576460			−0.288230	+	0.957561i	$0.593067\pi$
$488$		0			0
$489$	−6.65885e7	+	1.15335e8i	−0.0257524	+	0.0446045i
$490$		0			0
$491$	−8.09386e8	+	1.40190e9i	−0.308582	+	0.534479i	−0.978052	−	0.208359i	$-0.933188\pi$
$491$	−8.09386e8	+	1.40190e9i	−0.308582	+	0.534479i	0.669471	+	0.742839i	$0.266521\pi$
$492$		0			0
$493$	−2.66413e9			−1.00136
$494$		0			0
$495$	−3.22809e9			−1.19626
$496$		0			0
$497$	−2.59350e9	+	4.49207e9i	−0.947630	+	1.64134i
$498$		0			0
$499$	−2.50841e9	+	4.34470e9i	−0.903747	+	1.56534i	−0.0811572	+	0.996701i	$0.525862\pi$
$499$	−2.50841e9	+	4.34470e9i	−0.903747	+	1.56534i	−0.822590	+	0.568635i	$0.807472\pi$
$500$		0			0
$501$	−2.74658e8			−0.0975799
$502$		0			0
$503$	−2.82205e9	−	4.88794e9i	−0.988729	−	1.71253i	−0.624022	−	0.781407i	$-0.714502\pi$
$503$	−2.82205e9	−	4.88794e9i	−0.988729	−	1.71253i	−0.364707	−	0.931122i	$-0.618831\pi$
$504$		0			0
$505$	3.53770e9			1.22236
$506$		0			0
$507$	−1.04833e8	−	1.81577e8i	−0.0357250	−	0.0618775i
$508$		0			0
$509$	−6.82723e8	−	1.18251e9i	−0.229473	−	0.397460i	0.728179	−	0.685387i	$-0.240367\pi$
$509$	−6.82723e8	−	1.18251e9i	−0.229473	−	0.397460i	−0.957652	+	0.287928i	$0.907034\pi$
$510$		0			0
$511$	2.45212e9	−	4.24720e9i	0.812959	−	1.40809i
$512$		0			0
$513$	4.01872e8	−	1.86930e8i	0.131425	−	0.0611318i
$514$		0			0
$515$	−1.36447e9	+	2.36334e9i	−0.440190	+	0.762431i
$516$		0			0
$517$	2.75539e9	+	4.77248e9i	0.876934	+	1.51889i
$518$		0			0
$519$	7.31181e7	+	1.26644e8i	0.0229583	+	0.0397649i
$520$		0			0
$521$	2.42435e8			0.0751039			0.0375520	−	0.999295i	$-0.488044\pi$
$521$	2.42435e8			0.0751039			0.0375520	+	0.999295i	$0.488044\pi$
$522$		0			0
$523$	−1.75419e8	−	3.03835e8i	−0.0536193	−	0.0928714i	0.837970	−	0.545716i	$-0.183742\pi$
$523$	−1.75419e8	−	3.03835e8i	−0.0536193	−	0.0928714i	−0.891589	+	0.452845i	$0.850409\pi$
$524$		0			0
$525$	1.67605e8			0.0505510
$526$		0			0
$527$	3.03235e9	−	5.25218e9i	0.902488	−	1.56316i
$528$		0			0
$529$	1.18496e9	−	2.05242e9i	0.348025	−	0.602797i
$530$		0			0
$531$	3.04398e9			0.882288
$532$		0			0
$533$	−1.70262e8			−0.0487049
$534$		0			0
$535$	−2.31568e9	+	4.01088e9i	−0.653794	+	1.13240i
$536$		0			0
$537$	−2.48037e8	+	4.29612e8i	−0.0691203	+	0.119720i
$538$		0			0
$539$	−9.38383e9			−2.58119
$540$		0			0
$541$	1.54651e9	+	2.67863e9i	0.419915	+	0.727314i	0.995930	−	0.0901249i	$-0.0287266\pi$
$541$	1.54651e9	+	2.67863e9i	0.419915	+	0.727314i	−0.576016	+	0.817439i	$0.695393\pi$
$542$		0			0
$543$	2.25516e8			0.0604474
$544$		0			0
$545$	1.65929e9	+	2.87397e9i	0.439070	+	0.760491i
$546$		0			0
$547$	−5.69954e8	−	9.87190e8i	−0.148897	−	0.257896i	0.781923	−	0.623375i	$-0.214239\pi$
$547$	−5.69954e8	−	9.87190e8i	−0.148897	−	0.257896i	−0.930820	+	0.365478i	$0.880905\pi$
$548$		0			0
$549$	8.12249e8	−	1.40686e9i	0.209501	−	0.362866i
$550$		0			0
$551$	−3.22287e9	−	2.26126e9i	−0.820751	−	0.575863i
$552$		0			0
$553$	−5.49470e9	+	9.51711e9i	−1.38168	+	2.39313i
$554$		0			0
$555$	−1.24890e8	−	2.16315e8i	−0.0310099	−	0.0537108i
$556$		0			0
$557$	−2.79227e9	−	4.83635e9i	−0.684643	−	1.18584i	−0.973549	−	0.228479i	$-0.926625\pi$
$557$	−2.79227e9	−	4.83635e9i	−0.684643	−	1.18584i	0.288906	−	0.957358i	$-0.406709\pi$
