LMFDB - Embedded newform 75.9.d.c.74.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(74,75)]

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, 1]))

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

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 943 x^{18} + 318815 x^{16} + 48938090 x^{14} + 3842259173 x^{12} + 159675554657 x^{10} + \cdots + 336685801 $ x^20 + 943x^18 + 318815x^16 + 48938090x^14 + 3842259173x^12 + 159675554657x^10 + 3390679484573x^8 + 32981662033730x^6 + 124592584990175x^4 + 133865237486743*x^2 + 336685801
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{22}\cdot 5^{32} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.12
Root			$-5.77599i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.9.d.c.74.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.97572 q^{2} +(-75.3023 + 29.8424i) q^{3} -192.388 q^{4} +(-600.590 + 238.014i) q^{6} -3211.86i q^{7} -3576.22 q^{8} +(4779.87 - 4494.40i) q^{9} +O(q^{10})$	\(q+7.97572 q^{2} +(-75.3023 + 29.8424i) q^{3} -192.388 q^{4} +(-600.590 + 238.014i) q^{6} -3211.86i q^{7} -3576.22 q^{8} +(4779.87 - 4494.40i) q^{9} -6481.82i q^{11} +(14487.3 - 5741.31i) q^{12} +10510.3i q^{13} -25616.9i q^{14} +20728.4 q^{16} +57738.3 q^{17} +(38122.9 - 35846.0i) q^{18} -228667. q^{19} +(95849.5 + 241860. i) q^{21} -51697.1i q^{22} -113904. q^{23} +(269297. - 106723. i) q^{24} +83827.4i q^{26} +(-225811. + 481081. i) q^{27} +617923. i q^{28} +1.11666e6i q^{29} -304758. q^{31} +1.08084e6 q^{32} +(193433. + 488096. i) q^{33} +460504. q^{34} +(-919589. + 864668. i) q^{36} +630294. i q^{37} -1.82379e6 q^{38} +(-313653. - 791452. i) q^{39} +4.62696e6i q^{41} +(764469. + 1.92901e6i) q^{42} -5.44386e6i q^{43} +1.24702e6i q^{44} -908469. q^{46} +6.69692e6 q^{47} +(-1.56090e6 + 618586. i) q^{48} -4.55125e6 q^{49} +(-4.34782e6 + 1.72305e6i) q^{51} -2.02206e6i q^{52} +1.25196e7 q^{53} +(-1.80101e6 + 3.83696e6i) q^{54} +1.14863e7i q^{56} +(1.72192e7 - 6.82398e6i) q^{57} +8.90615e6i q^{58} +5.55982e6i q^{59} -1.31704e7 q^{61} -2.43067e6 q^{62} +(-1.44354e7 - 1.53523e7i) q^{63} +3.31396e6 q^{64} +(1.54276e6 + 3.89291e6i) q^{66} +1.70044e7i q^{67} -1.11081e7 q^{68} +(8.57726e6 - 3.39918e6i) q^{69} +1.08797e7i q^{71} +(-1.70938e7 + 1.60729e7i) q^{72} +2.67184e7i q^{73} +5.02705e6i q^{74} +4.39928e7 q^{76} -2.08187e7 q^{77} +(-2.50161e6 - 6.31239e6i) q^{78} +3.07855e6 q^{79} +(2.64753e6 - 4.29652e7i) q^{81} +3.69033e7i q^{82} +2.32739e7 q^{83} +(-1.84403e7 - 4.65310e7i) q^{84} -4.34187e7i q^{86} +(-3.33237e7 - 8.40869e7i) q^{87} +2.31804e7i q^{88} +3.21685e7i q^{89} +3.37577e7 q^{91} +2.19138e7 q^{92} +(2.29490e7 - 9.09471e6i) q^{93} +5.34128e7 q^{94} +(-8.13894e7 + 3.22547e7i) q^{96} -9.03610e7i q^{97} -3.62994e7 q^{98} +(-2.91319e7 - 3.09822e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9}+O(q^{10}) $ 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9	$ 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9} + 560772 q^{16} + 463032 q^{19} + 579144 q^{21} - 2272668 q^{24} + 1763240 q^{31} + 2222552 q^{34} - 1337324 q^{36} - 3653584 q^{39} - 50849208 q^{46} - 18708428 q^{49} - 55465384 q^{51} + 15959596 q^{54} + 44834040 q^{61} + 45870004 q^{64} - 54839600 q^{66} - 67125264 q^{69} + 397844872 q^{76} - 324621848 q^{79} - 187150780 q^{81} + 394693536 q^{84} + 888576928 q^{91} + 184100072 q^{94} - 721614812 q^{96} + 67930400 q^{99}+O(q^{100}) $ 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9 + 560772 * q^16 + 463032 * q^19 + 579144 * q^21 - 2272668 * q^24 + 1763240 * q^31 + 2222552 * q^34 - 1337324 * q^36 - 3653584 * q^39 - 50849208 * q^46 - 18708428 * q^49 - 55465384 * q^51 + 15959596 * q^54 + 44834040 * q^61 + 45870004 * q^64 - 54839600 * q^66 - 67125264 * q^69 + 397844872 * q^76 - 324621848 * q^79 - 187150780 * q^81 + 394693536 * q^84 + 888576928 * q^91 + 184100072 * q^94 - 721614812 * q^96 + 67930400 * 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 $

$2$	7.97572			0.498482			0.249241	−	0.968441i	$-0.419819\pi$
$2$	7.97572			0.498482			0.249241	+	0.968441i	$0.419819\pi$
$3$	−75.3023	+	29.8424i	−0.929658	+	0.368424i
$3$	−75.3023	+	29.8424i	−0.929658	+	0.368424i
$4$	−192.388			−0.751515
$5$		0			0
$5$		0			0
$6$	−600.590	+	238.014i	−0.463418	+	0.183653i
$7$		−	3211.86i		−	1.33772i	−0.743389	−	0.668859i	$-0.766783\pi$
$7$		−	3211.86i		−	1.33772i	0.743389	−	0.668859i	$-0.233217\pi$
$8$	−3576.22			−0.873099
$9$	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	$-0.0710491\pi$
$12$	14487.3	−	5741.31i	0.698652	−	0.276877i
$13$			10510.3i			0.367996i	0.982927	+	0.183998i	$0.0589040\pi$
$13$			10510.3i			0.367996i	−0.982927	+	0.183998i	$0.941096\pi$
$14$		−	25616.9i		−	0.666829i
$15$		0			0
$16$	20728.4			0.316291
$17$	57738.3			0.691302			0.345651	−	0.938363i	$-0.387658\pi$
$17$	57738.3			0.691302			0.345651	+	0.938363i	$0.387658\pi$
$18$	38122.9	−	35846.0i	0.363158	−	0.341469i
$19$	−228667.			−1.75465			−0.877324	−	0.479899i	$-0.840673\pi$
$19$	−228667.			−1.75465			−0.877324	+	0.479899i	$0.840673\pi$
$20$		0			0
$21$	95849.5	+	241860.i	0.492848	+	1.24362i
$22$		−	51697.1i		−	0.220686i
$23$	−113904.			−0.407033			−0.203516	−	0.979072i	$-0.565237\pi$
$23$	−113904.			−0.407033			−0.203516	+	0.979072i	$0.565237\pi$
$24$	269297.	−	106723.i	0.811684	−	0.321671i
$25$		0			0
$26$			83827.4i			0.183439i
$27$	−225811.	+	481081.i	−0.424904	+	0.905238i
$28$			617923.i			1.00532i
$29$			1.11666e6i			1.57880i	0.613876	+	0.789402i	$0.289609\pi$
$29$			1.11666e6i			1.57880i	−0.613876	+	0.789402i	$0.710391\pi$
$30$		0			0
$31$	−304758.			−0.329996			−0.164998	−	0.986294i	$-0.552762\pi$
$31$	−304758.			−0.329996			−0.164998	+	0.986294i	$0.552762\pi$
$32$	1.08084e6			1.03076
$33$	193433.	+	488096.i	0.163108	+	0.411575i
$34$	460504.			0.344602
$35$		0			0
$36$	−919589.	+	864668.i	−0.547499	+	0.514801i
$37$			630294.i			0.336307i	0.985761	+	0.168154i	$0.0537805\pi$
$37$			630294.i			0.336307i	−0.985761	+	0.168154i	$0.946220\pi$
$38$	−1.82379e6			−0.874661
$39$	−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	$0.694690\pi$
$42$	764469.	+	1.92901e6i	0.245676	+	0.619922i
$43$		−	5.44386e6i		−	1.59233i	−0.605080	−	0.796165i	$-0.706859\pi$
$43$		−	5.44386e6i		−	1.59233i	0.605080	−	0.796165i	$-0.293141\pi$
$44$			1.24702e6i			0.332709i
$45$		0			0
$46$	−908469.			−0.202898
$47$	6.69692e6			1.37241			0.686205	−	0.727408i	$-0.259275\pi$
$47$	6.69692e6			1.37241			0.686205	+	0.727408i	$0.259275\pi$
$48$	−1.56090e6	+	618586.i	−0.294042	+	0.116529i
$49$	−4.55125e6			−0.789489
$50$		0			0
$51$	−4.34782e6	+	1.72305e6i	−0.642675	+	0.254693i
$52$		−	2.02206e6i		−	0.276554i
$53$	1.25196e7			1.58667			0.793335	−	0.608786i	$-0.208343\pi$
$53$	1.25196e7			1.58667			0.793335	+	0.608786i	$0.208343\pi$
$54$	−1.80101e6	+	3.83696e6i	−0.211807	+	0.451245i
$55$		0			0
$56$			1.14863e7i			1.16796i
