LMFDB - Embedded newform 38.9.b.a.37.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,9,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	38.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:	$15.4803871823$
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} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.9
Root			$-61.5968i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.9.b.a.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} -74.3247i q^{3} -128.000 q^{4} -154.845 q^{5} +840.888 q^{6} +1585.07 q^{7} -1448.15i q^{8} +1036.83 q^{9} +O(q^{10})$	\(q+11.3137i q^{2} -74.3247i q^{3} -128.000 q^{4} -154.845 q^{5} +840.888 q^{6} +1585.07 q^{7} -1448.15i q^{8} +1036.83 q^{9} -1751.87i q^{10} -25549.2 q^{11} +9513.57i q^{12} +10369.3i q^{13} +17933.0i q^{14} +11508.8i q^{15} +16384.0 q^{16} -132967. q^{17} +11730.4i q^{18} +(13145.5 + 129656. i) q^{19} +19820.1 q^{20} -117810. i q^{21} -289057. i q^{22} -334889. q^{23} -107634. q^{24} -366648. q^{25} -117316. q^{26} -564707. i q^{27} -202889. q^{28} -605325. i q^{29} -130207. q^{30} +141494. i q^{31} +185364. i q^{32} +1.89894e6i q^{33} -1.50435e6i q^{34} -245440. q^{35} -132715. q^{36} +2.45048e6i q^{37} +(-1.46689e6 + 148724. i) q^{38} +770699. q^{39} +224239. i q^{40} +2.38955e6i q^{41} +1.33287e6 q^{42} +1.53889e6 q^{43} +3.27030e6 q^{44} -160548. q^{45} -3.78883e6i q^{46} +5.01339e6 q^{47} -1.21774e6i q^{48} -3.25235e6 q^{49} -4.14815e6i q^{50} +9.88272e6i q^{51} -1.32728e6i q^{52} -1.18301e7i q^{53} +6.38893e6 q^{54} +3.95616e6 q^{55} -2.29543e6i q^{56} +(9.63667e6 - 977036. i) q^{57} +6.84847e6 q^{58} +7.32369e6i q^{59} -1.47313e6i q^{60} -1.12872e7 q^{61} -1.60082e6 q^{62} +1.64346e6 q^{63} -2.09715e6 q^{64} -1.60564e6i q^{65} -2.14841e7 q^{66} +1.87052e7i q^{67} +1.70197e7 q^{68} +2.48905e7i q^{69} -2.77684e6i q^{70} -2.17638e7i q^{71} -1.50150e6i q^{72} -8.89170e6 q^{73} -2.77241e7 q^{74} +2.72510e7i q^{75} +(-1.68262e6 - 1.65960e7i) q^{76} -4.04974e7 q^{77} +8.71947e6i q^{78} -5.26495e7i q^{79} -2.53698e6 q^{80} -3.51690e7 q^{81} -2.70346e7 q^{82} +3.76587e7 q^{83} +1.50797e7i q^{84} +2.05892e7 q^{85} +1.74106e7i q^{86} -4.49906e7 q^{87} +3.69992e7i q^{88} -1.06654e8i q^{89} -1.81640e6i q^{90} +1.64362e7i q^{91} +4.28658e7 q^{92} +1.05165e7 q^{93} +5.67200e7i q^{94} +(-2.03551e6 - 2.00766e7i) q^{95} +1.37771e7 q^{96} +2.45621e7i q^{97} -3.67961e7i q^{98} -2.64903e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$			11.3137i			0.707107i
$2$			11.3137i			0.707107i
$3$		−	74.3247i		−	0.917589i	−0.888542	−	0.458795i	$-0.848281\pi$
$3$		−	74.3247i		−	0.917589i	0.888542	−	0.458795i	$-0.151719\pi$
$4$	−128.000			−0.500000
$5$	−154.845			−0.247752			−0.123876	−	0.992298i	$-0.539532\pi$
$5$	−154.845			−0.247752			−0.123876	+	0.992298i	$0.539532\pi$
$6$	840.888			0.648834
$7$	1585.07			0.660172			0.330086	−	0.943951i	$-0.392922\pi$
$7$	1585.07			0.660172			0.330086	+	0.943951i	$0.392922\pi$
$8$		−	1448.15i		−	0.353553i
$9$	1036.83			0.158030
$10$		−	1751.87i		−	0.175187i
$11$	−25549.2			−1.74505			−0.872524	−	0.488572i	$-0.837518\pi$
$11$	−25549.2			−1.74505			−0.872524	+	0.488572i	$0.837518\pi$
$12$			9513.57i			0.458795i
$13$			10369.3i			0.363060i	0.983385	+	0.181530i	$0.0581049\pi$
$13$			10369.3i			0.363060i	−0.983385	+	0.181530i	$0.941895\pi$
$14$			17933.0i			0.466812i
$15$			11508.8i			0.227334i
$16$	16384.0			0.250000
$17$	−132967.			−1.59202			−0.796008	−	0.605286i	$-0.793059\pi$
$17$	−132967.			−1.59202			−0.796008	+	0.605286i	$0.793059\pi$
$18$			11730.4i			0.111744i
$19$	13145.5	+	129656.i	0.100870	+	0.994900i
$19$	13145.5	+	129656.i	0.100870	+	0.994900i
$20$	19820.1			0.123876
$21$		−	117810.i		−	0.605767i
$22$		−	289057.i		−	1.23393i
$23$	−334889.			−1.19671			−0.598355	−	0.801231i	$-0.704179\pi$
$23$	−334889.			−1.19671			−0.598355	+	0.801231i	$0.704179\pi$
$24$	−107634.			−0.324417
$25$	−366648.			−0.938619
$26$	−117316.			−0.256722
$27$		−	564707.i		−	1.06260i
$28$	−202889.			−0.330086
$29$		−	605325.i		−	0.855848i	−0.903815	−	0.427924i	$-0.859245\pi$
$29$		−	605325.i		−	0.855848i	0.903815	−	0.427924i	$-0.140755\pi$
$30$	−130207.			−0.160750
$31$			141494.i			0.153211i	0.997061	+	0.0766055i	$0.0244082\pi$
$31$			141494.i			0.153211i	−0.997061	+	0.0766055i	$0.975592\pi$
$32$			185364.i			0.176777i
$33$			1.89894e6i			1.60124i
$34$		−	1.50435e6i		−	1.12573i
$35$	−245440.			−0.163559
$36$	−132715.			−0.0790149
$37$			2.45048e6i			1.30751i	0.756706	+	0.653755i	$0.226807\pi$
$37$			2.45048e6i			1.30751i	−0.756706	+	0.653755i	$0.773193\pi$
$38$	−1.46689e6	+	148724.i	−0.703500	+	0.0713260i
$39$	770699.			0.333140
$40$			224239.i			0.0875934i
$41$			2.38955e6i			0.845629i	0.906216	+	0.422815i	$0.138958\pi$
$41$			2.38955e6i			0.845629i	−0.906216	+	0.422815i	$0.861042\pi$
$42$	1.33287e6			0.428342
$43$	1.53889e6			0.450126			0.225063	−	0.974344i	$-0.427741\pi$
$43$	1.53889e6			0.450126			0.225063	+	0.974344i	$0.427741\pi$
$44$	3.27030e6			0.872524
$45$	−160548.			−0.0391521
$46$		−	3.78883e6i		−	0.846202i
$47$	5.01339e6			1.02740			0.513701	−	0.857970i	$-0.328274\pi$
$47$	5.01339e6			1.02740			0.513701	+	0.857970i	$0.328274\pi$
$48$		−	1.21774e6i		−	0.229397i
$49$	−3.25235e6			−0.564173
$50$		−	4.14815e6i		−	0.663704i
$51$			9.88272e6i			1.46082i
$52$		−	1.32728e6i		−	0.181530i
$53$		−	1.18301e7i		−	1.49929i	−0.661840	−	0.749645i	$-0.730224\pi$
$53$		−	1.18301e7i		−	1.49929i	0.661840	−	0.749645i	$-0.269776\pi$
$54$	6.38893e6			0.751369
$55$	3.95616e6			0.432338
$56$		−	2.29543e6i		−	0.233406i
$57$	9.63667e6	−	977036.i	0.912909	−	0.0925574i
$58$	6.84847e6			0.605176
$59$			7.32369e6i			0.604396i	0.953245	+	0.302198i	$0.0977204\pi$
$59$			7.32369e6i			0.604396i	−0.953245	+	0.302198i	$0.902280\pi$
$60$		−	1.47313e6i		−	0.113667i
$61$	−1.12872e7			−0.815203			−0.407602	−	0.913160i	$-0.633635\pi$
$61$	−1.12872e7			−0.815203			−0.407602	+	0.913160i	$0.633635\pi$
$62$	−1.60082e6			−0.108337
$63$	1.64346e6			0.104327
$64$	−2.09715e6			−0.125000
$65$		−	1.60564e6i		−	0.0899486i
$66$	−2.14841e7			−1.13225
$67$			1.87052e7i			0.928245i	0.885771	+	0.464122i	$0.153630\pi$
$67$			1.87052e7i			0.928245i	−0.885771	+	0.464122i	$0.846370\pi$
$68$	1.70197e7			0.796008
$69$			2.48905e7i			1.09809i
$70$		−	2.77684e6i		−	0.115653i
$71$		−	2.17638e7i		−	0.856450i	−0.903672	−	0.428225i	$-0.859139\pi$
$71$		−	2.17638e7i		−	0.856450i	0.903672	−	0.428225i	$-0.140861\pi$
$72$		−	1.50150e6i		−	0.0558720i
$73$	−8.89170e6			−0.313107			−0.156554	−	0.987669i	$-0.550038\pi$
$73$	−8.89170e6			−0.313107			−0.156554	+	0.987669i	$0.550038\pi$
$74$	−2.77241e7			−0.924549
$75$			2.72510e7i			0.861267i
$76$	−1.68262e6	−	1.65960e7i	−0.0504351	−	0.497450i
$77$	−4.04974e7			−1.15203
$78$			8.71947e6i			0.235565i
$79$		−	5.26495e7i		−	1.35172i	−0.737031	−	0.675859i	$-0.763773\pi$
