LMFDB - Embedded newform 72.8.d.c.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(37,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.d (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:	$22.4917218349$
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} - 206x^{10} + 24336x^{8} - 1510912x^{6} + 398721024x^{4} - 55297703936x^{2} + 4398046511104 $ x^12 - 206x^10 + 24336x^8 - 1510912x^6 + 398721024x^4 - 55297703936*x^2 + 4398046511104
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{6}\cdot 5^{2}\cdot 13^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.3
Root			$10.2927 - 4.69692i$ of defining polynomial
Character	$\chi$	$=$	72.37
Dual form			72.8.d.c.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-10.2927 - 4.69692i) q^{2} +(83.8779 + 96.6877i) q^{4} -316.260i q^{5} +1193.16 q^{7} +(-409.193 - 1389.14i) q^{8} +O(q^{10})$	\(q+(-10.2927 - 4.69692i) q^{2} +(83.8779 + 96.6877i) q^{4} -316.260i q^{5} +1193.16 q^{7} +(-409.193 - 1389.14i) q^{8} +(-1485.45 + 3255.16i) q^{10} +2202.76i q^{11} -2138.97i q^{13} +(-12280.8 - 5604.17i) q^{14} +(-2313.01 + 16219.9i) q^{16} +29183.7 q^{17} +44710.6i q^{19} +(30578.5 - 26527.2i) q^{20} +(10346.2 - 22672.3i) q^{22} -30861.6 q^{23} -21895.6 q^{25} +(-10046.6 + 22015.7i) q^{26} +(100080. + 115364. i) q^{28} -189611. i q^{29} +133272. q^{31} +(99990.6 - 156082. i) q^{32} +(-300378. - 137073. i) q^{34} -377349. i q^{35} -557840. i q^{37} +(210002. - 460192. i) q^{38} +(-439330. + 129411. i) q^{40} -185292. q^{41} -372296. i q^{43} +(-212980. + 184763. i) q^{44} +(317648. + 144954. i) q^{46} +1.00993e6 q^{47} +600086. q^{49} +(225364. + 102842. i) q^{50} +(206812. - 179412. i) q^{52} +245619. i q^{53} +696647. q^{55} +(-488232. - 1.65747e6i) q^{56} +(-890589. + 1.95160e6i) q^{58} -81380.0i q^{59} +775123. i q^{61} +(-1.37172e6 - 625966. i) q^{62} +(-1.76227e6 + 1.13685e6i) q^{64} -676470. q^{65} -4.37188e6i q^{67} +(2.44787e6 + 2.82170e6i) q^{68} +(-1.77238e6 + 3.88392e6i) q^{70} -2.32062e6 q^{71} +5.44398e6 q^{73} +(-2.62013e6 + 5.74166e6i) q^{74} +(-4.32297e6 + 3.75023e6i) q^{76} +2.62825e6i q^{77} +3.51878e6 q^{79} +(5.12971e6 + 731512. i) q^{80} +(1.90715e6 + 870301. i) q^{82} +4.53923e6i q^{83} -9.22964e6i q^{85} +(-1.74865e6 + 3.83192e6i) q^{86} +(3.05995e6 - 901355. i) q^{88} -7.05346e6 q^{89} -2.55213e6i q^{91} +(-2.58860e6 - 2.98394e6i) q^{92} +(-1.03949e7 - 4.74355e6i) q^{94} +1.41402e7 q^{95} -5.75365e6 q^{97} +(-6.17648e6 - 2.81856e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 412 q^{4} + 136 q^{7}+O(q^{10}) $ 12 * q + 412 * q^4 + 136 * q^7	$ 12 q + 412 q^{4} + 136 q^{7} + 4680 q^{10} - 12472 q^{16} + 32624 q^{22} + 229820 q^{25} - 157288 q^{28} - 37224 q^{31} - 74432 q^{34} - 937520 q^{40} - 1264256 q^{46} + 2668188 q^{49} - 1539680 q^{52} + 6928960 q^{55} - 4035448 q^{58} - 3530192 q^{64} - 10228720 q^{70} + 13619048 q^{73} - 2441920 q^{76} + 20470552 q^{79} - 2507200 q^{82} - 26170912 q^{88} - 22132608 q^{94} + 27442456 q^{97}+O(q^{100}) $ 12 * q + 412 * q^4 + 136 * q^7 + 4680 * q^10 - 12472 * q^16 + 32624 * q^22 + 229820 * q^25 - 157288 * q^28 - 37224 * q^31 - 74432 * q^34 - 937520 * q^40 - 1264256 * q^46 + 2668188 * q^49 - 1539680 * q^52 + 6928960 * q^55 - 4035448 * q^58 - 3530192 * q^64 - 10228720 * q^70 + 13619048 * q^73 - 2441920 * q^76 + 20470552 * q^79 - 2507200 * q^82 - 26170912 * q^88 - 22132608 * q^94 + 27442456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−10.2927	−	4.69692i	−0.909752	−	0.415153i
$2$	−10.2927	−	4.69692i	−0.909752	−	0.415153i
$3$		0			0
$3$		0			0
$4$	83.8779	+	96.6877i	0.655296	+	0.755372i
$5$		−	316.260i		−	1.13149i	−0.824581	−	0.565744i	$-0.808589\pi$
$5$		−	316.260i		−	1.13149i	0.824581	−	0.565744i	$-0.191411\pi$
$6$		0			0
$7$	1193.16			1.31479			0.657393	−	0.753548i	$-0.271659\pi$
$7$	1193.16			1.31479			0.657393	+	0.753548i	$0.271659\pi$
$8$	−409.193	−	1389.14i	−0.282561	−	0.959249i
$9$		0			0
$10$	−1485.45	+	3255.16i	−0.469740	+	1.02937i
$11$			2202.76i			0.498992i	0.968376	+	0.249496i	$0.0802650\pi$
$11$			2202.76i			0.498992i	−0.968376	+	0.249496i	$0.919735\pi$
$12$		0			0
$13$		−	2138.97i		−	0.270024i	−0.990844	−	0.135012i	$-0.956893\pi$
$13$		−	2138.97i		−	0.270024i	0.990844	−	0.135012i	$-0.0431073\pi$
$14$	−12280.8	−	5604.17i	−1.19613	−	0.545838i
$15$		0			0
$16$	−2313.01	+	16219.9i	−0.141175	+	0.989985i
$17$	29183.7			1.44068			0.720342	−	0.693619i	$-0.243985\pi$
$17$	29183.7			1.44068			0.720342	+	0.693619i	$0.243985\pi$
$18$		0			0
$19$			44710.6i			1.49545i	0.664006	+	0.747727i	$0.268855\pi$
$19$			44710.6i			1.49545i	−0.664006	+	0.747727i	$0.731145\pi$
$20$	30578.5	−	26527.2i	0.854694	−	0.741459i
$21$		0			0
$22$	10346.2	−	22672.3i	0.207158	−	0.453959i
$23$	−30861.6			−0.528897			−0.264448	−	0.964400i	$-0.585190\pi$
$23$	−30861.6			−0.528897			−0.264448	+	0.964400i	$0.585190\pi$
$24$		0			0
$25$	−21895.6			−0.280263
$26$	−10046.6	+	22015.7i	−0.112101	+	0.245655i
$27$		0			0
$28$	100080.	+	115364.i	0.861574	+	0.993153i
$29$		−	189611.i		−	1.44368i	−0.692060	−	0.721840i	$-0.743297\pi$
$29$		−	189611.i		−	1.44368i	0.692060	−	0.721840i	$-0.256703\pi$
$30$		0			0
$31$	133272.			0.803473			0.401737	−	0.915755i	$-0.368407\pi$
$31$	133272.			0.803473			0.401737	+	0.915755i	$0.368407\pi$
$32$	99990.6	−	156082.i	0.539429	−	0.842031i
$33$		0			0
$34$	−300378.	−	137073.i	−1.31067	−	0.598105i
$35$		−	377349.i		−	1.48766i
$36$		0			0
$37$		−	557840.i		−	1.81052i	−0.424855	−	0.905261i	$-0.639675\pi$
$37$		−	557840.i		−	1.81052i	0.424855	−	0.905261i	$-0.360325\pi$
$38$	210002.	−	460192.i	0.620843	−	1.36049i
$39$		0			0
$40$	−439330.	+	129411.i	−1.08538	+	0.319715i
$41$	−185292.			−0.419868			−0.209934	−	0.977716i	$-0.567325\pi$
$41$	−185292.			−0.419868			−0.209934	+	0.977716i	$0.567325\pi$
$42$		0			0
$43$		−	372296.i		−	0.714084i	−0.934088	−	0.357042i	$-0.883785\pi$
$43$		−	372296.i		−	0.714084i	0.934088	−	0.357042i	$-0.116215\pi$
$44$	−212980.	+	184763.i	−0.376925	+	0.326988i
$45$		0			0
$46$	317648.	+	144954.i	0.481165	+	0.219573i
$47$	1.00993e6			1.41889			0.709444	−	0.704762i	$-0.248946\pi$
$47$	1.00993e6			1.41889			0.709444	+	0.704762i	$0.248946\pi$
$48$		0			0
$49$	600086.			0.728664
$50$	225364.	+	102842.i	0.254970	+	0.116352i
$51$		0			0
$52$	206812.	−	179412.i	0.203969	−	0.176945i
$53$			245619.i			0.226619i	0.993560	+	0.113309i	$0.0361452\pi$
$53$			245619.i			0.226619i	−0.993560	+	0.113309i	$0.963855\pi$
$54$		0			0
$55$	696647.			0.564603
$56$	−488232.	−	1.65747e6i	−0.371508	−	1.26121i
$57$		0			0
$58$	−890589.	+	1.95160e6i	−0.599348	+	1.31339i
$59$		−	81380.0i		−	0.0515864i	−0.999667	−	0.0257932i	$-0.991789\pi$
$59$		−	81380.0i		−	0.0515864i	0.999667	−	0.0257932i	$-0.00821115\pi$
$60$		0			0
$61$			775123.i			0.437236i	0.975810	+	0.218618i	$0.0701549\pi$
$61$			775123.i			0.437236i	−0.975810	+	0.218618i	$0.929845\pi$
$62$	−1.37172e6	−	625966.i	−0.730961	−	0.333564i
$63$		0			0
$64$	−1.76227e6	+	1.13685e6i	−0.840318	+	0.542093i
