LMFDB - Embedded newform 13.12.b.a.12.9

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			12.9
Root			$37.0312i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.12.b.a.12.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+37.0312i q^{2} -143.287 q^{3} +676.691 q^{4} +5266.14i q^{5} -5306.08i q^{6} +16188.3i q^{7} +100899. i q^{8} -156616. q^{9} +O(q^{10})$	\(q+37.0312i q^{2} -143.287 q^{3} +676.691 q^{4} +5266.14i q^{5} -5306.08i q^{6} +16188.3i q^{7} +100899. i q^{8} -156616. q^{9} -195012. q^{10} -447110. i q^{11} -96960.9 q^{12} +(-1.33862e6 - 16101.7i) q^{13} -599473. q^{14} -754569. i q^{15} -2.35053e6 q^{16} -6.84716e6 q^{17} -5.79967e6i q^{18} +1.68794e7i q^{19} +3.56355e6i q^{20} -2.31957e6i q^{21} +1.65570e7 q^{22} +1.49304e7 q^{23} -1.44574e7i q^{24} +2.10958e7 q^{25} +(596266. - 4.95707e7i) q^{26} +4.78238e7 q^{27} +1.09545e7i q^{28} +3.19957e7 q^{29} +2.79426e7 q^{30} +2.68660e7i q^{31} +1.19597e8i q^{32} +6.40650e7i q^{33} -2.53558e8i q^{34} -8.52500e7 q^{35} -1.05981e8 q^{36} +5.43137e8i q^{37} -6.25066e8 q^{38} +(1.91806e8 + 2.30716e6i) q^{39} -5.31346e8 q^{40} -9.19462e8i q^{41} +8.58965e7 q^{42} -2.47741e8 q^{43} -3.02556e8i q^{44} -8.24762e8i q^{45} +5.52889e8i q^{46} +1.05508e9i q^{47} +3.36800e8 q^{48} +1.71526e9 q^{49} +7.81204e8i q^{50} +9.81107e8 q^{51} +(-9.05832e8 - 1.08959e7i) q^{52} +3.13264e9 q^{53} +1.77097e9i q^{54} +2.35455e9 q^{55} -1.63338e9 q^{56} -2.41860e9i q^{57} +1.18484e9i q^{58} -2.01659e9i q^{59} -5.10610e8i q^{60} +1.47077e9 q^{61} -9.94880e8 q^{62} -2.53535e9i q^{63} -9.24272e9 q^{64} +(8.47940e7 - 7.04936e9i) q^{65} -2.37240e9 q^{66} +1.34287e10i q^{67} -4.63341e9 q^{68} -2.13932e9 q^{69} -3.15691e9i q^{70} -2.43380e9i q^{71} -1.58023e10i q^{72} +2.78909e10i q^{73} -2.01130e10 q^{74} -3.02276e9 q^{75} +1.14222e10i q^{76} +7.23797e9 q^{77} +(-8.54370e7 + 7.10282e9i) q^{78} +1.03269e10 q^{79} -1.23782e10i q^{80} +2.08915e10 q^{81} +3.40488e10 q^{82} -3.76857e10i q^{83} -1.56963e9i q^{84} -3.60581e10i q^{85} -9.17413e9i q^{86} -4.58456e9 q^{87} +4.51128e10 q^{88} +9.91225e10i q^{89} +3.05419e10 q^{90} +(2.60660e8 - 2.16700e10i) q^{91} +1.01032e10 q^{92} -3.84954e9i q^{93} -3.90707e10 q^{94} -8.88896e10 q^{95} -1.71367e10i q^{96} -4.38508e10i q^{97} +6.35183e10i q^{98} +7.00246e10i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			37.0312i			0.818281i	0.912471	+	0.409141i	$0.134171\pi$
$2$			37.0312i			0.818281i	−0.912471	+	0.409141i	$0.865829\pi$
$3$	−143.287			−0.340439			−0.170219	−	0.985406i	$-0.554448\pi$
$3$	−143.287			−0.340439			−0.170219	+	0.985406i	$0.554448\pi$
$4$	676.691			0.330416
$5$			5266.14i			0.753629i	0.926289	+	0.376815i	$0.122981\pi$
$5$			5266.14i			0.753629i	−0.926289	+	0.376815i	$0.877019\pi$
$6$		−	5306.08i		−	0.278575i
$7$			16188.3i			0.364051i	0.983294	+	0.182026i	$0.0582654\pi$
$7$			16188.3i			0.364051i	−0.983294	+	0.182026i	$0.941735\pi$
$8$			100899.i			1.08865i
$9$	−156616.			−0.884101
$10$	−195012.			−0.616681
$11$		−	447110.i		−	0.837057i	−0.908204	−	0.418528i	$-0.862546\pi$
$11$		−	447110.i		−	0.837057i	0.908204	−	0.418528i	$-0.137454\pi$
$12$	−96960.9			−0.112486
$13$	−1.33862e6	−	16101.7i	−0.999928	−	0.0120277i
$13$	−1.33862e6	−	16101.7i	−0.999928	−	0.0120277i
$14$	−599473.			−0.297896
$15$		−	754569.i		−	0.256565i
$16$	−2.35053e6			−0.560410
$17$	−6.84716e6			−1.16961			−0.584805	−	0.811174i	$-0.698829\pi$
$17$	−6.84716e6			−1.16961			−0.584805	+	0.811174i	$0.698829\pi$
$18$		−	5.79967e6i		−	0.723444i
$19$			1.68794e7i			1.56392i	0.623331	+	0.781958i	$0.285779\pi$
$19$			1.68794e7i			1.56392i	−0.623331	+	0.781958i	$0.714221\pi$
$20$			3.56355e6i			0.249011i
$21$		−	2.31957e6i		−	0.123937i
$22$	1.65570e7			0.684948
$23$	1.49304e7			0.483690			0.241845	−	0.970315i	$-0.422247\pi$
$23$	1.49304e7			0.483690			0.241845	+	0.970315i	$0.422247\pi$
$24$		−	1.44574e7i		−	0.370620i
$25$	2.10958e7			0.432043
$26$	596266.	−	4.95707e7i	0.00984207	−	0.818222i
$27$	4.78238e7			0.641421
$28$			1.09545e7i			0.120288i
$29$	3.19957e7			0.289670			0.144835	−	0.989456i	$-0.453735\pi$
$29$	3.19957e7			0.289670			0.144835	+	0.989456i	$0.453735\pi$
$30$	2.79426e7			0.209942
$31$			2.68660e7i			0.168544i	0.996443	+	0.0842721i	$0.0268565\pi$
$31$			2.68660e7i			0.168544i	−0.996443	+	0.0842721i	$0.973144\pi$
$32$			1.19597e8i			0.630081i
$33$			6.40650e7i			0.284967i
$34$		−	2.53558e8i		−	0.957070i
$35$	−8.52500e7			−0.274360
$36$	−1.05981e8			−0.292121
$37$			5.43137e8i			1.28766i	0.765170	+	0.643828i	$0.222655\pi$
$37$			5.43137e8i			1.28766i	−0.765170	+	0.643828i	$0.777345\pi$
$38$	−6.25066e8			−1.27972
$39$	1.91806e8	+	2.30716e6i	0.340414	+	0.00409471i
$40$	−5.31346e8			−0.820442
$41$		−	9.19462e8i		−	1.23943i	−0.784826	−	0.619716i	$-0.787248\pi$
$41$		−	9.19462e8i		−	1.23943i	0.784826	−	0.619716i	$-0.212752\pi$
$42$	8.58965e7			0.101416
$43$	−2.47741e8			−0.256993			−0.128496	−	0.991710i	$-0.541015\pi$
$43$	−2.47741e8			−0.256993			−0.128496	+	0.991710i	$0.541015\pi$
$44$		−	3.02556e8i		−	0.276577i
$45$		−	8.24762e8i		−	0.666285i
$46$			5.52889e8i			0.395795i
$47$			1.05508e9i			0.671036i	0.942034	+	0.335518i	$0.108911\pi$
$47$			1.05508e9i			0.671036i	−0.942034	+	0.335518i	$0.891089\pi$
$48$	3.36800e8			0.190785
$49$	1.71526e9			0.867467
$50$			7.81204e8i			0.353533i
$51$	9.81107e8			0.398181
$52$	−9.05832e8	−	1.08959e7i	−0.330392	−	0.00397415i
$53$	3.13264e9			1.02895			0.514474	−	0.857506i	$-0.327987\pi$
$53$	3.13264e9			1.02895			0.514474	+	0.857506i	$0.327987\pi$
$54$			1.77097e9i			0.524863i
$55$	2.35455e9			0.630831
$56$	−1.63338e9			−0.396326
$57$		−	2.41860e9i		−	0.532418i
$58$			1.18484e9i			0.237031i
$59$		−	2.01659e9i		−	0.367224i	−0.982999	−	0.183612i	$-0.941221\pi$
$59$		−	2.01659e9i		−	0.367224i	0.982999	−	0.183612i	$-0.0587790\pi$
$60$		−	5.10610e8i		−	0.0847730i
$61$	1.47077e9			0.222961			0.111481	−	0.993767i	$-0.464441\pi$
$61$	1.47077e9			0.222961			0.111481	+	0.993767i	$0.464441\pi$
$62$	−9.94880e8			−0.137917
$63$		−	2.53535e9i		−	0.321858i
$64$	−9.24272e9			−1.07599
$65$	8.47940e7	−	7.04936e9i	0.00906445	−	0.753575i
$66$	−2.37240e9			−0.233183
$67$			1.34287e10i			1.21513i	0.794269	+	0.607566i	$0.207854\pi$
$67$			1.34287e10i			1.21513i	−0.794269	+	0.607566i	$0.792146\pi$
$68$	−4.63341e9			−0.386458
$69$	−2.13932e9			−0.164667
$70$		−	3.15691e9i		−	0.224503i
$71$		−	2.43380e9i		−	0.160090i	−0.996791	−	0.0800449i	$-0.974494\pi$
$71$		−	2.43380e9i		−	0.160090i	0.996791	−	0.0800449i	$-0.0255064\pi$
$72$		−	1.58023e10i		−	0.962481i
$73$			2.78909e10i			1.57466i	0.616531	+	0.787331i	$0.288538\pi$
$73$			2.78909e10i			1.57466i	−0.616531	+	0.787331i	$0.711462\pi$
$74$	−2.01130e10			−1.05366
$75$	−3.02276e9			−0.147084
$76$			1.14222e10i			0.516742i
$77$	7.23797e9			0.304732
$78$	−8.54370e7	+	7.10282e9i	−0.00335062	+	0.278555i
$79$	1.03269e10			0.377591			0.188795	−	0.982016i	$-0.439542\pi$
$79$	1.03269e10			0.377591			0.188795	+	0.982016i	$0.439542\pi$
$80$		−	1.23782e10i		−	0.422341i
