LMFDB - Embedded newform 1386.2.k.b.793.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(793,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("1386.793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.k (of order $3$, degree $2$, 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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			793.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1386.793
Dual form			1386.2.k.b.991.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} +(0.500000 + 0.866025i) q^{11} +2.00000 q^{13} +(2.50000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.00000 + 1.73205i) q^{19} +3.00000 q^{20} -1.00000 q^{22} +(-1.50000 + 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(-1.00000 + 1.73205i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(-1.00000 - 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{32} -3.00000 q^{34} +(7.50000 + 2.59808i) q^{35} +(-4.00000 + 6.92820i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(-1.50000 + 2.59808i) q^{40} -9.00000 q^{41} -4.00000 q^{43} +(0.500000 - 0.866025i) q^{44} +(-1.50000 - 2.59808i) q^{46} +(-1.50000 + 2.59808i) q^{47} +(-6.50000 + 2.59808i) q^{49} +4.00000 q^{50} +(-1.00000 - 1.73205i) q^{52} +(-3.00000 - 5.19615i) q^{53} -3.00000 q^{55} +(-0.500000 - 2.59808i) q^{56} +(-3.00000 - 5.19615i) q^{59} +(-2.50000 + 4.33013i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +(-5.50000 - 9.52628i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-6.00000 + 5.19615i) q^{70} +(-1.00000 - 1.73205i) q^{73} +(-4.00000 - 6.92820i) q^{74} +2.00000 q^{76} +(2.00000 - 1.73205i) q^{77} +(6.50000 - 11.2583i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(4.50000 - 7.79423i) q^{82} +9.00000 q^{83} -9.00000 q^{85} +(2.00000 - 3.46410i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-6.00000 + 10.3923i) q^{89} +(-1.00000 - 5.19615i) q^{91} +3.00000 q^{92} +(-1.50000 - 2.59808i) q^{94} +(-3.00000 - 5.19615i) q^{95} +5.00000 q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 3 q^{5} - q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - 3 * q^5 - q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - 3 q^{5} - q^{7} + 2 q^{8} - 3 q^{10} + q^{11} + 4 q^{13} + 5 q^{14} - q^{16} + 3 q^{17} - 2 q^{19} + 6 q^{20} - 2 q^{22} - 3 q^{23} - 4 q^{25} - 2 q^{26} - 4 q^{28} - 2 q^{31} - q^{32} - 6 q^{34} + 15 q^{35} - 8 q^{37} - 2 q^{38} - 3 q^{40} - 18 q^{41} - 8 q^{43} + q^{44} - 3 q^{46} - 3 q^{47} - 13 q^{49} + 8 q^{50} - 2 q^{52} - 6 q^{53} - 6 q^{55} - q^{56} - 6 q^{59} - 5 q^{61} + 4 q^{62} + 2 q^{64} - 6 q^{65} - 11 q^{67} + 3 q^{68} - 12 q^{70} - 2 q^{73} - 8 q^{74} + 4 q^{76} + 4 q^{77} + 13 q^{79} - 3 q^{80} + 9 q^{82} + 18 q^{83} - 18 q^{85} + 4 q^{86} + q^{88} - 12 q^{89} - 2 q^{91} + 6 q^{92} - 3 q^{94} - 6 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 3 * q^5 - q^7 + 2 * q^8 - 3 * q^10 + q^11 + 4 * q^13 + 5 * q^14 - q^16 + 3 * q^17 - 2 * q^19 + 6 * q^20 - 2 * q^22 - 3 * q^23 - 4 * q^25 - 2 * q^26 - 4 * q^28 - 2 * q^31 - q^32 - 6 * q^34 + 15 * q^35 - 8 * q^37 - 2 * q^38 - 3 * q^40 - 18 * q^41 - 8 * q^43 + q^44 - 3 * q^46 - 3 * q^47 - 13 * q^49 + 8 * q^50 - 2 * q^52 - 6 * q^53 - 6 * q^55 - q^56 - 6 * q^59 - 5 * q^61 + 4 * q^62 + 2 * q^64 - 6 * q^65 - 11 * q^67 + 3 * q^68 - 12 * q^70 - 2 * q^73 - 8 * q^74 + 4 * q^76 + 4 * q^77 + 13 * q^79 - 3 * q^80 + 9 * q^82 + 18 * q^83 - 18 * q^85 + 4 * q^86 + q^88 - 12 * q^89 - 2 * q^91 + 6 * q^92 - 3 * q^94 - 6 * q^95 + 10 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	0.306851	+	0.951757i	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.50000	−	2.59808i	−0.474342	−	0.821584i
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$	2.50000	+	0.866025i	0.668153	+	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	0.728219	+	0.685344i	$0.240348\pi$
$20$	3.00000			0.670820
$21$		0			0
$22$	−1.00000			−0.213201
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	−0.978961	−	0.204046i	$-0.934591\pi$
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	0.666190	+	0.745782i	$0.267924\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$	−1.00000	+	1.73205i	−0.196116	+	0.339683i
$27$		0			0
$28$	−2.00000	+	1.73205i	−0.377964	+	0.327327i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−3.00000			−0.514496
$35$	7.50000	+	2.59808i	1.26773	+	0.439155i
$36$		0			0
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	0.323640	+	0.946180i	$0.395093\pi$
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	−0.981236	+	0.192809i	$0.938240\pi$
$38$	−1.00000	−	1.73205i	−0.162221	−	0.280976i
$39$		0			0
$40$	−1.50000	+	2.59808i	−0.237171	+	0.410792i
$41$	−9.00000			−1.40556			−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000			−1.40556			−0.702782	+	0.711405i	$0.748059\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$	0.500000	−	0.866025i	0.0753778	−	0.130558i
$45$		0			0
$46$	−1.50000	−	2.59808i	−0.221163	−	0.383065i
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	−0.954441	−	0.298401i	$-0.903547\pi$
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	0.735643	+	0.677369i	$0.236880\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$	4.00000			0.565685
$51$		0			0
$52$	−1.00000	−	1.73205i	−0.138675	−	0.240192i
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$	−0.500000	−	2.59808i	−0.0668153	−	0.347183i
$57$		0			0
$58$		0			0
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	0.601958	−	0.798528i	$-0.294388\pi$
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	−0.992524	+	0.122047i	$0.961054\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$	2.00000			0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$	−3.00000	+	5.19615i	−0.372104	+	0.644503i
$66$		0			0
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	0.305424	−	0.952217i	$-0.401202\pi$
$68$	1.50000	−	2.59808i	0.181902	−	0.315063i
$69$		0			0
$70$	−6.00000	+	5.19615i	−0.717137	+	0.621059i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−1.00000	−	1.73205i	−0.117041	−	0.202721i	0.801553	−	0.597924i	$-0.204008\pi$
$73$	−1.00000	−	1.73205i	−0.117041	−	0.202721i	−0.918594	+	0.395203i	$0.870674\pi$
$74$	−4.00000	−	6.92820i	−0.464991	−	0.805387i
$75$		0			0
$76$	2.00000			0.229416
$77$	2.00000	−	1.73205i	0.227921	−	0.197386i
$78$		0			0
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$	−1.50000	−	2.59808i	−0.167705	−	0.290474i
$81$		0			0
$82$	4.50000	−	7.79423i	0.496942	−	0.860729i
$83$	9.00000			0.987878			0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000			0.987878			0.493939	+	0.869496i	$0.335557\pi$
$84$		0			0
$85$	−9.00000			−0.976187
