LMFDB - Embedded newform 10.8.b.a.9.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10,8,Mod(9,10)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("10.9");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 10 = 2 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	10.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.12385025484$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{31})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 15x^{2} + 64 $ x^4 - 15*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.3
Root			$2.78388 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	10.9
Dual form			10.8.b.a.9.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{2} -83.6776i q^{3} -64.0000 q^{4} +(237.711 - 147.033i) q^{5} +669.421 q^{6} -185.744i q^{7} -512.000i q^{8} -4814.95 q^{9} +O(q^{10})$	\(q+8.00000i q^{2} -83.6776i q^{3} -64.0000 q^{4} +(237.711 - 147.033i) q^{5} +669.421 q^{6} -185.744i q^{7} -512.000i q^{8} -4814.95 q^{9} +(1176.26 + 1901.68i) q^{10} +3561.16 q^{11} +5355.37i q^{12} +6094.86i q^{13} +1485.95 q^{14} +(-12303.4 - 19891.1i) q^{15} +4096.00 q^{16} +12470.2i q^{17} -38519.6i q^{18} +50642.8 q^{19} +(-15213.5 + 9410.11i) q^{20} -15542.6 q^{21} +28489.3i q^{22} +11442.3i q^{23} -42843.0 q^{24} +(34887.6 - 69902.6i) q^{25} -48758.8 q^{26} +219900. i q^{27} +11887.6i q^{28} -101926. q^{29} +(159128. - 98427.0i) q^{30} -29042.3 q^{31} +32768.0i q^{32} -297989. i q^{33} -99761.2 q^{34} +(-27310.4 - 44153.2i) q^{35} +308157. q^{36} -149393. i q^{37} +405143. i q^{38} +510003. q^{39} +(-75280.9 - 121708. i) q^{40} -374382. q^{41} -124341. i q^{42} -174226. i q^{43} -227914. q^{44} +(-1.14456e6 + 707956. i) q^{45} -91538.6 q^{46} +428201. i q^{47} -342744. i q^{48} +789042. q^{49} +(559221. + 279101. i) q^{50} +1.04347e6 q^{51} -390071. i q^{52} +1.71297e6i q^{53} -1.75920e6 q^{54} +(846525. - 523607. i) q^{55} -95100.7 q^{56} -4.23767e6i q^{57} -815412. i q^{58} -134952. q^{59} +(787416. + 1.27303e6i) q^{60} -1.39437e6 q^{61} -232338. i q^{62} +894345. i q^{63} -262144. q^{64} +(896145. + 1.44881e6i) q^{65} +2.38391e6 q^{66} +2.60763e6i q^{67} -798090. i q^{68} +957466. q^{69} +(353226. - 218483. i) q^{70} -4.91925e6 q^{71} +2.46525e6i q^{72} -119096. i q^{73} +1.19515e6 q^{74} +(-5.84928e6 - 2.91932e6i) q^{75} -3.24114e6 q^{76} -661462. i q^{77} +4.08003e6i q^{78} +4.70584e6 q^{79} +(973663. - 602247. i) q^{80} +7.87046e6 q^{81} -2.99506e6i q^{82} -9.19870e6i q^{83} +994725. q^{84} +(1.83352e6 + 2.96429e6i) q^{85} +1.39380e6 q^{86} +8.52897e6i q^{87} -1.82331e6i q^{88} -6.43438e6 q^{89} +(-5.66365e6 - 9.15651e6i) q^{90} +1.13208e6 q^{91} -732308. i q^{92} +2.43019e6i q^{93} -3.42561e6 q^{94} +(1.20383e7 - 7.44617e6i) q^{95} +2.74195e6 q^{96} +1.26986e7i q^{97} +6.31234e6i q^{98} -1.71468e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9}+O(q^{10}) $ 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9	$ 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9} - 640 q^{10} + 17808 q^{11} - 6528 q^{14} - 34960 q^{15} + 16384 q^{16} + 63600 q^{19} - 3840 q^{20} - 63952 q^{21} - 57344 q^{24} + 86100 q^{25} - 56064 q^{26} + 169560 q^{29} + 410240 q^{30} - 394112 q^{31} - 698368 q^{34} - 276720 q^{35} + 434432 q^{36} + 1163424 q^{39} + 40960 q^{40} + 232488 q^{41} - 1139712 q^{44} - 2879420 q^{45} + 1146496 q^{46} + 2520108 q^{49} + 2361600 q^{50} + 361088 q^{51} - 5116160 q^{54} - 526480 q^{55} + 417792 q^{56} + 2093520 q^{59} + 2237440 q^{60} - 5251432 q^{61} - 1048576 q^{64} + 2761440 q^{65} + 2401792 q^{66} + 6514864 q^{69} + 2679680 q^{70} - 7832352 q^{71} - 3126528 q^{74} - 7397600 q^{75} - 4070400 q^{76} + 7727040 q^{79} + 245760 q^{80} + 16428404 q^{81} + 4092928 q^{84} - 7995520 q^{85} + 4411776 q^{86} - 33470040 q^{89} - 15579520 q^{90} + 5340768 q^{91} + 15540352 q^{94} + 31904400 q^{95} + 3670016 q^{96} - 19109776 q^{99}+O(q^{100}) $ 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9 - 640 * q^10 + 17808 * q^11 - 6528 * q^14 - 34960 * q^15 + 16384 * q^16 + 63600 * q^19 - 3840 * q^20 - 63952 * q^21 - 57344 * q^24 + 86100 * q^25 - 56064 * q^26 + 169560 * q^29 + 410240 * q^30 - 394112 * q^31 - 698368 * q^34 - 276720 * q^35 + 434432 * q^36 + 1163424 * q^39 + 40960 * q^40 + 232488 * q^41 - 1139712 * q^44 - 2879420 * q^45 + 1146496 * q^46 + 2520108 * q^49 + 2361600 * q^50 + 361088 * q^51 - 5116160 * q^54 - 526480 * q^55 + 417792 * q^56 + 2093520 * q^59 + 2237440 * q^60 - 5251432 * q^61 - 1048576 * q^64 + 2761440 * q^65 + 2401792 * q^66 + 6514864 * q^69 + 2679680 * q^70 - 7832352 * q^71 - 3126528 * q^74 - 7397600 * q^75 - 4070400 * q^76 + 7727040 * q^79 + 245760 * q^80 + 16428404 * q^81 + 4092928 * q^84 - 7995520 * q^85 + 4411776 * q^86 - 33470040 * q^89 - 15579520 * q^90 + 5340768 * q^91 + 15540352 * q^94 + 31904400 * q^95 + 3670016 * q^96 - 19109776 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$			8.00000i			0.707107i
$2$			8.00000i			0.707107i
$3$		−	83.6776i		−	1.78931i	−0.446760	−	0.894654i	$-0.647422\pi$
$3$		−	83.6776i		−	1.78931i	0.446760	−	0.894654i	$-0.352578\pi$
$4$	−64.0000			−0.500000
$5$	237.711	−	147.033i	0.850459	−	0.526041i
$5$	237.711	−	147.033i	0.850459	−	0.526041i
$6$	669.421			1.26523
$7$		−	185.744i		−	0.204678i	−0.994750	−	0.102339i	$-0.967367\pi$
$7$		−	185.744i		−	0.204678i	0.994750	−	0.102339i	$-0.0326326\pi$
$8$		−	512.000i		−	0.353553i
$9$	−4814.95			−2.20162
$10$	1176.26	+	1901.68i	0.371967	+	0.601365i
$11$	3561.16			0.806709			0.403354	−	0.915044i	$-0.367844\pi$
$11$	3561.16			0.806709			0.403354	+	0.915044i	$0.367844\pi$
$12$			5355.37i			0.894654i
$13$			6094.86i			0.769417i	0.923038	+	0.384708i	$0.125698\pi$
$13$			6094.86i			0.769417i	−0.923038	+	0.384708i	$0.874302\pi$
$14$	1485.95			0.144729
$15$	−12303.4	−	19891.1i	−0.941249	−	1.52173i
$16$	4096.00			0.250000
$17$			12470.2i			0.615603i	0.951451	+	0.307801i	$0.0995933\pi$
$17$			12470.2i			0.615603i	−0.951451	+	0.307801i	$0.900407\pi$
$18$		−	38519.6i		−	1.55678i
$19$	50642.8			1.69387			0.846936	−	0.531695i	$-0.178445\pi$
$19$	50642.8			1.69387			0.846936	+	0.531695i	$0.178445\pi$
$20$	−15213.5	+	9410.11i	−0.425230	+	0.263021i
$21$	−15542.6			−0.366231
$22$			28489.3i			0.570429i
$23$			11442.3i			0.196095i	0.995182	+	0.0980475i	$0.0312597\pi$
$23$			11442.3i			0.196095i	−0.995182	+	0.0980475i	$0.968740\pi$
$24$	−42843.0			−0.632616
$25$	34887.6	−	69902.6i	0.446562	−	0.894753i
$26$	−48758.8			−0.544060
$27$			219900.i			2.15007i
$28$			11887.6i			0.102339i
$29$	−101926.			−0.776058			−0.388029	−	0.921647i	$-0.626844\pi$
$29$	−101926.			−0.776058			−0.388029	+	0.921647i	$0.626844\pi$
$30$	159128.	−	98427.0i	1.07603	−	0.665564i
$31$	−29042.3			−0.175092			−0.0875458	−	0.996160i	$-0.527902\pi$
$31$	−29042.3			−0.175092			−0.0875458	+	0.996160i	$0.527902\pi$
$32$			32768.0i			0.176777i
$33$		−	297989.i		−	1.44345i
$34$	−99761.2			−0.435297
$35$	−27310.4	−	44153.2i	−0.107669	−	0.174070i
$36$	308157.			1.10081
$37$		−	149393.i		−	0.484870i	−0.970168	−	0.242435i	$-0.922054\pi$
$37$		−	149393.i		−	0.484870i	0.970168	−	0.242435i	$-0.0779461\pi$
$38$			405143.i			1.19775i
$39$	510003.			1.37672
$40$	−75280.9	−	121708.i	−0.185984	−	0.300683i
$41$	−374382.			−0.848343			−0.424171	−	0.905582i	$-0.639435\pi$
$41$	−374382.			−0.848343			−0.424171	+	0.905582i	$0.639435\pi$
$42$		−	124341.i		−	0.258965i
$43$		−	174226.i		−	0.334174i	−0.985942	−	0.167087i	$-0.946564\pi$
$43$		−	174226.i		−	0.334174i	0.985942	−	0.167087i	$-0.0534360\pi$
$44$	−227914.			−0.403354
$45$	−1.14456e6	+	707956.i	−1.87239	+	1.15814i
$46$	−91538.6			−0.138660
$47$			428201.i			0.601597i	0.953688	+	0.300798i	$0.0972531\pi$
$47$			428201.i			0.601597i	−0.953688	+	0.300798i	$0.902747\pi$
$48$		−	342744.i		−	0.447327i
$49$	789042.			0.958107