$81$	2.08915e10			0.665736
$82$	3.40488e10			1.01420
$83$		−	3.76857e10i		−	1.05014i	−0.851059	−	0.525070i	$-0.824039\pi$
$83$		−	3.76857e10i		−	1.05014i	0.851059	−	0.525070i	$-0.175961\pi$
$84$		−	1.56963e9i		−	0.0409508i
$85$		−	3.60581e10i		−	0.881453i
$86$		−	9.17413e9i		−	0.210292i
$87$	−4.58456e9			−0.0986148
$88$	4.51128e10			0.911265
$89$			9.91225e10i			1.88160i	0.338963	+	0.940800i	$0.389924\pi$
$89$			9.91225e10i			1.88160i	−0.338963	+	0.940800i	$0.610076\pi$
$90$	3.05419e10			0.545208
$91$	2.60660e8	−	2.16700e10i	0.00437871	−	0.364025i
$92$	1.01032e10			0.159819
$93$		−	3.84954e9i		−	0.0573790i
$94$	−3.90707e10			−0.549096
$95$	−8.88896e10			−1.17861
$96$		−	1.71367e10i		−	0.214504i
$97$		−	4.38508e10i		−	0.518481i	−0.965813	−	0.259240i	$-0.916528\pi$
$97$		−	4.38508e10i		−	0.518481i	0.965813	−	0.259240i	$-0.0834722\pi$
$98$			6.35183e10i			0.709832i
$99$			7.00246e10i			0.740043i
$100$	1.42754e10			0.142754
$101$	−6.46974e10			−0.612519			−0.306259	−	0.951948i	$-0.599077\pi$
$101$	−6.46974e10			−0.612519			−0.306259	+	0.951948i	$0.599077\pi$
$102$			3.63316e10i			0.325824i
$103$	−6.05203e9			−0.0514395			−0.0257197	−	0.999669i	$-0.508188\pi$
$103$	−6.05203e9			−0.0514395			−0.0257197	+	0.999669i	$0.508188\pi$
$104$	1.62464e9	−	1.35065e11i	0.0130940	−	1.08858i
$105$	1.22152e10			0.0934027
$106$			1.16005e11i			0.841969i
$107$	−1.83061e11			−1.26179			−0.630893	−	0.775869i	$-0.717312\pi$
$107$	−1.83061e11			−1.26179			−0.630893	+	0.775869i	$0.717312\pi$
$108$	3.23620e10			0.211936
$109$		−	2.18877e11i		−	1.36255i	−0.732026	−	0.681277i	$-0.761425\pi$
$109$		−	2.18877e11i		−	1.36255i	0.732026	−	0.681277i	$-0.238575\pi$
$110$			8.71917e10i			0.516197i
$111$		−	7.78243e10i		−	0.438368i
$112$		−	3.80511e10i		−	0.204018i
$113$	−2.05501e11			−1.04926			−0.524630	−	0.851330i	$-0.675796\pi$
$113$	−2.05501e11			−1.04926			−0.524630	+	0.851330i	$0.675796\pi$
$114$	8.95637e10			0.435668
$115$			7.86255e10i			0.364523i
$116$	2.16512e10			0.0957114
$117$	2.09649e11	+	2.52179e9i	0.884037	+	0.0106337i
$118$	7.46767e10			0.300493
$119$		−	1.10844e11i		−	0.425798i
$120$	7.61349e10			0.279310
$121$	8.54040e10			0.299336
$122$			5.44642e10i			0.182445i
$123$			1.31747e11i			0.421951i
$124$			1.81800e10i			0.0556897i
$125$			3.68230e11i			1.07923i
$126$	9.38870e10			0.263371
$127$	5.86461e11			1.57514			0.787569	−	0.616226i	$-0.211339\pi$
$127$	5.86461e11			1.57514			0.787569	+	0.616226i	$0.211339\pi$
$128$		−	9.73333e10i		−	0.250384i
$129$	3.54980e10			0.0874903
$130$	2.61046e11	+	3.14002e9i	0.616636	+	0.00741727i
$131$	−1.08205e11			−0.245051			−0.122525	−	0.992465i	$-0.539099\pi$
$131$	−1.08205e11			−0.245051			−0.122525	+	0.992465i	$0.539099\pi$
$132$			4.33522e10i			0.0941575i
$133$	−2.73250e11			−0.569346
$134$	−4.97281e11			−0.994319
$135$			2.51847e11i			0.483394i
$136$		−	6.90868e11i		−	1.27330i
$137$		−	4.94496e11i		−	0.875386i	−0.899125	−	0.437693i	$-0.855796\pi$
$137$		−	4.94496e11i		−	0.875386i	0.899125	−	0.437693i	$-0.144204\pi$
$138$		−	7.92217e10i		−	0.134744i
$139$	−4.85658e11			−0.793869			−0.396934	−	0.917847i	$-0.629926\pi$
$139$	−4.85658e11			−0.793869			−0.396934	+	0.917847i	$0.629926\pi$
$140$	−5.76880e10			−0.0906528
$141$		−	1.51178e11i		−	0.228447i
$142$	9.01264e10			0.130998
$143$	−7.19925e9	+	5.98510e11i	−0.0100679	+	0.836996i
$144$	3.68130e11			0.495459
$145$			1.68494e11i			0.218304i
$146$	−1.03283e12			−1.28852
$147$	−2.45775e11			−0.295319
$148$			3.67536e11i			0.425462i
$149$			3.26280e11i			0.363971i	0.983301	+	0.181985i	$0.0582524\pi$
$149$			3.26280e11i			0.363971i	−0.983301	+	0.181985i	$0.941748\pi$
$150$		−	1.11936e11i		−	0.120356i
$151$		−	9.77744e11i		−	1.01357i	−0.862074	−	0.506783i	$-0.830835\pi$
$151$		−	9.77744e11i		−	1.01357i	0.862074	−	0.506783i	$-0.169165\pi$
$152$	−1.70311e12			−1.70256
$153$	1.07237e12			1.03405
$154$			2.68030e11i			0.249356i
$155$	−1.41480e11			−0.127020
$156$	1.29794e11	+	1.56124e9i	0.112478	+	0.00135296i
$157$	1.82267e12			1.52497			0.762483	−	0.647008i	$-0.223980\pi$
$157$	1.82267e12			1.52497			0.762483	+	0.647008i	$0.223980\pi$
$158$			3.82418e11i			0.308976i
$159$	−4.48866e11			−0.350294
$160$	−6.29817e11			−0.474848
$161$			2.41698e11i			0.176088i
$162$			7.73638e11i			0.544760i
$163$			2.23968e12i			1.52459i	0.647228	+	0.762296i	$0.275928\pi$
$163$			2.23968e12i			1.52459i	−0.647228	+	0.762296i	$0.724072\pi$
$164$		−	6.22192e11i		−	0.409528i
$165$	−3.37376e11			−0.214759
$166$	1.39555e12			0.859310
$167$		−	2.94541e12i		−	1.75471i	−0.479843	−	0.877355i	$-0.659306\pi$
$167$		−	2.94541e12i		−	1.75471i	0.479843	−	0.877355i	$-0.340694\pi$
$168$	2.34042e11			0.134925
$169$	1.79164e12	+	4.31081e10i	0.999711	+	0.0240537i
$170$	1.33527e12			0.721276
$171$		−	2.64359e12i		−	1.38266i
$172$	−1.67644e11			−0.0849144
$173$	6.82552e11			0.334875			0.167437	−	0.985883i	$-0.446451\pi$
$173$	6.82552e11			0.334875			0.167437	+	0.985883i	$0.446451\pi$
$174$		−	1.69772e11i		−	0.0806947i
$175$			3.41506e11i			0.157286i
$176$			1.05095e12i			0.469095i
$177$			2.88951e11i			0.125017i
$178$	−3.67062e12			−1.53968
$179$	3.04733e12			1.23945			0.619723	−	0.784820i	$-0.287245\pi$
$179$	3.04733e12			1.23945			0.619723	+	0.784820i	$0.287245\pi$
$180$		−	5.58109e11i		−	0.220151i
$181$	2.76647e11			0.105851			0.0529254	−	0.998598i	$-0.483145\pi$
$181$	2.76647e11			0.105851			0.0529254	+	0.998598i	$0.483145\pi$
$182$	8.02466e11	+	9.65254e9i	0.297875	+	0.00358302i
$183$	−2.10741e11			−0.0759047
$184$			1.50645e12i			0.526571i
$185$	−2.86024e12			−0.970415
$186$	1.42553e11			0.0469522
$187$			3.06143e12i			0.979030i
$188$			7.13961e11i			0.221721i
$189$			7.74187e11i			0.233510i
$190$		−	3.29169e12i		−	0.964437i
$191$	−6.98096e12			−1.98716			−0.993578	−	0.113153i	$-0.963905\pi$
$191$	−6.98096e12			−1.98716			−0.993578	+	0.113153i	$0.963905\pi$
$192$	1.32436e12			0.366310
$193$			2.26651e12i			0.609245i	0.952473	+	0.304622i	$0.0985302\pi$
$193$			2.26651e12i			0.609245i	−0.952473	+	0.304622i	$0.901470\pi$
$194$	1.62385e12			0.424263
$195$	−1.21499e10	+	1.01008e12i	−0.00308589	+	0.256546i
$196$	1.16071e12			0.286625
$197$		−	3.61395e12i		−	0.867798i	−0.900962	−	0.433899i	$-0.857138\pi$
$197$		−	3.61395e12i		−	0.867798i	0.900962	−	0.433899i	$-0.142862\pi$
$198$	−2.59309e12			−0.605563
$199$	−7.55025e12			−1.71502			−0.857511	−	0.514466i	$-0.827990\pi$
$199$	−7.55025e12			−1.71502			−0.857511	+	0.514466i	$0.827990\pi$
$200$			2.12854e12i			0.470345i
$201$		−	1.92416e12i		−	0.413678i
$202$		−	2.39582e12i		−	0.501212i
$203$			5.17957e11i			0.105455i
$204$	6.63907e11			0.131565
$205$	4.84202e12			0.934072
$206$		−	2.24114e11i		−	0.0420920i
$207$	−2.33833e12			−0.427631
$208$	3.14646e12	+	3.78476e10i	0.560369	+	0.00674046i
$209$	7.54697e12			1.30909
$210$			4.52344e11i			0.0764297i
$211$	−7.88114e12			−1.29729			−0.648643	−	0.761093i	$-0.724663\pi$
$211$	−7.88114e12			−1.29729			−0.648643	+	0.761093i	$0.724663\pi$
$212$	2.11983e12			0.339981
