LMFDB - Embedded newform 138.6.d.a.137.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,6,Mod(137,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("138.137");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	138.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:	$22.1329671342$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			137.1
Character	$\chi$	$=$	138.137
Dual form			138.6.d.a.137.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} +(-11.7474 + 10.2469i) q^{3} -16.0000 q^{4} -107.494 q^{5} +(40.9875 + 46.9896i) q^{6} -192.413i q^{7} +64.0000i q^{8} +(33.0032 - 240.748i) q^{9} +O(q^{10})$	\(q-4.00000i q^{2} +(-11.7474 + 10.2469i) q^{3} -16.0000 q^{4} -107.494 q^{5} +(40.9875 + 46.9896i) q^{6} -192.413i q^{7} +64.0000i q^{8} +(33.0032 - 240.748i) q^{9} +429.977i q^{10} -525.803 q^{11} +(187.959 - 163.950i) q^{12} -1096.97 q^{13} -769.652 q^{14} +(1262.78 - 1101.48i) q^{15} +256.000 q^{16} +665.814 q^{17} +(-962.994 - 132.013i) q^{18} -365.592i q^{19} +1719.91 q^{20} +(1971.63 + 2260.35i) q^{21} +2103.21i q^{22} +(-1075.29 + 2297.85i) q^{23} +(-655.800 - 751.834i) q^{24} +8430.00 q^{25} +4387.88i q^{26} +(2079.22 + 3166.35i) q^{27} +3078.61i q^{28} -2165.14i q^{29} +(-4405.92 - 5051.11i) q^{30} -4020.58 q^{31} -1024.00i q^{32} +(6176.82 - 5387.84i) q^{33} -2663.26i q^{34} +20683.3i q^{35} +(-528.051 + 3851.97i) q^{36} -11723.5i q^{37} -1462.37 q^{38} +(12886.6 - 11240.5i) q^{39} -6879.63i q^{40} +10135.8i q^{41} +(9041.41 - 7886.52i) q^{42} -3679.94i q^{43} +8412.85 q^{44} +(-3547.65 + 25879.1i) q^{45} +(9191.39 + 4301.15i) q^{46} +8076.34i q^{47} +(-3007.34 + 2623.20i) q^{48} -20215.7 q^{49} -33720.0i q^{50} +(-7821.59 + 6822.52i) q^{51} +17551.5 q^{52} -12175.6 q^{53} +(12665.4 - 8316.87i) q^{54} +56520.8 q^{55} +12314.4 q^{56} +(3746.18 + 4294.76i) q^{57} -8660.56 q^{58} -3846.00i q^{59} +(-20204.5 + 17623.7i) q^{60} -20774.9i q^{61} +16082.3i q^{62} +(-46323.1 - 6350.24i) q^{63} -4096.00 q^{64} +117918. q^{65} +(-21551.3 - 24707.3i) q^{66} -20138.2i q^{67} -10653.0 q^{68} +(-10913.9 - 38012.1i) q^{69} +82733.1 q^{70} -57543.7i q^{71} +(15407.9 + 2112.20i) q^{72} -34442.0 q^{73} -46893.9 q^{74} +(-99030.7 + 86381.2i) q^{75} +5849.48i q^{76} +101171. i q^{77} +(-44962.0 - 51546.2i) q^{78} +13257.0i q^{79} -27518.5 q^{80} +(-56870.6 - 15890.9i) q^{81} +40543.0 q^{82} -7450.77 q^{83} +(-31546.1 - 36165.7i) q^{84} -71571.2 q^{85} -14719.8 q^{86} +(22185.9 + 25434.8i) q^{87} -33651.4i q^{88} -18913.7 q^{89} +(103516. + 14190.6i) q^{90} +211071. i q^{91} +(17204.6 - 36765.6i) q^{92} +(47231.4 - 41198.4i) q^{93} +32305.4 q^{94} +39299.1i q^{95} +(10492.8 + 12029.3i) q^{96} -41309.0i q^{97} +80863.0i q^{98} +(-17353.2 + 126586. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9}+O(q^{10}) $ 40 * q - 4 * q^3 - 640 * q^4 + 80 * q^6 + 504 * q^9	$ 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9} + 64 q^{12} - 1048 q^{13} + 10240 q^{16} + 1280 q^{18} - 1280 q^{24} + 30480 q^{25} + 1700 q^{27} - 22576 q^{31} - 8064 q^{36} + 55608 q^{39} + 1088 q^{46} - 1024 q^{48} - 23224 q^{49} + 16768 q^{52} + 25456 q^{54} + 210400 q^{55} - 83168 q^{58} - 163840 q^{64} + 99076 q^{69} + 167520 q^{70} - 20480 q^{72} + 241160 q^{73} - 255604 q^{75} - 233440 q^{78} + 78512 q^{81} - 8832 q^{82} - 460296 q^{85} - 4136 q^{87} + 500704 q^{93} - 138272 q^{94} + 20480 q^{96}+O(q^{100}) $ 40 * q - 4 * q^3 - 640 * q^4 + 80 * q^6 + 504 * q^9 + 64 * q^12 - 1048 * q^13 + 10240 * q^16 + 1280 * q^18 - 1280 * q^24 + 30480 * q^25 + 1700 * q^27 - 22576 * q^31 - 8064 * q^36 + 55608 * q^39 + 1088 * q^46 - 1024 * q^48 - 23224 * q^49 + 16768 * q^52 + 25456 * q^54 + 210400 * q^55 - 83168 * q^58 - 163840 * q^64 + 99076 * q^69 + 167520 * q^70 - 20480 * q^72 + 241160 * q^73 - 255604 * q^75 - 233440 * q^78 + 78512 * q^81 - 8832 * q^82 - 460296 * q^85 - 4136 * q^87 + 500704 * q^93 - 138272 * q^94 + 20480 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	4.00000i		−	0.707107i
$2$		−	4.00000i		−	0.707107i
$3$	−11.7474	+	10.2469i	−0.753597	+	0.657337i
$3$	−11.7474	+	10.2469i	−0.753597	+	0.657337i
$4$	−16.0000			−0.500000
$5$	−107.494			−1.92291			−0.961457	−	0.274954i	$-0.911337\pi$
$5$	−107.494			−1.92291			−0.961457	+	0.274954i	$0.911337\pi$
$6$	40.9875	+	46.9896i	0.464808	+	0.532873i
$7$		−	192.413i		−	1.48419i	−0.670295	−	0.742094i	$-0.733833\pi$
$7$		−	192.413i		−	1.48419i	0.670295	−	0.742094i	$-0.266167\pi$
$8$			64.0000i			0.353553i
$9$	33.0032	−	240.748i	0.135816	−	0.990734i
$10$			429.977i			1.35971i
$11$	−525.803			−1.31021			−0.655106	−	0.755537i	$-0.727376\pi$
$11$	−525.803			−1.31021			−0.655106	+	0.755537i	$0.727376\pi$
$12$	187.959	−	163.950i	0.376798	−	0.328669i
$13$	−1096.97			−1.80026			−0.900132	−	0.435617i	$-0.856530\pi$
$13$	−1096.97			−1.80026			−0.900132	+	0.435617i	$0.856530\pi$
$14$	−769.652			−1.04948
$15$	1262.78	−	1101.48i	1.44910	−	1.26400i
$16$	256.000			0.250000
$17$	665.814			0.558767			0.279384	−	0.960180i	$-0.409870\pi$
$17$	665.814			0.558767			0.279384	+	0.960180i	$0.409870\pi$
$18$	−962.994	−	132.013i	−0.700555	−	0.0960361i
$19$		−	365.592i		−	0.232334i	−0.993230	−	0.116167i	$-0.962939\pi$
$19$		−	365.592i		−	0.232334i	0.993230	−	0.116167i	$-0.0370608\pi$
$20$	1719.91			0.961457
$21$	1971.63	+	2260.35i	0.975613	+	1.11848i
$22$			2103.21i			0.926459i
$23$	−1075.29	+	2297.85i	−0.423843	+	0.905736i
$23$	−1075.29	+	2297.85i	−0.423843	+	0.905736i
$24$	−655.800	−	751.834i	−0.232404	−	0.266437i
$25$	8430.00			2.69760
$26$			4387.88i			1.27298i
$27$	2079.22	+	3166.35i	0.548896	+	0.835890i
$28$			3078.61i			0.742094i
$29$		−	2165.14i		−	0.478069i	−0.971011	−	0.239035i	$-0.923169\pi$
$29$		−	2165.14i		−	0.478069i	0.971011	−	0.239035i	$-0.0768310\pi$
$30$	−4405.92	−	5051.11i	−0.893785	−	1.02467i
$31$	−4020.58			−0.751424			−0.375712	−	0.926737i	$-0.622602\pi$
$31$	−4020.58			−0.751424			−0.375712	+	0.926737i	$0.622602\pi$
$32$		−	1024.00i		−	0.176777i
$33$	6176.82	−	5387.84i	0.987371	−	0.861251i
$34$		−	2663.26i		−	0.395108i
$35$			20683.3i			2.85397i
$36$	−528.051	+	3851.97i	−0.0679078	+	0.495367i
$37$		−	11723.5i		−	1.40784i	−0.710281	−	0.703918i	$-0.751432\pi$
$37$		−	11723.5i		−	1.40784i	0.710281	−	0.703918i	$-0.248568\pi$
$38$	−1462.37			−0.164285
$39$	12886.6	−	11240.5i	1.35667	−	1.18338i
$40$		−	6879.63i		−	0.679853i
$41$			10135.8i			0.941665i	0.882223	+	0.470832i	$0.156046\pi$
$41$			10135.8i			0.941665i	−0.882223	+	0.470832i	$0.843954\pi$
$42$	9041.41	−	7886.52i	0.790885	−	0.689862i
$43$		−	3679.94i		−	0.303508i	−0.988418	−	0.151754i	$-0.951508\pi$
$43$		−	3679.94i		−	0.303508i	0.988418	−	0.151754i	$-0.0484921\pi$
$44$	8412.85			0.655106
$45$	−3547.65	+	25879.1i	−0.261162	+	1.90510i
$46$	9191.39	+	4301.15i	0.640452	+	0.299702i
$47$			8076.34i			0.533298i	0.963794	+	0.266649i	$0.0859164\pi$
$47$			8076.34i			0.533298i	−0.963794	+	0.266649i	$0.914084\pi$
$48$	−3007.34	+	2623.20i	−0.188399	+	0.164334i
$49$	−20215.7			−1.20282
$50$		−	33720.0i		−	1.90749i
$51$	−7821.59	+	6822.52i	−0.421085	+	0.367298i
$52$	17551.5			0.900132
$53$	−12175.6			−0.595388			−0.297694	−	0.954661i	$-0.596218\pi$
$53$	−12175.6			−0.595388			−0.297694	+	0.954661i	$0.596218\pi$
$54$	12665.4	−	8316.87i	0.591064	−	0.388128i
$55$	56520.8			2.51942
$56$	12314.4			0.524740
$57$	3746.18	+	4294.76i	0.152722	+	0.175086i
$58$	−8660.56			−0.338046
$59$		−	3846.00i		−	0.143840i	−0.997410	−	0.0719199i	$-0.977087\pi$
$59$		−	3846.00i		−	0.143840i	0.997410	−	0.0719199i	$-0.0229126\pi$
$60$	−20204.5	+	17623.7i	−0.724551	+	0.632002i
$61$		−	20774.9i		−	0.714848i	−0.933942	−	0.357424i	$-0.883655\pi$
$61$		−	20774.9i		−	0.714848i	0.933942	−	0.357424i	$-0.116345\pi$
$62$			16082.3i			0.531337i
$63$	−46323.1	−	6350.24i	−1.47044	−	0.201576i