$558$		0			0
$559$	3.38289e8			0.0819119
$560$		0			0
$561$	2.41658e8	+	4.18564e8i	0.0577872	+	0.100090i
$562$		0			0
$563$	−4.86661e9			−1.14934			−0.574668	−	0.818387i	$-0.694869\pi$
$563$	−4.86661e9			−1.14934			−0.574668	+	0.818387i	$0.694869\pi$
$564$		0			0
$565$	5.17185e8	−	8.95791e8i	0.120636	−	0.208947i
$566$		0			0
$567$	−3.45786e9	+	5.98920e9i	−0.796649	+	1.37984i
$568$		0			0
$569$	−4.00581e9			−0.911586			−0.455793	−	0.890086i	$-0.650644\pi$
$569$	−4.00581e9			−0.911586			−0.455793	+	0.890086i	$0.650644\pi$
$570$		0			0
$571$	2.28175e8			0.0512911			0.0256456	−	0.999671i	$-0.491836\pi$
$571$	2.28175e8			0.0512911			0.0256456	+	0.999671i	$0.491836\pi$
$572$		0			0
$573$	−1.11165e8	+	1.92544e8i	−0.0246848	+	0.0427552i
$574$		0			0
$575$	−5.39999e8	+	9.35306e8i	−0.118456	+	0.205171i
$576$		0			0
$577$	1.70670e9			0.369863			0.184932	−	0.982751i	$-0.440794\pi$
$577$	1.70670e9			0.369863			0.184932	+	0.982751i	$0.440794\pi$
$578$		0			0
$579$	2.61761e8	+	4.53384e8i	0.0560442	+	0.0970714i
$580$		0			0
$581$	5.32414e9			1.12625
$582$		0			0
$583$	−1.32926e9	−	2.30235e9i	−0.277825	−	0.481207i
$584$		0			0
$585$	2.35189e8	+	4.07360e8i	0.0485704	+	0.0841264i
$586$		0			0
$587$	3.59312e9	−	6.22346e9i	0.733226	−	1.26998i	−0.222271	−	0.974985i	$-0.571347\pi$
$587$	3.59312e9	−	6.22346e9i	0.733226	−	1.26998i	0.955497	−	0.295000i	$-0.0953195\pi$
$588$		0			0
$589$	8.12625e9	−	3.77990e9i	1.63865	−	0.762215i
$590$		0			0
$591$	−8.96319e7	+	1.55247e8i	−0.0178610	+	0.0309362i
$592$		0			0
$593$	2.87585e9	+	4.98111e9i	0.566336	+	0.980923i	0.996924	+	0.0783741i	$0.0249729\pi$
$593$	2.87585e9	+	4.98111e9i	0.566336	+	0.980923i	−0.430588	+	0.902549i	$0.641694\pi$
$594$		0			0
$595$	−3.13687e9	−	5.43323e9i	−0.610503	−	1.05742i
$596$		0			0
$597$	−4.30221e8			−0.0827525
$598$		0			0
$599$	3.75580e9	+	6.50524e9i	0.714017	+	1.23671i	0.963337	+	0.268293i	$0.0864596\pi$
$599$	3.75580e9	+	6.50524e9i	0.714017	+	1.23671i	−0.249320	+	0.968421i	$0.580207\pi$
$600$		0			0
$601$	−5.91228e9			−1.11095			−0.555475	−	0.831534i	$-0.687463\pi$
$601$	−5.91228e9			−1.11095			−0.555475	+	0.831534i	$0.687463\pi$
$602$		0			0
$603$	5.33342e9	−	9.23776e9i	0.990593	−	1.71576i
$604$		0			0
$605$	3.15917e9	−	5.47185e9i	0.580002	−	1.00459i
$606$		0			0
$607$	−6.87987e9			−1.24859			−0.624295	−	0.781188i	$-0.714614\pi$
$607$	−6.87987e9			−1.24859			−0.624295	+	0.781188i	$0.714614\pi$
$608$		0			0
$609$	−6.57421e8			−0.117946
$610$		0			0
$611$	4.01500e8	−	6.95419e8i	0.0712101	−	0.123340i
$612$		0			0
$613$	1.06237e8	−	1.84009e8i	0.0186280	−	0.0322646i	−0.856561	−	0.516046i	$-0.827404\pi$
$613$	1.06237e8	−	1.84009e8i	0.0186280	−	0.0322646i	0.875189	+	0.483781i	$0.160737\pi$
$614$		0			0
$615$	1.19220e8			0.0206674
$616$		0			0
$617$	1.47586e9	+	2.55626e9i	0.252956	+	0.438133i	0.964338	−	0.264672i	$-0.0852638\pi$
$617$	1.47586e9	+	2.55626e9i	0.252956	+	0.438133i	−0.711382	+	0.702805i	$0.751930\pi$
$618$		0			0
$619$	6.14884e9			1.04202			0.521010	−	0.853550i	$-0.325555\pi$
$619$	6.14884e9			1.04202			0.521010	+	0.853550i	$0.325555\pi$
$620$		0			0
$621$	2.38451e8	+	4.13010e8i	0.0399558	+	0.0692054i
$622$		0			0
$623$	7.55323e9	+	1.30826e10i	1.25148	+	2.16763i