$213$			3.48731e11i			0.0545008i
$214$		−	6.77898e12i		−	1.03250i
$215$		−	1.30464e12i		−	0.193677i
$216$			4.82535e12i			0.698286i
$217$	−4.34916e11			−0.0613587
$218$	8.10526e12			1.11495
$219$		−	3.99640e12i		−	0.536076i
$220$	1.59330e12			0.208436
$221$	9.16573e12	+	1.10251e11i	1.16953	+	0.0140678i
$222$	2.88193e12			0.358708
$223$			1.04892e13i			1.27370i	0.770988	+	0.636849i	$0.219763\pi$
$223$			1.04892e13i			1.27370i	−0.770988	+	0.636849i	$0.780237\pi$
$224$	−1.93608e12			−0.229382
$225$	−3.30394e12			−0.381970
$226$		−	7.60995e12i		−	0.858590i
$227$			1.15289e13i			1.26954i	0.772702	+	0.634769i	$0.218905\pi$
$227$			1.15289e13i			1.26954i	−0.772702	+	0.634769i	$0.781095\pi$
$228$		−	1.63665e12i		−	0.175919i
$229$			1.27585e12i			0.133876i	0.997757	+	0.0669381i	$0.0213230\pi$
$229$			1.27585e12i			0.133876i	−0.997757	+	0.0669381i	$0.978677\pi$
$230$	−2.91159e12			−0.298282
$231$	−1.03710e12			−0.103743
$232$			3.22832e12i			0.315350i
$233$	7.27631e12			0.694151			0.347075	−	0.937837i	$-0.387175\pi$
$233$	7.27631e12			0.694151			0.347075	+	0.937837i	$0.387175\pi$
$234$	−9.33847e10	+	7.76355e12i	−0.00870139	+	0.723391i
$235$	−5.55618e12			−0.505712
$236$		−	1.36461e12i		−	0.121337i
$237$	−1.47971e12			−0.128547
$238$	4.10468e12			0.348423
$239$		−	1.50484e13i		−	1.24825i	−0.781325	−	0.624125i	$-0.785456\pi$
$239$		−	1.50484e13i		−	1.24825i	0.781325	−	0.624125i	$-0.214544\pi$
$240$			1.77364e12i			0.143781i
$241$		−	1.40977e13i		−	1.11700i	−0.829504	−	0.558501i	$-0.811377\pi$
$241$		−	1.40977e13i		−	1.11700i	0.829504	−	0.558501i	$-0.188623\pi$
$242$			3.16261e12i			0.244941i
$243$	−1.14653e13			−0.868064
$244$	9.95255e11			0.0736699
$245$			9.03283e12i			0.653748i
$246$	−4.87874e12			−0.345274
$247$	2.71788e11	−	2.25951e13i	0.0188104	−	1.56380i
$248$	−2.71074e12			−0.183486
$249$			5.39986e12i			0.357508i
$250$	−1.36360e13			−0.883113
$251$	−7.86696e12			−0.498427			−0.249213	−	0.968449i	$-0.580172\pi$
$251$	−7.86696e12			−0.498427			−0.249213	+	0.968449i	$0.580172\pi$
$252$		−	1.71565e12i		−	0.106347i
$253$		−	6.67552e12i		−	0.404876i
$254$			2.17173e13i			1.28891i
$255$			5.16665e12i			0.300081i
$256$	−1.53247e13			−0.871109
$257$	1.15670e13			0.643560			0.321780	−	0.946814i	$-0.395719\pi$
$257$	1.15670e13			0.643560			0.321780	+	0.946814i	$0.395719\pi$
$258$			1.31453e12i			0.0715917i
$259$	−8.79247e12			−0.468773
$260$	5.73794e10	−	4.77024e12i	0.00299504	−	0.248993i
$261$	−5.01104e12			−0.256097
$262$		−	4.00697e12i		−	0.200520i
$263$	1.58415e13			0.776319			0.388160	−	0.921592i	$-0.373111\pi$
$263$	1.58415e13			0.776319			0.388160	+	0.921592i	$0.373111\pi$
$264$	−6.46407e12			−0.310230
$265$			1.64969e13i			0.775445i
$266$		−	1.01188e13i		−	0.465885i
$267$		−	1.42029e13i		−	0.640570i
$268$			9.08710e12i			0.401498i
$269$	2.84481e13			1.23145			0.615723	−	0.787963i	$-0.288864\pi$
$269$	2.84481e13			1.23145			0.615723	+	0.787963i	$0.288864\pi$
$270$	−9.32620e12			−0.395552
$271$			4.04902e13i			1.68275i	0.540453	+	0.841374i	$0.318253\pi$
$271$			4.04902e13i			1.68275i	−0.540453	+	0.841374i	$0.681747\pi$
$272$	1.60944e13			0.655461
$273$	−3.73491e10	+	3.10502e12i	−0.00149068	+	0.123928i
$274$	1.83118e13			0.716312
$275$		−	9.43217e12i		−	0.361644i
$276$	−1.44766e12			−0.0544086
$277$	2.83125e13			1.04313			0.521565	−	0.853211i	$-0.325348\pi$
$277$	2.83125e13			1.04313			0.521565	+	0.853211i	$0.325348\pi$
$278$		−	1.79845e13i		−	0.649608i
$279$		−	4.20764e12i		−	0.149010i
$280$		−	8.60161e12i		−	0.298683i
$281$		−	3.39651e12i		−	0.115651i	−0.998327	−	0.0578254i	$-0.981583\pi$
$281$		−	3.39651e12i		−	0.115651i	0.998327	−	0.0578254i	$-0.0184167\pi$
$282$	5.59832e12			0.186934
$283$	−4.66025e12			−0.152610			−0.0763052	−	0.997085i	$-0.524312\pi$
$283$	−4.66025e12			−0.152610			−0.0763052	+	0.997085i	$0.524312\pi$
$284$		−	1.64693e12i		−	0.0528962i
$285$	1.27367e13			0.401246
$286$	−2.21635e13	−	2.66597e11i	−0.684898	−	0.00823837i
$287$	1.48845e13			0.451217
$288$		−	1.87308e13i		−	0.557056i
$289$	1.26116e13			0.367988
$290$	−6.23954e12			−0.178634
$291$			6.28324e12i			0.176511i
$292$			1.88736e13i			0.520293i
$293$		−	2.66585e12i		−	0.0721214i	−0.999350	−	0.0360607i	$-0.988519\pi$
$293$		−	2.66585e12i		−	0.0721214i	0.999350	−	0.0360607i	$-0.0114810\pi$
$294$		−	9.10133e12i		−	0.241654i
$295$	1.06197e13			0.276751
$296$	−5.48017e13			−1.40181
$297$		−	2.13825e13i		−	0.536906i
$298$	−1.20825e13			−0.297830
$299$	−1.99861e13	−	2.40405e11i	−0.483655	−	0.00581770i
$300$	−2.04547e12			−0.0485989
$301$		−	4.01050e12i		−	0.0935585i
$302$	3.62070e13			0.829382
$303$	9.27028e12			0.208525
$304$		−	3.96756e13i		−	0.876434i
$305$			7.74527e12i			0.168030i
$306$			3.97113e13i			0.846147i
$307$			3.69671e13i			0.773667i	0.922150	+	0.386833i	$0.126431\pi$
$307$			3.69671e13i			0.773667i	−0.922150	+	0.386833i	$0.873569\pi$
$308$	4.89787e12			0.100688
$309$	8.67176e11			0.0175120
$310$		−	5.23918e12i		−	0.103938i
$311$	−7.88387e13			−1.53659			−0.768294	−	0.640097i	$-0.778894\pi$
$311$	−7.88387e13			−1.53659			−0.768294	+	0.640097i	$0.778894\pi$
$312$	−2.32789e11	+	1.93530e13i	−0.00445772	+	0.370594i
$313$	−7.62878e13			−1.43536			−0.717681	−	0.696372i	$-0.754796\pi$
$313$	−7.62878e13			−1.43536			−0.717681	+	0.696372i	$0.754796\pi$
$314$			6.74957e13i			1.24785i
$315$	1.33515e13			0.242562
$316$	6.98813e12			0.124762
$317$			6.93621e13i			1.21702i	0.793548	+	0.608508i	$0.208232\pi$
$317$			6.93621e13i			1.21702i	−0.793548	+	0.608508i	$0.791768\pi$
$318$		−	1.66220e13i		−	0.286639i
$319$		−	1.43056e13i		−	0.242470i
$320$		−	4.86735e13i		−	0.810900i
$321$	2.62303e13			0.429561
$322$	−8.95035e12			−0.144090
$323$		−	1.15576e14i		−	1.82917i
$324$	1.41371e13			0.219970
$325$	−2.82393e13	−	3.39679e11i	−0.432012	−	0.00519650i
$326$	−8.29379e13			−1.24755
$327$			3.13621e13i			0.463866i
$328$	9.27724e13			1.34931
$329$	−1.70799e13			−0.244291
$330$		−	1.24934e13i		−	0.175733i
$331$			2.88136e13i			0.398606i	0.979938	+	0.199303i	$0.0638677\pi$
$331$			2.88136e13i			0.398606i	−0.979938	+	0.199303i	$0.936132\pi$
$332$		−	2.55016e13i		−	0.346983i
$333$		−	8.50639e13i		−	1.13842i
$334$	1.09072e14			1.43585
$335$	−7.07176e13			−0.915759
$336$			5.45222e12i			0.0694556i
$337$	5.86242e13			0.734704			0.367352	−	0.930082i	$-0.380264\pi$
$337$	5.86242e13			0.734704			0.367352	+	0.930082i	$0.380264\pi$
$338$	−1.59635e12	+	6.63466e13i	−0.0196827	+	0.818045i
$339$	2.94456e13			0.357209
$340$		−	2.44002e13i		−	0.291246i
$341$	1.20121e13			0.141081
$342$	9.78952e13			1.13140
$343$			5.97769e13i			0.679854i
$344$		−	2.49967e13i		−	0.279776i
$345$		−	1.12660e13i		−	0.124098i
$346$			2.52757e13i			0.274022i
$347$	1.02722e14			1.09610			0.548052	−	0.836444i	$-0.315369\pi$
$347$	1.02722e14			1.09610			0.548052	+	0.836444i	$0.315369\pi$
$348$	−3.10234e12			−0.0325839
$349$			1.14493e13i			0.118369i	0.998247	+	0.0591845i	$0.0188500\pi$
$349$			1.14493e13i			0.118369i	−0.998247	+	0.0591845i	$0.981150\pi$
$350$	−1.26464e13			−0.128704