$57$	1.72192e7	−	6.82398e6i	1.63122	−	0.646455i
$58$			8.90615e6i			0.787006i
$59$			5.55982e6i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$59$			5.55982e6i			0.458831i	−0.973329	+	0.229416i	$0.926319\pi$
$60$		0			0
$61$	−1.31704e7			−0.951219			−0.475610	−	0.879656i	$-0.657773\pi$
$61$	−1.31704e7			−0.951219			−0.475610	+	0.879656i	$0.657773\pi$
$62$	−2.43067e6			−0.164497
$63$	−1.44354e7	−	1.53523e7i	−0.916359	−	0.974564i
$64$	3.31396e6			0.197527
$65$		0			0
$66$	1.54276e6	+	3.89291e6i	0.0813063	+	0.205163i
$67$			1.70044e7i			0.843841i	0.906633	+	0.421921i	$0.138644\pi$
$67$			1.70044e7i			0.843841i	−0.906633	+	0.421921i	$0.861356\pi$
$68$	−1.11081e7			−0.519524
$69$	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	$0.931328\pi$
$72$	−1.70938e7	+	1.60729e7i	−0.636077	+	0.598088i
$73$			2.67184e7i			0.940847i	0.882441	+	0.470424i	$0.155899\pi$
$73$			2.67184e7i			0.940847i	−0.882441	+	0.470424i	$0.844101\pi$
$74$			5.02705e6i			0.167643i
$75$		0			0
$76$	4.39928e7			1.31864
$77$	−2.08187e7			−0.592230
$78$	−2.50161e6	−	6.31239e6i	−0.0675835	−	0.170536i
$79$	3.07855e6			0.0790383			0.0395192	−	0.999219i	$-0.487417\pi$
$79$	3.07855e6			0.0790383			0.0395192	+	0.999219i	$0.487417\pi$
$80$		0			0
$81$	2.64753e6	−	4.29652e7i	0.0615035	−	0.998107i
$82$			3.69033e7i			0.816225i
$83$	2.32739e7			0.490407			0.245203	−	0.969472i	$-0.421145\pi$
$83$	2.32739e7			0.490407			0.245203	+	0.969472i	$0.421145\pi$
$84$	−1.84403e7	−	4.65310e7i	−0.370383	−	0.934599i
$85$		0			0
$86$		−	4.34187e7i		−	0.793748i
$87$	−3.33237e7	−	8.40869e7i	−0.581670	−	1.46775i
$88$			2.31804e7i			0.386536i
$89$			3.21685e7i			0.512709i	0.966583	+	0.256354i	$0.0825214\pi$
$89$			3.21685e7i			0.512709i	−0.966583	+	0.256354i	$0.917479\pi$
$90$		0			0
$91$	3.37577e7			0.492274
$92$	2.19138e7			0.305891
$93$	2.29490e7	−	9.09471e6i	0.306784	−	0.121579i
$94$	5.34128e7			0.684122
$95$		0			0
$96$	−8.13894e7	+	3.22547e7i	−0.958258	+	0.379759i
$97$		−	9.03610e7i		−	1.02069i	−0.859970	−	0.510345i	$-0.829518\pi$
$97$		−	9.03610e7i		−	1.02069i	0.859970	−	0.510345i	$-0.170482\pi$
$98$	−3.62994e7			−0.393546
$99$	−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	$-0.204195\pi$
$102$	−3.46770e7	+	1.37425e7i	−0.320362	+	0.126960i
$103$			9.24945e6i			0.0821802i	0.999155	+	0.0410901i	$0.0130831\pi$
$103$			9.24945e6i			0.0821802i	−0.999155	+	0.0410901i	$0.986917\pi$
$104$		−	3.75872e7i		−	0.321297i
$105$		0			0
$106$	9.98527e7			0.790927
$107$	−1.21294e8			−0.925348			−0.462674	−	0.886528i	$-0.653110\pi$
$107$	−1.21294e8			−0.925348			−0.462674	+	0.886528i	$0.653110\pi$
$108$	4.34434e7	−	9.25541e7i	0.319322	−	0.680301i
$109$	−1.29780e8			−0.919394			−0.459697	−	0.888076i	$-0.652042\pi$
$109$	−1.29780e8			−0.919394			−0.459697	+	0.888076i	$0.652042\pi$
$110$		0			0
$111$	−1.88095e7	−	4.74626e7i	−0.123904	−	0.312651i
$112$		−	6.65768e7i		−	0.423108i
$113$	1.54559e8			0.947942			0.473971	−	0.880541i	$-0.342820\pi$
$113$	1.54559e8			0.947942			0.473971	+	0.880541i	$0.342820\pi$
$114$	1.37335e8	−	5.44261e7i	0.813135	−	0.322246i
$115$		0			0
$116$		−	2.14832e8i		−	1.18650i
$117$	4.72376e7	+	5.02380e7i	0.252083	+	0.268095i
$118$			4.43436e7i			0.228719i
$119$		−	1.85447e8i		−	0.924767i
$120$		0			0
$121$	1.72345e8			0.804002
$122$	−1.05044e8			−0.474166
$123$	−1.38079e8	−	3.48420e8i	−0.603265	−	1.52224i
$124$	5.86319e7			0.247997
$125$		0			0
$126$	−1.15132e8	−	1.22445e8i	−0.456789	−	0.485803i
$127$			2.78135e8i			1.06916i	0.845119	+	0.534578i	$0.179530\pi$
$127$			2.78135e8i			1.06916i	−0.845119	+	0.534578i	$0.820470\pi$
$128$	−2.50263e8			−0.932301
$129$	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	$0.609319\pi$
$132$	−3.72141e7	−	9.39037e7i	−0.122578	−	0.309305i
$133$			7.34448e8i			2.34722i
$134$			1.35622e8i			0.420640i
$135$		0			0
$136$	−2.06484e8			−0.603576
$137$	5.75001e7			0.163225			0.0816124	−	0.996664i	$-0.473993\pi$
$137$	5.75001e7			0.163225			0.0816124	+	0.996664i	$0.473993\pi$
$138$	6.84098e7	−	2.71109e7i	0.188626	−	0.0747527i
$139$	−2.24212e8			−0.600620			−0.300310	−	0.953842i	$-0.597090\pi$
$139$	−2.24212e8			−0.600620			−0.300310	+	0.953842i	$0.597090\pi$
$140$		0			0
$141$	−5.04294e8	+	1.99852e8i	−1.27587	+	0.505629i
$142$			8.67732e7i			0.213419i
$143$	6.81260e7			0.162918
$144$	9.90792e7	−	9.31618e7i	0.230426	−	0.216665i
$145$		0			0
$146$			2.13098e8i			0.468996i
$147$	3.42719e8	−	1.35820e8i	0.733954	−	0.290867i
$148$		−	1.21261e8i		−	0.252740i
$149$			1.53808e8i			0.312057i	0.987753	+	0.156028i	$0.0498691\pi$
$149$			1.53808e8i			0.312057i	−0.987753	+	0.156028i	$0.950131\pi$
$150$		0			0
$151$	8.78222e8			1.68926			0.844630	−	0.535351i	$-0.179821\pi$
$151$	8.78222e8			1.68926			0.844630	+	0.535351i	$0.179821\pi$
$152$	8.17764e8			1.53198
$153$	2.75981e8	−	2.59499e8i	0.503633	−	0.473554i
$154$	−1.66044e8			−0.295216
$155$		0			0
$156$	6.03431e7	+	1.52266e8i	0.101889	+	0.257101i
$157$			2.80886e8i			0.462308i	0.972917	+	0.231154i	$0.0742501\pi$
$157$			2.80886e8i			0.462308i	−0.972917	+	0.231154i	$0.925750\pi$
$158$	2.45536e7			0.0393992
$159$	−9.42753e8	+	3.73614e8i	−1.47506	+	0.584568i
$160$		0			0
$161$			3.65845e8i			0.544495i
$162$	2.11159e7	−	3.42678e8i	0.0306584	−	0.497539i
$163$			5.02070e8i			0.711237i	0.934631	+	0.355618i	$0.115730\pi$
$163$			5.02070e8i			0.711237i	−0.934631	+	0.355618i	$0.884270\pi$
$164$		−	8.90171e8i		−	1.23055i
$165$		0			0
$166$	1.85626e8			0.244459
$167$	2.33934e8			0.300765			0.150383	−	0.988628i	$-0.451949\pi$
$167$	2.33934e8			0.300765			0.150383	+	0.988628i	$0.451949\pi$
$168$	−3.42778e8	−	8.64945e8i	−0.430305	−	1.08580i
$169$	7.05264e8			0.864579
$170$		0			0
$171$	−1.09300e9	+	1.02772e9i	−1.27831	+	1.20196i
$172$			1.04733e9i			1.19666i
$173$	−6.37704e8			−0.711925			−0.355963	−	0.934500i	$-0.615847\pi$
$173$	−6.37704e8			−0.711925			−0.355963	+	0.934500i	$0.615847\pi$
$174$	−2.65781e8	−	6.70654e8i	−0.289952	−	0.731646i
$175$		0			0
$176$		−	1.34358e8i		−	0.140027i
$177$	−1.65918e8	−	4.18667e8i	−0.169045	−	0.426556i
$178$			2.56567e8i			0.255576i
$179$			1.57028e9i			1.52955i	0.644297	+	0.764775i	$0.277150\pi$
$179$			1.57028e9i			1.52955i	−0.644297	+	0.764775i	$0.722850\pi$
$180$		0			0
$181$	−9.00318e8			−0.838845			−0.419423	−	0.907791i	$-0.637767\pi$
$181$	−9.00318e8			−0.838845			−0.419423	+	0.907791i	$0.637767\pi$
$182$	2.69242e8			0.245390
$183$	9.91763e8	−	3.93037e8i	0.884308	−	0.350452i
$184$	4.07347e8			0.355380
$185$		0			0
$186$	1.83035e8	−	7.25369e7i	0.152926	−	0.0606048i
$187$		−	3.74249e8i		−	0.306051i
$188$	−1.28841e9			−1.03139
$189$	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	$-0.200013\pi$
$192$	−2.49548e8	+	9.88963e7i	−0.183633	+	0.0727738i
$193$			8.89688e8i			0.641222i	0.947211	+	0.320611i	$0.103888\pi$
$193$			8.89688e8i			0.641222i	−0.947211	+	0.320611i	$0.896112\pi$
$194$		−	7.20694e8i		−	0.508796i
