LMFDB - Embedded newform 13.12.b.a.12.5

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			12.5
Root			$-25.1535i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.12.b.a.12.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.1535i q^{2} -638.702 q^{3} +1415.30 q^{4} +9287.93i q^{5} +16065.6i q^{6} -25239.3i q^{7} -87114.1i q^{8} +230793. q^{9} +O(q^{10})$	\(q-25.1535i q^{2} -638.702 q^{3} +1415.30 q^{4} +9287.93i q^{5} +16065.6i q^{6} -25239.3i q^{7} -87114.1i q^{8} +230793. q^{9} +233624. q^{10} -809580. i q^{11} -903957. q^{12} +(738773. + 1.11641e6i) q^{13} -634856. q^{14} -5.93222e6i q^{15} +707321. q^{16} +1.09516e7 q^{17} -5.80525e6i q^{18} -1.08406e7i q^{19} +1.31452e7i q^{20} +1.61204e7i q^{21} -2.03637e7 q^{22} -5.65347e6 q^{23} +5.56399e7i q^{24} -3.74375e7 q^{25} +(2.80816e7 - 1.85827e7i) q^{26} -3.42639e7 q^{27} -3.57213e7i q^{28} +7.64007e7 q^{29} -1.49216e8 q^{30} +1.22306e8i q^{31} -1.96201e8i q^{32} +5.17080e8i q^{33} -2.75471e8i q^{34} +2.34421e8 q^{35} +3.26642e8 q^{36} -5.45819e7i q^{37} -2.72678e8 q^{38} +(-4.71856e8 - 7.13054e8i) q^{39} +8.09109e8 q^{40} -9.95844e8i q^{41} +4.05484e8 q^{42} -3.96262e8 q^{43} -1.14580e9i q^{44} +2.14359e9i q^{45} +1.42204e8i q^{46} +7.01264e8i q^{47} -4.51767e8 q^{48} +1.34030e9 q^{49} +9.41682e8i q^{50} -6.99482e9 q^{51} +(1.04559e9 + 1.58006e9i) q^{52} +6.66216e8 q^{53} +8.61856e8i q^{54} +7.51932e9 q^{55} -2.19870e9 q^{56} +6.92389e9i q^{57} -1.92174e9i q^{58} -2.42983e9i q^{59} -8.39589e9i q^{60} -8.91257e9 q^{61} +3.07641e9 q^{62} -5.82506e9i q^{63} -3.48655e9 q^{64} +(-1.03692e10 + 6.86167e9i) q^{65} +1.30064e10 q^{66} +6.44060e9i q^{67} +1.54999e10 q^{68} +3.61088e9 q^{69} -5.89650e9i q^{70} -1.38964e10i q^{71} -2.01053e10i q^{72} +1.48127e10i q^{73} -1.37292e9 q^{74} +2.39114e10 q^{75} -1.53427e10i q^{76} -2.04332e10 q^{77} +(-1.79358e10 + 1.18688e10i) q^{78} +2.96608e10 q^{79} +6.56954e9i q^{80} -1.89999e10 q^{81} -2.50489e10 q^{82} +1.25290e10i q^{83} +2.28153e10i q^{84} +1.01718e11i q^{85} +9.96736e9i q^{86} -4.87973e10 q^{87} -7.05258e10 q^{88} -9.03892e10i q^{89} +5.39187e10 q^{90} +(2.81775e10 - 1.86461e10i) q^{91} -8.00138e9 q^{92} -7.81170e10i q^{93} +1.76392e10 q^{94} +1.00686e11 q^{95} +1.25314e11i q^{96} +5.24967e10i q^{97} -3.37133e10i q^{98} -1.86845e11i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	25.1535i		−	0.555818i	−0.960607	−	0.277909i	$-0.910359\pi$
$2$		−	25.1535i		−	0.555818i	0.960607	−	0.277909i	$-0.0896414\pi$
$3$	−638.702			−1.51751			−0.758755	−	0.651377i	$-0.774192\pi$
$3$	−638.702			−1.51751			−0.758755	+	0.651377i	$0.774192\pi$
$4$	1415.30			0.691066
$5$			9287.93i			1.32918i	0.747208	+	0.664590i	$0.231394\pi$
$5$			9287.93i			1.32918i	−0.747208	+	0.664590i	$0.768606\pi$
$6$			16065.6i			0.843459i
$7$		−	25239.3i		−	0.567595i	−0.958884	−	0.283797i	$-0.908406\pi$
$7$		−	25239.3i		−	0.567595i	0.958884	−	0.283797i	$-0.0915943\pi$
$8$		−	87114.1i		−	0.939925i
$9$	230793.			1.30283
$10$	233624.			0.738782
$11$		−	809580.i		−	1.51565i	−0.652456	−	0.757827i	$-0.726261\pi$
$11$		−	809580.i		−	1.51565i	0.652456	−	0.757827i	$-0.273739\pi$
$12$	−903957.			−1.04870
$13$	738773.	+	1.11641e6i	0.551852	+	0.833942i
$13$	738773.	+	1.11641e6i	0.551852	+	0.833942i
$14$	−634856.			−0.315479
$15$		−	5.93222e6i		−	2.01704i
$16$	707321.			0.168638
$17$	1.09516e7			1.87072			0.935361	−	0.353693i	$-0.115074\pi$
$17$	1.09516e7			1.87072			0.935361	+	0.353693i	$0.115074\pi$
$18$		−	5.80525e6i		−	0.724139i
$19$		−	1.08406e7i		−	1.00440i	−0.864751	−	0.502201i	$-0.832524\pi$
$19$		−	1.08406e7i		−	1.00440i	0.864751	−	0.502201i	$-0.167476\pi$
$20$			1.31452e7i			0.918551i
$21$			1.61204e7i			0.861330i
$22$	−2.03637e7			−0.842428
$23$	−5.65347e6			−0.183152			−0.0915761	−	0.995798i	$-0.529190\pi$
$23$	−5.65347e6			−0.183152			−0.0915761	+	0.995798i	$0.529190\pi$
$24$			5.56399e7i			1.42635i
$25$	−3.74375e7			−0.766719
$26$	2.80816e7	−	1.85827e7i	0.463520	−	0.306729i
$27$	−3.42639e7			−0.459554
$28$		−	3.57213e7i		−	0.392245i
$29$	7.64007e7			0.691685			0.345842	−	0.938293i	$-0.387593\pi$
$29$	7.64007e7			0.691685			0.345842	+	0.938293i	$0.387593\pi$
$30$	−1.49216e8			−1.12111
$31$			1.22306e8i			0.767287i	0.923481	+	0.383643i	$0.125331\pi$
$31$			1.22306e8i			0.767287i	−0.923481	+	0.383643i	$0.874669\pi$
$32$		−	1.96201e8i		−	1.03366i
$33$			5.17080e8i			2.30002i
$34$		−	2.75471e8i		−	1.03978i
$35$	2.34421e8			0.754435
$36$	3.26642e8			0.900344
$37$		−	5.45819e7i		−	0.129401i	−0.997905	−	0.0647007i	$-0.979391\pi$
$37$		−	5.45819e7i		−	0.129401i	0.997905	−	0.0647007i	$-0.0206093\pi$
$38$	−2.72678e8			−0.558265
$39$	−4.71856e8	−	7.13054e8i	−0.837440	−	1.26551i
$40$	8.09109e8			1.24933
$41$		−	9.95844e8i		−	1.34239i	−0.741279	−	0.671197i	$-0.765780\pi$
$41$		−	9.95844e8i		−	1.34239i	0.741279	−	0.671197i	$-0.234220\pi$
$42$	4.05484e8			0.478743
$43$	−3.96262e8			−0.411060			−0.205530	−	0.978651i	$-0.565892\pi$
$43$	−3.96262e8			−0.411060			−0.205530	+	0.978651i	$0.565892\pi$
$44$		−	1.14580e9i		−	1.04742i
$45$			2.14359e9i			1.73170i
$46$			1.42204e8i			0.101799i
$47$			7.01264e8i			0.446008i	0.974817	+	0.223004i	$0.0715864\pi$
$47$			7.01264e8i			0.446008i	−0.974817	+	0.223004i	$0.928414\pi$
$48$	−4.51767e8			−0.255910
$49$	1.34030e9			0.677836
$50$			9.41682e8i			0.426157i
$51$	−6.99482e9			−2.83884
$52$	1.04559e9	+	1.58006e9i	0.381366	+	0.576309i
$53$	6.66216e8			0.218825			0.109413	−	0.993996i	$-0.465103\pi$
$53$	6.66216e8			0.218825			0.109413	+	0.993996i	$0.465103\pi$
$54$			8.61856e8i			0.255428i
$55$	7.51932e9			2.01458
$56$	−2.19870e9			−0.533497
$57$			6.92389e9i			1.52419i
$58$		−	1.92174e9i		−	0.384451i
$59$		−	2.42983e9i		−	0.442476i	−0.975220	−	0.221238i	$-0.928990\pi$
$59$		−	2.42983e9i		−	0.442476i	0.975220	−	0.221238i	$-0.0710098\pi$
$60$		−	8.39589e9i		−	1.39391i
$61$	−8.91257e9			−1.35110			−0.675552	−	0.737312i	$-0.736095\pi$
$61$	−8.91257e9			−1.35110			−0.675552	+	0.737312i	$0.736095\pi$
$62$	3.07641e9			0.426472
$63$		−	5.82506e9i		−	0.739482i
$64$	−3.48655e9			−0.405887
$65$	−1.03692e10	+	6.86167e9i	−1.10846	+	0.733510i
$66$	1.30064e10			1.27839
$67$			6.44060e9i			0.582794i	0.956602	+	0.291397i	$0.0941200\pi$
$67$			6.44060e9i			0.582794i	−0.956602	+	0.291397i	$0.905880\pi$
$68$	1.54999e10			1.29279
$69$	3.61088e9			0.277935
$70$		−	5.89650e9i		−	0.419329i
$71$		−	1.38964e10i		−	0.914073i	−0.889448	−	0.457037i	$-0.848911\pi$
$71$		−	1.38964e10i		−	0.914073i	0.889448	−	0.457037i	$-0.151089\pi$
$72$		−	2.01053e10i		−	1.22457i
$73$			1.48127e10i			0.836296i	0.908379	+	0.418148i	$0.137321\pi$
$73$			1.48127e10i			0.836296i	−0.908379	+	0.418148i	$0.862679\pi$
$74$	−1.37292e9			−0.0719236
$75$	2.39114e10			1.16350
$76$		−	1.53427e10i		−	0.694108i
$77$	−2.04332e10			−0.860277
$78$	−1.79358e10	+	1.18688e10i	−0.703396	+	0.465465i
$79$	2.96608e10			1.08451			0.542256	−	0.840213i	$-0.317570\pi$
$79$	2.96608e10			1.08451			0.542256	+	0.840213i	$0.317570\pi$
