LMFDB - Embedded newform 378.2.m.a.251.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(125,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("378.125");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 $ x^16 - 6x^14 + 9x^12 + 54x^10 - 288x^8 + 486x^6 + 729x^4 - 4374*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			251.3
Root			$-0.0967785 - 1.72934i$ of defining polynomial
Character	$\chi$	$=$	378.251
Dual form			378.2.m.a.125.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.183299 + 0.317483i) q^{5} +(2.53871 - 0.744936i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.183299 + 0.317483i) q^{5} +(2.53871 - 0.744936i) q^{7} -1.00000i q^{8} -0.366598i q^{10} +(-0.579764 - 0.334727i) q^{11} +(-0.867380 + 0.500782i) q^{13} +(-2.57106 - 0.624224i) q^{14} +(-0.500000 + 0.866025i) q^{16} +4.98906 q^{17} +6.35722i q^{19} +(-0.183299 + 0.317483i) q^{20} +(0.334727 + 0.579764i) q^{22} +(6.66371 - 3.84729i) q^{23} +(2.43280 - 4.21374i) q^{25} +1.00156 q^{26} +(1.91449 + 1.82612i) q^{28} +(-1.58394 - 0.914490i) q^{29} +(5.47837 - 3.16294i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-4.32065 - 2.49453i) q^{34} +(0.701849 + 0.669453i) q^{35} -5.16789 q^{37} +(3.17861 - 5.50552i) q^{38} +(0.317483 - 0.183299i) q^{40} +(2.15928 + 3.73998i) q^{41} +(2.24922 - 3.89576i) q^{43} -0.669453i q^{44} -7.69459 q^{46} +(-4.16450 + 7.21313i) q^{47} +(5.89014 - 3.78236i) q^{49} +(-4.21374 + 2.43280i) q^{50} +(-0.867380 - 0.500782i) q^{52} -0.245420i q^{55} +(-0.744936 - 2.53871i) q^{56} +(0.914490 + 1.58394i) q^{58} +(-4.36348 - 7.55776i) q^{59} +(-4.29351 - 2.47886i) q^{61} -6.32588 q^{62} -1.00000 q^{64} +(-0.317980 - 0.183586i) q^{65} +(5.44537 + 9.43166i) q^{67} +(2.49453 + 4.32065i) q^{68} +(-0.273092 - 0.930688i) q^{70} +5.49843i q^{71} -4.07314i q^{73} +(4.47552 + 2.58394i) q^{74} +(-5.50552 + 3.17861i) q^{76} +(-1.72120 - 0.417889i) q^{77} +(-4.17784 + 7.23623i) q^{79} -0.366598 q^{80} -4.31856i q^{82} +(-8.50712 + 14.7348i) q^{83} +(0.914490 + 1.58394i) q^{85} +(-3.89576 + 2.24922i) q^{86} +(-0.334727 + 0.579764i) q^{88} -10.7113 q^{89} +(-1.82898 + 1.91749i) q^{91} +(6.66371 + 3.84729i) q^{92} +(7.21313 - 4.16450i) q^{94} +(-2.01831 + 1.16527i) q^{95} +(-14.9093 - 8.60787i) q^{97} +(-6.99219 + 0.330547i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} + 2 q^{7}+O(q^{10}) $ 16 * q + 8 * q^4 + 2 * q^7	$ 16 q + 8 q^{4} + 2 q^{7} + 12 q^{11} + 6 q^{14} - 8 q^{16} + 48 q^{23} - 8 q^{25} + 4 q^{28} + 12 q^{29} - 8 q^{37} + 4 q^{43} + 24 q^{46} - 8 q^{49} - 60 q^{50} + 6 q^{56} - 12 q^{58} - 16 q^{64} - 84 q^{65} - 28 q^{67} - 36 q^{74} - 78 q^{77} - 4 q^{79} - 12 q^{85} + 24 q^{86} + 24 q^{91} + 48 q^{92} - 12 q^{95}+O(q^{100}) $ 16 * q + 8 * q^4 + 2 * q^7 + 12 * q^11 + 6 * q^14 - 8 * q^16 + 48 * q^23 - 8 * q^25 + 4 * q^28 + 12 * q^29 - 8 * q^37 + 4 * q^43 + 24 * q^46 - 8 * q^49 - 60 * q^50 + 6 * q^56 - 12 * q^58 - 16 * q^64 - 84 * q^65 - 28 * q^67 - 36 * q^74 - 78 * q^77 - 4 * q^79 - 12 * q^85 + 24 * q^86 + 24 * q^91 + 48 * q^92 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	0.183299	+	0.317483i	0.0819738	+	0.141983i	0.904098	−	0.427326i	$-0.140544\pi$
$5$	0.183299	+	0.317483i	0.0819738	+	0.141983i	−0.822124	+	0.569309i	$0.807211\pi$
$6$		0			0
$7$	2.53871	−	0.744936i	0.959544	−	0.281559i
$7$	2.53871	−	0.744936i	0.959544	−	0.281559i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	0.366598i		−	0.115929i
$11$	−0.579764	−	0.334727i	−0.174805	−	0.100924i	0.410044	−	0.912066i	$-0.365513\pi$
$11$	−0.579764	−	0.334727i	−0.174805	−	0.100924i	−0.584850	+	0.811142i	$0.698847\pi$
$12$		0			0
$13$	−0.867380	+	0.500782i	−0.240568	+	0.138892i	−0.615438	−	0.788185i	$-0.711021\pi$
$13$	−0.867380	+	0.500782i	−0.240568	+	0.138892i	0.374870	+	0.927077i	$0.377687\pi$
$14$	−2.57106	−	0.624224i	−0.687144	−	0.166831i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	4.98906			1.21003			0.605013	−	0.796216i	$-0.293168\pi$
$17$	4.98906			1.21003			0.605013	+	0.796216i	$0.293168\pi$
$18$		0			0
$19$			6.35722i			1.45845i	0.684275	+	0.729224i	$0.260119\pi$
$19$			6.35722i			1.45845i	−0.684275	+	0.729224i	$0.739881\pi$
$20$	−0.183299	+	0.317483i	−0.0409869	+	0.0709914i
$21$		0			0
$22$	0.334727	+	0.579764i	0.0713640	+	0.123606i
$23$	6.66371	−	3.84729i	1.38948	−	0.802216i	0.396223	−	0.918154i	$-0.370321\pi$
$23$	6.66371	−	3.84729i	1.38948	−	0.802216i	0.993256	+	0.115938i	$0.0369875\pi$
$24$		0			0
$25$	2.43280	−	4.21374i	0.486561	−	0.842748i
$26$	1.00156			0.196423
$27$		0			0
$28$	1.91449	+	1.82612i	0.361805	+	0.345105i
$29$	−1.58394	−	0.914490i	−0.294131	−	0.169817i	0.345672	−	0.938355i	$-0.387651\pi$
$29$	−1.58394	−	0.914490i	−0.294131	−	0.169817i	−0.639803	+	0.768539i	$0.720984\pi$
$30$		0			0
$31$	5.47837	−	3.16294i	0.983944	−	0.568081i	0.0804857	−	0.996756i	$-0.474353\pi$
$31$	5.47837	−	3.16294i	0.983944	−	0.568081i	0.903459	+	0.428675i	$0.141020\pi$
$32$	0.866025	−	0.500000i	0.153093	−	0.0883883i
$33$		0			0
$34$	−4.32065	−	2.49453i	−0.740986	−	0.427809i
$35$	0.701849	+	0.669453i	0.118634	+	0.113158i
$36$		0			0
$37$	−5.16789			−0.849595			−0.424798	−	0.905288i	$-0.639655\pi$
$37$	−5.16789			−0.849595			−0.424798	+	0.905288i	$0.639655\pi$
$38$	3.17861	−	5.50552i	0.515639	−	0.893113i
$39$		0			0
$40$	0.317483	−	0.183299i	0.0501985	−	0.0289821i
$41$	2.15928	+	3.73998i	0.337223	+	0.584087i	0.983909	−	0.178669i	$-0.0571790\pi$
$41$	2.15928	+	3.73998i	0.337223	+	0.584087i	−0.646686	+	0.762756i	$0.723846\pi$
$42$		0			0
$43$	2.24922	−	3.89576i	0.343002	−	0.594098i	−0.641986	−	0.766716i	$-0.721889\pi$
$43$	2.24922	−	3.89576i	0.343002	−	0.594098i	0.984989	+	0.172618i	$0.0552228\pi$
$44$		−	0.669453i		−	0.100924i
$45$		0			0
$46$	−7.69459			−1.13450
$47$	−4.16450	+	7.21313i	−0.607455	+	1.05214i	0.384203	+	0.923249i	$0.374476\pi$
$47$	−4.16450	+	7.21313i	−0.607455	+	1.05214i	−0.991658	+	0.128895i	$0.958857\pi$
$48$		0			0
$49$	5.89014	−	3.78236i	0.841449	−	0.540337i
$50$	−4.21374	+	2.43280i	−0.595913	+	0.344050i
$51$		0			0
$52$	−0.867380	−	0.500782i	−0.120284	−	0.0694460i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		−	0.245420i		−	0.0330925i
$56$	−0.744936	−	2.53871i	−0.0995462	−	0.339250i
$57$		0			0
$58$	0.914490	+	1.58394i	0.120078	+	0.207982i
$59$	−4.36348	−	7.55776i	−0.568076	−	0.983937i	−0.996756	−	0.0804804i	$-0.974355\pi$
$59$	−4.36348	−	7.55776i	−0.568076	−	0.983937i	0.428680	−	0.903456i	$-0.358979\pi$
$60$		0			0
$61$	−4.29351	−	2.47886i	−0.549727	−	0.317385i	0.199285	−	0.979942i	$-0.436138\pi$
$61$	−4.29351	−	2.47886i	−0.549727	−	0.317385i	−0.749012	+	0.662556i	$0.769471\pi$
$62$	−6.32588			−0.803387
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−0.317980	−	0.183586i	−0.0394406	−	0.0227710i
$66$		0			0
$67$	5.44537	+	9.43166i	0.665258	+	1.15226i	0.979215	+	0.202823i	$0.0650117\pi$
$67$	5.44537	+	9.43166i	0.665258	+	1.15226i	−0.313958	+	0.949437i	$0.601655\pi$
$68$	2.49453	+	4.32065i	0.302506	+	0.523956i
$69$		0			0
$70$	−0.273092	−	0.930688i	−0.0326407	−	0.111238i
$71$			5.49843i			0.652544i	0.945276	+	0.326272i	$0.105793\pi$
$71$			5.49843i			0.652544i	−0.945276	+	0.326272i	$0.894207\pi$
$72$		0			0
$73$		−	4.07314i		−	0.476725i	−0.971176	−	0.238363i	$-0.923389\pi$
$73$		−	4.07314i		−	0.476725i	0.971176	−	0.238363i	$-0.0766106\pi$
$74$	4.47552	+	2.58394i	0.520269	+	0.300377i
$75$		0			0
$76$	−5.50552	+	3.17861i	−0.631526	+	0.364612i
$77$	−1.72120	−	0.417889i	−0.196149	−	0.0476229i