$195$		0			0
$196$	8.75605e8			0.593313
$197$	−9.87151e8			−0.655418			−0.327709	−	0.944779i	$-0.606277\pi$
$197$	−9.87151e8			−0.655418			−0.327709	+	0.944779i	$0.606277\pi$
$198$	−2.32347e8	−	2.47105e8i	−0.151174	−	0.160776i
$199$	−1.36346e9			−0.869418			−0.434709	−	0.900571i	$-0.643149\pi$
$199$	−1.36346e9			−0.869418			−0.434709	+	0.900571i	$0.643149\pi$
$200$		0			0
$201$	−5.07450e8	−	1.28047e9i	−0.310892	−	0.784484i
$202$		−	9.93284e8i		−	0.596579i
$203$	3.58655e9			2.11199
$204$	8.36469e8	−	3.31493e8i	0.482980	−	0.191405i
$205$		0			0
$206$			7.37710e7i			0.0409654i
$207$	−5.44448e8	+	5.11931e8i	−0.296534	+	0.278824i
$208$			2.17863e8i			0.116394i
$209$			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	$0.423155\pi$
$212$	−2.40862e9			−1.19241
$213$	−3.24675e8	−	8.19264e8i	−0.157736	−	0.398021i
$214$	−9.67409e8			−0.461270
$215$		0			0
$216$	8.07550e8	−	1.72045e9i	0.370983	−	0.790363i
$217$			9.78842e8i			0.441442i
$218$	−1.03509e9			−0.458302
$219$	−7.97340e8	−	2.01196e9i	−0.346631	−	0.874666i
$220$		0			0
$221$			6.06848e8i			0.254396i
$222$	−1.50019e8	−	3.78548e8i	−0.0617638	−	0.155851i
$223$		−	1.01919e9i		−	0.412130i	−0.978538	−	0.206065i	$-0.933934\pi$
$223$		−	1.01919e9i		−	0.412130i	0.978538	−	0.206065i	$-0.0660658\pi$
$224$		−	3.47149e9i		−	1.37887i
$225$		0			0
$226$	1.23272e9			0.472532
$227$	1.43749e8			0.0541377			0.0270689	−	0.999634i	$-0.491383\pi$
$227$	1.43749e8			0.0541377			0.0270689	+	0.999634i	$0.491383\pi$
$228$	−3.31276e9	+	1.31285e9i	−1.22589	+	0.485821i
$229$	6.14128e7			0.0223314			0.0111657	−	0.999938i	$-0.496446\pi$
$229$	6.14128e7			0.0223314			0.0111657	+	0.999938i	$0.496446\pi$
$230$		0			0
$231$	1.56769e9	−	6.21279e8i	0.550571	−	0.218192i
$232$		−	3.99341e9i		−	1.37845i
$233$	1.11728e9			0.379087			0.189543	−	0.981872i	$-0.439299\pi$
$233$	1.11728e9			0.379087			0.189543	+	0.981872i	$0.439299\pi$
$234$	3.76754e8	+	4.00684e8i	0.125659	+	0.133641i
$235$		0			0
$236$		−	1.06964e9i		−	0.344819i
$237$	−2.31822e8	+	9.18712e7i	−0.0734786	+	0.0291196i
$238$		−	1.47907e9i		−	0.460980i
$239$		−	1.81745e9i		−	0.557021i	−0.960433	−	0.278511i	$-0.910159\pi$
$239$		−	1.81745e9i		−	0.557021i	0.960433	−	0.278511i	$-0.0898407\pi$
$240$		0			0
$241$	2.33083e8			0.0690942			0.0345471	−	0.999403i	$-0.489001\pi$
$241$	2.33083e8			0.0690942			0.0345471	+	0.999403i	$0.489001\pi$
$242$	1.37457e9			0.400781
$243$	1.08282e9	+	3.31439e9i	0.310550	+	0.950557i
$244$	2.53383e9			0.714856
$245$		0			0
$246$	−1.10128e9	−	2.77890e9i	−0.300717	−	0.758810i
$247$		−	2.40337e9i		−	0.645703i
$248$	1.08988e9			0.288120
$249$	−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	$-0.172729\pi$
$252$	2.77719e9	+	2.95359e9i	0.688658	+	0.732400i
$253$			7.38307e8i			0.180200i
$254$			2.21833e9i			0.532956i
$255$		0			0
$256$	−2.84440e9			−0.662263
$257$	−6.47154e9			−1.48346			−0.741729	−	0.670700i	$-0.765994\pi$
$257$	−6.47154e9			−1.48346			−0.741729	+	0.670700i	$0.765994\pi$
$258$	1.29572e9	+	3.26952e9i	0.292436	+	0.737914i
$259$	2.02442e9			0.449884
$260$		0			0
$261$	5.01871e9	+	5.33748e9i	1.08151	+	1.15020i
$262$			4.42337e9i			0.938746i
$263$	−1.19014e9			−0.248757			−0.124379	−	0.992235i	$-0.539694\pi$
$263$	−1.19014e9			−0.248757			−0.124379	+	0.992235i	$0.539694\pi$
$264$	−6.91757e8	−	1.74553e9i	−0.142409	−	0.359346i
$265$		0			0
$266$			5.85775e9i			1.17005i
$267$	−9.59984e8	−	2.42236e9i	−0.188894	−	0.476644i
$268$		−	3.27143e9i		−	0.634160i
$269$			9.23705e9i			1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$269$			9.23705e9i			1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$270$		0			0
$271$	−5.47412e9			−1.01493			−0.507466	−	0.861672i	$-0.669418\pi$
$271$	−5.47412e9			−1.01493			−0.507466	+	0.861672i	$0.669418\pi$
$272$	1.19682e9			0.218653
$273$	−2.54203e9	+	1.00741e9i	−0.457647	+	0.181366i
$274$	4.58604e8			0.0813647
$275$		0			0
$276$	−1.65016e9	+	6.53961e8i	−0.284374	+	0.112698i
$277$			7.77657e9i			1.32090i	0.750871	+	0.660449i	$0.229634\pi$
$277$			7.77657e9i			1.32090i	−0.750871	+	0.660449i	$0.770366\pi$
$278$	−1.78825e9			−0.299398
$279$	−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	$0.911652\pi$
$282$	−4.02210e9	+	1.59396e9i	−0.635999	+	0.252047i
$283$		−	9.09925e9i		−	1.41860i	−0.704907	−	0.709300i	$-0.749011\pi$
$283$		−	9.09925e9i		−	1.41860i	0.704907	−	0.709300i	$-0.250989\pi$
$284$		−	2.09312e9i		−	0.321751i
$285$		0			0
$286$	5.43354e8			0.0812117
$287$	1.48611e10			2.19041
$288$	5.16625e9	−	4.85770e9i	0.750940	−	0.706091i
$289$	−3.64205e9			−0.522101
$290$		0			0
$291$	2.69659e9	+	6.80439e9i	0.376047	+	0.948893i
$292$		−	5.14030e9i		−	0.707061i
$293$	−2.26766e9			−0.307686			−0.153843	−	0.988095i	$-0.549165\pi$
$293$	−2.26766e9			−0.307686			−0.153843	+	0.988095i	$0.549165\pi$
$294$	2.73343e9	−	1.08326e9i	0.365863	−	0.144992i
$295$		0			0
$296$		−	2.25407e9i		−	0.293630i
$297$	3.11828e9	+	1.46367e9i	0.400764	+	0.188112i
$298$			1.22673e9i			0.155555i
$299$		−	1.19717e9i		−	0.149786i
$300$		0			0
$301$	−1.74849e10			−2.13009
$302$	7.00445e9			0.842066
$303$	3.71652e9	+	9.37803e9i	0.440927	+	1.11261i
$304$	−4.73992e9			−0.554979
$305$		0			0
$306$	2.20115e9	−	2.06969e9i	0.251052	−	0.236058i
$307$		−	1.05438e10i		−	1.18698i	−0.804842	−	0.593489i	$-0.797750\pi$
$307$		−	1.05438e10i		−	1.18698i	0.804842	−	0.593489i	$-0.202250\pi$
$308$	4.00527e9			0.445070
$309$	−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	$0.874186\pi$
$312$	1.12169e9	+	2.83040e9i	0.118374	+	0.298696i
$313$			2.32153e9i			0.241878i	0.992660	+	0.120939i	$0.0385906\pi$
$313$			2.32153e9i			0.241878i	−0.992660	+	0.120939i	$0.961409\pi$
$314$			2.24026e9i			0.230452i
$315$		0			0
$316$	−5.92276e8			−0.0593985
$317$	1.17812e9			0.116668			0.0583340	−	0.998297i	$-0.481421\pi$
$317$	1.17812e9			0.116668			0.0583340	+	0.998297i	$0.481421\pi$
$318$	−7.51913e9	+	2.97984e9i	−0.735291	+	0.291397i
$319$	7.23798e9			0.698963
$320$		0			0
$321$	9.13374e9	−	3.61971e9i	0.860257	−	0.340921i
$322$			2.91788e9i			0.271421i
$323$	−1.32029e10			−1.21299
$324$	−5.09352e8	+	8.26599e9i	−0.0462209	+	0.750093i
$325$		0			0
$326$			4.00437e9i			0.354539i
$327$	9.77273e9	−	3.87294e9i	0.854722	−	0.338727i
$328$		−	1.65470e10i		−	1.42963i
$329$		−	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	$0.451033\pi$
$332$	−4.47762e9			−0.368548
$333$	2.83279e9	+	3.01272e9i	0.230376	+	0.245009i
$334$	1.86579e9			0.149926
$335$		0			0
$336$	1.98681e9	+	5.01339e9i	0.155883	+	0.393346i
$337$			3.59368e9i			0.278625i	0.990248	+	0.139312i	$0.0444892\pi$
$337$			3.59368e9i			0.278625i	−0.990248	+	0.139312i	$0.955511\pi$
$338$	5.62498e9			0.430977
$339$	−1.16387e10	+	4.61242e9i	−0.881261	+	0.349245i
$340$		0			0
$341$			1.97539e9i			0.146095i
$342$	−8.71745e9	+	8.19682e9i	−0.637214	+	0.599157i
$343$		−	3.89777e9i		−	0.281605i
$344$			1.94684e10i			1.39026i
