LMFDB - Embedded newform 15.8.b.a.4.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,8,Mod(4,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("15.4");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	15.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.68577538226$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000 $ x^8 + 162x^6 + 7361x^4 + 87300*x^2 + 160000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{12}\cdot 5^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4.3
Root			$-3.83609i$ of defining polynomial
Character	$\chi$	$=$	15.4
Dual form			15.8.b.a.4.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.75511i q^{2} +27.0000i q^{3} +51.3480 q^{4} +(231.605 - 156.474i) q^{5} +236.388 q^{6} -536.160i q^{7} -1570.21i q^{8} -729.000 q^{9} +O(q^{10})$	\(q-8.75511i q^{2} +27.0000i q^{3} +51.3480 q^{4} +(231.605 - 156.474i) q^{5} +236.388 q^{6} -536.160i q^{7} -1570.21i q^{8} -729.000 q^{9} +(-1369.95 - 2027.73i) q^{10} +5830.22 q^{11} +1386.40i q^{12} +5815.44i q^{13} -4694.14 q^{14} +(4224.81 + 6253.33i) q^{15} -7174.84 q^{16} +9476.54i q^{17} +6382.48i q^{18} -52968.3 q^{19} +(11892.4 - 8034.65i) q^{20} +14476.3 q^{21} -51044.3i q^{22} +74966.7i q^{23} +42395.7 q^{24} +(29156.5 - 72480.4i) q^{25} +50914.8 q^{26} -19683.0i q^{27} -27530.7i q^{28} +22107.7 q^{29} +(54748.6 - 36988.7i) q^{30} -158843. q^{31} -138171. i q^{32} +157416. i q^{33} +82968.2 q^{34} +(-83895.3 - 124177. i) q^{35} -37432.7 q^{36} +264554. i q^{37} +463743. i q^{38} -157017. q^{39} +(-245698. - 363668. i) q^{40} -77139.7 q^{41} -126742. i q^{42} +845105. i q^{43} +299370. q^{44} +(-168840. + 114070. i) q^{45} +656342. q^{46} -893848. i q^{47} -193721. i q^{48} +536076. q^{49} +(-634575. - 255268. i) q^{50} -255867. q^{51} +298611. i q^{52} -153425. i q^{53} -172327. q^{54} +(1.35031e6 - 912281. i) q^{55} -841885. q^{56} -1.43014e6i q^{57} -193555. i q^{58} +155813. q^{59} +(216935. + 321096. i) q^{60} +721961. q^{61} +1.39068e6i q^{62} +390861. i q^{63} -2.12808e6 q^{64} +(909967. + 1.34688e6i) q^{65} +1.37820e6 q^{66} +2.87480e6i q^{67} +486601. i q^{68} -2.02410e6 q^{69} +(-1.08718e6 + 734513. i) q^{70} -1.89425e6 q^{71} +1.14468e6i q^{72} -4.67419e6i q^{73} +2.31620e6 q^{74} +(1.95697e6 + 787225. i) q^{75} -2.71981e6 q^{76} -3.12593e6i q^{77} +1.37470e6i q^{78} -2.70860e6 q^{79} +(-1.66173e6 + 1.12268e6i) q^{80} +531441. q^{81} +675367. i q^{82} -4.54782e6i q^{83} +743330. q^{84} +(1.48284e6 + 2.19481e6i) q^{85} +7.39899e6 q^{86} +596908. i q^{87} -9.15469e6i q^{88} +1.17121e7 q^{89} +(998695. + 1.47821e6i) q^{90} +3.11800e6 q^{91} +3.84939e6i q^{92} -4.28875e6i q^{93} -7.82574e6 q^{94} +(-1.22677e7 + 8.28818e6i) q^{95} +3.73061e6 q^{96} -4.83792e6i q^{97} -4.69340e6i q^{98} -4.25023e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10}) $ 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9	$ 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9} - 6686 q^{10} + 10752 q^{11} - 13524 q^{14} - 17496 q^{15} + 86530 q^{16} - 30464 q^{19} + 87444 q^{20} + 64152 q^{21} - 110322 q^{24} + 127616 q^{25} - 793524 q^{26} + 240072 q^{29} - 172044 q^{30} + 233728 q^{31} + 184748 q^{34} + 593520 q^{35} + 485514 q^{36} - 454896 q^{39} - 1147102 q^{40} + 507648 q^{41} + 2578572 q^{44} + 323676 q^{45} + 662408 q^{46} - 3267160 q^{49} - 5117736 q^{50} + 264384 q^{51} - 354294 q^{54} + 3525696 q^{55} + 705660 q^{56} + 1091424 q^{59} + 5307606 q^{60} - 6433520 q^{61} - 568594 q^{64} - 2555592 q^{65} - 2382372 q^{66} - 5940864 q^{69} + 12097800 q^{70} - 1381824 q^{71} + 23961276 q^{74} + 7768224 q^{75} + 13115664 q^{76} - 14380160 q^{79} - 31251876 q^{80} + 4251528 q^{81} - 36423756 q^{84} - 1452008 q^{85} - 19837608 q^{86} + 45778896 q^{89} + 4874094 q^{90} + 24075648 q^{91} - 45728896 q^{94} - 25774656 q^{95} + 33586002 q^{96} - 7838208 q^{99}+O(q^{100}) $ 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9 - 6686 * q^10 + 10752 * q^11 - 13524 * q^14 - 17496 * q^15 + 86530 * q^16 - 30464 * q^19 + 87444 * q^20 + 64152 * q^21 - 110322 * q^24 + 127616 * q^25 - 793524 * q^26 + 240072 * q^29 - 172044 * q^30 + 233728 * q^31 + 184748 * q^34 + 593520 * q^35 + 485514 * q^36 - 454896 * q^39 - 1147102 * q^40 + 507648 * q^41 + 2578572 * q^44 + 323676 * q^45 + 662408 * q^46 - 3267160 * q^49 - 5117736 * q^50 + 264384 * q^51 - 354294 * q^54 + 3525696 * q^55 + 705660 * q^56 + 1091424 * q^59 + 5307606 * q^60 - 6433520 * q^61 - 568594 * q^64 - 2555592 * q^65 - 2382372 * q^66 - 5940864 * q^69 + 12097800 * q^70 - 1381824 * q^71 + 23961276 * q^74 + 7768224 * q^75 + 13115664 * q^76 - 14380160 * q^79 - 31251876 * q^80 + 4251528 * q^81 - 36423756 * q^84 - 1452008 * q^85 - 19837608 * q^86 + 45778896 * q^89 + 4874094 * q^90 + 24075648 * q^91 - 45728896 * q^94 - 25774656 * q^95 + 33586002 * q^96 - 7838208 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	8.75511i		−	0.773850i	−0.922111	−	0.386925i	$-0.873537\pi$
$2$		−	8.75511i		−	0.773850i	0.922111	−	0.386925i	$-0.126463\pi$
$3$			27.0000i			0.577350i
$3$			27.0000i			0.577350i
$4$	51.3480			0.401156
$5$	231.605	−	156.474i	0.828614	−	0.559820i
$5$	231.605	−	156.474i	0.828614	−	0.559820i
$6$	236.388			0.446783
$7$		−	536.160i		−	0.590814i	−0.955371	−	0.295407i	$-0.904545\pi$
$7$		−	536.160i		−	0.590814i	0.955371	−	0.295407i	$-0.0954553\pi$
$8$		−	1570.21i		−	1.08428i
$9$	−729.000			−0.333333
$10$	−1369.95	−	2027.73i	−0.433217	−	0.641223i
$11$	5830.22			1.32072			0.660360	−	0.750949i	$-0.270404\pi$
$11$	5830.22			1.32072			0.660360	+	0.750949i	$0.270404\pi$
$12$			1386.40i			0.231608i
$13$			5815.44i			0.734143i	0.930193	+	0.367071i	$0.119640\pi$
$13$			5815.44i			0.734143i	−0.930193	+	0.367071i	$0.880360\pi$
$14$	−4694.14			−0.457202
$15$	4224.81	+	6253.33i	0.323212	+	0.478401i
$16$	−7174.84			−0.437918
$17$			9476.54i			0.467820i	0.972258	+	0.233910i	$0.0751521\pi$
$17$			9476.54i			0.467820i	−0.972258	+	0.233910i	$0.924848\pi$
$18$			6382.48i			0.257950i
$19$	−52968.3			−1.77165			−0.885825	−	0.464019i	$-0.846407\pi$
$19$	−52968.3			−1.77165			−0.885825	+	0.464019i	$0.846407\pi$
$20$	11892.4	−	8034.65i	0.332404	−	0.224575i
$21$	14476.3			0.341107
$22$		−	51044.3i		−	1.02204i
$23$			74966.7i			1.28476i	0.766387	+	0.642379i	$0.222052\pi$
$23$			74966.7i			1.28476i	−0.766387	+	0.642379i	$0.777948\pi$
$24$	42395.7			0.626012
$25$	29156.5	−	72480.4i	0.373203	−	0.927750i
$26$	50914.8			0.568117
$27$		−	19683.0i		−	0.192450i
$28$		−	27530.7i		−	0.237009i
$29$	22107.7			0.168326			0.0841628	−	0.996452i	$-0.473178\pi$
$29$	22107.7			0.168326			0.0841628	+	0.996452i	$0.473178\pi$
$30$	54748.6	−	36988.7i	0.370210	−	0.250118i
$31$	−158843.			−0.957637			−0.478819	−	0.877914i	$-0.658935\pi$
$31$	−158843.			−0.957637			−0.478819	+	0.877914i	$0.658935\pi$
$32$		−	138171.i		−	0.745402i
$33$			157416.i			0.762518i
$34$	82968.2			0.362022
$35$	−83895.3	−	124177.i	−0.330750	−	0.489557i
$36$	−37432.7			−0.133719
$37$			264554.i			0.858636i	0.903153	+	0.429318i	$0.141246\pi$
$37$			264554.i			0.858636i	−0.903153	+	0.429318i	$0.858754\pi$
$38$			463743.i			1.37099i
$39$	−157017.			−0.423858
$40$	−245698.	−	363668.i	−0.607004	−	0.898454i
$41$	−77139.7			−0.174797			−0.0873986	−	0.996173i	$-0.527855\pi$
$41$	−77139.7			−0.174797			−0.0873986	+	0.996173i	$0.527855\pi$
$42$		−	126742.i		−	0.263966i
$43$			845105.i			1.62096i	0.585769	+	0.810478i	$0.300793\pi$
$43$			845105.i			1.62096i	−0.585769	+	0.810478i	$0.699207\pi$
$44$	299370.			0.529815
$45$	−168840.	+	114070.i	−0.276205	+	0.186607i
$46$	656342.			0.994209
$47$		−	893848.i		−	1.25580i	−0.778293	−	0.627901i	$-0.783914\pi$
$47$		−	893848.i		−	1.25580i	0.778293	−	0.627901i	$-0.216086\pi$
$48$		−	193721.i		−	0.252832i
$49$	536076.			0.650938
