LMFDB - Embedded newform 75.9.c.g.26.6

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} - 4 x^{9} - 433 x^{8} - 2220 x^{7} + 49747 x^{6} + 744964 x^{5} + 4580249 x^{4} + 16418988 x^{3} + \cdots + 53656344 $ x^10 - 4x^9 - 433x^8 - 2220x^7 + 49747x^6 + 744964x^5 + 4580249x^4 + 16418988x^3 + 38943804x^2 + 57910464*x + 53656344
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5}\cdot 3^{11}\cdot 5^{10} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.6
Root			$-6.39402 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	75.26
Dual form			75.9.c.g.26.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.97572i q^{2} +(29.8424 + 75.3023i) q^{3} +192.388 q^{4} +(-600.590 + 238.014i) q^{6} +3211.86 q^{7} +3576.22i q^{8} +(-4779.87 + 4494.40i) q^{9} +O(q^{10})$	\(q+7.97572i q^{2} +(29.8424 + 75.3023i) q^{3} +192.388 q^{4} +(-600.590 + 238.014i) q^{6} +3211.86 q^{7} +3576.22i q^{8} +(-4779.87 + 4494.40i) q^{9} -6481.82i q^{11} +(5741.31 + 14487.3i) q^{12} +10510.3 q^{13} +25616.9i q^{14} +20728.4 q^{16} +57738.3i q^{17} +(-35846.0 - 38122.9i) q^{18} +228667. q^{19} +(95849.5 + 241860. i) q^{21} +51697.1 q^{22} +113904. i q^{23} +(-269297. + 106723. i) q^{24} +83827.4i q^{26} +(-481081. - 225811. i) q^{27} +617923. q^{28} -1.11666e6i q^{29} -304758. q^{31} +1.08084e6i q^{32} +(488096. - 193433. i) q^{33} -460504. q^{34} +(-919589. + 864668. i) q^{36} -630294. q^{37} +1.82379e6i q^{38} +(313653. + 791452. i) q^{39} +4.62696e6i q^{41} +(-1.92901e6 + 764469. i) q^{42} -5.44386e6 q^{43} -1.24702e6i q^{44} -908469. q^{46} +6.69692e6i q^{47} +(618586. + 1.56090e6i) q^{48} +4.55125e6 q^{49} +(-4.34782e6 + 1.72305e6i) q^{51} +2.02206e6 q^{52} -1.25196e7i q^{53} +(1.80101e6 - 3.83696e6i) q^{54} +1.14863e7i q^{56} +(6.82398e6 + 1.72192e7i) q^{57} +8.90615e6 q^{58} -5.55982e6i q^{59} -1.31704e7 q^{61} -2.43067e6i q^{62} +(-1.53523e7 + 1.44354e7i) q^{63} -3.31396e6 q^{64} +(1.54276e6 + 3.89291e6i) q^{66} -1.70044e7 q^{67} +1.11081e7i q^{68} +(-8.57726e6 + 3.39918e6i) q^{69} +1.08797e7i q^{71} +(-1.60729e7 - 1.70938e7i) q^{72} +2.67184e7 q^{73} -5.02705e6i q^{74} +4.39928e7 q^{76} -2.08187e7i q^{77} +(-6.31239e6 + 2.50161e6i) q^{78} -3.07855e6 q^{79} +(2.64753e6 - 4.29652e7i) q^{81} -3.69033e7 q^{82} -2.32739e7i q^{83} +(1.84403e7 + 4.65310e7i) q^{84} -4.34187e7i q^{86} +(8.40869e7 - 3.33237e7i) q^{87} +2.31804e7 q^{88} -3.21685e7i q^{89} +3.37577e7 q^{91} +2.19138e7i q^{92} +(-9.09471e6 - 2.29490e7i) q^{93} -5.34128e7 q^{94} +(-8.13894e7 + 3.22547e7i) q^{96} +9.03610e7 q^{97} +3.62994e7i q^{98} +(2.91319e7 + 3.09822e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 112 q^{3} - 786 q^{4} - 5282 q^{6} - 7156 q^{7} + 3922 q^{9}+O(q^{10}) $ 10 * q + 112 * q^3 - 786 * q^4 - 5282 * q^6 - 7156 * q^7 + 3922 * q^9	$ 10 q + 112 q^{3} - 786 q^{4} - 5282 q^{6} - 7156 q^{7} + 3922 q^{9} + 3812 q^{12} + 55464 q^{13} + 280386 q^{16} + 419800 q^{18} - 231516 q^{19} + 289572 q^{21} - 1129940 q^{22} + 1136334 q^{24} + 335512 q^{27} + 3340724 q^{28} + 881620 q^{31} - 1266460 q^{33} - 1111276 q^{34} - 668662 q^{36} - 4672616 q^{37} + 1826792 q^{39} + 5392860 q^{42} - 7731336 q^{43} - 25424604 q^{46} - 22413388 q^{48} + 9354214 q^{49} - 27732692 q^{51} - 21064016 q^{52} - 7979798 q^{54} + 2856304 q^{57} + 4351100 q^{58} + 22417020 q^{61} - 8830596 q^{63} - 22935002 q^{64} - 27419800 q^{66} + 46646024 q^{67} + 33562632 q^{69} - 54175560 q^{72} + 129964884 q^{73} + 198922436 q^{76} - 60388360 q^{78} + 162310924 q^{79} - 93575390 q^{81} - 202877560 q^{82} - 197346768 q^{84} + 168322540 q^{87} + 484775700 q^{88} + 444288464 q^{91} - 463412376 q^{93} - 92050036 q^{94} - 360807406 q^{96} + 258825724 q^{97} - 33965200 q^{99}+O(q^{100}) $ 10 * q + 112 * q^3 - 786 * q^4 - 5282 * q^6 - 7156 * q^7 + 3922 * q^9 + 3812 * q^12 + 55464 * q^13 + 280386 * q^16 + 419800 * q^18 - 231516 * q^19 + 289572 * q^21 - 1129940 * q^22 + 1136334 * q^24 + 335512 * q^27 + 3340724 * q^28 + 881620 * q^31 - 1266460 * q^33 - 1111276 * q^34 - 668662 * q^36 - 4672616 * q^37 + 1826792 * q^39 + 5392860 * q^42 - 7731336 * q^43 - 25424604 * q^46 - 22413388 * q^48 + 9354214 * q^49 - 27732692 * q^51 - 21064016 * q^52 - 7979798 * q^54 + 2856304 * q^57 + 4351100 * q^58 + 22417020 * q^61 - 8830596 * q^63 - 22935002 * q^64 - 27419800 * q^66 + 46646024 * q^67 + 33562632 * q^69 - 54175560 * q^72 + 129964884 * q^73 + 198922436 * q^76 - 60388360 * q^78 + 162310924 * q^79 - 93575390 * q^81 - 202877560 * q^82 - 197346768 * q^84 + 168322540 * q^87 + 484775700 * q^88 + 444288464 * q^91 - 463412376 * q^93 - 92050036 * q^94 - 360807406 * q^96 + 258825724 * q^97 - 33965200 * 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 $

$2$			7.97572i			0.498482i	0.968441	+	0.249241i	$0.0801812\pi$
$2$			7.97572i			0.498482i	−0.968441	+	0.249241i	$0.919819\pi$
$3$	29.8424	+	75.3023i	0.368424	+	0.929658i
$3$	29.8424	+	75.3023i	0.368424	+	0.929658i
$4$	192.388			0.751515
$5$		0			0
$5$		0			0
$6$	−600.590	+	238.014i	−0.463418	+	0.183653i
$7$	3211.86			1.33772			0.668859	−	0.743389i	$-0.266783\pi$
$7$	3211.86			1.33772			0.668859	+	0.743389i	$0.266783\pi$
$8$			3576.22i			0.873099i
$9$	−4779.87	+	4494.40i	−0.728527	+	0.685017i
$10$		0			0
$11$		−	6481.82i		−	0.442717i	−0.975193	−	0.221358i	$-0.928951\pi$
$11$		−	6481.82i		−	0.442717i	0.975193	−	0.221358i	$-0.0710491\pi$
$12$	5741.31	+	14487.3i	0.276877	+	0.698652i
$13$	10510.3			0.367996			0.183998	−	0.982927i	$-0.441096\pi$
$13$	10510.3			0.367996			0.183998	+	0.982927i	$0.441096\pi$
$14$			25616.9i			0.666829i
$15$		0			0
$16$	20728.4			0.316291
$17$			57738.3i			0.691302i	0.938363	+	0.345651i	$0.112342\pi$
$17$			57738.3i			0.691302i	−0.938363	+	0.345651i	$0.887658\pi$
$18$	−35846.0	−	38122.9i	−0.341469	−	0.363158i
$19$	228667.			1.75465			0.877324	−	0.479899i	$-0.159327\pi$
$19$	228667.			1.75465			0.877324	+	0.479899i	$0.159327\pi$
$20$		0			0
$21$	95849.5	+	241860.i	0.492848	+	1.24362i
$22$	51697.1			0.220686
$23$			113904.i			0.407033i	0.979072	+	0.203516i	$0.0652370\pi$
$23$			113904.i			0.407033i	−0.979072	+	0.203516i	$0.934763\pi$
$24$	−269297.	+	106723.i	−0.811684	+	0.321671i
$25$		0			0
$26$			83827.4i			0.183439i
$27$	−481081.	−	225811.i	−0.905238	−	0.424904i
$28$	617923.			1.00532
$29$		−	1.11666e6i		−	1.57880i	−0.613876	−	0.789402i	$-0.710391\pi$
$29$		−	1.11666e6i		−	1.57880i	0.613876	−	0.789402i	$-0.289609\pi$
$30$		0			0
$31$	−304758.			−0.329996			−0.164998	−	0.986294i	$-0.552762\pi$
$31$	−304758.			−0.329996			−0.164998	+	0.986294i	$0.552762\pi$
$32$			1.08084e6i			1.03076i
$33$	488096.	−	193433.i	0.411575	−	0.163108i
$34$	−460504.			−0.344602
$35$		0			0
$36$	−919589.	+	864668.i	−0.547499	+	0.514801i
$37$	−630294.			−0.336307			−0.168154	−	0.985761i	$-0.553780\pi$
$37$	−630294.			−0.336307			−0.168154	+	0.985761i	$0.553780\pi$
$38$			1.82379e6i			0.874661i
$39$	313653.	+	791452.i	0.135579	+	0.342110i
$40$		0			0
$41$			4.62696e6i			1.63742i	0.574207	+	0.818710i	$0.305310\pi$
$41$			4.62696e6i			1.63742i	−0.574207	+	0.818710i	$0.694690\pi$
$42$	−1.92901e6	+	764469.i	−0.619922	+	0.245676i
$43$	−5.44386e6			−1.59233			−0.796165	−	0.605080i	$-0.793141\pi$
$43$	−5.44386e6			−1.59233			−0.796165	+	0.605080i	$0.793141\pi$
$44$		−	1.24702e6i		−	0.332709i
$45$		0			0
$46$	−908469.			−0.202898
$47$			6.69692e6i			1.37241i	0.727408	+	0.686205i	$0.240725\pi$
$47$			6.69692e6i			1.37241i	−0.727408	+	0.686205i	$0.759275\pi$
$48$	618586.	+	1.56090e6i	0.116529	+	0.294042i
$49$	4.55125e6			0.789489
$50$		0			0