$345$		0			0
$346$	−5.08614e9			−0.354882
$347$	−2.36195e10			−1.62912			−0.814561	−	0.580078i	$-0.803022\pi$
$347$	−2.36195e10			−1.62912			−0.814561	+	0.580078i	$0.803022\pi$
$348$	6.41108e9	+	1.61773e10i	0.437134	+	1.10304i
$349$	−2.24057e9			−0.151028			−0.0755139	−	0.997145i	$-0.524060\pi$
$349$	−2.24057e9			−0.151028			−0.0755139	+	0.997145i	$0.524060\pi$
$350$		0			0
$351$	−5.05632e9	−	2.37335e9i	−0.333124	−	0.156363i
$352$		−	7.00578e9i		−	0.456337i
$353$	1.21859e10			0.784797			0.392399	−	0.919795i	$-0.371645\pi$
$353$	1.21859e10			0.784797			0.392399	+	0.919795i	$0.371645\pi$
$354$	−1.32332e9	−	3.33917e9i	−0.0842657	−	0.212631i
$355$		0			0
$356$		−	6.18883e9i		−	0.385309i
$357$	5.53418e9	+	1.39646e10i	0.340707	+	0.859717i
$358$			1.25241e10i			0.762454i
$359$		−	1.53929e10i		−	0.926706i	−0.886174	−	0.463353i	$-0.846646\pi$
$359$		−	1.53929e10i		−	0.926706i	0.886174	−	0.463353i	$-0.153354\pi$
$360$		0			0
$361$	3.53052e10			2.07879
$362$	−7.18068e9			−0.418149
$363$	−1.29780e10	+	5.14318e9i	−0.747447	+	0.296214i
$364$	−6.49457e9			−0.369952
$365$		0			0
$366$	7.91002e9	−	3.13475e9i	0.440812	−	0.174694i
$367$			3.67051e9i			0.202331i	0.994870	+	0.101165i	$0.0322571\pi$
$367$			3.67051e9i			0.202331i	−0.994870	+	0.101165i	$0.967743\pi$
$368$	−2.36106e9			−0.128741
$369$	2.07954e10	+	2.21162e10i	1.12166	+	1.19290i
$370$		0			0
$371$		−	4.02112e10i		−	2.12252i
$372$	−4.41511e9	+	1.74971e9i	−0.230553	+	0.0913682i
$373$		−	1.40817e10i		−	0.727478i	−0.931501	−	0.363739i	$-0.881500\pi$
$373$		−	1.40817e10i		−	0.727478i	0.931501	−	0.363739i	$-0.118500\pi$
$374$		−	2.98490e9i		−	0.152561i
$375$		0			0
$376$	−2.39496e10			−1.19825
$377$	−1.17364e10			−0.580993
$378$	1.23238e10	+	5.78459e9i	0.603639	+	0.283338i
$379$	2.08709e10			1.01154			0.505770	−	0.862668i	$-0.331208\pi$
$379$	2.08709e10			1.01154			0.505770	+	0.862668i	$0.331208\pi$
$380$		0			0
$381$	−8.30022e9	−	2.09442e10i	−0.393903	−	0.993950i
$382$		−	1.24789e10i		−	0.586034i
$383$	−9.93413e9			−0.461673			−0.230837	−	0.972993i	$-0.574146\pi$
$383$	−9.93413e9			−0.461673			−0.230837	+	0.972993i	$0.574146\pi$
$384$	1.88453e10	−	7.46843e9i	0.866721	−	0.343482i
$385$		0			0
$386$			7.09590e9i			0.319638i
$387$	−2.44669e10	−	2.60209e10i	−1.09077	−	1.16006i
$388$			1.73844e10i			0.767065i
$389$		−	2.35323e10i		−	1.02770i	−0.857881	−	0.513849i	$-0.828219\pi$
$389$		−	2.35323e10i		−	1.02770i	0.857881	−	0.513849i	$-0.171781\pi$
$390$		0			0
$391$	−6.57664e9			−0.281383
$392$	1.62762e10			0.689302
$393$	−1.65507e10	−	4.17630e10i	−0.693819	−	1.75074i
$394$	−7.87324e9			−0.326714
$395$		0			0
$396$	5.60462e9	+	5.96061e9i	0.227911	+	0.242387i
$397$		−	1.45076e10i		−	0.584027i	−0.956414	−	0.292014i	$-0.905675\pi$
$397$		−	1.45076e10i		−	0.584027i	0.956414	−	0.292014i	$-0.0943252\pi$
$398$	−1.08745e10			−0.433390
$399$	−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	$-0.108379\pi$
$402$	−4.04728e9	−	1.02126e10i	−0.154974	−	0.391051i
$403$		−	3.20311e9i		−	0.121437i
$404$			2.39597e10i			0.899407i
$405$		0			0
$406$	2.86053e10			1.05279
$407$	4.08545e9			0.148889
$408$	1.55487e10	−	6.16198e9i	0.561119	−	0.222372i
$409$	−1.11042e10			−0.396819			−0.198409	−	0.980119i	$-0.563578\pi$
$409$	−1.11042e10			−0.396819			−0.198409	+	0.980119i	$0.563578\pi$
$410$		0			0
$411$	−4.32989e9	+	1.71594e9i	−0.151743	+	0.0601360i
$412$		−	1.77948e9i		−	0.0617597i
$413$	1.78574e10			0.613787
$414$	−4.34236e9	+	4.08302e9i	−0.147817	+	0.138989i
$415$		0			0
$416$			1.13599e10i			0.379317i
$417$	1.68837e10	−	6.69102e9i	0.558371	−	0.221283i
$418$			1.18214e10i			0.387227i
$419$			2.66208e10i			0.863705i	0.901944	+	0.431852i	$0.142140\pi$
$419$			2.66208e10i			0.863705i	−0.901944	+	0.431852i	$0.857860\pi$
$420$		0			0
$421$	−2.03614e10			−0.648155			−0.324078	−	0.946031i	$-0.605054\pi$
$421$	−2.03614e10			−0.648155			−0.324078	+	0.946031i	$0.605054\pi$
$422$	7.55908e9			0.238352
$423$	3.20104e10	−	3.00986e10i	0.999838	−	0.940124i
$424$	−4.47727e10			−1.38532
$425$		0			0
$426$	−2.58952e9	−	6.53422e9i	−0.0786286	−	0.198406i
$427$			4.23016e10i			1.27246i
$428$	2.33356e10			0.695414
$429$	−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	$0.941308\pi$
$432$	−4.68072e9	+	9.97205e9i	−0.134393	+	0.286319i
$433$			3.63896e10i			1.03520i	0.855622	+	0.517601i	$0.173175\pi$
$433$			3.63896e10i			1.03520i	−0.855622	+	0.517601i	$0.826825\pi$
$434$			7.80696e9i			0.220051i
$435$		0			0
$436$	2.49681e10			0.690939
$437$	2.60462e10			0.714198
$438$	−6.35936e9	−	1.60468e10i	−0.172789	−	0.436005i
$439$	5.90206e10			1.58908			0.794539	−	0.607213i	$-0.207713\pi$
$439$	5.90206e10			1.58908			0.794539	+	0.607213i	$0.207713\pi$
$440$		0			0
$441$	−2.17543e10	+	2.04551e10i	−0.575164	+	0.540813i
$442$			4.84005e9i			0.126812i
$443$	−1.59909e10			−0.415200			−0.207600	−	0.978214i	$-0.566565\pi$
$443$	−1.59909e10			−0.415200			−0.207600	+	0.978214i	$0.566565\pi$
$444$	3.61872e9	+	9.13123e9i	0.0931156	+	0.234962i
$445$		0			0
$446$		−	8.12874e9i		−	0.205439i
$447$	−4.58999e9	−	1.15821e10i	−0.114969	−	0.290106i
$448$		−	1.06440e10i		−	0.264236i
$449$		−	1.61660e9i		−	0.0397756i	−0.999802	−	0.0198878i	$-0.993669\pi$
$449$		−	1.61660e9i		−	0.0397756i	0.999802	−	0.0198878i	$-0.00633090\pi$
$450$		0			0
$451$	2.99911e10			0.724913
$452$	−2.97354e10			−0.712393
$453$	−6.61321e10	+	2.62082e10i	−1.57043	+	0.622364i
$454$	1.14650e9			0.0269867
$455$		0			0
$456$	−6.15795e10	+	2.44040e10i	−1.42422	+	0.564419i
$457$			5.01622e10i			1.15004i	0.818140	+	0.575019i	$0.195005\pi$
$457$			5.01622e10i			1.15004i	−0.818140	+	0.575019i	$0.804995\pi$
$458$	4.89811e8			0.0111318
$459$	−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	$0.706689\pi$
$462$	1.25035e10	−	4.95514e9i	0.274450	−	0.108765i
$463$			6.19020e10i			1.34704i	0.739168	+	0.673521i	$0.235219\pi$
$463$			6.19020e10i			1.34704i	−0.739168	+	0.673521i	$0.764781\pi$
$464$			2.31466e10i			0.499361i
$465$		0			0
$466$	8.91112e9			0.188968
$467$	4.98272e9			0.104761			0.0523804	−	0.998627i	$-0.483319\pi$
$467$	4.98272e9			0.104761			0.0523804	+	0.998627i	$0.483319\pi$
$468$	−9.08794e9	−	9.66518e9i	−0.189445	−	0.201477i
$469$	5.46156e10			1.12882
$470$		0			0
$471$	−8.38229e9	−	2.11513e10i	−0.170325	−	0.429788i
$472$		−	1.98831e10i		−	0.400605i
$473$	−3.52861e10			−0.704951
$474$	−1.84895e9	+	7.32739e8i	−0.0366278	+	0.0145156i
$475$		0			0
$476$			3.56778e10i			0.694977i
$477$	5.98419e10	−	5.62680e10i	1.15593	−	1.08690i
$478$		−	1.44955e10i		−	0.277665i
$479$		−	1.00172e9i		−	0.0190284i	−0.999955	−	0.00951420i	$-0.996971\pi$
$479$		−	1.00172e9i		−	0.0190284i	0.999955	−	0.00951420i	$-0.00302851\pi$
$480$		0			0
$481$	−6.62460e9			−0.123760
$482$	1.85900e9			0.0344422
$483$	−1.09177e10	−	2.75490e10i	−0.200605	−	0.506194i
$484$	−3.31571e10			−0.604220
$485$		0			0
$486$	8.63626e9	+	2.64346e10i	0.154803	+	0.473836i
