LMFDB - Embedded newform 75.9.c.e.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(26,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("75.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.c (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:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$10$
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} + 1634x^{8} + 776307x^{6} + 148116566x^{4} + 10575941812x^{2} + 105274575720 $ x^10 + 1634x^8 + 776307x^6 + 148116566x^4 + 10575941812x^2 + 105274575720
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8}\cdot 3^{11}\cdot 5^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$3.43232i$ of defining polynomial
Character	$\chi$	$=$	75.26
Dual form			75.9.c.e.26.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.4751i q^{2} +(-29.7353 + 75.3446i) q^{3} -612.784 q^{4} +(2220.79 + 876.451i) q^{6} -3164.62 q^{7} +10516.3i q^{8} +(-4792.63 - 4480.79i) q^{9} +O(q^{10})$	\(q-29.4751i q^{2} +(-29.7353 + 75.3446i) q^{3} -612.784 q^{4} +(2220.79 + 876.451i) q^{6} -3164.62 q^{7} +10516.3i q^{8} +(-4792.63 - 4480.79i) q^{9} +20143.2i q^{11} +(18221.3 - 46170.0i) q^{12} +31424.2 q^{13} +93277.7i q^{14} +153096. q^{16} +26183.5i q^{17} +(-132072. + 141263. i) q^{18} -127845. q^{19} +(94100.9 - 238437. i) q^{21} +593723. q^{22} -378595. i q^{23} +(-792344. - 312704. i) q^{24} -926234. i q^{26} +(480113. - 227861. i) q^{27} +1.93923e6 q^{28} -759951. i q^{29} -832390. q^{31} -1.82036e6i q^{32} +(-1.51768e6 - 598962. i) q^{33} +771762. q^{34} +(2.93685e6 + 2.74575e6i) q^{36} +1.09633e6 q^{37} +3.76824e6i q^{38} +(-934408. + 2.36765e6i) q^{39} -22335.5i q^{41} +(-7.02798e6 - 2.77364e6i) q^{42} +4.00943e6 q^{43} -1.23434e7i q^{44} -1.11591e7 q^{46} +3.13980e6i q^{47} +(-4.55234e6 + 1.15349e7i) q^{48} +4.25004e6 q^{49} +(-1.97278e6 - 778572. i) q^{51} -1.92563e7 q^{52} -6.05943e6i q^{53} +(-6.71625e6 - 1.41514e7i) q^{54} -3.32800e7i q^{56} +(3.80150e6 - 9.63242e6i) q^{57} -2.23997e7 q^{58} +1.74124e6i q^{59} +7.86006e6 q^{61} +2.45348e7i q^{62} +(1.51669e7 + 1.41800e7i) q^{63} -1.44627e7 q^{64} +(-1.76545e7 + 4.47338e7i) q^{66} +9.88860e6 q^{67} -1.60448e7i q^{68} +(2.85251e7 + 1.12576e7i) q^{69} +5.43494e6i q^{71} +(4.71211e7 - 5.04006e7i) q^{72} +5.05573e7 q^{73} -3.23144e7i q^{74} +7.83412e7 q^{76} -6.37455e7i q^{77} +(6.97867e7 + 2.75418e7i) q^{78} -4.35502e7 q^{79} +(2.89185e6 + 4.29495e7i) q^{81} -658341. q^{82} -4.69277e7i q^{83} +(-5.76636e7 + 1.46111e8i) q^{84} -1.18178e8i q^{86} +(5.72582e7 + 2.25973e7i) q^{87} -2.11831e8 q^{88} -1.88276e7i q^{89} -9.94458e7 q^{91} +2.31997e8i q^{92} +(2.47513e7 - 6.27161e7i) q^{93} +9.25460e7 q^{94} +(1.37154e8 + 5.41288e7i) q^{96} +3.52136e7 q^{97} -1.25271e8i q^{98} +(9.02572e7 - 9.65387e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9}+O(q^{10}) $ 10 * q - 25 * q^3 - 1554 * q^4 + 2257 * q^6 + 1960 * q^7 - 11207 * q^9	$ 10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9} - 5915 q^{12} - 16920 q^{13} + 44634 q^{16} - 224875 q^{18} - 143934 q^{19} + 673428 q^{21} + 818990 q^{22} - 1016859 q^{24} - 260830 q^{27} + 3810100 q^{28} - 3014060 q^{31} - 4677515 q^{33} + 4977146 q^{34} + 4500527 q^{36} + 3016760 q^{37} - 7513282 q^{39} - 4001760 q^{42} + 11747340 q^{43} - 13938636 q^{46} - 14748755 q^{48} + 8953546 q^{49} + 6209287 q^{51} - 38918320 q^{52} - 8886272 q^{54} + 14759525 q^{57} - 48407900 q^{58} + 1520220 q^{61} + 74748240 q^{63} - 4536998 q^{64} + 10465295 q^{66} - 16269290 q^{67} + 11394978 q^{69} + 172231185 q^{72} - 52090170 q^{73} - 29529046 q^{76} + 198205810 q^{78} + 8549896 q^{79} + 22612945 q^{81} - 295714190 q^{82} - 136883292 q^{84} + 318901610 q^{87} - 310673250 q^{88} - 107224264 q^{91} + 79679130 q^{93} + 356118596 q^{94} + 525424001 q^{96} - 402167800 q^{97} - 382421335 q^{99}+O(q^{100}) $ 10 * q - 25 * q^3 - 1554 * q^4 + 2257 * q^6 + 1960 * q^7 - 11207 * q^9 - 5915 * q^12 - 16920 * q^13 + 44634 * q^16 - 224875 * q^18 - 143934 * q^19 + 673428 * q^21 + 818990 * q^22 - 1016859 * q^24 - 260830 * q^27 + 3810100 * q^28 - 3014060 * q^31 - 4677515 * q^33 + 4977146 * q^34 + 4500527 * q^36 + 3016760 * q^37 - 7513282 * q^39 - 4001760 * q^42 + 11747340 * q^43 - 13938636 * q^46 - 14748755 * q^48 + 8953546 * q^49 + 6209287 * q^51 - 38918320 * q^52 - 8886272 * q^54 + 14759525 * q^57 - 48407900 * q^58 + 1520220 * q^61 + 74748240 * q^63 - 4536998 * q^64 + 10465295 * q^66 - 16269290 * q^67 + 11394978 * q^69 + 172231185 * q^72 - 52090170 * q^73 - 29529046 * q^76 + 198205810 * q^78 + 8549896 * q^79 + 22612945 * q^81 - 295714190 * q^82 - 136883292 * q^84 + 318901610 * q^87 - 310673250 * q^88 - 107224264 * q^91 + 79679130 * q^93 + 356118596 * q^94 + 525424001 * q^96 - 402167800 * q^97 - 382421335 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	29.4751i		−	1.84220i	−0.389330	−	0.921098i	$-0.627293\pi$
$2$		−	29.4751i		−	1.84220i	0.389330	−	0.921098i	$-0.372707\pi$
$3$	−29.7353	+	75.3446i	−0.367102	+	0.930181i
$3$	−29.7353	+	75.3446i	−0.367102	+	0.930181i
$4$	−612.784			−2.39369
$5$		0			0
$5$		0			0
$6$	2220.79	+	876.451i	1.71358	+	0.676274i
$7$	−3164.62			−1.31804			−0.659022	−	0.752124i	$-0.729030\pi$
$7$	−3164.62			−1.31804			−0.659022	+	0.752124i	$0.729030\pi$
$8$			10516.3i			2.56745i
$9$	−4792.63	−	4480.79i	−0.730472	−	0.682942i
$10$		0			0
$11$			20143.2i			1.37581i	0.725803	+	0.687903i	$0.241468\pi$
$11$			20143.2i			1.37581i	−0.725803	+	0.687903i	$0.758532\pi$
$12$	18221.3	−	46170.0i	0.878728	−	2.22656i
$13$	31424.2			1.10025			0.550125	−	0.835083i	$-0.314580\pi$
$13$	31424.2			1.10025			0.550125	+	0.835083i	$0.314580\pi$
$14$			93277.7i			2.42810i
$15$		0			0
$16$	153096.			2.33606
$17$			26183.5i			0.313496i	0.987639	+	0.156748i	$0.0501010\pi$
$17$			26183.5i			0.313496i	−0.987639	+	0.156748i	$0.949899\pi$
$18$	−132072.	+	141263.i	−1.25811	+	1.34567i
$19$	−127845.			−0.980999			−0.490499	−	0.871442i	$-0.663186\pi$
$19$	−127845.			−0.980999			−0.490499	+	0.871442i	$0.663186\pi$
$20$		0			0
$21$	94100.9	−	238437.i	0.483857	−	1.22602i
$22$	593723.			2.53450
$23$		−	378595.i		−	1.35289i	−0.736492	−	0.676446i	$-0.763519\pi$
$23$		−	378595.i		−	1.35289i	0.736492	−	0.676446i	$-0.236481\pi$
$24$	−792344.	−	312704.i	−2.38819	−	0.942515i
$25$		0			0
$26$		−	926234.i		−	2.02688i
$27$	480113.	−	227861.i	0.903418	−	0.428762i
$28$	1.93923e6			3.15499
$29$		−	759951.i		−	1.07447i	−0.843433	−	0.537234i	$-0.819469\pi$
$29$		−	759951.i		−	1.07447i	0.843433	−	0.537234i	$-0.180531\pi$
$30$		0			0
$31$	−832390.			−0.901322			−0.450661	−	0.892695i	$-0.648812\pi$
$31$	−832390.			−0.901322			−0.450661	+	0.892695i	$0.648812\pi$
$32$		−	1.82036e6i		−	1.73603i
$33$	−1.51768e6	−	598962.i	−1.27975	−	0.505061i
$34$	771762.			0.577521
$35$		0			0
$36$	2.93685e6	+	2.74575e6i	1.74852	+	1.63475i
$37$	1.09633e6			0.584969			0.292484	−	0.956270i	$-0.405518\pi$
$37$	1.09633e6			0.584969			0.292484	+	0.956270i	$0.405518\pi$
$38$			3.76824e6i			1.80719i
$39$	−934408.	+	2.36765e6i	−0.403904	+	1.02343i
$40$		0			0
$41$		−	22335.5i		−	0.00790423i	−0.999992	−	0.00395212i	$-0.998742\pi$
$41$		−	22335.5i		−	0.00790423i	0.999992	−	0.00395212i	$-0.00125800\pi$
$42$	−7.02798e6	−	2.77364e6i	−2.25857	−	0.891359i
$43$	4.00943e6			1.17276			0.586379	−	0.810037i	$-0.300553\pi$
$43$	4.00943e6			1.17276			0.586379	+	0.810037i	$0.300553\pi$
$44$		−	1.23434e7i		−	3.29325i
$45$		0			0
$46$	−1.11591e7			−2.49229
$47$			3.13980e6i			0.643443i	0.946834	+	0.321722i	$0.104262\pi$