$79$		−	5.26495e7i		−	1.35172i	0.737031	−	0.675859i	$-0.236227\pi$
$80$	−2.53698e6			−0.0619379
$81$	−3.51690e7			−0.816997
$82$	−2.70346e7			−0.597950
$83$	3.76587e7			0.793511			0.396756	−	0.917924i	$-0.370136\pi$
$83$	3.76587e7			0.793511			0.396756	+	0.917924i	$0.370136\pi$
$84$			1.50797e7i			0.302883i
$85$	2.05892e7			0.394424
$86$			1.74106e7i			0.318287i
$87$	−4.49906e7			−0.785317
$88$			3.69992e7i			0.616967i
$89$		−	1.06654e8i		−	1.69988i	−0.526883	−	0.849938i	$-0.676640\pi$
$89$		−	1.06654e8i		−	1.69988i	0.526883	−	0.849938i	$-0.323360\pi$
$90$		−	1.81640e6i		−	0.0276847i
$91$			1.64362e7i			0.239682i
$92$	4.28658e7			0.598355
$93$	1.05165e7			0.140585
$94$			5.67200e7i			0.726482i
$95$	−2.03551e6	−	2.00766e7i	−0.0249907	−	0.246488i
$96$	1.37771e7			0.162208
$97$			2.45621e7i			0.277446i	0.990331	+	0.138723i	$0.0442998\pi$
$97$			2.45621e7i			0.277446i	−0.990331	+	0.138723i	$0.955700\pi$
$98$		−	3.67961e7i		−	0.398931i
$99$	−2.64903e7			−0.275769
$100$	4.69310e7			0.469310
$101$	4.35309e7			0.418323			0.209162	−	0.977881i	$-0.432927\pi$
$101$	4.35309e7			0.418323			0.209162	+	0.977881i	$0.432927\pi$
$102$	−1.11810e8			−1.03295
$103$		−	2.50911e7i		−	0.222931i	−0.993768	−	0.111465i	$-0.964446\pi$
$103$		−	2.50911e7i		−	0.222931i	0.993768	−	0.111465i	$-0.0355544\pi$
$104$	1.50164e7			0.128361
$105$			1.82423e7i			0.150080i
$106$	1.33843e8			1.06016
$107$			2.68160e7i			0.204578i	0.994755	+	0.102289i	$0.0326167\pi$
$107$			2.68160e7i			0.204578i	−0.994755	+	0.102289i	$0.967383\pi$
$108$			7.22825e7i			0.531298i
$109$		−	1.10533e8i		−	0.783041i	−0.920169	−	0.391521i	$-0.871949\pi$
$109$		−	1.10533e8i		−	0.783041i	0.920169	−	0.391521i	$-0.128051\pi$
$110$			4.47589e7i			0.305709i
$111$	1.82132e8			1.19976
$112$	2.59698e7			0.165043
$113$		−	1.51426e8i		−	0.928723i	−0.885646	−	0.464361i	$-0.846284\pi$
$113$		−	1.51426e8i		−	0.928723i	0.885646	−	0.464361i	$-0.153716\pi$
$114$	1.10539e7	+	1.09026e8i	0.0654479	+	0.645524i
$115$	5.18557e7			0.296487
$116$			7.74816e7i			0.427924i
$117$			1.07513e7i			0.0573743i
$118$	−8.28581e7			−0.427373
$119$	−2.10762e8			−1.05100
$120$	1.66665e7			0.0803748
$121$	4.38405e8			2.04519
$122$		−	1.27700e8i		−	0.576436i
$123$	1.77602e8			0.775941
$124$		−	1.81112e7i		−	0.0766055i
$125$	1.17260e8			0.480296
$126$			1.85936e7i			0.0737702i
$127$			4.59385e8i			1.76588i	0.469482	+	0.882942i	$0.344441\pi$
$127$			4.59385e8i			1.76588i	−0.469482	+	0.882942i	$0.655559\pi$
$128$		−	2.37266e7i		−	0.0883883i
$129$		−	1.14378e8i		−	0.413031i
$130$	1.81657e7			0.0636033
$131$	−3.08184e8			−1.04646			−0.523232	−	0.852190i	$-0.675274\pi$
$131$	−3.08184e8			−1.04646			−0.523232	+	0.852190i	$0.675274\pi$
$132$		−	2.43064e8i		−	0.800618i
$133$	2.08366e7	+	2.05515e8i	0.0665916	+	0.656805i
$134$	−2.11625e8			−0.656368
$135$			8.74419e7i			0.263260i
$136$			1.92556e8i			0.562863i
$137$	−4.53360e8			−1.28695			−0.643473	−	0.765468i	$-0.722507\pi$
$137$	−4.53360e8			−1.28695			−0.643473	+	0.765468i	$0.722507\pi$
$138$	−2.81604e8			−0.776466
$139$	−3.86640e8			−1.03573			−0.517866	−	0.855462i	$-0.673274\pi$
$139$	−3.86640e8			−1.03573			−0.517866	+	0.855462i	$0.673274\pi$
$140$	3.14163e7			0.0817793
$141$		−	3.72619e8i		−	0.942732i
$142$	2.46230e8			0.605602
$143$		−	2.64929e8i		−	0.633556i
$144$	1.69875e7			0.0395075
$145$			9.37314e7i			0.212038i
$146$		−	1.00598e8i		−	0.221400i
$147$			2.41730e8i			0.517679i
$148$		−	3.13662e8i		−	0.653755i
$149$	5.42202e8			1.10006			0.550029	−	0.835145i	$-0.314617\pi$
$149$	5.42202e8			1.10006			0.550029	+	0.835145i	$0.314617\pi$
$150$	−3.08310e8			−0.609008
$151$		−	1.75171e7i		−	0.0336942i	−0.999858	−	0.0168471i	$-0.994637\pi$
$151$		−	1.75171e7i		−	0.0336942i	0.999858	−	0.0168471i	$-0.00536285\pi$
$152$	1.87762e8	−	1.90367e7i	0.351750	−	0.0356630i
$153$	−1.37864e8			−0.251586
$154$		−	4.58176e8i		−	0.814609i
$155$		−	2.19095e7i		−	0.0379583i
$156$	−9.86495e7			−0.166570
$157$	−6.98783e8			−1.15012			−0.575061	−	0.818111i	$-0.695022\pi$
$157$	−6.98783e8			−1.15012			−0.575061	+	0.818111i	$0.695022\pi$
$158$	5.95662e8			0.955809
$159$	−8.79271e8			−1.37573
$160$		−	2.87026e7i		−	0.0437967i
$161$	−5.30823e8			−0.790035
$162$		−	3.97892e8i		−	0.577704i
$163$	3.03342e8			0.429716			0.214858	−	0.976645i	$-0.431071\pi$
$163$	3.03342e8			0.429716			0.214858	+	0.976645i	$0.431071\pi$
$164$		−	3.05862e8i		−	0.422815i
$165$		−	2.94041e8i		−	0.396709i
$166$			4.26060e8i			0.561097i
$167$			1.38262e9i			1.77761i	0.458286	+	0.888805i	$0.348464\pi$
$167$			1.38262e9i			1.77761i	−0.458286	+	0.888805i	$0.651536\pi$
$168$	−1.70607e8			−0.214171
$169$	7.08207e8			0.868188
$170$			2.32940e8i			0.278900i
$171$	1.36297e7	+	1.34432e8i	0.0159405	+	0.157224i
$172$	−1.96978e8			−0.225063
$173$		−	6.31095e8i		−	0.704547i	−0.935897	−	0.352274i	$-0.885409\pi$
$173$		−	6.31095e8i		−	0.704547i	0.935897	−	0.352274i	$-0.114591\pi$
$174$		−	5.09011e8i		−	0.555303i
$175$	−5.81164e8			−0.619650
$176$	−4.18599e8			−0.436262
$177$	5.44331e8			0.554588
$178$	1.20665e9			1.20199
$179$			1.63061e9i			1.58832i	0.607711	+	0.794158i	$0.292088\pi$
$179$			1.63061e9i			1.58832i	−0.607711	+	0.794158i	$0.707912\pi$
$180$	2.05502e7			0.0195761
$181$			8.11994e8i			0.756551i	0.925693	+	0.378276i	$0.123483\pi$
$181$			8.11994e8i			0.756551i	−0.925693	+	0.378276i	$0.876517\pi$
$182$	−1.85954e8			−0.169481
$183$			8.38916e8i			0.748022i
$184$			4.84971e8i			0.423101i
$185$		−	3.79444e8i		−	0.323938i
$186$			1.18980e8i			0.0994084i
$187$	3.39720e9			2.77814
$188$	−6.41714e8			−0.513701
$189$		−	8.95101e8i		−	0.701496i
$190$	2.27141e8	−	2.30292e7i	0.174293	−	0.0176711i
$191$	−1.14327e9			−0.859041			−0.429521	−	0.903057i	$-0.641317\pi$
$191$	−1.14327e9			−0.859041			−0.429521	+	0.903057i	$0.641317\pi$
$192$			1.55870e8i			0.114699i
$193$		−	1.21153e9i		−	0.873185i	−0.899659	−	0.436592i	$-0.856185\pi$
$193$		−	1.21153e9i		−	0.873185i	0.899659	−	0.436592i	$-0.143815\pi$
$194$	−2.77889e8			−0.196184
$195$	−1.19339e8			−0.0825359
$196$	4.16300e8			0.282087
$197$	4.99652e8			0.331744			0.165872	−	0.986147i	$-0.446956\pi$
$197$	4.99652e8			0.331744			0.165872	+	0.986147i	$0.446956\pi$
$198$		−	2.99704e8i		−	0.194998i
$199$	2.04764e9			1.30570			0.652848	−	0.757489i	$-0.273574\pi$
$199$	2.04764e9			1.30570			0.652848	+	0.757489i	$0.273574\pi$
$200$			5.30963e8i			0.331852i
$201$	1.39026e9			0.851747
$202$			4.92496e8i			0.295799i
$203$		−	9.59484e8i		−	0.565007i
$204$		−	1.26499e9i		−	0.730408i
$205$		−	3.70009e8i		−	0.209506i
$206$	2.83873e8			0.157636
$207$	−3.47224e8			−0.189116
$208$			1.69891e8i			0.0907649i
$209$	−3.35857e8	−	3.31262e9i	−0.176023	−	1.73615i
$210$	−2.06388e8			−0.106122
$211$		−	6.24079e8i		−	0.314854i	−0.987531	−	0.157427i	$-0.949680\pi$
$211$		−	6.24079e8i		−	0.314854i	0.987531	−	0.157427i	$-0.0503199\pi$