$487$			8.29709e10i			1.47506i	0.675314	+	0.737530i	$0.264008\pi$
$487$			8.29709e10i			1.47506i	−0.675314	+	0.737530i	$0.735992\pi$
$488$	4.71003e10			0.830509
$489$	−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	$-0.0473183\pi$
$492$	2.65648e10	+	6.70319e10i	0.453363	+	1.14399i
$493$			6.44739e10i			1.09143i
$494$		−	1.91686e10i		−	0.321871i
$495$		0			0
$496$	−6.31717e9			−0.104375
$497$	3.49440e10			0.572726
$498$	−1.39781e10	+	5.53952e9i	−0.227263	+	0.0900647i
$499$	−7.02839e10			−1.13358			−0.566792	−	0.823861i	$-0.691816\pi$
$499$	−7.02839e10			−1.13358			−0.566792	+	0.823861i	$0.691816\pi$
$500$		0			0
$501$	−1.76158e10	+	6.98114e9i	−0.279609	+	0.110809i
$502$		−	3.26951e10i		−	0.514834i
$503$	2.10506e10			0.328846			0.164423	−	0.986390i	$-0.447424\pi$
$503$	2.10506e10			0.328846			0.164423	+	0.986390i	$0.447424\pi$
$504$	5.16240e10	+	5.49030e10i	0.800073	+	0.850891i
$505$		0			0
$506$			5.88853e9i			0.0898266i
$507$	−5.31080e10	+	2.10467e10i	−0.803763	+	0.318532i
$508$		−	5.35099e10i		−	0.803488i
$509$			3.73005e10i			0.555704i	0.960624	+	0.277852i	$0.0896224\pi$
$509$			3.73005e10i			0.555704i	−0.960624	+	0.277852i	$0.910378\pi$
$510$		0			0
$511$	8.58158e10			1.25859
$512$	4.13811e10			0.602175
$513$	5.16357e10	−	1.10007e11i	0.745557	−	1.58837i
$514$	−5.16152e10			−0.739477
$515$		0			0
$516$	−3.12549e10	−	7.88665e10i	−0.440879	−	1.11248i
$517$		−	4.34082e10i		−	0.607589i
$518$	1.61462e10			0.224259
$519$	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	$0.629702\pi$
$522$	4.00278e10	+	4.25702e10i	0.539113	+	0.573355i
$523$			4.33058e9i			0.0578815i	0.999581	+	0.0289407i	$0.00921341\pi$
$523$			4.33058e9i			0.0578815i	−0.999581	+	0.0289407i	$0.990787\pi$
$524$		−	1.06699e11i		−	1.41526i
$525$		0			0
$526$	−9.49223e9			−0.124001
$527$	−1.75962e10			−0.228127
$528$	4.00956e9	+	1.01175e10i	0.0515894	+	0.130177i
$529$	−6.53368e10			−0.834325
$530$		0			0
$531$	2.49880e10	+	2.65752e10i	0.314307	+	0.334271i
$532$		−	1.41299e11i		−	1.76397i
$533$	−4.86308e10			−0.602564
$534$	−7.65656e9	−	1.93201e10i	−0.0941605	−	0.237598i
$535$		0			0
$536$		−	6.08112e10i		−	0.736757i
$537$	−4.68607e10	−	1.18245e11i	−0.563523	−	1.42196i
$538$			7.36721e10i			0.879374i
$539$			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	$0.758809\pi$
$542$	−4.36600e10			−0.505926
$543$	6.77960e10	−	2.68676e10i	0.779839	−	0.309051i
$544$	6.24055e10			0.712570
$545$		0			0
$546$	−2.02745e10	+	8.03481e9i	−0.228129	+	0.0904077i
$547$			3.16725e10i			0.353780i	0.984231	+	0.176890i	$0.0566036\pi$
$547$			3.16725e10i			0.353780i	−0.984231	+	0.176890i	$0.943396\pi$
$548$	−1.10623e10			−0.122666
$549$	−6.29529e10	+	5.91931e10i	−0.692989	+	0.651601i
$550$		0			0
$551$		−	2.55343e11i		−	2.77025i
$552$	−3.06741e10	+	1.21562e10i	−0.330382	+	0.130931i
$553$		−	9.88787e9i		−	0.105731i
$554$			6.20237e10i			0.658444i
$555$		0			0
$556$	4.31357e10			0.451375
$557$	3.02144e10			0.313902			0.156951	−	0.987606i	$-0.449834\pi$
$557$	3.02144e10			0.313902			0.156951	+	0.987606i	$0.449834\pi$
$558$	−1.16183e10	+	1.09244e10i	−0.119841	+	0.112683i
$559$	5.72167e10			0.585971
$560$		0			0
$561$	1.11685e10	+	2.81818e10i	0.112757	+	0.284523i
$562$			2.72508e10i			0.273171i
$563$	−6.11519e10			−0.608662			−0.304331	−	0.952566i	$-0.598433\pi$
$563$	−6.11519e10			−0.608662			−0.304331	+	0.952566i	$0.598433\pi$
$564$	9.70200e10	−	3.84491e10i	0.958837	−	0.379988i
$565$		0			0
$566$		−	7.25730e10i		−	0.707147i
$567$	−1.37998e11	−	8.50348e9i	−1.33519	−	0.0822744i
$568$		−	3.89081e10i		−	0.373806i
$569$		−	3.99163e10i		−	0.380804i	−0.981706	−	0.190402i	$-0.939021\pi$
$569$		−	3.99163e10i		−	0.380804i	0.981706	−	0.190402i	$-0.0609791\pi$
$570$		0			0
$571$	9.07730e10			0.853911			0.426955	−	0.904273i	$-0.359586\pi$
$571$	9.07730e10			0.853911			0.426955	+	0.904273i	$0.359586\pi$
$572$	−1.31066e10			−0.122435
$573$	4.66917e10	+	1.17819e11i	0.433133	+	1.09294i
$574$	1.18528e11			1.09188
$575$		0			0
$576$	1.58403e10	−	1.48942e10i	0.143904	−	0.135309i
$577$			1.33041e11i			1.20028i	0.799894	+	0.600141i	$0.204889\pi$
$577$			1.33041e11i			1.20028i	−0.799894	+	0.600141i	$0.795111\pi$
$578$	−2.90480e10			−0.260258
$579$	−2.65504e10	−	6.69956e10i	−0.236242	−	0.596117i
$580$		0			0
$581$		−	7.47525e10i		−	0.656026i
$582$	2.15072e10	+	5.42699e10i	0.187453	+	0.473006i
$583$		−	8.11497e10i		−	0.702445i
$584$		−	9.55508e10i		−	0.821453i
$585$		0			0
$586$	−1.80862e10			−0.153376
$587$	−1.98669e11			−1.67331			−0.836657	−	0.547728i	$-0.815493\pi$
$587$	−1.98669e11			−1.67331			−0.836657	+	0.547728i	$0.815493\pi$
$588$	−6.59350e10	+	2.61301e10i	−0.551578	+	0.218591i
$589$	6.96883e10			0.579027
$590$		0			0
$591$	7.43347e10	−	2.94589e10i	0.609315	−	0.241472i
$592$			1.30650e10i			0.106371i
$593$	1.33920e11			1.08300			0.541498	−	0.840702i	$-0.317857\pi$
$593$	1.33920e11			1.08300			0.541498	+	0.840702i	$0.317857\pi$
$594$	2.48705e10	+	1.16738e10i	0.199774	+	0.0937706i
$595$		0			0
$596$		−	2.95908e10i		−	0.234515i
$597$	1.02671e11	−	4.06887e10i	0.808261	−	0.320315i
$598$		−	9.54831e9i		−	0.0746658i
$599$			1.14673e10i			0.0890745i	0.999008	+	0.0445372i	$0.0141813\pi$
$599$			1.14673e10i			0.0890745i	−0.999008	+	0.0445372i	$0.985819\pi$
$600$		0			0
$601$	4.71067e10			0.361065			0.180532	−	0.983569i	$-0.442218\pi$
$601$	4.71067e10			0.361065			0.180532	+	0.983569i	$0.442218\pi$
$602$	−1.39455e11			−1.06181
$603$	7.64243e10	+	8.12785e10i	0.578046	+	0.614761i
$604$	−1.68959e11			−1.26950
$605$		0			0
$606$	2.96419e10	+	7.47965e10i	0.219794	+	0.554614i
$607$		−	1.36835e11i		−	1.00796i	−0.863716	−	0.503979i	$-0.831869\pi$
$607$		−	1.36835e11i		−	1.00796i	0.863716	−	0.503979i	$-0.168131\pi$
$608$	−2.47152e11			−1.80863
$609$	−2.70075e11	+	1.07031e11i	−1.96343	+	0.778110i
$610$		0			0
$611$			7.03868e10i			0.505041i
$612$	−5.30955e10	+	4.99244e10i	−0.378488	+	0.355883i
$613$		−	8.50169e10i		−	0.602093i	−0.953610	−	0.301046i	$-0.902664\pi$
$613$		−	8.50169e10i		−	0.602093i	0.953610	−	0.301046i	$-0.0973359\pi$
$614$		−	8.40942e10i		−	0.591688i
$615$		0			0
$616$	7.44521e10			0.517076
$617$	1.76659e11			1.21898			0.609489	−	0.792795i	$-0.291375\pi$
$617$	1.76659e11			1.21898			0.609489	+	0.792795i	$0.291375\pi$
$618$	−2.20150e9	−	5.55512e9i	−0.0150926	−	0.0380838i
$619$	1.22476e11			0.834234			0.417117	−	0.908853i	$-0.363040\pi$
$619$	1.22476e11			0.834234			0.417117	+	0.908853i	$0.363040\pi$
$620$		0			0
$621$	2.57209e10	−	5.47972e10i	0.172950	−	0.368461i
$622$			5.74582e10i			0.383876i
$623$	1.03321e11			0.685860
$624$	−6.50154e9	−	1.64056e10i	−0.0428823	−	0.108206i
$625$		0			0
$626$			1.85159e10i			0.120572i
$627$	−4.42318e10	−	1.11612e11i	−0.286196	−	0.722169i
$628$		−	5.40390e10i		−	0.347431i
$629$			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	$0.809657\pi$
$632$	−1.10096e10			−0.0690083