$78$		0			0
$79$	−4.17784	+	7.23623i	−0.470044	+	0.814140i	−0.999413	−	0.0342518i	$-0.989095\pi$
$79$	−4.17784	+	7.23623i	−0.470044	+	0.814140i	0.529370	+	0.848391i	$0.322429\pi$
$80$	−0.366598			−0.0409869
$81$		0			0
$82$		−	4.31856i		−	0.476905i
$83$	−8.50712	+	14.7348i	−0.933778	+	1.61735i	−0.156980	+	0.987602i	$0.550176\pi$
$83$	−8.50712	+	14.7348i	−0.933778	+	1.61735i	−0.776798	+	0.629750i	$0.783158\pi$
$84$		0			0
$85$	0.914490	+	1.58394i	0.0991904	+	0.171803i
$86$	−3.89576	+	2.24922i	−0.420090	+	0.242539i
$87$		0			0
$88$	−0.334727	+	0.579764i	−0.0356820	+	0.0618030i
$89$	−10.7113			−1.13540			−0.567699	−	0.823236i	$-0.692166\pi$
$89$	−10.7113			−1.13540			−0.567699	+	0.823236i	$0.692166\pi$
$90$		0			0
$91$	−1.82898	+	1.91749i	−0.191729	+	0.201007i
$92$	6.66371	+	3.84729i	0.694740	+	0.401108i
$93$		0			0
$94$	7.21313	−	4.16450i	0.743978	−	0.429536i
$95$	−2.01831	+	1.16527i	−0.207074	+	0.119555i
$96$		0			0
$97$	−14.9093	−	8.60787i	−1.51381	−	0.873997i	−0.999869	−	0.0161687i	$-0.994853\pi$
$97$	−14.9093	−	8.60787i	−1.51381	−	0.873997i	−0.513937	−	0.857828i	$-0.671814\pi$
$98$	−6.99219	+	0.330547i	−0.706318	+	0.0333902i
$99$		0			0
$100$	4.86561			0.486561
$101$	7.86586	−	13.6241i	0.782683	−	1.35565i	−0.147691	−	0.989034i	$-0.547184\pi$
$101$	7.86586	−	13.6241i	0.782683	−	1.35565i	0.930374	−	0.366613i	$-0.119483\pi$
$102$		0			0
$103$	−9.91124	+	5.72226i	−0.976584	+	0.563831i	−0.901237	−	0.433327i	$-0.857340\pi$
$103$	−9.91124	+	5.72226i	−0.976584	+	0.563831i	−0.0753467	+	0.997157i	$0.524006\pi$
$104$	0.500782	+	0.867380i	0.0491057	+	0.0850537i
$105$		0			0
$106$		0			0
$107$			11.0618i			1.06938i	0.845048	+	0.534690i	$0.179572\pi$
$107$			11.0618i			1.06938i	−0.845048	+	0.534690i	$0.820428\pi$
$108$		0			0
$109$	−10.5633			−1.01178			−0.505891	−	0.862597i	$-0.668836\pi$
$109$	−10.5633			−1.01178			−0.505891	+	0.862597i	$0.668836\pi$
$110$	−0.122710	+	0.212540i	−0.0117000	+	0.0202649i
$111$		0			0
$112$	−0.624224	+	2.57106i	−0.0589836	+	0.242942i
$113$	−3.60226	+	2.07976i	−0.338872	+	0.195648i	−0.659773	−	0.751465i	$-0.729348\pi$
$113$	−3.60226	+	2.07976i	−0.338872	+	0.195648i	0.320901	+	0.947113i	$0.396014\pi$
$114$		0			0
$115$	2.44290	+	1.41041i	0.227802	+	0.131521i
$116$		−	1.82898i		−	0.169817i
$117$		0			0
$118$			8.72695i			0.803381i
$119$	12.6658	−	3.71653i	1.16107	−	0.340694i
$120$		0			0
$121$	−5.27592	−	9.13815i	−0.479629	−	0.830741i
$122$	2.47886	+	4.29351i	0.224425	+	0.388716i
$123$		0			0
$124$	5.47837	+	3.16294i	0.491972	+	0.284040i
$125$	3.61671			0.323489
$126$		0			0
$127$	−1.66945			−0.148140			−0.0740700	−	0.997253i	$-0.523599\pi$
$127$	−1.66945			−0.148140			−0.0740700	+	0.997253i	$0.523599\pi$
$128$	0.866025	+	0.500000i	0.0765466	+	0.0441942i
$129$		0			0
$130$	0.183586	+	0.317980i	0.0161015	+	0.0278887i
$131$	−6.76607	−	11.7192i	−0.591154	−	1.02391i	−0.994077	−	0.108675i	$-0.965339\pi$
$131$	−6.76607	−	11.7192i	−0.591154	−	1.02391i	0.402923	−	0.915234i	$-0.367994\pi$
$132$		0			0
$133$	4.73572	+	16.1392i	0.410639	+	1.39944i
$134$		−	10.8907i		−	0.940817i
$135$		0			0
$136$		−	4.98906i		−	0.427809i
$137$	−7.78428	−	4.49425i	−0.665056	−	0.383970i	0.129145	−	0.991626i	$-0.458777\pi$
$137$	−7.78428	−	4.49425i	−0.665056	−	0.383970i	−0.794201	+	0.607656i	$0.792110\pi$
$138$		0			0
$139$	−8.05336	+	4.64961i	−0.683077	+	0.394375i	−0.801014	−	0.598646i	$-0.795706\pi$
$139$	−8.05336	+	4.64961i	−0.683077	+	0.394375i	0.117936	+	0.993021i	$0.462372\pi$
$140$	−0.228839	+	0.942545i	−0.0193405	+	0.0796596i
$141$		0			0
$142$	2.74922	−	4.76178i	0.230709	−	0.399600i
$143$	0.670501			0.0560701
$144$		0			0
$145$		−	0.670501i		−	0.0556821i
$146$	−2.03657	+	3.52744i	−0.168548	+	0.291933i
$147$		0			0
$148$	−2.58394	−	4.47552i	−0.212399	−	0.367886i
$149$	2.45268	−	1.41606i	0.200931	−	0.116008i	−0.396158	−	0.918182i	$-0.629657\pi$
$149$	2.45268	−	1.41606i	0.200931	−	0.116008i	0.597090	+	0.802174i	$0.296324\pi$
$150$		0			0
$151$	8.27592	−	14.3343i	0.673484	−	1.16651i	−0.303425	−	0.952855i	$-0.598130\pi$
$151$	8.27592	−	14.3343i	0.673484	−	1.16651i	0.976909	−	0.213654i	$-0.0685365\pi$
$152$	6.35722			0.515639
$153$		0			0
$154$	1.28166	+	1.22250i	0.103279	+	0.0985122i
$155$	2.00836	+	1.15953i	0.161315	+	0.0931355i
$156$		0			0
$157$	2.45480	−	1.41728i	0.195914	−	0.113111i	−0.398834	−	0.917023i	$-0.630585\pi$
$157$	2.45480	−	1.41728i	0.195914	−	0.113111i	0.594748	+	0.803912i	$0.297252\pi$
$158$	7.23623	−	4.17784i	0.575684	−	0.332371i
$159$		0			0
$160$	0.317483	+	0.183299i	0.0250993	+	0.0144911i
$161$	14.0513	−	14.7312i	1.10739	−	1.16098i
$162$		0			0
$163$	24.7281			1.93685			0.968426	−	0.249300i	$-0.0802005\pi$
$163$	24.7281			1.93685			0.968426	+	0.249300i	$0.0802005\pi$
$164$	−2.15928	+	3.73998i	−0.168611	+	0.292044i
$165$		0			0
$166$	14.7348	−	8.50712i	1.14364	−	0.660281i
$167$	9.67422	+	16.7562i	0.748614	+	1.29664i	0.948487	+	0.316815i	$0.102614\pi$
$167$	9.67422	+	16.7562i	0.748614	+	1.29664i	−0.199874	+	0.979822i	$0.564053\pi$
$168$		0			0
$169$	−5.99843	+	10.3896i	−0.461418	+	0.799199i
$170$		−	1.82898i		−	0.140276i
$171$		0			0
$172$	4.49843			0.343002
$173$	2.41827	−	4.18856i	0.183858	−	0.318451i	−0.759333	−	0.650702i	$-0.774475\pi$
$173$	2.41827	−	4.18856i	0.183858	−	0.318451i	0.943191	+	0.332251i	$0.107808\pi$
$174$		0			0
$175$	3.03723	−	12.5098i	0.229593	−	0.945649i
$176$	0.579764	−	0.334727i	0.0437013	−	0.0252310i
$177$		0			0
$178$	9.27628	+	5.35566i	0.695286	+	0.401424i
$179$			3.65796i			0.273409i	0.990612	+	0.136704i	$0.0436511\pi$
$179$			3.65796i			0.273409i	−0.990612	+	0.136704i	$0.956349\pi$
$180$		0			0
$181$		−	5.66796i		−	0.421296i	−0.977562	−	0.210648i	$-0.932443\pi$
$181$		−	5.66796i		−	0.421296i	0.977562	−	0.210648i	$-0.0675574\pi$
$182$	2.54269	−	0.746101i	0.188476	−	0.0553047i
$183$		0			0
$184$	−3.84729	−	6.66371i	−0.283626	−	0.491255i
$185$	−0.947269	−	1.64072i	−0.0696446	−	0.120628i
$186$		0			0
$187$	−2.89248	−	1.66997i	−0.211519	−	0.122120i
$188$	−8.32901			−0.607455
$189$		0			0
$190$	2.33055			0.169076
$191$	−23.7098	−	13.6888i	−1.71558	−	0.990490i	−0.926583	−	0.376091i	$-0.877268\pi$
$191$	−23.7098	−	13.6888i	−1.71558	−	0.990490i	−0.788996	−	0.614398i	$-0.789399\pi$
$192$		0			0
$193$	5.01413	+	8.68473i	0.360925	+	0.625141i	0.988113	−	0.153727i	$-0.0491276\pi$
$193$	5.01413	+	8.68473i	0.360925	+	0.625141i	−0.627188	+	0.778868i	$0.715794\pi$
$194$	8.60787	+	14.9093i	0.618009	+	1.07042i
$195$		0			0
$196$	6.22069	+	3.20983i	0.444335	+	0.229274i
$197$		−	18.8258i		−	1.34129i	−0.741780	−	0.670643i	$-0.766018\pi$
$197$		−	18.8258i		−	1.34129i	0.741780	−	0.670643i	$-0.233982\pi$
$198$		0			0
$199$			5.36406i			0.380248i	0.981760	+	0.190124i	$0.0608890\pi$
$199$			5.36406i			0.380248i	−0.981760	+	0.190124i	$0.939111\pi$
$200$	−4.21374	−	2.43280i	−0.297956	−	0.172025i
$201$		0			0
$202$	−13.6241	+	7.86586i	−0.958587	+	0.553440i
$203$	−4.70242	−	1.14169i	−0.330045	−	0.0801312i
$204$		0			0
$205$	−0.791588	+	1.37107i	−0.0552869	+	0.0957597i
$206$	11.4445			0.797377
$207$		0			0
$208$		−	1.00156i		−	0.0694460i
$209$	2.12793	−	3.68569i	0.147192	−	0.254944i
$210$		0			0
$211$	−0.828981	−	1.43584i	−0.0570694	−	0.0988471i	0.836079	−	0.548609i	$-0.184842\pi$
$211$	−0.828981	−	1.43584i	−0.0570694	−	0.0988471i	−0.893149	+	0.449762i	$0.851509\pi$
$212$		0			0
$213$		0			0
$214$	5.53088	−	9.57976i	0.378083	−	0.654859i
$215$	1.64912			0.112469
$216$		0			0
