LMFDB - Embedded newform 336.6.q.l.289.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(193,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.q (of order $3$, degree $2$, not 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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 564x^{8} + 117814x^{6} + 11067780x^{4} + 427918225x^{2} + 3489248448 $ x^10 + 564x^8 + 117814x^6 + 11067780x^4 + 427918225x^2 + 3489248448
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.3
Root			$-3.29825i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.6.q.l.193.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 7.79423i) q^{3} +(8.04054 - 13.9266i) q^{5} +(77.3451 - 104.042i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(-4.50000 - 7.79423i) q^{3} +(8.04054 - 13.9266i) q^{5} +(77.3451 - 104.042i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(329.284 + 570.336i) q^{11} +196.937 q^{13} -144.730 q^{15} +(-228.522 - 395.812i) q^{17} +(-517.763 + 896.792i) q^{19} +(-1158.98 - 134.656i) q^{21} +(-1123.07 + 1945.21i) q^{23} +(1433.20 + 2482.37i) q^{25} +729.000 q^{27} -8257.88 q^{29} +(2898.57 + 5020.47i) q^{31} +(2963.55 - 5133.02i) q^{33} +(-827.058 - 1913.71i) q^{35} +(4153.56 - 7194.18i) q^{37} +(-886.214 - 1534.97i) q^{39} +14585.4 q^{41} +796.436 q^{43} +(651.284 + 1128.06i) q^{45} +(-11043.2 + 19127.4i) q^{47} +(-4842.48 - 16094.3i) q^{49} +(-2056.70 + 3562.31i) q^{51} +(-7065.39 - 12237.6i) q^{53} +10590.5 q^{55} +9319.73 q^{57} +(17627.9 + 30532.3i) q^{59} +(-15771.0 + 27316.2i) q^{61} +(4165.87 + 9639.31i) q^{63} +(1583.48 - 2742.66i) q^{65} +(24864.0 + 43065.7i) q^{67} +20215.2 q^{69} +7903.78 q^{71} +(-8024.92 - 13899.6i) q^{73} +(12898.8 - 22341.4i) q^{75} +(84807.4 + 9853.34i) q^{77} +(9819.88 - 17008.5i) q^{79} +(-3280.50 - 5681.99i) q^{81} +60468.5 q^{83} -7349.77 q^{85} +(37160.5 + 64363.8i) q^{87} +(19131.6 - 33136.9i) q^{89} +(15232.1 - 20489.7i) q^{91} +(26087.1 - 45184.3i) q^{93} +(8326.19 + 14421.4i) q^{95} +57264.8 q^{97} -53344.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 45 q^{3} - 6 q^{5} + 97 q^{7} - 405 q^{9}+O(q^{10}) $ 10 * q - 45 * q^3 - 6 * q^5 + 97 * q^7 - 405 * q^9	$ 10 q - 45 q^{3} - 6 q^{5} + 97 q^{7} - 405 q^{9} - 424 q^{11} + 374 q^{13} + 108 q^{15} - 952 q^{17} - 139 q^{19} - 1044 q^{21} - 4288 q^{23} - 5605 q^{25} + 7290 q^{27} - 4216 q^{29} - 8131 q^{31} - 3816 q^{33} - 20106 q^{35} - 5425 q^{37} - 1683 q^{39} + 29364 q^{41} + 46862 q^{43} - 486 q^{45} + 17190 q^{47} + 23255 q^{49} - 8568 q^{51} + 15064 q^{53} + 1176 q^{55} + 2502 q^{57} + 83242 q^{59} + 14954 q^{61} + 1539 q^{63} - 23250 q^{65} - 39501 q^{67} + 77184 q^{69} + 56020 q^{71} - 90395 q^{73} - 50445 q^{75} + 63448 q^{77} + 43067 q^{79} - 32805 q^{81} + 75672 q^{83} - 75272 q^{85} + 18972 q^{87} - 72608 q^{89} - 288287 q^{91} - 73179 q^{93} - 190138 q^{95} + 183000 q^{97} + 68688 q^{99}+O(q^{100}) $ 10 * q - 45 * q^3 - 6 * q^5 + 97 * q^7 - 405 * q^9 - 424 * q^11 + 374 * q^13 + 108 * q^15 - 952 * q^17 - 139 * q^19 - 1044 * q^21 - 4288 * q^23 - 5605 * q^25 + 7290 * q^27 - 4216 * q^29 - 8131 * q^31 - 3816 * q^33 - 20106 * q^35 - 5425 * q^37 - 1683 * q^39 + 29364 * q^41 + 46862 * q^43 - 486 * q^45 + 17190 * q^47 + 23255 * q^49 - 8568 * q^51 + 15064 * q^53 + 1176 * q^55 + 2502 * q^57 + 83242 * q^59 + 14954 * q^61 + 1539 * q^63 - 23250 * q^65 - 39501 * q^67 + 77184 * q^69 + 56020 * q^71 - 90395 * q^73 - 50445 * q^75 + 63448 * q^77 + 43067 * q^79 - 32805 * q^81 + 75672 * q^83 - 75272 * q^85 + 18972 * q^87 - 72608 * q^89 - 288287 * q^91 - 73179 * q^93 - 190138 * q^95 + 183000 * q^97 + 68688 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−4.50000	−	7.79423i	−0.288675	−	0.500000i
$3$	−4.50000	−	7.79423i	−0.288675	−	0.500000i
$4$		0			0
$5$	8.04054	−	13.9266i	0.143834	−	0.249127i	−0.785104	−	0.619364i	$-0.787390\pi$
$5$	8.04054	−	13.9266i	0.143834	−	0.249127i	0.928937	+	0.370237i	$0.120724\pi$
$6$		0			0
$7$	77.3451	−	104.042i	0.596606	−	0.802534i
$7$	77.3451	−	104.042i	0.596606	−	0.802534i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	329.284	+	570.336i	0.820519	+	1.42118i	0.905297	+	0.424780i	$0.139649\pi$
$11$	329.284	+	570.336i	0.820519	+	1.42118i	−0.0847778	+	0.996400i	$0.527018\pi$
$12$		0			0
$13$	196.937			0.323197			0.161599	−	0.986857i	$-0.448335\pi$
$13$	196.937			0.323197			0.161599	+	0.986857i	$0.448335\pi$
$14$		0			0
$15$	−144.730			−0.166085
$16$		0			0
$17$	−228.522	−	395.812i	−0.191781	−	0.332175i	0.754059	−	0.656806i	$-0.228093\pi$
$17$	−228.522	−	395.812i	−0.191781	−	0.332175i	−0.945841	+	0.324632i	$0.894760\pi$
$18$		0			0
$19$	−517.763	+	896.792i	−0.329039	+	0.569912i	−0.982321	−	0.187202i	$-0.940058\pi$
$19$	−517.763	+	896.792i	−0.329039	+	0.569912i	0.653283	+	0.757114i	$0.273391\pi$
$20$		0			0
$21$	−1158.98	−	134.656i	−0.573492	−	0.0666312i
$22$		0			0
$23$	−1123.07	+	1945.21i	−0.442677	+	0.766739i	−0.997887	−	0.0649713i	$-0.979304\pi$
$23$	−1123.07	+	1945.21i	−0.442677	+	0.766739i	0.555210	+	0.831710i	$0.312638\pi$
$24$		0			0
$25$	1433.20	+	2482.37i	0.458624	+	0.794360i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	−8257.88			−1.82336			−0.911682	−	0.410896i	$-0.865216\pi$
$29$	−8257.88			−1.82336			−0.911682	+	0.410896i	$0.865216\pi$
$30$		0			0
$31$	2898.57	+	5020.47i	0.541726	+	0.938297i	0.998805	+	0.0488711i	$0.0155624\pi$
$31$	2898.57	+	5020.47i	0.541726	+	0.938297i	−0.457079	+	0.889426i	$0.651104\pi$
$32$		0			0
$33$	2963.55	−	5133.02i	0.473727	−	0.820519i
$34$		0			0
$35$	−827.058	−	1913.71i	−0.114121	−	0.264062i
$36$		0			0
$37$	4153.56	−	7194.18i	0.498788	−	0.863926i	−0.501211	−	0.865325i	$-0.667112\pi$
$37$	4153.56	−	7194.18i	0.498788	−	0.863926i	0.999999	+	0.00139879i	$0.000445249\pi$
$38$		0			0
$39$	−886.214	−	1534.97i	−0.0932991	−	0.161599i
$40$		0			0
$41$	14585.4			1.35506			0.677531	−	0.735494i	$-0.263050\pi$
$41$	14585.4			1.35506			0.677531	+	0.735494i	$0.263050\pi$
$42$		0			0
$43$	796.436			0.0656871			0.0328435	−	0.999461i	$-0.489544\pi$
$43$	796.436			0.0656871			0.0328435	+	0.999461i	$0.489544\pi$
$44$		0			0
$45$	651.284	+	1128.06i	0.0479445	+	0.0830424i
$46$		0			0
$47$	−11043.2	+	19127.4i	−0.729207	+	1.26302i	0.228011	+	0.973658i	$0.426778\pi$
$47$	−11043.2	+	19127.4i	−0.729207	+	1.26302i	−0.957219	+	0.289366i	$0.906556\pi$
$48$		0			0
$49$	−4842.48	−	16094.3i	−0.288123	−	0.957593i
$50$		0			0
$51$	−2056.70	+	3562.31i	−0.110725	+	0.191781i
$52$		0			0
$53$	−7065.39	−	12237.6i	−0.345499	−	0.598421i	0.639945	−	0.768420i	$-0.278957\pi$
$53$	−7065.39	−	12237.6i	−0.345499	−	0.598421i	−0.985444	+	0.169999i	$0.945624\pi$
$54$		0			0
$55$	10590.5			0.472073
$56$		0			0
$57$	9319.73			0.379941
$58$		0			0
$59$	17627.9	+	30532.3i	0.659279	+	1.14190i	0.980803	+	0.195003i	$0.0624718\pi$
$59$	17627.9	+	30532.3i	0.659279	+	1.14190i	−0.321523	+	0.946902i	$0.604195\pi$