$633$	−7.13686e10	+	2.82835e10i	−0.444521	+	0.176164i
$634$	9.39633e9			0.0581569
$635$		0			0
$636$	1.81374e11	−	7.18788e10i	1.10853	−	0.439312i
$637$		−	4.78351e10i		−	0.290529i
$638$	5.77280e10			0.348421
$639$	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	$0.759397\pi$
$642$	7.28481e10	−	2.88698e10i	0.428823	−	0.169943i
$643$			2.66842e11i			1.56103i	0.625138	+	0.780515i	$0.285043\pi$
$643$			2.66842e11i			1.56103i	−0.625138	+	0.780515i	$0.714957\pi$
$644$		−	7.03842e10i		−	0.409196i
$645$		0			0
$646$	−1.05302e11			−0.604655
$647$	1.98677e11			1.13378			0.566892	−	0.823792i	$-0.308146\pi$
$647$	1.98677e11			1.13378			0.566892	+	0.823792i	$0.308146\pi$
$648$	−9.46812e9	+	1.53653e11i	−0.0536987	+	0.871447i
$649$	3.60378e10			0.203132
$650$		0			0
$651$	−2.92109e10	−	7.37090e10i	−0.162638	−	0.410390i
$652$		−	9.65923e10i		−	0.534505i
$653$	2.49774e11			1.37371			0.686855	−	0.726795i	$-0.258991\pi$
$653$	2.49774e11			1.37371			0.686855	+	0.726795i	$0.258991\pi$
$654$	7.79445e10	−	3.08895e10i	0.426064	−	0.168850i
$655$		0			0
$656$			9.59096e10i			0.517901i
$657$	1.20083e11	+	1.27710e11i	0.644496	+	0.685433i
$658$		−	1.71554e11i		−	0.915162i
$659$			1.55774e11i			0.825947i	0.910743	+	0.412974i	$0.135510\pi$
$659$			1.55774e11i			0.825947i	−0.910743	+	0.412974i	$0.864490\pi$
$660$		0			0
$661$	−7.07460e10			−0.370592			−0.185296	−	0.982683i	$-0.559324\pi$
$661$	−7.07460e10			−0.370592			−0.185296	+	0.982683i	$0.559324\pi$
$662$	2.93395e10			0.152764
$663$	−1.81098e10	−	4.56970e10i	−0.0937258	−	0.236502i
$664$	−8.32324e10			−0.428174
$665$		0			0
$666$	2.25935e10	+	2.40286e10i	0.114838	+	0.122133i
$667$		−	1.27192e11i		−	0.642625i
$668$	−4.50061e10			−0.226030
$669$	3.04149e10	+	7.67471e10i	0.151839	+	0.383140i
$670$		0			0
$671$			8.53683e10i			0.421121i
$672$	1.03598e11	+	2.61411e11i	0.508010	+	1.28188i
$673$		−	3.05166e11i		−	1.48757i	−0.668421	−	0.743783i	$-0.733029\pi$
$673$		−	3.05166e11i		−	1.48757i	0.668421	−	0.743783i	$-0.266971\pi$
$674$			2.86622e10i			0.138890i
$675$		0			0
$676$	−1.35684e11			−0.649745
$677$	6.05189e10			0.288095			0.144048	−	0.989571i	$-0.453988\pi$
$677$	6.05189e10			0.288095			0.144048	+	0.989571i	$0.453988\pi$
$678$	−9.28268e10	+	3.67873e10i	−0.439293	+	0.174092i
$679$	−2.90227e11			−1.36540
$680$		0			0
$681$	−1.08246e10	+	4.28980e9i	−0.0503296	+	0.0199457i
$682$			1.57551e10i			0.0728257i
$683$	−1.55016e11			−0.712352			−0.356176	−	0.934419i	$-0.615920\pi$
$683$	−1.55016e11			−0.712352			−0.356176	+	0.934419i	$0.615920\pi$
$684$	2.10280e11	−	1.97721e11i	0.960668	−	0.903294i
$685$		0			0
$686$		−	3.10875e10i		−	0.140375i
$687$	−4.62452e9	+	1.83270e9i	−0.0207606	+	0.00822744i
$688$		−	1.12843e11i		−	0.503639i
$689$			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	$0.698245\pi$
$692$	1.22686e11			0.535023
$693$	−9.95106e10	+	9.35674e10i	−0.431456	+	0.405688i
$694$	−1.88383e11			−0.812088
$695$		0			0
$696$	1.19173e11	+	3.00713e11i	0.507856	+	1.28149i
$697$			2.67152e11i			1.13195i
$698$	−1.78702e10			−0.0752847
$699$	−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	$0.600771\pi$
$702$	−4.03277e10	−	1.89292e10i	−0.166056	−	0.0779441i
$703$		−	1.44128e11i		−	0.590101i
$704$		−	2.14805e10i		−	0.0874486i
$705$		0			0
$706$	9.71910e10			0.391208
$707$	−4.00000e11			−1.60097
$708$	3.19207e10	+	8.05466e10i	0.127040	+	0.320563i
$709$	3.22572e11			1.27656			0.638280	−	0.769804i	$-0.279646\pi$
$709$	3.22572e11			1.27656			0.638280	+	0.769804i	$0.279646\pi$
$710$		0			0
$711$	1.47151e10	−	1.38362e10i	0.0575816	−	0.0541426i
$712$		−	1.15041e11i		−	0.447646i
$713$	3.47133e10			0.134319
$714$	4.41391e10	+	1.11378e11i	0.169836	+	0.428554i
$715$		0			0
$716$		−	3.02102e11i		−	1.14948i
$717$	5.42371e10	+	1.36858e11i	0.205220	+	0.517839i
$718$		−	1.22769e11i		−	0.461946i
$719$			1.31701e11i			0.492805i	0.969168	+	0.246402i	$0.0792485\pi$
$719$			1.31701e11i			0.492805i	−0.969168	+	0.246402i	$0.920752\pi$
$720$		0			0
$721$	2.97079e10			0.109934
$722$	2.81584e11			1.03624
$723$	−1.75516e10	+	6.95574e9i	−0.0642340	+	0.0254560i
$724$	1.73210e11			0.630405
$725$		0			0
$726$	−1.03509e11	+	4.10205e10i	−0.372589	+	0.147657i
$727$			1.82998e11i			0.655102i	0.944834	+	0.327551i	$0.106223\pi$
$727$			1.82998e11i			0.655102i	−0.944834	+	0.327551i	$0.893777\pi$
$728$	−1.20725e11			−0.429805
$729$	−1.80448e11	−	2.17267e11i	−0.638913	−	0.769279i
$730$		0			0
$731$		−	3.14319e11i		−	1.10078i
$732$	−1.90803e11	+	7.56155e10i	−0.664571	+	0.263370i
$733$		−	1.52831e10i		−	0.0529413i	−0.999650	−	0.0264707i	$-0.991573\pi$
$733$		−	1.52831e10i		−	0.0529413i	0.999650	−	0.0264707i	$-0.00842686\pi$
$734$			2.92750e10i			0.100858i
$735$		0			0
$736$	−1.23112e11			−0.419555
$737$	1.10219e11			0.373583
$738$	1.65858e11	+	1.76393e11i	0.559128	+	0.594642i
$739$	2.80725e11			0.941245			0.470623	−	0.882335i	$-0.344029\pi$
$739$	2.80725e11			0.941245			0.470623	+	0.882335i	$0.344029\pi$
$740$		0			0
$741$	7.17222e10	+	1.80979e11i	0.237893	+	0.600283i
$742$		−	3.20713e11i		−	1.05804i
$743$	2.05247e11			0.673475			0.336737	−	0.941599i	$-0.390677\pi$
$743$	2.05247e11			0.673475			0.336737	+	0.941599i	$0.390677\pi$
$744$	−8.20706e10	+	3.25247e10i	−0.267853	+	0.106150i
$745$		0			0
$746$		−	1.12312e11i		−	0.362635i
$747$	1.11246e11	−	1.04602e11i	0.357275	−	0.335937i
$748$			7.20010e10i			0.230002i
$749$			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	$0.405664\pi$
$752$	1.38817e11			0.434081
$753$	1.22334e11	+	3.08689e11i	0.380510	+	0.960154i
$754$	−9.36066e10			−0.289615
$755$		0			0
$756$	−2.97271e11	−	1.39534e11i	−0.910050	−	0.427163i
$757$		−	1.02442e11i		−	0.311956i	−0.987761	−	0.155978i	$-0.950147\pi$
$757$		−	1.02442e11i		−	0.311956i	0.987761	−	0.155978i	$-0.0498529\pi$
$758$	1.66460e11			0.504235
$759$	−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	$0.900118\pi$
$762$	−6.62002e10	−	1.67045e11i	−0.196354	−	0.495466i
$763$			4.16835e11i			1.22989i
$764$			3.01013e11i			0.883509i
$765$		0			0
$766$	−7.92318e10			−0.230136
$767$	−5.84356e10			−0.168848
$768$	2.14190e11	−	8.48835e10i	0.615678	−	0.243994i
$769$	−4.39556e11			−1.25692			−0.628462	−	0.777841i	$-0.716315\pi$
$769$	−4.39556e11			−1.25692			−0.628462	+	0.777841i	$0.716315\pi$
$770$		0			0
$771$	4.87322e11	−	1.93126e11i	1.37911	−	0.546542i
$772$		−	1.71165e11i		−	0.481888i
$773$	−4.81386e11			−1.34827			−0.674133	−	0.738610i	$-0.735482\pi$
$773$	−4.81386e11			−1.34827			−0.674133	+	0.738610i	$0.735482\pi$
$774$	−1.95141e11	−	2.07535e11i	−0.543731	−	0.578267i
$775$		0			0
$776$			3.23150e11i			0.891164i
$777$	−1.52443e11	+	6.04134e10i	−0.418238	+	0.165748i
$778$		−	1.87687e11i		−	0.512289i
$779$		−	1.05803e12i		−	2.87309i
$780$		0			0
$781$	7.05201e10			0.189543
$782$	−5.24534e10			−0.140264
$783$	−5.37203e11	−	2.52154e11i	−1.42919	−	0.670840i