$64$	−4096.00			−0.125000
$65$	117918.			3.46176
$66$	−21551.3	−	24707.3i	−0.608996	−	0.698176i
$67$		−	20138.2i		−	0.548067i	−0.961720	−	0.274034i	$-0.911642\pi$
$67$		−	20138.2i		−	0.548067i	0.961720	−	0.274034i	$-0.0883580\pi$
$68$	−10653.0			−0.279384
$69$	−10913.9	−	38012.1i	−0.275968	−	0.961167i
$70$	82733.1			2.01806
$71$		−	57543.7i		−	1.35473i	−0.735649	−	0.677363i	$-0.763122\pi$
$71$		−	57543.7i		−	1.35473i	0.735649	−	0.677363i	$-0.236878\pi$
$72$	15407.9	+	2112.20i	0.350277	+	0.0480181i
$73$	−34442.0			−0.756452			−0.378226	−	0.925713i	$-0.623466\pi$
$73$	−34442.0			−0.756452			−0.378226	+	0.925713i	$0.623466\pi$
$74$	−46893.9			−0.995491
$75$	−99030.7	+	86381.2i	−2.03290	+	1.77323i
$76$			5849.48i			0.116167i
$77$			101171.i			1.94460i
$78$	−44962.0	−	51546.2i	−0.836777	−	0.959313i
$79$			13257.0i			0.238989i	0.992835	+	0.119495i	$0.0381274\pi$
$79$			13257.0i			0.238989i	−0.992835	+	0.119495i	$0.961873\pi$
$80$	−27518.5			−0.480729
$81$	−56870.6	−	15890.9i	−0.963108	−	0.269114i
$82$	40543.0			0.665858
$83$	−7450.77			−0.118715			−0.0593576	−	0.998237i	$-0.518905\pi$
$83$	−7450.77			−0.118715			−0.0593576	+	0.998237i	$0.518905\pi$
$84$	−31546.1	−	36165.7i	−0.487806	−	0.559240i
$85$	−71571.2			−1.07446
$86$	−14719.8			−0.214612
$87$	22185.9	+	25434.8i	0.314253	+	0.360271i
$88$		−	33651.4i		−	0.463230i
$89$	−18913.7			−0.253106			−0.126553	−	0.991960i	$-0.540391\pi$
$89$	−18913.7			−0.253106			−0.126553	+	0.991960i	$0.540391\pi$
$90$	103516.	+	14190.6i	1.34711	+	0.184669i
$91$			211071.i			2.67193i
$92$	17204.6	−	36765.6i	0.211921	−	0.452868i
$93$	47231.4	−	41198.4i	0.566270	−	0.493939i
$94$	32305.4			0.377099
$95$			39299.1i			0.446759i
$96$	10492.8	+	12029.3i	0.116202	+	0.133218i
$97$		−	41309.0i		−	0.445775i	−0.974844	−	0.222887i	$-0.928452\pi$
$97$		−	41309.0i		−	0.445775i	0.974844	−	0.222887i	$-0.0715482\pi$
$98$			80863.0i			0.850520i
$99$	−17353.2	+	126586.i	−0.177947	+	1.29807i
$100$	−134880.			−1.34880
$101$			112029.i			1.09277i	0.837535	+	0.546384i	$0.183996\pi$
$101$			112029.i			1.09277i	−0.837535	+	0.546384i	$0.816004\pi$
$102$	27290.1	+	31286.4i	0.259719	+	0.297752i
$103$		−	92392.1i		−	0.858107i	−0.903279	−	0.429054i	$-0.858847\pi$
$103$		−	92392.1i		−	0.858107i	0.903279	−	0.429054i	$-0.141153\pi$
$104$		−	70206.1i		−	0.636490i
$105$	−211939.	−	242975.i	−1.87602	−	2.15074i
$106$			48702.4i			0.421003i
$107$	−105386.			−0.889862			−0.444931	−	0.895565i	$-0.646772\pi$
$107$	−105386.			−0.889862			−0.444931	+	0.895565i	$0.646772\pi$
$108$	−33267.5	−	50661.6i	−0.274448	−	0.417945i
$109$			67041.1i			0.540474i	0.962794	+	0.270237i	$0.0871021\pi$
$109$			67041.1i			0.540474i	−0.962794	+	0.270237i	$0.912898\pi$
$110$		−	226083.i		−	1.78150i
$111$	120129.	+	137721.i	0.925423	+	1.06094i
$112$		−	49257.7i		−	0.371047i
$113$	191967.			1.41426			0.707131	−	0.707082i	$-0.249989\pi$
$113$	191967.			1.41426			0.707131	+	0.707082i	$0.249989\pi$
$114$	17179.1	−	14984.7i	0.123805	−	0.107991i
$115$	115587.	−	247005.i	0.815013	−	1.74165i
$116$			34642.2i			0.239035i
$117$	−36203.5	+	264094.i	−0.244504	+	1.78358i
$118$	−15384.0			−0.101710
$119$		−	128111.i		−	0.829316i
$120$	70494.7	+	80817.8i	0.446893	+	0.512335i
$121$	115418.			0.716653
$122$	−83099.4			−0.505474
$123$	−103860.	−	119069.i	−0.618991	−	0.709636i
$124$	64329.3			0.375712
$125$	−570257.			−3.26434
$126$	−25401.0	+	185292.i	−0.142536	+	1.03976i
$127$	130291.			0.716809			0.358405	−	0.933566i	$-0.383321\pi$
$127$	130291.			0.716809			0.358405	+	0.933566i	$0.383321\pi$
$128$			16384.0i			0.0883883i
$129$	37707.9	+	43229.8i	0.199507	+	0.228722i
$130$		−	471672.i		−	2.44783i
$131$		−	131558.i		−	0.669791i	−0.942255	−	0.334895i	$-0.891299\pi$
$131$		−	131558.i		−	0.669791i	0.942255	−	0.334895i	$-0.108701\pi$
$132$	−98829.1	+	86205.4i	−0.493685	+	0.430625i
$133$	−70344.7			−0.344828
$134$	−80552.9			−0.387542
$135$	−223504.	−	340364.i	−1.05548	−	1.60735i
$136$			42612.1i			0.197554i
$137$	−133840.			−0.609233			−0.304616	−	0.952475i	$-0.598528\pi$
$137$	−133840.			−0.609233			−0.304616	+	0.952475i	$0.598528\pi$
$138$	−152048.	+	43655.7i	−0.679648	+	0.195139i
$139$	−87285.3			−0.383181			−0.191590	−	0.981475i	$-0.561365\pi$
$139$	−87285.3			−0.383181			−0.191590	+	0.981475i	$0.561365\pi$
$140$		−	330932.i		−	1.42698i
$141$	−82757.2	−	94876.0i	−0.350557	−	0.401891i
$142$	−230175.			−0.957937
$143$	576790.			2.35873
$144$	8448.82	−	61631.6i	0.0339539	−	0.247684i
$145$			232740.i			0.919287i
$146$			137768.i			0.534893i
$147$	237483.	−	207148.i	0.906439	−	0.790656i
$148$			187576.i			0.703918i
$149$	−271122.			−1.00046			−0.500230	−	0.865893i	$-0.666751\pi$
$149$	−271122.			−1.00046			−0.500230	+	0.865893i	$0.666751\pi$
$150$	345525.	+	396123.i	1.25387	+	1.43748i
$151$	95889.0			0.342237			0.171118	−	0.985250i	$-0.445262\pi$
$151$	95889.0			0.342237			0.171118	+	0.985250i	$0.445262\pi$
$152$	23397.9			0.0821426
$153$	21974.0	−	160294.i	0.0758893	−	0.553590i
$154$	404685.			1.37504
$155$	432189.			1.44492
$156$	−206185.	+	179848.i	−0.678337	+	0.591690i
$157$			542326.i			1.75595i	0.478710	+	0.877973i	$0.341105\pi$
$157$			542326.i			1.75595i	−0.478710	+	0.877973i	$0.658895\pi$
$158$	53028.1			0.168991
$159$	143032.	−	124762.i	0.448683	−	0.391371i
$160$			110074.i			0.339926i
$161$	442136.	+	206899.i	1.34428	+	0.629062i
$162$	−63563.7	+	227482.i	−0.190293	+	0.681020i
$163$	535276.			1.57801			0.789004	−	0.614388i	$-0.210597\pi$
$163$	535276.			1.57801			0.789004	+	0.614388i	$0.210597\pi$
$164$		−	162172.i		−	0.470832i
$165$	−663972.	+	579161.i	−1.89863	+	1.65611i
$166$			29803.1i			0.0839443i
$167$		−	195562.i		−	0.542616i	−0.962493	−	0.271308i	$-0.912544\pi$
$167$		−	195562.i		−	0.542616i	0.962493	−	0.271308i	$-0.0874562\pi$
$168$	−144663.	+	126184.i	−0.395442	+	0.344931i
$169$	832050.			2.24095
$170$			286285.i			0.759759i
$171$	−88015.8	−	12065.7i	−0.230181	−	0.0315546i
$172$			58879.1i			0.151754i
$173$		−	700856.i		−	1.78038i	−0.455585	−	0.890192i	$-0.650570\pi$
$173$		−	700856.i		−	1.78038i	0.455585	−	0.890192i	$-0.349430\pi$
$174$	101739.	−	88743.7i	0.254750	−	0.222210i
$175$		−	1.62204e6i		−	4.00375i
$176$	−134606.			−0.327553
$177$	39409.4	+	45180.5i	0.0945512	+	0.108397i
$178$			75654.8i			0.178973i
$179$			460655.i			1.07459i	0.843394	+	0.537296i	$0.180554\pi$
$179$			460655.i			1.07459i	−0.843394	+	0.537296i	$0.819446\pi$
$180$	56762.4	−	414065.i	0.130581	−	0.952549i
$181$			517198.i			1.17344i	0.809791	+	0.586719i	$0.199581\pi$
$181$			517198.i			1.17344i	−0.809791	+	0.586719i	$0.800419\pi$
$182$	844285.			1.88934
$183$	212877.	+	244051.i	0.469896	+	0.538707i
$184$	−147062.	−	68818.3i	−0.320226	−	0.149851i
$185$			1.26021e6i			2.70715i
$186$	−164794.	−	188926.i	−0.349267	−	0.400413i
$187$	−350087.			−0.732103
$188$		−	129221.i		−	0.266649i
$189$	609247.	−	400068.i	1.24062	−	0.814666i
$190$	157196.			0.315906
$191$	−451349.			−0.895219			−0.447609	−	0.894229i	$-0.647724\pi$
$191$	−451349.			−0.895219			−0.447609	+	0.894229i	$0.647724\pi$
$192$	48117.4	−	41971.2i	0.0941996	−	0.0821672i
$193$	−360534.			−0.696712			−0.348356	−	0.937362i	$-0.613260\pi$
$193$	−360534.			−0.696712			−0.348356	+	0.937362i	$0.613260\pi$
$194$	−165236.			−0.315210
$195$	−1.38523e6	+	1.20829e6i	−2.60877	+	2.27554i
$196$	323452.			0.601408
$197$		−	241803.i		−	0.443911i	−0.975057	−	0.221955i	$-0.928756\pi$
$197$		−	241803.i		−	0.443911i	0.975057	−	0.221955i	$-0.0712439\pi$
$198$	506345.	+	69412.7i	0.917875	+	0.125828i
$199$		−	314759.i		−	0.563438i	−0.959497	−	0.281719i	$-0.909095\pi$
$199$		−	314759.i		−	0.563438i	0.959497	−	0.281719i	$-0.0909046\pi$
$200$			539520.i			0.953746i
$201$	206354.	+	236572.i	0.360265	+	0.413022i
$202$	448117.			0.772704
$203$	−416601.			−0.709545