$351$	−6.40179e13	−	7.70046e11i	−0.641375	−	0.00771485i
$352$	5.34732e13			0.527414
$353$		−	1.33798e14i		−	1.29924i	−0.760258	−	0.649621i	$-0.774928\pi$
$353$		−	1.33798e14i		−	1.29924i	0.760258	−	0.649621i	$-0.225072\pi$
$354$	−1.07002e13			−0.102299
$355$	1.28167e13			0.120648
$356$			6.70753e13i			0.621710i
$357$			1.58825e13i			0.144958i
$358$			1.12846e14i			1.01422i
$359$			1.72020e13i			0.152251i	0.997098	+	0.0761253i	$0.0242549\pi$
$359$			1.72020e13i			0.152251i	−0.997098	+	0.0761253i	$0.975745\pi$
$360$	8.32173e13			0.725354
$361$	−1.68425e14			−1.44583
$362$			1.02446e13i			0.0866158i
$363$	−1.22373e13			−0.101906
$364$	1.76386e11	−	1.46639e13i	0.00144680	−	0.120280i
$365$	−1.46878e14			−1.18671
$366$		−	7.80400e12i		−	0.0621114i
$367$	1.09729e14			0.860319			0.430159	−	0.902753i	$-0.358457\pi$
$367$	1.09729e14			0.860319			0.430159	+	0.902753i	$0.358457\pi$
$368$	−3.50943e13			−0.271065
$369$			1.44002e14i			1.09578i
$370$		−	1.05918e14i		−	0.794072i
$371$			5.07122e13i			0.374590i
$372$		−	2.60495e12i		−	0.0189589i
$373$	1.05991e14			0.760100			0.380050	−	0.924966i	$-0.375907\pi$
$373$	1.05991e14			0.760100			0.380050	+	0.924966i	$0.375907\pi$
$374$	−1.13369e14			−0.801122
$375$		−	5.27625e13i		−	0.367412i
$376$	−1.06456e14			−0.730526
$377$	−4.28301e13	−	5.15186e11i	−0.289649	−	0.00348407i
$378$	−2.86691e13			−0.191077
$379$		−	1.76509e14i		−	1.15944i	−0.814814	−	0.579722i	$-0.803161\pi$
$379$		−	1.76509e14i		−	1.15944i	0.814814	−	0.579722i	$-0.196839\pi$
$380$	−6.01508e13			−0.389432
$381$	−8.40321e13			−0.536238
$382$		−	2.58513e14i		−	1.62605i
$383$			1.13316e14i			0.702583i	0.936266	+	0.351292i	$0.114257\pi$
$383$			1.13316e14i			0.702583i	−0.936266	+	0.351292i	$0.885743\pi$
$384$			1.39466e13i			0.0852405i
$385$			3.81162e13i			0.229655i
$386$	−8.39314e13			−0.498533
$387$	3.88001e13			0.227208
$388$		−	2.96734e13i		−	0.171314i
$389$	9.60873e13			0.546944			0.273472	−	0.961880i	$-0.411828\pi$
$389$	9.60873e13			0.546944			0.273472	+	0.961880i	$0.411828\pi$
$390$	−3.74045e13	−	4.49924e11i	−0.209927	−	0.00252513i
$391$	−1.02231e14			−0.565729
$392$			1.73068e14i			0.944371i
$393$	1.55044e13			0.0834248
$394$	1.33829e14			0.710103
$395$			5.43830e13i			0.284564i
$396$			4.73850e13i			0.244522i
$397$		−	1.05375e14i		−	0.536276i	−0.963381	−	0.268138i	$-0.913592\pi$
$397$		−	1.05375e14i		−	0.536276i	0.963381	−	0.268138i	$-0.0864083\pi$
$398$		−	2.79595e14i		−	1.40337i
$399$	3.91531e13			0.193827
$400$	−4.95864e13			−0.242121
$401$			1.95797e14i			0.942999i	0.881866	+	0.471499i	$0.156287\pi$
$401$			1.95797e14i			0.942999i	−0.881866	+	0.471499i	$0.843713\pi$
$402$	7.12539e13			0.338505
$403$	4.32589e11	−	3.59634e13i	0.00202721	−	0.168532i
$404$	−4.37802e13			−0.202386
$405$			1.10018e14i			0.501719i
$406$	−1.91806e13			−0.0862915
$407$	2.42842e14			1.07784
$408$			9.89923e13i			0.433481i
$409$			1.68257e14i			0.726936i	0.931607	+	0.363468i	$0.118407\pi$
$409$			1.68257e14i			0.726936i	−0.931607	+	0.363468i	$0.881593\pi$
$410$			1.79306e14i			0.764334i
$411$			7.08547e13i			0.298015i
$412$	−4.09536e12			−0.0169964
$413$	3.26452e13			0.133688
$414$		−	8.65912e13i		−	0.349923i
$415$	1.98458e14			0.791416
$416$	1.92572e12	−	1.60095e14i	0.00757845	−	0.630036i
$417$	6.95883e13			0.270264
$418$			2.79473e14i			1.07120i
$419$	3.97425e14			1.50341			0.751706	−	0.659498i	$-0.229231\pi$
$419$	3.97425e14			1.50341			0.751706	+	0.659498i	$0.229231\pi$
$420$	8.26592e12			0.0308617
$421$		−	4.38503e14i		−	1.61593i	−0.589233	−	0.807963i	$-0.700570\pi$
$421$		−	4.38503e14i		−	1.61593i	0.589233	−	0.807963i	$-0.299430\pi$
$422$		−	2.91848e14i		−	1.06154i
$423$		−	1.65242e14i		−	0.593264i
$424$			3.16079e14i			1.12017i
$425$	−1.44447e14			−0.505322
$426$	−1.29139e13			−0.0445970
$427$			2.38092e13i			0.0811694i
$428$	−1.23876e14			−0.416914
$429$	1.03156e12	−	8.57586e13i	0.00342750	−	0.284946i
$430$	4.83123e13			0.158482
$431$		−	1.83108e14i		−	0.593037i	−0.955027	−	0.296519i	$-0.904174\pi$
$431$		−	1.83108e14i		−	0.593037i	0.955027	−	0.296519i	$-0.0958257\pi$
$432$	−1.12411e14			−0.359459
$433$	−1.30715e14			−0.412706			−0.206353	−	0.978478i	$-0.566159\pi$
$433$	−1.30715e14			−0.412706			−0.206353	+	0.978478i	$0.566159\pi$
$434$		−	1.61054e13i		−	0.0502087i
$435$		−	2.41430e13i		−	0.0743190i
$436$		−	1.48112e14i		−	0.450209i
$437$			2.52016e14i			0.756451i
$438$	1.47991e14			0.438661
$439$	3.54362e14			1.03727			0.518636	−	0.854995i	$-0.326440\pi$
$439$	3.54362e14			1.03727			0.518636	+	0.854995i	$0.326440\pi$
$440$			2.37570e14i			0.686756i
$441$	−2.68638e14			−0.766928
$442$	−4.08273e12	+	3.39418e14i	−0.0115114	+	0.957001i
$443$	3.54802e14			0.988021			0.494010	−	0.869456i	$-0.335530\pi$
$443$	3.54802e14			0.988021			0.494010	+	0.869456i	$0.335530\pi$
$444$		−	5.26630e13i		−	0.144844i
$445$	−5.21993e14			−1.41803
$446$	−3.88428e14			−1.04224
$447$		−	4.67517e13i		−	0.123910i
$448$		−	1.49624e14i		−	0.391717i
$449$		−	6.30849e14i		−	1.63144i	−0.578449	−	0.815718i	$-0.696342\pi$
$449$		−	6.30849e14i		−	1.63144i	0.578449	−	0.815718i	$-0.303658\pi$
$450$		−	1.22349e14i		−	0.312559i
$451$	−4.11101e14			−1.03747
$452$	−1.39061e14			−0.346692
$453$			1.40098e14i			0.345057i
$454$	−4.26929e14			−1.03884
$455$	1.14117e14	+	1.37267e12i	0.274340	+	0.00329993i
$456$	2.44033e14			0.579619
$457$			2.37509e14i			0.557366i	0.960383	+	0.278683i	$0.0898979\pi$
$457$			2.37509e14i			0.557366i	−0.960383	+	0.278683i	$0.910102\pi$
$458$	−4.72461e13			−0.109548
$459$	−3.27457e14			−0.750213
$460$			5.32052e13i			0.120444i
$461$			2.34445e14i			0.524428i	0.965010	+	0.262214i	$0.0844527\pi$
$461$			2.34445e14i			0.524428i	−0.965010	+	0.262214i	$0.915547\pi$
$462$		−	3.84052e13i		−	0.0848906i
$463$			5.27378e14i			1.15193i	0.817474	+	0.575966i	$0.195374\pi$
$463$			5.27378e14i			1.15193i	−0.817474	+	0.575966i	$0.804626\pi$
$464$	−7.52069e13			−0.162334
$465$	2.02723e13			0.0432425
$466$			2.69450e14i			0.568011i
$467$	5.77169e14			1.20243			0.601215	−	0.799087i	$-0.294684\pi$
$467$	5.77169e14			1.20243			0.601215	+	0.799087i	$0.294684\pi$
$468$	1.41868e14	+	1.70647e12i	0.292100	+	0.00351355i
$469$	−2.17388e14			−0.442370
$470$		−	2.05752e14i		−	0.413815i
$471$	−2.61165e14			−0.519158
$472$	2.03471e14			0.399780
$473$			1.10767e14i			0.215117i
$474$		−	5.47954e13i		−	0.105187i
$475$			3.56086e14i			0.675679i
$476$		−	7.50072e13i		−	0.140690i
$477$	−4.90621e14			−0.909694
$478$	5.57259e14			1.02142
$479$		−	9.38721e13i		−	0.170095i	−0.996377	−	0.0850474i	$-0.972896\pi$
$479$		−	9.38721e13i		−	0.170095i	0.996377	−	0.0850474i	$-0.0271042\pi$
$480$	9.02445e13			0.161657
$481$	8.74544e12	−	7.27053e14i	0.0154876	−	1.28756i
$482$	5.22054e14			0.914022
$483$		−	3.46321e13i		−	0.0599472i
$484$	5.77922e13			0.0989053
$485$	2.30925e14			0.390742
$486$		−	4.24574e14i		−	0.710321i
$487$		−	5.12071e14i		−	0.847073i	−0.905879	−	0.423536i	$-0.860789\pi$
$487$		−	5.12071e14i		−	0.847073i	0.905879	−	0.423536i	$-0.139211\pi$
