LMFDB - Embedded newform 76.8.i.a.17.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(5,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 16]))

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

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

chi := DirichletCharacter("76.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.i (of order $9$, degree $6$, 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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			17.10
Character	$\chi$	$=$	76.17
Dual form			76.8.i.a.9.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(41.0960 + 34.4837i) q^{3} +(-501.471 + 182.521i) q^{5} +(762.203 + 1320.17i) q^{7} +(119.992 + 680.510i) q^{9} +O(q^{10})$	\(q+(41.0960 + 34.4837i) q^{3} +(-501.471 + 182.521i) q^{5} +(762.203 + 1320.17i) q^{7} +(119.992 + 680.510i) q^{9} +(1179.77 - 2043.43i) q^{11} +(-4529.47 + 3800.68i) q^{13} +(-26902.5 - 9791.70i) q^{15} +(2725.34 - 15456.2i) q^{17} +(-28940.1 - 7506.34i) q^{19} +(-14200.9 + 80537.5i) q^{21} +(-72281.9 - 26308.5i) q^{23} +(158313. - 132840. i) q^{25} +(40127.8 - 69503.4i) q^{27} +(28287.6 + 160427. i) q^{29} +(77803.7 + 134760. i) q^{31} +(118949. - 43293.8i) q^{33} +(-623182. - 522912. i) q^{35} -461293. q^{37} -317204. q^{39} +(-217549. - 182545. i) q^{41} +(-538197. + 195888. i) q^{43} +(-184380. - 319355. i) q^{45} +(-142560. - 808501. i) q^{47} +(-750135. + 1.29927e6i) q^{49} +(644985. - 541207. i) q^{51} +(988509. + 359788. i) q^{53} +(-218655. + 1.24005e6i) q^{55} +(-930476. - 1.30644e6i) q^{57} +(71094.7 - 403198. i) q^{59} +(2.00476e6 + 729674. i) q^{61} +(-806933. + 677097. i) q^{63} +(1.57770e6 - 2.73265e6i) q^{65} +(650715. + 3.69039e6i) q^{67} +(-2.06329e6 - 3.57372e6i) q^{69} +(-3.22198e6 + 1.17270e6i) q^{71} +(-1.68175e6 - 1.41115e6i) q^{73} +1.10868e7 q^{75} +3.59691e6 q^{77} +(-604000. - 506816. i) q^{79} +(5.46592e6 - 1.98943e6i) q^{81} +(787110. + 1.36331e6i) q^{83} +(1.45439e6 + 8.24825e6i) q^{85} +(-4.36960e6 + 7.56837e6i) q^{87} +(-1.75845e6 + 1.47552e6i) q^{89} +(-8.46993e6 - 3.08280e6i) q^{91} +(-1.44959e6 + 8.22106e6i) q^{93} +(1.58827e7 - 1.51795e6i) q^{95} +(-2.04280e6 + 1.15853e7i) q^{97} +(1.53214e6 + 557652. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{5}{9}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	41.0960	+	34.4837i	0.878771	+	0.737376i	0.965926	−	0.258819i	$-0.0833332\pi$
$3$	41.0960	+	34.4837i	0.878771	+	0.737376i	−0.0871553	+	0.996195i	$0.527778\pi$
$4$		0			0
$5$	−501.471	+	182.521i	−1.79412	+	0.653006i	−0.795209	+	0.606336i	$0.792639\pi$
$5$	−501.471	+	182.521i	−1.79412	+	0.653006i	−0.998910	+	0.0466700i	$0.985139\pi$
$6$		0			0
$7$	762.203	+	1320.17i	0.839900	+	1.45475i	0.889978	+	0.456004i	$0.150720\pi$
$7$	762.203	+	1320.17i	0.839900	+	1.45475i	−0.0500784	+	0.998745i	$0.515947\pi$
$8$		0			0
$9$	119.992	+	680.510i	0.0548661	+	0.311161i
$10$		0			0
$11$	1179.77	−	2043.43i	0.267254	−	0.462897i	−0.700898	−	0.713262i	$-0.747217\pi$
$11$	1179.77	−	2043.43i	0.267254	−	0.462897i	0.968152	+	0.250364i	$0.0805504\pi$
$12$		0			0
$13$	−4529.47	+	3800.68i	−0.571802	+	0.479799i	−0.882243	−	0.470794i	$-0.843968\pi$
$13$	−4529.47	+	3800.68i	−0.571802	+	0.479799i	0.310442	+	0.950592i	$0.399523\pi$
$14$		0			0
$15$	−26902.5	−	9791.70i	−2.05813	−	0.749098i
$16$		0			0
$17$	2725.34	−	15456.2i	0.134539	−	0.763010i	−0.840640	−	0.541594i	$-0.817821\pi$
$17$	2725.34	−	15456.2i	0.134539	−	0.763010i	0.975179	−	0.221416i	$-0.0710678\pi$
$18$		0			0
$19$	−28940.1	−	7506.34i	−0.967970	−	0.251067i
$19$	−28940.1	−	7506.34i	−0.967970	−	0.251067i
$20$		0			0
$21$	−14200.9	+	80537.5i	−0.334618	+	1.89771i
$22$		0			0
$23$	−72281.9	−	26308.5i	−1.23875	−	0.450867i	−0.362162	−	0.932115i	$-0.617961\pi$
$23$	−72281.9	−	26308.5i	−1.23875	−	0.450867i	−0.876584	+	0.481248i	$0.840184\pi$
$24$		0			0
$25$	158313.	−	132840.i	2.02640	−	1.70035i
$26$		0			0
$27$	40127.8	−	69503.4i	0.392349	−	0.679568i
$28$		0			0
$29$	28287.6	+	160427.i	0.215379	+	1.22147i	0.880248	+	0.474514i	$0.157376\pi$
$29$	28287.6	+	160427.i	0.215379	+	1.22147i	−0.664869	+	0.746960i	$0.731513\pi$
$30$		0			0
$31$	77803.7	+	134760.i	0.469066	+	0.812447i	0.999375	−	0.0353581i	$-0.0112572\pi$
$31$	77803.7	+	134760.i	0.469066	+	0.812447i	−0.530308	+	0.847805i	$0.677924\pi$
$32$		0			0
$33$	118949.	−	43293.8i	0.576184	−	0.209714i
$34$		0			0
$35$	−623182.	−	522912.i	−2.45684	−	2.06153i
$36$		0			0
$37$	−461293.			−1.49717			−0.748584	−	0.663040i	$-0.769266\pi$
$37$	−461293.			−1.49717			−0.748584	+	0.663040i	$0.769266\pi$
$38$		0			0
$39$	−317204.			−0.856275
$40$		0			0
$41$	−217549.	−	182545.i	−0.492962	−	0.413644i	0.362124	−	0.932130i	$-0.382052\pi$
$41$	−217549.	−	182545.i	−0.492962	−	0.413644i	−0.855086	+	0.518486i	$0.826496\pi$
$42$		0			0
$43$	−538197.	+	195888.i	−1.03229	+	0.375723i	−0.801952	−	0.597388i	$-0.796205\pi$
$43$	−538197.	+	195888.i	−1.03229	+	0.375723i	−0.230338	+	0.973111i	$0.573983\pi$
$44$		0			0
$45$	−184380.	−	319355.i	−0.301627	−	0.522433i
$46$		0			0
$47$	−142560.	−	808501.i	−0.200289	−	1.13589i	−0.904683	−	0.426085i	$-0.859892\pi$
$47$	−142560.	−	808501.i	−0.200289	−	1.13589i	0.704394	−	0.709809i	$-0.251219\pi$
$48$		0			0
$49$	−750135.	+	1.29927e6i	−0.910863	+	1.57766i
$50$		0			0
$51$	644985.	−	541207.i	0.680854	−	0.571305i
$52$		0			0
$53$	988509.	+	359788.i	0.912042	+	0.331956i	0.755068	−	0.655646i	$-0.227604\pi$
$53$	988509.	+	359788.i	0.912042	+	0.331956i	0.156974	+	0.987603i	$0.449826\pi$
$54$		0			0
$55$	−218655.	+	1.24005e6i	−0.177211	+	1.00501i
$56$		0			0
$57$	−930476.	−	1.30644e6i	−0.665492	−	0.934388i
$58$		0			0
$59$	71094.7	−	403198.i	0.0450667	−	0.255586i	−0.953948	−	0.299973i	$-0.903022\pi$
$59$	71094.7	−	403198.i	0.0450667	−	0.255586i	0.999014	+	0.0443871i	$0.0141335\pi$
$60$		0			0
$61$	2.00476e6	+	729674.i	1.13086	+	0.411599i	0.838604	−	0.544741i	$-0.183372\pi$
$61$	2.00476e6	+	729674.i	1.13086	+	0.411599i	0.292255	+	0.956340i	$0.405594\pi$
$62$		0			0
$63$	−806933.	+	677097.i	−0.406580	+	0.341161i
$64$		0			0
$65$	1.57770e6	−	2.73265e6i	0.712569	−	1.23421i
$66$		0			0
$67$	650715.	+	3.69039e6i	0.264319	+	1.49903i	0.770966	+	0.636876i	$0.219774\pi$
$67$	650715.	+	3.69039e6i	0.264319	+	1.49903i	−0.506647	+	0.862154i	$0.669115\pi$
$68$		0			0
$69$	−2.06329e6	−	3.57372e6i	−0.756116	−	1.30963i
$70$		0			0
$71$	−3.22198e6	+	1.17270e6i	−1.06836	+	0.388852i	−0.815564	−	0.578668i	$-0.803573\pi$
$71$	−3.22198e6	+	1.17270e6i	−1.06836	+	0.388852i	−0.252798	+	0.967519i	$0.581351\pi$
$72$		0			0
$73$	−1.68175e6	−	1.41115e6i	−0.505977	−	0.424565i	0.353734	−	0.935346i	$-0.384912\pi$
$73$	−1.68175e6	−	1.41115e6i	−0.505977	−	0.424565i	−0.859711	+	0.510781i	$0.829356\pi$
$74$		0			0
$75$	1.10868e7			3.03454
$76$		0			0
$77$	3.59691e6			0.897866
$78$		0			0
$79$	−604000.	−	506816.i	−0.137829	−	0.115653i	0.571267	−	0.820765i	$-0.306452\pi$
$79$	−604000.	−	506816.i	−0.137829	−	0.115653i	−0.709096	+	0.705112i	$0.750897\pi$