$60$		0			0
$61$	−15771.0	+	27316.2i	−0.542669	+	0.939930i	0.456081	+	0.889938i	$0.349253\pi$
$61$	−15771.0	+	27316.2i	−0.542669	+	0.939930i	−0.998750	+	0.0499916i	$0.984081\pi$
$62$		0			0
$63$	4165.87	+	9639.31i	0.132237	+	0.305981i
$64$		0			0
$65$	1583.48	−	2742.66i	0.0464867	−	0.0805172i
$66$		0			0
$67$	24864.0	+	43065.7i	0.676681	+	1.17205i	0.975975	+	0.217884i	$0.0699155\pi$
$67$	24864.0	+	43065.7i	0.676681	+	1.17205i	−0.299294	+	0.954161i	$0.596751\pi$
$68$		0			0
$69$	20215.2			0.511159
$70$		0			0
$71$	7903.78			0.186075			0.0930377	−	0.995663i	$-0.470342\pi$
$71$	7903.78			0.186075			0.0930377	+	0.995663i	$0.470342\pi$
$72$		0			0
$73$	−8024.92	−	13899.6i	−0.176252	−	0.305277i	0.764342	−	0.644811i	$-0.223064\pi$
$73$	−8024.92	−	13899.6i	−0.176252	−	0.305277i	−0.940594	+	0.339534i	$0.889731\pi$
$74$		0			0
$75$	12898.8	−	22341.4i	0.264787	−	0.458624i
$76$		0			0
$77$	84807.4	+	9853.34i	1.63007	+	0.189390i
$78$		0			0
$79$	9819.88	−	17008.5i	0.177027	−	0.306619i	−0.763834	−	0.645413i	$-0.776685\pi$
$79$	9819.88	−	17008.5i	0.177027	−	0.306619i	0.940861	+	0.338794i	$0.110019\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	60468.5			0.963461			0.481731	−	0.876319i	$-0.340008\pi$
$83$	60468.5			0.963461			0.481731	+	0.876319i	$0.340008\pi$
$84$		0			0
$85$	−7349.77			−0.110338
$86$		0			0
$87$	37160.5	+	64363.8i	0.526360	+	0.911682i
$88$		0			0
$89$	19131.6	−	33136.9i	0.256022	−	0.443443i	−0.709151	−	0.705057i	$-0.750921\pi$
$89$	19131.6	−	33136.9i	0.256022	−	0.443443i	0.965173	+	0.261614i	$0.0842548\pi$
$90$		0			0
$91$	15232.1	−	20489.7i	0.192821	−	0.259377i
$92$		0			0
$93$	26087.1	−	45184.3i	0.312766	−	0.541726i
$94$		0			0
$95$	8326.19	+	14421.4i	0.0946536	+	0.163945i
$96$		0			0
$97$	57264.8			0.617957			0.308978	−	0.951069i	$-0.400013\pi$
$97$	57264.8			0.617957			0.308978	+	0.951069i	$0.400013\pi$
$98$		0			0
$99$	−53344.0			−0.547012
$100$		0			0
$101$	74144.1	+	128421.i	0.723224	+	1.25266i	0.959701	+	0.281024i	$0.0906740\pi$
$101$	74144.1	+	128421.i	0.723224	+	1.25266i	−0.236477	+	0.971637i	$0.575993\pi$
$102$		0			0
$103$	78360.5	−	135724.i	0.727787	−	1.26056i	−0.230030	−	0.973183i	$-0.573882\pi$
$103$	78360.5	−	135724.i	0.727787	−	1.26056i	0.957817	−	0.287380i	$-0.0927842\pi$
$104$		0			0
$105$	−11194.1	+	15058.0i	−0.0990871	+	0.133289i
$106$		0			0
$107$	25219.6	−	43681.7i	0.212951	−	0.368842i	−0.739686	−	0.672952i	$-0.765026\pi$
$107$	25219.6	−	43681.7i	0.212951	−	0.368842i	0.952637	+	0.304111i	$0.0983592\pi$
$108$		0			0
$109$	83620.6	+	144835.i	0.674135	+	1.16764i	0.976721	+	0.214514i	$0.0688169\pi$
$109$	83620.6	+	144835.i	0.674135	+	1.16764i	−0.302585	+	0.953122i	$0.597850\pi$
$110$		0			0
$111$	−74764.1			−0.575951
$112$		0			0
$113$	−14819.7			−0.109180			−0.0545901	−	0.998509i	$-0.517385\pi$
$113$	−14819.7			−0.109180			−0.0545901	+	0.998509i	$0.517385\pi$
$114$		0			0
$115$	18060.2	+	31281.1i	0.127344	+	0.220566i
$116$		0			0
$117$	−7975.93	+	13814.7i	−0.0538662	+	0.0932991i
$118$		0			0
$119$	−58856.1	−	6838.19i	−0.380999	−	0.0442664i
$120$		0			0
$121$	−136330.	+	236130.i	−0.846502	+	1.46618i
$122$		0			0
$123$	−65634.4	−	113682.i	−0.391173	−	0.677531i
$124$		0			0
$125$	96348.2			0.551529
$126$		0			0
$127$	279603.			1.53827			0.769135	−	0.639086i	$-0.220687\pi$
$127$	279603.			1.53827			0.769135	+	0.639086i	$0.220687\pi$
$128$		0			0
$129$	−3583.96	−	6207.61i	−0.0189622	−	0.0328435i
$130$		0			0
$131$	−69541.3	+	120449.i	−0.354050	+	0.613233i	−0.986955	−	0.160997i	$-0.948529\pi$
$131$	−69541.3	+	120449.i	−0.354050	+	0.613233i	0.632905	+	0.774230i	$0.281862\pi$
$132$		0			0
$133$	53257.6	+	123232.i	0.261067	+	0.604077i
$134$		0			0
$135$	5861.56	−	10152.5i	0.0276808	−	0.0479445i
$136$		0			0
$137$	68351.7	+	118389.i	0.311134	+	0.538900i	0.978608	−	0.205733i	$-0.0659578\pi$
$137$	68351.7	+	118389.i	0.311134	+	0.538900i	−0.667474	+	0.744633i	$0.732625\pi$
$138$		0			0
$139$	344653.			1.51302			0.756511	−	0.653981i	$-0.226902\pi$
$139$	344653.			1.51302			0.756511	+	0.653981i	$0.226902\pi$
$140$		0			0
$141$	198778.			0.842016
$142$		0			0
$143$	64848.0	+	112320.i	0.265190	+	0.459322i
$144$		0			0
$145$	−66397.8	+	115004.i	−0.262261	+	0.454250i
$146$		0			0
$147$	−103651.	+	110168.i	−0.395623	+	0.420495i
$148$		0			0
$149$	37483.4	−	64923.2i	0.138316	−	0.239571i	−0.788543	−	0.614980i	$-0.789164\pi$
$149$	37483.4	−	64923.2i	0.138316	−	0.239571i	0.926859	+	0.375409i	$0.122498\pi$
$150$		0			0
$151$	−269536.	−	466850.i	−0.961998	−	1.66623i	−0.717473	−	0.696586i	$-0.754702\pi$
$151$	−269536.	−	466850.i	−0.961998	−	1.66623i	−0.244524	−	0.969643i	$-0.578632\pi$
$152$		0			0
$153$	37020.6			0.127854
$154$		0			0
$155$	93224.4			0.311674
$156$		0			0
$157$	−179892.	−	311582.i	−0.582456	−	1.00884i	−0.995187	−	0.0979906i	$-0.968758\pi$
$157$	−179892.	−	311582.i	−0.582456	−	1.00884i	0.412731	−	0.910853i	$-0.364575\pi$
$158$		0			0
$159$	−63588.5	+	110139.i	−0.199474	+	0.345499i
$160$		0			0
$161$	115520.	+	267299.i	0.351231	+	0.812704i
$162$		0			0
$163$	−60711.4	+	105155.i	−0.178979	+	0.310000i	−0.941531	−	0.336926i	$-0.890613\pi$
$163$	−60711.4	+	105155.i	−0.178979	+	0.310000i	0.762552	+	0.646927i	$0.223946\pi$
$164$		0			0
$165$	−47657.1	−	82544.6i	−0.136276	−	0.236036i
$166$		0			0
$167$	−305821.			−0.848546			−0.424273	−	0.905534i	$-0.639470\pi$
$167$	−305821.			−0.848546			−0.424273	+	0.905534i	$0.639470\pi$
$168$		0			0
$169$	−332509.			−0.895543
$170$		0			0
$171$	−41938.8	−	72640.1i	−0.109680	−	0.189971i
$172$		0			0
$173$	−145173.	+	251446.i	−0.368782	+	0.638749i	−0.989375	−	0.145384i	$-0.953558\pi$
$173$	−145173.	+	251446.i	−0.368782	+	0.638749i	0.620594	+	0.784132i	$0.286892\pi$
$174$		0			0
$175$	369122.	+	42886.4i	0.911119	+	0.105858i
$176$		0			0
$177$	158651.	−	274791.i	0.380635	−	0.659279i
$178$		0			0
$179$	91921.2	+	159212.i	0.214429	+	0.371402i	0.953096	−	0.302669i	$-0.0978776\pi$
$179$	91921.2	+	159212.i	0.214429	+	0.371402i	−0.738667	+	0.674071i	$0.764544\pi$
$180$		0			0
$181$	−306853.			−0.696201			−0.348100	−	0.937457i	$-0.613173\pi$
$181$	−306853.			−0.696201			−0.348100	+	0.937457i	$0.613173\pi$
$182$		0			0
$183$	283878.			0.626620
$184$		0			0
$185$	−66793.7	−	115690.i	−0.143485	−	0.248523i
$186$		0			0
$187$	150497.	−	260669.i	0.314720	−	0.545111i
$188$		0			0
$189$	56384.5	−	75846.6i	0.114817	−	0.154448i
$190$		0			0
$191$	−198166.	+	343234.i	−0.393048	+	0.680779i	−0.992850	−	0.119369i	$-0.961913\pi$
$191$	−198166.	+	343234.i	−0.393048	+	0.680779i	0.599802	+	0.800149i	$0.295246\pi$
$192$		0			0
$193$	−199509.	−	345559.i	−0.385539	−	0.667773i	0.606305	−	0.795232i	$-0.292651\pi$
$193$	−199509.	−	345559.i	−0.385539	−	0.667773i	−0.991844	+	0.127459i	$0.959318\pi$
$194$		0			0
$195$	−28502.6			−0.0536782
$196$		0			0
$197$	456921.			0.838833			0.419416	−	0.907794i	$-0.362235\pi$