$212$			1.51426e9i			0.749645i
$213$	−1.61759e9			−0.785870
$214$	−3.03389e8			−0.144659
$215$	−2.38289e8			−0.111520
$216$	−8.17783e8			−0.375684
$217$			2.24277e8i			0.101146i
$218$	1.25053e9			0.553694
$219$			6.60873e8i			0.287304i
$220$	−5.06389e8			−0.216169
$221$		−	1.37878e9i		−	0.577997i
$222$			2.06058e9i			0.848356i
$223$		−	2.29270e9i		−	0.927102i	−0.886070	−	0.463551i	$-0.846575\pi$
$223$		−	2.29270e9i		−	0.927102i	0.886070	−	0.463551i	$-0.153425\pi$
$224$			2.93815e8i			0.116703i
$225$	−3.80153e8			−0.148330
$226$	1.71319e9			0.656706
$227$			3.90434e9i			1.47043i	0.677835	+	0.735214i	$0.262918\pi$
$227$			3.90434e9i			1.47043i	−0.677835	+	0.735214i	$0.737082\pi$
$228$	−1.23349e9	+	1.25061e8i	−0.456455	+	0.0462787i
$229$	−3.02177e8			−0.109880			−0.0549401	−	0.998490i	$-0.517497\pi$
$229$	−3.02177e8			−0.109880			−0.0549401	+	0.998490i	$0.517497\pi$
$230$			5.86681e8i			0.209648i
$231$			3.00996e9i			1.05709i
$232$	−8.76604e8			−0.302588
$233$	−4.59616e9			−1.55945			−0.779726	−	0.626121i	$-0.784641\pi$
$233$	−4.59616e9			−1.55945			−0.779726	+	0.626121i	$0.784641\pi$
$234$	−1.21637e8			−0.0405697
$235$	−7.76297e8			−0.254540
$236$		−	9.37432e8i		−	0.302198i
$237$	−3.91316e9			−1.24032
$238$		−	2.38450e9i		−	0.743172i
$239$	−1.85947e9			−0.569897			−0.284949	−	0.958543i	$-0.591977\pi$
$239$	−1.85947e9			−0.569897			−0.284949	+	0.958543i	$0.591977\pi$
$240$			1.88560e8i			0.0568335i
$241$			4.35931e8i			0.129226i	0.997910	+	0.0646130i	$0.0205813\pi$
$241$			4.35931e8i			0.129226i	−0.997910	+	0.0646130i	$0.979419\pi$
$242$			4.95998e9i			1.44617i
$243$		−	1.09111e9i		−	0.312928i
$244$	1.44476e9			0.407602
$245$	5.03609e8			0.139775
$246$			2.00934e9i			0.548673i
$247$	−1.34445e9	+	1.36310e8i	−0.361208	+	0.0366219i
$248$	2.04905e8			0.0541683
$249$		−	2.79897e9i		−	0.728117i
$250$			1.32664e9i			0.339621i
$251$	1.73172e9			0.436297			0.218148	−	0.975916i	$-0.429998\pi$
$251$	1.73172e9			0.436297			0.218148	+	0.975916i	$0.429998\pi$
$252$	−2.10362e8			−0.0521634
$253$	8.55615e9			2.08832
$254$	−5.19735e9			−1.24867
$255$		−	1.53029e9i		−	0.361920i
$256$	2.68435e8			0.0625000
$257$		−	7.49270e9i		−	1.71754i	−0.512364	−	0.858768i	$-0.671230\pi$
$257$		−	7.49270e9i		−	1.71754i	0.512364	−	0.858768i	$-0.328770\pi$
$258$	1.29404e9			0.292057
$259$			3.88419e9i			0.863181i
$260$			2.05522e8i			0.0449743i
$261$		−	6.27621e8i		−	0.135249i
$262$		−	3.48670e9i		−	0.739962i
$263$	−4.66900e9			−0.975890			−0.487945	−	0.872874i	$-0.662253\pi$
$263$	−4.66900e9			−0.975890			−0.487945	+	0.872874i	$0.662253\pi$
$264$	2.74996e9			0.566123
$265$			1.83183e9i			0.371452i
$266$	−2.32513e9	+	2.35739e8i	−0.464431	+	0.0470874i
$267$	−7.92703e9			−1.55979
$268$		−	2.39426e9i		−	0.464122i
$269$		−	2.06571e9i		−	0.394512i	−0.980352	−	0.197256i	$-0.936797\pi$
$269$		−	2.06571e9i		−	0.394512i	0.980352	−	0.197256i	$-0.0632030\pi$
$270$	−9.89292e8			−0.186153
$271$	1.24856e9			0.231490			0.115745	−	0.993279i	$-0.463074\pi$
$271$	1.24856e9			0.231490			0.115745	+	0.993279i	$0.463074\pi$
$272$	−2.17853e9			−0.398004
$273$	1.22161e9			0.219929
$274$		−	5.12918e9i		−	0.910009i
$275$	9.36758e9			1.63793
$276$		−	3.18599e9i		−	0.549044i
$277$	−1.05391e8			−0.0179013			−0.00895067	−	0.999960i	$-0.502849\pi$
$277$	−1.05391e8			−0.0179013			−0.00895067	+	0.999960i	$0.502849\pi$
$278$		−	4.37433e9i		−	0.732374i
$279$			1.46705e8i			0.0242119i
$280$			3.55435e8i			0.0578267i
$281$			9.25009e9i			1.48361i	0.670614	+	0.741807i	$0.266031\pi$
$281$			9.25009e9i			1.48361i	−0.670614	+	0.741807i	$0.733969\pi$
$282$	4.21570e9			0.666612
$283$	7.95944e9			1.24090			0.620450	−	0.784246i	$-0.286950\pi$
$283$	7.95944e9			1.24090			0.620450	+	0.784246i	$0.286950\pi$
$284$			2.78577e9i			0.428225i
$285$	−1.49219e9	+	1.51289e8i	−0.226175	+	0.0229312i
$286$	2.99733e9			0.447992
$287$			3.78760e9i			0.558261i
$288$			1.92191e8i			0.0279360i
$289$	1.07044e10			1.53451
$290$	−1.06045e9			−0.149933
$291$	1.82557e9			0.254582
$292$	1.13814e9			0.156554
$293$			1.46284e9i			0.198484i	0.995063	+	0.0992420i	$0.0316418\pi$
$293$			1.46284e9i			0.198484i	−0.995063	+	0.0992420i	$0.968358\pi$
$294$	−2.73486e9			−0.366055
$295$		−	1.13403e9i		−	0.149740i
$296$	3.54868e9			0.462274
$297$			1.44278e10i			1.85428i
$298$			6.13431e9i			0.777859i
$299$		−	3.47258e9i		−	0.434477i
$300$		−	3.48813e9i		−	0.430633i
$301$	2.43926e9			0.297161
$302$	1.98184e8			0.0238254
$303$		−	3.23542e9i		−	0.383849i
$304$	2.15376e8	+	2.12429e9i	0.0252175	+	0.248725i
$305$	1.74776e9			0.201968
$306$		−	1.55976e9i		−	0.177898i
$307$			1.59981e9i			0.180101i	0.995937	+	0.0900504i	$0.0287028\pi$
$307$			1.59981e9i			0.180101i	−0.995937	+	0.0900504i	$0.971297\pi$
$308$	5.18367e9			0.576015
$309$	−1.86489e9			−0.204559
$310$	2.47878e8			0.0268405
$311$	−1.36371e10			−1.45775			−0.728873	−	0.684649i	$-0.759955\pi$
$311$	−1.36371e10			−1.45775			−0.728873	+	0.684649i	$0.759955\pi$
$312$		−	1.11609e9i		−	0.117783i
$313$	−6.84472e9			−0.713146			−0.356573	−	0.934268i	$-0.616055\pi$
$313$	−6.84472e9			−0.713146			−0.356573	+	0.934268i	$0.616055\pi$
$314$		−	7.90583e9i		−	0.813259i
$315$	−2.54481e8			−0.0258471
$316$			6.73914e9i			0.675859i
$317$			4.91136e9i			0.486368i	0.969980	+	0.243184i	$0.0781919\pi$
$317$			4.91136e9i			0.486368i	−0.969980	+	0.243184i	$0.921808\pi$
$318$		−	9.94782e9i		−	0.972790i
$319$			1.54656e10i			1.49349i
$320$	3.24733e8			0.0309689
$321$	1.99310e9			0.187719
$322$		−	6.00557e9i		−	0.558639i
$323$	−1.74791e9	−	1.72400e10i	−0.160587	−	1.58390i
$324$	4.50164e9			0.408498
$325$		−	3.80190e9i		−	0.340775i
$326$			3.43192e9i			0.303855i
$327$	−8.21531e9			−0.718510
$328$	3.46043e9			0.298975
$329$	7.94658e9			0.678261
$330$	3.32669e9			0.280516
$331$			1.69481e10i			1.41192i	0.708252	+	0.705960i	$0.249484\pi$
$331$			1.69481e10i			1.41192i	−0.708252	+	0.705960i	$0.750516\pi$
$332$	−4.82031e9			−0.396756
$333$			2.54074e9i			0.206626i
$334$	−1.56425e10			−1.25696
$335$		−	2.89640e9i		−	0.229974i
$336$		−	1.93020e9i		−	0.151442i
$337$		−	2.45355e10i		−	1.90229i	−0.308750	−	0.951143i	$-0.599911\pi$
$337$		−	2.45355e10i		−	1.90229i	0.308750	−	0.951143i	$-0.400089\pi$
$338$			8.01245e9i			0.613901i
$339$	−1.12547e10			−0.852186
$340$	−2.63542e9			−0.197212
$341$		−	3.61505e9i		−	0.267360i
$342$	−1.52092e9	+	1.54202e8i	−0.111174	+	0.0112716i
$343$	−1.42928e10			−1.03262
$344$		−	2.22855e9i		−	0.159144i
$345$		−	3.85416e9i		−	0.272053i
$346$	7.14002e9			0.498190
$347$	1.97537e10			1.36248			0.681239	−	0.732061i	$-0.261441\pi$
$347$	1.97537e10			1.36248			0.681239	+	0.732061i	$0.261441\pi$
$348$	5.75880e9			0.392658
$349$	−7.25657e9			−0.489136			−0.244568	−	0.969632i	$-0.578646\pi$
$349$	−7.25657e9			−0.489136			−0.244568	+	0.969632i	$0.578646\pi$
$350$		−	6.57512e9i		−	0.438159i