$51$	−4.34782e6	+	1.72305e6i	−0.642675	+	0.254693i
$52$	2.02206e6			0.276554
$53$		−	1.25196e7i		−	1.58667i	−0.608786	−	0.793335i	$-0.708343\pi$
$53$		−	1.25196e7i		−	1.58667i	0.608786	−	0.793335i	$-0.291657\pi$
$54$	1.80101e6	−	3.83696e6i	0.211807	−	0.451245i
$55$		0			0
$56$			1.14863e7i			1.16796i
$57$	6.82398e6	+	1.72192e7i	0.646455	+	1.63122i
$58$	8.90615e6			0.787006
$59$		−	5.55982e6i		−	0.458831i	−0.973329	−	0.229416i	$-0.926319\pi$
$59$		−	5.55982e6i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$60$		0			0
$61$	−1.31704e7			−0.951219			−0.475610	−	0.879656i	$-0.657773\pi$
$61$	−1.31704e7			−0.951219			−0.475610	+	0.879656i	$0.657773\pi$
$62$		−	2.43067e6i		−	0.164497i
$63$	−1.53523e7	+	1.44354e7i	−0.974564	+	0.916359i
$64$	−3.31396e6			−0.197527
$65$		0			0
$66$	1.54276e6	+	3.89291e6i	0.0813063	+	0.205163i
$67$	−1.70044e7			−0.843841			−0.421921	−	0.906633i	$-0.638644\pi$
$67$	−1.70044e7			−0.843841			−0.421921	+	0.906633i	$0.638644\pi$
$68$			1.11081e7i			0.519524i
$69$	−8.57726e6	+	3.39918e6i	−0.378401	+	0.149961i
$70$		0			0
$71$			1.08797e7i			0.428137i	0.976819	+	0.214068i	$0.0686715\pi$
$71$			1.08797e7i			0.428137i	−0.976819	+	0.214068i	$0.931328\pi$
$72$	−1.60729e7	−	1.70938e7i	−0.598088	−	0.636077i
$73$	2.67184e7			0.940847			0.470424	−	0.882441i	$-0.344101\pi$
$73$	2.67184e7			0.940847			0.470424	+	0.882441i	$0.344101\pi$
$74$		−	5.02705e6i		−	0.167643i
$75$		0			0
$76$	4.39928e7			1.31864
$77$		−	2.08187e7i		−	0.592230i
$78$	−6.31239e6	+	2.50161e6i	−0.170536	+	0.0675835i
$79$	−3.07855e6			−0.0790383			−0.0395192	−	0.999219i	$-0.512583\pi$
$79$	−3.07855e6			−0.0790383			−0.0395192	+	0.999219i	$0.512583\pi$
$80$		0			0
$81$	2.64753e6	−	4.29652e7i	0.0615035	−	0.998107i
$82$	−3.69033e7			−0.816225
$83$		−	2.32739e7i		−	0.490407i	−0.969472	−	0.245203i	$-0.921145\pi$
$83$		−	2.32739e7i		−	0.490407i	0.969472	−	0.245203i	$-0.0788548\pi$
$84$	1.84403e7	+	4.65310e7i	0.370383	+	0.934599i
$85$		0			0
$86$		−	4.34187e7i		−	0.793748i
$87$	8.40869e7	−	3.33237e7i	1.46775	−	0.581670i
$88$	2.31804e7			0.386536
$89$		−	3.21685e7i		−	0.512709i	−0.966583	−	0.256354i	$-0.917479\pi$
$89$		−	3.21685e7i		−	0.512709i	0.966583	−	0.256354i	$-0.0825214\pi$
$90$		0			0
$91$	3.37577e7			0.492274
$92$			2.19138e7i			0.305891i
$93$	−9.09471e6	−	2.29490e7i	−0.121579	−	0.306784i
$94$	−5.34128e7			−0.684122
$95$		0			0
$96$	−8.13894e7	+	3.22547e7i	−0.958258	+	0.379759i
$97$	9.03610e7			1.02069			0.510345	−	0.859970i	$-0.329518\pi$
$97$	9.03610e7			1.02069			0.510345	+	0.859970i	$0.329518\pi$
$98$			3.62994e7i			0.393546i
$99$	2.91319e7	+	3.09822e7i	0.303269	+	0.322531i
$100$		0			0
$101$		−	1.24539e8i		−	1.19679i	−0.801201	−	0.598395i	$-0.795805\pi$
$101$		−	1.24539e8i		−	1.19679i	0.801201	−	0.598395i	$-0.204195\pi$
$102$	−1.37425e7	−	3.46770e7i	−0.126960	−	0.320362i
$103$	9.24945e6			0.0821802			0.0410901	−	0.999155i	$-0.486917\pi$
$103$	9.24945e6			0.0821802			0.0410901	+	0.999155i	$0.486917\pi$
$104$			3.75872e7i			0.321297i
$105$		0			0
$106$	9.98527e7			0.790927
$107$		−	1.21294e8i		−	0.925348i	−0.886528	−	0.462674i	$-0.846890\pi$
$107$		−	1.21294e8i		−	0.925348i	0.886528	−	0.462674i	$-0.153110\pi$
$108$	−9.25541e7	−	4.34434e7i	−0.680301	−	0.319322i
$109$	1.29780e8			0.919394			0.459697	−	0.888076i	$-0.347958\pi$
$109$	1.29780e8			0.919394			0.459697	+	0.888076i	$0.347958\pi$
$110$		0			0
$111$	−1.88095e7	−	4.74626e7i	−0.123904	−	0.312651i
$112$	6.65768e7			0.423108
$113$		−	1.54559e8i		−	0.947942i	−0.880541	−	0.473971i	$-0.842820\pi$
$113$		−	1.54559e8i		−	0.947942i	0.880541	−	0.473971i	$-0.157180\pi$
$114$	−1.37335e8	+	5.44261e7i	−0.813135	+	0.322246i
$115$		0			0
$116$		−	2.14832e8i		−	1.18650i
$117$	−5.02380e7	+	4.72376e7i	−0.268095	+	0.252083i
$118$	4.43436e7			0.228719
$119$			1.85447e8i			0.924767i
$120$		0			0
$121$	1.72345e8			0.804002
$122$		−	1.05044e8i		−	0.474166i
$123$	−3.48420e8	+	1.38079e8i	−1.52224	+	0.603265i
$124$	−5.86319e7			−0.247997
$125$		0			0
$126$	−1.15132e8	−	1.22445e8i	−0.456789	−	0.485803i
$127$	−2.78135e8			−1.06916			−0.534578	−	0.845119i	$-0.679530\pi$
$127$	−2.78135e8			−1.06916			−0.534578	+	0.845119i	$0.679530\pi$
$128$			2.50263e8i			0.932301i
$129$	−1.62458e8	−	4.09935e8i	−0.586653	−	1.48032i
$130$		0			0
$131$			5.54604e8i			1.88321i	0.336723	+	0.941604i	$0.390681\pi$
$131$			5.54604e8i			1.88321i	−0.336723	+	0.941604i	$0.609319\pi$
$132$	9.39037e7	−	3.72141e7i	0.309305	−	0.122578i
$133$	7.34448e8			2.34722
$134$		−	1.35622e8i		−	0.420640i
$135$		0			0
$136$	−2.06484e8			−0.603576
$137$			5.75001e7i			0.163225i	0.996664	+	0.0816124i	$0.0260070\pi$
$137$			5.75001e7i			0.163225i	−0.996664	+	0.0816124i	$0.973993\pi$
$138$	−2.71109e7	−	6.84098e7i	−0.0747527	−	0.188626i
$139$	2.24212e8			0.600620			0.300310	−	0.953842i	$-0.402910\pi$
$139$	2.24212e8			0.600620			0.300310	+	0.953842i	$0.402910\pi$
$140$		0			0
$141$	−5.04294e8	+	1.99852e8i	−1.27587	+	0.505629i
$142$	−8.67732e7			−0.213419
$143$		−	6.81260e7i		−	0.162918i
$144$	−9.90792e7	+	9.31618e7i	−0.230426	+	0.216665i
$145$		0			0
$146$			2.13098e8i			0.468996i
$147$	1.35820e8	+	3.42719e8i	0.290867	+	0.733954i
$148$	−1.21261e8			−0.252740
$149$		−	1.53808e8i		−	0.312057i	−0.987753	−	0.156028i	$-0.950131\pi$
$149$		−	1.53808e8i		−	0.312057i	0.987753	−	0.156028i	$-0.0498691\pi$
$150$		0			0
$151$	8.78222e8			1.68926			0.844630	−	0.535351i	$-0.179821\pi$
$151$	8.78222e8			1.68926			0.844630	+	0.535351i	$0.179821\pi$
$152$			8.17764e8i			1.53198i
$153$	−2.59499e8	−	2.75981e8i	−0.473554	−	0.503633i
$154$	1.66044e8			0.295216
$155$		0			0
$156$	6.03431e7	+	1.52266e8i	0.101889	+	0.257101i
$157$	−2.80886e8			−0.462308			−0.231154	−	0.972917i	$-0.574250\pi$
$157$	−2.80886e8			−0.462308			−0.231154	+	0.972917i	$0.574250\pi$
$158$		−	2.45536e7i		−	0.0393992i
$159$	9.42753e8	−	3.73614e8i	1.47506	−	0.584568i
$160$		0			0
$161$			3.65845e8i			0.544495i
$162$	3.42678e8	+	2.11159e7i	0.497539	+	0.0306584i
$163$	5.02070e8			0.711237			0.355618	−	0.934631i	$-0.384270\pi$
$163$	5.02070e8			0.711237			0.355618	+	0.934631i	$0.384270\pi$
$164$			8.90171e8i			1.23055i
$165$		0			0
$166$	1.85626e8			0.244459
$167$			2.33934e8i			0.300765i	0.988628	+	0.150383i	$0.0480505\pi$
$167$			2.33934e8i			0.300765i	−0.988628	+	0.150383i	$0.951949\pi$
$168$	−8.64945e8	+	3.42778e8i	−1.08580	+	0.430305i
$169$	−7.05264e8			−0.864579
$170$		0			0
$171$	−1.09300e9	+	1.02772e9i	−1.27831	+	1.20196i
$172$	−1.04733e9			−1.19666
$173$			6.37704e8i			0.711925i	0.934500	+	0.355963i	$0.115847\pi$
$173$			6.37704e8i			0.711925i	−0.934500	+	0.355963i	$0.884153\pi$
$174$	2.65781e8	+	6.70654e8i	0.289952	+	0.731646i
$175$		0			0
$176$		−	1.34358e8i		−	0.140027i
$177$	4.18667e8	−	1.65918e8i	0.426556	−	0.169045i
$178$	2.56567e8			0.255576
$179$		−	1.57028e9i		−	1.52955i	−0.644297	−	0.764775i	$-0.722850\pi$
$179$		−	1.57028e9i		−	1.52955i	0.644297	−	0.764775i	$-0.277150\pi$
$180$		0			0
$181$	−9.00318e8			−0.838845			−0.419423	−	0.907791i	$-0.637767\pi$
$181$	−9.00318e8			−0.838845			−0.419423	+	0.907791i	$0.637767\pi$
$182$			2.69242e8i			0.245390i
$183$	−3.93037e8	−	9.91763e8i	−0.350452	−	0.884308i
$184$	−4.07347e8			−0.355380
$185$		0			0
$186$	1.83035e8	−	7.25369e7i	0.152926	−	0.0606048i
$187$	3.74249e8			0.306051
$188$			1.28841e9i			1.03139i
$189$	−1.54516e9	−	7.25275e8i	−1.21095	−	0.568402i
$190$		0			0
$191$		−	1.56461e9i		−	1.17564i	−0.808993	−	0.587819i	$-0.799987\pi$
$191$		−	1.56461e9i		−	1.17564i	0.808993	−	0.587819i	$-0.200013\pi$