$80$		0			0
$81$	5.46592e6	−	1.98943e6i	1.14279	−	0.415941i
$82$		0			0
$83$	787110.	+	1.36331e6i	0.151099	+	0.261711i	0.931632	−	0.363404i	$-0.118385\pi$
$83$	787110.	+	1.36331e6i	0.151099	+	0.261711i	−0.780533	+	0.625115i	$0.785052\pi$
$84$		0			0
$85$	1.45439e6	+	8.24825e6i	0.256871	+	1.45679i
$86$		0			0
$87$	−4.36960e6	+	7.56837e6i	−0.711417	+	1.23221i
$88$		0			0
$89$	−1.75845e6	+	1.47552e6i	−0.264403	+	0.221860i	−0.765345	−	0.643621i	$-0.777432\pi$
$89$	−1.75845e6	+	1.47552e6i	−0.264403	+	0.221860i	0.500942	+	0.865481i	$0.332987\pi$
$90$		0			0
$91$	−8.46993e6	−	3.08280e6i	−1.17824	−	0.428845i
$92$		0			0
$93$	−1.44959e6	+	8.22106e6i	−0.186877	+	1.05983i
$94$		0			0
$95$	1.58827e7	−	1.51795e6i	1.90060	−	0.181645i
$96$		0			0
$97$	−2.04280e6	+	1.15853e7i	−0.227261	+	1.28886i	0.631055	+	0.775738i	$0.282622\pi$
$97$	−2.04280e6	+	1.15853e7i	−0.227261	+	1.28886i	−0.858316	+	0.513122i	$0.828489\pi$
$98$		0			0
$99$	1.53214e6	+	557652.i	0.158699	+	0.0577617i
$100$		0			0
$101$	2.67777e6	−	2.24692e6i	0.258612	−	0.217001i	−0.504258	−	0.863553i	$-0.668234\pi$
$101$	2.67777e6	−	2.24692e6i	0.258612	−	0.217001i	0.762870	+	0.646552i	$0.223790\pi$
$102$		0			0
$103$	2.47517e6	−	4.28712e6i	0.223190	−	0.386577i	−0.732585	−	0.680676i	$-0.761686\pi$
$103$	2.47517e6	−	4.28712e6i	0.223190	−	0.386577i	0.955775	+	0.294099i	$0.0950196\pi$
$104$		0			0
$105$	−7.57839e6	−	4.29792e7i	−0.638873	−	3.62323i
$106$		0			0
$107$	9.53220e6	+	1.65103e7i	0.752229	+	1.30290i	0.946740	+	0.321999i	$0.104355\pi$
$107$	9.53220e6	+	1.65103e7i	0.752229	+	1.30290i	−0.194511	+	0.980900i	$0.562312\pi$
$108$		0			0
$109$	−1.14314e7	+	4.16068e6i	−0.845484	+	0.307731i	−0.728198	−	0.685367i	$-0.759642\pi$
$109$	−1.14314e7	+	4.16068e6i	−0.845484	+	0.307731i	−0.117286	+	0.993098i	$0.537420\pi$
$110$		0			0
$111$	−1.89573e7	−	1.59071e7i	−1.31567	−	1.10398i
$112$		0			0
$113$	−7.21637e6			−0.470483			−0.235242	−	0.971937i	$-0.575588\pi$
$113$	−7.21637e6			−0.470483			−0.235242	+	0.971937i	$0.575588\pi$
$114$		0			0
$115$	4.10492e7			2.51688
$116$		0			0
$117$	−3.12990e6	−	2.62630e6i	−0.180667	−	0.151598i
$118$		0			0
$119$	2.24821e7	−	8.18280e6i	1.22299	−	0.445131i
$120$		0			0
$121$	6.95986e6	+	1.20548e7i	0.357151	+	0.618603i
$122$		0			0
$123$	−2.64557e6	−	1.50038e7i	−0.128189	−	0.726997i
$124$		0			0
$125$	−3.42973e7	+	5.94047e7i	−1.57063	+	2.72042i
$126$		0			0
$127$	−1.67316e6	+	1.40395e6i	−0.0724810	+	0.0608188i	−0.678308	−	0.734777i	$-0.737287\pi$
$127$	−1.67316e6	+	1.40395e6i	−0.0724810	+	0.0608188i	0.605827	+	0.795596i	$0.292842\pi$
$128$		0			0
$129$	−2.88727e7	−	1.05088e7i	−1.18420	−	0.431012i
$130$		0			0
$131$	1.17740e6	−	6.67735e6i	0.0457587	−	0.259510i	−0.953343	−	0.301889i	$-0.902383\pi$
$131$	1.17740e6	−	6.67735e6i	0.0457587	−	0.259510i	0.999102	+	0.0423792i	$0.0134937\pi$
$132$		0			0
$133$	−1.21485e7	−	4.39273e7i	−0.447757	−	1.61902i
$134$		0			0
$135$	−7.43714e6	+	4.21781e7i	−0.260158	+	1.47543i
$136$		0			0
$137$	−1.68689e6	−	613978.i	−0.0560486	−	0.0204000i	0.313844	−	0.949475i	$-0.398383\pi$
$137$	−1.68689e6	−	613978.i	−0.0560486	−	0.0204000i	−0.369892	+	0.929075i	$0.620605\pi$
$138$		0			0
$139$	−1.72243e7	+	1.44529e7i	−0.543988	+	0.456460i	−0.872899	−	0.487901i	$-0.837763\pi$
$139$	−1.72243e7	+	1.44529e7i	−0.543988	+	0.456460i	0.328911	+	0.944361i	$0.393318\pi$
$140$		0			0
$141$	2.20214e7	−	3.81422e7i	0.661573	−	1.14588i
$142$		0			0
$143$	2.42266e6	+	1.37396e7i	0.0692813	+	0.392914i
$144$		0			0
$145$	−4.34666e7	−	7.52864e7i	−1.18404	−	2.05083i
$146$		0			0
$147$	−7.56312e7	+	2.75275e7i	−1.96377	+	0.714753i
$148$		0			0
$149$	2.00067e6	+	1.67876e6i	0.0495476	+	0.0415754i	0.667225	−	0.744856i	$-0.267482\pi$
$149$	2.00067e6	+	1.67876e6i	0.0495476	+	0.0415754i	−0.617678	+	0.786431i	$0.711926\pi$
$150$		0			0
$151$	2.81266e7			0.664811			0.332406	−	0.943137i	$-0.392140\pi$
$151$	2.81266e7			0.664811			0.332406	+	0.943137i	$0.392140\pi$
$152$		0			0
$153$	1.08451e7			0.244801
$154$		0			0
$155$	−6.36128e7	−	5.33775e7i	−1.37209	−	1.15132i
$156$		0			0
$157$	1.69339e6	−	616342.i	0.0349227	−	0.0127108i	−0.324500	−	0.945886i	$-0.605196\pi$
$157$	1.69339e6	−	616342.i	0.0349227	−	0.0127108i	0.359422	+	0.933175i	$0.382974\pi$
$158$		0			0
$159$	2.82170e7	+	4.88733e7i	0.556700	+	0.964232i
$160$		0			0
$161$	−2.03617e7	−	1.15477e8i	−0.384525	−	2.18075i
$162$		0			0
$163$	−2.46490e7	+	4.26933e7i	−0.445803	+	0.772153i	−0.998108	−	0.0614891i	$-0.980415\pi$
$163$	−2.46490e7	+	4.26933e7i	−0.445803	+	0.772153i	0.552305	+	0.833642i	$0.313748\pi$
$164$		0			0
$165$	−5.17474e7	+	4.34213e7i	−0.896799	+	0.752504i
$166$		0			0
$167$	−1.83064e6	−	666300.i	−0.0304156	−	0.0110704i	0.326768	−	0.945105i	$-0.394041\pi$
$167$	−1.83064e6	−	666300.i	−0.0304156	−	0.0110704i	−0.357183	+	0.934034i	$0.616263\pi$
$168$		0			0
$169$	−4.82521e6	+	2.73652e7i	−0.0768977	+	0.436108i
$170$		0			0
$171$	1.63555e6	−	2.05947e7i	0.0250137	−	0.314970i
$172$		0			0
$173$	−1.29051e7	+	7.31884e7i	−0.189496	+	1.07468i	0.730546	+	0.682863i	$0.239266\pi$
$173$	−1.29051e7	+	7.31884e7i	−0.189496	+	1.07468i	−0.920042	+	0.391820i	$0.871845\pi$
$174$		0			0
$175$	2.96038e8	+	1.07749e8i	4.17556	+	1.51978i
$176$		0			0
$177$	1.68255e7	−	1.41182e7i	0.228066	−	0.191370i
$178$		0			0
$179$	9.36612e6	−	1.62226e7i	0.122060	−	0.211414i	−0.798520	−	0.601969i	$-0.794383\pi$
$179$	9.36612e6	−	1.62226e7i	0.122060	−	0.211414i	0.920580	+	0.390554i	$0.127717\pi$
$180$		0			0
$181$	−3.25580e6	−	1.84645e7i	−0.0408115	−	0.231453i	0.957579	−	0.288172i	$-0.0930474\pi$
$181$	−3.25580e6	−	1.84645e7i	−0.0408115	−	0.231453i	−0.998390	+	0.0567183i	$0.981936\pi$
$182$		0			0
$183$	5.72260e7	+	9.91183e7i	0.690262	+	1.19557i
$184$		0			0
$185$	2.31325e8	−	8.41954e7i	2.68610	−	0.977659i
$186$		0			0
$187$	−2.83682e7	−	2.38038e7i	−0.317239	−	0.266195i
$188$		0			0
$189$	1.22342e8			1.31813
$190$		0			0
$191$	−1.19715e8			−1.24317			−0.621585	−	0.783347i	$-0.713511\pi$
$191$	−1.19715e8			−1.24317			−0.621585	+	0.783347i	$0.713511\pi$
$192$		0			0
$193$	−1.11093e8	−	9.32182e7i	−1.11234	−	0.933362i	−0.114145	−	0.993464i	$-0.536413\pi$
$193$	−1.11093e8	−	9.32182e7i	−1.11234	−	0.933362i	−0.998192	+	0.0601019i	$0.980857\pi$
$194$		0			0
$195$	1.59069e8	−	5.78964e7i	1.53626	−	0.559152i
$196$		0			0
$197$	−5.19862e6	−	9.00427e6i	−0.0484458	−	0.0839106i	0.840786	−	0.541368i	$-0.182093\pi$
$197$	−5.19862e6	−	9.00427e6i	−0.0484458	−	0.0839106i	−0.889231	+	0.457458i	$0.848760\pi$
$198$		0			0
$199$	−1.25062e7	−	7.09261e7i	−0.112497	−	0.637999i	−0.987959	−	0.154714i	$-0.950554\pi$
$199$	−1.25062e7	−	7.09261e7i	−0.112497	−	0.637999i	0.875463	−	0.483285i	$-0.160557\pi$
$200$		0			0
$201$	−1.00516e8	+	1.74099e8i	−0.873073	+	1.51221i
$202$		0			0
$203$	−1.90230e8	+	1.59622e8i	−1.59604	+	1.33924i
$204$		0			0
$205$	1.42413e8	+	5.18340e7i	1.15454	+	0.420220i
$206$		0			0
$207$	9.22991e6	−	5.23454e7i	0.0723271	−	0.410187i
$208$		0			0