$50$	−634575.	−	255268.i	−0.717939	−	0.288803i
$51$	−255867.			−0.270096
$52$			298611.i			0.294506i
$53$		−	153425.i		−	0.141557i	−0.997492	−	0.0707784i	$-0.977452\pi$
$53$		−	153425.i		−	0.141557i	0.997492	−	0.0707784i	$-0.0225483\pi$
$54$	−172327.			−0.148928
$55$	1.35031e6	−	912281.i	1.09437	−	0.739366i
$56$	−841885.			−0.640611
$57$		−	1.43014e6i		−	1.02286i
$58$		−	193555.i		−	0.130259i
$59$	155813.			0.0987690			0.0493845	−	0.998780i	$-0.484274\pi$
$59$	155813.			0.0987690			0.0493845	+	0.998780i	$0.484274\pi$
$60$	216935.	+	321096.i	0.129659	+	0.191913i
$61$	721961.			0.407248			0.203624	−	0.979049i	$-0.434728\pi$
$61$	721961.			0.407248			0.203624	+	0.979049i	$0.434728\pi$
$62$			1.39068e6i			0.741068i
$63$			390861.i			0.196938i
$64$	−2.12808e6			−1.01475
$65$	909967.	+	1.34688e6i	0.410988	+	0.608321i
$66$	1.37820e6			0.590075
$67$			2.87480e6i			1.16774i	0.811848	+	0.583869i	$0.198462\pi$
$67$			2.87480e6i			1.16774i	−0.811848	+	0.583869i	$0.801538\pi$
$68$			486601.i			0.187669i
$69$	−2.02410e6			−0.741755
$70$	−1.08718e6	+	734513.i	−0.378844	+	0.255951i
$71$	−1.89425e6			−0.628107			−0.314053	−	0.949405i	$-0.601687\pi$
$71$	−1.89425e6			−0.628107			−0.314053	+	0.949405i	$0.601687\pi$
$72$			1.14468e6i			0.361428i
$73$		−	4.67419e6i		−	1.40630i	−0.711044	−	0.703148i	$-0.751777\pi$
$73$		−	4.67419e6i		−	1.40630i	0.711044	−	0.703148i	$-0.248223\pi$
$74$	2.31620e6			0.664455
$75$	1.95697e6	+	787225.i	0.535637	+	0.215469i
$76$	−2.71981e6			−0.710708
$77$		−	3.12593e6i		−	0.780300i
$78$			1.37470e6i			0.328002i
$79$	−2.70860e6			−0.618087			−0.309044	−	0.951048i	$-0.600009\pi$
$79$	−2.70860e6			−0.618087			−0.309044	+	0.951048i	$0.600009\pi$
$80$	−1.66173e6	+	1.12268e6i	−0.362865	+	0.245155i
$81$	531441.			0.111111
$82$			675367.i			0.135267i
$83$		−	4.54782e6i		−	0.873031i	−0.899697	−	0.436515i	$-0.856212\pi$
$83$		−	4.54782e6i		−	0.873031i	0.899697	−	0.436515i	$-0.143788\pi$
$84$	743330.			0.136837
$85$	1.48284e6	+	2.19481e6i	0.261895	+	0.387642i
$86$	7.39899e6			1.25438
$87$			596908.i			0.0971829i
$88$		−	9.15469e6i		−	1.43204i
$89$	1.17121e7			1.76104			0.880518	−	0.474012i	$-0.157195\pi$
$89$	1.17121e7			1.76104			0.880518	+	0.474012i	$0.157195\pi$
$90$	998695.	+	1.47821e6i	0.144406	+	0.213741i
$91$	3.11800e6			0.433742
$92$			3.84939e6i			0.515388i
$93$		−	4.28875e6i		−	0.552892i
$94$	−7.82574e6			−0.971802
$95$	−1.22677e7	+	8.28818e6i	−1.46801	+	0.991805i
$96$	3.73061e6			0.430358
$97$		−	4.83792e6i		−	0.538217i	−0.963110	−	0.269109i	$-0.913271\pi$
$97$		−	4.83792e6i		−	0.538217i	0.963110	−	0.269109i	$-0.0867291\pi$
$98$		−	4.69340e6i		−	0.503729i
$99$	−4.25023e6			−0.440240
$100$	1.49713e6	−	3.72172e6i	0.149713	−	0.372172i
$101$	−1.95309e7			−1.88624			−0.943121	−	0.332448i	$-0.892125\pi$
$101$	−1.95309e7			−1.88624			−0.943121	+	0.332448i	$0.892125\pi$
$102$			2.24014e6i			0.209014i
$103$			1.84497e7i			1.66364i	0.555044	+	0.831821i	$0.312701\pi$
$103$			1.84497e7i			1.66364i	−0.555044	+	0.831821i	$0.687299\pi$
$104$	9.13147e6			0.796020
$105$	3.35278e6	−	2.26517e6i	0.282646	−	0.190958i
$106$	−1.34325e6			−0.109544
$107$			1.06308e7i			0.838923i	0.907773	+	0.419462i	$0.137781\pi$
$107$			1.06308e7i			0.838923i	−0.907773	+	0.419462i	$0.862219\pi$
$108$		−	1.01068e6i		−	0.0772025i
$109$	1.26870e7			0.938351			0.469176	−	0.883105i	$-0.344551\pi$
$109$	1.26870e7			0.938351			0.469176	+	0.883105i	$0.344551\pi$
$110$	−7.98712e6	−	1.18221e7i	−0.572158	−	0.846876i
$111$	−7.14297e6			−0.495734
$112$			3.84686e6i			0.258728i
$113$			6.54277e6i			0.426567i	0.976990	+	0.213284i	$0.0684158\pi$
$113$			6.54277e6i			0.426567i	−0.976990	+	0.213284i	$0.931584\pi$
$114$	−1.25211e7			−0.791543
$115$	1.17304e7	+	1.73626e7i	0.719233	+	1.06457i
$116$	1.13519e6			0.0675249
$117$		−	4.23945e6i		−	0.244714i
$118$		−	1.36416e6i		−	0.0764324i
$119$	5.08094e6			0.276395
$120$	9.81905e6	−	6.63385e6i	0.518722	−	0.350454i
$121$	1.45043e7			0.744301
$122$		−	6.32085e6i		−	0.315149i
$123$		−	2.08277e6i		−	0.100919i
$124$	−8.15625e6			−0.384162
$125$	−4.58856e6	−	2.13491e7i	−0.210132	−	0.977673i
$126$	3.42203e6			0.152401
$127$		−	3.28179e7i		−	1.42167i	−0.703360	−	0.710834i	$-0.748318\pi$
$127$		−	3.28179e7i		−	1.42167i	0.703360	−	0.710834i	$-0.251682\pi$
$128$			945743.i			0.0398601i
$129$	−2.28178e7			−0.935860
$130$	1.17921e7	−	7.96687e6i	0.470749	−	0.318043i
$131$	2.08304e6			0.0809557			0.0404778	−	0.999180i	$-0.487112\pi$
$131$	2.08304e6			0.0809557			0.0404778	+	0.999180i	$0.487112\pi$
$132$			8.08299e6i			0.305889i
$133$			2.83994e7i			1.04672i
$134$	2.51692e7			0.903654
$135$	−3.07989e6	−	4.55868e6i	−0.107737	−	0.159467i
$136$	1.48802e7			0.507250
$137$		−	4.20215e7i		−	1.39621i	−0.715997	−	0.698103i	$-0.754028\pi$
$137$		−	4.20215e7i		−	1.39621i	0.715997	−	0.698103i	$-0.245972\pi$
$138$			1.77212e7i			0.574007i
$139$	−4.89465e7			−1.54586			−0.772930	−	0.634492i	$-0.781209\pi$
$139$	−4.89465e7			−1.54586			−0.772930	+	0.634492i	$0.781209\pi$
$140$	−4.30786e6	−	6.37624e6i	−0.132682	−	0.196389i
$141$	2.41339e7			0.725037
$142$			1.65844e7i			0.486060i
$143$			3.39053e7i			0.969597i
$144$	5.23046e6			0.145973
$145$	5.12025e6	−	3.45929e6i	0.139477	−	0.0942321i
$146$	−4.09231e7			−1.08826
$147$			1.44740e7i			0.375819i
$148$			1.35843e7i			0.344447i
$149$	3.77290e7			0.934379			0.467189	−	0.884157i	$-0.345267\pi$
$149$	3.77290e7			0.934379			0.467189	+	0.884157i	$0.345267\pi$
$150$	6.89224e6	−	1.71335e7i	0.166741	−	0.414502i
$151$	−6.39975e6			−0.151267			−0.0756334	−	0.997136i	$-0.524098\pi$
$151$	−6.39975e6			−0.151267			−0.0756334	+	0.997136i	$0.524098\pi$
$152$			8.31714e7i			1.92097i
$153$		−	6.90840e6i		−	0.155940i
$154$	−2.73679e7			−0.603836
$155$	−3.67887e7	+	2.48548e7i	−0.793512	+	0.536105i
$156$	−8.06249e6			−0.170033
$157$		−	6.53032e7i		−	1.34675i	−0.739303	−	0.673373i	$-0.764845\pi$
$157$		−	6.53032e7i		−	1.34675i	0.739303	−	0.673373i	$-0.235155\pi$
$158$			2.37141e7i			0.478307i
$159$	4.14248e6			0.0817279
$160$	−2.16202e7	−	3.20010e7i	−0.417291	−	0.617651i
$161$	4.01941e7			0.759053
$162$		−	4.65283e6i		−	0.0859833i
$163$		−	3.69087e6i		−	0.0667532i	−0.999443	−	0.0333766i	$-0.989374\pi$
$163$		−	3.69087e6i		−	0.0667532i	0.999443	−	0.0333766i	$-0.0106261\pi$
$164$	−3.96097e6			−0.0701210
$165$	2.46316e7	+	3.64583e7i	0.426873	+	0.631833i
$166$	−3.98166e7			−0.675595
$167$		−	3.63073e7i		−	0.603235i	−0.953429	−	0.301618i	$-0.902473\pi$
$167$		−	3.63073e7i		−	0.603235i	0.953429	−	0.301618i	$-0.0975266\pi$
$168$		−	2.27309e7i		−	0.369857i
$169$	2.89292e7			0.461034
$170$	1.92158e7	−	1.29824e7i	0.299977	−	0.202667i
$171$	3.86139e7			0.590550
$172$			4.33945e7i			0.650256i
$173$		−	6.98352e7i		−	1.02545i	−0.858554	−	0.512724i	$-0.828637\pi$
$173$		−	6.98352e7i		−	1.02545i	0.858554	−	0.512724i	$-0.171363\pi$
$174$	5.22599e6			0.0752050
$175$	−3.88611e7	−	1.56325e7i	−0.548128	−	0.220494i
$176$	−4.18309e7			−0.578367
$177$			4.20694e6i			0.0570243i
$178$		−	1.02540e8i		−	1.36278i
$179$	7.44273e7			0.969944			0.484972	−	0.874530i	$-0.338830\pi$
$179$	7.44273e7			0.969944			0.484972	+	0.874530i	$0.338830\pi$
$180$	−8.66958e6	+	5.85726e6i	−0.110801	+	0.0748584i
$181$	−2.23241e7			−0.279833			−0.139917	−	0.990163i	$-0.544683\pi$
$181$	−2.23241e7			−0.279833			−0.139917	+	0.990163i	$0.544683\pi$
$182$		−	2.72985e7i		−	0.335651i
$183$			1.94929e7i			0.235125i
$184$	1.17714e8			1.39304
$185$	4.13960e7	+	6.12720e7i	0.480682	+	0.711478i
$186$	−3.75485e7			−0.427856
$187$			5.52503e7i			0.617859i
$188$		−	4.58973e7i		−	0.503772i
$189$	−1.05532e7			−0.113702
$190$	7.25640e7	+	1.07405e8i	0.767509	+	1.13602i
$191$	−8.28691e6			−0.0860550			−0.0430275	−	0.999074i	$-0.513700\pi$
$191$	−8.28691e6			−0.0860550			−0.0430275	+	0.999074i	$0.513700\pi$
