LMFDB - Embedded newform 13.12.b.a.12.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,12,Mod(12,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("13.12");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 13 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	13.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:	$9.98846134727$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 $ x^12 + 18433x^10 + 121088056x^8 + 340607607312x^6 + 380893885719552x^4 + 134825856231997440*x^2 + 1497425476589715456
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4}\cdot 13^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			12.7
Root			$3.38741i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.12.b.a.12.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.38741i q^{2} +494.534 q^{3} +2036.53 q^{4} +9091.88i q^{5} +1675.19i q^{6} -45027.4i q^{7} +13836.0i q^{8} +67417.3 q^{9} +O(q^{10})$	\(q+3.38741i q^{2} +494.534 q^{3} +2036.53 q^{4} +9091.88i q^{5} +1675.19i q^{6} -45027.4i q^{7} +13836.0i q^{8} +67417.3 q^{9} -30797.9 q^{10} +810639. i q^{11} +1.00713e6 q^{12} +(1.11466e6 - 741417. i) q^{13} +152526. q^{14} +4.49625e6i q^{15} +4.12394e6 q^{16} -851428. q^{17} +228370. i q^{18} -7.56605e6i q^{19} +1.85158e7i q^{20} -2.22676e7i q^{21} -2.74596e6 q^{22} +3.69275e6 q^{23} +6.84236e6i q^{24} -3.38341e7 q^{25} +(2.51148e6 + 3.77580e6i) q^{26} -5.42651e7 q^{27} -9.16995e7i q^{28} -9.33625e7 q^{29} -1.52306e7 q^{30} -2.61508e8i q^{31} +4.23055e7i q^{32} +4.00889e8i q^{33} -2.88413e6i q^{34} +4.09384e8 q^{35} +1.37297e8 q^{36} +5.38221e8i q^{37} +2.56293e7 q^{38} +(5.51236e8 - 3.66656e8i) q^{39} -1.25795e8 q^{40} -6.71292e8i q^{41} +7.54295e7 q^{42} -1.09294e9 q^{43} +1.65089e9i q^{44} +6.12950e8i q^{45} +1.25088e7i q^{46} -1.79698e9i q^{47} +2.03943e9 q^{48} -5.01432e7 q^{49} -1.14610e8i q^{50} -4.21060e8 q^{51} +(2.27003e9 - 1.50991e9i) q^{52} -5.82497e9 q^{53} -1.83818e8i q^{54} -7.37023e9 q^{55} +6.22997e8 q^{56} -3.74167e9i q^{57} -3.16257e8i q^{58} -4.74847e9i q^{59} +9.15672e9i q^{60} +6.50495e9 q^{61} +8.85834e8 q^{62} -3.03563e9i q^{63} +8.30252e9 q^{64} +(6.74087e9 + 1.01343e10i) q^{65} -1.35797e9 q^{66} +8.11940e9i q^{67} -1.73395e9 q^{68} +1.82619e9 q^{69} +1.38675e9i q^{70} -3.16640e9i q^{71} +9.32783e8i q^{72} -8.22937e9i q^{73} -1.82317e9 q^{74} -1.67321e10 q^{75} -1.54085e10i q^{76} +3.65010e10 q^{77} +(1.24201e9 + 1.86726e9i) q^{78} -3.75500e9 q^{79} +3.74943e10i q^{80} -3.87787e10 q^{81} +2.27394e9 q^{82} -6.60119e9i q^{83} -4.53486e10i q^{84} -7.74108e9i q^{85} -3.70222e9i q^{86} -4.61710e10 q^{87} -1.12160e10 q^{88} +5.68420e10i q^{89} -2.07631e9 q^{90} +(-3.33841e10 - 5.01902e10i) q^{91} +7.52038e9 q^{92} -1.29325e11i q^{93} +6.08709e9 q^{94} +6.87896e10 q^{95} +2.09215e10i q^{96} +1.62006e11i q^{97} -1.69856e8i q^{98} +5.46511e10i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9}+O(q^{10}) $ 12 * q - 488 * q^3 - 12290 * q^4 + 654644 * q^9	$ 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9} - 333446 q^{10} + 2740298 q^{12} + 3208868 q^{13} - 5367450 q^{14} + 19025698 q^{16} + 12198768 q^{17} - 111171128 q^{22} + 5810592 q^{23} + 6102388 q^{25} - 64543986 q^{26} - 52613336 q^{27} - 244463112 q^{29} + 426504126 q^{30} - 562027560 q^{35} - 1357546052 q^{36} + 3171817788 q^{38} - 2199109744 q^{39} + 4092185498 q^{40} + 1280452314 q^{42} + 2294519976 q^{43} - 14206061378 q^{48} - 3573617796 q^{49} + 7713246552 q^{51} - 5597650396 q^{52} - 4602062760 q^{53} - 6178744976 q^{55} + 20017912662 q^{56} - 13775649944 q^{61} + 239765256 q^{62} - 3560815378 q^{64} - 7598401512 q^{65} + 37979507040 q^{66} + 40844682210 q^{68} - 25419983328 q^{69} + 19351803414 q^{74} + 68016370832 q^{75} - 80478036048 q^{77} + 89375282178 q^{78} + 18046097296 q^{79} - 132677486692 q^{81} - 255687836096 q^{82} + 94507900752 q^{87} + 239343029120 q^{88} - 190413561204 q^{90} + 104793638664 q^{91} - 135236877012 q^{92} - 78363161402 q^{94} + 145093149648 q^{95}+O(q^{100}) $ 12 * q - 488 * q^3 - 12290 * q^4 + 654644 * q^9 - 333446 * q^10 + 2740298 * q^12 + 3208868 * q^13 - 5367450 * q^14 + 19025698 * q^16 + 12198768 * q^17 - 111171128 * q^22 + 5810592 * q^23 + 6102388 * q^25 - 64543986 * q^26 - 52613336 * q^27 - 244463112 * q^29 + 426504126 * q^30 - 562027560 * q^35 - 1357546052 * q^36 + 3171817788 * q^38 - 2199109744 * q^39 + 4092185498 * q^40 + 1280452314 * q^42 + 2294519976 * q^43 - 14206061378 * q^48 - 3573617796 * q^49 + 7713246552 * q^51 - 5597650396 * q^52 - 4602062760 * q^53 - 6178744976 * q^55 + 20017912662 * q^56 - 13775649944 * q^61 + 239765256 * q^62 - 3560815378 * q^64 - 7598401512 * q^65 + 37979507040 * q^66 + 40844682210 * q^68 - 25419983328 * q^69 + 19351803414 * q^74 + 68016370832 * q^75 - 80478036048 * q^77 + 89375282178 * q^78 + 18046097296 * q^79 - 132677486692 * q^81 - 255687836096 * q^82 + 94507900752 * q^87 + 239343029120 * q^88 - 190413561204 * q^90 + 104793638664 * q^91 - 135236877012 * q^92 - 78363161402 * q^94 + 145093149648 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$			3.38741i			0.0748518i	0.999299	+	0.0374259i	$0.0119158\pi$
$2$			3.38741i			0.0748518i	−0.999299	+	0.0374259i	$0.988084\pi$
$3$	494.534			1.17498			0.587489	−	0.809232i	$-0.300117\pi$
$3$	494.534			1.17498			0.587489	+	0.809232i	$0.300117\pi$
$4$	2036.53			0.994397
$5$			9091.88i			1.30112i	0.759453	+	0.650562i	$0.225467\pi$
$5$			9091.88i			1.30112i	−0.759453	+	0.650562i	$0.774533\pi$
$6$			1675.19i			0.0879492i
$7$		−	45027.4i		−	1.01260i	−0.862357	−	0.506300i	$-0.831013\pi$
$7$		−	45027.4i		−	1.01260i	0.862357	−	0.506300i	$-0.168987\pi$
$8$			13836.0i			0.149284i
$9$	67417.3			0.380573
$10$	−30797.9			−0.0973915
$11$			810639.i			1.51764i	0.651303	+	0.758818i	$0.274223\pi$
$11$			810639.i			1.51764i	−0.651303	+	0.758818i	$0.725777\pi$
$12$	1.00713e6			1.16839
$13$	1.11466e6	−	741417.i	0.832632	−	0.553827i
$13$	1.11466e6	−	741417.i	0.832632	−	0.553827i
$14$	152526.			0.0757950
$15$			4.49625e6i			1.52879i
$16$	4.12394e6			0.983223
$17$	−851428.			−0.145438			−0.0727192	−	0.997352i	$-0.523168\pi$
$17$	−851428.			−0.145438			−0.0727192	+	0.997352i	$0.523168\pi$
$18$			228370.i			0.0284866i
$19$		−	7.56605e6i		−	0.701011i	−0.936561	−	0.350505i	$-0.886010\pi$
$19$		−	7.56605e6i		−	0.701011i	0.936561	−	0.350505i	$-0.113990\pi$
$20$			1.85158e7i			1.29383i
$21$		−	2.22676e7i		−	1.18978i
$22$	−2.74596e6			−0.113598
$23$	3.69275e6			0.119632			0.0598159	−	0.998209i	$-0.480949\pi$
$23$	3.69275e6			0.119632			0.0598159	+	0.998209i	$0.480949\pi$
$24$			6.84236e6i			0.175406i
$25$	−3.38341e7			−0.692923
$26$	2.51148e6	+	3.77580e6i	0.0414550	+	0.0623240i
$27$	−5.42651e7			−0.727813
$28$		−	9.16995e7i		−	1.00693i
$29$	−9.33625e7			−0.845247			−0.422623	−	0.906305i	$-0.638891\pi$
$29$	−9.33625e7			−0.845247			−0.422623	+	0.906305i	$0.638891\pi$
$30$	−1.52306e7			−0.114433
$31$		−	2.61508e8i		−	1.64057i	−0.571953	−	0.820286i	$-0.693814\pi$
$31$		−	2.61508e8i		−	1.64057i	0.571953	−	0.820286i	$-0.306186\pi$
$32$			4.23055e7i			0.222880i
$33$			4.00889e8i			1.78319i
$34$		−	2.88413e6i		−	0.0108863i
$35$	4.09384e8			1.31752
$36$	1.37297e8			0.378440
$37$			5.38221e8i			1.27600i	0.770036	+	0.638000i	$0.220238\pi$
$37$			5.38221e8i			1.27600i	−0.770036	+	0.638000i	$0.779762\pi$
$38$	2.56293e7			0.0524719
$39$	5.51236e8	−	3.66656e8i	0.978324	−	0.650734i
$40$	−1.25795e8			−0.194237
$41$		−	6.71292e8i		−	0.904899i	−0.891790	−	0.452449i	$-0.850550\pi$
$41$		−	6.71292e8i		−	0.904899i	0.891790	−	0.452449i	$-0.149450\pi$
$42$	7.54295e7			0.0890574