$209$	−4.94813e7	+	5.02811e7i	−0.374912	+	0.380972i
$210$		0			0
$211$	7.26730e6	−	4.12149e7i	0.0532580	−	0.302041i	−0.946530	−	0.322615i	$-0.895438\pi$
$211$	7.26730e6	−	4.12149e7i	0.0532580	−	0.302041i	0.999788	+	0.0205738i	$0.00654929\pi$
$212$		0			0
$213$	−1.72850e8	−	6.29121e7i	−1.22557	−	0.446073i
$214$		0			0
$215$	2.34137e8	−	1.96464e8i	1.60670	−	1.34818i
$216$		0			0
$217$	−1.18604e8	+	2.05429e8i	−0.787937	+	1.36475i
$218$		0			0
$219$	−2.04514e7	−	1.15986e8i	−0.131573	−	0.746190i
$220$		0			0
$221$	4.63995e7	+	8.03663e7i	0.289161	+	0.500842i
$222$		0			0
$223$	−2.49267e8	+	9.07257e7i	−1.50521	+	0.547852i	−0.957404	−	0.288753i	$-0.906759\pi$
$223$	−2.49267e8	+	9.07257e7i	−1.50521	+	0.547852i	−0.547807	+	0.836605i	$0.684537\pi$
$224$		0			0
$225$	1.09395e8	+	9.17935e7i	0.640265	+	0.537246i
$226$		0			0
$227$	−2.04314e8			−1.15933			−0.579665	−	0.814855i	$-0.696817\pi$
$227$	−2.04314e8			−1.15933			−0.579665	+	0.814855i	$0.696817\pi$
$228$		0			0
$229$	−8.35323e7			−0.459653			−0.229827	−	0.973232i	$-0.573816\pi$
$229$	−8.35323e7			−0.459653			−0.229827	+	0.973232i	$0.573816\pi$
$230$		0			0
$231$	1.47819e8	+	1.24034e8i	0.789018	+	0.662065i
$232$		0			0
$233$	3.51666e8	−	1.27996e8i	1.82132	−	0.662905i	0.826294	−	0.563239i	$-0.190445\pi$
$233$	3.51666e8	−	1.27996e8i	1.82132	−	0.662905i	0.995021	−	0.0996661i	$-0.0317774\pi$
$234$		0			0
$235$	2.19058e8	+	3.79420e8i	1.10109	+	1.90714i
$236$		0			0
$237$	−7.34512e6	−	4.16562e7i	−0.0358410	−	0.203264i
$238$		0			0
$239$	1.33536e8	−	2.31291e8i	0.632712	−	1.09589i	−0.354283	−	0.935138i	$-0.615275\pi$
$239$	1.33536e8	−	2.31291e8i	0.632712	−	1.09589i	0.986995	−	0.160751i	$-0.0513916\pi$
$240$		0			0
$241$	−1.52551e8	+	1.28005e8i	−0.702028	+	0.589071i	−0.922350	−	0.386356i	$-0.873734\pi$
$241$	−1.52551e8	+	1.28005e8i	−0.702028	+	0.589071i	0.220322	+	0.975427i	$0.429289\pi$
$242$		0			0
$243$	1.28297e8	+	4.66962e7i	0.573580	+	0.208766i
$244$		0			0
$245$	1.39027e8	−	7.88463e8i	0.603974	−	3.42531i
$246$		0			0
$247$	1.59612e8	−	7.59920e7i	0.673949	−	0.320870i
$248$		0			0
$249$	−1.46650e7	+	8.31693e7i	−0.0601983	+	0.341401i
$250$		0			0
$251$	−1.33482e8	−	4.85833e7i	−0.532799	−	0.193923i	0.0615889	−	0.998102i	$-0.480383\pi$
$251$	−1.33482e8	−	4.85833e7i	−0.532799	−	0.193923i	−0.594388	+	0.804179i	$0.702605\pi$
$252$		0			0
$253$	−1.39036e8	+	1.16665e8i	−0.539765	+	0.452917i
$254$		0			0
$255$	−2.24660e8	+	3.89123e8i	−0.848468	+	1.46959i
$256$		0			0
$257$	2.56047e7	+	1.45211e8i	0.0940922	+	0.533623i	0.995022	+	0.0996579i	$0.0317749\pi$
$257$	2.56047e7	+	1.45211e8i	0.0940922	+	0.533623i	−0.900930	+	0.433965i	$0.857114\pi$
$258$		0			0
$259$	−3.51598e8	−	6.08986e8i	−1.25747	−	2.17800i
$260$		0			0
$261$	−1.05778e8	+	3.85000e7i	−0.368258	+	0.134035i
$262$		0			0
$263$	2.29123e8	+	1.92257e8i	0.776648	+	0.651685i	0.942402	−	0.334482i	$-0.108561\pi$
$263$	2.29123e8	+	1.92257e8i	0.776648	+	0.651685i	−0.165754	+	0.986167i	$0.553006\pi$
$264$		0			0
$265$	−5.61378e8			−1.85308
$266$		0			0
$267$	−1.23147e8			−0.395944
$268$		0			0
$269$	−3.13560e8	−	2.63108e8i	−0.982173	−	0.824141i	0.00224311	−	0.999997i	$-0.499286\pi$
$269$	−3.13560e8	−	2.63108e8i	−0.982173	−	0.824141i	−0.984416	+	0.175857i	$0.943730\pi$
$270$		0			0
$271$	−2.06392e8	+	7.51206e7i	−0.629942	+	0.229280i	−0.637206	−	0.770693i	$-0.719910\pi$
$271$	−2.06392e8	+	7.51206e7i	−0.629942	+	0.229280i	0.00726383	+	0.999974i	$0.497688\pi$
$272$		0			0
$273$	−2.41774e8	−	4.18765e8i	−0.719185	−	1.24566i
$274$		0			0
$275$	−8.46760e7	−	4.80221e8i	−0.245525	−	1.39244i
$276$		0			0
$277$	7.04975e7	−	1.22105e8i	0.199294	−	0.345188i	−0.749006	−	0.662564i	$-0.769468\pi$
$277$	7.04975e7	−	1.22105e8i	0.199294	−	0.345188i	0.948300	+	0.317376i	$0.102802\pi$
$278$		0			0
$279$	−8.23696e7	+	6.91163e7i	−0.227066	+	0.190531i
$280$		0			0
$281$	3.02353e8	+	1.10048e8i	0.812910	+	0.295875i	0.714825	−	0.699303i	$-0.246506\pi$
$281$	3.02353e8	+	1.10048e8i	0.812910	+	0.295875i	0.0980848	+	0.995178i	$0.468728\pi$
$282$		0			0
$283$	−2.88359e7	+	1.63536e8i	−0.0756277	+	0.428906i	0.923361	+	0.383934i	$0.125431\pi$
$283$	−2.88359e7	+	1.63536e8i	−0.0756277	+	0.428906i	−0.998988	+	0.0449720i	$0.985680\pi$
$284$		0			0
$285$	7.05059e8	+	4.85311e8i	1.80413	+	1.24183i
$286$		0			0
$287$	7.51750e7	−	4.26339e8i	0.187710	−	1.06456i
$288$		0			0
$289$	1.54127e8	+	5.60977e7i	0.375609	+	0.136711i
$290$		0			0
$291$	−4.83454e8	+	4.05666e8i	−1.15008	+	0.965036i
$292$		0			0
$293$	−2.42364e8	+	4.19786e8i	−0.562899	+	0.974970i	0.434343	+	0.900748i	$0.356981\pi$
$293$	−2.42364e8	+	4.19786e8i	−0.562899	+	0.974970i	−0.997242	+	0.0742222i	$0.976353\pi$
$294$		0			0
$295$	3.79400e7	+	2.15169e8i	0.0860440	+	0.487980i
$296$		0			0
$297$	−9.46834e7	−	1.63996e8i	−0.209713	−	0.363234i
$298$		0			0
$299$	4.27389e8	−	1.55557e8i	0.924643	−	0.336542i
$300$		0			0
$301$	−6.68821e8	−	5.61208e8i	−1.41360	−	1.18615i
$302$		0			0
$303$	1.87528e8			0.387272
$304$		0			0
$305$	−1.13851e9			−2.29767
$306$		0			0
$307$	3.25664e8	+	2.73264e8i	0.642370	+	0.539013i	0.904745	−	0.425953i	$-0.140061\pi$
$307$	3.25664e8	+	2.73264e8i	0.642370	+	0.539013i	−0.262375	+	0.964966i	$0.584506\pi$
$308$		0			0
$309$	2.49556e8	−	9.08308e7i	0.481186	−	0.175137i
$310$		0			0
$311$	2.29110e8	+	3.96830e8i	0.431900	+	0.748072i	0.997037	−	0.0769246i	$-0.0245101\pi$
$311$	2.29110e8	+	3.96830e8i	0.431900	+	0.748072i	−0.565137	+	0.824997i	$0.691177\pi$
$312$		0			0
$313$	9.05665e7	+	5.13628e8i	0.166941	+	0.946768i	0.947041	+	0.321113i	$0.104057\pi$
$313$	9.05665e7	+	5.13628e8i	0.166941	+	0.946768i	−0.780100	+	0.625655i	$0.784832\pi$
$314$		0			0
$315$	2.81070e8	−	4.86827e8i	0.506672	−	0.877582i
$316$		0			0
$317$	4.41612e8	−	3.70557e8i	0.778635	−	0.653352i	−0.164270	−	0.986415i	$-0.552527\pi$
$317$	4.41612e8	−	3.70557e8i	0.778635	−	0.653352i	0.942904	+	0.333063i	$0.108082\pi$
$318$		0			0
$319$	3.61193e8	+	1.31464e8i	0.622978	+	0.226745i
$320$		0			0
$321$	−1.77599e8	+	1.00721e9i	−0.299690	+	1.69963i
$322$		0			0
$323$	−1.94890e8	+	4.26845e8i	−0.321797	+	0.704792i
$324$		0			0
$325$	−2.12190e8	+	1.20339e9i	−0.342873	+	1.94453i
$326$		0			0
$327$	−6.13259e8	−	2.23208e8i	−0.969900	−	0.353015i
$328$		0			0
$329$	9.58701e8	−	8.04446e8i	1.48422	−	1.24541i
$330$		0			0
$331$	4.05969e8	−	7.03159e8i	0.615311	−	1.06575i	−0.375019	−	0.927017i	$-0.622364\pi$
$331$	4.05969e8	−	7.03159e8i	0.615311	−	1.06575i	0.990330	−	0.138733i	$-0.0443029\pi$
$332$		0			0
$333$	−5.53515e7	−	3.13914e8i	−0.0821438	−	0.465861i
$334$		0			0
$335$	−9.99887e8	−	1.73186e9i	−1.45310	−	2.51684i
$336$		0			0
$337$	1.24017e9	−	4.51386e8i	1.76513	−	0.642456i	0.765136	−	0.643869i	$-0.222672\pi$
$337$	1.24017e9	−	4.51386e8i	1.76513	−	0.642456i	0.999999	+	0.00141265i	$0.000449662\pi$
$338$		0			0
$339$	−2.96564e8	−	2.48847e8i	−0.413447	−	0.346923i
$340$		0			0
$341$	3.67163e8			0.501439
$342$		0			0
$343$	−1.03161e9			−1.38033
$344$		0			0
$345$	1.68696e9	+	1.41553e9i	2.21176	+	1.85589i