$197$	456921.			0.838833			0.419416	+	0.907794i	$0.362235\pi$
$198$		0			0
$199$	−303084.	−	524958.i	−0.542539	−	0.939705i	−0.998757	−	0.0498373i	$-0.984130\pi$
$199$	−303084.	−	524958.i	−0.542539	−	0.939705i	0.456218	−	0.889868i	$-0.349204\pi$
$200$		0			0
$201$	223776.	−	387591.i	0.390682	−	0.676681i
$202$		0			0
$203$	−638706.	+	859166.i	−1.08783	+	1.46331i
$204$		0			0
$205$	117275.	−	203126.i	0.194904	−	0.337583i
$206$		0			0
$207$	−90968.6	−	157562.i	−0.147559	−	0.255580i
$208$		0			0
$209$	−681964.			−1.07993
$210$		0			0
$211$	−736564.			−1.13895			−0.569475	−	0.822009i	$-0.692853\pi$
$211$	−736564.			−1.13895			−0.569475	+	0.822009i	$0.692853\pi$
$212$		0			0
$213$	−35567.0	−	61603.9i	−0.0537153	−	0.0930377i
$214$		0			0
$215$	6403.78	−	11091.7i	0.00944801	−	0.0163644i
$216$		0			0
$217$	746530.	+	86735.6i	1.07621	+	0.125040i
$218$		0			0
$219$	−72224.3	+	125096.i	−0.101759	+	0.176252i
$220$		0			0
$221$	−45004.3	−	77949.8i	−0.0619832	−	0.107358i
$222$		0			0
$223$	1.05069e6			1.41486			0.707428	−	0.706785i	$-0.249855\pi$
$223$	1.05069e6			1.41486			0.707428	+	0.706785i	$0.249855\pi$
$224$		0			0
$225$	−232178.			−0.305749
$226$		0			0
$227$	553302.	+	958347.i	0.712685	+	1.23441i	0.963846	+	0.266461i	$0.0858545\pi$
$227$	553302.	+	958347.i	0.712685	+	1.23441i	−0.251160	+	0.967945i	$0.580812\pi$
$228$		0			0
$229$	41471.7	−	71831.1i	0.0522592	−	0.0905156i	−0.838712	−	0.544575i	$-0.816691\pi$
$229$	41471.7	−	71831.1i	0.0522592	−	0.0905156i	0.890972	+	0.454059i	$0.150024\pi$
$230$		0			0
$231$	−304834.	−	705348.i	−0.375866	−	0.869708i
$232$		0			0
$233$	584755.	−	1.01282e6i	0.705641	−	1.22221i	−0.260819	−	0.965388i	$-0.583992\pi$
$233$	584755.	−	1.01282e6i	0.705641	−	1.22221i	0.966460	−	0.256818i	$-0.0826742\pi$
$234$		0			0
$235$	177587.	+	307590.i	0.209769	+	0.363331i
$236$		0			0
$237$	−176758.			−0.204413
$238$		0			0
$239$	−786315.			−0.890434			−0.445217	−	0.895423i	$-0.646873\pi$
$239$	−786315.			−0.890434			−0.445217	+	0.895423i	$0.646873\pi$
$240$		0			0
$241$	−612502.	−	1.06088e6i	−0.679305	−	1.17659i	−0.975190	−	0.221368i	$-0.928948\pi$
$241$	−612502.	−	1.06088e6i	−0.679305	−	1.17659i	0.295885	−	0.955223i	$-0.404385\pi$
$242$		0			0
$243$	−29524.5	+	51137.9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	−263075.	−	61967.2i	−0.280004	−	0.0659548i
$246$		0			0
$247$	−101966.	+	176611.i	−0.106344	+	0.184194i
$248$		0			0
$249$	−272108.	−	471306.i	−0.278127	−	0.481731i
$250$		0			0
$251$	927766.			0.929510			0.464755	−	0.885439i	$-0.346142\pi$
$251$	927766.			0.929510			0.464755	+	0.885439i	$0.346142\pi$
$252$		0			0
$253$	−1.47923e6			−1.45290
$254$		0			0
$255$	33073.9	+	57285.8i	0.0318519	+	0.0551691i
$256$		0			0
$257$	−724479.	+	1.25484e6i	−0.684216	+	1.18510i	0.289467	+	0.957188i	$0.406522\pi$
$257$	−724479.	+	1.25484e6i	−0.684216	+	1.18510i	−0.973683	+	0.227909i	$0.926811\pi$
$258$		0			0
$259$	−427239.	−	988579.i	−0.395751	−	0.915718i
$260$		0			0
$261$	334444.	−	579274.i	0.303894	−	0.526360i
$262$		0			0
$263$	−930214.	−	1.61118e6i	−0.829265	−	1.43633i	−0.898616	−	0.438737i	$-0.855426\pi$
$263$	−930214.	−	1.61118e6i	−0.829265	−	1.43633i	0.0693507	−	0.997592i	$-0.477907\pi$
$264$		0			0
$265$	−227238.			−0.198777
$266$		0			0
$267$	−344369.			−0.295628
$268$		0			0
$269$	237843.	+	411957.i	0.200406	+	0.347113i	0.948659	−	0.316300i	$-0.102441\pi$
$269$	237843.	+	411957.i	0.200406	+	0.347113i	−0.748253	+	0.663413i	$0.769107\pi$
$270$		0			0
$271$	−843959.	+	1.46178e6i	−0.698069	+	1.20909i	0.271066	+	0.962561i	$0.412624\pi$
$271$	−843959.	+	1.46178e6i	−0.698069	+	1.20909i	−0.969135	+	0.246530i	$0.920710\pi$
$272$		0			0
$273$	−228246.	−	26518.7i	−0.185351	−	0.0215350i
$274$		0			0
$275$	−943858.	+	1.63481e6i	−0.752619	+	1.30357i
$276$		0			0
$277$	−211682.	−	366643.i	−0.165762	−	0.287108i	0.771164	−	0.636637i	$-0.219675\pi$
$277$	−211682.	−	366643.i	−0.165762	−	0.287108i	−0.936925	+	0.349529i	$0.886342\pi$
$278$		0			0
$279$	−469569.			−0.361151
$280$		0			0
$281$	−536458.			−0.405294			−0.202647	−	0.979252i	$-0.564954\pi$
$281$	−536458.			−0.405294			−0.202647	+	0.979252i	$0.564954\pi$
$282$		0			0
$283$	−426378.	−	738509.i	−0.316467	−	0.548138i	0.663281	−	0.748371i	$-0.269163\pi$
$283$	−426378.	−	738509.i	−0.316467	−	0.548138i	−0.979748	+	0.200233i	$0.935830\pi$
$284$		0			0
$285$	74935.7	−	129792.i	0.0546483	−	0.0946536i
$286$		0			0
$287$	1.12811e6	−	1.51750e6i	0.808438	−	1.08748i
$288$		0			0
$289$	605484.	−	1.04873e6i	0.426440	−	0.738616i
$290$		0			0
$291$	−257691.	−	446335.i	−0.178389	−	0.308978i
$292$		0			0
$293$	−619938.			−0.421871			−0.210935	−	0.977500i	$-0.567651\pi$
$293$	−619938.			−0.421871			−0.210935	+	0.977500i	$0.567651\pi$
$294$		0			0
$295$	566950.			0.379306
$296$		0			0
$297$	240048.	+	415775.i	0.157909	+	0.273506i
$298$		0			0
$299$	−221173.	+	383083.i	−0.143072	+	0.247808i
$300$		0			0
$301$	61600.4	−	82862.8i	0.0391893	−	0.0527161i
$302$		0			0
$303$	667297.	−	1.15579e6i	0.417554	−	0.723224i
$304$		0			0
$305$	253615.	+	439274.i	0.156108	+	0.270387i
$306$		0			0
$307$	14985.2			0.00907437			0.00453719	−	0.999990i	$-0.498556\pi$
$307$	14985.2			0.00907437			0.00453719	+	0.999990i	$0.498556\pi$
$308$		0			0
$309$	−1.41049e6			−0.840376
$310$		0			0
$311$	429953.	+	744700.i	0.252069	+	0.436597i	0.964095	−	0.265556i	$-0.0855556\pi$
$311$	429953.	+	744700.i	0.252069	+	0.436597i	−0.712026	+	0.702153i	$0.752222\pi$
$312$		0			0
$313$	−692471.	+	1.19940e6i	−0.399522	+	0.691993i	−0.993667	−	0.112365i	$-0.964157\pi$
$313$	−692471.	+	1.19940e6i	−0.399522	+	0.691993i	0.594145	+	0.804358i	$0.297491\pi$
$314$		0			0
$315$	167739.	+	19488.7i	0.0952483	+	0.0110664i
$316$		0			0
$317$	−6734.82	+	11665.0i	−0.00376424	+	0.00651986i	−0.867901	−	0.496737i	$-0.834532\pi$
$317$	−6734.82	+	11665.0i	−0.00376424	+	0.00651986i	0.864137	+	0.503256i	$0.167865\pi$
$318$		0			0
$319$	−2.71918e6	−	4.70977e6i	−1.49610	−	2.59133i
$320$		0			0
$321$	−453953.			−0.245894
$322$		0			0
$323$	473281.			0.252414
$324$		0			0
$325$	282249.	+	488870.i	0.148226	+	0.256735i
$326$		0			0
$327$	752586.	−	1.30352e6i	0.389212	−	0.674135i
$328$		0			0
$329$	1.13592e6	+	2.62837e6i	0.578571	+	1.33874i
$330$		0			0
$331$	−1.53155e6	+	2.65272e6i	−0.768353	+	1.33083i	0.170103	+	0.985426i	$0.445590\pi$
$331$	−1.53155e6	+	2.65272e6i	−0.768353	+	1.33083i	−0.938456	+	0.345400i	$0.887743\pi$
$332$		0			0
$333$	336438.	+	582728.i	0.166263	+	0.287975i
$334$		0			0
$335$	799680.			0.389318
$336$		0			0
$337$	3.83901e6			1.84138			0.920692	−	0.390290i	$-0.127625\pi$
$337$	3.83901e6			1.84138			0.920692	+	0.390290i	$0.127625\pi$
$338$		0			0
$339$	66688.7	+	115508.i	0.0315176	+	0.0545901i
$340$		0			0
$341$	−1.90890e6	+	3.30632e6i	−0.888993	+	1.53978i
$342$		0			0
$343$	−2.04902e6	−	740991.i	−0.940398	−	0.340077i
$344$		0			0