$47$			3.13980e6i			0.643443i	−0.946834	+	0.321722i	$0.895738\pi$
$48$	−4.55234e6	+	1.15349e7i	−0.857571	+	2.17295i
$49$	4.25004e6			0.737240
$50$		0			0
$51$	−1.97278e6	−	778572.i	−0.291608	−	0.115085i
$52$	−1.92563e7			−2.63365
$53$		−	6.05943e6i		−	0.767941i	−0.923345	−	0.383971i	$-0.874556\pi$
$53$		−	6.05943e6i		−	0.767941i	0.923345	−	0.383971i	$-0.125444\pi$
$54$	−6.71625e6	−	1.41514e7i	−0.789863	−	1.66427i
$55$		0			0
$56$		−	3.32800e7i		−	3.38401i
$57$	3.80150e6	−	9.63242e6i	0.360127	−	0.912506i
$58$	−2.23997e7			−1.97938
$59$			1.74124e6i			0.143698i	0.997416	+	0.0718489i	$0.0228899\pi$
$59$			1.74124e6i			0.143698i	−0.997416	+	0.0718489i	$0.977110\pi$
$60$		0			0
$61$	7.86006e6			0.567684			0.283842	−	0.958871i	$-0.408391\pi$
$61$	7.86006e6			0.567684			0.283842	+	0.958871i	$0.408391\pi$
$62$			2.45348e7i			1.66041i
$63$	1.51669e7	+	1.41800e7i	0.962794	+	0.900148i
$64$	−1.44627e7			−0.862046
$65$		0			0
$66$	−1.76545e7	+	4.47338e7i	−0.930422	+	2.35755i
$67$	9.88860e6			0.490722			0.245361	−	0.969432i	$-0.421094\pi$
$67$	9.88860e6			0.490722			0.245361	+	0.969432i	$0.421094\pi$
$68$		−	1.60448e7i		−	0.750411i
$69$	2.85251e7	+	1.12576e7i	1.25843	+	0.496649i
$70$		0			0
$71$			5.43494e6i			0.213876i	0.994266	+	0.106938i	$0.0341046\pi$
$71$			5.43494e6i			0.213876i	−0.994266	+	0.106938i	$0.965895\pi$
$72$	4.71211e7	−	5.04006e7i	1.75342	−	1.87545i
$73$	5.05573e7			1.78030			0.890148	−	0.455672i	$-0.150601\pi$
$73$	5.05573e7			1.78030			0.890148	+	0.455672i	$0.150601\pi$
$74$		−	3.23144e7i		−	1.07763i
$75$		0			0
$76$	7.83412e7			2.34821
$77$		−	6.37455e7i		−	1.81337i
$78$	6.97867e7	+	2.75418e7i	1.88536	+	0.744070i
$79$	−4.35502e7			−1.11810			−0.559051	−	0.829133i	$-0.688834\pi$
$79$	−4.35502e7			−1.11810			−0.559051	+	0.829133i	$0.688834\pi$
$80$		0			0
$81$	2.89185e6	+	4.29495e7i	0.0671793	+	0.997741i
$82$	−658341.			−0.0145611
$83$		−	4.69277e7i		−	0.988819i	−0.869229	−	0.494409i	$-0.835384\pi$
$83$		−	4.69277e7i		−	0.988819i	0.869229	−	0.494409i	$-0.164616\pi$
$84$	−5.76636e7	+	1.46111e8i	−1.15820	+	2.93471i
$85$		0			0
$86$		−	1.18178e8i		−	2.16045i
$87$	5.72582e7	+	2.25973e7i	0.999449	+	0.394439i
$88$	−2.11831e8			−3.53231
$89$		−	1.88276e7i		−	0.300078i	−0.988680	−	0.150039i	$-0.952060\pi$
$89$		−	1.88276e7i		−	0.300078i	0.988680	−	0.150039i	$-0.0479399\pi$
$90$		0			0
$91$	−9.94458e7			−1.45018
$92$			2.31997e8i			3.23840i
$93$	2.47513e7	−	6.27161e7i	0.330877	−	0.838392i
$94$	9.25460e7			1.18535
$95$		0			0
$96$	1.37154e8	+	5.41288e7i	1.61482	+	0.637299i
$97$	3.52136e7			0.397762			0.198881	−	0.980024i	$-0.436269\pi$
$97$	3.52136e7			0.397762			0.198881	+	0.980024i	$0.436269\pi$
$98$		−	1.25271e8i		−	1.35814i
$99$	9.02572e7	−	9.65387e7i	0.939596	−	1.00499i
$100$		0			0
$101$		−	4.96736e7i		−	0.477353i	−0.971099	−	0.238677i	$-0.923286\pi$
$101$		−	4.96736e7i		−	0.477353i	0.971099	−	0.238677i	$-0.0767136\pi$
$102$	−2.29485e7	+	5.81481e7i	−0.212009	+	0.537199i
$103$	1.30485e8			1.15934			0.579670	−	0.814851i	$-0.303181\pi$
$103$	1.30485e8			1.15934			0.579670	+	0.814851i	$0.303181\pi$
$104$			3.30466e8i			2.82483i
$105$		0			0
$106$	−1.78602e8			−1.41470
$107$			8.06696e7i			0.615425i	0.951479	+	0.307712i	$0.0995635\pi$
$107$			8.06696e7i			0.615425i	−0.951479	+	0.307712i	$0.900437\pi$
$108$	−2.94206e8	+	1.39630e8i	−2.16250	+	1.02632i
$109$	2.38708e8			1.69107			0.845533	−	0.533923i	$-0.179283\pi$
$109$	2.38708e8			1.69107			0.845533	+	0.533923i	$0.179283\pi$
$110$		0			0
$111$	−3.25995e7	+	8.26023e7i	−0.214743	+	0.544127i
$112$	−4.84490e8			−3.07902
$113$			9.23822e7i			0.566597i	0.959032	+	0.283299i	$0.0914288\pi$
$113$			9.23822e7i			0.566597i	−0.959032	+	0.283299i	$0.908571\pi$
$114$	−2.83917e8	−	1.12050e8i	−1.68102	−	0.663424i
$115$		0			0
$116$			4.65686e8i			2.57194i
$117$	−1.50605e8	−	1.40805e8i	−0.803702	−	0.751407i
$118$	5.13232e7			0.264720
$119$		−	8.28608e7i		−	0.413201i
$120$		0			0
$121$	−1.91388e8			−0.892841
$122$		−	2.31676e8i		−	1.04578i
$123$	1.68286e6	+	664151.i	0.00735236	+	0.00290166i
$124$	5.10075e8			2.15748
$125$		0			0
$126$	4.17958e8	−	4.47046e8i	1.65825	−	1.77366i
$127$	1.90148e7			0.0730931			0.0365466	−	0.999332i	$-0.488364\pi$
$127$	1.90148e7			0.0730931			0.0365466	+	0.999332i	$0.488364\pi$
$128$		−	3.97199e7i		−	0.147968i
$129$	−1.19221e8	+	3.02089e8i	−0.430522	+	1.09088i
$130$		0			0
$131$			4.89021e8i			1.66051i	0.557380	+	0.830257i	$0.311806\pi$
$131$			4.89021e8i			1.66051i	−0.557380	+	0.830257i	$0.688194\pi$
$132$	9.30010e8	+	3.67035e8i	3.06332	+	1.20896i
$133$	4.04580e8			1.29300
$134$		−	2.91468e8i		−	0.904007i
$135$		0			0
$136$	−2.75352e8			−0.804884
$137$		−	1.33995e8i		−	0.380370i	−0.981748	−	0.190185i	$-0.939091\pi$
$137$		−	1.33995e8i		−	0.380370i	0.981748	−	0.190185i	$-0.0609087\pi$
$138$	3.31820e8	−	8.40780e8i	0.914926	−	2.31828i
$139$	−5.44407e6			−0.0145836			−0.00729180	−	0.999973i	$-0.502321\pi$
$139$	−5.44407e6			−0.0145836			−0.00729180	+	0.999973i	$0.502321\pi$
$140$		0			0
$141$	−2.36567e8	−	9.33627e7i	−0.598519	−	0.236209i
$142$	1.60196e8			0.394001
$143$			6.32984e8i			1.51373i
$144$	−7.33731e8	−	6.85989e8i	−1.70642	−	1.59539i
$145$		0			0
$146$		−	1.49018e9i		−	3.27965i
$147$	−1.26376e8	+	3.20218e8i	−0.270642	+	0.685766i
$148$	−6.71811e8			−1.40023
$149$		−	1.55067e8i		−	0.314612i	−0.987550	−	0.157306i	$-0.949719\pi$
$149$		−	1.55067e8i		−	0.314612i	0.987550	−	0.157306i	$-0.0502809\pi$
$150$		0			0
$151$	9.60323e8			1.84718			0.923591	−	0.383379i	$-0.125240\pi$
$151$	9.60323e8			1.84718			0.923591	+	0.383379i	$0.125240\pi$
$152$		−	1.34445e9i		−	2.51866i
$153$	1.17323e8	−	1.25488e8i	0.214099	−	0.229000i
$154$	−1.87891e9			−3.34059
$155$		0			0
$156$	5.72590e8	−	1.45086e9i	0.966820	−	2.44977i
$157$	−2.42826e8			−0.399666			−0.199833	−	0.979830i	$-0.564040\pi$
$157$	−2.42826e8			−0.399666			−0.199833	+	0.979830i	$0.564040\pi$
$158$			1.28365e9i			2.05976i
$159$	4.56545e8	+	1.80179e8i	0.714324	+	0.281913i
$160$		0			0
$161$			1.19811e9i			1.78317i
$162$	1.26594e9	−	8.52377e7i	1.83803	−	0.123757i
$163$	−9.42493e8			−1.33514			−0.667571	−	0.744546i	$-0.732666\pi$
$163$	−9.42493e8			−1.33514			−0.667571	+	0.744546i	$0.732666\pi$
$164$			1.36868e7i			0.0189203i
$165$		0			0
$166$	−1.38320e9			−1.82160
$167$		−	1.18718e9i		−	1.52634i	−0.646197	−	0.763170i	$-0.723642\pi$
$167$		−	1.18718e9i		−	1.52634i	0.646197	−	0.763170i	$-0.276358\pi$
$168$	2.50747e9	+	9.89590e8i	3.14774	+	1.24228i
$169$	1.71751e8			0.210549
$170$		0			0
$171$	6.12712e8	+	5.72845e8i	0.716592	+	0.669966i
$172$	−2.45691e9			−2.80722
$173$		−	2.03038e8i		−	0.226669i	−0.993557	−	0.113335i	$-0.963847\pi$
$173$		−	2.03038e8i		−	0.226669i	0.993557	−	0.113335i	$-0.0361532\pi$
$174$	6.66060e8	−	1.68769e9i	0.726635	−	1.84118i
$175$		0			0
$176$			3.08383e9i			3.21396i
$177$	−1.31193e8	−	5.17762e7i	−0.133665	−	0.0527517i
$178$	−5.54946e8			−0.552803
$179$			1.00612e9i			0.980030i	0.871714	+	0.490015i	$0.163009\pi$
$179$			1.00612e9i			0.980030i	−0.871714	+	0.490015i	$0.836991\pi$
$180$		0			0
$181$	8.87701e8			0.827089			0.413545	−	0.910484i	$-0.364291\pi$
$181$	8.87701e8			0.827089			0.413545	+	0.910484i	$0.364291\pi$
$182$			2.93118e9i			2.67151i
$183$	−2.33721e8	+	5.92213e8i	−0.208398	+	0.528048i
$184$	3.98140e9			3.47348
$185$		0			0
$186$	−1.84857e9	−	7.29549e8i	−1.54448	−	0.609541i
$187$	−5.27418e8			−0.431309
$188$		−	1.92402e9i		−	1.54020i
$189$	−1.51938e9	+	7.21096e8i	−1.19074	+	0.565127i
$190$		0			0