$351$	5.85564e9			0.385786
$352$		−	4.73590e9i		−	0.308484i
$353$	−1.86135e10			−1.19875			−0.599375	−	0.800468i	$-0.704584\pi$
$353$	−1.86135e10			−1.19875			−0.599375	+	0.800468i	$0.704584\pi$
$354$			6.15841e9i			0.392153i
$355$			3.37002e9i			0.212187i
$356$			1.36517e10i			0.849938i
$357$			1.56648e10i			0.964390i
$358$	−1.84482e10			−1.12311
$359$	1.90002e10			1.14388			0.571940	−	0.820296i	$-0.306191\pi$
$359$	1.90002e10			1.14388			0.571940	+	0.820296i	$0.306191\pi$
$360$			2.32499e8i			0.0138424i
$361$	−1.66380e10	+	3.40879e9i	−0.979650	+	0.200711i
$362$	−9.18666e9			−0.534963
$363$		−	3.25843e10i		−	1.87664i
$364$		−	2.10383e9i		−	0.119841i
$365$	1.37683e9			0.0775729
$366$	−9.49125e9			−0.528931
$367$	−1.69812e10			−0.936061			−0.468030	−	0.883712i	$-0.655036\pi$
$367$	−1.69812e10			−0.936061			−0.468030	+	0.883712i	$0.655036\pi$
$368$	−5.48682e9			−0.299178
$369$			2.47756e9i			0.133635i
$370$	4.29292e9			0.229058
$371$		−	1.87516e10i		−	0.989789i
$372$	−1.34611e9			−0.0702924
$373$			3.35760e10i			1.73458i	0.497805	+	0.867289i	$0.334140\pi$
$373$			3.35760e10i			1.73458i	−0.497805	+	0.867289i	$0.665860\pi$
$374$			3.84349e10i			1.96444i
$375$		−	8.71530e9i		−	0.440714i
$376$		−	7.26016e9i		−	0.363241i
$377$	6.27683e9			0.310724
$378$	1.01269e10			0.496032
$379$			3.39145e9i			0.164372i	0.996617	+	0.0821860i	$0.0261902\pi$
$379$			3.39145e9i			0.164372i	−0.996617	+	0.0821860i	$0.973810\pi$
$380$	2.60545e8	+	2.56980e9i	0.0124954	+	0.123244i
$381$	3.41437e10			1.62036
$382$		−	1.29346e10i		−	0.607434i
$383$		−	4.19332e10i		−	1.94878i	−0.224864	−	0.974390i	$-0.572194\pi$
$383$		−	4.19332e10i		−	1.94878i	0.224864	−	0.974390i	$-0.427806\pi$
$384$	−1.76347e9			−0.0811042
$385$	6.27081e9			0.285417
$386$	1.37069e10			0.617435
$387$	1.59558e9			0.0711334
$388$		−	3.14395e9i		−	0.138723i
$389$	−1.91799e10			−0.837620			−0.418810	−	0.908074i	$-0.637553\pi$
$389$	−1.91799e10			−0.837620			−0.418810	+	0.908074i	$0.637553\pi$
$390$		−	1.35016e9i		−	0.0583617i
$391$	4.45291e10			1.90518
$392$			4.70990e9i			0.199465i
$393$			2.29057e10i			0.960224i
$394$			5.65291e9i			0.234578i
$395$			8.15250e9i			0.334890i
$396$	3.39076e9			0.137885
$397$	−3.73055e10			−1.50180			−0.750899	−	0.660417i	$-0.770379\pi$
$397$	−3.73055e10			−1.50180			−0.750899	+	0.660417i	$0.770379\pi$
$398$			2.31664e10i			0.923266i
$399$	1.52748e10	−	1.54867e9i	0.602677	−	0.0611038i
$400$	−6.00716e9			−0.234655
$401$		−	1.19174e10i		−	0.460895i	−0.973085	−	0.230448i	$-0.925981\pi$
$401$		−	1.19174e10i		−	0.460895i	0.973085	−	0.230448i	$-0.0740191\pi$
$402$			1.57290e10i			0.602276i
$403$	−1.46720e9			−0.0556247
$404$	−5.57195e9			−0.209162
$405$	5.44574e9			0.202412
$406$	1.08553e10			0.399520
$407$		−	6.26080e10i		−	2.28167i
$408$	1.43117e10			0.516477
$409$		−	1.57301e10i		−	0.562133i	−0.959688	−	0.281066i	$-0.909312\pi$
$409$		−	1.57301e10i		−	0.562133i	0.959688	−	0.281066i	$-0.0906881\pi$
$410$	4.18617e9			0.148143
$411$			3.36958e10i			1.18089i
$412$			3.21166e9i			0.111465i
$413$			1.16086e10i			0.399005i
$414$		−	3.92839e9i		−	0.133725i
$415$	−5.83125e9			−0.196594
$416$	−1.92210e9			−0.0641805
$417$			2.87369e10i			0.950377i
$418$	3.74780e10	−	3.79979e9i	1.22764	−	0.124467i
$419$	3.53760e10			1.14776			0.573882	−	0.818938i	$-0.305437\pi$
$419$	3.53760e10			1.14776			0.573882	+	0.818938i	$0.305437\pi$
$420$		−	2.33501e9i		−	0.0750398i
$421$			2.22556e10i			0.708454i	0.935159	+	0.354227i	$0.115256\pi$
$421$			2.22556e10i			0.708454i	−0.935159	+	0.354227i	$0.884744\pi$
$422$	7.06065e9			0.222636
$423$	5.19805e9			0.162360
$424$	−1.71319e10			−0.530079
$425$	4.87520e10			1.49430
$426$		−	1.83010e10i		−	0.555694i
$427$	−1.78910e10			−0.538174
$428$		−	3.43245e9i		−	0.102289i
$429$	−1.96908e10			−0.581345
$430$		−	2.69594e9i		−	0.0788562i
$431$			6.16218e10i			1.78577i	0.450285	+	0.892885i	$0.351322\pi$
$431$			6.16218e10i			1.78577i	−0.450285	+	0.892885i	$0.648678\pi$
$432$		−	9.25216e9i		−	0.265649i
$433$		−	4.72412e9i		−	0.134391i	−0.997740	−	0.0671954i	$-0.978595\pi$
$433$		−	4.72412e9i		−	0.134391i	0.997740	−	0.0671954i	$-0.0214051\pi$
$434$	−2.53741e9			−0.0715207
$435$	6.96656e9			0.194563
$436$			1.41482e10i			0.391521i
$437$	−4.40228e9	−	4.34204e10i	−0.120712	−	1.19061i
$438$	−7.47693e9			−0.203155
$439$		−	4.36060e10i		−	1.17405i	−0.809568	−	0.587027i	$-0.800298\pi$
$439$		−	4.36060e10i		−	1.17405i	0.809568	−	0.587027i	$-0.199702\pi$
$440$		−	5.72914e9i		−	0.152855i
$441$	−3.37214e9			−0.0891562
$442$	1.55991e10			0.408706
$443$	−4.41307e10			−1.14585			−0.572923	−	0.819610i	$-0.694190\pi$
$443$	−4.41307e10			−1.14585			−0.572923	+	0.819610i	$0.694190\pi$
$444$	−2.33128e10			−0.599878
$445$			1.65148e10i			0.421147i
$446$	2.59389e10			0.655560
$447$		−	4.02990e10i		−	1.00940i
$448$	−3.32414e9			−0.0825215
$449$			2.32868e10i			0.572961i	0.958086	+	0.286481i	$0.0924854\pi$
$449$			2.32868e10i			0.572961i	−0.958086	+	0.286481i	$0.907515\pi$
$450$		−	4.30094e9i		−	0.104885i
$451$		−	6.10511e10i		−	1.47566i
$452$			1.93825e10i			0.464361i
$453$	−1.30196e9			−0.0309174
$454$	−4.41725e10			−1.03975
$455$		−	2.54505e9i		−	0.0593815i
$456$	−1.41490e9	−	1.39554e10i	−0.0327240	−	0.322762i
$457$	−4.60367e10			−1.05545			−0.527727	−	0.849414i	$-0.676956\pi$
$457$	−4.60367e10			−1.05545			−0.527727	+	0.849414i	$0.676956\pi$
$458$		−	3.41875e9i		−	0.0776971i
$459$			7.50873e10i			1.69167i
$460$	−6.63754e9			−0.148243
$461$	3.61112e10			0.799537			0.399769	−	0.916616i	$-0.369091\pi$
$461$	3.61112e10			0.799537			0.399769	+	0.916616i	$0.369091\pi$
$462$	−3.40538e10			−0.747476
$463$	3.50543e9			0.0762812			0.0381406	−	0.999272i	$-0.487857\pi$
$463$	3.50543e9			0.0762812			0.0381406	+	0.999272i	$0.487857\pi$
$464$		−	9.91764e9i		−	0.213962i
$465$	−1.62842e9			−0.0348301
$466$		−	5.19997e10i		−	1.10270i
$467$	4.73893e9			0.0996352			0.0498176	−	0.998758i	$-0.484136\pi$
$467$	4.73893e9			0.0996352			0.0498176	+	0.998758i	$0.484136\pi$
$468$		−	1.37616e9i		−	0.0286871i
$469$			2.96490e10i			0.612801i
$470$		−	8.78280e9i		−	0.179987i
$471$			5.19369e10i			1.05534i
$472$	1.06058e10			0.213686
$473$	−3.93175e10			−0.785492
$474$		−	4.42724e10i		−	0.877040i
$475$	−4.81977e9	−	4.75382e10i	−0.0946787	−	0.933832i
$476$	2.69775e10			0.525502
$477$		−	1.22659e10i		−	0.236933i
$478$		−	2.10375e10i		−	0.402978i
$479$	−7.54648e9			−0.143352			−0.0716758	−	0.997428i	$-0.522835\pi$
$479$	−7.54648e9			−0.143352			−0.0716758	+	0.997428i	$0.522835\pi$
$480$	−2.13331e9			−0.0401874
$481$	−2.54099e10			−0.474704
$482$	−4.93200e9			−0.0913765
$483$			3.94533e10i			0.724927i
$484$	−5.61158e10			−1.02259
$485$		−	3.80332e9i		−	0.0687378i
$486$	1.23445e10			0.221274
$487$			8.92286e10i			1.58631i	0.609020	+	0.793155i	$0.291563\pi$
$487$			8.92286e10i			1.58631i	−0.609020	+	0.793155i	$0.708437\pi$