$345$	162541.	−	281530.i	0.0735219	−	0.127344i
$346$		0			0
$347$	1.03482e6	+	1.79236e6i	0.461362	+	0.799102i	0.999029	−	0.0440551i	$-0.0140277\pi$
$347$	1.03482e6	+	1.79236e6i	0.461362	+	0.799102i	−0.537667	+	0.843157i	$0.680694\pi$
$348$		0			0
$349$	−1.90413e6			−0.836822			−0.418411	−	0.908258i	$-0.637413\pi$
$349$	−1.90413e6			−0.836822			−0.418411	+	0.908258i	$0.637413\pi$
$350$		0			0
$351$	143567.			0.0621994
$352$		0			0
$353$	1.09918e6	+	1.90383e6i	0.469496	+	0.813190i	0.999392	−	0.0348723i	$-0.0111024\pi$
$353$	1.09918e6	+	1.90383e6i	0.469496	+	0.813190i	−0.529896	+	0.848063i	$0.677769\pi$
$354$		0			0
$355$	63550.7	−	110073.i	0.0267639	−	0.0463564i
$356$		0			0
$357$	211554.	+	489510.i	0.0878518	+	0.203278i
$358$		0			0
$359$	−650737.	+	1.12711e6i	−0.266483	+	0.461562i	−0.967951	−	0.251139i	$-0.919195\pi$
$359$	−650737.	+	1.12711e6i	−0.266483	+	0.461562i	0.701468	+	0.712701i	$0.252528\pi$
$360$		0			0
$361$	701893.	+	1.21571e6i	0.283467	+	0.490979i
$362$		0			0
$363$	2.45394e6			0.977456
$364$		0			0
$365$	−258099.			−0.101404
$366$		0			0
$367$	−21910.6	−	37950.2i	−0.00849158	−	0.0147078i	0.861748	−	0.507336i	$-0.169370\pi$
$367$	−21910.6	−	37950.2i	−0.00849158	−	0.0147078i	−0.870240	+	0.492628i	$0.836036\pi$
$368$		0			0
$369$	−590710.	+	1.02314e6i	−0.225844	+	0.391173i
$370$		0			0
$371$	−1.81970e6	−	211422.i	−0.686380	−	0.0797470i
$372$		0			0
$373$	2.35050e6	−	4.07118e6i	0.874757	−	1.51512i	0.0177351	−	0.999843i	$-0.494354\pi$
$373$	2.35050e6	−	4.07118e6i	0.874757	−	1.51512i	0.857022	−	0.515280i	$-0.172312\pi$
$374$		0			0
$375$	−433567.	−	750960.i	−0.159213	−	0.275765i
$376$		0			0
$377$	−1.62628e6			−0.589307
$378$		0			0
$379$	1.68913e6			0.604039			0.302019	−	0.953302i	$-0.402339\pi$
$379$	1.68913e6			0.604039			0.302019	+	0.953302i	$0.402339\pi$
$380$		0			0
$381$	−1.25821e6	−	2.17929e6i	−0.444060	−	0.769135i
$382$		0			0
$383$	196558.	−	340449.i	0.0684691	−	0.118592i	−0.829758	−	0.558123i	$-0.811522\pi$
$383$	196558.	−	340449.i	0.0684691	−	0.118592i	0.898228	+	0.439531i	$0.144855\pi$
$384$		0			0
$385$	819121.	−	1.10185e6i	0.281641	−	0.378855i
$386$		0			0
$387$	−32255.7	+	55868.5i	−0.0109478	+	0.0189622i
$388$		0			0
$389$	646075.	+	1.11903e6i	0.216476	+	0.374947i	0.953728	−	0.300671i	$-0.0972105\pi$
$389$	646075.	+	1.11903e6i	0.216476	+	0.374947i	−0.737252	+	0.675617i	$0.763877\pi$
$390$		0			0
$391$	1.02658e6			0.339588
$392$		0			0
$393$	1.25174e6			0.408822
$394$		0			0
$395$	−157914.	−	273516.i	−0.0509247	−	0.0882042i
$396$		0			0
$397$	1.02840e6	−	1.78124e6i	0.327481	−	0.567213i	−0.654531	−	0.756036i	$-0.727134\pi$
$397$	1.02840e6	−	1.78124e6i	0.327481	−	0.567213i	0.982011	+	0.188822i	$0.0604670\pi$
$398$		0			0
$399$	720835.	−	969644.i	0.226675	−	0.304916i
$400$		0			0
$401$	−1.64721e6	+	2.85305e6i	−0.511550	+	0.886030i	0.488361	+	0.872642i	$0.337595\pi$
$401$	−1.64721e6	+	2.85305e6i	−0.511550	+	0.886030i	−0.999910	+	0.0133882i	$0.995738\pi$
$402$		0			0
$403$	570835.	+	988715.i	0.175085	+	0.303255i
$404$		0			0
$405$	−105508.			−0.0319630
$406$		0			0
$407$	5.47080e6			1.63706
$408$		0			0
$409$	−680966.	−	1.17947e6i	−0.201288	−	0.348640i	0.747656	−	0.664086i	$-0.231179\pi$
$409$	−680966.	−	1.17947e6i	−0.201288	−	0.348640i	−0.948944	+	0.315446i	$0.897846\pi$
$410$		0			0
$411$	615165.	−	1.06550e6i	0.179633	−	0.311134i
$412$		0			0
$413$	4.54007e6	+	527488.i	1.30975	+	0.152173i
$414$		0			0
$415$	486200.	−	842123.i	0.138578	−	0.240024i
$416$		0			0
$417$	−1.55094e6	−	2.68631e6i	−0.436772	−	0.756511i
$418$		0			0
$419$	4.27003e6			1.18822			0.594109	−	0.804385i	$-0.297505\pi$
$419$	4.27003e6			1.18822			0.594109	+	0.804385i	$0.297505\pi$
$420$		0			0
$421$	5.54530e6			1.52482			0.762412	−	0.647092i	$-0.224015\pi$
$421$	5.54530e6			1.52482			0.762412	+	0.647092i	$0.224015\pi$
$422$		0			0
$423$	−894501.	−	1.54932e6i	−0.243069	−	0.421008i
$424$		0			0
$425$	655035.	−	1.13455e6i	0.175911	−	0.304686i
$426$		0			0
$427$	1.62222e6	+	3.75362e6i	0.430567	+	0.996278i
$428$		0			0
$429$	583632.	−	1.01088e6i	0.153107	−	0.265190i
$430$		0			0
$431$	−715024.	−	1.23846e6i	−0.185408	−	0.321135i	0.758306	−	0.651898i	$-0.226027\pi$
$431$	−715024.	−	1.23846e6i	−0.185408	−	0.321135i	−0.943714	+	0.330763i	$0.892694\pi$
$432$		0			0
$433$	−576083.			−0.147661			−0.0738304	−	0.997271i	$-0.523522\pi$
$433$	−576083.			−0.147661			−0.0738304	+	0.997271i	$0.523522\pi$
$434$		0			0
$435$	1.19516e6			0.302833
$436$		0			0
$437$	−1.16297e6	−	2.01432e6i	−0.291316	−	0.504573i
$438$		0			0
$439$	1.13780e6	−	1.97072e6i	0.281776	−	0.488050i	−0.690047	−	0.723765i	$-0.742410\pi$
$439$	1.13780e6	−	1.97072e6i	0.281776	−	0.488050i	0.971822	+	0.235715i	$0.0757433\pi$
$440$		0			0
$441$	1.32510e6	+	312127.i	0.324454	+	0.0764249i
$442$		0			0
$443$	−2.89207e6	+	5.00922e6i	−0.700164	+	1.21272i	0.268245	+	0.963351i	$0.413556\pi$
$443$	−2.89207e6	+	5.00922e6i	−0.700164	+	1.21272i	−0.968409	+	0.249369i	$0.919777\pi$
$444$		0			0
$445$	−307657.	−	532878.i	−0.0736490	−	0.127564i
$446$		0			0
$447$	−674702.			−0.159714
$448$		0			0
$449$	−2.02731e6			−0.474573			−0.237287	−	0.971440i	$-0.576258\pi$
$449$	−2.02731e6			−0.474573			−0.237287	+	0.971440i	$0.576258\pi$
$450$		0			0
$451$	4.80274e6	+	8.31859e6i	1.11185	+	1.92579i
$452$		0			0
$453$	−2.42582e6	+	4.20165e6i	−0.555410	+	0.961998i
$454$		0			0
$455$	−162878.	−	376879.i	−0.0368837	−	0.0853442i
$456$		0			0
$457$	1.82644e6	−	3.16349e6i	0.409086	−	0.708558i	−0.585701	−	0.810527i	$-0.699181\pi$
$457$	1.82644e6	−	3.16349e6i	0.409086	−	0.708558i	0.994788	+	0.101969i	$0.0325142\pi$
$458$		0			0
$459$	−166593.	−	288547.i	−0.0369083	−	0.0639270i
$460$		0			0
$461$	466198.			0.102169			0.0510844	−	0.998694i	$-0.483732\pi$
$461$	466198.			0.102169			0.0510844	+	0.998694i	$0.483732\pi$
$462$		0			0
$463$	1.42394e6			0.308702			0.154351	−	0.988016i	$-0.450671\pi$
$463$	1.42394e6			0.308702			0.154351	+	0.988016i	$0.450671\pi$
$464$		0			0
$465$	−419510.	−	726612.i	−0.0899724	−	0.155837i
$466$		0			0
$467$	−4.24826e6	+	7.35820e6i	−0.901403	+	1.56128i	−0.0757283	+	0.997128i	$0.524128\pi$
$467$	−4.24826e6	+	7.35820e6i	−0.901403	+	1.56128i	−0.825674	+	0.564147i	$0.809205\pi$
$468$		0			0
$469$	6.40375e6	+	744019.i	1.34432	+	0.156189i
$470$		0			0
$471$	−1.61903e6	+	2.80424e6i	−0.336281	+	0.582456i
$472$		0			0
$473$	262253.	+	454236.i	0.0538975	+	0.0933532i
$474$		0			0
$475$	−2.96823e6			−0.603620
$476$		0			0
$477$	1.14459e6			0.230333
$478$		0			0
$479$	3.13759e6	+	5.43446e6i	0.624823	+	1.08222i	0.988575	+	0.150729i	$0.0481620\pi$
$479$	3.13759e6	+	5.43446e6i	0.624823	+	1.08222i	−0.363753	+	0.931496i	$0.618505\pi$
$480$		0			0
$481$	817988.	−	1.41680e6i	0.161207	−	0.279219i
$482$		0			0
$483$	1.56355e6	−	2.10323e6i	0.304961	−	0.410223i
$484$		0			0
$485$	460440.	−	797505.i	0.0888829	−	0.153950i
$486$		0			0