$191$		−	2.16005e9i		−	1.62305i	−0.584320	−	0.811523i	$-0.698639\pi$
$191$		−	2.16005e9i		−	1.62305i	0.584320	−	0.811523i	$-0.301361\pi$
$192$	4.30053e8	−	1.08969e9i	0.316459	−	0.801859i
$193$	−1.50434e9			−1.08422			−0.542108	−	0.840309i	$-0.682374\pi$
$193$	−1.50434e9			−1.08422			−0.542108	+	0.840309i	$0.682374\pi$
$194$		−	1.03793e9i		−	0.732755i
$195$		0			0
$196$	−2.60436e9			−1.76472
$197$		−	3.65705e8i		−	0.242809i	−0.992603	−	0.121405i	$-0.961260\pi$
$197$		−	3.65705e8i		−	0.242809i	0.992603	−	0.121405i	$-0.0387399\pi$
$198$	−2.84549e9	−	2.66034e9i	−1.85139	−	1.73092i
$199$	−1.84128e9			−1.17411			−0.587053	−	0.809549i	$-0.699712\pi$
$199$	−1.84128e9			−1.17411			−0.587053	+	0.809549i	$0.699712\pi$
$200$		0			0
$201$	−2.94040e8	+	7.45053e8i	−0.180145	+	0.456460i
$202$	−1.46414e9			−0.879379
$203$			2.40496e9i			1.41620i
$204$	1.20889e9	+	4.77097e8i	0.698018	+	0.275477i
$205$		0			0
$206$		−	3.84606e9i		−	2.13573i
$207$	−1.69640e9	+	1.81446e9i	−0.923947	+	0.988250i
$208$	4.81092e9			2.57024
$209$		−	2.57520e9i		−	1.34966i
$210$		0			0
$211$	−8.21239e8			−0.414323			−0.207162	−	0.978307i	$-0.566423\pi$
$211$	−8.21239e8			−0.414323			−0.207162	+	0.978307i	$0.566423\pi$
$212$			3.71312e9i			1.83821i
$213$	−4.09494e8	−	1.61609e8i	−0.198943	−	0.0785142i
$214$	2.37775e9			1.13373
$215$		0			0
$216$	2.39625e9	+	5.04900e9i	1.10082	+	2.31948i
$217$	2.63420e9			1.18798
$218$		−	7.03595e9i		−	3.11528i
$219$	−1.50333e9	+	3.80922e9i	−0.653550	+	1.65600i
$220$		0			0
$221$			8.22795e8i			0.344923i
$222$	2.43471e9	+	9.60876e8i	1.00239	+	0.395599i
$223$	3.76056e8			0.152066			0.0760332	−	0.997105i	$-0.475775\pi$
$223$	3.76056e8			0.152066			0.0760332	+	0.997105i	$0.475775\pi$
$224$			5.76074e9i			2.28816i
$225$		0			0
$226$	2.72298e9			1.04378
$227$		−	4.58046e9i		−	1.72507i	−0.506000	−	0.862533i	$-0.668876\pi$
$227$		−	4.58046e9i		−	1.72507i	0.506000	−	0.862533i	$-0.331124\pi$
$228$	−2.32950e9	+	5.90259e9i	−0.862031	+	2.18426i
$229$	1.66628e9			0.605906			0.302953	−	0.953005i	$-0.402027\pi$
$229$	1.66628e9			0.605906			0.302953	+	0.953005i	$0.402027\pi$
$230$		0			0
$231$	4.80289e9	+	1.89549e9i	1.68676	+	0.665693i
$232$	7.99184e9			2.75864
$233$			4.25125e9i			1.44243i	0.692714	+	0.721213i	$0.256415\pi$
$233$			4.25125e9i			1.44243i	−0.692714	+	0.721213i	$0.743585\pi$
$234$	−4.15025e9	+	4.43909e9i	−1.38424	+	1.48058i
$235$		0			0
$236$		−	1.06700e9i		−	0.343968i
$237$	1.29498e9	−	3.28127e9i	0.410458	−	1.04004i
$238$	−2.44233e9			−0.761198
$239$			1.85631e9i			0.568930i	0.958686	+	0.284465i	$0.0918159\pi$
$239$			1.85631e9i			0.568930i	−0.958686	+	0.284465i	$0.908184\pi$
$240$		0			0
$241$	5.00083e9			1.48243			0.741215	−	0.671268i	$-0.234250\pi$
$241$	5.00083e9			1.48243			0.741215	+	0.671268i	$0.234250\pi$
$242$			5.64120e9i			1.64479i
$243$	−3.32200e9	−	1.05923e9i	−0.952741	−	0.303784i
$244$	−4.81652e9			−1.35886
$245$		0			0
$246$	1.95760e7	−	4.96025e7i	0.00534543	−	0.0135445i
$247$	−4.01742e9			−1.07934
$248$		−	8.75363e9i		−	2.31410i
$249$	3.53575e9	+	1.39541e9i	0.919780	+	0.362997i
$250$		0			0
$251$		−	1.91602e8i		−	0.0482732i	−0.999709	−	0.0241366i	$-0.992316\pi$
$251$		−	1.91602e8i		−	0.0482732i	0.999709	−	0.0241366i	$-0.00768367\pi$
$252$	−9.29401e9	−	8.68928e9i	−2.30463	−	2.15467i
$253$	7.62609e9			1.86132
$254$		−	5.60464e8i		−	0.134652i
$255$		0			0
$256$	−4.87321e9			−1.13463
$257$			2.54083e9i			0.582429i	0.956658	+	0.291215i	$0.0940594\pi$
$257$			2.54083e9i			0.582429i	−0.956658	+	0.291215i	$0.905941\pi$
$258$	8.90411e9	+	3.51407e9i	2.00961	+	0.793106i
$259$	−3.46946e9			−0.771015
$260$		0			0
$261$	−3.40518e9	+	3.64216e9i	−0.733800	+	0.784869i
$262$	1.44140e10			3.05899
$263$			1.61586e9i			0.337740i	0.985638	+	0.168870i	$0.0540117\pi$
$263$			1.61586e9i			0.337740i	−0.985638	+	0.168870i	$0.945988\pi$
$264$	6.29885e9	−	1.59603e10i	1.29672	−	3.28569i
$265$		0			0
$266$		−	1.19251e10i		−	2.38196i
$267$	1.41856e9	+	5.59843e8i	0.279127	+	0.110159i
$268$	−6.05958e9			−1.17464
$269$		−	7.37100e9i		−	1.40772i	−0.710337	−	0.703861i	$-0.751458\pi$
$269$		−	7.37100e9i		−	1.40772i	0.710337	−	0.703861i	$-0.248542\pi$
$270$		0			0
$271$	−3.90024e9			−0.723126			−0.361563	−	0.932348i	$-0.617757\pi$
$271$	−3.90024e9			−0.723126			−0.361563	+	0.932348i	$0.617757\pi$
$272$			4.00858e9i			0.732343i
$273$	2.95705e9	−	7.49271e9i	0.532363	−	1.34893i
$274$	−3.94952e9			−0.700716
$275$		0			0
$276$	−1.74797e10	−	6.89848e9i	−3.01230	−	1.18882i
$277$	−8.18524e9			−1.39031			−0.695157	−	0.718858i	$-0.744665\pi$
$277$	−8.18524e9			−1.39031			−0.695157	+	0.718858i	$0.744665\pi$
$278$			1.60465e8i			0.0268659i
$279$	3.98933e9	+	3.72976e9i	0.658391	+	0.615551i
$280$		0			0
$281$			7.15119e9i			1.14697i	0.819215	+	0.573486i	$0.194409\pi$
$281$			7.15119e9i			1.14697i	−0.819215	+	0.573486i	$0.805591\pi$
$282$	−2.75188e9	+	6.97285e9i	−0.435144	+	1.10259i
$283$	−7.42098e9			−1.15695			−0.578476	−	0.815699i	$-0.696352\pi$
$283$	−7.42098e9			−1.15695			−0.578476	+	0.815699i	$0.696352\pi$
$284$		−	3.33045e9i		−	0.511952i
$285$		0			0
$286$	1.86573e10			2.78859
$287$			7.06834e7i			0.0104181i
$288$	−8.15662e9	+	8.72429e9i	−1.18561	+	1.26812i
$289$	6.29018e9			0.901720
$290$		0			0
$291$	−1.04708e9	+	2.65315e9i	−0.146019	+	0.369990i
$292$	−3.09807e10			−4.26147
$293$			1.73588e8i			0.0235531i	0.999931	+	0.0117766i	$0.00374868\pi$
$293$			1.73588e8i			0.0235531i	−0.999931	+	0.0117766i	$0.996251\pi$
$294$	9.43846e9	+	3.72495e9i	1.26332	+	0.498576i
$295$		0			0
$296$			1.15293e10i			1.50188i
$297$	4.58985e9	+	9.67100e9i	0.589893	+	1.24293i
$298$	−4.57063e9			−0.579577
$299$		−	1.18970e10i		−	1.48852i
$300$		0			0
$301$	−1.26883e10			−1.54575
$302$		−	2.83057e10i		−	3.40287i
$303$	3.74264e9	+	1.47706e9i	0.444025	+	0.175237i
$304$	−1.95725e10			−2.29167
$305$		0			0
$306$	−3.69877e9	−	3.45810e9i	−0.421863	−	0.394413i
$307$	1.38868e10			1.56332			0.781661	−	0.623703i	$-0.214373\pi$
$307$	1.38868e10			1.56332			0.781661	+	0.623703i	$0.214373\pi$
$308$			3.90623e10i			4.34065i
$309$	−3.88000e9	+	9.83133e9i	−0.425596	+	1.07840i
$310$		0			0
$311$		−	9.61567e9i		−	1.02787i	−0.857829	−	0.513935i	$-0.828187\pi$
$311$		−	9.61567e9i		−	1.02787i	0.857829	−	0.513935i	$-0.171813\pi$
$312$	−2.48988e10	−	9.82648e9i	−2.62761	−	1.03700i
$313$	1.68395e10			1.75450			0.877248	−	0.480037i	$-0.159377\pi$
$313$	1.68395e10			1.75450			0.877248	+	0.480037i	$0.159377\pi$
$314$			7.15734e9i			0.736263i
$315$		0			0
$316$	2.66869e10			2.67639
$317$		−	1.96153e10i		−	1.94248i	−0.238094	−	0.971242i	$-0.576523\pi$
$317$		−	1.96153e10i		−	1.94248i	0.238094	−	0.971242i	$-0.423477\pi$
$318$	5.31079e9	−	1.34567e10i	0.519339	−	1.31593i
$319$	1.53078e10			1.47826
$320$		0			0
$321$	−6.07802e9	−	2.39873e9i	−0.572456	−	0.225924i
$322$	3.53144e10			3.28495
$323$		−	3.34742e9i		−	0.307539i
$324$	−1.77208e9	−	2.63188e10i	−0.160806	−	2.38828i
$325$		0			0
$326$			2.77801e10i			2.45960i
$327$	−7.09804e9	+	1.79853e10i	−0.620794	+	1.57300i
$328$	2.34886e8			0.0202937
$329$		−	9.93628e9i		−	0.848087i
$330$		0			0
$331$	−1.12041e10			−0.933396			−0.466698	−	0.884417i	$-0.654557\pi$
$331$	−1.12041e10			−0.933396			−0.466698	+	0.884417i	$0.654557\pi$
$332$			2.87565e10i			2.36692i
$333$	−5.25428e9	−	4.91240e9i	−0.427304	−	0.399500i
$334$	−3.49924e10			−2.81182
$335$		0			0
$336$	1.44065e10	−	3.65038e10i	1.13032	−	2.86405i
$337$	−8.17992e8			−0.0634205			−0.0317102	−	0.999497i	$-0.510095\pi$