$488$			1.48398e14i			0.242728i
$489$		−	3.20916e14i		−	0.519031i
$490$	−3.34497e14			−0.534950
$491$	1.08104e15			1.70959			0.854795	−	0.518967i	$-0.173683\pi$
$491$	1.08104e15			1.70959			0.854795	+	0.518967i	$0.173683\pi$
$492$			8.91519e13i			0.139419i
$493$	−2.19080e14			−0.338801
$494$	8.36725e14	+	1.00646e13i	1.27963	+	0.0153922i
$495$	−3.68760e14			−0.557718
$496$		−	6.31493e13i		−	0.0944538i
$497$	3.93991e13			0.0582809
$498$	−1.99963e14			−0.292542
$499$			6.34438e14i			0.917986i	0.888440	+	0.458993i	$0.151790\pi$
$499$			6.34438e14i			0.917986i	−0.888440	+	0.458993i	$0.848210\pi$
$500$			2.49178e14i			0.356594i
$501$			4.22038e14i			0.597371i
$502$		−	2.91323e14i		−	0.407853i
$503$	−3.95519e14			−0.547701			−0.273850	−	0.961772i	$-0.588297\pi$
$503$	−3.95519e14			−0.547701			−0.273850	+	0.961772i	$0.588297\pi$
$504$	2.55813e14			0.350392
$505$		−	3.40706e14i		−	0.461612i
$506$	2.47202e14			0.331303
$507$	−2.56719e14	−	6.17683e12i	−0.340340	−	0.00818883i
$508$	3.96853e14			0.520450
$509$			5.60027e14i			0.726542i	0.931683	+	0.363271i	$0.118340\pi$
$509$			5.60027e14i			0.726542i	−0.931683	+	0.363271i	$0.881660\pi$
$510$	−1.91327e14			−0.245550
$511$	−4.51507e14			−0.573258
$512$		−	7.66831e14i		−	0.963196i
$513$			8.07239e14i			1.00313i
$514$			4.28340e14i			0.526613i
$515$		−	3.18709e13i		−	0.0387663i
$516$	2.40212e13			0.0289082
$517$	4.71735e14			0.561695
$518$		−	3.25596e14i		−	0.383588i
$519$	−9.78007e13			−0.114004
$520$	7.11270e14	+	8.55559e12i	0.820382	+	0.00986806i
$521$	1.68889e14			0.192750			0.0963750	−	0.995345i	$-0.469275\pi$
$521$	1.68889e14			0.192750			0.0963750	+	0.995345i	$0.469275\pi$
$522$		−	1.85565e14i		−	0.209560i
$523$	−1.18986e15			−1.32965			−0.664825	−	0.746999i	$-0.731494\pi$
$523$	−1.18986e15			−1.32965			−0.664825	+	0.746999i	$0.731494\pi$
$524$	−7.32215e13			−0.0809686
$525$		−	4.89333e13i		−	0.0535462i
$526$			5.86630e14i			0.635247i
$527$		−	1.83956e14i		−	0.197131i
$528$		−	1.50587e14i		−	0.159698i
$529$	−7.29894e14			−0.766044
$530$	−6.10901e14			−0.634532
$531$			3.15830e14i			0.324663i
$532$	−1.84906e14			−0.188121
$533$	−1.48049e13	+	1.23081e15i	−0.0149076	+	1.23934i
$534$	5.25952e14			0.524166
$535$		−	9.64028e14i		−	0.950920i
$536$	−1.35494e15			−1.32286
$537$	−4.36642e14			−0.421956
$538$			1.05347e15i			1.00767i
$539$		−	7.66913e14i		−	0.726119i
$540$			1.70423e14i			0.159721i
$541$		−	1.55433e15i		−	1.44197i	−0.692948	−	0.720987i	$-0.743689\pi$
$541$		−	1.55433e15i		−	1.44197i	0.692948	−	0.720987i	$-0.256311\pi$
$542$	−1.49940e15			−1.37696
$543$	−3.96399e13			−0.0360358
$544$		−	8.18902e14i		−	0.736950i
$545$	1.15264e15			1.02686
$546$	−1.14983e14	−	1.38308e12i	−0.101408	−	0.00121980i
$547$	1.40215e14			0.122424			0.0612118	−	0.998125i	$-0.480504\pi$
$547$	1.40215e14			0.122424			0.0612118	+	0.998125i	$0.480504\pi$
$548$		−	3.34621e14i		−	0.289241i
$549$	−2.30345e14			−0.197120
$550$	3.49284e14			0.295927
$551$			5.40070e14i			0.453019i
$552$		−	2.15855e14i		−	0.179265i
$553$			1.67175e14i			0.137462i
$554$			1.04844e15i			0.853574i
$555$	4.09834e14			0.330367
$556$	−3.28640e14			−0.262307
$557$		−	6.89987e14i		−	0.545303i	−0.962113	−	0.272651i	$-0.912099\pi$
$557$		−	6.89987e14i		−	0.545303i	0.962113	−	0.272651i	$-0.0879005\pi$
$558$	1.55814e14			0.121932
$559$	3.31630e14	+	3.98905e12i	0.256974	+	0.00309104i
$560$	2.00383e14			0.153754
$561$		−	4.38663e14i		−	0.333300i
$562$	1.25777e14			0.0946348
$563$	−7.54180e14			−0.561926			−0.280963	−	0.959719i	$-0.590654\pi$
$563$	−7.54180e14			−0.561926			−0.280963	+	0.959719i	$0.590654\pi$
$564$		−	1.02301e14i		−	0.0754824i
$565$		−	1.08220e15i		−	0.790753i
$566$		−	1.72575e14i		−	0.124878i
$567$			3.38199e14i			0.242362i
$568$	2.45567e14			0.174282
$569$	2.05848e15			1.44687			0.723434	−	0.690393i	$-0.242563\pi$
$569$	2.05848e15			1.44687			0.723434	+	0.690393i	$0.242563\pi$
$570$			4.71655e14i			0.328332i
$571$	1.50848e15			1.04002			0.520009	−	0.854161i	$-0.325928\pi$
$571$	1.50848e15			1.04002			0.520009	+	0.854161i	$0.325928\pi$
$572$	−4.87167e12	+	4.05007e14i	−0.00332659	+	0.276557i
$573$	1.00028e15			0.676505
$574$			5.51192e14i			0.369222i
$575$	3.14969e14			0.208975
$576$	1.44756e15			0.951287
$577$			2.36925e15i			1.54221i	0.636706	+	0.771107i	$0.280297\pi$
$577$			2.36925e15i			1.54221i	−0.636706	+	0.771107i	$0.719703\pi$
$578$			4.67024e14i			0.301118i
$579$		−	3.24760e14i		−	0.207411i
$580$			1.14019e14i			0.0721309i
$581$	6.10068e14			0.382305
$582$	−2.32676e14			−0.144436
$583$		−	1.40064e15i		−	0.861288i
$584$	−2.81415e15			−1.71426
$585$	−1.32801e13	+	1.10404e15i	−0.00801390	+	0.666236i
$586$	9.87197e13			0.0590156
$587$			1.64925e15i			0.976738i	0.872637	+	0.488369i	$0.162408\pi$
$587$			1.64925e15i			0.976738i	−0.872637	+	0.488369i	$0.837592\pi$
$588$	−1.66314e14			−0.0975782
$589$	−4.53483e14			−0.263589
$590$			3.93258e14i			0.226460i
$591$			5.17832e14i			0.295432i
$592$		−	1.27666e15i		−	0.721615i
$593$		−	2.18612e15i		−	1.22426i	−0.790759	−	0.612128i	$-0.790314\pi$
$593$		−	2.18612e15i		−	1.22426i	0.790759	−	0.612128i	$-0.209686\pi$
$594$	7.91820e14			0.439340
$595$	5.83720e14			0.320894
$596$			2.20791e14i			0.120262i
$597$	1.08185e15			0.583860
$598$	8.90247e12	−	7.40108e14i	0.00476051	−	0.395766i
$599$	−3.27562e15			−1.73559			−0.867794	−	0.496924i	$-0.834463\pi$
$599$	−3.27562e15			−1.73559			−0.867794	+	0.496924i	$0.834463\pi$
$600$		−	3.04992e14i		−	0.160124i
$601$	3.53256e15			1.83772			0.918861	−	0.394583i	$-0.129111\pi$
$601$	3.53256e15			1.83772			0.918861	+	0.394583i	$0.129111\pi$
$602$	1.48514e14			0.0765572
$603$		−	2.10315e15i		−	1.07430i
$604$		−	6.61631e14i		−	0.334898i
$605$			4.49750e14i			0.225588i
$606$			3.43290e14i			0.170632i
$607$	−2.05879e15			−1.01408			−0.507041	−	0.861922i	$-0.669261\pi$
$607$	−2.05879e15			−1.01408			−0.507041	+	0.861922i	$0.669261\pi$
$608$	−2.01874e15			−0.985394
$609$		−	7.42164e13i		−	0.0359009i
$610$	−2.86816e14			−0.137496
$611$	1.69885e13	−	1.41235e15i	0.00807104	−	0.670987i
$612$	7.25666e14			0.341668
$613$			8.47376e14i			0.395406i	0.980262	+	0.197703i	$0.0633482\pi$
$613$			8.47376e14i			0.395406i	−0.980262	+	0.197703i	$0.936652\pi$
$614$	−1.36893e15			−0.633077
$615$	−6.93797e14			−0.317994
$616$			7.30300e14i			0.331747i
$617$			2.17604e15i			0.979714i	0.871803	+	0.489857i	$0.162951\pi$
$617$			2.17604e15i			0.979714i	−0.871803	+	0.489857i	$0.837049\pi$
$618$			3.21126e13i			0.0143297i
$619$			1.76976e15i			0.782739i	0.920234	+	0.391369i	$0.127998\pi$
$619$			1.76976e15i			0.782739i	−0.920234	+	0.391369i	$0.872002\pi$
$620$	−9.57385e13			−0.0419694
$621$	7.14027e14			0.310249
$622$		−	2.91949e15i		−	1.25736i
$623$	−1.60463e15			−0.684999
$624$	−4.50847e14	−	5.42306e12i	−0.190771	−	0.00229472i
$625$	−9.09081e14			−0.381296
$626$		−	2.82503e15i		−	1.17453i
$627$	−1.08138e15			−0.445664
$628$	1.23339e15			0.503873
$629$		−	3.71894e15i		−	1.50606i
$630$			4.94422e14i			0.198484i
