LMFDB - Embedded newform 15.8.b.a.4.4

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			4.4
Root			$7.26440i$ of defining polynomial
Character	$\chi$	$=$	15.4
Dual form			15.8.b.a.4.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.02250i q^{2} -27.0000i q^{3} +118.865 q^{4} +(-272.800 - 60.8703i) q^{5} -81.6074 q^{6} -1505.28i q^{7} -746.147i q^{8} -729.000 q^{9} +O(q^{10})$	\(q-3.02250i q^{2} -27.0000i q^{3} +118.865 q^{4} +(-272.800 - 60.8703i) q^{5} -81.6074 q^{6} -1505.28i q^{7} -746.147i q^{8} -729.000 q^{9} +(-183.980 + 824.537i) q^{10} +1596.37 q^{11} -3209.34i q^{12} +956.100i q^{13} -4549.69 q^{14} +(-1643.50 + 7365.60i) q^{15} +12959.4 q^{16} +32470.6i q^{17} +2203.40i q^{18} +39168.1 q^{19} +(-32426.2 - 7235.32i) q^{20} -40642.5 q^{21} -4825.04i q^{22} -59361.5i q^{23} -20146.0 q^{24} +(70714.6 + 33210.8i) q^{25} +2889.81 q^{26} +19683.0i q^{27} -178924. i q^{28} -66150.5 q^{29} +(22262.5 + 4967.46i) q^{30} -19664.1 q^{31} -134677. i q^{32} -43102.1i q^{33} +98142.2 q^{34} +(-91626.6 + 410640. i) q^{35} -86652.2 q^{36} +376045. i q^{37} -118386. i q^{38} +25814.7 q^{39} +(-45418.2 + 203549. i) q^{40} +385003. q^{41} +122842. i q^{42} -466410. i q^{43} +189752. q^{44} +(198871. + 44374.4i) q^{45} -179420. q^{46} +468903. i q^{47} -349905. i q^{48} -1.44232e6 q^{49} +(100380. - 213735. i) q^{50} +876706. q^{51} +113646. i q^{52} -1.60516e6i q^{53} +59491.8 q^{54} +(-435491. - 97171.8i) q^{55} -1.12316e6 q^{56} -1.05754e6i q^{57} +199940. i q^{58} +2.04044e6 q^{59} +(-195354. + 875508. i) q^{60} -378667. q^{61} +59434.6i q^{62} +1.09735e6i q^{63} +1.25175e6 q^{64} +(58198.1 - 260824. i) q^{65} -130276. q^{66} -4644.40i q^{67} +3.85960e6i q^{68} -1.60276e6 q^{69} +(1.24116e6 + 276941. i) q^{70} -2.79333e6 q^{71} +543941. i q^{72} +2.01174e6i q^{73} +1.13660e6 q^{74} +(896692. - 1.90929e6i) q^{75} +4.65570e6 q^{76} -2.40299e6i q^{77} -78024.8i q^{78} -1.76767e6 q^{79} +(-3.53533e6 - 788844. i) q^{80} +531441. q^{81} -1.16367e6i q^{82} +3.06625e6i q^{83} -4.83095e6 q^{84} +(1.97649e6 - 8.85797e6i) q^{85} -1.40972e6 q^{86} +1.78606e6i q^{87} -1.19113e6i q^{88} +6.14397e6 q^{89} +(134122. - 601087. i) q^{90} +1.43920e6 q^{91} -7.05598e6i q^{92} +530930. i q^{93} +1.41726e6 q^{94} +(-1.06851e7 - 2.38417e6i) q^{95} -3.63627e6 q^{96} +3.02969e6i q^{97} +4.35940e6i q^{98} -1.16376e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.02250i		−	0.267153i	−0.991038	−	0.133577i	$-0.957354\pi$
$2$		−	3.02250i		−	0.267153i	0.991038	−	0.133577i	$-0.0426463\pi$
$3$		−	27.0000i		−	0.577350i
$3$		−	27.0000i		−	0.577350i
$4$	118.865			0.928629
$5$	−272.800	−	60.8703i	−0.975999	−	0.217776i
$5$	−272.800	−	60.8703i	−0.975999	−	0.217776i
$6$	−81.6074			−0.154241
$7$		−	1505.28i		−	1.65872i	−0.558714	−	0.829361i	$-0.688705\pi$
$7$		−	1505.28i		−	1.65872i	0.558714	−	0.829361i	$-0.311295\pi$
$8$		−	746.147i		−	0.515240i
$9$	−729.000			−0.333333
$10$	−183.980	+	824.537i	−0.0581796	+	0.260741i
$11$	1596.37			0.361627			0.180813	−	0.983517i	$-0.442127\pi$
$11$	1596.37			0.361627			0.180813	+	0.983517i	$0.442127\pi$
$12$		−	3209.34i		−	0.536144i
$13$			956.100i			0.120698i	0.998177	+	0.0603492i	$0.0192214\pi$
$13$			956.100i			0.120698i	−0.998177	+	0.0603492i	$0.980779\pi$
$14$	−4549.69			−0.443133
$15$	−1643.50	+	7365.60i	−0.125733	+	0.563493i
$16$	12959.4			0.790981
$17$			32470.6i			1.60295i	0.598031	+	0.801473i	$0.295950\pi$
$17$			32470.6i			1.60295i	−0.598031	+	0.801473i	$0.704050\pi$
$18$			2203.40i			0.0890511i
$19$	39168.1			1.31007			0.655036	−	0.755598i	$-0.272653\pi$
$19$	39168.1			1.31007			0.655036	+	0.755598i	$0.272653\pi$
$20$	−32426.2	−	7235.32i	−0.906341	−	0.202233i
$21$	−40642.5			−0.957663
$22$		−	4825.04i		−	0.0966098i
$23$		−	59361.5i		−	1.01732i	−0.860968	−	0.508660i	$-0.830141\pi$
$23$		−	59361.5i		−	1.01732i	0.860968	−	0.508660i	$-0.169859\pi$
$24$	−20146.0			−0.297474
$25$	70714.6	+	33210.8i	0.905147	+	0.425098i
$26$	2889.81			0.0322450
$27$			19683.0i			0.192450i
$28$		−	178924.i		−	1.54034i
$29$	−66150.5			−0.503663			−0.251831	−	0.967771i	$-0.581033\pi$
$29$	−66150.5			−0.503663			−0.251831	+	0.967771i	$0.581033\pi$
$30$	22262.5	+	4967.46i	0.150539	+	0.0335900i
$31$	−19664.1			−0.118552			−0.0592758	−	0.998242i	$-0.518879\pi$
$31$	−19664.1			−0.118552			−0.0592758	+	0.998242i	$0.518879\pi$
$32$		−	134677.i		−	0.726553i
$33$		−	43102.1i		−	0.208785i
$34$	98142.2			0.428232
$35$	−91626.6	+	410640.i	−0.361230	+	1.61891i
$36$	−86652.2			−0.309543
$37$			376045.i			1.22049i	0.792213	+	0.610245i	$0.208929\pi$
$37$			376045.i			1.22049i	−0.792213	+	0.610245i	$0.791071\pi$
$38$		−	118386.i		−	0.349990i
$39$	25814.7			0.0696853
$40$	−45418.2	+	203549.i	−0.112207	+	0.502873i
$41$	385003.			0.872409			0.436205	−	0.899848i	$-0.356322\pi$
$41$	385003.			0.872409			0.436205	+	0.899848i	$0.356322\pi$
$42$			122842.i			0.255843i
$43$		−	466410.i		−	0.894600i	−0.894384	−	0.447300i	$-0.852386\pi$
$43$		−	466410.i		−	0.894600i	0.894384	−	0.447300i	$-0.147614\pi$
$44$	189752.			0.335817
$45$	198871.	+	44374.4i	0.325333	+	0.0725920i
$46$	−179420.			−0.271780
$47$			468903.i			0.658779i	0.944194	+	0.329390i	$0.106843\pi$
$47$			468903.i			0.658779i	−0.944194	+	0.329390i	$0.893157\pi$
$48$		−	349905.i		−	0.456673i
$49$	−1.44232e6			−1.75136
$50$	100380.	−	213735.i	0.113566	−	0.241813i
$51$	876706.			0.925461
$52$			113646.i			0.112084i
$53$		−	1.60516e6i		−	1.48100i	−0.672058	−	0.740498i	$-0.734590\pi$
$53$		−	1.60516e6i		−	1.48100i	0.672058	−	0.740498i	$-0.265410\pi$
$54$	59491.8			0.0514137
$55$	−435491.	−	97171.8i	−0.352947	−	0.0787536i
$56$	−1.12316e6			−0.854639
$57$		−	1.05754e6i		−	0.756371i
$58$			199940.i			0.134555i
$59$	2.04044e6			1.29342			0.646712	−	0.762734i	$-0.276144\pi$
$59$	2.04044e6			1.29342			0.646712	+	0.762734i	$0.276144\pi$
$60$	−195354.	+	875508.i	−0.116759	+	0.523276i
$61$	−378667.			−0.213601			−0.106800	−	0.994280i	$-0.534061\pi$
$61$	−378667.			−0.213601			−0.106800	+	0.994280i	$0.534061\pi$
$62$			59434.6i			0.0316715i
$63$			1.09735e6i			0.552907i
$64$	1.25175e6			0.596880
$65$	58198.1	−	260824.i	0.0262852	−	0.117801i
$66$	−130276.			−0.0557777
$67$		−	4644.40i		−	0.00188655i	−1.00000		0.000943274i	$-0.999700\pi$
$67$		−	4644.40i		−	0.00188655i	1.00000		0.000943274i	$-0.000300253\pi$
$68$			3.85960e6i			1.48854i
$69$	−1.60276e6			−0.587350
$70$	1.24116e6	+	276941.i	0.432497	+	0.0965038i
$71$	−2.79333e6			−0.926229			−0.463115	−	0.886298i	$-0.653268\pi$
$71$	−2.79333e6			−0.926229			−0.463115	+	0.886298i	$0.653268\pi$
$72$			543941.i			0.171747i
$73$			2.01174e6i			0.605259i	0.953108	+	0.302630i	$0.0978645\pi$
$73$			2.01174e6i			0.605259i	−0.953108	+	0.302630i	$0.902136\pi$
$74$	1.13660e6			0.326058
$75$	896692.	−	1.90929e6i	0.245431	−	0.522587i
$76$	4.65570e6			1.21657
$77$		−	2.40299e6i		−	0.599838i
$78$		−	78024.8i		−	0.0186167i
$79$	−1.76767e6			−0.403372			−0.201686	−	0.979450i	$-0.564642\pi$
$79$	−1.76767e6			−0.403372			−0.201686	+	0.979450i	$0.564642\pi$
$80$	−3.53533e6	−	788844.i	−0.771996	−	0.172257i
$81$	531441.			0.111111
$82$		−	1.16367e6i		−	0.233067i
$83$			3.06625e6i			0.588619i	0.955710	+	0.294310i	$0.0950897\pi$