$192$	−9.88963e7	−	2.49548e8i	−0.0727738	−	0.183633i
$193$	8.89688e8			0.641222			0.320611	−	0.947211i	$-0.396112\pi$
$193$	8.89688e8			0.641222			0.320611	+	0.947211i	$0.396112\pi$
$194$			7.20694e8i			0.508796i
$195$		0			0
$196$	8.75605e8			0.593313
$197$		−	9.87151e8i		−	0.655418i	−0.944779	−	0.327709i	$-0.893723\pi$
$197$		−	9.87151e8i		−	0.655418i	0.944779	−	0.327709i	$-0.106277\pi$
$198$	−2.47105e8	+	2.32347e8i	−0.160776	+	0.151174i
$199$	1.36346e9			0.869418			0.434709	−	0.900571i	$-0.356851\pi$
$199$	1.36346e9			0.869418			0.434709	+	0.900571i	$0.356851\pi$
$200$		0			0
$201$	−5.07450e8	−	1.28047e9i	−0.310892	−	0.784484i
$202$	9.93284e8			0.596579
$203$		−	3.58655e9i		−	2.11199i
$204$	−8.36469e8	+	3.31493e8i	−0.482980	+	0.191405i
$205$		0			0
$206$			7.37710e7i			0.0409654i
$207$	−5.11931e8	−	5.44448e8i	−0.278824	−	0.296534i
$208$	2.17863e8			0.116394
$209$		−	1.48218e9i		−	0.776812i
$210$		0			0
$211$	9.47762e8			0.478156			0.239078	−	0.971000i	$-0.423155\pi$
$211$	9.47762e8			0.478156			0.239078	+	0.971000i	$0.423155\pi$
$212$		−	2.40862e9i		−	1.19241i
$213$	−8.19264e8	+	3.24675e8i	−0.398021	+	0.157736i
$214$	9.67409e8			0.461270
$215$		0			0
$216$	8.07550e8	−	1.72045e9i	0.370983	−	0.790363i
$217$	−9.78842e8			−0.441442
$218$			1.03509e9i			0.458302i
$219$	7.97340e8	+	2.01196e9i	0.346631	+	0.874666i
$220$		0			0
$221$			6.06848e8i			0.254396i
$222$	3.78548e8	−	1.50019e8i	0.155851	−	0.0617638i
$223$	−1.01919e9			−0.412130			−0.206065	−	0.978538i	$-0.566066\pi$
$223$	−1.01919e9			−0.412130			−0.206065	+	0.978538i	$0.566066\pi$
$224$			3.47149e9i			1.37887i
$225$		0			0
$226$	1.23272e9			0.472532
$227$			1.43749e8i			0.0541377i	0.999634	+	0.0270689i	$0.00861734\pi$
$227$			1.43749e8i			0.0541377i	−0.999634	+	0.0270689i	$0.991383\pi$
$228$	1.31285e9	+	3.31276e9i	0.485821	+	1.22589i
$229$	−6.14128e7			−0.0223314			−0.0111657	−	0.999938i	$-0.503554\pi$
$229$	−6.14128e7			−0.0223314			−0.0111657	+	0.999938i	$0.503554\pi$
$230$		0			0
$231$	1.56769e9	−	6.21279e8i	0.550571	−	0.218192i
$232$	3.99341e9			1.37845
$233$		−	1.11728e9i		−	0.379087i	−0.981872	−	0.189543i	$-0.939299\pi$
$233$		−	1.11728e9i		−	0.379087i	0.981872	−	0.189543i	$-0.0607008\pi$
$234$	−3.76754e8	−	4.00684e8i	−0.125659	−	0.133641i
$235$		0			0
$236$		−	1.06964e9i		−	0.344819i
$237$	−9.18712e7	−	2.31822e8i	−0.0291196	−	0.0734786i
$238$	−1.47907e9			−0.460980
$239$			1.81745e9i			0.557021i	0.960433	+	0.278511i	$0.0898407\pi$
$239$			1.81745e9i			0.557021i	−0.960433	+	0.278511i	$0.910159\pi$
$240$		0			0
$241$	2.33083e8			0.0690942			0.0345471	−	0.999403i	$-0.489001\pi$
$241$	2.33083e8			0.0690942			0.0345471	+	0.999403i	$0.489001\pi$
$242$			1.37457e9i			0.400781i
$243$	3.31439e9	−	1.08282e9i	0.950557	−	0.310550i
$244$	−2.53383e9			−0.714856
$245$		0			0
$246$	−1.10128e9	−	2.77890e9i	−0.300717	−	0.758810i
$247$	2.40337e9			0.645703
$248$		−	1.08988e9i		−	0.288120i
$249$	1.75258e9	−	6.94548e8i	0.455911	−	0.180678i
$250$		0			0
$251$		−	4.09933e9i		−	1.03280i	−0.856346	−	0.516402i	$-0.827271\pi$
$251$		−	4.09933e9i		−	1.03280i	0.856346	−	0.516402i	$-0.172729\pi$
$252$	−2.95359e9	+	2.77719e9i	−0.732400	+	0.688658i
$253$	7.38307e8			0.180200
$254$		−	2.21833e9i		−	0.532956i
$255$		0			0
$256$	−2.84440e9			−0.662263
$257$		−	6.47154e9i		−	1.48346i	−0.670700	−	0.741729i	$-0.734006\pi$
$257$		−	6.47154e9i		−	1.48346i	0.670700	−	0.741729i	$-0.265994\pi$
$258$	3.26952e9	−	1.29572e9i	0.737914	−	0.292436i
$259$	−2.02442e9			−0.449884
$260$		0			0
$261$	5.01871e9	+	5.33748e9i	1.08151	+	1.15020i
$262$	−4.42337e9			−0.938746
$263$			1.19014e9i			0.248757i	0.992235	+	0.124379i	$0.0396937\pi$
$263$			1.19014e9i			0.248757i	−0.992235	+	0.124379i	$0.960306\pi$
$264$	6.91757e8	+	1.74553e9i	0.142409	+	0.359346i
$265$		0			0
$266$			5.85775e9i			1.17005i
$267$	2.42236e9	−	9.59984e8i	0.476644	−	0.188894i
$268$	−3.27143e9			−0.634160
$269$		−	9.23705e9i		−	1.76410i	−0.471153	−	0.882052i	$-0.656162\pi$
$269$		−	9.23705e9i		−	1.76410i	0.471153	−	0.882052i	$-0.343838\pi$
$270$		0			0
$271$	−5.47412e9			−1.01493			−0.507466	−	0.861672i	$-0.669418\pi$
$271$	−5.47412e9			−1.01493			−0.507466	+	0.861672i	$0.669418\pi$
$272$			1.19682e9i			0.218653i
$273$	1.00741e9	+	2.54203e9i	0.181366	+	0.457647i
$274$	−4.58604e8			−0.0813647
$275$		0			0
$276$	−1.65016e9	+	6.53961e8i	−0.284374	+	0.112698i
$277$	−7.77657e9			−1.32090			−0.660449	−	0.750871i	$-0.729634\pi$
$277$	−7.77657e9			−1.32090			−0.660449	+	0.750871i	$0.729634\pi$
$278$			1.78825e9i			0.299398i
$279$	1.45670e9	−	1.36971e9i	0.240411	−	0.226053i
$280$		0			0
$281$			3.41673e9i			0.548006i	0.961729	+	0.274003i	$0.0883478\pi$
$281$			3.41673e9i			0.548006i	−0.961729	+	0.274003i	$0.911652\pi$
$282$	−1.59396e9	−	4.02210e9i	−0.252047	−	0.635999i
$283$	−9.09925e9			−1.41860			−0.709300	−	0.704907i	$-0.750989\pi$
$283$	−9.09925e9			−1.41860			−0.709300	+	0.704907i	$0.750989\pi$
$284$			2.09312e9i			0.321751i
$285$		0			0
$286$	5.43354e8			0.0812117
$287$			1.48611e10i			2.19041i
$288$	−4.85770e9	−	5.16625e9i	−0.706091	−	0.750940i
$289$	3.64205e9			0.522101
$290$		0			0
$291$	2.69659e9	+	6.80439e9i	0.376047	+	0.948893i
$292$	5.14030e9			0.707061
$293$			2.26766e9i			0.307686i	0.988095	+	0.153843i	$0.0491650\pi$
$293$			2.26766e9i			0.307686i	−0.988095	+	0.153843i	$0.950835\pi$
$294$	−2.73343e9	+	1.08326e9i	−0.365863	+	0.144992i
$295$		0			0
$296$		−	2.25407e9i		−	0.293630i
$297$	−1.46367e9	+	3.11828e9i	−0.188112	+	0.400764i
$298$	1.22673e9			0.155555
$299$			1.19717e9i			0.149786i
$300$		0			0
$301$	−1.74849e10			−2.13009
$302$			7.00445e9i			0.842066i
$303$	9.37803e9	−	3.71652e9i	1.11261	−	0.440927i
$304$	4.73992e9			0.554979
$305$		0			0
$306$	2.20115e9	−	2.06969e9i	0.251052	−	0.236058i
$307$	1.05438e10			1.18698			0.593489	−	0.804842i	$-0.297750\pi$
$307$	1.05438e10			1.18698			0.593489	+	0.804842i	$0.297750\pi$
$308$		−	4.00527e9i		−	0.445070i
$309$	2.76025e8	+	6.96505e8i	0.0302772	+	0.0763994i
$310$		0			0
$311$			7.20415e9i			0.770089i	0.922898	+	0.385045i	$0.125814\pi$
$311$			7.20415e9i			0.770089i	−0.922898	+	0.385045i	$0.874186\pi$
$312$	−2.83040e9	+	1.12169e9i	−0.298696	+	0.118374i
$313$	2.32153e9			0.241878			0.120939	−	0.992660i	$-0.461409\pi$
$313$	2.32153e9			0.241878			0.120939	+	0.992660i	$0.461409\pi$
$314$		−	2.24026e9i		−	0.230452i
$315$		0			0
$316$	−5.92276e8			−0.0593985
$317$			1.17812e9i			0.116668i	0.998297	+	0.0583340i	$0.0185788\pi$
$317$			1.17812e9i			0.116668i	−0.998297	+	0.0583340i	$0.981421\pi$
$318$	2.97984e9	+	7.51913e9i	0.291397	+	0.735291i
$319$	−7.23798e9			−0.698963
$320$		0			0
$321$	9.13374e9	−	3.61971e9i	0.860257	−	0.340921i
$322$	−2.91788e9			−0.271421
$323$			1.32029e10i			1.21299i
$324$	5.09352e8	−	8.26599e9i	0.0462209	−	0.750093i
$325$		0			0
$326$			4.00437e9i			0.354539i
$327$	3.87294e9	+	9.77273e9i	0.338727	+	0.854722i
$328$	−1.65470e10			−1.42963
$329$			2.15096e10i			1.83590i
$330$		0			0
$331$	3.67861e9			0.306458			0.153229	−	0.988191i	$-0.451033\pi$
$331$	3.67861e9			0.306458			0.153229	+	0.988191i	$0.451033\pi$
$332$		−	4.47762e9i		−	0.368548i
$333$	3.01272e9	−	2.83279e9i	0.245009	−	0.230376i
$334$	−1.86579e9			−0.149926
$335$		0			0
$336$	1.98681e9	+	5.01339e9i	0.155883	+	0.393346i
$337$	−3.59368e9			−0.278625			−0.139312	−	0.990248i	$-0.544489\pi$
$337$	−3.59368e9			−0.278625			−0.139312	+	0.990248i	$0.544489\pi$
$338$		−	5.62498e9i		−	0.430977i
$339$	1.16387e10	−	4.61242e9i	0.881261	−	0.349245i
$340$		0			0
$341$			1.97539e9i			0.146095i