$192$		−	5.74581e7i		−	0.585865i
$193$		−	1.26638e7i		−	0.126798i	−0.997988	−	0.0633990i	$-0.979806\pi$
$193$		−	1.26638e7i		−	0.126798i	0.997988	−	0.0633990i	$-0.0201941\pi$
$194$	−4.23565e7			−0.416500
$195$	−3.63658e7	+	2.45691e7i	−0.351214	+	0.237284i
$196$	2.75264e7			0.261128
$197$			2.35528e7i			0.219488i	0.993960	+	0.109744i	$0.0350031\pi$
$197$			2.35528e7i			0.219488i	−0.993960	+	0.109744i	$0.964997\pi$
$198$			3.72113e7i			0.340680i
$199$	−1.39990e8			−1.25925			−0.629626	−	0.776898i	$-0.716792\pi$
$199$	−1.39990e8			−1.25925			−0.629626	+	0.776898i	$0.716792\pi$
$200$	−1.13810e8	−	4.57819e7i	−1.00594	−	0.404658i
$201$	−7.76195e7			−0.674194
$202$			1.70995e8i			1.45967i
$203$		−	1.18533e7i		−	0.0994493i
$204$	−1.31382e7			−0.108351
$205$	−1.78659e7	+	1.20704e7i	−0.144839	+	0.0978550i
$206$	1.61530e8			1.28741
$207$		−	5.46507e7i		−	0.428252i
$208$		−	4.17248e7i		−	0.321494i
$209$	−3.08817e8			−2.33985
$210$	−1.98319e7	−	2.93540e7i	−0.147773	−	0.218726i
$211$	1.10945e8			0.813055			0.406527	−	0.913639i	$-0.366740\pi$
$211$	1.10945e8			0.813055			0.406527	+	0.913639i	$0.366740\pi$
$212$		−	7.87807e6i		−	0.0567864i
$213$		−	5.11448e7i		−	0.362638i
$214$	9.30737e7			0.649201
$215$	1.32237e8	+	1.95730e8i	0.907444	+	1.34315i
$216$	−3.09065e7			−0.208671
$217$			8.51650e7i			0.565786i
$218$		−	1.11076e8i		−	0.726143i
$219$	1.26203e8			0.811925
$220$	6.93355e7	−	4.68438e7i	0.439012	−	0.296601i
$221$	−5.51102e7			−0.343446
$222$			6.25375e7i			0.383623i
$223$		−	4.58840e7i		−	0.277073i	−0.990357	−	0.138536i	$-0.955760\pi$
$223$		−	4.58840e7i		−	0.277073i	0.990357	−	0.138536i	$-0.0442398\pi$
$224$	−7.40815e7			−0.440394
$225$	−2.12551e7	+	5.28382e7i	−0.124401	+	0.309250i
$226$	5.72827e7			0.330099
$227$		−	6.60981e7i		−	0.375058i	−0.982259	−	0.187529i	$-0.939952\pi$
$227$		−	6.60981e7i		−	0.375058i	0.982259	−	0.187529i	$-0.0600479\pi$
$228$		−	7.34349e7i		−	0.410328i
$229$	1.41509e8			0.778684			0.389342	−	0.921093i	$-0.372702\pi$
$229$	1.41509e8			0.778684			0.389342	+	0.921093i	$0.372702\pi$
$230$	1.52012e8	−	1.02701e8i	0.823816	−	0.556578i
$231$	8.44001e7			0.450507
$232$		−	3.47138e7i		−	0.182513i
$233$			1.55558e8i			0.805651i	0.915277	+	0.402826i	$0.131972\pi$
$233$			1.55558e8i			0.805651i	−0.915277	+	0.402826i	$0.868028\pi$
$234$	−3.71169e7			−0.189372
$235$	−1.39864e8	−	2.07019e8i	−0.703023	−	1.04057i
$236$	8.00066e6			0.0396218
$237$		−	7.31322e7i		−	0.356853i
$238$		−	4.44842e7i		−	0.213888i
$239$	8.55969e6			0.0405570			0.0202785	−	0.999794i	$-0.493545\pi$
$239$	8.55969e6			0.0405570			0.0202785	+	0.999794i	$0.493545\pi$
$240$	−3.03124e7	−	4.48666e7i	−0.141540	−	0.209500i
$241$	−2.16028e8			−0.994147			−0.497073	−	0.867708i	$-0.665592\pi$
$241$	−2.16028e8			−0.994147			−0.497073	+	0.867708i	$0.665592\pi$
$242$		−	1.26987e8i		−	0.575978i
$243$			1.43489e7i			0.0641500i
$244$	3.70712e7			0.163370
$245$	1.24158e8	−	8.38822e7i	0.539377	−	0.364408i
$246$	−1.82349e7			−0.0780963
$247$		−	3.08034e8i		−	1.30064i
$248$			2.49417e8i			1.03835i
$249$	1.22791e8			0.504045
$250$	−1.86913e8	+	4.01734e7i	−0.756572	+	0.162610i
$251$	7.51840e7			0.300101			0.150051	−	0.988678i	$-0.452056\pi$
$251$	7.51840e7			0.300101			0.150051	+	0.988678i	$0.452056\pi$
$252$			2.00699e7i			0.0790029i
$253$			4.37073e8i			1.69680i
$254$	−2.87325e8			−1.10016
$255$	−5.92599e7	+	4.00366e7i	−0.223805	+	0.151205i
$256$	−2.64114e8			−0.983901
$257$			2.50348e8i			0.919981i	0.887924	+	0.459991i	$0.152147\pi$
$257$			2.50348e8i			0.919981i	−0.887924	+	0.459991i	$0.847853\pi$
$258$			1.99773e8i			0.724215i
$259$	1.41843e8			0.507294
$260$	4.67250e7	+	6.91597e7i	0.164870	+	0.244032i
$261$	−1.61165e7			−0.0561086
$262$		−	1.82372e7i		−	0.0626476i
$263$		−	2.25906e8i		−	0.765743i	−0.923802	−	0.382872i	$-0.874935\pi$
$263$		−	2.25906e8i		−	0.765743i	0.923802	−	0.382872i	$-0.125065\pi$
$264$	2.47177e8			0.826787
$265$	−2.40071e7	−	3.55340e7i	−0.0792464	−	0.117296i
$266$	2.48640e8			0.810002
$267$			3.16226e8i			1.01674i
$268$			1.47615e8i			0.468445i
$269$	2.55932e8			0.801662			0.400831	−	0.916152i	$-0.368721\pi$
$269$	2.55932e8			0.801662			0.400831	+	0.916152i	$0.368721\pi$
$270$	−3.99117e7	+	2.69648e7i	−0.123403	+	0.0833726i
$271$	−1.62476e8			−0.495903			−0.247951	−	0.968772i	$-0.579757\pi$
$271$	−1.62476e8			−0.495903			−0.247951	+	0.968772i	$0.579757\pi$
$272$		−	6.79927e7i		−	0.204867i
$273$			8.41861e7i			0.250421i
$274$	−3.67903e8			−1.08045
$275$	1.69989e8	−	4.22577e8i	0.492897	−	1.22530i
$276$	−1.03934e8			−0.297560
$277$		−	2.30349e8i		−	0.651190i	−0.945509	−	0.325595i	$-0.894435\pi$
$277$		−	2.30349e8i		−	0.651190i	0.945509	−	0.325595i	$-0.105565\pi$
$278$			4.28532e8i			1.19626i
$279$	1.15796e8			0.319212
$280$	−1.94984e8	+	1.31733e8i	−0.530819	+	0.358627i
$281$	−4.07719e8			−1.09620			−0.548098	−	0.836414i	$-0.684648\pi$
$281$	−4.07719e8			−1.09620			−0.548098	+	0.836414i	$0.684648\pi$
$282$		−	2.11295e8i		−	0.561070i
$283$		−	2.04032e8i		−	0.535113i	−0.963542	−	0.267556i	$-0.913784\pi$
$283$		−	2.04032e8i		−	0.535113i	0.963542	−	0.267556i	$-0.0862162\pi$
$284$	−9.72660e7			−0.251969
$285$	−2.23781e8	−	3.31228e8i	−0.572619	−	0.847559i
$286$	2.96845e8			0.750323
$287$			4.13592e7i			0.103273i
$288$			1.00726e8i			0.248467i
$289$	3.20534e8			0.781145
$290$	−3.02865e7	−	4.48283e7i	−0.0729215	−	0.107934i
$291$	1.30624e8			0.310740
$292$		−	2.40010e8i		−	0.564144i
$293$		−	9.60811e6i		−	0.0223152i	−0.999938	−	0.0111576i	$-0.996448\pi$
$293$		−	9.60811e6i		−	0.0223152i	0.999938	−	0.0111576i	$-0.00355165\pi$
$294$	1.26722e8			0.290828
$295$	3.60869e7	−	2.43807e7i	0.0818414	−	0.0552929i
$296$	4.15407e8			0.931006
$297$		−	1.14756e8i		−	0.254173i
$298$		−	3.30321e8i		−	0.723069i
$299$	−4.35964e8			−0.943195
$300$	1.00487e8	+	4.04224e7i	0.214874	+	0.0864366i
$301$	4.53112e8			0.957684
$302$			5.60305e7i			0.117058i
$303$		−	5.27335e8i		−	1.08902i
$304$	3.80039e8			0.775837
$305$	1.67210e8	−	1.12968e8i	0.337452	−	0.227986i
$306$	−6.04838e7			−0.120674
$307$			1.74442e8i			0.344085i	0.985090	+	0.172043i	$0.0550367\pi$
$307$			1.74442e8i			0.344085i	−0.985090	+	0.172043i	$0.944963\pi$
$308$		−	1.60510e8i		−	0.313022i
$309$	−4.98143e8			−0.960504
$310$	2.17607e8	+	3.22089e8i	0.414865	+	0.614059i
$311$	1.93660e8			0.365072			0.182536	−	0.983199i	$-0.441569\pi$
$311$	1.93660e8			0.365072			0.182536	+	0.983199i	$0.441569\pi$
$312$			2.46550e8i			0.459582i
$313$			8.09094e8i			1.49140i	0.666282	+	0.745700i	$0.267885\pi$
$313$			8.09094e8i			1.49140i	−0.666282	+	0.745700i	$0.732115\pi$
$314$	−5.71737e8			−1.04218
$315$	6.11597e7	+	9.05251e7i	0.110250	+	0.163186i
$316$	−1.39081e8			−0.247950
$317$			2.03639e8i			0.359049i	0.983754	+	0.179524i	$0.0574559\pi$
$317$			2.03639e8i			0.359049i	−0.983754	+	0.179524i	$0.942544\pi$
$318$		−	3.62679e7i		−	0.0632451i
$319$	1.28893e8			0.222311
$320$	−4.92873e8	+	3.32990e8i	−0.840834	+	0.568076i
$321$	−2.87031e8			−0.484352
$322$		−	3.51904e8i		−	0.587393i
$323$		−	5.01956e8i		−	0.828813i
$324$	2.72884e7			0.0445729
$325$	4.21505e8	+	1.69558e8i	0.681101	+	0.273984i
$326$	−3.23140e7			−0.0516569
$327$			3.42548e8i			0.541757i
$328$			1.21126e8i			0.189530i
$329$	−4.79245e8			−0.741946
$330$	3.19196e8	−	2.15652e8i	0.488944	−	0.330336i
$331$	−2.04595e8			−0.310097			−0.155049	−	0.987907i	$-0.549553\pi$
$331$	−2.04595e8			−0.310097			−0.155049	+	0.987907i	$0.549553\pi$
$332$		−	2.33521e8i		−	0.350222i
$333$		−	1.92860e8i		−	0.286212i
$334$	−3.17875e8			−0.466813
$335$	4.49832e8	+	6.65816e8i	0.653723	+	0.967604i
$336$	−1.03865e8			−0.149377
$337$			1.27274e9i			1.81149i	0.423823	+	0.905745i	$0.360688\pi$
$337$			1.27274e9i			1.81149i	−0.423823	+	0.905745i	$0.639312\pi$
$338$		−	2.53279e8i		−	0.356771i