$43$	−1.09294e9			−1.13375			−0.566876	−	0.823803i	$-0.691848\pi$
$43$	−1.09294e9			−1.13375			−0.566876	+	0.823803i	$0.691848\pi$
$44$			1.65089e9i			1.50913i
$45$			6.12950e8i			0.495172i
$46$			1.25088e7i			0.00895466i
$47$		−	1.79698e9i		−	1.14289i	−0.820641	−	0.571445i	$-0.806383\pi$
$47$		−	1.79698e9i		−	1.14289i	0.820641	−	0.571445i	$-0.193617\pi$
$48$	2.03943e9			1.15527
$49$	−5.01432e7			−0.0253591
$50$		−	1.14610e8i		−	0.0518665i
$51$	−4.21060e8			−0.170887
$52$	2.27003e9	−	1.50991e9i	0.827967	−	0.550724i
$53$	−5.82497e9			−1.91327			−0.956636	−	0.291287i	$-0.905917\pi$
$53$	−5.82497e9			−1.91327			−0.956636	+	0.291287i	$0.905917\pi$
$54$		−	1.83818e8i		−	0.0544782i
$55$	−7.37023e9			−1.97463
$56$	6.22997e8			0.151165
$57$		−	3.74167e9i		−	0.823672i
$58$		−	3.16257e8i		−	0.0632683i
$59$		−	4.74847e9i		−	0.864704i	−0.901705	−	0.432352i	$-0.857684\pi$
$59$		−	4.74847e9i		−	0.864704i	0.901705	−	0.432352i	$-0.142316\pi$
$60$			9.15672e9i			1.52023i
$61$	6.50495e9			0.986120			0.493060	−	0.869995i	$-0.335878\pi$
$61$	6.50495e9			0.986120			0.493060	+	0.869995i	$0.335878\pi$
$62$	8.85834e8			0.122800
$63$		−	3.03563e9i		−	0.385368i
$64$	8.30252e9			0.966540
$65$	6.74087e9	+	1.01343e10i	0.720597	+	1.08336i
$66$	−1.35797e9			−0.133475
$67$			8.11940e9i			0.734704i	0.930082	+	0.367352i	$0.119736\pi$
$67$			8.11940e9i			0.734704i	−0.930082	+	0.367352i	$0.880264\pi$
$68$	−1.73395e9			−0.144623
$69$	1.82619e9			0.140565
$70$			1.38675e9i			0.0986186i
$71$		−	3.16640e9i		−	0.208279i	−0.994563	−	0.104139i	$-0.966791\pi$
$71$		−	3.16640e9i		−	0.208279i	0.994563	−	0.104139i	$-0.0332088\pi$
$72$			9.32783e8i			0.0568135i
$73$		−	8.22937e9i		−	0.464613i	−0.972643	−	0.232306i	$-0.925373\pi$
$73$		−	8.22937e9i		−	0.464613i	0.972643	−	0.232306i	$-0.0746272\pi$
$74$	−1.82317e9			−0.0955110
$75$	−1.67321e10			−0.814169
$76$		−	1.54085e10i		−	0.697083i
$77$	3.65010e10			1.53676
$78$	1.24201e9	+	1.86726e9i	0.0487087	+	0.0732293i
$79$	−3.75500e9			−0.137297			−0.0686485	−	0.997641i	$-0.521869\pi$
$79$	−3.75500e9			−0.137297			−0.0686485	+	0.997641i	$0.521869\pi$
$80$			3.74943e10i			1.27929i
$81$	−3.87787e10			−1.23574
$82$	2.27394e9			0.0677333
$83$		−	6.60119e9i		−	0.183947i	−0.995761	−	0.0919735i	$-0.970682\pi$
$83$		−	6.60119e9i		−	0.183947i	0.995761	−	0.0919735i	$-0.0293175\pi$
$84$		−	4.53486e10i		−	1.18312i
$85$		−	7.74108e9i		−	0.189233i
$86$		−	3.70222e9i		−	0.0848634i
$87$	−4.61710e10			−0.993146
$88$	−1.12160e10			−0.226559
$89$			5.68420e10i			1.07901i	0.841983	+	0.539503i	$0.181388\pi$
$89$			5.68420e10i			1.07901i	−0.841983	+	0.539503i	$0.818612\pi$
$90$	−2.07631e9			−0.0370645
$91$	−3.33841e10	−	5.01902e10i	−0.560805	−	0.843123i
$92$	7.52038e9			0.118962
$93$		−	1.29325e11i		−	1.92764i
$94$	6.08709e9			0.0855474
$95$	6.87896e10			0.912101
$96$			2.09215e10i			0.261879i
$97$			1.62006e11i			1.91551i	0.287578	+	0.957757i	$0.407150\pi$
$97$			1.62006e11i			1.91551i	−0.287578	+	0.957757i	$0.592850\pi$
$98$		−	1.69856e8i		−	0.00189818i
$99$			5.46511e10i			0.577571i
$100$	−6.89040e10			−0.689040
$101$	1.31282e11			1.24290			0.621450	−	0.783454i	$-0.286544\pi$
$101$	1.31282e11			1.24290			0.621450	+	0.783454i	$0.286544\pi$
$102$		−	1.42630e9i		−	0.0127912i
$103$	−5.50452e10			−0.467859			−0.233929	−	0.972254i	$-0.575158\pi$
$103$	−5.50452e10			−0.467859			−0.233929	+	0.972254i	$0.575158\pi$
$104$	1.02582e10	+	1.54223e10i	0.0826777	+	0.124299i
$105$	2.02454e11			1.54805
$106$		−	1.97316e10i		−	0.143212i
$107$	1.85072e11			1.27564			0.637822	−	0.770184i	$-0.279836\pi$
$107$	1.85072e11			1.27564			0.637822	+	0.770184i	$0.279836\pi$
$108$	−1.10512e11			−0.723735
$109$		−	2.74414e10i		−	0.170829i	−0.996346	−	0.0854143i	$-0.972779\pi$
$109$		−	2.74414e10i		−	0.170829i	0.996346	−	0.0854143i	$-0.0272214\pi$
$110$		−	2.49660e10i		−	0.147805i
$111$			2.66169e11i			1.49927i
$112$		−	1.85690e11i		−	0.995612i
$113$	−1.64428e11			−0.839547			−0.419773	−	0.907629i	$-0.637890\pi$
$113$	−1.64428e11			−0.839547			−0.419773	+	0.907629i	$0.637890\pi$
$114$	1.26746e10			0.0616533
$115$			3.35740e10i			0.155656i
$116$	−1.90135e11			−0.840511
$117$	7.51472e10	−	4.99844e10i	0.316877	−	0.210771i
$118$	1.60850e10			0.0647247
$119$			3.83376e10i			0.147271i
$120$	−6.22099e10			−0.228225
$121$	−3.71823e11			−1.30322
$122$			2.20349e10i			0.0738129i
$123$		−	3.31977e11i		−	1.06324i
$124$		−	5.32567e11i		−	1.63138i
$125$			1.36324e11i			0.399546i
$126$	1.02829e10			0.0288455
$127$	5.48728e11			1.47379			0.736897	−	0.676005i	$-0.236290\pi$
$127$	5.48728e11			1.47379			0.736897	+	0.676005i	$0.236290\pi$
$128$			1.14766e11i			0.295228i
$129$	−5.40494e11			−1.33213
$130$	−3.43291e10	+	2.28341e10i	−0.0810912	+	0.0539380i
$131$	3.22742e11			0.730910			0.365455	−	0.930829i	$-0.380913\pi$
$131$	3.22742e11			0.730910			0.365455	+	0.930829i	$0.380913\pi$
$132$			8.16420e11i			1.77320i
$133$	−3.40680e11			−0.709843
$134$	−2.75037e10			−0.0549940
$135$		−	4.93372e11i		−	0.946975i
$136$		−	1.17803e10i		−	0.0217117i
$137$			1.94892e11i			0.345009i	0.985009	+	0.172505i	$0.0551859\pi$
$137$			1.94892e11i			0.345009i	−0.985009	+	0.172505i	$0.944814\pi$
$138$			6.18606e9i			0.0105215i
$139$	−6.64756e11			−1.08663			−0.543314	−	0.839530i	$-0.682831\pi$
$139$	−6.64756e11			−1.08663			−0.543314	+	0.839530i	$0.682831\pi$
$140$	8.33721e11			1.31014
$141$		−	8.88667e11i		−	1.34287i
$142$	1.07259e10			0.0155901
$143$	6.01021e11	+	9.03584e11i	0.840508	+	1.26363i
$144$	2.78025e11			0.374188
$145$		−	8.48840e11i		−	1.09977i
$146$	2.78762e10			0.0347771
$147$	−2.47976e10			−0.0297964
$148$			1.09610e12i			1.26885i
$149$			4.29539e11i			0.479157i	0.970877	+	0.239578i	$0.0770092\pi$
$149$			4.29539e11i			0.479157i	−0.970877	+	0.239578i	$0.922991\pi$
$150$		−	5.66785e10i		−	0.0609420i
$151$			1.20700e12i			1.25122i	0.780136	+	0.625610i	$0.215150\pi$
$151$			1.20700e12i			1.25122i	−0.780136	+	0.625610i	$0.784850\pi$
$152$	1.04684e11			0.104650
$153$	−5.74010e10			−0.0553499
$154$			1.23644e11i			0.115029i
$155$	2.37760e12			2.13459
$156$	1.12261e12	−	7.46705e11i	0.972842	−	0.647089i
$157$	−9.61496e10			−0.0804450			−0.0402225	−	0.999191i	$-0.512807\pi$
$157$	−9.61496e10			−0.0804450			−0.0402225	+	0.999191i	$0.512807\pi$
$158$		−	1.27197e10i		−	0.0102769i
$159$	−2.88065e12			−2.24805
$160$	−3.84636e11			−0.289995
$161$		−	1.66275e11i		−	0.121139i
$162$		−	1.31359e11i		−	0.0924972i
$163$		−	1.50626e12i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$163$		−	1.50626e12i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$164$		−	1.36710e12i		−	0.899829i
$165$	−3.64483e12			−2.32015
$166$	2.23609e10			0.0137688
$167$			1.03474e12i			0.616441i	0.951315	+	0.308221i	$0.0997335\pi$
$167$			1.03474e12i			0.616441i	−0.951315	+	0.308221i	$0.900266\pi$
$168$	3.08094e11			0.177616
$169$	6.92762e11	−	1.65285e12i	0.386551	−	0.922268i
$170$	2.62222e10			0.0141645
$171$		−	5.10083e11i		−	0.266786i
$172$	−2.22579e12			−1.12740
$173$	−1.11263e11			−0.0545881			−0.0272941	−	0.999627i	$-0.508689\pi$
$173$	−1.11263e11			−0.0545881			−0.0272941	+	0.999627i	$0.508689\pi$
$174$		−	1.56400e11i		−	0.0743388i
$175$			1.52346e12i			0.701653i
$176$			3.34302e12i			1.49217i
$177$		−	2.34828e12i		−	1.01601i
$178$	−1.92547e11			−0.0807656
$179$	2.06269e12			0.838964			0.419482	−	0.907764i	$-0.362212\pi$
$179$	2.06269e12			0.838964			0.419482	+	0.907764i	$0.362212\pi$
$180$			1.24829e12i			0.492398i
$181$	3.46724e12			1.32663			0.663317	−	0.748338i	$-0.269148\pi$
$181$	3.46724e12			1.32663			0.663317	+	0.748338i	$0.269148\pi$
$182$	1.70015e11	−	1.13086e11i	0.0631093	−	0.0419773i