$337$	−8.17992e8			−0.0634205			−0.0317102	+	0.999497i	$0.510095\pi$
$338$		−	5.06239e9i		−	0.387872i
$339$	−6.96050e9	−	2.74701e9i	−0.527038	−	0.207999i
$340$		0			0
$341$		−	1.67670e10i		−	1.24004i
$342$	1.68847e10	−	1.80598e10i	1.23421	−	1.32010i
$343$	4.79365e9			0.346330
$344$			4.21642e10i			3.01099i
$345$		0			0
$346$	−5.98458e9			−0.417570
$347$			1.44473e10i			0.996479i	0.867039	+	0.498239i	$0.166020\pi$
$347$			1.44473e10i			0.996479i	−0.867039	+	0.498239i	$0.833980\pi$
$348$	−3.50869e10	−	1.38473e10i	−2.39237	−	0.944165i
$349$	−1.45400e9			−0.0980080			−0.0490040	−	0.998799i	$-0.515605\pi$
$349$	−1.45400e9			−0.0980080			−0.0490040	+	0.998799i	$0.515605\pi$
$350$		0			0
$351$	1.50872e10	−	7.16037e9i	0.993985	−	0.471745i
$352$	3.66677e10			2.38844
$353$		−	1.22501e10i		−	0.788934i	−0.918910	−	0.394467i	$-0.870929\pi$
$353$		−	1.22501e10i		−	0.788934i	0.918910	−	0.394467i	$-0.129071\pi$
$354$	−1.52611e9	+	3.86693e9i	−0.0971791	+	0.246237i
$355$		0			0
$356$			1.15372e10i			0.718294i
$357$	6.24312e9	+	2.46389e9i	0.384352	+	0.151687i
$358$	2.96557e10			1.80541
$359$			1.08842e10i			0.655271i	0.944804	+	0.327635i	$0.106252\pi$
$359$			1.08842e10i			0.655271i	−0.944804	+	0.327635i	$0.893748\pi$
$360$		0			0
$361$	−6.39283e8			−0.0376413
$362$		−	2.61651e10i		−	1.52366i
$363$	5.69099e9	−	1.44201e10i	0.327764	−	0.830504i
$364$	6.09388e10			3.47127
$365$		0			0
$366$	1.74556e10	+	6.88896e9i	0.972769	+	0.383910i
$367$	2.34324e10			1.29167			0.645836	−	0.763476i	$-0.276509\pi$
$367$	2.34324e10			1.29167			0.645836	+	0.763476i	$0.276509\pi$
$368$		−	5.79612e10i		−	3.16043i
$369$	−1.00080e8	+	1.07046e8i	−0.00539814	+	0.00577382i
$370$		0			0
$371$			1.91758e10i			1.01218i
$372$	−1.51672e10	+	3.84314e10i	−0.792017	+	2.00685i
$373$	1.29023e10			0.666548			0.333274	−	0.942830i	$-0.391847\pi$
$373$	1.29023e10			0.666548			0.333274	+	0.942830i	$0.391847\pi$
$374$			1.55457e10i			0.794556i
$375$		0			0
$376$	−3.30190e10			−1.65201
$377$		−	2.38809e10i		−	1.18218i
$378$	2.12544e10	+	4.47839e10i	1.04107	+	2.19358i
$379$	−5.12576e9			−0.248429			−0.124214	−	0.992255i	$-0.539641\pi$
$379$	−5.12576e9			−0.248429			−0.124214	+	0.992255i	$0.539641\pi$
$380$		0			0
$381$	−5.65410e8	+	1.43266e9i	−0.0268326	+	0.0679898i
$382$	−6.36679e10			−2.98997
$383$			3.66239e10i			1.70204i	0.525136	+	0.851018i	$0.324014\pi$
$383$			3.66239e10i			1.70204i	−0.525136	+	0.851018i	$0.675986\pi$
$384$	2.99268e9	+	1.18108e9i	0.137637	+	0.0543194i
$385$		0			0
$386$			4.43406e10i			1.99734i
$387$	−1.92157e10	−	1.79654e10i	−0.856667	−	0.800926i
$388$	−2.15783e10			−0.952118
$389$		−	3.16164e10i		−	1.38075i	−0.723453	−	0.690373i	$-0.757446\pi$
$389$		−	3.16164e10i		−	1.38075i	0.723453	−	0.690373i	$-0.242554\pi$
$390$		0			0
$391$	9.91292e9			0.424126
$392$			4.46946e10i			1.89282i
$393$	−3.68451e10	−	1.45412e10i	−1.54458	−	0.609578i
$394$	−1.07792e10			−0.447303
$395$		0			0
$396$	−5.53082e10	+	5.91574e10i	−2.24910	+	2.40563i
$397$	5.05225e8			0.0203387			0.0101693	−	0.999948i	$-0.496763\pi$
$397$	5.05225e8			0.0203387			0.0101693	+	0.999948i	$0.496763\pi$
$398$			5.42719e10i			2.16293i
$399$	−1.20303e10	+	3.04830e10i	−0.474663	+	1.20272i
$400$		0			0
$401$		−	3.67187e10i		−	1.42007i	−0.704167	−	0.710035i	$-0.748679\pi$
$401$		−	3.67187e10i		−	1.42007i	0.704167	−	0.710035i	$-0.251321\pi$
$402$	2.19605e10	+	8.66688e9i	0.840889	+	0.331863i
$403$	−2.61572e10			−0.991679
$404$			3.04392e10i			1.14264i
$405$		0			0
$406$	7.08865e10			2.60891
$407$			2.20835e10i			0.804804i
$408$	8.18768e9	−	2.07463e10i	0.295474	−	0.748687i
$409$	7.11197e9			0.254154			0.127077	−	0.991893i	$-0.459441\pi$
$409$	7.11197e9			0.254154			0.127077	+	0.991893i	$0.459441\pi$
$410$		0			0
$411$	1.00958e10	+	3.98437e9i	0.353813	+	0.139635i
$412$	−7.99590e10			−2.77510
$413$		−	5.51036e9i		−	0.189400i
$414$	5.34816e10	+	5.00017e10i	1.82055	+	1.70209i
$415$		0			0
$416$		−	5.72033e10i		−	1.91006i
$417$	1.61881e8	−	4.10182e8i	0.00535367	−	0.0135654i
$418$	−7.59044e10			−2.48635
$419$		−	2.61481e10i		−	0.848369i	−0.905576	−	0.424184i	$-0.860561\pi$
$419$		−	2.61481e10i		−	0.848369i	0.905576	−	0.424184i	$-0.139439\pi$
$420$		0			0
$421$	5.80276e9			0.184717			0.0923584	−	0.995726i	$-0.470559\pi$
$421$	5.80276e9			0.184717			0.0923584	+	0.995726i	$0.470559\pi$
$422$			2.42061e10i			0.763265i
$423$	1.40688e10	−	1.50479e10i	0.439435	−	0.470017i
$424$	6.37225e10			1.97165
$425$		0			0
$426$	−4.76346e9	+	1.20699e10i	−0.144639	+	0.366492i
$427$	−2.48741e10			−0.748232
$428$		−	4.94331e10i		−	1.47314i
$429$	−4.76919e10	−	1.88219e10i	−1.40804	−	0.555693i
$430$		0			0
$431$		−	4.45744e10i		−	1.29175i	−0.763445	−	0.645873i	$-0.776494\pi$
$431$		−	4.45744e10i		−	1.29175i	0.763445	−	0.645873i	$-0.223506\pi$
$432$	7.35033e10	−	3.48846e10i	2.11043	−	1.00161i
$433$	−3.20554e10			−0.911905			−0.455952	−	0.890004i	$-0.650701\pi$
$433$	−3.20554e10			−0.911905			−0.455952	+	0.890004i	$0.650701\pi$
$434$		−	7.76434e10i		−	2.18850i
$435$		0			0
$436$	−1.46276e11			−4.04789
$437$			4.84013e10i			1.32718i
$438$	1.12277e11	+	4.43110e10i	3.05067	+	1.20397i
$439$	−2.57341e10			−0.692867			−0.346434	−	0.938074i	$-0.612607\pi$
$439$	−2.57341e10			−0.692867			−0.346434	+	0.938074i	$0.612607\pi$
$440$		0			0
$441$	−2.03689e10	−	1.90435e10i	−0.538533	−	0.503492i
$442$	2.42520e10			0.635417
$443$		−	7.09037e10i		−	1.84100i	−0.390743	−	0.920500i	$-0.627782\pi$
$443$		−	7.09037e10i		−	1.84100i	0.390743	−	0.920500i	$-0.372218\pi$
$444$	1.99765e10	−	5.06174e10i	0.514028	−	1.30247i
$445$		0			0
$446$		−	1.10843e10i		−	0.280136i
$447$	1.16835e10	+	4.61097e9i	0.292646	+	0.115495i
$448$	4.57691e10			1.13621
$449$			6.41882e10i			1.57932i	0.613545	+	0.789660i	$0.289743\pi$
$449$			6.41882e10i			1.57932i	−0.613545	+	0.789660i	$0.710257\pi$
$450$		0			0
$451$	4.49907e8			0.0108747
$452$		−	5.66103e10i		−	1.35626i
$453$	−2.85555e10	+	7.23552e10i	−0.678104	+	1.71821i
$454$	−1.35010e11			−3.17791
$455$		0			0
$456$	1.01297e11	+	3.99776e10i	2.34281	+	0.924607i
$457$	1.93552e10			0.443745			0.221873	−	0.975076i	$-0.428783\pi$
$457$	1.93552e10			0.443745			0.221873	+	0.975076i	$0.428783\pi$
$458$		−	4.91138e10i		−	1.11620i
$459$	5.96620e9	+	1.25710e10i	0.134415	+	0.283218i
$460$		0			0
$461$		−	5.65186e10i		−	1.25138i	−0.780073	−	0.625688i	$-0.784818\pi$
$461$		−	5.65186e10i		−	1.25138i	0.780073	−	0.625688i	$-0.215182\pi$
$462$	5.58699e10	−	1.41566e11i	1.22634	−	3.10735i
$463$	2.73707e9			0.0595610			0.0297805	−	0.999556i	$-0.490519\pi$
$463$	2.73707e9			0.0595610			0.0297805	+	0.999556i	$0.490519\pi$
$464$		−	1.16345e11i		−	2.51002i
$465$		0			0
$466$	1.25306e11			2.65723
$467$			1.99286e10i			0.418995i	0.977809	+	0.209498i	$0.0671829\pi$
$467$			1.99286e10i			0.418995i	−0.977809	+	0.209498i	$0.932817\pi$
$468$	9.22881e10	+	8.62832e10i	1.92381	+	1.79863i
$469$	−3.12937e10			−0.646793
$470$		0			0
$471$	7.22050e9	−	1.82957e10i	0.146718	−	0.371762i
$472$	−1.83113e10			−0.368937
$473$			8.07625e10i			1.61349i
$474$	−9.67160e10	−	3.81696e10i	−1.91595	−	0.756144i
$475$		0			0
$476$			5.07758e10i			0.989075i
$477$	−2.71510e10	+	2.90406e10i	−0.524460	+	0.560960i
$478$	5.47150e10			1.04808
$479$			3.16925e10i			0.602024i	0.953620	+	0.301012i	$0.0973245\pi$
$479$			3.16925e10i			0.602024i	−0.953620	+	0.301012i	$0.902676\pi$
$480$		0			0
$481$	3.44512e10			0.643612
$482$		−	1.47400e11i		−	2.73093i
$483$	−9.02711e10	−	3.56261e10i	−1.65867	−	0.654605i
$484$	1.17280e11			2.13718
$485$		0			0