$346$		0			0
$347$	−8.56307e8	+	3.11670e8i	−1.10021	+	0.400445i	−0.827394	−	0.561621i	$-0.810178\pi$
$347$	−8.56307e8	+	3.11670e8i	−1.10021	+	0.400445i	−0.272818	+	0.962066i	$0.587956\pi$
$348$		0			0
$349$	5.66481e8	+	9.81173e8i	0.713339	+	1.23554i	0.963597	+	0.267360i	$0.0861514\pi$
$349$	5.66481e8	+	9.81173e8i	0.713339	+	1.23554i	−0.250258	+	0.968179i	$0.580515\pi$
$350$		0			0
$351$	8.24022e7	+	4.67326e8i	0.101710	+	0.576826i
$352$		0			0
$353$	3.17094e8	−	5.49223e8i	0.383687	−	0.664565i	−0.607899	−	0.794014i	$-0.707988\pi$
$353$	3.17094e8	−	5.49223e8i	0.383687	−	0.664565i	0.991586	+	0.129449i	$0.0413209\pi$
$354$		0			0
$355$	1.40169e9	−	1.17615e9i	1.66284	−	1.39529i
$356$		0			0
$357$	1.20610e9	+	4.38983e8i	1.40295	+	0.510634i
$358$		0			0
$359$	1.05667e8	−	5.99268e8i	0.120534	−	0.683582i	−0.863327	−	0.504646i	$-0.831623\pi$
$359$	1.05667e8	−	5.99268e8i	0.120534	−	0.683582i	0.983861	−	0.178937i	$-0.0572657\pi$
$360$		0			0
$361$	7.81182e8	+	4.34467e8i	0.873930	+	0.486051i
$362$		0			0
$363$	−1.29672e8	+	7.35407e8i	−0.142290	+	0.806965i
$364$		0			0
$365$	1.10091e9	+	4.00699e8i	1.18503	+	0.431314i
$366$		0			0
$367$	1.14399e9	−	9.59925e8i	1.20807	−	1.01369i	0.208709	−	0.977978i	$-0.433074\pi$
$367$	1.14399e9	−	9.59925e8i	1.20807	−	1.01369i	0.999362	−	0.0357145i	$-0.0113707\pi$
$368$		0			0
$369$	9.81196e7	−	1.69948e8i	0.101663	−	0.176086i
$370$		0			0
$371$	2.78462e8	+	1.57924e9i	0.283111	+	1.60560i
$372$		0			0
$373$	6.47685e8	+	1.12182e9i	0.646224	+	1.11929i	0.984017	+	0.178073i	$0.0569862\pi$
$373$	6.47685e8	+	1.12182e9i	0.646224	+	1.11929i	−0.337793	+	0.941220i	$0.609680\pi$
$374$		0			0
$375$	−3.45798e9	+	1.25860e9i	−3.38620	+	1.23248i
$376$		0			0
$377$	−7.37858e8	−	6.19136e8i	−0.709215	−	0.595102i
$378$		0			0
$379$	1.15086e8			0.108589			0.0542945	−	0.998525i	$-0.482709\pi$
$379$	1.15086e8			0.108589			0.0542945	+	0.998525i	$0.482709\pi$
$380$		0			0
$381$	−1.17173e8			−0.108540
$382$		0			0
$383$	−6.95206e8	−	5.83347e8i	−0.632292	−	0.530556i	0.269348	−	0.963043i	$-0.413192\pi$
$383$	−6.95206e8	−	5.83347e8i	−0.632292	−	0.530556i	−0.901640	+	0.432487i	$0.857636\pi$
$384$		0			0
$385$	−1.80375e9	+	6.56510e8i	−1.61088	+	0.586312i
$386$		0			0
$387$	−1.97883e8	−	3.42744e8i	−0.173548	−	0.300594i
$388$		0			0
$389$	−3.58138e7	−	2.03110e8i	−0.0308480	−	0.174948i	0.965491	−	0.260436i	$-0.0838662\pi$
$389$	−3.58138e7	−	2.03110e8i	−0.0308480	−	0.174948i	−0.996339	+	0.0854880i	$0.972755\pi$
$390$		0			0
$391$	−6.03620e8	+	1.04550e9i	−0.510676	+	0.884516i
$392$		0			0
$393$	2.78646e8	−	2.33812e8i	0.231568	−	0.194309i
$394$		0			0
$395$	3.95393e8	+	1.43911e8i	0.322804	+	0.117491i
$396$		0			0
$397$	−4.32500e7	+	2.45283e8i	−0.0346912	+	0.196744i	−0.997228	−	0.0744079i	$-0.976293\pi$
$397$	−4.32500e7	+	2.45283e8i	−0.0346912	+	0.196744i	0.962537	+	0.271151i	$0.0874044\pi$
$398$		0			0
$399$	1.01552e9	−	2.22416e9i	0.800354	−	1.75292i
$400$		0			0
$401$	3.10645e8	−	1.76175e9i	0.240579	−	1.36439i	−0.589959	−	0.807433i	$-0.700856\pi$
$401$	3.10645e8	−	1.76175e9i	0.240579	−	1.36439i	0.830539	−	0.556961i	$-0.188033\pi$
$402$		0			0
$403$	−8.64588e8	−	3.14684e8i	−0.658024	−	0.239501i
$404$		0			0
$405$	−2.37789e9	+	1.99529e9i	−1.77869	+	1.49249i
$406$		0			0
$407$	−5.44221e8	+	9.42618e8i	−0.400124	+	0.693035i
$408$		0			0
$409$	−4.30039e8	−	2.43887e9i	−0.310796	−	1.76261i	−0.594880	−	0.803814i	$-0.702801\pi$
$409$	−4.30039e8	−	2.43887e9i	−0.310796	−	1.76261i	0.284084	−	0.958799i	$-0.408310\pi$
$410$		0			0
$411$	−4.81523e7	−	8.34022e7i	−0.0342114	−	0.0592559i
$412$		0			0
$413$	5.86480e8	−	2.13461e8i	0.409664	−	0.149106i
$414$		0			0
$415$	−6.43546e8	−	5.40000e8i	−0.441989	−	0.370873i
$416$		0			0
$417$	−1.20624e9			−0.814623
$418$		0			0
$419$	1.94824e9			1.29388			0.646940	−	0.762541i	$-0.276049\pi$
$419$	1.94824e9			1.29388			0.646940	+	0.762541i	$0.276049\pi$
$420$		0			0
$421$	−2.13449e8	−	1.79105e8i	−0.139414	−	0.116982i	0.570414	−	0.821357i	$-0.306783\pi$
$421$	−2.13449e8	−	1.79105e8i	−0.139414	−	0.116982i	−0.709828	+	0.704375i	$0.751227\pi$
$422$		0			0
$423$	5.33087e8	−	1.94028e8i	0.342457	−	0.124644i
$424$		0			0
$425$	−1.62174e9	−	2.80894e9i	−1.02476	−	1.77493i
$426$		0			0
$427$	5.64739e8	+	3.20279e9i	0.351035	+	1.99082i
$428$		0			0
$429$	−3.74229e8	+	6.48184e8i	−0.228843	+	0.396367i
$430$		0			0
$431$	1.21644e9	−	1.02071e9i	0.731846	−	0.614092i	−0.198788	−	0.980042i	$-0.563701\pi$
$431$	1.21644e9	−	1.02071e9i	0.731846	−	0.614092i	0.930634	+	0.365951i	$0.119256\pi$
$432$		0			0
$433$	−2.22721e9	−	8.10639e8i	−1.31842	−	0.479866i	−0.415469	−	0.909607i	$-0.636383\pi$
$433$	−2.22721e9	−	8.10639e8i	−1.31842	−	0.479866i	−0.902952	+	0.429741i	$0.858605\pi$
$434$		0			0
$435$	8.09846e8	−	4.59286e9i	0.471726	−	2.67529i
$436$		0			0
$437$	1.89436e9	+	1.30394e9i	1.08587	+	0.747434i
$438$		0			0
$439$	−2.60758e8	+	1.47883e9i	−0.147100	+	0.834244i	0.818557	+	0.574425i	$0.194774\pi$
$439$	−2.60758e8	+	1.47883e9i	−0.147100	+	0.834244i	−0.965657	+	0.259820i	$0.916337\pi$
$440$		0			0
$441$	−9.74177e8	−	3.54572e8i	−0.540883	−	0.196865i
$442$		0			0
$443$	−2.68462e9	+	2.25266e9i	−1.46713	+	1.23107i	−0.548382	+	0.836228i	$0.684756\pi$
$443$	−2.68462e9	+	2.25266e9i	−1.46713	+	1.23107i	−0.918749	+	0.394841i	$0.870800\pi$
$444$		0			0
$445$	6.12502e8	−	1.06089e9i	0.329494	−	0.570701i
$446$		0			0
$447$	2.43297e7	+	1.37981e8i	0.0128843	+	0.0730704i
$448$		0			0
$449$	8.15324e8	+	1.41218e9i	0.425077	+	0.736256i	0.996428	−	0.0844506i	$-0.0269135\pi$
$449$	8.15324e8	+	1.41218e9i	0.425077	+	0.736256i	−0.571350	+	0.820706i	$0.693580\pi$
$450$		0			0
$451$	−6.29676e8	+	2.29183e8i	−0.323221	+	0.117643i
$452$		0			0
$453$	1.15589e9	+	9.69909e8i	0.584216	+	0.490216i
$454$		0			0
$455$	4.81010e9			2.39395
$456$		0			0
$457$	1.19201e9			0.584216			0.292108	−	0.956385i	$-0.405643\pi$
$457$	1.19201e9			0.584216			0.292108	+	0.956385i	$0.405643\pi$
$458$		0			0
$459$	−9.64893e8	−	8.09641e8i	−0.465731	−	0.390794i
$460$		0			0
$461$	−2.41982e9	+	8.80744e8i	−1.15035	+	0.418694i	−0.845641	−	0.533751i	$-0.820782\pi$
$461$	−2.41982e9	+	8.80744e8i	−1.15035	+	0.418694i	−0.304710	+	0.952445i	$0.598560\pi$
$462$		0			0
$463$	1.37855e9	+	2.38771e9i	0.645487	+	1.11802i	0.984189	+	0.177123i	$0.0566789\pi$
$463$	1.37855e9	+	2.38771e9i	0.645487	+	1.11802i	−0.338702	+	0.940894i	$0.609988\pi$
$464$		0			0
$465$	−7.73583e8	−	4.38721e9i	−0.356797	−	2.02350i
$466$		0			0
$467$	8.05473e8	−	1.39512e9i	0.365967	−	0.633874i	−0.622964	−	0.782251i	$-0.714072\pi$
$467$	8.05473e8	−	1.39512e9i	0.365967	−	0.633874i	0.988931	+	0.148377i	$0.0474049\pi$
$468$		0			0
$469$	−4.37598e9	+	3.67188e9i	−1.95871	+	1.64355i
$470$		0			0
$471$	9.08452e7	+	3.30650e7i	0.0400617	+	0.0145812i
$472$		0			0
$473$	−2.34668e8	+	1.33087e9i	−0.101962	+	0.578258i
$474$		0			0
$475$	−5.57872e9	+	2.65605e9i	−2.38840	+	1.13713i
$476$		0			0
$477$	−1.26226e8	+	7.15862e8i	−0.0532517	+	0.302006i
$478$		0			0