$50$	559221.	+	279101.i	0.632686	+	0.315767i
$51$	1.04347e6			1.10150
$52$		−	390071.i		−	0.384708i
$53$			1.71297e6i			1.58047i	0.612806	+	0.790233i	$0.290041\pi$
$53$			1.71297e6i			1.58047i	−0.612806	+	0.790233i	$0.709959\pi$
$54$	−1.75920e6			−1.52033
$55$	846525.	−	523607.i	0.686073	−	0.424362i
$56$	−95100.7			−0.0723645
$57$		−	4.23767e6i		−	3.03086i
$58$		−	815412.i		−	0.548756i
$59$	−134952.			−0.0855458			−0.0427729	−	0.999085i	$-0.513619\pi$
$59$	−134952.			−0.0855458			−0.0427729	+	0.999085i	$0.513619\pi$
$60$	787416.	+	1.27303e6i	0.470625	+	0.760867i
$61$	−1.39437e6			−0.786545			−0.393273	−	0.919422i	$-0.628657\pi$
$61$	−1.39437e6			−0.786545			−0.393273	+	0.919422i	$0.628657\pi$
$62$		−	232338.i		−	0.123808i
$63$			894345.i			0.450623i
$64$	−262144.			−0.125000
$65$	896145.	+	1.44881e6i	0.404745	+	0.654358i
$66$	2.38391e6			1.02067
$67$			2.60763e6i			1.05922i	0.848243	+	0.529608i	$0.177661\pi$
$67$			2.60763e6i			1.05922i	−0.848243	+	0.529608i	$0.822339\pi$
$68$		−	798090.i		−	0.307801i
$69$	957466.			0.350874
$70$	353226.	−	218483.i	0.123086	−	0.0761334i
$71$	−4.91925e6			−1.63115			−0.815576	−	0.578650i	$-0.803580\pi$
$71$	−4.91925e6			−1.63115			−0.815576	+	0.578650i	$0.803580\pi$
$72$			2.46525e6i			0.778391i
$73$		−	119096.i		−	0.0358316i	−0.999839	−	0.0179158i	$-0.994297\pi$
$73$		−	119096.i		−	0.0358316i	0.999839	−	0.0179158i	$-0.00570309\pi$
$74$	1.19515e6			0.342855
$75$	−5.84928e6	−	2.91932e6i	−1.60099	−	0.799036i
$76$	−3.24114e6			−0.846936
$77$		−	661462.i		−	0.165115i
$78$			4.08003e6i			0.973491i
$79$	4.70584e6			1.07385			0.536924	−	0.843631i	$-0.319586\pi$
$79$	4.70584e6			1.07385			0.536924	+	0.843631i	$0.319586\pi$
$80$	973663.	−	602247.i	0.212615	−	0.131510i
$81$	7.87046e6			1.64552
$82$		−	2.99506e6i		−	0.599869i
$83$		−	9.19870e6i		−	1.76585i	−0.469517	−	0.882923i	$-0.655572\pi$
$83$		−	9.19870e6i		−	1.76585i	0.469517	−	0.882923i	$-0.344428\pi$
$84$	994725.			0.183116
$85$	1.83352e6	+	2.96429e6i	0.323832	+	0.523545i
$86$	1.39380e6			0.236296
$87$			8.52897e6i			1.38861i
$88$		−	1.82331e6i		−	0.285215i
$89$	−6.43438e6			−0.967480			−0.483740	−	0.875212i	$-0.660722\pi$
$89$	−6.43438e6			−0.967480			−0.483740	+	0.875212i	$0.660722\pi$
$90$	−5.66365e6	−	9.15651e6i	−0.818931	−	1.32398i
$91$	1.13208e6			0.157482
$92$		−	732308.i		−	0.0980475i
$93$			2.43019e6i			0.313293i
$94$	−3.42561e6			−0.425393
$95$	1.20383e7	−	7.44617e6i	1.44057	−	0.891046i
$96$	2.74195e6			0.316308
$97$			1.26986e7i			1.41271i	0.707858	+	0.706355i	$0.249662\pi$
$97$			1.26986e7i			1.41271i	−0.707858	+	0.706355i	$0.750338\pi$
$98$			6.31234e6i			0.677484i
$99$	−1.71468e7			−1.77607
$100$	−2.23281e6	+	4.47376e6i	−0.223281	+	0.447376i
$101$	6.66850e6			0.644026			0.322013	−	0.946735i	$-0.395641\pi$
$101$	6.66850e6			0.644026			0.322013	+	0.946735i	$0.395641\pi$
$102$			8.34779e6i			0.778880i
$103$		−	2.10469e6i		−	0.189783i	−0.995488	−	0.0948916i	$-0.969750\pi$
$103$		−	2.10469e6i		−	0.189783i	0.995488	−	0.0948916i	$-0.0302505\pi$
$104$	3.12057e6			0.272030
$105$	−3.69464e6	+	2.28527e6i	−0.311465	+	0.192653i
$106$	−1.37038e7			−1.11756
$107$		−	7.43589e6i		−	0.586799i	−0.955990	−	0.293400i	$-0.905213\pi$
$107$		−	7.43589e6i		−	0.586799i	0.955990	−	0.293400i	$-0.0947867\pi$
$108$		−	1.40736e7i		−	1.07504i
$109$	9.12137e6			0.674633			0.337316	−	0.941391i	$-0.390481\pi$
$109$	9.12137e6			0.674633			0.337316	+	0.941391i	$0.390481\pi$
$110$	4.18886e6	+	6.77220e6i	0.300069	+	0.485127i
$111$	−1.25009e7			−0.867582
$112$		−	760805.i		−	0.0511694i
$113$			2.93903e6i			0.191615i	0.995400	+	0.0958075i	$0.0305433\pi$
$113$			2.93903e6i			0.191615i	−0.995400	+	0.0958075i	$0.969457\pi$
$114$	3.39014e7			2.14314
$115$	1.68240e6	+	2.71996e6i	0.103154	+	0.166771i
$116$	6.52329e6			0.388029
$117$		−	2.93464e7i		−	1.69397i
$118$		−	1.07962e6i		−	0.0604900i
$119$	2.31625e6			0.126000
$120$	−1.01842e7	+	6.29933e6i	−0.538014	+	0.332782i
$121$	−6.80533e6			−0.349221
$122$		−	1.11550e7i		−	0.556171i
$123$			3.13274e7i			1.51795i
$124$	1.85871e6			0.0875458
$125$	−1.98482e6	−	2.17462e7i	−0.0908942	−	0.995861i
$126$	−7.15476e6			−0.318639
$127$		−	1.76254e7i		−	0.763529i	−0.924260	−	0.381764i	$-0.875317\pi$
$127$		−	1.76254e7i		−	0.763529i	0.924260	−	0.381764i	$-0.124683\pi$
$128$		−	2.09715e6i		−	0.0883883i
$129$	−1.45788e7			−0.597939
$130$	−1.15905e7	+	7.16916e6i	−0.462701	+	0.286198i
$131$	3.95489e6			0.153704			0.0768520	−	0.997043i	$-0.475513\pi$
$131$	3.95489e6			0.153704			0.0768520	+	0.997043i	$0.475513\pi$
$132$			1.90713e7i			0.721725i
$133$		−	9.40658e6i		−	0.346698i
$134$	−2.08610e7			−0.748978
$135$	3.23326e7	+	5.22727e7i	1.13103	+	1.82855i
$136$	6.38472e6			0.217648
$137$			2.95868e7i			0.983051i	0.870863	+	0.491525i	$0.163560\pi$
$137$			2.95868e7i			0.983051i	−0.870863	+	0.491525i	$0.836440\pi$
$138$			7.65973e6i			0.248106i
$139$	−4.26561e6			−0.134719			−0.0673596	−	0.997729i	$-0.521457\pi$
$139$	−4.26561e6			−0.134719			−0.0673596	+	0.997729i	$0.521457\pi$
$140$	1.74787e6	+	2.82580e6i	0.0538344	+	0.0870350i
$141$	3.58309e7			1.07644
$142$		−	3.93540e7i		−	1.15340i
$143$			2.17047e7i			0.620695i
$144$	−1.97220e7			−0.550406
$145$	−2.42290e7	+	1.49865e7i	−0.660005	+	0.408238i
$146$	952767.			0.0253368
$147$		−	6.60252e7i		−	1.71435i
$148$			9.56118e6i			0.242435i
$149$	−7.09039e7			−1.75597			−0.877987	−	0.478685i	$-0.841114\pi$
$149$	−7.09039e7			−1.75597			−0.877987	+	0.478685i	$0.841114\pi$
$150$	2.33545e7	−	4.67943e7i	0.565004	−	1.13207i
$151$	3.36796e7			0.796063			0.398032	−	0.917372i	$-0.369693\pi$
$151$	3.36796e7			0.796063			0.398032	+	0.917372i	$0.369693\pi$
$152$		−	2.59291e7i		−	0.598874i
$153$		−	6.00432e7i		−	1.35533i
$154$	5.29170e6			0.116754
$155$	−6.90366e6	+	4.27017e6i	−0.148908	+	0.0921053i
$156$	−3.26402e7			−0.688362
$157$			7.51572e7i			1.54997i	0.631982	+	0.774983i	$0.282241\pi$
$157$			7.51572e7i			1.54997i	−0.631982	+	0.774983i	$0.717759\pi$
$158$			3.76467e7i			0.759325i
$159$	1.43338e8			2.82794
$160$	4.81798e6	+	7.78930e6i	0.0929918	+	0.150341i
$161$	2.12534e6			0.0401363
$162$			6.29637e7i			1.16356i
$163$		−	8.50523e7i		−	1.53826i	−0.639093	−	0.769130i	$-0.720690\pi$
$163$		−	8.50523e7i		−	1.53826i	0.639093	−	0.769130i	$-0.279310\pi$
$164$	2.39604e7			0.424171
$165$	−4.38142e7	−	7.08352e7i	−0.759314	−	1.22760i
$166$	7.35896e7			1.24864
$167$			5.10526e7i			0.848223i	0.905610	+	0.424111i	$0.139413\pi$
$167$			5.10526e7i			0.848223i	−0.905610	+	0.424111i	$0.860587\pi$
$168$			7.95780e6i			0.129482i
$169$	2.56012e7			0.407998
$170$	−2.37143e7	+	1.46682e7i	−0.370202	+	0.228984i
$171$	−2.43843e8			−3.72927
$172$			1.11504e7i			0.167087i
$173$			7.71282e7i			1.13254i	0.824221	+	0.566268i	$0.191613\pi$
$173$			7.71282e7i			1.13254i	−0.824221	+	0.566268i	$0.808387\pi$
$174$	−6.82317e7			−0.981893
$175$	−1.29839e7	−	6.48015e6i	−0.183136	−	0.0914012i
$176$	1.45865e7			0.201677
$177$			1.12925e7i			0.153068i
$178$		−	5.14751e7i		−	0.684111i
$179$	9.92044e6			0.129284			0.0646421	−	0.997909i	$-0.479409\pi$
$179$	9.92044e6			0.129284			0.0646421	+	0.997909i	$0.479409\pi$
$180$	7.32521e7	−	4.53092e7i	0.936195	−	0.579072i
$181$	5.60330e7			0.702374			0.351187	−	0.936305i	$-0.385778\pi$
$181$	5.60330e7			0.702374			0.351187	+	0.936305i	$0.385778\pi$
$182$			9.05664e6i			0.111357i
$183$			1.16678e8i			1.40737i
$184$	5.85847e6			0.0693301
$185$	−2.19657e7	−	3.55124e7i	−0.255062	−	0.412362i
$186$	−1.94415e7			−0.221531
$187$			4.44082e7i			0.496612i
$188$		−	2.74049e7i		−	0.300798i
$189$	4.08451e7			0.440072
$190$	5.95693e7	+	9.63067e7i	0.630065	+	1.01864i
$191$	8.36160e7			0.868305			0.434153	−	0.900839i	$-0.357048\pi$
$191$	8.36160e7			0.868305			0.434153	+	0.900839i	$0.357048\pi$
$192$			2.19356e7i			0.223663i