$339$	−1.76655e8			−0.246279
$340$	7.61406e7	+	1.12699e8i	0.105061	+	0.155505i
$341$	−9.26088e8			−1.26477
$342$		−	3.38069e8i		−	0.456997i
$343$		−	7.28973e8i		−	0.975398i
$344$	1.32699e9			1.75758
$345$	−4.68791e8	+	3.16720e8i	−0.614629	+	0.415249i
$346$	−6.11416e8			−0.793542
$347$			1.14893e8i			0.147618i	0.997272	+	0.0738091i	$0.0235156\pi$
$347$			1.14893e8i			0.147618i	−0.997272	+	0.0738091i	$0.976484\pi$
$348$			3.06500e7i			0.0389855i
$349$	−4.53912e8			−0.571587			−0.285794	−	0.958291i	$-0.592257\pi$
$349$	−4.53912e8			−0.571587			−0.285794	+	0.958291i	$0.592257\pi$
$350$	−1.36865e8	+	3.40233e8i	−0.170629	+	0.424169i
$351$	1.14465e8			0.141286
$352$		−	8.05565e8i		−	0.984467i
$353$		−	8.07883e8i		−	0.977546i	−0.872411	−	0.488773i	$-0.837445\pi$
$353$		−	8.07883e8i		−	0.977546i	0.872411	−	0.488773i	$-0.162555\pi$
$354$	3.68322e7			0.0441283
$355$	−4.38718e8	+	2.96402e8i	−0.520458	+	0.351627i
$356$	6.01391e8			0.706451
$357$			1.37185e8i			0.159577i
$358$		−	6.51620e8i		−	0.750591i
$359$	1.05589e9			1.20445			0.602225	−	0.798327i	$-0.294281\pi$
$359$	1.05589e9			1.20445			0.602225	+	0.798327i	$0.294281\pi$
$360$	1.79114e8	+	2.65114e8i	0.202335	+	0.299485i
$361$	1.91176e9			2.13875
$362$			1.95450e8i			0.216549i
$363$			3.91617e8i			0.429723i
$364$	1.60103e8			0.173998
$365$	−7.31392e8	−	1.08257e9i	−0.787273	−	1.16528i
$366$	1.70663e8			0.181951
$367$			6.63835e8i			0.701018i	0.936559	+	0.350509i	$0.113991\pi$
$367$			6.63835e8i			0.701018i	−0.936559	+	0.350509i	$0.886009\pi$
$368$		−	5.37874e8i		−	0.562618i
$369$	5.62348e7			0.0582657
$370$	5.36444e8	−	3.62427e8i	0.550577	−	0.371975i
$371$	−8.22604e7			−0.0836338
$372$		−	2.20219e8i		−	0.221796i
$373$		−	8.17776e8i		−	0.815931i	−0.912997	−	0.407966i	$-0.866238\pi$
$373$		−	8.17776e8i		−	0.815931i	0.912997	−	0.407966i	$-0.133762\pi$
$374$	4.83723e8			0.478130
$375$	5.76425e8	−	1.23891e8i	0.564460	−	0.121320i
$376$	−1.40353e9			−1.36165
$377$			1.28566e8i			0.123575i
$378$			9.23948e7i			0.0879885i
$379$	−1.24877e9			−1.17827			−0.589137	−	0.808033i	$-0.700532\pi$
$379$	−1.24877e9			−1.17827			−0.589137	+	0.808033i	$0.700532\pi$
$380$	−6.29921e8	+	4.25581e8i	−0.588903	+	0.397869i
$381$	8.86084e8			0.820801
$382$			7.25529e7i			0.0665937i
$383$			1.76845e9i			1.60841i	0.594349	+	0.804207i	$0.297410\pi$
$383$			1.76845e9i			1.60841i	−0.594349	+	0.804207i	$0.702590\pi$
$384$	−2.55351e7			−0.0230132
$385$	−4.89128e8	−	7.23980e8i	−0.436828	−	0.646568i
$386$	−1.10873e8			−0.0981226
$387$		−	6.16082e8i		−	0.540319i
$388$		−	2.48417e8i		−	0.215909i
$389$	1.64047e9			1.41301			0.706506	−	0.707707i	$-0.250270\pi$
$389$	1.64047e9			1.41301			0.706506	+	0.707707i	$0.250270\pi$
$390$	2.15105e8	+	3.18387e8i	0.183622	+	0.271787i
$391$	−7.10425e8			−0.601035
$392$		−	8.41752e8i		−	0.705802i
$393$			5.62420e7i			0.0467398i
$394$	2.06208e8			0.169851
$395$	−6.27324e8	+	4.23827e8i	−0.512156	+	0.346018i
$396$	−2.18241e8			−0.176605
$397$		−	2.89394e8i		−	0.232126i	−0.993242	−	0.116063i	$-0.962973\pi$
$397$		−	2.89394e8i		−	0.232126i	0.993242	−	0.116063i	$-0.0370274\pi$
$398$			1.22563e9i			0.974472i
$399$	−7.66785e8			−0.604322
$400$	−2.09193e8	+	5.20036e8i	−0.163432	+	0.406278i
$401$	−1.40144e9			−1.08535			−0.542673	−	0.839944i	$-0.682588\pi$
$401$	−1.40144e9			−1.08535			−0.542673	+	0.839944i	$0.682588\pi$
$402$			6.79568e8i			0.521725i
$403$		−	9.23739e8i		−	0.703043i
$404$	−1.00287e9			−0.756678
$405$	1.23084e8	−	8.31569e7i	0.0920682	−	0.0622022i
$406$	−1.03777e8			−0.0769588
$407$			1.54241e9i			1.13402i
$408$			4.01765e8i			0.292861i
$409$	−2.85819e8			−0.206566			−0.103283	−	0.994652i	$-0.532935\pi$
$409$	−2.85819e8			−0.206566			−0.103283	+	0.994652i	$0.532935\pi$
$410$	1.05678e8	+	1.56418e8i	0.0757251	+	0.112084i
$411$	1.13458e9			0.806100
$412$			9.47356e8i			0.667380i
$413$		−	8.35405e7i		−	0.0583541i
$414$	−4.78473e8			−0.331403
$415$	−7.11617e8	−	1.05330e9i	−0.488740	−	0.723406i
$416$	8.03522e8			0.547232
$417$		−	1.32156e9i		−	0.892502i
$418$			2.70373e9i			1.81070i
$419$	1.44224e9			0.957827			0.478914	−	0.877862i	$-0.341031\pi$
$419$	1.44224e9			0.957827			0.478914	+	0.877862i	$0.341031\pi$
$420$	1.72159e8	−	1.16312e8i	0.113385	−	0.0766042i
$421$	2.04694e9			1.33696			0.668480	−	0.743730i	$-0.266945\pi$
$421$	2.04694e9			1.33696			0.668480	+	0.743730i	$0.266945\pi$
$422$		−	9.71338e8i		−	0.629183i
$423$			6.51615e8i			0.418601i
$424$	−2.40910e8			−0.153488
$425$	6.86864e8	+	2.76303e8i	0.434020	+	0.174592i
$426$	−4.47778e8			−0.280627
$427$		−	3.87086e8i		−	0.240608i
$428$			5.45869e8i			0.336539i
$429$	−9.15443e8			−0.559797
$430$	1.71364e9	−	1.15775e9i	1.03939	−	0.702225i
$431$	−1.83156e9			−1.10192			−0.550961	−	0.834531i	$-0.685739\pi$
$431$	−1.83156e9			−1.10192			−0.550961	+	0.834531i	$0.685739\pi$
$432$			1.41222e8i			0.0842773i
$433$			2.10470e9i			1.24590i	0.782262	+	0.622950i	$0.214066\pi$
$433$			2.10470e9i			1.24590i	−0.782262	+	0.622950i	$0.785934\pi$
$434$	7.45629e8			0.437833
$435$	9.34008e7	+	1.38247e8i	0.0544049	+	0.0805271i
$436$	6.51451e8			0.376425
$437$		−	3.97086e9i		−	2.27614i
$438$		−	1.10492e9i		−	0.628309i
$439$	−1.23565e9			−0.697059			−0.348529	−	0.937298i	$-0.613319\pi$
$439$	−1.23565e9			−0.697059			−0.348529	+	0.937298i	$0.613319\pi$
$440$	−1.43247e9	−	2.12027e9i	−0.801683	−	1.18661i
$441$	−3.90799e8			−0.216979
$442$			4.82496e8i			0.265776i
$443$		−	1.12007e9i		−	0.612114i	−0.952013	−	0.306057i	$-0.900990\pi$
$443$		−	1.12007e9i		−	0.612114i	0.952013	−	0.306057i	$-0.0990098\pi$
$444$	−3.66777e8			−0.198867
$445$	2.71257e9	−	1.83264e9i	1.45922	−	0.985864i
$446$	−4.01720e8			−0.214413
$447$			1.01868e9i			0.539464i
$448$			1.14099e9i			0.599527i
$449$	−1.60454e8			−0.0836542			−0.0418271	−	0.999125i	$-0.513318\pi$
$449$	−1.60454e8			−0.0836542			−0.0418271	+	0.999125i	$0.513318\pi$
$450$	4.62605e8	+	1.86091e8i	0.239313	+	0.0962677i
$451$	−4.49742e8			−0.230858
$452$			3.35958e8i			0.171120i
$453$		−	1.72793e8i		−	0.0873340i
$454$	−5.78697e8			−0.290239
$455$	7.22144e8	−	4.87888e8i	0.359405	−	0.242818i
$456$	−2.24563e9			−1.10907
$457$		−	1.05598e8i		−	0.0517544i	−0.999665	−	0.0258772i	$-0.991762\pi$
$457$		−	1.05598e8i		−	0.0517544i	0.999665	−	0.0258772i	$-0.00823789\pi$
$458$		−	1.23893e9i		−	0.602585i
$459$	1.86527e8			0.0900319
$460$	6.02331e8	+	8.91537e8i	0.288525	+	0.427058i
$461$	2.40108e9			1.14144			0.570721	−	0.821144i	$-0.306664\pi$
$461$	2.40108e9			1.14144			0.570721	+	0.821144i	$0.306664\pi$
$462$		−	7.38933e8i		−	0.348625i
$463$		−	1.64082e9i		−	0.768295i	−0.923272	−	0.384147i	$-0.874495\pi$
$463$		−	1.64082e9i		−	0.768295i	0.923272	−	0.384147i	$-0.125505\pi$
$464$	−1.58619e8			−0.0737128
$465$	−6.71080e8	−	9.93295e8i	−0.309520	−	0.458134i
$466$	1.36193e9			0.623453
$467$			1.58089e9i			0.718277i	0.933284	+	0.359139i	$0.116929\pi$
$467$			1.58089e9i			0.718277i	−0.933284	+	0.359139i	$0.883071\pi$
$468$		−	2.17687e8i		−	0.0981686i
$469$	1.54135e9			0.689916
$470$	−1.81248e9	+	1.22453e9i	−0.805249	+	0.544034i
$471$	1.76319e9			0.777544
$472$		−	2.44659e8i		−	0.107094i
$473$			4.92715e9i			2.14083i
$474$	−6.40281e8			−0.276151
$475$	−1.54437e9	+	3.83916e9i	−0.661185	+	1.64365i
$476$	2.60896e8			0.110877
$477$			1.11847e8i			0.0471856i
$478$		−	7.49411e7i		−	0.0313850i
$479$	7.41778e8			0.308389			0.154195	−	0.988040i	$-0.450722\pi$
$479$	7.41778e8			0.308389			0.154195	+	0.988040i	$0.450722\pi$
$480$	8.64026e8	−	5.83744e8i	0.356601	−	0.240923i
$481$	−1.53850e9			−0.630361
$482$			1.89135e9i			0.769321i
$483$			1.08524e9i			0.438240i
$484$	7.44768e8			0.298581
$485$	−7.57011e8	−	1.12049e9i	−0.301305	−	0.445975i
$486$	1.25626e8			0.0496425
$487$			6.33772e7i			0.0248646i	0.999923	+	0.0124323i	$0.00395743\pi$