$86$	2.00000	−	3.46410i	0.215666	−	0.373544i
$87$		0			0
$88$	0.500000	+	0.866025i	0.0533002	+	0.0923186i
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	0.350304	+	0.936636i	$0.386078\pi$
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	−0.986303	+	0.164946i	$0.947255\pi$
$90$		0			0
$91$	−1.00000	−	5.19615i	−0.104828	−	0.544705i
$92$	3.00000			0.312772
$93$		0			0
$94$	−1.50000	−	2.59808i	−0.154713	−	0.267971i
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	5.00000			0.507673			0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000			0.507673			0.253837	+	0.967247i	$0.418307\pi$
$98$	1.00000	−	6.92820i	0.101015	−	0.699854i
$99$		0			0
$100$	−2.00000	+	3.46410i	−0.200000	+	0.346410i
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	0.993258	+	0.115924i	$0.0369830\pi$
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	−0.396236	+	0.918149i	$0.629684\pi$
$102$		0			0
$103$	−10.0000	+	17.3205i	−0.985329	+	1.70664i	−0.344865	+	0.938652i	$0.612075\pi$
$103$	−10.0000	+	17.3205i	−0.985329	+	1.70664i	−0.640464	+	0.767988i	$0.721258\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$	6.00000			0.582772
$107$	−1.50000	+	2.59808i	−0.145010	+	0.251166i	−0.929377	−	0.369132i	$-0.879655\pi$
$107$	−1.50000	+	2.59808i	−0.145010	+	0.251166i	0.784366	+	0.620298i	$0.212988\pi$
$108$		0			0
$109$	0.500000	+	0.866025i	0.0478913	+	0.0829502i	0.888977	−	0.457951i	$-0.151417\pi$
$109$	0.500000	+	0.866025i	0.0478913	+	0.0829502i	−0.841086	+	0.540901i	$0.818083\pi$
$110$	1.50000	−	2.59808i	0.143019	−	0.247717i
$111$		0			0
$112$	2.50000	+	0.866025i	0.236228	+	0.0818317i
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	−4.50000	−	7.79423i	−0.419627	−	0.726816i
$116$		0			0
$117$		0			0
$118$	6.00000			0.552345
$119$	6.00000	−	5.19615i	0.550019	−	0.476331i
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$	−2.50000	−	4.33013i	−0.226339	−	0.392031i
$123$		0			0
$124$	−1.00000	+	1.73205i	−0.0898027	+	0.155543i
$125$	−3.00000			−0.268328
$126$		0			0
$127$	−19.0000			−1.68598			−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000			−1.68598			−0.842989	+	0.537931i	$0.819206\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−3.00000	−	5.19615i	−0.263117	−	0.455733i
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		0			0
$133$	5.00000	+	1.73205i	0.433555	+	0.150188i
$134$	11.0000			0.950255
$135$		0			0
$136$	1.50000	+	2.59808i	0.128624	+	0.222783i
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	0.708942	−	0.705266i	$-0.249173\pi$
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	−0.965250	+	0.261329i	$0.915839\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$	−1.50000	−	7.79423i	−0.126773	−	0.658733i
$141$		0			0
$142$		0			0
$143$	1.00000	+	1.73205i	0.0836242	+	0.144841i
$144$		0			0
$145$		0			0
$146$	2.00000			0.165521
$147$		0			0
$148$	8.00000			0.657596
$149$	−6.00000	+	10.3923i	−0.491539	+	0.851371i	−0.999953	−	0.00974235i	$-0.996899\pi$
$149$	−6.00000	+	10.3923i	−0.491539	+	0.851371i	0.508413	+	0.861113i	$0.330232\pi$
$150$		0			0
$151$	−2.50000	−	4.33013i	−0.203447	−	0.352381i	0.746190	−	0.665733i	$-0.231881\pi$
$151$	−2.50000	−	4.33013i	−0.203447	−	0.352381i	−0.949637	+	0.313353i	$0.898548\pi$
$152$	−1.00000	+	1.73205i	−0.0811107	+	0.140488i
$153$		0			0
$154$	0.500000	+	2.59808i	0.0402911	+	0.209359i
$155$	6.00000			0.481932
$156$		0			0
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	0.934731	−	0.355357i	$-0.115641\pi$
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	−0.775113	+	0.631822i	$0.782307\pi$
$158$	6.50000	+	11.2583i	0.517112	+	0.895665i
$159$		0			0
$160$	3.00000			0.237171
$161$	7.50000	+	2.59808i	0.591083	+	0.204757i
$162$		0			0
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	−0.206518	−	0.978443i	$-0.566213\pi$
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	0.950615	−	0.310372i	$-0.100454\pi$
$164$	4.50000	+	7.79423i	0.351391	+	0.608627i
$165$		0			0
$166$	−4.50000	+	7.79423i	−0.349268	+	0.604949i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	4.50000	−	7.79423i	0.345134	−	0.597790i
$171$		0			0
$172$	2.00000	+	3.46410i	0.152499	+	0.264135i
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	−0.729155	−	0.684349i	$-0.760087\pi$
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	0.957241	+	0.289292i	$0.0934200\pi$
$174$		0			0
$175$	−8.00000	+	6.92820i	−0.604743	+	0.523723i
$176$	−1.00000			−0.0753778
$177$		0			0
$178$	−6.00000	−	10.3923i	−0.449719	−	0.778936i
$179$	−3.00000	−	5.19615i	−0.224231	−	0.388379i	0.731858	−	0.681457i	$-0.238654\pi$
$179$	−3.00000	−	5.19615i	−0.224231	−	0.388379i	−0.956088	+	0.293079i	$0.905320\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$	5.00000	+	1.73205i	0.370625	+	0.128388i
$183$		0			0
$184$	−1.50000	+	2.59808i	−0.110581	+	0.191533i
$185$	−12.0000	−	20.7846i	−0.882258	−	1.52811i
$186$		0			0
$187$	−1.50000	+	2.59808i	−0.109691	+	0.189990i
$188$	3.00000			0.218797
$189$		0			0
$190$	6.00000			0.435286
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	−0.997225	−	0.0744412i	$-0.976283\pi$
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	0.563081	+	0.826402i	$0.309616\pi$
$192$		0			0
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	0.995948	+	0.0899262i	$0.0286631\pi$
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	−0.420096	+	0.907480i	$0.638004\pi$
$194$	−2.50000	+	4.33013i	−0.179490	+	0.310885i
$195$		0			0
$196$	5.50000	+	4.33013i	0.392857	+	0.309295i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	5.00000	+	8.66025i	0.354441	+	0.613909i	0.987022	−	0.160585i	$-0.0513380\pi$
$199$	5.00000	+	8.66025i	0.354441	+	0.613909i	−0.632581	+	0.774494i	$0.718005\pi$
$200$	−2.00000	−	3.46410i	−0.141421	−	0.244949i
$201$		0			0
$202$	−12.0000			−0.844317
$203$		0			0
$204$		0			0
$205$	13.5000	−	23.3827i	0.942881	−	1.63312i
$206$	−10.0000	−	17.3205i	−0.696733	−	1.20678i
$207$		0			0
$208$	−1.00000	+	1.73205i	−0.0693375	+	0.120096i
$209$	−2.00000			−0.138343
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$	−3.00000	+	5.19615i	−0.206041	+	0.356873i
$213$		0			0
$214$	−1.50000	−	2.59808i	−0.102538	−	0.177601i
$215$	6.00000	−	10.3923i	0.409197	−	0.708749i
$216$		0			0
$217$	−4.00000	+	3.46410i	−0.271538	+	0.235159i
$218$	−1.00000			−0.0677285
$219$		0			0
$220$	1.50000	+	2.59808i	0.101130	+	0.175162i
$221$	3.00000	+	5.19615i	0.201802	+	0.349531i
$222$		0			0
$223$	26.0000			1.74109			0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000			1.74109			0.870544	+	0.492090i	$0.163767\pi$
$224$	−2.00000	+	1.73205i	−0.133631	+	0.115728i
$225$		0			0
$226$	9.00000	−	15.5885i	0.598671	−	1.03693i