$204$	125145.	−	109160.i	0.210543	−	0.183649i
$205$		−	1.08953e6i		−	1.81074i
$206$	−369568.			−0.606773
$207$	517715.	+	334710.i	0.839779	+	0.542928i
$208$	−280824.			−0.450066
$209$			192230.i			0.304407i
$210$	−971899.	+	847756.i	−1.52080	+	1.32655i
$211$	214122.			0.331098			0.165549	−	0.986202i	$-0.447060\pi$
$211$	214122.			0.331098			0.165549	+	0.986202i	$0.447060\pi$
$212$	194809.			0.297694
$213$	589643.	+	675989.i	0.890512	+	1.02092i
$214$			421543.i			0.629228i
$215$			395572.i			0.583620i
$216$	−202646.	+	133070.i	−0.295532	+	0.194064i
$217$			773612.i			1.11525i
$218$	268164.			0.382173
$219$	404605.	−	352923.i	0.570060	−	0.497244i
$220$	−904332.			−1.25971
$221$	−730378.			−1.00593
$222$	550882.	−	480516.i	0.750199	−	0.654373i
$223$	160098.			0.215588			0.107794	−	0.994173i	$-0.465621\pi$
$223$	160098.			0.215588			0.107794	+	0.994173i	$0.465621\pi$
$224$	−197031.			−0.262370
$225$	278217.	−	2.02951e6i	0.366376	−	2.67260i
$226$		−	767867.i		−	1.00003i
$227$	420238.			0.541291			0.270645	−	0.962679i	$-0.412763\pi$
$227$	420238.			0.541291			0.270645	+	0.962679i	$0.412763\pi$
$228$	−59938.9	−	68716.2i	−0.0763610	−	0.0875432i
$229$		−	100486.i		−	0.126625i	−0.997994	−	0.0633124i	$-0.979834\pi$
$229$		−	100486.i		−	0.126625i	0.997994	−	0.0633124i	$-0.0201664\pi$
$230$	−988021.	−	462348.i	−1.23153	−	0.576301i
$231$	−1.03669e6	−	1.18850e6i	−1.27826	−	1.46544i
$232$	138569.			0.169023
$233$		−	1.34829e6i		−	1.62703i	−0.581546	−	0.813514i	$-0.697552\pi$
$233$		−	1.34829e6i		−	1.62703i	0.581546	−	0.813514i	$-0.302448\pi$
$234$	1.05638e6	+	144814.i	1.26118	+	0.172890i
$235$		−	868159.i		−	1.02549i
$236$			61535.9i			0.0719199i
$237$	−135843.	−	155736.i	−0.157097	−	0.180101i
$238$	−512445.			−0.586415
$239$		−	539225.i		−	0.610626i	−0.952252	−	0.305313i	$-0.901239\pi$
$239$		−	539225.i		−	0.610626i	0.952252	−	0.305313i	$-0.0987611\pi$
$240$	323271.	−	281979.i	0.362275	−	0.316001i
$241$			1.51483e6i			1.68005i	0.542548	+	0.840025i	$0.317460\pi$
$241$			1.51483e6i			1.68005i	−0.542548	+	0.840025i	$0.682540\pi$
$242$		−	461671.i		−	0.506750i
$243$	830914.	−	396068.i	0.902694	−	0.430283i
$244$			332398.i			0.357424i
$245$	2.17307e6			2.31291
$246$	−476275.	+	415439.i	−0.501788	+	0.437693i
$247$			401044.i			0.418263i
$248$		−	257317.i		−	0.265668i
$249$	87527.3	−	76347.1i	0.0894633	−	0.0780359i
$250$			2.28103e6i			2.30824i
$251$	−30311.8			−0.0303688			−0.0151844	−	0.999885i	$-0.504834\pi$
$251$	−30311.8			−0.0303688			−0.0151844	+	0.999885i	$0.504834\pi$
$252$	741170.	+	101604.i	0.735218	+	0.100788i
$253$	565389.	−	1.20821e6i	0.555323	−	1.18671i
$254$		−	521162.i		−	0.506861i
$255$	840776.	−	733381.i	0.809711	−	0.706284i
$256$	65536.0			0.0625000
$257$			1.54379e6i			1.45799i	0.684517	+	0.728997i	$0.260013\pi$
$257$			1.54379e6i			1.45799i	−0.684517	+	0.728997i	$0.739987\pi$
$258$	172919.	−	150832.i	0.161731	−	0.141073i
$259$	−2.25575e6			−2.08950
$260$	−1.88669e6			−1.73088
$261$	−521254.	−	71456.5i	−0.473640	−	0.0649293i
$262$	−526232.			−0.473613
$263$	635855.			0.566851			0.283425	−	0.958994i	$-0.408529\pi$
$263$	635855.			0.566851			0.283425	+	0.958994i	$0.408529\pi$
$264$	344821.	+	395317.i	0.304498	+	0.349088i
$265$	1.30880e6			1.14488
$266$			281379.i			0.243830i
$267$	222187.	−	193806.i	0.190740	−	0.166376i
$268$			322211.i			0.274034i
$269$			702799.i			0.592176i	0.955161	+	0.296088i	$0.0956821\pi$
$269$			702799.i			0.592176i	−0.955161	+	0.296088i	$0.904318\pi$
$270$	−1.36146e6	+	894015.i	−1.13657	+	0.746337i
$271$	−1.07699e6			−0.890817			−0.445409	−	0.895327i	$-0.646942\pi$
$271$	−1.07699e6			−0.890817			−0.445409	+	0.895327i	$0.646942\pi$
$272$	170448.			0.139692
$273$	−2.16282e6	−	2.47954e6i	−1.75636	−	2.01356i
$274$			535358.i			0.430793i
$275$	−4.43252e6			−3.53443
$276$	174623.	+	608193.i	0.137984	+	0.480583i
$277$	−301096.			−0.235779			−0.117890	−	0.993027i	$-0.537613\pi$
$277$	−301096.			−0.235779			−0.117890	+	0.993027i	$0.537613\pi$
$278$			349141.i			0.270950i
$279$	−132692.	+	967949.i	−0.102055	+	0.744461i
$280$	−1.32373e6			−1.00903
$281$	337816.			0.255220			0.127610	−	0.991824i	$-0.459269\pi$
$281$	337816.			0.255220			0.127610	+	0.991824i	$0.459269\pi$
$282$	−379504.	+	331029.i	−0.284180	+	0.247881i
$283$			1.49769e6i			1.11162i	0.831309	+	0.555811i	$0.187592\pi$
$283$			1.49769e6i			1.11162i	−0.831309	+	0.555811i	$0.812408\pi$
$284$			920699.i			0.677363i
$285$	−402693.	−	461662.i	−0.293671	−	0.336676i
$286$		−	2.30716e6i		−	1.66787i
$287$	1.95025e6			1.39761
$288$	−246526.	−	33795.3i	−0.175139	−	0.0240090i
$289$	−976548.			−0.687779
$290$	930960.			0.650034
$291$	423288.	+	485274.i	0.293024	+	0.335934i
$292$	551073.			0.378226
$293$	−10578.7			−0.00719883			−0.00359942	−	0.999994i	$-0.501146\pi$
$293$	−10578.7			−0.00719883			−0.00359942	+	0.999994i	$0.501146\pi$
$294$	−828592.	−	949930.i	−0.559078	−	0.640949i
$295$			413422.i			0.276591i
$296$	750303.			0.497745
$297$	−1.09326e6	−	1.66488e6i	−0.719170	−	1.09519i
$298$			1.08449e6i			0.707432i
$299$	1.17956e6	−	2.52067e6i	0.763029	−	1.63056i
$300$	1.58449e6	−	1.38210e6i	1.01645	−	0.886617i
$301$	−708069.			−0.450463
$302$		−	383556.i		−	0.241998i
$303$	−1.14795e6	−	1.31605e6i	−0.718317	−	0.823506i
$304$		−	93591.7i		−	0.0580836i
$305$			2.23318e6i			1.37459i
$306$	−641175.	−	87896.0i	−0.391447	−	0.0536618i
$307$	532300.			0.322337			0.161169	−	0.986927i	$-0.448474\pi$
$307$	532300.			0.322337			0.161169	+	0.986927i	$0.448474\pi$
$308$		−	1.61874e6i		−	0.972300i
$309$	946730.	+	1.08537e6i	0.564066	+	0.646667i
$310$		−	1.72876e6i		−	1.02172i
$311$		−	1.02100e6i		−	0.598586i	−0.954161	−	0.299293i	$-0.903249\pi$
$311$		−	1.02100e6i		−	0.598586i	0.954161	−	0.299293i	$-0.0967507\pi$
$312$	719393.	+	824739.i	0.418388	+	0.479656i
$313$			2.58546e6i			1.49168i	0.666123	+	0.745842i	$0.267953\pi$
$313$			2.58546e6i			1.49168i	−0.666123	+	0.745842i	$0.732047\pi$
$314$	2.16930e6			1.24164
$315$	4.97946e6	+	682614.i	2.82752	+	0.387613i
$316$		−	212112.i		−	0.119495i
$317$			1.88233e6i			1.05208i	0.850460	+	0.526039i	$0.176324\pi$
$317$			1.88233e6i			1.05208i	−0.850460	+	0.526039i	$0.823676\pi$
$318$	−499047.	−	572126.i	−0.276741	−	0.317266i
$319$			1.13844e6i			0.626372i
$320$	440296.			0.240364
$321$	1.23801e6	−	1.07988e6i	0.670597	−	0.584940i
$322$	827596.	−	1.76854e6i	0.444814	−	0.950552i
$323$		−	243417.i		−	0.129821i
$324$	909929.	+	254255.i	0.481554	+	0.134557i
$325$	−9.24746e6			−4.85639
$326$		−	2.14111e6i		−	1.11582i
$327$	−686962.	−	787559.i	−0.355274	−	0.407299i
$328$	−648688.			−0.332929
$329$	1.55399e6			0.791515
$330$	2.31664e6	+	2.65589e6i	1.17105	+	1.34253i
$331$	2.81060e6			1.41003			0.705016	−	0.709191i	$-0.250940\pi$
$331$	2.81060e6			1.41003			0.705016	+	0.709191i	$0.250940\pi$
$332$	119212.			0.0593576
$333$	−2.82241e6	−	386912.i	−1.39479	−	0.191206i
$334$	−782246.			−0.383687
$335$			2.16474e6i			1.05389i
$336$	504738.	+	578650.i	0.243903	+	0.279620i
$337$			1.28117e6i			0.614516i	0.951626	+	0.307258i	$0.0994115\pi$
$337$			1.28117e6i			0.614516i	−0.951626	+	0.307258i	$0.900588\pi$
$338$		−	3.32820e6i		−	1.58459i
$339$	−2.25511e6	+	1.96706e6i	−1.06578	+	0.929648i
$340$	1.14514e6			0.537231
$341$	2.11403e6			0.984523
$342$	−48262.9	+	352063.i	−0.0223125	+	0.162763i
$343$			655886.i			0.301018i
$344$	235516.			0.107306
$345$	1.17318e6	+	4.08608e6i	0.530662	+	1.84824i
$346$	−2.80343e6			−1.25892
$347$			2.57596e6i			1.14846i	0.818694	+	0.574230i	$0.194699\pi$
$347$			2.57596e6i			1.14846i	−0.818694	+	0.574230i	$0.805301\pi$
$348$	−354975.	−	406957.i	−0.157126	−	0.180136i
$349$	−26106.9			−0.0114734			−0.00573669	−	0.999984i	$-0.501826\pi$
$349$	−26106.9			−0.0114734			−0.00573669	+	0.999984i	$0.501826\pi$