$183$	3.21692e12			1.15867
$184$			5.10927e10i			0.0178592i
$185$	−4.89344e12			−1.66023
$186$	4.38075e11			0.144287
$187$		−	6.90200e11i		−	0.220722i
$188$		−	3.65959e12i		−	1.13649i
$189$			2.44342e12i			0.736984i
$190$			2.33018e11i			0.0682725i
$191$	−4.63569e12			−1.31957			−0.659783	−	0.751456i	$-0.729352\pi$
$191$	−4.63569e12			−1.31957			−0.659783	+	0.751456i	$0.729352\pi$
$192$	4.10588e12			1.13566
$193$		−	1.13783e12i		−	0.305852i	−0.988238	−	0.152926i	$-0.951130\pi$
$193$		−	1.13783e12i		−	0.305852i	0.988238	−	0.152926i	$-0.0488696\pi$
$194$	−5.48779e11			−0.143380
$195$	3.33359e12	+	5.01177e12i	0.846686	+	1.27292i
$196$	−1.02118e11			−0.0252170
$197$		−	2.32298e12i		−	0.557803i	−0.960320	−	0.278901i	$-0.910030\pi$
$197$		−	2.32298e12i		−	0.557803i	0.960320	−	0.278901i	$-0.0899702\pi$
$198$	−1.85126e11			−0.0432322
$199$	2.94354e11			0.0668618			0.0334309	−	0.999441i	$-0.489357\pi$
$199$	2.94354e11			0.0668618			0.0334309	+	0.999441i	$0.489357\pi$
$200$		−	4.68127e11i		−	0.103442i
$201$			4.01532e12i			0.863261i
$202$			4.44704e11i			0.0930333i
$203$			4.20387e12i			0.855897i
$204$	−8.57500e11			−0.169929
$205$	6.10330e12			1.17739
$206$		−	1.86461e11i		−	0.0350201i
$207$	2.48955e11			0.0455286
$208$	4.59678e12	−	3.05756e12i	0.818663	−	0.544536i
$209$	6.13333e12			1.06388
$210$			6.85796e11i			0.115875i
$211$	8.00209e11			0.131719			0.0658597	−	0.997829i	$-0.479021\pi$
$211$	8.00209e11			0.131719			0.0658597	+	0.997829i	$0.479021\pi$
$212$	−1.18627e13			−1.90255
$213$		−	1.56589e12i		−	0.244723i
$214$			6.26914e11i			0.0954843i
$215$		−	9.93684e12i		−	1.47515i
$216$		−	7.50809e11i		−	0.108651i
$217$	−1.17750e13			−1.66124
$218$	9.29552e10			0.0127868
$219$		−	4.06971e12i		−	0.545910i
$220$	−1.50097e13			−1.96357
$221$	−9.49050e11	+	6.31263e11i	−0.121097	+	0.0805477i
$222$	−9.01622e11			−0.112223
$223$			1.48493e13i			1.80314i	0.432631	+	0.901571i	$0.357585\pi$
$223$			1.48493e13i			1.80314i	−0.432631	+	0.901571i	$0.642415\pi$
$224$	1.90491e12			0.225689
$225$	−2.28100e12			−0.263707
$226$		−	5.56985e11i		−	0.0628416i
$227$		−	1.57088e13i		−	1.72982i	−0.501928	−	0.864910i	$-0.667376\pi$
$227$		−	1.57088e13i		−	1.72982i	0.501928	−	0.864910i	$-0.332624\pi$
$228$		−	7.62001e12i		−	0.819057i
$229$			2.72068e12i			0.285485i	0.989760	+	0.142742i	$0.0455920\pi$
$229$			2.72068e12i			0.285485i	−0.989760	+	0.142742i	$0.954408\pi$
$230$	−1.13729e11			−0.0116511
$231$	1.80510e13			1.80566
$232$		−	1.29176e12i		−	0.126182i
$233$	−8.42380e12			−0.803620			−0.401810	−	0.915723i	$-0.631619\pi$
$233$	−8.42380e12			−0.803620			−0.401810	+	0.915723i	$0.631619\pi$
$234$	1.69317e11	+	2.54554e11i	0.0157766	+	0.0237188i
$235$	1.63379e13			1.48704
$236$		−	9.67037e12i		−	0.859859i
$237$	−1.85698e12			−0.161321
$238$	−1.29865e11			−0.0110235
$239$			2.27825e13i			1.88979i	0.327374	+	0.944895i	$0.393836\pi$
$239$			2.27825e13i			1.88979i	−0.327374	+	0.944895i	$0.606164\pi$
$240$			1.85422e13i			1.50314i
$241$			6.32011e12i			0.500761i	0.968147	+	0.250381i	$0.0805558\pi$
$241$			6.32011e12i			0.500761i	−0.968147	+	0.250381i	$0.919444\pi$
$242$		−	1.25952e12i		−	0.0975483i
$243$	−9.56452e12			−0.724150
$244$	1.32475e13			0.980595
$245$		−	4.55896e11i		−	0.0329953i
$246$	1.12454e12			0.0795852
$247$	−5.60960e12	−	8.43356e12i	−0.388239	−	0.583684i
$248$	3.61821e12			0.244912
$249$		−	3.26452e12i		−	0.216134i
$250$	−4.61784e11			−0.0299067
$251$	5.09290e12			0.322671			0.161335	−	0.986900i	$-0.448420\pi$
$251$	5.09290e12			0.322671			0.161335	+	0.986900i	$0.448420\pi$
$252$		−	6.18214e12i		−	0.383209i
$253$			2.99349e12i			0.181558i
$254$			1.85877e12i			0.110316i
$255$		−	3.82823e12i		−	0.222345i
$256$	1.66148e13			0.944442
$257$	−2.23129e13			−1.24144			−0.620718	−	0.784034i	$-0.713159\pi$
$257$	−2.23129e13			−1.24144			−0.620718	+	0.784034i	$0.713159\pi$
$258$		−	1.83087e12i		−	0.0997126i
$259$	2.42347e13			1.29208
$260$	1.37280e13	+	2.06388e13i	0.716560	+	1.07729i
$261$	−6.29425e12			−0.321678
$262$			1.09326e12i			0.0547099i
$263$	1.02420e12			0.0501911			0.0250956	−	0.999685i	$-0.492011\pi$
$263$	1.02420e12			0.0501911			0.0250956	+	0.999685i	$0.492011\pi$
$264$	−5.54668e12			−0.266202
$265$		−	5.29599e13i		−	2.48940i
$266$		−	1.15402e12i		−	0.0531331i
$267$			2.81103e13i			1.26781i
$268$			1.65354e13i			0.730588i
$269$	−3.39292e13			−1.46871			−0.734355	−	0.678766i	$-0.762515\pi$
$269$	−3.39292e13			−1.46871			−0.734355	+	0.678766i	$0.762515\pi$
$270$	1.67125e12			0.0708828
$271$			4.37089e13i			1.81651i	0.418414	+	0.908257i	$0.362586\pi$
$271$			4.37089e13i			1.81651i	−0.418414	+	0.908257i	$0.637414\pi$
$272$	−3.51123e12			−0.142998
$273$	−1.65096e13	−	2.48208e13i	−0.658934	−	0.990651i
$274$	−6.60178e11			−0.0258246
$275$		−	2.74272e13i		−	1.05160i
$276$	3.71909e12			0.139777
$277$	1.69270e13			0.623651			0.311826	−	0.950139i	$-0.399060\pi$
$277$	1.69270e13			0.623651			0.311826	+	0.950139i	$0.399060\pi$
$278$		−	2.25180e12i		−	0.0813361i
$279$		−	1.76302e13i		−	0.624357i
$280$			5.66422e12i			0.196685i
$281$		−	4.31608e13i		−	1.46962i	−0.678274	−	0.734809i	$-0.737272\pi$
$281$		−	4.31608e13i		−	1.46962i	0.678274	−	0.734809i	$-0.262728\pi$
$282$	3.01028e12			0.100516
$283$	−4.77458e13			−1.56354			−0.781772	−	0.623565i	$-0.785684\pi$
$283$	−4.77458e13			−1.56354			−0.781772	+	0.623565i	$0.785684\pi$
$284$		−	6.44846e12i		−	0.207112i
$285$	3.40188e13			1.07170
$286$	−3.06081e12	+	2.03590e12i	−0.0945852	+	0.0629136i
$287$	−3.02265e13			−0.916301
$288$			2.85212e12i			0.0848222i
$289$	−3.35470e13			−0.978848
$290$	2.87537e12			0.0823198
$291$			8.01173e13i			2.25069i
$292$		−	1.67593e13i		−	0.462010i
$293$		−	1.64453e13i		−	0.444908i	−0.974943	−	0.222454i	$-0.928593\pi$
$293$		−	1.64453e13i		−	0.444908i	0.974943	−	0.222454i	$-0.0714067\pi$
$294$		−	8.39994e10i		−	0.00223031i
$295$	4.31725e13			1.12509
$296$	−7.44680e12			−0.190487
$297$		−	4.39894e13i		−	1.10456i
$298$	−1.45502e12			−0.0358658
$299$	4.11615e12	−	2.73787e12i	0.0996093	−	0.0662553i
$300$	−3.40754e13			−0.809607
$301$			4.92121e13i			1.14804i
$302$	−4.08860e12			−0.0936561
$303$	6.49232e13			1.46038
$304$		−	3.12019e13i		−	0.689250i
$305$			5.91422e13i			1.28306i
$306$		−	1.94441e11i		−	0.00414304i
$307$			5.79282e12i			0.121235i	0.998161	+	0.0606176i	$0.0193070\pi$
$307$			5.79282e12i			0.121235i	−0.998161	+	0.0606176i	$0.980693\pi$
$308$	7.43352e13			1.52815
$309$	−2.72217e13			−0.549724
$310$			8.05389e12i			0.159778i
$311$	6.88588e13			1.34208			0.671039	−	0.741422i	$-0.265848\pi$
$311$	6.88588e13			1.34208			0.671039	+	0.741422i	$0.265848\pi$
$312$	5.07304e12	+	7.62688e12i	0.0971444	+	0.146048i
$313$	−2.94167e13			−0.553477			−0.276738	−	0.960945i	$-0.589254\pi$
$313$	−2.94167e13			−0.553477			−0.276738	+	0.960945i	$0.589254\pi$
$314$		−	3.25698e11i		−	0.00602146i
$315$	2.75996e13			0.501411
$316$	−7.64715e12			−0.136528
$317$			2.48320e13i			0.435698i	0.975983	+	0.217849i	$0.0699040\pi$
$317$			2.48320e13i			0.435698i	−0.975983	+	0.217849i	$0.930096\pi$
$318$		−	9.75793e12i		−	0.168271i
$319$		−	7.56832e13i		−	1.28278i
$320$			7.54855e13i			1.25759i
$321$	9.15244e13			1.49885
$322$	5.63241e11			0.00906749
$323$			6.44195e12i			0.101954i
$324$	−7.89739e13			−1.22881
$325$	−3.77134e13	+	2.50852e13i	−0.576949	+	0.383759i
$326$	5.10231e12			0.0767486
$327$		−	1.35707e13i		−	0.200720i
$328$	9.28796e12			0.135087
$329$	−8.09132e13			−1.15729
$330$		−	1.23465e13i		−	0.173667i
$331$		−	3.37169e13i		−	0.466438i	−0.972424	−	0.233219i	$-0.925074\pi$
$331$		−	3.37169e13i		−	0.466438i	0.972424	−	0.233219i	$-0.0749259\pi$