$227$	−7.50000	−	12.9904i	−0.497792	−	0.862202i	0.502204	−	0.864749i	$-0.332523\pi$
$227$	−7.50000	−	12.9904i	−0.497792	−	0.862202i	−0.999997	+	0.00254715i	$0.999189\pi$
$228$		0			0
$229$	−10.0000	+	17.3205i	−0.660819	+	1.14457i	0.319582	+	0.947559i	$0.396457\pi$
$229$	−10.0000	+	17.3205i	−0.660819	+	1.14457i	−0.980401	+	0.197013i	$0.936876\pi$
$230$	9.00000			0.593442
$231$		0			0
$232$		0			0
$233$	−10.5000	+	18.1865i	−0.687878	+	1.19144i	0.284645	+	0.958633i	$0.408124\pi$
$233$	−10.5000	+	18.1865i	−0.687878	+	1.19144i	−0.972523	+	0.232806i	$0.925209\pi$
$234$		0			0
$235$	−4.50000	−	7.79423i	−0.293548	−	0.508439i
$236$	−3.00000	+	5.19615i	−0.195283	+	0.338241i
$237$		0			0
$238$	1.50000	+	7.79423i	0.0972306	+	0.505225i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	2.00000	+	3.46410i	0.128831	+	0.223142i	0.923224	−	0.384262i	$-0.125544\pi$
$241$	2.00000	+	3.46410i	0.128831	+	0.223142i	−0.794393	+	0.607404i	$0.792211\pi$
$242$	−0.500000	−	0.866025i	−0.0321412	−	0.0556702i
$243$		0			0
$244$	5.00000			0.320092
$245$	3.00000	−	20.7846i	0.191663	−	1.32788i
$246$		0			0
$247$	−2.00000	+	3.46410i	−0.127257	+	0.220416i
$248$	−1.00000	−	1.73205i	−0.0635001	−	0.109985i
$249$		0			0
$250$	1.50000	−	2.59808i	0.0948683	−	0.164317i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−3.00000			−0.188608
$254$	9.50000	−	16.4545i	0.596083	−	1.03245i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	12.0000	−	20.7846i	0.748539	−	1.29651i	−0.199983	−	0.979799i	$-0.564089\pi$
$257$	12.0000	−	20.7846i	0.748539	−	1.29651i	0.948523	−	0.316709i	$-0.102578\pi$
$258$		0			0
$259$	20.0000	+	6.92820i	1.24274	+	0.430498i
$260$	6.00000			0.372104
$261$		0			0
$262$	−6.00000	−	10.3923i	−0.370681	−	0.642039i
$263$	15.0000	+	25.9808i	0.924940	+	1.60204i	0.791658	+	0.610964i	$0.209218\pi$
$263$	15.0000	+	25.9808i	0.924940	+	1.60204i	0.133281	+	0.991078i	$0.457449\pi$
$264$		0			0
$265$	18.0000			1.10573
$266$	−4.00000	+	3.46410i	−0.245256	+	0.212398i
$267$		0			0
$268$	−5.50000	+	9.52628i	−0.335966	+	0.581910i
$269$	−7.50000	−	12.9904i	−0.457283	−	0.792038i	0.541533	−	0.840679i	$-0.317844\pi$
$269$	−7.50000	−	12.9904i	−0.457283	−	0.792038i	−0.998816	+	0.0486418i	$0.984511\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$	−3.00000			−0.181902
$273$		0			0
$274$	6.00000			0.362473
$275$	2.00000	−	3.46410i	0.120605	−	0.208893i
$276$		0			0
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	0.575408	−	0.817867i	$-0.304843\pi$
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	−0.995997	+	0.0893846i	$0.971510\pi$
$278$	−1.00000	+	1.73205i	−0.0599760	+	0.103882i
$279$		0			0
$280$	7.50000	+	2.59808i	0.448211	+	0.155265i
$281$	−27.0000			−1.61068			−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000			−1.61068			−0.805342	+	0.592810i	$0.798019\pi$
$282$		0			0
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	0.999608	−	0.0280052i	$-0.00891551\pi$
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	−0.524057	+	0.851683i	$0.675582\pi$
$284$		0			0
$285$		0			0
$286$	−2.00000			−0.118262
$287$	4.50000	+	23.3827i	0.265627	+	1.38024i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$	−1.00000	+	1.73205i	−0.0585206	+	0.101361i
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		0			0
$295$	18.0000			1.04800
$296$	−4.00000	+	6.92820i	−0.232495	+	0.402694i
$297$		0			0
$298$	−6.00000	−	10.3923i	−0.347571	−	0.602010i
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	2.00000	+	10.3923i	0.115278	+	0.599002i
$302$	5.00000			0.287718
$303$		0			0
$304$	−1.00000	−	1.73205i	−0.0573539	−	0.0993399i
$305$	−7.50000	−	12.9904i	−0.429449	−	0.743827i
$306$		0			0
$307$	8.00000			0.456584			0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000			0.456584			0.228292	+	0.973593i	$0.426686\pi$
$308$	−2.50000	−	0.866025i	−0.142451	−	0.0493464i
$309$		0			0
$310$	−3.00000	+	5.19615i	−0.170389	+	0.295122i
$311$	1.50000	+	2.59808i	0.0850572	+	0.147323i	0.905416	−	0.424526i	$-0.139559\pi$
$311$	1.50000	+	2.59808i	0.0850572	+	0.147323i	−0.820358	+	0.571850i	$0.806226\pi$
$312$		0			0
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	−0.689412	−	0.724370i	$-0.742131\pi$
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	0.972028	+	0.234863i	$0.0754642\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	−13.0000			−0.731307
$317$	7.50000	−	12.9904i	0.421242	−	0.729612i	−0.574819	−	0.818280i	$-0.694928\pi$
$317$	7.50000	−	12.9904i	0.421242	−	0.729612i	0.996061	+	0.0886679i	$0.0282610\pi$
$318$		0			0
$319$		0			0
$320$	−1.50000	+	2.59808i	−0.0838525	+	0.145237i
$321$		0			0
$322$	−6.00000	+	5.19615i	−0.334367	+	0.289570i
$323$	−6.00000			−0.333849
$324$		0			0
$325$	−4.00000	−	6.92820i	−0.221880	−	0.384308i
$326$	9.50000	+	16.4545i	0.526156	+	0.911330i
$327$		0			0
$328$	−9.00000			−0.496942
$329$	7.50000	+	2.59808i	0.413488	+	0.143237i
$330$		0			0
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	−0.477500	−	0.878632i	$-0.658457\pi$
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	0.999667	−	0.0257885i	$-0.00820965\pi$
$332$	−4.50000	−	7.79423i	−0.246970	−	0.427764i
$333$		0			0
$334$		0			0
$335$	33.0000			1.80298
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$	4.50000	−	7.79423i	0.244768	−	0.423950i
$339$		0			0
$340$	4.50000	+	7.79423i	0.244047	+	0.422701i
$341$	1.00000	−	1.73205i	0.0541530	−	0.0937958i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$	−4.00000			−0.215666
$345$		0			0
$346$	3.00000	+	5.19615i	0.161281	+	0.279347i
$347$	10.5000	+	18.1865i	0.563670	+	0.976304i	0.997172	+	0.0751519i	$0.0239442\pi$
$347$	10.5000	+	18.1865i	0.563670	+	0.976304i	−0.433503	+	0.901152i	$0.642722\pi$
$348$		0			0
$349$	23.0000			1.23116			0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000			1.23116			0.615581	+	0.788074i	$0.288921\pi$
$350$	−2.00000	−	10.3923i	−0.106904	−	0.555492i
$351$		0			0
$352$	0.500000	−	0.866025i	0.0266501	−	0.0461593i
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	0.920677	+	0.390324i	$0.127637\pi$
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	−0.122308	+	0.992492i	$0.539030\pi$
$354$		0			0
$355$		0			0
$356$	12.0000			0.635999
$357$		0			0
$358$	6.00000			0.317110
$359$	−6.00000	+	10.3923i	−0.316668	+	0.548485i	−0.979791	−	0.200026i	$-0.935897\pi$
$359$	−6.00000	+	10.3923i	−0.316668	+	0.548485i	0.663123	+	0.748511i	$0.269231\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$	−10.0000	+	17.3205i	−0.525588	+	0.910346i
$363$		0			0
$364$	−4.00000	+	3.46410i	−0.209657	+	0.181568i
$365$	6.00000			0.314054
$366$		0			0