$83$			3.06625e6i			0.588619i	−0.955710	+	0.294310i	$0.904910\pi$
$84$	−4.83095e6			−0.889314
$85$	1.97649e6	−	8.85797e6i	0.349083	−	1.56447i
$86$	−1.40972e6			−0.238995
$87$			1.78606e6i			0.290790i
$88$		−	1.19113e6i		−	0.186324i
$89$	6.14397e6			0.923813			0.461907	−	0.886929i	$-0.347166\pi$
$89$	6.14397e6			0.923813			0.461907	+	0.886929i	$0.347166\pi$
$90$	134122.	−	601087.i	0.0193932	−	0.0869138i
$91$	1.43920e6			0.200205
$92$		−	7.05598e6i		−	0.944713i
$93$			530930.i			0.0684459i
$94$	1.41726e6			0.175995
$95$	−1.06851e7	−	2.38417e6i	−1.27863	−	0.285302i
$96$	−3.63627e6			−0.419476
$97$			3.02969e6i			0.337053i	0.985697	+	0.168526i	$0.0539008\pi$
$97$			3.02969e6i			0.337053i	−0.985697	+	0.168526i	$0.946099\pi$
$98$			4.35940e6i			0.467881i
$99$	−1.16376e6			−0.120542
$100$	8.40546e6	+	3.94759e6i	0.840546	+	0.394759i
$101$	7.17509e6			0.692951			0.346475	−	0.938059i	$-0.387378\pi$
$101$	7.17509e6			0.692951			0.346475	+	0.938059i	$0.387378\pi$
$102$		−	2.64984e6i		−	0.247240i
$103$			1.45694e7i			1.31374i	0.754002	+	0.656872i	$0.228121\pi$
$103$			1.45694e7i			1.31374i	−0.754002	+	0.656872i	$0.771879\pi$
$104$	713391.			0.0621886
$105$	1.10873e7	+	2.47392e6i	0.934678	+	0.208556i
$106$	−4.85160e6			−0.395653
$107$			1.64442e7i			1.29769i	0.760923	+	0.648843i	$0.224747\pi$
$107$			1.64442e7i			1.29769i	−0.760923	+	0.648843i	$0.775253\pi$
$108$			2.33961e6i			0.178715i
$109$	−2.34832e7			−1.73686			−0.868430	−	0.495812i	$-0.834871\pi$
$109$	−2.34832e7			−1.73686			−0.868430	+	0.495812i	$0.834871\pi$
$110$	−293701.	+	1.31627e6i	−0.0210393	+	0.0942910i
$111$	1.01532e7			0.704650
$112$		−	1.95075e7i		−	1.31202i
$113$		−	1.25747e7i		−	0.819826i	−0.912125	−	0.409913i	$-0.865559\pi$
$113$		−	1.25747e7i		−	0.819826i	0.912125	−	0.409913i	$-0.134441\pi$
$114$	−3.19641e6			−0.202067
$115$	−3.61335e6	+	1.61938e7i	−0.221548	+	0.992903i
$116$	−7.86294e6			−0.467716
$117$		−	696997.i		−	0.0402328i
$118$		−	6.16721e6i		−	0.345543i
$119$	4.88772e7			2.65884
$120$	5.49582e6	+	1.22629e6i	0.290334	+	0.0647827i
$121$	−1.69388e7			−0.869226
$122$			1.14452e6i			0.0570642i
$123$		−	1.03951e7i		−	0.503686i
$124$	−2.33736e6			−0.110091
$125$	−1.72694e7	−	1.33643e7i	−0.790846	−	0.612015i
$126$	3.31673e6			0.147711
$127$			2.41070e7i			1.04431i	0.852850	+	0.522155i	$0.174872\pi$
$127$			2.41070e7i			1.04431i	−0.852850	+	0.522155i	$0.825128\pi$
$128$		−	2.10220e7i		−	0.886012i
$129$	−1.25931e7			−0.516497
$130$	−788339.	−	175903.i	−0.0314711	−	0.00702219i
$131$	−2.08040e7			−0.808533			−0.404266	−	0.914641i	$-0.632473\pi$
$131$	−2.08040e7			−0.808533			−0.404266	+	0.914641i	$0.632473\pi$
$132$		−	5.12331e6i		−	0.193884i
$133$		−	5.89589e7i		−	2.17304i
$134$	−14037.7			−0.000503998
$135$	1.19811e6	−	5.36952e6i	0.0419110	−	0.187831i
$136$	2.42278e7			0.825901
$137$			9.94308e6i			0.330369i	0.986263	+	0.165184i	$0.0528219\pi$
$137$			9.94308e6i			0.330369i	−0.986263	+	0.165184i	$0.947178\pi$
$138$			4.84434e6i			0.156912i
$139$	−1.47414e7			−0.465571			−0.232785	−	0.972528i	$-0.574784\pi$
$139$	−1.47414e7			−0.465571			−0.232785	+	0.972528i	$0.574784\pi$
$140$	−1.08912e7	+	4.88105e7i	−0.335448	+	1.50337i
$141$	1.26604e7			0.380347
$142$			8.44284e6i			0.247445i
$143$			1.52629e6i			0.0436478i
$144$	−9.44743e6			−0.263660
$145$	1.80458e7	+	4.02660e6i	0.491574	+	0.109686i
$146$	6.08047e6			0.161697
$147$			3.89425e7i			1.01115i
$148$			4.46985e7i			1.13338i
$149$	2.21899e7			0.549545			0.274772	−	0.961509i	$-0.411398\pi$
$149$	2.21899e7			0.549545			0.274772	+	0.961509i	$0.411398\pi$
$150$	−5.77084e6	−	2.71025e6i	−0.139611	−	0.0655676i
$151$	4.90759e7			1.15998			0.579988	−	0.814625i	$-0.303057\pi$
$151$	4.90759e7			1.15998			0.579988	+	0.814625i	$0.303057\pi$
$152$		−	2.92252e7i		−	0.675001i
$153$		−	2.36711e7i		−	0.534315i
$154$	−7.26302e6			−0.160249
$155$	5.36436e6	+	1.19696e6i	0.115706	+	0.0258177i
$156$	3.06845e6			0.0647118
$157$		−	6.21368e6i		−	0.128145i	−0.997945	−	0.0640723i	$-0.979591\pi$
$157$		−	6.21368e6i		−	0.128145i	0.997945	−	0.0640723i	$-0.0204088\pi$
$158$			5.34277e6i			0.107762i
$159$	−4.33394e7			−0.855054
$160$	−8.19780e6	+	3.67398e7i	−0.158226	+	0.709115i
$161$	−8.93555e7			−1.68745
$162$		−	1.60628e6i		−	0.0296837i
$163$		−	1.54205e7i		−	0.278895i	−0.990229	−	0.139448i	$-0.955467\pi$
$163$		−	1.54205e7i		−	0.278895i	0.990229	−	0.139448i	$-0.0445327\pi$
$164$	4.57632e7			0.810145
$165$	−2.62364e6	+	1.17583e7i	−0.0454684	+	0.203774i
$166$	9.26773e6			0.157252
$167$			7.74611e6i			0.128699i	0.997927	+	0.0643496i	$0.0204973\pi$
$167$			7.74611e6i			0.128699i	−0.997927	+	0.0643496i	$0.979503\pi$
$168$			3.03253e7i			0.493426i
$169$	6.18344e7			0.985432
$170$	−2.67732e7	−	5.97394e6i	−0.417954	−	0.0932588i
$171$	−2.85536e7			−0.436691
$172$		−	5.54397e7i		−	0.830751i
$173$		−	1.00086e7i		−	0.146965i	−0.997297	−	0.0734824i	$-0.976589\pi$
$173$		−	1.00086e7i		−	0.146965i	0.997297	−	0.0734824i	$-0.0234113\pi$
$174$	5.39837e6			0.0776855
$175$	4.99915e7	−	1.06445e8i	0.705120	−	1.50139i
$176$	2.06881e7			0.286040
$177$		−	5.50918e7i		−	0.746758i
$178$		−	1.85701e7i		−	0.246800i
$179$	−1.09231e8			−1.42350			−0.711751	−	0.702431i	$-0.752098\pi$
$179$	−1.09231e8			−1.42350			−0.711751	+	0.702431i	$0.752098\pi$
$180$	2.36387e7	+	5.27454e6i	0.302114	+	0.0674111i
$181$	−1.27824e7			−0.160228			−0.0801138	−	0.996786i	$-0.525528\pi$
$181$	−1.27824e7			−0.160228			−0.0801138	+	0.996786i	$0.525528\pi$
$182$		−	4.34996e6i		−	0.0534854i
$183$			1.02240e7i			0.123322i
$184$	−4.42924e7			−0.524164
$185$	2.28900e7	−	1.02585e8i	0.265794	−	1.19120i
$186$	1.60473e6			0.0182855
$187$			5.18352e7i			0.579668i
$188$			5.57359e7i			0.611762i
$189$	2.96284e7			0.319221
$190$	−7.20616e6	+	3.22956e7i	−0.0762195	+	0.341590i
$191$	1.54482e8			1.60421			0.802103	−	0.597185i	$-0.203714\pi$
$191$	1.54482e8			1.60421			0.802103	+	0.597185i	$0.203714\pi$
$192$		−	3.37972e7i		−	0.344609i
$193$			1.51365e8i			1.51556i	0.652507	+	0.757782i	$0.273717\pi$
$193$			1.51365e8i			1.51556i	−0.652507	+	0.757782i	$0.726283\pi$
$194$	9.15724e6			0.0900448
$195$	−7.04225e6	−	1.57135e6i	−0.0680127	−	0.0151758i
$196$	−1.71440e8			−1.62636
$197$			1.23829e8i			1.15396i	0.816758	+	0.576981i	$0.195769\pi$
$197$			1.23829e8i			1.15396i	−0.816758	+	0.576981i	$0.804231\pi$
$198$			3.51745e6i			0.0322033i
$199$	−4.77920e7			−0.429902			−0.214951	−	0.976625i	$-0.568959\pi$
$199$	−4.77920e7			−0.429902			−0.214951	+	0.976625i	$0.568959\pi$
$200$	2.47801e7	−	5.27635e7i	0.219028	−	0.466368i
$201$	−125399.			−0.00108920
$202$		−	2.16867e7i		−	0.185124i
$203$			9.95748e7i			0.835436i
$204$	1.04209e8			0.859410
$205$	−1.05029e8	−	2.34352e7i	−0.851470	−	0.189990i
$206$	4.40359e7			0.350971
$207$			4.32745e7i			0.339107i
$208$			1.23905e7i			0.0954701i
$209$	6.25270e7			0.473757
$210$	7.47741e6	−	3.35112e7i	0.0557165	−	0.249702i
$211$	6.28256e7			0.460413			0.230207	−	0.973142i	$-0.426060\pi$
$211$	6.28256e7			0.460413			0.230207	+	0.973142i	$0.426060\pi$
$212$		−	1.90797e8i		−	1.37530i
$213$			7.54200e7i			0.534759i
$214$	4.97025e7			0.346681
$215$	−2.83905e7	+	1.27237e8i	−0.194822	+	0.873128i
$216$	1.46864e7			0.0991580
$217$			2.95999e7i			0.196644i