$65$	−676470.			−0.305528
$66$		0			0
$67$		−	4.37188e6i		−	1.77585i	−0.459986	−	0.887926i	$-0.652146\pi$
$67$		−	4.37188e6i		−	1.77585i	0.459986	−	0.887926i	$-0.347854\pi$
$68$	2.44787e6	+	2.82170e6i	0.944075	+	1.08825i
$69$		0			0
$70$	−1.77238e6	+	3.88392e6i	−0.617608	+	1.35340i
$71$	−2.32062e6			−0.769484			−0.384742	−	0.923024i	$-0.625710\pi$
$71$	−2.32062e6			−0.769484			−0.384742	+	0.923024i	$0.625710\pi$
$72$		0			0
$73$	5.44398e6			1.63790			0.818949	−	0.573866i	$-0.194557\pi$
$73$	5.44398e6			1.63790			0.818949	+	0.573866i	$0.194557\pi$
$74$	−2.62013e6	+	5.74166e6i	−0.751644	+	1.64713i
$75$		0			0
$76$	−4.32297e6	+	3.75023e6i	−1.12962	+	0.979965i
$77$			2.62825e6i			0.656068i
$78$		0			0
$79$	3.51878e6			0.802966			0.401483	−	0.915866i	$-0.368495\pi$
$79$	3.51878e6			0.802966			0.401483	+	0.915866i	$0.368495\pi$
$80$	5.12971e6	+	731512.i	1.12015	+	0.159737i
$81$		0			0
$82$	1.90715e6	+	870301.i	0.381976	+	0.174309i
$83$			4.53923e6i			0.871383i	0.900096	+	0.435691i	$0.143496\pi$
$83$			4.53923e6i			0.871383i	−0.900096	+	0.435691i	$0.856504\pi$
$84$		0			0
$85$		−	9.22964e6i		−	1.63012i
$86$	−1.74865e6	+	3.83192e6i	−0.296454	+	0.649639i
$87$		0			0
$88$	3.05995e6	−	901355.i	0.478658	−	0.140996i
$89$	−7.05346e6			−1.06056			−0.530282	−	0.847821i	$-0.677914\pi$
$89$	−7.05346e6			−1.06056			−0.530282	+	0.847821i	$0.677914\pi$
$90$		0			0
$91$		−	2.55213e6i		−	0.355024i
$92$	−2.58860e6	−	2.98394e6i	−0.346584	−	0.399514i
$93$		0			0
$94$	−1.03949e7	−	4.74355e6i	−1.29084	−	0.589056i
$95$	1.41402e7			1.69209
$96$		0			0
$97$	−5.75365e6			−0.640092			−0.320046	−	0.947402i	$-0.603698\pi$
$97$	−5.75365e6			−0.640092			−0.320046	+	0.947402i	$0.603698\pi$
$98$	−6.17648e6	−	2.81856e6i	−0.662903	−	0.302507i
$99$		0			0
$100$	−1.83655e6	−	2.11703e6i	−0.183655	−	0.211703i
$101$			1.38615e7i			1.33871i	0.742944	+	0.669354i	$0.233429\pi$
$101$			1.38615e7i			1.33871i	−0.742944	+	0.669354i	$0.766571\pi$
$102$		0			0
$103$	7.40590e6			0.667802			0.333901	−	0.942608i	$-0.391635\pi$
$103$	7.40590e6			0.667802			0.333901	+	0.942608i	$0.391635\pi$
$104$	−2.97133e6	+	875249.i	−0.259020	+	0.0762983i
$105$		0			0
$106$	1.15365e6	−	2.52807e6i	0.0940816	−	0.206167i
$107$		−	1.83880e7i		−	1.45108i	−0.688181	−	0.725539i	$-0.741591\pi$
$107$		−	1.83880e7i		−	1.45108i	0.688181	−	0.725539i	$-0.258409\pi$
$108$		0			0
$109$		−	1.30010e7i		−	0.961576i	−0.876837	−	0.480788i	$-0.840351\pi$
$109$		−	1.30010e7i		−	0.961576i	0.876837	−	0.480788i	$-0.159649\pi$
$110$	−7.17035e6	−	3.27210e6i	−0.513649	−	0.234397i
$111$		0			0
$112$	−2.75978e6	+	1.93529e7i	−0.185615	+	1.30162i
$113$	−2.57409e7			−1.67822			−0.839112	−	0.543959i	$-0.816925\pi$
$113$	−2.57409e7			−1.67822			−0.839112	+	0.543959i	$0.816925\pi$
$114$		0			0
$115$			9.76030e6i			0.598440i
$116$	1.83331e7	−	1.59042e7i	1.09052	−	0.946037i
$117$		0			0
$118$	−382235.	+	837617.i	−0.0214163	+	0.0469308i
$119$	3.48208e7			1.89419
$120$		0			0
$121$	1.46350e7			0.751007
$122$	3.64069e6	−	7.97808e6i	0.181520	−	0.397776i
$123$		0			0
$124$	1.11785e7	+	1.28857e7i	0.526513	+	0.606921i
$125$		−	1.77831e7i		−	0.814373i
$126$		0			0
$127$	−1.01595e7			−0.440109			−0.220054	−	0.975488i	$-0.570623\pi$
$127$	−1.01595e7			−0.440109			−0.220054	+	0.975488i	$0.570623\pi$
$128$	2.34782e7	−	3.42397e6i	0.989533	−	0.144310i
$129$		0			0
$130$	6.96268e6	+	3.17733e6i	0.277955	+	0.126841i
$131$		−	2.50477e7i		−	0.973462i	−0.873552	−	0.486731i	$-0.838189\pi$
$131$		−	2.50477e7i		−	0.973462i	0.873552	−	0.486731i	$-0.161811\pi$
$132$		0			0
$133$			5.33469e7i			1.96620i
$134$	−2.05344e7	+	4.49983e7i	−0.737251	+	1.61558i
$135$		0			0
$136$	−1.19417e7	−	4.05403e7i	−0.407082	−	1.38198i
$137$	−2.23725e7			−0.743348			−0.371674	−	0.928363i	$-0.621216\pi$
$137$	−2.23725e7			−0.743348			−0.371674	+	0.928363i	$0.621216\pi$
$138$		0			0
$139$			1.36116e7i			0.429889i	0.976626	+	0.214944i	$0.0689570\pi$
$139$			1.36116e7i			0.429889i	−0.976626	+	0.214944i	$0.931043\pi$
$140$	3.64850e7	−	3.16512e7i	1.12374	−	0.974860i
$141$		0			0
$142$	2.38854e7	+	1.08998e7i	0.700040	+	0.319454i
$143$	4.71164e6			0.134740
$144$		0			0
$145$	−5.99665e7			−1.63351
$146$	−5.60331e7	−	2.55700e7i	−1.49008	−	0.679978i
$147$		0			0
$148$	5.39363e7	−	4.67905e7i	1.36762	−	1.18643i
$149$			1.07544e7i			0.266339i	0.991093	+	0.133169i	$0.0425154\pi$
$149$			1.07544e7i			0.266339i	−0.991093	+	0.133169i	$0.957485\pi$
$150$		0			0
$151$	−4.91328e7			−1.16132			−0.580660	−	0.814146i	$-0.697205\pi$
$151$	−4.91328e7			−1.16132			−0.580660	+	0.814146i	$0.697205\pi$
$152$	6.21094e7	−	1.82953e7i	1.43451	−	0.422558i
$153$		0			0
$154$	1.23447e7	−	2.70517e7i	0.272369	−	0.596859i
$155$		−	4.21485e7i		−	0.909120i
$156$		0			0
$157$			2.83533e7i			0.584729i	0.956307	+	0.292365i	$0.0944421\pi$
$157$			2.83533e7i			0.584729i	−0.956307	+	0.292365i	$0.905558\pi$
$158$	−3.62176e7	−	1.65274e7i	−0.730500	−	0.333354i
$159$		0			0
$160$	−4.93626e7	−	3.16231e7i	−0.952747	−	0.610357i
$161$	−3.68228e7			−0.695386
$162$		0			0
$163$		−	1.43798e7i		−	0.260073i	−0.991509	−	0.130037i	$-0.958490\pi$
$163$		−	1.43798e7i		−	0.260073i	0.991509	−	0.130037i	$-0.0415095\pi$
$164$	−1.55419e7	−	1.79154e7i	−0.275138	−	0.317157i
$165$		0			0
$166$	2.13204e7	−	4.67208e7i	0.361757	−	0.792742i
$167$	1.16629e8			1.93775			0.968876	−	0.247547i	$-0.0796245\pi$
$167$	1.16629e8			1.93775			0.968876	+	0.247547i	$0.0796245\pi$
$168$		0			0
$169$	5.81733e7			0.927087
$170$	−4.33509e7	+	9.49976e7i	−0.676748	+	1.48300i
$171$		0			0
$172$	3.59965e7	−	3.12274e7i	0.539399	−	0.467936i
$173$			1.20656e8i			1.77169i	0.463985	+	0.885843i	$0.346419\pi$
$173$			1.20656e8i			1.77169i	−0.463985	+	0.885843i	$0.653581\pi$
$174$		0			0
$175$	−2.61249e7			−0.368486
$176$	−3.57286e7	−	5.09501e6i	−0.493995	−	0.0704450i
$177$		0			0
$178$	7.25989e7	+	3.31295e7i	0.964851	+	0.440297i
$179$			1.08890e8i			1.41907i	0.704670	+	0.709535i	$0.251095\pi$
$179$			1.08890e8i			1.41907i	−0.704670	+	0.709535i	$0.748905\pi$
$180$		0			0
$181$		−	1.29863e7i		−	0.162784i	−0.996682	−	0.0813918i	$-0.974063\pi$
$181$		−	1.29863e7i		−	0.162784i	0.996682	−	0.0813918i	$-0.0259365\pi$
$182$	−1.19871e7	+	2.62682e7i	−0.147389	+	0.322983i
$183$		0			0
$184$	1.26283e7	+	4.28711e7i	0.149446	+	0.507344i
$185$	−1.76423e8			−2.04858
$186$		0			0
$187$			6.42848e7i			0.718890i
$188$	8.47106e7	+	9.76476e7i	0.929791	+	1.07179i
$189$		0			0
$190$	−1.45540e8	−	6.64154e7i	−1.53938	−	0.702475i
$191$	9.02408e7			0.937101			0.468550	−	0.883437i	$-0.344776\pi$
$191$	9.02408e7			0.937101			0.468550	+	0.883437i	$0.344776\pi$
$192$		0			0
$193$	−1.84834e8			−1.85068			−0.925340	−	0.379138i	$-0.876221\pi$
$193$	−1.84834e8			−1.85068			−0.925340	+	0.379138i	$0.876221\pi$
$194$	5.92204e7	+	2.70244e7i	0.582325	+	0.265736i
$195$		0			0
$196$	5.03339e7	+	5.80209e7i	0.477490	+	0.550412i
$197$			5.92205e7i			0.551874i	0.961176	+	0.275937i	$0.0889881\pi$
$197$			5.92205e7i			0.551874i	−0.961176	+	0.275937i	$0.911012\pi$
$198$		0			0