$487$	−3.16086e6	−	5.47477e6i	−0.603925	−	1.04603i	−0.992220	−	0.124494i	$-0.960269\pi$
$487$	−3.16086e6	−	5.47477e6i	−0.603925	−	1.04603i	0.388295	−	0.921535i	$-0.373064\pi$
$488$		0			0
$489$	1.09281e6			0.206667
$490$		0			0
$491$	3.37661e6			0.632088			0.316044	−	0.948745i	$-0.397645\pi$
$491$	3.37661e6			0.632088			0.316044	+	0.948745i	$0.397645\pi$
$492$		0			0
$493$	1.88711e6	+	3.26857e6i	0.349687	+	0.605675i
$494$		0			0
$495$	−428914.	+	742901.i	−0.0786788	+	0.136276i
$496$		0			0
$497$	611318.	−	822325.i	0.111014	−	0.149332i
$498$		0			0
$499$	−2.44738e6	+	4.23899e6i	−0.439998	+	0.762099i	−0.997689	−	0.0679502i	$-0.978354\pi$
$499$	−2.44738e6	+	4.23899e6i	−0.439998	+	0.762099i	0.557691	+	0.830049i	$0.311687\pi$
$500$		0			0
$501$	1.37619e6	+	2.38364e6i	0.244954	+	0.424273i
$502$		0			0
$503$	154938.			0.0273048			0.0136524	−	0.999907i	$-0.495654\pi$
$503$	154938.			0.0273048			0.0136524	+	0.999907i	$0.495654\pi$
$504$		0			0
$505$	2.38463e6			0.416096
$506$		0			0
$507$	1.49629e6	+	2.59165e6i	0.258521	+	0.447772i
$508$		0			0
$509$	177049.	−	306657.i	0.0302899	−	0.0524636i	−0.850483	−	0.526002i	$-0.823690\pi$
$509$	177049.	−	306657.i	0.0302899	−	0.0524636i	0.880773	+	0.473539i	$0.157024\pi$
$510$		0			0
$511$	−2.06683e6	−	240134.i	−0.350148	−	0.0406819i
$512$		0			0
$513$	−377449.	+	653761.i	−0.0633235	+	0.109680i
$514$		0			0
$515$	−1.26012e6	−	2.18259e6i	−0.209360	−	0.362623i
$516$		0			0
$517$	−1.45454e7			−2.39331
$518$		0			0
$519$	2.61311e6			0.425832
$520$		0			0
$521$	−307527.	−	532652.i	−0.0496350	−	0.0859704i	0.840140	−	0.542369i	$-0.182472\pi$
$521$	−307527.	−	532652.i	−0.0496350	−	0.0859704i	−0.889775	+	0.456399i	$0.849139\pi$
$522$		0			0
$523$	−3.41075e6	+	5.90760e6i	−0.545250	+	0.944401i	0.453341	+	0.891337i	$0.350232\pi$
$523$	−3.41075e6	+	5.90760e6i	−0.545250	+	0.944401i	−0.998591	+	0.0530640i	$0.983101\pi$
$524$		0			0
$525$	−1.32678e6	−	3.07001e6i	−0.210088	−	0.486118i
$526$		0			0
$527$	1.32478e6	−	2.29458e6i	0.207786	−	0.359895i
$528$		0			0
$529$	695605.	+	1.20482e6i	0.108074	+	0.187191i
$530$		0			0
$531$	−2.85571e6			−0.439519
$532$		0			0
$533$	2.87240e6			0.437953
$534$		0			0
$535$	−405559.	−	702449.i	−0.0612590	−	0.106104i
$536$		0			0
$537$	827291.	−	1.43291e6i	0.123801	−	0.214429i
$538$		0			0
$539$	7.58459e6	−	8.06142e6i	1.12450	−	1.19520i
$540$		0			0
$541$	−4.92707e6	+	8.53394e6i	−0.723762	+	1.25359i	0.235719	+	0.971821i	$0.424255\pi$
$541$	−4.92707e6	+	8.53394e6i	−0.723762	+	1.25359i	−0.959481	+	0.281772i	$0.909078\pi$
$542$		0			0
$543$	1.38084e6	+	2.39168e6i	0.200976	+	0.348100i
$544$		0			0
$545$	2.68942e6			0.387853
$546$		0			0
$547$	3.96635e6			0.566790			0.283395	−	0.959003i	$-0.408539\pi$
$547$	3.96635e6			0.566790			0.283395	+	0.959003i	$0.408539\pi$
$548$		0			0
$549$	−1.27745e6	−	2.21261e6i	−0.180890	−	0.313310i
$550$		0			0
$551$	4.27562e6	−	7.40560e6i	0.599957	−	1.03916i
$552$		0			0
$553$	−1.01008e6	−	2.33721e6i	−0.140457	−	0.325001i
$554$		0			0
$555$	−601144.	+	1.04121e6i	−0.0828411	+	0.143485i
$556$		0			0
$557$	−389384.	−	674433.i	−0.0531790	−	0.0921087i	0.838210	−	0.545347i	$-0.183602\pi$
$557$	−389384.	−	674433.i	−0.0531790	−	0.0921087i	−0.891389	+	0.453238i	$0.850269\pi$
$558$		0			0
$559$	156847.			0.0212299
$560$		0			0
$561$	−2.70895e6			−0.363407
$562$		0			0
$563$	3.11182e6	+	5.38983e6i	0.413755	+	0.716645i	0.995297	−	0.0968716i	$-0.0308836\pi$
$563$	3.11182e6	+	5.38983e6i	0.413755	+	0.716645i	−0.581542	+	0.813517i	$0.697550\pi$
$564$		0			0
$565$	−119159.	+	206389.i	−0.0157038	+	0.0271997i
$566$		0			0
$567$	−844896.	−	98164.2i	−0.110369	−	0.0128232i
$568$		0			0
$569$	4.06684e6	−	7.04397e6i	0.526594	−	0.912088i	−0.472926	−	0.881102i	$-0.656802\pi$
$569$	4.06684e6	−	7.04397e6i	0.526594	−	0.912088i	0.999520	−	0.0309854i	$-0.00986454\pi$
$570$		0			0
$571$	−3.97068e6	−	6.87742e6i	−0.509653	−	0.882746i	−0.999937	−	0.0111829i	$-0.996440\pi$
$571$	−3.97068e6	−	6.87742e6i	−0.509653	−	0.882746i	0.490284	−	0.871563i	$-0.336893\pi$
$572$		0			0
$573$	3.56699e6			0.453853
$574$		0			0
$575$	−6.43833e6			−0.812088
$576$		0			0
$577$	−1.48710e6	−	2.57573e6i	−0.185952	−	0.322078i	0.757945	−	0.652318i	$-0.226203\pi$
$577$	−1.48710e6	−	2.57573e6i	−0.185952	−	0.322078i	−0.943897	+	0.330241i	$0.892870\pi$
$578$		0			0
$579$	−1.79558e6	+	3.11003e6i	−0.222591	+	0.385539i
$580$		0			0
$581$	4.67694e6	−	6.29127e6i	0.574807	−	0.773211i
$582$		0			0
$583$	4.65304e6	−	8.05929e6i	0.566976	−	0.982032i
$584$		0			0
$585$	128262.	+	222156.i	0.0154956	+	0.0268391i
$586$		0			0
$587$	−1.57136e7			−1.88226			−0.941130	−	0.338044i	$-0.890235\pi$
$587$	−1.57136e7			−1.88226			−0.941130	+	0.338044i	$0.890235\pi$
$588$		0			0
$589$	−6.00309e6			−0.712995
$590$		0			0
$591$	−2.05614e6	−	3.56134e6i	−0.242150	−	0.419416i
$592$		0			0
$593$	−6.99245e6	+	1.21113e7i	−0.816569	+	1.41434i	0.0916269	+	0.995793i	$0.470793\pi$
$593$	−6.99245e6	+	1.21113e7i	−0.816569	+	1.41434i	−0.908196	+	0.418545i	$0.862540\pi$
$594$		0			0
$595$	−568468.	+	764684.i	−0.0658285	+	0.0885503i
$596$		0			0
$597$	−2.72776e6	+	4.72462e6i	−0.313235	+	0.542539i
$598$		0			0
$599$	−3.28236e6	−	5.68522e6i	−0.373783	−	0.647411i	0.616361	−	0.787463i	$-0.288606\pi$
$599$	−3.28236e6	−	5.68522e6i	−0.373783	−	0.647411i	−0.990144	+	0.140053i	$0.955273\pi$
$600$		0			0
$601$	−7.39552e6			−0.835185			−0.417592	−	0.908634i	$-0.637126\pi$
$601$	−7.39552e6			−0.835185			−0.417592	+	0.908634i	$0.637126\pi$
$602$		0			0
$603$	−4.02797e6			−0.451120
$604$		0			0
$605$	2.19233e6	+	3.79723e6i	0.243511	+	0.421773i
$606$		0			0
$607$	1.53084e6	−	2.65149e6i	0.168639	−	0.292091i	−0.769303	−	0.638885i	$-0.779396\pi$
$607$	1.53084e6	−	2.65149e6i	0.168639	−	0.292091i	0.937942	+	0.346793i	$0.112729\pi$
$608$		0			0
$609$	9.57072e6	+	1.11197e6i	1.04569	+	0.121493i
$610$		0			0
$611$	−2.17481e6	+	3.76689e6i	−0.235678	+	0.408206i
$612$		0			0
$613$	−549054.	−	950989.i	−0.0590152	−	0.102217i	0.835008	−	0.550237i	$-0.185463\pi$
$613$	−549054.	−	950989.i	−0.0590152	−	0.102217i	−0.894024	+	0.448020i	$0.852129\pi$
$614$		0			0
$615$	−2.11094e6			−0.225055
$616$		0			0
$617$	−5.20105e6			−0.550019			−0.275010	−	0.961441i	$-0.588681\pi$
$617$	−5.20105e6			−0.550019			−0.275010	+	0.961441i	$0.588681\pi$
$618$		0			0
$619$	−3.62680e6	−	6.28180e6i	−0.380449	−	0.658957i	0.610677	−	0.791880i	$-0.290897\pi$
$619$	−3.62680e6	−	6.28180e6i	−0.380449	−	0.658957i	−0.991126	+	0.132922i	$0.957564\pi$
$620$		0			0
$621$	−818717.	+	1.41806e6i	−0.0851932	+	0.147559i
$622$		0			0
$623$	−1.96790e6	−	4.55347e6i	−0.203134	−	0.470027i
$624$		0			0
$625$	−3.70406e6	+	6.41561e6i	−0.379295	+	0.656959i
$626$		0			0
$627$	3.06884e6	+	5.31538e6i	0.311749	+	0.539965i
$628$		0			0
$629$	−3.79672e6			−0.382633
$630$		0			0
$631$	2.90713e6			0.290664			0.145332	−	0.989383i	$-0.453575\pi$