$350$	−6.48817e6			−2.83108
$351$	−2.28084e6	−	3.47339e6i	−0.988159	−	1.50482i
$352$			538422.i			0.231615i
$353$		−	1.49802e6i		−	0.639853i	−0.947442	−	0.319927i	$-0.896342\pi$
$353$		−	1.49802e6i		−	0.639853i	0.947442	−	0.319927i	$-0.103658\pi$
$354$	180722.	−	157638.i	0.0766483	−	0.0668578i
$355$			6.18561e6i			2.60502i
$356$	302619.			0.126553
$357$	1.31274e6	+	1.50498e6i	0.545140	+	0.624970i
$358$	1.84262e6			0.759851
$359$	−2.55014e6			−1.04430			−0.522152	−	0.852852i	$-0.674871\pi$
$359$	−2.55014e6			−1.04430			−0.522152	+	0.852852i	$0.674871\pi$
$360$	−1.65626e6	−	227050.i	−0.673554	−	0.0923346i
$361$	2.34244e6			0.946021
$362$	2.06879e6			0.829746
$363$	−1.35586e6	+	1.18267e6i	−0.540067	+	0.471083i
$364$		−	3.37714e6i		−	1.33597i
$365$	3.70232e6			1.45459
$366$	976203.	−	851509.i	0.380923	−	0.332267i
$367$		−	4.09525e6i		−	1.58714i	−0.608479	−	0.793570i	$-0.708220\pi$
$367$		−	4.09525e6i		−	1.58714i	0.608479	−	0.793570i	$-0.291780\pi$
$368$	−275273.	+	588249.i	−0.105961	+	0.226434i
$369$	2.44017e6	+	334512.i	0.932940	+	0.127893i
$370$	5.04082e6			1.91424
$371$			2.34274e6i			0.883669i
$372$	−755703.	+	659174.i	−0.283135	+	0.246969i
$373$			1.56014e6i			0.580620i	0.956933	+	0.290310i	$0.0937584\pi$
$373$			1.56014e6i			0.580620i	−0.956933	+	0.290310i	$0.906242\pi$
$374$			1.40035e6i			0.517675i
$375$	6.69904e6	−	5.84335e6i	2.46000	−	2.14577i
$376$	−516886.			−0.188549
$377$			2.37509e6i			0.860652i
$378$	−1.60027e6	−	2.43699e6i	−0.576056	−	0.877250i
$379$		−	4.72690e6i		−	1.69036i	−0.534485	−	0.845178i	$-0.679494\pi$
$379$		−	4.72690e6i		−	1.69036i	0.534485	−	0.845178i	$-0.320506\pi$
$380$		−	628785.i		−	0.223379i
$381$	−1.53058e6	+	1.33507e6i	−0.540185	+	0.471185i
$382$			1.80540e6i			0.633015i
$383$	−3.06184e6			−1.06656			−0.533281	−	0.845938i	$-0.679041\pi$
$383$	−3.06184e6			−1.06656			−0.533281	+	0.845938i	$0.679041\pi$
$384$	−167885.	−	192470.i	−0.0581009	−	0.0666092i
$385$		−	1.08753e7i		−	3.73930i
$386$			1.44214e6i			0.492650i
$387$	−885940.	−	121450.i	−0.300696	−	0.0412211i
$388$			660944.i			0.222887i
$389$	−5.71202e6			−1.91389			−0.956943	−	0.290276i	$-0.906253\pi$
$389$	−5.71202e6			−1.91389			−0.956943	+	0.290276i	$0.906253\pi$
$390$	4.83316e6	+	5.54092e6i	1.60905	+	1.84468i
$391$	−715941.	+	1.52994e6i	−0.236829	+	0.506095i
$392$		−	1.29381e6i		−	0.425260i
$393$	1.34806e6	+	1.54547e6i	0.440278	+	0.504752i
$394$	−967211.			−0.313892
$395$		−	1.42505e6i		−	0.459556i
$396$	277651.	−	2.02538e6i	0.0889736	−	0.649035i
$397$	2.66168e6			0.847578			0.423789	−	0.905761i	$-0.360700\pi$
$397$	2.66168e6			0.847578			0.423789	+	0.905761i	$0.360700\pi$
$398$	−1.25904e6			−0.398411
$399$	826368.	−	720813.i	0.259861	−	0.226668i
$400$	2.15808e6			0.674400
$401$	−1.16925e6			−0.363117			−0.181559	−	0.983380i	$-0.558114\pi$
$401$	−1.16925e6			−0.363117			−0.181559	+	0.983380i	$0.558114\pi$
$402$	946287.	−	825415.i	0.292050	−	0.254746i
$403$	4.41046e6			1.35276
$404$		−	1.79247e6i		−	0.546384i
$405$	6.11326e6	+	1.70818e6i	1.85197	+	0.517484i
$406$			1.66640e6i			0.501724i
$407$			6.16424e6i			1.84456i
$408$	−436641.	−	500582.i	−0.129860	−	0.148876i
$409$	−2.01066e6			−0.594333			−0.297166	−	0.954826i	$-0.596042\pi$
$409$	−2.01066e6			−0.594333			−0.297166	+	0.954826i	$0.596042\pi$
$410$	−4.35814e6			−1.28039
$411$	1.57227e6	−	1.37144e6i	0.459116	−	0.400471i
$412$			1.47827e6i			0.429054i
$413$	−740019.			−0.213485
$414$	1.33884e6	−	2.07086e6i	0.383908	−	0.593813i
$415$	800915.			0.228279
$416$			1.12330e6i			0.318245i
$417$	1.02538e6	−	894401.i	0.288764	−	0.251879i
$418$	768918.			0.215248
$419$	2.01268e6			0.560068			0.280034	−	0.959990i	$-0.409654\pi$
$419$	2.01268e6			0.560068			0.280034	+	0.959990i	$0.409654\pi$
$420$	3.39102e6	+	3.88760e6i	0.938010	+	1.07537i
$421$			5.38411e6i			1.48050i	0.672332	+	0.740250i	$0.265293\pi$
$421$			5.38411e6i			1.48050i	−0.672332	+	0.740250i	$0.734707\pi$
$422$		−	856490.i		−	0.234121i
$423$	1.94437e6	+	266545.i	0.528356	+	0.0724302i
$424$		−	779238.i		−	0.210502i
$425$	5.61282e6			1.50733
$426$	2.70396e6	−	2.35857e6i	0.721898	−	0.629687i
$427$	−3.99735e6			−1.06097
$428$	1.68617e6			0.444931
$429$	−6.77579e6	+	5.91029e6i	−1.77753	+	1.55048i
$430$	1.58229e6			0.412681
$431$	1.98569e6			0.514893			0.257447	−	0.966293i	$-0.417119\pi$
$431$	1.98569e6			0.514893			0.257447	+	0.966293i	$0.417119\pi$
$432$	532279.	+	810585.i	0.137224	+	0.208973i
$433$			3.71038e6i			0.951041i	0.879705	+	0.475520i	$0.157740\pi$
$433$			3.71038e6i			0.951041i	−0.879705	+	0.475520i	$0.842260\pi$
$434$	3.09445e6			0.788604
$435$	−2.38486e6	−	2.73409e6i	−0.604281	−	0.692771i
$436$		−	1.07266e6i		−	0.270237i
$437$	840076.	+	393117.i	0.210433	+	0.0984731i
$438$	−1.41169e6	−	1.61842e6i	−0.351605	−	0.403093i
$439$	3.27132e6			0.810143			0.405071	−	0.914285i	$-0.367247\pi$
$439$	3.27132e6			0.810143			0.405071	+	0.914285i	$0.367247\pi$
$440$			3.61733e6i			0.890751i
$441$	−667184.	+	4.86691e6i	−0.163361	+	1.19167i
$442$			2.92151e6i			0.711299i
$443$			505087.i			0.122280i	0.998129	+	0.0611402i	$0.0194737\pi$
$443$			505087.i			0.122280i	−0.998129	+	0.0611402i	$0.980526\pi$
$444$	−1.92206e6	−	2.20353e6i	−0.462712	−	0.530470i
$445$	2.03311e6			0.486700
$446$		−	640392.i		−	0.152443i
$447$	3.18498e6	−	2.77816e6i	0.753943	−	0.657639i
$448$			788123.i			0.185524i
$449$		−	5.18927e6i		−	1.21476i	−0.794412	−	0.607380i	$-0.792221\pi$
$449$		−	5.18927e6i		−	1.21476i	0.794412	−	0.607380i	$-0.207779\pi$
$450$	−8.11804e6	−	1.11287e6i	−1.88982	−	0.259067i
$451$		−	5.32941e6i		−	1.23378i
$452$	−3.07147e6			−0.707131
$453$	−1.12645e6	+	982563.i	−0.257908	+	0.224965i
$454$		−	1.68095e6i		−	0.382750i
$455$		−	2.26889e7i		−	5.13790i
$456$	−274865.	+	239756.i	−0.0619024	+	0.0539954i
$457$		−	2.63136e6i		−	0.589373i	−0.955594	−	0.294686i	$-0.904785\pi$
$457$		−	2.63136e6i		−	0.589373i	0.955594	−	0.294686i	$-0.0952152\pi$
$458$	−401946.			−0.0895372
$459$	1.38437e6	+	2.10820e6i	0.306705	+	0.467068i
$460$	−1.84939e6	+	3.95208e6i	−0.407506	+	0.870826i
$461$			1.70117e6i			0.372817i	0.982472	+	0.186408i	$0.0596848\pi$
$461$			1.70117e6i			0.372817i	−0.982472	+	0.186408i	$0.940315\pi$
$462$	−4.75400e6	+	4.14676e6i	−1.03623	+	0.903865i
$463$	7.14600e6			1.54921			0.774605	−	0.632445i	$-0.217949\pi$
$463$	7.14600e6			1.54921			0.774605	+	0.632445i	$0.217949\pi$
$464$		−	554276.i		−	0.119517i
$465$	−5.07710e6	+	4.42859e6i	−1.08889	+	0.949802i
$466$	−5.39318e6			−1.15048
$467$	6.75731e6			1.43378			0.716889	−	0.697187i	$-0.245565\pi$
$467$	6.75731e6			1.43378			0.716889	+	0.697187i	$0.245565\pi$
$468$	579256.	−	4.22550e6i	0.122252	−	0.891792i
$469$	−3.87485e6			−0.813436
$470$	−3.47264e6			−0.725128
$471$	−5.55714e6	−	6.37092e6i	−1.15425	−	1.32328i
$472$	246144.			0.0508550
$473$			1.93492e6i			0.397659i
$474$	−622943.	+	543372.i	−0.127351	+	0.111084i
$475$		−	3.08195e6i		−	0.626745i
$476$			2.04978e6i			0.414658i
$477$	−401833.	+	2.93125e6i	−0.0808630	+	0.589871i
$478$	−2.15690e6			−0.431778
$479$	−6.70521e6			−1.33528			−0.667642	−	0.744483i	$-0.732696\pi$
$479$	−6.70521e6			−1.33528			−0.667642	+	0.744483i	$0.732696\pi$
$480$	−1.12791e6	−	1.29308e6i	−0.223446	−	0.256167i
$481$			1.28603e7i			2.53448i
$482$	6.05933e6			1.18797
$483$	−7.31401e6	+	2.09998e6i	−1.42655	+	0.409588i
$484$	−1.84668e6			−0.358327
$485$			4.44048e6i			0.857186i
$486$	−1.58427e6	−	3.32366e6i	−0.304256	−	0.638301i
$487$	−9.22373e6			−1.76232			−0.881159	−	0.472820i	$-0.843236\pi$
$487$	−9.22373e6			−1.76232			−0.881159	+	0.472820i	$0.843236\pi$
$488$	1.32959e6			0.252737
$489$	−6.28811e6	+	5.48491e6i	−1.18918	+	1.03728i
$490$		−	8.69230e6i		−	1.63548i
$491$		−	2.94821e6i		−	0.551893i	−0.961173	−	0.275946i	$-0.911009\pi$
$491$		−	2.94821e6i		−	0.551893i	0.961173	−	0.275946i	$-0.0889912\pi$