$80$			6.56954e9i			0.224151i
$81$	−1.89999e10			−0.605457
$82$	−2.50489e10			−0.746127
$83$			1.25290e10i			0.349129i	0.984646	+	0.174564i	$0.0558517\pi$
$83$			1.25290e10i			0.349129i	−0.984646	+	0.174564i	$0.944148\pi$
$84$			2.28153e10i			0.595236i
$85$			1.01718e11i			2.48653i
$86$			9.96736e9i			0.228475i
$87$	−4.87973e10			−1.04964
$88$	−7.05258e10			−1.42460
$89$		−	9.03892e10i		−	1.71582i	−0.513800	−	0.857910i	$-0.671763\pi$
$89$		−	9.03892e10i		−	1.71582i	0.513800	−	0.857910i	$-0.328237\pi$
$90$	5.39187e10			0.962511
$91$	2.81775e10	−	1.86461e10i	0.473341	−	0.313228i
$92$	−8.00138e9			−0.126570
$93$		−	7.81170e10i		−	1.16436i
$94$	1.76392e10			0.247900
$95$	1.00686e11			1.33503
$96$			1.25314e11i			1.56859i
$97$			5.24967e10i			0.620708i	0.950621	+	0.310354i	$0.100448\pi$
$97$			5.24967e10i			0.620708i	−0.950621	+	0.310354i	$0.899552\pi$
$98$		−	3.37133e10i		−	0.376754i
$99$		−	1.86845e11i		−	1.97464i
$100$	−5.29854e10			−0.529854
$101$	6.66125e10			0.630650			0.315325	−	0.948984i	$-0.397887\pi$
$101$	6.66125e10			0.630650			0.315325	+	0.948984i	$0.397887\pi$
$102$			1.75944e11i			1.57788i
$103$	9.97201e10			0.847574			0.423787	−	0.905762i	$-0.360700\pi$
$103$	9.97201e10			0.847574			0.423787	+	0.905762i	$0.360700\pi$
$104$	9.72552e10	−	6.43575e10i	0.783843	−	0.518699i
$105$	−1.49725e11			−1.14486
$106$		−	1.67576e10i		−	0.121627i
$107$	−2.29206e11			−1.57985			−0.789923	−	0.613207i	$-0.789879\pi$
$107$	−2.29206e11			−1.57985			−0.789923	+	0.613207i	$0.789879\pi$
$108$	−4.84938e10			−0.317582
$109$		−	1.26351e11i		−	0.786559i	−0.919419	−	0.393279i	$-0.871340\pi$
$109$		−	1.26351e11i		−	0.786559i	0.919419	−	0.393279i	$-0.128660\pi$
$110$		−	1.89137e11i		−	1.11974i
$111$			3.48615e10i			0.196368i
$112$		−	1.78523e10i		−	0.0957183i
$113$	2.83959e11			1.44985			0.724927	−	0.688825i	$-0.241873\pi$
$113$	2.83959e11			1.44985			0.724927	+	0.688825i	$0.241873\pi$
$114$	1.74160e11			0.847172
$115$		−	5.25090e10i		−	0.243442i
$116$	1.08130e11			0.478000
$117$	1.70504e11	+	2.57660e11i	0.718971	+	1.08649i
$118$	−6.11187e10			−0.245936
$119$		−	2.76411e11i		−	1.06181i
$120$	−5.16780e11			−1.89587
$121$	−3.70108e11			−1.29720
$122$			2.24182e11i			0.750969i
$123$			6.36048e11i			2.03710i
$124$			1.73100e11i			0.530246i
$125$			1.05796e11i			0.310072i
$126$	−1.46520e11			−0.411017
$127$	−1.13296e11			−0.304294			−0.152147	−	0.988358i	$-0.548619\pi$
$127$	−1.13296e11			−0.304294			−0.152147	+	0.988358i	$0.548619\pi$
$128$		−	3.14121e11i		−	0.808058i
$129$	2.53093e11			0.623788
$130$	1.72595e11	+	2.60820e11i	0.407698	+	0.616102i
$131$	2.66543e11			0.603636			0.301818	−	0.953366i	$-0.402406\pi$
$131$	2.66543e11			0.603636			0.301818	+	0.953366i	$0.402406\pi$
$132$			7.31825e11i			1.58946i
$133$	−2.73609e11			−0.570093
$134$	1.62003e11			0.323927
$135$		−	3.18241e11i		−	0.610829i
$136$		−	9.54041e11i		−	1.75834i
$137$		−	5.54368e11i		−	0.981375i	−0.871336	−	0.490688i	$-0.836746\pi$
$137$		−	5.54368e11i		−	0.981375i	0.871336	−	0.490688i	$-0.163254\pi$
$138$		−	9.08262e10i		−	0.154481i
$139$	−4.49505e11			−0.734772			−0.367386	−	0.930069i	$-0.619747\pi$
$139$	−4.49505e11			−0.734772			−0.367386	+	0.930069i	$0.619747\pi$
$140$	3.31777e11			0.521365
$141$		−	4.47898e11i		−	0.676822i
$142$	−3.49542e11			−0.508059
$143$	9.03824e11	−	5.98095e11i	1.26397	−	0.836416i
$144$	1.63245e11			0.219708
$145$			7.09604e11i			0.919373i
$146$	3.72592e11			0.464828
$147$	−8.56055e11			−1.02862
$148$		−	7.72499e10i		−	0.0894249i
$149$		−	1.82945e11i		−	0.204078i	−0.994780	−	0.102039i	$-0.967463\pi$
$149$		−	1.82945e11i		−	0.204078i	0.994780	−	0.102039i	$-0.0325367\pi$
$150$		−	6.01454e11i		−	0.646697i
$151$			1.63643e12i			1.69639i	0.529687	+	0.848193i	$0.322309\pi$
$151$			1.63643e12i			1.69639i	−0.529687	+	0.848193i	$0.677691\pi$
$152$	−9.44366e11			−0.944062
$153$	2.52756e12			2.43724
$154$			5.13967e11i			0.478157i
$155$	−1.13597e12			−1.01986
$156$	−6.67819e11	−	1.00919e12i	−0.578726	−	0.874554i
$157$	1.12984e12			0.945300			0.472650	−	0.881250i	$-0.343297\pi$
$157$	1.12984e12			0.945300			0.472650	+	0.881250i	$0.343297\pi$
$158$		−	7.46073e11i		−	0.602792i
$159$	−4.25513e11			−0.332069
$160$	1.82230e12			1.37392
$161$			1.42690e11i			0.103956i
$162$			4.77913e11i			0.336524i
$163$			1.19794e12i			0.815461i	0.913102	+	0.407730i	$0.133680\pi$
$163$			1.19794e12i			0.815461i	−0.913102	+	0.407730i	$0.866320\pi$
$164$		−	1.40942e12i		−	0.927683i
$165$	−4.80260e12			−3.05714
$166$	3.15147e11			0.194052
$167$			1.60628e12i			0.956932i	0.878106	+	0.478466i	$0.158807\pi$
$167$			1.60628e12i			0.956932i	−0.878106	+	0.478466i	$0.841193\pi$
$168$	1.40431e12			0.809586
$169$	−7.00590e11	+	1.64955e12i	−0.390919	+	0.920425i
$170$	2.55856e12			1.38206
$171$		−	2.50193e12i		−	1.30857i
$172$	−5.60831e11			−0.284070
$173$	−3.48270e12			−1.70869			−0.854344	−	0.519707i	$-0.826041\pi$
$173$	−3.48270e12			−1.70869			−0.854344	+	0.519707i	$0.826041\pi$
$174$			1.22742e12i			0.583408i
$175$			9.44896e11i			0.435186i
$176$		−	5.72632e11i		−	0.255597i
$177$			1.55194e12i			0.671462i
$178$	−2.27360e12			−0.953684
$179$	1.99582e12			0.811762			0.405881	−	0.913926i	$-0.366965\pi$
$179$	1.99582e12			0.811762			0.405881	+	0.913926i	$0.366965\pi$
$180$			3.03383e12i			1.19672i
$181$	−2.10025e12			−0.803598			−0.401799	−	0.915728i	$-0.631615\pi$
$181$	−2.10025e12			−0.803598			−0.401799	+	0.915728i	$0.631615\pi$
$182$	−4.69014e11	−	7.08761e11i	−0.174098	−	0.263092i
$183$	5.69248e12			2.05031
$184$			4.92497e11i			0.172149i
$185$	5.06952e11			0.171998
$186$	−1.96491e12			−0.647175
$187$		−	8.86621e12i		−	2.83537i
$188$			9.92501e11i			0.308221i
$189$			8.64797e11i			0.260840i
$190$		−	2.53261e12i		−	0.742034i
$191$	1.56924e11			0.0446689			0.0223344	−	0.999751i	$-0.492890\pi$
$191$	1.56924e11			0.0446689			0.0223344	+	0.999751i	$0.492890\pi$
$192$	2.22686e12			0.615938
$193$			8.94216e11i			0.240368i	0.992752	+	0.120184i	$0.0383485\pi$
$193$			8.94216e11i			0.240368i	−0.992752	+	0.120184i	$0.961651\pi$
$194$	1.32047e12			0.345001
$195$	6.62280e12	−	4.38256e12i	1.68210	−	1.11311i
$196$	1.89694e12			0.468430
$197$			3.78108e12i			0.907928i	0.891020	+	0.453964i	$0.149991\pi$
$197$			3.78108e12i			0.907928i	−0.891020	+	0.453964i	$0.850009\pi$
$198$	−4.69981e12			−1.09754
$199$	2.20196e12			0.500170			0.250085	−	0.968224i	$-0.419541\pi$
$199$	2.20196e12			0.500170			0.250085	+	0.968224i	$0.419541\pi$
$200$			3.26133e12i			0.720659i
$201$		−	4.11362e12i		−	0.884395i
$202$		−	1.67554e12i		−	0.350527i
$203$		−	1.92830e12i		−	0.392597i
$204$	−9.89980e12			−1.96183
$205$	9.24933e12			1.78428
$206$		−	2.50831e12i		−	0.471097i
$207$	−1.30478e12			−0.238617
$208$	5.22549e11	+	7.89661e11i	0.0930634	+	0.140635i
$209$	−8.77631e12			−1.52232
$210$			3.76610e12i			0.636336i
$211$	−9.45222e12			−1.55590			−0.777948	−	0.628329i	$-0.783739\pi$
$211$	−9.45222e12			−1.55590			−0.777948	+	0.628329i	$0.783739\pi$