$342$	−8.19682e9	−	8.71745e9i	−0.599157	−	0.637214i
$343$	−3.89777e9			−0.281605
$344$		−	1.94684e10i		−	1.39026i
$345$		0			0
$346$	−5.08614e9			−0.354882
$347$		−	2.36195e10i		−	1.62912i	−0.580078	−	0.814561i	$-0.696978\pi$
$347$		−	2.36195e10i		−	1.62912i	0.580078	−	0.814561i	$-0.303022\pi$
$348$	1.61773e10	−	6.41108e9i	1.10304	−	0.437134i
$349$	2.24057e9			0.151028			0.0755139	−	0.997145i	$-0.475940\pi$
$349$	2.24057e9			0.151028			0.0755139	+	0.997145i	$0.475940\pi$
$350$		0			0
$351$	−5.05632e9	−	2.37335e9i	−0.333124	−	0.156363i
$352$	7.00578e9			0.456337
$353$		−	1.21859e10i		−	0.784797i	−0.919795	−	0.392399i	$-0.871645\pi$
$353$		−	1.21859e10i		−	0.784797i	0.919795	−	0.392399i	$-0.128355\pi$
$354$	1.32332e9	+	3.33917e9i	0.0842657	+	0.212631i
$355$		0			0
$356$		−	6.18883e9i		−	0.385309i
$357$	−1.39646e10	+	5.53418e9i	−0.859717	+	0.340707i
$358$	1.25241e10			0.762454
$359$			1.53929e10i			0.926706i	0.886174	+	0.463353i	$0.153354\pi$
$359$			1.53929e10i			0.926706i	−0.886174	+	0.463353i	$0.846646\pi$
$360$		0			0
$361$	3.53052e10			2.07879
$362$		−	7.18068e9i		−	0.418149i
$363$	5.14318e9	+	1.29780e10i	0.296214	+	0.747447i
$364$	6.49457e9			0.369952
$365$		0			0
$366$	7.91002e9	−	3.13475e9i	0.440812	−	0.174694i
$367$	−3.67051e9			−0.202331			−0.101165	−	0.994870i	$-0.532257\pi$
$367$	−3.67051e9			−0.202331			−0.101165	+	0.994870i	$0.532257\pi$
$368$			2.36106e9i			0.128741i
$369$	−2.07954e10	−	2.21162e10i	−1.12166	−	1.19290i
$370$		0			0
$371$		−	4.02112e10i		−	2.12252i
$372$	−1.74971e9	−	4.41511e9i	−0.0913682	−	0.230553i
$373$	−1.40817e10			−0.727478			−0.363739	−	0.931501i	$-0.618500\pi$
$373$	−1.40817e10			−0.727478			−0.363739	+	0.931501i	$0.618500\pi$
$374$			2.98490e9i			0.152561i
$375$		0			0
$376$	−2.39496e10			−1.19825
$377$		−	1.17364e10i		−	0.580993i
$378$	5.78459e9	−	1.23238e10i	0.283338	−	0.603639i
$379$	−2.08709e10			−1.01154			−0.505770	−	0.862668i	$-0.668792\pi$
$379$	−2.08709e10			−1.01154			−0.505770	+	0.862668i	$0.668792\pi$
$380$		0			0
$381$	−8.30022e9	−	2.09442e10i	−0.393903	−	0.993950i
$382$	1.24789e10			0.586034
$383$			9.93413e9i			0.461673i	0.972993	+	0.230837i	$0.0741463\pi$
$383$			9.93413e9i			0.461673i	−0.972993	+	0.230837i	$0.925854\pi$
$384$	−1.88453e10	+	7.46843e9i	−0.866721	+	0.343482i
$385$		0			0
$386$			7.09590e9i			0.319638i
$387$	2.60209e10	−	2.44669e10i	1.16006	−	1.09077i
$388$	1.73844e10			0.767065
$389$			2.35323e10i			1.02770i	0.857881	+	0.513849i	$0.171781\pi$
$389$			2.35323e10i			1.02770i	−0.857881	+	0.513849i	$0.828219\pi$
$390$		0			0
$391$	−6.57664e9			−0.281383
$392$			1.62762e10i			0.689302i
$393$	−4.17630e10	+	1.65507e10i	−1.75074	+	0.693819i
$394$	7.87324e9			0.326714
$395$		0			0
$396$	5.60462e9	+	5.96061e9i	0.227911	+	0.242387i
$397$	1.45076e10			0.584027			0.292014	−	0.956414i	$-0.405675\pi$
$397$	1.45076e10			0.584027			0.292014	+	0.956414i	$0.405675\pi$
$398$			1.08745e10i			0.433390i
$399$	2.19177e10	+	5.53056e10i	0.864774	+	2.18211i
$400$		0			0
$401$		−	1.72695e10i		−	0.667887i	−0.942593	−	0.333943i	$-0.891621\pi$
$401$		−	1.72695e10i		−	0.667887i	0.942593	−	0.333943i	$-0.108379\pi$
$402$	1.02126e10	−	4.04728e9i	0.391051	−	0.154974i
$403$	−3.20311e9			−0.121437
$404$		−	2.39597e10i		−	0.899407i
$405$		0			0
$406$	2.86053e10			1.05279
$407$			4.08545e9i			0.148889i
$408$	−6.16198e9	−	1.55487e10i	−0.222372	−	0.561119i
$409$	1.11042e10			0.396819			0.198409	−	0.980119i	$-0.436422\pi$
$409$	1.11042e10			0.396819			0.198409	+	0.980119i	$0.436422\pi$
$410$		0			0
$411$	−4.32989e9	+	1.71594e9i	−0.151743	+	0.0601360i
$412$	1.77948e9			0.0617597
$413$		−	1.78574e10i		−	0.613787i
$414$	4.34236e9	−	4.08302e9i	0.147817	−	0.138989i
$415$		0			0
$416$			1.13599e10i			0.379317i
$417$	6.69102e9	+	1.68837e10i	0.221283	+	0.558371i
$418$	1.18214e10			0.387227
$419$		−	2.66208e10i		−	0.863705i	−0.901944	−	0.431852i	$-0.857860\pi$
$419$		−	2.66208e10i		−	0.863705i	0.901944	−	0.431852i	$-0.142140\pi$
$420$		0			0
$421$	−2.03614e10			−0.648155			−0.324078	−	0.946031i	$-0.605054\pi$
$421$	−2.03614e10			−0.648155			−0.324078	+	0.946031i	$0.605054\pi$
$422$			7.55908e9i			0.238352i
$423$	−3.00986e10	−	3.20104e10i	−0.940124	−	0.999838i
$424$	4.47727e10			1.38532
$425$		0			0
$426$	−2.58952e9	−	6.53422e9i	−0.0786286	−	0.198406i
$427$	−4.23016e10			−1.27246
$428$		−	2.33356e10i		−	0.695414i
$429$	5.13004e9	−	2.03304e9i	0.151458	−	0.0600229i
$430$		0			0
$431$			1.26533e10i			0.366687i	0.983049	+	0.183343i	$0.0586920\pi$
$431$			1.26533e10i			0.366687i	−0.983049	+	0.183343i	$0.941308\pi$
$432$	−9.97205e9	−	4.68072e9i	−0.286319	−	0.134393i
$433$	3.63896e10			1.03520			0.517601	−	0.855622i	$-0.326825\pi$
$433$	3.63896e10			1.03520			0.517601	+	0.855622i	$0.326825\pi$
$434$		−	7.80696e9i		−	0.220051i
$435$		0			0
$436$	2.49681e10			0.690939
$437$			2.60462e10i			0.714198i
$438$	−1.60468e10	+	6.35936e9i	−0.436005	+	0.172789i
$439$	−5.90206e10			−1.58908			−0.794539	−	0.607213i	$-0.792287\pi$
$439$	−5.90206e10			−1.58908			−0.794539	+	0.607213i	$0.792287\pi$
$440$		0			0
$441$	−2.17543e10	+	2.04551e10i	−0.575164	+	0.540813i
$442$	−4.84005e9			−0.126812
$443$			1.59909e10i			0.415200i	0.978214	+	0.207600i	$0.0665652\pi$
$443$			1.59909e10i			0.415200i	−0.978214	+	0.207600i	$0.933435\pi$
$444$	−3.61872e9	−	9.13123e9i	−0.0931156	−	0.234962i
$445$		0			0
$446$		−	8.12874e9i		−	0.205439i
$447$	1.15821e10	−	4.58999e9i	0.290106	−	0.114969i
$448$	−1.06440e10			−0.264236
$449$			1.61660e9i			0.0397756i	0.999802	+	0.0198878i	$0.00633090\pi$
$449$			1.61660e9i			0.0397756i	−0.999802	+	0.0198878i	$0.993669\pi$
$450$		0			0
$451$	2.99911e10			0.724913
$452$		−	2.97354e10i		−	0.712393i
$453$	2.62082e10	+	6.61321e10i	0.622364	+	1.57043i
$454$	−1.14650e9			−0.0269867
$455$		0			0
$456$	−6.15795e10	+	2.44040e10i	−1.42422	+	0.564419i
$457$	−5.01622e10			−1.15004			−0.575019	−	0.818140i	$-0.695005\pi$
$457$	−5.01622e10			−1.15004			−0.575019	+	0.818140i	$0.695005\pi$
$458$		−	4.89811e8i		−	0.0111318i
$459$	1.30380e10	−	2.77768e10i	0.293737	−	0.625793i
$460$		0			0
$461$			7.19471e10i			1.59298i	0.604654	+	0.796488i	$0.293311\pi$
$461$			7.19471e10i			1.59298i	−0.604654	+	0.796488i	$0.706689\pi$
$462$	4.95514e9	+	1.25035e10i	0.108765	+	0.274450i
$463$	6.19020e10			1.34704			0.673521	−	0.739168i	$-0.264781\pi$
$463$	6.19020e10			1.34704			0.673521	+	0.739168i	$0.264781\pi$
$464$		−	2.31466e10i		−	0.499361i
$465$		0			0
$466$	8.91112e9			0.188968
$467$			4.98272e9i			0.104761i	0.998627	+	0.0523804i	$0.0166808\pi$
$467$			4.98272e9i			0.104761i	−0.998627	+	0.0523804i	$0.983319\pi$
$468$	−9.66518e9	+	9.08794e9i	−0.201477	+	0.189445i
$469$	−5.46156e10			−1.12882
$470$		0			0
$471$	−8.38229e9	−	2.11513e10i	−0.170325	−	0.429788i
$472$	1.98831e10			0.400605
$473$			3.52861e10i			0.704951i
$474$	1.84895e9	−	7.32739e8i	0.0366278	−	0.0145156i
$475$		0			0
$476$			3.56778e10i			0.694977i
$477$	5.62680e10	+	5.98419e10i	1.08690	+	1.15593i
$478$	−1.44955e10			−0.277665
$479$			1.00172e9i			0.0190284i	0.999955	+	0.00951420i	$0.00302851\pi$
$479$			1.00172e9i			0.0190284i	−0.999955	+	0.00951420i	$0.996971\pi$
$480$		0			0
$481$	−6.62460e9			−0.123760
$482$			1.85900e9i			0.0344422i
$483$	−2.75490e10	+	1.09177e10i	−0.506194	+	0.200605i
$484$	3.31571e10			0.604220
$485$		0			0
$486$	8.63626e9	+	2.64346e10i	0.154803	+	0.473836i
$487$	−8.29709e10			−1.47506			−0.737530	−	0.675314i	$-0.764008\pi$
$487$	−8.29709e10			−1.47506			−0.737530	+	0.675314i	$0.764008\pi$
$488$		−	4.71003e10i		−	0.830509i