$492$	1.66176e6	+	1.90510e6i	0.309496	+	0.354818i
$493$		−	1.44158e6i		−	0.267130i
$494$	1.60418e6			0.295757
$495$	1.86537e6	−	1.36073e7i	0.342177	−	2.49608i
$496$	−1.02927e6			−0.187856
$497$	−1.10721e7			−2.01067
$498$	−305389.	−	350109.i	−0.0551797	−	0.0632601i
$499$	1.69625e6			0.304958			0.152479	−	0.988307i	$-0.451274\pi$
$499$	1.69625e6			0.304958			0.152479	+	0.988307i	$0.451274\pi$
$500$	9.12411e6			1.63217
$501$	2.00390e6	+	2.29734e6i	0.356682	+	0.408913i
$502$			121247.i			0.0214740i
$503$	−3.24172e6			−0.571288			−0.285644	−	0.958336i	$-0.592207\pi$
$503$	−3.24172e6			−0.571288			−0.285644	+	0.958336i	$0.592207\pi$
$504$	406415.	−	2.96468e6i	0.0712679	−	0.519878i
$505$		−	1.20425e7i		−	2.10130i
$506$	−4.83286e6	−	2.26155e6i	−0.839127	−	0.392673i
$507$	−9.77443e6	+	8.52591e6i	−1.68877	+	1.47306i
$508$	−2.08465e6			−0.358405
$509$		−	3.32600e6i		−	0.569021i	−0.958673	−	0.284511i	$-0.908169\pi$
$509$		−	3.32600e6i		−	0.569021i	0.958673	−	0.284511i	$-0.0918311\pi$
$510$	−2.93352e6	−	3.36310e6i	−0.499418	−	0.572552i
$511$			6.62709e6i			1.12272i
$512$		−	262144.i		−	0.0441942i
$513$	1.15759e6	−	760146.i	0.194206	−	0.127527i
$514$	6.17516e6			1.03096
$515$			9.93161e6i			1.65007i
$516$	−603326.	−	691677.i	−0.0997535	−	0.114361i
$517$		−	4.24656e6i		−	0.698733i
$518$			9.02300e6i			1.47750i
$519$	7.18159e6	+	8.23325e6i	1.17031	+	1.34169i
$520$			7.54675e6i			1.22392i
$521$	944343.			0.152418			0.0762089	−	0.997092i	$-0.475718\pi$
$521$	944343.			0.152418			0.0762089	+	0.997092i	$0.475718\pi$
$522$	−285826.	+	2.08502e6i	−0.0459119	+	0.334914i
$523$			759859.i			0.121473i	0.998154	+	0.0607364i	$0.0193449\pi$
$523$			759859.i			0.121473i	−0.998154	+	0.0607364i	$0.980655\pi$
$524$			2.10493e6i			0.334895i
$525$	1.66209e7	+	1.90548e7i	2.63181	+	3.01721i
$526$		−	2.54342e6i		−	0.400824i
$527$	−2.67696e6			−0.419871
$528$	1.58127e6	−	1.37929e6i	0.246843	−	0.215313i
$529$	−4.12386e6	−	4.94169e6i	−0.640715	−	0.767779i
$530$		−	5.23522e6i		−	0.809553i
$531$	−925917.	−	126930.i	−0.142507	−	0.0195357i
$532$	1.12552e6			0.172414
$533$		−	1.11186e7i		−	1.69525i
$534$	−775226.	−	888748.i	−0.117645	−	0.134873i
$535$	1.13284e7			1.71113
$536$	1.28885e6			0.193771
$537$	−4.72028e6	−	5.41151e6i	−0.706369	−	0.809809i
$538$	2.81120e6			0.418731
$539$	1.06295e7			1.57594
$540$	3.57606e6	+	5.44583e6i	0.527740	+	0.803673i
$541$	−6.37429e6			−0.936350			−0.468175	−	0.883636i	$-0.655088\pi$
$541$	−6.37429e6			−0.936350			−0.468175	+	0.883636i	$0.655088\pi$
$542$			4.30796e6i			0.629903i
$543$	−5.29966e6	−	6.07573e6i	−0.771344	−	0.884299i
$544$		−	681794.i		−	0.0987770i
$545$		−	7.20653e6i		−	1.03929i
$546$	−9.91816e6	+	8.65128e6i	−1.42380	+	1.24193i
$547$	7.43194e6			1.06202			0.531012	−	0.847364i	$-0.321812\pi$
$547$	7.43194e6			1.06202			0.531012	+	0.847364i	$0.321812\pi$
$548$	2.14143e6			0.304616
$549$	−5.00151e6	−	685637.i	−0.708224	−	0.0970875i
$550$			1.77301e7i			2.49922i
$551$	−791559.			−0.111072
$552$	2.43277e6	−	698491.i	0.339824	−	0.0975693i
$553$	2.55082e6			0.354705
$554$			1.20438e6i			0.166721i
$555$	−1.29132e7	−	1.48042e7i	−1.77951	−	2.04010i
$556$	1.39656e6			0.191590
$557$	6.53039e6			0.891869			0.445935	−	0.895066i	$-0.352871\pi$
$557$	6.53039e6			0.891869			0.445935	+	0.895066i	$0.352871\pi$
$558$	3.87180e6	+	530768.i	0.526413	+	0.0721638i
$559$			4.03679e6i			0.546394i
$560$			5.29492e6i			0.713492i
$561$	4.11262e6	−	3.58730e6i	0.551710	−	0.481239i
$562$		−	1.35126e6i		−	0.180468i
$563$	8.07931e6			1.07425			0.537123	−	0.843504i	$-0.319511\pi$
$563$	8.07931e6			1.07425			0.537123	+	0.843504i	$0.319511\pi$
$564$	1.32412e6	+	1.51802e6i	0.175278	+	0.200946i
$565$	−2.06353e7			−2.71951
$566$	5.99078e6			0.786035
$567$	−3.05762e6	+	1.09426e7i	−0.399416	+	1.42943i
$568$	3.68280e6			0.478968
$569$	−8.11855e6			−1.05123			−0.525615	−	0.850723i	$-0.676165\pi$
$569$	−8.11855e6			−1.05123			−0.525615	+	0.850723i	$0.676165\pi$
$570$	−1.84665e6	+	1.61077e6i	−0.238066	+	0.207657i
$571$		−	1.43018e7i		−	1.83569i	−0.396937	−	0.917846i	$-0.629927\pi$
$571$		−	1.43018e7i		−	1.83569i	0.396937	−	0.917846i	$-0.370073\pi$
$572$	−9.22864e6			−1.17936
$573$	5.30218e6	−	4.62492e6i	0.674634	−	0.588461i
$574$		−	7.80100e6i		−	0.988259i
$575$	−9.06467e6	+	1.93709e7i	−1.14336	+	2.44331i
$576$	−135181.	+	986105.i	−0.0169769	+	0.123842i
$577$	1.23884e7			1.54908			0.774541	−	0.632524i	$-0.217981\pi$
$577$	1.23884e7			1.54908			0.774541	+	0.632524i	$0.217981\pi$
$578$			3.90619e6i			0.486333i
$579$	4.23534e6	−	3.69435e6i	0.525040	−	0.457975i
$580$		−	3.72384e6i		−	0.459643i
$581$			1.43363e6i			0.176196i
$582$	1.94109e6	−	1.69315e6i	0.237541	−	0.207199i
$583$	6.40196e6			0.780084
$584$		−	2.20429e6i		−	0.267446i
$585$	3.89167e6	−	2.83885e7i	0.470160	−	3.42968i
$586$			42314.7i			0.00509034i
$587$		−	7.01309e6i		−	0.840068i	−0.907508	−	0.420034i	$-0.862018\pi$
$587$		−	7.01309e6i		−	0.840068i	0.907508	−	0.420034i	$-0.137982\pi$
$588$	−3.79972e6	+	3.31437e6i	−0.453219	+	0.395328i
$589$			1.46989e6i			0.174581i
$590$	1.65369e6			0.195580
$591$	2.47772e6	+	2.84056e6i	0.291799	+	0.334530i
$592$		−	3.00121e6i		−	0.351959i
$593$		−	374140.i		−	0.0436915i	−0.999761	−	0.0218458i	$-0.993046\pi$
$593$		−	374140.i		−	0.0436915i	0.999761	−	0.0218458i	$-0.00695427\pi$
$594$	−6.65950e6	+	4.37303e6i	−0.774418	+	0.508530i
$595$			1.37712e7i			1.59470i
$596$	4.33796e6			0.500230
$597$	3.22530e6	+	3.69761e6i	0.370369	+	0.424605i
$598$	−1.00827e7	−	4.71823e6i	−1.15298	−	0.539543i
$599$		−	2.30047e6i		−	0.261969i	−0.991384	−	0.130984i	$-0.958186\pi$
$599$		−	2.30047e6i		−	0.261969i	0.991384	−	0.130984i	$-0.0418137\pi$
$600$	−5.52839e6	−	6.33796e6i	−0.626933	−	0.718740i
$601$	−8.31550e6			−0.939079			−0.469539	−	0.882912i	$-0.655580\pi$
$601$	−8.31550e6			−0.939079			−0.469539	+	0.882912i	$0.655580\pi$
$602$			2.83227e6i			0.318525i
$603$	−4.84824e6	−	664625.i	−0.542989	−	0.0744361i
$604$	−1.53422e6			−0.171118
$605$	−1.24067e7			−1.37806
$606$	−5.26421e6	+	4.59180e6i	−0.582307	+	0.507927i
$607$	1.58877e7			1.75021			0.875103	−	0.483937i	$-0.160794\pi$
$607$	1.58877e7			1.75021			0.875103	+	0.483937i	$0.160794\pi$
$608$	−374367.			−0.0410713
$609$	4.89398e6	−	4.26886e6i	0.534711	−	0.466411i
$610$	8.93271e6			0.971983
$611$		−	8.85950e6i		−	0.960077i
$612$	−351584.	+	2.56470e6i	−0.0379446	+	0.276795i
$613$		−	1.65539e7i		−	1.77930i	−0.456648	−	0.889648i	$-0.650950\pi$
$613$		−	1.65539e7i		−	1.77930i	0.456648	−	0.889648i	$-0.349050\pi$
$614$		−	2.12920e6i		−	0.227927i
$615$	1.11643e7	+	1.27992e7i	1.19027	+	1.36457i
$616$	−6.47496e6			−0.687520
$617$	−2.54420e6			−0.269053			−0.134527	−	0.990910i	$-0.542951\pi$
$617$	−2.54420e6			−0.269053			−0.134527	+	0.990910i	$0.542951\pi$
$618$	4.34147e6	−	3.78692e6i	0.457262	−	0.398855i
$619$			3.65132e6i			0.383021i	0.981491	+	0.191511i	$0.0613387\pi$
$619$			3.65132e6i			0.383021i	−0.981491	+	0.191511i	$0.938661\pi$
$620$	−6.91503e6			−0.722462
$621$	−9.51154e6	+	1.37299e6i	−0.989742	+	0.142869i
$622$	−4.08401e6			−0.423264
$623$			3.63924e6i			0.375657i
$624$	3.29896e6	−	2.87757e6i	0.339168	−	0.295845i
$625$	3.49555e7			3.57945
$626$	1.03418e7			1.05478
$627$	−1.96975e6	−	2.25820e6i	−0.200098	−	0.229400i
$628$		−	8.67721e6i		−	0.877973i
$629$		−	7.80566e6i		−	0.786653i
$630$	2.73046e6	−	1.99179e7i	0.274084	−	1.99936i
$631$			1.41202e7i			1.41178i	0.708319	+	0.705892i	$0.249454\pi$
$631$			1.41202e7i			1.41178i	−0.708319	+	0.705892i	$0.750546\pi$
$632$	−848450.			−0.0844955
$633$	−2.51538e6	+	2.19409e6i	−0.249514	+	0.217643i
$634$	7.52933e6			0.743932
$635$	−1.40055e7			−1.37836
$636$	−2.28851e6	+	1.99619e6i	−0.224341	+	0.195685i
$637$	2.21761e7			2.16539
$638$	4.55375e6			0.442912