$624$		0			0
$625$	1.17683e9	−	2.03832e9i	0.192811	−	0.333959i
$626$		0			0
$627$	−6.29284e7	+	7.11463e8i	−0.0101955	+	0.115270i
$628$		0			0
$629$	3.52258e9	−	6.10129e9i	0.564396	−	0.977563i
$630$		0			0
$631$	5.54760e8	+	9.60872e8i	0.0879027	+	0.152252i	0.906624	−	0.421939i	$-0.138650\pi$
$631$	5.54760e8	+	9.60872e8i	0.0879027	+	0.152252i	−0.818722	+	0.574190i	$0.805317\pi$
$632$		0			0
$633$	−2.54312e7	−	4.40482e7i	−0.00398524	−	0.00690263i
$634$		0			0
$635$	−8.83193e7			−0.0136882
$636$		0			0
$637$	6.83679e8	+	1.18417e9i	0.104801	+	0.181520i
$638$		0			0
$639$	−7.68074e9			−1.16453
$640$		0			0
$641$	−1.40587e9	+	2.43504e9i	−0.210835	+	0.365177i	−0.951976	−	0.306173i	$-0.900952\pi$
$641$	−1.40587e9	+	2.43504e9i	−0.210835	+	0.365177i	0.741141	+	0.671349i	$0.234285\pi$
$642$		0			0
$643$	3.10266e9	−	5.37397e9i	0.460253	−	0.797181i	−0.538721	−	0.842484i	$-0.681092\pi$
$643$	3.10266e9	−	5.37397e9i	0.460253	−	0.797181i	0.998973	+	0.0453036i	$0.0144255\pi$
$644$		0			0
$645$	−2.36875e8			−0.0347585
$646$		0			0
$647$	6.28338e9			0.912070			0.456035	−	0.889962i	$-0.349269\pi$
$647$	6.28338e9			0.912070			0.456035	+	0.889962i	$0.349269\pi$
$648$		0			0
$649$	−4.91833e9	+	8.51880e9i	−0.706255	+	1.22327i
$650$		0			0
$651$	7.48285e8	−	1.29607e9i	0.106300	−	0.184117i
$652$		0			0
$653$	8.16900e9			1.14808			0.574041	−	0.818827i	$-0.305375\pi$
$653$	8.16900e9			1.14808			0.574041	+	0.818827i	$0.305375\pi$
$654$		0			0
$655$	−2.75344e8	−	4.76911e8i	−0.0382853	−	0.0663120i
$656$		0			0
$657$	7.26204e9			0.999033
$658$		0			0
$659$	−8.61016e8	−	1.49132e9i	−0.117196	−	0.202989i	0.801460	−	0.598049i	$-0.204057\pi$
$659$	−8.61016e8	−	1.49132e9i	−0.117196	−	0.202989i	−0.918655	+	0.395060i	$0.870724\pi$
$660$		0			0
$661$	−3.42887e9	−	5.93897e9i	−0.461791	−	0.799845i	0.537259	−	0.843417i	$-0.319460\pi$
$661$	−3.42887e9	−	5.93897e9i	−0.461791	−	0.799845i	−0.999050	+	0.0435717i	$0.986126\pi$
$662$		0			0
$663$	3.52131e7	−	6.09908e7i	0.00469252	−	0.00812769i
$664$		0			0
$665$	8.16849e8	−	9.23523e9i	0.107712	−	1.21779i
$666$		0			0
$667$	2.11811e9	−	3.66868e9i	0.276381	−	0.478706i
$668$		0			0
$669$	−6.75502e7	−	1.17000e8i	−0.00872239	−	0.0151076i
$670$		0			0
$671$	2.62480e9	+	4.54629e9i	0.335403	+	0.580936i
$672$		0			0
$673$	−4.13559e9			−0.522980			−0.261490	−	0.965206i	$-0.584214\pi$
$673$	−4.13559e9			−0.522980			−0.261490	+	0.965206i	$0.584214\pi$
$674$		0			0
$675$	2.48843e8	+	4.31009e8i	0.0311432	+	0.0539415i
$676$		0			0
$677$	1.12249e10			1.39034			0.695169	−	0.718847i	$-0.255330\pi$
$677$	1.12249e10			1.39034			0.695169	+	0.718847i	$0.255330\pi$
$678$		0			0
$679$	2.76376e9	−	4.78698e9i	0.338810	−	0.586836i
$680$		0			0
$681$	1.49554e8	−	2.59034e8i	0.0181460	−	0.0314299i
$682$		0			0
$683$	−1.36707e10			−1.64179			−0.820896	−	0.571077i	$-0.806526\pi$
$683$	−1.36707e10			−1.64179			−0.820896	+	0.571077i	$0.806526\pi$
$684$		0			0
$685$	−1.02751e9			−0.122143
$686$		0			0
$687$	−1.45360e8	+	2.51772e8i	−0.0171040	+	0.0296250i
$688$		0			0
$689$	−1.93693e8	+	3.35485e8i	−0.0225603	+	0.0390757i
$690$		0			0
$691$	3.00409e9			0.346369			0.173185	−	0.984889i	$-0.444594\pi$