$631$			4.92537e15i			1.96010i	0.198759	+	0.980048i	$0.436309\pi$
$631$			4.92537e15i			1.96010i	−0.198759	+	0.980048i	$0.563691\pi$
$632$			1.04197e15i			0.411066i
$633$	1.12926e15			0.441647
$634$	−2.56856e15			−0.995861
$635$			3.08839e15i			1.18707i
$636$	−3.03744e14			−0.115743
$637$	−2.29609e15	−	2.76187e13i	−0.867404	−	0.0104337i
$638$	5.29754e14			0.198409
$639$			3.81171e14i			0.141536i
$640$	5.12571e14			0.188697
$641$	−1.22830e14			−0.0448318			−0.0224159	−	0.999749i	$-0.507136\pi$
$641$	−1.22830e14			−0.0448318			−0.0224159	+	0.999749i	$0.507136\pi$
$642$			9.71338e14i			0.351502i
$643$		−	3.16693e15i		−	1.13626i	−0.822939	−	0.568130i	$-0.807667\pi$
$643$		−	3.16693e15i		−	1.13626i	0.822939	−	0.568130i	$-0.192333\pi$
$644$			1.63555e14i			0.0581823i
$645$			1.86937e14i			0.0659353i
$646$	4.27992e15			1.49678
$647$	−2.65721e15			−0.921408			−0.460704	−	0.887554i	$-0.652403\pi$
$647$	−2.65721e15			−0.921408			−0.460704	+	0.887554i	$0.652403\pi$
$648$			2.10792e15i			0.724757i
$649$	−9.01638e14			−0.307388
$650$	1.25787e13	−	1.04573e15i	0.00425220	−	0.353507i
$651$	6.23177e13			0.0208889
$652$			1.51557e15i			0.503749i
$653$	4.35077e15			1.43398			0.716992	−	0.697082i	$-0.245519\pi$
$653$	4.35077e15			1.43398			0.716992	+	0.697082i	$0.245519\pi$
$654$	−1.16138e15			−0.379573
$655$		−	5.69824e14i		−	0.184677i
$656$			2.16122e15i			0.694589i
$657$		−	4.36816e15i		−	1.39216i
$658$		−	6.32489e14i		−	0.199899i
$659$	4.68870e15			1.46954			0.734771	−	0.678315i	$-0.237289\pi$
$659$	4.68870e15			1.46954			0.734771	+	0.678315i	$0.237289\pi$
$660$	−2.28299e14			−0.0709598
$661$			4.44733e15i			1.37085i	0.728142	+	0.685427i	$0.240384\pi$
$661$			4.44733e15i			1.37085i	−0.728142	+	0.685427i	$0.759616\pi$
$662$	−1.06700e15			−0.326171
$663$	−1.31333e15	−	1.57975e13i	−0.398152	−	0.00478921i
$664$	3.80243e15			1.14324
$665$		−	1.43897e15i		−	0.429075i
$666$	3.15002e15			0.931546
$667$	4.77708e14			0.140110
$668$		−	1.99313e15i		−	0.579783i
$669$		−	1.50297e15i		−	0.433616i
$670$		−	2.61876e15i		−	0.749348i
$671$		−	6.57595e14i		−	0.186631i
$672$	2.77415e14			0.0780906
$673$	1.30699e15			0.364913			0.182457	−	0.983214i	$-0.441595\pi$
$673$	1.30699e15			0.364913			0.182457	+	0.983214i	$0.441595\pi$
$674$			2.17092e15i			0.601195i
$675$	1.00888e15			0.277122
$676$	1.21239e15	+	2.91709e13i	0.330320	+	0.00794773i
$677$	−6.24197e15			−1.68688			−0.843439	−	0.537224i	$-0.819473\pi$
$677$	−6.24197e15			−1.68688			−0.843439	+	0.537224i	$0.819473\pi$
$678$			1.09041e15i			0.292297i
$679$	7.09870e14			0.188754
$680$	3.63821e15			0.959597
$681$		−	1.65194e15i		−	0.432200i
$682$			4.44821e14i			0.115444i
$683$		−	1.60803e15i		−	0.413980i	−0.978343	−	0.206990i	$-0.933633\pi$
$683$		−	1.60803e15i		−	0.413980i	0.978343	−	0.206990i	$-0.0663668\pi$
$684$		−	1.78889e15i		−	0.456853i
$685$	2.60409e15			0.659716
$686$	−2.21361e15			−0.556312
$687$		−	1.82812e14i		−	0.0455767i
$688$	5.82321e14			0.144021
$689$	−4.19341e15	−	5.04409e13i	−1.02887	−	0.0123759i
$690$	4.17193e14			0.101547
$691$		−	1.96543e15i		−	0.474601i	−0.971436	−	0.237301i	$-0.923737\pi$
$691$		−	1.96543e15i		−	0.474601i	0.971436	−	0.237301i	$-0.0762627\pi$
$692$	4.61877e14			0.110648
$693$	−1.13358e15			−0.269414
$694$			3.80392e15i			0.896922i
$695$		−	2.55754e15i		−	0.598283i
$696$		−	4.62576e14i		−	0.107357i
$697$			6.29570e15i			1.44965i
$698$	−4.23980e14			−0.0968591
$699$	−1.04260e15			−0.236316
$700$			2.31094e14i			0.0519697i
$701$	3.16026e15			0.705138			0.352569	−	0.935786i	$-0.385308\pi$
$701$	3.16026e15			0.705138			0.352569	+	0.935786i	$0.385308\pi$
$702$	2.85157e13	−	2.37066e15i	0.00631292	−	0.524825i
$703$	−9.16785e15			−2.01378
$704$			4.13251e15i			0.900668i
$705$	7.96128e14			0.172164
$706$	4.95471e15			1.06315
$707$		−	1.04734e15i		−	0.222988i
$708$			1.95530e14i			0.0413077i
$709$			5.15618e15i			1.08087i	0.841385	+	0.540436i	$0.181741\pi$
$709$			5.15618e15i			1.08087i	−0.841385	+	0.540436i	$0.818259\pi$
$710$			4.74619e14i			0.0987243i
$711$	−1.61736e15			−0.333829
$712$	−1.00013e16			−2.04841
$713$			4.01119e14i			0.0815232i
$714$	−5.88147e14			−0.118617
$715$	−3.15184e15	−	3.79123e13i	−0.630785	−	0.00758746i
$716$	2.06210e15			0.409533
$717$			2.15623e15i			0.424953i
$718$	−6.37010e14			−0.124584
$719$	−6.74859e14			−0.130980			−0.0654899	−	0.997853i	$-0.520861\pi$
$719$	−6.74859e14			−0.130980			−0.0654899	+	0.997853i	$0.520861\pi$
$720$			1.93863e15i			0.373392i
$721$		−	9.79722e13i		−	0.0187266i
$722$		−	6.23699e15i		−	1.18310i
$723$			2.02001e15i			0.380271i
$724$	1.87205e14			0.0349748
$725$	6.74977e14			0.125150
$726$		−	4.53161e14i		−	0.0833874i
$727$	−8.65251e14			−0.158017			−0.0790083	−	0.996874i	$-0.525175\pi$
$727$	−8.65251e14			−0.158017			−0.0790083	+	0.996874i	$0.525175\pi$
$728$	2.18647e15	+	2.63002e13i	0.396297	+	0.00476691i
$729$	−2.05804e15			−0.370214
$730$		−	5.43905e15i		−	0.971064i
$731$	1.69632e15			0.300581
$732$	−1.42607e14			−0.0250801
$733$		−	1.01376e16i		−	1.76956i	−0.466009	−	0.884780i	$-0.654309\pi$
$733$		−	1.01376e16i		−	1.76956i	0.466009	−	0.884780i	$-0.345691\pi$
$734$			4.06341e15i			0.703983i
$735$		−	1.29429e15i		−	0.222561i
$736$			1.78563e15i			0.304764i
$737$	6.00412e15			1.01713
$738$	−5.33258e15			−0.896659
$739$		−	1.18425e16i		−	1.97652i	−0.152793	−	0.988258i	$-0.548827\pi$
$739$		−	1.18425e16i		−	1.97652i	0.152793	−	0.988258i	$-0.451173\pi$
$740$	−1.93550e15			−0.320640
$741$	−3.89436e13	+	3.23759e15i	−0.00640378	+	0.532379i
$742$	−1.87793e15			−0.306520
$743$			8.20805e15i			1.32985i	0.746911	+	0.664923i	$0.231536\pi$
$743$			8.20805e15i			1.32985i	−0.746911	+	0.664923i	$0.768464\pi$
$744$	3.88413e14			0.0624659
$745$	−1.71824e15			−0.274299
$746$			3.92498e15i			0.621976i
$747$			5.90218e15i			0.928430i
$748$			2.07165e15i			0.323487i
$749$		−	2.96346e15i		−	0.459355i
$750$	1.95386e15			0.300646
$751$	8.13107e15			1.24202			0.621009	−	0.783803i	$-0.286723\pi$
$751$	8.13107e15			1.24202			0.621009	+	0.783803i	$0.286723\pi$
$752$		−	2.47999e15i		−	0.376055i
$753$	1.12723e15			0.169684
$754$	1.90780e13	−	1.58605e15i	0.00285095	−	0.237014i
$755$	5.14894e15			0.763853
$756$			5.23886e14i			0.0771555i
$757$	2.28113e15			0.333521			0.166760	−	0.985997i	$-0.446669\pi$
$757$	2.28113e15			0.333521			0.166760	+	0.985997i	$0.446669\pi$
$758$	6.53632e15			0.948752
$759$			9.56514e14i			0.137836i
$760$		−	8.96883e15i		−	1.28310i
$761$		−	6.48483e14i		−	0.0921050i	−0.998939	−	0.0460525i	$-0.985336\pi$
$761$		−	6.48483e14i		−	0.0921050i	0.998939	−	0.0460525i	$-0.0146641\pi$
$762$		−	3.11181e15i		−	0.438794i
$763$	3.54324e15			0.496039
$764$	−4.72396e15			−0.656587
$765$			5.64727e15i			0.779293i
$766$	−4.19622e15			−0.574911
$767$	−3.24706e13	+	2.69944e15i	−0.00441688	+	0.367198i
$768$	2.19583e15			0.296559
$769$		−	3.11073e15i		−	0.417126i	−0.978009	−	0.208563i	$-0.933121\pi$
$769$		−	3.11073e15i		−	0.417126i	0.978009	−	0.208563i	$-0.0668786\pi$
$770$	−1.41149e15			−0.187922