$639$	−1.38535e7	−	1.89912e6i	−1.34217	−	0.183993i
$640$		−	1.76118e6i		−	0.169963i
$641$	−1.70992e7			−1.64373			−0.821865	−	0.569682i	$-0.807066\pi$
$641$	−1.70992e7			−1.64373			−0.821865	+	0.569682i	$0.807066\pi$
$642$	−4.31950e6	−	4.95204e6i	−0.413615	−	0.474184i
$643$		−	8.69003e6i		−	0.828884i	−0.910076	−	0.414442i	$-0.863977\pi$
$643$		−	8.69003e6i		−	0.828884i	0.910076	−	0.414442i	$-0.136023\pi$
$644$	−7.07417e6	−	3.31038e6i	−0.672142	−	0.314531i
$645$	−4.05338e6	−	4.64695e6i	−0.383635	−	0.439814i
$646$	−973667.			−0.0917971
$647$			1.20680e7i			1.13338i	0.823932	+	0.566689i	$0.191776\pi$
$647$			1.20680e7i			1.13338i	−0.823932	+	0.566689i	$0.808224\pi$
$648$	1.01702e6	−	3.63972e6i	0.0951463	−	0.340510i
$649$			2.02224e6i			0.188460i
$650$			3.69898e7i			3.43399i
$651$	−7.92711e6	−	9.08794e6i	−0.733098	−	0.840452i
$652$	−8.56442e6			−0.789004
$653$		−	8.02078e6i		−	0.736094i	−0.929807	−	0.368047i	$-0.880026\pi$
$653$		−	8.02078e6i		−	0.736094i	0.929807	−	0.368047i	$-0.119974\pi$
$654$	−3.15024e6	+	2.74785e6i	−0.288004	+	0.251216i
$655$			1.41417e7i			1.28795i
$656$			2.59475e6i			0.235416i
$657$	−1.13670e6	+	8.29186e6i	−0.102738	+	0.749443i
$658$		−	6.21597e6i		−	0.559685i
$659$	1.74345e7			1.56386			0.781928	−	0.623369i	$-0.214237\pi$
$659$	1.74345e7			1.56386			0.781928	+	0.623369i	$0.214237\pi$
$660$	1.06236e7	−	9.26658e6i	0.949315	−	0.828056i
$661$			1.41203e6i			0.125702i	0.998023	+	0.0628509i	$0.0200193\pi$
$661$			1.41203e6i			0.125702i	−0.998023	+	0.0628509i	$0.979981\pi$
$662$		−	1.12424e7i		−	0.997044i
$663$	8.58005e6	−	7.48409e6i	0.758065	−	0.661234i
$664$		−	476850.i		−	0.0419721i
$665$	7.56165e6			0.663075
$666$	−1.54765e6	+	1.12896e7i	−0.135203	+	0.986267i
$667$	4.97516e6	+	2.32815e6i	0.433005	+	0.202626i
$668$			3.12899e6i			0.271308i
$669$	−1.88074e6	+	1.64050e6i	−0.162466	+	0.141714i
$670$	8.65897e6			0.745211
$671$			1.09235e7i			0.936601i
$672$	2.31460e6	−	2.01895e6i	0.197721	−	0.172466i
$673$	−9.62733e6			−0.819347			−0.409674	−	0.912232i	$-0.634357\pi$
$673$	−9.62733e6			−0.819347			−0.409674	+	0.912232i	$0.634357\pi$
$674$	5.12470e6			0.434529
$675$	1.75278e7	+	2.66923e7i	1.48070	+	2.25490i
$676$	−1.33128e7			−1.12048
$677$	8.02836e6			0.673217			0.336609	−	0.941645i	$-0.390720\pi$
$677$	8.02836e6			0.673217			0.336609	+	0.941645i	$0.390720\pi$
$678$	7.86824e6	+	9.02045e6i	0.657360	+	0.753623i
$679$	−7.94838e6			−0.661614
$680$		−	4.58056e6i		−	0.379880i
$681$	−4.93671e6	+	4.30613e6i	−0.407915	+	0.355811i
$682$		−	8.45614e6i		−	0.696163i
$683$		−	7.90761e6i		−	0.648625i	−0.945950	−	0.324312i	$-0.894867\pi$
$683$		−	7.90761e6i		−	0.648625i	0.945950	−	0.324312i	$-0.105133\pi$
$684$	1.40825e6	+	193051.i	0.115091	+	0.0157773i
$685$	1.43870e7			1.17150
$686$	2.62354e6			0.212852
$687$	1.02967e6	+	1.18045e6i	0.0832352	+	0.0954240i
$688$		−	942065.i		−	0.0758770i
$689$	1.33563e7			1.07186
$690$	1.63443e7	−	4.69273e6i	1.30690	−	0.375235i
$691$	−3.74097e6			−0.298050			−0.149025	−	0.988833i	$-0.547613\pi$
$691$	−3.74097e6			−0.298050			−0.149025	+	0.988833i	$0.547613\pi$
$692$			1.12137e7i			0.890192i
$693$	2.43568e7	+	3.33898e6i	1.92658	+	0.264107i
$694$	1.03039e7			0.812084
$695$	9.38266e6			0.736824
$696$	−1.62783e6	+	1.41990e6i	−0.127375	+	0.111105i
$697$			6.74853e6i			0.526171i
$698$			104427.i			0.00811290i
$699$	1.38158e7	+	1.58390e7i	1.06951	+	1.22612i
$700$			2.59527e7i			2.00187i
$701$	−9.94431e6			−0.764328			−0.382164	−	0.924095i	$-0.624821\pi$
$701$	−9.94431e6			−0.764328			−0.382164	+	0.924095i	$0.624821\pi$
$702$	−1.38936e7	+	9.12335e6i	−1.06407	+	0.698734i
$703$	−4.28602e6			−0.327089
$704$	2.15369e6			0.163776
$705$	8.89592e6	+	1.01986e7i	0.674090	+	0.772803i
$706$	−5.99207e6			−0.452445
$707$	2.15559e7			1.62187
$708$	−630551.	−	722888.i	−0.0472756	−	0.0541986i
$709$		−	1.32750e7i		−	0.991786i	−0.868384	−	0.495893i	$-0.834841\pi$
$709$		−	1.32750e7i		−	0.991786i	0.868384	−	0.495893i	$-0.165159\pi$
$710$	2.47424e7			1.84203
$711$	3.19161e6	+	437524.i	0.236775	+	0.0324585i
$712$		−	1.21048e6i		−	0.0894863i
$713$	4.32328e6	−	9.23869e6i	0.318485	−	0.680591i
$714$	6.01990e6	−	5.25096e6i	0.441920	−	0.385472i
$715$	−6.20016e7			−4.53563
$716$		−	7.37049e6i		−	0.537296i
$717$	5.52537e6	+	6.33449e6i	0.401387	+	0.460166i
$718$			1.02005e7i			0.738435i
$719$			1.73762e7i			1.25352i	0.779212	+	0.626760i	$0.215620\pi$
$719$			1.73762e7i			1.25352i	−0.779212	+	0.626760i	$0.784380\pi$
$720$	−908199.	+	6.62504e6i	−0.0652904	+	0.476274i
$721$	−1.77774e7			−1.27359
$722$		−	9.36976e6i		−	0.668938i
$723$	−1.55223e7	−	1.77954e7i	−1.10436	−	1.26608i
$724$		−	8.27516e6i		−	0.586719i
$725$		−	1.82521e7i		−	1.28964i
$726$	4.73068e6	+	5.42344e6i	0.333106	+	0.381885i
$727$		−	2.23406e7i		−	1.56769i	−0.620958	−	0.783844i	$-0.713256\pi$
$727$		−	2.23406e7i		−	1.56769i	0.620958	−	0.783844i	$-0.286744\pi$
$728$	−1.35086e7			−0.944671
$729$	−5.70263e6	+	1.31670e7i	−0.397426	+	0.917634i
$730$		−	1.48093e7i		−	1.02855i
$731$		−	2.45016e6i		−	0.169590i
$732$	−3.40604e6	−	3.90481e6i	−0.234948	−	0.269353i
$733$		−	3.81239e6i		−	0.262082i	−0.991377	−	0.131041i	$-0.958168\pi$
$733$		−	3.81239e6i		−	0.262082i	0.991377	−	0.131041i	$-0.0418320\pi$
$734$	−1.63810e7			−1.12228
$735$	−2.55280e7	+	2.22672e7i	−1.74300	+	1.52036i
$736$	2.35300e6	+	1.10109e6i	0.160113	+	0.0749255i
$737$			1.05887e7i			0.718084i
$738$	1.33805e6	−	9.76067e6i	0.0904339	−	0.659688i
$739$	−2.01949e7			−1.36029			−0.680143	−	0.733079i	$-0.738082\pi$
$739$	−2.01949e7			−1.36029			−0.680143	+	0.733079i	$0.738082\pi$
$740$		−	2.01633e7i		−	1.35357i
$741$	−4.10945e6	−	4.71123e6i	−0.274940	−	0.315202i
$742$	9.37096e6			0.624848
$743$	−3.71077e6			−0.246599			−0.123300	−	0.992369i	$-0.539348\pi$
$743$	−3.71077e6			−0.246599			−0.123300	+	0.992369i	$0.539348\pi$
$744$	2.63670e6	+	3.02281e6i	0.174634	+	0.200207i
$745$	2.91441e7			1.92380
$746$	6.24057e6			0.410560
$747$	−245899.	+	1.79376e6i	−0.0161234	+	0.117615i
$748$	5.60139e6			0.366051
$749$			2.02776e7i			1.32072i
$750$	−2.33734e7	−	2.67962e7i	−1.51729	−	1.73948i
$751$		−	2.64124e7i		−	1.70886i	−0.519564	−	0.854432i	$-0.673905\pi$
$751$		−	2.64124e7i		−	1.70886i	0.519564	−	0.854432i	$-0.326095\pi$
$752$			2.06754e6i			0.133324i
$753$	356085.	−	310601.i	0.0228858	−	0.0199625i
$754$	9.50037e6			0.608573
$755$	−1.03075e7			−0.658092
$756$	−9.74794e6	+	6.40109e6i	−0.620310	+	0.407333i
$757$		−	1.30839e7i		−	0.829846i	−0.909856	−	0.414923i	$-0.863808\pi$
$757$		−	1.30839e7i		−	0.829846i	0.909856	−	0.414923i	$-0.136192\pi$
$758$	−1.89076e7			−1.19526
$759$	5.73857e6	+	1.99869e7i	0.361576	+	1.25933i
$760$	−2.51514e6			−0.157953
$761$			2.38346e7i			1.49192i	0.665989	+	0.745962i	$0.268010\pi$
$761$			2.38346e7i			1.49192i	−0.665989	+	0.745962i	$0.731990\pi$
$762$	5.34028e6	+	6.12230e6i	0.333178	+	0.381968i
$763$	1.28996e7			0.802166
$764$	7.22159e6			0.447609
$765$	−2.36208e6	+	1.72306e7i	−0.145929	+	1.06451i
$766$			1.22474e7i			0.754173i
$767$			4.21894e6i			0.258950i
$768$	−769878.	+	671539.i	−0.0470998	+	0.0410836i
$769$			2.74943e7i			1.67659i	0.545219	+	0.838294i	$0.316447\pi$
$769$			2.74943e7i			1.67659i	−0.545219	+	0.838294i	$0.683553\pi$
$770$	−4.35013e7			−2.64409
$771$	−1.58190e7	−	1.81355e7i	−0.958393	−	1.09874i
$772$	5.76855e6			0.348356
$773$	2.30897e7			1.38985			0.694927	−	0.719081i	$-0.255437\pi$
$773$	2.30897e7			1.38985			0.694927	+	0.719081i	$0.255437\pi$
$774$	−485799.	+	3.54376e6i	−0.0291477	+	0.212624i
$775$	−3.38935e7			−2.02704
$776$	2.64378e6			0.157605
$777$	2.64992e7	−	2.31144e7i	1.57464	−	1.37350i
$778$			2.28481e7i			1.35332i
$779$	3.70556e6			0.218781
$780$	2.21637e7	−	1.93326e7i	1.30438	−	1.13777i
$781$			3.02566e7i			1.77498i
$782$	6.11976e6	+	2.86376e6i	0.357864	+	0.167464i