$217$	11.5518	−	12.1108i	0.784189	−	0.822137i
$218$	9.14811	+	5.28166i	0.619588	+	0.357719i
$219$		0			0
$220$	0.212540	−	0.122710i	0.0143295	−	0.00827312i
$221$	−4.32741	+	2.49843i	−0.291093	+	0.168063i
$222$		0			0
$223$	14.7546	+	8.51860i	0.988044	+	0.570448i	0.904689	−	0.426072i	$-0.140103\pi$
$223$	14.7546	+	8.51860i	0.988044	+	0.570448i	0.0833551	+	0.996520i	$0.473436\pi$
$224$	1.82612	−	1.91449i	0.122013	−	0.127917i
$225$		0			0
$226$	4.15953			0.276688
$227$	−2.55512	+	4.42560i	−0.169589	+	0.293737i	−0.938276	−	0.345889i	$-0.887577\pi$
$227$	−2.55512	+	4.42560i	−0.169589	+	0.293737i	0.768686	+	0.639626i	$0.220911\pi$
$228$		0			0
$229$	−13.2215	+	7.63345i	−0.873703	+	0.504433i	−0.868577	−	0.495554i	$-0.834965\pi$
$229$	−13.2215	+	7.63345i	−0.873703	+	0.504433i	−0.00512595	+	0.999987i	$0.501632\pi$
$230$	−1.41041	−	2.44290i	−0.0929997	−	0.161080i
$231$		0			0
$232$	−0.914490	+	1.58394i	−0.0600392	+	0.103991i
$233$			10.1930i			0.667767i	0.942614	+	0.333883i	$0.108359\pi$
$233$			10.1930i			0.667767i	−0.942614	+	0.333883i	$0.891641\pi$
$234$		0			0
$235$	−3.05340			−0.199182
$236$	4.36348	−	7.55776i	0.284038	−	0.491968i
$237$		0			0
$238$	−12.8272	−	3.11429i	−0.831462	−	0.201870i
$239$	16.6117	−	9.59076i	1.07452	−	0.620375i	0.145108	−	0.989416i	$-0.453647\pi$
$239$	16.6117	−	9.59076i	1.07452	−	0.620375i	0.929413	+	0.369041i	$0.120314\pi$
$240$		0			0
$241$	17.9140	+	10.3426i	1.15394	+	0.666227i	0.949844	−	0.312724i	$-0.101241\pi$
$241$	17.9140	+	10.3426i	1.15394	+	0.666227i	0.204095	+	0.978951i	$0.434575\pi$
$242$			10.5518i			0.678297i
$243$		0			0
$244$		−	4.95771i		−	0.317385i
$245$	2.28049	+	1.17672i	0.145695	+	0.0751778i
$246$		0			0
$247$	−3.18359	−	5.51413i	−0.202567	−	0.350856i
$248$	−3.16294	−	5.47837i	−0.200847	−	0.347877i
$249$		0			0
$250$	−3.13216	−	1.80836i	−0.198096	−	0.114370i
$251$	1.81200			0.114373			0.0571864	−	0.998364i	$-0.481787\pi$
$251$	1.81200			0.114373			0.0571864	+	0.998364i	$0.481787\pi$
$252$		0			0
$253$	−5.15117			−0.323851
$254$	1.44579	+	0.834727i	0.0907169	+	0.0523754i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−3.22773	−	5.59059i	−0.201340	−	0.348731i	0.747620	−	0.664126i	$-0.231196\pi$
$257$	−3.22773	−	5.59059i	−0.201340	−	0.348731i	−0.948960	+	0.315395i	$0.897863\pi$
$258$		0			0
$259$	−13.1198	+	3.84974i	−0.815224	+	0.239211i
$260$		−	0.367172i		−	0.0227710i
$261$		0			0
$262$			13.5321i			0.836018i
$263$	7.63888	+	4.41031i	0.471034	+	0.271951i	0.716672	−	0.697410i	$-0.245664\pi$
$263$	7.63888	+	4.41031i	0.471034	+	0.271951i	−0.245639	+	0.969361i	$0.578998\pi$
$264$		0			0
$265$		0			0
$266$	3.96833	−	16.3448i	0.243314	−	1.00216i
$267$		0			0
$268$	−5.44537	+	9.43166i	−0.332629	+	0.576130i
$269$	−14.2653			−0.869773			−0.434886	−	0.900485i	$-0.643212\pi$
$269$	−14.2653			−0.869773			−0.434886	+	0.900485i	$0.643212\pi$
$270$		0			0
$271$		−	3.05281i		−	0.185445i	−0.995692	−	0.0927226i	$-0.970443\pi$
$271$		−	3.05281i		−	0.185445i	0.995692	−	0.0927226i	$-0.0295570\pi$
$272$	−2.49453	+	4.32065i	−0.151253	+	0.261978i
$273$		0			0
$274$	4.49425	+	7.78428i	0.271508	+	0.470265i
$275$	−2.82090	+	1.62865i	−0.170107	+	0.0982112i
$276$		0			0
$277$	−0.632828	+	1.09609i	−0.0380230	+	0.0658577i	−0.884411	−	0.466710i	$-0.845439\pi$
$277$	−0.632828	+	1.09609i	−0.0380230	+	0.0658577i	0.846388	+	0.532567i	$0.178773\pi$
$278$	9.29922			0.557730
$279$		0			0
$280$	0.669453	−	0.701849i	0.0400075	−	0.0419435i
$281$	−9.11639	−	5.26335i	−0.543838	−	0.313985i	0.202795	−	0.979221i	$-0.434998\pi$
$281$	−9.11639	−	5.26335i	−0.543838	−	0.313985i	−0.746633	+	0.665236i	$0.768331\pi$
$282$		0			0
$283$	−17.2094	+	9.93588i	−1.02300	+	0.590627i	−0.914970	−	0.403522i	$-0.867786\pi$
$283$	−17.2094	+	9.93588i	−1.02300	+	0.590627i	−0.108025	+	0.994148i	$0.534453\pi$
$284$	−4.76178	+	2.74922i	−0.282560	+	0.163136i
$285$		0			0
$286$	−0.580671	−	0.335250i	−0.0343358	−	0.0198238i
$287$	8.26784	+	7.88623i	0.488035	+	0.465509i
$288$		0			0
$289$	7.89074			0.464161
$290$	−0.335250	+	0.580671i	−0.0196866	+	0.0340982i
$291$		0			0
$292$	3.52744	−	2.03657i	0.206428	−	0.119181i
$293$	−6.70606	−	11.6152i	−0.391772	−	0.678569i	0.600911	−	0.799316i	$-0.294804\pi$
$293$	−6.70606	−	11.6152i	−0.391772	−	0.678569i	−0.992683	+	0.120747i	$0.961471\pi$
$294$		0			0
$295$	1.59964	−	2.77066i	0.0931348	−	0.161314i
$296$			5.16789i			0.300377i
$297$		0			0
$298$	−2.83211			−0.164060
$299$	−3.85331	+	6.67413i	−0.222843	+	0.385975i
$300$		0			0
$301$	2.80803	−	11.5657i	0.161852	−	0.666638i
$302$	−14.3343	+	8.27592i	−0.824847	+	0.476225i
$303$		0			0
$304$	−5.50552	−	3.17861i	−0.315763	−	0.182306i
$305$		−	1.81749i		−	0.104069i
$306$		0			0
$307$		−	0.653728i		−	0.0373102i	−0.999826	−	0.0186551i	$-0.994062\pi$
$307$		−	0.653728i		−	0.0373102i	0.999826	−	0.0186551i	$-0.00593845\pi$
$308$	−0.498700	−	1.69955i	−0.0284160	−	0.0968409i
$309$		0			0
$310$	−1.15953	−	2.00836i	−0.0658567	−	0.114067i
$311$	−4.62246	−	8.00634i	−0.262116	−	0.453998i	0.704688	−	0.709517i	$-0.251087\pi$
$311$	−4.62246	−	8.00634i	−0.262116	−	0.453998i	−0.966804	+	0.255519i	$0.917754\pi$
$312$		0			0
$313$	−5.33830	−	3.08207i	−0.301739	−	0.174209i	0.341485	−	0.939887i	$-0.389070\pi$
$313$	−5.33830	−	3.08207i	−0.301739	−	0.174209i	−0.643224	+	0.765678i	$0.722403\pi$
$314$	−2.83456			−0.159963
$315$		0			0
$316$	−8.35568			−0.470044
$317$	17.8876	+	10.3274i	1.00467	+	0.580045i	0.909626	−	0.415428i	$-0.136368\pi$
$317$	17.8876	+	10.3274i	1.00467	+	0.580045i	0.0950420	+	0.995473i	$0.469701\pi$
$318$		0			0
$319$	0.612209	+	1.06038i	0.0342771	+	0.0593697i
$320$	−0.183299	−	0.317483i	−0.0102467	−	0.0177479i
$321$		0			0
$322$	−19.5344	+	5.73197i	−1.08861	+	0.319430i
$323$			31.7166i			1.76476i
$324$		0			0
$325$			4.87322i			0.270318i
$326$	−21.4151	−	12.3640i	−1.18608	−	0.684781i
$327$		0			0
$328$	3.73998	−	2.15928i	0.206506	−	0.119226i
$329$	−5.19917	+	21.4144i	−0.286639	+	1.18061i
$330$		0			0
$331$	−5.35568	+	9.27631i	−0.294375	+	0.509872i	−0.974839	−	0.222909i	$-0.928445\pi$
$331$	−5.35568	+	9.27631i	−0.294375	+	0.509872i	0.680464	+	0.732781i	$0.261778\pi$
$332$	−17.0142			−0.933778
$333$		0			0
$334$		−	19.3484i		−	1.05870i
$335$	−1.99626	+	3.45763i	−0.109067	+	0.188910i
$336$		0			0
$337$	3.77592	+	6.54008i	0.205687	+	0.356261i	0.950351	−	0.311179i	$-0.100724\pi$
$337$	3.77592	+	6.54008i	0.205687	+	0.356261i	−0.744664	+	0.667439i	$0.767390\pi$
$338$	10.3896	−	5.99843i	0.565119	−	0.326272i
$339$		0			0
$340$	−0.914490	+	1.58394i	−0.0495952	+	0.0859014i
$341$	−4.23488			−0.229332
$342$		0			0
$343$	12.1358	−	13.9901i	0.655270	−	0.755394i
$344$	−3.89576	−	2.24922i	−0.210045	−	0.121270i
$345$		0			0
$346$	−4.18856	+	2.41827i	−0.225179	+	0.130007i
$347$	9.46737	−	5.46599i	0.508235	−	0.293430i	−0.223873	−	0.974618i	$-0.571870\pi$
$347$	9.46737	−	5.46599i	0.508235	−	0.293430i	0.732108	+	0.681189i	$0.238537\pi$
$348$		0			0
$349$	1.02562	+	0.592145i	0.0549004	+	0.0316968i	0.527199	−	0.849742i	$-0.323242\pi$
$349$	1.02562	+	0.592145i	0.0549004	+	0.0316968i	−0.472299	+	0.881439i	$0.656576\pi$
$350$	−8.88520	+	9.31516i	−0.474934	+	0.497916i
$351$		0			0
$352$	−0.669453			−0.0356820
$353$	−16.7912	+	29.0832i	−0.893706	+	1.54794i	−0.0583086	+	0.998299i	$0.518571\pi$
$353$	−16.7912	+	29.0832i	−0.893706	+	1.54794i	−0.835398	+	0.549646i	$0.814763\pi$
$354$		0			0
$355$	−1.74566	+	1.00786i	−0.0926501	+	0.0534915i
$356$	−5.35566	−	9.27628i	−0.283849	−	0.491642i