$479$	−1.42155e8	−	5.17403e7i	−0.0591002	−	0.0215107i	0.312301	−	0.949983i	$-0.398900\pi$
$479$	−1.42155e8	−	5.17403e7i	−0.0591002	−	0.0215107i	−0.371401	+	0.928473i	$0.621123\pi$
$480$		0			0
$481$	2.08941e9	−	1.75322e9i	0.856083	−	0.718339i
$482$		0			0
$483$	3.14529e9	−	5.44780e9i	1.27012	−	2.19992i
$484$		0			0
$485$	−1.09015e9	−	6.18254e9i	−0.433900	−	2.46077i
$486$		0			0
$487$	1.17579e9	+	2.03654e9i	0.461296	+	0.798989i	0.999026	−	0.0441286i	$-0.0140511\pi$
$487$	1.17579e9	+	2.03654e9i	0.461296	+	0.798989i	−0.537729	+	0.843117i	$0.680718\pi$
$488$		0			0
$489$	−2.48520e9	+	9.04538e8i	−0.961126	+	0.349821i
$490$		0			0
$491$	5.63902e8	+	4.73170e8i	0.214990	+	0.180398i	0.743923	−	0.668266i	$-0.232963\pi$
$491$	5.63902e8	+	4.73170e8i	0.214990	+	0.180398i	−0.528933	+	0.848664i	$0.677408\pi$
$492$		0			0
$493$	2.55667e9			0.960973
$494$		0			0
$495$	−8.70106e8			−0.322444
$496$		0			0
$497$	−4.00397e9	−	3.35973e9i	−1.46300	−	1.22760i
$498$		0			0
$499$	2.01228e9	−	7.32411e8i	0.724999	−	0.263878i	0.0469525	−	0.998897i	$-0.485049\pi$
$499$	2.01228e9	−	7.32411e8i	0.724999	−	0.263878i	0.678046	+	0.735019i	$0.262827\pi$
$500$		0			0
$501$	−5.22557e7	−	9.05096e7i	−0.0185653	−	0.0321560i
$502$		0			0
$503$	1.98414e8	+	1.12526e9i	0.0695158	+	0.394244i	0.999636	+	0.0269879i	$0.00859156\pi$
$503$	1.98414e8	+	1.12526e9i	0.0695158	+	0.394244i	−0.930120	+	0.367256i	$0.880297\pi$
$504$		0			0
$505$	−9.32717e8	+	1.61551e9i	−0.322278	+	0.558201i
$506$		0			0
$507$	−1.14195e9	+	9.58208e8i	−0.389151	+	0.326537i
$508$		0			0
$509$	1.80774e9	+	6.57962e8i	0.607607	+	0.221151i	0.627456	−	0.778652i	$-0.284096\pi$
$509$	1.80774e9	+	6.57962e8i	0.607607	+	0.221151i	−0.0198488	+	0.999803i	$0.506318\pi$
$510$		0			0
$511$	5.81135e8	−	3.29578e9i	0.192666	−	1.09266i
$512$		0			0
$513$	−1.68302e9	+	1.71022e9i	−0.550399	+	0.559295i
$514$		0			0
$515$	−4.58739e8	+	2.60164e9i	−0.147993	+	0.839309i
$516$		0			0
$517$	−1.82030e9	−	6.62535e8i	−0.579330	−	0.210859i
$518$		0			0
$519$	−3.05415e9	+	2.56274e9i	−0.958970	+	0.804671i
$520$		0			0
$521$	2.46959e9	−	4.27745e9i	0.765054	−	1.32511i	−0.175164	−	0.984539i	$-0.556046\pi$
$521$	2.46959e9	−	4.27745e9i	0.765054	−	1.32511i	0.940218	−	0.340573i	$-0.110621\pi$
$522$		0			0
$523$	9.52906e8	+	5.40420e9i	0.291269	+	1.65187i	0.681991	+	0.731360i	$0.261114\pi$
$523$	9.52906e8	+	5.40420e9i	0.291269	+	1.65187i	−0.390722	+	0.920509i	$0.627775\pi$
$524$		0			0
$525$	8.45042e9	+	1.46366e10i	2.54871	+	4.41450i
$526$		0			0
$527$	2.29491e9	−	8.35279e8i	0.683013	−	0.248596i
$528$		0			0
$529$	1.92430e9	+	1.61468e9i	0.565167	+	0.474232i
$530$		0			0
$531$	2.82911e8			0.0820010
$532$		0			0
$533$	1.67918e9			0.480342
$534$		0			0
$535$	−7.79359e9	−	6.53960e9i	−2.20039	−	1.84635i
$536$		0			0
$537$	9.44325e8	−	3.43706e8i	0.263155	−	0.0957805i
$538$		0			0
$539$	1.76998e9	+	3.06569e9i	0.486863	+	0.843272i
$540$		0			0
$541$	7.96283e7	+	4.51594e8i	0.0216211	+	0.122619i	0.993708	−	0.112002i	$-0.0357262\pi$
$541$	7.96283e7	+	4.51594e8i	0.0216211	+	0.122619i	−0.972087	+	0.234621i	$0.924615\pi$
$542$		0			0
$543$	5.02925e8	−	8.71092e8i	0.134804	−	0.233488i
$544$		0			0
$545$	4.97309e9	−	4.17292e9i	1.31595	−	1.10421i
$546$		0			0
$547$	4.91880e9	+	1.79030e9i	1.28500	+	0.467702i	0.892083	−	0.451871i	$-0.149243\pi$
$547$	4.91880e9	+	1.79030e9i	1.28500	+	0.467702i	0.392918	+	0.919573i	$0.371465\pi$
$548$		0			0
$549$	−2.55994e8	+	1.45182e9i	−0.0660278	+	0.374463i
$550$		0			0
$551$	3.85574e8	−	4.85510e9i	0.0981921	−	1.23642i
$552$		0			0
$553$	2.08715e8	−	1.18368e9i	0.0524826	−	0.297644i
$554$		0			0
$555$	1.24099e10	+	4.51684e9i	3.08137	+	1.12153i
$556$		0			0
$557$	9.92160e8	−	8.32521e8i	0.243270	−	0.204128i	−0.512998	−	0.858390i	$-0.671465\pi$
$557$	9.92160e8	−	8.32521e8i	0.243270	−	0.204128i	0.756268	+	0.654262i	$0.227021\pi$
$558$		0			0
$559$	1.69324e9	−	2.93278e9i	0.409994	−	0.710130i
$560$		0			0
$561$	−3.44981e8	−	1.95648e9i	−0.0824944	−	0.467849i
$562$		0			0
$563$	6.64007e8	+	1.15009e9i	0.156817	+	0.271615i	0.933719	−	0.358006i	$-0.116543\pi$
$563$	6.64007e8	+	1.15009e9i	0.156817	+	0.271615i	−0.776902	+	0.629621i	$0.783210\pi$
$564$		0			0
$565$	3.61880e9	−	1.31714e9i	0.844103	−	0.307228i
$566$		0			0
$567$	6.79254e9	+	5.69962e9i	1.56492	+	1.31312i
$568$		0			0
$569$	−3.64123e9			−0.828619			−0.414310	−	0.910136i	$-0.635977\pi$
$569$	−3.64123e9			−0.828619			−0.414310	+	0.910136i	$0.635977\pi$
$570$		0			0
$571$	4.17455e9			0.938391			0.469195	−	0.883094i	$-0.344544\pi$
$571$	4.17455e9			0.938391			0.469195	+	0.883094i	$0.344544\pi$
$572$		0			0
$573$	−4.91980e9	−	4.12820e9i	−1.09246	−	0.916684i
$574$		0			0
$575$	−1.49380e10	+	5.43697e9i	−3.27683	+	1.19267i
$576$		0			0
$577$	−4.15225e8	−	7.19191e8i	−0.0899846	−	0.155858i	0.817520	−	0.575900i	$-0.195348\pi$
$577$	−4.15225e8	−	7.19191e8i	−0.0899846	−	0.155858i	−0.907504	+	0.420043i	$0.862015\pi$
$578$		0			0
$579$	−1.35098e9	−	7.66180e9i	−0.289251	−	1.64042i
$580$		0			0
$581$	−1.19988e9	+	2.07824e9i	−0.253816	+	0.439623i
$582$		0			0
$583$	1.90142e9	−	1.59548e9i	0.397409	−	0.333465i
$584$		0			0
$585$	2.04891e9	+	7.45742e8i	0.423133	+	0.154008i
$586$		0			0
$587$	1.20245e9	−	6.81942e9i	0.245377	−	1.39160i	−0.574240	−	0.818687i	$-0.694702\pi$
$587$	1.20245e9	−	6.81942e9i	0.245377	−	1.39160i	0.819617	−	0.572912i	$-0.194187\pi$
$588$		0			0
$589$	−1.24009e9	−	4.48398e9i	−0.250063	−	0.904191i
$590$		0			0
$591$	9.68577e7	−	5.49307e8i	0.0193009	−	0.109461i
$592$		0			0
$593$	−9.11406e9	−	3.31725e9i	−1.79482	−	0.653261i	−0.998851	−	0.0479219i	$-0.984740\pi$
$593$	−9.11406e9	−	3.31725e9i	−1.79482	−	0.653261i	−0.795968	−	0.605339i	$-0.793038\pi$
$594$		0			0
$595$	−9.78059e9	+	8.20689e9i	−1.90351	+	1.59724i
$596$		0			0
$597$	1.93184e9	−	3.34604e9i	0.371587	−	0.643607i
$598$		0			0
$599$	1.15150e9	+	6.53049e9i	0.218913	+	1.24152i	0.873986	+	0.485950i	$0.161526\pi$
$599$	1.15150e9	+	6.53049e9i	0.218913	+	1.24152i	−0.655074	+	0.755565i	$0.727362\pi$
$600$		0			0
$601$	2.42937e9	+	4.20780e9i	0.456493	+	0.790669i	0.998773	−	0.0495293i	$-0.0157721\pi$
$601$	2.42937e9	+	4.20780e9i	0.456493	+	0.790669i	−0.542280	+	0.840198i	$0.682439\pi$
$602$		0			0
$603$	−2.43327e9	+	8.85636e8i	−0.451938	+	0.164492i
$604$		0			0
$605$	−5.69042e9	−	4.77483e9i	−1.04472	−	0.876626i
$606$		0			0
$607$	2.78852e9			0.506074			0.253037	−	0.967457i	$-0.418571\pi$
$607$	2.78852e9			0.506074			0.253037	+	0.967457i	$0.418571\pi$
$608$		0			0
$609$	−1.33221e10			−2.39008
$610$		0			0
$611$	3.71857e9	+	3.12025e9i	0.659526	+	0.553408i
$612$		0			0
$613$	7.46559e8	−	2.71725e8i	0.130904	−	0.0476451i	−0.275738	−	0.961233i	$-0.588922\pi$
$613$	7.46559e8	−	2.71725e8i	0.130904	−	0.0476451i	0.406641	+	0.913588i	$0.366700\pi$
$614$		0			0
$615$	4.06518e9	+	7.04109e9i	0.704720	+	1.22061i
$616$		0			0
$617$	1.78818e9	+	1.01413e10i	0.306488	+	1.73818i	0.616418	+	0.787419i	$0.288583\pi$
$617$	1.78818e9	+	1.01413e10i	0.306488	+	1.73818i	−0.309930	+	0.950759i	$0.600306\pi$