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	0.966507	−	0.256639i	$-0.0826151\pi$
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	−0.705509	+	0.708700i	$0.749282\pi$
$368$	−1.50000	−	2.59808i	−0.0781929	−	0.135434i
$369$		0			0
$370$	24.0000			1.24770
$371$	−12.0000	+	10.3923i	−0.623009	+	0.539542i
$372$		0			0
$373$	−11.5000	+	19.9186i	−0.595447	+	1.03135i	0.398036	+	0.917370i	$0.369692\pi$
$373$	−11.5000	+	19.9186i	−0.595447	+	1.03135i	−0.993484	+	0.113975i	$0.963641\pi$
$374$	−1.50000	−	2.59808i	−0.0775632	−	0.134343i
$375$		0			0
$376$	−1.50000	+	2.59808i	−0.0773566	+	0.133986i
$377$		0			0
$378$		0			0
$379$	−1.00000			−0.0513665			−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000			−0.0513665			−0.0256833	+	0.999670i	$0.508176\pi$
$380$	−3.00000	+	5.19615i	−0.153897	+	0.266557i
$381$		0			0
$382$	−6.00000	−	10.3923i	−0.306987	−	0.531717i
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$	1.50000	+	7.79423i	0.0764471	+	0.397231i
$386$	−16.0000			−0.814379
$387$		0			0
$388$	−2.50000	−	4.33013i	−0.126918	−	0.219829i
$389$	10.5000	+	18.1865i	0.532371	+	0.922094i	0.999286	+	0.0377914i	$0.0120322\pi$
$389$	10.5000	+	18.1865i	0.532371	+	0.922094i	−0.466915	+	0.884302i	$0.654634\pi$
$390$		0			0
$391$	−9.00000			−0.455150
$392$	−6.50000	+	2.59808i	−0.328300	+	0.131223i
$393$		0			0
$394$		0			0
$395$	19.5000	+	33.7750i	0.981151	+	1.69940i
$396$		0			0
$397$	14.0000	−	24.2487i	0.702640	−	1.21701i	−0.264897	−	0.964277i	$-0.585338\pi$
$397$	14.0000	−	24.2487i	0.702640	−	1.21701i	0.967537	−	0.252731i	$-0.0813288\pi$
$398$	−10.0000			−0.501255
$399$		0			0
$400$	4.00000			0.200000
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	−0.976050	−	0.217545i	$-0.930195\pi$
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	0.676425	+	0.736512i	$0.263528\pi$
$402$		0			0
$403$	−2.00000	−	3.46410i	−0.0996271	−	0.172559i
$404$	6.00000	−	10.3923i	0.298511	−	0.517036i
$405$		0			0
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	−1.00000	−	1.73205i	−0.0494468	−	0.0856444i	0.840243	−	0.542211i	$-0.182412\pi$
$409$	−1.00000	−	1.73205i	−0.0494468	−	0.0856444i	−0.889689	+	0.456566i	$0.849079\pi$
$410$	13.5000	+	23.3827i	0.666717	+	1.15479i
$411$		0			0
$412$	20.0000			0.985329
$413$	−12.0000	+	10.3923i	−0.590481	+	0.511372i
$414$		0			0
$415$	−13.5000	+	23.3827i	−0.662689	+	1.14781i
$416$	−1.00000	−	1.73205i	−0.0490290	−	0.0849208i
$417$		0			0
$418$	1.00000	−	1.73205i	0.0489116	−	0.0847174i
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$	−10.0000	+	17.3205i	−0.486792	+	0.843149i
$423$		0			0
$424$	−3.00000	−	5.19615i	−0.145693	−	0.252347i
$425$	6.00000	−	10.3923i	0.291043	−	0.504101i
$426$		0			0
$427$	12.5000	+	4.33013i	0.604917	+	0.209550i
$428$	3.00000			0.145010
$429$		0			0
$430$	6.00000	+	10.3923i	0.289346	+	0.501161i
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	0.959985	+	0.280052i	$0.0903517\pi$
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	−0.237460	+	0.971397i	$0.576315\pi$
$432$		0			0
$433$	35.0000			1.68199			0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000			1.68199			0.840996	+	0.541041i	$0.181970\pi$
$434$	−1.00000	−	5.19615i	−0.0480015	−	0.249423i
$435$		0			0
$436$	0.500000	−	0.866025i	0.0239457	−	0.0414751i
$437$	−3.00000	−	5.19615i	−0.143509	−	0.248566i
$438$		0			0
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	0.0586141	+	0.998281i	$0.481332\pi$
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	−0.893843	+	0.448379i	$0.852001\pi$
$440$	−3.00000			−0.143019
$441$		0			0
$442$	−6.00000			−0.285391
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	−0.426414	−	0.904528i	$-0.640223\pi$
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	0.996551	−	0.0829786i	$-0.0264433\pi$
$444$		0			0
$445$	−18.0000	−	31.1769i	−0.853282	−	1.47793i
$446$	−13.0000	+	22.5167i	−0.615568	+	1.06619i
$447$		0			0
$448$	−0.500000	−	2.59808i	−0.0236228	−	0.122748i
$449$	−36.0000			−1.69895			−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000			−1.69895			−0.849473	+	0.527633i	$0.823080\pi$
$450$		0			0
$451$	−4.50000	−	7.79423i	−0.211897	−	0.367016i
$452$	9.00000	+	15.5885i	0.423324	+	0.733219i
$453$		0			0
$454$	15.0000			0.703985
$455$	15.0000	+	5.19615i	0.703211	+	0.243599i
$456$		0			0
$457$	20.0000	−	34.6410i	0.935561	−	1.62044i	0.161929	−	0.986802i	$-0.448228\pi$
$457$	20.0000	−	34.6410i	0.935561	−	1.62044i	0.773631	−	0.633636i	$-0.218438\pi$
$458$	−10.0000	−	17.3205i	−0.467269	−	0.809334i
$459$		0			0
$460$	−4.50000	+	7.79423i	−0.209814	+	0.363408i
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$	−4.00000			−0.185896			−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000			−0.185896			−0.0929479	+	0.995671i	$0.529629\pi$
$464$		0			0
$465$		0			0
$466$	−10.5000	−	18.1865i	−0.486403	−	0.842475i
$467$	21.0000	−	36.3731i	0.971764	−	1.68314i	0.281539	−	0.959550i	$-0.409155\pi$
$467$	21.0000	−	36.3731i	0.971764	−	1.68314i	0.690225	−	0.723595i	$-0.257512\pi$
$468$		0			0
$469$	−22.0000	+	19.0526i	−1.01587	+	0.879765i
$470$	9.00000			0.415139
$471$		0			0
$472$	−3.00000	−	5.19615i	−0.138086	−	0.239172i
$473$	−2.00000	−	3.46410i	−0.0919601	−	0.159280i
$474$		0			0
$475$	8.00000			0.367065
$476$	−7.50000	−	2.59808i	−0.343762	−	0.119083i
$477$		0			0
$478$		0			0
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	0.969920	−	0.243426i	$-0.0782712\pi$
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	−0.695773	+	0.718262i	$0.744938\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$	−4.00000			−0.182195
$483$		0			0
$484$	1.00000			0.0454545
$485$	−7.50000	+	12.9904i	−0.340557	+	0.589863i
$486$		0			0
$487$	−19.0000	−	32.9090i	−0.860972	−	1.49125i	−0.870992	−	0.491298i	$-0.836523\pi$
$487$	−19.0000	−	32.9090i	−0.860972	−	1.49125i	0.0100195	−	0.999950i	$-0.496811\pi$
$488$	−2.50000	+	4.33013i	−0.113170	+	0.196016i
$489$		0			0
$490$	16.5000	+	12.9904i	0.745394	+	0.586846i
$491$	9.00000			0.406164			0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000			0.406164			0.203082	+	0.979162i	$0.434904\pi$
$492$		0			0
$493$		0			0
$494$	−2.00000	−	3.46410i	−0.0899843	−	0.155857i
$495$		0			0
$496$	2.00000			0.0898027
$497$		0			0
$498$		0			0
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	−0.998233	−	0.0594153i	$-0.981076\pi$
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	0.550572	+	0.834788i	$0.314410\pi$
$500$	1.50000	+	2.59808i	0.0670820	+	0.116190i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−36.0000			−1.60198
$506$	1.50000	−	2.59808i	0.0666831	−	0.115499i
$507$		0			0
$508$	9.50000	+	16.4545i	0.421494	+	0.730050i