$218$			7.09779e7i			0.464008i
$219$	5.43169e7			0.349447
$220$	−5.17644e7	−	1.15503e7i	−0.327757	−	0.0731329i
$221$	−3.10451e7			−0.193473
$222$		−	3.06881e7i		−	0.188250i
$223$		−	5.17882e6i		−	0.0312726i	−0.999878	−	0.0156363i	$-0.995023\pi$
$223$		−	5.17882e6i		−	0.0312726i	0.999878	−	0.0156363i	$-0.00497739\pi$
$224$	−2.02726e8			−1.20515
$225$	−5.15510e7	−	2.42107e7i	−0.301716	−	0.141699i
$226$	−3.80069e7			−0.219019
$227$		−	2.43478e8i		−	1.38156i	−0.723067	−	0.690778i	$-0.757268\pi$
$227$		−	2.43478e8i		−	1.38156i	0.723067	−	0.690778i	$-0.242732\pi$
$228$		−	1.25704e8i		−	0.702388i
$229$	1.09734e8			0.603832			0.301916	−	0.953335i	$-0.402374\pi$
$229$	1.09734e8			0.603832			0.301916	+	0.953335i	$0.402374\pi$
$230$	4.89457e7	+	1.09213e7i	0.265257	+	0.0591873i
$231$	−6.48806e7			−0.346317
$232$			4.93580e7i			0.259507i
$233$		−	1.85771e8i		−	0.962129i	−0.876685	−	0.481064i	$-0.840250\pi$
$233$		−	1.85771e8i		−	0.962129i	0.876685	−	0.481064i	$-0.159750\pi$
$234$	−2.10667e6			−0.0107483
$235$	2.85422e7	−	1.27917e8i	0.143466	−	0.642968i
$236$	2.42535e8			1.20111
$237$			4.77270e7i			0.232887i
$238$		−	1.47731e8i		−	0.710318i
$239$	−3.06316e8			−1.45137			−0.725684	−	0.688028i	$-0.758477\pi$
$239$	−3.06316e8			−1.45137			−0.725684	+	0.688028i	$0.758477\pi$
$240$	−2.12988e7	+	9.54540e7i	−0.0994525	+	0.445712i
$241$	−5.78177e7			−0.266073			−0.133037	−	0.991111i	$-0.542473\pi$
$241$	−5.78177e7			−0.266073			−0.133037	+	0.991111i	$0.542473\pi$
$242$			5.11973e7i			0.232217i
$243$		−	1.43489e7i		−	0.0641500i
$244$	−4.50101e7			−0.198356
$245$	3.93464e8	+	8.77942e7i	1.70932	+	0.381403i
$246$	−3.14191e7			−0.134561
$247$			3.74486e7i			0.158124i
$248$			1.46723e7i			0.0610826i
$249$	8.27888e7			0.339840
$250$	−4.03936e7	+	5.21967e7i	−0.163502	+	0.211277i
$251$	−3.90109e8			−1.55714			−0.778571	−	0.627557i	$-0.784055\pi$
$251$	−3.90109e8			−1.55714			−0.778571	+	0.627557i	$0.784055\pi$
$252$			1.30436e8i			0.513446i
$253$		−	9.47632e7i		−	0.367890i
$254$	7.28632e7			0.278991
$255$	−2.39165e8	−	5.33653e7i	−0.903249	−	0.201543i
$256$	9.66848e7			0.360179
$257$			5.32200e8i			1.95573i	0.209237	+	0.977865i	$0.432902\pi$
$257$			5.32200e8i			1.95573i	−0.209237	+	0.977865i	$0.567098\pi$
$258$			3.80625e7i			0.137984i
$259$	5.66053e8			2.02445
$260$	6.91768e6	−	3.10027e7i	0.0244092	−	0.109394i
$261$	4.82237e7			0.167888
$262$			6.28800e7i			0.216002i
$263$		−	3.12888e8i		−	1.06058i	−0.847816	−	0.530291i	$-0.822083\pi$
$263$		−	3.12888e8i		−	1.06058i	0.847816	−	0.530291i	$-0.177917\pi$
$264$	−3.21605e7			−0.107574
$265$	−9.77068e7	+	4.37889e8i	−0.322526	+	1.44545i
$266$	−1.78203e8			−0.580536
$267$		−	1.65887e8i		−	0.533364i
$268$		−	552054.i		−	0.00175190i
$269$	−2.95534e8			−0.925710			−0.462855	−	0.886434i	$-0.653175\pi$
$269$	−2.95534e8			−0.925710			−0.462855	+	0.886434i	$0.653175\pi$
$270$	−1.62294e7	−	3.62128e6i	−0.0501797	−	0.0111967i
$271$	3.96274e8			1.20949			0.604746	−	0.796419i	$-0.293275\pi$
$271$	3.96274e8			1.20949			0.604746	+	0.796419i	$0.293275\pi$
$272$			4.20800e8i			1.26790i
$273$		−	3.88583e7i		−	0.115588i
$274$	3.00529e7			0.0882591
$275$	1.12887e8	+	5.30169e7i	0.327325	+	0.153727i
$276$	−1.90511e8			−0.545430
$277$		−	4.00402e8i		−	1.13192i	−0.824432	−	0.565961i	$-0.808505\pi$
$277$		−	4.00402e8i		−	1.13192i	0.824432	−	0.565961i	$-0.191495\pi$
$278$			4.45557e7i			0.124379i
$279$	1.43351e7			0.0395172
$280$	3.06397e8	+	6.83669e7i	0.834127	+	0.186120i
$281$	3.58146e8			0.962916			0.481458	−	0.876469i	$-0.340107\pi$
$281$	3.58146e8			0.962916			0.481458	+	0.876469i	$0.340107\pi$
$282$		−	3.82659e7i		−	0.101611i
$283$		−	3.95688e8i		−	1.03777i	−0.854845	−	0.518884i	$-0.826348\pi$
$283$		−	3.95688e8i		−	1.03777i	0.854845	−	0.518884i	$-0.173652\pi$
$284$	−3.32028e8			−0.860123
$285$	−6.43727e7	+	2.88497e8i	−0.164719	+	0.738217i
$286$	4.61322e6			0.0116606
$287$		−	5.79536e8i		−	1.44708i
$288$			9.81793e7i			0.242184i
$289$	−6.44000e8			−1.56943
$290$	1.21704e7	−	5.45435e7i	0.0293029	−	0.131326i
$291$	8.18017e7			0.194597
$292$			2.39124e8i			0.562061i
$293$		−	1.93280e8i		−	0.448900i	−0.974486	−	0.224450i	$-0.927941\pi$
$293$		−	1.93280e8i		−	0.448900i	0.974486	−	0.224450i	$-0.0720586\pi$
$294$	1.17704e8			0.270131
$295$	−5.56631e8	−	1.24202e8i	−1.26238	−	0.281677i
$296$	2.80585e8			0.628845
$297$			3.14214e7i			0.0695951i
$298$		−	6.70688e7i		−	0.146813i
$299$	5.67555e7			0.122789
$300$	1.06585e8	−	2.26947e8i	0.227914	−	0.485289i
$301$	−7.02077e8			−1.48389
$302$		−	1.48332e8i		−	0.309892i
$303$		−	1.93727e8i		−	0.400075i
$304$	5.07597e8			1.03624
$305$	1.03300e8	+	2.30496e7i	0.208474	+	0.0465171i
$306$	−7.15457e7			−0.142744
$307$		−	2.11267e8i		−	0.416722i	−0.978052	−	0.208361i	$-0.933187\pi$
$307$		−	2.11267e8i		−	0.416722i	0.978052	−	0.208361i	$-0.0668129\pi$
$308$		−	2.85630e8i		−	0.557027i
$309$	3.93373e8			0.758490
$310$	3.61780e6	−	1.62138e7i	0.00689729	−	0.0309113i
$311$	2.59442e8			0.489079			0.244540	−	0.969639i	$-0.421363\pi$
$311$	2.59442e8			0.489079			0.244540	+	0.969639i	$0.421363\pi$
$312$		−	1.92616e7i		−	0.0359046i
$313$			9.29772e8i			1.71384i	0.515446	+	0.856922i	$0.327626\pi$
$313$			9.29772e8i			1.71384i	−0.515446	+	0.856922i	$0.672374\pi$
$314$	−1.87808e7			−0.0342343
$315$	6.67958e7	−	2.99356e8i	0.120410	−	0.539637i
$316$	−2.10113e8			−0.374583
$317$		−	1.63510e8i		−	0.288295i	−0.989556	−	0.144147i	$-0.953956\pi$
$317$		−	1.63510e8i		−	0.288295i	0.989556	−	0.144147i	$-0.0460439\pi$
$318$			1.30993e8i			0.228431i
$319$	−1.05601e8			−0.182138
$320$	−3.41477e8	−	7.61942e7i	−0.582554	−	0.129986i
$321$	4.43993e8			0.749219
$322$			2.70077e8i			0.450808i
$323$			1.27181e9i			2.09997i
$324$	6.31695e7			0.103181
$325$	−3.17529e7	+	6.76102e7i	−0.0513087	+	0.109250i
$326$	−4.66083e7			−0.0745079
$327$			6.34047e8i			1.00278i
$328$		−	2.87269e8i		−	0.449500i
$329$	7.05828e8			1.09273
$330$	3.55393e7	+	7.92993e6i	0.0544390	+	0.0121470i
$331$	−2.57948e8			−0.390962			−0.195481	−	0.980707i	$-0.562627\pi$
$331$	−2.57948e8			−0.390962			−0.195481	+	0.980707i	$0.562627\pi$
$332$			3.64469e8i			0.546609i
$333$		−	2.74137e8i		−	0.406830i
$334$	2.34126e7			0.0343825
$335$	−282706.	+	1.26699e6i	−0.000410845	+	0.00184127i
$336$	−5.26704e8			−0.757493
$337$			2.34282e8i			0.333452i	0.986003	+	0.166726i	$0.0533196\pi$
$337$			2.34282e8i			0.333452i	−0.986003	+	0.166726i	$0.946680\pi$
$338$		−	1.86894e8i		−	0.263261i
$339$	−3.39516e8			−0.473327
$340$	2.34935e8	−	1.05290e9i	0.324169	−	1.45282i
$341$	−3.13912e7			−0.0428715
$342$			8.63030e7i			0.116663i
$343$			9.31426e8i			1.24629i
$344$	−3.48011e8			−0.460933
$345$	4.37233e8	+	9.75605e7i	0.573253	+	0.127911i
$346$	−3.02511e7			−0.0392622
$347$		−	6.73974e8i		−	0.865944i	−0.901407	−	0.432972i	$-0.857465\pi$
$347$		−	6.73974e8i		−	0.865944i	0.901407	−	0.432972i	$-0.142535\pi$
$348$			2.12299e8i			0.270036i
$349$	−8.77464e8			−1.10494			−0.552472	−	0.833531i	$-0.686315\pi$
$349$	−8.77464e8			−1.10494			−0.552472	+	0.833531i	$0.686315\pi$
$350$	−3.21730e8	−	1.51099e8i	−0.401101	−	0.188375i
$351$	−1.88189e7			−0.0232284
$352$		−	2.14994e8i		−	0.262741i
$353$			6.75088e8i			0.816863i	0.912789	+	0.408431i	$0.133924\pi$