$618$		0			0
$619$	7.70266e8	−	1.33414e9i	0.130534	−	0.226091i	−0.793349	−	0.608768i	$-0.791664\pi$
$619$	7.70266e8	−	1.33414e9i	0.130534	−	0.226091i	0.923883	+	0.382676i	$0.124998\pi$
$620$		0			0
$621$	−4.72904e9	+	3.96814e9i	−0.792415	+	0.664915i
$622$		0			0
$623$	−3.28824e9	−	1.19682e9i	−0.544823	−	0.198299i
$624$		0			0
$625$	3.55290e9	−	2.01495e10i	0.582108	−	3.30130i
$626$		0			0
$627$	−3.76736e9	+	3.60056e8i	−0.610381	+	0.0583356i
$628$		0			0
$629$	−1.25718e9	+	7.12981e9i	−0.201428	+	1.14235i
$630$		0			0
$631$	−1.77877e9	−	6.47419e8i	−0.281849	−	0.102585i	0.197227	−	0.980358i	$-0.436806\pi$
$631$	−1.77877e9	−	6.47419e8i	−0.281849	−	0.102585i	−0.479077	+	0.877773i	$0.659028\pi$
$632$		0			0
$633$	1.71990e9	−	1.44317e9i	0.269519	−	0.226154i
$634$		0			0
$635$	5.82792e8	−	1.00943e9i	0.0903245	−	0.156447i
$636$		0			0
$637$	−1.54040e9	−	8.73603e9i	−0.236126	−	1.33914i
$638$		0			0
$639$	−1.18465e9	−	2.05187e9i	−0.179612	−	0.311098i
$640$		0			0
$641$	−6.89613e9	+	2.50999e9i	−1.03419	+	0.376416i	−0.802677	−	0.596415i	$-0.796591\pi$
$641$	−6.89613e9	+	2.50999e9i	−1.03419	+	0.376416i	−0.231518	+	0.972831i	$0.574369\pi$
$642$		0			0
$643$	−7.84777e7	−	6.58506e7i	−0.0116415	−	0.00976836i	0.636948	−	0.770907i	$-0.280196\pi$
$643$	−7.84777e7	−	6.58506e7i	−0.0116415	−	0.00976836i	−0.648590	+	0.761138i	$0.724641\pi$
$644$		0			0
$645$	1.63969e10			2.40604
$646$		0			0
$647$	1.84317e9			0.267547			0.133774	−	0.991012i	$-0.457290\pi$
$647$	1.84317e9			0.267547			0.133774	+	0.991012i	$0.457290\pi$
$648$		0			0
$649$	−7.40030e8	−	6.20959e8i	−0.106266	−	0.0891675i
$650$		0			0
$651$	−1.19581e10	+	4.35240e9i	−1.69875	+	0.618294i
$652$		0			0
$653$	−2.11868e9	−	3.66966e9i	−0.297762	−	0.515739i	0.677862	−	0.735189i	$-0.262907\pi$
$653$	−2.11868e9	−	3.66966e9i	−0.297762	−	0.515739i	−0.975624	+	0.219451i	$0.929573\pi$
$654$		0			0
$655$	6.28324e8	+	3.56340e9i	0.0873653	+	0.495473i
$656$		0			0
$657$	7.58507e8	−	1.31377e9i	0.104347	−	0.180735i
$658$		0			0
$659$	1.25764e8	−	1.05528e8i	0.0171181	−	0.0143638i	−0.634188	−	0.773179i	$-0.718666\pi$
$659$	1.25764e8	−	1.05528e8i	0.0171181	−	0.0143638i	0.651307	+	0.758815i	$0.274221\pi$
$660$		0			0
$661$	1.88744e9	+	6.86973e8i	0.254196	+	0.0925197i	0.465975	−	0.884798i	$-0.345704\pi$
$661$	1.88744e9	+	6.86973e8i	0.254196	+	0.0925197i	−0.211779	+	0.977318i	$0.567926\pi$
$662$		0			0
$663$	−8.64489e8	+	4.90276e9i	−0.115203	+	0.653346i
$664$		0			0
$665$	1.41098e10	+	1.98109e10i	1.86056	+	2.61233i
$666$		0			0
$667$	2.17590e9	−	1.23402e10i	0.283922	−	1.61020i
$668$		0			0
$669$	−1.33724e10	−	4.86717e9i	−1.72671	−	0.628470i
$670$		0			0
$671$	3.85620e9	−	3.23574e9i	0.492755	−	0.413470i
$672$		0			0
$673$	6.93202e9	−	1.20066e10i	0.876611	−	1.51833i	0.0215740	−	0.999767i	$-0.493132\pi$
$673$	6.93202e9	−	1.20066e10i	0.876611	−	1.51833i	0.855037	−	0.518567i	$-0.173534\pi$
$674$		0			0
$675$	−2.88010e9	−	1.63338e10i	−0.360449	−	2.04421i
$676$		0			0
$677$	−1.35582e8	−	2.34835e8i	−0.0167935	−	0.0290873i	0.857507	−	0.514473i	$-0.172012\pi$
$677$	−1.35582e8	−	2.34835e8i	−0.0167935	−	0.0290873i	−0.874300	+	0.485386i	$0.838679\pi$
$678$		0			0
$679$	−1.68516e10	+	6.13349e9i	−2.06584	+	0.751906i
$680$		0			0
$681$	−8.39648e9	−	7.04549e9i	−1.01878	−	0.854862i
$682$		0			0
$683$	−6.71865e9			−0.806881			−0.403440	−	0.915006i	$-0.632186\pi$
$683$	−6.71865e9			−0.806881			−0.403440	+	0.915006i	$0.632186\pi$
$684$		0			0
$685$	9.57991e8			0.113879
$686$		0			0
$687$	−3.43285e9	−	2.88050e9i	−0.403930	−	0.338937i
$688$		0			0
$689$	−5.84486e9	+	2.12735e9i	−0.680780	+	0.247784i
$690$		0			0
$691$	−4.48039e9	−	7.76026e9i	−0.516586	−	0.894753i	−0.999815	−	0.0192590i	$-0.993869\pi$
$691$	−4.48039e9	−	7.76026e9i	−0.516586	−	0.894753i	0.483229	−	0.875494i	$-0.339464\pi$
$692$		0			0
$693$	4.31601e8	+	2.44773e9i	0.0492624	+	0.279381i
$694$		0			0
$695$	5.99953e9	−	1.03915e10i	0.677907	−	1.17417i
$696$		0			0
$697$	−3.41434e9	+	2.86497e9i	−0.381937	+	0.320483i
$698$		0			0
$699$	1.88659e10	+	6.86662e9i	2.08933	+	0.760453i
$700$		0			0
$701$	−9.63421e8	+	5.46383e9i	−0.105634	+	0.599080i	0.885331	+	0.464961i	$0.153932\pi$
$701$	−9.63421e8	+	5.46383e9i	−0.105634	+	0.599080i	−0.990965	+	0.134119i	$0.957180\pi$
$702$		0			0
$703$	1.33498e10	+	3.46262e9i	1.44921	+	0.375890i
$704$		0			0
$705$	−4.08137e9	+	2.31466e10i	−0.438676	+	2.48785i
$706$		0			0
$707$	5.00733e9	+	1.82252e9i	0.532890	+	0.193956i
$708$		0			0
$709$	−1.18565e10	+	9.94882e9i	−1.24939	+	1.04836i	−0.252655	+	0.967556i	$0.581304\pi$
$709$	−1.18565e10	+	9.94882e9i	−1.24939	+	1.04836i	−0.996730	+	0.0808021i	$0.974252\pi$
$710$		0			0
$711$	2.72418e8	−	4.71842e8i	0.0284245	−	0.0492326i
$712$		0			0
$713$	−2.07847e9	−	1.17876e10i	−0.214749	−	1.21790i
$714$		0			0
$715$	−3.72265e9	−	6.44782e9i	−0.380874	−	0.659693i
$716$		0			0
$717$	1.34636e10	−	4.90034e9i	1.36409	−	0.496489i
$718$		0			0
$719$	−3.78645e9	−	3.17720e9i	−0.379910	−	0.318782i	0.432757	−	0.901511i	$-0.357541\pi$
$719$	−3.78645e9	−	3.17720e9i	−0.379910	−	0.318782i	−0.812667	+	0.582728i	$0.801985\pi$
$720$		0			0
$721$	7.54633e9			0.749829
$722$		0			0
$723$	−1.06833e10			−1.05129
$724$		0			0
$725$	2.57894e10	+	2.16399e10i	2.51338	+	2.10898i
$726$		0			0
$727$	4.59451e8	−	1.67227e8i	0.0443475	−	0.0161412i	−0.319751	−	0.947502i	$-0.603599\pi$
$727$	4.59451e8	−	1.67227e8i	0.0443475	−	0.0161412i	0.364099	+	0.931360i	$0.381377\pi$
$728$		0			0
$729$	−2.69834e9	−	4.67367e9i	−0.257959	−	0.446798i
$730$		0			0
$731$	1.56090e9	+	8.85232e9i	0.147797	+	0.838197i
$732$		0			0
$733$	4.92654e9	−	8.53302e9i	0.462038	−	0.800274i	−0.537024	−	0.843567i	$-0.680452\pi$
$733$	4.92654e9	−	8.53302e9i	0.462038	−	0.800274i	0.999062	+	0.0432930i	$0.0137849\pi$
$734$		0			0
$735$	3.29026e10	−	2.76085e10i	3.05650	−	2.56471i
$736$		0			0
$737$	8.30874e9	+	3.02413e9i	0.764537	+	0.278269i
$738$		0			0
$739$	−3.43141e8	+	1.94605e9i	−0.0312764	+	0.177378i	−0.996444	−	0.0842538i	$-0.973149\pi$
$739$	−3.43141e8	+	1.94605e9i	−0.0312764	+	0.177378i	0.965168	+	0.261631i	$0.0842605\pi$
$740$		0			0
$741$	9.17991e9	+	2.38104e9i	0.828848	+	0.214983i
$742$		0			0
$743$	−2.24099e9	+	1.27093e10i	−0.200437	+	1.13674i	0.704022	+	0.710178i	$0.251386\pi$
$743$	−2.24099e9	+	1.27093e10i	−0.200437	+	1.13674i	−0.904460	+	0.426559i	$0.859726\pi$
$744$		0			0
$745$	−1.30968e9	−	4.76686e8i	−0.116043	−	0.0422363i
$746$		0			0
$747$	−8.33302e8	+	6.99224e8i	−0.0731443	+	0.0613753i
$748$		0			0
$749$	−1.45309e10	+	2.51683e10i	−1.26359	+	2.18861i
$750$		0			0
$751$	3.79895e9	+	2.15449e10i	0.327283	+	1.85611i	0.493120	+	0.869961i	$0.335856\pi$
$751$	3.79895e9	+	2.15449e10i	0.327283	+	1.85611i	−0.165837	+	0.986153i	$0.553033\pi$
$752$		0			0
$753$	−3.81023e9	−	6.59951e9i	−0.325214	−	0.563287i
$754$		0			0
$755$	−1.41047e10	+	5.13369e9i	−1.19275	+	0.434126i
$756$		0			0
$757$	−5.75825e9	−	4.83175e9i	−0.482453	−	0.404826i	0.368859	−	0.929485i	$-0.379748\pi$