$486$	−3.12209e10	+	9.79165e10i	−0.559630	+	1.75514i
$487$	4.71599e10			0.838411			0.419205	−	0.907891i	$-0.362309\pi$
$487$	4.71599e10			0.838411			0.419205	+	0.907891i	$0.362309\pi$
$488$			8.26585e10i			1.45750i
$489$	2.80253e10	−	7.10118e10i	0.490134	−	1.24192i
$490$		0			0
$491$			6.61089e10i			1.13745i	0.822526	+	0.568727i	$0.192564\pi$
$491$			6.61089e10i			1.13745i	−0.822526	+	0.568727i	$0.807436\pi$
$492$	−1.03123e9	−	4.06981e8i	−0.0175993	−	0.00694567i
$493$	1.98981e10			0.336841
$494$			1.18414e11i			1.98836i
$495$		0			0
$496$	−1.27435e11			−2.10554
$497$		−	1.71995e10i		−	0.281898i
$498$	4.11298e10	−	1.04217e11i	0.668713	−	1.69442i
$499$	7.91261e10			1.27620			0.638098	−	0.769955i	$-0.279721\pi$
$499$	7.91261e10			1.27620			0.638098	+	0.769955i	$0.279721\pi$
$500$		0			0
$501$	8.94478e10	+	3.53012e10i	1.41977	+	0.560323i
$502$	−5.64751e9			−0.0889287
$503$		−	6.52715e10i		−	1.01965i	−0.860278	−	0.509826i	$-0.829710\pi$
$503$		−	6.52715e10i		−	1.01965i	0.860278	−	0.509826i	$-0.170290\pi$
$504$	−1.49121e11	+	1.59499e11i	−2.31108	+	2.47192i
$505$		0			0
$506$		−	2.24780e11i		−	3.42891i
$507$	−5.10707e9	+	1.29405e10i	−0.0772929	+	0.195848i
$508$	−1.16520e10			−0.174962
$509$		−	4.80429e10i		−	0.715745i	−0.933771	−	0.357872i	$-0.883502\pi$
$509$		−	4.80429e10i		−	0.715745i	0.933771	−	0.357872i	$-0.116498\pi$
$510$		0			0
$511$	−1.59995e11			−2.34651
$512$			1.33470e11i			1.94225i
$513$	−6.13800e10	+	2.91309e10i	−0.886252	+	0.420615i
$514$	7.48914e10			1.07295
$515$		0			0
$516$	7.30569e10	−	1.85115e11i	1.03053	−	2.61122i
$517$	−6.32455e10			−0.885253
$518$			1.02263e11i			1.42036i
$519$	1.52978e10	+	6.03739e9i	0.210844	+	0.0832108i
$520$		0			0
$521$			6.97814e10i			0.947085i	0.880771	+	0.473542i	$0.157025\pi$
$521$			6.97814e10i			0.947085i	−0.880771	+	0.473542i	$0.842975\pi$
$522$	1.07353e11	+	1.00368e11i	1.44588	+	1.35180i
$523$	1.13309e11			1.51445			0.757227	−	0.653152i	$-0.226554\pi$
$523$	1.13309e11			1.51445			0.757227	+	0.653152i	$0.226554\pi$
$524$		−	2.99665e11i		−	3.97475i
$525$		0			0
$526$	4.76278e10			0.622183
$527$		−	2.17948e10i		−	0.282560i
$528$	−2.32350e11	−	9.16986e10i	−2.98956	−	1.17985i
$529$	−6.50228e10			−0.830315
$530$		0			0
$531$	7.80211e9	−	8.34511e9i	0.0981373	−	0.104967i
$532$	−2.47921e11			−3.09504
$533$		−	7.01875e8i		−	0.00869663i
$534$	1.65015e10	−	4.18122e10i	0.202935	−	0.514207i
$535$		0			0
$536$			1.03991e11i			1.25990i
$537$	−7.58061e10	−	2.99174e10i	−0.911605	−	0.359771i
$538$	−2.17261e11			−2.59330
$539$			8.56093e10i			1.01430i
$540$		0			0
$541$	−3.53511e10			−0.412680			−0.206340	−	0.978480i	$-0.566155\pi$
$541$	−3.53511e10			−0.412680			−0.206340	+	0.978480i	$0.566155\pi$
$542$			1.14960e11i			1.33214i
$543$	−2.63960e10	+	6.68835e10i	−0.303626	+	0.769342i
$544$	4.76632e10			0.544237
$545$		0			0
$546$	−2.20849e11	−	8.71594e10i	−2.48499	−	0.980717i
$547$	−3.44410e10			−0.384703			−0.192352	−	0.981326i	$-0.561611\pi$
$547$	−3.44410e10			−0.384703			−0.192352	+	0.981326i	$0.561611\pi$
$548$			8.21100e10i			0.910487i
$549$	−3.76703e10	−	3.52192e10i	−0.414677	−	0.387695i
$550$		0			0
$551$			9.71557e10i			1.05405i
$552$	−1.18388e11	+	2.99977e11i	−1.27512	+	3.23096i
$553$	1.37820e11			1.47371
$554$			2.41261e11i			2.56123i
$555$		0			0
$556$	3.33604e9			0.0349086
$557$			1.45667e9i			0.0151335i	0.999971	+	0.00756676i	$0.00240860\pi$
$557$			1.45667e9i			0.0151335i	−0.999971	+	0.00756676i	$0.997591\pi$
$558$	1.09935e11	−	1.17586e11i	1.13397	−	1.21288i
$559$	1.25993e11			1.29033
$560$		0			0
$561$	1.56829e10	−	3.97381e10i	0.158334	−	0.401195i
$562$	2.10782e11			2.11295
$563$		−	1.15715e11i		−	1.15175i	−0.817539	−	0.575873i	$-0.804662\pi$
$563$		−	1.15715e11i		−	1.15175i	0.817539	−	0.575873i	$-0.195338\pi$
$564$	1.44964e11	+	5.72112e10i	1.43267	+	0.565412i
$565$		0			0
$566$			2.18734e11i			2.13133i
$567$	−9.15161e9	−	1.35919e11i	−0.0885453	−	1.31507i
$568$	−5.71553e10			−0.549115
$569$		−	3.28384e10i		−	0.313280i	−0.987656	−	0.156640i	$-0.949934\pi$
$569$		−	3.28384e10i		−	0.313280i	0.987656	−	0.156640i	$-0.0500663\pi$
$570$		0			0
$571$	1.02275e11			0.962111			0.481055	−	0.876690i	$-0.340254\pi$
$571$	1.02275e11			0.962111			0.481055	+	0.876690i	$0.340254\pi$
$572$		−	3.87882e11i		−	3.62340i
$573$	1.62748e11	+	6.42298e10i	1.50973	+	0.595824i
$574$	2.08340e9			0.0191922
$575$		0			0
$576$	6.93145e10	+	6.48044e10i	0.629701	+	0.588728i
$577$	1.74070e10			0.157043			0.0785217	−	0.996912i	$-0.474980\pi$
$577$	1.74070e10			0.157043			0.0785217	+	0.996912i	$0.474980\pi$
$578$		−	1.85404e11i		−	1.66115i
$579$	4.47319e10	−	1.13344e11i	0.398018	−	1.00852i
$580$		0			0
$581$			1.48508e11i			1.30331i
$582$	7.82021e10	+	3.08630e10i	0.681595	+	0.268996i
$583$	1.22056e11			1.05654
$584$			5.31674e11i			4.57082i
$585$		0			0
$586$	5.11652e9			0.0433895
$587$			5.82932e10i			0.490982i	0.969399	+	0.245491i	$0.0789492\pi$
$587$			5.82932e10i			0.490982i	−0.969399	+	0.245491i	$0.921051\pi$
$588$	7.74413e10	−	1.96224e11i	0.647833	−	1.64151i
$589$	1.06417e11			0.884196
$590$		0			0
$591$	2.75539e10	+	1.08743e10i	0.225857	+	0.0891359i
$592$	1.67843e11			1.36652
$593$			9.05975e10i			0.732652i	0.930487	+	0.366326i	$0.119384\pi$
$593$			9.05975e10i			0.732652i	−0.930487	+	0.366326i	$0.880616\pi$
$594$	2.85054e11	−	1.35287e11i	2.28972	−	1.08670i
$595$		0			0
$596$			9.50228e10i			0.753083i
$597$	5.47509e10	−	1.38730e11i	0.431016	−	1.09213i
$598$	−3.50667e11			−2.74214
$599$			2.21600e11i			1.72133i	0.509175	+	0.860663i	$0.329951\pi$
$599$			2.21600e11i			1.72133i	−0.509175	+	0.860663i	$0.670049\pi$
$600$		0			0
$601$	−9.42710e10			−0.722571			−0.361285	−	0.932455i	$-0.617662\pi$
$601$	−9.42710e10			−0.722571			−0.361285	+	0.932455i	$0.617662\pi$
$602$			3.73990e11i			2.84757i
$603$	−4.73924e10	−	4.43087e10i	−0.358459	−	0.335135i
$604$	−5.88471e11			−4.42158
$605$		0			0
$606$	4.35365e10	−	1.10315e11i	0.322822	−	0.817981i
$607$	2.52304e11			1.85853			0.929265	−	0.369415i	$-0.120442\pi$
$607$	2.52304e11			1.85853			0.929265	+	0.369415i	$0.120442\pi$
$608$			2.32723e11i			1.70304i
$609$	−1.81201e11	−	7.15120e10i	−1.31732	−	0.519888i
$610$		0			0
$611$			9.86657e10i			0.707948i
$612$	−7.18934e10	+	7.68968e10i	−0.512487	+	0.548154i
$613$	−1.39397e11			−0.987213			−0.493607	−	0.869685i	$-0.664322\pi$
$613$	−1.39397e11			−0.987213			−0.493607	+	0.869685i	$0.664322\pi$
$614$		−	4.09315e11i		−	2.87995i
$615$		0			0
$616$	6.70365e11			4.65574
$617$		−	4.85857e10i		−	0.335249i	−0.985851	−	0.167625i	$-0.946390\pi$
$617$		−	4.85857e10i		−	0.335249i	0.985851	−	0.167625i	$-0.0536096\pi$
$618$	2.89780e11	+	1.14364e11i	1.98662	+	0.784032i
$619$	1.67889e11			1.14356			0.571781	−	0.820406i	$-0.306253\pi$
$619$	1.67889e11			1.14356			0.571781	+	0.820406i	$0.306253\pi$
$620$		0			0
$621$	−8.62671e10	−	1.81768e11i	−0.580068	−	1.22223i
$622$	−2.83423e11			−1.89354
$623$			5.95822e10i			0.395516i
$624$	−1.43054e11	+	3.62477e11i	−0.943542	+	2.39079i
$625$		0			0
$626$		−	4.96347e11i		−	3.23213i
$627$	1.94027e11	+	7.65742e10i	1.25543	+	0.495464i
$628$	1.48800e11			0.956676
$629$			2.87056e10i			0.183385i
$630$		0			0
$631$	−7.74846e10			−0.488763			−0.244381	−	0.969679i	$-0.578585\pi$
$631$	−7.74846e10			−0.488763			−0.244381	+	0.969679i	$0.578585\pi$
$632$		−	4.57985e11i		−	2.87067i
$633$	2.44197e10	−	6.18759e10i	0.152099	−	0.385396i
$634$	−5.78163e11			−3.57844
$635$		0			0
$636$	−2.79764e11	−	1.10411e11i	−1.70987	−	0.674811i
$637$	1.33554e11			0.811148