$353$			6.75088e8i			0.816863i	−0.912789	+	0.408431i	$0.866076\pi$
$354$	−1.66515e8			−0.199499
$355$	7.62021e8	+	1.70031e8i	0.903999	+	0.201711i
$356$	7.30300e8			0.857880
$357$		−	1.31969e9i		−	1.53508i
$358$			3.30149e8i			0.380294i
$359$	7.40244e8			0.844393			0.422196	−	0.906504i	$-0.361259\pi$
$359$	7.40244e8			0.844393			0.422196	+	0.906504i	$0.361259\pi$
$360$	3.31098e7	−	1.48387e8i	0.0374023	−	0.167624i
$361$	6.40271e8			0.716290
$362$			3.86348e7i			0.0428054i
$363$			4.57346e8i			0.501848i
$364$	1.71069e8			0.185916
$365$	1.22455e8	−	5.48802e8i	0.131811	−	0.590732i
$366$	3.09020e7			0.0329460
$367$		−	1.05986e9i		−	1.11922i	−0.828755	−	0.559611i	$-0.810951\pi$
$367$		−	1.05986e9i		−	1.11922i	0.828755	−	0.559611i	$-0.189049\pi$
$368$		−	7.69291e8i		−	0.804680i
$369$	−2.80667e8			−0.290803
$370$	−3.10063e8	−	6.91849e7i	−0.318232	−	0.0710077i
$371$	−2.41622e9			−2.45656
$372$			6.31088e7i			0.0635608i
$373$		−	1.28239e9i		−	1.27950i	−0.768585	−	0.639748i	$-0.779039\pi$
$373$		−	1.28239e9i		−	1.27950i	0.768585	−	0.639748i	$-0.220961\pi$
$374$	1.56672e8			0.154860
$375$	−3.60837e8	+	4.66274e8i	−0.353347	+	0.456595i
$376$	3.49870e8			0.339429
$377$		−	6.32465e7i		−	0.0607913i
$378$		−	8.95516e7i		−	0.0852810i
$379$	2.97355e8			0.280568			0.140284	−	0.990111i	$-0.455198\pi$
$379$	2.97355e8			0.280568			0.140284	+	0.990111i	$0.455198\pi$
$380$	−1.27008e9	−	2.83394e8i	−1.18737	−	0.264940i
$381$	6.50888e8			0.602933
$382$		−	4.66920e8i		−	0.428569i
$383$			8.10508e8i			0.737160i	0.929596	+	0.368580i	$0.120156\pi$
$383$			8.10508e8i			0.737160i	−0.929596	+	0.368580i	$0.879844\pi$
$384$	−5.67594e8			−0.511539
$385$	−1.46270e8	+	6.55535e8i	−0.130630	+	0.585441i
$386$	4.57500e8			0.404888
$387$			3.40013e8i			0.298200i
$388$			3.60123e8i			0.312997i
$389$	3.38899e8			0.291908			0.145954	−	0.989291i	$-0.453375\pi$
$389$	3.38899e8			0.291908			0.145954	+	0.989291i	$0.453375\pi$
$390$	−4.74939e6	+	2.12852e7i	−0.00405426	+	0.0181698i
$391$	1.92750e9			1.63071
$392$			1.07618e9i			0.902368i
$393$			5.61708e8i			0.466807i
$394$	3.74273e8			0.308285
$395$	4.82220e8	+	1.07598e8i	0.393691	+	0.0878448i
$396$	−1.38329e8			−0.111939
$397$			1.00265e9i			0.804235i	0.915588	+	0.402117i	$0.131726\pi$
$397$			1.00265e9i			0.804235i	−0.915588	+	0.402117i	$0.868274\pi$
$398$			1.44451e8i			0.114850i
$399$	−1.59189e9			−1.25461
$400$	9.16421e8	+	4.30393e8i	0.715954	+	0.336245i
$401$	−1.09336e9			−0.846755			−0.423377	−	0.905953i	$-0.639156\pi$
$401$	−1.09336e9			−0.846755			−0.423377	+	0.905953i	$0.639156\pi$
$402$			379017.i			0.000290983i
$403$		−	1.88008e7i		−	0.0143090i
$404$	8.52863e8			0.643494
$405$	−1.44977e8	−	3.23490e7i	−0.108444	−	0.0241973i
$406$	3.00964e8			0.223190
$407$			6.00309e8i			0.441362i
$408$		−	6.54151e8i		−	0.476834i
$409$	−1.09511e9			−0.791453			−0.395726	−	0.918368i	$-0.629507\pi$
$409$	−1.09511e9			−0.791453			−0.395726	+	0.918368i	$0.629507\pi$
$410$	−7.08328e7	+	3.17449e8i	−0.0507564	+	0.227473i
$411$	2.68463e8			0.190738
$412$			1.73178e9i			1.21998i
$413$		−	3.07142e9i		−	2.14543i
$414$	1.30797e8			0.0905935
$415$	1.86644e8	−	8.36473e8i	0.128187	−	0.574492i
$416$	1.28764e8			0.0876938
$417$			3.98016e8i			0.268797i
$418$		−	1.88988e8i		−	0.126566i
$419$	1.75196e9			1.16352			0.581762	−	0.813359i	$-0.302364\pi$
$419$	1.75196e9			1.16352			0.581762	+	0.813359i	$0.302364\pi$
$420$	1.31788e9	+	2.94061e8i	0.867969	+	0.193671i
$421$	−4.58026e8			−0.299160			−0.149580	−	0.988750i	$-0.547792\pi$
$421$	−4.58026e8			−0.299160			−0.149580	+	0.988750i	$0.547792\pi$
$422$		−	1.89890e8i		−	0.123001i
$423$		−	3.41830e8i		−	0.219593i
$424$	−1.19769e9			−0.763068
$425$	−1.07837e9	+	2.29614e9i	−0.681410	+	1.45090i
$426$	2.27957e8			0.142863
$427$			5.69999e8i			0.354304i
$428$			1.95463e9i			1.20507i
$429$	4.12099e7			0.0252000
$430$	3.84573e8	+	8.58103e7i	0.233259	+	0.0520475i
$431$	−7.60129e8			−0.457317			−0.228658	−	0.973507i	$-0.573434\pi$
$431$	−7.60129e8			−0.457317			−0.228658	+	0.973507i	$0.573434\pi$
$432$			2.55081e8i			0.152224i
$433$		−	2.48461e9i		−	1.47079i	−0.677638	−	0.735396i	$-0.736996\pi$
$433$		−	2.48461e9i		−	1.47079i	0.677638	−	0.735396i	$-0.263004\pi$
$434$	8.94656e7			0.0525342
$435$	1.08718e8	−	4.87238e8i	0.0633271	−	0.283811i
$436$	−2.79132e9			−1.61290
$437$		−	2.32508e9i		−	1.33276i
$438$		−	1.64173e8i		−	0.0933558i
$439$	−1.69934e8			−0.0958637			−0.0479318	−	0.998851i	$-0.515263\pi$
$439$	−1.69934e8			−0.0958637			−0.0479318	+	0.998851i	$0.515263\pi$
$440$	−7.25044e7	+	3.24940e8i	−0.0405770	+	0.181852i
$441$	1.05145e9			0.583785
$442$			9.38337e7i			0.0516870i
$443$		−	1.53246e9i		−	0.837481i	−0.908106	−	0.418741i	$-0.862472\pi$
$443$		−	1.53246e9i		−	0.837481i	0.908106	−	0.418741i	$-0.137528\pi$
$444$	1.20686e9			0.654359
$445$	−1.67607e9	−	3.73985e8i	−0.901640	−	0.201184i
$446$	−1.56530e7			−0.00835458
$447$		−	5.99127e8i		−	0.317280i
$448$		−	1.88423e9i		−	0.990057i
$449$	3.45325e9			1.80039			0.900194	−	0.435489i	$-0.143425\pi$
$449$	3.45325e9			1.80039			0.900194	+	0.435489i	$0.143425\pi$
$450$	−7.31767e7	+	1.55813e8i	−0.0378555	+	0.0806044i
$451$	6.14609e8			0.315486
$452$		−	1.49468e9i		−	0.761315i
$453$		−	1.32505e9i		−	0.669713i
$454$	−7.35910e8			−0.369087
$455$	−3.92612e8	−	8.76042e7i	−0.195400	−	0.0435999i
$456$	−7.89080e8			−0.389712
$457$		−	7.56279e8i		−	0.370659i	−0.982676	−	0.185330i	$-0.940665\pi$
$457$		−	7.56279e8i		−	0.370659i	0.982676	−	0.185330i	$-0.0593353\pi$
$458$		−	3.31670e8i		−	0.161316i
$459$	−6.39118e8			−0.308487
$460$	−4.29499e8	+	1.92487e9i	−0.205736	+	0.922038i
$461$	−8.78272e8			−0.417518			−0.208759	−	0.977967i	$-0.566942\pi$
$461$	−8.78272e8			−0.417518			−0.208759	+	0.977967i	$0.566942\pi$
$462$			1.96101e8i			0.0925196i
$463$			3.54973e9i			1.66212i	0.556185	+	0.831058i	$0.312264\pi$
$463$			3.54973e9i			1.66212i	−0.556185	+	0.831058i	$0.687736\pi$
$464$	−8.57273e8			−0.398388
$465$	3.23179e7	−	1.44838e8i	0.0149059	−	0.0668031i
$466$	−5.61493e8			−0.257036
$467$			3.03363e9i			1.37833i	0.724605	+	0.689165i	$0.242022\pi$
$467$			3.03363e9i			1.37833i	−0.724605	+	0.689165i	$0.757978\pi$
$468$		−	8.28482e7i		−	0.0373613i
$469$	−6.99111e6			−0.00312926
$470$	−3.86627e8	−	8.62688e7i	−0.171771	−	0.0383275i
$471$	−1.67769e8			−0.0739843
$472$		−	1.52246e9i		−	0.666423i
$473$		−	7.44566e8i		−	0.323511i
$474$	1.44255e8			0.0622165
$475$	2.76976e9	+	1.30081e9i	1.18581	+	0.556910i
$476$	5.80977e9			2.46908
$477$			1.17017e9i			0.493666i
$478$			9.25840e8i			0.387738i
$479$	−3.00343e9			−1.24866			−0.624328	−	0.781162i	$-0.714627\pi$
$479$	−3.00343e9			−1.24866			−0.624328	+	0.781162i	$0.714627\pi$
$480$	9.91974e8	+	2.21341e8i	0.409408	+	0.0913518i
$481$	−3.59537e8			−0.147311
$482$			1.74754e8i			0.0710823i
$483$			2.41260e9i			0.974249i
$484$	−2.01342e9			−0.807189
$485$	1.84418e8	−	8.26500e8i	0.0734020	−	0.328963i
$486$	−4.33695e7			−0.0171379
$487$			3.28649e8i			0.128938i	0.997920	+	0.0644690i	$0.0205354\pi$
$487$			3.28649e8i			0.128938i	−0.997920	+	0.0644690i	$0.979465\pi$
$488$			2.82541e8i			0.110056i
$489$	−4.16353e8			−0.161020
$490$	2.65358e8	−	1.18924e9i	0.101893	−	0.456651i