$691$	3.00409e9			0.346369			0.173185	+	0.984889i	$0.444594\pi$
$692$		0			0
$693$	−1.12341e10	−	1.94580e10i	−1.28225	−	2.22092i
$694$		0			0
$695$	4.76107e9			0.537970
$696$		0			0
$697$	1.68133e9	+	2.91216e9i	0.188079	+	0.325762i
$698$		0			0
$699$	5.79875e8	+	1.00437e9i	0.0642191	+	0.111231i
$700$		0			0
$701$	−3.14452e9	+	5.44646e9i	−0.344779	+	0.597174i	−0.985314	−	0.170755i	$-0.945379\pi$
$701$	−3.14452e9	+	5.44646e9i	−0.344779	+	0.597174i	0.640535	+	0.767929i	$0.278713\pi$
$702$		0			0
$703$	9.44001e9	−	4.39100e9i	1.02478	−	0.476672i
$704$		0			0
$705$	−2.81137e8	+	4.86943e8i	−0.0302173	+	0.0523379i
$706$		0			0
$707$	1.23116e10	+	2.13243e10i	1.31022	+	2.26938i
$708$		0			0
$709$	1.68408e9	+	2.91692e9i	0.177460	+	0.307371i	0.941010	−	0.338379i	$-0.109878\pi$
$709$	1.68408e9	+	2.91692e9i	0.177460	+	0.307371i	−0.763550	+	0.645749i	$0.776545\pi$
$710$		0			0
$711$	−1.62728e10			−1.69792
$712$		0			0
$713$	4.82173e9	+	8.35147e9i	0.498183	+	0.862879i
$714$		0			0
$715$	−1.52004e9			−0.155519
$716$		0			0
$717$	2.70213e8	−	4.68023e8i	0.0273772	−	0.0474187i
$718$		0			0
$719$	7.06416e9	−	1.22355e10i	0.708777	−	1.22764i	−0.256534	−	0.966535i	$-0.582581\pi$
$719$	7.06416e9	−	1.22355e10i	0.708777	−	1.22764i	0.965311	−	0.261103i	$-0.0840861\pi$
$720$		0			0
$721$	−1.89941e10			−1.88732
$722$		0			0
$723$	−6.94462e8			−0.0683384
$724$		0			0
$725$	2.21042e9	−	3.82856e9i	0.215423	−	0.373123i
$726$		0			0
$727$	−6.08855e9	+	1.05457e10i	−0.587683	+	1.01790i	0.406852	+	0.913494i	$0.366627\pi$
$727$	−6.08855e9	+	1.05457e10i	−0.587683	+	1.01790i	−0.994535	+	0.104403i	$0.966707\pi$
$728$		0			0
$729$	−1.01304e10			−0.968458
$730$		0			0
$731$	−3.34060e9	−	5.78609e9i	−0.316311	−	0.547866i
$732$		0			0
$733$	−1.02435e10			−0.960697			−0.480349	−	0.877078i	$-0.659490\pi$
$733$	−1.02435e10			−0.960697			−0.480349	+	0.877078i	$0.659490\pi$
$734$		0			0
$735$	−4.78723e8	−	8.29172e8i	−0.0444711	−	0.0770263i
$736$		0			0
$737$	1.72351e10	+	2.98520e10i	1.58590	+	2.74686i
$738$		0			0
$739$	6.14551e8	−	1.06443e9i	0.0560148	−	0.0970204i	−0.836658	−	0.547725i	$-0.815494\pi$
$739$	6.14551e8	−	1.06443e9i	0.0560148	−	0.0970204i	0.892673	+	0.450705i	$0.148827\pi$
$740$		0			0
$741$	9.43659e7	−	4.38941e7i	0.00852023	−	0.00396317i
$742$		0			0
$743$	7.23473e9	−	1.25309e10i	0.647085	−	1.12078i	−0.336731	−	0.941601i	$-0.609321\pi$
$743$	7.23473e9	−	1.25309e10i	0.647085	−	1.12078i	0.983816	−	0.179183i	$-0.0573454\pi$
$744$		0			0
$745$	2.38388e9	+	4.12901e9i	0.211222	+	0.365847i
$746$		0			0
$747$	3.94191e9	+	6.82759e9i	0.346007	+	0.599301i
$748$		0			0
$749$	−3.22353e10			−2.80315
$750$		0			0
$751$	−4.75603e9	−	8.23768e9i	−0.409736	−	0.709684i	0.585124	−	0.810944i	$-0.301046\pi$
$751$	−4.75603e9	−	8.23768e9i	−0.409736	−	0.709684i	−0.994860	+	0.101260i	$0.967713\pi$
$752$		0			0
$753$	9.04417e8			0.0771945
$754$		0			0
$755$	−1.41097e9	+	2.44387e9i	−0.119317	+	0.206664i
$756$		0			0
$757$	7.10988e8	−	1.23147e9i	0.0595699	−	0.103178i	−0.834703	−	0.550701i	$-0.814360\pi$
$757$	7.10988e8	−	1.23147e9i	0.0595699	−	0.103178i	0.894272	+	0.447523i	$0.147694\pi$
$758$		0			0
$759$	−7.68521e8			−0.0637983
$760$		0			0
$761$	−7.63024e9			−0.627613			−0.313807	−	0.949487i	$-0.601604\pi$