$783$	6.85559e6	−	4.50179e6i	0.399614	−	0.262411i
$784$	−5.17523e6			−0.300704
$785$		−	5.82969e7i		−	3.37653i
$786$	6.18186e6	−	5.39223e6i	0.356913	−	0.311324i
$787$			1.46279e7i			0.841869i	0.907091	+	0.420935i	$0.138298\pi$
$787$			1.46279e7i			0.841869i	−0.907091	+	0.420935i	$0.861702\pi$
$788$			3.86884e6i			0.221955i
$789$	−7.46965e6	+	6.51553e6i	−0.427177	+	0.372612i
$790$	−5.70021e6			−0.324955
$791$		−	3.69369e7i		−	2.09903i
$792$	−8.10152e6	−	1.11060e6i	−0.458937	−	0.0629138i
$793$			2.27894e7i			1.28692i
$794$		−	1.06467e7i		−	0.599328i
$795$	−1.53751e7	+	1.34112e7i	−0.862778	+	0.752573i
$796$			5.03615e6i			0.281719i
$797$	−7.24279e6			−0.403887			−0.201943	−	0.979397i	$-0.564726\pi$
$797$	−7.24279e6			−0.403887			−0.201943	+	0.979397i	$0.564726\pi$
$798$	−2.88325e6	−	3.30547e6i	−0.160279	−	0.183750i
$799$			5.37734e6i			0.297989i
$800$		−	8.63232e6i		−	0.476873i
$801$	−624213.	+	4.55345e6i	−0.0343757	+	0.250760i
$802$			4.67701e6i			0.256763i
$803$	1.81097e7			0.991112
$804$	−3.30166e6	−	3.78515e6i	−0.180133	−	0.206511i
$805$	−4.75270e7	−	2.22404e7i	−2.58494	−	1.20963i
$806$		−	1.76418e7i		−	0.956547i
$807$	−7.20149e6	−	8.25607e6i	−0.389259	−	0.446262i
$808$	−7.16987e6			−0.386352
$809$		−	5.10431e6i		−	0.274199i	−0.990557	−	0.137099i	$-0.956222\pi$
$809$		−	5.10431e6i		−	0.274199i	0.990557	−	0.137099i	$-0.0437780\pi$
$810$	6.83273e6	−	2.44530e7i	0.365916	−	1.30954i
$811$	−2.82748e7			−1.50955			−0.754775	−	0.655984i	$-0.772254\pi$
$811$	−2.82748e7			−1.50955			−0.754775	+	0.655984i	$0.772254\pi$
$812$	6.66561e6			0.354773
$813$	1.26518e7	−	1.10358e7i	0.671317	−	0.585567i
$814$	2.46570e7			1.30430
$815$	−5.75391e7			−3.03437
$816$	−2.00233e6	+	1.74656e6i	−0.105271	+	0.0918246i
$817$	−1.34536e6			−0.0705153
$818$			8.04263e6i			0.420257i
$819$	5.08151e7	+	6.96602e6i	2.64718	+	0.362890i
$820$			1.74326e7i			0.905371i
$821$		−	7.74746e6i		−	0.401145i	−0.979679	−	0.200572i	$-0.935720\pi$
$821$		−	7.74746e6i		−	0.401145i	0.979679	−	0.200572i	$-0.0642802\pi$
$822$	−5.48575e6	−	6.28907e6i	−0.283176	−	0.324644i
$823$	2.24967e7			1.15776			0.578882	−	0.815412i	$-0.303489\pi$
$823$	2.24967e7			1.15776			0.578882	+	0.815412i	$0.303489\pi$
$824$	5.91309e6			0.303387
$825$	5.20706e7	−	4.54195e7i	2.66353	−	2.32331i
$826$			2.96008e6i			0.150957i
$827$	1.29376e7			0.657794			0.328897	−	0.944366i	$-0.393323\pi$
$827$	1.29376e7			0.657794			0.328897	+	0.944366i	$0.393323\pi$
$828$	−8.28344e6	−	5.35536e6i	−0.419890	−	0.271464i
$829$	−2.87039e7			−1.45062			−0.725311	−	0.688421i	$-0.758304\pi$
$829$	−2.87039e7			−1.45062			−0.725311	+	0.688421i	$0.758304\pi$
$830$		−	3.20366e6i		−	0.161418i
$831$	3.53710e6	−	3.08529e6i	0.177682	−	0.154986i
$832$	4.49319e6			0.225033
$833$	−1.34599e7			−0.672094
$834$	−3.57760e6	−	4.10150e6i	−0.178105	−	0.204187i
$835$			2.10217e7i			1.04340i
$836$		−	3.07567e6i		−	0.152203i
$837$	−8.35966e6	−	1.27306e7i	−0.412454	−	0.628108i
$838$		−	8.05074e6i		−	0.396028i
$839$	−5.07889e6			−0.249095			−0.124547	−	0.992214i	$-0.539748\pi$
$839$	−5.07889e6			−0.249095			−0.124547	+	0.992214i	$0.539748\pi$
$840$	1.55504e7	−	1.35641e7i	0.760402	−	0.663273i
$841$	1.58233e7			0.771450
$842$	2.15364e7			1.04687
$843$	−3.96846e6	+	3.46156e6i	−0.192333	+	0.167765i
$844$	−3.42596e6			−0.165549
$845$	−8.94406e7			−4.30916
$846$	1.06618e6	−	7.77746e6i	0.0512159	−	0.373604i
$847$		−	2.22079e7i		−	1.06365i
$848$	−3.11695e6			−0.148847
$849$	−1.53467e7	−	1.75940e7i	−0.730710	−	0.837714i
$850$		−	2.24513e7i		−	1.06584i
$851$	2.69388e7	+	1.26061e7i	1.27513	+	0.596701i
$852$	−9.43428e6	−	1.08158e7i	−0.445256	−	0.510459i
$853$	1.78715e7			0.840987			0.420494	−	0.907296i	$-0.361857\pi$
$853$	1.78715e7			0.840987			0.420494	+	0.907296i	$0.361857\pi$
$854$			1.59894e7i			0.750218i
$855$	9.46119e6	+	1.29699e6i	0.442619	+	0.0606768i
$856$		−	6.74469e6i		−	0.314614i
$857$		−	1.84032e7i		−	0.855938i	−0.903793	−	0.427969i	$-0.859229\pi$
$857$		−	1.84032e7i		−	0.855938i	0.903793	−	0.427969i	$-0.140771\pi$
$858$	2.36412e7	+	2.71032e7i	1.09635	+	1.25690i
$859$	−1.83557e7			−0.848765			−0.424383	−	0.905483i	$-0.639509\pi$
$859$	−1.83557e7			−0.848765			−0.424383	+	0.905483i	$0.639509\pi$
$860$		−	6.32916e6i		−	0.291810i
$861$	−2.29104e7	+	1.99840e7i	−1.05323	+	0.918700i
$862$		−	7.94274e6i		−	0.364085i
$863$		−	4.87260e6i		−	0.222707i	−0.993781	−	0.111353i	$-0.964481\pi$
$863$		−	4.87260e6i		−	0.222707i	0.993781	−	0.111353i	$-0.0355185\pi$
$864$	3.24234e6	−	2.12912e6i	0.147766	−	0.0970321i
$865$			7.53380e7i			3.42353i
$866$	1.48415e7			0.672487
$867$	1.14719e7	−	1.00066e7i	0.518308	−	0.452103i
$868$		−	1.23778e7i		−	0.557627i
$869$		−	6.97058e6i		−	0.313126i
$870$	−1.09364e7	+	9.53943e6i	−0.489863	+	0.427291i
$871$			2.20910e7i			0.986666i
$872$	−4.29063e6			−0.191086
$873$	−9.94507e6	−	1.36333e6i	−0.441644	−	0.0605431i
$874$	1.57247e6	−	3.36030e6i	0.0696310	−	0.148799i
$875$			1.09725e8i			4.84490i
$876$	−6.47367e6	+	5.64677e6i	−0.285030	+	0.248622i
$877$	2.14525e7			0.941842			0.470921	−	0.882175i	$-0.343922\pi$
$877$	2.14525e7			0.941842			0.470921	+	0.882175i	$0.343922\pi$
$878$		−	1.30853e7i		−	0.572857i
$879$	124272.	−	108398.i	0.00542501	−	0.00473206i
$880$	1.44693e7			0.629856
$881$	−2.89857e7			−1.25819			−0.629093	−	0.777330i	$-0.716573\pi$
$881$	−2.89857e7			−1.25819			−0.629093	+	0.777330i	$0.716573\pi$
$882$	1.94676e7	+	2.66874e6i	0.842639	+	0.115514i
$883$	−1.65402e7			−0.713904			−0.356952	−	0.934123i	$-0.616184\pi$
$883$	−1.65402e7			−0.713904			−0.356952	+	0.934123i	$0.616184\pi$
$884$	1.16861e7			0.502964
$885$	−4.23629e6	−	4.85664e6i	−0.181814	−	0.208438i
$886$	2.02035e6			0.0864654
$887$		−	636415.i		−	0.0271601i	−0.999908	−	0.0135801i	$-0.995677\pi$
$887$		−	636415.i		−	0.0271601i	0.999908	−	0.0135801i	$-0.00432280\pi$
$888$	−8.81411e6	+	7.68826e6i	−0.375099	+	0.327187i
$889$		−	2.50696e7i		−	1.06388i
$890$		−	8.13246e6i		−	0.344149i
$891$	2.99027e7	+	8.35550e6i	1.26188	+	0.352597i
$892$	−2.56157e6			−0.107794
$893$	2.95265e6			0.123903
$894$	−1.11126e7	−	1.27399e7i	−0.465021	−	0.533118i
$895$		−	4.95178e7i		−	2.06635i
$896$	3.15249e6			0.131185
$897$	1.19722e7	+	4.16981e7i	0.496815	+	1.73035i
$898$	−2.07571e7			−0.858965
$899$			8.70512e6i			0.359233i
$900$	−4.45147e6	+	3.24721e7i	−0.183188	+	1.33630i
$901$	−8.10668e6			−0.332683
$902$	−2.13176e7			−0.872414
$903$	8.31797e6	−	7.25549e6i	0.339467	−	0.296106i
$904$			1.22859e7i			0.500017i
$905$		−	5.55957e7i		−	2.25642i
$906$	3.93025e6	+	4.50579e6i	0.159074	+	0.182369i
$907$			1.04906e7i			0.423432i	0.977331	+	0.211716i	$0.0679051\pi$
$907$			1.04906e7i			0.423432i	−0.977331	+	0.211716i	$0.932095\pi$
$908$	−6.72381e6			−0.270645
$909$	2.69709e7	+	3.69732e6i	1.08264	+	0.148415i
$910$	−9.07557e7			−3.63304
$911$	4.44172e7			1.77319			0.886595	−	0.462547i	$-0.153064\pi$
$911$	4.44172e7			1.77319			0.886595	+	0.462547i	$0.153064\pi$
$912$	959022.	+	1.09946e6i	0.0381805	+	0.0437716i
$913$	3.91764e6			0.155542
$914$	−1.05254e7			−0.416749
$915$	−2.28831e7	−	2.62340e7i	−0.903570	−	1.03589i
$916$			1.60778e6i			0.0633124i
$917$	−2.53135e7			−0.994096
$918$	8.43280e6	−	5.53749e6i	0.330267	−	0.216873i
$919$			2.84538e7i			1.11135i	0.831399	+	0.555676i	$0.187541\pi$
$919$			2.84538e7i			1.11135i	−0.831399	+	0.555676i	$0.812459\pi$
$920$	1.58083e7	+	7.39757e6i	0.615767	+	0.288151i
$921$	−6.25315e6	+	5.45441e6i	−0.242912	+	0.211884i
$922$	6.80468e6			0.263621
$923$			6.31237e7i			2.43887i
$924$	1.65870e7	+	1.90160e7i	0.639129	+	0.732722i
$925$		−	9.88290e7i		−	3.79778i
$926$		−	2.85840e7i		−	1.09546i
$927$	−2.22432e7	−	3.04923e6i	−0.850156	−	0.116544i
$928$	−2.21710e6			−0.0845115
$929$			2.38380e7i			0.906214i	0.891456	+	0.453107i	$0.149684\pi$