$631$	2.90713e6			0.290664			0.145332	+	0.989383i	$0.453575\pi$
$632$		0			0
$633$	3.31454e6	+	5.74095e6i	0.328786	+	0.569475i
$634$		0			0
$635$	2.24816e6	−	3.89393e6i	0.221255	−	0.383225i
$636$		0			0
$637$	−953662.	−	3.16955e6i	−0.0931206	−	0.309492i
$638$		0			0
$639$	−320103.	+	554435.i	−0.0310126	+	0.0537153i
$640$		0			0
$641$	−8.33394e6	−	1.44348e7i	−0.801135	−	1.38761i	−0.918870	−	0.394561i	$-0.870897\pi$
$641$	−8.33394e6	−	1.44348e7i	−0.801135	−	1.38761i	0.117735	−	0.993045i	$-0.462437\pi$
$642$		0			0
$643$	1.48288e7			1.41442			0.707209	−	0.707005i	$-0.249954\pi$
$643$	1.48288e7			1.41442			0.707209	+	0.707005i	$0.249954\pi$
$644$		0			0
$645$	−115268.			−0.0109096
$646$		0			0
$647$	−4.73788e6	−	8.20624e6i	−0.444962	−	0.770697i	0.553087	−	0.833123i	$-0.313450\pi$
$647$	−4.73788e6	−	8.20624e6i	−0.444962	−	0.770697i	−0.998050	+	0.0624262i	$0.980116\pi$
$648$		0			0
$649$	−1.16091e7	+	2.01076e7i	−1.08190	+	1.87391i
$650$		0			0
$651$	−2.68335e6	−	6.20894e6i	−0.248156	−	0.574202i
$652$		0			0
$653$	4.45856e6	−	7.72244e6i	0.409177	−	0.708715i	−0.585621	−	0.810585i	$-0.699149\pi$
$653$	4.45856e6	−	7.72244e6i	0.409177	−	0.708715i	0.994798	+	0.101870i	$0.0324825\pi$
$654$		0			0
$655$	1.11830e6	+	1.93695e6i	0.101849	+	0.176407i
$656$		0			0
$657$	1.30004e6			0.117501
$658$		0			0
$659$	−1.79221e6			−0.160759			−0.0803795	−	0.996764i	$-0.525613\pi$
$659$	−1.79221e6			−0.160759			−0.0803795	+	0.996764i	$0.525613\pi$
$660$		0			0
$661$	9.73580e6	+	1.68629e7i	0.866699	+	1.50117i	0.865350	+	0.501167i	$0.167096\pi$
$661$	9.73580e6	+	1.68629e7i	0.866699	+	1.50117i	0.00134840	+	0.999999i	$0.499571\pi$
$662$		0			0
$663$	−405039.	+	701548.i	−0.0357860	+	0.0619832i
$664$		0			0
$665$	2.14442e6	+	249149.i	0.188042	+	0.0218477i
$666$		0			0
$667$	9.27417e6	−	1.60633e7i	0.807161	−	1.39804i
$668$		0			0
$669$	−4.72810e6	−	8.18932e6i	−0.408434	−	0.707428i
$670$		0			0
$671$	−2.07725e7			−1.78108
$672$		0			0
$673$	1.01698e7			0.865512			0.432756	−	0.901511i	$-0.357541\pi$
$673$	1.01698e7			0.865512			0.432756	+	0.901511i	$0.357541\pi$
$674$		0			0
$675$	1.04480e6	+	1.80965e6i	0.0882622	+	0.152875i
$676$		0			0
$677$	1.54240e6	−	2.67151e6i	0.129337	−	0.224019i	−0.794083	−	0.607810i	$-0.792048\pi$
$677$	1.54240e6	−	2.67151e6i	0.129337	−	0.224019i	0.923420	+	0.383791i	$0.125382\pi$
$678$		0			0
$679$	4.42915e6	−	5.95794e6i	0.368677	−	0.495932i
$680$		0			0
$681$	4.97972e6	−	8.62513e6i	0.411469	−	0.712685i
$682$		0			0
$683$	−9.43644e6	−	1.63444e7i	−0.774027	−	1.34065i	−0.935340	−	0.353751i	$-0.884906\pi$
$683$	−9.43644e6	−	1.63444e7i	−0.774027	−	1.34065i	0.161312	−	0.986903i	$-0.448427\pi$
$684$		0			0
$685$	2.19834e6			0.179006
$686$		0			0
$687$	−746490.			−0.0603438
$688$		0			0
$689$	−1.39143e6	−	2.41003e6i	−0.111664	−	0.193408i
$690$		0			0
$691$	−9.66264e6	+	1.67362e7i	−0.769841	+	1.33340i	0.167808	+	0.985820i	$0.446331\pi$
$691$	−9.66264e6	+	1.67362e7i	−0.769841	+	1.33340i	−0.937649	+	0.347583i	$0.887002\pi$
$692$		0			0
$693$	−4.12589e6	+	5.55001e6i	−0.326351	+	0.438996i
$694$		0			0
$695$	2.77120e6	−	4.79986e6i	0.217623	−	0.376935i
$696$		0			0
$697$	−3.33309e6	−	5.77308e6i	−0.259875	−	0.450118i
$698$		0			0
$699$	−1.05256e7			−0.814804
$700$		0			0
$701$	2.02964e7			1.56000			0.779999	−	0.625780i	$-0.215219\pi$
$701$	2.02964e7			1.56000			0.779999	+	0.625780i	$0.215219\pi$
$702$		0			0
$703$	4.30112e6	+	7.44975e6i	0.328241	+	0.568530i
$704$		0			0
$705$	1.59828e6	−	2.76831e6i	0.121110	−	0.209769i
$706$		0			0
$707$	1.90959e7	+	2.21865e6i	1.43678	+	0.166933i
$708$		0			0
$709$	−8.14916e6	+	1.41148e7i	−0.608831	+	1.05453i	0.382602	+	0.923913i	$0.375028\pi$
$709$	−8.14916e6	+	1.41148e7i	−0.608831	+	1.05453i	−0.991433	+	0.130614i	$0.958305\pi$
$710$		0			0
$711$	795410.	+	1.37769e6i	0.0590089	+	0.102206i
$712$		0			0
$713$	−1.30212e7			−0.959238
$714$		0			0
$715$	2.08565e6			0.152573
$716$		0			0
$717$	3.53842e6	+	6.12872e6i	0.257046	+	0.445217i
$718$		0			0
$719$	−510536.	+	884274.i	−0.0368302	+	0.0637918i	−0.883853	−	0.467765i	$-0.845059\pi$
$719$	−510536.	+	884274.i	−0.0368302	+	0.0637918i	0.847023	+	0.531557i	$0.178393\pi$
$720$		0			0
$721$	−8.06024e6	−	1.86504e7i	−0.577444	−	1.33613i
$722$		0			0
$723$	−5.51252e6	+	9.54796e6i	−0.392197	+	0.679305i
$724$		0			0
$725$	−1.18352e7	−	2.04991e7i	−0.836238	−	1.44841i
$726$		0			0
$727$	−4.36229e6			−0.306111			−0.153055	−	0.988218i	$-0.548911\pi$
$727$	−4.36229e6			−0.306111			−0.153055	+	0.988218i	$0.548911\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−182003.	−	315239.i	−0.0125975	−	0.0218196i
$732$		0			0
$733$	8.26829e6	−	1.43211e7i	0.568402	−	0.984502i	−0.428322	−	0.903626i	$-0.640895\pi$
$733$	8.26829e6	−	1.43211e7i	0.568402	−	0.984502i	0.996724	−	0.0808755i	$-0.0257716\pi$
$734$		0			0
$735$	700852.	+	2.32932e6i	0.0478528	+	0.159042i
$736$		0			0
$737$	−1.63746e7	+	2.83617e7i	−1.11046	+	1.92337i
$738$		0			0
$739$	−1.23971e7	−	2.14725e7i	−0.835045	−	1.44634i	−0.893993	−	0.448080i	$-0.852108\pi$
$739$	−1.23971e7	−	2.14725e7i	−0.835045	−	1.44634i	0.0589480	−	0.998261i	$-0.481225\pi$
$740$		0			0
$741$	1.83540e6			0.122796
$742$		0			0
$743$	−3.16423e6			−0.210279			−0.105140	−	0.994457i	$-0.533529\pi$
$743$	−3.16423e6			−0.210279			−0.105140	+	0.994457i	$0.533529\pi$
$744$		0			0
$745$	−602774.	−	1.04404e6i	−0.0397891	−	0.0689167i
$746$		0			0
$747$	−2.44898e6	+	4.24175e6i	−0.160577	+	0.278127i
$748$		0			0
$749$	−2.59412e6	−	6.00247e6i	−0.168960	−	0.390954i
$750$		0			0
$751$	−1.12862e7	+	1.95482e7i	−0.730207	+	1.26476i	0.226587	+	0.973991i	$0.427243\pi$
$751$	−1.12862e7	+	1.95482e7i	−0.730207	+	1.26476i	−0.956794	+	0.290765i	$0.906090\pi$
$752$		0			0
$753$	−4.17495e6	−	7.23122e6i	−0.268326	−	0.464755i
$754$		0			0
$755$	−8.66886e6			−0.553470
$756$		0			0
$757$	−1.36178e7			−0.863711			−0.431855	−	0.901943i	$-0.642141\pi$
$757$	−1.36178e7			−0.863711			−0.431855	+	0.901943i	$0.642141\pi$
$758$		0			0
$759$	6.65655e6	+	1.15295e7i	0.419416	+	0.726449i
$760$		0			0
$761$	1.54920e7	−	2.68329e7i	0.969716	−	1.67960i	0.273345	−	0.961916i	$-0.411870\pi$
$761$	1.54920e7	−	2.68329e7i	0.969716	−	1.67960i	0.696372	−	0.717681i	$-0.254797\pi$
$762$		0			0
$763$	2.15366e7	+	2.50223e6i	1.33926	+	0.155602i
$764$		0			0
$765$	297666.	−	515572.i	0.0183897	−	0.0318519i
$766$		0			0
$767$	3.47157e6	+	6.01293e6i	0.213077	+	0.369061i
$768$		0			0
$769$	−1.12439e7			−0.685649			−0.342825	−	0.939399i	$-0.611384\pi$
$769$	−1.12439e7			−0.685649			−0.342825	+	0.939399i	$0.611384\pi$
$770$		0			0
$771$	1.30406e7			0.790065
$772$		0			0
$773$	−8.91365e6	−	1.54389e7i	−0.536546	−	0.929325i	−0.999087	−	0.0427267i	$-0.986396\pi$
$773$	−8.91365e6	−	1.54389e7i	−0.536546	−	0.929325i	0.462541	−	0.886598i	$-0.346938\pi$
$774$		0			0
$775$	−8.30846e6	+	1.43907e7i	−0.496897	+	0.860651i
$776$		0			0