$193$		−	1.84081e8i		−	1.84314i	−0.388216	−	0.921569i	$-0.626908\pi$
$193$		−	1.84081e8i		−	1.84314i	0.388216	−	0.921569i	$-0.373092\pi$
$194$	−1.01588e8			−0.998937
$195$	1.21233e8	−	7.49873e7i	1.17085	−	0.724213i
$196$	−5.04987e7			−0.479054
$197$		−	3.32048e7i		−	0.309435i	−0.987959	−	0.154717i	$-0.950553\pi$
$197$		−	3.32048e7i		−	0.309435i	0.987959	−	0.154717i	$-0.0494467\pi$
$198$		−	1.37174e8i		−	1.25587i
$199$	−1.57568e8			−1.41737			−0.708683	−	0.705527i	$-0.750710\pi$
$199$	−1.57568e8			−1.41737			−0.708683	+	0.705527i	$0.750710\pi$
$200$	−3.57901e7	−	1.78625e7i	−0.316343	−	0.157883i
$201$	2.18200e8			1.89526
$202$			5.33480e7i			0.455395i
$203$			1.89322e7i			0.158842i
$204$	−6.67823e7			−0.550752
$205$	−8.89945e7	+	5.50465e7i	−0.721481	+	0.446263i
$206$	1.68375e7			0.134197
$207$		−	5.50942e7i		−	0.431727i
$208$			2.49645e7i			0.192354i
$209$	1.80347e8			1.36646
$210$	−1.82822e7	−	2.95571e7i	−0.136226	−	0.220239i
$211$	−1.86392e8			−1.36597			−0.682983	−	0.730435i	$-0.739318\pi$
$211$	−1.86392e8			−1.36597			−0.682983	+	0.730435i	$0.739318\pi$
$212$		−	1.09630e8i		−	0.790233i
$213$			4.11631e8i			2.91863i
$214$	5.94871e7			0.414930
$215$	−2.56169e7	−	4.14152e7i	−0.175789	−	0.284201i
$216$	1.12589e8			0.760165
$217$			5.39442e6i			0.0358373i
$218$			7.29710e7i			0.477037i
$219$	−9.96566e6			−0.0641138
$220$	−5.41776e7	+	3.35109e7i	−0.343036	+	0.212181i
$221$	−7.60038e7			−0.473655
$222$		−	1.00007e8i		−	0.613473i
$223$		−	2.57497e8i		−	1.55491i	−0.628938	−	0.777456i	$-0.716510\pi$
$223$		−	2.57497e8i		−	1.55491i	0.628938	−	0.777456i	$-0.283490\pi$
$224$	6.08644e6			0.0361822
$225$	−1.67982e8	+	3.36577e8i	−0.983160	+	1.96991i
$226$	−2.35122e7			−0.135492
$227$		−	1.10040e8i		−	0.624398i	−0.950017	−	0.312199i	$-0.898934\pi$
$227$		−	1.10040e8i		−	0.624398i	0.950017	−	0.312199i	$-0.101066\pi$
$228$			2.71211e8i			1.51543i
$229$	−4.69036e7			−0.258096			−0.129048	−	0.991638i	$-0.541192\pi$
$229$	−4.69036e7			−0.258096			−0.129048	+	0.991638i	$0.541192\pi$
$230$	−2.17597e7	+	1.34592e7i	−0.117925	+	0.0729409i
$231$	−5.53496e7			−0.295442
$232$			5.21863e7i			0.274378i
$233$		−	1.60419e8i		−	0.830824i	−0.909633	−	0.415412i	$-0.863637\pi$
$233$		−	1.60419e8i		−	0.830824i	0.909633	−	0.415412i	$-0.136363\pi$
$234$	2.34771e8			1.19781
$235$	6.29597e7	+	1.01788e8i	0.316464	+	0.511633i
$236$	8.63696e6			0.0427729
$237$		−	3.93774e8i		−	1.92144i
$238$			1.85300e7i			0.0890956i
$239$	2.51647e8			1.19234			0.596169	−	0.802859i	$-0.296689\pi$
$239$	2.51647e8			1.19234			0.596169	+	0.802859i	$0.296689\pi$
$240$	−5.03946e7	−	8.14738e7i	−0.235312	−	0.380433i
$241$	2.30623e8			1.06131			0.530655	−	0.847588i	$-0.321946\pi$
$241$	2.30623e8			1.06131			0.530655	+	0.847588i	$0.321946\pi$
$242$		−	5.44426e7i		−	0.246936i
$243$		−	1.77660e8i		−	0.794267i
$244$	8.92397e7			0.393273
$245$	1.87564e8	−	1.16015e8i	0.814831	−	0.504004i
$246$	−2.50619e8			−1.07335
$247$			3.08661e8i			1.30329i
$248$			1.48697e7i			0.0619042i
$249$	−7.69725e8			−3.15964
$250$	1.73970e8	−	1.58786e7i	0.704180	−	0.0642719i
$251$	−2.64559e8			−1.05600			−0.528002	−	0.849243i	$-0.677058\pi$
$251$	−2.64559e8			−1.05600			−0.528002	+	0.849243i	$0.677058\pi$
$252$		−	5.72381e7i		−	0.225311i
$253$			4.07479e7i			0.158192i
$254$	1.41003e8			0.539896
$255$	2.48045e8	−	1.53425e8i	0.936783	−	0.579436i
$256$	1.67772e7			0.0625000
$257$		−	1.54580e8i		−	0.568050i	−0.958817	−	0.284025i	$-0.908330\pi$
$257$		−	1.54580e8i		−	0.568050i	0.958817	−	0.284025i	$-0.0916699\pi$
$258$		−	1.16630e8i		−	0.422807i
$259$	−2.77489e7			−0.0992421
$260$	−5.73533e7	−	9.27240e7i	−0.202372	−	0.327179i
$261$	4.90771e8			1.70859
$262$			3.16391e7i			0.108685i
$263$			4.30058e8i			1.45775i	0.684649	+	0.728873i	$0.259955\pi$
$263$			4.30058e8i			1.45775i	−0.684649	+	0.728873i	$0.740045\pi$
$264$	−1.52571e8			−0.510337
$265$	2.51864e8	+	4.07192e8i	0.831390	+	1.34412i
$266$	7.52526e7			0.245152
$267$			5.38414e8i			1.73112i
$268$		−	1.66888e8i		−	0.529608i
$269$	−1.73129e6			−0.00542297			−0.00271149	−	0.999996i	$-0.500863\pi$
$269$	−1.73129e6			−0.00542297			−0.00271149	+	0.999996i	$0.500863\pi$
$270$	−4.18181e8	+	2.58661e8i	−1.29298	+	0.799756i
$271$	−2.91507e8			−0.889727			−0.444864	−	0.895598i	$-0.646748\pi$
$271$	−2.91507e8			−0.889727			−0.444864	+	0.895598i	$0.646748\pi$
$272$			5.10778e7i			0.153901i
$273$		−	9.47298e7i		−	0.281785i
$274$	−2.36694e8			−0.695122
$275$	1.24240e8	−	2.48934e8i	0.360245	−	0.721805i
$276$	−6.12778e7			−0.175437
$277$		−	2.01058e8i		−	0.568383i	−0.958767	−	0.284192i	$-0.908275\pi$
$277$		−	2.01058e8i		−	0.568383i	0.958767	−	0.284192i	$-0.0917252\pi$
$278$		−	3.41249e7i		−	0.0952609i
$279$	1.39837e8			0.385485
$280$	−2.26064e7	+	1.39829e7i	−0.0615430	+	0.0380667i
$281$	−3.53690e8			−0.950935			−0.475468	−	0.879733i	$-0.657721\pi$
$281$	−3.53690e8			−0.950935			−0.475468	+	0.879733i	$0.657721\pi$
$282$			2.86647e8i			0.761159i
$283$			9.83086e7i			0.257833i	0.991655	+	0.128917i	$0.0411500\pi$
$283$			9.83086e7i			0.257833i	−0.991655	+	0.128917i	$0.958850\pi$
$284$	3.14832e8			0.815576
$285$	−6.23078e8	−	1.00734e9i	−1.59436	−	2.57762i
$286$	−1.73638e8			−0.438898
$287$			6.95390e7i			0.173637i
$288$		−	1.57776e8i		−	0.389196i
$289$	2.54834e8			0.621033
$290$	−1.19892e8	−	1.93832e8i	−0.288668	−	0.466694i
$291$	1.06258e9			2.52777
$292$			7.62214e6i			0.0179158i
$293$			2.63252e8i			0.611414i	0.952126	+	0.305707i	$0.0988928\pi$
$293$			2.63252e8i			0.611414i	−0.952126	+	0.305707i	$0.901107\pi$
$294$	5.28202e8			1.21223
$295$	−3.20796e7	+	1.98425e7i	−0.0727532	+	0.0450006i
$296$	−7.64894e7			−0.171427
$297$			7.83100e8i			1.73448i
$298$		−	5.67231e8i		−	1.24166i
$299$	−6.97393e7			−0.150879
$300$	3.74354e8	+	1.86836e8i	0.800494	+	0.399518i
$301$	−3.23613e7			−0.0683979
$302$			2.69437e8i			0.562902i
$303$		−	5.58004e8i		−	1.15236i
$304$	2.07433e8			0.423468
$305$	−3.31457e8	+	2.05018e8i	−0.668925	+	0.413755i
$306$	4.80345e8			0.958360
$307$		−	4.04138e8i		−	0.797160i	−0.917133	−	0.398580i	$-0.869503\pi$
$307$		−	4.04138e8i		−	0.797160i	0.917133	−	0.398580i	$-0.130497\pi$
$308$			4.23336e7i			0.0825576i
$309$	−1.76115e8			−0.339581
$310$	−3.41614e7	−	5.52293e7i	−0.0651283	−	0.105294i
$311$	−5.84260e8			−1.10140			−0.550700	−	0.834703i	$-0.685639\pi$
$311$	−5.84260e8			−1.10140			−0.550700	+	0.834703i	$0.685639\pi$
$312$		−	2.61122e8i		−	0.486745i
$313$		−	8.58721e8i		−	1.58288i	−0.611250	−	0.791438i	$-0.709333\pi$
$313$		−	8.58721e8i		−	1.58288i	0.611250	−	0.791438i	$-0.290667\pi$
$314$	−6.01258e8			−1.09599
$315$	1.31498e8	+	2.12595e8i	0.237046	+	0.383236i
$316$	−3.01174e8			−0.536924
$317$			1.01493e8i			0.178950i	0.995989	+	0.0894748i	$0.0285188\pi$
$317$			1.01493e8i			0.178950i	−0.995989	+	0.0894748i	$0.971481\pi$
$318$			1.14670e9i			1.99966i
$319$	−3.62976e8			−0.626052
$320$	−6.23144e7	+	3.85438e7i	−0.106307	+	0.0657551i
$321$	−6.22218e8			−1.04996
$322$			1.70027e7i			0.0283806i
$323$			6.31524e8i			1.04275i
$324$	−5.03710e8			−0.822759
$325$	4.26046e8	+	2.12635e8i	0.688438	+	0.343592i
$326$	6.80419e8			1.08771
$327$		−	7.63255e8i		−	1.20713i
$328$			1.91684e8i			0.299934i
$329$	7.95356e7			0.123133
$330$	5.66682e8	−	3.50514e8i	0.868041	−	0.536916i
$331$	5.45528e8			0.826835			0.413418	−	0.910542i	$-0.364335\pi$
$331$	5.45528e8			0.826835			0.413418	+	0.910542i	$0.364335\pi$
$332$			5.88717e8i			0.882923i
$333$			7.19321e8i			1.06750i
$334$	−4.08420e8			−0.599784
$335$	3.83408e8	+	6.19861e8i	0.557191	+	0.900819i
$336$	−6.36624e7			−0.0915578
$337$			6.98565e8i			0.994266i	0.867674	+	0.497133i	$0.165614\pi$
$337$			6.98565e8i			0.994266i	−0.867674	+	0.497133i	$0.834386\pi$
$338$			2.04810e8i			0.288498i
$339$	2.45931e8			0.342858
$340$	−1.17346e8	−	1.89714e8i	−0.161916	−	0.261773i
$341$	−1.03424e8			−0.141248