$357$		0			0
$358$	1.82898	−	3.16789i	0.0966646	−	0.167428i
$359$		−	10.1281i		−	0.534542i	−0.963621	−	0.267271i	$-0.913878\pi$
$359$		−	10.1281i		−	0.534542i	0.963621	−	0.267271i	$-0.0861219\pi$
$360$		0			0
$361$	−21.4143			−1.12707
$362$	−2.83398	+	4.90860i	−0.148951	+	0.257990i
$363$		0			0
$364$	−2.57508	−	0.625201i	−0.134971	−	0.0327694i
$365$	1.29315	−	0.746603i	0.0676868	−	0.0390790i
$366$		0			0
$367$	−15.5903	−	9.00104i	−0.813805	−	0.469850i	0.0344706	−	0.999406i	$-0.489025\pi$
$367$	−15.5903	−	9.00104i	−0.813805	−	0.469850i	−0.848275	+	0.529555i	$0.822359\pi$
$368$			7.69459i			0.401108i
$369$		0			0
$370$			1.89454i			0.0984923i
$371$		0			0
$372$		0			0
$373$	−8.20451	−	14.2106i	−0.424814	−	0.735799i	0.571589	−	0.820540i	$-0.306327\pi$
$373$	−8.20451	−	14.2106i	−0.424814	−	0.735799i	−0.996403	+	0.0847411i	$0.972994\pi$
$374$	1.66997	+	2.89248i	0.0863522	+	0.149566i
$375$		0			0
$376$	7.21313	+	4.16450i	0.371989	+	0.214768i
$377$	1.83184			0.0943447
$378$		0			0
$379$	−2.91372			−0.149668			−0.0748339	−	0.997196i	$-0.523843\pi$
$379$	−2.91372			−0.149668			−0.0748339	+	0.997196i	$0.523843\pi$
$380$	−2.01831	−	1.16527i	−0.103537	−	0.0597773i
$381$		0			0
$382$	13.6888	+	23.7098i	0.700382	+	1.21310i
$383$	4.28721	+	7.42567i	0.219066	+	0.379434i	0.954523	−	0.298138i	$-0.0963655\pi$
$383$	4.28721	+	7.42567i	0.219066	+	0.379434i	−0.735456	+	0.677572i	$0.763032\pi$
$384$		0			0
$385$	−0.182822	−	0.623052i	−0.00931749	−	0.0317537i
$386$		−	10.0283i		−	0.510425i
$387$		0			0
$388$		−	17.2157i		−	0.873997i
$389$	30.7906	+	17.7770i	1.56115	+	0.901328i	0.997142	+	0.0755559i	$0.0240731\pi$
$389$	30.7906	+	17.7770i	1.56115	+	0.901328i	0.564004	+	0.825772i	$0.309260\pi$
$390$		0			0
$391$	33.2456	−	19.1944i	1.68130	−	0.970702i
$392$	−3.78236	−	5.89014i	−0.191038	−	0.297497i
$393$		0			0
$394$	−9.41292	+	16.3037i	−0.474216	+	0.821367i
$395$	−3.06318			−0.154125
$396$		0			0
$397$			3.58034i			0.179692i	0.995956	+	0.0898460i	$0.0286375\pi$
$397$			3.58034i			0.179692i	−0.995956	+	0.0898460i	$0.971363\pi$
$398$	2.68203	−	4.64541i	0.134438	−	0.232853i
$399$		0			0
$400$	2.43280	+	4.21374i	0.121640	+	0.210687i
$401$	−0.165300	+	0.0954357i	−0.00825467	+	0.00476583i	−0.504122	−	0.863633i	$-0.668184\pi$
$401$	−0.165300	+	0.0954357i	−0.00825467	+	0.00476583i	0.495867	+	0.868398i	$0.334850\pi$
$402$		0			0
$403$	−3.16789	+	5.48694i	−0.157804	+	0.273324i
$404$	15.7317			0.782683
$405$		0			0
$406$	3.50157	+	3.33994i	0.173780	+	0.165759i
$407$	2.99615	+	1.72983i	0.148514	+	0.0857445i
$408$		0			0
$409$	−3.00832	+	1.73685i	−0.148752	+	0.0858819i	−0.572529	−	0.819885i	$-0.694037\pi$
$409$	−3.00832	+	1.73685i	−0.148752	+	0.0858819i	0.423777	+	0.905767i	$0.360704\pi$
$410$	1.37107	−	0.791588i	0.0677124	−	0.0390938i
$411$		0			0
$412$	−9.91124	−	5.72226i	−0.488292	−	0.281915i
$413$	−16.7077	−	15.9365i	−0.822130	−	0.784183i
$414$		0			0
$415$	−6.23739			−0.306182
$416$	−0.500782	+	0.867380i	−0.0245529	+	0.0425268i
$417$		0			0
$418$	−3.68569	+	2.12793i	−0.180273	+	0.104081i
$419$	−0.703955	−	1.21929i	−0.0343905	−	0.0595660i	0.848318	−	0.529487i	$-0.177616\pi$
$419$	−0.703955	−	1.21929i	−0.0343905	−	0.0595660i	−0.882708	+	0.469921i	$0.844282\pi$
$420$		0			0
$421$	15.1930	−	26.3151i	0.740463	−	1.28252i	−0.211822	−	0.977308i	$-0.567940\pi$
$421$	15.1930	−	26.3151i	0.740463	−	1.28252i	0.952285	−	0.305211i	$-0.0987268\pi$
$422$			1.65796i			0.0807083i
$423$		0			0
$424$		0			0
$425$	12.1374	−	21.0226i	0.588751	−	1.01975i
$426$		0			0
$427$	−12.7466	−	3.09472i	−0.616850	−	0.149764i
$428$	−9.57976	+	5.53088i	−0.463055	+	0.267345i
$429$		0			0
$430$	−1.42818	−	0.824559i	−0.0688728	−	0.0397638i
$431$		−	27.2747i		−	1.31378i	−0.753988	−	0.656888i	$-0.771873\pi$
$431$		−	27.2747i		−	1.31378i	0.753988	−	0.656888i	$-0.228127\pi$
$432$		0			0
$433$		−	8.15047i		−	0.391686i	−0.980635	−	0.195843i	$-0.937256\pi$
$433$		−	8.15047i		−	0.391686i	0.980635	−	0.195843i	$-0.0627444\pi$
$434$	−16.0596	+	4.71237i	−0.770885	+	0.226201i
$435$		0			0
$436$	−5.28166	−	9.14811i	−0.252946	−	0.438115i
$437$	24.4581	+	42.3627i	1.16999	+	2.02648i
$438$		0			0
$439$	10.6005	+	6.12020i	0.505934	+	0.292101i	0.731161	−	0.682205i	$-0.238979\pi$
$439$	10.6005	+	6.12020i	0.505934	+	0.292101i	−0.225226	+	0.974306i	$0.572312\pi$
$440$	−0.245420			−0.0117000
$441$		0			0
$442$	4.99687			0.237677
$443$	−6.93544	−	4.00418i	−0.329513	−	0.190244i	0.326112	−	0.945331i	$-0.394261\pi$
$443$	−6.93544	−	4.00418i	−0.329513	−	0.190244i	−0.655625	+	0.755087i	$0.727595\pi$
$444$		0			0
$445$	−1.96337	−	3.40067i	−0.0930729	−	0.161207i
$446$	−8.51860	−	14.7546i	−0.403367	−	0.698653i
$447$		0			0
$448$	−2.53871	+	0.744936i	−0.119943	+	0.0351949i
$449$			14.5183i			0.685163i	0.939488	+	0.342581i	$0.111301\pi$
$449$			14.5183i			0.685163i	−0.939488	+	0.342581i	$0.888699\pi$
$450$		0			0
$451$		−	2.89108i		−	0.136135i
$452$	−3.60226	−	2.07976i	−0.169436	−	0.0978239i
$453$		0			0
$454$	4.42560	−	2.55512i	0.207704	−	0.119918i
$455$	−0.944020	−	0.229197i	−0.0442563	−	0.0107449i
$456$		0			0
$457$	−4.97751	+	8.62130i	−0.232838	+	0.403287i	−0.958642	−	0.284614i	$-0.908135\pi$
$457$	−4.97751	+	8.62130i	−0.232838	+	0.403287i	0.725804	+	0.687901i	$0.241468\pi$
$458$	15.2669			0.713375
$459$		0			0
$460$			2.82082i			0.131521i
$461$	16.1635	−	27.9960i	0.752810	−	1.30391i	−0.193645	−	0.981072i	$-0.562031\pi$
$461$	16.1635	−	27.9960i	0.752810	−	1.30391i	0.946456	−	0.322834i	$-0.104636\pi$
$462$		0			0
$463$	−4.72516	−	8.18421i	−0.219597	−	0.380353i	0.735088	−	0.677972i	$-0.237141\pi$
$463$	−4.72516	−	8.18421i	−0.219597	−	0.380353i	−0.954685	+	0.297619i	$0.903807\pi$
$464$	1.58394	−	0.914490i	0.0735327	−	0.0424541i
$465$		0			0
$466$	5.09651	−	8.82741i	0.236091	−	0.408922i
$467$	−20.6623			−0.956138			−0.478069	−	0.878322i	$-0.658663\pi$
$467$	−20.6623			−0.956138			−0.478069	+	0.878322i	$0.658663\pi$
$468$		0			0
$469$	20.8502	+	19.8878i	0.962773	+	0.918335i
$470$	2.64432	+	1.52670i	0.121973	+	0.0704214i
$471$		0			0
$472$	−7.55776	+	4.36348i	−0.347874	+	0.200845i
$473$	−2.60803	+	1.50575i	−0.119917	+	0.0692343i
$474$		0			0
$475$	26.7877	+	15.4659i	1.22910	+	0.709623i
$476$	9.55151	+	9.11064i	0.437793	+	0.417586i
$477$		0			0
$478$	−19.1815			−0.877343
$479$	5.08042	−	8.79955i	0.232131	−	0.402062i	−0.726304	−	0.687373i	$-0.758764\pi$
$479$	5.08042	−	8.79955i	0.232131	−	0.402062i	0.958435	+	0.285311i	$0.0920970\pi$
$480$		0			0
$481$	4.48252	−	2.58799i	0.204386	−	0.118002i
$482$	−10.3426	−	17.9140i	−0.471094	−	0.815958i
$483$		0			0
$484$	5.27592	−	9.13815i	0.239814	−	0.415371i
$485$		−	6.31126i		−	0.286579i
$486$		0			0
$487$	−31.2296			−1.41515			−0.707575	−	0.706638i	$-0.750211\pi$
$487$	−31.2296			−1.41515			−0.707575	+	0.706638i	$0.750211\pi$
$488$	−2.47886	+	4.29351i	−0.112213	+	0.194358i
$489$		0			0
$490$	−1.38661	−	2.15931i	−0.0626404	−	0.0975479i
$491$	−17.8314	+	10.2950i	−0.804720	+	0.464605i	−0.845119	−	0.534578i	$-0.820471\pi$
$491$	−17.8314	+	10.2950i	−0.804720	+	0.464605i	0.0403987	+	0.999184i	$0.487137\pi$
$492$		0			0
$493$	−7.90239	−	4.56245i	−0.355906	−	0.205482i
$494$			6.36717i			0.286473i
$495$		0			0
$496$			6.32588i			0.284040i
$497$	4.09598	+	13.9590i	0.183730	+	0.626145i
$498$		0			0
$499$	12.5766	+	21.7834i	0.563007	+	0.975157i	0.997232	+	0.0743527i	$0.0236891\pi$
$499$	12.5766	+	21.7834i	0.563007	+	0.975157i	−0.434225	+	0.900805i	$0.642978\pi$