$199$	3.52395e7			0.316988			0.158494	−	0.987360i	$-0.449336\pi$
$199$	3.52395e7			0.316988			0.158494	+	0.987360i	$0.449336\pi$
$200$	8.95950e6	+	3.04160e7i	0.0791915	+	0.268842i
$201$		0			0
$202$	6.51064e7	−	1.42672e8i	0.555769	−	1.21789i
$203$		−	2.26236e8i		−	1.89813i
$204$		0			0
$205$			5.86004e7i			0.475075i
$206$	−7.62265e7	−	3.47849e7i	−0.607534	−	0.277240i
$207$		0			0
$208$	3.46938e7	+	4.94744e6i	0.267319	+	0.0381205i
$209$	−9.84870e7			−0.746220
$210$		0			0
$211$			2.14798e8i			1.57413i	0.616869	+	0.787066i	$0.288401\pi$
$211$			2.14798e8i			1.57413i	−0.616869	+	0.787066i	$0.711599\pi$
$212$	−2.37483e7	+	2.06020e7i	−0.171182	+	0.148502i
$213$		0			0
$214$	−8.63669e7	+	1.89261e8i	−0.602419	+	1.32012i
$215$	−1.17743e8			−0.807977
$216$		0			0
$217$	1.59014e8			1.05640
$218$	−6.10646e7	+	1.33815e8i	−0.399201	+	0.874795i
$219$		0			0
$220$	5.84333e7	+	6.73572e7i	0.369982	+	0.426486i
$221$		−	6.24229e7i		−	0.389019i
$222$		0			0
$223$	7.82914e7			0.472767			0.236383	−	0.971660i	$-0.424038\pi$
$223$	7.82914e7			0.472767			0.236383	+	0.971660i	$0.424038\pi$
$224$	1.19305e8	−	1.86231e8i	0.709234	−	1.10709i
$225$		0			0
$226$	2.64943e8	+	1.20903e8i	1.52677	+	0.696720i
$227$			2.67847e8i			1.51983i	0.650021	+	0.759917i	$0.274760\pi$
$227$			2.67847e8i			1.51983i	−0.650021	+	0.759917i	$0.725240\pi$
$228$		0			0
$229$		−	2.73358e8i		−	1.50421i	−0.659045	−	0.752104i	$-0.729039\pi$
$229$		−	2.73358e8i		−	1.50421i	0.659045	−	0.752104i	$-0.270961\pi$
$230$	4.58433e7	−	1.00459e8i	0.248444	−	0.544432i
$231$		0			0
$232$	−2.63397e8	+	7.75875e7i	−1.38485	+	0.407928i
$233$	3.26079e8			1.68879			0.844397	−	0.535718i	$-0.179959\pi$
$233$	3.26079e8			1.68879			0.844397	+	0.535718i	$0.179959\pi$
$234$		0			0
$235$		−	3.19400e8i		−	1.60545i
$236$	7.86844e6	−	6.82598e6i	0.0389670	−	0.0338044i
$237$		0			0
$238$	−3.58399e8	−	1.63550e8i	−1.72325	−	0.786380i
$239$	8.21057e7			0.389028			0.194514	−	0.980900i	$-0.437687\pi$
$239$	8.21057e7			0.389028			0.194514	+	0.980900i	$0.437687\pi$
$240$		0			0
$241$	1.42032e8			0.653622			0.326811	−	0.945090i	$-0.394026\pi$
$241$	1.42032e8			0.653622			0.326811	+	0.945090i	$0.394026\pi$
$242$	−1.50633e8	−	6.87394e7i	−0.683230	−	0.311783i
$243$		0			0
$244$	−7.49448e7	+	6.50157e7i	−0.330276	+	0.286519i
$245$		−	1.89783e8i		−	0.824473i
$246$		0			0
$247$	9.56345e7			0.403808
$248$	−5.45337e7	−	1.85133e8i	−0.227030	−	0.770731i
$249$		0			0
$250$	−8.35260e7	+	1.83036e8i	−0.338089	+	0.740877i
$251$		−	2.23144e8i		−	0.890690i	−0.895359	−	0.445345i	$-0.853081\pi$
$251$		−	2.23144e8i		−	0.890690i	0.895359	−	0.445345i	$-0.146919\pi$
$252$		0			0
$253$		−	6.79808e7i		−	0.263915i
$254$	1.04568e8	+	4.77184e7i	0.400390	+	0.182712i
$255$		0			0
$256$	−2.57735e8	−	7.50335e7i	−0.960139	−	0.279521i
$257$	2.58377e8			0.949484			0.474742	−	0.880125i	$-0.342541\pi$
$257$	2.58377e8			0.949484			0.474742	+	0.880125i	$0.342541\pi$
$258$		0			0
$259$		−	6.65592e8i		−	2.38045i
$260$	−5.67409e7	−	6.54063e7i	−0.200212	−	0.230788i
$261$		0			0
$262$	−1.17647e8	+	2.57808e8i	−0.404136	+	0.885608i
$263$	−2.71648e8			−0.920793			−0.460396	−	0.887714i	$-0.652293\pi$
$263$	−2.71648e8			−0.920793			−0.460396	+	0.887714i	$0.652293\pi$
$264$		0			0
$265$	7.76795e7			0.256416
$266$	2.50566e8	−	5.49082e8i	0.816275	−	1.78876i
$267$		0			0
$268$	4.22707e8	−	3.66704e8i	1.34143	−	1.16371i
$269$		−	1.66488e8i		−	0.521494i	−0.965407	−	0.260747i	$-0.916031\pi$
$269$		−	1.66488e8i		−	0.521494i	0.965407	−	0.260747i	$-0.0839689\pi$
$270$		0			0
$271$	4.36637e7			0.133269			0.0666344	−	0.997777i	$-0.478774\pi$
$271$	4.36637e7			0.133269			0.0666344	+	0.997777i	$0.478774\pi$
$272$	−6.75020e7	+	4.73357e8i	−0.203388	+	1.42626i
$273$		0			0
$274$	2.30272e8	+	1.05082e8i	0.676262	+	0.308603i
$275$		−	4.82308e7i		−	0.139849i
$276$		0			0
$277$			2.76808e8i			0.782528i	0.920278	+	0.391264i	$0.127962\pi$
$277$			2.76808e8i			0.782528i	−0.920278	+	0.391264i	$0.872038\pi$
$278$	6.39324e7	−	1.40099e8i	0.178470	−	0.391092i
$279$		0			0
$280$	−5.24191e8	+	1.54408e8i	−1.42704	+	0.420356i
$281$	−1.48932e8			−0.400421			−0.200211	−	0.979753i	$-0.564163\pi$
$281$	−1.48932e8			−0.400421			−0.200211	+	0.979753i	$0.564163\pi$
$282$		0			0
$283$			2.28883e8i			0.600290i	0.953894	+	0.300145i	$0.0970351\pi$
$283$			2.28883e8i			0.600290i	−0.953894	+	0.300145i	$0.902965\pi$
$284$	−1.94649e8	−	2.24375e8i	−0.504240	−	0.581247i
$285$		0			0
$286$	−4.84953e7	−	2.21302e7i	−0.122580	−	0.0559376i
$287$	−2.21083e8			−0.552037
$288$		0			0
$289$	4.41349e8			1.07557
$290$	6.17215e8	+	2.81658e8i	1.48608	+	0.678155i
$291$		0			0
$292$	4.56630e8	+	5.26366e8i	1.07331	+	1.23722i
$293$		−	2.99647e8i		−	0.695942i	−0.937505	−	0.347971i	$-0.886871\pi$
$293$		−	2.99647e8i		−	0.695942i	0.937505	−	0.347971i	$-0.113129\pi$
$294$		0			0
$295$	−2.57373e7			−0.0583694
$296$	−7.74919e8	+	2.28264e8i	−1.73674	+	0.511584i
$297$		0			0
$298$	5.05126e7	−	1.10691e8i	0.110571	−	0.242302i
$299$			6.60119e7i			0.142815i
$300$		0			0
$301$		−	4.44209e8i		−	0.938868i
$302$	5.05707e8	+	2.30773e8i	1.05651	+	0.482125i
$303$		0			0
$304$	−7.25202e8	−	1.03416e8i	−1.48048	−	0.211120i
$305$	2.45141e8			0.494727
$306$		0			0
$307$			3.47105e8i			0.684662i	0.939579	+	0.342331i	$0.111216\pi$
$307$			3.47105e8i			0.684662i	−0.939579	+	0.342331i	$0.888784\pi$
$308$	−2.54119e8	+	2.20452e8i	−0.495576	+	0.429919i
$309$		0			0
$310$	−1.97968e8	+	4.33820e8i	−0.377424	+	0.827073i
$311$	−2.88699e8			−0.544231			−0.272115	−	0.962265i	$-0.587723\pi$
$311$	−2.88699e8			−0.544231			−0.272115	+	0.962265i	$0.587723\pi$
$312$		0			0
$313$	−5.29567e8			−0.976149			−0.488074	−	0.872802i	$-0.662300\pi$
$313$	−5.29567e8			−0.976149			−0.488074	+	0.872802i	$0.662300\pi$
$314$	1.33173e8	−	2.91831e8i	0.242752	−	0.531958i
$315$		0			0
$316$	2.95148e8	+	3.40223e8i	0.526180	+	0.606538i
$317$			4.77983e8i			0.842763i	0.906884	+	0.421381i	$0.138455\pi$
$317$			4.77983e8i			0.842763i	−0.906884	+	0.421381i	$0.861545\pi$
$318$		0			0
$319$	4.17669e8			0.720385
$320$	3.59541e8	+	5.57338e8i	0.613372	+	0.950809i
$321$		0			0
$322$	3.79005e8	+	1.72954e8i	0.632629	+	0.288692i
$323$			1.30482e9i			2.15448i
$324$		0			0
$325$			4.68338e7i			0.0756777i
$326$	−6.75407e7	+	1.48006e8i	−0.107970	+	0.236602i
$327$		0			0
$328$	7.58200e7	+	2.57397e8i	0.118638	+	0.402758i
$329$	1.20501e9			1.86553
$330$		0			0
$331$			3.00324e8i			0.455189i	0.973756	+	0.227595i	$0.0730861\pi$
$331$			3.00324e8i			0.455189i	−0.973756	+	0.227595i	$0.926914\pi$
$332$	−4.38888e8	+	3.80741e8i	−0.658218	+	0.571014i
$333$		0			0
$334$	−1.20042e9	−	5.47796e8i	−1.76287	−	0.804464i
$335$	−1.38265e9			−2.00935
$336$		0			0
$337$	−1.42727e8			−0.203143			−0.101572	−	0.994828i	$-0.532387\pi$
$337$	−1.42727e8			−0.203143			−0.101572	+	0.994828i	$0.532387\pi$
$338$	−5.98759e8	−	2.73236e8i	−0.843419	−	0.384883i
$339$		0			0
$340$	8.92392e8	−	7.74163e8i	1.23134	−	1.06821i
$341$			2.93566e8i			0.400927i
$342$		0			0
$343$	−2.66620e8			−0.356749