$342$		−	1.95074e9i		−	2.63699i
$343$		−	2.99527e8i		−	0.400781i
$344$	−8.92035e7			−0.118148
$345$	2.27600e8	−	1.40779e8i	0.298404	−	0.184574i
$346$	−6.17025e8			−0.800823
$347$		−	5.05448e8i		−	0.649417i	−0.945814	−	0.324708i	$-0.894734\pi$
$347$		−	5.05448e8i		−	0.649417i	0.945814	−	0.324708i	$-0.105266\pi$
$348$		−	5.45854e8i		−	0.694303i
$349$	1.05055e9			1.32290			0.661449	−	0.749990i	$-0.269942\pi$
$349$	1.05055e9			1.32290			0.661449	+	0.749990i	$0.269942\pi$
$350$	5.18412e7	−	1.03872e8i	0.0646304	−	0.129497i
$351$	−1.34026e9			−1.65430
$352$			1.16692e8i			0.142607i
$353$		−	1.35289e9i		−	1.63701i	−0.574501	−	0.818504i	$-0.694804\pi$
$353$		−	1.35289e9i		−	1.63701i	0.574501	−	0.818504i	$-0.305196\pi$
$354$	−9.03400e7			−0.108235
$355$	−1.16936e9	+	7.23291e8i	−1.38723	+	0.858053i
$356$	4.11800e8			0.483740
$357$		−	1.93818e8i		−	0.225453i
$358$			7.93635e7i			0.0914177i
$359$	4.43730e8			0.506161			0.253080	−	0.967445i	$-0.418556\pi$
$359$	4.43730e8			0.506161			0.253080	+	0.967445i	$0.418556\pi$
$360$	3.62473e8	+	5.86017e8i	0.409466	+	0.661990i
$361$	1.67083e9			1.86920
$362$			4.48264e8i			0.496654i
$363$			5.69454e8i			0.624864i
$364$	−7.24531e7			−0.0787412
$365$	−1.75110e7	−	2.83103e7i	−0.0188489	−	0.0304733i
$366$	−9.33421e8			−0.995162
$367$		−	1.51050e9i		−	1.59510i	−0.603251	−	0.797551i	$-0.706128\pi$
$367$		−	1.51050e9i		−	1.59510i	0.603251	−	0.797551i	$-0.293872\pi$
$368$			4.68677e7i			0.0490238i
$369$	1.80263e9			1.86773
$370$	2.84099e8	−	1.75726e8i	0.291584	−	0.180356i
$371$	3.18174e8			0.323486
$372$		−	1.55532e8i		−	0.156646i
$373$			1.64359e9i			1.63988i	0.572448	+	0.819941i	$0.305994\pi$
$373$			1.64359e9i			1.63988i	−0.572448	+	0.819941i	$0.694006\pi$
$374$	−3.55266e8			−0.351158
$375$	−1.81967e9	+	1.66085e8i	−1.78190	+	0.162638i
$376$	2.19239e8			0.212697
$377$		−	6.21227e8i		−	0.597112i
$378$			3.26761e8i			0.311178i
$379$	−3.65881e7			−0.0345225			−0.0172613	−	0.999851i	$-0.505495\pi$
$379$	−3.65881e7			−0.0345225			−0.0172613	+	0.999851i	$0.505495\pi$
$380$	−7.70454e8	+	4.76555e8i	−0.720284	+	0.445523i
$381$	−1.47485e9			−1.36619
$382$			6.68928e8i			0.613985i
$383$			5.68889e8i			0.517406i	0.965957	+	0.258703i	$0.0832952\pi$
$383$			5.68889e8i			0.517406i	−0.965957	+	0.258703i	$0.916705\pi$
$384$	−1.75485e8			−0.158154
$385$	−9.72567e7	−	1.57236e8i	−0.0868574	−	0.140424i
$386$	1.47265e9			1.30329
$387$			8.38887e8i			0.735724i
$388$		−	8.12707e8i		−	0.706355i
$389$	−2.10183e9			−1.81040			−0.905199	−	0.424988i	$-0.860278\pi$
$389$	−2.10183e9			−1.81040			−0.905199	+	0.424988i	$0.860278\pi$
$390$	5.99898e8	+	9.69865e8i	0.512096	+	0.827914i
$391$	−1.42688e8			−0.120717
$392$		−	4.03990e8i		−	0.338742i
$393$		−	3.30936e8i		−	0.275024i
$394$	2.65638e8			0.218803
$395$	1.11863e9	−	6.91914e8i	0.913263	−	0.564888i
$396$	1.09739e9			0.888034
$397$			1.62761e9i			1.30552i	0.757564	+	0.652760i	$0.226389\pi$
$397$			1.62761e9i			1.30552i	−0.757564	+	0.652760i	$0.773611\pi$
$398$		−	1.26054e9i		−	1.00223i
$399$	−7.87120e8			−0.620349
$400$	1.42900e8	−	2.86321e8i	0.111640	−	0.223688i
$401$	−6.33359e8			−0.490506			−0.245253	−	0.969459i	$-0.578871\pi$
$401$	−6.33359e8			−0.490506			−0.245253	+	0.969459i	$0.578871\pi$
$402$			1.74560e9i			1.34015i
$403$		−	1.77009e8i		−	0.134718i
$404$	−4.26784e8			−0.322013
$405$	1.87089e9	−	1.15722e9i	1.39945	−	0.865610i
$406$	−1.51457e8			−0.112318
$407$		−	5.32013e8i		−	0.391149i
$408$		−	5.34258e8i		−	0.389440i
$409$	−1.41470e9			−1.02243			−0.511213	−	0.859454i	$-0.670804\pi$
$409$	−1.41470e9			−1.02243			−0.511213	+	0.859454i	$0.670804\pi$
$410$	−4.40372e8	−	7.11956e8i	−0.315556	−	0.510164i
$411$	2.47575e9			1.75898
$412$			1.34700e8i			0.0948916i
$413$			2.50665e7i			0.0175093i
$414$	4.40753e8			0.305277
$415$	−1.35251e9	−	2.18663e9i	−0.928908	−	1.50178i
$416$	−1.99716e8			−0.136015
$417$			3.56936e8i			0.241054i
$418$			1.44278e9i			0.966234i
$419$	8.14907e7			0.0541202			0.0270601	−	0.999634i	$-0.491385\pi$
$419$	8.14907e7			0.0541202			0.0270601	+	0.999634i	$0.491385\pi$
$420$	2.36457e8	−	1.46257e8i	0.155732	−	0.0963264i
$421$	2.06996e7			0.0135199			0.00675996	−	0.999977i	$-0.497848\pi$
$421$	2.06996e7			0.0135199			0.00675996	+	0.999977i	$0.497848\pi$
$422$		−	1.49114e9i		−	0.965883i
$423$		−	2.06177e9i		−	1.32449i
$424$	8.77043e8			0.558779
$425$	8.71696e8	+	4.35054e8i	0.550812	+	0.274905i
$426$	−3.29305e9			−2.06379
$427$			2.58995e8i			0.160988i
$428$			4.75897e8i			0.293400i
$429$	1.81620e9			1.11062
$430$	3.31322e8	−	2.04935e8i	0.200960	−	0.124302i
$431$	2.21145e9			1.33047			0.665237	−	0.746633i	$-0.268331\pi$
$431$	2.21145e9			1.33047			0.665237	+	0.746633i	$0.268331\pi$
$432$			9.00712e8i			0.537518i
$433$			1.16437e9i			0.689261i	0.938738	+	0.344630i	$0.111996\pi$
$433$			1.16437e9i			0.689261i	−0.938738	+	0.344630i	$0.888004\pi$
$434$	−4.31553e7			−0.0253408
$435$	1.25404e9	+	2.02743e9i	0.730464	+	1.18095i
$436$	−5.83768e8			−0.337316
$437$			5.79472e8i			0.332160i
$438$		−	7.97253e7i		−	0.0453353i
$439$	−7.24010e8			−0.408431			−0.204216	−	0.978926i	$-0.565464\pi$
$439$	−7.24010e8			−0.408431			−0.204216	+	0.978926i	$0.565464\pi$
$440$	−2.68087e8	−	4.33421e8i	−0.150035	−	0.242563i
$441$	−3.79920e9			−2.10939
$442$		−	6.08030e8i		−	0.334925i
$443$		−	1.52679e9i		−	0.834385i	−0.908818	−	0.417193i	$-0.863014\pi$
$443$		−	1.52679e9i		−	0.834385i	0.908818	−	0.417193i	$-0.136986\pi$
$444$	8.00057e8			0.433791
$445$	−1.52952e9	+	9.46066e8i	−0.822802	+	0.508934i
$446$	2.05998e9			1.09949
$447$			5.93307e9i			3.14198i
$448$			4.86915e7i			0.0255847i
$449$	2.71929e9			1.41773			0.708864	−	0.705345i	$-0.249208\pi$
$449$	2.71929e9			1.41773			0.708864	+	0.705345i	$0.249208\pi$
$450$	−2.69262e9	−	1.34386e9i	−1.39294	−	0.695199i
$451$	−1.33323e9			−0.684366
$452$		−	1.88098e8i		−	0.0958075i
$453$		−	2.81823e9i		−	1.42440i
$454$	8.80323e8			0.441516
$455$	2.69107e8	−	1.66453e8i	0.133932	−	0.0828422i
$456$	−2.16969e9			−1.07157
$457$		−	2.20279e9i		−	1.07961i	−0.841790	−	0.539805i	$-0.818498\pi$
$457$		−	2.20279e9i		−	1.07961i	0.841790	−	0.539805i	$-0.181502\pi$
$458$		−	3.75229e8i		−	0.182502i
$459$	−2.74219e9			−1.32359
$460$	−1.07673e8	−	1.74077e8i	−0.0515770	−	0.0833854i
$461$	−2.41318e9			−1.14719			−0.573597	−	0.819137i	$-0.694453\pi$
$461$	−2.41318e9			−1.14719			−0.573597	+	0.819137i	$0.694453\pi$
$462$		−	4.42797e8i		−	0.208909i
$463$			3.44274e9i			1.61202i	0.591902	+	0.806010i	$0.298377\pi$
$463$			3.44274e9i			1.61202i	−0.591902	+	0.806010i	$0.701623\pi$
$464$	−4.17491e8			−0.194014
$465$	3.57318e8	+	5.77682e8i	0.164805	+	0.266443i
$466$	1.28335e9			0.587482
$467$			2.27849e9i			1.03523i	0.855613	+	0.517616i	$0.173180\pi$
$467$			2.27849e9i			1.03523i	−0.855613	+	0.517616i	$0.826820\pi$
$468$			1.87817e9i			0.846983i
$469$	4.84350e8			0.216798
$470$	−8.14304e8	+	5.03677e8i	−0.361779	+	0.223774i
$471$	6.28898e9			2.77337
$472$			6.90957e7i			0.0302450i
$473$		−	6.20444e8i		−	0.269581i
$474$	3.15019e9			1.35867
$475$	1.76681e9	−	3.54007e9i	0.756418	−	1.51560i
$476$	−1.48240e8			−0.0630001
$477$		−	8.24788e9i		−	3.47959i
$478$			2.01318e9i			0.843110i
$479$	1.54719e8			0.0643235			0.0321618	−	0.999483i	$-0.489761\pi$
$479$	1.54719e8			0.0643235			0.0321618	+	0.999483i	$0.489761\pi$
$480$	6.51790e8	−	4.03157e8i	0.269007	−	0.166391i
$481$	9.10531e8			0.373067
$482$			1.84498e9i			0.750459i
$483$		−	1.77843e8i		−	0.0718162i
$484$	4.35541e8			0.174610
$485$	1.86711e9	+	3.01858e9i	0.743144	+	1.20145i
$486$	1.42128e9			0.561632
$487$		−	2.63948e9i		−	1.03554i	−0.855520	−	0.517770i	$-0.826762\pi$
$487$		−	2.63948e9i		−	1.03554i	0.855520	−	0.517770i	$-0.173238\pi$
$488$			7.13917e8i			0.278086i
$489$	−7.11698e9			−2.75242
$490$	9.28122e8	+	1.50051e9i	0.356384	+	0.576172i