$332$		−	1.34435e13i		−	0.182916i
$333$			3.62854e13i			0.485611i
$334$	−3.50509e12			−0.0461418
$335$	−7.38206e13			−0.955941
$336$		−	9.18302e13i		−	1.16982i
$337$	−8.98229e13			−1.12570			−0.562850	−	0.826559i	$-0.690295\pi$
$337$	−8.98229e13			−1.12570			−0.562850	+	0.826559i	$0.690295\pi$
$338$	5.59888e12	+	2.34667e12i	0.0690335	+	0.0289341i
$339$	−8.13154e13			−0.986449
$340$		−	1.57649e13i		−	0.188173i
$341$	2.11988e14			2.48979
$342$	1.72786e12			0.0199694
$343$		−	8.67761e13i		−	0.986922i
$344$		−	1.51218e13i		−	0.169251i
$345$			1.66035e13i			0.182892i
$346$		−	3.76894e11i		−	0.00408602i
$347$	8.75394e13			0.934096			0.467048	−	0.884232i	$-0.345318\pi$
$347$	8.75394e13			0.934096			0.467048	+	0.884232i	$0.345318\pi$
$348$	−9.40284e13			−0.987582
$349$			3.87065e13i			0.400169i	0.979779	+	0.200085i	$0.0641217\pi$
$349$			3.87065e13i			0.400169i	−0.979779	+	0.200085i	$0.935878\pi$
$350$	−5.16059e12			−0.0525201
$351$	−6.04870e13	+	4.02331e13i	−0.606000	+	0.403083i
$352$	−3.42945e13			−0.338251
$353$			3.24210e13i			0.314822i	0.987533	+	0.157411i	$0.0503148\pi$
$353$			3.24210e13i			0.314822i	−0.987533	+	0.157411i	$0.949685\pi$
$354$	7.95458e12			0.0760500
$355$	2.87885e13			0.270996
$356$			1.15760e14i			1.07296i
$357$			1.89593e13i			0.173040i
$358$			6.98719e12i			0.0627980i
$359$		−	1.45534e14i		−	1.28809i	−0.764990	−	0.644043i	$-0.777256\pi$
$359$		−	1.45534e14i		−	1.28809i	0.764990	−	0.644043i	$-0.222744\pi$
$360$	−8.48075e12			−0.0739214
$361$	5.92451e13			0.508584
$362$			1.17449e13i			0.0993011i
$363$	−1.83879e14			−1.53125
$364$	−6.79876e13	−	1.02214e14i	−0.557663	−	0.838399i
$365$	7.48205e13			0.604518
$366$			1.08970e13i			0.0867285i
$367$	6.95668e13			0.545430			0.272715	−	0.962095i	$-0.412078\pi$
$367$	6.95668e13			0.545430			0.272715	+	0.962095i	$0.412078\pi$
$368$	1.52287e13			0.117625
$369$		−	4.52567e13i		−	0.344380i
$370$		−	1.65761e13i		−	0.124272i
$371$			2.62284e14i			1.93738i
$372$		−	2.63373e14i		−	1.91684i
$373$	1.20982e14			0.867604			0.433802	−	0.901008i	$-0.357172\pi$
$373$	1.20982e14			0.867604			0.433802	+	0.901008i	$0.357172\pi$
$374$	2.33799e12			0.0165215
$375$			6.74168e13i			0.469457i
$376$	2.48629e13			0.170615
$377$	−1.04067e14	+	6.92206e13i	−0.703779	+	0.468120i
$378$	−8.27685e12			−0.0551646
$379$		−	3.22918e13i		−	0.212117i	−0.994360	−	0.106059i	$-0.966177\pi$
$379$		−	3.22918e13i		−	0.212117i	0.994360	−	0.106059i	$-0.0338231\pi$
$380$	1.40092e14			0.906991
$381$	2.71365e14			1.73168
$382$		−	1.57030e13i		−	0.0987719i
$383$		−	6.60313e13i		−	0.409408i	−0.978824	−	0.204704i	$-0.934377\pi$
$383$		−	6.60313e13i		−	0.409408i	0.978824	−	0.204704i	$-0.0656232\pi$
$384$			5.67556e13i			0.346886i
$385$			3.31862e14i			1.99951i
$386$	3.85428e12			0.0228936
$387$	−7.36828e13			−0.431475
$388$			3.29929e14i			1.90478i
$389$	1.13990e14			0.648851			0.324426	−	0.945911i	$-0.394829\pi$
$389$	1.13990e14			0.648851			0.324426	+	0.945911i	$0.394829\pi$
$390$	−1.69769e13	+	1.12922e13i	−0.0952804	+	0.0633760i
$391$	−3.14411e12			−0.0173991
$392$		−	6.93779e11i		−	0.00378572i
$393$	1.59607e14			0.858803
$394$	7.86887e12			0.0417525
$395$		−	3.41400e13i		−	0.178640i
$396$			1.11298e14i			0.574335i
$397$		−	2.68777e14i		−	1.36787i	−0.729543	−	0.683935i	$-0.760267\pi$
$397$		−	2.68777e14i		−	1.36787i	0.729543	−	0.683935i	$-0.239733\pi$
$398$			9.97097e11i			0.00500473i
$399$	−1.68478e14			−0.834050
$400$	−1.39530e14			−0.681297
$401$		−	3.43265e14i		−	1.65324i	−0.562762	−	0.826619i	$-0.690261\pi$
$401$		−	3.43265e14i		−	1.65324i	0.562762	−	0.826619i	$-0.309739\pi$
$402$	−1.36015e13			−0.0646167
$403$	−1.93886e14	−	2.91492e14i	−0.908593	−	1.36599i
$404$	2.67358e14			1.23594
$405$		−	3.52572e14i		−	1.60785i
$406$	−1.42402e13			−0.0640655
$407$	−4.36303e14			−1.93650
$408$		−	5.82577e12i		−	0.0255107i
$409$			2.66850e14i			1.15289i	0.817135	+	0.576446i	$0.195561\pi$
$409$			2.66850e14i			1.15289i	−0.817135	+	0.576446i	$0.804439\pi$
$410$			2.06744e13i			0.0881294i
$411$			9.63807e13i			0.405378i
$412$	−1.12101e14			−0.465237
$413$	−2.13811e14			−0.875599
$414$			8.43313e11i			0.00340790i
$415$	6.00172e13			0.239338
$416$	3.13660e13	+	4.71561e13i	0.123437	+	0.185577i
$417$	−3.28745e14			−1.27676
$418$			2.07761e13i			0.0796333i
$419$	−2.13925e14			−0.809253			−0.404626	−	0.914482i	$-0.632598\pi$
$419$	−2.13925e14			−0.809253			−0.404626	+	0.914482i	$0.632598\pi$
$420$	4.12304e14			1.53938
$421$			1.24393e14i			0.458398i	0.973380	+	0.229199i	$0.0736107\pi$
$421$			1.24393e14i			0.458398i	−0.973380	+	0.229199i	$0.926389\pi$
$422$			2.71063e12i			0.00985944i
$423$		−	1.21147e14i		−	0.434953i
$424$		−	8.05940e13i		−	0.285621i
$425$	2.88073e13			0.100777
$426$	5.30432e12			0.0183180
$427$		−	2.92901e14i		−	0.998545i
$428$	3.76903e14			1.26850
$429$	2.97226e14	+	4.46854e14i	0.987578	+	1.48474i
$430$	3.36601e13			0.110418
$431$			3.23913e14i			1.04907i	0.851389	+	0.524535i	$0.175761\pi$
$431$			3.23913e14i			1.04907i	−0.851389	+	0.524535i	$0.824239\pi$
$432$	−2.23786e14			−0.715603
$433$	−2.35022e13			−0.0742036			−0.0371018	−	0.999311i	$-0.511813\pi$
$433$	−2.35022e13			−0.0742036			−0.0371018	+	0.999311i	$0.511813\pi$
$434$		−	3.98868e13i		−	0.124347i
$435$		−	4.19781e14i		−	1.29221i
$436$		−	5.58851e13i		−	0.169871i
$437$		−	2.79395e13i		−	0.0838632i
$438$	1.37858e13			0.0408623
$439$	4.02014e14			1.17676			0.588379	−	0.808586i	$-0.299767\pi$
$439$	4.02014e14			1.17676			0.588379	+	0.808586i	$0.299767\pi$
$440$		−	1.01974e14i		−	0.294781i
$441$	−3.38052e12			−0.00965098
$442$	−2.13835e12	−	3.21482e12i	−0.00602914	−	0.00906430i
$443$	−3.08170e14			−0.858164			−0.429082	−	0.903265i	$-0.641163\pi$
$443$	−3.08170e14			−0.858164			−0.429082	+	0.903265i	$0.641163\pi$
$444$			5.42059e14i			1.49087i
$445$	−5.16800e14			−1.40392
$446$	−5.03007e13			−0.134969
$447$			2.12422e14i			0.562999i
$448$		−	3.73841e14i		−	0.978719i
$449$		−	1.99777e14i		−	0.516643i	−0.966059	−	0.258322i	$-0.916831\pi$
$449$		−	1.99777e14i		−	0.516643i	0.966059	−	0.258322i	$-0.0831694\pi$
$450$		−	7.72669e12i		−	0.0197390i
$451$	5.44175e14			1.37331
$452$	−3.34862e14			−0.834843
$453$			5.96902e14i			1.47016i
$454$	5.32121e13			0.129480
$455$	4.56323e14	−	3.03524e14i	1.09701	−	0.729677i
$456$	5.17696e13			0.122961
$457$		−	4.57271e13i		−	0.107309i	−0.998560	−	0.0536543i	$-0.982913\pi$
$457$		−	4.57271e13i		−	0.107309i	0.998560	−	0.0536543i	$-0.0170869\pi$
$458$	−9.21606e12			−0.0213690
$459$	4.62028e13			0.105852
$460$			6.83744e13i			0.154784i
$461$			3.40488e14i			0.761635i	0.924650	+	0.380817i	$0.124357\pi$
$461$			3.40488e14i			0.761635i	−0.924650	+	0.380817i	$0.875643\pi$
$462$			6.11461e13i			0.135157i
$463$			3.63954e14i			0.794971i	0.917608	+	0.397486i	$0.130117\pi$
$463$			3.63954e14i			0.794971i	−0.917608	+	0.397486i	$0.869883\pi$
$464$	−3.85021e14			−0.831066
$465$	1.17580e15			2.50809
$466$		−	2.85349e13i		−	0.0601524i
$467$	−1.10274e14			−0.229738			−0.114869	−	0.993381i	$-0.536645\pi$
$467$	−1.10274e14			−0.229738			−0.114869	+	0.993381i	$0.536645\pi$
$468$	1.53039e14	−	1.01794e14i	0.315102	−	0.209591i
$469$	3.65596e14			0.743962
$470$			5.53431e13i			0.111308i
$471$	−4.75493e13			−0.0945211
$472$	6.56995e13			0.129087
$473$		−	8.85976e14i		−	1.72062i
$474$		−	6.29034e12i		−	0.0120752i
$475$			2.55991e14i			0.485746i
$476$			7.80755e13i			0.146446i
$477$	−3.92704e14			−0.728139
$478$	−7.71737e13			−0.141454
$479$		−	1.66407e14i		−	0.301526i	−0.988570	−	0.150763i	$-0.951827\pi$
$479$		−	1.66407e14i		−	0.301526i	0.988570	−	0.150763i	$-0.0481731\pi$
$480$	−1.90216e14			−0.340738