$489$	1.49830e10	+	3.78070e10i	0.262037	+	0.661207i
$490$		0			0
$491$		−	1.72161e10i		−	0.296216i	−0.988971	−	0.148108i	$-0.952682\pi$
$491$		−	1.72161e10i		−	0.296216i	0.988971	−	0.148108i	$-0.0473183\pi$
$492$	−6.70319e10	+	2.65648e10i	−1.14399	+	0.453363i
$493$	6.44739e10			1.09143
$494$			1.91686e10i			0.321871i
$495$		0			0
$496$	−6.31717e9			−0.104375
$497$			3.49440e10i			0.572726i
$498$	5.53952e9	+	1.39781e10i	0.0900647	+	0.227263i
$499$	7.02839e10			1.13358			0.566792	−	0.823861i	$-0.308184\pi$
$499$	7.02839e10			1.13358			0.566792	+	0.823861i	$0.308184\pi$
$500$		0			0
$501$	−1.76158e10	+	6.98114e9i	−0.279609	+	0.110809i
$502$	3.26951e10			0.514834
$503$		−	2.10506e10i		−	0.328846i	−0.986390	−	0.164423i	$-0.947424\pi$
$503$		−	2.10506e10i		−	0.328846i	0.986390	−	0.164423i	$-0.0525762\pi$
$504$	−5.16240e10	−	5.49030e10i	−0.800073	−	0.850891i
$505$		0			0
$506$			5.88853e9i			0.0898266i
$507$	−2.10467e10	−	5.31080e10i	−0.318532	−	0.803763i
$508$	−5.35099e10			−0.803488
$509$		−	3.73005e10i		−	0.555704i	−0.960624	−	0.277852i	$-0.910378\pi$
$509$		−	3.73005e10i		−	0.555704i	0.960624	−	0.277852i	$-0.0896224\pi$
$510$		0			0
$511$	8.58158e10			1.25859
$512$			4.13811e10i			0.602175i
$513$	−1.10007e11	−	5.16357e10i	−1.58837	−	0.745557i
$514$	5.16152e10			0.739477
$515$		0			0
$516$	−3.12549e10	−	7.88665e10i	−0.440879	−	1.11248i
$517$	4.34082e10			0.607589
$518$		−	1.61462e10i		−	0.224259i
$519$	−4.80205e10	+	1.90306e10i	−0.661847	+	0.262291i
$520$		0			0
$521$			1.35295e11i			1.83625i	0.396288	+	0.918126i	$0.370298\pi$
$521$			1.35295e11i			1.83625i	−0.396288	+	0.918126i	$0.629702\pi$
$522$	−4.25702e10	+	4.00278e10i	−0.573355	+	0.539113i
$523$	4.33058e9			0.0578815			0.0289407	−	0.999581i	$-0.490787\pi$
$523$	4.33058e9			0.0578815			0.0289407	+	0.999581i	$0.490787\pi$
$524$			1.06699e11i			1.41526i
$525$		0			0
$526$	−9.49223e9			−0.124001
$527$		−	1.75962e10i		−	0.228127i
$528$	1.01175e10	−	4.00956e9i	0.130177	−	0.0515894i
$529$	6.53368e10			0.834325
$530$		0			0
$531$	2.49880e10	+	2.65752e10i	0.314307	+	0.334271i
$532$	1.41299e11			1.76397
$533$			4.86308e10i			0.602564i
$534$	7.65656e9	+	1.93201e10i	0.0941605	+	0.237598i
$535$		0			0
$536$		−	6.08112e10i		−	0.736757i
$537$	1.18245e11	−	4.68607e10i	1.42196	−	0.563523i
$538$	7.36721e10			0.879374
$539$		−	2.95003e10i		−	0.349520i
$540$		0			0
$541$	−1.24450e11			−1.45280			−0.726402	−	0.687270i	$-0.758809\pi$
$541$	−1.24450e11			−1.45280			−0.726402	+	0.687270i	$0.758809\pi$
$542$		−	4.36600e10i		−	0.505926i
$543$	−2.68676e10	−	6.77960e10i	−0.309051	−	0.779839i
$544$	−6.24055e10			−0.712570
$545$		0			0
$546$	−2.02745e10	+	8.03481e9i	−0.228129	+	0.0904077i
$547$	−3.16725e10			−0.353780			−0.176890	−	0.984231i	$-0.556604\pi$
$547$	−3.16725e10			−0.353780			−0.176890	+	0.984231i	$0.556604\pi$
$548$			1.10623e10i			0.122666i
$549$	6.29529e10	−	5.91931e10i	0.692989	−	0.651601i
$550$		0			0
$551$		−	2.55343e11i		−	2.77025i
$552$	−1.21562e10	−	3.06741e10i	−0.130931	−	0.330382i
$553$	−9.88787e9			−0.105731
$554$		−	6.20237e10i		−	0.658444i
$555$		0			0
$556$	4.31357e10			0.451375
$557$			3.02144e10i			0.313902i	0.987606	+	0.156951i	$0.0501664\pi$
$557$			3.02144e10i			0.313902i	−0.987606	+	0.156951i	$0.949834\pi$
$558$	1.09244e10	+	1.16183e10i	0.112683	+	0.119841i
$559$	−5.72167e10			−0.585971
$560$		0			0
$561$	1.11685e10	+	2.81818e10i	0.112757	+	0.284523i
$562$	−2.72508e10			−0.273171
$563$			6.11519e10i			0.608662i	0.952566	+	0.304331i	$0.0984329\pi$
$563$			6.11519e10i			0.608662i	−0.952566	+	0.304331i	$0.901567\pi$
$564$	−9.70200e10	+	3.84491e10i	−0.958837	+	0.379988i
$565$		0			0
$566$		−	7.25730e10i		−	0.707147i
$567$	8.50348e9	−	1.37998e11i	0.0822744	−	1.33519i
$568$	−3.89081e10			−0.373806
$569$			3.99163e10i			0.380804i	0.981706	+	0.190402i	$0.0609791\pi$
$569$			3.99163e10i			0.380804i	−0.981706	+	0.190402i	$0.939021\pi$
$570$		0			0
$571$	9.07730e10			0.853911			0.426955	−	0.904273i	$-0.359586\pi$
$571$	9.07730e10			0.853911			0.426955	+	0.904273i	$0.359586\pi$
$572$		−	1.31066e10i		−	0.122435i
$573$	1.17819e11	−	4.66917e10i	1.09294	−	0.433133i
$574$	−1.18528e11			−1.09188
$575$		0			0
$576$	1.58403e10	−	1.48942e10i	0.143904	−	0.135309i
$577$	−1.33041e11			−1.20028			−0.600141	−	0.799894i	$-0.704889\pi$
$577$	−1.33041e11			−1.20028			−0.600141	+	0.799894i	$0.704889\pi$
$578$			2.90480e10i			0.260258i
$579$	2.65504e10	+	6.69956e10i	0.236242	+	0.596117i
$580$		0			0
$581$		−	7.47525e10i		−	0.656026i
$582$	−5.42699e10	+	2.15072e10i	−0.473006	+	0.187453i
$583$	−8.11497e10			−0.702445
$584$			9.55508e10i			0.821453i
$585$		0			0
$586$	−1.80862e10			−0.153376
$587$		−	1.98669e11i		−	1.67331i	−0.547728	−	0.836657i	$-0.684507\pi$
$587$		−	1.98669e11i		−	1.67331i	0.547728	−	0.836657i	$-0.315493\pi$
$588$	2.61301e10	+	6.59350e10i	0.218591	+	0.551578i
$589$	−6.96883e10			−0.579027
$590$		0			0
$591$	7.43347e10	−	2.94589e10i	0.609315	−	0.241472i
$592$	−1.30650e10			−0.106371
$593$		−	1.33920e11i		−	1.08300i	−0.840702	−	0.541498i	$-0.817857\pi$
$593$		−	1.33920e11i		−	1.08300i	0.840702	−	0.541498i	$-0.182143\pi$
$594$	−2.48705e10	−	1.16738e10i	−0.199774	−	0.0937706i
$595$		0			0
$596$		−	2.95908e10i		−	0.234515i
$597$	4.06887e10	+	1.02671e11i	0.320315	+	0.808261i
$598$	−9.54831e9			−0.0746658
$599$		−	1.14673e10i		−	0.0890745i	−0.999008	−	0.0445372i	$-0.985819\pi$
$599$		−	1.14673e10i		−	0.0890745i	0.999008	−	0.0445372i	$-0.0141813\pi$
$600$		0			0
$601$	4.71067e10			0.361065			0.180532	−	0.983569i	$-0.442218\pi$
$601$	4.71067e10			0.361065			0.180532	+	0.983569i	$0.442218\pi$
$602$		−	1.39455e11i		−	1.06181i
$603$	8.12785e10	−	7.64243e10i	0.614761	−	0.578046i
$604$	1.68959e11			1.26950
$605$		0			0
$606$	2.96419e10	+	7.47965e10i	0.219794	+	0.554614i
$607$	1.36835e11			1.00796			0.503979	−	0.863716i	$-0.331869\pi$
$607$	1.36835e11			1.00796			0.503979	+	0.863716i	$0.331869\pi$
$608$			2.47152e11i			1.80863i
$609$	2.70075e11	−	1.07031e11i	1.96343	−	0.778110i
$610$		0			0
$611$			7.03868e10i			0.505041i
$612$	−4.99244e10	−	5.30955e10i	−0.355883	−	0.378488i
$613$	−8.50169e10			−0.602093			−0.301046	−	0.953610i	$-0.597336\pi$
$613$	−8.50169e10			−0.602093			−0.301046	+	0.953610i	$0.597336\pi$
$614$			8.40942e10i			0.591688i
$615$		0			0
$616$	7.44521e10			0.517076
$617$			1.76659e11i			1.21898i	0.792795	+	0.609489i	$0.208625\pi$
$617$			1.76659e11i			1.21898i	−0.792795	+	0.609489i	$0.791375\pi$
$618$	−5.55512e9	+	2.20150e9i	−0.0380838	+	0.0150926i
$619$	−1.22476e11			−0.834234			−0.417117	−	0.908853i	$-0.636960\pi$
$619$	−1.22476e11			−0.834234			−0.417117	+	0.908853i	$0.636960\pi$
$620$		0			0
$621$	2.57209e10	−	5.47972e10i	0.172950	−	0.368461i
$622$	−5.74582e10			−0.383876
$623$		−	1.03321e11i		−	0.685860i
$624$	6.50154e9	+	1.64056e10i	0.0428823	+	0.108206i
$625$		0			0
$626$			1.85159e10i			0.120572i
$627$	1.11612e11	−	4.42318e10i	0.722169	−	0.286196i
$628$	−5.40390e10			−0.347431
$629$		−	3.63921e10i		−	0.232490i
$630$		0			0
$631$	−2.62046e11			−1.65295			−0.826475	−	0.562974i	$-0.809657\pi$
$631$	−2.62046e11			−1.65295			−0.826475	+	0.562974i	$0.809657\pi$
$632$		−	1.10096e10i		−	0.0690083i
$633$	2.82835e10	+	7.13686e10i	0.176164	+	0.444521i
$634$	−9.39633e9			−0.0581569
$635$		0			0
$636$	1.81374e11	−	7.18788e10i	1.10853	−	0.439312i
$637$	4.78351e10			0.290529
$638$		−	5.77280e10i		−	0.348421i
$639$	−4.88976e10	−	5.20034e10i	−0.293281	−	0.311909i
$640$		0			0
$641$			2.31601e11i			1.37185i	0.727671	+	0.685926i	$0.240603\pi$