$491$	−6.97275e8			−0.265839			−0.132919	−	0.991127i	$-0.542435\pi$
$491$	−6.97275e8			−0.265839			−0.132919	+	0.991127i	$0.542435\pi$
$492$		−	1.23561e9i		−	0.467737i
$493$		−	2.14794e9i		−	0.807344i
$494$	1.13188e8			0.0422433
$495$	3.17473e8	+	7.08382e7i	0.117649	+	0.0262512i
$496$	−2.54835e8			−0.0937721
$497$			4.20474e9i			1.53636i
$498$		−	2.50229e8i		−	0.0907893i
$499$	3.18226e9			1.14652			0.573262	−	0.819372i	$-0.305678\pi$
$499$	3.18226e9			1.14652			0.573262	+	0.819372i	$0.305678\pi$
$500$	−2.05272e9	−	1.58854e9i	−0.734403	−	0.568335i
$501$	2.09145e8			0.0743046
$502$			1.17910e9i			0.415996i
$503$			4.07550e9i			1.42788i	0.700205	+	0.713942i	$0.253092\pi$
$503$			4.07550e9i			1.42788i	−0.700205	+	0.713942i	$0.746908\pi$
$504$	8.18782e8			0.284880
$505$	−1.95736e9	−	4.36749e8i	−0.676319	−	0.150908i
$506$	−2.86421e8			−0.0982830
$507$		−	1.66953e9i		−	0.568939i
$508$			2.86546e9i			0.969777i
$509$	−4.18725e8			−0.140740			−0.0703699	−	0.997521i	$-0.522418\pi$
$509$	−4.18725e8			−0.140740			−0.0703699	+	0.997521i	$0.522418\pi$
$510$	−1.61296e8	+	7.22876e8i	−0.0538430	+	0.241306i
$511$	3.02822e9			1.00396
$512$		−	2.98305e9i		−	0.982235i
$513$			7.70946e8i			0.252124i
$514$	1.60857e9			0.522480
$515$	8.86842e8	−	3.97452e9i	0.286102	−	1.28221i
$516$	−1.49687e9			−0.479634
$517$			7.48544e8i			0.238232i
$518$		−	1.71089e9i		−	0.540839i
$519$	−2.70233e8			−0.0848502
$520$	−1.94613e8	−	4.34243e7i	−0.0606960	−	0.0135432i
$521$	−2.02201e9			−0.626400			−0.313200	−	0.949687i	$-0.601401\pi$
$521$	−2.02201e9			−0.626400			−0.313200	+	0.949687i	$0.601401\pi$
$522$		−	1.45756e8i		−	0.0448518i
$523$			1.87087e8i			0.0571857i	0.999591	+	0.0285928i	$0.00910262\pi$
$523$			1.87087e8i			0.0571857i	−0.999591	+	0.0285928i	$0.990897\pi$
$524$	−2.47286e9			−0.750827
$525$	−2.87402e9	−	1.34977e9i	−0.866826	−	0.407101i
$526$	−9.45704e8			−0.283338
$527$		−	6.38504e8i		−	0.190032i
$528$		−	5.58579e8i		−	0.165145i
$529$	−1.18962e8			−0.0349392
$530$	1.32352e9	+	2.95318e8i	0.386157	+	0.0861638i
$531$	−1.48748e9			−0.431141
$532$		−	7.00812e9i		−	2.01795i
$533$			3.68101e8i			0.105298i
$534$	−5.01393e8			−0.142490
$535$	1.00096e9	−	4.48598e9i	0.282605	−	1.26654i
$536$	−3.46541e6			−0.000972025
$537$			2.94922e9i			0.821860i
$538$			8.93251e8i			0.247307i
$539$	−2.30248e9			−0.633337
$540$	1.42413e8	−	6.38246e8i	0.0389198	−	0.174425i
$541$	3.94897e9			1.07224			0.536122	−	0.844140i	$-0.319889\pi$
$541$	3.94897e9			1.07224			0.536122	+	0.844140i	$0.319889\pi$
$542$		−	1.19774e9i		−	0.323120i
$543$			3.45125e8i			0.0925075i
$544$	4.37303e9			1.16463
$545$	6.40622e9	+	1.42943e9i	1.69517	+	0.378247i
$546$	−1.17449e8			−0.0308798
$547$		−	1.71845e9i		−	0.448932i	−0.974482	−	0.224466i	$-0.927936\pi$
$547$		−	1.71845e9i		−	0.448932i	0.974482	−	0.224466i	$-0.0720638\pi$
$548$			1.18188e9i			0.306790i
$549$	2.76048e8			0.0712003
$550$	1.60243e8	−	3.41201e8i	0.0410687	−	0.0874461i
$551$	−2.59099e9			−0.659835
$552$			1.19589e9i			0.302626i
$553$			2.66083e9i			0.669082i
$554$	−1.21021e9			−0.302397
$555$	−2.76980e9	−	6.18030e8i	−0.687738	−	0.153456i
$556$	−1.75222e9			−0.432342
$557$		−	3.22406e9i		−	0.790516i	−0.918570	−	0.395258i	$-0.870655\pi$
$557$		−	3.22406e9i		−	0.790516i	0.918570	−	0.395258i	$-0.129345\pi$
$558$		−	4.33278e7i		−	0.0105572i
$559$	4.45935e8			0.107977
$560$	−1.18743e9	+	5.32166e9i	−0.285726	+	1.28053i
$561$	1.39955e9			0.334671
$562$		−	1.08250e9i		−	0.257246i
$563$			5.69907e9i			1.34594i	0.739671	+	0.672968i	$0.234981\pi$
$563$			5.69907e9i			1.34594i	−0.739671	+	0.672968i	$0.765019\pi$
$564$	1.50487e9			0.353201
$565$	−7.65423e8	+	3.43037e9i	−0.178539	+	0.800149i
$566$	−1.19596e9			−0.277243
$567$		−	7.99966e8i		−	0.184302i
$568$			2.08424e9i			0.477230i
$569$	−3.70804e9			−0.843823			−0.421912	−	0.906637i	$-0.638641\pi$
$569$	−3.70804e9			−0.843823			−0.421912	+	0.906637i	$0.638641\pi$
$570$	8.71980e8	+	1.94566e8i	0.197217	+	0.0440054i
$571$	−3.34811e9			−0.752615			−0.376308	−	0.926495i	$-0.622806\pi$
$571$	−3.34811e9			−0.752615			−0.376308	+	0.926495i	$0.622806\pi$
$572$			1.81422e8i			0.0405326i
$573$		−	4.17101e9i		−	0.926189i
$574$	−1.75164e9			−0.386593
$575$	1.97144e9	−	4.19773e9i	0.432461	−	0.920824i
$576$	−9.12524e8			−0.198960
$577$			8.37348e9i			1.81464i	0.420441	+	0.907320i	$0.361876\pi$
$577$			8.37348e9i			1.81464i	−0.420441	+	0.907320i	$0.638124\pi$
$578$			1.94649e9i			0.419280i
$579$	4.08685e9			0.875012
$580$	2.14501e9	+	4.78620e8i	0.456490	+	0.101857i
$581$	4.61556e9			0.976355
$582$		−	2.47245e8i		−	0.0519874i
$583$		−	2.56244e9i		−	0.535568i
$584$	1.50105e9			0.311854
$585$	−4.24264e7	+	1.90141e8i	−0.00876174	+	0.0392672i
$586$	−5.84188e8			−0.119925
$587$		−	9.32230e8i		−	0.190235i	−0.995466	−	0.0951174i	$-0.969677\pi$
$587$		−	9.32230e8i		−	0.190235i	0.995466	−	0.0951174i	$-0.0303226\pi$
$588$			4.62889e9i			0.938979i
$589$	−7.70205e8			−0.155311
$590$	−3.75400e8	+	1.68241e9i	−0.0752509	+	0.337249i
$591$	3.34339e9			0.666240
$592$			4.87334e9i			0.965385i
$593$		−	4.69858e9i		−	0.925283i	−0.886545	−	0.462642i	$-0.846902\pi$
$593$		−	4.69858e9i		−	0.925283i	0.886545	−	0.462642i	$-0.153098\pi$
$594$	9.49712e7			0.0185926
$595$	−1.33337e10	−	2.97517e9i	−2.59502	−	0.579032i
$596$	2.63759e9			0.510323
$597$			1.29038e9i			0.248204i
$598$		−	1.71543e8i		−	0.0328035i
$599$	2.66487e9			0.506620			0.253310	−	0.967385i	$-0.418481\pi$
$599$	2.66487e9			0.506620			0.253310	+	0.967385i	$0.418481\pi$
$600$	−1.42461e9	−	6.69064e8i	−0.269258	−	0.126456i
$601$	−1.21738e9			−0.228753			−0.114376	−	0.993437i	$-0.536487\pi$
$601$	−1.21738e9			−0.228753			−0.114376	+	0.993437i	$0.536487\pi$
$602$			2.12203e9i			0.396427i
$603$			3.38577e6i			0.000628849i
$604$	5.83339e9			1.07719
$605$	4.62089e9	+	1.03107e9i	0.848364	+	0.189297i
$606$	−5.85540e8			−0.106881
$607$		−	2.74917e9i		−	0.498932i	−0.968384	−	0.249466i	$-0.919745\pi$
$607$		−	2.74917e9i		−	0.498932i	0.968384	−	0.249466i	$-0.0802551\pi$
$608$		−	5.27503e9i		−	0.951837i
$609$	2.68852e9			0.482339
$610$	6.96672e7	−	3.12225e8i	0.0124272	−	0.0556946i
$611$	−4.48318e8			−0.0795136
$612$		−	2.81365e9i		−	0.496181i
$613$		−	7.89654e9i		−	1.38460i	−0.721609	−	0.692301i	$-0.756597\pi$
$613$		−	7.89654e9i		−	1.38460i	0.721609	−	0.692301i	$-0.243403\pi$
$614$	−6.38552e8			−0.111329
$615$	−6.32751e8	+	2.83578e9i	−0.109691	+	0.491597i
$616$	−1.79298e9			−0.309060
$617$			3.64303e9i			0.624403i	0.950016	+	0.312202i	$0.101066\pi$
$617$			3.64303e9i			0.624403i	−0.950016	+	0.312202i	$0.898934\pi$
$618$		−	1.18897e9i		−	0.202633i
$619$	−7.22253e9			−1.22397			−0.611987	−	0.790868i	$-0.709630\pi$
$619$	−7.22253e9			−1.22397			−0.611987	+	0.790868i	$0.709630\pi$
$620$	6.37632e8	+	1.42276e8i	0.107448	+	0.0239751i
$621$	1.16841e9			0.195783
$622$		−	7.84163e8i		−	0.130659i
$623$		−	9.24838e9i		−	1.53235i
$624$	3.34544e8			0.0551197
$625$	3.89760e9	+	4.69698e9i	0.638583	+	0.769553i
$626$	2.81023e9			0.457859
$627$		−	1.68823e9i		−	0.273524i
$628$		−	7.38586e8i		−	0.118999i
$629$	−1.22104e10			−1.95638
$630$	−9.04803e8	−	2.01890e8i	−0.144166	−	0.0321679i
$631$	−1.55879e9			−0.246993			−0.123496	−	0.992345i	$-0.539411\pi$
$631$	−1.55879e9			−0.246993			−0.123496	+	0.992345i	$0.539411\pi$