$784$	−9.43402e10			−0.249708
$785$		0			0
$786$	−1.32004e11	−	3.33090e11i	−0.345857	−	0.872712i
$787$			1.76591e10i			0.0460331i	0.999735	+	0.0230166i	$0.00732705\pi$
$787$			1.76591e10i			0.0460331i	−0.999735	+	0.0230166i	$0.992673\pi$
$788$	1.89916e11			0.492557
$789$	8.96203e10	−	3.55166e10i	0.231259	−	0.0916481i
$790$		0			0
$791$		−	4.96423e11i		−	1.26808i
$792$	1.04182e11	+	1.10799e11i	0.264784	+	0.281602i
$793$		−	1.38426e11i		−	0.350045i
$794$		−	1.15708e11i		−	0.291127i
$795$		0			0
$796$	2.62312e11			0.653381
$797$	−2.22982e11			−0.552633			−0.276317	−	0.961067i	$-0.589114\pi$
$797$	−2.22982e11			−0.552633			−0.276317	+	0.961067i	$0.589114\pi$
$798$	−1.74809e11	−	4.41102e11i	−0.431074	−	1.08775i
$799$	3.86669e11			0.948750
$800$		0			0
$801$	1.44578e11	+	1.53761e11i	0.351214	+	0.373522i
$802$		−	1.37737e11i		−	0.332930i
$803$	1.73184e11			0.416529
$804$	9.76273e10	+	2.46346e11i	0.233640	+	0.589552i
$805$		0			0
$806$		−	2.55471e10i		−	0.0605343i
$807$	−2.75655e11	−	6.95571e11i	−0.649938	−	1.64001i
$808$			4.45377e11i			1.04492i
$809$		−	1.32706e11i		−	0.309810i	−0.987929	−	0.154905i	$-0.950493\pi$
$809$		−	1.32706e11i		−	0.309810i	0.987929	−	0.154905i	$-0.0495072\pi$
$810$		0			0
$811$	7.48071e11			1.72926			0.864628	−	0.502412i	$-0.167554\pi$
$811$	7.48071e11			1.72926			0.864628	+	0.502412i	$0.167554\pi$
$812$	−6.90009e11			−1.58720
$813$	4.12214e11	−	1.63361e11i	0.943540	−	0.373926i
$814$	3.25844e10			0.0742185
$815$		0			0
$816$	−9.01236e10	+	3.57161e10i	−0.203272	+	0.0805569i
$817$			1.24483e12i			2.79398i
$818$	−8.85636e10			−0.197807
$819$	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	$-0.179887\pi$
$822$	−3.45339e10	+	1.36858e10i	−0.0756413	+	0.0299767i
$823$			2.67253e11i			0.582537i	0.956641	+	0.291269i	$0.0940774\pi$
$823$			2.67253e11i			0.582537i	−0.956641	+	0.291269i	$0.905923\pi$
$824$		−	3.30780e10i		−	0.0717515i
$825$		0			0
$826$	1.42425e11			0.305962
$827$	−9.99227e10			−0.213620			−0.106810	−	0.994279i	$-0.534064\pi$
$827$	−9.99227e10			−0.213620			−0.106810	+	0.994279i	$0.534064\pi$
$828$	1.04745e11	−	9.84894e10i	0.222850	−	0.209541i
$829$	1.37202e11			0.290497			0.145249	−	0.989395i	$-0.453602\pi$
$829$	1.37202e11			0.290497			0.145249	+	0.989395i	$0.453602\pi$
$830$		0			0
$831$	−2.32071e11	−	5.85594e11i	−0.486651	−	1.22798i
$832$			3.48308e10i			0.0726892i
$833$	−2.62781e11			−0.545775
$834$	1.34659e11	−	5.33656e10i	0.278338	−	0.110306i
$835$		0			0
$836$		−	2.85154e11i		−	0.583786i
$837$	6.88179e10	−	1.46613e11i	0.140217	−	0.298725i
$838$			2.12320e11i			0.430542i
$839$		−	2.29368e11i		−	0.462899i	−0.972847	−	0.231449i	$-0.925653\pi$
$839$		−	2.29368e11i		−	0.462899i	0.972847	−	0.231449i	$-0.0743467\pi$
$840$		0			0
$841$	−7.46680e11			−1.49262
$842$	−1.62397e11			−0.323094
$843$	−1.01963e11	−	2.57287e11i	−0.201899	−	0.509458i
$844$	−1.82338e11			−0.359341
$845$		0			0
$846$	2.55306e11	−	2.40058e11i	0.498401	−	0.468635i
$847$		−	5.53548e11i		−	1.07553i
$848$	2.59511e11			0.501849
$849$	2.71543e11	+	6.85194e11i	0.522646	+	1.31881i
$850$		0			0
$851$		−	7.17933e10i		−	0.136888i
$852$	6.24636e10	+	1.57617e11i	0.118541	+	0.299119i
$853$			7.54864e11i			1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$853$			7.54864e11i			1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$854$			3.37385e11i			0.634300i
$855$		0			0
$856$	4.33774e11			0.807921
$857$	9.33249e9			0.0173011			0.00865056	−	0.999963i	$-0.497246\pi$
$857$	9.33249e9			0.0173011			0.00865056	+	0.999963i	$0.497246\pi$
$858$	−4.09158e10	+	1.62150e10i	−0.0754991	+	0.0299204i
$859$	−3.51970e11			−0.646447			−0.323224	−	0.946323i	$-0.604767\pi$
$859$	−3.51970e11			−0.646447			−0.323224	+	0.946323i	$0.604767\pi$
$860$		0			0
$861$	−1.11908e12	+	4.43492e11i	−2.03633	+	0.806999i
$862$			1.00919e11i			0.182787i
$863$	4.14848e11			0.747905			0.373952	−	0.927448i	$-0.378002\pi$
$863$	4.14848e11			0.747905			0.373952	+	0.927448i	$0.378002\pi$
$864$	−2.44065e11	+	5.19969e11i	−0.437976	+	0.933088i
$865$		0			0
$866$			2.90233e11i			0.516030i
$867$	2.74255e11	−	1.08687e11i	0.485375	−	0.192355i
$868$		−	1.88317e11i		−	0.331750i
$869$		−	1.99546e10i		−	0.0349916i
$870$		0			0
$871$	−1.78721e11			−0.310530
$872$	4.64121e11			0.802723
$873$	−4.06118e11	−	4.31913e11i	−0.699190	−	0.743601i
$874$	2.07737e11			0.356015
$875$		0			0
$876$	1.53399e11	+	3.87076e11i	0.260498	+	0.657325i
$877$		−	6.49541e11i		−	1.09801i	−0.835818	−	0.549007i	$-0.815006\pi$
$877$		−	6.49541e11i		−	1.09801i	0.835818	−	0.549007i	$-0.184994\pi$
$878$	4.70731e11			0.792127
$879$	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	$0.935254\pi$
$882$	−1.73506e11	+	1.63144e11i	−0.286709	+	0.269586i
$883$		−	5.16877e11i		−	0.850245i	−0.905136	−	0.425122i	$-0.860231\pi$
$883$		−	5.16877e11i		−	0.850245i	0.905136	−	0.425122i	$-0.139769\pi$
$884$		−	1.16750e11i		−	0.191183i
$885$		0			0
$886$	−1.27539e11			−0.206970
$887$	−6.37860e11			−1.03046			−0.515229	−	0.857052i	$-0.672293\pi$
$887$	−6.37860e11			−1.03046			−0.515229	+	0.857052i	$0.672293\pi$
$888$	6.72667e10	+	1.69736e11i	0.108180	+	0.272975i
$889$	8.93332e11			1.43023
$890$		0			0
$891$	−2.78493e11	−	1.71608e10i	−0.441879	−	0.0272286i
$892$			1.96079e11i			0.309722i
$893$	−1.53137e12			−2.40810
$894$	−3.66085e10	−	9.23754e10i	−0.0573101	−	0.144613i
$895$		0			0
$896$			8.03809e11i			1.24716i
$897$	3.57265e10	+	9.01498e10i	0.0551849	+	0.139250i
$898$		−	1.28935e10i		−	0.0198274i
$899$		−	3.40311e11i		−	0.521000i
$900$		0			0
$901$	7.22859e11			1.09687
$902$	2.39200e11			0.361356
$903$	1.31665e12	−	5.21791e11i	1.98025	−	0.784776i
$904$	−5.52738e11			−0.827647
$905$		0			0
$906$	−5.27451e11	+	2.09029e11i	−0.782833	+	0.310238i
$907$			5.28792e11i			0.781368i	0.920525	+	0.390684i	$0.127761\pi$
$907$			5.28792e11i			0.781368i	−0.920525	+	0.390684i	$0.872239\pi$
$908$	−2.76555e10			−0.0406853
$909$	−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	$-0.295908\pi$
$912$	3.56927e11	−	1.41450e11i	0.515940	−	0.204468i
$913$		−	1.50857e11i		−	0.217111i
$914$			4.00080e11i			0.573273i
$915$		0			0
$916$	−1.18151e10			−0.0167824
$917$	1.78131e12			2.51920
$918$	−1.03987e11	+	2.21540e11i	−0.146423	+	0.311947i
$919$	−8.75563e11			−1.22751			−0.613756	−	0.789496i	$-0.710342\pi$
$919$	−8.75563e11			−1.22751			−0.613756	+	0.789496i	$0.710342\pi$
$920$		0			0
$921$	3.14651e11	+	7.93971e11i	0.437312	+	1.10348i
$922$			5.73829e11i			0.794070i
$923$	−1.14349e11			−0.157553
$924$	−3.01606e11	+	1.19527e11i	−0.413763	+	0.163975i
$925$		0			0
$926$			4.93713e11i			0.671476i
$927$	4.15707e10	+	4.42111e10i	0.0562948	+	0.0598705i
$928$			1.20692e12i			1.62738i
$929$			2.94492e11i			0.395377i	0.980265	+	0.197688i	$0.0633434\pi$
$929$			2.94492e11i			0.395377i	−0.980265	+	0.197688i	$0.936657\pi$
$930$		0			0
$931$	1.04072e12			1.38527
$932$	−2.14951e11			−0.284890