$641$			2.31601e11i			1.37185i	−0.727671	+	0.685926i	$0.759397\pi$
$642$	2.88698e10	+	7.28481e10i	0.169943	+	0.428823i
$643$	2.66842e11			1.56103			0.780515	−	0.625138i	$-0.214957\pi$
$643$	2.66842e11			1.56103			0.780515	+	0.625138i	$0.214957\pi$
$644$			7.03842e10i			0.409196i
$645$		0			0
$646$	−1.05302e11			−0.604655
$647$			1.98677e11i			1.13378i	0.823792	+	0.566892i	$0.191854\pi$
$647$			1.98677e11i			1.13378i	−0.823792	+	0.566892i	$0.808146\pi$
$648$	1.53653e11	+	9.46812e9i	0.871447	+	0.0536987i
$649$	−3.60378e10			−0.203132
$650$		0			0
$651$	−2.92109e10	−	7.37090e10i	−0.162638	−	0.410390i
$652$	9.65923e10			0.534505
$653$		−	2.49774e11i		−	1.37371i	−0.726795	−	0.686855i	$-0.758991\pi$
$653$		−	2.49774e11i		−	1.37371i	0.726795	−	0.686855i	$-0.241009\pi$
$654$	−7.79445e10	+	3.08895e10i	−0.426064	+	0.168850i
$655$		0			0
$656$			9.59096e10i			0.517901i
$657$	−1.27710e11	+	1.20083e11i	−0.685433	+	0.644496i
$658$	−1.71554e11			−0.915162
$659$		−	1.55774e11i		−	0.825947i	−0.910743	−	0.412974i	$-0.864490\pi$
$659$		−	1.55774e11i		−	0.825947i	0.910743	−	0.412974i	$-0.135510\pi$
$660$		0			0
$661$	−7.07460e10			−0.370592			−0.185296	−	0.982683i	$-0.559324\pi$
$661$	−7.07460e10			−0.370592			−0.185296	+	0.982683i	$0.559324\pi$
$662$			2.93395e10i			0.152764i
$663$	−4.56970e10	+	1.81098e10i	−0.236502	+	0.0937258i
$664$	8.32324e10			0.428174
$665$		0			0
$666$	2.25935e10	+	2.40286e10i	0.114838	+	0.122133i
$667$	1.27192e11			0.642625
$668$			4.50061e10i			0.226030i
$669$	−3.04149e10	−	7.67471e10i	−0.151839	−	0.383140i
$670$		0			0
$671$			8.53683e10i			0.421121i
$672$	−2.61411e11	+	1.03598e11i	−1.28188	+	0.508010i
$673$	−3.05166e11			−1.48757			−0.743783	−	0.668421i	$-0.766971\pi$
$673$	−3.05166e11			−1.48757			−0.743783	+	0.668421i	$0.766971\pi$
$674$		−	2.86622e10i		−	0.138890i
$675$		0			0
$676$	−1.35684e11			−0.649745
$677$			6.05189e10i			0.288095i	0.989571	+	0.144048i	$0.0460119\pi$
$677$			6.05189e10i			0.288095i	−0.989571	+	0.144048i	$0.953988\pi$
$678$	3.67873e10	+	9.28268e10i	0.174092	+	0.439293i
$679$	2.90227e11			1.36540
$680$		0			0
$681$	−1.08246e10	+	4.28980e9i	−0.0503296	+	0.0199457i
$682$	−1.57551e10			−0.0728257
$683$			1.55016e11i			0.712352i	0.934419	+	0.356176i	$0.115920\pi$
$683$			1.55016e11i			0.712352i	−0.934419	+	0.356176i	$0.884080\pi$
$684$	−2.10280e11	+	1.97721e11i	−0.960668	+	0.903294i
$685$		0			0
$686$		−	3.10875e10i		−	0.140375i
$687$	−1.83270e9	−	4.62452e9i	−0.00822744	−	0.0207606i
$688$	−1.12843e11			−0.503639
$689$		−	1.31585e11i		−	0.583888i
$690$		0			0
$691$	−2.65978e11			−1.16663			−0.583315	−	0.812246i	$-0.698245\pi$
$691$	−2.65978e11			−1.16663			−0.583315	+	0.812246i	$0.698245\pi$
$692$			1.22686e11i			0.535023i
$693$	9.35674e10	+	9.95106e10i	0.405688	+	0.431456i
$694$	1.88383e11			0.812088
$695$		0			0
$696$	1.19173e11	+	3.00713e11i	0.507856	+	1.28149i
$697$	−2.67152e11			−1.13195
$698$			1.78702e10i			0.0752847i
$699$	8.41338e10	−	3.33423e10i	0.352421	−	0.139665i
$700$		0			0
$701$			4.58950e11i			1.90061i	0.311319	+	0.950306i	$0.399229\pi$
$701$			4.58950e11i			1.90061i	−0.311319	+	0.950306i	$0.600771\pi$
$702$	1.89292e10	−	4.03277e10i	0.0779441	−	0.166056i
$703$	−1.44128e11			−0.590101
$704$			2.14805e10i			0.0874486i
$705$		0			0
$706$	9.71910e10			0.391208
$707$		−	4.00000e11i		−	1.60097i
$708$	8.05466e10	−	3.19207e10i	0.320563	−	0.127040i
$709$	−3.22572e11			−1.27656			−0.638280	−	0.769804i	$-0.720354\pi$
$709$	−3.22572e11			−1.27656			−0.638280	+	0.769804i	$0.720354\pi$
$710$		0			0
$711$	1.47151e10	−	1.38362e10i	0.0575816	−	0.0541426i
$712$	1.15041e11			0.447646
$713$		−	3.47133e10i		−	0.134319i
$714$	−4.41391e10	−	1.11378e11i	−0.169836	−	0.428554i
$715$		0			0
$716$		−	3.02102e11i		−	1.14948i
$717$	−1.36858e11	+	5.42371e10i	−0.517839	+	0.205220i
$718$	−1.22769e11			−0.461946
$719$		−	1.31701e11i		−	0.492805i	−0.969168	−	0.246402i	$-0.920752\pi$
$719$		−	1.31701e11i		−	0.492805i	0.969168	−	0.246402i	$-0.0792485\pi$
$720$		0			0
$721$	2.97079e10			0.109934
$722$			2.81584e11i			1.03624i
$723$	6.95574e9	+	1.75516e10i	0.0254560	+	0.0642340i
$724$	−1.73210e11			−0.630405
$725$		0			0
$726$	−1.03509e11	+	4.10205e10i	−0.372589	+	0.147657i
$727$	−1.82998e11			−0.655102			−0.327551	−	0.944834i	$-0.606223\pi$
$727$	−1.82998e11			−0.655102			−0.327551	+	0.944834i	$0.606223\pi$
$728$			1.20725e11i			0.429805i
$729$	1.80448e11	+	2.17267e11i	0.638913	+	0.769279i
$730$		0			0
$731$		−	3.14319e11i		−	1.10078i
$732$	−7.56155e10	−	1.90803e11i	−0.263370	−	0.664571i
$733$	−1.52831e10			−0.0529413			−0.0264707	−	0.999650i	$-0.508427\pi$
$733$	−1.52831e10			−0.0529413			−0.0264707	+	0.999650i	$0.508427\pi$
$734$		−	2.92750e10i		−	0.100858i
$735$		0			0
$736$	−1.23112e11			−0.419555
$737$			1.10219e11i			0.373583i
$738$	1.76393e11	−	1.65858e11i	0.594642	−	0.559128i
$739$	−2.80725e11			−0.941245			−0.470623	−	0.882335i	$-0.655971\pi$
$739$	−2.80725e11			−0.941245			−0.470623	+	0.882335i	$0.655971\pi$
$740$		0			0
$741$	7.17222e10	+	1.80979e11i	0.237893	+	0.600283i
$742$	3.20713e11			1.05804
$743$		−	2.05247e11i		−	0.673475i	−0.941599	−	0.336737i	$-0.890677\pi$
$743$		−	2.05247e11i		−	0.673475i	0.941599	−	0.336737i	$-0.109323\pi$
$744$	8.20706e10	−	3.25247e10i	0.267853	−	0.106150i
$745$		0			0
$746$		−	1.12312e11i		−	0.362635i
$747$	1.04602e11	+	1.11246e11i	0.335937	+	0.357275i
$748$	7.20010e10			0.230002
$749$		−	3.89580e11i		−	1.23785i
$750$		0			0
$751$	1.85798e11			0.584094			0.292047	−	0.956404i	$-0.405664\pi$
$751$	1.85798e11			0.584094			0.292047	+	0.956404i	$0.405664\pi$
$752$			1.38817e11i			0.434081i
$753$	3.08689e11	−	1.22334e11i	0.960154	−	0.380510i
$754$	9.36066e10			0.289615
$755$		0			0
$756$	−2.97271e11	−	1.39534e11i	−0.910050	−	0.427163i
$757$	1.02442e11			0.311956			0.155978	−	0.987761i	$-0.450147\pi$
$757$	1.02442e11			0.311956			0.155978	+	0.987761i	$0.450147\pi$
$758$		−	1.66460e11i		−	0.504235i
$759$	2.20328e10	+	5.55962e10i	0.0663901	+	0.167524i
$760$		0			0
$761$			2.07040e11i			0.617327i	0.951171	+	0.308664i	$0.0998817\pi$
$761$			2.07040e11i			0.617327i	−0.951171	+	0.308664i	$0.900118\pi$
$762$	1.67045e11	−	6.62002e10i	0.495466	−	0.196354i
$763$	4.16835e11			1.22989
$764$		−	3.01013e11i		−	0.883509i
$765$		0			0
$766$	−7.92318e10			−0.230136
$767$		−	5.84356e10i		−	0.168848i
$768$	−8.48835e10	−	2.14190e11i	−0.243994	−	0.615678i
$769$	4.39556e11			1.25692			0.628462	−	0.777841i	$-0.283685\pi$
$769$	4.39556e11			1.25692			0.628462	+	0.777841i	$0.283685\pi$
$770$		0			0
$771$	4.87322e11	−	1.93126e11i	1.37911	−	0.546542i
$772$	1.71165e11			0.481888
$773$			4.81386e11i			1.34827i	0.738610	+	0.674133i	$0.235482\pi$
$773$			4.81386e11i			1.34827i	−0.738610	+	0.674133i	$0.764518\pi$
$774$	1.95141e11	+	2.07535e11i	0.543731	+	0.578267i
$775$		0			0
$776$			3.23150e11i			0.891164i
$777$	−6.04134e10	−	1.52443e11i	−0.165748	−	0.418238i
$778$	−1.87687e11			−0.512289
$779$			1.05803e12i			2.87309i
$780$		0			0
$781$	7.05201e10			0.189543
$782$		−	5.24534e10i		−	0.140264i
$783$	−2.52154e11	+	5.37203e11i	−0.670840	+	1.42919i
$784$	9.43402e10			0.249708
$785$		0			0
$786$	−1.32004e11	−	3.33090e11i	−0.345857	−	0.872712i
$787$	−1.76591e10			−0.0460331			−0.0230166	−	0.999735i	$-0.507327\pi$
$787$	−1.76591e10			−0.0460331			−0.0230166	+	0.999735i	$0.507327\pi$
$788$		−	1.89916e11i		−	0.492557i
$789$	−8.96203e10	+	3.55166e10i	−0.231259	+	0.0916481i
$790$		0			0
$791$		−	4.96423e11i		−	1.26808i
$792$	−1.10799e11	+	1.04182e11i	−0.281602	+	0.264784i