$771$	−1.65740e15			−0.219093
$772$			1.53373e15i			0.201304i
$773$			7.17757e14i			0.0935385i	0.998906	+	0.0467693i	$0.0148926\pi$
$773$			7.17757e14i			0.0935385i	−0.998906	+	0.0467693i	$0.985107\pi$
$774$			1.43681e15i			0.185920i
$775$			5.66761e14i			0.0728183i
$776$	4.42448e15			0.564446
$777$	1.25985e15			0.159588
$778$			3.55822e15i			0.447554i
$779$	1.55200e16			1.93837
$780$	−8.22171e12	+	6.83513e14i	−0.00101963	+	0.0847669i
$781$	−1.08818e15			−0.134004
$782$		−	3.78572e15i		−	0.462926i
$783$	1.53016e15			0.185800
$784$	−4.03178e15			−0.486137
$785$			9.59845e15i			1.14926i
$786$			5.74145e14i			0.0682650i
$787$			1.47799e16i			1.74506i	0.488557	+	0.872532i	$0.337524\pi$
$787$			1.47799e16i			1.74506i	−0.488557	+	0.872532i	$0.662476\pi$
$788$		−	2.44553e15i		−	0.286734i
$789$	−2.26988e15			−0.264289
$790$	−2.01387e15			−0.232853
$791$		−	3.32672e15i		−	0.381984i
$792$	−7.06538e15			−0.805651
$793$	−1.96880e15	−	2.36819e13i	−0.222945	−	0.00268172i
$794$	3.90215e15			0.438824
$795$		−	2.36379e15i		−	0.263992i
$796$	−5.10919e15			−0.566670
$797$	−1.57649e14			−0.0173648			−0.00868240	−	0.999962i	$-0.502764\pi$
$797$	−1.57649e14			−0.0173648			−0.00868240	+	0.999962i	$0.502764\pi$
$798$			1.44989e15i			0.158605i
$799$		−	7.22427e15i		−	0.784850i
$800$			2.52301e15i			0.272222i
$801$		−	1.55242e16i		−	1.66352i
$802$	−7.25058e15			−0.771638
$803$	1.24703e16			1.31808
$804$		−	1.30206e15i		−	0.136686i
$805$	−1.27281e15			−0.132705
$806$	1.33177e15	+	1.60193e13i	0.137907	+	0.00165882i
$807$	−4.07623e15			−0.419232
$808$		−	6.52787e15i		−	0.666821i
$809$	−7.98389e15			−0.810023			−0.405012	−	0.914312i	$-0.632733\pi$
$809$	−7.98389e15			−0.810023			−0.405012	+	0.914312i	$0.632733\pi$
$810$	−4.07409e15			−0.410547
$811$			6.94088e15i			0.694704i	0.937735	+	0.347352i	$0.112919\pi$
$811$			6.94088e15i			0.694704i	−0.937735	+	0.347352i	$0.887081\pi$
$812$			3.50497e14i			0.0348439i
$813$		−	5.80171e15i		−	0.572873i
$814$			8.99273e15i			0.881977i
$815$	−1.17945e16			−1.14898
$816$	−2.30612e15			−0.223144
$817$		−	4.18172e15i		−	0.401915i
$818$	−6.23077e15			−0.594838
$819$	−4.08235e13	+	3.39387e15i	−0.00387123	+	0.321835i
$820$	3.27655e15			0.308632
$821$		−	7.52996e15i		−	0.704539i	−0.935899	−	0.352270i	$-0.885410\pi$
$821$		−	7.52996e15i		−	0.704539i	0.935899	−	0.352270i	$-0.114590\pi$
$822$	−2.62383e15			−0.243860
$823$	−1.12488e16			−1.03850			−0.519251	−	0.854622i	$-0.673789\pi$
$823$	−1.12488e16			−1.03850			−0.519251	+	0.854622i	$0.673789\pi$
$824$		−	6.10641e14i		−	0.0559998i
$825$			1.35151e15i			0.123118i
$826$			1.20889e15i			0.109395i
$827$		−	1.66909e16i		−	1.50038i	−0.661224	−	0.750188i	$-0.729963\pi$
$827$		−	1.66909e16i		−	1.50038i	0.661224	−	0.750188i	$-0.270037\pi$
$828$	−1.58233e15			−0.141296
$829$	6.61191e14			0.0586512			0.0293256	−	0.999570i	$-0.490664\pi$
$829$	6.61191e14			0.0586512			0.0293256	+	0.999570i	$0.490664\pi$
$830$			7.34915e15i			0.647601i
$831$	−4.05680e15			−0.355122
$832$	1.23725e16	+	1.48824e14i	1.07592	+	0.0129418i
$833$	−1.17447e16			−1.01460
$834$			2.57694e15i			0.221152i
$835$	1.55110e16			1.32240
$836$	5.10697e15			0.432543
$837$			1.28483e15i			0.108108i
$838$			1.47171e16i			1.23021i
$839$		−	4.53131e15i		−	0.376299i	−0.982140	−	0.188149i	$-0.939751\pi$
$839$		−	4.53131e15i		−	0.376299i	0.982140	−	0.188149i	$-0.0602489\pi$
$840$			1.23250e15i			0.101683i
$841$	−1.11768e16			−0.916092
$842$	1.62383e16			1.32228
$843$			4.86675e14i			0.0393720i
$844$	−5.33310e15			−0.428644
$845$	−2.27014e14	+	9.43505e15i	−0.0181276	+	0.753411i
$846$	6.11910e15			0.485456
$847$			1.38255e15i			0.108974i
$848$	−7.36336e15			−0.576632
$849$	6.67753e14			0.0519545
$850$		−	5.34903e15i		−	0.413495i
$851$			8.10923e15i			0.622826i
$852$			2.35983e14i			0.0180079i
$853$			1.83259e16i			1.38946i	0.719270	+	0.694731i	$0.244477\pi$
$853$			1.83259e16i			1.38946i	−0.719270	+	0.694731i	$0.755523\pi$
$854$	−8.81684e14			−0.0664194
$855$	1.39215e16			1.04201
$856$		−	1.84706e16i		−	1.37365i
$857$	1.43741e15			0.106215			0.0531076	−	0.998589i	$-0.483087\pi$
$857$	1.43741e15			0.106215			0.0531076	+	0.998589i	$0.483087\pi$
$858$	3.17574e15	+	3.81998e13i	0.233166	+	0.00280466i
$859$	1.66971e16			1.21809			0.609043	−	0.793137i	$-0.291554\pi$
$859$	1.66971e16			1.21809			0.609043	+	0.793137i	$0.291554\pi$
$860$		−	8.82837e14i		−	0.0639940i
$861$	−2.13276e15			−0.153612
$862$	6.78070e15			0.485271
$863$		−	6.56479e14i		−	0.0466833i	−0.999728	−	0.0233416i	$-0.992569\pi$
$863$		−	6.56479e14i		−	0.0466833i	0.999728	−	0.0233416i	$-0.00743055\pi$
$864$			5.71960e15i			0.404148i
$865$			3.59442e15i			0.252371i
$866$		−	4.84051e15i		−	0.337709i
$867$	−1.80708e15			−0.125277
$868$	−2.94304e14			−0.0202739
$869$		−	4.61727e15i		−	0.316065i
$870$	8.94043e14			0.0608139
$871$	2.16226e14	−	1.79759e16i	0.0146153	−	1.21504i
$872$	2.20843e16			1.48335
$873$			6.86773e15i			0.458390i
$874$	−9.33246e15			−0.618989
$875$	−5.96102e15			−0.392895
$876$		−	2.70433e15i		−	0.177128i
$877$		−	1.38289e16i		−	0.900097i	−0.893004	−	0.450049i	$-0.851407\pi$
$877$		−	1.38289e16i		−	0.900097i	0.893004	−	0.450049i	$-0.148593\pi$
$878$			1.31225e16i			0.848780i
$879$			3.81981e14i			0.0245529i
$880$	−5.53443e15			−0.353524
$881$	−8.15095e15			−0.517417			−0.258709	−	0.965955i	$-0.583297\pi$
$881$	−8.15095e15			−0.517417			−0.258709	+	0.965955i	$0.583297\pi$
$882$		−	9.94797e15i		−	0.627563i
$883$	2.00627e16			1.25778			0.628891	−	0.777494i	$-0.283509\pi$
$883$	2.00627e16			1.25778			0.628891	+	0.777494i	$0.283509\pi$
$884$	6.20237e15	+	7.46059e13i	0.386430	+	0.00464821i
$885$	−1.52166e15			−0.0942168
$886$			1.31388e16i			0.808479i
$887$	−1.58308e16			−0.968104			−0.484052	−	0.875039i	$-0.660835\pi$
$887$	−1.58308e16			−0.968104			−0.484052	+	0.875039i	$0.660835\pi$
$888$	7.85236e15			0.477231
$889$			9.49382e15i			0.573431i
$890$		−	1.93300e16i		−	1.16035i
$891$		−	9.34081e15i		−	0.557259i
$892$			7.09797e15i			0.420850i
$893$	−1.78091e16			−1.04944
$894$	1.73127e15			0.101393
$895$			1.60477e16i			0.934083i
$896$	1.57566e15			0.0911526
$897$	2.86374e15	+	3.44468e13i	0.164655	+	0.00198057i
$898$	2.33611e16			1.33497
$899$			8.59597e14i			0.0488221i
$900$	−2.23575e15			−0.126209
$901$	−2.14497e16			−1.20347
$902$		−	1.52236e16i		−	0.848946i
$903$			5.74652e14i			0.0318510i
$904$		−	2.07348e16i		−	1.14228i
$905$			1.45687e15i			0.0797723i
$906$	−5.18799e15			−0.282354
$907$	3.23895e16			1.75212			0.876059	−	0.482203i	$-0.160163\pi$
$907$	3.23895e16			1.75212			0.876059	+	0.482203i	$0.160163\pi$
$908$			7.80151e15i			0.419475i
$909$	1.01326e16			0.541528
$910$	−5.08317e13	+	4.22590e15i	−0.00270027	+	0.224487i
$911$	−2.16663e16			−1.14402			−0.572009	−	0.820247i	$-0.693836\pi$
$911$	−2.16663e16			−1.14402			−0.572009	+	0.820247i	$0.693836\pi$
$912$			5.68499e15i			0.298372i
$913$	−1.68497e16			−0.879026
$914$	−8.79523e15			−0.456082
$915$		−	1.10979e15i		−	0.0572040i
$916$			8.63354e14i			0.0442348i