$212$	9.42897e11			0.151223
$213$			8.87565e12i			1.38711i
$214$			5.76532e12i			0.878107i
$215$		−	3.68045e12i		−	0.546373i
$216$			2.98487e12i			0.431946i
$217$	3.08691e12			0.435508
$218$	−3.17815e12			−0.437184
$219$		−	9.46093e12i		−	1.26909i
$220$	1.06421e13			1.39221
$221$	8.09076e12	+	1.22265e13i	1.03236	+	1.56007i
$222$	8.76889e11			0.109145
$223$		−	2.58107e12i		−	0.313417i	−0.987645	−	0.156709i	$-0.949912\pi$
$223$		−	2.58107e12i		−	0.313417i	0.987645	−	0.156709i	$-0.0500883\pi$
$224$	−4.95198e12			−0.586699
$225$	−8.64031e12			−0.998908
$226$		−	7.14256e12i		−	0.805856i
$227$		−	7.72832e12i		−	0.851027i	−0.904952	−	0.425513i	$-0.860094\pi$
$227$		−	7.72832e12i		−	0.851027i	0.904952	−	0.425513i	$-0.139906\pi$
$228$			9.79941e12i			1.05331i
$229$			1.52739e13i			1.60271i	0.598192	+	0.801353i	$0.295886\pi$
$229$			1.52739e13i			1.60271i	−0.598192	+	0.801353i	$0.704114\pi$
$230$	−1.32078e12			−0.135310
$231$	1.30507e13			1.30548
$232$		−	6.65557e12i		−	0.650132i
$233$	−3.77378e12			−0.360014			−0.180007	−	0.983665i	$-0.557612\pi$
$233$	−3.77378e12			−0.360014			−0.180007	+	0.983665i	$0.557612\pi$
$234$	6.48105e12	−	4.28876e12i	0.603890	−	0.399617i
$235$	−6.51329e12			−0.592826
$236$		−	3.43895e12i		−	0.305780i
$237$	−1.89444e13			−1.64576
$238$	−6.95270e12			−0.590175
$239$			1.60079e13i			1.32784i	0.747805	+	0.663919i	$0.231108\pi$
$239$			1.60079e13i			1.32784i	−0.747805	+	0.663919i	$0.768892\pi$
$240$		−	4.19598e12i		−	0.340151i
$241$		−	1.89827e13i		−	1.50406i	−0.659132	−	0.752028i	$-0.729076\pi$
$241$		−	1.89827e13i		−	1.50406i	0.659132	−	0.752028i	$-0.270924\pi$
$242$			9.30949e12i			0.721010i
$243$	1.82050e13			1.37834
$244$	−1.26140e13			−0.933703
$245$			1.24486e13i			0.900966i
$246$	1.59988e13			1.13226
$247$	1.21025e13	−	8.00872e12i	0.837613	−	0.554281i
$248$	1.06546e13			0.721192
$249$		−	8.00227e12i		−	0.529806i
$250$	2.66112e12			0.172344
$251$	−1.01879e13			−0.645473			−0.322736	−	0.946489i	$-0.604603\pi$
$251$	−1.01879e13			−0.645473			−0.322736	+	0.946489i	$0.604603\pi$
$252$		−	8.24423e12i		−	0.511031i
$253$			4.57694e12i			0.277595i
$254$			2.84979e12i			0.169132i
$255$		−	6.49674e13i		−	3.77333i
$256$	−1.50417e13			−0.855021
$257$	4.75191e12			0.264384			0.132192	−	0.991224i	$-0.457798\pi$
$257$	4.75191e12			0.264384			0.132192	+	0.991224i	$0.457798\pi$
$258$		−	6.36617e12i		−	0.346713i
$259$	−1.37761e12			−0.0734475
$260$	−1.46755e13	+	9.71134e12i	−0.766019	+	0.506904i
$261$	1.76328e13			0.901150
$262$		−	6.70448e12i		−	0.335512i
$263$	−3.41999e12			−0.167598			−0.0837989	−	0.996483i	$-0.526705\pi$
$263$	−3.41999e12			−0.167598			−0.0837989	+	0.996483i	$0.526705\pi$
$264$	4.50450e13			2.16184
$265$			6.18776e12i			0.290858i
$266$			6.88220e12i			0.316868i
$267$			5.77318e13i			2.60377i
$268$			9.11540e12i			0.402749i
$269$	−2.29606e13			−0.993905			−0.496953	−	0.867778i	$-0.665548\pi$
$269$	−2.29606e13			−0.993905			−0.496953	+	0.867778i	$0.665548\pi$
$270$	−8.00485e12			−0.339510
$271$			5.51961e12i			0.229392i	0.993401	+	0.114696i	$0.0365893\pi$
$271$			5.51961e12i			0.229392i	−0.993401	+	0.114696i	$0.963411\pi$
$272$	7.74631e12			0.315476
$273$	−1.79970e13	+	1.19093e13i	−0.718300	+	0.475327i
$274$	−1.39443e13			−0.545466
$275$			3.03086e13i			1.16208i
$276$	5.11049e12			0.192072
$277$	−1.44914e13			−0.533915			−0.266957	−	0.963708i	$-0.586018\pi$
$277$	−1.44914e13			−0.533915			−0.266957	+	0.963708i	$0.586018\pi$
$278$			1.13066e13i			0.408400i
$279$			2.82273e13i			0.999648i
$280$		−	2.04214e13i		−	0.709113i
$281$		−	1.56226e13i		−	0.531948i	−0.963980	−	0.265974i	$-0.914306\pi$
$281$		−	1.56226e13i		−	0.531948i	0.963980	−	0.265974i	$-0.0856936\pi$
$282$	−1.12662e13			−0.376190
$283$	1.17544e13			0.384924			0.192462	−	0.981304i	$-0.438353\pi$
$283$	1.17544e13			0.384924			0.192462	+	0.981304i	$0.438353\pi$
$284$		−	1.96676e13i		−	0.631685i
$285$	−6.43086e13			−2.02592
$286$	−1.50442e13	−	2.27343e13i	−0.464895	−	0.702536i
$287$	−2.51344e13			−0.761936
$288$		−	4.52819e13i		−	1.34668i
$289$	8.56662e13			2.49960
$290$	1.78490e13			0.511005
$291$		−	3.35297e13i		−	0.941930i
$292$			2.09645e13i			0.577936i
$293$			2.44993e13i			0.662799i	0.943491	+	0.331399i	$0.107521\pi$
$293$			2.44993e13i			0.662799i	−0.943491	+	0.331399i	$0.892479\pi$
$294$			2.15327e13i			0.571727i
$295$	2.25681e13			0.588131
$296$	−4.75485e12			−0.121628
$297$			2.77394e13i			0.696524i
$298$	−4.60171e12			−0.113431
$299$	−4.17663e12	−	6.31160e12i	−0.101073	−	0.152738i
$300$	3.38419e13			0.804058
$301$			1.00014e13i			0.233316i
$302$	4.11619e13			0.942883
$303$	−4.25455e13			−0.957017
$304$		−	7.66776e12i		−	0.169381i
$305$		−	8.27793e13i		−	1.79586i
$306$		−	6.35769e13i		−	1.35466i
$307$			8.49589e13i			1.77807i	0.457843	+	0.889033i	$0.348622\pi$
$307$			8.49589e13i			1.77807i	−0.457843	+	0.889033i	$0.651378\pi$
$308$	−2.89192e13			−0.594508
$309$	−6.36914e13			−1.28620
$310$			2.85735e13i			0.566858i
$311$	2.90036e13			0.565289			0.282644	−	0.959225i	$-0.408788\pi$
$311$	2.90036e13			0.565289			0.282644	+	0.959225i	$0.408788\pi$
$312$	−6.21171e13	+	4.11053e13i	−1.18949	+	0.787131i
$313$	4.29452e13			0.808017			0.404009	−	0.914755i	$-0.367617\pi$
$313$	4.29452e13			0.808017			0.404009	+	0.914755i	$0.367617\pi$
$314$		−	2.84195e13i		−	0.525415i
$315$	5.41027e13			0.982904
$316$	4.19791e13			0.749470
$317$			1.21340e13i			0.212901i	0.994318	+	0.106450i	$0.0339485\pi$
$317$			1.21340e13i			0.212901i	−0.994318	+	0.106450i	$0.966052\pi$
$318$			1.07031e13i			0.184570i
$319$		−	6.18524e13i		−	1.04835i
$320$		−	3.23828e13i		−	0.539497i
$321$	1.46394e14			2.39743
$322$	3.58914e12			0.0577807
$323$		−	1.18722e14i		−	1.87896i
$324$	−2.68906e13			−0.418411
$325$	−2.76578e13	−	4.17956e13i	−0.423115	−	0.639400i
$326$	3.01323e13			0.453248
$327$			8.07003e13i			1.19361i
$328$	−8.67520e13			−1.26175
$329$	1.76994e13			0.253152
$330$			1.20802e14i			1.69921i
$331$		−	9.04525e13i		−	1.25131i	−0.780098	−	0.625657i	$-0.784831\pi$
$331$		−	9.04525e13i		−	1.25131i	0.780098	−	0.625657i	$-0.215169\pi$
$332$			1.77323e13i			0.241271i
$333$		−	1.25971e13i		−	0.168589i
$334$	4.04035e13			0.531880
$335$	−5.98198e13			−0.774638
$336$			1.14023e13i			0.145253i
$337$	−1.34962e14			−1.69140			−0.845701	−	0.533657i	$-0.820818\pi$
$337$	−1.34962e14			−1.69140			−0.845701	+	0.533657i	$0.820818\pi$
$338$	4.14919e13	+	1.76223e13i	0.511589	+	0.217280i
$339$	−1.81365e14			−2.20017
$340$			1.43962e14i			1.71835i
$341$	9.90163e13			1.16294
$342$	−6.29322e13			−0.727326
$343$		−	8.37347e13i		−	0.952331i
$344$			3.45200e13i			0.386366i
$345$			3.35376e13i			0.369426i
$346$			8.76021e13i			0.949720i
$347$	9.61562e13			1.02604			0.513021	−	0.858376i	$-0.328526\pi$
$347$	9.61562e13			1.02604			0.513021	+	0.858376i	$0.328526\pi$
$348$	−6.90629e13			−0.725369
$349$			1.77772e14i			1.83791i	0.394361	+	0.918956i	$0.370966\pi$
$349$			1.77772e14i			1.83791i	−0.394361	+	0.918956i	$0.629034\pi$