$487$			6.33772e7i			0.0248646i	−0.999923	+	0.0124323i	$0.996043\pi$
$488$		−	1.13363e9i		−	0.441573i
$489$	9.96534e7			0.0385400
$490$	−7.34398e8	−	1.08701e9i	−0.281997	−	0.417397i
$491$	−3.94003e9			−1.50215			−0.751077	−	0.660215i	$-0.770465\pi$
$491$	−3.94003e9			−1.50215			−0.751077	+	0.660215i	$0.770465\pi$
$492$		−	1.06946e8i		−	0.0404844i
$493$			2.09504e8i			0.0787461i
$494$	−2.69687e9			−1.00650
$495$	−9.84374e8	+	6.65053e8i	−0.364789	+	0.246455i
$496$	1.13967e9			0.419366
$497$			1.01562e9i			0.371095i
$498$		−	1.07505e9i		−	0.390055i
$499$	3.16502e8			0.114031			0.0570157	−	0.998373i	$-0.481841\pi$
$499$	3.16502e8			0.114031			0.0570157	+	0.998373i	$0.481841\pi$
$500$	−2.35613e8	−	1.09623e9i	−0.0842956	−	0.392200i
$501$	9.80298e8			0.348278
$502$		−	6.58245e8i		−	0.232233i
$503$		−	3.82978e9i		−	1.34179i	−0.741551	−	0.670896i	$-0.765910\pi$
$503$		−	3.82978e9i		−	1.34179i	0.741551	−	0.670896i	$-0.234090\pi$
$504$	6.13734e8			0.213537
$505$	−4.52345e9	+	3.05609e9i	−1.56297	+	1.05596i
$506$	3.82662e9			1.31307
$507$			7.81089e8i			0.266178i
$508$		−	1.68513e9i		−	0.570311i
$509$	2.44585e8			0.0822085			0.0411043	−	0.999155i	$-0.486912\pi$
$509$	2.44585e8			0.0822085			0.0411043	+	0.999155i	$0.486912\pi$
$510$	3.50525e8	+	5.18827e8i	0.117010	+	0.173192i
$511$	−2.50612e9			−0.830860
$512$			2.43340e9i			0.801252i
$513$			1.04257e9i			0.340954i
$514$	2.19183e9			0.711928
$515$	2.88691e9	+	4.27304e9i	0.931340	+	1.37852i
$516$	−1.17165e9			−0.375426
$517$		−	5.21133e9i		−	1.65856i
$518$		−	1.24186e9i		−	0.392570i
$519$	1.88555e9			0.592042
$520$	2.11489e9	−	1.42884e9i	0.659593	−	0.445628i
$521$	−7.10895e8			−0.220229			−0.110114	−	0.993919i	$-0.535122\pi$
$521$	−7.10895e8			−0.220229			−0.110114	+	0.993919i	$0.535122\pi$
$522$			1.41102e8i			0.0434196i
$523$		−	3.12678e9i		−	0.955744i	−0.878429	−	0.477872i	$-0.841408\pi$
$523$		−	3.12678e9i		−	0.955744i	0.878429	−	0.477872i	$-0.158592\pi$
$524$	1.06960e8			0.0324759
$525$	4.22078e8	−	1.04925e9i	0.127302	−	0.316462i
$526$	−1.97783e9			−0.592570
$527$		−	1.50528e9i		−	0.448002i
$528$		−	1.12944e9i		−	0.333920i
$529$	−2.21518e9			−0.650601
$530$	−3.11104e8	+	2.10185e8i	−0.0907695	+	0.0613248i
$531$	−1.13587e8			−0.0329230
$532$			1.45825e9i			0.419897i
$533$		−	4.48601e8i		−	0.128326i
$534$	2.76859e9			0.786801
$535$	1.66345e9	+	2.46214e9i	0.469646	+	0.695144i
$536$	4.51404e9			1.26616
$537$			2.00954e9i			0.559997i
$538$		−	2.24071e9i		−	0.620366i
$539$	3.12544e9			0.859707
$540$	−1.58146e8	−	2.34079e8i	−0.0432195	−	0.0639711i
$541$	1.36632e7			0.00370991			0.00185495	−	0.999998i	$-0.499410\pi$
$541$	1.36632e7			0.00370991			0.00185495	+	0.999998i	$0.499410\pi$
$542$			1.42250e9i			0.383755i
$543$		−	6.02751e8i		−	0.161562i
$544$	1.30938e9			0.348714
$545$	2.93836e9	−	1.98519e9i	0.777531	−	0.525308i
$546$	7.37059e8			0.193788
$547$		−	2.64940e9i		−	0.692136i	−0.938209	−	0.346068i	$-0.887517\pi$
$547$		−	2.64940e9i		−	0.692136i	0.938209	−	0.346068i	$-0.112483\pi$
$548$		−	2.15772e9i		−	0.560097i
$549$	−5.26309e8			−0.135749
$550$	−3.69971e9	−	1.48827e9i	−0.948197	−	0.381428i
$551$	−1.17101e9			−0.298214
$552$			3.17827e9i			0.804274i
$553$			1.45224e9i			0.365175i
$554$	−2.01673e9			−0.503923
$555$	−1.65435e9	+	1.11769e9i	−0.410772	+	0.277522i
$556$	−2.51330e9			−0.620131
$557$		−	1.23692e9i		−	0.303284i	−0.988435	−	0.151642i	$-0.951544\pi$
$557$		−	1.23692e9i		−	0.303284i	0.988435	−	0.151642i	$-0.0484561\pi$
$558$		−	1.01381e9i		−	0.247023i
$559$	−4.91466e9			−1.19001
$560$	6.01936e8	+	8.90952e8i	0.144841	+	0.214386i
$561$	−1.49176e9			−0.356721
$562$			3.56962e9i			0.848292i
$563$		−	1.73095e9i		−	0.408795i	−0.978888	−	0.204397i	$-0.934477\pi$
$563$		−	1.73095e9i		−	0.408795i	0.978888	−	0.204397i	$-0.0655234\pi$
$564$	1.23923e9			0.290853
$565$	1.02378e9	+	1.51534e9i	0.238801	+	0.353460i
$566$	−1.78632e9			−0.414097
$567$		−	2.84937e8i		−	0.0656461i
$568$			2.97438e9i			0.681046i
$569$	−5.70834e9			−1.29902			−0.649511	−	0.760352i	$-0.725027\pi$
$569$	−5.70834e9			−1.29902			−0.649511	+	0.760352i	$0.725027\pi$
$570$	−2.89994e9	+	1.95923e9i	−0.655883	+	0.443121i
$571$	3.41952e9			0.768668			0.384334	−	0.923194i	$-0.374431\pi$
$571$	3.41952e9			0.768668			0.384334	+	0.923194i	$0.374431\pi$
$572$			1.74097e9i			0.388960i
$573$		−	2.23747e8i		−	0.0496839i
$574$	3.62105e8			0.0799176
$575$	5.43362e9	+	2.18577e9i	1.19193	+	0.479475i
$576$	1.55137e9			0.338249
$577$			3.59458e9i			0.778991i	0.921028	+	0.389496i	$0.127351\pi$
$577$			3.59458e9i			0.778991i	−0.921028	+	0.389496i	$0.872649\pi$
$578$		−	2.80631e9i		−	0.604489i
$579$	3.41922e8			0.0732068
$580$	2.62914e8	−	1.77628e8i	0.0559521	−	0.0378018i
$581$	−2.43836e9			−0.515799
$582$		−	1.14363e9i		−	0.240466i
$583$		−	8.94502e8i		−	0.186957i
$584$	−7.33948e9			−1.52483
$585$	−6.63366e8	−	9.81877e8i	−0.136996	−	0.202774i
$586$	−8.41201e7			−0.0172686
$587$			6.09879e8i			0.124454i	0.998062	+	0.0622272i	$0.0198204\pi$
$587$			6.09879e8i			0.124454i	−0.998062	+	0.0622272i	$0.980180\pi$
$588$			7.43213e8i			0.150762i
$589$	8.41361e9			1.69660
$590$	−2.13456e8	−	3.15945e8i	−0.0427884	−	0.0633330i
$591$	−6.35926e8			−0.126722
$592$		−	1.89814e9i		−	0.376012i
$593$			8.96159e9i			1.76479i	0.470507	+	0.882396i	$0.344071\pi$
$593$			8.96159e9i			1.76479i	−0.470507	+	0.882396i	$0.655929\pi$
$594$	−1.00470e9			−0.196692
$595$	1.17677e9	−	7.95037e8i	0.229025	−	0.154731i
$596$	1.93731e9			0.374832
$597$		−	3.77974e9i		−	0.727030i
$598$			3.81692e9i			0.729892i
$599$	−7.37621e9			−1.40230			−0.701148	−	0.713016i	$-0.747329\pi$
$599$	−7.37621e9			−1.40230			−0.701148	+	0.713016i	$0.747329\pi$
$600$	1.23611e9	−	3.07286e9i	0.233630	−	0.580783i
$601$	4.41191e9			0.829022			0.414511	−	0.910044i	$-0.363953\pi$
$601$	4.41191e9			0.829022			0.414511	+	0.910044i	$0.363953\pi$
$602$		−	3.96704e9i		−	0.741104i
$603$		−	2.09573e9i		−	0.389246i
$604$	−3.28614e8			−0.0606816
$605$	3.35927e9	−	2.26956e9i	0.616739	−	0.416675i
$606$	−4.61687e9			−0.842740
$607$			2.08098e9i			0.377666i	0.982009	+	0.188833i	$0.0604704\pi$
$607$			2.08098e9i			0.377666i	−0.982009	+	0.188833i	$0.939530\pi$
$608$			7.31865e9i			1.32059i
$609$	3.20038e8			0.0574171
$610$	−9.89051e8	−	1.46394e9i	−0.176427	−	0.261137i
$611$	5.19811e9			0.921938
$612$		−	3.54732e8i		−	0.0625562i
$613$			7.91735e8i			0.138825i	0.997588	+	0.0694126i	$0.0221125\pi$
$613$			7.91735e8i			0.138825i	−0.997588	+	0.0694126i	$0.977888\pi$
$614$	1.52726e9			0.266270
$615$	−3.25901e8	−	4.82380e8i	−0.0564966	−	0.0836231i
$616$	−4.90837e9			−0.846068
$617$		−	6.13159e9i		−	1.05093i	−0.850814	−	0.525467i	$-0.823891\pi$
$617$		−	6.13159e9i		−	1.05093i	0.850814	−	0.525467i	$-0.176109\pi$
$618$			4.36130e9i			0.743286i
$619$	7.99222e9			1.35441			0.677205	−	0.735795i	$-0.263191\pi$
$619$	7.99222e9			1.35441			0.677205	+	0.735795i	$0.263191\pi$
$620$	−1.88902e9	+	1.27624e9i	−0.318322	+	0.215062i
$621$	1.47557e9			0.247252
$622$		−	1.69552e9i		−	0.282511i
$623$		−	6.27954e9i		−	1.04045i
$624$	1.12657e9			0.185615
$625$	−4.40331e9	−	4.22655e9i	−0.721439	−	0.692478i
$626$	7.08371e9			1.15412
$627$		−	8.33805e9i		−	1.35092i
$628$		−	3.35319e9i		−	0.540255i
$629$	−2.50706e9			−0.401687
$630$	7.92558e8	−	5.35460e8i	0.126281	−	0.0853169i
$631$	1.14433e10			1.81322			0.906609	−	0.421973i	$-0.138662\pi$
$631$	1.14433e10			1.81322			0.906609	+	0.421973i	$0.138662\pi$
$632$			4.25307e9i			0.670183i
$633$			2.99552e9i			0.469417i
$634$	1.78288e9			0.277850
$635$	−5.13517e9	−	7.60079e9i	−0.795878	−	1.17801i
$636$	2.12708e8			0.0327856
$637$			3.11751e9i			0.477882i
$638$		−	1.12847e9i		−	0.172035i
$639$	1.38091e9			0.209369
$640$	1.47985e8	+	2.19038e8i	0.0223145	+	0.0330286i