$491$	−1.72366e9			−0.657151			−0.328576	−	0.944478i	$-0.606569\pi$
$491$	−1.72366e9			−0.657151			−0.328576	+	0.944478i	$0.606569\pi$
$492$		−	2.00495e9i		−	0.758973i
$493$		−	1.27104e9i		−	0.477743i
$494$	−2.46929e9			−0.921568
$495$	−4.07597e9	+	2.52114e9i	−1.51047	+	0.934285i
$496$	−1.18957e8			−0.0437729
$497$			9.13718e8i			0.333860i
$498$		−	6.15780e9i		−	2.23420i
$499$	−2.37910e9			−0.857158			−0.428579	−	0.903504i	$-0.640986\pi$
$499$	−2.37910e9			−0.857158			−0.428579	+	0.903504i	$0.640986\pi$
$500$	1.27028e8	+	1.39176e9i	0.0454471	+	0.497930i
$501$	4.27196e9			1.51773
$502$		−	2.11648e9i		−	0.746707i
$503$		−	4.58013e8i		−	0.160468i	−0.996776	−	0.0802342i	$-0.974433\pi$
$503$		−	4.58013e8i		−	0.160468i	0.996776	−	0.0802342i	$-0.0255668\pi$
$504$	4.57905e8			0.159319
$505$	1.58517e9	−	9.80489e8i	0.547717	−	0.338784i
$506$	−3.25983e8			−0.111858
$507$		−	2.14225e9i		−	0.730033i
$508$			1.12802e9i			0.381764i
$509$	−1.88963e9			−0.635132			−0.317566	−	0.948236i	$-0.602866\pi$
$509$	−1.88963e9			−0.635132			−0.317566	+	0.948236i	$0.602866\pi$
$510$	1.22740e9	+	1.98436e9i	0.409723	+	0.662406i
$511$	−2.21213e7			−0.00733394
$512$			1.34218e8i			0.0441942i
$513$			1.11364e10i			3.64195i
$514$	1.23664e9			0.401672
$515$	−3.09459e8	−	5.00307e8i	−0.0998338	−	0.161403i
$516$	9.33042e8			0.298970
$517$			1.52489e9i			0.485313i
$518$		−	2.21991e8i		−	0.0701747i
$519$	6.45390e9			2.02645
$520$	7.41792e8	−	4.58826e8i	0.231350	−	0.143099i
$521$	−4.36991e9			−1.35376			−0.676878	−	0.736096i	$-0.736667\pi$
$521$	−4.36991e9			−1.35376			−0.676878	+	0.736096i	$0.736667\pi$
$522$			3.92616e9i			1.20815i
$523$			7.59971e8i			0.232296i	0.993232	+	0.116148i	$0.0370547\pi$
$523$			7.59971e8i			0.232296i	−0.993232	+	0.116148i	$0.962945\pi$
$524$	−2.53113e8			−0.0768520
$525$	−5.42244e8	+	1.08647e9i	−0.163545	+	0.327687i
$526$	−3.44046e9			−1.03078
$527$		−	3.62162e8i		−	0.107787i
$528$		−	1.22056e9i		−	0.360863i
$529$	3.27390e9			0.961547
$530$	−3.25754e9	+	2.01491e9i	−0.950438	+	0.587881i
$531$	6.49789e8			0.188340
$532$			6.02021e8i			0.173349i
$533$		−	2.28180e9i		−	0.652729i
$534$	−4.30731e9			−1.22409
$535$	−1.09332e9	−	1.76759e9i	−0.308681	−	0.499049i
$536$	1.33511e9			0.374489
$537$		−	8.30119e8i		−	0.231329i
$538$		−	1.38503e7i		−	0.00383462i
$539$	2.80990e9			0.772913
$540$	−2.06929e9	−	3.34545e9i	−0.565513	−	0.914274i
$541$	−3.02681e9			−0.821854			−0.410927	−	0.911668i	$-0.634795\pi$
$541$	−3.02681e9			−0.821854			−0.410927	+	0.911668i	$0.634795\pi$
$542$		−	2.33206e9i		−	0.629132i
$543$		−	4.68871e9i		−	1.25676i
$544$	−4.08622e8			−0.108824
$545$	2.16825e9	−	1.34114e9i	0.573748	−	0.354884i
$546$	7.57838e8			0.199252
$547$		−	1.76298e9i		−	0.460566i	−0.973124	−	0.230283i	$-0.926035\pi$
$547$		−	1.76298e9i		−	0.460566i	0.973124	−	0.230283i	$-0.0739651\pi$
$548$		−	1.89355e9i		−	0.491525i
$549$	6.71382e9			1.73168
$550$	1.99147e9	+	9.93923e8i	0.510393	+	0.254732i
$551$	−5.16185e9			−1.31454
$552$		−	4.90223e8i		−	0.124053i
$553$		−	8.74080e8i		−	0.219793i
$554$	1.60846e9			0.401908
$555$	−2.97159e9	+	1.83804e9i	−0.737843	+	0.456384i
$556$	2.72999e8			0.0673596
$557$			6.96721e9i			1.70831i	0.520021	+	0.854153i	$0.325924\pi$
$557$			6.96721e9i			1.70831i	−0.520021	+	0.854153i	$0.674076\pi$
$558$			1.11870e9i			0.272579i
$559$	1.06188e9			0.257119
$560$	−1.11863e8	−	1.80851e8i	−0.0269172	−	0.0435175i
$561$	3.71597e9			0.888592
$562$		−	2.82952e9i		−	0.672413i
$563$			2.44613e9i			0.577698i	0.957375	+	0.288849i	$0.0932725\pi$
$563$			2.44613e9i			0.577698i	−0.957375	+	0.288849i	$0.906728\pi$
$564$	−2.29318e9			−0.538221
$565$	4.32134e8	+	6.98638e8i	0.100797	+	0.162961i
$566$	−7.86469e8			−0.182316
$567$		−	1.46189e9i		−	0.336801i
$568$			2.51865e9i			0.576699i
$569$	4.26784e9			0.971215			0.485607	−	0.874177i	$-0.338599\pi$
$569$	4.26784e9			0.971215			0.485607	+	0.874177i	$0.338599\pi$
$570$	8.05872e9	−	4.98462e9i	1.82265	−	1.12738i
$571$	3.15003e9			0.708090			0.354045	−	0.935228i	$-0.384806\pi$
$571$	3.15003e9			0.708090			0.354045	+	0.935228i	$0.384806\pi$
$572$		−	1.38910e9i		−	0.310348i
$573$		−	6.99679e9i		−	1.55367i
$574$	−5.56312e8			−0.122780
$575$	7.99847e8	+	3.99195e8i	0.175457	+	0.0875685i
$576$	1.26221e9			0.275203
$577$			1.59513e8i			0.0345685i	0.999851	+	0.0172843i	$0.00550202\pi$
$577$			1.59513e8i			0.0345685i	−0.999851	+	0.0172843i	$0.994498\pi$
$578$			2.03867e9i			0.439137i
$579$	−1.54034e10			−3.29794
$580$	1.55066e9	−	9.59139e8i	0.330003	−	0.204119i
$581$	−1.70860e9			−0.361429
$582$			8.50068e9i			1.78741i
$583$			6.10017e9i			1.27498i
$584$	−6.09771e7			−0.0126684
$585$	−4.31489e9	−	6.97595e9i	−0.891095	−	1.44065i
$586$	−2.10602e9			−0.432335
$587$			3.17099e9i			0.647085i	0.946214	+	0.323543i	$0.104874\pi$
$587$			3.17099e9i			0.647085i	−0.946214	+	0.323543i	$0.895126\pi$
$588$			4.22561e9i			0.857174i
$589$	−1.47078e9			−0.296583
$590$	−1.58740e8	−	2.56637e8i	−0.0318202	−	0.0514443i
$591$	−2.77850e9			−0.553674
$592$		−	6.11915e8i		−	0.121218i
$593$		−	5.65837e9i		−	1.11429i	−0.830414	−	0.557147i	$-0.811896\pi$
$593$		−	5.65837e9i		−	1.11429i	0.830414	−	0.557147i	$-0.188104\pi$
$594$	−6.26480e9			−1.22646
$595$	5.50597e8	−	3.40565e8i	0.107158	−	0.0662813i
$596$	4.53785e9			0.877987
$597$			1.31849e10i			2.53610i
$598$		−	5.57914e8i		−	0.106687i
$599$	3.67540e9			0.698732			0.349366	−	0.936986i	$-0.386397\pi$
$599$	3.67540e9			0.698732			0.349366	+	0.936986i	$0.386397\pi$
$600$	−1.49469e9	+	2.99483e9i	−0.282502	+	0.566035i
$601$	5.94912e9			1.11787			0.558936	−	0.829211i	$-0.311210\pi$
$601$	5.94912e9			1.11787			0.558936	+	0.829211i	$0.311210\pi$
$602$		−	2.58890e8i		−	0.0483646i
$603$		−	1.25556e10i		−	2.33199i
$604$	−2.15549e9			−0.398032
$605$	−1.61770e9	+	1.00061e9i	−0.296998	+	0.183704i
$606$	4.46403e9			0.814842
$607$			5.91556e9i			1.07358i	0.843715	+	0.536792i	$0.180364\pi$
$607$			5.91556e9i			1.07358i	−0.843715	+	0.536792i	$0.819636\pi$
$608$			1.65946e9i			0.299437i
$609$	1.58420e9			0.284217
$610$	−1.64015e9	−	2.65165e9i	−0.292569	−	0.473001i
$611$	−2.60982e9			−0.462879
$612$			3.84276e9i			0.677663i
$613$		−	2.91525e8i		−	0.0511169i	−0.999673	−	0.0255585i	$-0.991864\pi$
$613$		−	2.91525e8i		−	0.0511169i	0.999673	−	0.0255585i	$-0.00813639\pi$
$614$	3.23311e9			0.563677
$615$	4.60616e9	+	7.44685e9i	0.798502	+	1.29095i
$616$	−3.38669e8			−0.0583771
$617$		−	6.88662e9i		−	1.18034i	−0.807278	−	0.590171i	$-0.799060\pi$
$617$		−	6.88662e9i		−	1.18034i	0.807278	−	0.590171i	$-0.200940\pi$
$618$		−	1.40892e9i		−	0.240120i
$619$	−5.67213e9			−0.961234			−0.480617	−	0.876931i	$-0.659587\pi$
$619$	−5.67213e9			−0.961234			−0.480617	+	0.876931i	$0.659587\pi$
$620$	4.41834e8	−	2.73291e8i	0.0744541	−	0.0460527i
$621$	−2.51617e9			−0.421619
$622$		−	4.67408e9i		−	0.778807i
$623$			1.19514e9i			0.198021i
$624$	2.08897e9			0.344181
$625$	−3.66922e9	−	4.87747e9i	−0.601165	−	0.799125i
$626$	6.86976e9			1.11926
$627$		−	1.50910e10i		−	2.44502i
$628$		−	4.81006e9i		−	0.774983i
$629$	1.86296e9			0.298487
$630$	−1.70076e9	+	1.05199e9i	−0.270989	+	0.167617i
$631$	−1.36737e9			−0.216662			−0.108331	−	0.994115i	$-0.534551\pi$
$631$	−1.36737e9			−0.216662			−0.108331	+	0.994115i	$0.534551\pi$
$632$		−	2.40939e9i		−	0.379662i
$633$			1.55969e10i			2.44413i
$634$	−8.11948e8			−0.126536
$635$	−2.59151e9	−	4.18974e9i	−0.401647	−	0.649350i
$636$	−9.17361e9			−1.41397
$637$			4.80910e9i			0.737184i
$638$		−	2.90381e9i		−	0.442686i
$639$	2.36859e10			3.59118
$640$	−3.08350e8	−	4.98515e8i	−0.0464959	−	0.0751707i
$641$	6.50671e9			0.975794			0.487897	−	0.872901i	$-0.337764\pi$
$641$	6.50671e9			0.975794			0.487897	+	0.872901i	$0.337764\pi$
$642$		−	4.97774e9i		−	0.742437i
$643$			1.00325e10i			1.48824i	0.668048	+	0.744118i	$0.267130\pi$