$350$	2.37674e13			0.241884
$351$	−2.53132e13	−	3.82526e13i	−0.253605	−	0.383241i
$352$	−1.58840e14			−1.56667
$353$		−	9.37066e13i		−	0.909933i	−0.890509	−	0.454966i	$-0.849651\pi$
$353$		−	9.37066e13i		−	0.909933i	0.890509	−	0.454966i	$-0.150349\pi$
$354$	3.90366e13			0.373211
$355$	1.29069e14			1.21497
$356$		−	1.27928e14i		−	1.18574i
$357$			1.76545e14i			1.61131i
$358$		−	5.02017e13i		−	0.451192i
$359$			3.23925e13i			0.286698i	0.989672	+	0.143349i	$0.0457872\pi$
$359$			3.23925e13i			0.286698i	−0.989672	+	0.143349i	$0.954213\pi$
$360$	1.86737e14			1.62767
$361$	−1.02770e12			−0.00882218
$362$			5.28286e13i			0.446654i
$363$	2.36388e14			1.96852
$364$	3.98797e13	−	2.63899e13i	0.327110	−	0.216461i
$365$	−1.37580e14			−1.11159
$366$		−	1.43186e14i		−	1.13960i
$367$	−1.22124e14			−0.957499			−0.478749	−	0.877952i	$-0.658910\pi$
$367$	−1.22124e14			−0.957499			−0.478749	+	0.877952i	$0.658910\pi$
$368$	−3.99882e12			−0.0308865
$369$		−	2.29834e14i		−	1.74892i
$370$		−	1.27516e13i		−	0.0955995i
$371$		−	1.68148e13i		−	0.124204i
$372$		−	1.10559e14i		−	0.804653i
$373$	−3.80298e13			−0.272726			−0.136363	−	0.990659i	$-0.543541\pi$
$373$	−3.80298e13			−0.272726			−0.136363	+	0.990659i	$0.543541\pi$
$374$	−2.23016e14			−1.57595
$375$		−	6.75718e13i		−	0.470537i
$376$	6.10899e13			0.419215
$377$	5.64427e13	+	8.52946e13i	0.381707	+	0.576825i
$378$	2.17526e13			0.144980
$379$			8.21824e13i			0.539838i	0.962883	+	0.269919i	$0.0869969\pi$
$379$			8.21824e13i			0.539838i	−0.962883	+	0.269919i	$0.913003\pi$
$380$	1.42502e14			0.922594
$381$	7.23623e13			0.461770
$382$		−	3.94718e12i		−	0.0248278i
$383$			7.83489e13i			0.485780i	0.970054	+	0.242890i	$0.0780954\pi$
$383$			7.83489e13i			0.485780i	−0.970054	+	0.242890i	$0.921905\pi$
$384$			2.00630e14i			1.22624i
$385$		−	1.89782e14i		−	1.14346i
$386$	2.24926e13			0.133601
$387$	−9.14545e13			−0.535544
$388$			7.42987e13i			0.428950i
$389$	1.76093e14			1.00235			0.501176	−	0.865345i	$-0.332901\pi$
$389$	1.76093e14			1.00235			0.501176	+	0.865345i	$0.332901\pi$
$390$	−1.10237e14	−	1.66586e14i	−0.618686	−	0.934940i
$391$	−6.19147e13			−0.342627
$392$		−	1.16759e14i		−	0.637116i
$393$	−1.70242e14			−0.916024
$394$	9.51072e13			0.504643
$395$			2.75488e14i			1.44151i
$396$		−	2.64443e14i		−	1.36461i
$397$			5.25583e13i			0.267481i	0.991016	+	0.133741i	$0.0426989\pi$
$397$			5.25583e13i			0.267481i	−0.991016	+	0.133741i	$0.957301\pi$
$398$		−	5.53870e13i		−	0.278004i
$399$	1.74754e14			0.865121
$400$	−2.64803e13			−0.129298
$401$		−	6.98075e13i		−	0.336208i	−0.985769	−	0.168104i	$-0.946236\pi$
$401$		−	6.98075e13i		−	0.336208i	0.985769	−	0.168104i	$-0.0537644\pi$
$402$	−1.03472e14			−0.491563
$403$	−1.36544e14	+	9.03562e13i	−0.639873	+	0.423429i
$404$	9.42769e13			0.435821
$405$		−	1.76470e14i		−	0.804762i
$406$	−4.85034e13			−0.218212
$407$	−4.41884e13			−0.196128
$408$			6.09348e14i			2.66830i
$409$		−	2.03685e12i		−	0.00879996i	−0.999990	−	0.00439998i	$-0.998599\pi$
$409$		−	2.03685e12i		−	0.00879996i	0.999990	−	0.00439998i	$-0.00140056\pi$
$410$		−	2.32653e14i		−	0.991738i
$411$			3.54076e14i			1.48925i
$412$	1.41134e14			0.585730
$413$	−6.13273e13			−0.251147
$414$			3.28198e13i			0.132628i
$415$	−1.16368e14			−0.464055
$416$	2.19041e14	−	1.44948e14i	0.862011	−	0.570426i
$417$	2.87099e14			1.11502
$418$			2.20754e14i			0.846136i
$419$	2.11635e13			0.0800591			0.0400296	−	0.999198i	$-0.487255\pi$
$419$	2.11635e13			0.0800591			0.0400296	+	0.999198i	$0.487255\pi$
$420$	−2.11906e14			−0.791176
$421$		−	4.53231e14i		−	1.67020i	−0.550098	−	0.835100i	$-0.685410\pi$
$421$		−	4.53231e14i		−	1.67020i	0.550098	−	0.835100i	$-0.314590\pi$
$422$			2.37756e14i			0.864795i
$423$			1.61847e14i			0.581075i
$424$		−	5.80368e13i		−	0.205679i
$425$	−4.10001e14			−1.43432
$426$	2.23253e14			0.770984
$427$			2.24947e14i			0.766880i
$428$	−3.24395e14			−1.09178
$429$	−5.77274e14	+	3.82005e14i	−1.91808	+	1.26927i
$430$	−9.25761e13			−0.303684
$431$			2.32051e14i			0.751551i	0.926711	+	0.375775i	$0.122624\pi$
$431$			2.32051e14i			0.751551i	−0.926711	+	0.375775i	$0.877376\pi$
$432$	−2.42356e13			−0.0774984
$433$	3.22668e14			1.01876			0.509381	−	0.860541i	$-0.329874\pi$
$433$	3.22668e14			1.01876			0.509381	+	0.860541i	$0.329874\pi$
$434$		−	7.76466e13i		−	0.242063i
$435$		−	4.53225e14i		−	1.39516i
$436$		−	1.78824e14i		−	0.543564i
$437$			6.12868e13i			0.183958i
$438$	−2.37975e14			−0.705381
$439$	−1.43942e13			−0.0421341			−0.0210671	−	0.999778i	$-0.506706\pi$
$439$	−1.43942e13			−0.0421341			−0.0210671	+	0.999778i	$0.506706\pi$
$440$		−	6.55038e14i		−	1.89355i
$441$	3.09333e14			0.883108
$442$	3.07539e14	−	2.03511e14i	0.867118	−	0.573805i
$443$	−2.12647e13			−0.0592159			−0.0296079	−	0.999562i	$-0.509426\pi$
$443$	−2.12647e13			−0.0592159			−0.0296079	+	0.999562i	$0.509426\pi$
$444$			4.93397e13i			0.135703i
$445$	8.39529e14			2.28063
$446$	−6.49228e13			−0.174203
$447$			1.16848e14i			0.309691i
$448$			8.79980e13i			0.230380i
$449$			1.13580e14i			0.293729i	0.989157	+	0.146865i	$0.0469182\pi$
$449$			1.13580e14i			0.293729i	−0.989157	+	0.146865i	$0.953082\pi$
$450$			2.17334e14i			0.555211i
$451$	−8.06215e14			−2.03460
$452$	4.01888e14			1.00195
$453$		−	1.04519e15i		−	2.57428i
$454$	−1.94394e14			−0.473016
$455$	1.73184e14	+	2.61710e14i	0.416337	+	0.629156i
$456$	6.03169e14			1.43262
$457$		−	2.88690e14i		−	0.677473i	−0.940881	−	0.338737i	$-0.890000\pi$
$457$		−	2.88690e14i		−	0.677473i	0.940881	−	0.338737i	$-0.110000\pi$
$458$	3.84191e14			0.890813
$459$	−3.75245e14			−0.859697
$460$		−	7.43162e13i		−	0.168235i
$461$			3.07352e14i			0.687514i	0.939059	+	0.343757i	$0.111700\pi$
$461$			3.07352e14i			0.687514i	−0.939059	+	0.343757i	$0.888300\pi$
$462$		−	3.28271e14i		−	0.725608i
$463$		−	4.18580e14i		−	0.914288i	−0.889393	−	0.457144i	$-0.848872\pi$
$463$		−	4.18580e14i		−	0.914288i	0.889393	−	0.457144i	$-0.151128\pi$
$464$	5.40398e13			0.116645
$465$	7.25545e14			1.54765
$466$			9.49237e13i			0.200102i
$467$	−4.56891e14			−0.951853			−0.475926	−	0.879485i	$-0.657887\pi$
$467$	−4.56891e14			−0.951853			−0.475926	+	0.879485i	$0.657887\pi$
$468$	2.41314e14	+	3.64667e14i	0.496857	+	0.750835i
$469$	1.62556e14			0.330791
$470$			1.63832e14i			0.329503i
$471$	−7.21633e14			−1.43450
$472$	−2.11672e14			−0.415895
$473$			3.20806e14i			0.623025i
$474$			4.76518e14i			0.914742i
$475$			4.05844e14i			0.770094i
$476$		−	3.91206e14i		−	0.733782i
$477$	1.53758e14			0.285093
$478$	4.02653e14			0.738036
$479$			8.50737e14i			1.54152i	0.637124	+	0.770762i	$0.280124\pi$
$479$			8.50737e14i			1.54152i	−0.637124	+	0.770762i	$0.719876\pi$
$480$	−1.16391e15			−2.08493
$481$	6.09358e13	−	4.03236e13i	0.107913	−	0.0714104i
$482$	−4.77480e14			−0.835981
$483$		−	9.11362e13i		−	0.157754i
$484$	−5.23815e14			−0.896454
$485$	−4.87585e14			−0.825032
$486$		−	4.57919e14i		−	0.766107i
$487$		−	6.78950e14i		−	1.12313i	−0.827434	−	0.561563i	$-0.810200\pi$