$500$	1.80836	+	3.13216i	0.0808722	+	0.140075i
$501$		0			0
$502$	−1.56924	−	0.906002i	−0.0700387	−	0.0404369i
$503$	31.1553			1.38915			0.694574	−	0.719421i	$-0.255593\pi$
$503$	31.1553			1.38915			0.694574	+	0.719421i	$0.255593\pi$
$504$		0			0
$505$	5.76722			0.256638
$506$	4.46104	+	2.57558i	0.198317	+	0.114499i
$507$		0			0
$508$	−0.834727	−	1.44579i	−0.0370350	−	0.0641465i
$509$	2.41674	+	4.18591i	0.107120	+	0.185537i	0.914602	−	0.404354i	$-0.132504\pi$
$509$	2.41674	+	4.18591i	0.107120	+	0.185537i	−0.807482	+	0.589892i	$0.799170\pi$
$510$		0			0
$511$	−3.03423	−	10.3405i	−0.134226	−	0.457439i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			6.45545i			0.284738i
$515$	−3.63344	−	2.09777i	−0.160109	−	0.0924387i
$516$		0			0
$517$	4.82886	−	2.78794i	0.212373	−	0.122613i
$518$	13.2869	+	3.22592i	0.583795	+	0.141739i
$519$		0			0
$520$	−0.183586	+	0.317980i	−0.00805077	+	0.0139443i
$521$	−17.5322			−0.768101			−0.384050	−	0.923312i	$-0.625471\pi$
$521$	−17.5322			−0.768101			−0.384050	+	0.923312i	$0.625471\pi$
$522$		0			0
$523$			19.1019i			0.835267i	0.908616	+	0.417633i	$0.137140\pi$
$523$			19.1019i			0.835267i	−0.908616	+	0.417633i	$0.862860\pi$
$524$	6.76607	−	11.7192i	0.295577	−	0.511955i
$525$		0			0
$526$	−4.41031	−	7.63888i	−0.192299	−	0.333071i
$527$	27.3319	−	15.7801i	1.19060	−	0.687392i
$528$		0			0
$529$	18.1033	−	31.3559i	0.787101	−	1.36330i
$530$		0			0
$531$		0			0
$532$	−11.6091	+	12.1708i	−0.503317	+	0.527673i
$533$	−3.74584	−	2.16266i	−0.162250	−	0.0936752i
$534$		0			0
$535$	−3.51192	+	2.02761i	−0.151834	+	0.0876612i
$536$	9.43166	−	5.44537i	0.407386	−	0.235204i
$537$		0			0
$538$	12.3541	+	7.13267i	0.532625	+	0.307511i
$539$	−4.68095	+	0.221286i	−0.201623	+	0.00953144i
$540$		0			0
$541$	13.6642			0.587471			0.293735	−	0.955887i	$-0.405102\pi$
$541$	13.6642			0.587471			0.293735	+	0.955887i	$0.405102\pi$
$542$	−1.52641	+	2.64381i	−0.0655648	+	0.113562i
$543$		0			0
$544$	4.32065	−	2.49453i	0.185247	−	0.106952i
$545$	−1.93625	−	3.35368i	−0.0829397	−	0.143656i
$546$		0			0
$547$	4.94380	−	8.56292i	0.211382	−	0.366124i	−0.740765	−	0.671764i	$-0.765537\pi$
$547$	4.94380	−	8.56292i	0.211382	−	0.366124i	0.952147	+	0.305640i	$0.0988703\pi$
$548$		−	8.98851i		−	0.383970i
$549$		0			0
$550$	3.25730			0.138892
$551$	5.81362	−	10.0695i	0.247669	−	0.428975i
$552$		0			0
$553$	−5.21582	+	21.4829i	−0.221799	+	0.913548i
$554$	1.09609	−	0.632828i	0.0465684	−	0.0268863i
$555$		0			0
$556$	−8.05336	−	4.64961i	−0.341539	−	0.197187i
$557$		−	12.5800i		−	0.533034i	−0.963830	−	0.266517i	$-0.914127\pi$
$557$		−	12.5800i		−	0.533034i	0.963830	−	0.266517i	$-0.0858728\pi$
$558$		0			0
$559$			4.50547i			0.190561i
$560$	−0.930688	+	0.273092i	−0.0393287	+	0.0115402i
$561$		0			0
$562$	5.26335	+	9.11639i	0.222021	+	0.384552i
$563$	−12.1666	−	21.0732i	−0.512763	−	0.888132i	−0.999890	−	0.0148007i	$-0.995289\pi$
$563$	−12.1666	−	21.0732i	−0.512763	−	0.888132i	0.487127	−	0.873331i	$-0.338045\pi$
$564$		0			0
$565$	−1.32058	−	0.762437i	−0.0555572	−	0.0320760i
$566$	19.8718			0.835272
$567$		0			0
$568$	5.49843			0.230709
$569$	8.18746	+	4.72703i	0.343236	+	0.198167i	0.661702	−	0.749767i	$-0.269834\pi$
$569$	8.18746	+	4.72703i	0.343236	+	0.198167i	−0.318466	+	0.947934i	$0.603168\pi$
$570$		0			0
$571$	15.7843	+	27.3392i	0.660551	+	1.14411i	0.980471	+	0.196664i	$0.0630108\pi$
$571$	15.7843	+	27.3392i	0.660551	+	1.14411i	−0.319920	+	0.947445i	$0.603656\pi$
$572$	0.335250	+	0.580671i	0.0140175	+	0.0242791i
$573$		0			0
$574$	−3.21705	−	10.9636i	−0.134277	−	0.457612i
$575$		−	37.4388i		−	1.56131i
$576$		0			0
$577$			33.5794i			1.39793i	0.715157	+	0.698964i	$0.246355\pi$
$577$			33.5794i			1.39793i	−0.715157	+	0.698964i	$0.753645\pi$
$578$	−6.83358	−	3.94537i	−0.284239	−	0.164106i
$579$		0			0
$580$	0.580671	−	0.335250i	0.0241110	−	0.0139205i
$581$	−10.6207	+	43.7446i	−0.440621	+	1.81483i
$582$		0			0
$583$		0			0
$584$	−4.07314			−0.168548
$585$		0			0
$586$			13.4121i			0.554049i
$587$	9.65855	−	16.7291i	0.398651	−	0.690484i	−0.594909	−	0.803793i	$-0.702812\pi$
$587$	9.65855	−	16.7291i	0.398651	−	0.690484i	0.993560	+	0.113310i	$0.0361452\pi$
$588$		0			0
$589$	20.1075	+	34.8272i	0.828516	+	1.43503i
$590$	−2.77066	+	1.59964i	−0.114066	+	0.0658562i
$591$		0			0
$592$	2.58394	−	4.47552i	0.106199	−	0.183943i
$593$	−0.733196			−0.0301088			−0.0150544	−	0.999887i	$-0.504792\pi$
$593$	−0.733196			−0.0301088			−0.0150544	+	0.999887i	$0.504792\pi$
$594$		0			0
$595$	3.50157	+	3.33994i	0.143550	+	0.136924i
$596$	2.45268	+	1.41606i	0.100466	+	0.0580039i
$597$		0			0
$598$	6.67413	−	3.85331i	0.272926	−	0.157574i
$599$	−26.6548	+	15.3892i	−1.08909	+	0.628785i	−0.933333	−	0.359011i	$-0.883114\pi$
$599$	−26.6548	+	15.3892i	−1.08909	+	0.628785i	−0.155754	+	0.987796i	$0.549781\pi$
$600$		0			0
$601$	−0.786931	−	0.454335i	−0.0320996	−	0.0185327i	0.483864	−	0.875143i	$-0.339233\pi$
$601$	−0.786931	−	0.454335i	−0.0320996	−	0.0185327i	−0.515964	+	0.856610i	$0.672566\pi$
$602$	−8.21470	+	8.61221i	−0.334806	+	0.351007i
$603$		0			0
$604$	16.5518			0.673484
$605$	1.93414	−	3.35003i	0.0786340	−	0.136198i
$606$		0			0
$607$	38.7783	−	22.3887i	1.57396	−	0.908728i	0.578287	−	0.815833i	$-0.303721\pi$
$607$	38.7783	−	22.3887i	1.57396	−	0.908728i	0.995676	−	0.0928949i	$-0.0296121\pi$
$608$	3.17861	+	5.50552i	0.128910	+	0.223278i
$609$		0			0
$610$	−0.908744	+	1.57399i	−0.0367940	+	0.0637290i
$611$		−	8.34204i		−	0.337483i
$612$		0			0
$613$	18.1480			0.732992			0.366496	−	0.930420i	$-0.380557\pi$
$613$	18.1480			0.732992			0.366496	+	0.930420i	$0.380557\pi$
$614$	−0.326864	+	0.566145i	−0.0131912	+	0.0228478i
$615$		0			0
$616$	−0.417889	+	1.72120i	−0.0168372	+	0.0693493i
$617$	19.7393	−	11.3965i	0.794674	−	0.458805i	−0.0469315	−	0.998898i	$-0.514944\pi$
$617$	19.7393	−	11.3965i	0.794674	−	0.458805i	0.841605	+	0.540093i	$0.181611\pi$
$618$		0			0
$619$	−38.4228	−	22.1834i	−1.54434	−	0.891626i	−0.998557	−	0.0537011i	$-0.982898\pi$
$619$	−38.4228	−	22.1834i	−1.54434	−	0.891626i	−0.545785	−	0.837925i	$-0.683768\pi$
$620$			2.31905i			0.0931355i
$621$		0			0
$622$			9.24493i			0.370688i
$623$	−27.1930	+	7.97924i	−1.08946	+	0.319682i
$624$		0			0
$625$	−11.5011	−	19.9204i	−0.460043	−	0.796818i
$626$	3.08207	+	5.33830i	0.123184	+	0.213361i
$627$		0			0
$628$	2.45480	+	1.41728i	0.0979571	+	0.0565555i
$629$	−25.7829			−1.02803
$630$		0			0
$631$	−32.5707			−1.29662			−0.648310	−	0.761377i	$-0.724524\pi$
$631$	−32.5707			−1.29662			−0.648310	+	0.761377i	$0.724524\pi$
$632$	7.23623	+	4.17784i	0.287842	+	0.166186i
$633$		0			0
$634$	−10.3274	−	17.8876i	−0.410154	−	0.710408i
$635$	−0.306009	−	0.530024i	−0.0121436	−	0.0210333i
$636$		0			0
$637$	−3.21486	+	6.23042i	−0.127377	+	0.246858i
$638$		−	1.22442i		−	0.0484751i
$639$		0			0
$640$			0.366598i			0.0144911i
$641$	10.2270	+	5.90456i	0.403942	+	0.233216i	0.688184	−	0.725537i	$-0.258408\pi$
$641$	10.2270	+	5.90456i	0.403942	+	0.233216i	−0.284241	+	0.958753i	$0.591742\pi$
$642$		0			0
$643$	25.3714	−	14.6482i	1.00055	−	0.577668i	0.0921392	−	0.995746i	$-0.470630\pi$
$643$	25.3714	−	14.6482i	1.00055	−	0.577668i	0.908411	+	0.418078i	$0.137296\pi$
$644$	19.7832	+	4.80315i	0.779569	+	0.189270i
$645$		0			0
$646$	15.8583	−	27.4674i	0.623936	−	1.08069i
$647$	28.1683			1.10741			0.553705	−	0.832713i	$-0.313214\pi$
$647$	28.1683			1.10741			0.553705	+	0.832713i	$0.313214\pi$
$648$		0			0
$649$			5.84229i			0.229330i