$643$			1.00325e10i			1.48824i	−0.668048	+	0.744118i	$0.732870\pi$
$644$	−1.36022e8			−0.0200681
$645$	−3.46553e9	+	2.14356e9i	−0.508523	+	0.314541i
$646$	−5.05219e9			−0.737337
$647$		−	7.56024e9i		−	1.09741i	−0.836015	−	0.548707i	$-0.815120\pi$
$647$		−	7.56024e9i		−	1.09741i	0.836015	−	0.548707i	$-0.184880\pi$
$648$		−	4.02968e9i		−	0.581779i
$649$	−4.80587e8			−0.0690106
$650$	−1.70108e9	+	3.40837e9i	−0.242956	+	0.486799i
$651$	4.51392e8			0.0641240
$652$			5.44335e9i			0.769130i
$653$		−	7.10083e9i		−	0.997960i	−0.866613	−	0.498980i	$-0.833708\pi$
$653$		−	7.10083e9i		−	0.997960i	0.866613	−	0.498980i	$-0.166292\pi$
$654$	6.10604e9			0.853567
$655$	9.40120e8	−	5.81499e8i	0.130719	−	0.0808546i
$656$	−1.53347e9			−0.212086
$657$			5.73440e8i			0.0788877i
$658$			6.36285e8i			0.0870685i
$659$	2.72098e8			0.0370362			0.0185181	−	0.999829i	$-0.494105\pi$
$659$	2.72098e8			0.0370362			0.0185181	+	0.999829i	$0.494105\pi$
$660$	2.80411e9	+	4.53345e9i	0.379657	+	0.613798i
$661$	9.94289e8			0.133908			0.0669541	−	0.997756i	$-0.478672\pi$
$661$	9.94289e8			0.133908			0.0669541	+	0.997756i	$0.478672\pi$
$662$			4.36422e9i			0.584661i
$663$			6.35982e9i			0.847515i
$664$	−4.70973e9			−0.624321
$665$	−1.38308e9	−	2.23604e9i	−0.182377	−	0.294852i
$666$	−5.75457e9			−0.754837
$667$		−	1.16628e9i		−	0.152181i
$668$		−	3.26736e9i		−	0.424111i
$669$	−2.15468e10			−2.78222
$670$	−4.95889e9	+	3.06726e9i	−0.636975	+	0.393993i
$671$	−4.96557e9			−0.634513
$672$		−	5.09299e8i		−	0.0647412i
$673$		−	4.39633e9i		−	0.555953i	−0.960588	−	0.277976i	$-0.910336\pi$
$673$		−	4.39633e9i		−	0.555953i	0.960588	−	0.277976i	$-0.0896637\pi$
$674$	−5.58852e9			−0.703052
$675$	1.53716e10	+	7.67181e9i	1.92378	+	0.960140i
$676$	−1.63848e9			−0.203999
$677$		−	5.81617e9i		−	0.720406i	−0.932874	−	0.360203i	$-0.882708\pi$
$677$		−	5.81617e9i		−	0.720406i	0.932874	−	0.360203i	$-0.117292\pi$
$678$			1.96745e9i			0.242437i
$679$	2.35867e9			0.289150
$680$	1.51772e9	−	9.38764e8i	0.185101	−	0.114492i
$681$	−9.20792e9			−1.11724
$682$		−	8.27394e8i		−	0.0998773i
$683$			1.01860e10i			1.22329i	0.791130	+	0.611647i	$0.209493\pi$
$683$			1.01860e10i			1.22329i	−0.791130	+	0.611647i	$0.790507\pi$
$684$	1.56059e10			1.86463
$685$	4.35023e9	+	7.03309e9i	0.517125	+	0.836044i
$686$	2.39622e9			0.283395
$687$			3.92478e9i			0.461814i
$688$		−	7.13628e8i		−	0.0835434i
$689$	−1.04403e10			−1.21604
$690$	1.12623e9	+	1.82080e9i	0.130514	+	0.211004i
$691$	1.04954e10			1.21011			0.605056	−	0.796183i	$-0.293151\pi$
$691$	1.04954e10			1.21011			0.605056	+	0.796183i	$0.293151\pi$
$692$		−	4.93620e9i		−	0.566268i
$693$			3.18490e9i			0.363522i
$694$	4.04359e9			0.459207
$695$	−1.01398e9	+	6.27185e8i	−0.114573	+	0.0708678i
$696$	4.36683e9			0.490946
$697$		−	4.66860e9i		−	0.522242i
$698$			8.40437e9i			0.935430i
$699$	−1.34235e10			−1.48660
$700$	8.30973e8	+	4.14730e8i	0.0915680	+	0.0457006i
$701$	1.44399e10			1.58325			0.791626	−	0.611006i	$-0.209235\pi$
$701$	1.44399e10			1.58325			0.791626	+	0.611006i	$0.209235\pi$
$702$		−	1.07221e10i		−	1.16977i
$703$		−	7.56571e9i		−	0.821308i
$704$	−9.33536e8			−0.100839
$705$	8.51738e9	−	5.26832e9i	0.915469	−	0.566252i
$706$	1.08231e10			1.15754
$707$		−	1.23863e9i		−	0.131818i
$708$		−	7.22720e8i		−	0.0765339i
$709$	5.52527e8			0.0582226			0.0291113	−	0.999576i	$-0.490732\pi$
$709$	5.52527e8			0.0582226			0.0291113	+	0.999576i	$0.490732\pi$
$710$	−5.78633e9	−	9.35486e9i	−0.606735	−	0.980919i
$711$	−2.26584e10			−2.36421
$712$			3.29440e9i			0.342056i
$713$		−	3.32311e8i		−	0.0343346i
$714$	1.55055e9			0.159419
$715$	3.19131e9	+	5.15945e9i	0.326511	+	0.527876i
$716$	−6.34908e8			−0.0646421
$717$		−	2.10572e10i		−	2.13346i
$718$			3.54984e9i			0.357910i
$719$	−3.15086e8			−0.0316139			−0.0158069	−	0.999875i	$-0.505032\pi$
$719$	−3.15086e8			−0.0316139			−0.0158069	+	0.999875i	$0.505032\pi$
$720$	−4.68813e9	+	2.89979e9i	−0.468097	+	0.289536i
$721$	−3.90932e8			−0.0388444
$722$			1.33666e10i			1.32173i
$723$		−	1.92980e10i		−	1.89901i
$724$	−3.58611e9			−0.351187
$725$	−3.55597e9	+	7.12492e9i	−0.346558	+	0.694380i
$726$	−4.55563e9			−0.441845
$727$			2.96166e9i			0.285867i	0.989732	+	0.142934i	$0.0456535\pi$
$727$			2.96166e9i			0.285867i	−0.989732	+	0.142934i	$0.954346\pi$
$728$		−	5.79625e8i		−	0.0556785i
$729$	2.34658e9			0.224330
$730$	2.26483e8	−	1.40088e8i	0.0215479	−	0.0133282i
$731$	2.17262e9			0.205718
$732$		−	7.46737e9i		−	0.703686i
$733$		−	2.04158e10i		−	1.91471i	−0.288914	−	0.957355i	$-0.593294\pi$
$733$		−	2.04158e10i		−	1.91471i	0.288914	−	0.957355i	$-0.406706\pi$
$734$	1.20840e10			1.12791
$735$	−9.70788e9	−	1.56949e10i	−0.901818	−	1.45798i
$736$	−3.74942e8			−0.0346650
$737$			9.28618e9i			0.854478i
$738$			1.44210e10i			1.32068i
$739$	1.24593e10			1.13563			0.567817	−	0.823155i	$-0.307788\pi$
$739$	1.24593e10			1.13563			0.567817	+	0.823155i	$0.307788\pi$
$740$	1.40581e9	+	2.27279e9i	0.127531	+	0.206181i
$741$	2.58280e10			2.33199
$742$			2.54539e9i			0.228739i
$743$			8.88793e9i			0.794950i	0.917613	+	0.397475i	$0.130114\pi$
$743$			8.88793e9i			0.794950i	−0.917613	+	0.397475i	$0.869886\pi$
$744$	1.24426e9			0.110766
$745$	−1.68546e10	+	1.04252e10i	−1.49338	+	0.923714i
$746$	−1.31487e10			−1.15957
$747$			4.42912e10i			3.88773i
$748$		−	2.84212e9i		−	0.248306i
$749$	−1.38117e9			−0.120105
$750$	−1.32868e9	−	1.45574e10i	−0.115002	−	1.25999i
$751$	7.69514e9			0.662944			0.331472	−	0.943465i	$-0.392455\pi$
$751$	7.69514e9			0.662944			0.331472	+	0.943465i	$0.392455\pi$
$752$			1.75391e9i			0.150399i
$753$			2.21377e10i			1.88951i
$754$	4.96982e9			0.422222
$755$	8.00600e9	−	4.95201e9i	0.677019	−	0.418762i
$756$	−2.61409e9			−0.220036
$757$			1.31095e9i			0.109838i	0.998491	+	0.0549189i	$0.0174900\pi$
$757$			1.31095e9i			0.109838i	−0.998491	+	0.0549189i	$0.982510\pi$
$758$		−	2.92705e8i		−	0.0244111i
$759$	3.40969e9			0.283053
$760$	−3.81244e9	−	6.16363e9i	−0.315032	−	0.509318i
$761$	−4.20324e9			−0.345731			−0.172865	−	0.984945i	$-0.555303\pi$
$761$	−4.20324e9			−0.345731			−0.172865	+	0.984945i	$0.555303\pi$
$762$		−	1.17988e10i		−	0.966041i
$763$		−	1.69424e9i		−	0.138082i
$764$	−5.35142e9			−0.434153
$765$	−8.82832e9	−	1.42729e10i	−0.712957	−	1.15265i
$766$	−4.55111e9			−0.365862
$767$		−	8.22516e8i		−	0.0658204i
$768$		−	1.40388e9i		−	0.111832i
$769$	−4.90370e9			−0.388850			−0.194425	−	0.980917i	$-0.562284\pi$
$769$	−4.90370e9			−0.388850			−0.194425	+	0.980917i	$0.562284\pi$
$770$	1.25789e9	−	7.78053e8i	0.0992946	−	0.0614175i
$771$	−1.29349e10			−1.01642
$772$			1.17812e10i			0.921569i
$773$			8.59620e9i			0.669389i	0.942327	+	0.334694i	$0.108633\pi$
$773$			8.59620e9i			0.669389i	−0.942327	+	0.334694i	$0.891367\pi$
$774$	−6.71109e9			−0.520235
$775$	−1.01322e9	+	2.03013e9i	−0.0781892	+	0.156664i
$776$	6.50166e9			0.499469
$777$			2.32196e9i			0.177575i
$778$		−	1.68146e10i		−	1.28014i
$779$	−1.89598e10			−1.43698
$780$	−7.75892e9	+	4.79919e9i	−0.585424	+	0.362107i
$781$	−1.75182e10			−1.31586
$782$		−	1.14150e9i		−	0.0853596i
$783$		−	2.24137e10i		−	1.66858i
$784$	3.23192e9			0.239527
$785$	1.10506e10	+	1.78657e10i	0.815345	+	1.31818i
$786$	2.64749e9			0.194471
$787$		−	1.11113e10i		−	0.812555i	−0.913750	−	0.406277i	$-0.866827\pi$
$787$		−	1.11113e10i		−	0.812555i	0.913750	−	0.406277i	$-0.133173\pi$
$788$			2.12511e9i			0.154717i
$789$	3.59862e10			2.60836
$790$	5.53531e9	+	8.94903e9i	0.399436	+	0.645775i
$791$	5.45906e8			0.0392193
$792$			8.77916e9i			0.627935i
$793$		−	8.49849e9i		−	0.605181i
$794$	−1.30209e10			−0.923143
$795$	3.40729e10	−	2.10753e10i	2.40505	−	1.48761i
$796$	1.00843e10			0.708683
$797$			6.73976e9i			0.471564i	0.971806	+	0.235782i	$0.0757650\pi$
$797$			6.73976e9i			0.471564i	−0.971806	+	0.235782i	$0.924235\pi$
$798$		−	6.29696e9i		−	0.438653i