$487$		−	6.78950e14i		−	1.12313i	0.827434	−	0.561563i	$-0.189800\pi$
$488$			7.76411e14i			1.26994i
$489$		−	7.65126e14i		−	1.23747i
$490$	3.13127e14			0.500774
$491$	1.96695e14			0.311060			0.155530	−	0.987831i	$-0.450291\pi$
$491$	1.96695e14			0.311060			0.155530	+	0.987831i	$0.450291\pi$
$492$			9.00200e14i			1.40777i
$493$	8.36711e14			1.29395
$494$	−2.01447e14	−	3.04421e14i	−0.308079	−	0.465560i
$495$	1.73541e15			2.62466
$496$			8.65094e13i			0.129394i
$497$	−3.50735e14			−0.518823
$498$	−2.01285e14			−0.294476
$499$			7.12395e14i			1.03078i	0.856954	+	0.515392i	$0.172354\pi$
$499$			7.12395e14i			1.03078i	−0.856954	+	0.515392i	$0.827646\pi$
$500$			1.49733e14i			0.214280i
$501$		−	1.02594e15i		−	1.45215i
$502$			2.56260e14i			0.358765i
$503$	−1.36394e15			−1.88874			−0.944368	−	0.328892i	$-0.893325\pi$
$503$	−1.36394e15			−1.88874			−0.944368	+	0.328892i	$0.893325\pi$
$504$	−5.07445e14			−0.695058
$505$			6.18692e14i			0.838247i
$506$	1.15126e14			0.154292
$507$	4.47468e14	−	1.05357e15i	0.593224	−	1.39675i
$508$	−1.60348e14			−0.210288
$509$			6.73249e12i			0.00873430i	0.999990	+	0.00436715i	$0.00139011\pi$
$509$			6.73249e12i			0.00873430i	−0.999990	+	0.00436715i	$0.998610\pi$
$510$	−1.63416e15			−2.09728
$511$	3.73864e14			0.474677
$512$		−	2.64970e14i		−	0.332822i
$513$			3.71440e14i			0.461576i
$514$		−	1.19527e14i		−	0.146950i
$515$			9.26193e14i			1.12658i
$516$	3.58204e14			0.431079
$517$	5.67729e14			0.675994
$518$			3.46516e13i			0.0408235i
$519$	2.22441e15			2.59295
$520$	5.97748e14	+	9.03299e14i	0.689445	+	1.04187i
$521$	−1.59359e15			−1.81873			−0.909364	−	0.416000i	$-0.863432\pi$
$521$	−1.59359e15			−1.81873			−0.909364	+	0.416000i	$0.863432\pi$
$522$		−	4.43525e14i		−	0.500876i
$523$	−2.42512e14			−0.271003			−0.135501	−	0.990777i	$-0.543265\pi$
$523$	−2.42512e14			−0.271003			−0.135501	+	0.990777i	$0.543265\pi$
$524$	3.77239e14			0.417153
$525$		−	6.03507e14i		−	0.660399i
$526$			8.60246e13i			0.0931539i
$527$			1.33945e15i			1.43538i
$528$			3.65741e14i			0.387871i
$529$	−9.20848e14			−0.966455
$530$	1.55644e14			0.161664
$531$		−	5.60788e14i		−	0.576473i
$532$	−3.87239e14			−0.393972
$533$	1.11177e15	−	7.35703e14i	1.11948	−	0.740803i
$534$	1.45215e15			1.44722
$535$		−	2.12885e15i		−	2.09990i
$536$	5.61067e14			0.547783
$537$	−1.27473e15			−1.23186
$538$			5.77538e14i			0.552431i
$539$		−	1.08508e15i		−	1.02736i
$540$		−	4.50407e14i		−	0.422124i
$541$			2.45829e13i			0.0228059i	0.999935	+	0.0114030i	$0.00362976\pi$
$541$			2.45829e13i			0.0228059i	−0.999935	+	0.0114030i	$0.996370\pi$
$542$	1.38837e14			0.127500
$543$	1.34143e15			1.21947
$544$		−	2.14872e15i		−	1.93369i
$545$	1.17353e15			1.04548
$546$	2.99560e14	+	4.52687e14i	0.264195	+	0.399244i
$547$	−1.94712e15			−1.70006			−0.850028	−	0.526737i	$-0.823415\pi$
$547$	−1.94712e15			−1.70006			−0.850028	+	0.526737i	$0.823415\pi$
$548$		−	7.84599e14i		−	0.678195i
$549$	−2.05696e15			−1.76027
$550$	7.62367e14			0.645906
$551$		−	8.28227e14i		−	0.694729i
$552$		−	3.14559e14i		−	0.261238i
$553$		−	7.48619e14i		−	0.615563i
$554$			3.64509e14i			0.296760i
$555$	−3.23791e14			−0.261008
$556$	−6.36185e14			−0.507776
$557$			1.17444e15i			0.928170i	0.885791	+	0.464085i	$0.153617\pi$
$557$			1.17444e15i			0.928170i	−0.885791	+	0.464085i	$0.846383\pi$
$558$	7.10015e14			0.555622
$559$	−2.92747e14	−	4.42391e14i	−0.226844	−	0.342801i
$560$	1.65811e14			0.127227
$561$			5.66287e15i			4.30270i
$562$	−3.92963e14			−0.295667
$563$	1.20559e15			0.898266			0.449133	−	0.893465i	$-0.351733\pi$
$563$	1.20559e15			0.898266			0.449133	+	0.893465i	$0.351733\pi$
$564$		−	6.33912e14i		−	0.467729i
$565$			2.63739e15i			1.92712i
$566$		−	2.95664e14i		−	0.213948i
$567$			4.79544e14i			0.343654i
$568$	−1.21057e15			−0.859161
$569$	−9.51790e14			−0.668996			−0.334498	−	0.942396i	$-0.608567\pi$
$569$	−9.51790e14			−0.668996			−0.334498	+	0.942396i	$0.608567\pi$
$570$			1.61758e15i			1.12604i
$571$	−1.03752e15			−0.715318			−0.357659	−	0.933852i	$-0.616425\pi$
$571$	−1.03752e15			−0.715318			−0.357659	+	0.933852i	$0.616425\pi$
$572$	1.27919e15	−	8.46486e14i	0.873485	−	0.578019i
$573$	−1.00227e14			−0.0677854
$574$			6.32218e14i			0.423498i
$575$	2.11652e14			0.140426
$576$	−8.04671e14			−0.528804
$577$		−	5.50976e14i		−	0.358645i	−0.983790	−	0.179323i	$-0.942609\pi$
$577$		−	5.50976e14i		−	0.358645i	0.983790	−	0.179323i	$-0.0573906\pi$
$578$		−	2.15480e15i		−	1.38933i
$579$		−	5.71137e14i		−	0.364761i
$580$			1.00430e15i			0.635348i
$581$	3.16222e14			0.198164
$582$	−8.43389e14			−0.523542
$583$		−	5.39355e14i		−	0.331663i
$584$	1.29040e15			0.786056
$585$	−2.39313e15	+	1.58363e15i	−1.44414	+	0.955642i
$586$	6.16242e14			0.368396
$587$			1.76515e15i			1.04537i	0.852525	+	0.522686i	$0.175070\pi$
$587$			1.76515e15i			1.04537i	−0.852525	+	0.522686i	$0.824930\pi$
$588$	−1.21158e15			−0.710846
$589$	1.32586e15			0.770664
$590$		−	5.67666e14i		−	0.326894i
$591$		−	2.41498e15i		−	1.37779i
$592$		−	3.86069e13i		−	0.0218220i
$593$		−	6.34735e14i		−	0.355461i	−0.984079	−	0.177730i	$-0.943125\pi$
$593$		−	6.34735e14i		−	0.355461i	0.984079	−	0.177730i	$-0.0568755\pi$
$594$	6.97741e14			0.387141
$595$	2.56729e15			1.41134
$596$		−	2.58923e14i		−	0.141032i
$597$	−1.40640e15			−0.759013
$598$	−1.58759e14	+	1.05057e14i	−0.0848947	+	0.0561781i
$599$	1.65843e15			0.878720			0.439360	−	0.898311i	$-0.355205\pi$
$599$	1.65843e15			0.878720			0.439360	+	0.898311i	$0.355205\pi$
$600$		−	2.08302e15i		−	1.09361i
$601$	8.05963e14			0.419282			0.209641	−	0.977778i	$-0.432771\pi$
$601$	8.05963e14			0.419282			0.209641	+	0.977778i	$0.432771\pi$
$602$	2.51569e14			0.129681
$603$			1.48645e15i			0.759283i
$604$			2.31605e15i			1.17232i
$605$		−	3.43753e15i		−	1.72422i
$606$			1.07017e15i			0.531927i
$607$	1.27550e15			0.628266			0.314133	−	0.949379i	$-0.398286\pi$
$607$	1.27550e15			0.628266			0.314133	+	0.949379i	$0.398286\pi$
$608$	−2.12693e15			−1.03821
$609$			1.23161e15i			0.595769i
$610$	−2.08219e15			−0.998173
$611$	−7.82899e14	+	5.18075e14i	−0.371945	+	0.246131i
$612$	3.57726e15			1.68429
$613$		−	2.84979e15i		−	1.32978i	−0.746941	−	0.664891i	$-0.768478\pi$
$613$		−	2.84979e15i		−	1.32978i	0.746941	−	0.664891i	$-0.231522\pi$
$614$	2.13701e15			0.988282
$615$	−5.90756e15			−2.70767
$616$			1.78002e15i			0.808596i
$617$			9.26977e14i			0.417350i	0.977985	+	0.208675i	$0.0669151\pi$
$617$			9.26977e14i			0.417350i	−0.977985	+	0.208675i	$0.933085\pi$
$618$			1.60206e15i			0.714895i
$619$		−	3.69032e15i		−	1.63217i	−0.577932	−	0.816085i	$-0.696140\pi$
$619$		−	3.69032e15i		−	1.63217i	0.577932	−	0.816085i	$-0.303860\pi$
$620$	−1.60774e15			−0.704792
$621$	1.93710e14			0.0841682
$622$		−	7.29542e14i		−	0.314198i
$623$	−2.28136e15			−0.973890
$624$	−3.33753e14	−	5.04358e14i	−0.141225	−	0.213414i
$625$	−2.81062e15			−1.17886
$626$		−	1.08022e15i		−	0.449111i
$627$	5.60544e15			2.31014
$628$	1.59907e15			0.653265