$632$			1.31894e9i			0.207833i
$633$		−	1.69629e9i		−	0.265820i
$634$	−4.94208e8			−0.0770189
$635$	1.46740e9	−	6.57638e9i	0.227426	−	1.01925i
$636$	−5.15152e9			−0.794028
$637$		−	1.37900e9i		−	0.211386i
$638$			3.19178e8i			0.0486588i
$639$	2.03634e9			0.308743
$640$	−1.27962e9	+	5.73480e9i	−0.192952	+	0.864746i
$641$	3.18649e9			0.477869			0.238935	−	0.971036i	$-0.423202\pi$
$641$	3.18649e9			0.477869			0.238935	+	0.971036i	$0.423202\pi$
$642$		−	1.34197e9i		−	0.200156i
$643$		−	1.65144e9i		−	0.244976i	−0.992470	−	0.122488i	$-0.960913\pi$
$643$		−	1.65144e9i		−	0.244976i	0.992470	−	0.122488i	$-0.0390873\pi$
$644$	−1.06212e10			−1.56701
$645$	3.43539e9	+	7.66544e8i	0.504101	+	0.112481i
$646$	3.84405e9			0.561015
$647$			3.95797e9i			0.574523i	0.957852	+	0.287262i	$0.0927449\pi$
$647$			3.95797e9i			0.574523i	−0.957852	+	0.287262i	$0.907255\pi$
$648$		−	3.96533e8i		−	0.0572489i
$649$	3.25730e9			0.467736
$650$	2.04352e8	+	9.59729e7i	0.0291865	+	0.0137073i
$651$	7.99197e8			0.113533
$652$		−	1.83295e9i		−	0.258990i
$653$			2.09998e9i			0.295134i	0.989052	+	0.147567i	$0.0471443\pi$
$653$			2.09998e9i			0.295134i	−0.989052	+	0.147567i	$0.952856\pi$
$654$	1.91640e9			0.267895
$655$	5.67533e9	+	1.26635e9i	0.789127	+	0.176079i
$656$	4.98942e9			0.690059
$657$		−	1.46656e9i		−	0.201753i
$658$		−	2.13336e9i		−	0.291927i
$659$	1.18536e10			1.61344			0.806718	−	0.590937i	$-0.201242\pi$
$659$	1.18536e10			1.61344			0.806718	+	0.590937i	$0.201242\pi$
$660$	−3.11857e8	+	1.39764e9i	−0.0422233	+	0.189231i
$661$	−1.14873e10			−1.54709			−0.773543	−	0.633744i	$-0.781517\pi$
$661$	−1.14873e10			−1.54709			−0.773543	+	0.633744i	$0.781517\pi$
$662$			7.79648e8i			0.104447i
$663$			8.38218e8i			0.111702i
$664$	2.28787e9			0.303280
$665$	−3.58884e9	+	1.60840e10i	−0.473237	+	2.12089i
$666$	−8.28578e8			−0.108686
$667$			3.92679e9i			0.512386i
$668$			9.20738e8i			0.119514i
$669$	−1.39828e8			−0.0180552
$670$	3.82948e6	+	854477.i	0.000491901	+	0.000109759i
$671$	−6.04494e8			−0.0772437
$672$			5.47359e9i			0.695793i
$673$			1.99478e9i			0.252257i	0.992014	+	0.126128i	$0.0402551\pi$
$673$			1.99478e9i			0.252257i	−0.992014	+	0.126128i	$0.959745\pi$
$674$	7.08115e8			0.0890829
$675$	−6.53688e8	+	1.39188e9i	−0.0818102	+	0.174196i
$676$	7.34992e9			0.915101
$677$		−	3.21860e9i		−	0.398664i	−0.979932	−	0.199332i	$-0.936123\pi$
$677$		−	3.21860e9i		−	0.398664i	0.979932	−	0.199332i	$-0.0638772\pi$
$678$			1.02619e9i			0.126451i
$679$	4.56053e9			0.559076
$680$	−6.60935e9	−	1.47475e9i	−0.806079	−	0.179862i
$681$	−6.57389e9			−0.797642
$682$			9.48799e7i			0.0114533i
$683$			1.87259e9i			0.224890i	0.993658	+	0.112445i	$0.0358683\pi$
$683$			1.87259e9i			0.224890i	−0.993658	+	0.112445i	$0.964132\pi$
$684$	−3.39401e9			−0.405524
$685$	6.05238e8	−	2.71247e9i	0.0719464	−	0.322439i
$686$	2.81523e9			0.332950
$687$		−	2.96281e9i		−	0.348622i
$688$		−	6.04442e9i		−	0.707611i
$689$	1.53470e9			0.178754
$690$	2.94876e8	−	1.32153e9i	0.0341718	−	0.153146i
$691$	3.08223e8			0.0355380			0.0177690	−	0.999842i	$-0.494344\pi$
$691$	3.08223e8			0.0355380			0.0177690	+	0.999842i	$0.494344\pi$
$692$		−	1.18967e9i		−	0.136476i
$693$			1.75178e9i			0.199946i
$694$	−2.03708e9			−0.231340
$695$	4.02144e9	+	8.97310e8i	0.454396	+	0.101390i
$696$	1.33267e9			0.149827
$697$			1.25013e10i			1.39842i
$698$			2.65213e9i			0.295190i
$699$	−5.01583e9			−0.555485
$700$	5.94221e9	−	1.26525e10i	0.654795	−	1.39423i
$701$	−1.39953e10			−1.53450			−0.767251	−	0.641346i	$-0.778376\pi$
$701$	−1.39953e10			−1.53450			−0.767251	+	0.641346i	$0.778376\pi$
$702$			5.68801e7i			0.00620555i
$703$			1.47290e10i			1.59893i
$704$	1.99826e9			0.215848
$705$	−3.45375e9	−	7.70640e8i	−0.371218	−	0.0828304i
$706$	2.04045e9			0.218228
$707$		−	1.08005e10i		−	1.14941i
$708$		−	6.54845e9i		−	0.693462i
$709$	1.02356e10			1.07858			0.539290	−	0.842120i	$-0.318693\pi$
$709$	1.02356e10			1.07858			0.539290	+	0.842120i	$0.318693\pi$
$710$	5.13918e8	−	2.30321e9i	0.0538877	−	0.241506i
$711$	1.28863e9			0.134457
$712$		−	4.58430e9i		−	0.475985i
$713$			1.16729e9i			0.120605i
$714$	−3.98874e9			−0.410102
$715$	9.29059e7	−	4.16373e8i	0.00950544	−	0.0426002i
$716$	−1.29836e10			−1.32191
$717$			8.27054e9i			0.837948i
$718$		−	2.23739e9i		−	0.225582i
$719$	−9.63078e9			−0.966297			−0.483148	−	0.875538i	$-0.660507\pi$
$719$	−9.63078e9			−0.966297			−0.483148	+	0.875538i	$0.660507\pi$
$720$	2.57726e9	+	5.75067e8i	0.257332	+	0.0574189i
$721$	2.19309e10			2.17913
$722$		−	1.93522e9i		−	0.191359i
$723$			1.56108e9i			0.153617i
$724$	−1.51937e9			−0.148792
$725$	−4.67781e9	−	2.19691e9i	−0.455889	−	0.214106i
$726$	1.38233e9			0.134070
$727$		−	1.62740e10i		−	1.57081i	−0.618980	−	0.785407i	$-0.712454\pi$
$727$		−	1.62740e10i		−	1.57081i	0.618980	−	0.785407i	$-0.287546\pi$
$728$		−	1.07385e9i		−	0.103154i
$729$	−3.87420e8			−0.0370370
$730$	−1.65875e9	−	3.70120e8i	−0.157816	−	0.0352137i
$731$	1.51446e10			1.43399
$732$			1.21527e9i			0.114521i
$733$		−	1.62212e10i		−	1.52132i	−0.649152	−	0.760659i	$-0.724876\pi$
$733$		−	1.62212e10i		−	1.52132i	0.649152	−	0.760659i	$-0.275124\pi$
$734$	−3.20342e9			−0.299004
$735$	2.37044e9	−	1.06235e10i	0.220203	−	0.986877i
$736$	−7.99461e9			−0.739137
$737$		−	7.41420e6i		−	0.000682226i
$738$			8.48315e8i			0.0776890i
$739$	−1.78796e10			−1.62968			−0.814841	−	0.579684i	$-0.803176\pi$
$739$	−1.78796e10			−1.62968			−0.814841	+	0.579684i	$0.803176\pi$
$740$	2.72081e9	−	1.21937e10i	0.246824	−	1.10618i
$741$	1.01111e9			0.0912927
$742$			7.30301e9i			0.656278i
$743$			1.10072e10i			0.984499i	0.870454	+	0.492249i	$0.163825\pi$
$743$			1.10072e10i			0.984499i	−0.870454	+	0.492249i	$0.836175\pi$
$744$	3.96152e8			0.0352660
$745$	−6.05340e9	−	1.35070e9i	−0.536355	−	0.119678i
$746$	−3.87602e9			−0.341822
$747$		−	2.23530e9i		−	0.196206i
$748$			6.16137e9i			0.538296i
$749$	2.47531e10			2.15250
$750$	1.40931e9	+	1.09063e9i	0.121981	+	0.0943979i
$751$	−8.82495e9			−0.760278			−0.380139	−	0.924929i	$-0.624124\pi$
$751$	−8.82495e9			−0.760278			−0.380139	+	0.924929i	$0.624124\pi$
$752$			6.07671e9i			0.521082i
$753$			1.05329e10i			0.899016i
$754$	−1.91162e8			−0.0162406
$755$	−1.33879e10	−	2.98727e9i	−1.13214	−	0.252615i
$756$	3.52176e9			0.296438
$757$		−	1.26747e10i		−	1.06194i	−0.847389	−	0.530972i	$-0.821827\pi$
$757$		−	1.26747e10i		−	1.06194i	0.847389	−	0.530972i	$-0.178173\pi$
$758$		−	8.98754e8i		−	0.0749547i
$759$	−2.55861e9			−0.212401
$760$	−1.77894e9	+	7.97263e9i	−0.146999	+	0.658801i
$761$	−4.76538e9			−0.391969			−0.195984	−	0.980607i	$-0.562790\pi$
$761$	−4.76538e9			−0.391969			−0.195984	+	0.980607i	$0.562790\pi$
$762$		−	1.96731e9i		−	0.161076i
$763$			3.53487e10i			2.88097i
$764$	1.83624e10			1.48971
$765$	−1.44086e9	+	6.45746e9i	−0.116361	+	0.521491i
$766$	2.44976e9			0.196935
$767$			1.95086e9i			0.156114i
$768$		−	2.61049e9i		−	0.207949i
$769$	3.33620e9			0.264552			0.132276	−	0.991213i	$-0.457772\pi$
$769$	3.33620e9			0.264552			0.132276	+	0.991213i	$0.457772\pi$
$770$	1.98135e9	+	4.42102e8i	0.156403	+	0.0348983i
$771$	1.43694e10			1.12914
$772$			1.79919e10i			1.40740i
$773$		−	1.22441e9i		−	0.0953453i	−0.998863	−	0.0476726i	$-0.984820\pi$