$799$	−5.33974e9			−0.370345
$800$	2.29057e9	+	1.14320e9i	0.158171	+	0.0789417i
$801$	3.09812e10			2.13002
$802$		−	5.06687e9i		−	0.346840i
$803$		−	4.24119e8i		−	0.0289057i
$804$	−1.39648e10			−0.947631
$805$	5.05215e8	−	3.12494e8i	0.0341343	−	0.0211133i
$806$	1.41607e9			0.0952603
$807$			1.44870e8i			0.00970336i
$808$		−	3.41427e9i		−	0.227697i
$809$	−2.01007e9			−0.133472			−0.0667362	−	0.997771i	$-0.521259\pi$
$809$	−2.01007e9			−0.133472			−0.0667362	+	0.997771i	$0.521259\pi$
$810$	9.25774e9	+	1.49671e10i	0.612079	+	0.989558i
$811$	−1.56045e10			−1.02725			−0.513626	−	0.858014i	$-0.671698\pi$
$811$	−1.56045e10			−1.02725			−0.513626	+	0.858014i	$0.671698\pi$
$812$		−	1.21166e9i		−	0.0794208i
$813$			2.43926e10i			1.59200i
$814$	4.25611e9			0.276584
$815$	−1.25055e10	−	2.02178e10i	−0.809188	−	1.30823i
$816$	4.27407e9			0.275376
$817$		−	8.82328e9i		−	0.566047i
$818$		−	1.13176e10i		−	0.722964i
$819$	−5.45091e9			−0.346717
$820$	5.69565e9	−	3.52297e9i	0.360740	−	0.223132i
$821$	−7.82354e9			−0.493403			−0.246702	−	0.969091i	$-0.579347\pi$
$821$	−7.82354e9			−0.493403			−0.246702	+	0.969091i	$0.579347\pi$
$822$			1.98060e10i			1.24379i
$823$		−	9.42050e9i		−	0.589080i	−0.955639	−	0.294540i	$-0.904834\pi$
$823$		−	9.42050e9i		−	0.589080i	0.955639	−	0.294540i	$-0.0951665\pi$
$824$	−1.07760e9			−0.0670985
$825$	−2.08302e10	−	1.03961e10i	−1.29153	−	0.644590i
$826$	−2.00532e8			−0.0123810
$827$		−	3.29188e9i		−	0.202383i	−0.994867	−	0.101192i	$-0.967734\pi$
$827$		−	3.29188e9i		−	0.202383i	0.994867	−	0.101192i	$-0.0322655\pi$
$828$			3.52603e9i			0.215864i
$829$	−1.72231e10			−1.04995			−0.524976	−	0.851117i	$-0.675926\pi$
$829$	−1.72231e10			−1.04995			−0.524976	+	0.851117i	$0.675926\pi$
$830$	1.74930e10	−	1.08201e10i	1.06192	−	0.656837i
$831$	−1.68240e10			−1.01701
$832$		−	1.59773e9i		−	0.0961771i
$833$			9.83948e9i			0.589813i
$834$	−2.85549e9			−0.170451
$835$	7.50641e9	+	1.21357e10i	0.446200	+	0.721379i
$836$	−1.15422e10			−0.683231
$837$		−	6.38642e9i		−	0.376459i
$838$			6.51926e8i			0.0382687i
$839$	9.34237e9			0.546123			0.273061	−	0.961997i	$-0.411964\pi$
$839$	9.34237e9			0.546123			0.273061	+	0.961997i	$0.411964\pi$
$840$	1.17006e9	+	1.89165e9i	0.0681130	+	0.110119i
$841$	−6.86087e9			−0.397735
$842$			1.65597e8i			0.00956003i
$843$			2.95960e10i			1.70152i
$844$	1.19291e10			0.682983
$845$	6.08569e9	−	3.76423e9i	0.346985	−	0.214623i
$846$	1.64941e10			0.936555
$847$			1.26405e9i			0.0714777i
$848$			7.01634e9i			0.395117i
$849$	8.22623e9			0.461343
$850$	−3.48043e9	+	6.97357e9i	−0.194387	+	0.389483i
$851$	1.70941e9			0.0950806
$852$		−	2.63444e10i		−	1.45932i
$853$			1.68999e9i			0.0932312i	0.998913	+	0.0466156i	$0.0148436\pi$
$853$			1.68999e9i			0.0932312i	−0.998913	+	0.0466156i	$0.985156\pi$
$854$	−2.07196e9			−0.113836
$855$	−5.79640e10	+	3.58529e10i	−3.17159	+	1.96175i
$856$	−3.80718e9			−0.207465
$857$		−	3.51222e9i		−	0.190611i	−0.995448	−	0.0953057i	$-0.969617\pi$
$857$		−	3.51222e9i		−	0.190611i	0.995448	−	0.0953057i	$-0.0303829\pi$
$858$			1.45296e10i			0.785323i
$859$	2.67283e10			1.43878			0.719392	−	0.694604i	$-0.244421\pi$
$859$	2.67283e10			1.43878			0.719392	+	0.694604i	$0.244421\pi$
$860$	1.63948e9	+	2.65058e9i	0.0878945	+	0.142101i
$861$	5.81886e9			0.310690
$862$			1.76916e10i			0.940787i
$863$			7.78348e9i			0.412227i	0.978528	+	0.206113i	$0.0660815\pi$
$863$			7.78348e9i			0.412227i	−0.978528	+	0.206113i	$0.933918\pi$
$864$	−7.20570e9			−0.380083
$865$	1.13404e10	+	1.83342e10i	0.595760	+	0.963175i
$866$	−9.31497e9			−0.487381
$867$		−	2.13239e10i		−	1.11122i
$868$		−	3.45243e8i		−	0.0179187i
$869$	1.67582e10			0.866282
$870$	−1.62194e10	+	1.00323e10i	−0.835060	+	0.516516i
$871$	−1.58931e10			−0.814978
$872$		−	4.67014e9i		−	0.238519i
$873$		−	6.11429e10i		−	3.11025i
$874$	−4.63577e9			−0.234873
$875$	−4.03922e9	+	3.68667e8i	−0.203830	+	0.0186040i
$876$	6.37802e8			0.0320569
$877$		−	9.46355e9i		−	0.473757i	−0.971539	−	0.236878i	$-0.923876\pi$
$877$		−	9.46355e9i		−	0.473757i	0.971539	−	0.236878i	$-0.0761243\pi$
$878$		−	5.79208e9i		−	0.288805i
$879$	2.20283e10			1.09401
$880$	3.46737e9	−	2.14470e9i	0.171518	−	0.106090i
$881$	2.29088e10			1.12872			0.564360	−	0.825529i	$-0.309123\pi$
$881$	2.29088e10			1.12872			0.564360	+	0.825529i	$0.309123\pi$
$882$		−	3.03936e10i		−	1.49156i
$883$			1.00814e10i			0.492784i	0.969170	+	0.246392i	$0.0792451\pi$
$883$			1.00814e10i			0.492784i	−0.969170	+	0.246392i	$0.920755\pi$
$884$	4.86424e9			0.236828
$885$	1.66037e9	+	2.68435e9i	0.0805199	+	0.130178i
$886$	1.22143e10			0.589999
$887$		−	2.86392e10i		−	1.37793i	−0.724792	−	0.688967i	$-0.758064\pi$
$887$		−	2.86392e10i		−	1.37793i	0.724792	−	0.688967i	$-0.241936\pi$
$888$			6.40045e9i			0.306736i
$889$	−3.27380e9			−0.156277
$890$	−7.56853e9	−	1.22362e10i	−0.359871	−	0.581809i
$891$	2.80280e10			1.32745
$892$			1.64798e10i			0.777456i
$893$			2.16853e10i			1.01903i
$894$	−4.74645e10			−2.22171
$895$	2.35819e9	−	1.45863e9i	0.109951	−	0.0680088i
$896$	−3.89532e8			−0.0180911
$897$			5.83562e9i			0.269969i
$898$			2.17543e10i			1.00249i
$899$	2.96018e9			0.135881
$900$	1.07509e10	−	2.15409e10i	0.491580	−	0.984954i
$901$	−2.13611e10			−0.972940
$902$		−	1.06659e10i		−	0.483920i
$903$			2.70791e9i			0.122385i
$904$	1.50478e9			0.0677461
$905$	1.33196e10	−	8.23869e9i	0.597341	−	0.369478i
$906$	2.25458e10			1.00720
$907$			3.43088e10i			1.52679i	0.645930	+	0.763397i	$0.276470\pi$
$907$			3.43088e10i			1.52679i	−0.645930	+	0.763397i	$0.723530\pi$
$908$			7.04258e9i			0.312199i
$909$	−3.21085e10			−1.41790
$910$	1.33162e9	+	2.15286e9i	0.0585783	+	0.0947045i
$911$	1.71466e10			0.751385			0.375693	−	0.926744i	$-0.377405\pi$
$911$	1.71466e10			0.751385			0.375693	+	0.926744i	$0.377405\pi$
$912$		−	1.73575e10i		−	0.757714i
$913$		−	3.27580e10i		−	1.42452i
$914$	1.76224e10			0.763400
$915$	1.71554e10	+	2.77355e10i	0.740335	+	1.19691i
$916$	3.00183e9			0.129048
$917$		−	7.34596e8i		−	0.0314598i
$918$		−	2.19375e10i		−	0.935920i
$919$	1.75219e10			0.744692			0.372346	−	0.928094i	$-0.378553\pi$
$919$	1.75219e10			0.744692			0.372346	+	0.928094i	$0.378553\pi$
$920$	1.39262e9	−	8.61388e8i	0.0589624	−	0.0364705i
$921$	−3.38173e10			−1.42637
$922$		−	1.93055e10i		−	0.811189i
$923$		−	2.99821e10i		−	1.25504i
$924$	3.54237e9			0.147721
$925$	−1.04430e10	−	5.21198e9i	−0.433839	−	0.216524i
$926$	−2.75419e10			−1.13987
$927$			1.01340e10i			0.417831i
$928$		−	3.33993e9i		−	0.137189i
$929$	−1.78228e10			−0.729324			−0.364662	−	0.931140i	$-0.618816\pi$
$929$	−1.78228e10			−0.729324			−0.364662	+	0.931140i	$0.618816\pi$
$930$	−4.62146e9	+	2.85855e9i	−0.188403	+	0.116535i
$931$	3.99594e10			1.62291
$932$			1.02668e10i			0.415412i
$933$			4.88895e10i			1.97074i
$934$	−1.82279e10			−0.732020
$935$	6.52947e9	+	1.05563e10i	0.261238	+	0.422348i
$936$	−1.50254e10			−0.598907
$937$			3.74390e10i			1.48674i	0.668879	+	0.743371i	$0.266774\pi$
$937$			3.74390e10i			1.48674i	−0.668879	+	0.743371i	$0.733226\pi$
$938$			3.87480e9i			0.153299i
$939$	−7.18557e10			−2.83225
$940$	−4.02942e9	−	6.51443e9i	−0.158232	−	0.255817i
$941$	−5.59371e9			−0.218845			−0.109422	−	0.993995i	$-0.534900\pi$
$941$	−5.59371e9			−0.218845			−0.109422	+	0.993995i	$0.534900\pi$
$942$			5.03118e10i			1.96107i
$943$		−	4.28380e9i		−	0.166356i
$944$	−5.52765e8			−0.0213865
$945$	9.70931e9	−	6.00557e9i	0.374263	−	0.231496i
$946$	4.96356e9			0.190622
$947$		−	4.72461e10i		−	1.80776i	−0.427785	−	0.903880i	$-0.640706\pi$
$947$		−	4.72461e10i		−	1.80776i	0.427785	−	0.903880i	$-0.359294\pi$
$948$			2.52015e10i			0.960722i
$949$	7.25872e8			0.0275695
$950$	2.83205e10	+	1.41345e10i	1.07169	+	0.534869i
$951$	8.49273e9			0.320196
$952$		−	1.18592e9i		−	0.0445478i
$953$		−	1.44402e10i		−	0.540442i	−0.962798	−	0.270221i	$-0.912903\pi$