$793$	−1.38426e11			−0.350045
$794$			1.15708e11i			0.291127i
$795$		0			0
$796$	2.62312e11			0.653381
$797$		−	2.22982e11i		−	0.552633i	−0.961067	−	0.276317i	$-0.910886\pi$
$797$		−	2.22982e11i		−	0.552633i	0.961067	−	0.276317i	$-0.0891139\pi$
$798$	−4.41102e11	+	1.74809e11i	−1.08775	+	0.431074i
$799$	−3.86669e11			−0.948750
$800$		0			0
$801$	1.44578e11	+	1.53761e11i	0.351214	+	0.373522i
$802$	1.37737e11			0.332930
$803$		−	1.73184e11i		−	0.416529i
$804$	−9.76273e10	−	2.46346e11i	−0.233640	−	0.589552i
$805$		0			0
$806$		−	2.55471e10i		−	0.0605343i
$807$	6.95571e11	−	2.75655e11i	1.64001	−	0.649938i
$808$	4.45377e11			1.04492
$809$			1.32706e11i			0.309810i	0.987929	+	0.154905i	$0.0495072\pi$
$809$			1.32706e11i			0.309810i	−0.987929	+	0.154905i	$0.950493\pi$
$810$		0			0
$811$	7.48071e11			1.72926			0.864628	−	0.502412i	$-0.167554\pi$
$811$	7.48071e11			1.72926			0.864628	+	0.502412i	$0.167554\pi$
$812$		−	6.90009e11i		−	1.58720i
$813$	−1.63361e11	−	4.12214e11i	−0.373926	−	0.943540i
$814$	−3.25844e10			−0.0742185
$815$		0			0
$816$	−9.01236e10	+	3.57161e10i	−0.203272	+	0.0805569i
$817$	−1.24483e12			−2.79398
$818$			8.85636e10i			0.197807i
$819$	−1.61357e11	+	1.51720e11i	−0.358635	+	0.337216i
$820$		0			0
$821$		−	4.86613e11i		−	1.07105i	−0.844519	−	0.535526i	$-0.820113\pi$
$821$		−	4.86613e11i		−	1.07105i	0.844519	−	0.535526i	$-0.179887\pi$
$822$	−1.36858e10	−	3.45339e10i	−0.0299767	−	0.0756413i
$823$	2.67253e11			0.582537			0.291269	−	0.956641i	$-0.405923\pi$
$823$	2.67253e11			0.582537			0.291269	+	0.956641i	$0.405923\pi$
$824$			3.30780e10i			0.0717515i
$825$		0			0
$826$	1.42425e11			0.305962
$827$		−	9.99227e10i		−	0.213620i	−0.994279	−	0.106810i	$-0.965936\pi$
$827$		−	9.99227e10i		−	0.213620i	0.994279	−	0.106810i	$-0.0340637\pi$
$828$	−9.84894e10	−	1.04745e11i	−0.209541	−	0.222850i
$829$	−1.37202e11			−0.290497			−0.145249	−	0.989395i	$-0.546398\pi$
$829$	−1.37202e11			−0.290497			−0.145249	+	0.989395i	$0.546398\pi$
$830$		0			0
$831$	−2.32071e11	−	5.85594e11i	−0.486651	−	1.22798i
$832$	−3.48308e10			−0.0726892
$833$			2.62781e11i			0.545775i
$834$	−1.34659e11	+	5.33656e10i	−0.278338	+	0.110306i
$835$		0			0
$836$		−	2.85154e11i		−	0.583786i
$837$	1.46613e11	+	6.88179e10i	0.298725	+	0.140217i
$838$	2.12320e11			0.430542
$839$			2.29368e11i			0.462899i	0.972847	+	0.231449i	$0.0743467\pi$
$839$			2.29368e11i			0.462899i	−0.972847	+	0.231449i	$0.925653\pi$
$840$		0			0
$841$	−7.46680e11			−1.49262
$842$		−	1.62397e11i		−	0.323094i
$843$	−2.57287e11	+	1.01963e11i	−0.509458	+	0.201899i
$844$	1.82338e11			0.359341
$845$		0			0
$846$	2.55306e11	−	2.40058e11i	0.498401	−	0.468635i
$847$	5.53548e11			1.07553
$848$		−	2.59511e11i		−	0.501849i
$849$	−2.71543e11	−	6.85194e11i	−0.522646	−	1.31881i
$850$		0			0
$851$		−	7.17933e10i		−	0.136888i
$852$	−1.57617e11	+	6.24636e10i	−0.299119	+	0.118541i
$853$	7.54864e11			1.42585			0.712923	−	0.701242i	$-0.247371\pi$
$853$	7.54864e11			1.42585			0.712923	+	0.701242i	$0.247371\pi$
$854$		−	3.37385e11i		−	0.634300i
$855$		0			0
$856$	4.33774e11			0.807921
$857$			9.33249e9i			0.0173011i	0.999963	+	0.00865056i	$0.00275359\pi$
$857$			9.33249e9i			0.0173011i	−0.999963	+	0.00865056i	$0.997246\pi$
$858$	1.62150e10	+	4.09158e10i	0.0299204	+	0.0754991i
$859$	3.51970e11			0.646447			0.323224	−	0.946323i	$-0.395233\pi$
$859$	3.51970e11			0.646447			0.323224	+	0.946323i	$0.395233\pi$
$860$		0			0
$861$	−1.11908e12	+	4.43492e11i	−2.03633	+	0.806999i
$862$	−1.00919e11			−0.182787
$863$		−	4.14848e11i		−	0.747905i	−0.927448	−	0.373952i	$-0.878002\pi$
$863$		−	4.14848e11i		−	0.747905i	0.927448	−	0.373952i	$-0.121998\pi$
$864$	2.44065e11	−	5.19969e11i	0.437976	−	0.933088i
$865$		0			0
$866$			2.90233e11i			0.516030i
$867$	1.08687e11	+	2.74255e11i	0.192355	+	0.485375i
$868$	−1.88317e11			−0.331750
$869$			1.99546e10i			0.0349916i
$870$		0			0
$871$	−1.78721e11			−0.310530
$872$			4.64121e11i			0.802723i
$873$	−4.31913e11	+	4.06118e11i	−0.743601	+	0.699190i
$874$	−2.07737e11			−0.356015
$875$		0			0
$876$	1.53399e11	+	3.87076e11i	0.260498	+	0.657325i
$877$	6.49541e11			1.09801			0.549007	−	0.835818i	$-0.315006\pi$
$877$	6.49541e11			1.09801			0.549007	+	0.835818i	$0.315006\pi$
$878$		−	4.70731e11i		−	0.792127i
$879$	−1.70760e11	+	6.76724e10i	−0.286043	+	0.113359i
$880$		0			0
$881$			2.43386e11i			0.404010i	0.979384	+	0.202005i	$0.0647458\pi$
$881$			2.43386e11i			0.404010i	−0.979384	+	0.202005i	$0.935254\pi$
$882$	−1.63144e11	−	1.73506e11i	−0.269586	−	0.286709i
$883$	−5.16877e11			−0.850245			−0.425122	−	0.905136i	$-0.639769\pi$
$883$	−5.16877e11			−0.850245			−0.425122	+	0.905136i	$0.639769\pi$
$884$			1.16750e11i			0.191183i
$885$		0			0
$886$	−1.27539e11			−0.206970
$887$		−	6.37860e11i		−	1.03046i	−0.857052	−	0.515229i	$-0.827707\pi$
$887$		−	6.37860e11i		−	1.03046i	0.857052	−	0.515229i	$-0.172293\pi$
$888$	1.69736e11	−	6.72667e10i	0.272975	−	0.108180i
$889$	−8.93332e11			−1.43023
$890$		0			0
$891$	−2.78493e11	−	1.71608e10i	−0.441879	−	0.0272286i
$892$	−1.96079e11			−0.309722
$893$			1.53137e12i			2.40810i
$894$	3.66085e10	+	9.23754e10i	0.0573101	+	0.144613i
$895$		0			0
$896$			8.03809e11i			1.24716i
$897$	−9.01498e10	+	3.57265e10i	−0.139250	+	0.0551849i
$898$	−1.28935e10			−0.0198274
$899$			3.40311e11i			0.521000i
$900$		0			0
$901$	7.22859e11			1.09687
$902$			2.39200e11i			0.361356i
$903$	−5.21791e11	−	1.31665e12i	−0.784776	−	1.98025i
$904$	5.52738e11			0.827647
$905$		0			0
$906$	−5.27451e11	+	2.09029e11i	−0.782833	+	0.310238i
$907$	−5.28792e11			−0.781368			−0.390684	−	0.920525i	$-0.627761\pi$
$907$	−5.28792e11			−0.781368			−0.390684	+	0.920525i	$0.627761\pi$
$908$			2.76555e10i			0.0406853i
$909$	5.59725e11	+	5.95277e11i	0.819822	+	0.871894i
$910$		0			0
$911$		−	1.10395e12i		−	1.60279i	−0.598137	−	0.801394i	$-0.704092\pi$
$911$		−	1.10395e12i		−	1.60279i	0.598137	−	0.801394i	$-0.295908\pi$
$912$	1.41450e11	+	3.56927e11i	0.204468	+	0.515940i
$913$	−1.50857e11			−0.217111
$914$		−	4.00080e11i		−	0.573273i
$915$		0			0
$916$	−1.18151e10			−0.0167824
$917$			1.78131e12i			2.51920i
$918$	2.21540e11	+	1.03987e11i	0.311947	+	0.146423i
$919$	8.75563e11			1.22751			0.613756	−	0.789496i	$-0.289658\pi$
$919$	8.75563e11			1.22751			0.613756	+	0.789496i	$0.289658\pi$
$920$		0			0
$921$	3.14651e11	+	7.93971e11i	0.437312	+	1.10348i
$922$	−5.73829e11			−0.794070
$923$			1.14349e11i			0.157553i
$924$	3.01606e11	−	1.19527e11i	0.413763	−	0.163975i
$925$		0			0
$926$			4.93713e11i			0.671476i
$927$	−4.42111e10	+	4.15707e10i	−0.0598705	+	0.0562948i
$928$	1.20692e12			1.62738
$929$		−	2.94492e11i		−	0.395377i	−0.980265	−	0.197688i	$-0.936657\pi$
$929$		−	2.94492e11i		−	0.395377i	0.980265	−	0.197688i	$-0.0633434\pi$
$930$		0			0
$931$	1.04072e12			1.38527
$932$		−	2.14951e11i		−	0.284890i
$933$	−5.42489e11	+	2.14989e11i	−0.715919	+	0.283720i
$934$	−3.97407e10			−0.0522214
$935$		0			0
$936$	−1.68932e11	−	1.79662e11i	−0.220094	−	0.234073i
$937$	6.27494e11			0.814050			0.407025	−	0.913417i	$-0.366566\pi$
$937$	6.27494e11			0.814050			0.407025	+	0.913417i	$0.366566\pi$
$938$		−	4.35599e11i		−	0.562698i
$939$	6.92799e10	+	1.74816e11i	0.0891138	+	0.224864i
$940$		0			0
$941$			4.89291e11i			0.624035i	0.950076	+	0.312017i	$0.101005\pi$
$941$			4.89291e11i			0.624035i	−0.950076	+	0.312017i	$0.898995\pi$
$942$	1.68697e11	−	6.68548e10i	0.214242	−	0.0849042i
$943$	−5.27031e11			−0.666483
$944$		−	1.15246e11i		−	0.145124i
$945$		0			0
$946$	−2.81432e11			−0.351406