$488$			1.63456e10i			0.288218i
$489$		−	2.25458e10i		−	0.394303i
$490$			5.69768e9i			0.0988357i
$491$	−3.26865e10			−0.562397			−0.281198	−	0.959650i	$-0.590732\pi$
$491$	−3.26865e10			−0.562397			−0.281198	+	0.959650i	$0.590732\pi$
$492$	−2.27331e10			−0.387970
$493$			8.04881e10i			1.36252i
$494$	−1.54217e9	−	1.52107e10i	−0.0258956	−	0.255413i
$495$	4.10188e9			0.0683223
$496$			2.31823e9i			0.0383027i
$497$		−	3.44973e10i		−	0.565404i
$498$	3.16668e10			0.514857
$499$	−9.51579e10			−1.53477			−0.767384	−	0.641188i	$-0.778442\pi$
$499$	−9.51579e10			−1.53477			−0.767384	+	0.641188i	$0.778442\pi$
$500$	−1.50092e10			−0.240148
$501$	1.02763e11			1.63112
$502$			1.95921e10i			0.308508i
$503$	1.90098e10			0.296966			0.148483	−	0.988915i	$-0.452561\pi$
$503$	1.90098e10			0.296966			0.148483	+	0.988915i	$0.452561\pi$
$504$		−	2.37998e9i		−	0.0368851i
$505$	−6.74053e9			−0.103640
$506$			9.68018e10i			1.47666i
$507$		−	5.26373e10i		−	0.796640i
$508$		−	5.88013e10i		−	0.882942i
$509$			1.06082e11i			1.58041i	0.612845	+	0.790203i	$0.290025\pi$
$509$			1.06082e11i			1.58041i	−0.612845	+	0.790203i	$0.709975\pi$
$510$	1.73132e10			0.255916
$511$	−1.40940e10			−0.206705
$512$			3.03700e9i			0.0441942i
$513$	7.32178e10	−	7.42335e9i	1.05718	−	0.107184i
$514$	8.47702e10			1.21448
$515$			3.88522e9i			0.0552315i
$516$			1.46404e10i			0.206516i
$517$	−1.28088e11			−1.79286
$518$	−4.39446e10			−0.610361
$519$	−4.69060e10			−0.646485
$520$	−2.32521e9			−0.0318016
$521$		−	5.55106e10i		−	0.753399i	−0.926335	−	0.376700i	$-0.877059\pi$
$521$		−	5.55106e10i		−	0.753399i	0.926335	−	0.376700i	$-0.122941\pi$
$522$	7.10072e9			0.0956358
$523$		−	1.05842e11i		−	1.41465i	−0.706888	−	0.707326i	$-0.749902\pi$
$523$		−	1.05842e11i		−	1.41465i	0.706888	−	0.707326i	$-0.250098\pi$
$524$	3.94475e10			0.523232
$525$			4.31948e10i			0.568584i
$526$		−	5.28237e10i		−	0.690059i
$527$		−	1.88139e10i		−	0.243914i
$528$			3.11122e10i			0.400309i
$529$	3.38395e10			0.432116
$530$	−2.07248e10			−0.262656
$531$			7.59345e9i			0.0955126i
$532$	−2.66708e9	−	2.63059e10i	−0.0332958	−	0.328402i
$533$	−2.47780e10			−0.307014
$534$		−	8.96841e10i		−	1.10294i
$535$		−	4.15232e9i		−	0.0506846i
$536$	2.70880e10			0.328184
$537$	1.21194e11			1.45742
$538$	2.33708e10			0.278962
$539$	8.30950e10			0.984509
$540$		−	1.11926e10i		−	0.131630i
$541$	6.69196e10			0.781204			0.390602	−	0.920560i	$-0.372267\pi$
$541$	6.69196e10			0.781204			0.390602	+	0.920560i	$0.372267\pi$
$542$			1.41259e10i			0.163688i
$543$	6.03512e10			0.694204
$544$		−	2.46472e10i		−	0.281431i
$545$			1.71154e10i			0.194000i
$546$			1.38210e10i			0.155514i
$547$		−	9.76775e9i		−	0.109105i	−0.998511	−	0.0545526i	$-0.982627\pi$
$547$		−	9.76775e9i		−	0.109105i	0.998511	−	0.0545526i	$-0.0173732\pi$
$548$	5.80300e10			0.643473
$549$	−1.17029e10			−0.128826
$550$			1.05982e11i			1.15819i
$551$	7.84842e10	−	7.95730e9i	0.851483	−	0.0863295i
$552$	3.60453e10			0.388233
$553$		−	8.34533e10i		−	0.892366i
$554$		−	1.19237e9i		−	0.0126582i
$555$	−2.82021e10			−0.297242
$556$	4.94899e10			0.517866
$557$	1.59399e10			0.165602			0.0828010	−	0.996566i	$-0.473613\pi$
$557$	1.59399e10			0.165602			0.0828010	+	0.996566i	$0.473613\pi$
$558$	−1.65978e9			−0.0171204
$559$			1.59573e10i			0.163423i
$560$	−4.02129e9			−0.0408896
$561$		−	2.52496e11i		−	2.54919i
$562$	−1.04653e11			−1.04907
$563$			1.84760e10i			0.183897i	0.995764	+	0.0919483i	$0.0293094\pi$
$563$			1.84760e10i			0.183897i	−0.995764	+	0.0919483i	$0.970691\pi$
$564$			4.76952e10i			0.471366i
$565$			2.34475e10i			0.230093i
$566$			9.00508e10i			0.877449i
$567$	−5.57455e10			−0.539358
$568$	−3.15174e10			−0.302801
$569$		−	5.39118e10i		−	0.514322i	−0.966369	−	0.257161i	$-0.917213\pi$
$569$		−	5.39118e10i		−	0.514322i	0.966369	−	0.257161i	$-0.0827870\pi$
$570$	−1.71164e9	−	1.68822e10i	−0.0162148	−	0.159930i
$571$	5.13331e9			0.0482896			0.0241448	−	0.999708i	$-0.492314\pi$
$571$	5.13331e9			0.0482896			0.0241448	+	0.999708i	$0.492314\pi$
$572$			3.39109e10i			0.316778i
$573$			8.49730e10i			0.788247i
$574$	−4.28519e10			−0.394750
$575$	1.22786e11			1.12326
$576$	−2.17440e9			−0.0197537
$577$	−4.51019e9			−0.0406903			−0.0203452	−	0.999793i	$-0.506477\pi$
$577$	−4.51019e9			−0.0406903			−0.0203452	+	0.999793i	$0.506477\pi$
$578$			1.21107e11i			1.08507i
$579$	−9.00469e10			−0.801225
$580$		−	1.19976e10i		−	0.106019i
$581$	5.96918e10			0.523854
$582$			2.06540e10i			0.180017i
$583$			3.02251e11i			2.61633i
$584$			1.28766e10i			0.110700i
$585$		−	1.66478e9i		−	0.0142146i
$586$	−1.65501e10			−0.140349
$587$	−5.32145e10			−0.448206			−0.224103	−	0.974565i	$-0.571945\pi$
$587$	−5.32145e10			−0.448206			−0.224103	+	0.974565i	$0.571945\pi$
$588$		−	3.09414e10i		−	0.258840i
$589$	−1.83455e10	+	1.86000e9i	−0.152430	+	0.0154544i
$590$	1.28301e10			0.105882
$591$		−	3.71365e10i		−	0.304404i
$592$			4.01487e10i			0.326877i
$593$	−2.81590e10			−0.227719			−0.113859	−	0.993497i	$-0.536321\pi$
$593$	−2.81590e10			−0.227719			−0.113859	+	0.993497i	$0.536321\pi$
$594$	−1.63232e11			−1.31117
$595$	3.26354e10			0.260388
$596$	−6.94018e10			−0.550029
$597$		−	1.52191e11i		−	1.19809i
$598$	3.92877e10			0.307222
$599$		−	1.53599e10i		−	0.119311i	−0.998219	−	0.0596554i	$-0.981000\pi$
$599$		−	1.53599e10i		−	0.119311i	0.998219	−	0.0596554i	$-0.0190002\pi$
$600$	3.94637e10			0.304504
$601$		−	1.08743e11i		−	0.833494i	−0.909023	−	0.416747i	$-0.863170\pi$
$601$		−	1.08743e11i		−	0.833494i	0.909023	−	0.416747i	$-0.136830\pi$
$602$			2.75970e10i			0.210124i
$603$			1.93941e10i			0.146690i
$604$			2.24219e9i			0.0168471i
$605$	−6.78846e10			−0.506699
$606$	3.66046e10			0.271422
$607$			1.12346e11i			0.827565i	0.910376	+	0.413783i	$0.135793\pi$
$607$			1.12346e11i			0.827565i	−0.910376	+	0.413783i	$0.864207\pi$
$608$	−2.40336e10	+	2.43670e9i	−0.175875	+	0.0178315i
$609$	−7.13134e10			−0.518444
$610$			1.97736e10i			0.142813i
$611$			5.19856e10i			0.373008i
$612$	1.76466e10			0.125793
$613$	−2.12496e10			−0.150490			−0.0752451	−	0.997165i	$-0.523974\pi$
$613$	−2.12496e10			−0.150490			−0.0752451	+	0.997165i	$0.523974\pi$
$614$	−1.80998e10			−0.127350
$615$	−2.75008e10			−0.192240
$616$			5.86465e10i			0.407304i
$617$	−2.05896e11			−1.42072			−0.710359	−	0.703840i	$-0.751467\pi$
$617$	−2.05896e11			−1.42072			−0.710359	+	0.703840i	$0.751467\pi$
$618$		−	2.10988e10i		−	0.144645i
$619$	1.59468e9			0.0108620			0.00543100	−	0.999985i	$-0.498271\pi$
$619$	1.59468e9			0.0108620			0.00543100	+	0.999985i	$0.498271\pi$
$620$			2.80442e9i			0.0189791i
$621$			1.89114e11i			1.27162i
$622$		−	1.54287e11i		−	1.03078i
$623$		−	1.69054e11i		−	1.12221i
$624$	1.26271e10			0.0832849
$625$	1.25065e11			0.819625
$626$		−	7.74391e10i		−	0.504270i
$627$	−2.46210e11	+	2.49625e10i	−1.59307	+	0.161517i