$650$	2.43661	−	4.22033i	0.0955717	−	0.165535i
$651$		0			0
$652$	12.3640	+	21.4151i	0.484213	+	0.838682i
$653$	−39.0555	+	22.5487i	−1.52836	+	0.882399i	−0.528929	+	0.848666i	$0.677406\pi$
$653$	−39.0555	+	22.5487i	−1.52836	+	0.882399i	−0.999431	+	0.0337326i	$0.989261\pi$
$654$		0			0
$655$	2.48043	−	4.29623i	0.0969184	−	0.167868i
$656$	−4.31856			−0.168611
$657$		0			0
$658$	15.2098	−	15.9458i	0.592939	−	0.621632i
$659$	−27.5435	−	15.9022i	−1.07294	−	0.619463i	−0.143958	−	0.989584i	$-0.545983\pi$
$659$	−27.5435	−	15.9022i	−1.07294	−	0.619463i	−0.928984	+	0.370121i	$0.879316\pi$
$660$		0			0
$661$	17.1234	−	9.88619i	0.666022	−	0.384528i	−0.128546	−	0.991704i	$-0.541031\pi$
$661$	17.1234	−	9.88619i	0.666022	−	0.384528i	0.794568	+	0.607175i	$0.207698\pi$
$662$	9.27631	−	5.35568i	0.360534	−	0.208154i
$663$		0			0
$664$	14.7348	+	8.50712i	0.571820	+	0.330140i
$665$	−4.25587	+	4.46181i	−0.165035	+	0.173022i
$666$		0			0
$667$	−14.0733			−0.544918
$668$	−9.67422	+	16.7562i	−0.374307	+	0.648318i
$669$		0			0
$670$	3.45763	−	1.99626i	0.133580	−	0.0771223i
$671$	1.65948	+	2.87430i	0.0640635	+	0.110961i
$672$		0			0
$673$	−0.945369	+	1.63743i	−0.0364413	+	0.0631182i	−0.883671	−	0.468109i	$-0.844936\pi$
$673$	−0.945369	+	1.63743i	−0.0364413	+	0.0631182i	0.847230	+	0.531227i	$0.178269\pi$
$674$		−	7.55183i		−	0.290886i
$675$		0			0
$676$	−11.9969			−0.461418
$677$	−10.5661	+	18.3010i	−0.406088	+	0.703364i	−0.994447	−	0.105235i	$-0.966441\pi$
$677$	−10.5661	+	18.3010i	−0.406088	+	0.703364i	0.588360	+	0.808599i	$0.299774\pi$
$678$		0			0
$679$	−44.2627	−	10.7465i	−1.69865	−	0.412412i
$680$	1.58394	−	0.914490i	0.0607415	−	0.0350691i
$681$		0			0
$682$	3.66751	+	2.11744i	0.140436	+	0.0810810i
$683$		−	8.71972i		−	0.333651i	−0.985986	−	0.166825i	$-0.946648\pi$
$683$		−	8.71972i		−	0.333651i	0.985986	−	0.166825i	$-0.0533516\pi$
$684$		0			0
$685$		−	3.29517i		−	0.125902i
$686$	−17.5049	+	6.04790i	−0.668342	+	0.230910i
$687$		0			0
$688$	2.24922	+	3.89576i	0.0857506	+	0.148524i
$689$		0			0
$690$		0			0
$691$	−15.7071	−	9.06850i	−0.597526	−	0.344982i	0.170542	−	0.985350i	$-0.445448\pi$
$691$	−15.7071	−	9.06850i	−0.597526	−	0.344982i	−0.768068	+	0.640369i	$0.778782\pi$
$692$	4.83654			0.183858
$693$		0			0
$694$	−10.9320			−0.414972
$695$	−2.95235	−	1.70454i	−0.111989	−	0.0646568i
$696$		0			0
$697$	10.7728	+	18.6590i	0.408048	+	0.706760i
$698$	−0.592145	−	1.02562i	−0.0224130	−	0.0388205i
$699$		0			0
$700$	12.3524	−	3.62456i	0.466876	−	0.136996i
$701$			35.6167i			1.34523i	0.739995	+	0.672613i	$0.234828\pi$
$701$			35.6167i			1.34523i	−0.739995	+	0.672613i	$0.765172\pi$
$702$		0			0
$703$		−	32.8534i		−	1.23909i
$704$	0.579764	+	0.334727i	0.0218507	+	0.0126155i
$705$		0			0
$706$	29.0832	−	16.7912i	1.09456	−	0.631946i
$707$	9.82012	−	40.4472i	0.369324	−	1.52117i
$708$		0			0
$709$	1.80385	−	3.12436i	0.0677449	−	0.117338i	−0.830163	−	0.557520i	$-0.811753\pi$
$709$	1.80385	−	3.12436i	0.0677449	−	0.117338i	0.897908	+	0.440183i	$0.145086\pi$
$710$	2.01572			0.0756485
$711$		0			0
$712$			10.7113i			0.401424i
$713$	24.3375	−	42.1538i	0.911447	−	1.57867i
$714$		0			0
$715$	0.122902	+	0.212873i	0.00459628	+	0.00796099i
$716$	−3.16789	+	1.82898i	−0.118390	+	0.0683522i
$717$		0			0
$718$	−5.06407	+	8.77122i	−0.188989	+	0.327339i
$719$	25.7829			0.961540			0.480770	−	0.876847i	$-0.340357\pi$
$719$	25.7829			0.961540			0.480770	+	0.876847i	$0.340357\pi$
$720$		0			0
$721$	−20.8991	+	21.9104i	−0.778323	+	0.815987i
$722$	18.5453	+	10.7072i	0.690186	+	0.398479i
$723$		0			0
$724$	4.90860	−	2.83398i	0.182427	−	0.105324i
$725$	−7.70685	+	4.44955i	−0.286225	+	0.165252i
$726$		0			0
$727$	1.32423	+	0.764544i	0.0491129	+	0.0283554i	0.524355	−	0.851499i	$-0.324306\pi$
$727$	1.32423	+	0.764544i	0.0491129	+	0.0283554i	−0.475242	+	0.879855i	$0.657640\pi$
$728$	1.91749	+	1.82898i	0.0710668	+	0.0677865i
$729$		0			0
$730$	−1.49321			−0.0552660
$731$	11.2215	−	19.4362i	0.415042	−	0.718873i
$732$		0			0
$733$	−17.9908	+	10.3870i	−0.664504	+	0.383651i	−0.793991	−	0.607930i	$-0.792000\pi$
$733$	−17.9908	+	10.3870i	−0.664504	+	0.383651i	0.129487	+	0.991581i	$0.458667\pi$
$734$	9.00104	+	15.5903i	0.332234	+	0.575447i
$735$		0			0
$736$	3.84729	−	6.66371i	0.141813	−	0.245628i
$737$		−	7.29084i		−	0.268562i
$738$		0			0
$739$	−11.8709			−0.436678			−0.218339	−	0.975873i	$-0.570064\pi$
$739$	−11.8709			−0.436678			−0.218339	+	0.975873i	$0.570064\pi$
$740$	0.947269	−	1.64072i	0.0348223	−	0.0603140i
$741$		0			0
$742$		0			0
$743$	37.5906	−	21.7029i	1.37907	−	0.796204i	0.387019	−	0.922072i	$-0.373505\pi$
$743$	37.5906	−	21.7029i	1.37907	−	0.796204i	0.992047	+	0.125868i	$0.0401716\pi$
$744$		0			0
$745$	0.899148	+	0.519124i	0.0329422	+	0.0190192i
$746$			16.4090i			0.600777i
$747$		0			0
$748$		−	3.33994i		−	0.122120i
$749$	8.24030	+	28.0826i	0.301094	+	1.02612i
$750$		0			0
$751$	−1.15691	−	2.00383i	−0.0422164	−	0.0731209i	0.844145	−	0.536115i	$-0.180109\pi$
$751$	−1.15691	−	2.00383i	−0.0422164	−	0.0731209i	−0.886362	+	0.462994i	$0.846775\pi$
$752$	−4.16450	−	7.21313i	−0.151864	−	0.263036i
$753$		0			0
$754$	−1.58642	−	0.915921i	−0.0577741	−	0.0333559i
$755$	6.06787			0.220832
$756$		0			0
$757$	−15.0946			−0.548624			−0.274312	−	0.961641i	$-0.588450\pi$
$757$	−15.0946			−0.548624			−0.274312	+	0.961641i	$0.588450\pi$
$758$	2.52336	+	1.45686i	0.0916525	+	0.0529156i
$759$		0			0
$760$	1.16527	+	2.01831i	0.0422689	+	0.0732119i
$761$	−11.6690	−	20.2112i	−0.422999	−	0.732656i	0.573232	−	0.819393i	$-0.305689\pi$
$761$	−11.6690	−	20.2112i	−0.422999	−	0.732656i	−0.996231	+	0.0867370i	$0.972356\pi$
$762$		0			0
$763$	−26.8173	+	7.86900i	−0.970850	+	0.284877i
$764$		−	27.3777i		−	0.990490i
$765$		0			0
$766$		−	8.57443i		−	0.309807i
$767$	7.56959	+	4.37030i	0.273322	+	0.157803i
$768$		0			0
$769$	−15.8266	+	9.13748i	−0.570721	+	0.329506i	−0.757437	−	0.652908i	$-0.773549\pi$
$769$	−15.8266	+	9.13748i	−0.570721	+	0.329506i	0.186716	+	0.982414i	$0.440216\pi$
$770$	−0.153197	+	0.630990i	−0.00552085	+	0.0227393i
$771$		0			0
$772$	−5.01413	+	8.68473i	−0.180463	+	0.312570i
$773$	0.438507			0.0157720			0.00788600	−	0.999969i	$-0.497490\pi$
$773$	0.438507			0.0157720			0.00788600	+	0.999969i	$0.497490\pi$
$774$		0			0
$775$		−	30.7792i		−	1.10562i
$776$	−8.60787	+	14.9093i	−0.309004	+	0.535211i
$777$		0			0
$778$	−17.7770	−	30.7906i	−0.637335	−	1.10390i
$779$	−23.7759	+	13.7270i	−0.851861	+	0.491822i
$780$		0			0
$781$	1.84047	−	3.18779i	0.0658573	−	0.114068i
$782$	−38.3888			−1.37278
$783$		0			0
$784$	0.330547	+	6.99219i	0.0118052	+	0.249721i
$785$	0.899924	+	0.519571i	0.0321197	+	0.0185443i
$786$		0			0
$787$	−33.1317	+	19.1286i	−1.18102	+	0.681861i	−0.956250	−	0.292551i	$-0.905496\pi$
$787$	−33.1317	+	19.1286i	−1.18102	+	0.681861i	−0.224769	+	0.974412i	$0.572163\pi$
$788$	16.3037	−	9.41292i	0.580794	−	0.335322i
$789$		0			0
$790$	2.65279	+	1.53159i	0.0943820	+	0.0544915i
$791$	−7.59581	+	7.96337i	−0.270076	+	0.283145i
$792$		0			0
$793$	4.96547			0.176329
$794$	1.79017	−	3.10066i	0.0635307	−	0.110038i
$795$		0			0
$796$	−4.64541	+	2.68203i	−0.164652	+	0.0950620i
$797$	−17.6613	−	30.5902i	−0.625594	−	1.08356i	−0.988426	−	0.151706i	$-0.951523\pi$
$797$	−17.6613	−	30.5902i	−0.625594	−	1.08356i	0.362832	−	0.931855i	$-0.381810\pi$
$798$		0			0
$799$	−20.7770	+	35.9868i	−0.735036	+	1.27312i
$800$		−	4.86561i		−	0.172025i
$801$		0			0
$802$	0.190871			0.00673991
$803$	−1.36339	+	2.36146i	−0.0481130	+	0.0833341i