$344$	−5.17172e8	+	1.52341e8i	−0.684984	+	0.201772i
$345$		0			0
$346$	5.66711e8	−	1.24187e9i	0.735521	−	1.61179i
$347$			3.44497e8i			0.442621i	0.975203	+	0.221310i	$0.0710334\pi$
$347$			3.44497e8i			0.442621i	−0.975203	+	0.221310i	$0.928967\pi$
$348$		0			0
$349$			4.86434e7i			0.0612541i	0.999531	+	0.0306271i	$0.00975042\pi$
$349$			4.86434e7i			0.0612541i	−0.999531	+	0.0306271i	$0.990250\pi$
$350$	2.68895e8	+	1.22707e8i	0.335231	+	0.152978i
$351$		0			0
$352$	3.43812e8	+	2.20256e8i	0.420167	+	0.269171i
$353$	9.56323e8			1.15716			0.578580	−	0.815626i	$-0.303607\pi$
$353$	9.56323e8			1.15716			0.578580	+	0.815626i	$0.303607\pi$
$354$		0			0
$355$			7.33920e8i			0.870662i
$356$	−5.91629e8	−	6.81983e8i	−0.694984	−	0.801121i
$357$		0			0
$358$	5.11449e8	−	1.12077e9i	0.589131	−	1.29100i
$359$	−3.16193e8			−0.360679			−0.180340	−	0.983604i	$-0.557720\pi$
$359$	−3.16193e8			−0.360679			−0.180340	+	0.983604i	$0.557720\pi$
$360$		0			0
$361$	−1.10517e9			−1.23638
$362$	−6.09957e7	+	1.33664e8i	−0.0675801	+	0.148093i
$363$		0			0
$364$	2.46759e8	−	2.14067e8i	0.268175	−	0.232646i
$365$		−	1.72172e9i		−	1.85326i
$366$		0			0
$367$	−1.18396e9			−1.25028			−0.625138	−	0.780514i	$-0.714957\pi$
$367$	−1.18396e9			−1.25028			−0.625138	+	0.780514i	$0.714957\pi$
$368$	7.13830e7	−	5.00572e8i	0.0746668	−	0.523600i
$369$		0			0
$370$	1.81586e9	+	8.28644e8i	1.86370	+	0.850475i
$371$			2.93062e8i			0.297956i
$372$		0			0
$373$			6.53550e8i			0.652076i	0.945357	+	0.326038i	$0.105714\pi$
$373$			6.53550e8i			0.652076i	−0.945357	+	0.326038i	$0.894286\pi$
$374$	3.01941e8	−	6.61662e8i	0.298450	−	0.654012i
$375$		0			0
$376$	−4.13255e8	−	1.40293e9i	−0.400923	−	1.36107i
$377$	−4.05572e8			−0.389828
$378$		0			0
$379$		−	2.25264e8i		−	0.212547i	−0.994337	−	0.106273i	$-0.966108\pi$
$379$		−	2.25264e8i		−	0.212547i	0.994337	−	0.106273i	$-0.0338919\pi$
$380$	1.18605e9	+	1.36718e9i	1.10882	+	1.27816i
$381$		0			0
$382$	−9.28819e8	−	4.23854e8i	−0.852529	−	0.389040i
$383$	−7.88929e8			−0.717534			−0.358767	−	0.933427i	$-0.616803\pi$
$383$	−7.88929e8			−0.717534			−0.358767	+	0.933427i	$0.616803\pi$
$384$		0			0
$385$	8.31211e8			0.742333
$386$	1.90243e9	+	8.68151e8i	1.68366	+	0.768316i
$387$		0			0
$388$	−4.82604e8	−	5.56307e8i	−0.419450	−	0.483508i
$389$			5.49026e8i			0.472900i	0.971644	+	0.236450i	$0.0759840\pi$
$389$			5.49026e8i			0.472900i	−0.971644	+	0.236450i	$0.924016\pi$
$390$		0			0
$391$	−9.00655e8			−0.761974
$392$	−2.45551e8	−	8.33604e8i	−0.205892	−	0.698970i
$393$		0			0
$394$	2.78154e8	−	6.09536e8i	0.229112	−	0.502068i
$395$		−	1.11285e9i		−	0.908546i
$396$		0			0
$397$			1.35509e9i			1.08693i	0.839431	+	0.543466i	$0.182888\pi$
$397$			1.35509e9i			1.08693i	−0.839431	+	0.543466i	$0.817112\pi$
$398$	−3.62708e8	−	1.65517e8i	−0.288381	−	0.131599i
$399$		0			0
$400$	5.06445e7	−	3.55144e8i	0.0395660	−	0.277456i
$401$	−1.38484e8			−0.107249			−0.0536244	−	0.998561i	$-0.517077\pi$
$401$	−1.38484e8			−0.107249			−0.0536244	+	0.998561i	$0.517077\pi$
$402$		0			0
$403$		−	2.85063e8i		−	0.216957i
$404$	−1.34024e9	+	1.16267e9i	−1.01122	+	0.877250i
$405$		0			0
$406$	−1.06261e9	+	2.32857e9i	−0.788015	+	1.72683i
$407$	1.22879e9			0.903436
$408$		0			0
$409$	−1.35047e9			−0.976010			−0.488005	−	0.872841i	$-0.662275\pi$
$409$	−1.35047e9			−0.976010			−0.488005	+	0.872841i	$0.662275\pi$
$410$	2.75242e8	−	6.03155e8i	0.197229	−	0.432200i
$411$		0			0
$412$	6.21191e8	+	7.16059e8i	0.437608	+	0.504439i
$413$		−	9.70993e7i		−	0.0678252i
$414$		0			0
$415$	1.43558e9			0.985958
$416$	−3.33854e8	−	2.13876e8i	−0.227368	−	0.145659i
$417$		0			0
$418$	1.01369e9	+	4.62586e8i	0.678875	+	0.309796i
$419$		−	2.80534e9i		−	1.86310i	−0.363612	−	0.931550i	$-0.618457\pi$
$419$		−	2.80534e9i		−	1.86310i	0.363612	−	0.931550i	$-0.381543\pi$
$420$		0			0
$421$			4.48841e8i			0.293160i	0.989199	+	0.146580i	$0.0468266\pi$
$421$			4.48841e8i			0.293160i	−0.989199	+	0.146580i	$0.953173\pi$
$422$	1.00889e9	−	2.21084e9i	0.653506	−	1.43207i
$423$		0			0
$424$	3.41199e8	−	1.00505e8i	0.217384	−	0.0640338i
$425$	−6.38993e8			−0.403771
$426$		0			0
$427$			9.24845e8i			0.574872i
$428$	1.77789e9	−	1.54234e9i	1.09610	−	0.950885i
$429$		0			0
$430$	1.21188e9	+	5.53027e8i	0.735058	+	0.335434i
$431$	2.67589e9			1.60990			0.804949	−	0.593344i	$-0.202192\pi$
$431$	2.67589e9			1.60990			0.804949	+	0.593344i	$0.202192\pi$
$432$		0			0
$433$	1.68907e9			0.999860			0.499930	−	0.866066i	$-0.333359\pi$
$433$	1.68907e9			0.999860			0.499930	+	0.866066i	$0.333359\pi$
$434$	−1.63668e9	−	7.46877e8i	−0.961058	−	0.438566i
$435$		0			0
$436$	1.25703e9	−	1.09049e9i	0.726348	−	0.630117i
$437$		−	1.37984e9i		−	0.790941i
$438$		0			0
$439$	−9.19152e8			−0.518515			−0.259258	−	0.965808i	$-0.583478\pi$
$439$	−9.19152e8			−0.518515			−0.259258	+	0.965808i	$0.583478\pi$
$440$	−2.85063e8	−	9.67741e8i	−0.159535	−	0.541595i
$441$		0			0
$442$	−2.93195e8	+	6.42498e8i	−0.161503	+	0.353911i
$443$			2.83982e9i			1.55195i	0.630764	+	0.775975i	$0.282742\pi$
$443$			2.83982e9i			1.55195i	−0.630764	+	0.775975i	$0.717258\pi$
$444$		0			0
$445$			2.23073e9i			1.20002i
$446$	−8.05827e8	−	3.67728e8i	−0.430100	−	0.196271i
$447$		0			0
$448$	−2.10267e9	+	1.35645e9i	−1.10484	+	0.712737i
$449$	−1.24346e9			−0.648291			−0.324145	−	0.946007i	$-0.605077\pi$
$449$	−1.24346e9			−0.648291			−0.324145	+	0.946007i	$0.605077\pi$
$450$		0			0
$451$		−	4.08154e8i		−	0.209511i
$452$	−2.15909e9	−	2.48883e9i	−1.09973	−	1.26768i
$453$		0			0
$454$	1.25805e9	−	2.75686e9i	0.630963	−	1.38267i
$455$	−8.07136e8			−0.401705
$456$		0			0
$457$	−1.93504e8			−0.0948384			−0.0474192	−	0.998875i	$-0.515100\pi$
$457$	−1.93504e8			−0.0948384			−0.0474192	+	0.998875i	$0.515100\pi$
$458$	−1.28394e9	+	2.81358e9i	−0.624476	+	1.36845i
$459$		0			0
$460$	−9.43700e8	+	8.18673e8i	−0.452045	+	0.392155i
$461$			1.97565e9i			0.939197i	0.882880	+	0.469599i	$0.155601\pi$
$461$			1.97565e9i			0.939197i	−0.882880	+	0.469599i	$0.844399\pi$
$462$		0			0
$463$	−1.13506e9			−0.531478			−0.265739	−	0.964045i	$-0.585616\pi$
$463$	−1.13506e9			−0.531478			−0.265739	+	0.964045i	$0.585616\pi$
$464$	3.07548e9	+	4.38572e8i	1.42922	+	0.203811i
$465$		0			0
$466$	−3.35622e9	−	1.53157e9i	−1.53638	−	0.701108i
$467$		−	2.02735e9i		−	0.921127i	−0.887627	−	0.460564i	$-0.847647\pi$
$467$		−	2.02735e9i		−	0.921127i	0.887627	−	0.460564i	$-0.152353\pi$
$468$		0			0
$469$		−	5.21635e9i		−	2.33487i
$470$	−1.50020e9	+	3.28748e9i	−0.666509	+	1.46056i
$471$		0			0
$472$	−1.13048e8	+	3.33001e7i	−0.0494843	+	0.0145763i
$473$	8.20081e8			0.356322
$474$		0			0
$475$		−	9.78964e8i		−	0.419121i
$476$	2.92069e9	+	3.36674e9i	1.24126	+	1.43082i
$477$		0			0
$478$	−8.45086e8	−	3.85644e8i	−0.353919	−	0.161506i
$479$	1.67866e9			0.697893			0.348947	−	0.937143i	$-0.386539\pi$
$479$	1.67866e9			0.697893			0.348947	+	0.937143i	$0.386539\pi$
$480$		0			0
$481$	−1.19320e9			−0.488884
$482$	−1.46189e9	−	6.67113e8i	−0.594634	−	0.271353i
$483$		0			0