$641$	−1.08654e10			−1.62945			−0.814726	−	0.579846i	$-0.803113\pi$
$641$	−1.08654e10			−1.62945			−0.814726	+	0.579846i	$0.803113\pi$
$642$			2.51299e9i			0.374816i
$643$		−	1.73452e7i		−	0.00257301i	−0.999999	−	0.00128651i	$-0.999590\pi$
$643$		−	1.73452e7i		−	0.00257301i	0.999999	−	0.00128651i	$-0.000409508\pi$
$644$	2.06389e9			0.304499
$645$	−5.28472e9	+	3.57041e9i	−0.775466	+	0.523913i
$646$	−4.39468e9			−0.641377
$647$		−	4.60457e9i		−	0.668381i	−0.942506	−	0.334190i	$-0.891537\pi$
$647$		−	4.60457e9i		−	0.668381i	0.942506	−	0.334190i	$-0.108463\pi$
$648$		−	8.34475e8i		−	0.120476i
$649$	9.08422e8			0.130446
$650$	1.48450e9	−	3.69033e9i	0.212023	−	0.527070i
$651$	−2.29946e9			−0.326657
$652$		−	1.89519e8i		−	0.0267784i
$653$			8.69457e9i			1.22195i	0.791652	+	0.610973i	$0.209222\pi$
$653$			8.69457e9i			1.22195i	−0.791652	+	0.610973i	$0.790778\pi$
$654$	2.99905e9			0.419239
$655$	4.82441e8	−	3.25942e8i	0.0670810	−	0.0453206i
$656$	5.53465e8			0.0765468
$657$			3.40749e9i			0.468765i
$658$			4.19585e9i			0.574155i
$659$	6.26635e8			0.0852934			0.0426467	−	0.999090i	$-0.486421\pi$
$659$	6.26635e8			0.0852934			0.0426467	+	0.999090i	$0.486421\pi$
$660$	1.26478e9	+	1.87206e9i	0.171243	+	0.253464i
$661$	−1.43479e10			−1.93234			−0.966170	−	0.257906i	$-0.916968\pi$
$661$	−1.43479e10			−1.93234			−0.966170	+	0.257906i	$0.916968\pi$
$662$			1.79126e9i			0.239969i
$663$		−	1.48798e9i		−	0.198289i
$664$	−7.14104e9			−0.946614
$665$	4.44379e9	+	6.57745e9i	0.585973	+	0.867324i
$666$	−1.68851e9			−0.221485
$667$			1.65734e9i			0.216258i
$668$		−	1.86431e9i		−	0.241991i
$669$	1.23887e9			0.159968
$670$	5.82930e9	−	3.93833e9i	0.748780	−	0.505884i
$671$	4.20919e9			0.537861
$672$		−	2.00020e9i		−	0.254262i
$673$			1.39752e10i			1.76728i	0.468169	+	0.883639i	$0.344914\pi$
$673$			1.39752e10i			1.76728i	−0.468169	+	0.883639i	$0.655086\pi$
$674$	1.11430e10			1.40182
$675$	−1.42663e9	−	5.73887e8i	−0.178546	−	0.0718229i
$676$	1.48546e9			0.184947
$677$		−	1.74930e9i		−	0.216672i	−0.994114	−	0.108336i	$-0.965448\pi$
$677$		−	1.74930e9i		−	0.216672i	0.994114	−	0.108336i	$-0.0345523\pi$
$678$			1.54663e9i			0.190583i
$679$	−2.59390e9			−0.317987
$680$	3.44632e9	−	2.32837e9i	0.420314	−	0.283969i
$681$	1.78465e9			0.216540
$682$			8.10800e9i			0.978743i
$683$			5.25111e9i			0.630636i	0.948986	+	0.315318i	$0.102111\pi$
$683$			5.25111e9i			0.630636i	−0.948986	+	0.315318i	$0.897889\pi$
$684$	1.98274e9			0.236903
$685$	−6.57529e9	−	9.73238e9i	−0.781625	−	1.15692i
$686$	−6.38224e9			−0.754812
$687$			3.82075e9i			0.449573i
$688$		−	6.06350e9i		−	0.709845i
$689$	8.92234e8			0.103923
$690$	2.77292e9	+	4.10432e9i	0.321341	+	0.475630i
$691$	−3.17886e9			−0.366521			−0.183260	−	0.983064i	$-0.558665\pi$
$691$	−3.17886e9			−0.366521			−0.183260	+	0.983064i	$0.558665\pi$
$692$		−	3.58590e9i		−	0.411364i
$693$			2.27880e9i			0.260100i
$694$	1.00590e9			0.114234
$695$	−1.13362e10	+	7.65888e9i	−1.28092	+	0.865403i
$696$	9.37272e8			0.105374
$697$		−	7.31017e8i		−	0.0817736i
$698$			3.97405e9i			0.442323i
$699$	−4.20007e9			−0.465143
$700$	−1.99544e9	−	8.02699e8i	−0.219885	−	0.0884524i
$701$	1.19964e10			1.31534			0.657672	−	0.753304i	$-0.271541\pi$
$701$	1.19964e10			1.31534			0.657672	+	0.753304i	$0.271541\pi$
$702$		−	1.00216e9i		−	0.109334i
$703$		−	1.40130e10i		−	1.52120i
$704$	−1.24072e10			−1.34020
$705$	5.58952e9	−	3.77634e9i	0.600776	−	0.405890i
$706$	−7.07311e9			−0.756474
$707$			1.04717e10i			1.11442i
$708$			2.16018e8i			0.0228756i
$709$	−6.04187e9			−0.636663			−0.318332	−	0.947979i	$-0.603123\pi$
$709$	−6.04187e9			−0.636663			−0.318332	+	0.947979i	$0.603123\pi$
$710$	2.59503e9	+	3.84102e9i	0.272106	+	0.402757i
$711$	1.97457e9			0.206029
$712$		−	1.83904e10i		−	1.90947i
$713$		−	1.19079e10i		−	1.23033i
$714$	1.20107e9			0.123488
$715$	5.30531e9	+	7.85263e9i	0.542800	+	0.803422i
$716$	3.82169e9			0.389099
$717$			2.31112e8i			0.0234156i
$718$		−	9.24445e9i		−	0.932063i
$719$	7.33783e9			0.736235			0.368118	−	0.929779i	$-0.380002\pi$
$719$	7.33783e9			0.736235			0.368118	+	0.929779i	$0.380002\pi$
$720$	1.21140e9	−	8.18434e8i	0.120955	−	0.0817184i
$721$	9.89200e9			0.982903
$722$		−	1.67377e10i		−	1.65507i
$723$		−	5.83276e9i		−	0.573971i
$724$	−1.14630e9			−0.112257
$725$	6.44583e8	−	1.60238e9i	0.0628196	−	0.156164i
$726$	3.42865e9			0.332541
$727$			5.42358e9i			0.523499i	0.965136	+	0.261749i	$0.0842994\pi$
$727$			5.42358e9i			0.523499i	−0.965136	+	0.261749i	$0.915701\pi$
$728$		−	4.89593e9i		−	0.470300i
$729$	−3.87420e8			−0.0370370
$730$	−9.47798e9	+	6.40342e9i	−0.901750	+	0.609231i
$731$	−8.00867e9			−0.758315
$732$			1.00092e9i			0.0943218i
$733$			2.45352e9i			0.230104i	0.993359	+	0.115052i	$0.0367035\pi$
$733$			2.45352e9i			0.230104i	−0.993359	+	0.115052i	$0.963296\pi$
$734$	5.81195e9			0.542483
$735$	2.26482e9	+	3.35226e9i	0.210391	+	0.311409i
$736$	1.03582e10			0.957661
$737$			1.67607e10i			1.54225i
$738$		−	4.92342e8i		−	0.0450889i
$739$	−1.09881e9			−0.100154			−0.0500770	−	0.998745i	$-0.515947\pi$
$739$	−1.09881e9			−0.100154			−0.0500770	+	0.998745i	$0.515947\pi$
$740$	2.12560e9	+	3.14620e9i	0.192828	+	0.285414i
$741$	8.31691e9			0.750928
$742$			7.20199e8i			0.0647200i
$743$		−	1.26814e10i		−	1.13425i	−0.823633	−	0.567123i	$-0.808056\pi$
$743$		−	1.26814e10i		−	1.13425i	0.823633	−	0.567123i	$-0.191944\pi$
$744$	−6.73425e9			−0.599492
$745$	8.73821e9	−	5.90362e9i	0.774240	−	0.523084i
$746$	−7.15972e9			−0.631408
$747$			3.31536e9i			0.291010i
$748$			2.83699e9i			0.247858i
$749$	5.69980e9			0.495648
$750$	−1.08468e9	−	5.04666e9i	−0.0938831	−	0.436807i
$751$	1.42471e9			0.122740			0.0613699	−	0.998115i	$-0.480453\pi$
$751$	1.42471e9			0.122740			0.0613699	+	0.998115i	$0.480453\pi$
$752$			6.41322e9i			0.549938i
$753$			2.02997e9i			0.173263i
$754$	1.12561e9			0.0956286
$755$	−1.48221e9	+	1.00140e9i	−0.125342	+	0.0846822i
$756$	−5.41887e8			−0.0456124
$757$			1.25591e9i			0.105226i	0.998615	+	0.0526128i	$0.0167549\pi$
$757$			1.25591e9i			0.105226i	−0.998615	+	0.0526128i	$0.983245\pi$
$758$			1.09332e10i			0.911808i
$759$	−1.18010e10			−0.979651
$760$	1.30142e10	+	1.92629e10i	1.07540	+	1.59175i
$761$	−6.61122e9			−0.543795			−0.271897	−	0.962326i	$-0.587651\pi$
$761$	−6.61122e9			−0.543795			−0.271897	+	0.962326i	$0.587651\pi$
$762$		−	7.75777e9i		−	0.635177i
$763$		−	6.80225e9i		−	0.554392i
$764$	−4.25516e8			−0.0345215
$765$	−1.08099e9	−	1.60002e9i	−0.0872983	−	0.129214i
$766$	1.54830e10			1.24467
$767$			9.06118e8i			0.0725106i
$768$		−	7.13108e9i		−	0.568056i
$769$	−1.61751e10			−1.28264			−0.641320	−	0.767273i	$-0.721613\pi$
$769$	−1.61751e10			−1.28264			−0.641320	+	0.767273i	$0.721613\pi$
$770$	−6.33853e9	+	4.28238e9i	−0.500347	+	0.338039i
$771$	−6.75941e9			−0.531151
$772$		−	6.50259e8i		−	0.0508658i
$773$		−	2.10944e9i		−	0.164263i	−0.996622	−	0.0821313i	$-0.973827\pi$
$773$		−	2.10944e9i		−	0.164263i	0.996622	−	0.0821313i	$-0.0261727\pi$
$774$	−5.39387e9			−0.418126
$775$	−4.63129e9	+	1.15130e10i	−0.357393	+	0.888448i
$776$	−7.59656e9			−0.583581
$777$			3.82977e9i			0.292887i
$778$		−	1.43625e10i		−	1.09346i
$779$	4.08595e9			0.309680
$780$	−1.86731e9	+	1.26157e9i	−0.140892	+	0.0951879i
$781$	−1.10439e10			−0.829553
$782$			6.21985e9i			0.465111i
$783$		−	4.35146e8i		−	0.0323943i
$784$	−3.84626e9			−0.285057
$785$	−1.02183e10	−	1.51245e10i	−0.753935	−	1.11593i
$786$	4.92405e8			0.0361696
$787$			1.72091e10i			1.25848i	0.777211	+	0.629240i	$0.216634\pi$
$787$			1.72091e10i			1.25848i	−0.777211	+	0.629240i	$0.783366\pi$
$788$			1.20939e9i			0.0880491i
$789$	6.09947e9			0.442102
$790$	3.71065e9	+	5.49230e9i	0.267766	+	0.396332i
$791$	3.50797e9			0.252022
$792$			6.67377e9i			0.477346i
$793$			4.19852e9i			0.298978i
$794$	−2.53368e9			−0.179630