$757$	−5.75825e9	−	4.83175e9i	−0.482453	−	0.404826i	−0.851313	+	0.524659i	$0.824193\pi$
$758$		0			0
$759$	−9.73685e9			−0.808299
$760$		0			0
$761$	−1.95823e10			−1.61071			−0.805354	−	0.592794i	$-0.798025\pi$
$761$	−1.95823e10			−1.61071			−0.805354	+	0.592794i	$0.798025\pi$
$762$		0			0
$763$	−1.42058e10	−	1.19201e10i	−1.15779	−	0.971504i
$764$		0			0
$765$	−5.43850e9	+	1.97945e9i	−0.439202	+	0.159856i
$766$		0			0
$767$	1.21040e9	+	2.09648e9i	0.0968605	+	0.167767i
$768$		0			0
$769$	1.68965e9	+	9.58246e9i	0.133984	+	0.759862i	0.975562	+	0.219726i	$0.0705162\pi$
$769$	1.68965e9	+	9.58246e9i	0.133984	+	0.759862i	−0.841577	+	0.540136i	$0.818373\pi$
$770$		0			0
$771$	−3.95517e9	+	6.85056e9i	−0.310796	+	0.538314i
$772$		0			0
$773$	1.42544e10	−	1.19609e10i	1.10999	−	0.931395i	0.111937	−	0.993715i	$-0.464294\pi$
$773$	1.42544e10	−	1.19609e10i	1.10999	−	0.931395i	0.998056	+	0.0623201i	$0.0198500\pi$
$774$		0			0
$775$	3.02188e10	+	1.09988e10i	2.33196	+	0.848765i
$776$		0			0
$777$	6.55078e9	−	3.71513e10i	0.500979	−	2.84119i
$778$		0			0
$779$	4.92563e9	+	6.91586e9i	0.373320	+	0.524162i
$780$		0			0
$781$	−1.40487e9	+	7.96740e9i	−0.105525	+	0.598464i
$782$		0			0
$783$	1.22853e10	+	4.47149e9i	0.914578	+	0.332879i
$784$		0			0
$785$	−7.36690e8	+	6.18156e8i	−0.0543552	+	0.0456094i
$786$		0			0
$787$	4.49814e9	−	7.79101e9i	0.328944	−	0.569747i	−0.653359	−	0.757048i	$-0.726641\pi$
$787$	4.49814e9	−	7.79101e9i	0.328944	−	0.569747i	0.982303	+	0.187301i	$0.0599740\pi$
$788$		0			0
$789$	2.78632e9	+	1.58020e10i	0.201958	+	1.14536i
$790$		0			0
$791$	−5.50033e9	−	9.52686e9i	−0.395159	−	0.684435i
$792$		0			0
$793$	−1.18538e10	+	4.31442e9i	−0.844112	+	0.307232i
$794$		0			0
$795$	−2.30704e10	−	1.93584e10i	−1.62843	−	1.36642i
$796$		0			0
$797$	1.17928e10			0.825110			0.412555	−	0.910933i	$-0.364636\pi$
$797$	1.17928e10			0.825110			0.412555	+	0.910933i	$0.364636\pi$
$798$		0			0
$799$	−1.28848e10			−0.893645
$800$		0			0
$801$	−1.21511e9	−	1.01960e9i	−0.0835412	−	0.0700994i
$802$		0			0
$803$	−4.86767e9	+	1.77169e9i	−0.331754	+	0.120749i
$804$		0			0
$805$	3.12878e10	+	5.41921e10i	2.11392	+	3.66142i
$806$		0			0
$807$	−3.81314e9	−	2.16254e10i	−0.255403	−	1.44846i
$808$		0			0
$809$	8.39656e9	−	1.45433e10i	0.557547	−	0.965700i	−0.440153	−	0.897923i	$-0.645076\pi$
$809$	8.39656e9	−	1.45433e10i	0.557547	−	0.965700i	0.997700	−	0.0677777i	$-0.0215909\pi$
$810$		0			0
$811$	5.76151e9	−	4.83448e9i	0.379283	−	0.318256i	−0.433138	−	0.901328i	$-0.642594\pi$
$811$	5.76151e9	−	4.83448e9i	0.379283	−	0.318256i	0.812421	+	0.583071i	$0.198149\pi$
$812$		0			0
$813$	−1.10723e10	−	4.03000e9i	−0.722641	−	0.263020i
$814$		0			0
$815$	4.56836e9	−	2.59084e10i	0.295603	−	1.67645i
$816$		0			0
$817$	1.70459e10	−	1.62911e9i	1.09356	−	0.104514i
$818$		0			0
$819$	1.08155e9	−	6.13378e9i	0.0687944	−	0.390153i
$820$		0			0
$821$	−7.73231e9	−	2.81433e9i	−0.487650	−	0.177490i	0.0864812	−	0.996253i	$-0.472438\pi$
$821$	−7.73231e9	−	2.81433e9i	−0.487650	−	0.177490i	−0.574131	+	0.818764i	$0.694660\pi$
$822$		0			0
$823$	−1.78424e8	+	1.49716e8i	−0.0111572	+	0.00936198i	−0.648349	−	0.761343i	$-0.724540\pi$
$823$	−1.78424e8	+	1.49716e8i	−0.0111572	+	0.00936198i	0.637192	+	0.770705i	$0.280096\pi$
$824$		0			0
$825$	1.30800e10	−	2.26551e10i	0.810993	−	1.40468i
$826$		0			0
$827$	4.97506e9	+	2.82150e10i	0.305864	+	1.73464i	0.619408	+	0.785070i	$0.287373\pi$
$827$	4.97506e9	+	2.82150e10i	0.305864	+	1.73464i	−0.313543	+	0.949574i	$0.601516\pi$
$828$		0			0
$829$	−6.37828e9	−	1.10475e10i	−0.388833	−	0.673478i	0.603460	−	0.797393i	$-0.293788\pi$
$829$	−6.37828e9	−	1.10475e10i	−0.388833	−	0.673478i	−0.992293	+	0.123915i	$0.960455\pi$
$830$		0			0
$831$	7.10781e9	−	2.58703e9i	0.429667	−	0.156386i
$832$		0			0
$833$	1.80374e10	+	1.51351e10i	1.08122	+	0.907254i
$834$		0			0
$835$	1.03963e9			0.0617982
$836$		0			0
$837$	1.24884e10			0.736150
$838$		0			0
$839$	9.59063e8	+	8.04750e8i	0.0560635	+	0.0470429i	0.670389	−	0.742010i	$-0.266127\pi$
$839$	9.59063e8	+	8.04750e8i	0.0560635	+	0.0470429i	−0.614325	+	0.789053i	$0.710572\pi$
$840$		0			0
$841$	−8.72701e9	+	3.17637e9i	−0.505917	+	0.184139i
$842$		0			0
$843$	8.63068e9	+	1.49488e10i	0.496190	+	0.859427i
$844$		0			0
$845$	−2.57500e9	−	1.46035e10i	−0.146818	−	0.832645i
$846$		0			0
$847$	−1.06096e10	+	1.83764e10i	−0.599941	+	1.03913i
$848$		0			0
$849$	−6.82438e9	+	5.72633e9i	−0.382724	+	0.321144i
$850$		0			0
$851$	3.33431e10	+	1.21359e10i	1.85461	+	0.675023i
$852$		0			0
$853$	−3.39595e9	+	1.92594e10i	−0.187344	+	1.06248i	0.735562	+	0.677457i	$0.236918\pi$
$853$	−3.39595e9	+	1.92594e10i	−0.187344	+	1.06248i	−0.922906	+	0.385024i	$0.874193\pi$
$854$		0			0
$855$	2.93877e9	+	1.06262e10i	0.160800	+	0.581428i
$856$		0			0
$857$	−1.07946e9	+	6.12190e9i	−0.0585830	+	0.332241i	−0.999987	−	0.00503941i	$-0.998396\pi$
$857$	−1.07946e9	+	6.12190e9i	−0.0585830	+	0.332241i	0.941404	+	0.337280i	$0.109507\pi$
$858$		0			0
$859$	−1.70298e10	−	6.19834e9i	−0.916713	−	0.333656i	−0.159783	−	0.987152i	$-0.551079\pi$
$859$	−1.70298e10	−	6.19834e9i	−0.916713	−	0.333656i	−0.756930	+	0.653496i	$0.773302\pi$
$860$		0			0
$861$	1.77911e10	−	1.49285e10i	0.949932	−	0.797087i
$862$		0			0
$863$	−1.00955e10	+	1.74860e10i	−0.534676	+	0.926087i	0.464503	+	0.885572i	$0.346233\pi$
$863$	−1.00955e10	+	1.74860e10i	−0.534676	+	0.926087i	−0.999179	+	0.0405149i	$0.987100\pi$
$864$		0			0
$865$	−6.88686e9	−	3.90573e10i	−0.361797	−	2.05185i
$866$		0			0
$867$	4.39956e9	+	7.62026e9i	0.229267	+	0.397103i
$868$		0			0
$869$	−1.74822e9	+	6.36301e8i	−0.0903707	+	0.0328923i
$870$		0			0
$871$	−1.69734e10	−	1.42423e10i	−0.870371	−	0.730328i
$872$		0			0
$873$	−8.12902e9			−0.413512
$874$		0			0
$875$	−1.04566e11			−5.27670
$876$		0			0
$877$	−7.68577e9	−	6.44912e9i	−0.384759	−	0.322851i	0.429808	−	0.902920i	$-0.358581\pi$
$877$	−7.68577e9	−	6.44912e9i	−0.384759	−	0.322851i	−0.814567	+	0.580069i	$0.803026\pi$
$878$		0			0
$879$	−2.44359e10	+	8.89396e9i	−1.21358	+	0.441707i
$880$		0			0
$881$	−1.35048e10	−	2.33911e10i	−0.665386	−	1.15248i	−0.979181	−	0.202991i	$-0.934934\pi$
$881$	−1.35048e10	−	2.33911e10i	−0.665386	−	1.15248i	0.313795	−	0.949491i	$-0.398400\pi$
$882$		0			0
$883$	−5.79350e9	−	3.28566e10i	−0.283190	−	1.60605i	−0.711678	−	0.702505i	$-0.752065\pi$
$883$	−5.79350e9	−	3.28566e10i	−0.283190	−	1.60605i	0.428488	−	0.903548i	$-0.359046\pi$
$884$		0			0
$885$	−5.86062e9	+	1.01509e10i	−0.284212	+	0.492269i
$886$		0			0
$887$	−1.25881e10	+	1.05626e10i	−0.605657	+	0.508206i	−0.893258	−	0.449544i	$-0.851587\pi$
$887$	−1.25881e10	+	1.05626e10i	−0.605657	+	0.508206i	0.287601	+	0.957750i	$0.407142\pi$
$888$		0			0
$889$	−3.12874e9	−	1.13877e9i	−0.149353	−	0.0543600i
$890$		0			0
$891$	2.38329e9	−	1.35163e10i	0.112877	−	0.640156i
$892$		0			0
$893$	−1.94317e9	+	2.44682e10i	−0.0913126	+	1.14980i
$894$		0			0
$895$	−1.73588e9	+	9.84468e9i	−0.0809356	+	0.459009i
$896$		0			0