$929$			2.38380e7i			0.906214i	−0.891456	+	0.453107i	$0.850316\pi$
$930$	1.77144e7	+	2.03084e7i	0.671611	+	0.769961i
$931$			7.39072e6i			0.279456i
$932$			2.15727e7i			0.813514i
$933$	1.04621e7	+	1.19941e7i	0.393473	+	0.451092i
$934$		−	2.70293e7i		−	1.01383i
$935$	3.76323e7			1.40777
$936$	−1.69020e7	−	2.31702e6i	−0.630592	−	0.0864452i
$937$			1.48633e6i			0.0553052i	0.999618	+	0.0276526i	$0.00880322\pi$
$937$			1.48633e6i			0.0553052i	−0.999618	+	0.0276526i	$0.991197\pi$
$938$			1.54994e7i			0.575186i
$939$	−2.64929e7	−	3.03724e7i	−0.980539	−	1.12413i
$940$			1.38906e7i			0.512743i
$941$	4.67537e7			1.72124			0.860620	−	0.509247i	$-0.170076\pi$
$941$	4.67537e7			1.72124			0.860620	+	0.509247i	$0.170076\pi$
$942$	−2.54837e7	+	2.22286e7i	−0.935697	+	0.816177i
$943$	−2.32904e7	−	1.08988e7i	−0.852900	−	0.399118i
$944$		−	984575.i		−	0.0359599i
$945$	−6.54905e7	+	4.30050e7i	−2.38561	+	1.56653i
$946$	7.73970e6			0.281188
$947$		−	5.39020e7i		−	1.95313i	−0.215233	−	0.976563i	$-0.569051\pi$
$947$		−	5.39020e7i		−	1.95313i	0.215233	−	0.976563i	$-0.430949\pi$
$948$	2.17349e6	+	2.49177e6i	0.0785483	+	0.0900507i
$949$	3.77819e7			1.36181
$950$	−1.23278e7			−0.443176
$951$	−1.92880e7	−	2.21125e7i	−0.691570	−	0.792843i
$952$	8.19912e6			0.293207
$953$	5.48502e7			1.95635			0.978173	−	0.207791i	$-0.0666275\pi$
$953$	5.48502e7			1.95635			0.978173	+	0.207791i	$0.0666275\pi$
$954$	1.17250e7	+	1.60733e6i	0.417102	+	0.0571788i
$955$	4.85174e7			1.72143
$956$			8.62760e6i			0.305313i
$957$	−1.16654e7	−	1.33737e7i	−0.411738	−	0.472032i
$958$			2.68208e7i			0.944188i
$959$			2.57525e7i			0.904216i
$960$	−5.17234e6	+	4.51166e6i	−0.181138	+	0.158000i
$961$	−1.24641e7			−0.435363
$962$	5.14412e7			1.79215
$963$	−3.47807e6	+	2.53715e7i	−0.120857	+	0.881617i
$964$		−	2.42373e7i		−	0.840025i
$965$	3.87553e7			1.33972
$966$	8.39992e6	+	2.92561e7i	0.289623	+	1.00873i
$967$	2.46542e7			0.847860			0.423930	−	0.905695i	$-0.360650\pi$
$967$	2.46542e7			0.847860			0.423930	+	0.905695i	$0.360650\pi$
$968$			7.38673e6i			0.253375i
$969$	2.49426e6	+	2.85952e6i	0.0853360	+	0.0978325i
$970$	1.77619e7			0.606122
$971$	−3.70476e7			−1.26099			−0.630496	−	0.776193i	$-0.717148\pi$
$971$	−3.70476e7			−1.26099			−0.630496	+	0.776193i	$0.717148\pi$
$972$	−1.32946e7	+	6.33709e6i	−0.451347	+	0.215142i
$973$			1.67948e7i			0.568713i
$974$			3.68949e7i			1.24615i
$975$	1.08634e8	−	9.47575e7i	3.65976	−	3.19229i
$976$		−	5.31836e6i		−	0.178712i
$977$	1.33333e7			0.446891			0.223446	−	0.974716i	$-0.428269\pi$
$977$	1.33333e7			0.446891			0.223446	+	0.974716i	$0.428269\pi$
$978$	2.19396e7	+	2.51524e7i	0.733470	+	0.840878i
$979$	9.94488e6			0.331622
$980$	−3.47692e7			−1.15646
$981$	1.61400e7	+	2.21257e6i	0.535466	+	0.0734048i
$982$	−1.17928e7			−0.390247
$983$	6.57597e6			0.217058			0.108529	−	0.994093i	$-0.465386\pi$
$983$	6.57597e6			0.217058			0.108529	+	0.994093i	$0.465386\pi$
$984$	7.62041e6	−	6.64703e6i	0.250894	−	0.218847i
$985$			2.59924e7i			0.853602i
$986$	−5.76633e6			−0.188889
$987$	−1.82554e7	+	1.59236e7i	−0.596483	+	0.520292i
$988$		−	6.41670e6i		−	0.209132i
$989$	8.45595e6	+	3.95699e6i	0.274898	+	0.128640i
$990$	−5.44291e7	−	7.46146e6i	−1.76499	−	0.241956i
$991$	−2.76808e6			−0.0895352			−0.0447676	−	0.998997i	$-0.514255\pi$
$991$	−2.76808e6			−0.0895352			−0.0447676	+	0.998997i	$0.514255\pi$
$992$			4.11708e6i			0.132834i
$993$	−3.30173e7	+	2.87999e7i	−1.06260	+	0.926867i
$994$			4.42886e7i			1.42176i
$995$			3.38348e7i			1.08344i
$996$	−1.40044e6	+	1.22155e6i	−0.0447317	+	0.0390179i
$997$	1.59147e7			0.507060			0.253530	−	0.967328i	$-0.418408\pi$
$997$	1.59147e7			0.507060			0.253530	+	0.967328i	$0.418408\pi$
$998$		−	6.78502e6i		−	0.215638i
$999$	3.71206e7	−	2.43757e7i	1.17680	−	0.772756i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.6.d.a.137.1	✓	40
3.2	odd	2	inner	138.6.d.a.137.4	yes	40
23.22	odd	2	inner	138.6.d.a.137.2	yes	40
69.68	even	2	inner	138.6.d.a.137.3	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.d.a.137.1	✓	40	1.1	even	1	trivial
138.6.d.a.137.2	yes	40	23.22	odd	2	inner
138.6.d.a.137.3	yes	40	69.68	even	2	inner
138.6.d.a.137.4	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	138.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +(-11.7474 + 10.2469i) q^{3} -16.0000 q^{4} -107.494 q^{5} +(40.9875 + 46.9896i) q^{6} -192.413i q^{7} +64.0000i q^{8} +(33.0032 - 240.748i) q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} +(-11.7474 + 10.2469i) q^{3} -16.0000 q^{4} -107.494 q^{5} +(40.9875 + 46.9896i) q^{6} -192.413i q^{7} +64.0000i q^{8} +(33.0032 - 240.748i) q^{9} +429.977i q^{10} -525.803 q^{11} +(187.959 - 163.950i) q^{12} -1096.97 q^{13} -769.652 q^{14} +(1262.78 - 1101.48i) q^{15} +256.000 q^{16} +665.814 q^{17} +(-962.994 - 132.013i) q^{18} -365.592i q^{19} +1719.91 q^{20} +(1971.63 + 2260.35i) q^{21} +2103.21i q^{22} +(-1075.29 + 2297.85i) q^{23} +(-655.800 - 751.834i) q^{24} +8430.00 q^{25} +4387.88i q^{26} +(2079.22 + 3166.35i) q^{27} +3078.61i q^{28} -2165.14i q^{29} +(-4405.92 - 5051.11i) q^{30} -4020.58 q^{31} -1024.00i q^{32} +(6176.82 - 5387.84i) q^{33} -2663.26i q^{34} +20683.3i q^{35} +(-528.051 + 3851.97i) q^{36} -11723.5i q^{37} -1462.37 q^{38} +(12886.6 - 11240.5i) q^{39} -6879.63i q^{40} +10135.8i q^{41} +(9041.41 - 7886.52i) q^{42} -3679.94i q^{43} +8412.85 q^{44} +(-3547.65 + 25879.1i) q^{45} +(9191.39 + 4301.15i) q^{46} +8076.34i q^{47} +(-3007.34 + 2623.20i) q^{48} -20215.7 q^{49} -33720.0i q^{50} +(-7821.59 + 6822.52i) q^{51} +17551.5 q^{52} -12175.6 q^{53} +(12665.4 - 8316.87i) q^{54} +56520.8 q^{55} +12314.4 q^{56} +(3746.18 + 4294.76i) q^{57} -8660.56 q^{58} -3846.00i q^{59} +(-20204.5 + 17623.7i) q^{60} -20774.9i q^{61} +16082.3i q^{62} +(-46323.1 - 6350.24i) q^{63} -4096.00 q^{64} +117918. q^{65} +(-21551.3 - 24707.3i) q^{66} -20138.2i q^{67} -10653.0 q^{68} +(-10913.9 - 38012.1i) q^{69} +82733.1 q^{70} -57543.7i q^{71} +(15407.9 + 2112.20i) q^{72} -34442.0 q^{73} -46893.9 q^{74} +(-99030.7 + 86381.2i) q^{75} +5849.48i q^{76} +101171. i q^{77} +(-44962.0 - 51546.2i) q^{78} +13257.0i q^{79} -27518.5 q^{80} +(-56870.6 - 15890.9i) q^{81} +40543.0 q^{82} -7450.77 q^{83} +(-31546.1 - 36165.7i) q^{84} -71571.2 q^{85} -14719.8 q^{86} +(22185.9 + 25434.8i) q^{87} -33651.4i q^{88} -18913.7 q^{89} +(103516. + 14190.6i) q^{90} +211071. i q^{91} +(17204.6 - 36765.6i) q^{92} +(47231.4 - 41198.4i) q^{93} +32305.4 q^{94} +39299.1i q^{95} +(10492.8 + 12029.3i) q^{96} -41309.0i q^{97} +80863.0i q^{98} +(-17353.2 + 126586. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9}+O(q^{10}) \) 40 * q - 4 * q^3 - 640 * q^4 + 80 * q^6 + 504 * q^9	\( 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9} + 64 q^{12} - 1048 q^{13} + 10240 q^{16} + 1280 q^{18} - 1280 q^{24} + 30480 q^{25} + 1700 q^{27} - 22576 q^{31} - 8064 q^{36} + 55608 q^{39} + 1088 q^{46} - 1024 q^{48} - 23224 q^{49} + 16768 q^{52} + 25456 q^{54} + 210400 q^{55} - 83168 q^{58} - 163840 q^{64} + 99076 q^{69} + 167520 q^{70} - 20480 q^{72} + 241160 q^{73} - 255604 q^{75} - 233440 q^{78} + 78512 q^{81} - 8832 q^{82} - 460296 q^{85} - 4136 q^{87} + 500704 q^{93} - 138272 q^{94} + 20480 q^{96}+O(q^{100}) \) 40 * q - 4 * q^3 - 640 * q^4 + 80 * q^6 + 504 * q^9 + 64 * q^12 - 1048 * q^13 + 10240 * q^16 + 1280 * q^18 - 1280 * q^24 + 30480 * q^25 + 1700 * q^27 - 22576 * q^31 - 8064 * q^36 + 55608 * q^39 + 1088 * q^46 - 1024 * q^48 - 23224 * q^49 + 16768 * q^52 + 25456 * q^54 + 210400 * q^55 - 83168 * q^58 - 163840 * q^64 + 99076 * q^69 + 167520 * q^70 - 20480 * q^72 + 241160 * q^73 - 255604 * q^75 - 233440 * q^78 + 78512 * q^81 - 8832 * q^82 - 460296 * q^85 - 4136 * q^87 + 500704 * q^93 - 138272 * q^94 + 20480 * q^96

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