$795$	9.59417e8	−	6.48192e8i	0.0677209	−	0.0457529i
$796$	−7.18823e9			−0.505157
$797$		−	9.52182e9i		−	0.666217i	−0.942889	−	0.333108i	$-0.891902\pi$
$797$		−	9.52182e9i		−	0.666217i	0.942889	−	0.333108i	$-0.108098\pi$
$798$			6.71329e9i			0.467655i
$799$	8.47058e9			0.587489
$800$	−1.00147e10	−	4.02857e9i	−0.691547	−	0.278186i
$801$	−8.53810e9			−0.587012
$802$			1.22697e10i			0.839895i
$803$		−	2.72516e10i		−	1.85732i
$804$	−3.98560e9			−0.270457
$805$	9.30915e9	−	6.28936e9i	0.628962	−	0.424933i
$806$	−8.08744e9			−0.544050
$807$			6.91016e9i			0.462840i
$808$			3.06677e10i			2.04522i
$809$	2.68450e10			1.78256			0.891278	−	0.453458i	$-0.149810\pi$
$809$	2.68450e10			1.78256			0.891278	+	0.453458i	$0.149810\pi$
$810$	−7.28049e8	−	1.07762e9i	−0.0481352	−	0.0712470i
$811$	1.76621e10			1.16270			0.581351	−	0.813653i	$-0.302524\pi$
$811$	1.76621e10			1.16270			0.581351	+	0.813653i	$0.302524\pi$
$812$		−	6.08641e8i		−	0.0398947i
$813$		−	4.38685e9i		−	0.286310i
$814$	1.35040e10			0.877559
$815$	−5.77526e8	−	8.54822e8i	−0.0373698	−	0.0553126i
$816$	1.83580e9			0.118280
$817$		−	4.47638e10i		−	2.87177i
$818$			2.50238e9i			0.159851i
$819$	−2.27302e9			−0.144581
$820$	−9.17379e8	+	6.19790e8i	−0.0581032	+	0.0392551i
$821$	7.60914e9			0.479882			0.239941	−	0.970787i	$-0.422872\pi$
$821$	7.60914e9			0.479882			0.239941	+	0.970787i	$0.422872\pi$
$822$		−	9.93339e9i		−	0.623801i
$823$			2.14232e10i			1.33963i	0.742529	+	0.669814i	$0.233626\pi$
$823$			2.14232e10i			1.33963i	−0.742529	+	0.669814i	$0.766374\pi$
$824$	2.89700e10			1.80386
$825$	1.14096e10	+	4.58970e9i	0.707426	+	0.284574i
$826$	−7.31406e8			−0.0451574
$827$			1.41079e10i			0.867345i	0.901071	+	0.433673i	$0.142783\pi$
$827$			1.41079e10i			0.867345i	−0.901071	+	0.433673i	$0.857217\pi$
$828$		−	2.80620e9i		−	0.171796i
$829$	−6.92118e9			−0.421929			−0.210965	−	0.977494i	$-0.567660\pi$
$829$	−6.92118e9			−0.421929			−0.210965	+	0.977494i	$0.567660\pi$
$830$	−9.22172e9	+	6.23029e9i	−0.559808	+	0.378212i
$831$	6.21943e9			0.375965
$832$		−	1.23757e10i		−	0.744969i
$833$			5.08014e9i			0.304522i
$834$	−1.15704e10			−0.690663
$835$	−5.68117e9	−	8.40895e9i	−0.337703	−	0.499849i
$836$	−1.58571e10			−0.938647
$837$			3.12650e9i			0.184297i
$838$		−	1.26269e10i		−	0.741215i
$839$	−2.84396e9			−0.166248			−0.0831240	−	0.996539i	$-0.526490\pi$
$839$	−2.84396e9			−0.166248			−0.0831240	+	0.996539i	$0.526490\pi$
$840$	−3.55680e9	−	5.26458e9i	−0.207053	−	0.306469i
$841$	−1.67611e10			−0.971666
$842$		−	1.79212e10i		−	1.03461i
$843$		−	1.10084e10i		−	0.632889i
$844$	5.69681e9			0.326162
$845$	6.70014e9	−	4.52668e9i	0.382020	−	0.258096i
$846$	5.70496e9			0.323934
$847$		−	7.77664e9i		−	0.439744i
$848$			1.10080e9i			0.0619902i
$849$	5.50886e9			0.308948
$850$	2.41906e9	−	6.01357e9i	0.135108	−	0.335866i
$851$	−1.98328e10			−1.10314
$852$		−	2.62618e9i		−	0.145474i
$853$		−	1.54319e10i		−	0.851330i	−0.904881	−	0.425665i	$-0.860040\pi$
$853$		−	1.54319e10i		−	0.851330i	0.904881	−	0.425665i	$-0.139960\pi$
$854$	−3.38899e9			−0.186195
$855$	8.94315e9	−	6.04208e9i	0.489338	−	0.330602i
$856$	1.66926e10			0.909631
$857$			1.53742e10i			0.834373i	0.908821	+	0.417186i	$0.136984\pi$
$857$			1.53742e10i			0.834373i	−0.908821	+	0.417186i	$0.863016\pi$
$858$			8.01481e9i			0.433199i
$859$	1.32076e10			0.710966			0.355483	−	0.934683i	$-0.384316\pi$
$859$	1.32076e10			0.710966			0.355483	+	0.934683i	$0.384316\pi$
$860$	6.79012e9	+	1.00504e10i	0.364027	+	0.538812i
$861$	−1.11670e9			−0.0596245
$862$			1.60355e10i			0.852722i
$863$			2.37085e10i			1.25564i	0.778357	+	0.627822i	$0.216053\pi$
$863$			2.37085e10i			1.25564i	−0.778357	+	0.627822i	$0.783947\pi$
$864$	−2.71961e9			−0.143453
$865$	−1.09274e10	−	1.61742e10i	−0.574066	−	0.849700i
$866$	1.84269e10			0.964139
$867$			8.65442e9i			0.450994i
$868$			4.37305e9i			0.226968i
$869$	−1.57917e10			−0.816320
$870$	1.21036e9	−	8.17735e8i	0.0623159	−	0.0421013i
$871$	−1.67182e10			−0.857286
$872$		−	1.99212e10i		−	1.01744i
$873$			3.52684e9i			0.179406i
$874$	−3.47653e10			−1.76139
$875$	−1.14465e10	+	2.46020e9i	−0.577623	+	0.124149i
$876$	6.48028e9			0.325709
$877$		−	3.40371e9i		−	0.170394i	−0.996364	−	0.0851970i	$-0.972848\pi$
$877$		−	3.40371e9i		−	0.170394i	0.996364	−	0.0851970i	$-0.0271520\pi$
$878$			1.08182e10i			0.539419i
$879$	2.59419e8			0.0128837
$880$	−9.68824e9	+	6.54547e9i	−0.479243	+	0.323781i
$881$	3.81831e10			1.88129			0.940644	−	0.339395i	$-0.110222\pi$
$881$	3.81831e10			1.88129			0.940644	+	0.339395i	$0.110222\pi$
$882$			3.42149e9i			0.167910i
$883$			2.44534e10i			1.19530i	0.801758	+	0.597649i	$0.203898\pi$
$883$			2.44534e10i			1.19530i	−0.801758	+	0.597649i	$0.796102\pi$
$884$	−2.82980e9			−0.137776
$885$	6.58279e8	+	9.74347e8i	0.0319233	+	0.0472511i
$886$	−9.80635e9			−0.473685
$887$			1.85474e8i			0.00892383i	0.999990	+	0.00446192i	$0.00142028\pi$
$887$			1.85474e8i			0.00892383i	−0.999990	+	0.00446192i	$0.998580\pi$
$888$			1.12160e10i			0.537516i
$889$	−1.75957e10			−0.839942
$890$	−1.60450e10	−	2.37489e10i	−0.762911	−	1.12922i
$891$	3.09842e9			0.146747
$892$		−	2.35605e9i		−	0.111150i
$893$			4.73455e10i			2.22484i
$894$	8.91868e9			0.417464
$895$	1.72377e10	−	1.16460e10i	0.803709	−	0.542994i
$896$	5.07069e8			0.0235499
$897$		−	1.17710e10i		−	0.544554i
$898$			1.40479e9i			0.0647358i
$899$	−3.51164e9			−0.161195
$900$	−1.09141e9	+	2.71314e9i	−0.0499042	+	0.124057i
$901$	1.45394e9			0.0662231
$902$			3.93754e9i			0.178650i
$903$			1.22340e10i			0.552919i
$904$	1.02735e10			0.462520
$905$	−5.17037e9	+	3.49316e9i	−0.231874	+	0.156656i
$906$	−1.51282e9			−0.0675834
$907$			2.45453e10i			1.09230i	0.837686	+	0.546152i	$0.183908\pi$
$907$			2.45453e10i			1.09230i	−0.837686	+	0.546152i	$0.816092\pi$
$908$		−	3.39401e9i		−	0.150457i
$909$	1.42380e10			0.628748
$910$	−4.27151e9	−	6.32246e9i	−0.187904	−	0.278126i
$911$	2.83465e9			0.124218			0.0621092	−	0.998069i	$-0.480217\pi$
$911$	2.83465e9			0.124218			0.0621092	+	0.998069i	$0.480217\pi$
$912$			1.02611e10i			0.447930i
$913$		−	2.65148e10i		−	1.15303i
$914$	−9.24519e8			−0.0400501
$915$	3.05015e9	+	4.51466e9i	0.131628	+	0.194828i
$916$	7.26622e9			0.312374
$917$		−	1.11684e9i		−	0.0478298i
$918$		−	1.63306e9i		−	0.0696712i
$919$	−4.04874e10			−1.72074			−0.860370	−	0.509669i	$-0.829768\pi$
$919$	−4.04874e10			−1.72074			−0.860370	+	0.509669i	$0.829768\pi$
$920$	2.72630e10	−	1.84192e10i	1.15429	−	0.779853i
$921$	−4.70992e9			−0.198658
$922$		−	2.10218e10i		−	0.883305i
$923$		−	1.10159e10i		−	0.461120i
$924$	4.33378e9			0.180723
$925$	1.91750e10	+	7.71348e9i	0.796599	+	0.320445i
$926$	−1.43656e10			−0.594545
$927$		−	1.34499e10i		−	0.554547i
$928$		−	3.05463e9i		−	0.125470i
$929$	1.05190e10			0.430447			0.215223	−	0.976565i	$-0.430952\pi$
$929$	1.05190e10			0.430447			0.215223	+	0.976565i	$0.430952\pi$
$930$	−8.69641e9	+	5.87538e9i	−0.354527	+	0.239522i
$931$	−2.83950e10			−1.15324
$932$			7.98760e9i			0.323192i
$933$			5.22882e9i			0.210775i
$934$	1.38409e10			0.555839
$935$	8.64527e9	+	1.27962e10i	0.345890	+	0.511967i
$936$	−6.65684e9			−0.265340
$937$			3.07454e10i			1.22093i	0.792043	+	0.610466i	$0.209018\pi$
$937$			3.07454e10i			1.22093i	−0.792043	+	0.610466i	$0.790982\pi$
$938$		−	1.34947e10i		−	0.533892i
$939$	−2.18456e10			−0.861060
$940$	−7.18175e9	−	1.06300e10i	−0.282022	−	0.417433i
$941$	−8.99712e7			−0.00351998			−0.00175999	−	0.999998i	$-0.500560\pi$
$941$	−8.99712e7			−0.00351998			−0.00175999	+	0.999998i	$0.500560\pi$
$942$		−	1.54369e10i		−	0.601703i
$943$		−	5.78291e9i		−	0.224572i
$944$	−1.11793e9			−0.0432527
$945$	−2.44418e9	+	1.65131e9i	−0.0942153	+	0.0636528i
$946$	4.31378e10			1.65668
$947$		−	4.18307e10i		−	1.60055i	−0.599630	−	0.800277i	$-0.704686\pi$