$761$	−7.63024e9			−0.627613			−0.313807	+	0.949487i	$0.601604\pi$
$762$		0			0
$763$	−1.15490e10	+	2.00035e10i	−0.941258	+	1.63031i
$764$		0			0
$765$	4.64498e9	−	8.04534e9i	0.375119	−	0.649725i
$766$		0			0
$767$	1.43334e9			0.114701
$768$		0			0
$769$	−8.54117e9	−	1.47937e10i	−0.677291	−	1.17310i	−0.975794	−	0.218693i	$-0.929821\pi$
$769$	−8.54117e9	−	1.47937e10i	−0.677291	−	1.17310i	0.298503	−	0.954409i	$-0.403513\pi$
$770$		0			0
$771$	−1.46240e7			−0.00114915
$772$		0			0
$773$	7.91151e7	+	1.37031e8i	0.00616072	+	0.0106707i	0.869089	−	0.494655i	$-0.164706\pi$
$773$	7.91151e7	+	1.37031e8i	0.00616072	+	0.0106707i	−0.862929	+	0.505326i	$0.831372\pi$
$774$		0			0
$775$	5.03186e9	+	8.71543e9i	0.388305	+	0.672563i
$776$		0			0
$777$	8.69259e8	−	1.50560e9i	0.0664777	−	0.115143i
$778$		0			0
$779$	−4.37823e8	+	4.94999e9i	−0.0331831	+	0.375166i
$780$		0			0
$781$	1.24102e10	−	2.14952e10i	0.932183	−	1.61459i
$782$		0			0
$783$	−9.76071e8	−	1.69060e9i	−0.0726633	−	0.125857i
$784$		0			0
$785$	−8.36260e8	−	1.44845e9i	−0.0617018	−	0.106871i
$786$		0			0
$787$	1.41324e10			1.03348			0.516741	−	0.856142i	$-0.327145\pi$
$787$	1.41324e10			1.03348			0.516741	+	0.856142i	$0.327145\pi$
$788$		0			0
$789$	−6.51453e8	−	1.12835e9i	−0.0472186	−	0.0817851i
$790$		0			0
$791$	7.19945e9			0.517227
$792$		0			0
$793$	3.82471e8	−	6.62459e8i	0.0272359	−	0.0471740i
$794$		0			0
$795$	1.35627e8	−	2.34912e8i	0.00957326	−	0.0165814i
$796$		0			0
$797$	−2.02765e10			−1.41869			−0.709347	−	0.704860i	$-0.751010\pi$
$797$	−2.02765e10			−1.41869			−0.709347	+	0.704860i	$0.751010\pi$
$798$		0			0
$799$	−1.58592e10			−1.09994
$800$		0			0
$801$	−1.11846e10	+	1.93723e10i	−0.768964	+	1.33188i
$802$		0			0
$803$	−1.17337e10	+	2.03234e10i	−0.799707	+	1.38513i
$804$		0			0
$805$	9.97587e9			0.674008
$806$		0			0
$807$	−8.25081e7	−	1.42908e8i	−0.00552636	−	0.00957194i
$808$		0			0
$809$	1.11112e10			0.737806			0.368903	−	0.929468i	$-0.379733\pi$
$809$	1.11112e10			0.737806			0.368903	+	0.929468i	$0.379733\pi$
$810$		0			0
$811$	−8.91509e9	−	1.54414e10i	−0.586885	−	1.01651i	−0.994638	−	0.103422i	$-0.967021\pi$
$811$	−8.91509e9	−	1.54414e10i	−0.586885	−	1.01651i	0.407753	−	0.913092i	$-0.366312\pi$
$812$		0			0
$813$	−2.60649e8	−	4.51457e8i	−0.0170113	−	0.0294645i
$814$		0			0
$815$	−4.13607e9	+	7.16389e9i	−0.267631	+	0.463551i
$816$		0			0
$817$	8.69900e8	−	9.83502e9i	0.0558074	−	0.630954i
$818$		0			0
$819$	−1.63697e9	+	2.83531e9i	−0.104123	+	0.180346i
$820$		0			0
$821$	8.83233e9	+	1.52980e10i	0.557024	+	0.964794i	0.997743	+	0.0671491i	$0.0213903\pi$
$821$	8.83233e9	+	1.52980e10i	0.557024	+	0.964794i	−0.440719	+	0.897645i	$0.645276\pi$
$822$		0			0
$823$	−7.69368e9	−	1.33258e10i	−0.481099	−	0.833288i	0.518666	−	0.854977i	$-0.326429\pi$
$823$	−7.69368e9	−	1.33258e10i	−0.481099	−	0.833288i	−0.999765	+	0.0216892i	$0.993096\pi$
$824$		0			0
$825$	−8.02013e8			−0.0497270
$826$		0			0
$827$	1.30057e10	+	2.25265e10i	0.799585	+	1.38492i	0.919887	+	0.392184i	$0.128280\pi$
$827$	1.30057e10	+	2.25265e10i	0.799585	+	1.38492i	−0.120302	+	0.992737i	$0.538386\pi$
$828$		0			0
$829$	−3.18060e10			−1.93896			−0.969480	−	0.245172i	$-0.921155\pi$