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	−0.791849	−	0.610718i	$-0.790881\pi$
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	0.924821	+	0.380402i	$0.124214\pi$
$510$		0			0
$511$	−4.00000	+	3.46410i	−0.176950	+	0.153243i
$512$	1.00000			0.0441942
$513$		0			0
$514$	12.0000	+	20.7846i	0.529297	+	0.916770i
$515$	−30.0000	−	51.9615i	−1.32196	−	2.28970i
$516$		0			0
$517$	−3.00000			−0.131940
$518$	−16.0000	+	13.8564i	−0.703000	+	0.608816i
$519$		0			0
$520$	−3.00000	+	5.19615i	−0.131559	+	0.227866i
$521$	−21.0000	−	36.3731i	−0.920027	−	1.59353i	−0.799370	−	0.600839i	$-0.794833\pi$
$521$	−21.0000	−	36.3731i	−0.920027	−	1.59353i	−0.120656	−	0.992694i	$-0.538500\pi$
$522$		0			0
$523$	5.00000	−	8.66025i	0.218635	−	0.378686i	−0.735756	−	0.677247i	$-0.763173\pi$
$523$	5.00000	−	8.66025i	0.218635	−	0.378686i	0.954391	+	0.298560i	$0.0965063\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$	−30.0000			−1.30806
$527$	3.00000	−	5.19615i	0.130682	−	0.226348i
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$	−9.00000	+	15.5885i	−0.390935	+	0.677119i
$531$		0			0
$532$	−1.00000	−	5.19615i	−0.0433555	−	0.225282i
$533$	−18.0000			−0.779667
$534$		0			0
$535$	−4.50000	−	7.79423i	−0.194552	−	0.336974i
$536$	−5.50000	−	9.52628i	−0.237564	−	0.411473i
$537$		0			0
$538$	15.0000			0.646696
$539$	−5.50000	−	4.33013i	−0.236902	−	0.186512i
$540$		0			0
$541$	−5.50000	+	9.52628i	−0.236463	+	0.409567i	−0.959697	−	0.281037i	$-0.909322\pi$
$541$	−5.50000	+	9.52628i	−0.236463	+	0.409567i	0.723234	+	0.690604i	$0.242655\pi$
$542$	−4.00000	−	6.92820i	−0.171815	−	0.297592i
$543$		0			0
$544$	1.50000	−	2.59808i	0.0643120	−	0.111392i
$545$	−3.00000			−0.128506
$546$		0			0
$547$	14.0000			0.598597			0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000			0.598597			0.299298	+	0.954160i	$0.403247\pi$
$548$	−3.00000	+	5.19615i	−0.128154	+	0.221969i
$549$		0			0
$550$	2.00000	+	3.46410i	0.0852803	+	0.147710i
$551$		0			0
$552$		0			0
$553$	−32.5000	−	11.2583i	−1.38204	−	0.478753i
$554$	14.0000			0.594803
$555$		0			0
$556$	−1.00000	−	1.73205i	−0.0424094	−	0.0734553i
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	−0.999952	−	0.00979220i	$-0.996883\pi$
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	0.491496	−	0.870880i	$-0.336450\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$	−6.00000	+	5.19615i	−0.253546	+	0.219578i
$561$		0			0
$562$	13.5000	−	23.3827i	0.569463	−	0.986339i
$563$	6.00000	+	10.3923i	0.252870	+	0.437983i	0.964315	−	0.264758i	$-0.0852922\pi$
$563$	6.00000	+	10.3923i	0.252870	+	0.437983i	−0.711445	+	0.702742i	$0.751959\pi$
$564$		0			0
$565$	27.0000	−	46.7654i	1.13590	−	1.96743i
$566$	−16.0000			−0.672530
$567$		0			0
$568$		0			0
$569$	21.0000	−	36.3731i	0.880366	−	1.52484i	0.0294311	−	0.999567i	$-0.490630\pi$
$569$	21.0000	−	36.3731i	0.880366	−	1.52484i	0.850935	−	0.525271i	$-0.176036\pi$
$570$		0			0
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	−0.978022	−	0.208502i	$-0.933141\pi$
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	0.308443	−	0.951243i	$-0.400192\pi$
$572$	1.00000	−	1.73205i	0.0418121	−	0.0724207i
$573$		0			0
$574$	−22.5000	−	7.79423i	−0.939132	−	0.325325i
$575$	12.0000			0.500435
$576$		0			0
$577$	−8.50000	−	14.7224i	−0.353860	−	0.612903i	0.633062	−	0.774101i	$-0.281798\pi$
$577$	−8.50000	−	14.7224i	−0.353860	−	0.612903i	−0.986922	+	0.161198i	$0.948464\pi$
$578$	4.00000	+	6.92820i	0.166378	+	0.288175i
$579$		0			0
$580$		0			0
$581$	−4.50000	−	23.3827i	−0.186691	−	0.970077i
$582$		0			0
$583$	3.00000	−	5.19615i	0.124247	−	0.215203i
$584$	−1.00000	−	1.73205i	−0.0413803	−	0.0716728i
$585$		0			0
$586$	9.00000	−	15.5885i	0.371787	−	0.643953i
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$	−9.00000	+	15.5885i	−0.370524	+	0.641767i
$591$		0			0
$592$	−4.00000	−	6.92820i	−0.164399	−	0.284747i
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	−0.797831	−	0.602881i	$-0.794019\pi$
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	0.921026	+	0.389501i	$0.127353\pi$
$594$		0			0
$595$	4.50000	+	23.3827i	0.184482	+	0.958597i
$596$	12.0000			0.491539
$597$		0			0
$598$	−3.00000	−	5.19615i	−0.122679	−	0.212486i
$599$	−7.50000	−	12.9904i	−0.306442	−	0.530773i	0.671140	−	0.741331i	$-0.265805\pi$
$599$	−7.50000	−	12.9904i	−0.306442	−	0.530773i	−0.977581	+	0.210558i	$0.932472\pi$
$600$		0			0
$601$	−40.0000			−1.63163			−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000			−1.63163			−0.815817	+	0.578310i	$0.803712\pi$
$602$	−10.0000	−	3.46410i	−0.407570	−	0.141186i
$603$		0			0
$604$	−2.50000	+	4.33013i	−0.101724	+	0.176190i
$605$	−1.50000	−	2.59808i	−0.0609837	−	0.105627i
$606$		0			0
$607$	−20.5000	+	35.5070i	−0.832069	+	1.44119i	0.0643251	+	0.997929i	$0.479511\pi$
$607$	−20.5000	+	35.5070i	−0.832069	+	1.44119i	−0.896394	+	0.443257i	$0.853823\pi$
$608$	2.00000			0.0811107
$609$		0			0
$610$	15.0000			0.607332
$611$	−3.00000	+	5.19615i	−0.121367	+	0.210214i
$612$		0			0
$613$	−8.50000	−	14.7224i	−0.343312	−	0.594633i	0.641734	−	0.766927i	$-0.278215\pi$
$613$	−8.50000	−	14.7224i	−0.343312	−	0.594633i	−0.985046	+	0.172294i	$0.944882\pi$
$614$	−4.00000	+	6.92820i	−0.161427	+	0.279600i
$615$		0			0
$616$	2.00000	−	1.73205i	0.0805823	−	0.0697863i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	15.5000	+	26.8468i	0.622998	+	1.07906i	0.988924	+	0.148420i	$0.0474187\pi$
$619$	15.5000	+	26.8468i	0.622998	+	1.07906i	−0.365927	+	0.930644i	$0.619248\pi$
$620$	−3.00000	−	5.19615i	−0.120483	−	0.208683i
$621$		0			0
$622$	−3.00000			−0.120289
$623$	30.0000	+	10.3923i	1.20192	+	0.416359i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	5.00000	+	8.66025i	0.199840	+	0.346133i
$627$		0			0
$628$	2.00000	−	3.46410i	0.0798087	−	0.138233i
$629$	−24.0000			−0.956943
$630$		0			0
$631$	−10.0000			−0.398094			−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000			−0.398094			−0.199047	+	0.979990i	$0.563785\pi$
$632$	6.50000	−	11.2583i	0.258556	−	0.447832i
$633$		0			0
$634$	7.50000	+	12.9904i	0.297863	+	0.515914i
$635$	28.5000	−	49.3634i	1.13099	−	1.95893i
$636$		0			0
$637$	−13.0000	+	5.19615i	−0.515079	+	0.205879i
$638$		0			0
$639$		0			0
$640$	−1.50000	−	2.59808i	−0.0592927	−	0.102698i
$641$	6.00000	+	10.3923i	0.236986	+	0.410471i	0.959848	−	0.280521i	$-0.0905072\pi$
$641$	6.00000	+	10.3923i	0.236986	+	0.410471i	−0.722862	+	0.690992i	$0.757174\pi$
$642$		0			0
$643$	20.0000			0.788723			0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000			0.788723			0.394362	+	0.918955i	$0.370966\pi$
$644$	−1.50000	−	7.79423i	−0.0591083	−	0.307136i
$645$		0			0
$646$	3.00000	−	5.19615i	0.118033	−	0.204440i