$638$		−	4.51200e11i		−	2.72324i
$639$	2.43528e10	−	2.60477e10i	0.146065	−	0.156230i
$640$		0			0
$641$		−	2.34664e11i		−	1.39000i	−0.719012	−	0.694998i	$-0.755405\pi$
$641$		−	2.34664e11i		−	1.39000i	0.719012	−	0.694998i	$-0.244595\pi$
$642$	−7.07030e10	+	1.79151e11i	−0.416196	+	1.05458i
$643$	3.28468e11			1.92154			0.960768	−	0.277353i	$-0.0894572\pi$
$643$	3.28468e11			1.92154			0.960768	+	0.277353i	$0.0894572\pi$
$644$		−	7.34182e11i		−	4.26835i
$645$		0			0
$646$	−9.86657e10			−0.566547
$647$			1.20987e11i			0.690435i	0.938523	+	0.345217i	$0.112195\pi$
$647$			1.20987e11i			0.690435i	−0.938523	+	0.345217i	$0.887805\pi$
$648$	−4.51668e11	+	3.04115e10i	−2.56165	+	0.172479i
$649$	−3.50740e10			−0.197700
$650$		0			0
$651$	−7.83286e10	+	1.98473e11i	−0.436110	+	1.10504i
$652$	5.77545e11			3.19592
$653$		−	6.71025e9i		−	0.0369051i	−0.999830	−	0.0184525i	$-0.994126\pi$
$653$		−	6.71025e9i		−	0.0369051i	0.999830	−	0.0184525i	$-0.00587396\pi$
$654$	5.30121e11	+	2.09216e11i	2.89777	+	1.14362i
$655$		0			0
$656$		−	3.41947e9i		−	0.0184647i
$657$	−2.42302e11	−	2.26536e11i	−1.30046	−	1.21584i
$658$	−2.92873e11			−1.56234
$659$			2.82641e11i			1.49863i	0.662214	+	0.749315i	$0.269617\pi$
$659$			2.82641e11i			1.49863i	−0.662214	+	0.749315i	$0.730383\pi$
$660$		0			0
$661$	−3.05260e10			−0.159906			−0.0799530	−	0.996799i	$-0.525477\pi$
$661$	−3.05260e10			−0.159906			−0.0799530	+	0.996799i	$0.525477\pi$
$662$			3.30243e11i			1.71950i
$663$	−6.19932e10	−	2.44660e10i	−0.320841	−	0.126622i
$664$	4.93504e11			2.53874
$665$		0			0
$666$	−1.44794e11	+	1.54871e11i	−0.735958	+	0.787177i
$667$	−2.87713e11			−1.45364
$668$			7.27486e11i			3.65358i
$669$	−1.11821e10	+	2.83338e10i	−0.0558239	+	0.141449i
$670$		0			0
$671$			1.58326e11i			0.781022i
$672$	−4.34041e11	−	1.71297e11i	−2.12840	−	0.839988i
$673$	1.35686e11			0.661417			0.330708	−	0.943733i	$-0.392712\pi$
$673$	1.35686e11			0.661417			0.330708	+	0.943733i	$0.392712\pi$
$674$			2.41104e10i			0.116833i
$675$		0			0
$676$	−1.05246e11			−0.503988
$677$		−	3.54740e11i		−	1.68871i	−0.535783	−	0.844356i	$-0.679984\pi$
$677$		−	3.54740e11i		−	1.68871i	0.535783	−	0.844356i	$-0.320016\pi$
$678$	−8.09685e10	+	2.05162e11i	−0.383175	+	0.970907i
$679$	−1.11438e11			−0.524268
$680$		0			0
$681$	3.45113e11	+	1.36201e11i	1.60462	+	0.633276i
$682$	−4.94209e11			−2.28440
$683$		−	3.26911e11i		−	1.50227i	−0.660151	−	0.751133i	$-0.729508\pi$
$683$		−	3.26911e11i		−	1.50227i	0.660151	−	0.751133i	$-0.270492\pi$
$684$	−3.75460e11	−	3.51030e11i	−1.71530	−	1.60369i
$685$		0			0
$686$		−	1.41293e11i		−	0.638007i
$687$	−4.95472e10	+	1.25545e11i	−0.222429	+	0.563603i
$688$	6.13826e11			2.73963
$689$		−	1.90413e11i		−	0.844927i
$690$		0			0
$691$	2.99967e11			1.31571			0.657856	−	0.753143i	$-0.271463\pi$
$691$	2.99967e11			1.31571			0.657856	+	0.753143i	$0.271463\pi$
$692$			1.24418e11i			0.542576i
$693$	−2.85630e11	+	3.05509e11i	−1.23843	+	1.32462i
$694$	4.25836e11			1.83571
$695$		0			0
$696$	−2.37640e11	+	6.02143e11i	−1.01270	+	2.56603i
$697$	5.84820e8			0.00247794
$698$			4.28567e10i			0.180550i
$699$	−3.20309e11	−	1.26412e11i	−1.34172	−	0.529517i
$700$		0			0
$701$			1.72997e11i			0.716416i	0.933642	+	0.358208i	$0.116612\pi$
$701$			1.72997e11i			0.716416i	−0.933642	+	0.358208i	$0.883388\pi$
$702$	−2.11053e11	−	4.44697e11i	−0.869047	−	1.83112i
$703$	−1.40160e11			−0.573854
$704$		−	2.91325e11i		−	1.18601i
$705$		0			0
$706$	−3.61073e11			−1.45337
$707$			1.57198e11i			0.629173i
$708$	8.03930e10	+	3.17276e10i	0.319952	+	0.126271i
$709$	2.67236e11			1.05757			0.528786	−	0.848755i	$-0.322648\pi$
$709$	2.67236e11			1.05757			0.528786	+	0.848755i	$0.322648\pi$
$710$		0			0
$711$	2.08720e11	+	1.95139e11i	0.816743	+	0.763600i
$712$	1.97996e11			0.770435
$713$			3.15138e11i			1.21939i
$714$	7.26235e10	−	1.84017e11i	0.279437	−	0.708051i
$715$		0			0
$716$		−	6.16537e11i		−	2.34589i
$717$	−1.39863e11	−	5.51979e10i	−0.529208	−	0.208855i
$718$	3.20815e11			1.20714
$719$			3.27548e11i			1.22563i	0.790227	+	0.612815i	$0.209963\pi$
$719$			3.27548e11i			1.22563i	−0.790227	+	0.612815i	$0.790037\pi$
$720$		0			0
$721$	−4.12935e11			−1.52806
$722$			1.88430e10i			0.0693427i
$723$	−1.48701e11	+	3.76786e11i	−0.544203	+	1.37893i
$724$	−5.43969e11			−1.97979
$725$		0			0
$726$	−4.25034e11	−	1.67743e11i	−1.52995	−	0.603805i
$727$	−8.40255e10			−0.300797			−0.150398	−	0.988625i	$-0.548056\pi$
$727$	−8.40255e10			−0.300797			−0.150398	+	0.988625i	$0.548056\pi$
$728$		−	1.04580e12i		−	3.72325i
$729$	1.78588e11	−	2.18799e11i	0.632327	−	0.774702i
$730$		0			0
$731$			1.04981e11i			0.367654i
$732$	1.43220e11	−	3.62899e11i	0.498839	−	1.26398i
$733$	4.23327e10			0.146643			0.0733213	−	0.997308i	$-0.476640\pi$
$733$	4.23327e10			0.146643			0.0733213	+	0.997308i	$0.476640\pi$
$734$		−	6.90673e11i		−	2.37951i
$735$		0			0
$736$	−6.89177e11			−2.34866
$737$			1.99188e11i			0.675138i
$738$	3.15518e9	+	2.94989e9i	0.0106365	+	0.00994443i
$739$	2.88845e11			0.968473			0.484236	−	0.874937i	$-0.339097\pi$
$739$	2.88845e11			0.968473			0.484236	+	0.874937i	$0.339097\pi$
$740$		0			0
$741$	1.19459e11	−	3.02691e11i	0.396229	−	1.00398i
$742$	5.65210e11			1.86464
$743$		−	3.03005e11i		−	0.994247i	−0.867680	−	0.497123i	$-0.834390\pi$
$743$		−	3.03005e11i		−	0.994247i	0.867680	−	0.497123i	$-0.165610\pi$
$744$	6.59539e11	+	2.60292e11i	2.15253	+	0.849510i
$745$		0			0
$746$		−	3.80297e11i		−	1.22791i
$747$	−2.10273e11	+	2.24907e11i	−0.675306	+	0.722305i
$748$	3.23194e11			1.03242
$749$		−	2.55289e11i		−	0.811157i
$750$		0			0
$751$	4.66239e11			1.46571			0.732856	−	0.680383i	$-0.238187\pi$
$751$	4.66239e11			1.46571			0.732856	+	0.680383i	$0.238187\pi$
$752$			4.80690e11i			1.50312i
$753$	1.44362e10	+	5.69735e9i	0.0449028	+	0.0177212i
$754$	−7.03892e11			−2.17781
$755$		0			0
$756$	9.31051e11	−	4.41876e11i	2.85027	−	1.35274i
$757$	1.41830e11			0.431901			0.215951	−	0.976404i	$-0.430715\pi$
$757$	1.41830e11			0.431901			0.215951	+	0.976404i	$0.430715\pi$
$758$			1.51083e11i			0.457655i
$759$	−2.26764e11	+	5.74585e11i	−0.683293	+	1.73136i
$760$		0			0
$761$		−	2.61107e11i		−	0.778538i	−0.921124	−	0.389269i	$-0.872728\pi$
$761$		−	2.61107e11i		−	0.778538i	0.921124	−	0.389269i	$-0.127272\pi$
$762$	4.22279e10	+	1.66655e10i	0.125251	+	0.0494310i
$763$	−7.55420e11			−2.22890
$764$			1.32365e12i			3.88507i
$765$		0			0
$766$	1.07949e12			3.13549
$767$			5.47170e10i			0.158103i
$768$	1.44906e11	−	3.67170e11i	0.416526	−	1.05541i
$769$	−6.57270e11			−1.87948			−0.939742	−	0.341884i	$-0.888935\pi$
$769$	−6.57270e11			−1.87948			−0.939742	+	0.341884i	$0.888935\pi$
$770$		0			0
$771$	−1.91438e11	−	7.55523e10i	−0.541764	−	0.213811i
$772$	9.21835e11			2.59528
$773$		−	2.64170e10i		−	0.0739887i	−0.999315	−	0.0369943i	$-0.988222\pi$
$773$		−	2.64170e10i		−	0.0739887i	0.999315	−	0.0369943i	$-0.0117784\pi$
$774$	−5.29532e11	+	5.66385e11i	−1.47546	+	1.57815i
$775$		0			0
$776$			3.70315e11i			1.02123i
$777$	1.03165e11	−	2.61405e11i	0.283041	−	0.717183i
$778$	−9.31898e11			−2.54361
$779$			2.85547e9i			0.00775404i
$780$		0			0
$781$	−1.09477e11			−0.294251
$782$		−	2.92185e11i		−	0.781323i
$783$	−1.73163e11	−	3.64862e11i	−0.460691	−	0.970693i
$784$	6.50663e11			1.72223
$785$		0			0
$786$	−4.28603e11	+	1.08602e12i	−1.12296	+	2.84542i
$787$	−1.72594e11			−0.449911			−0.224956	−	0.974369i	$-0.572224\pi$
$787$	−1.72594e11			−0.449911			−0.224956	+	0.974369i	$0.572224\pi$
$788$			2.24098e11i			0.581210i