$829$	−3.18060e10			−1.93896			−0.969480	+	0.245172i	$0.921155\pi$
$830$		0			0
$831$	1.87762e8	−	3.25214e8i	0.0113502	−	0.0196592i
$832$		0			0
$833$	1.35026e10	−	2.33873e10i	0.809397	−	1.40192i
$834$		0			0
$835$	−1.70601e10			−1.01410
$836$		0			0
$837$	4.44391e9			0.261955
$838$		0			0
$839$	3.95057e9	−	6.84259e9i	0.230937	−	0.399994i	−0.727147	−	0.686481i	$-0.759154\pi$
$839$	3.95057e9	−	6.84259e9i	0.230937	−	0.399994i	0.958084	+	0.286487i	$0.0924877\pi$
$840$		0			0
$841$	−4.52926e7	+	7.84491e7i	−0.00262568	+	0.00454781i
$842$		0			0
$843$	−5.49068e8			−0.0315667
$844$		0			0
$845$	−6.51162e9	−	1.12785e10i	−0.371270	−	0.643059i
$846$		0			0
$847$	4.39771e10			2.48676
$848$		0			0
$849$	5.03087e8	+	8.71372e8i	0.0282141	+	0.0488683i
$850$		0			0
$851$	5.60125e9	+	9.70165e9i	0.311553	+	0.539625i
$852$		0			0
$853$	7.16114e9	−	1.24035e10i	0.395058	−	0.684260i	−0.598051	−	0.801458i	$-0.704058\pi$
$853$	7.16114e9	−	1.24035e10i	0.395058	−	0.684260i	0.993109	+	0.117198i	$0.0373912\pi$
$854$		0			0
$855$	1.24479e10	−	5.79010e9i	0.681104	−	0.316814i
$856$		0			0
$857$	−9.17088e9	+	1.58844e10i	−0.497712	+	0.862062i	−0.999997	−	0.00263999i	$-0.999160\pi$
$857$	−9.17088e9	+	1.58844e10i	−0.497712	+	0.862062i	0.502285	+	0.864702i	$0.332493\pi$
$858$		0			0
$859$	6.27285e9	+	1.08649e10i	0.337667	+	0.584856i	0.983993	−	0.178205i	$-0.0570289\pi$
$859$	6.27285e9	+	1.08649e10i	0.337667	+	0.584856i	−0.646327	+	0.763061i	$0.723696\pi$
$860$		0			0
$861$	4.14899e8	+	7.18626e8i	0.0221529	+	0.0383700i
$862$		0			0
$863$	1.08051e10			0.572259			0.286130	−	0.958191i	$-0.407631\pi$
$863$	1.08051e10			0.572259			0.286130	+	0.958191i	$0.407631\pi$
$864$		0			0
$865$	4.54165e9	+	7.86637e9i	0.238593	+	0.413255i
$866$		0			0
$867$	3.50325e6			0.000182560
$868$		0			0
$869$	2.62929e10	−	4.55406e10i	1.35915	−	2.35412i
$870$		0			0
$871$	2.51139e9	−	4.34986e9i	0.128781	−	0.223055i
$872$		0			0
$873$	8.18498e9			0.416359
$874$		0			0
$875$	3.46372e10			1.74789
$876$		0			0
$877$	1.86220e9	−	3.22543e9i	0.0932241	−	0.161469i	−0.815642	−	0.578557i	$-0.803616\pi$
$877$	1.86220e9	−	3.22543e9i	0.0932241	−	0.161469i	0.908866	+	0.417088i	$0.136949\pi$
$878$		0			0
$879$	2.55336e8	−	4.42254e8i	0.0126809	−	0.0219640i
$880$		0			0
$881$	−1.34218e10			−0.661293			−0.330646	−	0.943755i	$-0.607267\pi$
$881$	−1.34218e10			−0.661293			−0.330646	+	0.943755i	$0.607267\pi$
$882$		0			0
$883$	1.07896e10	+	1.86881e10i	0.527401	+	0.913486i	0.999490	+	0.0319348i	$0.0101669\pi$
$883$	1.07896e10	+	1.86881e10i	0.527401	+	0.913486i	−0.472089	+	0.881551i	$0.656500\pi$
$884$		0			0
$885$	−1.00365e9			−0.0486721
$886$		0			0
$887$	1.09556e10	+	1.89756e10i	0.527111	+	0.912983i	0.999501	+	0.0315931i	$0.0100581\pi$
$887$	1.09556e10	+	1.89756e10i	0.527111	+	0.912983i	−0.472390	+	0.881390i	$0.656609\pi$
$888$		0			0
$889$	−3.07361e8	−	5.32365e8i	−0.0146721	−	0.0254128i
$890$		0			0
$891$	1.65463e10	−	2.86591e10i	0.783663	−	1.35734i
$892$		0			0
$893$	−1.91853e10	−	1.34610e10i	−0.901549	−	0.632553i
$894$		0			0
$895$	−1.54065e10	+	2.66849e10i	−0.718330	+	1.24418i
$896$		0			0
$897$	5.59922e7	+	9.69814e7i	0.00259032	+	0.00448657i
$898$		0			0
$899$	−1.97371e10	−	3.41857e10i	−0.905994	−	1.56923i