$777$	−5.78263e6	+	7.77860e6i	−0.343616	+	0.462220i
$778$		0			0
$779$	−7.55179e6	+	1.30801e7i	−0.445868	+	0.772266i
$780$		0			0
$781$	2.60259e6	+	4.50781e6i	0.152678	+	0.264447i
$782$		0			0
$783$	−6.01999e6			−0.350907
$784$		0			0
$785$	−5.78572e6			−0.335107
$786$		0			0
$787$	−1.51845e7	−	2.63003e7i	−0.873903	−	1.51364i	−0.857927	−	0.513771i	$-0.828248\pi$
$787$	−1.51845e7	−	2.63003e7i	−0.873903	−	1.51364i	−0.0159756	−	0.999872i	$-0.505085\pi$
$788$		0			0
$789$	−8.37192e6	+	1.45006e7i	−0.478776	+	0.829265i
$790$		0			0
$791$	−1.14623e6	+	1.54187e6i	−0.0651375	+	0.0876208i
$792$		0			0
$793$	−3.10589e6	+	5.37956e6i	−0.175389	+	0.303783i
$794$		0			0
$795$	1.02257e6	+	1.77115e6i	0.0573821	+	0.0993887i
$796$		0			0
$797$	1.71661e7			0.957250			0.478625	−	0.878019i	$-0.341135\pi$
$797$	1.71661e7			0.957250			0.478625	+	0.878019i	$0.341135\pi$
$798$		0			0
$799$	1.00945e7			0.559393
$800$		0			0
$801$	1.54966e6	+	2.68409e6i	0.0853406	+	0.147814i
$802$		0			0
$803$	5.28495e6	−	9.15380e6i	0.289236	−	0.500971i
$804$		0			0
$805$	4.65142e6	+	540424.i	0.252985	+	0.0293931i
$806$		0			0
$807$	2.14059e6	−	3.70761e6i	0.115704	−	0.200406i
$808$		0			0
$809$	−1.29741e7	−	2.24717e7i	−0.696954	−	1.20716i	−0.969517	−	0.245022i	$-0.921205\pi$
$809$	−1.29741e7	−	2.24717e7i	−0.696954	−	1.20716i	0.272563	−	0.962138i	$-0.412129\pi$
$810$		0			0
$811$	4.70517e6			0.251202			0.125601	−	0.992081i	$-0.459914\pi$
$811$	4.70517e6			0.251202			0.125601	+	0.992081i	$0.459914\pi$
$812$		0			0
$813$	1.51913e7			0.806060
$814$		0			0
$815$	976305.	+	1.69101e6i	0.0514863	+	0.0891768i
$816$		0			0
$817$	−412365.	+	714237.i	−0.0216136	+	0.0374358i
$818$		0			0
$819$	820412.	+	1.89833e6i	0.0427388	+	0.0988923i
$820$		0			0
$821$	1.36373e7	−	2.36205e7i	0.706108	−	1.22302i	−0.260182	−	0.965560i	$-0.583783\pi$
$821$	1.36373e7	−	2.36205e7i	0.706108	−	1.22302i	0.966290	−	0.257456i	$-0.0828841\pi$
$822$		0			0
$823$	7.56779e6	+	1.31078e7i	0.389466	+	0.674574i	0.992378	−	0.123233i	$-0.0393263\pi$
$823$	7.56779e6	+	1.31078e7i	0.389466	+	0.674574i	−0.602912	+	0.797808i	$0.705993\pi$
$824$		0			0
$825$	1.69894e7			0.869049
$826$		0			0
$827$	2.00520e7			1.01952			0.509759	−	0.860317i	$-0.329735\pi$
$827$	2.00520e7			1.01952			0.509759	+	0.860317i	$0.329735\pi$
$828$		0			0
$829$	−9.65650e6	−	1.67256e7i	−0.488015	−	0.845268i	0.511890	−	0.859051i	$-0.328946\pi$
$829$	−9.65650e6	−	1.67256e7i	−0.488015	−	0.845268i	−0.999905	+	0.0137837i	$0.995612\pi$
$830$		0			0
$831$	−1.90514e6	+	3.29979e6i	−0.0957025	+	0.165762i
$832$		0			0
$833$	−5.26369e6	+	5.59461e6i	−0.262832	+	0.279356i
$834$		0			0
$835$	−2.45896e6	+	4.25905e6i	−0.122049	+	0.211396i
$836$		0			0
$837$	2.11306e6	+	3.65993e6i	0.104255	+	0.180575i
$838$		0			0
$839$	−9.06208e6			−0.444450			−0.222225	−	0.974995i	$-0.571332\pi$
$839$	−9.06208e6			−0.444450			−0.222225	+	0.974995i	$0.571332\pi$
$840$		0			0
$841$	4.76814e7			2.32466
$842$		0			0
$843$	2.41406e6	+	4.18127e6i	0.116998	+	0.202647i
$844$		0			0
$845$	−2.67355e6	+	4.63073e6i	−0.128809	+	0.223104i
$846$		0			0
$847$	1.40230e7	+	3.24476e7i	0.671635	+	1.55408i
$848$		0			0
$849$	−3.83741e6	+	6.64658e6i	−0.182713	+	0.316467i
$850$		0			0
$851$	9.32947e6	+	1.61591e7i	0.441604	+	0.764880i
$852$		0			0
$853$	−3.60691e7			−1.69732			−0.848658	−	0.528942i	$-0.822589\pi$
$853$	−3.60691e7			−1.69732			−0.848658	+	0.528942i	$0.822589\pi$
$854$		0			0
$855$	−1.34884e6			−0.0631024
$856$		0			0
$857$	1.49712e7	+	2.59308e7i	0.696311	+	1.20605i	0.969737	+	0.244153i	$0.0785098\pi$
$857$	1.49712e7	+	2.59308e7i	0.696311	+	1.20605i	−0.273426	+	0.961893i	$0.588157\pi$
$858$		0			0
$859$	−1.63513e7	+	2.83214e7i	−0.756085	+	1.30958i	0.188748	+	0.982026i	$0.439557\pi$
$859$	−1.63513e7	+	2.83214e7i	−0.756085	+	1.30958i	−0.944833	+	0.327552i	$0.893776\pi$
$860$		0			0
$861$	−1.69042e7	−	1.96401e6i	−0.777118	−	0.0902894i
$862$		0			0
$863$	1.72889e7	−	2.99452e7i	0.790204	−	1.36867i	−0.135636	−	0.990759i	$-0.543308\pi$
$863$	1.72889e7	−	2.99452e7i	0.790204	−	1.36867i	0.925840	−	0.377915i	$-0.123359\pi$
$864$		0			0
$865$	2.33453e6	+	4.04353e6i	0.106086	+	0.183747i
$866$		0			0
$867$	−1.08987e7			−0.492410
$868$		0			0
$869$	1.29341e7			0.581014
$870$		0			0
$871$	4.89663e6	+	8.48121e6i	0.218701	+	0.378802i
$872$		0			0
$873$	−2.31922e6	+	4.01701e6i	−0.102993	+	0.178389i
$874$		0			0
$875$	7.45206e6	−	1.00243e7i	0.329046	−	0.442621i
$876$		0			0
$877$	−1.92175e7	+	3.32857e7i	−0.843718	+	1.46136i	0.0430114	+	0.999075i	$0.486305\pi$
$877$	−1.92175e7	+	3.32857e7i	−0.843718	+	1.46136i	−0.886730	+	0.462288i	$0.847029\pi$
$878$		0			0
$879$	2.78972e6	+	4.83194e6i	0.121784	+	0.210935i
$880$		0			0
$881$	3.75738e7			1.63097			0.815484	−	0.578780i	$-0.196471\pi$
$881$	3.75738e7			1.63097			0.815484	+	0.578780i	$0.196471\pi$
$882$		0			0
$883$	−2.19072e7			−0.945550			−0.472775	−	0.881183i	$-0.656748\pi$
$883$	−2.19072e7			−0.945550			−0.472775	+	0.881183i	$0.656748\pi$
$884$		0			0
$885$	−2.55127e6	−	4.41894e6i	−0.109496	−	0.189653i
$886$		0			0
$887$	−1.63761e7	+	2.83642e7i	−0.698878	+	1.21049i	0.269978	+	0.962867i	$0.412984\pi$
$887$	−1.63761e7	+	2.83642e7i	−0.698878	+	1.21049i	−0.968856	+	0.247626i	$0.920350\pi$
$888$		0			0
$889$	2.16259e7	−	2.90905e7i	0.917741	−	1.23452i
$890$		0			0
$891$	2.16043e6	−	3.74197e6i	0.0911687	−	0.157909i
$892$		0			0
$893$	−1.14355e7	−	1.98069e7i	−0.479875	−	0.831168i
$894$		0			0
$895$	2.95639e6			0.123368
$896$		0			0
$897$	3.98112e6			0.165205
$898$		0			0
$899$	−2.39361e7	−	4.14585e7i	−0.987764	−	1.71086i
$900$		0			0
$901$	−3.22920e6	+	5.59313e6i	−0.132520	+	0.229532i
$902$		0			0
$903$	−923054.	−	107245.i	−0.0376710	−	0.00437681i
$904$		0			0
$905$	−2.46727e6	+	4.27343e6i	−0.100137	+	0.173442i
$906$		0			0
$907$	2.69804e6	+	4.67314e6i	0.108901	+	0.188621i	0.915325	−	0.402716i	$-0.131934\pi$
$907$	2.69804e6	+	4.67314e6i	0.108901	+	0.188621i	−0.806425	+	0.591337i	$0.798600\pi$
$908$		0			0
$909$	−1.20113e7			−0.482149
$910$		0			0
$911$	1.97503e7			0.788455			0.394228	−	0.919013i	$-0.371012\pi$
$911$	1.97503e7			0.788455			0.394228	+	0.919013i	$0.371012\pi$
$912$		0			0
$913$	1.99113e7	+	3.44874e7i	0.790538	+	1.36925i
$914$		0			0
$915$	2.28253e6	−	3.95347e6i	0.0901290	−	0.156108i
$916$		0			0
$917$	7.15309e6	+	1.65514e7i	0.280912	+	0.649995i
$918$		0			0
$919$	1.36210e7	−	2.35923e7i	0.532010	−	0.921469i	−0.467291	−	0.884103i	$-0.654770\pi$
$919$	1.36210e7	−	2.35923e7i	0.532010	−	0.921469i	0.999302	−	0.0373654i	$-0.0118966\pi$
$920$		0			0
$921$	−67433.4	−	116798.i	−0.00261955	−	0.00453719i
$922$		0			0
$923$	1.55654e6			0.0601391
$924$		0			0
$925$	2.38115e7			0.915024
$926$		0			0
$927$	6.34720e6	+	1.09937e7i	0.242596	+	0.420188i
$928$		0			0
$929$	−1.20199e7	+	2.08190e7i	−0.456941	+	0.791445i	−0.998798	−	0.0490257i	$-0.984388\pi$