$917$		−	1.75166e15i		−	0.0892110i
$918$		−	1.21261e16i		−	0.613885i
$919$	−2.33762e16			−1.17635			−0.588177	−	0.808732i	$-0.700154\pi$
$919$	−2.33762e16			−1.17635			−0.588177	+	0.808732i	$0.700154\pi$
$920$	−7.93320e15			−0.396840
$921$		−	5.29689e15i		−	0.263386i
$922$	−8.68178e15			−0.429130
$923$	−3.91883e13	+	3.25793e15i	−0.00192552	+	0.160078i
$924$	−7.01800e14			−0.0342782
$925$			1.14579e16i			0.556322i
$926$	−1.95294e16			−0.942605
$927$	9.47844e14			0.0454777
$928$			3.82660e15i			0.182515i
$929$			1.37774e16i			0.653254i	0.945153	+	0.326627i	$0.105912\pi$
$929$			1.37774e16i			0.653254i	−0.945153	+	0.326627i	$0.894088\pi$
$930$			7.50706e14i			0.0353845i
$931$			2.89527e16i			1.35664i
$932$	4.92382e15			0.229358
$933$	1.12965e16			0.523114
$934$			2.13732e16i			0.983926i
$935$	−1.61220e16			−0.737826
$936$	−2.54445e14	+	2.11533e16i	−0.0115765	+	0.962411i
$937$	−5.30380e15			−0.239894			−0.119947	−	0.992780i	$-0.538272\pi$
$937$	−5.30380e15			−0.239894			−0.119947	+	0.992780i	$0.538272\pi$
$938$		−	8.05015e15i		−	0.361983i
$939$	1.09310e16			0.488653
$940$	−3.75982e15			−0.167095
$941$		−	8.71178e15i		−	0.384914i	−0.981305	−	0.192457i	$-0.938354\pi$
$941$		−	8.71178e15i		−	0.384914i	0.981305	−	0.192457i	$-0.0616457\pi$
$942$		−	9.67124e15i		−	0.424817i
$943$		−	1.37279e16i		−	0.599501i
$944$			4.74005e15i			0.205796i
$945$	−4.07698e15			−0.175980
$946$	−4.10185e15			−0.176027
$947$			3.10725e16i			1.32572i	0.748744	+	0.662860i	$0.230657\pi$
$947$			3.10725e16i			1.32572i	−0.748744	+	0.662860i	$0.769343\pi$
$948$	−1.00131e15			−0.0424738
$949$	4.49092e14	−	3.73353e16i	0.0189396	−	1.57455i
$950$	−1.31863e16			−0.552895
$951$		−	9.93867e15i		−	0.414319i
$952$	1.11840e16			0.463547
$953$	−4.00695e15			−0.165121			−0.0825605	−	0.996586i	$-0.526310\pi$
$953$	−4.00695e15			−0.165121			−0.0825605	+	0.996586i	$0.526310\pi$
$954$		−	1.81683e16i		−	0.744386i
$955$		−	3.67628e16i		−	1.49758i
$956$		−	1.01831e16i		−	0.412441i
$957$			2.04981e15i			0.0825462i
$958$	3.47619e15			0.139185
$959$	8.00506e15			0.318685
$960$			6.97427e15i			0.276062i
$961$	2.46867e16			0.971593
$962$	2.69236e16	+	3.23854e14i	1.05359	+	0.0126732i
$963$	2.86703e16			1.11555
$964$		−	9.53978e15i		−	0.369075i
$965$	−1.19357e16			−0.459145
$966$	1.28247e15			0.0490537
$967$		−	1.13777e16i		−	0.432724i	−0.976313	−	0.216362i	$-0.930581\pi$
$967$		−	1.13777e16i		−	0.432724i	0.976313	−	0.216362i	$-0.0694191\pi$
$968$			8.61714e15i			0.325873i
$969$			1.65605e16i			0.622721i
$970$			8.55141e15i			0.319737i
$971$	−8.13823e15			−0.302569			−0.151284	−	0.988490i	$-0.548341\pi$
$971$	−8.13823e15			−0.302569			−0.151284	+	0.988490i	$0.548341\pi$
$972$	−7.75849e15			−0.286822
$973$		−	7.86198e15i		−	0.289009i
$974$	1.89626e16			0.693144
$975$	4.04632e15	+	4.86716e13i	0.147074	+	0.00176909i
$976$	−3.45708e15			−0.124950
$977$		−	1.57114e16i		−	0.564671i	−0.959316	−	0.282335i	$-0.908891\pi$
$977$		−	1.57114e16i		−	0.564671i	0.959316	−	0.282335i	$-0.0911091\pi$
$978$	1.18839e16			0.424713
$979$	4.43187e16			1.57501
$980$			6.11244e15i			0.216009i
$981$			3.42795e16i			1.20464i
$982$			4.00320e16i			1.39892i
$983$		−	7.00897e15i		−	0.243562i	−0.992557	−	0.121781i	$-0.961139\pi$
$983$		−	7.00897e15i		−	0.243562i	0.992557	−	0.121781i	$-0.0388606\pi$
$984$	−1.32931e16			−0.459358
$985$	1.90316e16			0.653998
$986$		−	8.11278e15i		−	0.277234i
$987$	2.44733e15			0.0831663
$988$	1.83917e14	−	1.52899e16i	0.00621524	−	0.516705i
$989$	−3.69886e15			−0.124305
$990$		−	1.36556e16i		−	0.456370i
$991$	−1.80634e16			−0.600335			−0.300167	−	0.953887i	$-0.597043\pi$
$991$	−1.80634e16			−0.600335			−0.300167	+	0.953887i	$0.597043\pi$
$992$	−3.21310e15			−0.106197
$993$		−	4.12861e15i		−	0.135701i
$994$			1.45900e15i			0.0476902i
$995$		−	3.97607e16i		−	1.29249i
$996$			3.65404e15i			0.118126i
$997$	4.85471e16			1.56077			0.780387	−	0.625296i	$-0.215022\pi$
$997$	4.85471e16			1.56077			0.780387	+	0.625296i	$0.215022\pi$
$998$	−2.34940e16			−0.751171
$999$			2.59749e16i			0.825930i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.b.a.12.9	yes	12
3.2	odd	2		117.12.b.b.64.4		12
4.3	odd	2		208.12.f.b.129.8		12
13.5	odd	4		169.12.a.e.1.9		12
13.8	odd	4		169.12.a.e.1.4		12
13.12	even	2	inner	13.12.b.a.12.4	✓	12
39.38	odd	2		117.12.b.b.64.9		12
52.51	odd	2		208.12.f.b.129.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.b.a.12.4	✓	12	13.12	even	2	inner
13.12.b.a.12.9	yes	12	1.1	even	1	trivial
117.12.b.b.64.4		12	3.2	odd	2
117.12.b.b.64.9		12	39.38	odd	2
169.12.a.e.1.4		12	13.8	odd	4
169.12.a.e.1.9		12	13.5	odd	4
208.12.f.b.129.7		12	52.51	odd	2
208.12.f.b.129.8		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)

\(f(q)\)	\(=\)	\(q+37.0312i q^{2} -143.287 q^{3} +676.691 q^{4} +5266.14i q^{5} -5306.08i q^{6} +16188.3i q^{7} +100899. i q^{8} -156616. q^{9} +O(q^{10})\)	\(q+37.0312i q^{2} -143.287 q^{3} +676.691 q^{4} +5266.14i q^{5} -5306.08i q^{6} +16188.3i q^{7} +100899. i q^{8} -156616. q^{9} -195012. q^{10} -447110. i q^{11} -96960.9 q^{12} +(-1.33862e6 - 16101.7i) q^{13} -599473. q^{14} -754569. i q^{15} -2.35053e6 q^{16} -6.84716e6 q^{17} -5.79967e6i q^{18} +1.68794e7i q^{19} +3.56355e6i q^{20} -2.31957e6i q^{21} +1.65570e7 q^{22} +1.49304e7 q^{23} -1.44574e7i q^{24} +2.10958e7 q^{25} +(596266. - 4.95707e7i) q^{26} +4.78238e7 q^{27} +1.09545e7i q^{28} +3.19957e7 q^{29} +2.79426e7 q^{30} +2.68660e7i q^{31} +1.19597e8i q^{32} +6.40650e7i q^{33} -2.53558e8i q^{34} -8.52500e7 q^{35} -1.05981e8 q^{36} +5.43137e8i q^{37} -6.25066e8 q^{38} +(1.91806e8 + 2.30716e6i) q^{39} -5.31346e8 q^{40} -9.19462e8i q^{41} +8.58965e7 q^{42} -2.47741e8 q^{43} -3.02556e8i q^{44} -8.24762e8i q^{45} +5.52889e8i q^{46} +1.05508e9i q^{47} +3.36800e8 q^{48} +1.71526e9 q^{49} +7.81204e8i q^{50} +9.81107e8 q^{51} +(-9.05832e8 - 1.08959e7i) q^{52} +3.13264e9 q^{53} +1.77097e9i q^{54} +2.35455e9 q^{55} -1.63338e9 q^{56} -2.41860e9i q^{57} +1.18484e9i q^{58} -2.01659e9i q^{59} -5.10610e8i q^{60} +1.47077e9 q^{61} -9.94880e8 q^{62} -2.53535e9i q^{63} -9.24272e9 q^{64} +(8.47940e7 - 7.04936e9i) q^{65} -2.37240e9 q^{66} +1.34287e10i q^{67} -4.63341e9 q^{68} -2.13932e9 q^{69} -3.15691e9i q^{70} -2.43380e9i q^{71} -1.58023e10i q^{72} +2.78909e10i q^{73} -2.01130e10 q^{74} -3.02276e9 q^{75} +1.14222e10i q^{76} +7.23797e9 q^{77} +(-8.54370e7 + 7.10282e9i) q^{78} +1.03269e10 q^{79} -1.23782e10i q^{80} +2.08915e10 q^{81} +3.40488e10 q^{82} -3.76857e10i q^{83} -1.56963e9i q^{84} -3.60581e10i q^{85} -9.17413e9i q^{86} -4.58456e9 q^{87} +4.51128e10 q^{88} +9.91225e10i q^{89} +3.05419e10 q^{90} +(2.60660e8 - 2.16700e10i) q^{91} +1.01032e10 q^{92} -3.84954e9i q^{93} -3.90707e10 q^{94} -8.88896e10 q^{95} -1.71367e10i q^{96} -4.38508e10i q^{97} +6.35183e10i q^{98} +7.00246e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 488 q^{3} - 12290 q^{4} + 654644 q^{9}+O(q^{10}) \) 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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