$900$		0			0
$901$	7.65085e9			0.348476
$902$		0			0
$903$	−8.24352e8	−	1.42782e9i	−0.0372568	−	0.0645307i
$904$		0			0
$905$	1.40077e10			0.628197
$906$		0			0
$907$	8.70698e9	−	1.50809e10i	0.387474	−	0.671124i	−0.604635	−	0.796502i	$-0.706681\pi$
$907$	8.70698e9	−	1.50809e10i	0.387474	−	0.671124i	0.992109	+	0.125379i	$0.0400145\pi$
$908$		0			0
$909$	−1.82306e10	+	3.15763e10i	−0.805058	+	1.39440i
$910$		0			0
$911$	−6.25280e9			−0.274006			−0.137003	−	0.990571i	$-0.543747\pi$
$911$	−6.25280e9			−0.274006			−0.137003	+	0.990571i	$0.543747\pi$
$912$		0			0
$913$	−2.54767e10			−1.10789
$914$		0			0
$915$	−2.67812e8	+	4.63864e8i	−0.0115573	+	0.0200178i
$916$		0			0
$917$	1.91646e9	−	3.31940e9i	0.0820742	−	0.142157i
$918$		0			0
$919$	−2.35616e10			−1.00138			−0.500692	−	0.865625i	$-0.666921\pi$
$919$	−2.35616e10			−1.00138			−0.500692	+	0.865625i	$0.666921\pi$
$920$		0			0
$921$	−5.09471e7	−	8.82430e7i	−0.00214888	−	0.00372196i
$922$		0			0
$923$	−3.61670e9			−0.151393
$924$		0			0
$925$	5.84535e9	+	1.01244e10i	0.242837	+	0.420606i
$926$		0			0
$927$	−1.40629e10	−	2.43577e10i	−0.579824	−	1.00428i
$928$		0			0
$929$	1.25664e10	−	2.17656e10i	0.514227	−	0.890667i	−0.485637	−	0.874161i	$-0.661412\pi$
$929$	1.25664e10	−	2.17656e10i	0.514227	−	0.890667i	0.999864	−	0.0165065i	$-0.00525441\pi$
$930$		0			0
$931$	3.61851e10	−	1.68314e10i	1.46962	−	0.683592i
$932$		0			0
$933$	−5.06059e8	+	8.76520e8i	−0.0203993	+	0.0353326i
$934$		0			0
$935$	1.50103e10	+	2.59987e10i	0.600551	+	1.04019i
$936$		0			0
$937$	−4.80428e9	−	8.32126e9i	−0.190783	−	0.330446i	0.754727	−	0.656039i	$-0.227769\pi$
$937$	−4.80428e9	−	8.32126e9i	−0.190783	−	0.330446i	−0.945510	+	0.325593i	$0.894436\pi$
$938$		0			0
$939$	2.00964e9			0.0792116
$940$		0			0
$941$	2.30207e10	+	3.98730e10i	0.900648	+	1.55997i	0.826655	+	0.562709i	$0.190241\pi$
$941$	2.30207e10	+	3.98730e10i	0.900648	+	1.55997i	0.0739930	+	0.997259i	$0.476426\pi$
$942$		0			0
$943$	−5.34697e9			−0.207643
$944$		0			0
$945$	2.29855e9	−	3.98120e9i	0.0886017	−	0.153463i
$946$		0			0
$947$	7.97642e9	−	1.38156e10i	0.305199	−	0.528620i	−0.672107	−	0.740454i	$-0.734610\pi$
$947$	7.97642e9	−	1.38156e10i	0.305199	−	0.528620i	0.977306	+	0.211834i	$0.0679437\pi$
$948$		0			0
$949$	3.41954e9			0.129878
$950$		0			0
$951$	−7.89148e8			−0.0297527
$952$		0			0
$953$	−5.86372e9	+	1.01563e10i	−0.219456	+	0.380109i	−0.954642	−	0.297756i	$-0.903762\pi$
$953$	−5.86372e9	+	1.01563e10i	−0.219456	+	0.380109i	0.735186	+	0.677866i	$0.237095\pi$
$954$		0			0
$955$	−6.90492e9	+	1.19597e10i	−0.256535	+	0.444332i
$956$		0			0
$957$	3.14584e9			0.116023
$958$		0			0
$959$	−3.57584e9	−	6.19353e9i	−0.130922	−	0.226764i
$960$		0			0
$961$	6.23476e10			2.26615
$962$		0			0
$963$	−2.38665e10	−	4.13380e10i	−0.861186	−	1.49162i
$964$		0			0
$965$	1.62590e10	+	2.81615e10i	0.582437	+	1.00881i
$966$		0			0
$967$	3.66212e9	−	6.34298e9i	0.130239	−	0.225580i	−0.793530	−	0.608531i	$-0.791759\pi$
$967$	3.66212e9	−	6.34298e9i	0.130239	−	0.225580i	0.923769	+	0.382951i	$0.125092\pi$
$968$		0			0
$969$	−1.68263e9	−	1.18058e9i	−0.0594093	−	0.0416833i
$970$		0			0
$971$	−1.57337e10	+	2.72516e10i	−0.551525	+	0.955269i	0.446640	+	0.894714i	$0.352620\pi$