$804$		0			0
$805$	7.25250	+	1.76082i	0.255617	+	0.0620609i
$806$	5.48694	−	3.16789i	0.193269	−	0.111584i
$807$		0			0
$808$	−13.6241	−	7.86586i	−0.479293	−	0.276720i
$809$			21.7669i			0.765282i	0.923897	+	0.382641i	$0.124985\pi$
$809$			21.7669i			0.765282i	−0.923897	+	0.382641i	$0.875015\pi$
$810$		0			0
$811$			17.0184i			0.597598i	0.954316	+	0.298799i	$0.0965860\pi$
$811$			17.0184i			0.597598i	−0.954316	+	0.298799i	$0.903414\pi$
$812$	−1.36247	−	4.64326i	−0.0478134	−	0.162946i
$813$		0			0
$814$	−1.72983	−	2.99615i	−0.0606305	−	0.105015i
$815$	4.53263	+	7.85075i	0.158771	+	0.275000i
$816$		0			0
$817$	24.7662	+	14.2988i	0.866460	+	0.500251i
$818$	3.47371			0.121455
$819$		0			0
$820$	−1.58318			−0.0552869
$821$	21.4786	+	12.4007i	0.749608	+	0.432786i	0.825552	−	0.564326i	$-0.190864\pi$
$821$	21.4786	+	12.4007i	0.749608	+	0.432786i	−0.0759445	+	0.997112i	$0.524197\pi$
$822$		0			0
$823$	−10.6572	−	18.4588i	−0.371486	−	0.643433i	0.618308	−	0.785936i	$-0.287818\pi$
$823$	−10.6572	−	18.4588i	−0.371486	−	0.643433i	−0.989794	+	0.142503i	$0.954485\pi$
$824$	5.72226	+	9.91124i	0.199344	+	0.345274i
$825$		0			0
$826$	6.50102	+	22.1552i	0.226199	+	0.770879i
$827$			49.7585i			1.73027i	0.501537	+	0.865136i	$0.332768\pi$
$827$			49.7585i			1.73027i	−0.501537	+	0.865136i	$0.667232\pi$
$828$		0			0
$829$		−	43.1190i		−	1.49759i	−0.662804	−	0.748793i	$-0.730634\pi$
$829$		−	43.1190i		−	1.49759i	0.662804	−	0.748793i	$-0.269366\pi$
$830$	5.40174	+	3.11870i	0.187497	+	0.108252i
$831$		0			0
$832$	0.867380	−	0.500782i	0.0300710	−	0.0173615i
$833$	29.3863	−	18.8704i	1.01817	−	0.653821i
$834$		0			0
$835$	−3.54655	+	6.14281i	−0.122733	+	0.212581i
$836$	4.25587			0.147192
$837$		0			0
$838$			1.40791i			0.0486354i
$839$	−14.9985	+	25.9782i	−0.517807	+	0.896868i	0.481979	+	0.876183i	$0.339918\pi$
$839$	−14.9985	+	25.9782i	−0.517807	+	0.896868i	−0.999786	+	0.0206851i	$0.993415\pi$
$840$		0			0
$841$	−12.8274	−	22.2177i	−0.442325	−	0.766129i
$842$	−26.3151	+	15.1930i	−0.906878	+	0.523586i
$843$		0			0
$844$	0.828981	−	1.43584i	0.0285347	−	0.0494235i
$845$	−4.39803			−0.151297
$846$		0			0
$847$	−20.2014	−	19.2689i	−0.694128	−	0.662089i
$848$		0			0
$849$		0			0
$850$	−21.0226	+	12.1374i	−0.721069	+	0.416310i
$851$	−34.4373	+	19.8824i	−1.18050	+	0.681559i
$852$		0			0
$853$	−25.7693	−	14.8779i	−0.882325	−	0.509411i	−0.0109007	−	0.999941i	$-0.503470\pi$
$853$	−25.7693	−	14.8779i	−0.882325	−	0.509411i	−0.871424	+	0.490530i	$0.836803\pi$
$854$	9.49150	+	9.05340i	0.324792	+	0.309801i
$855$		0			0
$856$	11.0618			0.378083
$857$	−22.9296	+	39.7152i	−0.783260	+	1.35665i	0.146773	+	0.989170i	$0.453111\pi$
$857$	−22.9296	+	39.7152i	−0.783260	+	1.35665i	−0.930033	+	0.367476i	$0.880222\pi$
$858$		0			0
$859$	−3.24073	+	1.87104i	−0.110572	+	0.0638390i	−0.554266	−	0.832339i	$-0.687001\pi$
$859$	−3.24073	+	1.87104i	−0.110572	+	0.0638390i	0.443694	+	0.896178i	$0.353668\pi$
$860$	0.824559	+	1.42818i	0.0281172	+	0.0487005i
$861$		0			0
$862$	−13.6373	+	23.6206i	−0.464490	+	0.804520i
$863$			31.3944i			1.06868i	0.845270	+	0.534339i	$0.179439\pi$
$863$			31.3944i			1.06868i	−0.845270	+	0.534339i	$0.820561\pi$
$864$		0			0
$865$	1.77307			0.0602860
$866$	−4.07523	+	7.05851i	−0.138482	+	0.239858i
$867$		0			0
$868$	16.2642	+	3.94876i	0.552043	+	0.134030i
$869$	4.84432	−	2.79687i	0.164332	−	0.0948773i
$870$		0			0
$871$	−9.44641	−	5.45389i	−0.320080	−	0.184798i
$872$			10.5633i			0.357719i
$873$		0			0
$874$		−	48.9162i		−	1.65462i
$875$	9.18180	−	2.69422i	0.310402	−	0.0910812i
$876$		0			0
$877$	10.1962	+	17.6603i	0.344300	+	0.596344i	0.985226	−	0.171258i	$-0.0547831\pi$
$877$	10.1962	+	17.6603i	0.344300	+	0.596344i	−0.640927	+	0.767602i	$0.721450\pi$
$878$	−6.12020	−	10.6005i	−0.206547	−	0.357750i
$879$		0			0
$880$	0.212540	+	0.122710i	0.00716473	+	0.00413656i
$881$	−21.2010			−0.714280			−0.357140	−	0.934051i	$-0.616248\pi$
$881$	−21.2010			−0.714280			−0.357140	+	0.934051i	$0.616248\pi$
$882$		0			0
$883$	38.6157			1.29952			0.649761	−	0.760139i	$-0.274869\pi$
$883$	38.6157			1.29952			0.649761	+	0.760139i	$0.274869\pi$
$884$	−4.32741	−	2.49843i	−0.145547	−	0.0840314i
$885$		0			0
$886$	4.00418	+	6.93544i	0.134523	+	0.233001i
$887$	3.09606	+	5.36253i	0.103955	+	0.180056i	0.913311	−	0.407263i	$-0.133517\pi$
$887$	3.09606	+	5.36253i	0.103955	+	0.180056i	−0.809356	+	0.587319i	$0.800183\pi$
$888$		0			0
$889$	−4.23827	+	1.24364i	−0.142147	+	0.0417102i
$890$			3.92675i			0.131625i
$891$		0			0
$892$			17.0372i			0.570448i
$893$	−45.8555	−	26.4747i	−1.53450	−	0.885941i
$894$		0			0
$895$	−1.16134	+	0.670501i	−0.0388194	+	0.0224124i
$896$	2.57106	+	0.624224i	0.0858931	+	0.0208539i
$897$		0			0
$898$	7.25917	−	12.5733i	0.242242	−	0.419575i
$899$	−11.5699			−0.385878
$900$		0			0
$901$		0			0
$902$	−1.44554	+	2.50374i	−0.0481311	+	0.0833656i
$903$		0			0
$904$	2.07976	+	3.60226i	0.0691719	+	0.119809i
$905$	1.79948	−	1.03893i	0.0598168	−	0.0345353i
$906$		0			0
$907$	0.0645566	−	0.111815i	0.00214357	−	0.00371277i	−0.864952	−	0.501855i	$-0.832651\pi$
$907$	0.0645566	−	0.111815i	0.00214357	−	0.00371277i	0.867095	+	0.498142i	$0.165984\pi$
$908$	−5.11024			−0.169589
$909$		0			0
$910$	0.702947	+	0.670501i	0.0233025	+	0.0222269i
$911$	−29.6682	−	17.1290i	−0.982952	−	0.567508i	−0.0797919	−	0.996812i	$-0.525426\pi$
$911$	−29.6682	−	17.1290i	−0.982952	−	0.567508i	−0.903160	+	0.429304i	$0.858759\pi$
$912$		0			0
$913$	9.86424	−	5.69512i	0.326459	−	0.188481i
$914$	8.62130	−	4.97751i	0.285167	−	0.164641i
$915$		0			0
$916$	−13.2215	−	7.63345i	−0.436851	−	0.252216i
$917$	−25.9072	−	24.7114i	−0.855530	−	0.816041i
$918$		0			0
$919$	14.3054			0.471892			0.235946	−	0.971766i	$-0.424181\pi$
$919$	14.3054			0.471892			0.235946	+	0.971766i	$0.424181\pi$
$920$	1.41041	−	2.44290i	0.0464999	−	0.0805401i
$921$		0			0
$922$	−27.9960	+	16.1635i	−0.922001	+	0.532317i
$923$	−2.75352	−	4.76923i	−0.0906332	−	0.156981i
$924$		0			0
$925$	−12.5725	+	21.7761i	−0.413380	+	0.715995i
$926$			9.45032i			0.310557i
$927$		0			0
$928$	−1.82898			−0.0600392
$929$	−5.87364	+	10.1734i	−0.192708	+	0.333780i	−0.946147	−	0.323738i	$-0.895060\pi$
$929$	−5.87364	+	10.1734i	−0.192708	+	0.333780i	0.753439	+	0.657518i	$0.228394\pi$
$930$		0			0
$931$	24.0453	+	37.4450i	0.788053	+	1.22721i
$932$	−8.82741	+	5.09651i	−0.289152	+	0.166942i
$933$		0			0
$934$	17.8941	+	10.3312i	0.585512	+	0.338046i
$935$		−	1.22442i		−	0.0400427i
$936$		0			0
$937$		−	2.63611i		−	0.0861179i	−0.999073	−	0.0430589i	$-0.986290\pi$
$937$		−	2.63611i		−	0.0861179i	0.999073	−	0.0430589i	$-0.0137103\pi$
$938$	−8.11290	−	27.6485i	−0.264896	−	0.902755i
$939$		0			0
$940$	−1.52670	−	2.64432i	−0.0497954	−	0.0862482i
$941$	5.96557	+	10.3327i	0.194472	+	0.336836i	0.946727	−	0.322036i	$-0.104367\pi$
$941$	5.96557	+	10.3327i	0.194472	+	0.336836i	−0.752255	+	0.658872i	$0.771034\pi$
$942$		0			0
$943$	28.7776	+	16.6148i	0.937128	+	0.541051i
$944$	8.72695			0.284038
$945$		0			0
$946$	3.01149			0.0979121
$947$	−6.70267	−	3.86979i	−0.217807	−	0.125751i	0.387127	−	0.922026i	$-0.373467\pi$
$947$	−6.70267	−	3.86979i	−0.217807	−	0.125751i	−0.604935	+	0.796275i	$0.706801\pi$
$948$		0			0
$949$	2.03976	+	3.53296i	0.0662133	+	0.114685i
$950$	−15.4659	−	26.7877i	−0.501779	−	0.869107i
$951$		0			0
$952$	−3.71653	−	12.6658i	−0.120453	−	0.410501i
$953$			3.76685i			0.122020i	0.998137	+	0.0610102i	$0.0194322\pi$