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	0.680096	−	0.733123i	$-0.261938\pi$
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	−0.974951	+	0.222419i	$0.928605\pi$
$648$		0			0
$649$	3.00000	−	5.19615i	0.117760	−	0.203967i
$650$	8.00000			0.313786
$651$		0			0
$652$	−19.0000			−0.744097
$653$	7.50000	−	12.9904i	0.293498	−	0.508353i	−0.681137	−	0.732156i	$-0.738514\pi$
$653$	7.50000	−	12.9904i	0.293498	−	0.508353i	0.974634	+	0.223803i	$0.0718474\pi$
$654$		0			0
$655$	−18.0000	−	31.1769i	−0.703318	−	1.21818i
$656$	4.50000	−	7.79423i	0.175695	−	0.304314i
$657$		0			0
$658$	−6.00000	+	5.19615i	−0.233904	+	0.202567i
$659$	−45.0000			−1.75295			−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000			−1.75295			−0.876476	+	0.481446i	$0.840112\pi$
$660$		0			0
$661$	−7.00000	−	12.1244i	−0.272268	−	0.471583i	0.697174	−	0.716902i	$-0.254441\pi$
$661$	−7.00000	−	12.1244i	−0.272268	−	0.471583i	−0.969442	+	0.245319i	$0.921107\pi$
$662$	9.50000	+	16.4545i	0.369228	+	0.639522i
$663$		0			0
$664$	9.00000			0.349268
$665$	−12.0000	+	10.3923i	−0.465340	+	0.402996i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−16.5000	+	28.5788i	−0.637451	+	1.10410i
$671$	−5.00000			−0.193023
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$	11.0000	−	19.0526i	0.423704	−	0.733877i
$675$		0			0
$676$	4.50000	+	7.79423i	0.173077	+	0.299778i
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	−0.802600	−	0.596518i	$-0.796551\pi$
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	0.917899	+	0.396813i	$0.129884\pi$
$678$		0			0
$679$	−2.50000	−	12.9904i	−0.0959412	−	0.498525i
$680$	−9.00000			−0.345134
$681$		0			0
$682$	1.00000	+	1.73205i	0.0382920	+	0.0663237i
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	0.539750	−	0.841825i	$-0.318519\pi$
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	−0.998917	+	0.0465244i	$0.985185\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$	−18.5000	+	0.866025i	−0.706333	+	0.0330650i
$687$		0			0
$688$	2.00000	−	3.46410i	0.0762493	−	0.132068i
$689$	−6.00000	−	10.3923i	−0.228582	−	0.395915i
$690$		0			0
$691$	−11.5000	+	19.9186i	−0.437481	+	0.757739i	−0.997494	−	0.0707446i	$-0.977462\pi$
$691$	−11.5000	+	19.9186i	−0.437481	+	0.757739i	0.560014	+	0.828483i	$0.310796\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$	−21.0000			−0.797149
$695$	−3.00000	+	5.19615i	−0.113796	+	0.197101i
$696$		0			0
$697$	−13.5000	−	23.3827i	−0.511349	−	0.885682i
$698$	−11.5000	+	19.9186i	−0.435281	+	0.753930i
$699$		0			0
$700$	10.0000	+	3.46410i	0.377964	+	0.130931i
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	−8.00000	−	13.8564i	−0.301726	−	0.522604i
$704$	0.500000	+	0.866025i	0.0188445	+	0.0326396i
$705$		0			0
$706$	−30.0000			−1.12906
$707$	24.0000	−	20.7846i	0.902613	−	0.781686i
$708$		0			0
$709$	−7.00000	+	12.1244i	−0.262891	+	0.455340i	−0.967009	−	0.254743i	$-0.918009\pi$
$709$	−7.00000	+	12.1244i	−0.262891	+	0.455340i	0.704118	+	0.710083i	$0.251342\pi$
$710$		0			0
$711$		0			0
$712$	−6.00000	+	10.3923i	−0.224860	+	0.389468i
$713$	6.00000			0.224702
$714$		0			0
$715$	−6.00000			−0.224387
$716$	−3.00000	+	5.19615i	−0.112115	+	0.194189i
$717$		0			0
$718$	−6.00000	−	10.3923i	−0.223918	−	0.387837i
$719$	−19.5000	+	33.7750i	−0.727227	+	1.25959i	0.230823	+	0.972996i	$0.425858\pi$
$719$	−19.5000	+	33.7750i	−0.727227	+	1.25959i	−0.958051	+	0.286599i	$0.907475\pi$
$720$		0			0
$721$	50.0000	+	17.3205i	1.86210	+	0.645049i
$722$	−15.0000			−0.558242
$723$		0			0
$724$	−10.0000	−	17.3205i	−0.371647	−	0.643712i
$725$		0			0
$726$		0			0
$727$	−4.00000			−0.148352			−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000			−0.148352			−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−1.00000	−	5.19615i	−0.0370625	−	0.192582i
$729$		0			0
$730$	−3.00000	+	5.19615i	−0.111035	+	0.192318i
$731$	−6.00000	−	10.3923i	−0.221918	−	0.384373i
$732$		0			0
$733$	−23.5000	+	40.7032i	−0.867992	+	1.50341i	−0.00394730	+	0.999992i	$0.501256\pi$
$733$	−23.5000	+	40.7032i	−0.867992	+	1.50341i	−0.864045	+	0.503415i	$0.832077\pi$
$734$	−10.0000			−0.369107
$735$		0			0
$736$	3.00000			0.110581
$737$	5.50000	−	9.52628i	0.202595	−	0.350905i
$738$		0			0
$739$	5.00000	+	8.66025i	0.183928	+	0.318573i	0.943215	−	0.332184i	$-0.107785\pi$
$739$	5.00000	+	8.66025i	0.183928	+	0.318573i	−0.759287	+	0.650756i	$0.774452\pi$
$740$	−12.0000	+	20.7846i	−0.441129	+	0.764057i
$741$		0			0
$742$	−3.00000	−	15.5885i	−0.110133	−	0.572270i
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$	−18.0000	−	31.1769i	−0.659469	−	1.14223i
$746$	−11.5000	−	19.9186i	−0.421045	−	0.729271i
$747$		0			0
$748$	3.00000			0.109691
$749$	7.50000	+	2.59808i	0.274044	+	0.0949316i
$750$		0			0
$751$	−25.0000	+	43.3013i	−0.912263	+	1.58009i	−0.101403	+	0.994845i	$0.532333\pi$
$751$	−25.0000	+	43.3013i	−0.912263	+	1.58009i	−0.810860	+	0.585240i	$0.801000\pi$
$752$	−1.50000	−	2.59808i	−0.0546994	−	0.0947421i
$753$		0			0
$754$		0			0
$755$	15.0000			0.545906
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$	0.500000	−	0.866025i	0.0181608	−	0.0314555i
$759$		0			0
$760$	−3.00000	−	5.19615i	−0.108821	−	0.188484i
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	−0.697467	−	0.716617i	$-0.745690\pi$
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	0.969342	+	0.245716i	$0.0790230\pi$
$762$		0			0
$763$	2.00000	−	1.73205i	0.0724049	−	0.0627044i
$764$	12.0000			0.434145
$765$		0			0
$766$	12.0000	+	20.7846i	0.433578	+	0.750978i
$767$	−6.00000	−	10.3923i	−0.216647	−	0.375244i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$	−7.50000	−	2.59808i	−0.270281	−	0.0936282i
$771$		0			0
$772$	8.00000	−	13.8564i	0.287926	−	0.498703i
$773$	10.5000	+	18.1865i	0.377659	+	0.654124i	0.990721	−	0.135910i	$-0.0433959\pi$
$773$	10.5000	+	18.1865i	0.377659	+	0.654124i	−0.613062	+	0.790034i	$0.710063\pi$
$774$		0			0
$775$	−4.00000	+	6.92820i	−0.143684	+	0.248868i
$776$	5.00000			0.179490
$777$		0			0
$778$	−21.0000			−0.752886
$779$	9.00000	−	15.5885i	0.322458	−	0.558514i
$780$		0			0
$781$		0			0
$782$	4.50000	−	7.79423i	0.160920	−	0.278721i
$783$		0			0
$784$	1.00000	−	6.92820i	0.0357143	−	0.247436i
$785$	−12.0000			−0.428298
$786$		0			0
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	−0.975798	−	0.218675i	$-0.929827\pi$
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	0.298521	−	0.954403i	$-0.403507\pi$
$788$		0			0
$789$		0			0
$790$	−39.0000			−1.38756
$791$	9.00000	+	46.7654i	0.320003	+	1.66279i
$792$		0			0
$793$	−5.00000	+	8.66025i	−0.177555	+	0.307535i
$794$	14.0000	+	24.2487i	0.496841	+	0.860555i