$484$	1.22755e9	+	1.41502e9i	0.492132	+	0.567290i
$485$			1.81965e9i			0.724256i
$486$		0			0
$487$	3.64775e9			1.43111			0.715556	−	0.698555i	$-0.246173\pi$
$487$	3.64775e9			1.43111			0.715556	+	0.698555i	$0.246173\pi$
$488$	1.07676e9	−	3.17174e8i	0.419418	−	0.123546i
$489$		0			0
$490$	−8.91397e8	+	1.95338e9i	−0.342283	+	0.750066i
$491$			2.01669e9i			0.768870i	0.923152	+	0.384435i	$0.125604\pi$
$491$			2.01669e9i			0.768870i	−0.923152	+	0.384435i	$0.874396\pi$
$492$		0			0
$493$		−	5.53355e9i		−	2.07989i
$494$	−9.84334e8	−	4.49188e8i	−0.367365	−	0.167642i
$495$		0			0
$496$	−3.08258e8	+	2.16165e9i	−0.113430	+	0.795426i
$497$	−2.76887e9			−1.01171
$498$		0			0
$499$		−	1.19631e9i		−	0.431014i	−0.976502	−	0.215507i	$-0.930860\pi$
$499$		−	1.19631e9i		−	0.431014i	0.976502	−	0.215507i	$-0.0691404\pi$
$500$	1.71941e9	−	1.49161e9i	0.615155	−	0.533655i
$501$		0			0
$502$	−1.04809e9	+	2.29674e9i	−0.369773	+	0.810307i
$503$	−1.62416e9			−0.569038			−0.284519	−	0.958670i	$-0.591834\pi$
$503$	−1.62416e9			−0.569038			−0.284519	+	0.958670i	$0.591834\pi$
$504$		0			0
$505$	4.38385e9			1.51473
$506$	−3.19301e8	+	6.99704e8i	−0.109565	+	0.240097i
$507$		0			0
$508$	−8.52159e8	−	9.82300e8i	−0.288401	−	0.332446i
$509$			2.38008e9i			0.799981i	0.916519	+	0.399990i	$0.130987\pi$
$509$			2.38008e9i			0.799981i	−0.916519	+	0.399990i	$0.869013\pi$
$510$		0			0
$511$	6.49554e9			2.15349
$512$	2.30036e9	+	1.98286e9i	0.757444	+	0.652900i
$513$		0			0
$514$	−2.65939e9	−	1.21358e9i	−0.863795	−	0.394181i
$515$		−	2.34219e9i		−	0.755609i
$516$		0			0
$517$			2.22463e9i			0.708014i
$518$	−3.12623e9	+	6.85072e9i	−0.988251	+	2.16562i
$519$		0			0
$520$	2.76806e8	+	9.39712e8i	0.0863305	+	0.293078i
$521$	−5.54589e9			−1.71806			−0.859032	−	0.511922i	$-0.828934\pi$
$521$	−5.54589e9			−1.71806			−0.859032	+	0.511922i	$0.828934\pi$
$522$		0			0
$523$		−	3.10271e9i		−	0.948386i	−0.880421	−	0.474193i	$-0.842740\pi$
$523$		−	3.10271e9i		−	0.948386i	0.880421	−	0.474193i	$-0.157260\pi$
$524$	2.42181e9	−	2.10095e9i	0.735326	−	0.637905i
$525$		0			0
$526$	2.79598e9	+	1.27591e9i	0.837692	+	0.382270i
$527$	3.88935e9			1.15755
$528$		0			0
$529$	−2.45239e9			−0.720268
$530$	−7.99529e8	−	3.64854e8i	−0.233275	−	0.106452i
$531$		0			0
$532$	−5.15799e9	+	4.47462e9i	−1.48522	+	1.28845i
$533$			3.96333e8i			0.113374i
$534$		0			0
$535$	−5.81539e9			−1.64188
$536$	−6.07317e9	+	1.78894e9i	−1.70349	+	0.501787i
$537$		0			0
$538$	−7.81980e8	+	1.71360e9i	−0.216500	+	0.474430i
$539$			1.32185e9i			0.363597i
$540$		0			0
$541$		−	3.26721e9i		−	0.887128i	−0.896243	−	0.443564i	$-0.853714\pi$
$541$		−	3.26721e9i		−	0.887128i	0.896243	−	0.443564i	$-0.146286\pi$
$542$	−4.49416e8	−	2.05085e8i	−0.121241	−	0.0553269i
$543$		0			0
$544$	2.91809e9	−	4.55505e9i	0.777147	−	1.21310i
$545$	−4.11169e9			−1.08801
$546$		0			0
$547$			4.58100e9i			1.19675i	0.801214	+	0.598377i	$0.204188\pi$
$547$			4.58100e9i			1.19675i	−0.801214	+	0.598377i	$0.795812\pi$
$548$	−1.87656e9	−	2.16314e9i	−0.487113	−	0.561504i
$549$		0			0
$550$	−2.26536e8	+	4.96423e8i	−0.0580588	+	0.127228i
$551$	8.47763e9			2.15896
$552$		0			0
$553$	4.19846e9			1.05573
$554$	1.30015e9	−	2.84910e9i	0.324869	−	0.711906i
$555$		0			0
$556$	−1.31607e9	+	1.14171e9i	−0.324726	+	0.281704i
$557$			1.65111e9i			0.404838i	0.979299	+	0.202419i	$0.0648804\pi$
$557$			1.65111e9i			0.404838i	−0.979299	+	0.202419i	$0.935120\pi$
$558$		0			0
$559$	−7.96329e8			−0.192820
$560$	6.12056e9	+	8.72810e8i	1.47276	+	0.210020i
$561$		0			0
$562$	1.53291e9	+	6.99524e8i	0.364284	+	0.166236i
$563$		−	2.19386e9i		−	0.518118i	−0.965861	−	0.259059i	$-0.916588\pi$
$563$		−	2.19386e9i		−	0.518118i	0.965861	−	0.259059i	$-0.0834125\pi$
$564$		0			0
$565$			8.14084e9i			1.89889i
$566$	1.07505e9	−	2.35582e9i	0.249212	−	0.546115i
$567$		0			0
$568$	9.49580e8	+	3.22367e9i	0.217426	+	0.738127i
$569$	−1.95566e9			−0.445041			−0.222520	−	0.974928i	$-0.571428\pi$
$569$	−1.95566e9			−0.445041			−0.222520	+	0.974928i	$0.571428\pi$
$570$		0			0
$571$		−	1.49759e9i		−	0.336640i	−0.985732	−	0.168320i	$-0.946166\pi$
$571$		−	1.49759e9i		−	0.336640i	0.985732	−	0.168320i	$-0.0538342\pi$
$572$	3.95202e8	+	4.55557e8i	0.0882944	+	0.101779i
$573$		0			0
$574$	2.27553e9	+	1.03841e9i	0.502216	+	0.229180i
$575$	6.75732e8			0.148230
$576$		0			0
$577$	−5.22656e8			−0.113266			−0.0566332	−	0.998395i	$-0.518037\pi$
$577$	−5.22656e8			−0.113266			−0.0566332	+	0.998395i	$0.518037\pi$
$578$	−4.54266e9	−	2.07298e9i	−0.978504	−	0.446527i
$579$		0			0
$580$	−5.02986e9	−	5.79802e9i	−1.07043	−	1.23390i
$581$			5.41603e9i			1.14568i
$582$		0			0
$583$	−5.41041e8			−0.113081
$584$	−2.22764e9	−	7.56246e9i	−0.462807	−	1.57115i
$585$		0			0
$586$	−1.40742e9	+	3.08416e9i	−0.288922	+	0.633134i
$587$		−	4.68092e9i		−	0.955208i	−0.878575	−	0.477604i	$-0.841505\pi$
$587$		−	4.68092e9i		−	0.955208i	0.878575	−	0.477604i	$-0.158495\pi$
$588$		0			0
$589$			5.95865e9i			1.20156i
$590$	2.64905e8	+	1.20886e8i	0.0531017	+	0.0242322i
$591$		0			0
$592$	9.04812e9	+	1.29029e9i	1.79239	+	0.255600i
$593$	6.28546e9			1.23779			0.618893	−	0.785475i	$-0.287581\pi$
$593$	6.28546e9			1.23779			0.618893	+	0.785475i	$0.287581\pi$
$594$		0			0
$595$		−	1.10124e10i		−	2.14325i
$596$	−1.03982e9	+	9.02057e8i	−0.201185	+	0.174531i
$597$		0			0
$598$	3.10053e8	−	6.79438e8i	0.0592900	−	0.129926i
$599$	3.47628e9			0.660877			0.330438	−	0.943828i	$-0.392803\pi$
$599$	3.47628e9			0.660877			0.330438	+	0.943828i	$0.392803\pi$
$600$		0			0
$601$	1.06456e9			0.200037			0.100018	−	0.994986i	$-0.468110\pi$
$601$	1.06456e9			0.200037			0.100018	+	0.994986i	$0.468110\pi$
$602$	−2.08641e9	+	4.57209e9i	−0.389774	+	0.854136i
$603$		0			0
$604$	−4.12115e9	−	4.75053e9i	−0.761008	−	0.877229i
$605$		−	4.62847e9i		−	0.849755i
$606$		0			0
$607$	3.59754e9			0.652898			0.326449	−	0.945215i	$-0.394148\pi$
$607$	3.59754e9			0.652898			0.326449	+	0.945215i	$0.394148\pi$
$608$	6.97853e9	+	4.47064e9i	1.25922	+	0.806692i
$609$		0			0
$610$	−2.52315e9	−	1.15141e9i	−0.450079	−	0.205387i
$611$		−	2.16020e9i		−	0.383133i
$612$		0			0
$613$		−	7.53170e9i		−	1.32063i	−0.750988	−	0.660315i	$-0.770423\pi$
$613$		−	7.53170e9i		−	1.32063i	0.750988	−	0.660315i	$-0.229577\pi$
$614$	1.63032e9	−	3.57263e9i	0.284239	−	0.622872i
$615$		0			0
$616$	3.65101e9	−	1.07546e9i	0.629333	−	0.185380i
$617$	−2.41576e9			−0.414053			−0.207026	−	0.978335i	$-0.566379\pi$
$617$	−2.41576e9			−0.414053			−0.207026	+	0.978335i	$0.566379\pi$
$618$		0			0
$619$			8.34652e9i			1.41445i	0.706988	+	0.707226i	$0.250054\pi$
$619$			8.34652e9i			1.41445i	−0.706988	+	0.707226i	$0.749946\pi$
$620$	4.07524e9	−	3.53533e9i	0.686724	−	0.595742i
$621$		0			0
$622$	2.97148e9	+	1.35599e9i	0.495115	+	0.225939i
$623$	−8.41590e9			−1.39442
$624$		0			0
$625$	−7.33469e9			−1.20172
$626$	5.45066e9	+	2.48734e9i	0.888053	+	0.405251i
$627$		0			0
$628$	−2.74141e9	+	2.37821e9i	−0.441688	+	0.383171i
$629$		−	1.62798e10i		−	2.60839i
$630$		0			0
$631$	−5.01432e9			−0.794528			−0.397264	−	0.917704i	$-0.630040\pi$