$947$		−	4.18307e10i		−	1.60055i	0.599630	−	0.800277i	$-0.295314\pi$
$948$		−	3.75519e9i		−	0.143154i
$949$	2.71825e10			1.03242
$950$	3.36123e10	+	1.35211e10i	1.27194	+	0.511658i
$951$	−5.49825e9			−0.207297
$952$		−	7.97815e9i		−	0.299690i
$953$		−	3.89523e10i		−	1.45783i	−0.684602	−	0.728917i	$-0.740024\pi$
$953$		−	3.89523e10i		−	1.45783i	0.684602	−	0.728917i	$-0.259976\pi$
$954$	9.79232e8			0.0365146
$955$	−1.91929e9	+	1.29669e9i	−0.0713064	+	0.0481753i
$956$	4.39523e8			0.0162697
$957$			3.48010e9i			0.128351i
$958$		−	6.49435e9i		−	0.238647i
$959$	−2.25302e10			−0.824899
$960$	−8.99073e9	−	1.33076e10i	−0.327979	−	0.485456i
$961$	−2.28165e9			−0.0829309
$962$			1.34697e10i			0.487805i
$963$		−	7.74984e9i		−	0.279641i
$964$	−1.10926e10			−0.398808
$965$	−1.98156e9	−	2.93299e9i	−0.0709840	−	0.105067i
$966$	9.50142e9			0.339132
$967$		−	2.21029e9i		−	0.0786060i	−0.999227	−	0.0393030i	$-0.987486\pi$
$967$		−	2.21029e9i		−	0.0786060i	0.999227	−	0.0393030i	$-0.0125138\pi$
$968$		−	2.27749e10i		−	0.807035i
$969$	1.35528e10			0.478515
$970$	−9.80998e9	+	6.62772e9i	−0.345117	+	0.233165i
$971$	−1.45215e10			−0.509029			−0.254515	−	0.967069i	$-0.581916\pi$
$971$	−1.45215e10			−0.509029			−0.254515	+	0.967069i	$0.581916\pi$
$972$			7.36787e8i			0.0257342i
$973$			2.62432e10i			0.913316i
$974$	5.54875e8			0.0192415
$975$	−4.57806e9	+	1.13806e10i	−0.158185	+	0.393234i
$976$	−5.17996e9			−0.178341
$977$		−	1.45088e10i		−	0.497739i	−0.968537	−	0.248870i	$-0.919941\pi$
$977$		−	1.45088e10i		−	0.497739i	0.968537	−	0.248870i	$-0.0800591\pi$
$978$		−	8.72477e8i		−	0.0298241i
$979$	6.82840e10			2.32584
$980$	6.37524e9	−	4.30718e9i	0.216374	−	0.146185i
$981$	−9.24881e9			−0.312784
$982$			3.44954e10i			1.16244i
$983$			3.73720e10i			1.25490i	0.778657	+	0.627450i	$0.215901\pi$
$983$			3.73720e10i			1.25490i	−0.778657	+	0.627450i	$0.784099\pi$
$984$	−3.27039e9			−0.109425
$985$	3.68542e9	+	5.45495e9i	0.122874	+	0.181871i
$986$	1.83423e9			0.0609376
$987$		−	1.29396e10i		−	0.428363i
$988$		−	1.58169e10i		−	0.521761i
$989$	−6.33548e10			−2.08254
$990$	5.82261e9	+	8.61831e9i	0.190719	+	0.282292i
$991$	3.20279e10			1.04537			0.522687	−	0.852525i	$-0.324930\pi$
$991$	3.20279e10			1.04537			0.522687	+	0.852525i	$0.324930\pi$
$992$			2.19474e10i			0.713825i
$993$		−	5.52407e9i		−	0.179035i
$994$	8.89188e9			0.287172
$995$	−3.24225e10	+	2.19049e10i	−1.04343	+	0.704955i
$996$	6.30507e9			0.202201
$997$			1.33254e10i			0.425841i	0.977070	+	0.212920i	$0.0682975\pi$
$997$			1.33254e10i			0.425841i	−0.977070	+	0.212920i	$0.931703\pi$
$998$		−	2.77101e9i		−	0.0882432i
$999$	5.20722e9			0.165245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.b.a.4.3	✓	8
3.2	odd	2		45.8.b.d.19.6		8
4.3	odd	2		240.8.f.e.49.4		8
5.2	odd	4		75.8.a.j.1.3		4
5.3	odd	4		75.8.a.i.1.2		4
5.4	even	2	inner	15.8.b.a.4.6	yes	8
15.2	even	4		225.8.a.z.1.2		4
15.8	even	4		225.8.a.bb.1.3		4
15.14	odd	2		45.8.b.d.19.3		8
20.19	odd	2		240.8.f.e.49.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.b.a.4.3	✓	8	1.1	even	1	trivial
15.8.b.a.4.6	yes	8	5.4	even	2	inner
45.8.b.d.19.3		8	15.14	odd	2
45.8.b.d.19.6		8	3.2	odd	2
75.8.a.i.1.2		4	5.3	odd	4
75.8.a.j.1.3		4	5.2	odd	4
225.8.a.z.1.2		4	15.2	even	4
225.8.a.bb.1.3		4	15.8	even	4
240.8.f.e.49.4		8	4.3	odd	2
240.8.f.e.49.8		8	20.19	odd	2

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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	15.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.75511i q^{2} +27.0000i q^{3} +51.3480 q^{4} +(231.605 - 156.474i) q^{5} +236.388 q^{6} -536.160i q^{7} -1570.21i q^{8} -729.000 q^{9} +O(q^{10})\)	\(q-8.75511i q^{2} +27.0000i q^{3} +51.3480 q^{4} +(231.605 - 156.474i) q^{5} +236.388 q^{6} -536.160i q^{7} -1570.21i q^{8} -729.000 q^{9} +(-1369.95 - 2027.73i) q^{10} +5830.22 q^{11} +1386.40i q^{12} +5815.44i q^{13} -4694.14 q^{14} +(4224.81 + 6253.33i) q^{15} -7174.84 q^{16} +9476.54i q^{17} +6382.48i q^{18} -52968.3 q^{19} +(11892.4 - 8034.65i) q^{20} +14476.3 q^{21} -51044.3i q^{22} +74966.7i q^{23} +42395.7 q^{24} +(29156.5 - 72480.4i) q^{25} +50914.8 q^{26} -19683.0i q^{27} -27530.7i q^{28} +22107.7 q^{29} +(54748.6 - 36988.7i) q^{30} -158843. q^{31} -138171. i q^{32} +157416. i q^{33} +82968.2 q^{34} +(-83895.3 - 124177. i) q^{35} -37432.7 q^{36} +264554. i q^{37} +463743. i q^{38} -157017. q^{39} +(-245698. - 363668. i) q^{40} -77139.7 q^{41} -126742. i q^{42} +845105. i q^{43} +299370. q^{44} +(-168840. + 114070. i) q^{45} +656342. q^{46} -893848. i q^{47} -193721. i q^{48} +536076. q^{49} +(-634575. - 255268. i) q^{50} -255867. q^{51} +298611. i q^{52} -153425. i q^{53} -172327. q^{54} +(1.35031e6 - 912281. i) q^{55} -841885. q^{56} -1.43014e6i q^{57} -193555. i q^{58} +155813. q^{59} +(216935. + 321096. i) q^{60} +721961. q^{61} +1.39068e6i q^{62} +390861. i q^{63} -2.12808e6 q^{64} +(909967. + 1.34688e6i) q^{65} +1.37820e6 q^{66} +2.87480e6i q^{67} +486601. i q^{68} -2.02410e6 q^{69} +(-1.08718e6 + 734513. i) q^{70} -1.89425e6 q^{71} +1.14468e6i q^{72} -4.67419e6i q^{73} +2.31620e6 q^{74} +(1.95697e6 + 787225. i) q^{75} -2.71981e6 q^{76} -3.12593e6i q^{77} +1.37470e6i q^{78} -2.70860e6 q^{79} +(-1.66173e6 + 1.12268e6i) q^{80} +531441. q^{81} +675367. i q^{82} -4.54782e6i q^{83} +743330. q^{84} +(1.48284e6 + 2.19481e6i) q^{85} +7.39899e6 q^{86} +596908. i q^{87} -9.15469e6i q^{88} +1.17121e7 q^{89} +(998695. + 1.47821e6i) q^{90} +3.11800e6 q^{91} +3.84939e6i q^{92} -4.28875e6i q^{93} -7.82574e6 q^{94} +(-1.22677e7 + 8.28818e6i) q^{95} +3.73061e6 q^{96} -4.83792e6i q^{97} -4.69340e6i q^{98} -4.25023e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10}) \) 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9	\( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9} - 6686 q^{10} + 10752 q^{11} - 13524 q^{14} - 17496 q^{15} + 86530 q^{16} - 30464 q^{19} + 87444 q^{20} + 64152 q^{21} - 110322 q^{24} + 127616 q^{25} - 793524 q^{26} + 240072 q^{29} - 172044 q^{30} + 233728 q^{31} + 184748 q^{34} + 593520 q^{35} + 485514 q^{36} - 454896 q^{39} - 1147102 q^{40} + 507648 q^{41} + 2578572 q^{44} + 323676 q^{45} + 662408 q^{46} - 3267160 q^{49} - 5117736 q^{50} + 264384 q^{51} - 354294 q^{54} + 3525696 q^{55} + 705660 q^{56} + 1091424 q^{59} + 5307606 q^{60} - 6433520 q^{61} - 568594 q^{64} - 2555592 q^{65} - 2382372 q^{66} - 5940864 q^{69} + 12097800 q^{70} - 1381824 q^{71} + 23961276 q^{74} + 7768224 q^{75} + 13115664 q^{76} - 14380160 q^{79} - 31251876 q^{80} + 4251528 q^{81} - 36423756 q^{84} - 1452008 q^{85} - 19837608 q^{86} + 45778896 q^{89} + 4874094 q^{90} + 24075648 q^{91} - 45728896 q^{94} - 25774656 q^{95} + 33586002 q^{96} - 7838208 q^{99}+O(q^{100}) \) 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9 - 6686 * q^10 + 10752 * q^11 - 13524 * q^14 - 17496 * q^15 + 86530 * q^16 - 30464 * q^19 + 87444 * q^20 + 64152 * q^21 - 110322 * q^24 + 127616 * q^25 - 793524 * q^26 + 240072 * q^29 - 172044 * q^30 + 233728 * q^31 + 184748 * q^34 + 593520 * q^35 + 485514 * q^36 - 454896 * q^39 - 1147102 * q^40 + 507648 * q^41 + 2578572 * q^44 + 323676 * q^45 + 662408 * q^46 - 3267160 * q^49 - 5117736 * q^50 + 264384 * q^51 - 354294 * q^54 + 3525696 * q^55 + 705660 * q^56 + 1091424 * q^59 + 5307606 * q^60 - 6433520 * q^61 - 568594 * q^64 - 2555592 * q^65 - 2382372 * q^66 - 5940864 * q^69 + 12097800 * q^70 - 1381824 * q^71 + 23961276 * q^74 + 7768224 * q^75 + 13115664 * q^76 - 14380160 * q^79 - 31251876 * q^80 + 4251528 * q^81 - 36423756 * q^84 - 1452008 * q^85 - 19837608 * q^86 + 45778896 * q^89 + 4874094 * q^90 + 24075648 * q^91 - 45728896 * q^94 - 25774656 * q^95 + 33586002 * q^96 - 7838208 * q^99

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