$795$		0			0
$796$	5.00000	−	8.66025i	0.177220	−	0.306955i
$797$	−45.0000			−1.59398			−0.796991	−	0.603991i	$-0.793576\pi$
$797$	−45.0000			−1.59398			−0.796991	+	0.603991i	$0.793576\pi$
$798$		0			0
$799$	−9.00000			−0.318397
$800$	−2.00000	+	3.46410i	−0.0707107	+	0.122474i
$801$		0			0
$802$	−6.00000	−	10.3923i	−0.211867	−	0.366965i
$803$	1.00000	−	1.73205i	0.0352892	−	0.0611227i
$804$		0			0
$805$	−18.0000	+	15.5885i	−0.634417	+	0.549421i
$806$	4.00000			0.140894
$807$		0			0
$808$	6.00000	+	10.3923i	0.211079	+	0.365600i
$809$	19.5000	+	33.7750i	0.685583	+	1.18747i	0.973253	+	0.229736i	$0.0737862\pi$
$809$	19.5000	+	33.7750i	0.685583	+	1.18747i	−0.287670	+	0.957730i	$0.592880\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$		0			0
$813$		0			0
$814$	4.00000	−	6.92820i	0.140200	−	0.242833i
$815$	28.5000	+	49.3634i	0.998311	+	1.72913i
$816$		0			0
$817$	4.00000	−	6.92820i	0.139942	−	0.242387i
$818$	2.00000			0.0699284
$819$		0			0
$820$	−27.0000			−0.942881
$821$	−15.0000	+	25.9808i	−0.523504	+	0.906735i	0.476122	+	0.879379i	$0.342042\pi$
$821$	−15.0000	+	25.9808i	−0.523504	+	0.906735i	−0.999626	+	0.0273557i	$0.991291\pi$
$822$		0			0
$823$	17.0000	+	29.4449i	0.592583	+	1.02638i	0.993883	+	0.110437i	$0.0352250\pi$
$823$	17.0000	+	29.4449i	0.592583	+	1.02638i	−0.401300	+	0.915947i	$0.631442\pi$
$824$	−10.0000	+	17.3205i	−0.348367	+	0.603388i
$825$		0			0
$826$	−3.00000	−	15.5885i	−0.104383	−	0.542392i
$827$	45.0000			1.56480			0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000			1.56480			0.782402	+	0.622774i	$0.213994\pi$
$828$		0			0
$829$	26.0000	+	45.0333i	0.903017	+	1.56407i	0.823557	+	0.567234i	$0.191986\pi$
$829$	26.0000	+	45.0333i	0.903017	+	1.56407i	0.0794606	+	0.996838i	$0.474680\pi$
$830$	−13.5000	−	23.3827i	−0.468592	−	0.811625i
$831$		0			0
$832$	2.00000			0.0693375
$833$	−16.5000	−	12.9904i	−0.571691	−	0.450090i
$834$		0			0
$835$		0			0
$836$	1.00000	+	1.73205i	0.0345857	+	0.0599042i
$837$		0			0
$838$	−9.00000	+	15.5885i	−0.310900	+	0.538494i
$839$	9.00000			0.310715			0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000			0.310715			0.155357	+	0.987858i	$0.450347\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	−4.00000	+	6.92820i	−0.137849	+	0.238762i
$843$		0			0
$844$	−10.0000	−	17.3205i	−0.344214	−	0.596196i
$845$	13.5000	−	23.3827i	0.464414	−	0.804389i
$846$		0			0
$847$	2.50000	+	0.866025i	0.0859010	+	0.0297570i
$848$	6.00000			0.206041
$849$		0			0
$850$	6.00000	+	10.3923i	0.205798	+	0.356453i
$851$	−12.0000	−	20.7846i	−0.411355	−	0.712487i
$852$		0			0
$853$	−19.0000			−0.650548			−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000			−0.650548			−0.325274	+	0.945620i	$0.605456\pi$
$854$	−10.0000	+	8.66025i	−0.342193	+	0.296348i
$855$		0			0
$856$	−1.50000	+	2.59808i	−0.0512689	+	0.0888004i
$857$	19.5000	+	33.7750i	0.666107	+	1.15373i	0.978984	+	0.203938i	$0.0653741\pi$
$857$	19.5000	+	33.7750i	0.666107	+	1.15373i	−0.312877	+	0.949794i	$0.601293\pi$
$858$		0			0
$859$	3.50000	−	6.06218i	0.119418	−	0.206839i	−0.800119	−	0.599841i	$-0.795230\pi$
$859$	3.50000	−	6.06218i	0.119418	−	0.206839i	0.919537	+	0.393003i	$0.128564\pi$
$860$	−12.0000			−0.409197
$861$		0			0
$862$	−30.0000			−1.02180
$863$	16.5000	−	28.5788i	0.561667	−	0.972835i	−0.435685	−	0.900099i	$-0.643494\pi$
$863$	16.5000	−	28.5788i	0.561667	−	0.972835i	0.997351	−	0.0727356i	$-0.0231729\pi$
$864$		0			0
$865$	9.00000	+	15.5885i	0.306009	+	0.530023i
$866$	−17.5000	+	30.3109i	−0.594674	+	1.03001i
$867$		0			0
$868$	5.00000	+	1.73205i	0.169711	+	0.0587896i
$869$	13.0000			0.440995
$870$		0			0
$871$	−11.0000	−	19.0526i	−0.372721	−	0.645571i
$872$	0.500000	+	0.866025i	0.0169321	+	0.0293273i
$873$		0			0
$874$	6.00000			0.202953
$875$	1.50000	+	7.79423i	0.0507093	+	0.263493i
$876$		0			0
$877$	−5.50000	+	9.52628i	−0.185722	+	0.321680i	−0.943820	−	0.330461i	$-0.892796\pi$
$877$	−5.50000	+	9.52628i	−0.185722	+	0.321680i	0.758098	+	0.652141i	$0.226129\pi$
$878$	−17.5000	−	30.3109i	−0.590596	−	1.02294i
$879$		0			0
$880$	1.50000	−	2.59808i	0.0505650	−	0.0875811i
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$	−7.00000			−0.235569			−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000			−0.235569			−0.117784	+	0.993039i	$0.537579\pi$
$884$	3.00000	−	5.19615i	0.100901	−	0.174766i
$885$		0			0
$886$	12.0000	+	20.7846i	0.403148	+	0.698273i
$887$	−15.0000	+	25.9808i	−0.503651	+	0.872349i	0.496340	+	0.868128i	$0.334677\pi$
$887$	−15.0000	+	25.9808i	−0.503651	+	0.872349i	−0.999991	+	0.00422062i	$0.998657\pi$
$888$		0			0
$889$	9.50000	+	49.3634i	0.318620	+	1.65560i
$890$	36.0000			1.20672
$891$		0			0
$892$	−13.0000	−	22.5167i	−0.435272	−	0.753914i
$893$	−3.00000	−	5.19615i	−0.100391	−	0.173883i
$894$		0			0
$895$	18.0000			0.601674
$896$	2.50000	+	0.866025i	0.0835191	+	0.0289319i
$897$		0			0
$898$	18.0000	−	31.1769i	0.600668	−	1.04039i
$899$		0			0
$900$		0			0
$901$	9.00000	−	15.5885i	0.299833	−	0.519327i
$902$	9.00000			0.299667
$903$		0			0
$904$	−18.0000			−0.598671
$905$	−30.0000	+	51.9615i	−0.997234	+	1.72726i
$906$		0			0
$907$	−20.5000	−	35.5070i	−0.680691	−	1.17899i	−0.974770	−	0.223211i	$-0.928346\pi$
$907$	−20.5000	−	35.5070i	−0.680691	−	1.17899i	0.294079	−	0.955781i	$-0.404987\pi$
$908$	−7.50000	+	12.9904i	−0.248896	+	0.431101i
$909$		0			0
$910$	−12.0000	+	10.3923i	−0.397796	+	0.344502i
$911$	−27.0000			−0.894550			−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000			−0.894550			−0.447275	+	0.894397i	$0.647605\pi$
$912$		0			0
$913$	4.50000	+	7.79423i	0.148928	+	0.257951i
$914$	20.0000	+	34.6410i	0.661541	+	1.14582i
$915$		0			0
$916$	20.0000			0.660819
$917$	30.0000	+	10.3923i	0.990687	+	0.343184i
$918$		0			0
$919$	18.5000	−	32.0429i	0.610259	−	1.05700i	−0.380938	−	0.924601i	$-0.624399\pi$
$919$	18.5000	−	32.0429i	0.610259	−	1.05700i	0.991197	−	0.132398i	$-0.0422678\pi$
$920$	−4.50000	−	7.79423i	−0.148361	−	0.256968i
$921$		0			0
$922$	−9.00000	+	15.5885i	−0.296399	+	0.513378i
$923$		0			0
$924$		0			0
$925$	32.0000			1.05215
$926$	2.00000	−	3.46410i	0.0657241	−	0.113837i
$927$		0			0
$928$		0			0
$929$	3.00000	−	5.19615i	0.0984268	−	0.170480i	−0.812607	−	0.582812i	$-0.801952\pi$
$929$	3.00000	−	5.19615i	0.0984268	−	0.170480i	0.911034	+	0.412332i	$0.135286\pi$
$930$		0			0
$931$	2.00000	−	13.8564i	0.0655474	−	0.454125i
$932$	21.0000			0.687878
$933$		0			0
$934$	21.0000	+	36.3731i	0.687141	+	1.19016i
$935$	−4.50000	−	7.79423i	−0.147166	−	0.254899i
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$	−5.50000	−	28.5788i	−0.179581	−	0.933132i
$939$		0			0
$940$	−4.50000	+	7.79423i	−0.146774	+	0.254220i