$631$	−5.01432e9			−0.794528			−0.397264	+	0.917704i	$0.630040\pi$
$632$	−1.43986e9	−	4.88808e9i	−0.226887	−	0.770245i
$633$		0			0
$634$	2.24505e9	−	4.91972e9i	0.349875	−	0.766705i
$635$			3.21305e9i			0.497977i
$636$		0			0
$637$		−	1.28356e9i		−	0.196757i
$638$	−4.29892e9	−	1.96176e9i	−0.655371	−	0.299070i
$639$		0			0
$640$	−1.08287e9	−	7.42522e9i	−0.163285	−	1.11964i
$641$	−2.31712e9			−0.347493			−0.173747	−	0.984790i	$-0.555587\pi$
$641$	−2.31712e9			−0.347493			−0.173747	+	0.984790i	$0.555587\pi$
$642$		0			0
$643$			6.61604e9i			0.981431i	0.871320	+	0.490716i	$0.163265\pi$
$643$			6.61604e9i			0.981431i	−0.871320	+	0.490716i	$0.836735\pi$
$644$	−3.08862e9	−	3.56031e9i	−0.455684	−	0.525276i
$645$		0			0
$646$	6.12864e9	−	1.34301e10i	0.894438	−	1.96004i
$647$	−7.71651e9			−1.12010			−0.560049	−	0.828460i	$-0.689218\pi$
$647$	−7.71651e9			−1.12010			−0.560049	+	0.828460i	$0.689218\pi$
$648$		0			0
$649$	1.79261e8			0.0257412
$650$	2.19975e8	−	4.82045e8i	0.0314178	−	0.0688479i
$651$		0			0
$652$	1.39035e9	−	1.20615e9i	0.196452	−	0.170425i
$653$			4.65540e8i			0.0654275i	0.999465	+	0.0327138i	$0.0104150\pi$
$653$			4.65540e8i			0.0654275i	−0.999465	+	0.0327138i	$0.989585\pi$
$654$		0			0
$655$	−7.92160e9			−1.10146
$656$	4.28581e8	−	3.00542e9i	0.0592747	−	0.415663i
$657$		0			0
$658$	−1.24027e10	−	5.65981e9i	−1.69717	−	0.774482i
$659$		−	6.54492e9i		−	0.890851i	−0.895319	−	0.445426i	$-0.853052\pi$
$659$		−	6.54492e9i		−	0.890851i	0.895319	−	0.445426i	$-0.146948\pi$
$660$		0			0
$661$			2.92056e9i			0.393333i	0.980470	+	0.196666i	$0.0630116\pi$
$661$			2.92056e9i			0.393333i	−0.980470	+	0.196666i	$0.936988\pi$
$662$	1.41060e9	−	3.09113e9i	0.188973	−	0.414109i
$663$		0			0
$664$	6.30563e9	−	1.85742e9i	0.835873	−	0.246219i
$665$	1.68715e10			2.22473
$666$		0			0
$667$			5.85170e9i			0.763558i
$668$	9.78257e9	+	1.12766e10i	1.26980	+	1.46372i
$669$		0			0
$670$	1.42312e10	+	6.49421e9i	1.82801	+	0.834190i
$671$	−1.70741e9			−0.218177
$672$		0			0
$673$	−9.21338e9			−1.16511			−0.582554	−	0.812792i	$-0.697947\pi$
$673$	−9.21338e9			−1.16511			−0.582554	+	0.812792i	$0.697947\pi$
$674$	1.46904e9	+	6.70379e8i	0.184810	+	0.0843355i
$675$		0			0
$676$	4.87946e9	+	5.62464e9i	0.607516	+	0.700296i
$677$			9.01623e9i			1.11677i	0.829581	+	0.558386i	$0.188579\pi$
$677$			9.01623e9i			1.11677i	−0.829581	+	0.558386i	$0.811421\pi$
$678$		0			0
$679$	−6.86502e9			−0.841585
$680$	−1.28213e10	+	3.77670e9i	−1.56369	+	0.460608i
$681$		0			0
$682$	1.37886e9	−	3.02157e9i	0.166446	−	0.364744i
$683$			1.23626e10i			1.48470i	0.670012	+	0.742350i	$0.266289\pi$
$683$			1.23626e10i			1.48470i	−0.670012	+	0.742350i	$0.733711\pi$
$684$		0			0
$685$			7.07553e9i			0.841089i
$686$	2.74423e9	+	1.25229e9i	0.324553	+	0.148106i
$687$		0			0
$688$	6.03861e9	+	8.61123e8i	0.706932	+	0.100811i
$689$	5.25370e8			0.0611925
$690$		0			0
$691$			1.32041e10i			1.52243i	0.648500	+	0.761214i	$0.275397\pi$
$691$			1.32041e10i			1.52243i	−0.648500	+	0.761214i	$0.724603\pi$
$692$	−1.16659e10	+	1.01204e10i	−1.33828	+	1.16098i
$693$		0			0
$694$	1.61807e9	−	3.54579e9i	0.183755	−	0.402675i
$695$	4.30480e9			0.486414
$696$		0			0
$697$	−5.40750e9			−0.604897
$698$	2.28474e8	−	5.00671e8i	0.0254298	−	0.0557260i
$699$		0			0
$700$	−2.19130e9	−	2.52595e9i	−0.241467	−	0.278344i
$701$			8.01688e9i			0.879007i	0.898241	+	0.439504i	$0.144846\pi$
$701$			8.01688e9i			0.879007i	−0.898241	+	0.439504i	$0.855154\pi$
$702$		0			0
$703$	2.49414e10			2.70755
$704$	−2.50422e9	−	3.88188e9i	−0.270500	−	0.419312i
$705$		0			0
$706$	−9.84312e9	−	4.49178e9i	−1.05273	−	0.480398i
$707$			1.65390e10i			1.76012i
$708$		0			0
$709$		−	1.49980e8i		−	0.0158042i	−0.999969	−	0.00790209i	$-0.997485\pi$
$709$		−	1.49980e8i		−	0.0158042i	0.999969	−	0.00790209i	$-0.00251534\pi$
$710$	3.44716e9	−	7.55399e9i	0.361458	−	0.792086i
$711$		0			0
$712$	2.88622e9	+	9.79825e9i	0.299675	+	1.01735i
$713$	−4.11297e9			−0.424954
$714$		0			0
$715$		−	1.49010e9i		−	0.152456i
$716$	−1.05284e10	+	9.13349e9i	−1.07193	+	0.929911i
$717$		0			0
$718$	3.25447e9	+	1.48513e9i	0.328129	+	0.149737i
$719$	−4.43085e9			−0.444565			−0.222283	−	0.974982i	$-0.571351\pi$
$719$	−4.43085e9			−0.444565			−0.222283	+	0.974982i	$0.571351\pi$
$720$		0			0
$721$	8.83642e9			0.878017
$722$	1.13751e10	+	5.19089e9i	1.12480	+	0.513289i
$723$		0			0
$724$	1.25562e9	−	1.08926e9i	0.122962	−	0.106671i
$725$			4.15164e9i			0.404610i
$726$		0			0
$727$	−1.33493e10			−1.28851			−0.644255	−	0.764811i	$-0.722832\pi$
$727$	−1.33493e10			−1.28851			−0.644255	+	0.764811i	$0.722832\pi$
$728$	−3.54526e9	+	1.04431e9i	−0.340556	+	0.100316i
$729$		0			0
$730$	−8.08676e9	+	1.77210e10i	−0.769387	+	1.68601i
$731$		−	1.08650e10i		−	1.02877i
$732$		0			0
$733$			7.28787e9i			0.683497i	0.939791	+	0.341749i	$0.111019\pi$
$733$			7.28787e9i			0.683497i	−0.939791	+	0.341749i	$0.888981\pi$
$734$	1.21861e10	+	5.56097e9i	1.13744	+	0.519056i
$735$		0			0
$736$	−3.08587e9	+	4.81694e9i	−0.285302	+	0.445348i
$737$	9.63023e9			0.886136
$738$		0			0
$739$		−	9.20806e9i		−	0.839291i	−0.907688	−	0.419646i	$-0.862154\pi$
$739$		−	9.20806e9i		−	0.839291i	0.907688	−	0.419646i	$-0.137846\pi$
$740$	−1.47980e10	−	1.70579e10i	−1.34243	−	1.54744i
$741$		0			0
$742$	1.37649e9	−	3.01639e9i	0.123697	−	0.271066i
$743$	−1.72064e10			−1.53896			−0.769481	−	0.638669i	$-0.779485\pi$
$743$	−1.72064e10			−1.53896			−0.769481	+	0.638669i	$0.779485\pi$
$744$		0			0
$745$	3.40119e9			0.301359
$746$	3.06967e9	−	6.72677e9i	0.270711	−	0.593227i
$747$		0			0
$748$	−6.21555e9	+	5.39207e9i	−0.543030	+	0.471086i
$749$		−	2.19398e10i		−	1.90786i
$750$		0			0
$751$	−9.62266e9			−0.829001			−0.414501	−	0.910049i	$-0.636044\pi$
$751$	−9.62266e9			−0.829001			−0.414501	+	0.910049i	$0.636044\pi$
$752$	−2.33597e9	+	1.63809e10i	−0.200311	+	1.40468i
$753$		0			0
$754$	4.17441e9	+	1.90494e9i	0.354647	+	0.161838i
$755$			1.55387e10i			1.31402i
$756$		0			0
$757$		−	1.06615e10i		−	0.893269i	−0.894716	−	0.446635i	$-0.852622\pi$
$757$		−	1.06615e10i		−	0.893269i	0.894716	−	0.446635i	$-0.147378\pi$
$758$	−1.05805e9	+	2.31857e9i	−0.0882394	+	0.193365i
$759$		0			0
$760$	−5.78606e9	−	1.96427e10i	−0.478119	−	1.62313i
$761$	9.85241e8			0.0810394			0.0405197	−	0.999179i	$-0.487099\pi$
$761$	9.85241e8			0.0810394			0.0405197	+	0.999179i	$0.487099\pi$
$762$		0			0
$763$		−	1.55122e10i		−	1.26427i
$764$	7.56921e9	+	8.72518e9i	0.614078	+	0.707860i
$765$		0			0
$766$	8.12018e9	+	3.70554e9i	0.652777	+	0.297886i
$767$	−1.74069e8			−0.0139296
$768$		0			0
$769$	3.77074e9			0.299009			0.149505	−	0.988761i	$-0.452232\pi$
$769$	3.77074e9			0.299009			0.149505	+	0.988761i	$0.452232\pi$
$770$	−8.55537e9	−	3.90413e9i	−0.675338	−	0.308182i
$771$		0			0
$772$	−1.55035e10	−	1.78712e10i	−1.21274	−	1.39795i
$773$			5.93783e9i			0.462381i	0.972909	+	0.231190i	$0.0742620\pi$
$773$			5.93783e9i			0.462381i	−0.972909	+	0.231190i	$0.925738\pi$
$774$		0			0