$481$	3.99046e14	+	5.99932e14i	0.706684	+	1.06244i
$482$	−2.14088e13			−0.0374829
$483$		−	8.22287e13i		−	0.142336i
$484$	−7.57228e14			−1.29592
$485$	−1.47293e15			−2.49232
$486$		−	3.23989e13i		−	0.0542040i
$487$			4.21283e13i			0.0696891i	0.999393	+	0.0348446i	$0.0110936\pi$
$487$			4.21283e13i			0.0696891i	−0.999393	+	0.0348446i	$0.988906\pi$
$488$			9.00021e13i			0.147212i
$489$		−	7.44897e14i		−	1.20475i
$490$	1.54431e12			0.00246976
$491$	3.56448e13			0.0563700			0.0281850	−	0.999603i	$-0.491027\pi$
$491$	3.56448e13			0.0563700			0.0281850	+	0.999603i	$0.491027\pi$
$492$		−	6.76079e14i		−	1.05728i
$493$	7.94914e13			0.122931
$494$	2.85679e13	−	1.90020e13i	0.0436898	−	0.0290604i
$495$	−4.96881e14			−0.751491
$496$		−	1.07844e15i		−	1.61305i
$497$	−1.42575e14			−0.210903
$498$	1.10582e13			0.0161780
$499$			2.70054e14i			0.390748i	0.980729	+	0.195374i	$0.0625921\pi$
$499$			2.70054e14i			0.390748i	−0.980729	+	0.195374i	$0.937408\pi$
$500$			2.77627e14i			0.397307i
$501$			5.11716e14i			0.724305i
$502$			1.72517e13i			0.0241525i
$503$	−4.93264e14			−0.683054			−0.341527	−	0.939872i	$-0.610944\pi$
$503$	−4.93264e14			−0.683054			−0.341527	+	0.939872i	$0.610944\pi$
$504$	4.20008e13			0.0575294
$505$			1.19360e15i			1.61717i
$506$	−1.01402e13			−0.0135899
$507$	3.42595e14	−	8.17392e14i	0.454189	−	1.08364i
$508$	1.11750e15			1.46554
$509$		−	7.49682e14i		−	0.972589i	−0.873795	−	0.486294i	$-0.838348\pi$
$509$		−	7.49682e14i		−	0.972589i	0.873795	−	0.486294i	$-0.161652\pi$
$510$	1.29678e13			0.0166429
$511$	−3.70548e14			−0.470467
$512$			2.91321e14i			0.365921i
$513$			4.10573e14i			0.510205i
$514$		−	7.55830e13i		−	0.0929238i
$515$		−	5.00464e14i		−	0.608742i
$516$	−1.10073e15			−1.32467
$517$	1.45670e15			1.73449
$518$			8.20928e13i			0.0967145i
$519$	−5.50235e13			−0.0641398
$520$	−1.40218e14	+	9.32664e13i	−0.161728	+	0.107574i
$521$	1.28499e15			1.46654			0.733270	−	0.679938i	$-0.237993\pi$
$521$	1.28499e15			1.46654			0.733270	+	0.679938i	$0.237993\pi$
$522$		−	2.13212e13i		−	0.0240782i
$523$	1.71007e14			0.191097			0.0955486	−	0.995425i	$-0.469539\pi$
$523$	1.71007e14			0.191097			0.0955486	+	0.995425i	$0.469539\pi$
$524$	6.57273e14			0.726815
$525$			7.53405e14i			0.824427i
$526$			3.46937e12i			0.00375690i
$527$			2.22655e14i			0.238602i
$528$			1.65324e15i			1.75327i
$529$	−9.39173e14			−0.985688
$530$	1.79397e14			0.186336
$531$		−	3.20129e14i		−	0.329083i
$532$	−6.93803e14			−0.705866
$533$	−4.97707e14	−	7.48260e14i	−0.501157	−	0.753447i
$534$	−9.52211e13			−0.0948978
$535$			1.68265e15i			1.65977i
$536$	−1.12340e14			−0.109680
$537$	1.02007e15			0.985764
$538$		−	1.14932e14i		−	0.109936i
$539$		−	4.06480e13i		−	0.0384859i
$540$		−	1.00476e15i		−	0.941669i
$541$		−	1.59371e15i		−	1.47851i	−0.673424	−	0.739256i	$-0.735177\pi$
$541$		−	1.59371e15i		−	1.47851i	0.673424	−	0.739256i	$-0.264823\pi$
$542$	−1.48060e14			−0.135969
$543$	1.71467e15			1.55877
$544$		−	3.60201e13i		−	0.0324153i
$545$	2.49494e14			0.222269
$546$	8.40780e13	−	5.59247e13i	0.0741520	−	0.0493224i
$547$	1.12365e15			0.981073			0.490537	−	0.871421i	$-0.336801\pi$
$547$	1.12365e15			0.981073			0.490537	+	0.871421i	$0.336801\pi$
$548$			3.96902e14i			0.343076i
$549$	4.38546e14			0.375290
$550$	9.29072e13			0.0787145
$551$			7.06386e14i			0.592527i
$552$			2.52671e13i			0.0209841i
$553$			1.69078e14i			0.139027i
$554$			5.73387e13i			0.0466814i
$555$	−2.41997e15			−1.95074
$556$	−1.35379e15			−1.08054
$557$			5.45822e14i			0.431368i	0.976463	+	0.215684i	$0.0691981\pi$
$557$			5.45822e14i			0.431368i	−0.976463	+	0.215684i	$0.930802\pi$
$558$	5.97205e13			0.0467343
$559$	−1.21825e15	+	8.10321e14i	−0.943998	+	0.627903i
$560$	1.68827e15			1.29541
$561$		−	3.41328e14i		−	0.259344i
$562$	1.46203e14			0.110004
$563$	−1.07491e15			−0.800897			−0.400448	−	0.916319i	$-0.631146\pi$
$563$	−1.07491e15			−0.800897			−0.400448	+	0.916319i	$0.631146\pi$
$564$		−	1.80979e15i		−	1.33535i
$565$		−	1.49496e15i		−	1.09235i
$566$		−	1.61735e14i		−	0.117034i
$567$			1.74611e15i			1.25131i
$568$	4.38102e13			0.0310928
$569$	−3.71485e14			−0.261110			−0.130555	−	0.991441i	$-0.541676\pi$
$569$	−3.71485e14			−0.261110			−0.130555	+	0.991441i	$0.541676\pi$
$570$			1.15236e14i			0.0802186i
$571$	5.89713e14			0.406576			0.203288	−	0.979119i	$-0.434837\pi$
$571$	5.89713e14			0.406576			0.203288	+	0.979119i	$0.434837\pi$
$572$	1.22400e15	+	1.84017e15i	0.835799	+	1.25655i
$573$	−2.29251e15			−1.55046
$574$		−	1.02390e14i		−	0.0685868i
$575$	−1.24941e14			−0.0828956
$576$	5.59733e14			0.367839
$577$		−	4.47961e14i		−	0.291590i	−0.989315	−	0.145795i	$-0.953426\pi$
$577$		−	4.47961e14i		−	0.291590i	0.989315	−	0.145795i	$-0.0465740\pi$
$578$		−	1.13637e14i		−	0.0732686i
$579$		−	5.62695e14i		−	0.359369i
$580$		−	1.72869e15i		−	1.09361i
$581$	−2.97235e14			−0.186265
$582$	−2.71390e14			−0.168468
$583$		−	4.72195e15i		−	2.90365i
$584$	1.13861e14			0.0693594
$585$	4.54452e14	+	6.83229e14i	0.274240	+	0.412296i
$586$	5.57070e13			0.0333022
$587$			2.90281e14i			0.171913i	0.996299	+	0.0859567i	$0.0273947\pi$
$587$			2.90281e14i			0.171913i	−0.996299	+	0.0859567i	$0.972605\pi$
$588$	−5.05009e13			−0.0296294
$589$	−1.97858e15			−1.15006
$590$			1.46243e14i			0.0842148i
$591$		−	1.14879e15i		−	0.655406i
$592$			2.21959e15i			1.25459i
$593$			1.53135e15i			0.857581i	0.903404	+	0.428790i	$0.141060\pi$
$593$			1.53135e15i			0.857581i	−0.903404	+	0.428790i	$0.858940\pi$
$594$	1.49010e14			0.0826780
$595$	−3.48561e14			−0.191618
$596$			8.74767e14i			0.476472i
$597$	1.45568e14			0.0785611
$598$	9.27427e12	+	1.39431e13i	0.00495933	+	0.00745594i
$599$	−2.34047e14			−0.124009			−0.0620047	−	0.998076i	$-0.519749\pi$
$599$	−2.34047e14			−0.124009			−0.0620047	+	0.998076i	$0.519749\pi$
$600$		−	2.31505e14i		−	0.121543i
$601$	−2.19750e15			−1.14319			−0.571597	−	0.820535i	$-0.693676\pi$
$601$	−2.19750e15			−1.14319			−0.571597	+	0.820535i	$0.693676\pi$
$602$	−1.66701e14			−0.0859327
$603$			5.47388e14i			0.279608i
$604$			2.45808e15i			1.24421i
$605$		−	3.38057e15i		−	1.69565i
$606$			2.19921e14i			0.109312i
$607$	3.02645e15			1.49072			0.745360	−	0.666662i	$-0.232277\pi$
$607$	3.02645e15			1.49072			0.745360	+	0.666662i	$0.232277\pi$
$608$	3.20086e14			0.156241
$609$			2.07896e15i			1.00566i
$610$	−2.00339e14			−0.0960397
$611$	−1.33231e15	−	2.00301e15i	−0.632963	−	0.951606i
$612$	−1.16899e14			−0.0550397
$613$			1.31524e15i			0.613724i	0.951754	+	0.306862i	$0.0992791\pi$
$613$			1.31524e15i			0.613724i	−0.951754	+	0.306862i	$0.900721\pi$
$614$	−1.96226e13			−0.00907468
$615$	3.01829e15			1.38340
$616$			5.05026e14i			0.229414i
$617$			7.90565e14i			0.355934i	0.984036	+	0.177967i	$0.0569520\pi$
$617$			7.90565e14i			0.355934i	−0.984036	+	0.177967i	$0.943048\pi$
$618$		−	9.22111e13i		−	0.0411478i
$619$			5.75601e13i			0.0254579i	0.999919	+	0.0127290i	$0.00405186\pi$
$619$			5.75601e13i			0.0254579i	−0.999919	+	0.0127290i	$0.995948\pi$
$620$	4.84204e15			2.12263
$621$	−2.00387e14			−0.0870696
$622$			2.33253e14i			0.100457i
$623$	2.55945e15			1.09260
$624$	2.27326e15	−	1.51207e15i	0.961910	−	0.639817i
$625$	−2.89150e15			−1.21278
$626$		−	9.96462e13i		−	0.0414288i
$627$	3.03315e15			1.25003
$628$	−1.95811e14			−0.0799943
$629$		−	4.58256e14i		−	0.185579i
$630$			9.34910e13i			0.0375316i
$631$			3.34887e15i			1.33271i	0.745633	+	0.666356i	$0.232147\pi$
$631$			3.34887e15i			1.33271i	−0.745633	+	0.666356i	$0.767853\pi$
$632$		−	5.19540e13i		−	0.0204963i
$633$	3.95731e14			0.154767
$634$	−8.41160e13			−0.0326128
$635$			4.98897e15i			1.91759i