$629$		−	5.97760e14i		−	0.242074i
$630$		−	1.36087e15i		−	0.546316i
$631$			2.66062e15i			1.05882i	0.848367	+	0.529408i	$0.177586\pi$
$631$			2.66062e15i			1.05882i	−0.848367	+	0.529408i	$0.822414\pi$
$632$		−	2.58388e15i		−	1.01936i
$633$	6.03715e15			2.36109
$634$	3.05211e14			0.118334
$635$		−	1.05228e15i		−	0.404462i
$636$	−6.02230e14			−0.229482
$637$	9.90180e14	+	1.49633e15i	0.374065	+	0.565276i
$638$	−1.55580e15			−0.582694
$639$		−	3.20719e15i		−	1.19089i
$640$	2.91754e15			1.07405
$641$	1.06859e15			0.390023			0.195012	−	0.980801i	$-0.437526\pi$
$641$	1.06859e15			0.390023			0.195012	+	0.980801i	$0.437526\pi$
$642$		−	3.68232e15i		−	1.33253i
$643$			3.45048e15i			1.23799i	0.785393	+	0.618997i	$0.212461\pi$
$643$			3.45048e15i			1.23799i	−0.785393	+	0.618997i	$0.787539\pi$
$644$			2.01949e14i			0.0718406i
$645$			2.35071e15i			0.829127i
$646$	−2.98627e15			−1.04436
$647$	5.12730e15			1.77793			0.888965	−	0.457974i	$-0.151425\pi$
$647$	5.12730e15			1.77793			0.888965	+	0.457974i	$0.151425\pi$
$648$			1.65516e15i			0.569085i
$649$	−1.96714e15			−0.670640
$650$	−1.05130e15	+	6.95689e14i	−0.355390	+	0.235175i
$651$	−1.97162e15			−0.660887
$652$			1.69545e15i			0.563537i
$653$	2.17199e15			0.715874			0.357937	−	0.933746i	$-0.383480\pi$
$653$	2.17199e15			0.715874			0.357937	+	0.933746i	$0.383480\pi$
$654$	2.02989e15			0.663430
$655$			2.47563e15i			0.802341i
$656$		−	7.04381e14i		−	0.226379i
$657$			3.41868e15i			1.08955i
$658$		−	4.45202e14i		−	0.140707i
$659$	−3.20683e15			−1.00509			−0.502546	−	0.864551i	$-0.667603\pi$
$659$	−3.20683e15			−1.00509			−0.502546	+	0.864551i	$0.667603\pi$
$660$	−6.79714e15			−2.11268
$661$		−	8.32193e14i		−	0.256517i	−0.991741	−	0.128258i	$-0.959061\pi$
$661$		−	8.32193e14i		−	0.256517i	0.991741	−	0.128258i	$-0.0409387\pi$
$662$	−2.27519e15			−0.695503
$663$	−5.16758e15	−	7.80910e15i	−1.56662	−	2.36743i
$664$	1.09145e15			0.328155
$665$		−	2.54126e15i		−	0.757756i
$666$	−3.16861e14			−0.0937046
$667$	−4.31929e14			−0.126684
$668$			2.27338e15i			0.661303i
$669$			1.64853e15i			0.475613i
$670$			1.50468e15i			0.430558i
$671$			7.21544e15i			2.04781i
$672$	3.16284e15			0.890320
$673$	−4.73256e15			−1.32133			−0.660667	−	0.750679i	$-0.729727\pi$
$673$	−4.73256e15			−1.32133			−0.660667	+	0.750679i	$0.729727\pi$
$674$			3.39476e15i			0.940112i
$675$	1.28275e15			0.352349
$676$	−9.91547e14	+	2.33461e15i	−0.270151	+	0.636074i
$677$	−4.17475e15			−1.12822			−0.564109	−	0.825700i	$-0.690780\pi$
$677$	−4.17475e15			−1.12822			−0.564109	+	0.825700i	$0.690780\pi$
$678$			4.56197e15i			1.22289i
$679$	1.32498e15			0.352310
$680$	8.86106e15			2.33715
$681$			4.93610e15i			1.29144i
$682$		−	2.49060e15i		−	0.646384i
$683$		−	1.95274e15i		−	0.502725i	−0.967893	−	0.251362i	$-0.919121\pi$
$683$		−	1.95274e15i		−	0.502725i	0.967893	−	0.251362i	$-0.0808786\pi$
$684$		−	3.54099e15i		−	0.904307i
$685$	5.14893e15			1.30442
$686$	−2.10622e15			−0.529323
$687$		−	9.75545e15i		−	2.43212i
$688$	−2.80284e14			−0.0693206
$689$	4.92182e14	+	7.43771e14i	0.120759	+	0.182488i
$690$	8.43587e14			0.205334
$691$			3.36443e15i			0.812423i	0.913779	+	0.406212i	$0.133150\pi$
$691$			3.36443e15i			0.812423i	−0.913779	+	0.406212i	$0.866850\pi$
$692$	−4.92908e15			−1.18082
$693$	−4.71585e15			−1.12080
$694$		−	2.41866e15i		−	0.570293i
$695$		−	4.17497e15i		−	0.976644i
$696$			4.25093e15i			0.986581i
$697$		−	1.09061e16i		−	2.51125i
$698$	4.47159e15			1.02154
$699$	2.41032e15			0.546325
$700$			1.33731e15i			0.300742i
$701$	1.96043e15			0.437423			0.218711	−	0.975790i	$-0.429815\pi$
$701$	1.96043e15			0.437423			0.218711	+	0.975790i	$0.429815\pi$
$702$	−9.62186e14	+	6.36716e14i	−0.213012	+	0.140959i
$703$	−5.91699e14			−0.129971
$704$			2.82264e15i			0.615185i
$705$	4.16005e15			0.899618
$706$	−2.35704e15			−0.505757
$707$		−	1.68125e15i		−	0.357953i
$708$			2.19646e15i			0.464024i
$709$		−	1.86748e15i		−	0.391474i	−0.980656	−	0.195737i	$-0.937290\pi$
$709$		−	1.86748e15i		−	0.391474i	0.980656	−	0.195737i	$-0.0627098\pi$
$710$		−	3.24652e15i		−	0.675301i
$711$	6.84552e15			1.41294
$712$	−7.87417e15			−1.61274
$713$		−	6.91452e14i		−	0.140530i
$714$	4.44071e15			0.895595
$715$	5.55507e15	+	8.39465e15i	1.11175	+	1.68004i
$716$	2.82469e15			0.560981
$717$		−	1.02243e16i		−	2.01501i
$718$	8.14783e14			0.159352
$719$	−1.66595e15			−0.323334			−0.161667	−	0.986845i	$-0.551687\pi$
$719$	−1.66595e15			−0.323334			−0.161667	+	0.986845i	$0.551687\pi$
$720$			1.51621e15i			0.292031i
$721$		−	2.51687e15i		−	0.481079i
$722$			2.58502e13i			0.00490353i
$723$			1.21243e16i			2.28242i
$724$	−2.97249e15			−0.555339
$725$	−2.86025e15			−0.530328
$726$		−	5.94599e15i		−	1.09414i
$727$	9.12480e15			1.66642			0.833209	−	0.552958i	$-0.186501\pi$
$727$	9.12480e15			1.66642			0.833209	+	0.552958i	$0.186501\pi$
$728$	−1.62434e15	−	2.45465e15i	−0.294411	−	0.444905i
$729$	−8.26181e15			−1.48619
$730$			3.46061e15i			0.617841i
$731$	−4.33971e15			−0.768980
$732$	8.05658e15			1.41690
$733$		−	3.87654e15i		−	0.676663i	−0.941027	−	0.338331i	$-0.890138\pi$
$733$		−	3.87654e15i		−	0.676663i	0.941027	−	0.338331i	$-0.109862\pi$
$734$			3.07184e15i			0.532195i
$735$		−	7.95097e15i		−	1.36722i
$736$			1.10922e15i			0.189317i
$737$	5.21418e15			0.883313
$738$	−5.78112e15			−0.972080
$739$			1.39355e15i			0.232583i	0.993215	+	0.116292i	$0.0371007\pi$
$739$			1.39355e15i			0.232583i	−0.993215	+	0.116292i	$0.962899\pi$
$740$	7.17491e14			0.118862
$741$	−7.72992e15	+	5.11518e15i	−1.27109	+	0.841126i
$742$	−4.22951e14			−0.0690349
$743$			1.07016e13i			0.00173384i	1.00000		0.000866919i	$0.000275949\pi$
$743$			1.07016e13i			0.00173384i	−1.00000		0.000866919i	$0.999724\pi$
$744$	−6.80509e15			−1.09442
$745$	1.69918e15			0.271257
$746$			9.56582e14i			0.151586i
$747$			2.89160e15i			0.454857i
$748$		−	1.25484e16i		−	1.95943i
$749$			5.78499e15i			0.896712i
$750$	−1.69967e15			−0.261533
$751$	3.46539e14			0.0529337			0.0264668	−	0.999650i	$-0.491574\pi$
$751$	3.46539e14			0.0529337			0.0264668	+	0.999650i	$0.491574\pi$
$752$			4.96018e14i			0.0752142i
$753$	6.50701e15			0.979510
$754$	2.14545e15	−	1.41973e15i	0.320610	−	0.212160i
$755$	−1.51991e16			−2.25480
$756$			1.22395e15i			0.180258i
$757$	7.02514e15			1.02714			0.513568	−	0.858049i	$-0.328324\pi$
$757$	7.02514e15			1.02714			0.513568	+	0.858049i	$0.328324\pi$
$758$	2.06717e15			0.300052
$759$		−	2.92330e15i		−	0.421253i
$760$		−	8.77120e15i		−	1.25483i
$761$			3.91316e15i			0.555792i	0.960611	+	0.277896i	$0.0896371\pi$
$761$			3.91316e15i			0.555792i	−0.960611	+	0.277896i	$0.910363\pi$
$762$		−	1.82016e15i		−	0.256660i
$763$	−3.18900e15			−0.446447
$764$	2.22095e14			0.0308691
$765$			2.34758e16i			3.23953i
$766$	1.97075e15			0.270005
$767$	2.71269e15	−	1.79509e15i	0.369000	−	0.244181i
$768$	9.60715e15			1.29750
$769$			5.94587e15i			0.797298i	0.917104	+	0.398649i	$0.130521\pi$
$769$			5.94587e15i			0.797298i	−0.917104	+	0.398649i	$0.869479\pi$