$775$	−2.91805e9			−0.225184
$776$	2.35435e9	+	7.99264e9i	0.180865	+	0.614008i
$777$		0			0
$778$	2.57873e9	−	5.65094e9i	0.196326	−	0.430222i
$779$		−	8.28452e9i		−	0.627894i
$780$		0			0
$781$		−	5.11178e9i		−	0.383967i
$782$	9.27014e9	+	4.23031e9i	0.693207	+	0.316336i
$783$		0			0
$784$	−1.38800e9	+	9.73334e9i	−0.102869	+	0.721366i
$785$	8.96702e9			0.661614
$786$		0			0
$787$			3.44036e9i			0.251589i	0.992056	+	0.125795i	$0.0401480\pi$
$787$			3.44036e9i			0.251589i	−0.992056	+	0.125795i	$0.959852\pi$
$788$	−5.72589e9	+	4.96729e9i	−0.416870	+	0.361641i
$789$		0			0
$790$	−5.22697e9	+	1.14542e10i	−0.377186	+	0.826551i
$791$	−3.07130e10			−2.20651
$792$		0			0
$793$	1.65796e9			0.118064
$794$	6.36476e9	−	1.39475e10i	0.451243	−	0.988838i
$795$		0			0
$796$	2.95581e9	+	3.40722e9i	0.207721	+	0.239444i
$797$		−	1.01695e9i		−	0.0711537i	−0.999367	−	0.0355769i	$-0.988673\pi$
$797$		−	1.01695e9i		−	0.0711537i	0.999367	−	0.0355769i	$-0.0113269\pi$
$798$		0			0
$799$	2.94734e10			2.04417
$800$	−2.18935e9	+	3.41750e9i	−0.151182	+	0.235990i
$801$		0			0
$802$	1.42536e9	+	6.50446e8i	0.0975699	+	0.0445247i
$803$			1.19918e10i			0.817298i
$804$		0			0
$805$			1.16456e10i			0.786821i
$806$	−1.33892e9	+	2.93406e9i	−0.0900703	+	0.197377i
$807$		0			0
$808$	1.92556e10	−	5.67203e9i	1.28415	−	0.378267i
$809$	2.54544e10			1.69022			0.845108	−	0.534595i	$-0.179536\pi$
$809$	2.54544e10			1.69022			0.845108	+	0.534595i	$0.179536\pi$
$810$		0			0
$811$		−	1.46755e10i		−	0.966096i	−0.875594	−	0.483048i	$-0.839530\pi$
$811$		−	1.46755e10i		−	0.966096i	0.875594	−	0.483048i	$-0.160470\pi$
$812$	2.18743e10	−	1.89762e10i	1.43380	−	1.24384i
$813$		0			0
$814$	−1.26475e10	−	5.77153e9i	−0.821903	−	0.375064i
$815$	−4.54775e9			−0.294270
$816$		0			0
$817$	1.66456e10			1.06788
$818$	1.39000e10	+	6.34306e9i	0.887927	+	0.405194i
$819$		0			0
$820$	−5.66594e9	+	4.91528e9i	−0.358859	+	0.311315i
$821$		−	2.29026e10i		−	1.44439i	−0.691691	−	0.722193i	$-0.743134\pi$
$821$		−	2.29026e10i		−	1.44439i	0.691691	−	0.722193i	$-0.256866\pi$
$822$		0			0
$823$	−1.16697e10			−0.729729			−0.364864	−	0.931061i	$-0.618885\pi$
$823$	−1.16697e10			−0.729729			−0.364864	+	0.931061i	$0.618885\pi$
$824$	−3.03044e9	−	1.02878e10i	−0.188695	−	0.640589i
$825$		0			0
$826$	−4.56068e8	+	9.99410e8i	−0.0281578	+	0.0617040i
$827$		−	1.59065e10i		−	0.977928i	−0.872304	−	0.488964i	$-0.837375\pi$
$827$		−	1.59065e10i		−	0.977928i	0.872304	−	0.488964i	$-0.162625\pi$
$828$		0			0
$829$		−	2.22501e10i		−	1.35641i	−0.734873	−	0.678205i	$-0.762758\pi$
$829$		−	2.22501e10i		−	1.35641i	0.734873	−	0.678205i	$-0.237242\pi$
$830$	−1.47759e10	−	6.74280e9i	−0.896977	−	0.409324i
$831$		0			0
$832$	2.43169e9	+	3.76945e9i	0.146378	+	0.226906i
$833$	1.75127e10			1.04977
$834$		0			0
$835$		−	3.68851e10i		−	2.19254i
$836$	−8.26088e9	−	9.52248e9i	−0.488995	−	0.563674i
$837$		0			0
$838$	−1.31765e10	+	2.88744e10i	−0.773472	+	1.69496i
$839$	1.74931e10			1.02258			0.511292	−	0.859407i	$-0.329167\pi$
$839$	1.74931e10			1.02258			0.511292	+	0.859407i	$0.329167\pi$
$840$		0			0
$841$	−1.87025e10			−1.08421
$842$	2.10817e9	−	4.61977e9i	0.121706	−	0.266703i
$843$		0			0
$844$	−2.07683e10	+	1.80168e10i	−1.18906	+	1.03152i
$845$		−	1.83979e10i		−	1.04899i
$846$		0			0
$847$	1.74619e10			0.987414
$848$	−3.98392e9	−	5.68118e8i	−0.224349	−	0.0319929i
$849$		0			0
$850$	6.57694e9	+	3.00130e9i	0.367331	+	0.167627i
$851$			1.72158e10i			0.957580i
$852$		0			0
$853$		−	5.33580e9i		−	0.294359i	−0.989110	−	0.147180i	$-0.952980\pi$
$853$		−	5.33580e9i		−	0.294359i	0.989110	−	0.147180i	$-0.0470195\pi$
$854$	4.34392e9	−	9.51912e9i	0.238660	−	0.522991i
$855$		0			0
$856$	−2.55435e10	+	7.52422e9i	−1.39195	+	0.410019i
$857$	−1.18390e10			−0.642516			−0.321258	−	0.946992i	$-0.604106\pi$
$857$	−1.18390e10			−0.642516			−0.321258	+	0.946992i	$0.604106\pi$
$858$		0			0
$859$			3.05240e10i			1.64310i	0.570133	+	0.821552i	$0.306891\pi$
$859$			3.05240e10i			1.64310i	−0.570133	+	0.821552i	$0.693109\pi$
$860$	−9.87599e9	−	1.13843e10i	−0.529464	−	0.610323i
$861$		0			0
$862$	−2.75421e10	−	1.25685e10i	−1.46461	−	0.668354i
$863$	2.15426e9			0.114093			0.0570466	−	0.998372i	$-0.481832\pi$
$863$	2.15426e9			0.114093			0.0570466	+	0.998372i	$0.481832\pi$
$864$		0			0
$865$	3.81586e10			2.00464
$866$	−1.73850e10	−	7.93342e9i	−0.909625	−	0.415095i
$867$		0			0
$868$	1.33378e10	+	1.53747e10i	0.692252	+	0.797972i
$869$			7.75104e9i			0.400674i
$870$		0			0
$871$	−9.35131e9			−0.479522
$872$	−1.80602e10	+	5.31991e9i	−0.922391	+	0.271704i
$873$		0			0
$874$	−6.48101e9	+	1.42022e10i	−0.328362	+	0.719560i
$875$		−	2.12181e10i		−	1.07073i
$876$		0			0
$877$		−	1.98515e10i		−	0.993789i	−0.867811	−	0.496894i	$-0.834474\pi$
$877$		−	1.98515e10i		−	0.993789i	0.867811	−	0.496894i	$-0.165526\pi$
$878$	9.46053e9	+	4.31719e9i	0.471720	+	0.215263i
$879$		0			0
$880$	−1.61135e9	+	1.12996e10i	−0.0797077	+	0.558949i
$881$	−1.81176e10			−0.892659			−0.446330	−	0.894869i	$-0.647269\pi$
$881$	−1.81176e10			−0.892659			−0.446330	+	0.894869i	$0.647269\pi$
$882$		0			0
$883$			2.44143e10i			1.19339i	0.802469	+	0.596693i	$0.203519\pi$
$883$			2.44143e10i			1.19339i	−0.802469	+	0.596693i	$0.796481\pi$
$884$	6.03552e9	−	5.23590e9i	0.293854	−	0.254923i
$885$		0			0
$886$	1.33384e10	−	2.92293e10i	0.644297	−	1.41189i
$887$	8.86049e9			0.426310			0.213155	−	0.977018i	$-0.431626\pi$
$887$	8.86049e9			0.426310			0.213155	+	0.977018i	$0.431626\pi$
$888$		0			0
$889$	−1.21219e10			−0.578649
$890$	1.04776e10	−	2.29601e10i	0.498190	−	1.09172i
$891$		0			0
$892$	6.56691e9	+	7.56981e9i	0.309802	+	0.357115i
$893$			4.51545e10i			2.12188i
$894$		0			0
$895$	3.44377e10			1.60566
$896$	2.80132e10	−	4.08535e9i	1.30102	−	0.189737i
$897$		0			0
$898$	1.27985e10	+	5.84043e9i	0.589783	+	0.269140i
$899$		−	2.52698e10i		−	1.15996i
$900$		0			0
$901$			7.16807e9i			0.326487i
$902$	−1.91707e9	+	4.20100e9i	−0.0869791	+	0.190603i
$903$		0			0
$904$	1.05330e10	+	3.57578e10i	0.474201	+	1.60983i
$905$	−4.10706e9			−0.184188
$906$		0			0
$907$		−	2.38413e9i		−	0.106097i	−0.998592	−	0.0530487i	$-0.983106\pi$
$907$		−	2.38413e9i		−	0.106097i	0.998592	−	0.0530487i	$-0.0168939\pi$
$908$	−2.58975e10	+	2.24664e10i	−1.14804	+	0.995940i
$909$		0			0
$910$	8.30758e9	+	3.79105e9i	0.365452	+	0.166769i
$911$	1.97036e10			0.863440			0.431720	−	0.902008i	$-0.357907\pi$
$911$	1.97036e10			0.863440			0.431720	+	0.902008i	$0.357907\pi$
$912$		0			0
$913$	−9.99886e9			−0.434813
$914$	1.99168e9	+	9.08875e8i	0.0862794	+	0.0393724i
$915$		0			0
$916$	2.64303e10	−	2.29287e10i	1.13624	−	0.985701i
$917$		−	2.98859e10i		−	1.27989i
$918$		0			0
$919$	−2.34466e10			−0.996496			−0.498248	−	0.867035i	$-0.666023\pi$
$919$	−2.34466e10			−0.996496			−0.498248	+	0.867035i	$0.666023\pi$
$920$	1.35584e10	−	3.99384e9i	0.574053	−	0.169096i
$921$		0			0
$922$	9.27947e9	−	2.03347e10i	0.389911	−	0.854436i
$923$			4.96373e9i			0.207779i
$924$		0			0
$925$			1.22142e10i			0.507423i