$636$	−5.86652e15			−2.23546
$637$	−5.58925e13	+	3.71771e13i	−0.0211148	+	0.0140446i
$638$	2.56370e14			0.0960182
$639$		−	2.13470e14i		−	0.0792652i
$640$	−1.04344e15			−0.384128
$641$	−5.29783e14			−0.193365			−0.0966827	−	0.995315i	$-0.530823\pi$
$641$	−5.29783e14			−0.193365			−0.0966827	+	0.995315i	$0.530823\pi$
$642$			3.10030e14i			0.112192i
$643$		−	1.01506e15i		−	0.364194i	−0.983281	−	0.182097i	$-0.941712\pi$
$643$		−	1.01506e15i		−	0.364194i	0.983281	−	0.182097i	$-0.0582885\pi$
$644$		−	3.38623e14i		−	0.120460i
$645$		−	4.91411e15i		−	1.73327i
$646$	−2.18215e13			−0.00763143
$647$	1.92293e14			0.0666793			0.0333396	−	0.999444i	$-0.489386\pi$
$647$	1.92293e14			0.0666793			0.0333396	+	0.999444i	$0.489386\pi$
$648$		−	5.36541e14i		−	0.184476i
$649$	3.84929e15			1.31231
$650$	−8.49737e13	−	1.27751e14i	−0.0287251	−	0.0431857i
$651$	−5.82316e15			−1.95192
$652$		−	3.06753e15i		−	1.01959i
$653$	−2.34263e15			−0.772114			−0.386057	−	0.922475i	$-0.626163\pi$
$653$	−2.34263e15			−0.772114			−0.386057	+	0.922475i	$0.626163\pi$
$654$	4.59695e13			0.0150242
$655$			2.93433e15i			0.951004i
$656$		−	2.76836e15i		−	0.889717i
$657$		−	5.54802e14i		−	0.176819i
$658$		−	2.74086e14i		−	0.0866253i
$659$	5.91488e15			1.85386			0.926928	−	0.375239i	$-0.122440\pi$
$659$	5.91488e15			1.85386			0.926928	+	0.375239i	$0.122440\pi$
$660$	−7.42279e15			−2.30715
$661$		−	4.81720e15i		−	1.48486i	−0.669922	−	0.742432i	$-0.733672\pi$
$661$		−	4.81720e15i		−	1.48486i	0.669922	−	0.742432i	$-0.266328\pi$
$662$	1.14213e14			0.0349138
$663$	−4.69338e14	+	3.12181e14i	−0.142286	+	0.0946417i
$664$	9.13337e13			0.0274604
$665$		−	3.09742e15i		−	0.923594i
$666$	−1.22913e14			−0.0363489
$667$	−3.44764e14			−0.101118
$668$			2.10728e15i			0.612988i
$669$			7.34350e15i			2.11865i
$670$		−	2.50060e14i		−	0.0715540i
$671$			5.27316e15i			1.49657i
$672$	9.42042e14			0.265179
$673$	−1.84391e15			−0.514821			−0.257410	−	0.966302i	$-0.582869\pi$
$673$	−1.84391e15			−0.514821			−0.257410	+	0.966302i	$0.582869\pi$
$674$		−	3.04267e14i		−	0.0842607i
$675$	1.83601e15			0.504318
$676$	1.41083e15	−	3.36608e15i	0.384385	−	0.917101i
$677$	−2.85026e15			−0.770276			−0.385138	−	0.922859i	$-0.625846\pi$
$677$	−2.85026e15			−0.770276			−0.385138	+	0.922859i	$0.625846\pi$
$678$		−	2.75448e14i		−	0.0738375i
$679$	7.29470e15			1.93965
$680$	1.07105e14			0.0282495
$681$		−	7.76854e15i		−	2.03250i
$682$			7.18091e14i			0.186365i
$683$			4.69918e15i			1.20978i	0.796308	+	0.604892i	$0.206784\pi$
$683$			4.69918e15i			1.20978i	−0.796308	+	0.604892i	$0.793216\pi$
$684$		−	1.03880e15i		−	0.265291i
$685$	−1.77193e15			−0.448899
$686$	2.93946e14			0.0738729
$687$			1.34547e15i			0.335438i
$688$	−4.50720e15			−1.11473
$689$	−6.49285e15	+	4.31873e15i	−1.59305	+	1.05962i
$690$	−5.62429e13			−0.0136898
$691$		−	2.34559e15i		−	0.566399i	−0.959061	−	0.283199i	$-0.908604\pi$
$691$		−	2.34559e15i		−	0.566399i	0.959061	−	0.283199i	$-0.0913958\pi$
$692$	−2.26590e14			−0.0542823
$693$	2.46080e15			0.584848
$694$			2.96532e14i			0.0699188i
$695$		−	6.04388e15i		−	1.41384i
$696$		−	6.38819e14i		−	0.148261i
$697$			5.71556e14i			0.131607i
$698$	−1.31115e14			−0.0299534
$699$	−4.16586e15			−0.944236
$700$			3.10257e15i			0.697722i
$701$	2.09935e15			0.468421			0.234210	−	0.972186i	$-0.424750\pi$
$701$	2.09935e15			0.468421			0.234210	+	0.972186i	$0.424750\pi$
$702$	−1.36286e14	−	2.04894e14i	−0.0301715	−	0.0453602i
$703$	4.07221e15			0.894490
$704$			6.73034e15i			1.46686i
$705$	8.07965e15			1.74724
$706$	−1.09823e14			−0.0235650
$707$		−	5.91127e15i		−	1.25856i
$708$		−	4.78233e15i		−	1.01031i
$709$		−	6.10587e15i		−	1.27995i	−0.768396	−	0.639975i	$-0.778945\pi$
$709$		−	6.10587e15i		−	1.27995i	0.768396	−	0.639975i	$-0.221055\pi$
$710$			9.75185e13i			0.0202846i
$711$	−2.53152e14			−0.0522515
$712$	−7.86463e14			−0.161079
$713$		−	9.65683e14i		−	0.196265i
$714$	−6.42228e13			−0.0129524
$715$	−8.21528e15	+	5.46441e15i	−1.64414	+	1.09360i
$716$	4.20073e15			0.834263
$717$			1.12667e16i			2.22046i
$718$	4.92983e14			0.0964156
$719$	5.91981e15			1.14894			0.574472	−	0.818524i	$-0.305207\pi$
$719$	5.91981e15			1.14894			0.574472	+	0.818524i	$0.305207\pi$
$720$			2.52777e15i			0.486865i
$721$			2.47854e15i			0.473754i
$722$			2.00687e14i			0.0380685i
$723$			3.12551e15i			0.588384i
$724$	7.06112e15			1.31920
$725$	3.15884e15			0.585690
$726$		−	6.22875e14i		−	0.114617i
$727$	2.61053e15			0.476748			0.238374	−	0.971173i	$-0.423386\pi$
$727$	2.61053e15			0.476748			0.238374	+	0.971173i	$0.423386\pi$
$728$	6.94429e14	−	4.61901e14i	0.125865	−	0.0837194i
$729$	2.13955e15			0.384876
$730$			2.53447e14i			0.0452493i
$731$	9.30556e14			0.164891
$732$	6.55134e15			1.15218
$733$			9.10402e15i			1.58914i	0.607175	+	0.794568i	$0.292303\pi$
$733$			9.10402e15i			1.58914i	−0.607175	+	0.794568i	$0.707697\pi$
$734$			2.35651e14i			0.0408264i
$735$		−	2.25456e14i		−	0.0387688i
$736$			1.56224e14i			0.0266636i
$737$	−6.58190e15			−1.11501
$738$	1.53303e14			0.0257775
$739$		−	2.00804e15i		−	0.335141i	−0.985860	−	0.167570i	$-0.946408\pi$
$739$		−	2.00804e15i		−	0.335141i	0.985860	−	0.167570i	$-0.0535921\pi$
$740$	−9.96561e15			−1.65093
$741$	−2.77414e15	−	4.17068e15i	−0.456172	−	0.685815i
$742$	−8.88461e14			−0.145016
$743$		−	8.48953e15i		−	1.37545i	−0.725971	−	0.687725i	$-0.758609\pi$
$743$		−	8.48953e15i		−	1.37545i	0.725971	−	0.687725i	$-0.241391\pi$
$744$	1.78933e15			0.287766
$745$	−3.90531e15			−0.623442
$746$			4.09815e14i			0.0649417i
$747$		−	4.45035e14i		−	0.0700052i
$748$		−	1.40561e15i		−	0.219486i
$749$		−	8.33331e15i		−	1.29172i
$750$	−2.28368e14			−0.0351397
$751$	−5.75001e15			−0.878312			−0.439156	−	0.898411i	$-0.644722\pi$
$751$	−5.75001e15			−0.878312			−0.439156	+	0.898411i	$0.644722\pi$
$752$		−	7.41062e15i		−	1.12372i
$753$	2.51861e15			0.379131
$754$	−2.34478e14	−	3.52518e14i	−0.0350397	−	0.0526792i
$755$	−1.09739e16			−1.62799
$756$			4.97608e15i			0.732855i
$757$	4.57782e15			0.669317			0.334658	−	0.942339i	$-0.391379\pi$
$757$	4.57782e15			0.669317			0.334658	+	0.942339i	$0.391379\pi$
$758$	1.09385e14			0.0158774
$759$			1.48038e15i			0.213326i
$760$			9.51770e14i			0.136162i
$761$		−	3.71156e15i		−	0.527158i	−0.964638	−	0.263579i	$-0.915097\pi$
$761$		−	3.71156e15i		−	0.527158i	0.964638	−	0.263579i	$-0.0849029\pi$
$762$			9.19224e14i			0.129619i
$763$	−1.23562e15			−0.172981
$764$	−9.44070e15			−1.31217
$765$		−	5.21883e14i		−	0.0720170i
$766$	2.23675e14			0.0306450
$767$	−3.52059e15	−	5.29291e15i	−0.478896	−	0.719980i
$768$	8.21659e15			1.10970
$769$		−	2.37640e15i		−	0.318658i	−0.987226	−	0.159329i	$-0.949067\pi$
$769$		−	2.37640e15i		−	0.318658i	0.987226	−	0.159329i	$-0.0509331\pi$
$770$	−1.12415e15			−0.149667
$771$	−1.10345e16			−1.45866
$772$		−	2.31721e15i		−	0.304138i
$773$			4.48843e15i			0.584934i	0.956276	+	0.292467i	$0.0944762\pi$
$773$			4.48843e15i			0.584934i	−0.956276	+	0.292467i	$0.905524\pi$
$774$		−	2.49594e14i		−	0.0322967i
$775$			8.84788e15i			1.13679i
$776$	−2.24150e15			−0.285956
$777$	1.19849e16			1.51816
$778$			3.86132e14i			0.0485677i
$779$	−5.07903e15			−0.634344
$780$	6.78895e15	+	1.02066e16i	0.841942	+	1.26579i
$781$	2.56681e15			0.316091
$782$		−	1.06504e13i		−	0.00130235i
$783$	5.06633e15			0.615182
$784$	−2.06788e14			−0.0249337
$785$		−	8.74180e14i		−	0.104669i
$786$			5.40655e14i			0.0642830i
$787$			1.79432e15i			0.211856i	0.994374	+	0.105928i	$0.0337812\pi$
$787$			1.79432e15i			0.211856i	−0.994374	+	0.105928i	$0.966219\pi$
$788$		−	4.73080e15i		−	0.554677i
$789$	5.06501e14			0.0589735
$790$	1.15646e14			0.0133716