$789$	−1.21747e11	−	4.80482e10i	−0.314159	−	0.123985i
$790$		0			0
$791$		−	2.92355e11i		−	0.746800i
$792$	1.01523e12	+	9.49169e11i	2.58025	+	2.41236i
$793$	2.46996e11			0.624594
$794$		−	1.48916e10i		−	0.0374679i
$795$		0			0
$796$	1.12831e12			2.81044
$797$		−	7.06159e9i		−	0.0175013i	−0.999962	−	0.00875063i	$-0.997215\pi$
$797$		−	7.06159e9i		−	0.0175013i	0.999962	−	0.00875063i	$-0.00278545\pi$
$798$	8.98490e11	+	3.54595e11i	2.21565	+	0.874422i
$799$	−8.22108e10			−0.201717
$800$		0			0
$801$	−8.43623e10	+	9.02336e10i	−0.204936	+	0.219199i
$802$	−1.08229e12			−2.61605
$803$			1.01838e12i			2.44934i
$804$	1.80183e11	−	4.56557e11i	0.431211	−	1.09262i
$805$		0			0
$806$			7.70987e11i			1.82687i
$807$	5.55365e11	+	2.19179e11i	1.30944	+	0.516778i
$808$	5.22381e11			1.22558
$809$		−	5.66925e11i		−	1.32352i	−0.749714	−	0.661762i	$-0.769809\pi$
$809$		−	5.66925e11i		−	1.32352i	0.749714	−	0.661762i	$-0.230191\pi$
$810$		0			0
$811$	−5.11294e11			−1.18192			−0.590959	−	0.806702i	$-0.701250\pi$
$811$	−5.11294e11			−1.18192			−0.590959	+	0.806702i	$0.701250\pi$
$812$		−	1.47372e12i		−	3.38993i
$813$	1.15975e11	−	2.93862e11i	0.265461	−	0.672638i
$814$	6.50914e11			1.48261
$815$		0			0
$816$	−3.02025e11	−	1.19196e11i	−0.681212	−	0.268845i
$817$	−5.12584e11			−1.15047
$818$		−	2.09626e11i		−	0.468201i
$819$	4.76607e11	+	4.45595e11i	1.05931	+	0.990387i
$820$		0			0
$821$		−	4.92725e10i		−	0.108451i	−0.998529	−	0.0542253i	$-0.982731\pi$
$821$		−	4.92725e10i		−	0.108451i	0.998529	−	0.0542253i	$-0.0172689\pi$
$822$	1.17440e11	−	2.97575e11i	0.257234	−	0.651792i
$823$	−2.61102e11			−0.569130			−0.284565	−	0.958657i	$-0.591849\pi$
$823$	−2.61102e11			−0.569130			−0.284565	+	0.958657i	$0.591849\pi$
$824$			1.37221e12i			2.97655i
$825$		0			0
$826$	−1.62419e11			−0.348912
$827$			5.67337e11i			1.21288i	0.795128	+	0.606442i	$0.207404\pi$
$827$			5.67337e11i			1.21288i	−0.795128	+	0.606442i	$0.792596\pi$
$828$	1.03953e12	−	1.11187e12i	2.21164	−	2.36556i
$829$	−6.10109e11			−1.29178			−0.645891	−	0.763430i	$-0.723514\pi$
$829$	−6.10109e11			−1.29178			−0.645891	+	0.763430i	$0.723514\pi$
$830$		0			0
$831$	2.43390e11	−	6.16714e11i	0.510387	−	1.29324i
$832$	−4.54480e11			−0.948466
$833$			1.11281e11i			0.231121i
$834$	−1.20902e10	−	4.77147e9i	−0.0249901	−	0.00986251i
$835$		0			0
$836$			1.57804e12i			3.23067i
$837$	−3.99641e11	+	1.89670e11i	−0.814270	+	0.386452i
$838$	−7.70720e11			−1.56286
$839$			1.77752e10i			0.0358729i	0.999839	+	0.0179365i	$0.00570966\pi$
$839$			1.77752e10i			0.0358729i	−0.999839	+	0.0179365i	$0.994290\pi$
$840$		0			0
$841$	−7.72785e10			−0.154481
$842$		−	1.71037e11i		−	0.340285i
$843$	−5.38804e11	−	2.12642e11i	−1.06689	−	0.421056i
$844$	5.03242e11			0.991761
$845$		0			0
$846$	−4.43539e11	−	4.14679e11i	−0.865865	−	0.809525i
$847$	6.05672e11			1.17680
$848$		−	9.27672e11i		−	1.79395i
$849$	2.20665e11	−	5.59131e11i	0.424720	−	1.07617i
$850$		0			0
$851$		−	4.15063e11i		−	0.791399i
$852$	2.50931e11	+	9.90317e10i	0.476208	+	0.187939i
$853$	4.79452e11			0.905626			0.452813	−	0.891606i	$-0.350421\pi$
$853$	4.79452e11			0.905626			0.452813	+	0.891606i	$0.350421\pi$
$854$			7.33168e11i			1.37839i
$855$		0			0
$856$	−8.48344e11			−1.58007
$857$			9.38022e10i			0.173896i	0.996213	+	0.0869480i	$0.0277114\pi$
$857$			9.38022e10i			0.173896i	−0.996213	+	0.0869480i	$0.972289\pi$
$858$	−5.54779e11	+	1.40573e12i	−1.02370	+	2.59389i
$859$	4.46094e11			0.819320			0.409660	−	0.912238i	$-0.365647\pi$
$859$	4.46094e11			0.819320			0.409660	+	0.912238i	$0.365647\pi$
$860$		0			0
$861$	−5.32561e9	−	2.10179e9i	−0.00969074	−	0.00382451i
$862$	−1.31384e12			−2.37965
$863$		−	8.85309e11i		−	1.59607i	−0.602612	−	0.798035i	$-0.705873\pi$
$863$		−	8.85309e11i		−	1.59607i	0.602612	−	0.798035i	$-0.294127\pi$
$864$	−4.14789e11	−	8.73977e11i	−0.744341	−	1.56836i
$865$		0			0
$866$			9.44837e11i			1.67991i
$867$	−1.87040e11	+	4.73932e11i	−0.331023	+	0.838763i
$868$	−1.61420e12			−2.84366
$869$		−	8.77239e11i		−	1.53829i
$870$		0			0
$871$	3.10742e11			0.539917
$872$			2.51031e12i			4.34172i
$873$	−1.68766e11	−	1.57784e11i	−0.290554	−	0.271648i
$874$	1.42664e12			2.44494
$875$		0			0
$876$	9.21219e11	−	2.33423e12i	1.56440	−	3.96394i
$877$	−1.47075e11			−0.248622			−0.124311	−	0.992243i	$-0.539672\pi$
$877$	−1.47075e11			−0.248622			−0.124311	+	0.992243i	$0.539672\pi$
$878$			7.58515e11i			1.27640i
$879$	−1.30789e10	−	5.16168e9i	−0.0219087	−	0.00864640i
$880$		0			0
$881$		−	1.00879e11i		−	0.167455i	−0.996489	−	0.0837275i	$-0.973317\pi$
$881$		−	1.00879e11i		−	0.167455i	0.996489	−	0.0837275i	$-0.0266825\pi$
$882$	−5.61310e11	+	6.00375e11i	−0.927532	+	0.992084i
$883$	1.17637e11			0.193509			0.0967543	−	0.995308i	$-0.469154\pi$
$883$	1.17637e11			0.193509			0.0967543	+	0.995308i	$0.469154\pi$
$884$		−	5.04196e11i		−	0.825639i
$885$		0			0
$886$	−2.08990e12			−3.39148
$887$		−	1.03512e12i		−	1.67223i	−0.548552	−	0.836117i	$-0.684821\pi$
$887$		−	1.03512e12i		−	1.67223i	0.548552	−	0.836117i	$-0.315179\pi$
$888$	−8.68668e11	−	3.42826e11i	−1.39702	−	0.551342i
$889$	−6.01747e10			−0.0963400
$890$		0			0
$891$	−8.65139e11	+	5.82510e10i	−1.37270	+	0.0924257i
$892$	−2.30441e11			−0.364000
$893$		−	4.01407e11i		−	0.631217i
$894$	1.35909e11	−	3.44373e11i	0.212764	−	0.539111i
$895$		0			0
$896$			1.25699e11i			0.195029i
$897$	8.96378e11	+	3.53762e11i	1.38459	+	0.546438i
$898$	1.89196e12			2.90942
$899$			6.32575e11i			0.968441i
$900$		0			0
$901$	1.58657e11			0.240746
$902$		−	1.32611e10i		−	0.0200333i
$903$	3.77291e11	−	9.55997e11i	0.567447	−	1.43782i
$904$	−9.71516e11			−1.45471
$905$		0			0
$906$	2.13268e12	+	8.41677e11i	3.16529	+	1.24920i
$907$	5.01146e11			0.740516			0.370258	−	0.928929i	$-0.379269\pi$
$907$	5.01146e11			0.740516			0.370258	+	0.928929i	$0.379269\pi$
$908$			2.80684e12i			4.12927i
$909$	−2.22577e11	+	2.38067e11i	−0.326005	+	0.348693i
$910$		0			0
$911$		−	4.27267e11i		−	0.620334i	−0.950682	−	0.310167i	$-0.899615\pi$
$911$		−	4.27267e11i		−	0.620334i	0.950682	−	0.310167i	$-0.100385\pi$
$912$	5.81993e11	−	1.47468e12i	0.841276	−	2.13167i
$913$	9.45272e11			1.36042
$914$		−	5.70498e11i		−	0.817466i
$915$		0			0
$916$	−1.02107e12			−1.45035
$917$		−	1.54757e12i		−	2.18863i
$918$	3.70533e11	−	1.75855e11i	0.521742	−	0.247619i
$919$	−1.00999e11			−0.141598			−0.0707990	−	0.997491i	$-0.522555\pi$
$919$	−1.00999e11			−0.141598			−0.0707990	+	0.997491i	$0.522555\pi$
$920$		0			0
$921$	−4.12928e11	+	1.04630e12i	−0.573899	+	1.45417i
$922$	−1.66589e12			−2.30528
$923$			1.70789e11i			0.235317i
$924$	−2.94313e12	−	1.16153e12i	−4.03759	−	1.59346i
$925$		0			0
$926$		−	8.06756e10i		−	0.109723i
$927$	−6.25365e11	−	5.84674e11i	−0.846866	−	0.791763i
$928$	−1.38338e12			−1.86530
$929$		−	7.10245e11i		−	0.953554i	−0.879024	−	0.476777i	$-0.841805\pi$
$929$		−	7.10245e11i		−	0.953554i	0.879024	−	0.476777i	$-0.158195\pi$
$930$		0			0
$931$	−5.43345e11			−0.723231
$932$		−	2.60510e12i		−	3.45272i
$933$	7.24489e11	+	2.85925e11i	0.956105	+	0.377333i
$934$	5.87398e11			0.771872
$935$		0			0
$936$	1.48075e12	−	1.58380e12i	1.92920	−	2.06346i
$937$	9.65584e11			1.25266			0.626328	−	0.779560i	$-0.284557\pi$
$937$	9.65584e11			1.25266			0.626328	+	0.779560i	$0.284557\pi$
$938$			9.22386e11i			1.19152i
$939$	−5.00728e11	+	1.26877e12i	−0.644079	+	1.63200i
$940$		0			0
$941$		−	2.54695e11i		−	0.324835i	−0.986722	−	0.162417i	$-0.948071\pi$
$941$		−	2.54695e11i		−	0.324835i	0.986722	−	0.162417i	$-0.0519291\pi$