$628$	8.94442e10			0.575061
$629$		−	3.25833e11i		−	2.08158i
$630$		−	2.87912e9i		−	0.0182767i
$631$	2.84477e11			1.79444			0.897222	−	0.441580i	$-0.145582\pi$
$631$	2.84477e11			1.79444			0.897222	+	0.441580i	$0.145582\pi$
$632$	−7.62447e10			−0.477905
$633$	−4.63845e10			−0.288907
$634$	−5.55657e10			−0.343914
$635$		−	7.11334e10i		−	0.437501i
$636$	1.12547e11			0.687867
$637$		−	3.37247e10i		−	0.204829i
$638$	−1.74973e11			−1.05606
$639$		−	2.25655e10i		−	0.135345i
$640$			3.67393e9i			0.0218984i
$641$		−	5.30174e10i		−	0.314041i	−0.987595	−	0.157021i	$-0.949811\pi$
$641$		−	5.30174e10i		−	0.314041i	0.987595	−	0.157021i	$-0.0501889\pi$
$642$			2.25493e10i			0.132737i
$643$	5.79754e10			0.339156			0.169578	−	0.985517i	$-0.445759\pi$
$643$	5.79754e10			0.339156			0.169578	+	0.985517i	$0.445759\pi$
$644$	6.79453e10			0.395017
$645$			1.77108e10i			0.102329i
$646$	1.95048e11	−	1.97754e10i	1.11998	−	0.113552i
$647$	−2.75434e11			−1.57181			−0.785904	−	0.618348i	$-0.787802\pi$
$647$	−2.75434e11			−1.57181			−0.785904	+	0.618348i	$0.787802\pi$
$648$			5.09302e10i			0.288852i
$649$		−	1.87115e11i		−	1.05470i
$650$	4.30136e10			0.240964
$651$	1.66694e10			0.0928101
$652$	−3.88278e10			−0.214858
$653$	−2.69428e11			−1.48180			−0.740900	−	0.671615i	$-0.765601\pi$
$653$	−2.69428e11			−1.48180			−0.740900	+	0.671615i	$0.765601\pi$
$654$		−	9.29456e10i		−	0.508063i
$655$	4.77206e10			0.259263
$656$			3.91503e10i			0.211407i
$657$	−9.21921e9			−0.0494803
$658$			8.99053e10i			0.479603i
$659$			8.80932e10i			0.467090i	0.972346	+	0.233545i	$0.0750326\pi$
$659$			8.80932e10i			0.467090i	−0.972346	+	0.233545i	$0.924967\pi$
$660$			3.76372e10i			0.198354i
$661$			2.53825e10i			0.132962i	0.997788	+	0.0664811i	$0.0211772\pi$
$661$			2.53825e10i			0.132962i	−0.997788	+	0.0664811i	$0.978823\pi$
$662$	−1.91746e11			−0.998378
$663$	−1.02477e11			−0.530364
$664$		−	5.45356e10i		−	0.280549i
$665$	−3.22643e9	−	3.18229e10i	−0.0164982	−	0.162724i
$666$	−2.87452e10			−0.146106
$667$			2.02716e11i			1.02420i
$668$		−	1.76975e11i		−	0.888805i
$669$	−1.70404e11			−0.850699
$670$	3.27690e10			0.162616
$671$	2.88379e11			1.42257
$672$	2.18377e10			0.107085
$673$			1.21341e11i			0.591490i	0.955267	+	0.295745i	$0.0955679\pi$
$673$			1.21341e11i			0.591490i	−0.955267	+	0.295745i	$0.904432\pi$
$674$	2.77588e11			1.34512
$675$			2.07049e11i			0.997373i
$676$	−9.06505e10			−0.434094
$677$			3.38640e11i			1.61207i	0.591870	+	0.806034i	$0.298390\pi$
$677$			3.38640e11i			1.61207i	−0.591870	+	0.806034i	$0.701610\pi$
$678$		−	1.27332e11i		−	0.602587i
$679$			3.89328e10i			0.183162i
$680$		−	2.98164e10i		−	0.139450i
$681$	2.90189e11			1.34925
$682$	4.08996e10			0.189052
$683$			1.36488e11i			0.627208i	0.949554	+	0.313604i	$0.101536\pi$
$683$			1.36488e11i			0.627208i	−0.949554	+	0.313604i	$0.898464\pi$
$684$	−1.74460e9	−	1.72073e10i	−0.00797025	−	0.0786119i
$685$	7.02004e10			0.318843
$686$		−	1.61705e11i		−	0.730175i
$687$			2.24592e10i			0.100825i
$688$	2.52132e10			0.112532
$689$	1.22671e11			0.544332
$690$	4.36049e10			0.192371
$691$	3.85706e11			1.69178			0.845890	−	0.533358i	$-0.179070\pi$
$691$	3.85706e11			1.69178			0.845890	+	0.533358i	$0.179070\pi$
$692$			8.07801e10i			0.352274i
$693$	−4.19891e10			−0.182055
$694$			2.23487e11i			0.963418i
$695$	5.98692e10			0.256604
$696$			6.51534e10i			0.277651i
$697$		−	3.17730e11i		−	1.34626i
$698$		−	8.20988e10i		−	0.345872i
$699$			3.41609e11i			1.43094i
$700$	7.43890e10			0.309825
$701$	4.23246e11			1.75275			0.876377	−	0.481626i	$-0.159954\pi$
$701$	4.23246e11			1.75275			0.876377	+	0.481626i	$0.159954\pi$
$702$			6.62490e10i			0.272792i
$703$	−3.17721e11	+	3.22128e10i	−1.30084	+	0.131889i
$704$	5.35806e10			0.218131
$705$			5.76981e10i			0.233563i
$706$		−	2.10587e11i		−	0.847644i
$707$	6.89996e10			0.276165
$708$	−6.96744e10			−0.277294
$709$	4.73862e10			0.187529			0.0937643	−	0.995594i	$-0.470110\pi$
$709$	4.73862e10			0.187529			0.0937643	+	0.995594i	$0.470110\pi$
$710$	−3.81274e10			−0.150039
$711$		−	5.45888e10i		−	0.213612i
$712$	−1.54451e11			−0.600997
$713$		−	4.73846e10i		−	0.183349i
$714$	−1.77227e11			−0.681927
$715$			4.10229e10i			0.156965i
$716$		−	2.08718e11i		−	0.794158i
$717$			1.38204e11i			0.522932i
$718$			2.14963e11i			0.808845i
$719$	−1.26152e11			−0.472039			−0.236020	−	0.971748i	$-0.575843\pi$
$719$	−1.26152e11			−0.472039			−0.236020	+	0.971748i	$0.575843\pi$
$720$	−2.63042e9			−0.00978803
$721$		−	3.97711e10i		−	0.147173i
$722$	−3.85661e10	−	1.88237e11i	−0.141924	−	0.692717i
$723$	3.24005e10			0.118576
$724$		−	1.03935e11i		−	0.378276i
$725$			2.21941e11i			0.803315i
$726$	3.68649e11			1.32699
$727$	8.49520e10			0.304114			0.152057	−	0.988372i	$-0.451410\pi$
$727$	8.49520e10			0.304114			0.152057	+	0.988372i	$0.451410\pi$
$728$	2.38021e10			0.0847403
$729$	−3.11841e11			−1.10414
$730$			1.55771e10i			0.0548523i
$731$	−2.04622e11			−0.716608
$732$		−	1.07381e11i		−	0.374011i
$733$	5.23738e11			1.81426			0.907128	−	0.420855i	$-0.138270\pi$
$733$	5.23738e11			1.81426			0.907128	+	0.420855i	$0.138270\pi$
$734$		−	1.92120e11i		−	0.661895i
$735$		−	3.74306e10i		−	0.128256i
$736$		−	6.20762e10i		−	0.211551i
$737$		−	4.77903e11i		−	1.61983i
$738$	−2.80304e10			−0.0944940
$739$	3.44499e11			1.15508			0.577538	−	0.816364i	$-0.304014\pi$
$739$	3.44499e11			1.15508			0.577538	+	0.816364i	$0.304014\pi$
$740$			4.85689e10i			0.161969i
$741$	1.01312e10	+	9.99260e10i	0.0336039	+	0.331441i
$742$	2.12150e11			0.699887
$743$			4.53922e10i			0.148945i	0.997223	+	0.0744726i	$0.0237273\pi$
$743$			4.53922e10i			0.148945i	−0.997223	+	0.0744726i	$0.976273\pi$
$744$		−	1.52295e10i		−	0.0497042i
$745$	−8.39571e10			−0.272541
$746$	−3.79869e11			−1.22653
$747$	3.90458e10			0.125398
$748$	−4.34842e11			−1.38907
$749$			4.25054e10i			0.135057i
$750$	9.86024e10			0.311632
$751$			8.96270e10i			0.281760i	0.990027	+	0.140880i	$0.0449931\pi$
$751$			8.96270e10i			0.281760i	−0.990027	+	0.140880i	$0.955007\pi$
$752$	8.21394e10			0.256850
$753$		−	1.28709e11i		−	0.400341i
$754$			7.10142e10i			0.219715i
$755$			2.71243e9i			0.00834779i
$756$			1.14573e11i			0.350748i
$757$	−5.94573e11			−1.81060			−0.905298	−	0.424778i	$-0.860352\pi$
$757$	−5.94573e11			−1.81060			−0.905298	+	0.424778i	$0.860352\pi$
$758$	−3.83698e10			−0.116229
$759$		−	6.35934e11i		−	1.91622i
$760$	−2.90740e10	+	2.94774e9i	−0.0871466	+	0.00883556i
$761$	−4.29954e11			−1.28199			−0.640993	−	0.767547i	$-0.721477\pi$
$761$	−4.29954e11			−1.28199			−0.640993	+	0.767547i	$0.721477\pi$
$762$			3.86292e11i			1.14577i
$763$		−	1.75202e11i		−	0.516942i
$764$	1.46338e11			0.429521
$765$	2.13476e10			0.0623308
$766$	4.74420e11			1.37800
$767$	−7.59419e10			−0.219432
$768$		−	1.99514e10i		−	0.0573493i
$769$	3.94606e11			1.12839			0.564194	−	0.825642i	$-0.309187\pi$