$773$		−	1.22441e9i		−	0.0953453i	0.998863	−	0.0476726i	$-0.0151804\pi$
$774$	1.02769e9			0.0796651
$775$	−1.39054e9	−	6.53060e8i	−0.107307	−	0.0503961i
$776$	2.26060e9			0.173663
$777$		−	1.52834e10i		−	1.16882i
$778$		−	1.02432e9i		−	0.0779843i
$779$	1.50798e10			1.14292
$780$	−8.37073e8	−	1.86777e8i	−0.0631586	−	0.0140927i
$781$	−4.45921e9			−0.334949
$782$		−	5.82587e9i		−	0.435649i
$783$		−	1.30204e9i		−	0.0969300i
$784$	−1.86916e10			−1.38529
$785$	−3.78228e8	+	1.69509e9i	−0.0279068	+	0.125069i
$786$	1.69776e9			0.124709
$787$		−	1.72256e10i		−	1.25969i	−0.776721	−	0.629844i	$-0.783119\pi$
$787$		−	1.72256e10i		−	1.25969i	0.776721	−	0.629844i	$-0.216881\pi$
$788$			1.47189e10i			1.07160i
$789$	−8.44799e9			−0.612327
$790$	3.25216e8	−	1.45751e9i	0.0234680	−	0.105176i
$791$	−1.89284e10			−1.35986
$792$			8.68334e8i			0.0621082i
$793$		−	3.62043e8i		−	0.0257813i
$794$	3.03051e9			0.214854
$795$	1.18230e10	+	2.63808e9i	0.834531	+	0.186210i
$796$	−5.68077e9			−0.399219
$797$			1.01065e10i			0.707127i	0.935411	+	0.353564i	$0.115030\pi$
$797$			1.01065e10i			0.707127i	−0.935411	+	0.353564i	$0.884970\pi$
$798$			4.81148e9i			0.335173i
$799$	−1.52255e10			−1.05599
$800$	4.47272e9	−	9.52361e9i	0.308857	−	0.657637i
$801$	−4.47895e9			−0.307938
$802$			3.30467e9i			0.226213i
$803$			3.21149e9i			0.218878i
$804$	−1.49055e7			−0.00101146
$805$	2.43762e10	+	5.43909e9i	1.64695	+	0.367486i
$806$	−5.68254e7			−0.00382270
$807$			7.97943e9i			0.534459i
$808$		−	5.35367e9i		−	0.357036i
$809$	−9.46505e9			−0.628497			−0.314248	−	0.949341i	$-0.601753\pi$
$809$	−9.46505e9			−0.628497			−0.314248	+	0.949341i	$0.601753\pi$
$810$	−9.77746e7	+	4.38193e8i	−0.00646440	+	0.0289713i
$811$	1.67478e10			1.10251			0.551257	−	0.834335i	$-0.314148\pi$
$811$	1.67478e10			1.10251			0.551257	+	0.834335i	$0.314148\pi$
$812$			1.18359e10i			0.775810i
$813$		−	1.06994e10i		−	0.698300i
$814$	1.81443e9			0.117911
$815$	−9.38649e8	+	4.20671e9i	−0.0607367	+	0.272202i
$816$	1.13616e10			0.732022
$817$		−	1.82684e10i		−	1.17199i
$818$			3.30996e9i			0.211439i
$819$	−1.04917e9			−0.0667350
$820$	−1.24842e10	−	2.78562e9i	−0.790700	−	0.176430i
$821$	−3.34978e9			−0.211259			−0.105630	−	0.994406i	$-0.533686\pi$
$821$	−3.34978e9			−0.211259			−0.105630	+	0.994406i	$0.533686\pi$
$822$		−	8.11429e8i		−	0.0509564i
$823$		−	6.69632e9i		−	0.418732i	−0.977837	−	0.209366i	$-0.932860\pi$
$823$		−	6.69632e9i		−	0.418732i	0.977837	−	0.209366i	$-0.0671401\pi$
$824$	1.08709e10			0.676893
$825$	1.43146e9	−	3.04795e9i	0.0887543	−	0.188981i
$826$	−9.28336e9			−0.573159
$827$		−	5.81000e9i		−	0.357196i	−0.983922	−	0.178598i	$-0.942844\pi$
$827$		−	5.81000e9i		−	0.357196i	0.983922	−	0.178598i	$-0.0571562\pi$
$828$			5.14381e9i			0.314904i
$829$	−6.84048e9			−0.417009			−0.208505	−	0.978021i	$-0.566860\pi$
$829$	−6.84048e9			−0.417009			−0.208505	+	0.978021i	$0.566860\pi$
$830$	−2.52824e9	−	5.64129e8i	−0.153477	−	0.0342457i
$831$	−1.08108e10			−0.653516
$832$			1.19680e9i			0.0720424i
$833$		−	4.68329e10i		−	2.80733i
$834$	1.20300e9			0.0718101
$835$	4.71508e8	−	2.11314e9i	0.0280276	−	0.125610i
$836$	7.43224e9			0.439945
$837$		−	3.87048e8i		−	0.0228153i
$838$		−	5.29529e9i		−	0.310839i
$839$	1.04085e8			0.00608445			0.00304223	−	0.999995i	$-0.499032\pi$
$839$	1.04085e8			0.00608445			0.00304223	+	0.999995i	$0.499032\pi$
$840$	1.84591e9	−	8.27273e9i	0.107456	−	0.481583i
$841$	−1.28740e10			−0.746324
$842$			1.38438e9i			0.0799215i
$843$		−	9.66995e9i		−	0.555940i
$844$	7.46773e9			0.427553
$845$	−1.68684e10	−	3.76388e9i	−0.961780	−	0.214604i
$846$	−1.03318e9			−0.0586651
$847$			2.54975e10i			1.44180i
$848$		−	2.08020e10i		−	1.17144i
$849$	−1.06836e10			−0.599155
$850$	6.94009e9	+	3.25938e9i	0.387613	+	0.182041i
$851$	2.23226e10			1.24163
$852$			8.96476e9i			0.496593i
$853$		−	6.47246e9i		−	0.357066i	−0.983934	−	0.178533i	$-0.942865\pi$
$853$		−	6.47246e9i		−	0.357066i	0.983934	−	0.178533i	$-0.0571351\pi$
$854$	1.72282e9			0.0946536
$855$	7.78941e9	+	1.73806e9i	0.426210	+	0.0951008i
$856$	1.22698e10			0.668619
$857$		−	2.27261e10i		−	1.23337i	−0.787211	−	0.616684i	$-0.788476\pi$
$857$		−	2.27261e10i		−	1.23337i	0.787211	−	0.616684i	$-0.211524\pi$
$858$		−	1.24557e8i		−	0.00673228i
$859$	1.08363e10			0.583315			0.291657	−	0.956523i	$-0.405793\pi$
$859$	1.08363e10			0.583315			0.291657	+	0.956523i	$0.405793\pi$
$860$	−3.37463e9	+	1.51239e10i	−0.180918	+	0.810812i
$861$	−1.56475e10			−0.835474
$862$			2.29749e9i			0.122174i
$863$			3.23218e10i			1.71182i	0.517125	+	0.855910i	$0.327002\pi$
$863$			3.23218e10i			1.71182i	−0.517125	+	0.855910i	$0.672998\pi$
$864$	2.65084e9			0.139825
$865$	−6.09228e8	+	2.73035e9i	−0.0320054	+	0.143437i
$866$	−7.50973e9			−0.392927
$867$			1.73880e10i			0.906114i
$868$			3.51838e9i			0.182610i
$869$	−2.82186e9			−0.145870
$870$	−1.47267e9	−	3.28600e8i	−0.0758210	−	0.0169180i
$871$	4.44051e6			0.000227703
$872$			1.75219e10i			0.894899i
$873$		−	2.20865e9i		−	0.112351i
$874$	−7.02754e9			−0.356052
$875$	−2.01170e10	+	2.59952e10i	−1.01516	+	1.31179i
$876$	6.45635e9			0.324506
$877$			4.21367e9i			0.210941i	0.994422	+	0.105471i	$0.0336349\pi$
$877$			4.21367e9i			0.210941i	−0.994422	+	0.105471i	$0.966365\pi$
$878$			5.13624e8i			0.0256103i
$879$	−5.21856e9			−0.259173
$880$	−5.64372e9	−	1.25929e9i	−0.279175	−	0.0622926i
$881$	5.03021e9			0.247840			0.123920	−	0.992292i	$-0.460453\pi$
$881$	5.03021e9			0.247840			0.123920	+	0.992292i	$0.460453\pi$
$882$		−	3.17800e9i		−	0.155960i
$883$		−	2.81606e10i		−	1.37651i	−0.725469	−	0.688255i	$-0.758377\pi$
$883$		−	2.81606e10i		−	1.37651i	0.725469	−	0.688255i	$-0.241623\pi$
$884$	−3.69016e9			−0.179665
$885$	−3.35345e9	+	1.50290e10i	−0.162626	+	0.728835i
$886$	−4.63184e9			−0.223736
$887$		−	1.59422e10i		−	0.767038i	−0.923533	−	0.383519i	$-0.874712\pi$
$887$		−	1.59422e10i		−	0.767038i	0.923533	−	0.383519i	$-0.125288\pi$
$888$		−	7.57580e9i		−	0.363064i
$889$	3.62877e10			1.73222
$890$	−1.13037e9	+	5.06593e9i	−0.0537471	+	0.240876i
$891$	8.48379e8			0.0401807
$892$		−	6.15578e8i		−	0.0290406i
$893$			1.83660e10i			0.863049i
$894$	−1.81086e9			−0.0847624
$895$	2.97981e10	+	6.64889e9i	1.38934	+	0.310005i
$896$	−3.16440e10			−1.46965
$897$		−	1.53240e9i		−	0.0708922i
$898$		−	1.04374e10i		−	0.480980i
$899$	1.30079e9			0.0597101
$900$	−6.12758e9	−	2.87779e9i	−0.280182	−	0.131586i
$901$	5.21206e10			2.37396
$902$		−	1.85765e9i		−	0.0842833i
$903$			1.89561e10i			0.856725i
$904$	−9.38255e9			−0.422407
$905$	3.48704e9	+	7.78068e8i	0.156382	+	0.0348937i
$906$	−4.00496e9			−0.178916
$907$			2.20381e10i			0.980729i	0.871517	+	0.490364i	$0.163136\pi$
$907$			2.20381e10i			0.980729i	−0.871517	+	0.490364i	$0.836864\pi$
$908$		−	2.89408e10i		−	1.28295i
$909$	−5.23064e9			−0.230984
$910$	−2.64783e8	+	1.18667e9i	−0.0116479	+	0.0522017i
$911$	1.50904e10			0.661283			0.330641	−	0.943756i	$-0.392735\pi$
$911$	1.50904e10			0.661283			0.330641	+	0.943756i	$0.392735\pi$
$912$		−	1.37051e10i		−	0.598275i
$913$			4.89489e9i			0.212860i
$914$	−2.28585e9			−0.0990229
$915$	6.22338e8	−	2.78911e9i	0.0268567	−	0.120363i
$916$	1.30434e10			0.560736
$917$			3.13158e10i			1.34113i