$929$	−1.20199e7	+	2.08190e7i	−0.456941	+	0.791445i	0.541856	+	0.840471i	$0.317722\pi$
$930$		0			0
$931$	1.69405e7	+	3.99032e6i	0.640547	+	0.150881i
$932$		0			0
$933$	3.86957e6	−	6.70230e6i	0.145532	−	0.252069i
$934$		0			0
$935$	−2.42016e6	−	4.19184e6i	−0.0905346	−	0.156811i
$936$		0			0
$937$	4.57105e7			1.70086			0.850428	−	0.526092i	$-0.176343\pi$
$937$	4.57105e7			1.70086			0.850428	+	0.526092i	$0.176343\pi$
$938$		0			0
$939$	1.24645e7			0.461328
$940$		0			0
$941$	1.17091e7	+	2.02807e7i	0.431071	+	0.746637i	0.996966	−	0.0778403i	$-0.0248024\pi$
$941$	1.17091e7	+	2.02807e7i	0.431071	+	0.746637i	−0.565895	+	0.824478i	$0.691469\pi$
$942$		0			0
$943$	−1.63804e7	+	2.83717e7i	−0.599855	+	1.03898i
$944$		0			0
$945$	−602925.	−	1.39509e6i	−0.0219626	−	0.0508188i
$946$		0			0
$947$	2.37124e7	−	4.10710e7i	0.859212	−	1.48820i	−0.0134704	−	0.999909i	$-0.504288\pi$
$947$	2.37124e7	−	4.10710e7i	0.859212	−	1.48820i	0.872682	−	0.488289i	$-0.162379\pi$
$948$		0			0
$949$	−1.58040e6	−	2.73733e6i	−0.0569641	−	0.0986648i
$950$		0			0
$951$	121227.			0.00434657
$952$		0			0
$953$	−3.21894e7			−1.14810			−0.574051	−	0.818820i	$-0.694629\pi$
$953$	−3.21894e7			−1.14810			−0.574051	+	0.818820i	$0.694629\pi$
$954$		0			0
$955$	3.18672e6	+	5.51957e6i	0.113067	+	0.195838i
$956$		0			0
$957$	−2.44727e7	+	4.23879e7i	−0.863776	+	1.49610i
$958$		0			0
$959$	1.76040e7	+	2.04532e6i	0.618110	+	0.0718151i
$960$		0			0
$961$	−2.48886e6	+	4.31083e6i	−0.0869345	+	0.150575i
$962$		0			0
$963$	2.04279e6	+	3.53822e6i	0.0709836	+	0.122947i
$964$		0			0
$965$	−6.41663e6			−0.221814
$966$		0			0
$967$	−8.92543e6			−0.306947			−0.153473	−	0.988153i	$-0.549046\pi$
$967$	−8.92543e6			−0.306947			−0.153473	+	0.988153i	$0.549046\pi$
$968$		0			0
$969$	−2.12976e6	−	3.68886e6i	−0.0728655	−	0.126207i
$970$		0			0
$971$	1.82260e7	−	3.15683e7i	0.620358	−	1.07449i	−0.369061	−	0.929405i	$-0.620321\pi$
$971$	1.82260e7	−	3.15683e7i	0.620358	−	1.07449i	0.989419	−	0.145087i	$-0.0463462\pi$
$972$		0			0
$973$	2.66572e7	−	3.58584e7i	0.902678	−	1.21425i
$974$		0			0
$975$	2.54024e6	−	4.39983e6i	0.0855783	−	0.148226i
$976$		0			0
$977$	−2.57742e7	−	4.46423e7i	−0.863871	−	1.49627i	−0.868163	−	0.496278i	$-0.834700\pi$
$977$	−2.57742e7	−	4.46423e7i	−0.863871	−	1.49627i	0.00429197	−	0.999991i	$-0.498634\pi$
$978$		0			0
$979$	2.51989e7			0.840282
$980$		0			0
$981$	−1.35465e7			−0.449424
$982$		0			0
$983$	2.52525e7	+	4.37387e7i	0.833530	+	1.44372i	0.895222	+	0.445621i	$0.147017\pi$
$983$	2.52525e7	+	4.37387e7i	0.833530	+	1.44372i	−0.0616914	+	0.998095i	$0.519649\pi$
$984$		0			0
$985$	3.67389e6	−	6.36336e6i	0.120652	−	0.208976i
$986$		0			0
$987$	1.53745e7	−	2.06813e7i	0.502352	−	0.675747i
$988$		0			0
$989$	−894453.	+	1.54924e6i	−0.0290781	+	0.0503648i
$990$		0			0
$991$	−2.88819e7	−	5.00249e7i	−0.934203	−	1.61809i	−0.776050	−	0.630672i	$-0.782779\pi$
$991$	−2.88819e7	−	5.00249e7i	−0.934203	−	1.61809i	−0.158153	−	0.987415i	$-0.550554\pi$
$992$		0			0
$993$	2.75679e7			0.887217
$994$		0			0
$995$	−9.74785e6			−0.312141
$996$		0			0
$997$	1.52053e7	+	2.63364e7i	0.484460	+	0.839110i	0.999841	−	0.0178518i	$-0.00568269\pi$
$997$	1.52053e7	+	2.63364e7i	0.484460	+	0.839110i	−0.515380	+	0.856962i	$0.672349\pi$
$998$		0			0
$999$	3.02794e6	−	5.24455e6i	0.0959918	−	0.166263i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.q.l.289.3		10
4.3	odd	2		168.6.q.c.121.3	yes	10
7.4	even	3	inner	336.6.q.l.193.3		10
12.11	even	2		504.6.s.c.289.3		10
28.11	odd	6		168.6.q.c.25.3	✓	10
84.11	even	6		504.6.s.c.361.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.q.c.25.3	✓	10	28.11	odd	6
168.6.q.c.121.3	yes	10	4.3	odd	2
336.6.q.l.193.3		10	7.4	even	3	inner
336.6.q.l.289.3		10	1.1	even	1	trivial
504.6.s.c.289.3		10	12.11	even	2
504.6.s.c.361.3		10	84.11	even	6

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{3} +(8.04054 - 13.9266i) q^{5} +(77.3451 - 104.042i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 - 7.79423i) q^{3} +(8.04054 - 13.9266i) q^{5} +(77.3451 - 104.042i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(329.284 + 570.336i) q^{11} +196.937 q^{13} -144.730 q^{15} +(-228.522 - 395.812i) q^{17} +(-517.763 + 896.792i) q^{19} +(-1158.98 - 134.656i) q^{21} +(-1123.07 + 1945.21i) q^{23} +(1433.20 + 2482.37i) q^{25} +729.000 q^{27} -8257.88 q^{29} +(2898.57 + 5020.47i) q^{31} +(2963.55 - 5133.02i) q^{33} +(-827.058 - 1913.71i) q^{35} +(4153.56 - 7194.18i) q^{37} +(-886.214 - 1534.97i) q^{39} +14585.4 q^{41} +796.436 q^{43} +(651.284 + 1128.06i) q^{45} +(-11043.2 + 19127.4i) q^{47} +(-4842.48 - 16094.3i) q^{49} +(-2056.70 + 3562.31i) q^{51} +(-7065.39 - 12237.6i) q^{53} +10590.5 q^{55} +9319.73 q^{57} +(17627.9 + 30532.3i) q^{59} +(-15771.0 + 27316.2i) q^{61} +(4165.87 + 9639.31i) q^{63} +(1583.48 - 2742.66i) q^{65} +(24864.0 + 43065.7i) q^{67} +20215.2 q^{69} +7903.78 q^{71} +(-8024.92 - 13899.6i) q^{73} +(12898.8 - 22341.4i) q^{75} +(84807.4 + 9853.34i) q^{77} +(9819.88 - 17008.5i) q^{79} +(-3280.50 - 5681.99i) q^{81} +60468.5 q^{83} -7349.77 q^{85} +(37160.5 + 64363.8i) q^{87} +(19131.6 - 33136.9i) q^{89} +(15232.1 - 20489.7i) q^{91} +(26087.1 - 45184.3i) q^{93} +(8326.19 + 14421.4i) q^{95} +57264.8 q^{97} -53344.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 45 q^{3} - 6 q^{5} + 97 q^{7} - 405 q^{9}+O(q^{10}) \) 10 * q - 45 * q^3 - 6 * q^5 + 97 * q^7 - 405 * q^9	\( 10 q - 45 q^{3} - 6 q^{5} + 97 q^{7} - 405 q^{9} - 424 q^{11} + 374 q^{13} + 108 q^{15} - 952 q^{17} - 139 q^{19} - 1044 q^{21} - 4288 q^{23} - 5605 q^{25} + 7290 q^{27} - 4216 q^{29} - 8131 q^{31} - 3816 q^{33} - 20106 q^{35} - 5425 q^{37} - 1683 q^{39} + 29364 q^{41} + 46862 q^{43} - 486 q^{45} + 17190 q^{47} + 23255 q^{49} - 8568 q^{51} + 15064 q^{53} + 1176 q^{55} + 2502 q^{57} + 83242 q^{59} + 14954 q^{61} + 1539 q^{63} - 23250 q^{65} - 39501 q^{67} + 77184 q^{69} + 56020 q^{71} - 90395 q^{73} - 50445 q^{75} + 63448 q^{77} + 43067 q^{79} - 32805 q^{81} + 75672 q^{83} - 75272 q^{85} + 18972 q^{87} - 72608 q^{89} - 288287 q^{91} - 73179 q^{93} - 190138 q^{95} + 183000 q^{97} + 68688 q^{99}+O(q^{100}) \) 10 * q - 45 * q^3 - 6 * q^5 + 97 * q^7 - 405 * q^9 - 424 * q^11 + 374 * q^13 + 108 * q^15 - 952 * q^17 - 139 * q^19 - 1044 * q^21 - 4288 * q^23 - 5605 * q^25 + 7290 * q^27 - 4216 * q^29 - 8131 * q^31 - 3816 * q^33 - 20106 * q^35 - 5425 * q^37 - 1683 * q^39 + 29364 * q^41 + 46862 * q^43 - 486 * q^45 + 17190 * q^47 + 23255 * q^49 - 8568 * q^51 + 15064 * q^53 + 1176 * q^55 + 2502 * q^57 + 83242 * q^59 + 14954 * q^61 + 1539 * q^63 - 23250 * q^65 - 39501 * q^67 + 77184 * q^69 + 56020 * q^71 - 90395 * q^73 - 50445 * q^75 + 63448 * q^77 + 43067 * q^79 - 32805 * q^81 + 75672 * q^83 - 75272 * q^85 + 18972 * q^87 - 72608 * q^89 - 288287 * q^91 - 73179 * q^93 - 190138 * q^95 + 183000 * q^97 + 68688 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more