$953$			3.76685i			0.122020i	−0.998137	+	0.0610102i	$0.980568\pi$
$954$		0			0
$955$		−	10.0366i		−	0.324777i
$956$	16.6117	+	9.59076i	0.537260	+	0.310187i
$957$		0			0
$958$	−8.79955	+	5.08042i	−0.284301	+	0.164141i
$959$	−23.1100	−	5.61084i	−0.746260	−	0.181184i
$960$		0			0
$961$	4.50836	−	7.80871i	0.145431	−	0.251894i
$962$	−5.17597			−0.166880
$963$		0			0
$964$			20.6853i			0.666227i
$965$	−1.83817	+	3.18381i	−0.0591728	+	0.102490i
$966$		0			0
$967$	−2.28741	−	3.96191i	−0.0735581	−	0.127406i	0.826900	−	0.562349i	$-0.190102\pi$
$967$	−2.28741	−	3.96191i	−0.0735581	−	0.127406i	−0.900458	+	0.434942i	$0.856769\pi$
$968$	−9.13815	+	5.27592i	−0.293711	+	0.169574i
$969$		0			0
$970$	−3.15563	+	5.46571i	−0.101321	+	0.175493i
$971$	25.8445			0.829388			0.414694	−	0.909961i	$-0.363889\pi$
$971$	25.8445			0.829388			0.414694	+	0.909961i	$0.363889\pi$
$972$		0			0
$973$	−16.9815	+	17.8033i	−0.544403	+	0.570747i
$974$	27.0457	+	15.6148i	0.866599	+	0.500331i
$975$		0			0
$976$	4.29351	−	2.47886i	0.137432	−	0.0793463i
$977$	26.0950	−	15.0659i	0.834852	−	0.482002i	−0.0206590	−	0.999787i	$-0.506576\pi$
$977$	26.0950	−	15.0659i	0.834852	−	0.482002i	0.855511	+	0.517785i	$0.173243\pi$
$978$		0			0
$979$	6.21003	+	3.58536i	0.198474	+	0.114589i
$980$	0.121178	+	2.56332i	0.00387088	+	0.0818824i
$981$		0			0
$982$	20.5899			0.657051
$983$	6.30293	−	10.9170i	0.201032	−	0.348198i	−0.747829	−	0.663891i	$-0.768904\pi$
$983$	6.30293	−	10.9170i	0.201032	−	0.348198i	0.948861	+	0.315693i	$0.102237\pi$
$984$		0			0
$985$	5.97689	−	3.45076i	0.190440	−	0.109950i
$986$	4.56245	+	7.90239i	0.145298	+	0.251663i
$987$		0			0
$988$	3.18359	−	5.51413i	0.101283	−	0.175428i
$989$		−	34.6136i		−	1.10065i
$990$		0			0
$991$	51.6852			1.64184			0.820918	−	0.571046i	$-0.193462\pi$
$991$	51.6852			1.64184			0.820918	+	0.571046i	$0.193462\pi$
$992$	3.16294	−	5.47837i	0.100423	−	0.173938i
$993$		0			0
$994$	3.43226	−	14.1368i	0.108865	−	0.448392i
$995$	−1.70300	+	0.983227i	−0.0539887	+	0.0311704i
$996$		0			0
$997$	35.1469	+	20.2921i	1.11311	+	0.642656i	0.939634	−	0.342181i	$-0.111166\pi$
$997$	35.1469	+	20.2921i	1.11311	+	0.642656i	0.173479	+	0.984837i	$0.444499\pi$
$998$		−	25.1533i		−	0.796213i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.m.a.251.3		16
3.2	odd	2		126.2.m.a.83.6	yes	16
4.3	odd	2		3024.2.cc.b.2897.5		16
7.2	even	3		2646.2.l.b.521.7		16
7.3	odd	6		2646.2.t.a.1979.7		16
7.4	even	3		2646.2.t.a.1979.6		16
7.5	odd	6		2646.2.l.b.521.6		16
7.6	odd	2	inner	378.2.m.a.251.2		16
9.2	odd	6		1134.2.d.a.1133.5		16
9.4	even	3		126.2.m.a.41.7	yes	16
9.5	odd	6	inner	378.2.m.a.125.2		16
9.7	even	3		1134.2.d.a.1133.12		16
12.11	even	2		1008.2.cc.b.209.5		16
21.2	odd	6		882.2.l.a.227.4		16
21.5	even	6		882.2.l.a.227.1		16
21.11	odd	6		882.2.t.b.803.1		16
21.17	even	6		882.2.t.b.803.4		16
21.20	even	2		126.2.m.a.83.7	yes	16
28.27	even	2		3024.2.cc.b.2897.4		16
36.23	even	6		3024.2.cc.b.881.4		16
36.31	odd	6		1008.2.cc.b.545.4		16
63.4	even	3		882.2.l.a.509.5		16
63.5	even	6		2646.2.t.a.2285.6		16
63.13	odd	6		126.2.m.a.41.6	✓	16
63.20	even	6		1134.2.d.a.1133.4		16
63.23	odd	6		2646.2.t.a.2285.7		16
63.31	odd	6		882.2.l.a.509.8		16
63.32	odd	6		2646.2.l.b.1097.2		16
63.34	odd	6		1134.2.d.a.1133.13		16
63.40	odd	6		882.2.t.b.815.1		16
63.41	even	6	inner	378.2.m.a.125.3		16
63.58	even	3		882.2.t.b.815.4		16
63.59	even	6		2646.2.l.b.1097.3		16
84.83	odd	2		1008.2.cc.b.209.4		16
252.139	even	6		1008.2.cc.b.545.5		16
252.167	odd	6		3024.2.cc.b.881.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.m.a.41.6	✓	16	63.13	odd	6
126.2.m.a.41.7	yes	16	9.4	even	3
126.2.m.a.83.6	yes	16	3.2	odd	2
126.2.m.a.83.7	yes	16	21.20	even	2
378.2.m.a.125.2		16	9.5	odd	6	inner
378.2.m.a.125.3		16	63.41	even	6	inner
378.2.m.a.251.2		16	7.6	odd	2	inner
378.2.m.a.251.3		16	1.1	even	1	trivial
882.2.l.a.227.1		16	21.5	even	6
882.2.l.a.227.4		16	21.2	odd	6
882.2.l.a.509.5		16	63.4	even	3
882.2.l.a.509.8		16	63.31	odd	6
882.2.t.b.803.1		16	21.11	odd	6
882.2.t.b.803.4		16	21.17	even	6
882.2.t.b.815.1		16	63.40	odd	6
882.2.t.b.815.4		16	63.58	even	3
1008.2.cc.b.209.4		16	84.83	odd	2
1008.2.cc.b.209.5		16	12.11	even	2
1008.2.cc.b.545.4		16	36.31	odd	6
1008.2.cc.b.545.5		16	252.139	even	6
1134.2.d.a.1133.4		16	63.20	even	6
1134.2.d.a.1133.5		16	9.2	odd	6
1134.2.d.a.1133.12		16	9.7	even	3
1134.2.d.a.1133.13		16	63.34	odd	6
2646.2.l.b.521.6		16	7.5	odd	6
2646.2.l.b.521.7		16	7.2	even	3
2646.2.l.b.1097.2		16	63.32	odd	6
2646.2.l.b.1097.3		16	63.59	even	6
2646.2.t.a.1979.6		16	7.4	even	3
2646.2.t.a.1979.7		16	7.3	odd	6
2646.2.t.a.2285.6		16	63.5	even	6
2646.2.t.a.2285.7		16	63.23	odd	6
3024.2.cc.b.881.4		16	36.23	even	6
3024.2.cc.b.881.5		16	252.167	odd	6
3024.2.cc.b.2897.4		16	28.27	even	2
3024.2.cc.b.2897.5		16	4.3	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.183299 + 0.317483i) q^{5} +(2.53871 - 0.744936i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.183299 + 0.317483i) q^{5} +(2.53871 - 0.744936i) q^{7} -1.00000i q^{8} -0.366598i q^{10} +(-0.579764 - 0.334727i) q^{11} +(-0.867380 + 0.500782i) q^{13} +(-2.57106 - 0.624224i) q^{14} +(-0.500000 + 0.866025i) q^{16} +4.98906 q^{17} +6.35722i q^{19} +(-0.183299 + 0.317483i) q^{20} +(0.334727 + 0.579764i) q^{22} +(6.66371 - 3.84729i) q^{23} +(2.43280 - 4.21374i) q^{25} +1.00156 q^{26} +(1.91449 + 1.82612i) q^{28} +(-1.58394 - 0.914490i) q^{29} +(5.47837 - 3.16294i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-4.32065 - 2.49453i) q^{34} +(0.701849 + 0.669453i) q^{35} -5.16789 q^{37} +(3.17861 - 5.50552i) q^{38} +(0.317483 - 0.183299i) q^{40} +(2.15928 + 3.73998i) q^{41} +(2.24922 - 3.89576i) q^{43} -0.669453i q^{44} -7.69459 q^{46} +(-4.16450 + 7.21313i) q^{47} +(5.89014 - 3.78236i) q^{49} +(-4.21374 + 2.43280i) q^{50} +(-0.867380 - 0.500782i) q^{52} -0.245420i q^{55} +(-0.744936 - 2.53871i) q^{56} +(0.914490 + 1.58394i) q^{58} +(-4.36348 - 7.55776i) q^{59} +(-4.29351 - 2.47886i) q^{61} -6.32588 q^{62} -1.00000 q^{64} +(-0.317980 - 0.183586i) q^{65} +(5.44537 + 9.43166i) q^{67} +(2.49453 + 4.32065i) q^{68} +(-0.273092 - 0.930688i) q^{70} +5.49843i q^{71} -4.07314i q^{73} +(4.47552 + 2.58394i) q^{74} +(-5.50552 + 3.17861i) q^{76} +(-1.72120 - 0.417889i) q^{77} +(-4.17784 + 7.23623i) q^{79} -0.366598 q^{80} -4.31856i q^{82} +(-8.50712 + 14.7348i) q^{83} +(0.914490 + 1.58394i) q^{85} +(-3.89576 + 2.24922i) q^{86} +(-0.334727 + 0.579764i) q^{88} -10.7113 q^{89} +(-1.82898 + 1.91749i) q^{91} +(6.66371 + 3.84729i) q^{92} +(7.21313 - 4.16450i) q^{94} +(-2.01831 + 1.16527i) q^{95} +(-14.9093 - 8.60787i) q^{97} +(-6.99219 + 0.330547i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} + 2 q^{7}+O(q^{10}) \) 16 * q + 8 * q^4 + 2 * q^7	\( 16 q + 8 q^{4} + 2 q^{7} + 12 q^{11} + 6 q^{14} - 8 q^{16} + 48 q^{23} - 8 q^{25} + 4 q^{28} + 12 q^{29} - 8 q^{37} + 4 q^{43} + 24 q^{46} - 8 q^{49} - 60 q^{50} + 6 q^{56} - 12 q^{58} - 16 q^{64} - 84 q^{65} - 28 q^{67} - 36 q^{74} - 78 q^{77} - 4 q^{79} - 12 q^{85} + 24 q^{86} + 24 q^{91} + 48 q^{92} - 12 q^{95}+O(q^{100}) \) 16 * q + 8 * q^4 + 2 * q^7 + 12 * q^11 + 6 * q^14 - 8 * q^16 + 48 * q^23 - 8 * q^25 + 4 * q^28 + 12 * q^29 - 8 * q^37 + 4 * q^43 + 24 * q^46 - 8 * q^49 - 60 * q^50 + 6 * q^56 - 12 * q^58 - 16 * q^64 - 84 * q^65 - 28 * q^67 - 36 * q^74 - 78 * q^77 - 4 * q^79 - 12 * q^85 + 24 * q^86 + 24 * q^91 + 48 * q^92 - 12 * q^95

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