$971$	−1.57337e10	+	2.72516e10i	−0.551525	+	0.955269i	−0.998165	+	0.0605549i	$0.980713\pi$
$972$		0			0
$973$	1.65691e10	+	2.86984e10i	0.576637	+	0.998765i
$974$		0			0
$975$	5.84324e7	+	1.01208e8i	0.00201900	+	0.00349702i
$976$		0			0
$977$	1.22143e10			0.419024			0.209512	−	0.977806i	$-0.432812\pi$
$977$	1.22143e10			0.419024			0.209512	+	0.977806i	$0.432812\pi$
$978$		0			0
$979$	−3.61432e10	−	6.26018e10i	−1.23108	−	2.13230i
$980$		0			0
$981$	−3.42028e10			−1.15670
$982$		0			0
$983$	−8.79126e9	+	1.52269e10i	−0.295198	+	0.511298i	−0.975031	−	0.222069i	$-0.928719\pi$
$983$	−8.79126e9	+	1.52269e10i	−0.295198	+	0.511298i	0.679833	+	0.733367i	$0.262052\pi$
$984$		0			0
$985$	−5.56739e9	+	9.64300e9i	−0.185620	+	0.321503i
$986$		0			0
$987$	−3.91355e9			−0.129557
$988$		0			0
$989$	1.06238e10			0.349214
$990$		0			0
$991$	−2.78304e10	+	4.82036e10i	−0.908366	+	1.57334i	−0.0920329	+	0.995756i	$0.529336\pi$
$991$	−2.78304e10	+	4.82036e10i	−0.908366	+	1.57334i	−0.816334	+	0.577581i	$0.803997\pi$
$992$		0			0
$993$	−1.88375e9	+	3.26275e9i	−0.0610521	+	0.105745i
$994$		0			0
$995$	−2.67227e10			−0.860002
$996$		0			0
$997$	6.13955e9	+	1.06340e10i	0.196202	+	0.339832i	0.947294	−	0.320366i	$-0.103806\pi$
$997$	6.13955e9	+	1.06340e10i	0.196202	+	0.339832i	−0.751092	+	0.660198i	$0.770472\pi$
$998$		0			0
$999$	5.16235e9			0.163820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.e.a.49.6	yes	22
19.7	even	3	inner	76.8.e.a.45.6	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.e.a.45.6	✓	22	19.7	even	3	inner
76.8.e.a.49.6	yes	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.69911 - 2.94294i) q^{3} +(105.538 - 182.797i) q^{5} +1469.14 q^{7} +(1087.73 + 1884.00i) q^{9} +O(q^{10})\)	\(q+(1.69911 - 2.94294i) q^{3} +(105.538 - 182.797i) q^{5} +1469.14 q^{7} +(1087.73 + 1884.00i) q^{9} -7030.02 q^{11} +(512.187 + 887.134i) q^{13} +(-358.641 - 621.184i) q^{15} +(10115.7 - 17520.9i) q^{17} +(27108.5 - 12609.5i) q^{19} +(2496.22 - 4323.58i) q^{21} +(16084.9 + 27859.9i) q^{23} +(16785.9 + 29074.0i) q^{25} +14824.5 q^{27} +(-65841.6 - 114041. i) q^{29} +299767. q^{31} +(-11944.7 + 20688.9i) q^{33} +(155050. - 268555. i) q^{35} +348230. q^{37} +3481.04 q^{39} +(-83105.4 + 143943. i) q^{41} +(165120. - 285996. i) q^{43} +459186. q^{45} +(-391947. - 678872. i) q^{47} +1.33482e6 q^{49} +(-34375.2 - 59539.6i) q^{51} +(189084. + 327503. i) q^{53} +(-741935. + 1.28507e6i) q^{55} +(8951.38 - 101204. i) q^{57} +(699619. - 1.21178e6i) q^{59} +(-373370. - 646696. i) q^{61} +(1.59802e6 + 2.76785e6i) q^{63} +216221. q^{65} +(-2.45164e6 - 4.24636e6i) q^{67} +109320. q^{69} +(-1.76532e6 + 3.05763e6i) q^{71} +(1.66909e6 - 2.89095e6i) q^{73} +114084. q^{75} -1.03281e7 q^{77} +(-3.74009e6 + 6.47802e6i) q^{79} +(-2.35367e6 + 4.07667e6i) q^{81} +3.62399e6 q^{83} +(-2.13518e6 - 3.69824e6i) q^{85} -447487. q^{87} +(5.14127e6 + 8.90494e6i) q^{89} +(752473. + 1.30332e6i) q^{91} +(509336. - 882196. i) q^{93} +(556006. - 6.28615e6i) q^{95} +(1.88121e6 - 3.25836e6i) q^{97} +(-7.64673e6 - 1.32445e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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