$791$			7.40378e15i			0.850125i
$792$	−7.56150e14			−0.0862223
$793$	7.25079e15	−	4.82288e15i	0.821074	−	0.546140i
$794$	9.10458e14			0.102388
$795$		−	2.61905e16i		−	2.92499i
$796$	5.99459e14			0.0664872
$797$	1.29926e15			0.143111			0.0715556	−	0.997437i	$-0.477204\pi$
$797$	1.29926e15			0.143111			0.0715556	+	0.997437i	$0.477204\pi$
$798$		−	5.70704e14i		−	0.0624302i
$799$			1.53000e15i			0.166220i
$800$		−	1.43137e15i		−	0.154439i
$801$			3.83213e15i			0.410641i
$802$	1.16278e15			0.123748
$803$	6.67105e15			0.705113
$804$			8.17731e15i			0.858425i
$805$	1.51175e15			0.157617
$806$	9.87401e14	−	6.56772e14i	0.102247	−	0.0680099i
$807$	−1.67791e16			−1.72570
$808$			1.81641e15i			0.185545i
$809$	−1.71565e16			−1.74065			−0.870326	−	0.492476i	$-0.836092\pi$
$809$	−1.71565e16			−1.74065			−0.870326	+	0.492476i	$0.836092\pi$
$810$	1.19430e15			0.120350
$811$		−	8.99602e15i		−	0.900401i	−0.892928	−	0.450200i	$-0.851353\pi$
$811$		−	8.99602e15i		−	0.900401i	0.892928	−	0.450200i	$-0.148647\pi$
$812$			8.56130e15i			0.851102i
$813$			2.16155e16i			2.13436i
$814$		−	1.47793e15i		−	0.144951i
$815$	1.36947e16			1.33409
$816$	−1.73643e15			−0.168020
$817$			8.26921e15i			0.794772i
$818$	−9.03929e14			−0.0862961
$819$	−2.25067e15	−	3.38369e15i	−0.213427	−	0.320870i
$820$	1.24295e16			1.17079
$821$			1.60428e16i			1.50104i	0.660848	+	0.750520i	$0.270197\pi$
$821$			1.60428e16i			1.50104i	−0.660848	+	0.750520i	$0.729803\pi$
$822$	−3.26481e14			−0.0303433
$823$	−5.49308e15			−0.507128			−0.253564	−	0.967319i	$-0.581603\pi$
$823$	−5.49308e15			−0.507128			−0.253564	+	0.967319i	$0.581603\pi$
$824$		−	7.61603e14i		−	0.0698440i
$825$		−	1.35637e16i		−	1.23561i
$826$		−	7.24266e14i		−	0.0655402i
$827$		−	8.14813e15i		−	0.732450i	−0.930526	−	0.366225i	$-0.880650\pi$
$827$		−	8.14813e15i		−	0.732450i	0.930526	−	0.366225i	$-0.119350\pi$
$828$	5.07004e14			0.0452735
$829$	1.98376e16			1.75971			0.879853	−	0.475246i	$-0.157641\pi$
$829$	1.98376e16			1.75971			0.879853	+	0.475246i	$0.157641\pi$
$830$			2.03303e14i			0.0179149i
$831$	8.37100e15			0.732776
$832$	9.25446e15	−	6.15563e15i	0.804772	−	0.535296i
$833$	4.26934e13			0.00368819
$834$		−	1.11359e15i		−	0.0955681i
$835$	−9.40775e15			−0.802066
$836$	1.24907e16			1.05792
$837$			1.41907e16i			1.19403i
$838$		−	7.24651e14i		−	0.0605741i
$839$			2.34908e15i			0.195077i	0.995232	+	0.0975387i	$0.0310970\pi$
$839$			2.34908e15i			0.195077i	−0.995232	+	0.0975387i	$0.968903\pi$
$840$			2.80115e15i			0.231100i
$841$	−3.48395e15			−0.285558
$842$	−4.21368e14			−0.0343119
$843$		−	2.13445e16i		−	1.72677i
$844$	1.62965e15			0.130981
$845$	1.50275e16	+	6.29850e15i	1.19998	+	0.502951i
$846$	4.10375e14			0.0325570
$847$			1.67423e16i			1.31964i
$848$	−2.40218e16			−1.88117
$849$	−2.36120e16			−1.83713
$850$			9.75821e13i			0.00754338i
$851$			1.98751e15i			0.152650i
$852$		−	3.18898e15i		−	0.243352i
$853$		−	1.56183e16i		−	1.18417i	−0.805875	−	0.592086i	$-0.798305\pi$
$853$		−	1.56183e16i		−	1.18417i	0.805875	−	0.592086i	$-0.201695\pi$
$854$	9.92175e14			0.0747429
$855$	4.63761e15			0.347121
$856$			2.56064e15i			0.190434i
$857$	1.24481e16			0.919834			0.459917	−	0.887962i	$-0.347879\pi$
$857$	1.24481e16			0.919834			0.459917	+	0.887962i	$0.347879\pi$
$858$	−1.51368e15	+	1.00682e15i	−0.111135	+	0.0739220i
$859$	−1.17701e16			−0.858651			−0.429326	−	0.903150i	$-0.641249\pi$
$859$	−1.17701e16			−0.858651			−0.429326	+	0.903150i	$0.641249\pi$
$860$		−	2.02366e16i		−	1.46689i
$861$	−1.49481e16			−1.07663
$862$	−1.09723e15			−0.0785247
$863$			4.09282e15i			0.291047i	0.989355	+	0.145524i	$0.0464867\pi$
$863$			4.09282e15i			0.291047i	−0.989355	+	0.145524i	$0.953513\pi$
$864$		−	2.29571e15i		−	0.162215i
$865$		−	1.01159e15i		−	0.0710259i
$866$		−	7.96115e13i		−	0.00555428i
$867$	−1.65901e16			−1.15012
$868$	−2.39801e16			−1.65194
$869$		−	3.04395e15i		−	0.208367i
$870$	1.42197e15			0.0967240
$871$	6.01986e15	+	9.05035e15i	0.406899	+	0.611738i
$872$	3.79678e14			0.0255020
$873$			1.09220e16i			0.728993i
$874$	9.46426e13			0.00627731
$875$	6.13831e15			0.404580
$876$		−	8.28807e15i		−	0.542851i
$877$		−	1.66555e16i		−	1.08408i	−0.840354	−	0.542038i	$-0.817653\pi$
$877$		−	1.66555e16i		−	1.08408i	0.840354	−	0.542038i	$-0.182347\pi$
$878$			1.36179e15i			0.0880824i
$879$		−	8.13278e15i		−	0.522757i
$880$	−3.03943e16			−1.94150
$881$	2.33295e16			1.48094			0.740472	−	0.672087i	$-0.234602\pi$
$881$	2.33295e16			1.48094			0.740472	+	0.672087i	$0.234602\pi$
$882$		−	1.14512e13i		−	0.000722394i
$883$	5.66211e14			0.0354972			0.0177486	−	0.999842i	$-0.494350\pi$
$883$	5.66211e14			0.0354972			0.0177486	+	0.999842i	$0.494350\pi$
$884$	−1.93277e15	+	1.28558e15i	−0.120418	+	0.0800964i
$885$	2.13503e16			1.32195
$886$		−	1.04390e15i		−	0.0642352i
$887$	8.53270e15			0.521803			0.260901	−	0.965365i	$-0.415980\pi$
$887$	8.53270e15			0.521803			0.260901	+	0.965365i	$0.415980\pi$
$888$	−3.68270e15			−0.223818
$889$		−	2.47078e16i		−	1.49236i
$890$		−	1.75061e15i		−	0.105086i
$891$		−	3.14355e16i		−	1.87540i
$892$			3.02410e16i			1.79304i
$893$	−1.35960e16			−0.801177
$894$	−7.19559e14			−0.0421415
$895$			1.87538e16i			1.09160i
$896$	5.16760e15			0.298948
$897$	2.03558e15	−	1.35397e15i	0.117039	−	0.0778486i
$898$	6.76726e14			0.0386717
$899$			2.44150e16i			1.38669i
$900$	−4.64532e15			−0.262230
$901$	4.95954e15			0.278263
$902$			1.84334e15i			0.102795i
$903$			2.43371e16i			1.34892i
$904$		−	2.27502e15i		−	0.125331i
$905$			3.15237e16i			1.72612i
$906$	−2.02195e15			−0.110044
$907$	9.92538e15			0.536917			0.268458	−	0.963291i	$-0.413486\pi$
$907$	9.92538e15			0.536917			0.268458	+	0.963291i	$0.413486\pi$
$908$		−	3.19914e16i		−	1.72013i
$909$	8.85065e15			0.473014
$910$	1.02816e15	+	1.54575e15i	0.0546177	+	0.0821130i
$911$	1.06854e16			0.564208			0.282104	−	0.959384i	$-0.408968\pi$
$911$	1.06854e16			0.564208			0.282104	+	0.959384i	$0.408968\pi$
$912$		−	1.54304e16i		−	0.809853i
$913$	5.35118e15			0.279165
$914$	1.54896e14			0.00803225
$915$			2.92478e16i			1.50757i
$916$			5.54074e15i			0.283885i
$917$		−	1.45323e16i		−	0.740119i
$918$			1.56508e14i			0.00792321i
$919$	−6.16827e15			−0.310405			−0.155202	−	0.987883i	$-0.549603\pi$
$919$	−6.16827e15			−0.310405			−0.155202	+	0.987883i	$0.549603\pi$
$920$	−4.64529e14			−0.0232370
$921$			2.86475e15i			0.142449i
$922$	−1.15337e15			−0.0570098
$923$	−2.34762e15	−	3.52945e15i	−0.115350	−	0.173420i
$924$	3.67613e16			1.79554
$925$		−	1.82102e16i		−	0.884170i
$926$	−1.23286e15			−0.0595051
$927$	−3.71100e15			−0.178054
$928$		−	3.94975e15i		−	0.188389i
$929$			1.71984e16i			0.815460i	0.913103	+	0.407730i	$0.133680\pi$
$929$			1.71984e16i			0.815460i	−0.913103	+	0.407730i	$0.866320\pi$
$930$			3.98293e15i			0.187735i
$931$			3.79386e14i			0.0177770i
$932$	−1.71553e16			−0.799117
$933$	3.40531e16			1.57691
$934$		−	3.73544e14i		−	0.0171963i
$935$	6.27522e15			0.287187
$936$	6.91581e14	+	1.03973e15i	0.0314649	+	0.0473048i
$937$	−1.38540e16			−0.626626			−0.313313	−	0.949650i	$-0.601439\pi$
$937$	−1.38540e16			−0.626626			−0.313313	+	0.949650i	$0.601439\pi$
$938$			1.23842e15i			0.0556869i
$939$	−1.45476e16			−0.650323
$940$	3.32725e16			1.47871
$941$			3.01742e16i			1.33319i	0.745419	+	0.666597i	$0.232250\pi$
$941$			3.01742e16i			1.33319i	−0.745419	+	0.666597i	$0.767750\pi$
$942$		−	1.61069e14i		−	0.00707508i
$943$		−	2.47891e15i		−	0.108255i
$944$		−	1.95824e16i		−	0.850196i
$945$	−2.22153e16			−0.958907
$946$	3.00116e15			0.128792
$947$		−	1.47817e16i		−	0.630665i	−0.948981	−	0.315333i	$-0.897884\pi$