$941$	15.0000	+	25.9808i	0.488986	+	0.846949i	0.999920	−	0.0126715i	$-0.00403357\pi$
$941$	15.0000	+	25.9808i	0.488986	+	0.846949i	−0.510934	+	0.859620i	$0.670700\pi$
$942$		0			0
$943$	13.5000	−	23.3827i	0.439620	−	0.761445i
$944$	6.00000			0.195283
$945$		0			0
$946$	4.00000			0.130051
$947$	21.0000	−	36.3731i	0.682408	−	1.18197i	−0.291835	−	0.956469i	$-0.594266\pi$
$947$	21.0000	−	36.3731i	0.682408	−	1.18197i	0.974244	−	0.225497i	$-0.0724007\pi$
$948$		0			0
$949$	−2.00000	−	3.46410i	−0.0649227	−	0.112449i
$950$	−4.00000	+	6.92820i	−0.129777	+	0.224781i
$951$		0			0
$952$	6.00000	−	5.19615i	0.194461	−	0.168408i
$953$	45.0000			1.45769			0.728846	−	0.684677i	$-0.240057\pi$
$953$	45.0000			1.45769			0.728846	+	0.684677i	$0.240057\pi$
$954$		0			0
$955$	−18.0000	−	31.1769i	−0.582466	−	1.00886i
$956$		0			0
$957$		0			0
$958$	−12.0000			−0.387702
$959$	−12.0000	+	10.3923i	−0.387500	+	0.335585i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$	−8.00000	−	13.8564i	−0.257930	−	0.446748i
$963$		0			0
$964$	2.00000	−	3.46410i	0.0644157	−	0.111571i
$965$	−48.0000			−1.54517
$966$		0			0
$967$	47.0000			1.51142			0.755709	−	0.654907i	$-0.227292\pi$
$967$	47.0000			1.51142			0.755709	+	0.654907i	$0.227292\pi$
$968$	−0.500000	+	0.866025i	−0.0160706	+	0.0278351i
$969$		0			0
$970$	−7.50000	−	12.9904i	−0.240810	−	0.417096i
$971$	3.00000	−	5.19615i	0.0962746	−	0.166752i	−0.813865	−	0.581054i	$-0.802641\pi$
$971$	3.00000	−	5.19615i	0.0962746	−	0.166752i	0.910140	+	0.414301i	$0.135974\pi$
$972$		0			0
$973$	−1.00000	−	5.19615i	−0.0320585	−	0.166581i
$974$	38.0000			1.21760
$975$		0			0
$976$	−2.50000	−	4.33013i	−0.0800230	−	0.138604i
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	0.814038	−	0.580812i	$-0.197265\pi$
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	−0.910017	+	0.414572i	$0.863931\pi$
$978$		0			0
$979$	−12.0000			−0.383522
$980$	−19.5000	+	7.79423i	−0.622905	+	0.248978i
$981$		0			0
$982$	−4.50000	+	7.79423i	−0.143601	+	0.248724i
$983$	10.5000	+	18.1865i	0.334898	+	0.580060i	0.983465	−	0.181097i	$-0.0579648\pi$
$983$	10.5000	+	18.1865i	0.334898	+	0.580060i	−0.648567	+	0.761157i	$0.724631\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	4.00000			0.127257
$989$	6.00000	−	10.3923i	0.190789	−	0.330456i
$990$		0			0
$991$	−4.00000	−	6.92820i	−0.127064	−	0.220082i	0.795474	−	0.605988i	$-0.207222\pi$
$991$	−4.00000	−	6.92820i	−0.127064	−	0.220082i	−0.922538	+	0.385906i	$0.873889\pi$
$992$	−1.00000	+	1.73205i	−0.0317500	+	0.0549927i
$993$		0			0
$994$		0			0
$995$	−30.0000			−0.951064
$996$		0			0
$997$	−25.0000	−	43.3013i	−0.791758	−	1.37136i	−0.924878	−	0.380265i	$-0.875833\pi$
$997$	−25.0000	−	43.3013i	−0.791758	−	1.37136i	0.133120	−	0.991100i	$-0.457501\pi$
$998$	−10.0000	−	17.3205i	−0.316544	−	0.548271i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.k.b.793.1	✓	2
3.2	odd	2		1386.2.k.p.793.1	yes	2
7.2	even	3		9702.2.a.cd.1.1		1
7.4	even	3	inner	1386.2.k.b.991.1	yes	2
7.5	odd	6		9702.2.a.bc.1.1		1
21.2	odd	6		9702.2.a.c.1.1		1
21.5	even	6		9702.2.a.z.1.1		1
21.11	odd	6		1386.2.k.p.991.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.k.b.793.1	✓	2	1.1	even	1	trivial
1386.2.k.b.991.1	yes	2	7.4	even	3	inner
1386.2.k.p.793.1	yes	2	3.2	odd	2
1386.2.k.p.991.1	yes	2	21.11	odd	6
9702.2.a.c.1.1		1	21.2	odd	6
9702.2.a.z.1.1		1	21.5	even	6
9702.2.a.bc.1.1		1	7.5	odd	6
9702.2.a.cd.1.1		1	7.2	even	3

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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} +(0.500000 + 0.866025i) q^{11} +2.00000 q^{13} +(2.50000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.00000 + 1.73205i) q^{19} +3.00000 q^{20} -1.00000 q^{22} +(-1.50000 + 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(-1.00000 + 1.73205i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(-1.00000 - 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{32} -3.00000 q^{34} +(7.50000 + 2.59808i) q^{35} +(-4.00000 + 6.92820i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(-1.50000 + 2.59808i) q^{40} -9.00000 q^{41} -4.00000 q^{43} +(0.500000 - 0.866025i) q^{44} +(-1.50000 - 2.59808i) q^{46} +(-1.50000 + 2.59808i) q^{47} +(-6.50000 + 2.59808i) q^{49} +4.00000 q^{50} +(-1.00000 - 1.73205i) q^{52} +(-3.00000 - 5.19615i) q^{53} -3.00000 q^{55} +(-0.500000 - 2.59808i) q^{56} +(-3.00000 - 5.19615i) q^{59} +(-2.50000 + 4.33013i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +(-5.50000 - 9.52628i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-6.00000 + 5.19615i) q^{70} +(-1.00000 - 1.73205i) q^{73} +(-4.00000 - 6.92820i) q^{74} +2.00000 q^{76} +(2.00000 - 1.73205i) q^{77} +(6.50000 - 11.2583i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(4.50000 - 7.79423i) q^{82} +9.00000 q^{83} -9.00000 q^{85} +(2.00000 - 3.46410i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-6.00000 + 10.3923i) q^{89} +(-1.00000 - 5.19615i) q^{91} +3.00000 q^{92} +(-1.50000 - 2.59808i) q^{94} +(-3.00000 - 5.19615i) q^{95} +5.00000 q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 3 q^{5} - q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - 3 * q^5 - q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - 3 q^{5} - q^{7} + 2 q^{8} - 3 q^{10} + q^{11} + 4 q^{13} + 5 q^{14} - q^{16} + 3 q^{17} - 2 q^{19} + 6 q^{20} - 2 q^{22} - 3 q^{23} - 4 q^{25} - 2 q^{26} - 4 q^{28} - 2 q^{31} - q^{32} - 6 q^{34} + 15 q^{35} - 8 q^{37} - 2 q^{38} - 3 q^{40} - 18 q^{41} - 8 q^{43} + q^{44} - 3 q^{46} - 3 q^{47} - 13 q^{49} + 8 q^{50} - 2 q^{52} - 6 q^{53} - 6 q^{55} - q^{56} - 6 q^{59} - 5 q^{61} + 4 q^{62} + 2 q^{64} - 6 q^{65} - 11 q^{67} + 3 q^{68} - 12 q^{70} - 2 q^{73} - 8 q^{74} + 4 q^{76} + 4 q^{77} + 13 q^{79} - 3 q^{80} + 9 q^{82} + 18 q^{83} - 18 q^{85} + 4 q^{86} + q^{88} - 12 q^{89} - 2 q^{91} + 6 q^{92} - 3 q^{94} - 6 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 3 * q^5 - q^7 + 2 * q^8 - 3 * q^10 + q^11 + 4 * q^13 + 5 * q^14 - q^16 + 3 * q^17 - 2 * q^19 + 6 * q^20 - 2 * q^22 - 3 * q^23 - 4 * q^25 - 2 * q^26 - 4 * q^28 - 2 * q^31 - q^32 - 6 * q^34 + 15 * q^35 - 8 * q^37 - 2 * q^38 - 3 * q^40 - 18 * q^41 - 8 * q^43 + q^44 - 3 * q^46 - 3 * q^47 - 13 * q^49 + 8 * q^50 - 2 * q^52 - 6 * q^53 - 6 * q^55 - q^56 - 6 * q^59 - 5 * q^61 + 4 * q^62 + 2 * q^64 - 6 * q^65 - 11 * q^67 + 3 * q^68 - 12 * q^70 - 2 * q^73 - 8 * q^74 + 4 * q^76 + 4 * q^77 + 13 * q^79 - 3 * q^80 + 9 * q^82 + 18 * q^83 - 18 * q^85 + 4 * q^86 + q^88 - 12 * q^89 - 2 * q^91 + 6 * q^92 - 3 * q^94 - 6 * q^95 + 10 * q^97 + 2 * q^98

Introduction

L-functions

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