$770$	−4.77368e15			−0.635557
$771$	−3.03505e15			−0.401206
$772$			1.26559e15i			0.166110i
$773$			6.92376e15i			0.902308i	0.892446	+	0.451154i	$0.148987\pi$
$773$			6.92376e15i			0.902308i	−0.892446	+	0.451154i	$0.851013\pi$
$774$			2.30040e15i			0.297665i
$775$		−	4.57882e15i		−	0.588294i
$776$	4.57320e15			0.583419
$777$	8.79881e14			0.111457
$778$		−	4.42936e15i		−	0.557125i
$779$	−1.07955e16			−1.34830
$780$	9.37327e15	−	6.20265e15i	1.16244	−	0.769232i
$781$	−1.12502e16			−1.38542
$782$			1.55737e15i			0.190438i
$783$	−2.61779e15			−0.317866
$784$	9.48025e14			0.114309
$785$			1.04939e16i			1.25647i
$786$			4.28217e15i			0.509143i
$787$			9.36325e15i			1.10552i	0.833341	+	0.552759i	$0.186425\pi$
$787$			9.36325e15i			1.10552i	−0.833341	+	0.552759i	$0.813575\pi$
$788$			5.35137e15i			0.627438i
$789$	2.18436e15			0.254331
$790$	6.92947e15			0.801219
$791$		−	7.16694e15i		−	0.822930i
$792$	−1.62769e16			−1.85602
$793$	−6.58437e15	−	9.95010e15i	−0.745610	−	1.12674i
$794$	1.32202e15			0.148671
$795$		−	3.95214e15i		−	0.441380i
$796$	3.11644e15			0.345651
$797$	1.64665e15			0.181377			0.0906883	−	0.995879i	$-0.471093\pi$
$797$	1.64665e15			0.181377			0.0906883	+	0.995879i	$0.471093\pi$
$798$		−	4.39568e15i		−	0.480850i
$799$			7.67998e15i			0.834358i
$800$			7.34528e15i			0.792525i
$801$		−	2.08612e16i		−	2.23543i
$802$	−1.75590e15			−0.186870
$803$	1.19921e16			1.26753
$804$		−	5.82202e15i		−	0.611175i
$805$	−1.32529e15			−0.138176
$806$	2.27277e15	+	3.43455e15i	0.235349	+	0.355653i
$807$	1.46650e16			1.50826
$808$		−	5.80289e15i		−	0.592764i
$809$	7.15540e15			0.725967			0.362984	−	0.931796i	$-0.381758\pi$
$809$	7.15540e15			0.725967			0.362984	+	0.931796i	$0.381758\pi$
$810$	−4.43882e15			−0.447301
$811$		−	1.16278e16i		−	1.16381i	−0.813258	−	0.581904i	$-0.802308\pi$
$811$		−	1.16278e16i		−	1.16381i	0.813258	−	0.581904i	$-0.197692\pi$
$812$		−	2.72913e15i		−	0.271310i
$813$		−	3.52539e15i		−	0.348104i
$814$			1.11149e15i			0.109011i
$815$	−1.11264e16			−1.08389
$816$	−4.94758e15			−0.478737
$817$			4.29570e15i			0.412870i
$818$	−5.12338e13			−0.00489118
$819$	6.50317e15	−	4.30340e15i	0.616685	−	0.408084i
$820$	1.30906e16			1.23306
$821$		−	2.45685e15i		−	0.229875i	−0.993373	−	0.114937i	$-0.963333\pi$
$821$		−	2.45685e15i		−	0.229875i	0.993373	−	0.114937i	$-0.0366667\pi$
$822$	8.90623e15			0.827750
$823$	−9.65791e15			−0.891628			−0.445814	−	0.895125i	$-0.647086\pi$
$823$	−9.65791e15			−0.891628			−0.445814	+	0.895125i	$0.647086\pi$
$824$		−	8.68702e15i		−	0.796657i
$825$		−	1.93582e16i		−	1.76347i
$826$			1.54259e15i			0.139592i
$827$			1.55380e16i			1.39673i	0.715740	+	0.698367i	$0.246090\pi$
$827$			1.55380e16i			1.39673i	−0.715740	+	0.698367i	$0.753910\pi$
$828$	−1.84666e15			−0.164900
$829$	1.12675e16			0.999491			0.499745	−	0.866172i	$-0.333427\pi$
$829$	1.12675e16			0.999491			0.499745	+	0.866172i	$0.333427\pi$
$830$			2.92706e15i			0.257930i
$831$	9.25570e15			0.810221
$832$	−2.57577e15	−	3.89242e15i	−0.223990	−	0.338487i
$833$	1.46785e16			1.26804
$834$		−	7.22154e15i		−	0.619750i
$835$	−1.49190e16			−1.27193
$836$	−1.24211e16			−1.05203
$837$		−	4.19068e15i		−	0.352609i
$838$		−	5.32336e14i		−	0.0444983i
$839$			2.39925e15i			0.199244i	0.995025	+	0.0996220i	$0.0317634\pi$
$839$			2.39925e15i			0.199244i	−0.995025	+	0.0996220i	$0.968237\pi$
$840$			1.30432e16i			1.07609i
$841$	−6.36345e15			−0.521572
$842$	−1.14003e16			−0.928328
$843$			9.97820e15i			0.807236i
$844$	−1.33778e16			−1.07523
$845$	−1.53209e16	−	6.50703e15i	−1.22341	−	0.519602i
$846$	4.07101e15			0.322972
$847$			9.34126e15i			0.736286i
$848$	4.71228e14			0.0369023
$849$	−7.50756e15			−0.584126
$850$			1.03129e16i			0.797221i
$851$			3.08577e14i			0.0237001i
$852$			1.25617e16i			0.958588i
$853$			1.64529e16i			1.24745i	0.781643	+	0.623726i	$0.214382\pi$
$853$			1.64529e16i			1.24745i	−0.781643	+	0.623726i	$0.785618\pi$
$854$	5.65820e15			0.426246
$855$	2.32377e16			1.73932
$856$			1.99670e16i			1.48494i
$857$	3.30993e15			0.244582			0.122291	−	0.992494i	$-0.460976\pi$
$857$	3.30993e15			0.244582			0.122291	+	0.992494i	$0.460976\pi$
$858$	9.60874e15	+	1.45204e16i	0.705483	+	1.06610i
$859$	1.74823e16			1.27537			0.637685	−	0.770298i	$-0.279892\pi$
$859$	1.74823e16			1.27537			0.637685	+	0.770298i	$0.279892\pi$
$860$		−	5.20895e15i		−	0.377580i
$861$	1.60534e16			1.15624
$862$	5.83688e15			0.417726
$863$		−	2.44481e14i		−	0.0173854i	−0.999962	−	0.00869272i	$-0.997233\pi$
$863$		−	2.44481e14i		−	0.0173854i	0.999962	−	0.00869272i	$-0.00276701\pi$
$864$			6.72262e15i			0.475021i
$865$		−	3.23471e16i		−	2.27116i
$866$		−	8.11623e15i		−	0.566247i
$867$	−5.47151e16			−3.79317
$868$	4.36892e15			0.300965
$869$		−	2.40128e16i		−	1.64374i
$870$	−1.14002e16			−0.775454
$871$	−7.19036e15	+	4.75814e15i	−0.486016	+	0.321616i
$872$	−1.10069e16			−0.739307
$873$			1.21159e16i			0.808679i
$874$	1.54158e15			0.102247
$875$	2.67021e15			0.175995
$876$		−	1.33901e16i		−	0.877023i
$877$			1.53247e16i			0.997460i	0.866757	+	0.498730i	$0.166200\pi$
$877$			1.53247e16i			0.997460i	−0.866757	+	0.498730i	$0.833800\pi$
$878$			3.62065e14i			0.0234189i
$879$		−	1.56477e16i		−	1.00580i
$880$	5.31857e15			0.339735
$881$	2.84693e16			1.80721			0.903607	−	0.428362i	$-0.140909\pi$
$881$	2.84693e16			1.80721			0.903607	+	0.428362i	$0.140909\pi$
$882$		−	7.78080e15i		−	0.490848i
$883$	−1.20988e15			−0.0758506			−0.0379253	−	0.999281i	$-0.512075\pi$
$883$	−1.20988e15			−0.0758506			−0.0379253	+	0.999281i	$0.512075\pi$
$884$	1.14509e16	+	1.73042e16i	0.713430	+	1.07811i
$885$	−1.44143e16			−0.892493
$886$			5.34880e14i			0.0329133i
$887$	−3.13464e16			−1.91694			−0.958469	−	0.285198i	$-0.907941\pi$
$887$	−3.13464e16			−1.91694			−0.958469	+	0.285198i	$0.907941\pi$
$888$	3.03693e15			0.184571
$889$			2.85951e15i			0.172716i
$890$		−	2.11171e16i		−	1.26762i
$891$			1.53819e16i			0.917663i
$892$		−	3.65299e15i		−	0.216592i
$893$	7.60210e15			0.447972
$894$	2.93912e15			0.172132
$895$			1.85370e16i			1.07898i
$896$	−7.92821e15			−0.458649
$897$	2.66762e15	+	4.03123e15i	0.153379	+	0.231782i
$898$	2.85693e15			0.163260
$899$			9.34425e15i			0.530721i
$900$	−1.22287e16			−0.690312
$901$	7.29614e15			0.409361
$902$			2.02791e16i			1.13087i
$903$		−	6.38790e15i		−	0.354059i
$904$		−	2.47368e16i		−	1.36276i
$905$		−	1.95070e16i		−	1.06813i
$906$	−2.62902e16			−1.43083
$907$	−3.40850e15			−0.184384			−0.0921920	−	0.995741i	$-0.529387\pi$
$907$	−3.40850e15			−0.184384			−0.0921920	+	0.995741i	$0.529387\pi$
$908$		−	1.09379e16i		−	0.588116i
$909$	1.53737e16			0.821632
$910$	6.58292e15	−	4.35617e15i	0.349696	−	0.231407i
$911$	1.09436e16			0.577840			0.288920	−	0.957353i	$-0.406704\pi$
$911$	1.09436e16			0.577840			0.288920	+	0.957353i	$0.406704\pi$
$912$			4.89741e15i			0.257037i
$913$	1.01432e16			0.529158
$914$	−7.26155e15			−0.376552
$915$			5.28713e16i			2.72524i
$916$			2.16172e16i			1.10758i