$947$		−	1.47817e16i		−	0.630665i	0.948981	−	0.315333i	$-0.102116\pi$
$948$	−3.78178e15			−0.160417
$949$	−6.10140e15	−	9.17293e15i	−0.257315	−	0.386851i
$950$	−8.67145e14			−0.0363590
$951$			1.22803e16i			0.511935i
$952$	−5.30437e14			−0.0219852
$953$	−2.28957e16			−0.943501			−0.471750	−	0.881732i	$-0.656378\pi$
$953$	−2.28957e16			−0.943501			−0.471750	+	0.881732i	$0.656378\pi$
$954$		−	1.33025e15i		−	0.0545025i
$955$		−	4.21471e16i		−	1.71692i
$956$			4.63972e16i			1.87920i
$957$		−	3.74280e16i		−	1.50723i
$958$	5.63687e14			0.0225698
$959$	8.77548e15			0.349356
$960$			3.73302e16i			1.47764i
$961$	−4.29779e16			−1.69148
$962$	−2.03221e15	+	1.35173e15i	−0.0795255	+	0.0528966i
$963$	1.24770e16			0.485475
$964$			1.28711e16i			0.497956i
$965$	1.03450e16			0.397951
$966$	2.78542e14			0.0106541
$967$		−	3.65782e16i		−	1.39116i	−0.718448	−	0.695580i	$-0.755147\pi$
$967$		−	3.65782e16i		−	1.39116i	0.718448	−	0.695580i	$-0.244853\pi$
$968$		−	5.14453e15i		−	0.194550i
$969$			3.18577e15i			0.119793i
$970$		−	4.98943e15i		−	0.186555i
$971$	2.24568e16			0.834915			0.417458	−	0.908696i	$-0.362921\pi$
$971$	2.24568e16			0.834915			0.417458	+	0.908696i	$0.362921\pi$
$972$	−1.94784e16			−0.720093
$973$			2.99323e16i			1.10032i
$974$	−1.42706e14			−0.00521636
$975$	−1.86506e16	+	1.24055e16i	−0.677903	+	0.450909i
$976$	2.68260e16			0.969575
$977$		−	2.77848e16i		−	0.998590i	−0.866432	−	0.499295i	$-0.833592\pi$
$977$		−	2.77848e16i		−	0.998590i	0.866432	−	0.499295i	$-0.166408\pi$
$978$	2.52327e15			0.0901779
$979$	−4.60783e16			−1.63754
$980$		−	9.28444e14i		−	0.0328105i
$981$		−	1.85003e15i		−	0.0650127i
$982$			1.20744e14i			0.00421940i
$983$		−	6.91991e15i		−	0.240467i	−0.992746	−	0.120234i	$-0.961636\pi$
$983$		−	6.91991e15i		−	0.240467i	0.992746	−	0.120234i	$-0.0383644\pi$
$984$	4.59322e15			0.158724
$985$	2.11202e16			0.725770
$986$			2.69270e14i			0.00920163i
$987$	−4.00144e16			−1.35979
$988$	−1.14241e16	−	1.71752e16i	−0.386063	−	0.580413i
$989$	−4.03594e15			−0.135633
$990$		−	1.68314e15i		−	0.0562505i
$991$	4.49174e16			1.49283			0.746414	−	0.665482i	$-0.231774\pi$
$991$	4.49174e16			1.49283			0.746414	+	0.665482i	$0.231774\pi$
$992$	1.10632e16			0.365651
$993$		−	1.66742e16i		−	0.548054i
$994$		−	4.82959e14i		−	0.0157865i
$995$			2.67623e15i			0.0869954i
$996$		−	6.64827e15i		−	0.214923i
$997$	−4.98182e16			−1.60164			−0.800819	−	0.598907i	$-0.795602\pi$
$997$	−4.98182e16			−1.60164			−0.800819	+	0.598907i	$0.795602\pi$
$998$	−9.14782e14			−0.0292482
$999$		−	2.92066e16i		−	0.928690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.b.a.12.7	yes	12
3.2	odd	2		117.12.b.b.64.6		12
4.3	odd	2		208.12.f.b.129.4		12
13.5	odd	4		169.12.a.e.1.7		12
13.8	odd	4		169.12.a.e.1.6		12
13.12	even	2	inner	13.12.b.a.12.6	✓	12
39.38	odd	2		117.12.b.b.64.7		12
52.51	odd	2		208.12.f.b.129.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.b.a.12.6	✓	12	13.12	even	2	inner
13.12.b.a.12.7	yes	12	1.1	even	1	trivial
117.12.b.b.64.6		12	3.2	odd	2
117.12.b.b.64.7		12	39.38	odd	2
169.12.a.e.1.6		12	13.8	odd	4
169.12.a.e.1.7		12	13.5	odd	4
208.12.f.b.129.3		12	52.51	odd	2
208.12.f.b.129.4		12	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:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	13.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.38741i q^{2} +494.534 q^{3} +2036.53 q^{4} +9091.88i q^{5} +1675.19i q^{6} -45027.4i q^{7} +13836.0i q^{8} +67417.3 q^{9} +O(q^{10})\)	\(q+3.38741i q^{2} +494.534 q^{3} +2036.53 q^{4} +9091.88i q^{5} +1675.19i q^{6} -45027.4i q^{7} +13836.0i q^{8} +67417.3 q^{9} -30797.9 q^{10} +810639. i q^{11} +1.00713e6 q^{12} +(1.11466e6 - 741417. i) q^{13} +152526. q^{14} +4.49625e6i q^{15} +4.12394e6 q^{16} -851428. q^{17} +228370. i q^{18} -7.56605e6i q^{19} +1.85158e7i q^{20} -2.22676e7i q^{21} -2.74596e6 q^{22} +3.69275e6 q^{23} +6.84236e6i q^{24} -3.38341e7 q^{25} +(2.51148e6 + 3.77580e6i) q^{26} -5.42651e7 q^{27} -9.16995e7i q^{28} -9.33625e7 q^{29} -1.52306e7 q^{30} -2.61508e8i q^{31} +4.23055e7i q^{32} +4.00889e8i q^{33} -2.88413e6i q^{34} +4.09384e8 q^{35} +1.37297e8 q^{36} +5.38221e8i q^{37} +2.56293e7 q^{38} +(5.51236e8 - 3.66656e8i) q^{39} -1.25795e8 q^{40} -6.71292e8i q^{41} +7.54295e7 q^{42} -1.09294e9 q^{43} +1.65089e9i q^{44} +6.12950e8i q^{45} +1.25088e7i q^{46} -1.79698e9i q^{47} +2.03943e9 q^{48} -5.01432e7 q^{49} -1.14610e8i q^{50} -4.21060e8 q^{51} +(2.27003e9 - 1.50991e9i) q^{52} -5.82497e9 q^{53} -1.83818e8i q^{54} -7.37023e9 q^{55} +6.22997e8 q^{56} -3.74167e9i q^{57} -3.16257e8i q^{58} -4.74847e9i q^{59} +9.15672e9i q^{60} +6.50495e9 q^{61} +8.85834e8 q^{62} -3.03563e9i q^{63} +8.30252e9 q^{64} +(6.74087e9 + 1.01343e10i) q^{65} -1.35797e9 q^{66} +8.11940e9i q^{67} -1.73395e9 q^{68} +1.82619e9 q^{69} +1.38675e9i q^{70} -3.16640e9i q^{71} +9.32783e8i q^{72} -8.22937e9i q^{73} -1.82317e9 q^{74} -1.67321e10 q^{75} -1.54085e10i q^{76} +3.65010e10 q^{77} +(1.24201e9 + 1.86726e9i) q^{78} -3.75500e9 q^{79} +3.74943e10i q^{80} -3.87787e10 q^{81} +2.27394e9 q^{82} -6.60119e9i q^{83} -4.53486e10i q^{84} -7.74108e9i q^{85} -3.70222e9i q^{86} -4.61710e10 q^{87} -1.12160e10 q^{88} +5.68420e10i q^{89} -2.07631e9 q^{90} +(-3.33841e10 - 5.01902e10i) q^{91} +7.52038e9 q^{92} -1.29325e11i q^{93} +6.08709e9 q^{94} +6.87896e10 q^{95} +2.09215e10i q^{96} +1.62006e11i q^{97} -1.69856e8i q^{98} +5.46511e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9}+O(q^{10}) \) 12 * q - 488 * q^3 - 12290 * q^4 + 654644 * q^9	\( 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9} - 333446 q^{10} + 2740298 q^{12} + 3208868 q^{13} - 5367450 q^{14} + 19025698 q^{16} + 12198768 q^{17} - 111171128 q^{22} + 5810592 q^{23} + 6102388 q^{25} - 64543986 q^{26} - 52613336 q^{27} - 244463112 q^{29} + 426504126 q^{30} - 562027560 q^{35} - 1357546052 q^{36} + 3171817788 q^{38} - 2199109744 q^{39} + 4092185498 q^{40} + 1280452314 q^{42} + 2294519976 q^{43} - 14206061378 q^{48} - 3573617796 q^{49} + 7713246552 q^{51} - 5597650396 q^{52} - 4602062760 q^{53} - 6178744976 q^{55} + 20017912662 q^{56} - 13775649944 q^{61} + 239765256 q^{62} - 3560815378 q^{64} - 7598401512 q^{65} + 37979507040 q^{66} + 40844682210 q^{68} - 25419983328 q^{69} + 19351803414 q^{74} + 68016370832 q^{75} - 80478036048 q^{77} + 89375282178 q^{78} + 18046097296 q^{79} - 132677486692 q^{81} - 255687836096 q^{82} + 94507900752 q^{87} + 239343029120 q^{88} - 190413561204 q^{90} + 104793638664 q^{91} - 135236877012 q^{92} - 78363161402 q^{94} + 145093149648 q^{95}+O(q^{100}) \) 12 * q - 488 * q^3 - 12290 * q^4 + 654644 * q^9 - 333446 * q^10 + 2740298 * q^12 + 3208868 * q^13 - 5367450 * q^14 + 19025698 * q^16 + 12198768 * q^17 - 111171128 * q^22 + 5810592 * q^23 + 6102388 * q^25 - 64543986 * q^26 - 52613336 * q^27 - 244463112 * q^29 + 426504126 * q^30 - 562027560 * q^35 - 1357546052 * q^36 + 3171817788 * q^38 - 2199109744 * q^39 + 4092185498 * q^40 + 1280452314 * q^42 + 2294519976 * q^43 - 14206061378 * q^48 - 3573617796 * q^49 + 7713246552 * q^51 - 5597650396 * q^52 - 4602062760 * q^53 - 6178744976 * q^55 + 20017912662 * q^56 - 13775649944 * q^61 + 239765256 * q^62 - 3560815378 * q^64 - 7598401512 * q^65 + 37979507040 * q^66 + 40844682210 * q^68 - 25419983328 * q^69 + 19351803414 * q^74 + 68016370832 * q^75 - 80478036048 * q^77 + 89375282178 * q^78 + 18046097296 * q^79 - 132677486692 * q^81 - 255687836096 * q^82 + 94507900752 * q^87 + 239343029120 * q^88 - 190413561204 * q^90 + 104793638664 * q^91 - 135236877012 * q^92 - 78363161402 * q^94 + 145093149648 * q^95

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