$926$	1.16828e10	+	5.33129e9i	0.483513	+	0.220645i
$927$		0			0
$928$	−2.95949e10	−	1.89593e10i	−1.21562	−	0.778763i
$929$	−2.93528e10			−1.20114			−0.600571	−	0.799572i	$-0.705060\pi$
$929$	−2.93528e10			−1.20114			−0.600571	+	0.799572i	$0.705060\pi$
$930$		0			0
$931$			2.68302e10i			1.08968i
$932$	2.73508e10	+	3.15278e10i	1.10666	+	1.27567i
$933$		0			0
$934$	−9.52230e9	+	2.08668e10i	−0.382409	+	0.837997i
$935$	2.03307e10			0.813415
$936$		0			0
$937$	−2.54313e10			−1.00991			−0.504953	−	0.863147i	$-0.668490\pi$
$937$	−2.54313e10			−1.00991			−0.504953	+	0.863147i	$0.668490\pi$
$938$	−2.45008e10	+	5.36902e10i	−0.969327	+	2.12415i
$939$		0			0
$940$	3.08821e10	−	2.67906e10i	1.21271	−	1.05205i
$941$		−	3.53014e8i		−	0.0138111i	−0.999976	−	0.00690556i	$-0.997802\pi$
$941$		−	3.53014e8i		−	0.0138111i	0.999976	−	0.00690556i	$-0.00219812\pi$
$942$		0			0
$943$	5.71840e9			0.222067
$944$	1.31998e9	+	1.88232e8i	0.0510698	+	0.00728270i
$945$		0			0
$946$	−8.44082e9	−	3.85186e9i	−0.324165	−	0.147928i
$947$			1.56019e10i			0.596968i	0.954415	+	0.298484i	$0.0964810\pi$
$947$			1.56019e10i			0.596968i	−0.954415	+	0.298484i	$0.903519\pi$
$948$		0			0
$949$		−	1.16445e10i		−	0.442271i
$950$	−4.59812e9	+	1.00761e10i	−0.173999	+	0.381296i
$951$		0			0
$952$	−1.42484e10	−	4.83710e10i	−0.535226	−	1.81700i
$953$	−4.43492e10			−1.65982			−0.829909	−	0.557898i	$-0.811608\pi$
$953$	−4.43492e10			−1.65982			−0.829909	+	0.557898i	$0.811608\pi$
$954$		0			0
$955$		−	2.85396e10i		−	1.06032i
$956$	6.88685e9	+	7.93860e9i	0.254928	+	0.293861i
$957$		0			0
$958$	−1.72779e10	−	7.88455e9i	−0.634910	−	0.289733i
$959$	−2.66939e10			−0.977344
$960$		0			0
$961$	−9.75131e9			−0.354431
$962$	1.22812e10	+	5.60437e9i	0.444763	+	0.202962i
$963$		0			0
$964$	1.19133e10	+	1.37327e10i	0.428316	+	0.493728i
$965$			5.84557e10i			2.09402i
$966$		0			0
$967$	−3.48087e10			−1.23793			−0.618964	−	0.785419i	$-0.712447\pi$
$967$	−3.48087e10			−1.23793			−0.618964	+	0.785419i	$0.712447\pi$
$968$	−5.98853e9	−	2.03301e10i	−0.212205	−	0.720403i
$969$		0			0
$970$	8.54676e9	−	1.87291e10i	0.300677	−	0.658893i
$971$			2.46548e10i			0.864239i	0.901816	+	0.432119i	$0.142234\pi$
$971$			2.46548e10i			0.864239i	−0.901816	+	0.432119i	$0.857766\pi$
$972$		0			0
$973$			1.62408e10i			0.565212i
$974$	−3.75451e10	−	1.71332e10i	−1.30196	−	0.594131i
$975$		0			0
$976$	−1.25724e10	−	1.79286e9i	−0.432857	−	0.0617267i
$977$	6.03006e9			0.206867			0.103433	−	0.994636i	$-0.467017\pi$
$977$	6.03006e9			0.206867			0.103433	+	0.994636i	$0.467017\pi$
$978$		0			0
$979$		−	1.55371e10i		−	0.529214i
$980$	1.83497e10	−	1.59186e10i	0.622784	−	0.540274i
$981$		0			0
$982$	9.47221e9	−	2.07571e10i	0.319199	−	0.699481i
$983$	4.42805e10			1.48688			0.743439	−	0.668803i	$-0.233193\pi$
$983$	4.42805e10			1.48688			0.743439	+	0.668803i	$0.233193\pi$
$984$		0			0
$985$	1.87291e10			0.624438
$986$	−2.59907e10	+	5.69550e10i	−0.863472	+	1.89218i
$987$		0			0
$988$	8.02162e9	+	9.24668e9i	0.264614	+	0.305026i
$989$			1.14897e10i			0.377677i
$990$		0			0
$991$	1.23756e10			0.403932			0.201966	−	0.979392i	$-0.435267\pi$
$991$	1.23756e10			0.403932			0.201966	+	0.979392i	$0.435267\pi$
$992$	1.33259e10	−	2.08013e10i	0.433417	−	0.676549i
$993$		0			0
$994$	2.84990e10	+	1.30052e10i	0.920403	+	0.420013i
$995$		−	1.11448e10i		−	0.358668i
$996$		0			0
$997$			1.41900e10i			0.453470i	0.973957	+	0.226735i	$0.0728051\pi$
$997$			1.41900e10i			0.453470i	−0.973957	+	0.226735i	$0.927195\pi$
$998$	−5.61897e9	+	1.23132e10i	−0.178937	+	0.392116i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.d.c.37.3	✓	12
3.2	odd	2	inner	72.8.d.c.37.10	yes	12
4.3	odd	2		288.8.d.c.145.1		12
8.3	odd	2		288.8.d.c.145.12		12
8.5	even	2	inner	72.8.d.c.37.4	yes	12
12.11	even	2		288.8.d.c.145.11		12
24.5	odd	2	inner	72.8.d.c.37.9	yes	12
24.11	even	2		288.8.d.c.145.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.d.c.37.3	✓	12	1.1	even	1	trivial
72.8.d.c.37.4	yes	12	8.5	even	2	inner
72.8.d.c.37.9	yes	12	24.5	odd	2	inner
72.8.d.c.37.10	yes	12	3.2	odd	2	inner
288.8.d.c.145.1		12	4.3	odd	2
288.8.d.c.145.2		12	24.11	even	2
288.8.d.c.145.11		12	12.11	even	2
288.8.d.c.145.12		12	8.3	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-10.2927 - 4.69692i) q^{2} +(83.8779 + 96.6877i) q^{4} -316.260i q^{5} +1193.16 q^{7} +(-409.193 - 1389.14i) q^{8} +O(q^{10})\)	\(q+(-10.2927 - 4.69692i) q^{2} +(83.8779 + 96.6877i) q^{4} -316.260i q^{5} +1193.16 q^{7} +(-409.193 - 1389.14i) q^{8} +(-1485.45 + 3255.16i) q^{10} +2202.76i q^{11} -2138.97i q^{13} +(-12280.8 - 5604.17i) q^{14} +(-2313.01 + 16219.9i) q^{16} +29183.7 q^{17} +44710.6i q^{19} +(30578.5 - 26527.2i) q^{20} +(10346.2 - 22672.3i) q^{22} -30861.6 q^{23} -21895.6 q^{25} +(-10046.6 + 22015.7i) q^{26} +(100080. + 115364. i) q^{28} -189611. i q^{29} +133272. q^{31} +(99990.6 - 156082. i) q^{32} +(-300378. - 137073. i) q^{34} -377349. i q^{35} -557840. i q^{37} +(210002. - 460192. i) q^{38} +(-439330. + 129411. i) q^{40} -185292. q^{41} -372296. i q^{43} +(-212980. + 184763. i) q^{44} +(317648. + 144954. i) q^{46} +1.00993e6 q^{47} +600086. q^{49} +(225364. + 102842. i) q^{50} +(206812. - 179412. i) q^{52} +245619. i q^{53} +696647. q^{55} +(-488232. - 1.65747e6i) q^{56} +(-890589. + 1.95160e6i) q^{58} -81380.0i q^{59} +775123. i q^{61} +(-1.37172e6 - 625966. i) q^{62} +(-1.76227e6 + 1.13685e6i) q^{64} -676470. q^{65} -4.37188e6i q^{67} +(2.44787e6 + 2.82170e6i) q^{68} +(-1.77238e6 + 3.88392e6i) q^{70} -2.32062e6 q^{71} +5.44398e6 q^{73} +(-2.62013e6 + 5.74166e6i) q^{74} +(-4.32297e6 + 3.75023e6i) q^{76} +2.62825e6i q^{77} +3.51878e6 q^{79} +(5.12971e6 + 731512. i) q^{80} +(1.90715e6 + 870301. i) q^{82} +4.53923e6i q^{83} -9.22964e6i q^{85} +(-1.74865e6 + 3.83192e6i) q^{86} +(3.05995e6 - 901355. i) q^{88} -7.05346e6 q^{89} -2.55213e6i q^{91} +(-2.58860e6 - 2.98394e6i) q^{92} +(-1.03949e7 - 4.74355e6i) q^{94} +1.41402e7 q^{95} -5.75365e6 q^{97} +(-6.17648e6 - 2.81856e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 412 q^{4} + 136 q^{7}+O(q^{10}) \) 12 * q + 412 * q^4 + 136 * q^7	\( 12 q + 412 q^{4} + 136 q^{7} + 4680 q^{10} - 12472 q^{16} + 32624 q^{22} + 229820 q^{25} - 157288 q^{28} - 37224 q^{31} - 74432 q^{34} - 937520 q^{40} - 1264256 q^{46} + 2668188 q^{49} - 1539680 q^{52} + 6928960 q^{55} - 4035448 q^{58} - 3530192 q^{64} - 10228720 q^{70} + 13619048 q^{73} - 2441920 q^{76} + 20470552 q^{79} - 2507200 q^{82} - 26170912 q^{88} - 22132608 q^{94} + 27442456 q^{97}+O(q^{100}) \) 12 * q + 412 * q^4 + 136 * q^7 + 4680 * q^10 - 12472 * q^16 + 32624 * q^22 + 229820 * q^25 - 157288 * q^28 - 37224 * q^31 - 74432 * q^34 - 937520 * q^40 - 1264256 * q^46 + 2668188 * q^49 - 1539680 * q^52 + 6928960 * q^55 - 4035448 * q^58 - 3530192 * q^64 - 10228720 * q^70 + 13619048 * q^73 - 2441920 * q^76 + 20470552 * q^79 - 2507200 * q^82 - 26170912 * q^88 - 22132608 * q^94 + 27442456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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