$769$	3.94606e11			1.12839			0.564194	+	0.825642i	$0.309187\pi$
$770$			7.09461e10i			0.201821i
$771$	−5.56893e11			−1.57599
$772$			1.55076e11i			0.436592i
$773$			3.34209e10i			0.0936053i	0.998904	+	0.0468026i	$0.0149032\pi$
$773$			3.34209e10i			0.0936053i	−0.998904	+	0.0468026i	$0.985097\pi$
$774$			1.80519e10i			0.0502989i
$775$		−	5.18783e10i		−	0.143807i
$776$	3.55698e10			0.0980921
$777$	2.88692e11			0.792046
$778$		−	2.16995e11i		−	0.592287i
$779$	−3.09820e11	+	3.14118e10i	−0.841316	+	0.0852988i
$780$	1.52754e10			0.0412679
$781$			5.56049e11i			1.49455i
$782$			5.03789e11i			1.34717i
$783$	−3.41831e11			−0.909420
$784$	−5.32865e10			−0.141043
$785$	1.08203e11			0.284944
$786$	−2.59148e11			−0.678981
$787$		−	5.01627e11i		−	1.30762i	−0.756658	−	0.653811i	$-0.773169\pi$
$787$		−	5.01627e11i		−	1.30762i	0.756658	−	0.653811i	$-0.226831\pi$
$788$	−6.39554e10			−0.165872
$789$			3.47022e11i			0.895467i
$790$	−9.22350e10			−0.236803
$791$		−	2.40021e11i		−	0.613116i
$792$			3.83621e10i			0.0974992i
$793$		−	1.17041e11i		−	0.295967i
$794$		−	4.22064e11i		−	1.06193i
$795$	1.36150e11			0.340840
$796$	−2.62098e11			−0.652848
$797$			3.80292e11i			0.942505i	0.881998	+	0.471253i	$0.156198\pi$
$797$			3.80292e11i			0.942505i	−0.881998	+	0.471253i	$0.843802\pi$
$798$	1.75212e10	+	1.72815e11i	0.0432069	+	0.426157i
$799$	−6.66614e11			−1.63564
$800$		−	6.79633e10i		−	0.165926i
$801$		−	1.10582e11i		−	0.268631i
$802$	1.34829e11			0.325902
$803$	2.27176e11			0.546387
$804$	−1.77953e11			−0.425874
$805$	8.21951e10			0.195732
$806$		−	1.65994e10i		−	0.0393326i
$807$	−1.53533e11			−0.362000
$808$		−	6.30395e10i		−	0.147900i
$809$	−2.33585e11			−0.545319			−0.272660	−	0.962111i	$-0.587903\pi$
$809$	−2.33585e11			−0.545319			−0.272660	+	0.962111i	$0.587903\pi$
$810$			6.16115e10i			0.143127i
$811$		−	2.37656e10i		−	0.0549371i	−0.999623	−	0.0274685i	$-0.991255\pi$
$811$		−	2.37656e10i		−	0.0549371i	0.999623	−	0.0274685i	$-0.00874461\pi$
$812$			1.22814e11i			0.282503i
$813$		−	9.27991e10i		−	0.212413i
$814$	7.08329e11			1.61338
$815$	−4.69709e10			−0.106463
$816$			1.61918e11i			0.365204i
$817$	2.02295e10	+	1.99527e11i	0.0454043	+	0.447831i
$818$	1.77966e11			0.397488
$819$			1.70416e10i			0.0378769i
$820$			4.73611e10i			0.104753i
$821$	−5.15004e11			−1.13354			−0.566771	−	0.823875i	$-0.691808\pi$
$821$	−5.15004e11			−1.13354			−0.566771	+	0.823875i	$0.691808\pi$
$822$	−3.81225e11			−0.835015
$823$	2.28392e11			0.497831			0.248916	−	0.968525i	$-0.419926\pi$
$823$	2.28392e11			0.497831			0.248916	+	0.968525i	$0.419926\pi$
$824$	−3.63357e10			−0.0788179
$825$		−	6.96243e11i		−	1.50295i
$826$	−1.31336e11			−0.282139
$827$		−	6.97217e10i		−	0.149055i	−0.997219	−	0.0745274i	$-0.976255\pi$
$827$		−	6.97217e10i		−	0.149055i	0.997219	−	0.0745274i	$-0.0237448\pi$
$828$	4.44447e10			0.0945580
$829$		−	4.08026e11i		−	0.863913i	−0.901894	−	0.431956i	$-0.857823\pi$
$829$		−	4.08026e11i		−	0.863913i	0.901894	−	0.431956i	$-0.142177\pi$
$830$		−	6.59731e10i		−	0.139013i
$831$			7.83318e9i			0.0164261i
$832$		−	2.17461e10i		−	0.0453825i
$833$	4.32454e11			0.898173
$834$	−3.25121e11			−0.672018
$835$		−	2.14091e11i		−	0.440406i
$836$	4.29898e10	+	4.24015e11i	0.0880116	+	0.868073i
$837$	7.99024e10			0.162801
$838$			4.00234e11i			0.811591i
$839$			9.01495e11i			1.81935i	0.415325	+	0.909673i	$0.363668\pi$
$839$			9.01495e11i			1.81935i	−0.415325	+	0.909673i	$0.636332\pi$
$840$	2.64176e10			0.0530612
$841$	1.33828e11			0.267524
$842$	−2.51794e11			−0.500953
$843$	6.87511e11			1.36135
$844$			7.98821e10i			0.157427i
$845$	−1.09662e11			−0.215095
$846$			5.88092e10i			0.114806i
$847$	6.94903e11			1.35018
$848$		−	1.93825e11i		−	0.374823i
$849$		−	5.91583e11i		−	1.13864i
$850$			5.51566e11i			1.05663i
$851$		−	8.20639e11i		−	1.56471i
$852$	2.07052e11			0.392935
$853$	−5.99808e10			−0.113296			−0.0566482	−	0.998394i	$-0.518041\pi$
$853$	−5.99808e10			−0.113296			−0.0566482	+	0.998394i	$0.518041\pi$
$854$		−	2.02413e11i		−	0.380547i
$855$	−2.11049e9	−	2.08161e10i	−0.00394928	−	0.0389524i
$856$	3.88338e10			0.0723293
$857$		−	2.70130e10i		−	0.0500783i	−0.999686	−	0.0250391i	$-0.992029\pi$
$857$		−	2.70130e10i		−	0.0500783i	0.999686	−	0.0250391i	$-0.00797104\pi$
$858$		−	2.22776e11i		−	0.411073i
$859$	−7.98513e10			−0.146659			−0.0733296	−	0.997308i	$-0.523363\pi$
$859$	−7.98513e10			−0.146659			−0.0733296	+	0.997308i	$0.523363\pi$
$860$	3.05010e10			0.0557598
$861$	2.81513e11			0.512254
$862$	−6.97171e11			−1.26273
$863$			1.02525e12i			1.84836i	0.381954	+	0.924181i	$0.375251\pi$
$863$			1.02525e12i			1.84836i	−0.381954	+	0.924181i	$0.624749\pi$
$864$	1.04676e11			0.187842
$865$			9.77217e10i			0.174553i
$866$	5.34473e10			0.0950286
$867$		−	7.95602e11i		−	1.40805i
$868$		−	2.87075e10i		−	0.0505728i
$869$			1.34516e12i			2.35881i
$870$			7.88176e10i			0.137577i
$871$	−1.93960e11			−0.337008
$872$	−1.60068e11			−0.276847
$873$			2.54669e10i			0.0438448i
$874$	4.91246e11	−	4.98061e10i	0.841886	−	0.0853565i
$875$	1.85865e11			0.317078
$876$		−	8.45918e10i		−	0.143652i
$877$		−	9.55040e11i		−	1.61445i	−0.590247	−	0.807223i	$-0.700970\pi$
$877$		−	9.55040e11i		−	1.61445i	0.590247	−	0.807223i	$-0.299030\pi$
$878$	4.93345e11			0.830181
$879$	1.08725e11			0.182127
$880$	6.48178e10			0.108085
$881$	1.13499e12			1.88403			0.942014	−	0.335573i	$-0.108930\pi$
$881$	1.13499e12			1.88403			0.942014	+	0.335573i	$0.108930\pi$
$882$		−	3.81514e10i		−	0.0630430i
$883$	2.78656e10			0.0458379			0.0229190	−	0.999737i	$-0.492704\pi$
$883$	2.78656e10			0.0458379			0.0229190	+	0.999737i	$0.492704\pi$
$884$			1.76484e11i			0.288998i
$885$	−8.42868e10			−0.137400
$886$		−	4.99282e11i		−	0.810235i
$887$		−	6.71858e11i		−	1.08538i	−0.839932	−	0.542691i	$-0.817405\pi$
$887$		−	6.71858e11i		−	1.08538i	0.839932	−	0.542691i	$-0.182595\pi$
$888$		−	2.63755e11i		−	0.424178i
$889$			7.28159e11i			1.16579i
$890$	−1.86844e11			−0.297796
$891$	8.98542e11			1.42570
$892$			2.93465e11i			0.463551i
$893$	6.59035e10	+	6.50018e11i	0.103634	+	1.02216i
$894$	4.55931e11			0.713755
$895$		−	2.52491e11i		−	0.393508i
$896$		−	3.76083e10i		−	0.0583515i
$897$	−2.58098e11			−0.398672
$898$	−2.63461e11			−0.405145
$899$	8.56496e10			0.131125
$900$	4.86596e10			0.0741649
$901$			1.57301e12i			2.38689i
$902$	6.90714e11			1.04345
$903$		−	1.81297e11i		−	0.272672i
$904$	−2.19288e11			−0.328353
$905$		−	1.25733e11i		−	0.187437i
$906$		−	1.47299e10i		−	0.0218619i
$907$		−	6.08911e11i		−	0.899757i	−0.893090	−	0.449878i	$-0.851467\pi$
$907$		−	6.08911e11i		−	0.899757i	0.893090	−	0.449878i	$-0.148533\pi$
$908$		−	4.99755e11i		−	0.735214i
$909$	4.51343e10			0.0661075
$910$	2.87940e10			0.0419891
$911$		−	7.07571e10i		−	0.102730i	−0.998680	−	0.0513649i	$-0.983643\pi$
$911$		−	7.07571e10i		−	0.102730i	0.998680	−	0.0513649i	$-0.0163572\pi$