$947$			1.12362e12i			1.39708i	0.715571	+	0.698540i	$0.246166\pi$
$947$			1.12362e12i			1.39708i	−0.715571	+	0.698540i	$0.753834\pi$
$948$	−1.76749e10	−	4.45997e10i	−0.0218839	−	0.0552203i
$949$	2.80819e11			0.346228
$950$		0			0
$951$	−8.87149e10	+	3.51578e10i	−0.108461	+	0.0429833i
$952$	−6.63199e11			−0.807414
$953$		−	2.38597e11i		−	0.289263i	−0.989486	−	0.144632i	$-0.953800\pi$
$953$		−	2.38597e11i		−	0.289263i	0.989486	−	0.144632i	$-0.0461997\pi$
$954$	−4.77282e11	+	4.48777e11i	−0.576211	+	0.541798i
$955$		0			0
$956$			3.49656e11i			0.418610i
$957$	−2.15998e11	−	5.45036e11i	−0.257515	−	0.649797i
$958$	−7.98940e9			−0.00948532
$959$			1.84682e11i			0.218349i
$960$		0			0
$961$	−7.60013e11			−0.891102
$962$		−	5.28359e10i		−	0.0616920i
$963$	5.45145e11	+	5.79770e11i	0.633879	+	0.674141i
$964$	4.48423e10			0.0519254
$965$		0			0
$966$	−8.70763e10	−	2.19723e11i	−0.0999981	−	0.252329i
$967$	7.79476e11			0.891450			0.445725	−	0.895170i	$-0.352946\pi$
$967$	7.79476e11			0.891450			0.445725	+	0.895170i	$0.352946\pi$
$968$			6.16343e11i			0.701974i
$969$	−9.94205e11	+	3.94004e11i	−1.12767	+	0.446896i
$970$		0			0
$971$			8.66546e11i			0.974798i	0.873179	+	0.487399i	$0.162054\pi$
$971$			8.66546e11i			0.974798i	−0.873179	+	0.487399i	$0.837946\pi$
$972$	6.37648e11	−	2.08321e11i	0.714358	−	0.233383i
$973$	7.20137e11			0.803460
$974$		−	6.61752e11i		−	0.735291i
$975$		0			0
$976$	−2.73002e11			−0.300862
$977$			4.89741e11i			0.537512i	0.963208	+	0.268756i	$0.0866125\pi$
$977$			4.89741e11i			0.537512i	−0.963208	+	0.268756i	$0.913388\pi$
$978$	−3.01538e11	+	1.19500e11i	−0.329600	+	0.130621i
$979$	−2.08510e11			−0.226985
$980$		0			0
$981$	−6.20331e11	+	5.83283e11i	−0.669804	+	0.629801i
$982$	1.37311e11			0.147659
$983$			6.13109e11i			0.656635i	0.944567	+	0.328317i	$0.106482\pi$
$983$			6.13109e11i			0.656635i	−0.944567	+	0.328317i	$0.893518\pi$
$984$	−4.93802e11	−	1.24603e12i	−0.526711	−	1.32907i
$985$		0			0
$986$			5.14226e11i			0.544059i
$987$	−1.61972e12	+	6.41897e11i	−1.70676	+	0.676389i
$988$	4.62379e11			0.485256
$989$		−	6.20079e11i		−	0.648130i
$990$		0			0
$991$	3.85928e11			0.400140			0.200070	−	0.979782i	$-0.435883\pi$
$991$	3.85928e11			0.400140			0.200070	+	0.979782i	$0.435883\pi$
$992$		−	3.29394e11i		−	0.340148i
$993$	1.09778e11	+	2.77008e11i	0.112907	+	0.284901i
$994$	−2.78703e11			−0.285494
$995$		0			0
$996$	3.37175e11	−	1.33623e11i	0.342624	−	0.135782i
$997$	−1.56419e12			−1.58310			−0.791551	−	0.611104i	$-0.790726\pi$
$997$	−1.56419e12			−1.58310			−0.791551	+	0.611104i	$0.790726\pi$
$998$			5.60565e11i			0.565072i
$999$	3.03222e11	+	1.42328e11i	0.304438	+	0.142898i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.c.g.26.6		10
3.2	odd	2	inner	75.9.c.g.26.5		10
5.2	odd	4		75.9.d.c.74.9		20
5.3	odd	4		75.9.d.c.74.12		20
5.4	even	2		15.9.c.a.11.5	✓	10
15.2	even	4		75.9.d.c.74.11		20
15.8	even	4		75.9.d.c.74.10		20
15.14	odd	2		15.9.c.a.11.6	yes	10
20.19	odd	2		240.9.l.b.161.6		10
60.59	even	2		240.9.l.b.161.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.9.c.a.11.5	✓	10	5.4	even	2
15.9.c.a.11.6	yes	10	15.14	odd	2
75.9.c.g.26.5		10	3.2	odd	2	inner
75.9.c.g.26.6		10	1.1	even	1	trivial
75.9.d.c.74.9		20	5.2	odd	4
75.9.d.c.74.10		20	15.8	even	4
75.9.d.c.74.11		20	15.2	even	4
75.9.d.c.74.12		20	5.3	odd	4
240.9.l.b.161.5		10	60.59	even	2
240.9.l.b.161.6		10	20.19	odd	2

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)

\(f(q)\)	\(=\)	\(q+7.97572i q^{2} +(29.8424 + 75.3023i) q^{3} +192.388 q^{4} +(-600.590 + 238.014i) q^{6} +3211.86 q^{7} +3576.22i q^{8} +(-4779.87 + 4494.40i) q^{9} +O(q^{10})\)	\(q+7.97572i q^{2} +(29.8424 + 75.3023i) q^{3} +192.388 q^{4} +(-600.590 + 238.014i) q^{6} +3211.86 q^{7} +3576.22i q^{8} +(-4779.87 + 4494.40i) q^{9} -6481.82i q^{11} +(5741.31 + 14487.3i) q^{12} +10510.3 q^{13} +25616.9i q^{14} +20728.4 q^{16} +57738.3i q^{17} +(-35846.0 - 38122.9i) q^{18} +228667. q^{19} +(95849.5 + 241860. i) q^{21} +51697.1 q^{22} +113904. i q^{23} +(-269297. + 106723. i) q^{24} +83827.4i q^{26} +(-481081. - 225811. i) q^{27} +617923. q^{28} -1.11666e6i q^{29} -304758. q^{31} +1.08084e6i q^{32} +(488096. - 193433. i) q^{33} -460504. q^{34} +(-919589. + 864668. i) q^{36} -630294. q^{37} +1.82379e6i q^{38} +(313653. + 791452. i) q^{39} +4.62696e6i q^{41} +(-1.92901e6 + 764469. i) q^{42} -5.44386e6 q^{43} -1.24702e6i q^{44} -908469. q^{46} +6.69692e6i q^{47} +(618586. + 1.56090e6i) q^{48} +4.55125e6 q^{49} +(-4.34782e6 + 1.72305e6i) q^{51} +2.02206e6 q^{52} -1.25196e7i q^{53} +(1.80101e6 - 3.83696e6i) q^{54} +1.14863e7i q^{56} +(6.82398e6 + 1.72192e7i) q^{57} +8.90615e6 q^{58} -5.55982e6i q^{59} -1.31704e7 q^{61} -2.43067e6i q^{62} +(-1.53523e7 + 1.44354e7i) q^{63} -3.31396e6 q^{64} +(1.54276e6 + 3.89291e6i) q^{66} -1.70044e7 q^{67} +1.11081e7i q^{68} +(-8.57726e6 + 3.39918e6i) q^{69} +1.08797e7i q^{71} +(-1.60729e7 - 1.70938e7i) q^{72} +2.67184e7 q^{73} -5.02705e6i q^{74} +4.39928e7 q^{76} -2.08187e7i q^{77} +(-6.31239e6 + 2.50161e6i) q^{78} -3.07855e6 q^{79} +(2.64753e6 - 4.29652e7i) q^{81} -3.69033e7 q^{82} -2.32739e7i q^{83} +(1.84403e7 + 4.65310e7i) q^{84} -4.34187e7i q^{86} +(8.40869e7 - 3.33237e7i) q^{87} +2.31804e7 q^{88} -3.21685e7i q^{89} +3.37577e7 q^{91} +2.19138e7i q^{92} +(-9.09471e6 - 2.29490e7i) q^{93} -5.34128e7 q^{94} +(-8.13894e7 + 3.22547e7i) q^{96} +9.03610e7 q^{97} +3.62994e7i q^{98} +(2.91319e7 + 3.09822e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 112 q^{3} - 786 q^{4} - 5282 q^{6} - 7156 q^{7} + 3922 q^{9}+O(q^{10}) \) 10 * q + 112 * q^3 - 786 * q^4 - 5282 * q^6 - 7156 * q^7 + 3922 * q^9	\( 10 q + 112 q^{3} - 786 q^{4} - 5282 q^{6} - 7156 q^{7} + 3922 q^{9} + 3812 q^{12} + 55464 q^{13} + 280386 q^{16} + 419800 q^{18} - 231516 q^{19} + 289572 q^{21} - 1129940 q^{22} + 1136334 q^{24} + 335512 q^{27} + 3340724 q^{28} + 881620 q^{31} - 1266460 q^{33} - 1111276 q^{34} - 668662 q^{36} - 4672616 q^{37} + 1826792 q^{39} + 5392860 q^{42} - 7731336 q^{43} - 25424604 q^{46} - 22413388 q^{48} + 9354214 q^{49} - 27732692 q^{51} - 21064016 q^{52} - 7979798 q^{54} + 2856304 q^{57} + 4351100 q^{58} + 22417020 q^{61} - 8830596 q^{63} - 22935002 q^{64} - 27419800 q^{66} + 46646024 q^{67} + 33562632 q^{69} - 54175560 q^{72} + 129964884 q^{73} + 198922436 q^{76} - 60388360 q^{78} + 162310924 q^{79} - 93575390 q^{81} - 202877560 q^{82} - 197346768 q^{84} + 168322540 q^{87} + 484775700 q^{88} + 444288464 q^{91} - 463412376 q^{93} - 92050036 q^{94} - 360807406 q^{96} + 258825724 q^{97} - 33965200 q^{99}+O(q^{100}) \) 10 * q + 112 * q^3 - 786 * q^4 - 5282 * q^6 - 7156 * q^7 + 3922 * q^9 + 3812 * q^12 + 55464 * q^13 + 280386 * q^16 + 419800 * q^18 - 231516 * q^19 + 289572 * q^21 - 1129940 * q^22 + 1136334 * q^24 + 335512 * q^27 + 3340724 * q^28 + 881620 * q^31 - 1266460 * q^33 - 1111276 * q^34 - 668662 * q^36 - 4672616 * q^37 + 1826792 * q^39 + 5392860 * q^42 - 7731336 * q^43 - 25424604 * q^46 - 22413388 * q^48 + 9354214 * q^49 - 27732692 * q^51 - 21064016 * q^52 - 7979798 * q^54 + 2856304 * q^57 + 4351100 * q^58 + 22417020 * q^61 - 8830596 * q^63 - 22935002 * q^64 - 27419800 * q^66 + 46646024 * q^67 + 33562632 * q^69 - 54175560 * q^72 + 129964884 * q^73 + 198922436 * q^76 - 60388360 * q^78 + 162310924 * q^79 - 93575390 * q^81 - 202877560 * q^82 - 197346768 * q^84 + 168322540 * q^87 + 484775700 * q^88 + 444288464 * q^91 - 463412376 * q^93 - 92050036 * q^94 - 360807406 * q^96 + 258825724 * q^97 - 33965200 * q^99

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