$933$	−2.14989e11	−	5.42489e11i	−0.283720	−	0.715919i
$934$	3.97407e10			0.0522214
$935$		0			0
$936$	−1.68932e11	−	1.79662e11i	−0.220094	−	0.234073i
$937$		−	6.27494e11i		−	0.814050i	−0.913417	−	0.407025i	$-0.866566\pi$
$937$		−	6.27494e11i		−	0.814050i	0.913417	−	0.407025i	$-0.133434\pi$
$938$	4.35599e11			0.562698
$939$	−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	$0.898995\pi$
$942$	−6.68548e10	−	1.68697e11i	−0.0849042	−	0.214242i
$943$		−	5.27031e11i		−	0.666483i
$944$			1.15246e11i			0.145124i
$945$		0			0
$946$	−2.81432e11			−0.351406
$947$	1.12362e12			1.39708			0.698540	−	0.715571i	$-0.253834\pi$
$947$	1.12362e12			1.39708			0.698540	+	0.715571i	$0.253834\pi$
$948$	4.45997e10	−	1.76749e10i	0.0552203	−	0.0218839i
$949$	−2.80819e11			−0.346228
$950$		0			0
$951$	−8.87149e10	+	3.51578e10i	−0.108461	+	0.0429833i
$952$			6.63199e11i			0.807414i
$953$	2.38597e11			0.289263			0.144632	−	0.989486i	$-0.453800\pi$
$953$	2.38597e11			0.289263			0.144632	+	0.989486i	$0.453800\pi$
$954$	4.77282e11	−	4.48777e11i	0.576211	−	0.541798i
$955$		0			0
$956$			3.49656e11i			0.418610i
$957$	−5.45036e11	+	2.15998e11i	−0.649797	+	0.257515i
$958$		−	7.98940e9i		−	0.00948532i
$959$		−	1.84682e11i		−	0.218349i
$960$		0			0
$961$	−7.60013e11			−0.891102
$962$	−5.28359e10			−0.0616920
$963$	−5.79770e11	+	5.45145e11i	−0.674141	+	0.633879i
$964$	−4.48423e10			−0.0519254
$965$		0			0
$966$	−8.70763e10	−	2.19723e11i	−0.0999981	−	0.252329i
$967$		−	7.79476e11i		−	0.891450i	−0.895170	−	0.445725i	$-0.852946\pi$
$967$		−	7.79476e11i		−	0.891450i	0.895170	−	0.445725i	$-0.147054\pi$
$968$	−6.16343e11			−0.701974
$969$	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	$0.837946\pi$
$972$	−2.08321e11	−	6.37648e11i	−0.233383	−	0.714358i
$973$			7.20137e11i			0.803460i
$974$			6.61752e11i			0.735291i
$975$		0			0
$976$	−2.73002e11			−0.300862
$977$	4.89741e11			0.537512			0.268756	−	0.963208i	$-0.413388\pi$
$977$	4.89741e11			0.537512			0.268756	+	0.963208i	$0.413388\pi$
$978$	−1.19500e11	−	3.01538e11i	−0.130621	−	0.329600i
$979$	2.08510e11			0.226985
$980$		0			0
$981$	−6.20331e11	+	5.83283e11i	−0.669804	+	0.629801i
$982$		−	1.37311e11i		−	0.147659i
$983$	−6.13109e11			−0.656635			−0.328317	−	0.944567i	$-0.606482\pi$
$983$	−6.13109e11			−0.656635			−0.328317	+	0.944567i	$0.606482\pi$
$984$	4.93802e11	+	1.24603e12i	0.526711	+	1.32907i
$985$		0			0
$986$			5.14226e11i			0.544059i
$987$	6.41897e11	+	1.61972e12i	0.676389	+	1.70676i
$988$			4.62379e11i			0.485256i
$989$			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	$0.435883\pi$
$992$	−3.29394e11			−0.340148
$993$	−2.77008e11	+	1.09778e11i	−0.284901	+	0.112907i
$994$	2.78703e11			0.285494
$995$		0			0
$996$	3.37175e11	−	1.33623e11i	0.342624	−	0.135782i
$997$			1.56419e12i			1.58310i	0.611104	+	0.791551i	$0.290726\pi$
$997$			1.56419e12i			1.58310i	−0.611104	+	0.791551i	$0.709274\pi$
$998$	−5.60565e11			−0.565072
$999$	−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.d.c.74.12		20
3.2	odd	2	inner	75.9.d.c.74.10		20
5.2	odd	4		75.9.c.g.26.6		10
5.3	odd	4		15.9.c.a.11.5	✓	10
5.4	even	2	inner	75.9.d.c.74.9		20
15.2	even	4		75.9.c.g.26.5		10
15.8	even	4		15.9.c.a.11.6	yes	10
15.14	odd	2	inner	75.9.d.c.74.11		20
20.3	even	4		240.9.l.b.161.6		10
60.23	odd	4		240.9.l.b.161.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.9.c.a.11.5	✓	10	5.3	odd	4
15.9.c.a.11.6	yes	10	15.8	even	4
75.9.c.g.26.5		10	15.2	even	4
75.9.c.g.26.6		10	5.2	odd	4
75.9.d.c.74.9		20	5.4	even	2	inner
75.9.d.c.74.10		20	3.2	odd	2	inner
75.9.d.c.74.11		20	15.14	odd	2	inner
75.9.d.c.74.12		20	1.1	even	1	trivial
240.9.l.b.161.5		10	60.23	odd	4
240.9.l.b.161.6		10	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.97572 q^{2} +(-75.3023 + 29.8424i) q^{3} -192.388 q^{4} +(-600.590 + 238.014i) q^{6} -3211.86i q^{7} -3576.22 q^{8} +(4779.87 - 4494.40i) q^{9} +O(q^{10})\)	\(q+7.97572 q^{2} +(-75.3023 + 29.8424i) q^{3} -192.388 q^{4} +(-600.590 + 238.014i) q^{6} -3211.86i q^{7} -3576.22 q^{8} +(4779.87 - 4494.40i) q^{9} -6481.82i q^{11} +(14487.3 - 5741.31i) q^{12} +10510.3i q^{13} -25616.9i q^{14} +20728.4 q^{16} +57738.3 q^{17} +(38122.9 - 35846.0i) q^{18} -228667. q^{19} +(95849.5 + 241860. i) q^{21} -51697.1i q^{22} -113904. q^{23} +(269297. - 106723. i) q^{24} +83827.4i q^{26} +(-225811. + 481081. i) q^{27} +617923. i q^{28} +1.11666e6i q^{29} -304758. q^{31} +1.08084e6 q^{32} +(193433. + 488096. i) q^{33} +460504. q^{34} +(-919589. + 864668. i) q^{36} +630294. i q^{37} -1.82379e6 q^{38} +(-313653. - 791452. i) q^{39} +4.62696e6i q^{41} +(764469. + 1.92901e6i) q^{42} -5.44386e6i q^{43} +1.24702e6i q^{44} -908469. q^{46} +6.69692e6 q^{47} +(-1.56090e6 + 618586. i) q^{48} -4.55125e6 q^{49} +(-4.34782e6 + 1.72305e6i) q^{51} -2.02206e6i q^{52} +1.25196e7 q^{53} +(-1.80101e6 + 3.83696e6i) q^{54} +1.14863e7i q^{56} +(1.72192e7 - 6.82398e6i) q^{57} +8.90615e6i q^{58} +5.55982e6i q^{59} -1.31704e7 q^{61} -2.43067e6 q^{62} +(-1.44354e7 - 1.53523e7i) q^{63} +3.31396e6 q^{64} +(1.54276e6 + 3.89291e6i) q^{66} +1.70044e7i q^{67} -1.11081e7 q^{68} +(8.57726e6 - 3.39918e6i) q^{69} +1.08797e7i q^{71} +(-1.70938e7 + 1.60729e7i) q^{72} +2.67184e7i q^{73} +5.02705e6i q^{74} +4.39928e7 q^{76} -2.08187e7 q^{77} +(-2.50161e6 - 6.31239e6i) q^{78} +3.07855e6 q^{79} +(2.64753e6 - 4.29652e7i) q^{81} +3.69033e7i q^{82} +2.32739e7 q^{83} +(-1.84403e7 - 4.65310e7i) q^{84} -4.34187e7i q^{86} +(-3.33237e7 - 8.40869e7i) q^{87} +2.31804e7i q^{88} +3.21685e7i q^{89} +3.37577e7 q^{91} +2.19138e7 q^{92} +(2.29490e7 - 9.09471e6i) q^{93} +5.34128e7 q^{94} +(-8.13894e7 + 3.22547e7i) q^{96} -9.03610e7i q^{97} -3.62994e7 q^{98} +(-2.91319e7 - 3.09822e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9}+O(q^{10}) \) 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9	\( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9} + 560772 q^{16} + 463032 q^{19} + 579144 q^{21} - 2272668 q^{24} + 1763240 q^{31} + 2222552 q^{34} - 1337324 q^{36} - 3653584 q^{39} - 50849208 q^{46} - 18708428 q^{49} - 55465384 q^{51} + 15959596 q^{54} + 44834040 q^{61} + 45870004 q^{64} - 54839600 q^{66} - 67125264 q^{69} + 397844872 q^{76} - 324621848 q^{79} - 187150780 q^{81} + 394693536 q^{84} + 888576928 q^{91} + 184100072 q^{94} - 721614812 q^{96} + 67930400 q^{99}+O(q^{100}) \) 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9 + 560772 * q^16 + 463032 * q^19 + 579144 * q^21 - 2272668 * q^24 + 1763240 * q^31 + 2222552 * q^34 - 1337324 * q^36 - 3653584 * q^39 - 50849208 * q^46 - 18708428 * q^49 - 55465384 * q^51 + 15959596 * q^54 + 44834040 * q^61 + 45870004 * q^64 - 54839600 * q^66 - 67125264 * q^69 + 397844872 * q^76 - 324621848 * q^79 - 187150780 * q^81 + 394693536 * q^84 + 888576928 * q^91 + 184100072 * q^94 - 721614812 * q^96 + 67930400 * q^99

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