$912$	1.57887e11	−	1.60078e10i	0.228227	−	0.0231393i
$913$	−9.62151e11			−1.38471
$914$		−	5.20846e11i		−	0.746319i
$915$		−	1.29902e11i		−	0.185324i
$916$	3.86787e10			0.0549401
$917$	−4.88493e11			−0.690846
$918$	−8.49515e11			−1.19619
$919$	−2.81891e11			−0.395202			−0.197601	−	0.980283i	$-0.563315\pi$
$919$	−2.81891e11			−0.395202			−0.197601	+	0.980283i	$0.563315\pi$
$920$		−	7.50951e10i		−	0.104824i
$921$	1.18906e11			0.165259
$922$			4.08552e11i			0.565358i
$923$	2.25677e11			0.310943
$924$		−	3.85275e11i		−	0.528546i
$925$		−	8.98465e11i		−	1.22725i
$926$			3.96594e10i			0.0539389i
$927$		−	2.60153e10i		−	0.0352297i
$928$	1.12205e11			0.151294
$929$	−1.08860e12			−1.46152			−0.730762	−	0.682632i	$-0.760835\pi$
$929$	−1.08860e12			−1.46152			−0.730762	+	0.682632i	$0.760835\pi$
$930$		−	1.84235e10i		−	0.0246286i
$931$	−4.27537e10	−	4.21687e11i	−0.0569082	−	0.561296i
$932$	5.88309e11			0.779726
$933$			1.01358e12i			1.33761i
$934$			5.36149e10i			0.0704527i
$935$	−5.26038e11			−0.688289
$936$	1.55695e10			0.0202849
$937$	−8.13586e11			−1.05547			−0.527734	−	0.849410i	$-0.676958\pi$
$937$	−8.13586e11			−1.05547			−0.527734	+	0.849410i	$0.676958\pi$
$938$	−3.35441e11			−0.433316
$939$			5.08732e11i			0.654375i
$940$	9.93660e10			0.127270
$941$			1.38430e12i			1.76552i	0.469827	+	0.882759i	$0.344316\pi$
$941$			1.38430e12i			1.76552i	−0.469827	+	0.882759i	$0.655684\pi$
$942$	−5.87598e11			−0.746237
$943$		−	8.00232e11i		−	1.01197i
$944$			1.19991e11i			0.151099i
$945$			1.38602e11i			0.173797i
$946$		−	4.44827e11i		−	0.555427i
$947$	5.32762e11			0.662420			0.331210	−	0.943557i	$-0.392543\pi$
$947$	5.32762e11			0.662420			0.331210	+	0.943557i	$0.392543\pi$
$948$	5.00885e11			0.620161
$949$		−	9.22011e10i		−	0.113677i
$950$	5.37834e11	−	5.45295e10i	0.660319	−	0.0669479i
$951$	3.65036e11			0.446286
$952$			3.05216e11i			0.371586i
$953$		−	5.73373e11i		−	0.695129i	−0.937656	−	0.347564i	$-0.887009\pi$
$953$		−	5.73373e11i		−	0.695129i	0.937656	−	0.347564i	$-0.112991\pi$
$954$	1.38772e11			0.167537
$955$	1.77029e11			0.212829
$956$	2.38012e11			0.284949
$957$	1.14948e12			1.37042
$958$		−	8.53787e10i		−	0.101365i
$959$	−7.18608e11			−0.849606
$960$		−	2.41357e10i		−	0.0284168i
$961$	8.32871e11			0.976526
$962$		−	2.87480e11i		−	0.335666i
$963$			2.78038e10i			0.0323295i
$964$		−	5.57992e10i		−	0.0646130i
$965$			1.87600e11i			0.216333i
$966$	−4.46363e11			−0.512601
$967$	−1.36489e12			−1.56096			−0.780479	−	0.625182i	$-0.785025\pi$
$967$	−1.36489e12			−1.56096			−0.780479	+	0.625182i	$0.785025\pi$
$968$		−	6.34878e11i		−	0.723084i
$969$	−1.28136e12	+	1.29913e11i	−1.45337	+	0.147353i
$970$	4.30296e10			0.0486050
$971$			1.57922e12i			1.77650i	0.459361	+	0.888250i	$0.348078\pi$
$971$			1.57922e12i			1.77650i	−0.459361	+	0.888250i	$0.651922\pi$
$972$			1.39663e11i			0.156464i
$973$	−6.12853e11			−0.683762
$974$	−1.00951e12			−1.12169
$975$	−2.82575e11			−0.312691
$976$	−1.84929e11			−0.203801
$977$			5.51758e11i			0.605578i	0.953058	+	0.302789i	$0.0979178\pi$
$977$			5.51758e11i			0.605578i	−0.953058	+	0.302789i	$0.902082\pi$
$978$	2.55077e11			0.278814
$979$			2.72493e12i			2.96636i
$980$	−6.44619e10			−0.0698874
$981$		−	1.14604e11i		−	0.123744i
$982$		−	3.69806e11i		−	0.397675i
$983$		−	8.60977e11i		−	0.922099i	−0.887374	−	0.461050i	$-0.847473\pi$
$983$		−	8.60977e11i		−	0.922099i	0.887374	−	0.461050i	$-0.152527\pi$
$984$		−	2.57196e11i		−	0.274336i
$985$	−7.73684e10			−0.0821900
$986$	−9.10619e11			−0.963450
$987$		−	5.90628e11i		−	0.622365i
$988$	1.72090e11	−	1.74477e10i	0.180604	−	0.0183109i
$989$	−5.15358e11			−0.538671
$990$			4.64075e10i			0.0483112i
$991$		−	1.27354e11i		−	0.132044i	−0.997818	−	0.0660220i	$-0.978969\pi$
$991$		−	1.27354e11i		−	0.132044i	0.997818	−	0.0660220i	$-0.0210308\pi$
$992$	−2.62278e10			−0.0270841
$993$	1.25967e12			1.29556
$994$	3.90292e11			0.399801
$995$	−3.17067e11			−0.323488
$996$			3.58269e11i			0.364059i
$997$	−1.40891e12			−1.42595			−0.712975	−	0.701190i	$-0.752653\pi$
$997$	−1.40891e12			−1.42595			−0.712975	+	0.701190i	$0.752653\pi$
$998$		−	1.07659e12i		−	1.08525i
$999$	1.38381e12			1.38935

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.b.a.37.9	yes	12
3.2	odd	2		342.9.d.a.37.4		12
4.3	odd	2		304.9.e.e.113.9		12
19.18	odd	2	inner	38.9.b.a.37.4	✓	12
57.56	even	2		342.9.d.a.37.10		12
76.75	even	2		304.9.e.e.113.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.4	✓	12	19.18	odd	2	inner
38.9.b.a.37.9	yes	12	1.1	even	1	trivial
304.9.e.e.113.4		12	76.75	even	2
304.9.e.e.113.9		12	4.3	odd	2
342.9.d.a.37.4		12	3.2	odd	2
342.9.d.a.37.10		12	57.56	even	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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} -74.3247i q^{3} -128.000 q^{4} -154.845 q^{5} +840.888 q^{6} +1585.07 q^{7} -1448.15i q^{8} +1036.83 q^{9} +O(q^{10})\)	\(q+11.3137i q^{2} -74.3247i q^{3} -128.000 q^{4} -154.845 q^{5} +840.888 q^{6} +1585.07 q^{7} -1448.15i q^{8} +1036.83 q^{9} -1751.87i q^{10} -25549.2 q^{11} +9513.57i q^{12} +10369.3i q^{13} +17933.0i q^{14} +11508.8i q^{15} +16384.0 q^{16} -132967. q^{17} +11730.4i q^{18} +(13145.5 + 129656. i) q^{19} +19820.1 q^{20} -117810. i q^{21} -289057. i q^{22} -334889. q^{23} -107634. q^{24} -366648. q^{25} -117316. q^{26} -564707. i q^{27} -202889. q^{28} -605325. i q^{29} -130207. q^{30} +141494. i q^{31} +185364. i q^{32} +1.89894e6i q^{33} -1.50435e6i q^{34} -245440. q^{35} -132715. q^{36} +2.45048e6i q^{37} +(-1.46689e6 + 148724. i) q^{38} +770699. q^{39} +224239. i q^{40} +2.38955e6i q^{41} +1.33287e6 q^{42} +1.53889e6 q^{43} +3.27030e6 q^{44} -160548. q^{45} -3.78883e6i q^{46} +5.01339e6 q^{47} -1.21774e6i q^{48} -3.25235e6 q^{49} -4.14815e6i q^{50} +9.88272e6i q^{51} -1.32728e6i q^{52} -1.18301e7i q^{53} +6.38893e6 q^{54} +3.95616e6 q^{55} -2.29543e6i q^{56} +(9.63667e6 - 977036. i) q^{57} +6.84847e6 q^{58} +7.32369e6i q^{59} -1.47313e6i q^{60} -1.12872e7 q^{61} -1.60082e6 q^{62} +1.64346e6 q^{63} -2.09715e6 q^{64} -1.60564e6i q^{65} -2.14841e7 q^{66} +1.87052e7i q^{67} +1.70197e7 q^{68} +2.48905e7i q^{69} -2.77684e6i q^{70} -2.17638e7i q^{71} -1.50150e6i q^{72} -8.89170e6 q^{73} -2.77241e7 q^{74} +2.72510e7i q^{75} +(-1.68262e6 - 1.65960e7i) q^{76} -4.04974e7 q^{77} +8.71947e6i q^{78} -5.26495e7i q^{79} -2.53698e6 q^{80} -3.51690e7 q^{81} -2.70346e7 q^{82} +3.76587e7 q^{83} +1.50797e7i q^{84} +2.05892e7 q^{85} +1.74106e7i q^{86} -4.49906e7 q^{87} +3.69992e7i q^{88} -1.06654e8i q^{89} -1.81640e6i q^{90} +1.64362e7i q^{91} +4.28658e7 q^{92} +1.05165e7 q^{93} +5.67200e7i q^{94} +(-2.03551e6 - 2.00766e7i) q^{95} +1.37771e7 q^{96} +2.45621e7i q^{97} -3.67961e7i q^{98} -2.64903e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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