$942$	−5.39267e11	−	2.12825e11i	−0.684858	−	0.270284i
$943$	−8.45609e9			−0.0106936
$944$			2.66576e11i			0.335686i
$945$		0			0
$946$	2.38049e12			2.97236
$947$			1.42329e12i			1.76967i	0.465905	+	0.884835i	$0.345729\pi$
$947$			1.42329e12i			1.76967i	−0.465905	+	0.884835i	$0.654271\pi$
$948$	−7.93541e11	+	2.01071e12i	−0.982508	+	2.48953i
$949$	1.58872e12			1.95877
$950$		0			0
$951$	1.47791e12	+	5.83266e11i	1.80686	+	0.713090i
$952$	8.71387e11			1.06087
$953$			9.36372e11i			1.13521i	0.823300	+	0.567606i	$0.192130\pi$
$953$			9.36372e11i			1.13521i	−0.823300	+	0.567606i	$0.807870\pi$
$954$	8.55975e11	+	8.00279e11i	1.03340	+	0.966158i
$955$		0			0
$956$		−	1.13752e12i		−	1.36184i
$957$	−4.55182e11	+	1.15336e12i	−0.542672	+	1.37505i
$958$	9.34140e11			1.10905
$959$			4.24043e11i			0.501344i
$960$		0			0
$961$	−1.60019e11			−0.187619
$962$		−	1.01545e12i		−	1.18566i
$963$	3.61463e11	−	3.86620e11i	0.420300	−	0.449551i
$964$	−3.06443e12			−3.54848
$965$		0			0
$966$	−1.05008e12	+	2.66075e12i	−1.20591	+	3.05560i
$967$	1.17155e11			0.133984			0.0669921	−	0.997754i	$-0.478660\pi$
$967$	1.17155e11			0.133984			0.0669921	+	0.997754i	$0.478660\pi$
$968$		−	2.01269e12i		−	2.29232i
$969$	2.52210e11	+	9.95364e10i	0.286067	+	0.112898i
$970$		0			0
$971$			3.28921e11i			0.370011i	0.982737	+	0.185006i	$0.0592303\pi$
$971$			3.28921e11i			0.370011i	−0.982737	+	0.185006i	$0.940770\pi$
$972$	2.03567e12	+	6.49079e11i	2.28057	+	0.727164i
$973$	1.72284e10			0.0192218
$974$		−	1.39004e12i		−	1.54452i
$975$		0			0
$976$	1.20334e12			1.32614
$977$		−	1.00676e12i		−	1.10496i	−0.833525	−	0.552482i	$-0.813681\pi$
$977$		−	1.00676e12i		−	1.10496i	0.833525	−	0.552482i	$-0.186319\pi$
$978$	−2.09308e12	−	8.26049e11i	−2.28787	−	0.902922i
$979$	3.79247e11			0.412849
$980$		0			0
$981$	−1.14404e12	−	1.06960e12i	−1.23528	−	1.15490i
$982$	1.94857e12			2.09542
$983$		−	1.03088e12i		−	1.10406i	−0.833823	−	0.552032i	$-0.813852\pi$
$983$		−	1.03088e12i		−	1.10406i	0.833823	−	0.552032i	$-0.186148\pi$
$984$	−6.98439e9	+	1.76974e10i	−0.00744986	+	0.0188768i
$985$		0			0
$986$		−	5.86501e11i		−	0.620527i
$987$	7.48645e11	+	2.95458e11i	0.788874	+	0.311334i
$988$	2.46181e12			2.58361
$989$		−	1.51795e12i		−	1.58661i
$990$		0			0
$991$	−1.00159e11			−0.103847			−0.0519236	−	0.998651i	$-0.516535\pi$
$991$	−1.00159e11			−0.103847			−0.0519236	+	0.998651i	$0.516535\pi$
$992$			1.51525e12i			1.56472i
$993$	3.33158e11	−	8.44171e11i	0.342652	−	0.868227i
$994$	−5.06959e11			−0.519311
$995$		0			0
$996$	−2.16665e12	−	8.55083e11i	−2.20167	−	0.868903i
$997$	1.30987e12			1.32570			0.662852	−	0.748750i	$-0.269346\pi$
$997$	1.30987e12			1.32570			0.662852	+	0.748750i	$0.269346\pi$
$998$		−	2.33225e12i		−	2.35100i
$999$	5.26361e11	−	2.49810e11i	0.528471	−	0.250812i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.c.e.26.1	✓	10
3.2	odd	2	inner	75.9.c.e.26.10	yes	10
5.2	odd	4		75.9.d.d.74.20		20
5.3	odd	4		75.9.d.d.74.1		20
5.4	even	2		75.9.c.f.26.10	yes	10
15.2	even	4		75.9.d.d.74.2		20
15.8	even	4		75.9.d.d.74.19		20
15.14	odd	2		75.9.c.f.26.1	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.c.e.26.1	✓	10	1.1	even	1	trivial
75.9.c.e.26.10	yes	10	3.2	odd	2	inner
75.9.c.f.26.1	yes	10	15.14	odd	2
75.9.c.f.26.10	yes	10	5.4	even	2
75.9.d.d.74.1		20	5.3	odd	4
75.9.d.d.74.2		20	15.2	even	4
75.9.d.d.74.19		20	15.8	even	4
75.9.d.d.74.20		20	5.2	odd	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-29.4751i q^{2} +(-29.7353 + 75.3446i) q^{3} -612.784 q^{4} +(2220.79 + 876.451i) q^{6} -3164.62 q^{7} +10516.3i q^{8} +(-4792.63 - 4480.79i) q^{9} +O(q^{10})\)	\(q-29.4751i q^{2} +(-29.7353 + 75.3446i) q^{3} -612.784 q^{4} +(2220.79 + 876.451i) q^{6} -3164.62 q^{7} +10516.3i q^{8} +(-4792.63 - 4480.79i) q^{9} +20143.2i q^{11} +(18221.3 - 46170.0i) q^{12} +31424.2 q^{13} +93277.7i q^{14} +153096. q^{16} +26183.5i q^{17} +(-132072. + 141263. i) q^{18} -127845. q^{19} +(94100.9 - 238437. i) q^{21} +593723. q^{22} -378595. i q^{23} +(-792344. - 312704. i) q^{24} -926234. i q^{26} +(480113. - 227861. i) q^{27} +1.93923e6 q^{28} -759951. i q^{29} -832390. q^{31} -1.82036e6i q^{32} +(-1.51768e6 - 598962. i) q^{33} +771762. q^{34} +(2.93685e6 + 2.74575e6i) q^{36} +1.09633e6 q^{37} +3.76824e6i q^{38} +(-934408. + 2.36765e6i) q^{39} -22335.5i q^{41} +(-7.02798e6 - 2.77364e6i) q^{42} +4.00943e6 q^{43} -1.23434e7i q^{44} -1.11591e7 q^{46} +3.13980e6i q^{47} +(-4.55234e6 + 1.15349e7i) q^{48} +4.25004e6 q^{49} +(-1.97278e6 - 778572. i) q^{51} -1.92563e7 q^{52} -6.05943e6i q^{53} +(-6.71625e6 - 1.41514e7i) q^{54} -3.32800e7i q^{56} +(3.80150e6 - 9.63242e6i) q^{57} -2.23997e7 q^{58} +1.74124e6i q^{59} +7.86006e6 q^{61} +2.45348e7i q^{62} +(1.51669e7 + 1.41800e7i) q^{63} -1.44627e7 q^{64} +(-1.76545e7 + 4.47338e7i) q^{66} +9.88860e6 q^{67} -1.60448e7i q^{68} +(2.85251e7 + 1.12576e7i) q^{69} +5.43494e6i q^{71} +(4.71211e7 - 5.04006e7i) q^{72} +5.05573e7 q^{73} -3.23144e7i q^{74} +7.83412e7 q^{76} -6.37455e7i q^{77} +(6.97867e7 + 2.75418e7i) q^{78} -4.35502e7 q^{79} +(2.89185e6 + 4.29495e7i) q^{81} -658341. q^{82} -4.69277e7i q^{83} +(-5.76636e7 + 1.46111e8i) q^{84} -1.18178e8i q^{86} +(5.72582e7 + 2.25973e7i) q^{87} -2.11831e8 q^{88} -1.88276e7i q^{89} -9.94458e7 q^{91} +2.31997e8i q^{92} +(2.47513e7 - 6.27161e7i) q^{93} +9.25460e7 q^{94} +(1.37154e8 + 5.41288e7i) q^{96} +3.52136e7 q^{97} -1.25271e8i q^{98} +(9.02572e7 - 9.65387e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9}+O(q^{10}) \) 10 * q - 25 * q^3 - 1554 * q^4 + 2257 * q^6 + 1960 * q^7 - 11207 * q^9	\( 10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9} - 5915 q^{12} - 16920 q^{13} + 44634 q^{16} - 224875 q^{18} - 143934 q^{19} + 673428 q^{21} + 818990 q^{22} - 1016859 q^{24} - 260830 q^{27} + 3810100 q^{28} - 3014060 q^{31} - 4677515 q^{33} + 4977146 q^{34} + 4500527 q^{36} + 3016760 q^{37} - 7513282 q^{39} - 4001760 q^{42} + 11747340 q^{43} - 13938636 q^{46} - 14748755 q^{48} + 8953546 q^{49} + 6209287 q^{51} - 38918320 q^{52} - 8886272 q^{54} + 14759525 q^{57} - 48407900 q^{58} + 1520220 q^{61} + 74748240 q^{63} - 4536998 q^{64} + 10465295 q^{66} - 16269290 q^{67} + 11394978 q^{69} + 172231185 q^{72} - 52090170 q^{73} - 29529046 q^{76} + 198205810 q^{78} + 8549896 q^{79} + 22612945 q^{81} - 295714190 q^{82} - 136883292 q^{84} + 318901610 q^{87} - 310673250 q^{88} - 107224264 q^{91} + 79679130 q^{93} + 356118596 q^{94} + 525424001 q^{96} - 402167800 q^{97} - 382421335 q^{99}+O(q^{100}) \) 10 * q - 25 * q^3 - 1554 * q^4 + 2257 * q^6 + 1960 * q^7 - 11207 * q^9 - 5915 * q^12 - 16920 * q^13 + 44634 * q^16 - 224875 * q^18 - 143934 * q^19 + 673428 * q^21 + 818990 * q^22 - 1016859 * q^24 - 260830 * q^27 + 3810100 * q^28 - 3014060 * q^31 - 4677515 * q^33 + 4977146 * q^34 + 4500527 * q^36 + 3016760 * q^37 - 7513282 * q^39 - 4001760 * q^42 + 11747340 * q^43 - 13938636 * q^46 - 14748755 * q^48 + 8953546 * q^49 + 6209287 * q^51 - 38918320 * q^52 - 8886272 * q^54 + 14759525 * q^57 - 48407900 * q^58 + 1520220 * q^61 + 74748240 * q^63 - 4536998 * q^64 + 10465295 * q^66 - 16269290 * q^67 + 11394978 * q^69 + 172231185 * q^72 - 52090170 * q^73 - 29529046 * q^76 + 198205810 * q^78 + 8549896 * q^79 + 22612945 * q^81 - 295714190 * q^82 - 136883292 * q^84 + 318901610 * q^87 - 310673250 * q^88 - 107224264 * q^91 + 79679130 * q^93 + 356118596 * q^94 + 525424001 * q^96 - 402167800 * q^97 - 382421335 * q^99

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