$953$		−	1.44402e10i		−	0.540442i	0.962798	−	0.270221i	$-0.0870967\pi$
$954$	6.59830e10			2.46044
$955$	1.98764e10	−	1.22943e10i	0.738458	−	0.456764i
$956$	−1.61054e10			−0.596169
$957$			3.03730e10i			1.12020i
$958$			1.23775e9i			0.0454836i
$959$	5.49555e9			0.201209
$960$	3.22525e9	+	5.21432e9i	0.117656	+	0.190217i
$961$	−2.66692e10			−0.969343
$962$			7.28425e9i			0.263798i
$963$			3.58034e10i			1.29191i
$964$	−1.47598e10			−0.530655
$965$	−2.70659e10	−	4.37579e10i	−0.969566	−	1.56751i
$966$	1.42275e9			0.0507817
$967$		−	3.52615e10i		−	1.25403i	−0.779008	−	0.627014i	$-0.784277\pi$
$967$		−	3.52615e10i		−	1.25403i	0.779008	−	0.627014i	$-0.215723\pi$
$968$			3.48433e9i			0.123468i
$969$	5.28445e10			1.86580
$970$	−2.41486e10	+	1.49368e10i	−0.849555	+	0.525482i
$971$	6.30626e9			0.221057			0.110528	−	0.993873i	$-0.464746\pi$
$971$	6.30626e9			0.221057			0.110528	+	0.993873i	$0.464746\pi$
$972$			1.13702e10i			0.397133i
$973$			7.92310e8i			0.0275740i
$974$	2.11158e10			0.732237
$975$	1.77928e10	−	3.56505e10i	0.614792	−	1.23183i
$976$	−5.71134e9			−0.196636
$977$			2.25969e10i			0.775206i	0.921826	+	0.387603i	$0.126697\pi$
$977$			2.25969e10i			0.775206i	−0.921826	+	0.387603i	$0.873303\pi$
$978$		−	5.69358e10i		−	1.94625i
$979$	−2.29138e10			−0.780474
$980$	−1.20041e10	+	7.42497e9i	−0.407415	+	0.252002i
$981$	−4.39189e10			−1.48529
$982$		−	1.37892e10i		−	0.464676i
$983$			1.48266e10i			0.497855i	0.968522	+	0.248928i	$0.0800781\pi$
$983$			1.48266e10i			0.497855i	−0.968522	+	0.248928i	$0.919922\pi$
$984$	1.60396e10			0.536675
$985$	−4.88220e9	−	7.89313e9i	−0.162775	−	0.263161i
$986$	1.01683e10			0.337816
$987$		−	6.65535e9i		−	0.220324i
$988$		−	1.97543e10i		−	0.651647i
$989$	1.99354e9			0.0655298
$990$	−2.01691e10	−	3.26078e10i	−0.660639	−	1.06807i
$991$	−4.13851e10			−1.35079			−0.675393	−	0.737458i	$-0.736026\pi$
$991$	−4.13851e10			−1.35079			−0.675393	+	0.737458i	$0.736026\pi$
$992$		−	9.51658e8i		−	0.0309521i
$993$		−	4.56485e10i		−	1.47946i
$994$	−7.30975e9			−0.236075
$995$	−3.74556e10	+	2.31677e10i	−1.20541	+	0.745593i
$996$	4.92624e10			1.57982
$997$			4.69228e9i			0.149952i	0.997185	+	0.0749758i	$0.0238879\pi$
$997$			4.69228e9i			0.149952i	−0.997185	+	0.0749758i	$0.976112\pi$
$998$		−	1.90328e10i		−	0.606102i
$999$	3.28517e10			1.04251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.8.b.a.9.3	yes	4
3.2	odd	2		90.8.c.c.19.1		4
4.3	odd	2		80.8.c.d.49.4		4
5.2	odd	4		50.8.a.i.1.1		2
5.3	odd	4		50.8.a.j.1.2		2
5.4	even	2	inner	10.8.b.a.9.2	✓	4
8.3	odd	2		320.8.c.g.129.1		4
8.5	even	2		320.8.c.f.129.4		4
15.2	even	4		450.8.a.bi.1.2		2
15.8	even	4		450.8.a.bd.1.1		2
15.14	odd	2		90.8.c.c.19.3		4
20.3	even	4		400.8.a.t.1.1		2
20.7	even	4		400.8.a.bf.1.2		2
20.19	odd	2		80.8.c.d.49.1		4
40.19	odd	2		320.8.c.g.129.4		4
40.29	even	2		320.8.c.f.129.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.8.b.a.9.2	✓	4	5.4	even	2	inner
10.8.b.a.9.3	yes	4	1.1	even	1	trivial
50.8.a.i.1.1		2	5.2	odd	4
50.8.a.j.1.2		2	5.3	odd	4
80.8.c.d.49.1		4	20.19	odd	2
80.8.c.d.49.4		4	4.3	odd	2
90.8.c.c.19.1		4	3.2	odd	2
90.8.c.c.19.3		4	15.14	odd	2
320.8.c.f.129.1		4	40.29	even	2
320.8.c.f.129.4		4	8.5	even	2
320.8.c.g.129.1		4	8.3	odd	2
320.8.c.g.129.4		4	40.19	odd	2
400.8.a.t.1.1		2	20.3	even	4
400.8.a.bf.1.2		2	20.7	even	4
450.8.a.bd.1.1		2	15.8	even	4
450.8.a.bi.1.2		2	15.2	even	4

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:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	10.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} -83.6776i q^{3} -64.0000 q^{4} +(237.711 - 147.033i) q^{5} +669.421 q^{6} -185.744i q^{7} -512.000i q^{8} -4814.95 q^{9} +O(q^{10})\)	\(q+8.00000i q^{2} -83.6776i q^{3} -64.0000 q^{4} +(237.711 - 147.033i) q^{5} +669.421 q^{6} -185.744i q^{7} -512.000i q^{8} -4814.95 q^{9} +(1176.26 + 1901.68i) q^{10} +3561.16 q^{11} +5355.37i q^{12} +6094.86i q^{13} +1485.95 q^{14} +(-12303.4 - 19891.1i) q^{15} +4096.00 q^{16} +12470.2i q^{17} -38519.6i q^{18} +50642.8 q^{19} +(-15213.5 + 9410.11i) q^{20} -15542.6 q^{21} +28489.3i q^{22} +11442.3i q^{23} -42843.0 q^{24} +(34887.6 - 69902.6i) q^{25} -48758.8 q^{26} +219900. i q^{27} +11887.6i q^{28} -101926. q^{29} +(159128. - 98427.0i) q^{30} -29042.3 q^{31} +32768.0i q^{32} -297989. i q^{33} -99761.2 q^{34} +(-27310.4 - 44153.2i) q^{35} +308157. q^{36} -149393. i q^{37} +405143. i q^{38} +510003. q^{39} +(-75280.9 - 121708. i) q^{40} -374382. q^{41} -124341. i q^{42} -174226. i q^{43} -227914. q^{44} +(-1.14456e6 + 707956. i) q^{45} -91538.6 q^{46} +428201. i q^{47} -342744. i q^{48} +789042. q^{49} +(559221. + 279101. i) q^{50} +1.04347e6 q^{51} -390071. i q^{52} +1.71297e6i q^{53} -1.75920e6 q^{54} +(846525. - 523607. i) q^{55} -95100.7 q^{56} -4.23767e6i q^{57} -815412. i q^{58} -134952. q^{59} +(787416. + 1.27303e6i) q^{60} -1.39437e6 q^{61} -232338. i q^{62} +894345. i q^{63} -262144. q^{64} +(896145. + 1.44881e6i) q^{65} +2.38391e6 q^{66} +2.60763e6i q^{67} -798090. i q^{68} +957466. q^{69} +(353226. - 218483. i) q^{70} -4.91925e6 q^{71} +2.46525e6i q^{72} -119096. i q^{73} +1.19515e6 q^{74} +(-5.84928e6 - 2.91932e6i) q^{75} -3.24114e6 q^{76} -661462. i q^{77} +4.08003e6i q^{78} +4.70584e6 q^{79} +(973663. - 602247. i) q^{80} +7.87046e6 q^{81} -2.99506e6i q^{82} -9.19870e6i q^{83} +994725. q^{84} +(1.83352e6 + 2.96429e6i) q^{85} +1.39380e6 q^{86} +8.52897e6i q^{87} -1.82331e6i q^{88} -6.43438e6 q^{89} +(-5.66365e6 - 9.15651e6i) q^{90} +1.13208e6 q^{91} -732308. i q^{92} +2.43019e6i q^{93} -3.42561e6 q^{94} +(1.20383e7 - 7.44617e6i) q^{95} +2.74195e6 q^{96} +1.26986e7i q^{97} +6.31234e6i q^{98} -1.71468e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9}+O(q^{10}) \) 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9	\( 4 q - 256 q^{4} + 60 q^{5} + 896 q^{6} - 6788 q^{9} - 640 q^{10} + 17808 q^{11} - 6528 q^{14} - 34960 q^{15} + 16384 q^{16} + 63600 q^{19} - 3840 q^{20} - 63952 q^{21} - 57344 q^{24} + 86100 q^{25} - 56064 q^{26} + 169560 q^{29} + 410240 q^{30} - 394112 q^{31} - 698368 q^{34} - 276720 q^{35} + 434432 q^{36} + 1163424 q^{39} + 40960 q^{40} + 232488 q^{41} - 1139712 q^{44} - 2879420 q^{45} + 1146496 q^{46} + 2520108 q^{49} + 2361600 q^{50} + 361088 q^{51} - 5116160 q^{54} - 526480 q^{55} + 417792 q^{56} + 2093520 q^{59} + 2237440 q^{60} - 5251432 q^{61} - 1048576 q^{64} + 2761440 q^{65} + 2401792 q^{66} + 6514864 q^{69} + 2679680 q^{70} - 7832352 q^{71} - 3126528 q^{74} - 7397600 q^{75} - 4070400 q^{76} + 7727040 q^{79} + 245760 q^{80} + 16428404 q^{81} + 4092928 q^{84} - 7995520 q^{85} + 4411776 q^{86} - 33470040 q^{89} - 15579520 q^{90} + 5340768 q^{91} + 15540352 q^{94} + 31904400 q^{95} + 3670016 q^{96} - 19109776 q^{99}+O(q^{100}) \) 4 * q - 256 * q^4 + 60 * q^5 + 896 * q^6 - 6788 * q^9 - 640 * q^10 + 17808 * q^11 - 6528 * q^14 - 34960 * q^15 + 16384 * q^16 + 63600 * q^19 - 3840 * q^20 - 63952 * q^21 - 57344 * q^24 + 86100 * q^25 - 56064 * q^26 + 169560 * q^29 + 410240 * q^30 - 394112 * q^31 - 698368 * q^34 - 276720 * q^35 + 434432 * q^36 + 1163424 * q^39 + 40960 * q^40 + 232488 * q^41 - 1139712 * q^44 - 2879420 * q^45 + 1146496 * q^46 + 2520108 * q^49 + 2361600 * q^50 + 361088 * q^51 - 5116160 * q^54 - 526480 * q^55 + 417792 * q^56 + 2093520 * q^59 + 2237440 * q^60 - 5251432 * q^61 - 1048576 * q^64 + 2761440 * q^65 + 2401792 * q^66 + 6514864 * q^69 + 2679680 * q^70 - 7832352 * q^71 - 3126528 * q^74 - 7397600 * q^75 - 4070400 * q^76 + 7727040 * q^79 + 245760 * q^80 + 16428404 * q^81 + 4092928 * q^84 - 7995520 * q^85 + 4411776 * q^86 - 33470040 * q^89 - 15579520 * q^90 + 5340768 * q^91 + 15540352 * q^94 + 31904400 * q^95 + 3670016 * q^96 - 19109776 * q^99

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