$897$	2.29282e10	+	8.34517e9i	1.06071	+	0.386066i
$898$		0			0
$899$	−1.94182e10	+	1.62938e10i	−0.891355	+	0.747936i
$900$		0			0
$901$	8.25495e9	−	1.42980e10i	0.375991	−	0.651236i
$902$		0			0
$903$	−8.13340e9	−	4.61268e10i	−0.367591	−	2.08471i
$904$		0			0
$905$	5.00285e9	+	8.66519e9i	0.224361	+	0.388605i
$906$		0			0
$907$	3.79943e10	−	1.38288e10i	1.69080	−	0.615402i	0.696078	−	0.717966i	$-0.254927\pi$
$907$	3.79943e10	−	1.38288e10i	1.69080	−	0.615402i	0.994726	+	0.102564i	$0.0327047\pi$
$908$		0			0
$909$	1.85036e9	+	1.55264e9i	0.0817115	+	0.0685641i
$910$		0			0
$911$	−9.23644e9			−0.404753			−0.202377	−	0.979308i	$-0.564867\pi$
$911$	−9.23644e9			−0.404753			−0.202377	+	0.979308i	$0.564867\pi$
$912$		0			0
$913$	3.71445e9			0.161527
$914$		0			0
$915$	−4.67883e10	−	3.92601e10i	−2.01913	−	1.69425i
$916$		0			0
$917$	9.71268e9	−	3.53513e9i	0.415955	−	0.151395i
$918$		0			0
$919$	−9.51218e9	−	1.64756e10i	−0.404274	−	0.700223i	0.589963	−	0.807431i	$-0.299142\pi$
$919$	−9.51218e9	−	1.64756e10i	−0.404274	−	0.700223i	−0.994237	+	0.107207i	$0.965809\pi$
$920$		0			0
$921$	3.96033e9	+	2.24602e10i	0.167041	+	0.947337i
$922$		0			0
$923$	1.01368e10	−	1.75574e10i	0.424320	−	0.734944i
$924$		0			0
$925$	−7.30284e10	+	6.12781e10i	−3.03386	+	2.54571i
$926$		0			0
$927$	3.21443e9	+	1.16996e9i	0.132533	+	0.0482382i
$928$		0			0
$929$	7.23617e9	−	4.10383e10i	0.296110	−	1.67933i	−0.366545	−	0.930400i	$-0.619459\pi$
$929$	7.23617e9	−	4.10383e10i	0.296110	−	1.67933i	0.662655	−	0.748925i	$-0.269430\pi$
$930$		0			0
$931$	3.14617e10	−	3.19702e10i	1.27779	−	1.29844i
$932$		0			0
$933$	−4.26865e9	+	2.42087e10i	−0.172070	+	0.975857i
$934$		0			0
$935$	1.85705e10	+	6.75913e9i	0.742992	+	0.270427i
$936$		0			0
$937$	2.75739e10	−	2.31372e10i	1.09499	−	0.918804i	0.0979105	−	0.995195i	$-0.468784\pi$
$937$	2.75739e10	−	2.31372e10i	1.09499	−	0.918804i	0.997078	+	0.0763908i	$0.0243397\pi$
$938$		0			0
$939$	−1.39899e10	+	2.42312e10i	−0.551422	+	0.955090i
$940$		0			0
$941$	−4.20351e9	−	2.38393e10i	−0.164455	−	0.932673i	−0.949624	−	0.313391i	$-0.898535\pi$
$941$	−4.20351e9	−	2.38393e10i	−0.164455	−	0.932673i	0.785169	−	0.619282i	$-0.212576\pi$
$942$		0			0
$943$	1.09224e10	+	1.89181e10i	0.424156	+	0.734660i
$944$		0			0
$945$	−6.13511e10	+	2.23300e10i	−2.36489	+	0.860749i
$946$		0			0
$947$	3.69296e10	+	3.09876e10i	1.41302	+	1.18567i	0.954954	+	0.296752i	$0.0959036\pi$
$947$	3.69296e10	+	3.09876e10i	1.41302	+	1.18567i	0.458070	+	0.888916i	$0.348541\pi$
$948$		0			0
$949$	1.29807e10			0.493024
$950$		0			0
$951$	3.09267e10			1.16601
$952$		0			0
$953$	−1.26111e10	−	1.05820e10i	−0.471986	−	0.396044i	0.375532	−	0.926809i	$-0.377460\pi$
$953$	−1.26111e10	−	1.05820e10i	−0.471986	−	0.396044i	−0.847519	+	0.530766i	$0.821904\pi$
$954$		0			0
$955$	6.00335e10	−	2.18504e10i	2.23039	−	0.811797i
$956$		0			0
$957$	1.03103e10	+	1.78579e10i	0.380258	+	0.658626i
$958$		0			0
$959$	−4.75195e8	−	2.69496e9i	−0.0173983	−	0.0986706i
$960$		0			0
$961$	1.64948e9	−	2.85698e9i	0.0599535	−	0.103843i
$962$		0			0
$963$	−1.00916e10	+	8.46786e9i	−0.364140	+	0.305550i
$964$		0			0
$965$	7.27243e10	+	2.64695e10i	2.60516	+	0.948200i
$966$		0			0
$967$	2.53325e9	−	1.43668e10i	0.0900920	−	0.510937i	−0.906049	−	0.423172i	$-0.860917\pi$
$967$	2.53325e9	−	1.43668e10i	0.0900920	−	0.510937i	0.996141	−	0.0877647i	$-0.0279724\pi$
$968$		0			0
$969$	−2.27284e10	+	1.08211e10i	−0.802482	+	0.382065i
$970$		0			0
$971$	−2.44295e9	+	1.38547e10i	−0.0856344	+	0.485657i	0.911584	+	0.411115i	$0.134861\pi$
$971$	−2.44295e9	+	1.38547e10i	−0.0856344	+	0.485657i	−0.997218	+	0.0745417i	$0.976251\pi$
$972$		0			0
$973$	−3.22087e10	−	1.17230e10i	−1.12093	−	0.407985i
$974$		0			0
$975$	−5.02175e10	+	4.21375e10i	−1.73516	+	1.45597i
$976$		0			0
$977$	−1.03730e10	+	1.79665e10i	−0.355855	+	0.616359i	−0.987264	−	0.159091i	$-0.949144\pi$
$977$	−1.03730e10	+	1.79665e10i	−0.355855	+	0.616359i	0.631409	+	0.775450i	$0.282477\pi$
$978$		0			0
$979$	9.40537e8	+	5.33405e9i	0.0320359	+	0.181684i
$980$		0			0
$981$	−4.20306e9	−	7.27991e9i	−0.142142	−	0.246198i
$982$		0			0
$983$	2.01730e10	−	7.34237e9i	0.677381	−	0.246547i	0.0196585	−	0.999807i	$-0.493742\pi$
$983$	2.01730e10	−	7.34237e9i	0.677381	−	0.246547i	0.657723	+	0.753260i	$0.271520\pi$
$984$		0			0
$985$	4.25043e9	+	3.56653e9i	0.141712	+	0.118910i
$986$		0			0
$987$	6.71391e10			2.22262
$988$		0			0
$989$	4.40554e10			1.44815
$990$		0			0
$991$	−4.48523e9	−	3.76356e9i	−0.146395	−	0.122840i	0.566649	−	0.823959i	$-0.308240\pi$
$991$	−4.48523e9	−	3.76356e9i	−0.146395	−	0.122840i	−0.713044	+	0.701119i	$0.752684\pi$
$992$		0			0
$993$	4.09312e10	−	1.48977e10i	1.32658	−	0.482834i
$994$		0			0
$995$	1.92170e10	+	3.32848e10i	0.618449	+	1.07119i
$996$		0			0
$997$	−3.05809e9	−	1.73433e10i	−0.0977276	−	0.554241i	−0.993877	−	0.110489i	$-0.964758\pi$
$997$	−3.05809e9	−	1.73433e10i	−0.0977276	−	0.554241i	0.896150	−	0.443752i	$-0.146353\pi$
$998$		0			0
$999$	−1.85107e10	+	3.20614e10i	−0.587412	+	1.01743i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.i.a.17.10	yes	72
19.9	even	9	inner	76.8.i.a.9.10	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.i.a.9.10	✓	72	19.9	even	9	inner
76.8.i.a.17.10	yes	72	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(41.0960 + 34.4837i) q^{3} +(-501.471 + 182.521i) q^{5} +(762.203 + 1320.17i) q^{7} +(119.992 + 680.510i) q^{9} +O(q^{10})\)	\(q+(41.0960 + 34.4837i) q^{3} +(-501.471 + 182.521i) q^{5} +(762.203 + 1320.17i) q^{7} +(119.992 + 680.510i) q^{9} +(1179.77 - 2043.43i) q^{11} +(-4529.47 + 3800.68i) q^{13} +(-26902.5 - 9791.70i) q^{15} +(2725.34 - 15456.2i) q^{17} +(-28940.1 - 7506.34i) q^{19} +(-14200.9 + 80537.5i) q^{21} +(-72281.9 - 26308.5i) q^{23} +(158313. - 132840. i) q^{25} +(40127.8 - 69503.4i) q^{27} +(28287.6 + 160427. i) q^{29} +(77803.7 + 134760. i) q^{31} +(118949. - 43293.8i) q^{33} +(-623182. - 522912. i) q^{35} -461293. q^{37} -317204. q^{39} +(-217549. - 182545. i) q^{41} +(-538197. + 195888. i) q^{43} +(-184380. - 319355. i) q^{45} +(-142560. - 808501. i) q^{47} +(-750135. + 1.29927e6i) q^{49} +(644985. - 541207. i) q^{51} +(988509. + 359788. i) q^{53} +(-218655. + 1.24005e6i) q^{55} +(-930476. - 1.30644e6i) q^{57} +(71094.7 - 403198. i) q^{59} +(2.00476e6 + 729674. i) q^{61} +(-806933. + 677097. i) q^{63} +(1.57770e6 - 2.73265e6i) q^{65} +(650715. + 3.69039e6i) q^{67} +(-2.06329e6 - 3.57372e6i) q^{69} +(-3.22198e6 + 1.17270e6i) q^{71} +(-1.68175e6 - 1.41115e6i) q^{73} +1.10868e7 q^{75} +3.59691e6 q^{77} +(-604000. - 506816. i) q^{79} +(5.46592e6 - 1.98943e6i) q^{81} +(787110. + 1.36331e6i) q^{83} +(1.45439e6 + 8.24825e6i) q^{85} +(-4.36960e6 + 7.56837e6i) q^{87} +(-1.75845e6 + 1.47552e6i) q^{89} +(-8.46993e6 - 3.08280e6i) q^{91} +(-1.44959e6 + 8.22106e6i) q^{93} +(1.58827e7 - 1.51795e6i) q^{95} +(-2.04280e6 + 1.15853e7i) q^{97} +(1.53214e6 + 557652. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more