$918$			1.93173e9i			0.0824134i
$919$	−9.21105e9			−0.391476			−0.195738	−	0.980656i	$-0.562710\pi$
$919$	−9.21105e9			−0.391476			−0.195738	+	0.980656i	$0.562710\pi$
$920$	1.20830e10	+	2.69609e9i	0.511583	+	0.114150i
$921$	−5.70420e9			−0.240595
$922$			2.65457e9i			0.111541i
$923$		−	2.67071e9i		−	0.111794i
$924$	−7.71201e9			−0.321600
$925$	−1.24888e10	+	2.65919e10i	−0.518828	+	1.10472i
$926$	1.07290e10			0.444040
$927$		−	1.06211e10i		−	0.437915i
$928$			8.90892e9i			0.365938i
$929$	−4.08831e10			−1.67297			−0.836487	−	0.547987i	$-0.815394\pi$
$929$	−4.08831e10			−1.67297			−0.836487	+	0.547987i	$0.815394\pi$
$930$	−4.37771e8	−	9.76806e7i	−0.0178467	−	0.00398215i
$931$	−5.64929e10			−2.29440
$932$		−	2.20816e10i		−	0.893461i
$933$		−	7.00494e9i		−	0.282370i
$934$	9.16912e9			0.368225
$935$	3.15522e9	−	1.41406e10i	0.126238	−	0.565755i
$936$	−5.20062e8			−0.0207295
$937$		−	3.38015e10i		−	1.34229i	−0.741324	−	0.671147i	$-0.765802\pi$
$937$		−	3.38015e10i		−	1.34229i	0.741324	−	0.671147i	$-0.234198\pi$
$938$			2.11306e7i			0.000835992i
$939$	2.51038e10			0.989488
$940$	3.39266e9	−	1.52047e10i	0.133227	−	0.597079i
$941$	2.16815e10			0.848252			0.424126	−	0.905603i	$-0.360581\pi$
$941$	2.16815e10			0.848252			0.424126	+	0.905603i	$0.360581\pi$
$942$			5.07082e8i			0.0197652i
$943$		−	2.28543e10i		−	0.887519i
$944$	2.64429e10			1.02307
$945$	−8.08262e9	−	1.80349e9i	−0.311559	−	0.0695187i
$946$	−2.25045e9			−0.0864271
$947$			1.47060e10i			0.562692i	0.959606	+	0.281346i	$0.0907809\pi$
$947$			1.47060e10i			0.562692i	−0.959606	+	0.281346i	$0.909219\pi$
$948$			5.67305e9i			0.216266i
$949$	−1.92342e9			−0.0730538
$950$	3.93168e9	−	8.37159e9i	0.148780	−	0.316793i
$951$	−4.41476e9			−0.166447
$952$		−	3.64696e10i		−	1.36994i
$953$		−	1.26663e10i		−	0.474050i	−0.971504	−	0.237025i	$-0.923828\pi$
$953$		−	1.26663e10i		−	0.474050i	0.971504	−	0.237025i	$-0.0761724\pi$
$954$	3.53682e9			0.131884
$955$	−4.21426e10	−	9.40335e9i	−1.56570	−	0.349358i
$956$	−3.64102e10			−1.34778
$957$			2.85123e9i			0.105157i
$958$			9.07785e9i			0.333583i
$959$	1.49671e10			0.547989
$960$	−2.05724e9	+	9.21987e9i	−0.0750475	+	0.336338i
$961$	−2.71259e10			−0.985945
$962$			1.08670e9i			0.0393547i
$963$		−	1.19878e10i		−	0.432562i
$964$	−6.87247e9			−0.247083
$965$	9.21362e9	−	4.12923e10i	0.330054	−	1.47919i
$966$	7.29207e9			0.260274
$967$			2.68713e10i			0.955644i	0.878457	+	0.477822i	$0.158574\pi$
$967$			2.68713e10i			0.955644i	−0.878457	+	0.477822i	$0.841426\pi$
$968$			1.26388e10i			0.447860i
$969$	3.43389e10			1.21242
$970$	−2.49809e9	−	5.57404e8i	−0.0878836	−	0.0196096i
$971$	1.85571e9			0.0650494			0.0325247	−	0.999471i	$-0.489645\pi$
$971$	1.85571e9			0.0650494			0.0325247	+	0.999471i	$0.489645\pi$
$972$		−	1.70558e9i		−	0.0595716i
$973$			2.21898e10i			0.772252i
$974$	9.93340e8			0.0344462
$975$	1.82548e9	+	8.57327e8i	0.0630754	+	0.0296231i
$976$	−4.90731e9			−0.168954
$977$			3.13360e10i			1.07501i	0.843260	+	0.537506i	$0.180633\pi$
$977$			3.13360e10i			1.07501i	−0.843260	+	0.537506i	$0.819367\pi$
$978$			1.25843e9i			0.0430171i
$979$	9.80808e9			0.334075
$980$	4.67689e10	+	1.04356e10i	1.58732	+	0.354182i
$981$	1.71193e10			0.578953
$982$			2.10751e9i			0.0710198i
$983$			2.78355e9i			0.0934679i	0.998907	+	0.0467339i	$0.0148813\pi$
$983$			2.78355e9i			0.0934679i	−0.998907	+	0.0467339i	$0.985119\pi$
$984$	−7.75625e9			−0.259519
$985$	7.53752e9	−	3.37806e10i	0.251305	−	1.12626i
$986$	−6.49215e9			−0.215685
$987$		−	1.90574e10i		−	0.630889i
$988$			4.45132e9i			0.146838i
$989$	−2.76868e10			−0.910094
$990$	2.14108e8	−	9.59561e8i	0.00701310	−	0.0314303i
$991$	2.20283e10			0.718991			0.359495	−	0.933147i	$-0.382949\pi$
$991$	2.20283e10			0.718991			0.359495	+	0.933147i	$0.382949\pi$
$992$			2.64829e9i			0.0861341i
$993$			6.96461e9i			0.225722i
$994$	1.27088e10			0.410443
$995$	1.30376e10	+	2.90911e9i	0.419584	+	0.0936223i
$996$	9.84065e9			0.315585
$997$			5.02880e10i			1.60706i	0.595266	+	0.803528i	$0.297047\pi$
$997$			5.02880e10i			1.60706i	−0.595266	+	0.803528i	$0.702953\pi$
$998$		−	9.61836e9i		−	0.306298i
$999$	−7.40170e9			−0.234883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.b.a.4.4	✓	8
3.2	odd	2		45.8.b.d.19.5		8
4.3	odd	2		240.8.f.e.49.5		8
5.2	odd	4		75.8.a.i.1.3		4
5.3	odd	4		75.8.a.j.1.2		4
5.4	even	2	inner	15.8.b.a.4.5	yes	8
15.2	even	4		225.8.a.bb.1.2		4
15.8	even	4		225.8.a.z.1.3		4
15.14	odd	2		45.8.b.d.19.4		8
20.19	odd	2		240.8.f.e.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.b.a.4.4	✓	8	1.1	even	1	trivial
15.8.b.a.4.5	yes	8	5.4	even	2	inner
45.8.b.d.19.4		8	15.14	odd	2
45.8.b.d.19.5		8	3.2	odd	2
75.8.a.i.1.3		4	5.2	odd	4
75.8.a.j.1.2		4	5.3	odd	4
225.8.a.z.1.3		4	15.8	even	4
225.8.a.bb.1.2		4	15.2	even	4
240.8.f.e.49.1		8	20.19	odd	2
240.8.f.e.49.5		8	4.3	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)

\(f(q)\)	\(=\)	\(q-3.02250i q^{2} -27.0000i q^{3} +118.865 q^{4} +(-272.800 - 60.8703i) q^{5} -81.6074 q^{6} -1505.28i q^{7} -746.147i q^{8} -729.000 q^{9} +O(q^{10})\)	\(q-3.02250i q^{2} -27.0000i q^{3} +118.865 q^{4} +(-272.800 - 60.8703i) q^{5} -81.6074 q^{6} -1505.28i q^{7} -746.147i q^{8} -729.000 q^{9} +(-183.980 + 824.537i) q^{10} +1596.37 q^{11} -3209.34i q^{12} +956.100i q^{13} -4549.69 q^{14} +(-1643.50 + 7365.60i) q^{15} +12959.4 q^{16} +32470.6i q^{17} +2203.40i q^{18} +39168.1 q^{19} +(-32426.2 - 7235.32i) q^{20} -40642.5 q^{21} -4825.04i q^{22} -59361.5i q^{23} -20146.0 q^{24} +(70714.6 + 33210.8i) q^{25} +2889.81 q^{26} +19683.0i q^{27} -178924. i q^{28} -66150.5 q^{29} +(22262.5 + 4967.46i) q^{30} -19664.1 q^{31} -134677. i q^{32} -43102.1i q^{33} +98142.2 q^{34} +(-91626.6 + 410640. i) q^{35} -86652.2 q^{36} +376045. i q^{37} -118386. i q^{38} +25814.7 q^{39} +(-45418.2 + 203549. i) q^{40} +385003. q^{41} +122842. i q^{42} -466410. i q^{43} +189752. q^{44} +(198871. + 44374.4i) q^{45} -179420. q^{46} +468903. i q^{47} -349905. i q^{48} -1.44232e6 q^{49} +(100380. - 213735. i) q^{50} +876706. q^{51} +113646. i q^{52} -1.60516e6i q^{53} +59491.8 q^{54} +(-435491. - 97171.8i) q^{55} -1.12316e6 q^{56} -1.05754e6i q^{57} +199940. i q^{58} +2.04044e6 q^{59} +(-195354. + 875508. i) q^{60} -378667. q^{61} +59434.6i q^{62} +1.09735e6i q^{63} +1.25175e6 q^{64} +(58198.1 - 260824. i) q^{65} -130276. q^{66} -4644.40i q^{67} +3.85960e6i q^{68} -1.60276e6 q^{69} +(1.24116e6 + 276941. i) q^{70} -2.79333e6 q^{71} +543941. i q^{72} +2.01174e6i q^{73} +1.13660e6 q^{74} +(896692. - 1.90929e6i) q^{75} +4.65570e6 q^{76} -2.40299e6i q^{77} -78024.8i q^{78} -1.76767e6 q^{79} +(-3.53533e6 - 788844. i) q^{80} +531441. q^{81} -1.16367e6i q^{82} +3.06625e6i q^{83} -4.83095e6 q^{84} +(1.97649e6 - 8.85797e6i) q^{85} -1.40972e6 q^{86} +1.78606e6i q^{87} -1.19113e6i q^{88} +6.14397e6 q^{89} +(134122. - 601087. i) q^{90} +1.43920e6 q^{91} -7.05598e6i q^{92} +530930. i q^{93} +1.41726e6 q^{94} +(-1.06851e7 - 2.38417e6i) q^{95} -3.63627e6 q^{96} +3.02969e6i q^{97} +4.35940e6i q^{98} -1.16376e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10}) \) 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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