$917$		−	6.72737e15i		−	0.342621i
$918$			9.43872e15i			0.477835i
$919$	−6.25304e15			−0.314670			−0.157335	−	0.987545i	$-0.550290\pi$
$919$	−6.25304e15			−0.314670			−0.157335	+	0.987545i	$0.550290\pi$
$920$	−4.57428e15			−0.228817
$921$		−	5.42634e16i		−	2.69823i
$922$	7.73098e15			0.382133
$923$	1.55141e16	−	1.02663e16i	0.762284	−	0.504433i
$924$	1.84708e16			0.902171
$925$			2.04341e15i			0.0992146i
$926$	−1.05287e16			−0.508178
$927$	2.30147e16			1.10425
$928$		−	1.49899e16i		−	0.714965i
$929$		−	1.65981e16i		−	0.786996i	−0.919326	−	0.393498i	$-0.871265\pi$
$929$		−	1.65981e16i		−	0.786996i	0.919326	−	0.393498i	$-0.128735\pi$
$930$		−	1.82500e16i		−	0.860212i
$931$		−	1.45297e16i		−	0.680820i
$932$	−5.34105e15			−0.248793
$933$	−1.85247e16			−0.857831
$934$			1.14924e16i			0.529057i
$935$	8.23487e16			3.76871
$936$	2.24458e16	−	1.48533e16i	1.02122	−	0.675779i
$937$	3.21563e15			0.145445			0.0727225	−	0.997352i	$-0.476831\pi$
$937$	3.21563e15			0.145445			0.0727225	+	0.997352i	$0.476831\pi$
$938$		−	4.08885e15i		−	0.183859i
$939$	−2.74292e16			−1.22617
$940$	−9.21828e15			−0.409682
$941$		−	4.00810e16i		−	1.77091i	−0.464729	−	0.885453i	$-0.653848\pi$
$941$		−	4.00810e16i		−	1.77091i	0.464729	−	0.885453i	$-0.346152\pi$
$942$			1.81516e16i			0.797322i
$943$			5.62998e15i			0.245862i
$944$		−	1.71867e15i		−	0.0746185i
$945$	−8.03218e15			−0.346704
$946$	8.06937e15			0.346289
$947$			4.94025e14i			0.0210777i	0.999944	+	0.0105389i	$0.00335469\pi$
$947$			4.94025e14i			0.0210777i	−0.999944	+	0.0105389i	$0.996645\pi$
$948$	−2.68121e16			−1.13733
$949$	−1.65371e16	+	1.09433e16i	−0.697422	+	0.461511i
$950$	1.02084e16			0.428032
$951$		−	7.74999e15i		−	0.323078i
$952$	−2.40793e16			−0.998024
$953$	−5.84963e15			−0.241056			−0.120528	−	0.992710i	$-0.538459\pi$
$953$	−5.84963e15			−0.241056			−0.120528	+	0.992710i	$0.538459\pi$
$954$		−	3.86755e15i		−	0.158460i
$955$			1.45750e15i			0.0593730i
$956$			2.26560e16i			0.917624i
$957$			3.95053e16i			1.59089i
$958$	2.13990e16			0.856807
$959$	−1.39919e16			−0.557023
$960$			2.06829e16i			0.818692i
$961$	1.04498e16			0.411271
$962$	−1.01428e15	−	1.53275e15i	−0.0396912	−	0.0599802i
$963$	−5.28991e16			−2.05828
$964$		−	2.68662e16i		−	1.03940i
$965$	−8.30541e15			−0.319493
$966$	−2.29239e15			−0.0876828
$967$		−	3.46312e16i		−	1.31711i	−0.752532	−	0.658555i	$-0.771168\pi$
$967$		−	3.46312e16i		−	1.31711i	0.752532	−	0.658555i	$-0.228832\pi$
$968$			3.22416e16i			1.21928i
$969$			7.58279e16i			2.85133i
$970$			1.22645e16i			0.458568i
$971$	1.55279e16			0.577307			0.288653	−	0.957434i	$-0.406792\pi$
$971$	1.55279e16			0.577307			0.288653	+	0.957434i	$0.406792\pi$
$972$	2.57656e16			0.952524
$973$			1.13452e16i			0.417053i
$974$	−1.70779e16			−0.624254
$975$	1.76651e16	+	2.66950e16i	0.642082	+	0.970295i
$976$	−6.30405e15			−0.227848
$977$			1.74397e16i			0.626785i	0.949624	+	0.313393i	$0.101466\pi$
$977$			1.74397e16i			0.626785i	−0.949624	+	0.313393i	$0.898534\pi$
$978$	−1.92456e16			−0.687808
$979$	−7.31773e16			−2.60059
$980$			1.76186e16i			0.622627i
$981$		−	2.91608e16i		−	1.02476i
$982$		−	4.94756e15i		−	0.172893i
$983$		−	5.06070e16i		−	1.75860i	−0.476272	−	0.879298i	$-0.658012\pi$
$983$		−	5.06070e16i		−	1.75860i	0.476272	−	0.879298i	$-0.341988\pi$
$984$	5.54087e16			1.91472
$985$	−3.51184e16			−1.20680
$986$		−	2.10462e16i		−	0.719201i
$987$	−1.13046e16			−0.384161
$988$	1.71288e16	−	1.13348e16i	0.578846	−	0.383045i
$989$	2.24025e15			0.0752866
$990$		−	4.36515e16i		−	1.45883i
$991$	−1.03141e16			−0.342789			−0.171394	−	0.985202i	$-0.554827\pi$
$991$	−1.03141e16			−0.342789			−0.171394	+	0.985202i	$0.554827\pi$
$992$	2.39965e16			0.793112
$993$			5.77722e16i			1.89888i
$994$			8.82221e15i			0.288371i
$995$			2.04517e16i			0.664816i
$996$		−	1.13256e16i		−	0.366131i
$997$	−5.12991e15			−0.164925			−0.0824624	−	0.996594i	$-0.526278\pi$
$997$	−5.12991e15			−0.164925			−0.0824624	+	0.996594i	$0.526278\pi$
$998$	1.79192e16			0.572929
$999$			1.87019e15i			0.0594669i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.b.a.12.5	✓	12
3.2	odd	2		117.12.b.b.64.8		12
4.3	odd	2		208.12.f.b.129.12		12
13.5	odd	4		169.12.a.e.1.5		12
13.8	odd	4		169.12.a.e.1.8		12
13.12	even	2	inner	13.12.b.a.12.8	yes	12
39.38	odd	2		117.12.b.b.64.5		12
52.51	odd	2		208.12.f.b.129.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.b.a.12.5	✓	12	1.1	even	1	trivial
13.12.b.a.12.8	yes	12	13.12	even	2	inner
117.12.b.b.64.5		12	39.38	odd	2
117.12.b.b.64.8		12	3.2	odd	2
169.12.a.e.1.5		12	13.5	odd	4
169.12.a.e.1.8		12	13.8	odd	4
208.12.f.b.129.11		12	52.51	odd	2
208.12.f.b.129.12		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)

\(f(q)\)	\(=\)	\(q-25.1535i q^{2} -638.702 q^{3} +1415.30 q^{4} +9287.93i q^{5} +16065.6i q^{6} -25239.3i q^{7} -87114.1i q^{8} +230793. q^{9} +O(q^{10})\)	\(q-25.1535i q^{2} -638.702 q^{3} +1415.30 q^{4} +9287.93i q^{5} +16065.6i q^{6} -25239.3i q^{7} -87114.1i q^{8} +230793. q^{9} +233624. q^{10} -809580. i q^{11} -903957. q^{12} +(738773. + 1.11641e6i) q^{13} -634856. q^{14} -5.93222e6i q^{15} +707321. q^{16} +1.09516e7 q^{17} -5.80525e6i q^{18} -1.08406e7i q^{19} +1.31452e7i q^{20} +1.61204e7i q^{21} -2.03637e7 q^{22} -5.65347e6 q^{23} +5.56399e7i q^{24} -3.74375e7 q^{25} +(2.80816e7 - 1.85827e7i) q^{26} -3.42639e7 q^{27} -3.57213e7i q^{28} +7.64007e7 q^{29} -1.49216e8 q^{30} +1.22306e8i q^{31} -1.96201e8i q^{32} +5.17080e8i q^{33} -2.75471e8i q^{34} +2.34421e8 q^{35} +3.26642e8 q^{36} -5.45819e7i q^{37} -2.72678e8 q^{38} +(-4.71856e8 - 7.13054e8i) q^{39} +8.09109e8 q^{40} -9.95844e8i q^{41} +4.05484e8 q^{42} -3.96262e8 q^{43} -1.14580e9i q^{44} +2.14359e9i q^{45} +1.42204e8i q^{46} +7.01264e8i q^{47} -4.51767e8 q^{48} +1.34030e9 q^{49} +9.41682e8i q^{50} -6.99482e9 q^{51} +(1.04559e9 + 1.58006e9i) q^{52} +6.66216e8 q^{53} +8.61856e8i q^{54} +7.51932e9 q^{55} -2.19870e9 q^{56} +6.92389e9i q^{57} -1.92174e9i q^{58} -2.42983e9i q^{59} -8.39589e9i q^{60} -8.91257e9 q^{61} +3.07641e9 q^{62} -5.82506e9i q^{63} -3.48655e9 q^{64} +(-1.03692e10 + 6.86167e9i) q^{65} +1.30064e10 q^{66} +6.44060e9i q^{67} +1.54999e10 q^{68} +3.61088e9 q^{69} -5.89650e9i q^{70} -1.38964e10i q^{71} -2.01053e10i q^{72} +1.48127e10i q^{73} -1.37292e9 q^{74} +2.39114e10 q^{75} -1.53427e10i q^{76} -2.04332e10 q^{77} +(-1.79358e10 + 1.18688e10i) q^{78} +2.96608e10 q^{79} +6.56954e9i q^{80} -1.89999e10 q^{81} -2.50489e10 q^{82} +1.25290e10i q^{83} +2.28153e10i q^{84} +1.01718e11i q^{85} +9.96736e9i q^{86} -4.87973e10 q^{87} -7.05258e10 q^{88} -9.03892e10i q^{89} +5.39187e10 q^{90} +(2.81775e10 - 1.86461e10i) q^{91} -8.00138e9 q^{92} -7.81170e10i q^{93} +1.76392e10 q^{94} +1.00686e11 q^{95} +1.25314e11i q^{96} +5.24967e10i q^{97} -3.37133e10i q^{98} -1.86845e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9}+O(q^{10}) \) 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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